CLASSIFICATION NOTES NO 412

DET NORSKE VERITAS Veritasveien 1 N-1322 Hoslashvik Norway Tel +47 67 57 99 00 Fax +47 67 57 99 11

CALCULATION OF GEAR RATING FOR MARINE TRANSMISSIONS

MAY 2003

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Classification Notes

Classification Notes are publications that give practical information on classification of ships and other objects Examples of design solutions calculation methods specifications of test procedures as well as acceptable repair methods for some compo-nents are given as interpretations of the more general rule requirements

A list of Classification Notes is found in the latest edition of the Introduction booklets to the rdquoRules for Classification of Shipsrdquo and the rdquoRules for Classification of High Speed Light Craft and Naval Surface Craftrdquo In ldquoRules for Classification of Fixed Offshore Installationsrdquo only those Classification Notes that are relevant for this type of structure have been listed

The list of Classification Notes is also included in the current ldquoClassification Services ndash Publicationsrdquo issued by the Society which is available on request All publications may be ordered from the Societyrsquos Web site httpexchangednvcom

Provisions

It is assumed that the execution of the provisions of this Classification Note is entrusted to appropriately qualified and experienced people for whose use it has been prepared

DET NORSKE VERITAS

CONTENTS

1 Basic Principles and General Influence Factors 4 11 Scope and Basic Principles 4 12 Symbols Nomenclature and Units 4 13 Geometrical Definitions5 14 Bevel Gear Conversion Formulae and Specific

Formulae6 15 Nominal Tangential Load Ft Fbt Fmt and Fmbt 6 16 Application Factors KA and KAP 7 161 KA 7 162 KAP7 163 Frequent overloads8 17 Load Sharing Factor Kγ 8 171 General method8 172 Simplified method 8 18 Dynamic Factor Kv 8 181 Single resonance method 8 182 Multi-resonance method 10 19 Face Load Factors KHβ and KFβ 10 191 Relations between KHβ and KFβ10 192 Measurement of face load factors 10 193 Theoretical determination of KHβ11 194 Determination of fsh 12 195 Determination of fdefl13 196 Determination of fbe 13 197 Determination of fma 13 198 Comments to various gear types 13 199 Determination of KHβ for bevel gears 13 110 Transversal Load Distribution Factors KHα and KFα14 111 Tooth Stiffness Constants cacute and cγ 14 112 Running-in Allowances 15 2 Calculation of Surface Durability17 21 Scope and General Remarks 17 22 Basic Equations 17 221 Contact stress 17 222 Permissible contact stress 17 23 Zone Factors ZH ZBD and ZM18 231 Zone factor ZH 18 232 Zone factors ZBD 18 233 Zone factor ZM 18 234 Inner contact point 18 24 Elasticity Factor ZE 18 25 Contact Ratio Factor Zε18 26 Helix Angle Factor Zβ18 27 Bevel Gear Factor ZK18 28 Values of Endurance Limit σHlim and Static

Strength 5H10σ 3H10σ 18 29 Life Factor ZN 19 210 Influence Factors on Lubrication Film ZL ZV and

ZR 19 211 Work Hardening Factor ZW 20 212 Size Factor ZX20 213 Subsurface Fatigue20 3 Calculation of Tooth Strength 22 31 Scope and General Remarks 22 32 Tooth Root Stresses 22 321 Local tooth root stress22 322 Permissible tooth root stress 22

33 Tooth Form Factors YF YFa 23 331 Determination of parameters 23 332 Gearing with εαn gt 2 24 34 Stress Correction Factors YS YSa 24 35 Contact Ratio Factor Yε 25 36 Helix Angle Factor Yβ 25 37 Values of Endurance Limit σFE 25 38 Mean stress influence Factor YM 26 381 For idlers planets and PTO with ice class 26 382 For gears with periodical change of rotational

direction 26 383 For gears with shrinkage stresses and unidirectional

load 26 384 For shrink-fitted idlers and planets 26 385 Additional requirements for peak loads 27 39 Life Factor YN 27 310 Relative Notch Sensitivity Factor YδrelT 28 311 Relative Surface Condition Factor YRrelT 28 312 Size Factor YX 28 313 Case Depth Factor YC 28 314 Thin rim factor YB 29 315 Stresses in Thin Rims 29 3151 General 29 3152 Stress concentration factors at the 75ordm tangents 30 3153 Nominal rim stresses 30 3154 Root fillet stresses 30 316 Permissible Stresses in Thin Rims 31 3161 General 31 3162 For gt3middot106 cycles 31 3163 For le 103 cycles 31 3164 For 103 lt cycles lt 3106 32 4 Calculation of Scuffing Load Capacity 33 41 Introduction 33 42 General Criteria 33 43 Influence Factors 34 431 Coefficient of friction 34 432 Effective tip relief Ceff 34 433 Tip relief and extension 34 434 Bulk temperature 35 44 The Flash Temperature flaϑ 35 441 Basic formula 35 442 Geometrical relations 35 443 Load sharing factor XΓ 36 Appendix A Fatigue Damage Accumulation 40 A1 Stress Spectrum 40 A2 σminusN-curve 40 A3 Damage accumulation 40 Appendix B Application Factors for Diesel Driven

Gears 41 B1 Definitions 41 B2 Determination of decisive load 41 B3 Simplified procedure 41 Appendix C Calculation of Pinion-Rack 42 C1 Pinion tooth root stresses 42 C2 Rack tooth root stresses 42 C3 Surface hardened pinions 42

4 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

1 Basic Principles and General Influence Factors

11 Scope and Basic Principles The gear rating procedures given in this Classification Note are mainly based on the ISO-6336 Part 1-5 (cylindrical gears) and partly on ISO 10300 Part 1-3 (bevel gears) and ISO Technical Reports on Scuffing and Fatigue Damage Accumulation but especially applied for marine purposes such as marine propulsion and important auxiliaries onboard ships and mobile offshore units

The calculation procedures cover gear rating as limited by contact stresses (pitting spalling or case crushing) tooth root stresses (fatigue breakage or overload breakage) and scuff-ing resistance Even though no calculation procedures for other damages such as wear grey staining (micropitting) etc are given such damages may limit the gear rating

The Classification Note applies to enclosed parallel shaft gears epicyclic gears and bevel gears (with intersecting axis) However open gear trains may be considered with regard to tooth strength ie part 1 and 3 may apply Even pinion-rack tooth strength may be considered but since such gear trains often are designed with non-involute pinions the calculation procedure of pinion-racks is described in Appen-dix C

Steel is the only material considered

The methods applied throughout this document are only valid for a transverse contact ratio 1 lt εα lt 2 If εα gt 2 either special considerations are to be made or suggested simplifi-cation may be used

All influence factors are defined regarding their physical interpretation Some of the influence factors are determined by the gear geometry or have been established by conven-tions These factors are to be calculated in accordance with the equations provided Other factors are approximations which are clearly stated in the text by terms as laquomay be cal-culated asraquo These approximations are substitutes for exact evaluations where such are lacking or too extensive for prac-tical purposes or factors based on experience In principle any suitable method may replace these approximations

Bevel gears are calculated on basis of virtual (equivalent) cylindrical gears using the geometry of the midsection The virtual (helical) cylindrical gear is to be calculated by using all the factors as a real cylindrical gear with some exceptions These exceptions are mentioned in connection with the ap-plicable factors Wherever a factor or calculation procedure has no reference to either cylindrical gears or bevel gears it is generally valid ie combined for both cylindrical and bevel

In order to minimise the volume of this Classification Note such combinations are widely used and everywhere it is necessary to distinguish it is clearly pointed out by local headings such as

Cylindrical gears

Bevel gears

The permissible contact stresses tooth root stresses and scuffing load capacity depend on the safety factors as re-quired in the respective Rule sections

Terms as endurance limit and static strength are used throughout this Classification Note

Endurance limit is to be understood as the fatigue strength in the range of cycles beyond the lower knee of the σndashN curves regardless if it is constant or drops with higher number of cycles

Static strength is to be understood as the fatigue strength in the range of cycles less than at the upper knee of the σndashN curves

For gears that are subjected to a limited number of cycles at different load levels a cumulative fatigue calculation applies Information on this is given in Appendix A

When the term infinite life is used it means number of cy-cles in the range 108ndash1010

12 Symbols Nomenclature and Units The symbols are generally from ISO 701 ISOR31 and ISO 1328 with a few additional symbols Only SI units are used

The main symbols as influence factors (K Z Y and X with indeces) etc are presented in their respective headings Symbols which are not explained in their respective Secs are as follows

a = centre distance (mm) b = facewidth (mm) d = reference diameter (mm) da = tip diameter (mm) db = base diameter (mm) dw = working pitch diameter (mm) ha = addendum (mm) ha0 = addendum of tool ref to mn hfp = dedendum of basic rack ref to mn (= ha0) hFe = bending moment arm (mm) for tooth root

stresses for application of load at the outer point of single tooth pair contact

hFa = bending moment arm (mm) for tooth root stresses for application of load at tooth tip

HB = Brinell hardness HV = Vickers hardness HRC = Rockwell C hardness mn = normal module n = rev per minute NL = number of load cycles qs = notch parameter Ra = average roughness value (microm) Ry = peak to valley roughness (microm) Rz = mean peak to valley roughness (microm) san = tooth top land thickness (mm) sat = transverse top land thickness (mm) sFn = tooth root chord (mm) in the critical section spr = protuberance value of tool minus grinding stock

equal residual undercut of basic rack ref to mn T = torque (Nm) u = gear ratio (per stage) v = linear speed (ms) at reference diameter

Classification Notes- No 412 5 May 2003

DET NORSKE VERITAS

x = addendum modification coefficient z = number of teeth zn = virtual number of spur teeth αn = normal pressure angle at ref cylinder αt = transverse pressure angle at ref cylinder αa = transverse pressure angle at tip cylinder αwt = transverse pressure angle at pitch cylinder β = helix angle at ref cylinder βb = helix angle at base cylinder βa = helix angle at tip cylinder εα = transverse contact ratio εβ = overlap ratio εγ = total contact ratio ρa0 = tip radius of tool ref to mn ρfp = root radius of basic rack ref to mn ( = ρa0) ρC = effective radius (mm) of curvature at pitch point ρF = root fillet radius (mm) in the critical section σB = ultimate tensile strength (Nmm2) σy = yield strength resp 02 proof stress (Nmm2)

Index 1 refers to the pinion 2 to the wheel

Index n refers to normal section or virtual spur gear of a heli-cal gear

Index w refers to pitch point

Special additional symbols for bevel gears are as follows

Σ = angle between intersection axis Kϑ = angle modification (Klingelnberg)

m0 = tool module (Klingelnberg) δ = pitch cone angle xsm = tooth thickness modification coefficient (mid-

face) R = pitch cone distance (mm)

Index v refers to the virtual (equivalent) helical cylindrical gear

Index m refers to the midsection of the bevel gear

13 Geometrical Definitions For internal gearing z2 a da2 dw2 d2 and db2 are negative x2 is positive if da2 is increased ie the numeric value is de-creased

The pinion has the smaller number of teeth ie

11

2 ge=zz

u

For calculation of surface durability b is the common face-width on pitch diameter

For tooth strength calculations b1 or b2 are facewidths at the respective tooth roots If b1 or b2 differ much from b above they are not to be taken more than 1 module on either side of b

Cylindrical gears

tan αt = tan αn cos β tan βb = tan β cos αt

tan βa = tan β da d

cos αa = dbda

d = z mn cos β mt = mn cos β db = d cos αt = dw cos αwt

a = 05 (dw1 + dw2)

dw1dw2 = z1 z2

inv α = tan α - α (radians)

inv αwt = inv αt + 2 tan αn (x1 + x2)(z1 + z2)

zn = z (cos2 βb cos β)

1

aw1fw1α T

ξξε

+=

where ξfw1 is to be taken as the smaller of

bull wtfw1 αtanξ =

bull soi1

b1wtfw1 d

dacostan -tanαξ =

bull 1

2wt

a2

b2fw1 z

ztanα

d

dacostan ξ

minus=

and

2

1fw2aw1 z

zξξ = where ξfw2 is calculated as ξfw1

substituting the values for the wheel by the values for the pinion and visa versa

11 z

2πT =

( ) +

sdot+minusminussdot= minus

2sinαρρxhm

2d2d nfpfp1fpnsoi1

21

t

nfpfplfpn2

tanα)sinαρρx(hm

sdot+minusminus

nmsinbπ

β=εβ

(for double helix b is to be taken as the width of one helix)

εy = βα εε +

ρC = ( )2

b

wt

u1βcos

αsinua

+

v = 311 10dn

60π minus

6 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

pbt = βcos

αcosmπ tn

sat =

minus++

at

ninvααinv

z

αtanx22

π

d a

san = aβcossat

14 Bevel Gear Conversion Formulae and Specific Formulae Conversion of bevel gears to virtual equivalent helical cylin-drical gears is based on the bevel gear midsection The con-version formulae are

Number of teeth

zv12 = z12 cos δ12

(δ1 + δ2 = Σ)

Gear ratio

uv = 1v

2vzz

tan αvt = tan αn cos βm

tan βbm = tan βm cos αvt

Base pitch

pbtm = m

vtnmβcos

αcosmπ

Reference pitch diameters

dv12 = 21

21m

cosdδ

Centre distance

av = 05 (dv1 + dv2)

Tip diameters

dva 12 = dv 12 + 2 ham 12

Addenda

for gears with constant addenda (Klingelnberg)

ham 12 = mmn (1 + xm 12)

for gears with variable addenda (Gleason)

ham 12 = ha 12 ndash b2 tan (δa 12 ndash δ12)

(when ha is addendum at outer end and δa is the outer cone angle)

Addendum modification coefficients

xm 12 = mn

12am21am

m2

hh minus

Base circle

dvb 12 = dv 12 cos αvt

Transverse contact ratio)

εα = btmP

αsinadd05dd05 vtv2

2vb2

2va2

1vb2

1va minusminus+minus

Overlap ratio) (theoretical value for bevel gears with no crowning but used as approximations in the calculation pro-cedures)

εβ = nm

mmπβsinb

Total contact ratio)

εγ = 2β

2α εε +

( Note that index laquovraquo is left out in order to combine formu-lae for cylindrical and bevel gears)

Tangential speed at midsection

vmt = 3m11 10dn

60π minus

Effective radius of curvature (normal section)

ρvc = ( )2vbm

vtvv

u1βcosαsinua

+

Length of line of contact

lb = ( )( )( )

1εifε

ε1ε2ε

βcosεb

β2γ

2βα

2γ

bm

α ltminusminusminus

lb = 1εifβcosε

εbβ

bmγ

α ge

15 Nominal Tangential Load Ft Fbt Fmt and Fmbt The nominal tangential load (tangential to the reference cyl-inder with diameter d and perpendicular to an axial plane) is calculated from the nominal (rated) torque T transmitted by the gear set

Cylindrical gears

dT2000

Ft = t

tbt αcos

FF =

Bevel gears

mmt d

T2000F =

vt

mtmbt αcos

FF =

Classification Notes- No 412 7 May 2003

DET NORSKE VERITAS

16 Application Factors KA and KAP The application factor KA accounts for dynamic overloads from sources external to the gearing

It is distinguished between the influence of repetitive cyclic torques KA (161) and the influence of temporary occasional peak torques KAP (162)

Calculations are always to be made with KA In certain cases additional calculations with KAP may be necessary

For gears with a defined load spectrum the calculation with a KA may be replaced by a fatigue damage calculation as given in Appendix A

161 KA For gears designed for long or infinite life at nominal rated torque KA is defined as the ratio between the maximum re-petitive cyclic torque applied to the gear set and nominal rated torque

This definition is suitable for main propulsion gears and most of the auxiliary gears

KA can be determined by measurements or system analysis or may be ruled by conventions (ice classes) (For the pur-pose of a preliminary (but not binding) calculation before KA is determined it is advised to apply either the max values mentioned below or values known from similar plants)

a) For main propulsion gears KA can be taken from the (mandatory) torsional vibration analysis thereby con-sidering all permissible driving conditions) Unless specially agreed the rules do not allow KA in excess of 135 for diesel propulsion) With turbine or electric propulsion KA would normally not exceed 12 However special attention should be given to thrusters that are arranged in such a way that heavy vessel movements andor manoeuvring can cause severe load fluctuations This means eg thrusters positioned far from the rolling axis of vessels that could be susceptible to rolling If leading to propeller air suction the condi-tions may be even worse The above mentioned movements or manoeuvring will result in increased propeller excitation If the thruster is driven by a diesel engine the engine mean torque is limited to 100 However thrusters driven by electric motors can suffer temporary mean torque much above 100 unless a suitable load control system (limiting available e-motor torque) is provided

b) For main propulsion gears with ice class notation (see Rules Pt5 Ch1 Sec J500) KA ice has to be taken as the higher value of the applicable (rule defined) ice shock torque referred to nominal rated torque and the value under a) The Baltic ice class notations refer to a few millions ice shock loads Thus the life factors may be put YN=1 and ZN=12 (except for nitrided gears where ZN = 1 applies) Additionally the calculations with the normal KA (no ice class) are to fulfil the normal requirements For polar ice class notations KA ice applies to all criteria and for long or infinite life

c) For a power take off (PTO) branch from a main propul-sion gear with ice class ice shocks result in negative torques It is assumed that the PTO branch is unloaded when the ice shock load occurs The influence of these reverse shock loads may be taken into account as follows The negative torque (reversed load) expressed by means of an application factor based on rated forward load (T or Ft) is KAreverse = KA ice ndash1 (the minus 1 be-cause no mean torque assumed) KAice to be calculated as in the ice class rules This KAreverse should be used for back flank considera-tions such as pitting and scuffing The influence on tooth bending strength (forward di-rection) may be simplified by using the factor YM = 1 minus 03 middot KAreverse KA

d) For diesel driven auxiliaries KA can be taken from the torsional vibration analysis if available For units where no vibration analysis is required (lt 200 kW) or available it is advised to apply KA as the upper allow-able value 135)

e) For turbine or electro driven auxiliaries the same as for c) applies however the practical upper value is 12

) For diesel driven gears more information on KA for mis-firing and normal driving is given in Appendix B

162 KAP The peak overload factor KAP is defined as the ratio between the temporary occasional peak overload torque and the nominal rated torque

For plants where high temporary occasional peak torques can occur (ie in excess of the above mentioned KA) the gearing (if nitrided) has to be checked with regard to static strength Unless otherwise specified the same safety factors as for infinite life apply

The scuffing safety is to be specially considered whereby the KA applies in connection with the bulk temperature and the KAP applies for the flash temperature calculation and should replace KA in the formulae in 431 432 and 441

KAP can be evaluated from the torsional impact vibration calculation (as required by the rules)

If the overloads have a duration corresponding to several revolutions of the shafts the scuffing safety has to be consid-ered on basis of this overload both with respect to bulk and flash temperature This applies to plants with ice class nota-tions (Baltic and polar) and to plants with prime movers which have high temporary overload capacity such as eg electric motors (provided the driven member can have a con-siderable increase in demand torque as eg azimuth thrusters during manoeuvring)

For plants without additional ice class notation KAP should normally not exceed 15

8 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

163 Frequent overloads For plants where high overloads or shock loads occur regu-larly the influence of this is to be considered by means of cumulative fatigue (see Appendix A)

17 Load Sharing Factor Kγ The load sharing factor Kγ accounts for the maldistribution of load in multiple-path transmissions (dual tandem epicyclic double helix etc) Kγ is defined as the ratio between the max load through an actual path and the evenly shared load

171 General method Kγ mainly depends on accuracy and flexibility of the branches (eg quill shaft planet support external forces etc) and should be considered on basis of measurements or of relevant analysis as eg

Kγ = δ+δ f

δ = total compliance of a branch under full load (assuming even load share) referred to gear mesh

f = minusminusminusminus+++ 23

22

21 fff where f1 f2 etc are the

main individual errors that may contribute to a maldis-tribution between the branches Eg tooth pitch errors planet carrier pitch errors bearing clearance influences etc Compensating effects should also be considered

For double helical gears

An external axial force Fext applied from sources outside the actual gearing (eg thrust via or from a tooth coupling) will cause a maldistribution of forces between the two helices Expressed by a load sharing factor the

βsdot

plusmn=γ tanFF1K

t

ext

If the direction of Fext is known the calculation should be carried out separately for each helix and with the tangential force corrected with the pertinent Kγ If the direction of Fext is unknown both combinations are to be calculated and the higher σH or σF to be used

172 Simplified method If no relevant analysis is available the following may apply

For epicyclic gears

Kγ = 32501 minus+ pln

where npl = number of planets ( gt3 )

For multistage gears with locked paths and gear stages sepa-rated by quill shafts (see figure below)

Figure 10 Locked paths gear

Kγ = ( )φ201+

where φ = quill shaft twist (degrees) under full load

18 Dynamic Factor Kv The dynamic factor Kv accounts for the internally generated dynamic loads

Kv is defined as the ratio between the maximum load that dynamically acts on the tooth flanks and the maximum ex-ternally applied load Ft KA Kγ

In the following 2 different methods (181 and 182) are described In case of controversy between the methods the next following is decisive ie the methods are listed with increasing priority

It is important to observe the limitations for the method in 181 In particular the influence of lateral stiffness of shafts is often underestimated and resonances occur at considerably lower speed than determined in 1811

However for low speed gears with vmiddotz1 lt 300 calculations may be omitted and the dynamic factor simplified to Kv=105

181 Single resonance method For a single stage gear Kv may be determined on basis of the relative proximity (or resonance ratio) N between actual speed n1 and the lowest resonance speed nE1

N = 1E

1

nn

Note that for epicyclic gears n is the relative speed ie the speed that multiplied with z gives the mesh frequency

1811 Determination of critical speed It is not advised to apply this method for multimesh gears for N gt 085 as the influence of higher modes has to be consid-ered see 182 In case of significant lateral shaft flexibility (eg overhung mounted bevel gears) the influence of cou-pled bending and torsional vibrations between pinion and wheel should be considered if N ge 075 see 182

nE1 = redmγc

1zπ

31030 sdot

where

cγ is the actual mesh stiffness per unit facewidth see 111

Classification Notes- No 412 9 May 2003

DET NORSKE VERITAS

For gears with inactive ends of the facewidth as eg due to high crowning or end relief such as often applied for bevel gears the use of cγ in connection with determination of natu-ral frequencies may need correction cγ is defined as stiffness per unit facewidth but when used in connection with the total mesh stiffness it is not as simple as cγmiddotb as only a part of the facewidth is active Such corrections are given in 111

mred is the reduced mass of the gear pair per unit facewidth and referred to the plane of contact

For a single gear stage where no significant inertias are closely connected to neither pinion nor wheel mred is calcu-lated as

mred = 21

21mm

mm+

The individual masses per unit facewidth are calculated as

m12 = 221b

21

)2d(b

I

where I is the polar moment of inertia (kgmm2)

The inertia of bevel gears may be approximated as discs with diameter equal the midface pitch diameter and width equal to b However if the shape of the pinion or wheel body differs much from this idealised cylinder the inertia should be cor-rected accordingly

For all kind of gears if a significant inertia (eg a clutch) is very rigidly connected to the pinion or wheel it should be added to that particular inertia (pinion or wheel) If there is a shaft piece between these inertias the torsional shaft stiffness alters the system into a 3-mass (or more) system This can be calculated as in 182 but also simplified as a 2-mass system calculated with only pinion and wheel masses

1812 Factors used for determination of Kv Non-dimensional gear accuracy dependent parameters

( )bKKF

yfcB

At

pptp

γ

minus=

( )bKKF

yFcBAt

ff

γ

α minus=

Non-dimensional tip relief parameter

bKKFcC

1BγAt

ak sdotsdot

sdotminus=

For gears of quality grade (ISO 1328) Q = 7 or coarser Bk = 1

For gears with Q le 6 and excessive tip relief Bk is limited to max 1

For gears (all quality grades) with tip relief of more than 2middotCeff (see 432) the reduction of αε has to be considered (see 443)

where

fpt = the single pitch deviation (ISO 1328) max of pinion or wheel

Fα = the total profile form deviation (ISO 1328) max of pinion or wheel (Note Fα is pt not available for bevel gears thus use Fα = fpt)

yp and yf = the respective running-in allowances and may be calculated similarly to yα in 112 ie the value of fpt is replaced by Fα for yf

cacute = the single tooth stiffness see 111

Ca = the amount of tip relief see 433 In case of different tip relief on pinion and wheel the value that results in the greater value of Bk is to be used If Ca is zero by design the value of running-in tip relief Cay (see 112) may be used in the above formula

1813 Kv in the subcritical range Cylindrical gears N le 085

Bevel gears N le 075

Kv = 1 + N K

K = Cv1 Bp + Cv2 Bf + Cv3 Bk

Cv1 accounts for the pitch error influence

Cv1 = 032

Cv2 accounts for profile error influence

Cv2 = 034 for γε le 2

Cv2 = 03ε

057

γ minus for γε gt 2

Cv3 accounts for the cyclic mesh stiffness variation

Cv3 = 023 for γε le 2

Cv3 = 156ε

0096

γ minus for γε gt 2

1814 Kv in the main resonance range Cylindrical gears 085 lt N le 115

Bevel gears 075 lt N le 125

Running in this range should preferably be avoided and is only allowed for high precision gears

Kv = 1 + Cv1 Bp + Cv2 Bf + Cv4 Bk

Cv4 accounts for the resonance condition with the cyclic mesh stiffness variation

Cv4 = 090 for γε le 2

Cv4 = 144ε

ε005057

γ

γ

minus

minus for γε gt 2

10 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

1815 Kv in the supercritical range Cylindrical gears N ge 15

Bevel gears N ge 15

Special care should be taken as to influence of higher vibra-tion modes andor influence of coupled bending (ie lateral shaft vibrations) and torsional vibrations between pinion and wheel These influences are not covered by the following approach

Kv = Cv5 Bp + Cv6 Bf + Cv7

Cv5 accounts for the pitch error influence

Cv5 = 047

Cv6 accounts for the profile error influence

Cv6 = 047 for εγ le 2

Cv6 = 174ε

012

γ minus for εγ gt 2

Cv7 relates the maximum externally applied tooth loading to the maximum tooth loading of ideal accurate gears operating in the supercritical speed sector when the circumferential vibration becomes very soft

Cv7 = 075 for εγ le 15

Cv7 = [ ] 8750)2(sin 0125 +minusβεπ for 15 lt εγ le 25

Cv7 = 10 for εγ gt 25

1816 Kv in the intermediate range Cylindrical gears 115 lt N lt 15

Bevel gears 125 lt N lt 15

Comments raised in 1814 and 1815 should be observed

Kv is determined by linear interpolation between Kv for N = 115 respectively 125 and N = 15 as

Cylindrical gears

( ) ( ) ( )[ ]51Nv151Nv51Nvv KK350

N51KK === minussdot

minus

+=

Bevel gears

( ) ( ) ( )[ ]51Nv251Nv51Nvv KK250

N51KK === minussdot

minus

+=

182 Multi-resonance method For high speed gear (vgt40 ms) for multimesh medium speed gears for gears with significant lateral shaft flexibility etc it is advised to determine Kv on basis of relevant dy-namic analysis

Incorporating lateral shaft compliance requires transforma-tion of even a simple pinion-wheel system into a lumped multi-mass system It is advised to incorporate all relevant inertias and torsional shaft stiffnesses into an equivalent (to pinion speed) system Thereby the mesh stiffness appears as an equivalent torsional stiffness

cγ b (db12)2 (Nmrad)

The natural frequencies are found by solving the set of dif-ferential equations (one equation per inertia) Note that for a gear put on a laterally flexible shaft the coupling bending-torsionals is arranged by introducing the gear mass and the lateral stiffness with its relation to the torsional displacement and torque in that shaft

Only the natural frequency (ies) having high relative dis-placement and relative torque through the actual pinion-wheel flexible element need(s) to be considered as critical frequency (ies)

Kv may be determined by means of the method mentioned in 181 thereby using N as the least favourable ratio (in case of more than one pinion-wheel dominated natural frequency) Ie the N-ratio that results in the highest Kv has to be consid-ered

The level of the dynamic factor may also be determined on basis of simulation technique using numeric time integration with relevant tooth stiffness variation and pitchprofile er-rors

19 Face Load Factors KHβ and KFβ The face load factors KHβ for contact stresses and for scuff-ing KFβ for tooth root stresses account for non-uniform load distribution across the facewidth

KHβ is defined as the ratio between the maximum load per unit facewidth and the mean load per unit facewidth

KFβ is defined as the ratio between the maximum tooth root stress per unit facewidth and the mean tooth root stress per unit facewidth The mean tooth root stress relates to the con-sidered facewidth b1 respectively b2

Note that facewidth in this context is the design facewidth b even if the ends are unloaded as often applies to eg bevel gears

The plane of contact is considered

191 Relations between KHβ and KFβ

KFβ = expβHK 2(hb)hb1

1exp++

=

where hb is the ratio tooth heightfacewidth The maximum of h1b1 and h2b2 is to be used but not higher than 13 For double helical gears use only the facewidth of one helix

If the tooth root facewidth (b1 or b2) is considerably wider than b the value of KFβ(1or2) is to be specially considered as it may even exceed KHβ

Eg in pinion-rack lifting systems for jack up rigs where b = b2 asymp mn and b1 asymp 3 mn the typical KHβ asymp KFβ2 asymp 1 and KFβ1 asymp 13

192 Measurement of face load factors Primarily

KFβ may be determined by a number of strain gauges distrib-uted over the facewidth Such strain gauges must be put in

Classification Notes- No 412 11 May 2003

DET NORSKE VERITAS

exactly the same position relative to the root fillet Relations in 191 apply for conversion to KHβ

Secondarily

KHβ may be evaluated by observed contact patterns on vari-ous defined load levels It is imperative that the various test loads are well defined Usually it is also necessary to evalu-ate the elastic deflections Some teeth at each 90 degrees are to be painted with a suitable lacquer Always consider the poorest of the contact patterns

After having run the gear for a suitable time at test load 1 (the lowest) observe the contact pattern with respect to ex-tension over the facewidth Evaluate that KHβ by means of the methods mentioned in this section Proceed in the same way for the next higher test load etc until there is a full face contact pattern From these data the initial mesh misalign-ment (ie without elastic deflections) can be found by ex-trapolation and then also the KHβ at design load can be found by calculation and extrapolation See example

Figure 11 Example of experimental determination of KHβ

It must be considered that inaccurate gears may accumulate a larger observed contact pattern than the actual single mesh to mesh contact patterns This is particularly important for lapped bevel gears Ground or hard metal hobbed bevel gears are assumed to present an accumulated contact pattern that is practically equal the actual single mesh to mesh con-tact patterns As a rough guidance the (observed) accumu-lated contact pattern of lapped bevel gears may be reduced by 10 in order to assess the single mesh to mesh contact pattern which is used in 199

193 Theoretical determination of KHβ The methods described in 193 to 198 may be used for cy-lindrical gears The principles may to some extent also be used for bevel gears but a more practical approach is given in 199

General For gears where the tooth contact pattern cannot be verified during assembly or under load all assumptions are to be well on the safe side

KHβ is to be determined in the plane of contact

The influence parameters considered in this method are

bull mean mesh stiffness cγ (see 111) (if necessary also variable stiffness over b)

bull mean unit load Fmb = Fbt KA Kγ Kvb (for double helical gears see 17 for use of Kγ)

bull misalignment fsh due to elastic deflections of shafts and gear bodies (both pinion and wheel)

bull misalignment fdefl due to elastic deflections of and work-ing positions in bearings

bull misalignment fbe due to bearing clearance tolerances bull misalignment fma due to manufacturing tolerances bull helix modifications as crowning end relief helix correc-

tion bull running in amount yβ (see 112)

In practice several other parameters such as centrifugal ex-pansion thermal expansion housing deflection etc contrib-ute to KHβ However these parameters are not taken into account unless in special cases when being considered as particularly important

When all or most of the am parameters are to be considered the most practical way to determine KHβ is by means of a graphical approach described in 1931

If cγ can be considered constant over the facewidth and no helix modifications apply KHβ can be determined analyti-cally as described in 1932

1931 Graphical method The graphical method utilises the superposition principle and is as follows

bull Calculate the mean mesh deflection Mδ as a function of

γm c and bF see 111

bull Draw a base line with length b and draw up a rectangu-lar with height δM (The area δM b is proportional to the transmitted force)

bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the areas above and below this zero line are equal

Figure 12 fsh balanced around zero line

bull Superimpose these ordinates of the fsh curve to the previ-ous load distribution curve (The area under this new load distribution curve is still δM b)

bull Calculate the bearing deflections andor working posi-tions in the bearings and evaluate the influence fdefl in the plane of contact This is a straight line and is bal-anced around a zero line as indicated in Fig 14 but with one distinct direction Superimpose these ordinates to the previous load distribution curve

12 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

bull The amount of crowning end relief or helix correction (defined in the plane of contact) is to be balanced around a zero line similarly to fsh

Figure 13 Crowning Cc balanced aound zero line

bull Superimpose these ordinates to the previous load distri-bution curve In case of high crowning etc as eg often applied to bevel gears the new load distribution curve may cross the base line (the real zero line) The result is areas with negative load that is not real as the load in those areas should be zero Thus corrective actions must be made but for practical reasons it may be postponed to after next operation

bull The amount of initial mesh misalignment bema ff + (defined in the plane of contact) is to be bal-

anced around a zero line If the direction of bema ff + is known (due to initial contact check) or if the direction of fbe is known due to design (eg overhang bevel pin-ion) this should be taken into account If direction un-known the influence of bema ff + in both directions as well as equal zero should be considered

Figure 14 fma+fbe in both directions balanced around zero line

Superimpose these ordinates to the previous load distri-bution curve This results in up to 3 different curves of which the one with the highest peak is to be chosen for further evaluation

bull If the chosen load distribution curve crosses the base line (ie mathematically negative load) the curve is to be corrected by adding the negative areas and dividing this with the active facewidth The (constant) ordinates of this rectangular correction area are to be subtracted from the positive part of the load distribution curve It is advisable to check that the area covered under this new load distribution curve is still equal δM b

bull If cγ cannot be considered as constant over b then cor-rect the ordinates of the load distribution curve with the local (on various positions over the facewidth) ratio between local mesh stiffness and average mesh stiffness cγ (average over the active facewidth only) Note that the result is to be a curve that covers the same area δM b as before

bull The influence of running in yβ is to be determined as in 112 whereby the value for Fβx is to be taken as twice the distance between the peak of the load distribution curve and δM

bull Determine

KHβ = M

β

δycurveofpeak minus

1932 Simplified analytical method for cylindrical gears The analytical approach is similar to 1931 but has a more limited application as cγ is assumed constant over the face-width and no helix modification applies

bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the area above and below this zero line are equal see Fig 12 The max positive ordinate is frac12∆fsh

bull Calculate the initial mesh alignment as Fβx= deflbemash ffff plusmnplusmnplusmn∆ The negative signs may only be used if this is justified andor verified by a contact pattern test Otherwise al-ways use positive signs If a negative sign is justified the value of Fβx is not to be taken less than the largest of each of these elements

bull Calculate the effective mesh misalignment as Fβy = Fβx - yβ (yβ see 112)

bull Determine

2KforF2

bFc1K βH

m

βγγH le+=β

or

2KforF

bFc2K Hβ

m

βγγH gt=β

where cγ as used here is the effective mesh stiffness see 111

194 Determination of fsh fsh is the mesh misalignment due to elastic deflections Usu-ally it is sufficient to consider the combined mesh deflection of the pinion body and shaft and the wheel shaft The calcu-lation is to be made in the plane of contact (of the considered gear mesh) and to consider all forces (incl axial) acting on the shafts Forces from other meshes can be parted into components parallel respectively vertical to the considered plane of contact Forces vertical to this plane of contact have no influence on fsh

It is advised to use following diameters for toothed elements

d + 2 x mn for bending and shear deflection

d + 2 mn (x ndash ha0 + 02) for torsional deflection

Usually fsh is calculated on basis of an evenly distributed load If the analysis of KHβ shows a considerable maldis-

Classification Notes- No 412 13 May 2003

DET NORSKE VERITAS

tribution in term of hard end contact or if it is known by other reasons that there exists a hard end contact the load should be correspondingly distributed when calculating fsh In fact the whole KHβ procedure can be used iteratively 2-3 iterations will be enough even for almost triangular load distributions

195 Determination of fdefl fdefl is the mesh misalignment in the plane of contact due to bearing deflections and working positions (housing deflec-tion may be included if determined)

First the journal working positions in the bearings are to be determined The influence of external moments and forces must be considered This is of special importance for twin pinion single output gears with all 3 shafts in one plane

For rolling bearings fdefl is further determined on basis of the elastic deflection of the bearings An elastic bearing deflec-tion depends on the bearing load and size and number of rolling elements Note that the bearing clearance tolerances are not included here

For fluid film bearings fdefl is further determined on basis of the lift and angular shift of the shafts due to lubrication oil film thickness Note that fbe takes into account the influence of the bearing clearance tolerance

When working positions bearing deflections and oil film lift are combined for all bearings the angular misalignment as projected into the plane of the contact is to be determined fdefl is this angular misalignment (radians) times the face-width

196 Determination of fbe fbe is the mesh misalignment in the plane of contact due to tolerances in bearing clearances In principle fbe and fdefl could be combined But as fdefl can be determined by analy-sis and has a distinct direction and fbe is dependent on toler-ances and in most cases has no distinct direction (ie + toler-ance) it is practicable to separate these two influences

Due to different bearing clearance tolerances in both pinion and wheel shafts the two shaft axis will have an angular mis-alignment in the plane of contact that is superimposed to the working positions determined in 195 fbe is the facewidth times this angular misalignment Note that fbe may have a distinct direction or be given as a + tolerance or a combina-tion of both For combination of + tolerance it is adviced to use

fbe= 22be

21be ff ++plusmn

fbe is particularly important for overhang designs for gears with widely different kinds of bearings on each side and when the bearings have wide tolerances on clearances In general it shall be possible to replace standard bearings with-out causing the real load distribution to exceed the design premises For slow speed gears with journal bearings the expected wear should also be considered

197 Determination of fma fma is the mesh misalignment due to manufacturing toler-ances (helix slope deviation) of pinion fHβ1 wheel fHβ2 and housing bore

For gear without specifically approved requirements to as-sembly control the value of fma is to be determined as

fma= 22Hβ

21Hβ ff +

For gears with specially approved assembly control the value of fma will depend on those specific requirements

198 Comments to various gear types For double helical gears KHβ is to be determined for both helices Usually an even load share between the helices can be assumed If not the calculation is to be made as de-scribed in 171

For planetary gears the free floating sun pinion suffers only twist no bending It must be noted that the total twist is the sum of the twist due to each mesh If the value of 1K γ ne this must be taken into account when calculating the total sun pinion twist (ie twist calculated with the force per mesh without Kγ and multiplied with the number of planets)

When planets are mounted on spherical bearings the mesh misalignments sun-planet respectively planet-annulus will be balanced Ie the misalignment will be the average between the two theoretical individual misalignments The faceload distribution on the flanks of the planets can take full advan-tage of this However as the sun and annulus mesh with several planets with possibly different lead errors the sun and annulus cannot obtain the above mentioned advantage to the full extent

199 Determination of KHβ for bevel gears If a theoretical approach similar to 193-198 is not docu-mented the following may be used

testeff

H Kb

b851851K sdot

minussdot=β

beff b represents the relative active facewidth (regarding lapped gears see 192 last part)

Higher values than beff b = 090 are normally not to be used in the formula

For dual directional gears it may be difficult to obtain a high beff b in both directions In that case the smaller beff b is to be used

Ktest represents the influence of the bearing arrangement shaft stiffness bearing stiffness housing stiffness etc on the faceload distribution and the verification thereof Expected variations in length- and height-wise tooth profile is also accounted for to some extent

a) Ktest = 1 For ground or hard metal hobbed gears with the specified contact pattern verified at full rating or at full torque slow turning at a condition representative for the thermal expansion at normal operation

14 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

It also applies when the bearing arrangementsupport has insignificant elastic deflections and thermal axial expansion However each initial mesh contact must be verified to be within acceptance criteria that are cali-brated against a type test at full load Reproduction of the gear tooth length- and height-wise profile must also be verified This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon)

b) Ktest = 1 + 04middot(beffbndash06) For designs with possible influence of thermal expansion in the axial direction of the pinion The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load

c) Ktest = 12 if mesh is only checked by toolmakerrsquos blue or by spin test contact For gears in this category beffb gt 085 is not to be used in the calculation

110 Transversal Load Distribution Factors KHα and KFα The transverse load distribution factors KHα for contact stresses and for scuffing KFα for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairs of teeth in mesh

The following relations may be used

Cylindrical gears

( )

minus+==

tH

αptγγHαFα F

byfc0409

2ε

KK

valid for 2εγ le

( ) ( )tH

αptγ

γ

γHαFα F

byfcε

1ε20409KK

minusminus+==

valid for 2ε γ gt

where

FtH = Ft KA Kγ Kv KHβ

cγ = See 111

γα = See 112

fpt = Maximum single pitch deviation (microm) of pinion or wheel or maximum total profile form devia-tion Fα of pinion or wheel if this is larger than the maximum single pitch deviation

Note In case of adequate equivalent tip relief adapted to the load half of the above mentioned fpt can be introduced A tip relief is considered adequate when the average of Ca1 and Ca2 is within plusmn40 of the value of Ceff in 432

Limitations of KHα and KFα

If the calculated values for

KFα = KHα lt 1 use KFα = KHα = 10

If the calculated value of KHα gt 2εα

γ

Zε

ε use KHα = 2

εα

γ

Zε

ε

If the calculated value of KFα gt εα

γ

Yεε

use KFα = εα

γ

Yεε

where Yε = αnε

075025+ (for εαn see 331c)

Bevel gears

For ground or hard metal hobbed gears KFα = KHα = 1

For lapped gears KFα = KHα = 11

111 Tooth Stiffness Constants cacute and cγ The tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having 1 mm facewidth by an amount of 1 microm in the plane of contact cacute is the maximum stiffness of a single pair of teeth cγ is the mean value of the mesh stiffness in a transverse plane (brief term mesh stiffness)

Both valid for high unit load (Unit load = Ft middot KA middot Kγb)

Cylindrical gears The real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bending stiff-nesses For high unit loads the Hertzian stiffness has little importance and can be disregarded This approach is on the safe side for determination of KHβ and KHα However for moderate or low loads Kv may be underestimated due to determination of a too high resonance speed

The linear approach is described in A

An optional approach for inclusion of the non-linear stiffness is described in B

A The linear approach

BRCCq

βcos08acutec =

and

( )025ε075cc αγ +prime=

where

( )[ ]n02a01a

B α2000212

hh12051C minusminus

+minus+=

12n1n

x000635z

025791z

015551004723q minus++=

12

2n

22

1n

1 x005290z

x024188x000193z

x011654+minusminusminus

+ 000182 x22

Classification Notes- No 412 15 May 2003

DET NORSKE VERITAS

(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as

( )( )nR m5s

sR e5

bbln1C +=

where

bs = thickness of a central web

sR = average thickness of rim (net value from tooth root to inside of rim)

The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered

Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ

B The non-linear approach

In the following an example is given on how to consider the non-linearity

The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm

With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as

( )10δKbF

minus= for 500bFgt

minus=

500Fb10δK

bF for 500

bFlt

with γAt KK

bF

bF

sdotsdot= etc (Nmm) ie unit load incorporat-

ing the relevant factors as

KA middot Kγ for determination of Kv

KA middot Kγ middot Kv for determination of KHβ

KA middot Kγ middot Kv middot KHβ for determination of KHα

δ = mesh deflection (microm)

K = applicable stiffness (c or cγ)

Use of stiffnesses for KV KHβ and KHα

For calculation of Kv and KHα the stiffness is calculated as follows

When Fb lt 500

the stiffness is determined as δ∆

∆ bF

where the increment is chosen as eg ∆ Fb = 10 and thus

50010Fb10

K10Fb∆δ +

++

=

When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly

or an equivalent stiffness determined as δsdotb

F

Bevel gears

In lack of more detailed relationship between stiffness and geometry the following may be used

b085

b13cacute eff=

b085b

16c effγ =

beff not to be used in excess of 085 b in these formulae

Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)

112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements

yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload

Cay is defined as the running-in amount that compensates for lack of tip relief

The following relations may be used

For not surface hardened steel

ptHlim

α fσ160y =

yβ = βxlimH

f320σ

with the following upper limits

V lt 5 ms 5-10 ms gt 10 ms

yα max none

limH

12800σ limH

6400σ

yβ max none

limH

25600σ limH

12800σ

For surface hardened steel

yα = 0075 fpt but not more than 3 for any speed

yβ = 015 Fβx but not more than 6 for any speed

16 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

For all kinds of steel

5145189718

1C2

limHay +

minusσ

=

When pinion and wheel material differ the following ap-plies

bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv

bull Use ( )2β1ββ yy21y += in the calculation of KHβ

bull Use ( )2ay1aya CC21C += in the calculation of Kv

bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing

calculation if no design tip relief is foreseen

Classification Notes- No 412 17 May 2003

DET NORSKE VERITAS

2 Calculation of Surface Durability

21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213

Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted

Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213

For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength

For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied

22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater

Calculation of surface durability for helical gears is based on the contact stress at the pitch point

For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies

Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact

Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses

The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses

The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel

221 Contact stress Cylindrical gears

( )HαHβvγA

1

tβεEHDBH KKKKK

bud1uF

ZZZZZσ+

=

where

ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)

ZH = Zone factor for pitch point (see 231)

ZE = Elasticity factor (see 24)

Zε = Contact ratio factor (see 25)

Zβ = Helix angle factor (see 26)

Ft KA Kγ Kv KHβ KHα see 15 ndash 110

d1 b u see 12 ndash 15

Bevel gears

( )HαHβvγA

v1v

vmtKEMH KKKKK

bud1uF

ZZZ105σ+

sdot=

where

105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)

ZE KA etc see above

ZM = mid-zone factor see 233

ZK = bevel gear factor see 27

Fmt dv1 uv see 12 ndash 15

It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks

222 Permissible contact stress

XWRvLH

NHlimHP ZZZZZ

SZσ

σ =

where

σH lim = Endurance limit for contact stresses (see 28)

ZN = Life factor for contact stresses (see 29)

SH = Required safety factor according to the rules

ZLZvZR = Oil film influence factors (see 210)

ZW = Work hardening factor (see 211)

ZX = Size factor (see 212)

18 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

23 Zone Factors ZH ZBD and ZM

231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder

wtt2

wtbH sinααcos

cosαcosβ2Z =

232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel

For εβ ge 1 ZBD = 1

For internal gears ZD = 1

For εβ = 0 (spur gears)

( )

π

minusεminusminus

π

minusminus

α=

α2

2

2b

2a

1

2

1b

1a

wtB

z211

dd

z21

dd

tanZ

( )

π

minusεminusminus

π

minusminus

α=

α1

2

1b

1a

2

2

2b

2a

wtD

z211

dd

z21

dd

tanZ

If ZB lt 1 use ZB = 1

If ZD lt 1 use ZD = 1

For 0 lt εβ lt 1

ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)

233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears

εminusminus

εminusminus

αβ=

αα btm2

2vb2

2vabtm2

1vb2val

2v1vvtbmM

pddpdd

ddtancos2Z

This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint

234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel

24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses

For steel against steel ZE = 1898

25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses

αε1Z =ε for 1εβ ge

( )α

ββ

αε ε

εε1

3ε4

Z +minusminus

= for εβ lt 1

26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability

cosβZβ =

27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply

The following may be used ZK = 080

28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ

σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting

For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply

For this purpose pitting is defined by

bull for not surface hardened gears pitted area ge 2 of total active flank area

bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given

material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies

The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as

Classification Notes- No 412 19 May 2003

DET NORSKE VERITAS

the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99

σHlim σH105 σH10

3

Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade

1650 1500

2500 2400

3100 3100

Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)

1000

13 σHlim

13 σHlim

Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV

Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV

Carbon steel 15 HV + 250 16 σHlim 16 σHlim

These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15

29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply

If this is not documented by approved fatigue tests the fol-lowing method may be used

For all steels except nitrided

7L 105N sdotge ZN = 1 or

01570

L

7

N N105Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 5middot107 510N

logZ037

L

7

N N105Z

sdot=

NL = 105 WXRVLHlim

WstX10H1010NN ZZZZZσ

ZZσZZ

55

5=

103 lt NL lt 105 )Z(Zlog05

L

5

10NN

5N103N10

5N10ZZ

==

3L 10N le

WXRVLlimH

Wst10X10H10NN ZZZZZ

ZZZZ

33

3σ

σ==

(but not less than ZN105)

For nitrided steels

6L 102N sdotge ZN = 1 or

00980

L

6

N N102Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 2middot106 510N

Zlog76860

L

6

N N102Z

sdot=

5L 10N le

XWRVL

10XWst10NN ZZZZZ

ZZ13ZZ

55 ==

Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles

210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity

The following methods may be applied in connection with the endurance limit

20 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Surface hardened steels Not surface hardened steels

ZL ( )24013421

360910ν+

+

( )24013421

680830ν+

+

ZV ( )v3280

140930+

+ ( )v3280300850

++

ZR

080

ZrelR3

150

ZrelR3

where

ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits

For values of ν40 gt 500 use ν40 = 500

RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm

RZrel

= ( ) 31

cZ2Z1 ρ

10RR50

+

RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)

For 5

L 10N le ZL ZV ZR = 10

211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface

The following approximation may be used for the endurance limit

Surface hardened steel against not surface hardened steel 150

ZeqW R

31700

130HB21Z

minus

minus=

where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130

RZeq = equivalent roughness

033

c40

066

ZS

ZHZHZEQ ρvν

15000RRRR

=

If RZeq gt 16 then use RZeq = 16

If RZeq lt 15 then use RZeq = 15

where

RZH = surface roughness of the hard member before run in

RZS = surface roughness of the soft member before run in

ν40 = see 210

If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem

Through hardened pinion against softer wheel

( )

minussdotminus+= 000829

HBHB0008981u1Z

2

1W

For 21HBHB

2

1 le use ZW = 1

For 71HBHB

2

1 gt use 17HBHB

2

1 =

For u gt 20 use u = 20

For static strength (lt 105 cycles)

Surface hardened against not surface hardened

ZWst = 105

Through hardened pinion against softer wheel

ZWst = 1

212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality

ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213

213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented

The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent

Classification Notes- No 412 21 May 2003

DET NORSKE VERITAS

stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically

The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence

The real Hertzian stresses σHR are determined as

For helical gears with εβ gt 1

σHR = σH

For helical gears with εβ lt 1 and spur gears

ε

α

ββ ε

ε+εminus

sdotσ=σZ

1

HHR

For bevel gears

KHHR Z

1σσ sdot=

The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)

The coordinates tz and HV are to be compared with the de-sign specification such as

bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400

and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)

In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods

For high cycle fatigue (gt3 106 cycles) the following applies

sdot+

minussdotsdotsdot= o90

05azt

05azt

cosSσ04HV

H

HHHR

applicable to 05a

zt

Hge

For 05at

H

z lt the value for 05at

H

z = applies

56300ρSσ12a cHHR

Hsdotsdot

sdot=

Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength

If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows

reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue

For static strength (lt103 cycles) the following applies

sdot+

minussdotsdotsdot= o90

07at

06a

zt

cosSσ019HV

Hst

z

HstHHR

applicable to 06at

Hst

z ge

56300ρSσa cHHR

Hstsdotsdot

=

In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies

For limited life fatigue (103 lt cycles lt 3106 )

For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance

The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram

minussdotminus

= sdot logN3477

logSlogSlogS

310H6103HHN

36 10H103H Slog86281Slog86280 sdot+sdot sdot

22 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3 Calculation of Tooth Strength

31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding

For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36

For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning

Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα

In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314

A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired

It should be noted that this part 3 does not cover fractures caused by

bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining

Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears

Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied

32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level

321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance

Normally the stresses for pinion and wheel are calculated as

Cylindrical gears

FαFβvγAβSFn

tF K K K K K Y Y Y

m bFσ =

where

YF = Tooth form factor (see 33)

YS = Stress correction factor (see 34)

Yβ = Helix angle factor (see 36)

Ft KA Kγ Kv KFβ KFα see 15 ndash 110

b see 13

Bevel gears

FαFβvγASaFamn

mtF K K K K K Y Y Y

m bFσ ε=

where

YFa = Tooth form factor see 33

YSa = Stress correction factor see 34

Yε = Contact ratio factor see 35

Fmt KA etc see 15 ndash 110

b see 13

322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is

CXRrelTrelTF

NMFEFP YYYY

SYY

δσ

=σ

Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN

where

σFE = Local tooth root bending endurance limit of reference test gear (see 37)

YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)

YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)

SF = Required safety factor according to the rules

YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)

YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)

Classification Notes- No 412 23 May 2003

DET NORSKE VERITAS

YX = Size factor (see 312)

YC = Case depth factor (see 313)

33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress

YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears

YFa applies to load application at the tooth tip and is used for bevel gears

Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears

Figure 31 External tooth in normal section

Figure 32 Internal tooth in normal section

Definitions

n

2

n

Fn

enFn

Fe

F

α cosms

α cosmh6

Y

=

n

2

n

Fn

anFn

Fa

Fa

α cosms

α cosmh6

Y

=

In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth

YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)

The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing

mn with mnm

zn with zvn

αt with αvt

β with βm

with undercut without undercut

Fig 33 Dimensions and basic rack profile of the teeth (finished profile)

Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless

331 Determination of parameters

( )n

n

prnfPnfP m

α cossαsin 1ρ

αtan h4πE

minusminusminusminus=

For external gears fPfP ρρ =

For internal gears ( )0z

195fPfP0

fPfP 10363156ρhxρρ

sdotminus+

+=

where

z0 = number of teeth of pinion cutter

x0 = addendum modification coefficient of pinion cutter

hfP = addendum of pinion cutter

ρfP = tip radius of pinion cutter

xhρG fpfp +minus=

24 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

τmE

2π

z2H

nnminus

minus=

with

3πτ = for external gears

6πτ = for internal gears

Htan zG2

nminusϑ=ϑ

(to be solved iteratively suitable start value6π

=ϑ for exter-

nal gears and 3π for internal gears)

a) Tooth root chord sFn For external gears

minus

ϑ+

ϑminus= ρ

cosG3

3πsinz

ms

fpnn

Fn

For bevel gears with a tooth thickness modification

xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn

For internal gears

minus

ϑ+

ϑminus= ρ

cosG

6πsinz

ms

fPnn

Fn

b) Root fillet radius ρF at 30ordm tangent

( )G2coszcosG2ρ

mρ

2n

2

fpn

F

minusϑϑ+=

c) Determination of bending moment arm hF dn = zn mn

b2α

αn βcosε

ε =

dan = dn + 2 ha

pbn = π mn cos αn

dbn = dn cos αn

( )4

d1εpzz

2dd

zz2d

2bn

2

αnbn

2bn

2an

en +

minusminus

minus=

en

bnen d

dcos arcα =

ennnn

e α invα invα x tan 22π

z1

minus+

+=γ

αFen = αen ndash γe

For external gears

( )

minus=

n

enFenee

n

Fe

mdαtan sin γ γcos

21

mh

]ρcos

G3πcosz fpn +

ϑminus

ϑminusminus

For internal gears

( ) minus

sdotsdotminus=

n

enFenee

n

Fe

mdα tanγ sinγ cos

21

mh

minus

ϑminus

ϑminussdot ρ

cosG3

6πcosz fPn

332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as

250ε205for 0666ε2366Y αnαnDT leleminus=

205εfor 10Y αnDT lt=

34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section

Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm

YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip

YS can be determined as follows

( )

+

+=L23121

1

sS qL01312Y

where Fe

Fn

hsL = and

F

Fns ρ2

sq = (see 33)

YSa can be calculated by replacing hFe with hFa in the above formulae

Note a) Range of validity 1 lt qs lt 8

In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered

b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with

Classification Notes- No 412 25 May 2003

DET NORSKE VERITAS

g

g

ρt

0613

13

minus

where

tg = depth of the grinding notch

ρg = radius of the grinding notch

c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles

35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears

The following may be used

Yε = 0625

36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank

The following may be used (β input in degrees)

Yβ = 1 ndash εβ β120

When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula

However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)

37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM

σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision

If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules

σFE

Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)

bull of specially approved high grade

1050

bull of normal grade

minus CrNiMo steels with approved process

1000

minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850

Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

840

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

720

Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)

07 HV + 300

Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)

025 σB + 125

Alloyed quenched and tempered steel 04 σB + 200

Carbon steel 025 σB + 250

Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40

1) These values are valid for a root radius

bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)

bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)

bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for

6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked

2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)

26 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears

YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0

YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst

For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316

The following method may be used within a stress ratio ndash12 lt R lt 05

381 For idlers planets and PTO with ice class

M1M1R1

1Yor Y MstM

+minus

minus=

where

R = stress ratio = min stress divided by max stress

For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12

For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as

branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus

For a power take off (PTO) with ice class see 161 c

M considers the mean stress influence on the endurance (or static) strength amplitudes

M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress

Following M values may be used

Endurance limit

Static strength

Case hardened 08 ndash 015 Ys 1) 07

If shot peened 04 06

Nitrided 03 03

Induction or flame hardened

04 06

Not surface hardened steel 03 05

Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M

The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches

382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106

For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers

For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used

383 For gears with shrinkage stresses and unidirectional load For endurance strength

FE

fitM σ

σM1

M21Y+

minus=

σFE is the endurance limit for R = 0

For static strength YMst = 1 and σfit accounted for in 39b

σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor

n

Ffit m

ρ215scf minus=

384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is

( ) ( ) FE

fitM σ

σR1M1

M2

M1M1R1

1Yminussdot+

minus

+minus

minus=

Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12

For static strength

M1M1R1

1YMst

+minus

minus=

The effect of σfit is accounted for in 39 b

Classification Notes- No 412 27 May 2003

DET NORSKE VERITAS

385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed

F

y

Sσ225

for not surface hardened fillets

FSHV5 for surface hardened fillets

39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles

Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN

If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used

Determination of the σ ndash N ndash curve

a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF

b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ

FstS1

FPstσ minussdotsdotsdot=

where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability

σFst

Alloyed case hardened steel 1) 2300

Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

1250

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

1050

Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)

18 HV + 800

Steel with not surface hardened fillets the smaller value of 2)

18 σB or 225 σy

1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet

c) Calculate YN as

NL gt 3middot106

YN = 1 or 001

L

6

N N103Y

sdot= ie Yn = 092 for 1010

The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2

103ltNLlt3middot106exp

L

6

N N103Y

sdot=

cycles6103forσcycles310forσlog02876exp

FP

FPst

sdot=

NL lt 103 cycles6103forσcycles310forσ

NYFP

FPst

sdot=

or simply use σFPst as mentioned in b) directly

Guidance on number of load cycles NL for various applica-tions

bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)

bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied

28 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage

YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit

The following method may be used

For endurance limit

for not surface hardened fillets

( )

024

s024

relT σ103133q21σ1012201351

Ysdotsdotminus

+sdotsdotminus+= minus

minus

δ

for all surface hardened fillets except nitrided

( )

106q21002451

Y srelT

++=δ

for nitrided fillets

( )

1347q2101421

Y srelT

++=δ

For static strength

for not surface hardened fillets1)

( )( )( )025

02

02502s

relTst σ3000821σ300 1Y0821Y

+

minus+=δ

for surface hardened fillets except nitrided

YδrelTst = 044 YS + 012

for nitrided fillets

YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not

exceed the yield point and thereby alter the residual stress level See also 39b footnote 2

311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness

YRrelT differs for endurance limit and static strength

The following method may be used

For endurance limit

YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels

YδrelT = 43 ndash 326 (Ry + 1)0005

for nitrided steels

For static strength

YRrelTst = 1 for all Ry and all materials

For a fillet without any longitudinal machining trace Ry asymp Rz

312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength

The following may be used

For endurance limit

YX = 1 for mn le 5 generally

YX = 103 ndash 0006 mn for 5 lt mn lt 30

YX = 085 for mn ge 30

for not surface hard-ened steels

YX = 105 ndash 001 mn for 5 lt mn ge 25

YX = 08 for mn ge 25

for surface hardened steels

For static strength

YXst = 1 for all mn and all materials

313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength

YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength

In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength

The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses

The following simplified method for YC may be used

YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)

YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10

For endurance limit

+

+=nFFE

C m02ρt31

σconstY

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

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FOREWORD DET NORSKE VERITAS is an autonomous and independent Foundation with the objective of safeguarding life property and the environment at sea and ashore

DET NORSKE VERITAS AS is a fully owned subsidiary Society of the Foundation It undertakes classification and certifica-tion of ships mobile offshore units fixed offshore structures facilities and systems for shipping and other industries The So-ciety also carries out research and development associated with these functions

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Classification Notes

Classification Notes are publications that give practical information on classification of ships and other objects Examples of design solutions calculation methods specifications of test procedures as well as acceptable repair methods for some compo-nents are given as interpretations of the more general rule requirements

A list of Classification Notes is found in the latest edition of the Introduction booklets to the rdquoRules for Classification of Shipsrdquo and the rdquoRules for Classification of High Speed Light Craft and Naval Surface Craftrdquo In ldquoRules for Classification of Fixed Offshore Installationsrdquo only those Classification Notes that are relevant for this type of structure have been listed

The list of Classification Notes is also included in the current ldquoClassification Services ndash Publicationsrdquo issued by the Society which is available on request All publications may be ordered from the Societyrsquos Web site httpexchangednvcom

Provisions

It is assumed that the execution of the provisions of this Classification Note is entrusted to appropriately qualified and experienced people for whose use it has been prepared

DET NORSKE VERITAS

CONTENTS

1 Basic Principles and General Influence Factors 4 11 Scope and Basic Principles 4 12 Symbols Nomenclature and Units 4 13 Geometrical Definitions5 14 Bevel Gear Conversion Formulae and Specific

Formulae6 15 Nominal Tangential Load Ft Fbt Fmt and Fmbt 6 16 Application Factors KA and KAP 7 161 KA 7 162 KAP7 163 Frequent overloads8 17 Load Sharing Factor Kγ 8 171 General method8 172 Simplified method 8 18 Dynamic Factor Kv 8 181 Single resonance method 8 182 Multi-resonance method 10 19 Face Load Factors KHβ and KFβ 10 191 Relations between KHβ and KFβ10 192 Measurement of face load factors 10 193 Theoretical determination of KHβ11 194 Determination of fsh 12 195 Determination of fdefl13 196 Determination of fbe 13 197 Determination of fma 13 198 Comments to various gear types 13 199 Determination of KHβ for bevel gears 13 110 Transversal Load Distribution Factors KHα and KFα14 111 Tooth Stiffness Constants cacute and cγ 14 112 Running-in Allowances 15 2 Calculation of Surface Durability17 21 Scope and General Remarks 17 22 Basic Equations 17 221 Contact stress 17 222 Permissible contact stress 17 23 Zone Factors ZH ZBD and ZM18 231 Zone factor ZH 18 232 Zone factors ZBD 18 233 Zone factor ZM 18 234 Inner contact point 18 24 Elasticity Factor ZE 18 25 Contact Ratio Factor Zε18 26 Helix Angle Factor Zβ18 27 Bevel Gear Factor ZK18 28 Values of Endurance Limit σHlim and Static

Strength 5H10σ 3H10σ 18 29 Life Factor ZN 19 210 Influence Factors on Lubrication Film ZL ZV and

ZR 19 211 Work Hardening Factor ZW 20 212 Size Factor ZX20 213 Subsurface Fatigue20 3 Calculation of Tooth Strength 22 31 Scope and General Remarks 22 32 Tooth Root Stresses 22 321 Local tooth root stress22 322 Permissible tooth root stress 22

33 Tooth Form Factors YF YFa 23 331 Determination of parameters 23 332 Gearing with εαn gt 2 24 34 Stress Correction Factors YS YSa 24 35 Contact Ratio Factor Yε 25 36 Helix Angle Factor Yβ 25 37 Values of Endurance Limit σFE 25 38 Mean stress influence Factor YM 26 381 For idlers planets and PTO with ice class 26 382 For gears with periodical change of rotational

direction 26 383 For gears with shrinkage stresses and unidirectional

load 26 384 For shrink-fitted idlers and planets 26 385 Additional requirements for peak loads 27 39 Life Factor YN 27 310 Relative Notch Sensitivity Factor YδrelT 28 311 Relative Surface Condition Factor YRrelT 28 312 Size Factor YX 28 313 Case Depth Factor YC 28 314 Thin rim factor YB 29 315 Stresses in Thin Rims 29 3151 General 29 3152 Stress concentration factors at the 75ordm tangents 30 3153 Nominal rim stresses 30 3154 Root fillet stresses 30 316 Permissible Stresses in Thin Rims 31 3161 General 31 3162 For gt3middot106 cycles 31 3163 For le 103 cycles 31 3164 For 103 lt cycles lt 3106 32 4 Calculation of Scuffing Load Capacity 33 41 Introduction 33 42 General Criteria 33 43 Influence Factors 34 431 Coefficient of friction 34 432 Effective tip relief Ceff 34 433 Tip relief and extension 34 434 Bulk temperature 35 44 The Flash Temperature flaϑ 35 441 Basic formula 35 442 Geometrical relations 35 443 Load sharing factor XΓ 36 Appendix A Fatigue Damage Accumulation 40 A1 Stress Spectrum 40 A2 σminusN-curve 40 A3 Damage accumulation 40 Appendix B Application Factors for Diesel Driven

Gears 41 B1 Definitions 41 B2 Determination of decisive load 41 B3 Simplified procedure 41 Appendix C Calculation of Pinion-Rack 42 C1 Pinion tooth root stresses 42 C2 Rack tooth root stresses 42 C3 Surface hardened pinions 42

4 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

1 Basic Principles and General Influence Factors

11 Scope and Basic Principles The gear rating procedures given in this Classification Note are mainly based on the ISO-6336 Part 1-5 (cylindrical gears) and partly on ISO 10300 Part 1-3 (bevel gears) and ISO Technical Reports on Scuffing and Fatigue Damage Accumulation but especially applied for marine purposes such as marine propulsion and important auxiliaries onboard ships and mobile offshore units

The calculation procedures cover gear rating as limited by contact stresses (pitting spalling or case crushing) tooth root stresses (fatigue breakage or overload breakage) and scuff-ing resistance Even though no calculation procedures for other damages such as wear grey staining (micropitting) etc are given such damages may limit the gear rating

The Classification Note applies to enclosed parallel shaft gears epicyclic gears and bevel gears (with intersecting axis) However open gear trains may be considered with regard to tooth strength ie part 1 and 3 may apply Even pinion-rack tooth strength may be considered but since such gear trains often are designed with non-involute pinions the calculation procedure of pinion-racks is described in Appen-dix C

Steel is the only material considered

The methods applied throughout this document are only valid for a transverse contact ratio 1 lt εα lt 2 If εα gt 2 either special considerations are to be made or suggested simplifi-cation may be used

All influence factors are defined regarding their physical interpretation Some of the influence factors are determined by the gear geometry or have been established by conven-tions These factors are to be calculated in accordance with the equations provided Other factors are approximations which are clearly stated in the text by terms as laquomay be cal-culated asraquo These approximations are substitutes for exact evaluations where such are lacking or too extensive for prac-tical purposes or factors based on experience In principle any suitable method may replace these approximations

Bevel gears are calculated on basis of virtual (equivalent) cylindrical gears using the geometry of the midsection The virtual (helical) cylindrical gear is to be calculated by using all the factors as a real cylindrical gear with some exceptions These exceptions are mentioned in connection with the ap-plicable factors Wherever a factor or calculation procedure has no reference to either cylindrical gears or bevel gears it is generally valid ie combined for both cylindrical and bevel

In order to minimise the volume of this Classification Note such combinations are widely used and everywhere it is necessary to distinguish it is clearly pointed out by local headings such as

Cylindrical gears

Bevel gears

The permissible contact stresses tooth root stresses and scuffing load capacity depend on the safety factors as re-quired in the respective Rule sections

Terms as endurance limit and static strength are used throughout this Classification Note

Endurance limit is to be understood as the fatigue strength in the range of cycles beyond the lower knee of the σndashN curves regardless if it is constant or drops with higher number of cycles

Static strength is to be understood as the fatigue strength in the range of cycles less than at the upper knee of the σndashN curves

For gears that are subjected to a limited number of cycles at different load levels a cumulative fatigue calculation applies Information on this is given in Appendix A

When the term infinite life is used it means number of cy-cles in the range 108ndash1010

12 Symbols Nomenclature and Units The symbols are generally from ISO 701 ISOR31 and ISO 1328 with a few additional symbols Only SI units are used

The main symbols as influence factors (K Z Y and X with indeces) etc are presented in their respective headings Symbols which are not explained in their respective Secs are as follows

a = centre distance (mm) b = facewidth (mm) d = reference diameter (mm) da = tip diameter (mm) db = base diameter (mm) dw = working pitch diameter (mm) ha = addendum (mm) ha0 = addendum of tool ref to mn hfp = dedendum of basic rack ref to mn (= ha0) hFe = bending moment arm (mm) for tooth root

stresses for application of load at the outer point of single tooth pair contact

hFa = bending moment arm (mm) for tooth root stresses for application of load at tooth tip

HB = Brinell hardness HV = Vickers hardness HRC = Rockwell C hardness mn = normal module n = rev per minute NL = number of load cycles qs = notch parameter Ra = average roughness value (microm) Ry = peak to valley roughness (microm) Rz = mean peak to valley roughness (microm) san = tooth top land thickness (mm) sat = transverse top land thickness (mm) sFn = tooth root chord (mm) in the critical section spr = protuberance value of tool minus grinding stock

equal residual undercut of basic rack ref to mn T = torque (Nm) u = gear ratio (per stage) v = linear speed (ms) at reference diameter

Classification Notes- No 412 5 May 2003

DET NORSKE VERITAS

x = addendum modification coefficient z = number of teeth zn = virtual number of spur teeth αn = normal pressure angle at ref cylinder αt = transverse pressure angle at ref cylinder αa = transverse pressure angle at tip cylinder αwt = transverse pressure angle at pitch cylinder β = helix angle at ref cylinder βb = helix angle at base cylinder βa = helix angle at tip cylinder εα = transverse contact ratio εβ = overlap ratio εγ = total contact ratio ρa0 = tip radius of tool ref to mn ρfp = root radius of basic rack ref to mn ( = ρa0) ρC = effective radius (mm) of curvature at pitch point ρF = root fillet radius (mm) in the critical section σB = ultimate tensile strength (Nmm2) σy = yield strength resp 02 proof stress (Nmm2)

Index 1 refers to the pinion 2 to the wheel

Index n refers to normal section or virtual spur gear of a heli-cal gear

Index w refers to pitch point

Special additional symbols for bevel gears are as follows

Σ = angle between intersection axis Kϑ = angle modification (Klingelnberg)

m0 = tool module (Klingelnberg) δ = pitch cone angle xsm = tooth thickness modification coefficient (mid-

face) R = pitch cone distance (mm)

Index v refers to the virtual (equivalent) helical cylindrical gear

Index m refers to the midsection of the bevel gear

13 Geometrical Definitions For internal gearing z2 a da2 dw2 d2 and db2 are negative x2 is positive if da2 is increased ie the numeric value is de-creased

The pinion has the smaller number of teeth ie

11

2 ge=zz

u

For calculation of surface durability b is the common face-width on pitch diameter

For tooth strength calculations b1 or b2 are facewidths at the respective tooth roots If b1 or b2 differ much from b above they are not to be taken more than 1 module on either side of b

Cylindrical gears

tan αt = tan αn cos β tan βb = tan β cos αt

tan βa = tan β da d

cos αa = dbda

d = z mn cos β mt = mn cos β db = d cos αt = dw cos αwt

a = 05 (dw1 + dw2)

dw1dw2 = z1 z2

inv α = tan α - α (radians)

inv αwt = inv αt + 2 tan αn (x1 + x2)(z1 + z2)

zn = z (cos2 βb cos β)

1

aw1fw1α T

ξξε

+=

where ξfw1 is to be taken as the smaller of

bull wtfw1 αtanξ =

bull soi1

b1wtfw1 d

dacostan -tanαξ =

bull 1

2wt

a2

b2fw1 z

ztanα

d

dacostan ξ

minus=

and

2

1fw2aw1 z

zξξ = where ξfw2 is calculated as ξfw1

substituting the values for the wheel by the values for the pinion and visa versa

11 z

2πT =

( ) +

sdot+minusminussdot= minus

2sinαρρxhm

2d2d nfpfp1fpnsoi1

21

t

nfpfplfpn2

tanα)sinαρρx(hm

sdot+minusminus

nmsinbπ

β=εβ

(for double helix b is to be taken as the width of one helix)

εy = βα εε +

ρC = ( )2

b

wt

u1βcos

αsinua

+

v = 311 10dn

60π minus

6 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

pbt = βcos

αcosmπ tn

sat =

minus++

at

ninvααinv

z

αtanx22

π

d a

san = aβcossat

14 Bevel Gear Conversion Formulae and Specific Formulae Conversion of bevel gears to virtual equivalent helical cylin-drical gears is based on the bevel gear midsection The con-version formulae are

Number of teeth

zv12 = z12 cos δ12

(δ1 + δ2 = Σ)

Gear ratio

uv = 1v

2vzz

tan αvt = tan αn cos βm

tan βbm = tan βm cos αvt

Base pitch

pbtm = m

vtnmβcos

αcosmπ

Reference pitch diameters

dv12 = 21

21m

cosdδ

Centre distance

av = 05 (dv1 + dv2)

Tip diameters

dva 12 = dv 12 + 2 ham 12

Addenda

for gears with constant addenda (Klingelnberg)

ham 12 = mmn (1 + xm 12)

for gears with variable addenda (Gleason)

ham 12 = ha 12 ndash b2 tan (δa 12 ndash δ12)

(when ha is addendum at outer end and δa is the outer cone angle)

Addendum modification coefficients

xm 12 = mn

12am21am

m2

hh minus

Base circle

dvb 12 = dv 12 cos αvt

Transverse contact ratio)

εα = btmP

αsinadd05dd05 vtv2

2vb2

2va2

1vb2

1va minusminus+minus

Overlap ratio) (theoretical value for bevel gears with no crowning but used as approximations in the calculation pro-cedures)

εβ = nm

mmπβsinb

Total contact ratio)

εγ = 2β

2α εε +

( Note that index laquovraquo is left out in order to combine formu-lae for cylindrical and bevel gears)

Tangential speed at midsection

vmt = 3m11 10dn

60π minus

Effective radius of curvature (normal section)

ρvc = ( )2vbm

vtvv

u1βcosαsinua

+

Length of line of contact

lb = ( )( )( )

1εifε

ε1ε2ε

βcosεb

β2γ

2βα

2γ

bm

α ltminusminusminus

lb = 1εifβcosε

εbβ

bmγ

α ge

15 Nominal Tangential Load Ft Fbt Fmt and Fmbt The nominal tangential load (tangential to the reference cyl-inder with diameter d and perpendicular to an axial plane) is calculated from the nominal (rated) torque T transmitted by the gear set

Cylindrical gears

dT2000

Ft = t

tbt αcos

FF =

Bevel gears

mmt d

T2000F =

vt

mtmbt αcos

FF =

Classification Notes- No 412 7 May 2003

DET NORSKE VERITAS

16 Application Factors KA and KAP The application factor KA accounts for dynamic overloads from sources external to the gearing

It is distinguished between the influence of repetitive cyclic torques KA (161) and the influence of temporary occasional peak torques KAP (162)

Calculations are always to be made with KA In certain cases additional calculations with KAP may be necessary

For gears with a defined load spectrum the calculation with a KA may be replaced by a fatigue damage calculation as given in Appendix A

161 KA For gears designed for long or infinite life at nominal rated torque KA is defined as the ratio between the maximum re-petitive cyclic torque applied to the gear set and nominal rated torque

This definition is suitable for main propulsion gears and most of the auxiliary gears

KA can be determined by measurements or system analysis or may be ruled by conventions (ice classes) (For the pur-pose of a preliminary (but not binding) calculation before KA is determined it is advised to apply either the max values mentioned below or values known from similar plants)

a) For main propulsion gears KA can be taken from the (mandatory) torsional vibration analysis thereby con-sidering all permissible driving conditions) Unless specially agreed the rules do not allow KA in excess of 135 for diesel propulsion) With turbine or electric propulsion KA would normally not exceed 12 However special attention should be given to thrusters that are arranged in such a way that heavy vessel movements andor manoeuvring can cause severe load fluctuations This means eg thrusters positioned far from the rolling axis of vessels that could be susceptible to rolling If leading to propeller air suction the condi-tions may be even worse The above mentioned movements or manoeuvring will result in increased propeller excitation If the thruster is driven by a diesel engine the engine mean torque is limited to 100 However thrusters driven by electric motors can suffer temporary mean torque much above 100 unless a suitable load control system (limiting available e-motor torque) is provided

b) For main propulsion gears with ice class notation (see Rules Pt5 Ch1 Sec J500) KA ice has to be taken as the higher value of the applicable (rule defined) ice shock torque referred to nominal rated torque and the value under a) The Baltic ice class notations refer to a few millions ice shock loads Thus the life factors may be put YN=1 and ZN=12 (except for nitrided gears where ZN = 1 applies) Additionally the calculations with the normal KA (no ice class) are to fulfil the normal requirements For polar ice class notations KA ice applies to all criteria and for long or infinite life

c) For a power take off (PTO) branch from a main propul-sion gear with ice class ice shocks result in negative torques It is assumed that the PTO branch is unloaded when the ice shock load occurs The influence of these reverse shock loads may be taken into account as follows The negative torque (reversed load) expressed by means of an application factor based on rated forward load (T or Ft) is KAreverse = KA ice ndash1 (the minus 1 be-cause no mean torque assumed) KAice to be calculated as in the ice class rules This KAreverse should be used for back flank considera-tions such as pitting and scuffing The influence on tooth bending strength (forward di-rection) may be simplified by using the factor YM = 1 minus 03 middot KAreverse KA

d) For diesel driven auxiliaries KA can be taken from the torsional vibration analysis if available For units where no vibration analysis is required (lt 200 kW) or available it is advised to apply KA as the upper allow-able value 135)

e) For turbine or electro driven auxiliaries the same as for c) applies however the practical upper value is 12

) For diesel driven gears more information on KA for mis-firing and normal driving is given in Appendix B

162 KAP The peak overload factor KAP is defined as the ratio between the temporary occasional peak overload torque and the nominal rated torque

For plants where high temporary occasional peak torques can occur (ie in excess of the above mentioned KA) the gearing (if nitrided) has to be checked with regard to static strength Unless otherwise specified the same safety factors as for infinite life apply

The scuffing safety is to be specially considered whereby the KA applies in connection with the bulk temperature and the KAP applies for the flash temperature calculation and should replace KA in the formulae in 431 432 and 441

KAP can be evaluated from the torsional impact vibration calculation (as required by the rules)

If the overloads have a duration corresponding to several revolutions of the shafts the scuffing safety has to be consid-ered on basis of this overload both with respect to bulk and flash temperature This applies to plants with ice class nota-tions (Baltic and polar) and to plants with prime movers which have high temporary overload capacity such as eg electric motors (provided the driven member can have a con-siderable increase in demand torque as eg azimuth thrusters during manoeuvring)

For plants without additional ice class notation KAP should normally not exceed 15

8 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

163 Frequent overloads For plants where high overloads or shock loads occur regu-larly the influence of this is to be considered by means of cumulative fatigue (see Appendix A)

17 Load Sharing Factor Kγ The load sharing factor Kγ accounts for the maldistribution of load in multiple-path transmissions (dual tandem epicyclic double helix etc) Kγ is defined as the ratio between the max load through an actual path and the evenly shared load

171 General method Kγ mainly depends on accuracy and flexibility of the branches (eg quill shaft planet support external forces etc) and should be considered on basis of measurements or of relevant analysis as eg

Kγ = δ+δ f

δ = total compliance of a branch under full load (assuming even load share) referred to gear mesh

f = minusminusminusminus+++ 23

22

21 fff where f1 f2 etc are the

main individual errors that may contribute to a maldis-tribution between the branches Eg tooth pitch errors planet carrier pitch errors bearing clearance influences etc Compensating effects should also be considered

For double helical gears

An external axial force Fext applied from sources outside the actual gearing (eg thrust via or from a tooth coupling) will cause a maldistribution of forces between the two helices Expressed by a load sharing factor the

βsdot

plusmn=γ tanFF1K

t

ext

If the direction of Fext is known the calculation should be carried out separately for each helix and with the tangential force corrected with the pertinent Kγ If the direction of Fext is unknown both combinations are to be calculated and the higher σH or σF to be used

172 Simplified method If no relevant analysis is available the following may apply

For epicyclic gears

Kγ = 32501 minus+ pln

where npl = number of planets ( gt3 )

For multistage gears with locked paths and gear stages sepa-rated by quill shafts (see figure below)

Figure 10 Locked paths gear

Kγ = ( )φ201+

where φ = quill shaft twist (degrees) under full load

18 Dynamic Factor Kv The dynamic factor Kv accounts for the internally generated dynamic loads

Kv is defined as the ratio between the maximum load that dynamically acts on the tooth flanks and the maximum ex-ternally applied load Ft KA Kγ

In the following 2 different methods (181 and 182) are described In case of controversy between the methods the next following is decisive ie the methods are listed with increasing priority

It is important to observe the limitations for the method in 181 In particular the influence of lateral stiffness of shafts is often underestimated and resonances occur at considerably lower speed than determined in 1811

However for low speed gears with vmiddotz1 lt 300 calculations may be omitted and the dynamic factor simplified to Kv=105

181 Single resonance method For a single stage gear Kv may be determined on basis of the relative proximity (or resonance ratio) N between actual speed n1 and the lowest resonance speed nE1

N = 1E

1

nn

Note that for epicyclic gears n is the relative speed ie the speed that multiplied with z gives the mesh frequency

1811 Determination of critical speed It is not advised to apply this method for multimesh gears for N gt 085 as the influence of higher modes has to be consid-ered see 182 In case of significant lateral shaft flexibility (eg overhung mounted bevel gears) the influence of cou-pled bending and torsional vibrations between pinion and wheel should be considered if N ge 075 see 182

nE1 = redmγc

1zπ

31030 sdot

where

cγ is the actual mesh stiffness per unit facewidth see 111

Classification Notes- No 412 9 May 2003

DET NORSKE VERITAS

For gears with inactive ends of the facewidth as eg due to high crowning or end relief such as often applied for bevel gears the use of cγ in connection with determination of natu-ral frequencies may need correction cγ is defined as stiffness per unit facewidth but when used in connection with the total mesh stiffness it is not as simple as cγmiddotb as only a part of the facewidth is active Such corrections are given in 111

mred is the reduced mass of the gear pair per unit facewidth and referred to the plane of contact

For a single gear stage where no significant inertias are closely connected to neither pinion nor wheel mred is calcu-lated as

mred = 21

21mm

mm+

The individual masses per unit facewidth are calculated as

m12 = 221b

21

)2d(b

I

where I is the polar moment of inertia (kgmm2)

The inertia of bevel gears may be approximated as discs with diameter equal the midface pitch diameter and width equal to b However if the shape of the pinion or wheel body differs much from this idealised cylinder the inertia should be cor-rected accordingly

For all kind of gears if a significant inertia (eg a clutch) is very rigidly connected to the pinion or wheel it should be added to that particular inertia (pinion or wheel) If there is a shaft piece between these inertias the torsional shaft stiffness alters the system into a 3-mass (or more) system This can be calculated as in 182 but also simplified as a 2-mass system calculated with only pinion and wheel masses

1812 Factors used for determination of Kv Non-dimensional gear accuracy dependent parameters

( )bKKF

yfcB

At

pptp

γ

minus=

( )bKKF

yFcBAt

ff

γ

α minus=

Non-dimensional tip relief parameter

bKKFcC

1BγAt

ak sdotsdot

sdotminus=

For gears of quality grade (ISO 1328) Q = 7 or coarser Bk = 1

For gears with Q le 6 and excessive tip relief Bk is limited to max 1

For gears (all quality grades) with tip relief of more than 2middotCeff (see 432) the reduction of αε has to be considered (see 443)

where

fpt = the single pitch deviation (ISO 1328) max of pinion or wheel

Fα = the total profile form deviation (ISO 1328) max of pinion or wheel (Note Fα is pt not available for bevel gears thus use Fα = fpt)

yp and yf = the respective running-in allowances and may be calculated similarly to yα in 112 ie the value of fpt is replaced by Fα for yf

cacute = the single tooth stiffness see 111

Ca = the amount of tip relief see 433 In case of different tip relief on pinion and wheel the value that results in the greater value of Bk is to be used If Ca is zero by design the value of running-in tip relief Cay (see 112) may be used in the above formula

1813 Kv in the subcritical range Cylindrical gears N le 085

Bevel gears N le 075

Kv = 1 + N K

K = Cv1 Bp + Cv2 Bf + Cv3 Bk

Cv1 accounts for the pitch error influence

Cv1 = 032

Cv2 accounts for profile error influence

Cv2 = 034 for γε le 2

Cv2 = 03ε

057

γ minus for γε gt 2

Cv3 accounts for the cyclic mesh stiffness variation

Cv3 = 023 for γε le 2

Cv3 = 156ε

0096

γ minus for γε gt 2

1814 Kv in the main resonance range Cylindrical gears 085 lt N le 115

Bevel gears 075 lt N le 125

Running in this range should preferably be avoided and is only allowed for high precision gears

Kv = 1 + Cv1 Bp + Cv2 Bf + Cv4 Bk

Cv4 accounts for the resonance condition with the cyclic mesh stiffness variation

Cv4 = 090 for γε le 2

Cv4 = 144ε

ε005057

γ

γ

minus

minus for γε gt 2

10 Classification Notes- No 412

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DET NORSKE VERITAS

1815 Kv in the supercritical range Cylindrical gears N ge 15

Bevel gears N ge 15

Special care should be taken as to influence of higher vibra-tion modes andor influence of coupled bending (ie lateral shaft vibrations) and torsional vibrations between pinion and wheel These influences are not covered by the following approach

Kv = Cv5 Bp + Cv6 Bf + Cv7

Cv5 accounts for the pitch error influence

Cv5 = 047

Cv6 accounts for the profile error influence

Cv6 = 047 for εγ le 2

Cv6 = 174ε

012

γ minus for εγ gt 2

Cv7 relates the maximum externally applied tooth loading to the maximum tooth loading of ideal accurate gears operating in the supercritical speed sector when the circumferential vibration becomes very soft

Cv7 = 075 for εγ le 15

Cv7 = [ ] 8750)2(sin 0125 +minusβεπ for 15 lt εγ le 25

Cv7 = 10 for εγ gt 25

1816 Kv in the intermediate range Cylindrical gears 115 lt N lt 15

Bevel gears 125 lt N lt 15

Comments raised in 1814 and 1815 should be observed

Kv is determined by linear interpolation between Kv for N = 115 respectively 125 and N = 15 as

Cylindrical gears

( ) ( ) ( )[ ]51Nv151Nv51Nvv KK350

N51KK === minussdot

minus

+=

Bevel gears

( ) ( ) ( )[ ]51Nv251Nv51Nvv KK250

N51KK === minussdot

minus

+=

182 Multi-resonance method For high speed gear (vgt40 ms) for multimesh medium speed gears for gears with significant lateral shaft flexibility etc it is advised to determine Kv on basis of relevant dy-namic analysis

Incorporating lateral shaft compliance requires transforma-tion of even a simple pinion-wheel system into a lumped multi-mass system It is advised to incorporate all relevant inertias and torsional shaft stiffnesses into an equivalent (to pinion speed) system Thereby the mesh stiffness appears as an equivalent torsional stiffness

cγ b (db12)2 (Nmrad)

The natural frequencies are found by solving the set of dif-ferential equations (one equation per inertia) Note that for a gear put on a laterally flexible shaft the coupling bending-torsionals is arranged by introducing the gear mass and the lateral stiffness with its relation to the torsional displacement and torque in that shaft

Only the natural frequency (ies) having high relative dis-placement and relative torque through the actual pinion-wheel flexible element need(s) to be considered as critical frequency (ies)

Kv may be determined by means of the method mentioned in 181 thereby using N as the least favourable ratio (in case of more than one pinion-wheel dominated natural frequency) Ie the N-ratio that results in the highest Kv has to be consid-ered

The level of the dynamic factor may also be determined on basis of simulation technique using numeric time integration with relevant tooth stiffness variation and pitchprofile er-rors

19 Face Load Factors KHβ and KFβ The face load factors KHβ for contact stresses and for scuff-ing KFβ for tooth root stresses account for non-uniform load distribution across the facewidth

KHβ is defined as the ratio between the maximum load per unit facewidth and the mean load per unit facewidth

KFβ is defined as the ratio between the maximum tooth root stress per unit facewidth and the mean tooth root stress per unit facewidth The mean tooth root stress relates to the con-sidered facewidth b1 respectively b2

Note that facewidth in this context is the design facewidth b even if the ends are unloaded as often applies to eg bevel gears

The plane of contact is considered

191 Relations between KHβ and KFβ

KFβ = expβHK 2(hb)hb1

1exp++

=

where hb is the ratio tooth heightfacewidth The maximum of h1b1 and h2b2 is to be used but not higher than 13 For double helical gears use only the facewidth of one helix

If the tooth root facewidth (b1 or b2) is considerably wider than b the value of KFβ(1or2) is to be specially considered as it may even exceed KHβ

Eg in pinion-rack lifting systems for jack up rigs where b = b2 asymp mn and b1 asymp 3 mn the typical KHβ asymp KFβ2 asymp 1 and KFβ1 asymp 13

192 Measurement of face load factors Primarily

KFβ may be determined by a number of strain gauges distrib-uted over the facewidth Such strain gauges must be put in

Classification Notes- No 412 11 May 2003

DET NORSKE VERITAS

exactly the same position relative to the root fillet Relations in 191 apply for conversion to KHβ

Secondarily

KHβ may be evaluated by observed contact patterns on vari-ous defined load levels It is imperative that the various test loads are well defined Usually it is also necessary to evalu-ate the elastic deflections Some teeth at each 90 degrees are to be painted with a suitable lacquer Always consider the poorest of the contact patterns

After having run the gear for a suitable time at test load 1 (the lowest) observe the contact pattern with respect to ex-tension over the facewidth Evaluate that KHβ by means of the methods mentioned in this section Proceed in the same way for the next higher test load etc until there is a full face contact pattern From these data the initial mesh misalign-ment (ie without elastic deflections) can be found by ex-trapolation and then also the KHβ at design load can be found by calculation and extrapolation See example

Figure 11 Example of experimental determination of KHβ

It must be considered that inaccurate gears may accumulate a larger observed contact pattern than the actual single mesh to mesh contact patterns This is particularly important for lapped bevel gears Ground or hard metal hobbed bevel gears are assumed to present an accumulated contact pattern that is practically equal the actual single mesh to mesh con-tact patterns As a rough guidance the (observed) accumu-lated contact pattern of lapped bevel gears may be reduced by 10 in order to assess the single mesh to mesh contact pattern which is used in 199

193 Theoretical determination of KHβ The methods described in 193 to 198 may be used for cy-lindrical gears The principles may to some extent also be used for bevel gears but a more practical approach is given in 199

General For gears where the tooth contact pattern cannot be verified during assembly or under load all assumptions are to be well on the safe side

KHβ is to be determined in the plane of contact

The influence parameters considered in this method are

bull mean mesh stiffness cγ (see 111) (if necessary also variable stiffness over b)

bull mean unit load Fmb = Fbt KA Kγ Kvb (for double helical gears see 17 for use of Kγ)

bull misalignment fsh due to elastic deflections of shafts and gear bodies (both pinion and wheel)

bull misalignment fdefl due to elastic deflections of and work-ing positions in bearings

bull misalignment fbe due to bearing clearance tolerances bull misalignment fma due to manufacturing tolerances bull helix modifications as crowning end relief helix correc-

tion bull running in amount yβ (see 112)

In practice several other parameters such as centrifugal ex-pansion thermal expansion housing deflection etc contrib-ute to KHβ However these parameters are not taken into account unless in special cases when being considered as particularly important

When all or most of the am parameters are to be considered the most practical way to determine KHβ is by means of a graphical approach described in 1931

If cγ can be considered constant over the facewidth and no helix modifications apply KHβ can be determined analyti-cally as described in 1932

1931 Graphical method The graphical method utilises the superposition principle and is as follows

bull Calculate the mean mesh deflection Mδ as a function of

γm c and bF see 111

bull Draw a base line with length b and draw up a rectangu-lar with height δM (The area δM b is proportional to the transmitted force)

bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the areas above and below this zero line are equal

Figure 12 fsh balanced around zero line

bull Superimpose these ordinates of the fsh curve to the previ-ous load distribution curve (The area under this new load distribution curve is still δM b)

bull Calculate the bearing deflections andor working posi-tions in the bearings and evaluate the influence fdefl in the plane of contact This is a straight line and is bal-anced around a zero line as indicated in Fig 14 but with one distinct direction Superimpose these ordinates to the previous load distribution curve

12 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

bull The amount of crowning end relief or helix correction (defined in the plane of contact) is to be balanced around a zero line similarly to fsh

Figure 13 Crowning Cc balanced aound zero line

bull Superimpose these ordinates to the previous load distri-bution curve In case of high crowning etc as eg often applied to bevel gears the new load distribution curve may cross the base line (the real zero line) The result is areas with negative load that is not real as the load in those areas should be zero Thus corrective actions must be made but for practical reasons it may be postponed to after next operation

bull The amount of initial mesh misalignment bema ff + (defined in the plane of contact) is to be bal-

anced around a zero line If the direction of bema ff + is known (due to initial contact check) or if the direction of fbe is known due to design (eg overhang bevel pin-ion) this should be taken into account If direction un-known the influence of bema ff + in both directions as well as equal zero should be considered

Figure 14 fma+fbe in both directions balanced around zero line

Superimpose these ordinates to the previous load distri-bution curve This results in up to 3 different curves of which the one with the highest peak is to be chosen for further evaluation

bull If the chosen load distribution curve crosses the base line (ie mathematically negative load) the curve is to be corrected by adding the negative areas and dividing this with the active facewidth The (constant) ordinates of this rectangular correction area are to be subtracted from the positive part of the load distribution curve It is advisable to check that the area covered under this new load distribution curve is still equal δM b

bull If cγ cannot be considered as constant over b then cor-rect the ordinates of the load distribution curve with the local (on various positions over the facewidth) ratio between local mesh stiffness and average mesh stiffness cγ (average over the active facewidth only) Note that the result is to be a curve that covers the same area δM b as before

bull The influence of running in yβ is to be determined as in 112 whereby the value for Fβx is to be taken as twice the distance between the peak of the load distribution curve and δM

bull Determine

KHβ = M

β

δycurveofpeak minus

1932 Simplified analytical method for cylindrical gears The analytical approach is similar to 1931 but has a more limited application as cγ is assumed constant over the face-width and no helix modification applies

bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the area above and below this zero line are equal see Fig 12 The max positive ordinate is frac12∆fsh

bull Calculate the initial mesh alignment as Fβx= deflbemash ffff plusmnplusmnplusmn∆ The negative signs may only be used if this is justified andor verified by a contact pattern test Otherwise al-ways use positive signs If a negative sign is justified the value of Fβx is not to be taken less than the largest of each of these elements

bull Calculate the effective mesh misalignment as Fβy = Fβx - yβ (yβ see 112)

bull Determine

2KforF2

bFc1K βH

m

βγγH le+=β

or

2KforF

bFc2K Hβ

m

βγγH gt=β

where cγ as used here is the effective mesh stiffness see 111

194 Determination of fsh fsh is the mesh misalignment due to elastic deflections Usu-ally it is sufficient to consider the combined mesh deflection of the pinion body and shaft and the wheel shaft The calcu-lation is to be made in the plane of contact (of the considered gear mesh) and to consider all forces (incl axial) acting on the shafts Forces from other meshes can be parted into components parallel respectively vertical to the considered plane of contact Forces vertical to this plane of contact have no influence on fsh

It is advised to use following diameters for toothed elements

d + 2 x mn for bending and shear deflection

d + 2 mn (x ndash ha0 + 02) for torsional deflection

Usually fsh is calculated on basis of an evenly distributed load If the analysis of KHβ shows a considerable maldis-

Classification Notes- No 412 13 May 2003

DET NORSKE VERITAS

tribution in term of hard end contact or if it is known by other reasons that there exists a hard end contact the load should be correspondingly distributed when calculating fsh In fact the whole KHβ procedure can be used iteratively 2-3 iterations will be enough even for almost triangular load distributions

195 Determination of fdefl fdefl is the mesh misalignment in the plane of contact due to bearing deflections and working positions (housing deflec-tion may be included if determined)

First the journal working positions in the bearings are to be determined The influence of external moments and forces must be considered This is of special importance for twin pinion single output gears with all 3 shafts in one plane

For rolling bearings fdefl is further determined on basis of the elastic deflection of the bearings An elastic bearing deflec-tion depends on the bearing load and size and number of rolling elements Note that the bearing clearance tolerances are not included here

For fluid film bearings fdefl is further determined on basis of the lift and angular shift of the shafts due to lubrication oil film thickness Note that fbe takes into account the influence of the bearing clearance tolerance

When working positions bearing deflections and oil film lift are combined for all bearings the angular misalignment as projected into the plane of the contact is to be determined fdefl is this angular misalignment (radians) times the face-width

196 Determination of fbe fbe is the mesh misalignment in the plane of contact due to tolerances in bearing clearances In principle fbe and fdefl could be combined But as fdefl can be determined by analy-sis and has a distinct direction and fbe is dependent on toler-ances and in most cases has no distinct direction (ie + toler-ance) it is practicable to separate these two influences

Due to different bearing clearance tolerances in both pinion and wheel shafts the two shaft axis will have an angular mis-alignment in the plane of contact that is superimposed to the working positions determined in 195 fbe is the facewidth times this angular misalignment Note that fbe may have a distinct direction or be given as a + tolerance or a combina-tion of both For combination of + tolerance it is adviced to use

fbe= 22be

21be ff ++plusmn

fbe is particularly important for overhang designs for gears with widely different kinds of bearings on each side and when the bearings have wide tolerances on clearances In general it shall be possible to replace standard bearings with-out causing the real load distribution to exceed the design premises For slow speed gears with journal bearings the expected wear should also be considered

197 Determination of fma fma is the mesh misalignment due to manufacturing toler-ances (helix slope deviation) of pinion fHβ1 wheel fHβ2 and housing bore

For gear without specifically approved requirements to as-sembly control the value of fma is to be determined as

fma= 22Hβ

21Hβ ff +

For gears with specially approved assembly control the value of fma will depend on those specific requirements

198 Comments to various gear types For double helical gears KHβ is to be determined for both helices Usually an even load share between the helices can be assumed If not the calculation is to be made as de-scribed in 171

For planetary gears the free floating sun pinion suffers only twist no bending It must be noted that the total twist is the sum of the twist due to each mesh If the value of 1K γ ne this must be taken into account when calculating the total sun pinion twist (ie twist calculated with the force per mesh without Kγ and multiplied with the number of planets)

When planets are mounted on spherical bearings the mesh misalignments sun-planet respectively planet-annulus will be balanced Ie the misalignment will be the average between the two theoretical individual misalignments The faceload distribution on the flanks of the planets can take full advan-tage of this However as the sun and annulus mesh with several planets with possibly different lead errors the sun and annulus cannot obtain the above mentioned advantage to the full extent

199 Determination of KHβ for bevel gears If a theoretical approach similar to 193-198 is not docu-mented the following may be used

testeff

H Kb

b851851K sdot

minussdot=β

beff b represents the relative active facewidth (regarding lapped gears see 192 last part)

Higher values than beff b = 090 are normally not to be used in the formula

For dual directional gears it may be difficult to obtain a high beff b in both directions In that case the smaller beff b is to be used

Ktest represents the influence of the bearing arrangement shaft stiffness bearing stiffness housing stiffness etc on the faceload distribution and the verification thereof Expected variations in length- and height-wise tooth profile is also accounted for to some extent

a) Ktest = 1 For ground or hard metal hobbed gears with the specified contact pattern verified at full rating or at full torque slow turning at a condition representative for the thermal expansion at normal operation

14 Classification Notes- No 412

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DET NORSKE VERITAS

It also applies when the bearing arrangementsupport has insignificant elastic deflections and thermal axial expansion However each initial mesh contact must be verified to be within acceptance criteria that are cali-brated against a type test at full load Reproduction of the gear tooth length- and height-wise profile must also be verified This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon)

b) Ktest = 1 + 04middot(beffbndash06) For designs with possible influence of thermal expansion in the axial direction of the pinion The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load

c) Ktest = 12 if mesh is only checked by toolmakerrsquos blue or by spin test contact For gears in this category beffb gt 085 is not to be used in the calculation

110 Transversal Load Distribution Factors KHα and KFα The transverse load distribution factors KHα for contact stresses and for scuffing KFα for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairs of teeth in mesh

The following relations may be used

Cylindrical gears

( )

minus+==

tH

αptγγHαFα F

byfc0409

2ε

KK

valid for 2εγ le

( ) ( )tH

αptγ

γ

γHαFα F

byfcε

1ε20409KK

minusminus+==

valid for 2ε γ gt

where

FtH = Ft KA Kγ Kv KHβ

cγ = See 111

γα = See 112

fpt = Maximum single pitch deviation (microm) of pinion or wheel or maximum total profile form devia-tion Fα of pinion or wheel if this is larger than the maximum single pitch deviation

Note In case of adequate equivalent tip relief adapted to the load half of the above mentioned fpt can be introduced A tip relief is considered adequate when the average of Ca1 and Ca2 is within plusmn40 of the value of Ceff in 432

Limitations of KHα and KFα

If the calculated values for

KFα = KHα lt 1 use KFα = KHα = 10

If the calculated value of KHα gt 2εα

γ

Zε

ε use KHα = 2

εα

γ

Zε

ε

If the calculated value of KFα gt εα

γ

Yεε

use KFα = εα

γ

Yεε

where Yε = αnε

075025+ (for εαn see 331c)

Bevel gears

For ground or hard metal hobbed gears KFα = KHα = 1

For lapped gears KFα = KHα = 11

111 Tooth Stiffness Constants cacute and cγ The tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having 1 mm facewidth by an amount of 1 microm in the plane of contact cacute is the maximum stiffness of a single pair of teeth cγ is the mean value of the mesh stiffness in a transverse plane (brief term mesh stiffness)

Both valid for high unit load (Unit load = Ft middot KA middot Kγb)

Cylindrical gears The real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bending stiff-nesses For high unit loads the Hertzian stiffness has little importance and can be disregarded This approach is on the safe side for determination of KHβ and KHα However for moderate or low loads Kv may be underestimated due to determination of a too high resonance speed

The linear approach is described in A

An optional approach for inclusion of the non-linear stiffness is described in B

A The linear approach

BRCCq

βcos08acutec =

and

( )025ε075cc αγ +prime=

where

( )[ ]n02a01a

B α2000212

hh12051C minusminus

+minus+=

12n1n

x000635z

025791z

015551004723q minus++=

12

2n

22

1n

1 x005290z

x024188x000193z

x011654+minusminusminus

+ 000182 x22

Classification Notes- No 412 15 May 2003

DET NORSKE VERITAS

(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as

( )( )nR m5s

sR e5

bbln1C +=

where

bs = thickness of a central web

sR = average thickness of rim (net value from tooth root to inside of rim)

The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered

Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ

B The non-linear approach

In the following an example is given on how to consider the non-linearity

The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm

With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as

( )10δKbF

minus= for 500bFgt

minus=

500Fb10δK

bF for 500

bFlt

with γAt KK

bF

bF

sdotsdot= etc (Nmm) ie unit load incorporat-

ing the relevant factors as

KA middot Kγ for determination of Kv

KA middot Kγ middot Kv for determination of KHβ

KA middot Kγ middot Kv middot KHβ for determination of KHα

δ = mesh deflection (microm)

K = applicable stiffness (c or cγ)

Use of stiffnesses for KV KHβ and KHα

For calculation of Kv and KHα the stiffness is calculated as follows

When Fb lt 500

the stiffness is determined as δ∆

∆ bF

where the increment is chosen as eg ∆ Fb = 10 and thus

50010Fb10

K10Fb∆δ +

++

=

When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly

or an equivalent stiffness determined as δsdotb

F

Bevel gears

In lack of more detailed relationship between stiffness and geometry the following may be used

b085

b13cacute eff=

b085b

16c effγ =

beff not to be used in excess of 085 b in these formulae

Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)

112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements

yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload

Cay is defined as the running-in amount that compensates for lack of tip relief

The following relations may be used

For not surface hardened steel

ptHlim

α fσ160y =

yβ = βxlimH

f320σ

with the following upper limits

V lt 5 ms 5-10 ms gt 10 ms

yα max none

limH

12800σ limH

6400σ

yβ max none

limH

25600σ limH

12800σ

For surface hardened steel

yα = 0075 fpt but not more than 3 for any speed

yβ = 015 Fβx but not more than 6 for any speed

16 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

For all kinds of steel

5145189718

1C2

limHay +

minusσ

=

When pinion and wheel material differ the following ap-plies

bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv

bull Use ( )2β1ββ yy21y += in the calculation of KHβ

bull Use ( )2ay1aya CC21C += in the calculation of Kv

bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing

calculation if no design tip relief is foreseen

Classification Notes- No 412 17 May 2003

DET NORSKE VERITAS

2 Calculation of Surface Durability

21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213

Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted

Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213

For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength

For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied

22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater

Calculation of surface durability for helical gears is based on the contact stress at the pitch point

For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies

Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact

Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses

The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses

The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel

221 Contact stress Cylindrical gears

( )HαHβvγA

1

tβεEHDBH KKKKK

bud1uF

ZZZZZσ+

=

where

ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)

ZH = Zone factor for pitch point (see 231)

ZE = Elasticity factor (see 24)

Zε = Contact ratio factor (see 25)

Zβ = Helix angle factor (see 26)

Ft KA Kγ Kv KHβ KHα see 15 ndash 110

d1 b u see 12 ndash 15

Bevel gears

( )HαHβvγA

v1v

vmtKEMH KKKKK

bud1uF

ZZZ105σ+

sdot=

where

105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)

ZE KA etc see above

ZM = mid-zone factor see 233

ZK = bevel gear factor see 27

Fmt dv1 uv see 12 ndash 15

It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks

222 Permissible contact stress

XWRvLH

NHlimHP ZZZZZ

SZσ

σ =

where

σH lim = Endurance limit for contact stresses (see 28)

ZN = Life factor for contact stresses (see 29)

SH = Required safety factor according to the rules

ZLZvZR = Oil film influence factors (see 210)

ZW = Work hardening factor (see 211)

ZX = Size factor (see 212)

18 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

23 Zone Factors ZH ZBD and ZM

231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder

wtt2

wtbH sinααcos

cosαcosβ2Z =

232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel

For εβ ge 1 ZBD = 1

For internal gears ZD = 1

For εβ = 0 (spur gears)

( )

π

minusεminusminus

π

minusminus

α=

α2

2

2b

2a

1

2

1b

1a

wtB

z211

dd

z21

dd

tanZ

( )

π

minusεminusminus

π

minusminus

α=

α1

2

1b

1a

2

2

2b

2a

wtD

z211

dd

z21

dd

tanZ

If ZB lt 1 use ZB = 1

If ZD lt 1 use ZD = 1

For 0 lt εβ lt 1

ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)

233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears

εminusminus

εminusminus

αβ=

αα btm2

2vb2

2vabtm2

1vb2val

2v1vvtbmM

pddpdd

ddtancos2Z

This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint

234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel

24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses

For steel against steel ZE = 1898

25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses

αε1Z =ε for 1εβ ge

( )α

ββ

αε ε

εε1

3ε4

Z +minusminus

= for εβ lt 1

26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability

cosβZβ =

27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply

The following may be used ZK = 080

28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ

σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting

For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply

For this purpose pitting is defined by

bull for not surface hardened gears pitted area ge 2 of total active flank area

bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given

material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies

The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as

Classification Notes- No 412 19 May 2003

DET NORSKE VERITAS

the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99

σHlim σH105 σH10

3

Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade

1650 1500

2500 2400

3100 3100

Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)

1000

13 σHlim

13 σHlim

Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV

Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV

Carbon steel 15 HV + 250 16 σHlim 16 σHlim

These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15

29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply

If this is not documented by approved fatigue tests the fol-lowing method may be used

For all steels except nitrided

7L 105N sdotge ZN = 1 or

01570

L

7

N N105Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 5middot107 510N

logZ037

L

7

N N105Z

sdot=

NL = 105 WXRVLHlim

WstX10H1010NN ZZZZZσ

ZZσZZ

55

5=

103 lt NL lt 105 )Z(Zlog05

L

5

10NN

5N103N10

5N10ZZ

==

3L 10N le

WXRVLlimH

Wst10X10H10NN ZZZZZ

ZZZZ

33

3σ

σ==

(but not less than ZN105)

For nitrided steels

6L 102N sdotge ZN = 1 or

00980

L

6

N N102Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 2middot106 510N

Zlog76860

L

6

N N102Z

sdot=

5L 10N le

XWRVL

10XWst10NN ZZZZZ

ZZ13ZZ

55 ==

Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles

210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity

The following methods may be applied in connection with the endurance limit

20 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Surface hardened steels Not surface hardened steels

ZL ( )24013421

360910ν+

+

( )24013421

680830ν+

+

ZV ( )v3280

140930+

+ ( )v3280300850

++

ZR

080

ZrelR3

150

ZrelR3

where

ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits

For values of ν40 gt 500 use ν40 = 500

RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm

RZrel

= ( ) 31

cZ2Z1 ρ

10RR50

+

RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)

For 5

L 10N le ZL ZV ZR = 10

211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface

The following approximation may be used for the endurance limit

Surface hardened steel against not surface hardened steel 150

ZeqW R

31700

130HB21Z

minus

minus=

where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130

RZeq = equivalent roughness

033

c40

066

ZS

ZHZHZEQ ρvν

15000RRRR

=

If RZeq gt 16 then use RZeq = 16

If RZeq lt 15 then use RZeq = 15

where

RZH = surface roughness of the hard member before run in

RZS = surface roughness of the soft member before run in

ν40 = see 210

If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem

Through hardened pinion against softer wheel

( )

minussdotminus+= 000829

HBHB0008981u1Z

2

1W

For 21HBHB

2

1 le use ZW = 1

For 71HBHB

2

1 gt use 17HBHB

2

1 =

For u gt 20 use u = 20

For static strength (lt 105 cycles)

Surface hardened against not surface hardened

ZWst = 105

Through hardened pinion against softer wheel

ZWst = 1

212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality

ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213

213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented

The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent

Classification Notes- No 412 21 May 2003

DET NORSKE VERITAS

stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically

The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence

The real Hertzian stresses σHR are determined as

For helical gears with εβ gt 1

σHR = σH

For helical gears with εβ lt 1 and spur gears

ε

α

ββ ε

ε+εminus

sdotσ=σZ

1

HHR

For bevel gears

KHHR Z

1σσ sdot=

The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)

The coordinates tz and HV are to be compared with the de-sign specification such as

bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400

and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)

In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods

For high cycle fatigue (gt3 106 cycles) the following applies

sdot+

minussdotsdotsdot= o90

05azt

05azt

cosSσ04HV

H

HHHR

applicable to 05a

zt

Hge

For 05at

H

z lt the value for 05at

H

z = applies

56300ρSσ12a cHHR

Hsdotsdot

sdot=

Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength

If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows

reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue

For static strength (lt103 cycles) the following applies

sdot+

minussdotsdotsdot= o90

07at

06a

zt

cosSσ019HV

Hst

z

HstHHR

applicable to 06at

Hst

z ge

56300ρSσa cHHR

Hstsdotsdot

=

In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies

For limited life fatigue (103 lt cycles lt 3106 )

For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance

The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram

minussdotminus

= sdot logN3477

logSlogSlogS

310H6103HHN

36 10H103H Slog86281Slog86280 sdot+sdot sdot

22 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3 Calculation of Tooth Strength

31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding

For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36

For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning

Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα

In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314

A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired

It should be noted that this part 3 does not cover fractures caused by

bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining

Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears

Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied

32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level

321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance

Normally the stresses for pinion and wheel are calculated as

Cylindrical gears

FαFβvγAβSFn

tF K K K K K Y Y Y

m bFσ =

where

YF = Tooth form factor (see 33)

YS = Stress correction factor (see 34)

Yβ = Helix angle factor (see 36)

Ft KA Kγ Kv KFβ KFα see 15 ndash 110

b see 13

Bevel gears

FαFβvγASaFamn

mtF K K K K K Y Y Y

m bFσ ε=

where

YFa = Tooth form factor see 33

YSa = Stress correction factor see 34

Yε = Contact ratio factor see 35

Fmt KA etc see 15 ndash 110

b see 13

322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is

CXRrelTrelTF

NMFEFP YYYY

SYY

δσ

=σ

Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN

where

σFE = Local tooth root bending endurance limit of reference test gear (see 37)

YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)

YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)

SF = Required safety factor according to the rules

YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)

YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)

Classification Notes- No 412 23 May 2003

DET NORSKE VERITAS

YX = Size factor (see 312)

YC = Case depth factor (see 313)

33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress

YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears

YFa applies to load application at the tooth tip and is used for bevel gears

Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears

Figure 31 External tooth in normal section

Figure 32 Internal tooth in normal section

Definitions

n

2

n

Fn

enFn

Fe

F

α cosms

α cosmh6

Y

=

n

2

n

Fn

anFn

Fa

Fa

α cosms

α cosmh6

Y

=

In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth

YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)

The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing

mn with mnm

zn with zvn

αt with αvt

β with βm

with undercut without undercut

Fig 33 Dimensions and basic rack profile of the teeth (finished profile)

Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless

331 Determination of parameters

( )n

n

prnfPnfP m

α cossαsin 1ρ

αtan h4πE

minusminusminusminus=

For external gears fPfP ρρ =

For internal gears ( )0z

195fPfP0

fPfP 10363156ρhxρρ

sdotminus+

+=

where

z0 = number of teeth of pinion cutter

x0 = addendum modification coefficient of pinion cutter

hfP = addendum of pinion cutter

ρfP = tip radius of pinion cutter

xhρG fpfp +minus=

24 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

τmE

2π

z2H

nnminus

minus=

with

3πτ = for external gears

6πτ = for internal gears

Htan zG2

nminusϑ=ϑ

(to be solved iteratively suitable start value6π

=ϑ for exter-

nal gears and 3π for internal gears)

a) Tooth root chord sFn For external gears

minus

ϑ+

ϑminus= ρ

cosG3

3πsinz

ms

fpnn

Fn

For bevel gears with a tooth thickness modification

xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn

For internal gears

minus

ϑ+

ϑminus= ρ

cosG

6πsinz

ms

fPnn

Fn

b) Root fillet radius ρF at 30ordm tangent

( )G2coszcosG2ρ

mρ

2n

2

fpn

F

minusϑϑ+=

c) Determination of bending moment arm hF dn = zn mn

b2α

αn βcosε

ε =

dan = dn + 2 ha

pbn = π mn cos αn

dbn = dn cos αn

( )4

d1εpzz

2dd

zz2d

2bn

2

αnbn

2bn

2an

en +

minusminus

minus=

en

bnen d

dcos arcα =

ennnn

e α invα invα x tan 22π

z1

minus+

+=γ

αFen = αen ndash γe

For external gears

( )

minus=

n

enFenee

n

Fe

mdαtan sin γ γcos

21

mh

]ρcos

G3πcosz fpn +

ϑminus

ϑminusminus

For internal gears

( ) minus

sdotsdotminus=

n

enFenee

n

Fe

mdα tanγ sinγ cos

21

mh

minus

ϑminus

ϑminussdot ρ

cosG3

6πcosz fPn

332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as

250ε205for 0666ε2366Y αnαnDT leleminus=

205εfor 10Y αnDT lt=

34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section

Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm

YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip

YS can be determined as follows

( )

+

+=L23121

1

sS qL01312Y

where Fe

Fn

hsL = and

F

Fns ρ2

sq = (see 33)

YSa can be calculated by replacing hFe with hFa in the above formulae

Note a) Range of validity 1 lt qs lt 8

In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered

b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with

Classification Notes- No 412 25 May 2003

DET NORSKE VERITAS

g

g

ρt

0613

13

minus

where

tg = depth of the grinding notch

ρg = radius of the grinding notch

c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles

35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears

The following may be used

Yε = 0625

36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank

The following may be used (β input in degrees)

Yβ = 1 ndash εβ β120

When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula

However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)

37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM

σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision

If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules

σFE

Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)

bull of specially approved high grade

1050

bull of normal grade

minus CrNiMo steels with approved process

1000

minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850

Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

840

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

720

Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)

07 HV + 300

Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)

025 σB + 125

Alloyed quenched and tempered steel 04 σB + 200

Carbon steel 025 σB + 250

Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40

1) These values are valid for a root radius

bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)

bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)

bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for

6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked

2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)

26 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears

YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0

YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst

For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316

The following method may be used within a stress ratio ndash12 lt R lt 05

381 For idlers planets and PTO with ice class

M1M1R1

1Yor Y MstM

+minus

minus=

where

R = stress ratio = min stress divided by max stress

For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12

For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as

branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus

For a power take off (PTO) with ice class see 161 c

M considers the mean stress influence on the endurance (or static) strength amplitudes

M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress

Following M values may be used

Endurance limit

Static strength

Case hardened 08 ndash 015 Ys 1) 07

If shot peened 04 06

Nitrided 03 03

Induction or flame hardened

04 06

Not surface hardened steel 03 05

Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M

The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches

382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106

For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers

For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used

383 For gears with shrinkage stresses and unidirectional load For endurance strength

FE

fitM σ

σM1

M21Y+

minus=

σFE is the endurance limit for R = 0

For static strength YMst = 1 and σfit accounted for in 39b

σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor

n

Ffit m

ρ215scf minus=

384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is

( ) ( ) FE

fitM σ

σR1M1

M2

M1M1R1

1Yminussdot+

minus

+minus

minus=

Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12

For static strength

M1M1R1

1YMst

+minus

minus=

The effect of σfit is accounted for in 39 b

Classification Notes- No 412 27 May 2003

DET NORSKE VERITAS

385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed

F

y

Sσ225

for not surface hardened fillets

FSHV5 for surface hardened fillets

39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles

Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN

If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used

Determination of the σ ndash N ndash curve

a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF

b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ

FstS1

FPstσ minussdotsdotsdot=

where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability

σFst

Alloyed case hardened steel 1) 2300

Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

1250

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

1050

Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)

18 HV + 800

Steel with not surface hardened fillets the smaller value of 2)

18 σB or 225 σy

1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet

c) Calculate YN as

NL gt 3middot106

YN = 1 or 001

L

6

N N103Y

sdot= ie Yn = 092 for 1010

The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2

103ltNLlt3middot106exp

L

6

N N103Y

sdot=

cycles6103forσcycles310forσlog02876exp

FP

FPst

sdot=

NL lt 103 cycles6103forσcycles310forσ

NYFP

FPst

sdot=

or simply use σFPst as mentioned in b) directly

Guidance on number of load cycles NL for various applica-tions

bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)

bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied

28 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage

YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit

The following method may be used

For endurance limit

for not surface hardened fillets

( )

024

s024

relT σ103133q21σ1012201351

Ysdotsdotminus

+sdotsdotminus+= minus

minus

δ

for all surface hardened fillets except nitrided

( )

106q21002451

Y srelT

++=δ

for nitrided fillets

( )

1347q2101421

Y srelT

++=δ

For static strength

for not surface hardened fillets1)

( )( )( )025

02

02502s

relTst σ3000821σ300 1Y0821Y

+

minus+=δ

for surface hardened fillets except nitrided

YδrelTst = 044 YS + 012

for nitrided fillets

YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not

exceed the yield point and thereby alter the residual stress level See also 39b footnote 2

311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness

YRrelT differs for endurance limit and static strength

The following method may be used

For endurance limit

YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels

YδrelT = 43 ndash 326 (Ry + 1)0005

for nitrided steels

For static strength

YRrelTst = 1 for all Ry and all materials

For a fillet without any longitudinal machining trace Ry asymp Rz

312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength

The following may be used

For endurance limit

YX = 1 for mn le 5 generally

YX = 103 ndash 0006 mn for 5 lt mn lt 30

YX = 085 for mn ge 30

for not surface hard-ened steels

YX = 105 ndash 001 mn for 5 lt mn ge 25

YX = 08 for mn ge 25

for surface hardened steels

For static strength

YXst = 1 for all mn and all materials

313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength

YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength

In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength

The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses

The following simplified method for YC may be used

YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)

YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10

For endurance limit

+

+=nFFE

C m02ρt31

σconstY

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

DET NORSKE VERITAS

CONTENTS

1 Basic Principles and General Influence Factors 4 11 Scope and Basic Principles 4 12 Symbols Nomenclature and Units 4 13 Geometrical Definitions5 14 Bevel Gear Conversion Formulae and Specific

Formulae6 15 Nominal Tangential Load Ft Fbt Fmt and Fmbt 6 16 Application Factors KA and KAP 7 161 KA 7 162 KAP7 163 Frequent overloads8 17 Load Sharing Factor Kγ 8 171 General method8 172 Simplified method 8 18 Dynamic Factor Kv 8 181 Single resonance method 8 182 Multi-resonance method 10 19 Face Load Factors KHβ and KFβ 10 191 Relations between KHβ and KFβ10 192 Measurement of face load factors 10 193 Theoretical determination of KHβ11 194 Determination of fsh 12 195 Determination of fdefl13 196 Determination of fbe 13 197 Determination of fma 13 198 Comments to various gear types 13 199 Determination of KHβ for bevel gears 13 110 Transversal Load Distribution Factors KHα and KFα14 111 Tooth Stiffness Constants cacute and cγ 14 112 Running-in Allowances 15 2 Calculation of Surface Durability17 21 Scope and General Remarks 17 22 Basic Equations 17 221 Contact stress 17 222 Permissible contact stress 17 23 Zone Factors ZH ZBD and ZM18 231 Zone factor ZH 18 232 Zone factors ZBD 18 233 Zone factor ZM 18 234 Inner contact point 18 24 Elasticity Factor ZE 18 25 Contact Ratio Factor Zε18 26 Helix Angle Factor Zβ18 27 Bevel Gear Factor ZK18 28 Values of Endurance Limit σHlim and Static

Strength 5H10σ 3H10σ 18 29 Life Factor ZN 19 210 Influence Factors on Lubrication Film ZL ZV and

ZR 19 211 Work Hardening Factor ZW 20 212 Size Factor ZX20 213 Subsurface Fatigue20 3 Calculation of Tooth Strength 22 31 Scope and General Remarks 22 32 Tooth Root Stresses 22 321 Local tooth root stress22 322 Permissible tooth root stress 22

33 Tooth Form Factors YF YFa 23 331 Determination of parameters 23 332 Gearing with εαn gt 2 24 34 Stress Correction Factors YS YSa 24 35 Contact Ratio Factor Yε 25 36 Helix Angle Factor Yβ 25 37 Values of Endurance Limit σFE 25 38 Mean stress influence Factor YM 26 381 For idlers planets and PTO with ice class 26 382 For gears with periodical change of rotational

direction 26 383 For gears with shrinkage stresses and unidirectional

load 26 384 For shrink-fitted idlers and planets 26 385 Additional requirements for peak loads 27 39 Life Factor YN 27 310 Relative Notch Sensitivity Factor YδrelT 28 311 Relative Surface Condition Factor YRrelT 28 312 Size Factor YX 28 313 Case Depth Factor YC 28 314 Thin rim factor YB 29 315 Stresses in Thin Rims 29 3151 General 29 3152 Stress concentration factors at the 75ordm tangents 30 3153 Nominal rim stresses 30 3154 Root fillet stresses 30 316 Permissible Stresses in Thin Rims 31 3161 General 31 3162 For gt3middot106 cycles 31 3163 For le 103 cycles 31 3164 For 103 lt cycles lt 3106 32 4 Calculation of Scuffing Load Capacity 33 41 Introduction 33 42 General Criteria 33 43 Influence Factors 34 431 Coefficient of friction 34 432 Effective tip relief Ceff 34 433 Tip relief and extension 34 434 Bulk temperature 35 44 The Flash Temperature flaϑ 35 441 Basic formula 35 442 Geometrical relations 35 443 Load sharing factor XΓ 36 Appendix A Fatigue Damage Accumulation 40 A1 Stress Spectrum 40 A2 σminusN-curve 40 A3 Damage accumulation 40 Appendix B Application Factors for Diesel Driven

Gears 41 B1 Definitions 41 B2 Determination of decisive load 41 B3 Simplified procedure 41 Appendix C Calculation of Pinion-Rack 42 C1 Pinion tooth root stresses 42 C2 Rack tooth root stresses 42 C3 Surface hardened pinions 42

4 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

1 Basic Principles and General Influence Factors

11 Scope and Basic Principles The gear rating procedures given in this Classification Note are mainly based on the ISO-6336 Part 1-5 (cylindrical gears) and partly on ISO 10300 Part 1-3 (bevel gears) and ISO Technical Reports on Scuffing and Fatigue Damage Accumulation but especially applied for marine purposes such as marine propulsion and important auxiliaries onboard ships and mobile offshore units

The calculation procedures cover gear rating as limited by contact stresses (pitting spalling or case crushing) tooth root stresses (fatigue breakage or overload breakage) and scuff-ing resistance Even though no calculation procedures for other damages such as wear grey staining (micropitting) etc are given such damages may limit the gear rating

The Classification Note applies to enclosed parallel shaft gears epicyclic gears and bevel gears (with intersecting axis) However open gear trains may be considered with regard to tooth strength ie part 1 and 3 may apply Even pinion-rack tooth strength may be considered but since such gear trains often are designed with non-involute pinions the calculation procedure of pinion-racks is described in Appen-dix C

Steel is the only material considered

The methods applied throughout this document are only valid for a transverse contact ratio 1 lt εα lt 2 If εα gt 2 either special considerations are to be made or suggested simplifi-cation may be used

All influence factors are defined regarding their physical interpretation Some of the influence factors are determined by the gear geometry or have been established by conven-tions These factors are to be calculated in accordance with the equations provided Other factors are approximations which are clearly stated in the text by terms as laquomay be cal-culated asraquo These approximations are substitutes for exact evaluations where such are lacking or too extensive for prac-tical purposes or factors based on experience In principle any suitable method may replace these approximations

Bevel gears are calculated on basis of virtual (equivalent) cylindrical gears using the geometry of the midsection The virtual (helical) cylindrical gear is to be calculated by using all the factors as a real cylindrical gear with some exceptions These exceptions are mentioned in connection with the ap-plicable factors Wherever a factor or calculation procedure has no reference to either cylindrical gears or bevel gears it is generally valid ie combined for both cylindrical and bevel

In order to minimise the volume of this Classification Note such combinations are widely used and everywhere it is necessary to distinguish it is clearly pointed out by local headings such as

Cylindrical gears

Bevel gears

The permissible contact stresses tooth root stresses and scuffing load capacity depend on the safety factors as re-quired in the respective Rule sections

Terms as endurance limit and static strength are used throughout this Classification Note

Endurance limit is to be understood as the fatigue strength in the range of cycles beyond the lower knee of the σndashN curves regardless if it is constant or drops with higher number of cycles

Static strength is to be understood as the fatigue strength in the range of cycles less than at the upper knee of the σndashN curves

For gears that are subjected to a limited number of cycles at different load levels a cumulative fatigue calculation applies Information on this is given in Appendix A

When the term infinite life is used it means number of cy-cles in the range 108ndash1010

12 Symbols Nomenclature and Units The symbols are generally from ISO 701 ISOR31 and ISO 1328 with a few additional symbols Only SI units are used

The main symbols as influence factors (K Z Y and X with indeces) etc are presented in their respective headings Symbols which are not explained in their respective Secs are as follows

a = centre distance (mm) b = facewidth (mm) d = reference diameter (mm) da = tip diameter (mm) db = base diameter (mm) dw = working pitch diameter (mm) ha = addendum (mm) ha0 = addendum of tool ref to mn hfp = dedendum of basic rack ref to mn (= ha0) hFe = bending moment arm (mm) for tooth root

stresses for application of load at the outer point of single tooth pair contact

hFa = bending moment arm (mm) for tooth root stresses for application of load at tooth tip

HB = Brinell hardness HV = Vickers hardness HRC = Rockwell C hardness mn = normal module n = rev per minute NL = number of load cycles qs = notch parameter Ra = average roughness value (microm) Ry = peak to valley roughness (microm) Rz = mean peak to valley roughness (microm) san = tooth top land thickness (mm) sat = transverse top land thickness (mm) sFn = tooth root chord (mm) in the critical section spr = protuberance value of tool minus grinding stock

equal residual undercut of basic rack ref to mn T = torque (Nm) u = gear ratio (per stage) v = linear speed (ms) at reference diameter

Classification Notes- No 412 5 May 2003

DET NORSKE VERITAS

x = addendum modification coefficient z = number of teeth zn = virtual number of spur teeth αn = normal pressure angle at ref cylinder αt = transverse pressure angle at ref cylinder αa = transverse pressure angle at tip cylinder αwt = transverse pressure angle at pitch cylinder β = helix angle at ref cylinder βb = helix angle at base cylinder βa = helix angle at tip cylinder εα = transverse contact ratio εβ = overlap ratio εγ = total contact ratio ρa0 = tip radius of tool ref to mn ρfp = root radius of basic rack ref to mn ( = ρa0) ρC = effective radius (mm) of curvature at pitch point ρF = root fillet radius (mm) in the critical section σB = ultimate tensile strength (Nmm2) σy = yield strength resp 02 proof stress (Nmm2)

Index 1 refers to the pinion 2 to the wheel

Index n refers to normal section or virtual spur gear of a heli-cal gear

Index w refers to pitch point

Special additional symbols for bevel gears are as follows

Σ = angle between intersection axis Kϑ = angle modification (Klingelnberg)

m0 = tool module (Klingelnberg) δ = pitch cone angle xsm = tooth thickness modification coefficient (mid-

face) R = pitch cone distance (mm)

Index v refers to the virtual (equivalent) helical cylindrical gear

Index m refers to the midsection of the bevel gear

13 Geometrical Definitions For internal gearing z2 a da2 dw2 d2 and db2 are negative x2 is positive if da2 is increased ie the numeric value is de-creased

The pinion has the smaller number of teeth ie

11

2 ge=zz

u

For calculation of surface durability b is the common face-width on pitch diameter

For tooth strength calculations b1 or b2 are facewidths at the respective tooth roots If b1 or b2 differ much from b above they are not to be taken more than 1 module on either side of b

Cylindrical gears

tan αt = tan αn cos β tan βb = tan β cos αt

tan βa = tan β da d

cos αa = dbda

d = z mn cos β mt = mn cos β db = d cos αt = dw cos αwt

a = 05 (dw1 + dw2)

dw1dw2 = z1 z2

inv α = tan α - α (radians)

inv αwt = inv αt + 2 tan αn (x1 + x2)(z1 + z2)

zn = z (cos2 βb cos β)

1

aw1fw1α T

ξξε

+=

where ξfw1 is to be taken as the smaller of

bull wtfw1 αtanξ =

bull soi1

b1wtfw1 d

dacostan -tanαξ =

bull 1

2wt

a2

b2fw1 z

ztanα

d

dacostan ξ

minus=

and

2

1fw2aw1 z

zξξ = where ξfw2 is calculated as ξfw1

substituting the values for the wheel by the values for the pinion and visa versa

11 z

2πT =

( ) +

sdot+minusminussdot= minus

2sinαρρxhm

2d2d nfpfp1fpnsoi1

21

t

nfpfplfpn2

tanα)sinαρρx(hm

sdot+minusminus

nmsinbπ

β=εβ

(for double helix b is to be taken as the width of one helix)

εy = βα εε +

ρC = ( )2

b

wt

u1βcos

αsinua

+

v = 311 10dn

60π minus

6 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

pbt = βcos

αcosmπ tn

sat =

minus++

at

ninvααinv

z

αtanx22

π

d a

san = aβcossat

14 Bevel Gear Conversion Formulae and Specific Formulae Conversion of bevel gears to virtual equivalent helical cylin-drical gears is based on the bevel gear midsection The con-version formulae are

Number of teeth

zv12 = z12 cos δ12

(δ1 + δ2 = Σ)

Gear ratio

uv = 1v

2vzz

tan αvt = tan αn cos βm

tan βbm = tan βm cos αvt

Base pitch

pbtm = m

vtnmβcos

αcosmπ

Reference pitch diameters

dv12 = 21

21m

cosdδ

Centre distance

av = 05 (dv1 + dv2)

Tip diameters

dva 12 = dv 12 + 2 ham 12

Addenda

for gears with constant addenda (Klingelnberg)

ham 12 = mmn (1 + xm 12)

for gears with variable addenda (Gleason)

ham 12 = ha 12 ndash b2 tan (δa 12 ndash δ12)

(when ha is addendum at outer end and δa is the outer cone angle)

Addendum modification coefficients

xm 12 = mn

12am21am

m2

hh minus

Base circle

dvb 12 = dv 12 cos αvt

Transverse contact ratio)

εα = btmP

αsinadd05dd05 vtv2

2vb2

2va2

1vb2

1va minusminus+minus

Overlap ratio) (theoretical value for bevel gears with no crowning but used as approximations in the calculation pro-cedures)

εβ = nm

mmπβsinb

Total contact ratio)

εγ = 2β

2α εε +

( Note that index laquovraquo is left out in order to combine formu-lae for cylindrical and bevel gears)

Tangential speed at midsection

vmt = 3m11 10dn

60π minus

Effective radius of curvature (normal section)

ρvc = ( )2vbm

vtvv

u1βcosαsinua

+

Length of line of contact

lb = ( )( )( )

1εifε

ε1ε2ε

βcosεb

β2γ

2βα

2γ

bm

α ltminusminusminus

lb = 1εifβcosε

εbβ

bmγ

α ge

15 Nominal Tangential Load Ft Fbt Fmt and Fmbt The nominal tangential load (tangential to the reference cyl-inder with diameter d and perpendicular to an axial plane) is calculated from the nominal (rated) torque T transmitted by the gear set

Cylindrical gears

dT2000

Ft = t

tbt αcos

FF =

Bevel gears

mmt d

T2000F =

vt

mtmbt αcos

FF =

Classification Notes- No 412 7 May 2003

DET NORSKE VERITAS

16 Application Factors KA and KAP The application factor KA accounts for dynamic overloads from sources external to the gearing

It is distinguished between the influence of repetitive cyclic torques KA (161) and the influence of temporary occasional peak torques KAP (162)

Calculations are always to be made with KA In certain cases additional calculations with KAP may be necessary

For gears with a defined load spectrum the calculation with a KA may be replaced by a fatigue damage calculation as given in Appendix A

161 KA For gears designed for long or infinite life at nominal rated torque KA is defined as the ratio between the maximum re-petitive cyclic torque applied to the gear set and nominal rated torque

This definition is suitable for main propulsion gears and most of the auxiliary gears

KA can be determined by measurements or system analysis or may be ruled by conventions (ice classes) (For the pur-pose of a preliminary (but not binding) calculation before KA is determined it is advised to apply either the max values mentioned below or values known from similar plants)

a) For main propulsion gears KA can be taken from the (mandatory) torsional vibration analysis thereby con-sidering all permissible driving conditions) Unless specially agreed the rules do not allow KA in excess of 135 for diesel propulsion) With turbine or electric propulsion KA would normally not exceed 12 However special attention should be given to thrusters that are arranged in such a way that heavy vessel movements andor manoeuvring can cause severe load fluctuations This means eg thrusters positioned far from the rolling axis of vessels that could be susceptible to rolling If leading to propeller air suction the condi-tions may be even worse The above mentioned movements or manoeuvring will result in increased propeller excitation If the thruster is driven by a diesel engine the engine mean torque is limited to 100 However thrusters driven by electric motors can suffer temporary mean torque much above 100 unless a suitable load control system (limiting available e-motor torque) is provided

b) For main propulsion gears with ice class notation (see Rules Pt5 Ch1 Sec J500) KA ice has to be taken as the higher value of the applicable (rule defined) ice shock torque referred to nominal rated torque and the value under a) The Baltic ice class notations refer to a few millions ice shock loads Thus the life factors may be put YN=1 and ZN=12 (except for nitrided gears where ZN = 1 applies) Additionally the calculations with the normal KA (no ice class) are to fulfil the normal requirements For polar ice class notations KA ice applies to all criteria and for long or infinite life

c) For a power take off (PTO) branch from a main propul-sion gear with ice class ice shocks result in negative torques It is assumed that the PTO branch is unloaded when the ice shock load occurs The influence of these reverse shock loads may be taken into account as follows The negative torque (reversed load) expressed by means of an application factor based on rated forward load (T or Ft) is KAreverse = KA ice ndash1 (the minus 1 be-cause no mean torque assumed) KAice to be calculated as in the ice class rules This KAreverse should be used for back flank considera-tions such as pitting and scuffing The influence on tooth bending strength (forward di-rection) may be simplified by using the factor YM = 1 minus 03 middot KAreverse KA

d) For diesel driven auxiliaries KA can be taken from the torsional vibration analysis if available For units where no vibration analysis is required (lt 200 kW) or available it is advised to apply KA as the upper allow-able value 135)

e) For turbine or electro driven auxiliaries the same as for c) applies however the practical upper value is 12

) For diesel driven gears more information on KA for mis-firing and normal driving is given in Appendix B

162 KAP The peak overload factor KAP is defined as the ratio between the temporary occasional peak overload torque and the nominal rated torque

For plants where high temporary occasional peak torques can occur (ie in excess of the above mentioned KA) the gearing (if nitrided) has to be checked with regard to static strength Unless otherwise specified the same safety factors as for infinite life apply

The scuffing safety is to be specially considered whereby the KA applies in connection with the bulk temperature and the KAP applies for the flash temperature calculation and should replace KA in the formulae in 431 432 and 441

KAP can be evaluated from the torsional impact vibration calculation (as required by the rules)

If the overloads have a duration corresponding to several revolutions of the shafts the scuffing safety has to be consid-ered on basis of this overload both with respect to bulk and flash temperature This applies to plants with ice class nota-tions (Baltic and polar) and to plants with prime movers which have high temporary overload capacity such as eg electric motors (provided the driven member can have a con-siderable increase in demand torque as eg azimuth thrusters during manoeuvring)

For plants without additional ice class notation KAP should normally not exceed 15

8 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

163 Frequent overloads For plants where high overloads or shock loads occur regu-larly the influence of this is to be considered by means of cumulative fatigue (see Appendix A)

17 Load Sharing Factor Kγ The load sharing factor Kγ accounts for the maldistribution of load in multiple-path transmissions (dual tandem epicyclic double helix etc) Kγ is defined as the ratio between the max load through an actual path and the evenly shared load

171 General method Kγ mainly depends on accuracy and flexibility of the branches (eg quill shaft planet support external forces etc) and should be considered on basis of measurements or of relevant analysis as eg

Kγ = δ+δ f

δ = total compliance of a branch under full load (assuming even load share) referred to gear mesh

f = minusminusminusminus+++ 23

22

21 fff where f1 f2 etc are the

main individual errors that may contribute to a maldis-tribution between the branches Eg tooth pitch errors planet carrier pitch errors bearing clearance influences etc Compensating effects should also be considered

For double helical gears

An external axial force Fext applied from sources outside the actual gearing (eg thrust via or from a tooth coupling) will cause a maldistribution of forces between the two helices Expressed by a load sharing factor the

βsdot

plusmn=γ tanFF1K

t

ext

If the direction of Fext is known the calculation should be carried out separately for each helix and with the tangential force corrected with the pertinent Kγ If the direction of Fext is unknown both combinations are to be calculated and the higher σH or σF to be used

172 Simplified method If no relevant analysis is available the following may apply

For epicyclic gears

Kγ = 32501 minus+ pln

where npl = number of planets ( gt3 )

For multistage gears with locked paths and gear stages sepa-rated by quill shafts (see figure below)

Figure 10 Locked paths gear

Kγ = ( )φ201+

where φ = quill shaft twist (degrees) under full load

18 Dynamic Factor Kv The dynamic factor Kv accounts for the internally generated dynamic loads

Kv is defined as the ratio between the maximum load that dynamically acts on the tooth flanks and the maximum ex-ternally applied load Ft KA Kγ

In the following 2 different methods (181 and 182) are described In case of controversy between the methods the next following is decisive ie the methods are listed with increasing priority

It is important to observe the limitations for the method in 181 In particular the influence of lateral stiffness of shafts is often underestimated and resonances occur at considerably lower speed than determined in 1811

However for low speed gears with vmiddotz1 lt 300 calculations may be omitted and the dynamic factor simplified to Kv=105

181 Single resonance method For a single stage gear Kv may be determined on basis of the relative proximity (or resonance ratio) N between actual speed n1 and the lowest resonance speed nE1

N = 1E

1

nn

Note that for epicyclic gears n is the relative speed ie the speed that multiplied with z gives the mesh frequency

1811 Determination of critical speed It is not advised to apply this method for multimesh gears for N gt 085 as the influence of higher modes has to be consid-ered see 182 In case of significant lateral shaft flexibility (eg overhung mounted bevel gears) the influence of cou-pled bending and torsional vibrations between pinion and wheel should be considered if N ge 075 see 182

nE1 = redmγc

1zπ

31030 sdot

where

cγ is the actual mesh stiffness per unit facewidth see 111

Classification Notes- No 412 9 May 2003

DET NORSKE VERITAS

For gears with inactive ends of the facewidth as eg due to high crowning or end relief such as often applied for bevel gears the use of cγ in connection with determination of natu-ral frequencies may need correction cγ is defined as stiffness per unit facewidth but when used in connection with the total mesh stiffness it is not as simple as cγmiddotb as only a part of the facewidth is active Such corrections are given in 111

mred is the reduced mass of the gear pair per unit facewidth and referred to the plane of contact

For a single gear stage where no significant inertias are closely connected to neither pinion nor wheel mred is calcu-lated as

mred = 21

21mm

mm+

The individual masses per unit facewidth are calculated as

m12 = 221b

21

)2d(b

I

where I is the polar moment of inertia (kgmm2)

The inertia of bevel gears may be approximated as discs with diameter equal the midface pitch diameter and width equal to b However if the shape of the pinion or wheel body differs much from this idealised cylinder the inertia should be cor-rected accordingly

For all kind of gears if a significant inertia (eg a clutch) is very rigidly connected to the pinion or wheel it should be added to that particular inertia (pinion or wheel) If there is a shaft piece between these inertias the torsional shaft stiffness alters the system into a 3-mass (or more) system This can be calculated as in 182 but also simplified as a 2-mass system calculated with only pinion and wheel masses

1812 Factors used for determination of Kv Non-dimensional gear accuracy dependent parameters

( )bKKF

yfcB

At

pptp

γ

minus=

( )bKKF

yFcBAt

ff

γ

α minus=

Non-dimensional tip relief parameter

bKKFcC

1BγAt

ak sdotsdot

sdotminus=

For gears of quality grade (ISO 1328) Q = 7 or coarser Bk = 1

For gears with Q le 6 and excessive tip relief Bk is limited to max 1

For gears (all quality grades) with tip relief of more than 2middotCeff (see 432) the reduction of αε has to be considered (see 443)

where

fpt = the single pitch deviation (ISO 1328) max of pinion or wheel

Fα = the total profile form deviation (ISO 1328) max of pinion or wheel (Note Fα is pt not available for bevel gears thus use Fα = fpt)

yp and yf = the respective running-in allowances and may be calculated similarly to yα in 112 ie the value of fpt is replaced by Fα for yf

cacute = the single tooth stiffness see 111

Ca = the amount of tip relief see 433 In case of different tip relief on pinion and wheel the value that results in the greater value of Bk is to be used If Ca is zero by design the value of running-in tip relief Cay (see 112) may be used in the above formula

1813 Kv in the subcritical range Cylindrical gears N le 085

Bevel gears N le 075

Kv = 1 + N K

K = Cv1 Bp + Cv2 Bf + Cv3 Bk

Cv1 accounts for the pitch error influence

Cv1 = 032

Cv2 accounts for profile error influence

Cv2 = 034 for γε le 2

Cv2 = 03ε

057

γ minus for γε gt 2

Cv3 accounts for the cyclic mesh stiffness variation

Cv3 = 023 for γε le 2

Cv3 = 156ε

0096

γ minus for γε gt 2

1814 Kv in the main resonance range Cylindrical gears 085 lt N le 115

Bevel gears 075 lt N le 125

Running in this range should preferably be avoided and is only allowed for high precision gears

Kv = 1 + Cv1 Bp + Cv2 Bf + Cv4 Bk

Cv4 accounts for the resonance condition with the cyclic mesh stiffness variation

Cv4 = 090 for γε le 2

Cv4 = 144ε

ε005057

γ

γ

minus

minus for γε gt 2

10 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

1815 Kv in the supercritical range Cylindrical gears N ge 15

Bevel gears N ge 15

Special care should be taken as to influence of higher vibra-tion modes andor influence of coupled bending (ie lateral shaft vibrations) and torsional vibrations between pinion and wheel These influences are not covered by the following approach

Kv = Cv5 Bp + Cv6 Bf + Cv7

Cv5 accounts for the pitch error influence

Cv5 = 047

Cv6 accounts for the profile error influence

Cv6 = 047 for εγ le 2

Cv6 = 174ε

012

γ minus for εγ gt 2

Cv7 relates the maximum externally applied tooth loading to the maximum tooth loading of ideal accurate gears operating in the supercritical speed sector when the circumferential vibration becomes very soft

Cv7 = 075 for εγ le 15

Cv7 = [ ] 8750)2(sin 0125 +minusβεπ for 15 lt εγ le 25

Cv7 = 10 for εγ gt 25

1816 Kv in the intermediate range Cylindrical gears 115 lt N lt 15

Bevel gears 125 lt N lt 15

Comments raised in 1814 and 1815 should be observed

Kv is determined by linear interpolation between Kv for N = 115 respectively 125 and N = 15 as

Cylindrical gears

( ) ( ) ( )[ ]51Nv151Nv51Nvv KK350

N51KK === minussdot

minus

+=

Bevel gears

( ) ( ) ( )[ ]51Nv251Nv51Nvv KK250

N51KK === minussdot

minus

+=

182 Multi-resonance method For high speed gear (vgt40 ms) for multimesh medium speed gears for gears with significant lateral shaft flexibility etc it is advised to determine Kv on basis of relevant dy-namic analysis

Incorporating lateral shaft compliance requires transforma-tion of even a simple pinion-wheel system into a lumped multi-mass system It is advised to incorporate all relevant inertias and torsional shaft stiffnesses into an equivalent (to pinion speed) system Thereby the mesh stiffness appears as an equivalent torsional stiffness

cγ b (db12)2 (Nmrad)

The natural frequencies are found by solving the set of dif-ferential equations (one equation per inertia) Note that for a gear put on a laterally flexible shaft the coupling bending-torsionals is arranged by introducing the gear mass and the lateral stiffness with its relation to the torsional displacement and torque in that shaft

Only the natural frequency (ies) having high relative dis-placement and relative torque through the actual pinion-wheel flexible element need(s) to be considered as critical frequency (ies)

Kv may be determined by means of the method mentioned in 181 thereby using N as the least favourable ratio (in case of more than one pinion-wheel dominated natural frequency) Ie the N-ratio that results in the highest Kv has to be consid-ered

The level of the dynamic factor may also be determined on basis of simulation technique using numeric time integration with relevant tooth stiffness variation and pitchprofile er-rors

19 Face Load Factors KHβ and KFβ The face load factors KHβ for contact stresses and for scuff-ing KFβ for tooth root stresses account for non-uniform load distribution across the facewidth

KHβ is defined as the ratio between the maximum load per unit facewidth and the mean load per unit facewidth

KFβ is defined as the ratio between the maximum tooth root stress per unit facewidth and the mean tooth root stress per unit facewidth The mean tooth root stress relates to the con-sidered facewidth b1 respectively b2

Note that facewidth in this context is the design facewidth b even if the ends are unloaded as often applies to eg bevel gears

The plane of contact is considered

191 Relations between KHβ and KFβ

KFβ = expβHK 2(hb)hb1

1exp++

=

where hb is the ratio tooth heightfacewidth The maximum of h1b1 and h2b2 is to be used but not higher than 13 For double helical gears use only the facewidth of one helix

If the tooth root facewidth (b1 or b2) is considerably wider than b the value of KFβ(1or2) is to be specially considered as it may even exceed KHβ

Eg in pinion-rack lifting systems for jack up rigs where b = b2 asymp mn and b1 asymp 3 mn the typical KHβ asymp KFβ2 asymp 1 and KFβ1 asymp 13

192 Measurement of face load factors Primarily

KFβ may be determined by a number of strain gauges distrib-uted over the facewidth Such strain gauges must be put in

Classification Notes- No 412 11 May 2003

DET NORSKE VERITAS

exactly the same position relative to the root fillet Relations in 191 apply for conversion to KHβ

Secondarily

KHβ may be evaluated by observed contact patterns on vari-ous defined load levels It is imperative that the various test loads are well defined Usually it is also necessary to evalu-ate the elastic deflections Some teeth at each 90 degrees are to be painted with a suitable lacquer Always consider the poorest of the contact patterns

After having run the gear for a suitable time at test load 1 (the lowest) observe the contact pattern with respect to ex-tension over the facewidth Evaluate that KHβ by means of the methods mentioned in this section Proceed in the same way for the next higher test load etc until there is a full face contact pattern From these data the initial mesh misalign-ment (ie without elastic deflections) can be found by ex-trapolation and then also the KHβ at design load can be found by calculation and extrapolation See example

Figure 11 Example of experimental determination of KHβ

It must be considered that inaccurate gears may accumulate a larger observed contact pattern than the actual single mesh to mesh contact patterns This is particularly important for lapped bevel gears Ground or hard metal hobbed bevel gears are assumed to present an accumulated contact pattern that is practically equal the actual single mesh to mesh con-tact patterns As a rough guidance the (observed) accumu-lated contact pattern of lapped bevel gears may be reduced by 10 in order to assess the single mesh to mesh contact pattern which is used in 199

193 Theoretical determination of KHβ The methods described in 193 to 198 may be used for cy-lindrical gears The principles may to some extent also be used for bevel gears but a more practical approach is given in 199

General For gears where the tooth contact pattern cannot be verified during assembly or under load all assumptions are to be well on the safe side

KHβ is to be determined in the plane of contact

The influence parameters considered in this method are

bull mean mesh stiffness cγ (see 111) (if necessary also variable stiffness over b)

bull mean unit load Fmb = Fbt KA Kγ Kvb (for double helical gears see 17 for use of Kγ)

bull misalignment fsh due to elastic deflections of shafts and gear bodies (both pinion and wheel)

bull misalignment fdefl due to elastic deflections of and work-ing positions in bearings

bull misalignment fbe due to bearing clearance tolerances bull misalignment fma due to manufacturing tolerances bull helix modifications as crowning end relief helix correc-

tion bull running in amount yβ (see 112)

In practice several other parameters such as centrifugal ex-pansion thermal expansion housing deflection etc contrib-ute to KHβ However these parameters are not taken into account unless in special cases when being considered as particularly important

When all or most of the am parameters are to be considered the most practical way to determine KHβ is by means of a graphical approach described in 1931

If cγ can be considered constant over the facewidth and no helix modifications apply KHβ can be determined analyti-cally as described in 1932

1931 Graphical method The graphical method utilises the superposition principle and is as follows

bull Calculate the mean mesh deflection Mδ as a function of

γm c and bF see 111

bull Draw a base line with length b and draw up a rectangu-lar with height δM (The area δM b is proportional to the transmitted force)

bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the areas above and below this zero line are equal

Figure 12 fsh balanced around zero line

bull Superimpose these ordinates of the fsh curve to the previ-ous load distribution curve (The area under this new load distribution curve is still δM b)

bull Calculate the bearing deflections andor working posi-tions in the bearings and evaluate the influence fdefl in the plane of contact This is a straight line and is bal-anced around a zero line as indicated in Fig 14 but with one distinct direction Superimpose these ordinates to the previous load distribution curve

12 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

bull The amount of crowning end relief or helix correction (defined in the plane of contact) is to be balanced around a zero line similarly to fsh

Figure 13 Crowning Cc balanced aound zero line

bull Superimpose these ordinates to the previous load distri-bution curve In case of high crowning etc as eg often applied to bevel gears the new load distribution curve may cross the base line (the real zero line) The result is areas with negative load that is not real as the load in those areas should be zero Thus corrective actions must be made but for practical reasons it may be postponed to after next operation

bull The amount of initial mesh misalignment bema ff + (defined in the plane of contact) is to be bal-

anced around a zero line If the direction of bema ff + is known (due to initial contact check) or if the direction of fbe is known due to design (eg overhang bevel pin-ion) this should be taken into account If direction un-known the influence of bema ff + in both directions as well as equal zero should be considered

Figure 14 fma+fbe in both directions balanced around zero line

Superimpose these ordinates to the previous load distri-bution curve This results in up to 3 different curves of which the one with the highest peak is to be chosen for further evaluation

bull If the chosen load distribution curve crosses the base line (ie mathematically negative load) the curve is to be corrected by adding the negative areas and dividing this with the active facewidth The (constant) ordinates of this rectangular correction area are to be subtracted from the positive part of the load distribution curve It is advisable to check that the area covered under this new load distribution curve is still equal δM b

bull If cγ cannot be considered as constant over b then cor-rect the ordinates of the load distribution curve with the local (on various positions over the facewidth) ratio between local mesh stiffness and average mesh stiffness cγ (average over the active facewidth only) Note that the result is to be a curve that covers the same area δM b as before

bull The influence of running in yβ is to be determined as in 112 whereby the value for Fβx is to be taken as twice the distance between the peak of the load distribution curve and δM

bull Determine

KHβ = M

β

δycurveofpeak minus

1932 Simplified analytical method for cylindrical gears The analytical approach is similar to 1931 but has a more limited application as cγ is assumed constant over the face-width and no helix modification applies

bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the area above and below this zero line are equal see Fig 12 The max positive ordinate is frac12∆fsh

bull Calculate the initial mesh alignment as Fβx= deflbemash ffff plusmnplusmnplusmn∆ The negative signs may only be used if this is justified andor verified by a contact pattern test Otherwise al-ways use positive signs If a negative sign is justified the value of Fβx is not to be taken less than the largest of each of these elements

bull Calculate the effective mesh misalignment as Fβy = Fβx - yβ (yβ see 112)

bull Determine

2KforF2

bFc1K βH

m

βγγH le+=β

or

2KforF

bFc2K Hβ

m

βγγH gt=β

where cγ as used here is the effective mesh stiffness see 111

194 Determination of fsh fsh is the mesh misalignment due to elastic deflections Usu-ally it is sufficient to consider the combined mesh deflection of the pinion body and shaft and the wheel shaft The calcu-lation is to be made in the plane of contact (of the considered gear mesh) and to consider all forces (incl axial) acting on the shafts Forces from other meshes can be parted into components parallel respectively vertical to the considered plane of contact Forces vertical to this plane of contact have no influence on fsh

It is advised to use following diameters for toothed elements

d + 2 x mn for bending and shear deflection

d + 2 mn (x ndash ha0 + 02) for torsional deflection

Usually fsh is calculated on basis of an evenly distributed load If the analysis of KHβ shows a considerable maldis-

Classification Notes- No 412 13 May 2003

DET NORSKE VERITAS

tribution in term of hard end contact or if it is known by other reasons that there exists a hard end contact the load should be correspondingly distributed when calculating fsh In fact the whole KHβ procedure can be used iteratively 2-3 iterations will be enough even for almost triangular load distributions

195 Determination of fdefl fdefl is the mesh misalignment in the plane of contact due to bearing deflections and working positions (housing deflec-tion may be included if determined)

First the journal working positions in the bearings are to be determined The influence of external moments and forces must be considered This is of special importance for twin pinion single output gears with all 3 shafts in one plane

For rolling bearings fdefl is further determined on basis of the elastic deflection of the bearings An elastic bearing deflec-tion depends on the bearing load and size and number of rolling elements Note that the bearing clearance tolerances are not included here

For fluid film bearings fdefl is further determined on basis of the lift and angular shift of the shafts due to lubrication oil film thickness Note that fbe takes into account the influence of the bearing clearance tolerance

When working positions bearing deflections and oil film lift are combined for all bearings the angular misalignment as projected into the plane of the contact is to be determined fdefl is this angular misalignment (radians) times the face-width

196 Determination of fbe fbe is the mesh misalignment in the plane of contact due to tolerances in bearing clearances In principle fbe and fdefl could be combined But as fdefl can be determined by analy-sis and has a distinct direction and fbe is dependent on toler-ances and in most cases has no distinct direction (ie + toler-ance) it is practicable to separate these two influences

Due to different bearing clearance tolerances in both pinion and wheel shafts the two shaft axis will have an angular mis-alignment in the plane of contact that is superimposed to the working positions determined in 195 fbe is the facewidth times this angular misalignment Note that fbe may have a distinct direction or be given as a + tolerance or a combina-tion of both For combination of + tolerance it is adviced to use

fbe= 22be

21be ff ++plusmn

fbe is particularly important for overhang designs for gears with widely different kinds of bearings on each side and when the bearings have wide tolerances on clearances In general it shall be possible to replace standard bearings with-out causing the real load distribution to exceed the design premises For slow speed gears with journal bearings the expected wear should also be considered

197 Determination of fma fma is the mesh misalignment due to manufacturing toler-ances (helix slope deviation) of pinion fHβ1 wheel fHβ2 and housing bore

For gear without specifically approved requirements to as-sembly control the value of fma is to be determined as

fma= 22Hβ

21Hβ ff +

For gears with specially approved assembly control the value of fma will depend on those specific requirements

198 Comments to various gear types For double helical gears KHβ is to be determined for both helices Usually an even load share between the helices can be assumed If not the calculation is to be made as de-scribed in 171

For planetary gears the free floating sun pinion suffers only twist no bending It must be noted that the total twist is the sum of the twist due to each mesh If the value of 1K γ ne this must be taken into account when calculating the total sun pinion twist (ie twist calculated with the force per mesh without Kγ and multiplied with the number of planets)

When planets are mounted on spherical bearings the mesh misalignments sun-planet respectively planet-annulus will be balanced Ie the misalignment will be the average between the two theoretical individual misalignments The faceload distribution on the flanks of the planets can take full advan-tage of this However as the sun and annulus mesh with several planets with possibly different lead errors the sun and annulus cannot obtain the above mentioned advantage to the full extent

199 Determination of KHβ for bevel gears If a theoretical approach similar to 193-198 is not docu-mented the following may be used

testeff

H Kb

b851851K sdot

minussdot=β

beff b represents the relative active facewidth (regarding lapped gears see 192 last part)

Higher values than beff b = 090 are normally not to be used in the formula

For dual directional gears it may be difficult to obtain a high beff b in both directions In that case the smaller beff b is to be used

Ktest represents the influence of the bearing arrangement shaft stiffness bearing stiffness housing stiffness etc on the faceload distribution and the verification thereof Expected variations in length- and height-wise tooth profile is also accounted for to some extent

a) Ktest = 1 For ground or hard metal hobbed gears with the specified contact pattern verified at full rating or at full torque slow turning at a condition representative for the thermal expansion at normal operation

14 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

It also applies when the bearing arrangementsupport has insignificant elastic deflections and thermal axial expansion However each initial mesh contact must be verified to be within acceptance criteria that are cali-brated against a type test at full load Reproduction of the gear tooth length- and height-wise profile must also be verified This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon)

b) Ktest = 1 + 04middot(beffbndash06) For designs with possible influence of thermal expansion in the axial direction of the pinion The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load

c) Ktest = 12 if mesh is only checked by toolmakerrsquos blue or by spin test contact For gears in this category beffb gt 085 is not to be used in the calculation

110 Transversal Load Distribution Factors KHα and KFα The transverse load distribution factors KHα for contact stresses and for scuffing KFα for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairs of teeth in mesh

The following relations may be used

Cylindrical gears

( )

minus+==

tH

αptγγHαFα F

byfc0409

2ε

KK

valid for 2εγ le

( ) ( )tH

αptγ

γ

γHαFα F

byfcε

1ε20409KK

minusminus+==

valid for 2ε γ gt

where

FtH = Ft KA Kγ Kv KHβ

cγ = See 111

γα = See 112

fpt = Maximum single pitch deviation (microm) of pinion or wheel or maximum total profile form devia-tion Fα of pinion or wheel if this is larger than the maximum single pitch deviation

Note In case of adequate equivalent tip relief adapted to the load half of the above mentioned fpt can be introduced A tip relief is considered adequate when the average of Ca1 and Ca2 is within plusmn40 of the value of Ceff in 432

Limitations of KHα and KFα

If the calculated values for

KFα = KHα lt 1 use KFα = KHα = 10

If the calculated value of KHα gt 2εα

γ

Zε

ε use KHα = 2

εα

γ

Zε

ε

If the calculated value of KFα gt εα

γ

Yεε

use KFα = εα

γ

Yεε

where Yε = αnε

075025+ (for εαn see 331c)

Bevel gears

For ground or hard metal hobbed gears KFα = KHα = 1

For lapped gears KFα = KHα = 11

111 Tooth Stiffness Constants cacute and cγ The tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having 1 mm facewidth by an amount of 1 microm in the plane of contact cacute is the maximum stiffness of a single pair of teeth cγ is the mean value of the mesh stiffness in a transverse plane (brief term mesh stiffness)

Both valid for high unit load (Unit load = Ft middot KA middot Kγb)

Cylindrical gears The real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bending stiff-nesses For high unit loads the Hertzian stiffness has little importance and can be disregarded This approach is on the safe side for determination of KHβ and KHα However for moderate or low loads Kv may be underestimated due to determination of a too high resonance speed

The linear approach is described in A

An optional approach for inclusion of the non-linear stiffness is described in B

A The linear approach

BRCCq

βcos08acutec =

and

( )025ε075cc αγ +prime=

where

( )[ ]n02a01a

B α2000212

hh12051C minusminus

+minus+=

12n1n

x000635z

025791z

015551004723q minus++=

12

2n

22

1n

1 x005290z

x024188x000193z

x011654+minusminusminus

+ 000182 x22

Classification Notes- No 412 15 May 2003

DET NORSKE VERITAS

(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as

( )( )nR m5s

sR e5

bbln1C +=

where

bs = thickness of a central web

sR = average thickness of rim (net value from tooth root to inside of rim)

The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered

Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ

B The non-linear approach

In the following an example is given on how to consider the non-linearity

The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm

With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as

( )10δKbF

minus= for 500bFgt

minus=

500Fb10δK

bF for 500

bFlt

with γAt KK

bF

bF

sdotsdot= etc (Nmm) ie unit load incorporat-

ing the relevant factors as

KA middot Kγ for determination of Kv

KA middot Kγ middot Kv for determination of KHβ

KA middot Kγ middot Kv middot KHβ for determination of KHα

δ = mesh deflection (microm)

K = applicable stiffness (c or cγ)

Use of stiffnesses for KV KHβ and KHα

For calculation of Kv and KHα the stiffness is calculated as follows

When Fb lt 500

the stiffness is determined as δ∆

∆ bF

where the increment is chosen as eg ∆ Fb = 10 and thus

50010Fb10

K10Fb∆δ +

++

=

When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly

or an equivalent stiffness determined as δsdotb

F

Bevel gears

In lack of more detailed relationship between stiffness and geometry the following may be used

b085

b13cacute eff=

b085b

16c effγ =

beff not to be used in excess of 085 b in these formulae

Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)

112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements

yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload

Cay is defined as the running-in amount that compensates for lack of tip relief

The following relations may be used

For not surface hardened steel

ptHlim

α fσ160y =

yβ = βxlimH

f320σ

with the following upper limits

V lt 5 ms 5-10 ms gt 10 ms

yα max none

limH

12800σ limH

6400σ

yβ max none

limH

25600σ limH

12800σ

For surface hardened steel

yα = 0075 fpt but not more than 3 for any speed

yβ = 015 Fβx but not more than 6 for any speed

16 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

For all kinds of steel

5145189718

1C2

limHay +

minusσ

=

When pinion and wheel material differ the following ap-plies

bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv

bull Use ( )2β1ββ yy21y += in the calculation of KHβ

bull Use ( )2ay1aya CC21C += in the calculation of Kv

bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing

calculation if no design tip relief is foreseen

Classification Notes- No 412 17 May 2003

DET NORSKE VERITAS

2 Calculation of Surface Durability

21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213

Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted

Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213

For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength

For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied

22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater

Calculation of surface durability for helical gears is based on the contact stress at the pitch point

For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies

Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact

Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses

The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses

The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel

221 Contact stress Cylindrical gears

( )HαHβvγA

1

tβεEHDBH KKKKK

bud1uF

ZZZZZσ+

=

where

ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)

ZH = Zone factor for pitch point (see 231)

ZE = Elasticity factor (see 24)

Zε = Contact ratio factor (see 25)

Zβ = Helix angle factor (see 26)

Ft KA Kγ Kv KHβ KHα see 15 ndash 110

d1 b u see 12 ndash 15

Bevel gears

( )HαHβvγA

v1v

vmtKEMH KKKKK

bud1uF

ZZZ105σ+

sdot=

where

105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)

ZE KA etc see above

ZM = mid-zone factor see 233

ZK = bevel gear factor see 27

Fmt dv1 uv see 12 ndash 15

It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks

222 Permissible contact stress

XWRvLH

NHlimHP ZZZZZ

SZσ

σ =

where

σH lim = Endurance limit for contact stresses (see 28)

ZN = Life factor for contact stresses (see 29)

SH = Required safety factor according to the rules

ZLZvZR = Oil film influence factors (see 210)

ZW = Work hardening factor (see 211)

ZX = Size factor (see 212)

18 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

23 Zone Factors ZH ZBD and ZM

231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder

wtt2

wtbH sinααcos

cosαcosβ2Z =

232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel

For εβ ge 1 ZBD = 1

For internal gears ZD = 1

For εβ = 0 (spur gears)

( )

π

minusεminusminus

π

minusminus

α=

α2

2

2b

2a

1

2

1b

1a

wtB

z211

dd

z21

dd

tanZ

( )

π

minusεminusminus

π

minusminus

α=

α1

2

1b

1a

2

2

2b

2a

wtD

z211

dd

z21

dd

tanZ

If ZB lt 1 use ZB = 1

If ZD lt 1 use ZD = 1

For 0 lt εβ lt 1

ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)

233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears

εminusminus

εminusminus

αβ=

αα btm2

2vb2

2vabtm2

1vb2val

2v1vvtbmM

pddpdd

ddtancos2Z

This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint

234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel

24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses

For steel against steel ZE = 1898

25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses

αε1Z =ε for 1εβ ge

( )α

ββ

αε ε

εε1

3ε4

Z +minusminus

= for εβ lt 1

26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability

cosβZβ =

27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply

The following may be used ZK = 080

28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ

σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting

For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply

For this purpose pitting is defined by

bull for not surface hardened gears pitted area ge 2 of total active flank area

bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given

material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies

The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as

Classification Notes- No 412 19 May 2003

DET NORSKE VERITAS

the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99

σHlim σH105 σH10

3

Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade

1650 1500

2500 2400

3100 3100

Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)

1000

13 σHlim

13 σHlim

Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV

Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV

Carbon steel 15 HV + 250 16 σHlim 16 σHlim

These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15

29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply

If this is not documented by approved fatigue tests the fol-lowing method may be used

For all steels except nitrided

7L 105N sdotge ZN = 1 or

01570

L

7

N N105Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 5middot107 510N

logZ037

L

7

N N105Z

sdot=

NL = 105 WXRVLHlim

WstX10H1010NN ZZZZZσ

ZZσZZ

55

5=

103 lt NL lt 105 )Z(Zlog05

L

5

10NN

5N103N10

5N10ZZ

==

3L 10N le

WXRVLlimH

Wst10X10H10NN ZZZZZ

ZZZZ

33

3σ

σ==

(but not less than ZN105)

For nitrided steels

6L 102N sdotge ZN = 1 or

00980

L

6

N N102Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 2middot106 510N

Zlog76860

L

6

N N102Z

sdot=

5L 10N le

XWRVL

10XWst10NN ZZZZZ

ZZ13ZZ

55 ==

Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles

210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity

The following methods may be applied in connection with the endurance limit

20 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Surface hardened steels Not surface hardened steels

ZL ( )24013421

360910ν+

+

( )24013421

680830ν+

+

ZV ( )v3280

140930+

+ ( )v3280300850

++

ZR

080

ZrelR3

150

ZrelR3

where

ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits

For values of ν40 gt 500 use ν40 = 500

RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm

RZrel

= ( ) 31

cZ2Z1 ρ

10RR50

+

RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)

For 5

L 10N le ZL ZV ZR = 10

211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface

The following approximation may be used for the endurance limit

Surface hardened steel against not surface hardened steel 150

ZeqW R

31700

130HB21Z

minus

minus=

where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130

RZeq = equivalent roughness

033

c40

066

ZS

ZHZHZEQ ρvν

15000RRRR

=

If RZeq gt 16 then use RZeq = 16

If RZeq lt 15 then use RZeq = 15

where

RZH = surface roughness of the hard member before run in

RZS = surface roughness of the soft member before run in

ν40 = see 210

If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem

Through hardened pinion against softer wheel

( )

minussdotminus+= 000829

HBHB0008981u1Z

2

1W

For 21HBHB

2

1 le use ZW = 1

For 71HBHB

2

1 gt use 17HBHB

2

1 =

For u gt 20 use u = 20

For static strength (lt 105 cycles)

Surface hardened against not surface hardened

ZWst = 105

Through hardened pinion against softer wheel

ZWst = 1

212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality

ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213

213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented

The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent

Classification Notes- No 412 21 May 2003

DET NORSKE VERITAS

stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically

The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence

The real Hertzian stresses σHR are determined as

For helical gears with εβ gt 1

σHR = σH

For helical gears with εβ lt 1 and spur gears

ε

α

ββ ε

ε+εminus

sdotσ=σZ

1

HHR

For bevel gears

KHHR Z

1σσ sdot=

The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)

The coordinates tz and HV are to be compared with the de-sign specification such as

bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400

and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)

In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods

For high cycle fatigue (gt3 106 cycles) the following applies

sdot+

minussdotsdotsdot= o90

05azt

05azt

cosSσ04HV

H

HHHR

applicable to 05a

zt

Hge

For 05at

H

z lt the value for 05at

H

z = applies

56300ρSσ12a cHHR

Hsdotsdot

sdot=

Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength

If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows

reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue

For static strength (lt103 cycles) the following applies

sdot+

minussdotsdotsdot= o90

07at

06a

zt

cosSσ019HV

Hst

z

HstHHR

applicable to 06at

Hst

z ge

56300ρSσa cHHR

Hstsdotsdot

=

In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies

For limited life fatigue (103 lt cycles lt 3106 )

For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance

The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram

minussdotminus

= sdot logN3477

logSlogSlogS

310H6103HHN

36 10H103H Slog86281Slog86280 sdot+sdot sdot

22 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3 Calculation of Tooth Strength

31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding

For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36

For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning

Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα

In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314

A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired

It should be noted that this part 3 does not cover fractures caused by

bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining

Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears

Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied

32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level

321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance

Normally the stresses for pinion and wheel are calculated as

Cylindrical gears

FαFβvγAβSFn

tF K K K K K Y Y Y

m bFσ =

where

YF = Tooth form factor (see 33)

YS = Stress correction factor (see 34)

Yβ = Helix angle factor (see 36)

Ft KA Kγ Kv KFβ KFα see 15 ndash 110

b see 13

Bevel gears

FαFβvγASaFamn

mtF K K K K K Y Y Y

m bFσ ε=

where

YFa = Tooth form factor see 33

YSa = Stress correction factor see 34

Yε = Contact ratio factor see 35

Fmt KA etc see 15 ndash 110

b see 13

322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is

CXRrelTrelTF

NMFEFP YYYY

SYY

δσ

=σ

Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN

where

σFE = Local tooth root bending endurance limit of reference test gear (see 37)

YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)

YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)

SF = Required safety factor according to the rules

YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)

YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)

Classification Notes- No 412 23 May 2003

DET NORSKE VERITAS

YX = Size factor (see 312)

YC = Case depth factor (see 313)

33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress

YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears

YFa applies to load application at the tooth tip and is used for bevel gears

Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears

Figure 31 External tooth in normal section

Figure 32 Internal tooth in normal section

Definitions

n

2

n

Fn

enFn

Fe

F

α cosms

α cosmh6

Y

=

n

2

n

Fn

anFn

Fa

Fa

α cosms

α cosmh6

Y

=

In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth

YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)

The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing

mn with mnm

zn with zvn

αt with αvt

β with βm

with undercut without undercut

Fig 33 Dimensions and basic rack profile of the teeth (finished profile)

Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless

331 Determination of parameters

( )n

n

prnfPnfP m

α cossαsin 1ρ

αtan h4πE

minusminusminusminus=

For external gears fPfP ρρ =

For internal gears ( )0z

195fPfP0

fPfP 10363156ρhxρρ

sdotminus+

+=

where

z0 = number of teeth of pinion cutter

x0 = addendum modification coefficient of pinion cutter

hfP = addendum of pinion cutter

ρfP = tip radius of pinion cutter

xhρG fpfp +minus=

24 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

τmE

2π

z2H

nnminus

minus=

with

3πτ = for external gears

6πτ = for internal gears

Htan zG2

nminusϑ=ϑ

(to be solved iteratively suitable start value6π

=ϑ for exter-

nal gears and 3π for internal gears)

a) Tooth root chord sFn For external gears

minus

ϑ+

ϑminus= ρ

cosG3

3πsinz

ms

fpnn

Fn

For bevel gears with a tooth thickness modification

xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn

For internal gears

minus

ϑ+

ϑminus= ρ

cosG

6πsinz

ms

fPnn

Fn

b) Root fillet radius ρF at 30ordm tangent

( )G2coszcosG2ρ

mρ

2n

2

fpn

F

minusϑϑ+=

c) Determination of bending moment arm hF dn = zn mn

b2α

αn βcosε

ε =

dan = dn + 2 ha

pbn = π mn cos αn

dbn = dn cos αn

( )4

d1εpzz

2dd

zz2d

2bn

2

αnbn

2bn

2an

en +

minusminus

minus=

en

bnen d

dcos arcα =

ennnn

e α invα invα x tan 22π

z1

minus+

+=γ

αFen = αen ndash γe

For external gears

( )

minus=

n

enFenee

n

Fe

mdαtan sin γ γcos

21

mh

]ρcos

G3πcosz fpn +

ϑminus

ϑminusminus

For internal gears

( ) minus

sdotsdotminus=

n

enFenee

n

Fe

mdα tanγ sinγ cos

21

mh

minus

ϑminus

ϑminussdot ρ

cosG3

6πcosz fPn

332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as

250ε205for 0666ε2366Y αnαnDT leleminus=

205εfor 10Y αnDT lt=

34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section

Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm

YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip

YS can be determined as follows

( )

+

+=L23121

1

sS qL01312Y

where Fe

Fn

hsL = and

F

Fns ρ2

sq = (see 33)

YSa can be calculated by replacing hFe with hFa in the above formulae

Note a) Range of validity 1 lt qs lt 8

In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered

b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with

Classification Notes- No 412 25 May 2003

DET NORSKE VERITAS

g

g

ρt

0613

13

minus

where

tg = depth of the grinding notch

ρg = radius of the grinding notch

c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles

35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears

The following may be used

Yε = 0625

36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank

The following may be used (β input in degrees)

Yβ = 1 ndash εβ β120

When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula

However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)

37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM

σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision

If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules

σFE

Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)

bull of specially approved high grade

1050

bull of normal grade

minus CrNiMo steels with approved process

1000

minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850

Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

840

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

720

Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)

07 HV + 300

Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)

025 σB + 125

Alloyed quenched and tempered steel 04 σB + 200

Carbon steel 025 σB + 250

Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40

1) These values are valid for a root radius

bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)

bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)

bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for

6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked

2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)

26 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears

YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0

YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst

For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316

The following method may be used within a stress ratio ndash12 lt R lt 05

381 For idlers planets and PTO with ice class

M1M1R1

1Yor Y MstM

+minus

minus=

where

R = stress ratio = min stress divided by max stress

For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12

For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as

branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus

For a power take off (PTO) with ice class see 161 c

M considers the mean stress influence on the endurance (or static) strength amplitudes

M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress

Following M values may be used

Endurance limit

Static strength

Case hardened 08 ndash 015 Ys 1) 07

If shot peened 04 06

Nitrided 03 03

Induction or flame hardened

04 06

Not surface hardened steel 03 05

Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M

The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches

382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106

For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers

For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used

383 For gears with shrinkage stresses and unidirectional load For endurance strength

FE

fitM σ

σM1

M21Y+

minus=

σFE is the endurance limit for R = 0

For static strength YMst = 1 and σfit accounted for in 39b

σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor

n

Ffit m

ρ215scf minus=

384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is

( ) ( ) FE

fitM σ

σR1M1

M2

M1M1R1

1Yminussdot+

minus

+minus

minus=

Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12

For static strength

M1M1R1

1YMst

+minus

minus=

The effect of σfit is accounted for in 39 b

Classification Notes- No 412 27 May 2003

DET NORSKE VERITAS

385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed

F

y

Sσ225

for not surface hardened fillets

FSHV5 for surface hardened fillets

39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles

Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN

If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used

Determination of the σ ndash N ndash curve

a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF

b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ

FstS1

FPstσ minussdotsdotsdot=

where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability

σFst

Alloyed case hardened steel 1) 2300

Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

1250

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

1050

Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)

18 HV + 800

Steel with not surface hardened fillets the smaller value of 2)

18 σB or 225 σy

1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet

c) Calculate YN as

NL gt 3middot106

YN = 1 or 001

L

6

N N103Y

sdot= ie Yn = 092 for 1010

The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2

103ltNLlt3middot106exp

L

6

N N103Y

sdot=

cycles6103forσcycles310forσlog02876exp

FP

FPst

sdot=

NL lt 103 cycles6103forσcycles310forσ

NYFP

FPst

sdot=

or simply use σFPst as mentioned in b) directly

Guidance on number of load cycles NL for various applica-tions

bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)

bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied

28 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage

YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit

The following method may be used

For endurance limit

for not surface hardened fillets

( )

024

s024

relT σ103133q21σ1012201351

Ysdotsdotminus

+sdotsdotminus+= minus

minus

δ

for all surface hardened fillets except nitrided

( )

106q21002451

Y srelT

++=δ

for nitrided fillets

( )

1347q2101421

Y srelT

++=δ

For static strength

for not surface hardened fillets1)

( )( )( )025

02

02502s

relTst σ3000821σ300 1Y0821Y

+

minus+=δ

for surface hardened fillets except nitrided

YδrelTst = 044 YS + 012

for nitrided fillets

YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not

exceed the yield point and thereby alter the residual stress level See also 39b footnote 2

311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness

YRrelT differs for endurance limit and static strength

The following method may be used

For endurance limit

YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels

YδrelT = 43 ndash 326 (Ry + 1)0005

for nitrided steels

For static strength

YRrelTst = 1 for all Ry and all materials

For a fillet without any longitudinal machining trace Ry asymp Rz

312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength

The following may be used

For endurance limit

YX = 1 for mn le 5 generally

YX = 103 ndash 0006 mn for 5 lt mn lt 30

YX = 085 for mn ge 30

for not surface hard-ened steels

YX = 105 ndash 001 mn for 5 lt mn ge 25

YX = 08 for mn ge 25

for surface hardened steels

For static strength

YXst = 1 for all mn and all materials

313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength

YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength

In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength

The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses

The following simplified method for YC may be used

YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)

YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10

For endurance limit

+

+=nFFE

C m02ρt31

σconstY

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

4 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

1 Basic Principles and General Influence Factors

11 Scope and Basic Principles The gear rating procedures given in this Classification Note are mainly based on the ISO-6336 Part 1-5 (cylindrical gears) and partly on ISO 10300 Part 1-3 (bevel gears) and ISO Technical Reports on Scuffing and Fatigue Damage Accumulation but especially applied for marine purposes such as marine propulsion and important auxiliaries onboard ships and mobile offshore units

The calculation procedures cover gear rating as limited by contact stresses (pitting spalling or case crushing) tooth root stresses (fatigue breakage or overload breakage) and scuff-ing resistance Even though no calculation procedures for other damages such as wear grey staining (micropitting) etc are given such damages may limit the gear rating

The Classification Note applies to enclosed parallel shaft gears epicyclic gears and bevel gears (with intersecting axis) However open gear trains may be considered with regard to tooth strength ie part 1 and 3 may apply Even pinion-rack tooth strength may be considered but since such gear trains often are designed with non-involute pinions the calculation procedure of pinion-racks is described in Appen-dix C

Steel is the only material considered

The methods applied throughout this document are only valid for a transverse contact ratio 1 lt εα lt 2 If εα gt 2 either special considerations are to be made or suggested simplifi-cation may be used

All influence factors are defined regarding their physical interpretation Some of the influence factors are determined by the gear geometry or have been established by conven-tions These factors are to be calculated in accordance with the equations provided Other factors are approximations which are clearly stated in the text by terms as laquomay be cal-culated asraquo These approximations are substitutes for exact evaluations where such are lacking or too extensive for prac-tical purposes or factors based on experience In principle any suitable method may replace these approximations

Bevel gears are calculated on basis of virtual (equivalent) cylindrical gears using the geometry of the midsection The virtual (helical) cylindrical gear is to be calculated by using all the factors as a real cylindrical gear with some exceptions These exceptions are mentioned in connection with the ap-plicable factors Wherever a factor or calculation procedure has no reference to either cylindrical gears or bevel gears it is generally valid ie combined for both cylindrical and bevel

In order to minimise the volume of this Classification Note such combinations are widely used and everywhere it is necessary to distinguish it is clearly pointed out by local headings such as

Cylindrical gears

Bevel gears

The permissible contact stresses tooth root stresses and scuffing load capacity depend on the safety factors as re-quired in the respective Rule sections

Terms as endurance limit and static strength are used throughout this Classification Note

Endurance limit is to be understood as the fatigue strength in the range of cycles beyond the lower knee of the σndashN curves regardless if it is constant or drops with higher number of cycles

Static strength is to be understood as the fatigue strength in the range of cycles less than at the upper knee of the σndashN curves

For gears that are subjected to a limited number of cycles at different load levels a cumulative fatigue calculation applies Information on this is given in Appendix A

When the term infinite life is used it means number of cy-cles in the range 108ndash1010

12 Symbols Nomenclature and Units The symbols are generally from ISO 701 ISOR31 and ISO 1328 with a few additional symbols Only SI units are used

The main symbols as influence factors (K Z Y and X with indeces) etc are presented in their respective headings Symbols which are not explained in their respective Secs are as follows

a = centre distance (mm) b = facewidth (mm) d = reference diameter (mm) da = tip diameter (mm) db = base diameter (mm) dw = working pitch diameter (mm) ha = addendum (mm) ha0 = addendum of tool ref to mn hfp = dedendum of basic rack ref to mn (= ha0) hFe = bending moment arm (mm) for tooth root

stresses for application of load at the outer point of single tooth pair contact

hFa = bending moment arm (mm) for tooth root stresses for application of load at tooth tip

HB = Brinell hardness HV = Vickers hardness HRC = Rockwell C hardness mn = normal module n = rev per minute NL = number of load cycles qs = notch parameter Ra = average roughness value (microm) Ry = peak to valley roughness (microm) Rz = mean peak to valley roughness (microm) san = tooth top land thickness (mm) sat = transverse top land thickness (mm) sFn = tooth root chord (mm) in the critical section spr = protuberance value of tool minus grinding stock

equal residual undercut of basic rack ref to mn T = torque (Nm) u = gear ratio (per stage) v = linear speed (ms) at reference diameter

Classification Notes- No 412 5 May 2003

DET NORSKE VERITAS

x = addendum modification coefficient z = number of teeth zn = virtual number of spur teeth αn = normal pressure angle at ref cylinder αt = transverse pressure angle at ref cylinder αa = transverse pressure angle at tip cylinder αwt = transverse pressure angle at pitch cylinder β = helix angle at ref cylinder βb = helix angle at base cylinder βa = helix angle at tip cylinder εα = transverse contact ratio εβ = overlap ratio εγ = total contact ratio ρa0 = tip radius of tool ref to mn ρfp = root radius of basic rack ref to mn ( = ρa0) ρC = effective radius (mm) of curvature at pitch point ρF = root fillet radius (mm) in the critical section σB = ultimate tensile strength (Nmm2) σy = yield strength resp 02 proof stress (Nmm2)

Index 1 refers to the pinion 2 to the wheel

Index n refers to normal section or virtual spur gear of a heli-cal gear

Index w refers to pitch point

Special additional symbols for bevel gears are as follows

Σ = angle between intersection axis Kϑ = angle modification (Klingelnberg)

m0 = tool module (Klingelnberg) δ = pitch cone angle xsm = tooth thickness modification coefficient (mid-

face) R = pitch cone distance (mm)

Index v refers to the virtual (equivalent) helical cylindrical gear

Index m refers to the midsection of the bevel gear

13 Geometrical Definitions For internal gearing z2 a da2 dw2 d2 and db2 are negative x2 is positive if da2 is increased ie the numeric value is de-creased

The pinion has the smaller number of teeth ie

11

2 ge=zz

u

For calculation of surface durability b is the common face-width on pitch diameter

For tooth strength calculations b1 or b2 are facewidths at the respective tooth roots If b1 or b2 differ much from b above they are not to be taken more than 1 module on either side of b

Cylindrical gears

tan αt = tan αn cos β tan βb = tan β cos αt

tan βa = tan β da d

cos αa = dbda

d = z mn cos β mt = mn cos β db = d cos αt = dw cos αwt

a = 05 (dw1 + dw2)

dw1dw2 = z1 z2

inv α = tan α - α (radians)

inv αwt = inv αt + 2 tan αn (x1 + x2)(z1 + z2)

zn = z (cos2 βb cos β)

1

aw1fw1α T

ξξε

+=

where ξfw1 is to be taken as the smaller of

bull wtfw1 αtanξ =

bull soi1

b1wtfw1 d

dacostan -tanαξ =

bull 1

2wt

a2

b2fw1 z

ztanα

d

dacostan ξ

minus=

and

2

1fw2aw1 z

zξξ = where ξfw2 is calculated as ξfw1

substituting the values for the wheel by the values for the pinion and visa versa

11 z

2πT =

( ) +

sdot+minusminussdot= minus

2sinαρρxhm

2d2d nfpfp1fpnsoi1

21

t

nfpfplfpn2

tanα)sinαρρx(hm

sdot+minusminus

nmsinbπ

β=εβ

(for double helix b is to be taken as the width of one helix)

εy = βα εε +

ρC = ( )2

b

wt

u1βcos

αsinua

+

v = 311 10dn

60π minus

6 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

pbt = βcos

αcosmπ tn

sat =

minus++

at

ninvααinv

z

αtanx22

π

d a

san = aβcossat

14 Bevel Gear Conversion Formulae and Specific Formulae Conversion of bevel gears to virtual equivalent helical cylin-drical gears is based on the bevel gear midsection The con-version formulae are

Number of teeth

zv12 = z12 cos δ12

(δ1 + δ2 = Σ)

Gear ratio

uv = 1v

2vzz

tan αvt = tan αn cos βm

tan βbm = tan βm cos αvt

Base pitch

pbtm = m

vtnmβcos

αcosmπ

Reference pitch diameters

dv12 = 21

21m

cosdδ

Centre distance

av = 05 (dv1 + dv2)

Tip diameters

dva 12 = dv 12 + 2 ham 12

Addenda

for gears with constant addenda (Klingelnberg)

ham 12 = mmn (1 + xm 12)

for gears with variable addenda (Gleason)

ham 12 = ha 12 ndash b2 tan (δa 12 ndash δ12)

(when ha is addendum at outer end and δa is the outer cone angle)

Addendum modification coefficients

xm 12 = mn

12am21am

m2

hh minus

Base circle

dvb 12 = dv 12 cos αvt

Transverse contact ratio)

εα = btmP

αsinadd05dd05 vtv2

2vb2

2va2

1vb2

1va minusminus+minus

Overlap ratio) (theoretical value for bevel gears with no crowning but used as approximations in the calculation pro-cedures)

εβ = nm

mmπβsinb

Total contact ratio)

εγ = 2β

2α εε +

( Note that index laquovraquo is left out in order to combine formu-lae for cylindrical and bevel gears)

Tangential speed at midsection

vmt = 3m11 10dn

60π minus

Effective radius of curvature (normal section)

ρvc = ( )2vbm

vtvv

u1βcosαsinua

+

Length of line of contact

lb = ( )( )( )

1εifε

ε1ε2ε

βcosεb

β2γ

2βα

2γ

bm

α ltminusminusminus

lb = 1εifβcosε

εbβ

bmγ

α ge

15 Nominal Tangential Load Ft Fbt Fmt and Fmbt The nominal tangential load (tangential to the reference cyl-inder with diameter d and perpendicular to an axial plane) is calculated from the nominal (rated) torque T transmitted by the gear set

Cylindrical gears

dT2000

Ft = t

tbt αcos

FF =

Bevel gears

mmt d

T2000F =

vt

mtmbt αcos

FF =

Classification Notes- No 412 7 May 2003

DET NORSKE VERITAS

16 Application Factors KA and KAP The application factor KA accounts for dynamic overloads from sources external to the gearing

It is distinguished between the influence of repetitive cyclic torques KA (161) and the influence of temporary occasional peak torques KAP (162)

Calculations are always to be made with KA In certain cases additional calculations with KAP may be necessary

For gears with a defined load spectrum the calculation with a KA may be replaced by a fatigue damage calculation as given in Appendix A

161 KA For gears designed for long or infinite life at nominal rated torque KA is defined as the ratio between the maximum re-petitive cyclic torque applied to the gear set and nominal rated torque

This definition is suitable for main propulsion gears and most of the auxiliary gears

KA can be determined by measurements or system analysis or may be ruled by conventions (ice classes) (For the pur-pose of a preliminary (but not binding) calculation before KA is determined it is advised to apply either the max values mentioned below or values known from similar plants)

a) For main propulsion gears KA can be taken from the (mandatory) torsional vibration analysis thereby con-sidering all permissible driving conditions) Unless specially agreed the rules do not allow KA in excess of 135 for diesel propulsion) With turbine or electric propulsion KA would normally not exceed 12 However special attention should be given to thrusters that are arranged in such a way that heavy vessel movements andor manoeuvring can cause severe load fluctuations This means eg thrusters positioned far from the rolling axis of vessels that could be susceptible to rolling If leading to propeller air suction the condi-tions may be even worse The above mentioned movements or manoeuvring will result in increased propeller excitation If the thruster is driven by a diesel engine the engine mean torque is limited to 100 However thrusters driven by electric motors can suffer temporary mean torque much above 100 unless a suitable load control system (limiting available e-motor torque) is provided

b) For main propulsion gears with ice class notation (see Rules Pt5 Ch1 Sec J500) KA ice has to be taken as the higher value of the applicable (rule defined) ice shock torque referred to nominal rated torque and the value under a) The Baltic ice class notations refer to a few millions ice shock loads Thus the life factors may be put YN=1 and ZN=12 (except for nitrided gears where ZN = 1 applies) Additionally the calculations with the normal KA (no ice class) are to fulfil the normal requirements For polar ice class notations KA ice applies to all criteria and for long or infinite life

c) For a power take off (PTO) branch from a main propul-sion gear with ice class ice shocks result in negative torques It is assumed that the PTO branch is unloaded when the ice shock load occurs The influence of these reverse shock loads may be taken into account as follows The negative torque (reversed load) expressed by means of an application factor based on rated forward load (T or Ft) is KAreverse = KA ice ndash1 (the minus 1 be-cause no mean torque assumed) KAice to be calculated as in the ice class rules This KAreverse should be used for back flank considera-tions such as pitting and scuffing The influence on tooth bending strength (forward di-rection) may be simplified by using the factor YM = 1 minus 03 middot KAreverse KA

d) For diesel driven auxiliaries KA can be taken from the torsional vibration analysis if available For units where no vibration analysis is required (lt 200 kW) or available it is advised to apply KA as the upper allow-able value 135)

e) For turbine or electro driven auxiliaries the same as for c) applies however the practical upper value is 12

) For diesel driven gears more information on KA for mis-firing and normal driving is given in Appendix B

162 KAP The peak overload factor KAP is defined as the ratio between the temporary occasional peak overload torque and the nominal rated torque

For plants where high temporary occasional peak torques can occur (ie in excess of the above mentioned KA) the gearing (if nitrided) has to be checked with regard to static strength Unless otherwise specified the same safety factors as for infinite life apply

The scuffing safety is to be specially considered whereby the KA applies in connection with the bulk temperature and the KAP applies for the flash temperature calculation and should replace KA in the formulae in 431 432 and 441

KAP can be evaluated from the torsional impact vibration calculation (as required by the rules)

If the overloads have a duration corresponding to several revolutions of the shafts the scuffing safety has to be consid-ered on basis of this overload both with respect to bulk and flash temperature This applies to plants with ice class nota-tions (Baltic and polar) and to plants with prime movers which have high temporary overload capacity such as eg electric motors (provided the driven member can have a con-siderable increase in demand torque as eg azimuth thrusters during manoeuvring)

For plants without additional ice class notation KAP should normally not exceed 15

8 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

163 Frequent overloads For plants where high overloads or shock loads occur regu-larly the influence of this is to be considered by means of cumulative fatigue (see Appendix A)

17 Load Sharing Factor Kγ The load sharing factor Kγ accounts for the maldistribution of load in multiple-path transmissions (dual tandem epicyclic double helix etc) Kγ is defined as the ratio between the max load through an actual path and the evenly shared load

171 General method Kγ mainly depends on accuracy and flexibility of the branches (eg quill shaft planet support external forces etc) and should be considered on basis of measurements or of relevant analysis as eg

Kγ = δ+δ f

δ = total compliance of a branch under full load (assuming even load share) referred to gear mesh

f = minusminusminusminus+++ 23

22

21 fff where f1 f2 etc are the

main individual errors that may contribute to a maldis-tribution between the branches Eg tooth pitch errors planet carrier pitch errors bearing clearance influences etc Compensating effects should also be considered

For double helical gears

An external axial force Fext applied from sources outside the actual gearing (eg thrust via or from a tooth coupling) will cause a maldistribution of forces between the two helices Expressed by a load sharing factor the

βsdot

plusmn=γ tanFF1K

t

ext

If the direction of Fext is known the calculation should be carried out separately for each helix and with the tangential force corrected with the pertinent Kγ If the direction of Fext is unknown both combinations are to be calculated and the higher σH or σF to be used

172 Simplified method If no relevant analysis is available the following may apply

For epicyclic gears

Kγ = 32501 minus+ pln

where npl = number of planets ( gt3 )

For multistage gears with locked paths and gear stages sepa-rated by quill shafts (see figure below)

Figure 10 Locked paths gear

Kγ = ( )φ201+

where φ = quill shaft twist (degrees) under full load

18 Dynamic Factor Kv The dynamic factor Kv accounts for the internally generated dynamic loads

Kv is defined as the ratio between the maximum load that dynamically acts on the tooth flanks and the maximum ex-ternally applied load Ft KA Kγ

In the following 2 different methods (181 and 182) are described In case of controversy between the methods the next following is decisive ie the methods are listed with increasing priority

It is important to observe the limitations for the method in 181 In particular the influence of lateral stiffness of shafts is often underestimated and resonances occur at considerably lower speed than determined in 1811

However for low speed gears with vmiddotz1 lt 300 calculations may be omitted and the dynamic factor simplified to Kv=105

181 Single resonance method For a single stage gear Kv may be determined on basis of the relative proximity (or resonance ratio) N between actual speed n1 and the lowest resonance speed nE1

N = 1E

1

nn

Note that for epicyclic gears n is the relative speed ie the speed that multiplied with z gives the mesh frequency

1811 Determination of critical speed It is not advised to apply this method for multimesh gears for N gt 085 as the influence of higher modes has to be consid-ered see 182 In case of significant lateral shaft flexibility (eg overhung mounted bevel gears) the influence of cou-pled bending and torsional vibrations between pinion and wheel should be considered if N ge 075 see 182

nE1 = redmγc

1zπ

31030 sdot

where

cγ is the actual mesh stiffness per unit facewidth see 111

Classification Notes- No 412 9 May 2003

DET NORSKE VERITAS

For gears with inactive ends of the facewidth as eg due to high crowning or end relief such as often applied for bevel gears the use of cγ in connection with determination of natu-ral frequencies may need correction cγ is defined as stiffness per unit facewidth but when used in connection with the total mesh stiffness it is not as simple as cγmiddotb as only a part of the facewidth is active Such corrections are given in 111

mred is the reduced mass of the gear pair per unit facewidth and referred to the plane of contact

For a single gear stage where no significant inertias are closely connected to neither pinion nor wheel mred is calcu-lated as

mred = 21

21mm

mm+

The individual masses per unit facewidth are calculated as

m12 = 221b

21

)2d(b

I

where I is the polar moment of inertia (kgmm2)

The inertia of bevel gears may be approximated as discs with diameter equal the midface pitch diameter and width equal to b However if the shape of the pinion or wheel body differs much from this idealised cylinder the inertia should be cor-rected accordingly

For all kind of gears if a significant inertia (eg a clutch) is very rigidly connected to the pinion or wheel it should be added to that particular inertia (pinion or wheel) If there is a shaft piece between these inertias the torsional shaft stiffness alters the system into a 3-mass (or more) system This can be calculated as in 182 but also simplified as a 2-mass system calculated with only pinion and wheel masses

1812 Factors used for determination of Kv Non-dimensional gear accuracy dependent parameters

( )bKKF

yfcB

At

pptp

γ

minus=

( )bKKF

yFcBAt

ff

γ

α minus=

Non-dimensional tip relief parameter

bKKFcC

1BγAt

ak sdotsdot

sdotminus=

For gears of quality grade (ISO 1328) Q = 7 or coarser Bk = 1

For gears with Q le 6 and excessive tip relief Bk is limited to max 1

For gears (all quality grades) with tip relief of more than 2middotCeff (see 432) the reduction of αε has to be considered (see 443)

where

fpt = the single pitch deviation (ISO 1328) max of pinion or wheel

Fα = the total profile form deviation (ISO 1328) max of pinion or wheel (Note Fα is pt not available for bevel gears thus use Fα = fpt)

yp and yf = the respective running-in allowances and may be calculated similarly to yα in 112 ie the value of fpt is replaced by Fα for yf

cacute = the single tooth stiffness see 111

Ca = the amount of tip relief see 433 In case of different tip relief on pinion and wheel the value that results in the greater value of Bk is to be used If Ca is zero by design the value of running-in tip relief Cay (see 112) may be used in the above formula

1813 Kv in the subcritical range Cylindrical gears N le 085

Bevel gears N le 075

Kv = 1 + N K

K = Cv1 Bp + Cv2 Bf + Cv3 Bk

Cv1 accounts for the pitch error influence

Cv1 = 032

Cv2 accounts for profile error influence

Cv2 = 034 for γε le 2

Cv2 = 03ε

057

γ minus for γε gt 2

Cv3 accounts for the cyclic mesh stiffness variation

Cv3 = 023 for γε le 2

Cv3 = 156ε

0096

γ minus for γε gt 2

1814 Kv in the main resonance range Cylindrical gears 085 lt N le 115

Bevel gears 075 lt N le 125

Running in this range should preferably be avoided and is only allowed for high precision gears

Kv = 1 + Cv1 Bp + Cv2 Bf + Cv4 Bk

Cv4 accounts for the resonance condition with the cyclic mesh stiffness variation

Cv4 = 090 for γε le 2

Cv4 = 144ε

ε005057

γ

γ

minus

minus for γε gt 2

10 Classification Notes- No 412

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DET NORSKE VERITAS

1815 Kv in the supercritical range Cylindrical gears N ge 15

Bevel gears N ge 15

Special care should be taken as to influence of higher vibra-tion modes andor influence of coupled bending (ie lateral shaft vibrations) and torsional vibrations between pinion and wheel These influences are not covered by the following approach

Kv = Cv5 Bp + Cv6 Bf + Cv7

Cv5 accounts for the pitch error influence

Cv5 = 047

Cv6 accounts for the profile error influence

Cv6 = 047 for εγ le 2

Cv6 = 174ε

012

γ minus for εγ gt 2

Cv7 relates the maximum externally applied tooth loading to the maximum tooth loading of ideal accurate gears operating in the supercritical speed sector when the circumferential vibration becomes very soft

Cv7 = 075 for εγ le 15

Cv7 = [ ] 8750)2(sin 0125 +minusβεπ for 15 lt εγ le 25

Cv7 = 10 for εγ gt 25

1816 Kv in the intermediate range Cylindrical gears 115 lt N lt 15

Bevel gears 125 lt N lt 15

Comments raised in 1814 and 1815 should be observed

Kv is determined by linear interpolation between Kv for N = 115 respectively 125 and N = 15 as

Cylindrical gears

( ) ( ) ( )[ ]51Nv151Nv51Nvv KK350

N51KK === minussdot

minus

+=

Bevel gears

( ) ( ) ( )[ ]51Nv251Nv51Nvv KK250

N51KK === minussdot

minus

+=

182 Multi-resonance method For high speed gear (vgt40 ms) for multimesh medium speed gears for gears with significant lateral shaft flexibility etc it is advised to determine Kv on basis of relevant dy-namic analysis

Incorporating lateral shaft compliance requires transforma-tion of even a simple pinion-wheel system into a lumped multi-mass system It is advised to incorporate all relevant inertias and torsional shaft stiffnesses into an equivalent (to pinion speed) system Thereby the mesh stiffness appears as an equivalent torsional stiffness

cγ b (db12)2 (Nmrad)

The natural frequencies are found by solving the set of dif-ferential equations (one equation per inertia) Note that for a gear put on a laterally flexible shaft the coupling bending-torsionals is arranged by introducing the gear mass and the lateral stiffness with its relation to the torsional displacement and torque in that shaft

Only the natural frequency (ies) having high relative dis-placement and relative torque through the actual pinion-wheel flexible element need(s) to be considered as critical frequency (ies)

Kv may be determined by means of the method mentioned in 181 thereby using N as the least favourable ratio (in case of more than one pinion-wheel dominated natural frequency) Ie the N-ratio that results in the highest Kv has to be consid-ered

The level of the dynamic factor may also be determined on basis of simulation technique using numeric time integration with relevant tooth stiffness variation and pitchprofile er-rors

19 Face Load Factors KHβ and KFβ The face load factors KHβ for contact stresses and for scuff-ing KFβ for tooth root stresses account for non-uniform load distribution across the facewidth

KHβ is defined as the ratio between the maximum load per unit facewidth and the mean load per unit facewidth

KFβ is defined as the ratio between the maximum tooth root stress per unit facewidth and the mean tooth root stress per unit facewidth The mean tooth root stress relates to the con-sidered facewidth b1 respectively b2

Note that facewidth in this context is the design facewidth b even if the ends are unloaded as often applies to eg bevel gears

The plane of contact is considered

191 Relations between KHβ and KFβ

KFβ = expβHK 2(hb)hb1

1exp++

=

where hb is the ratio tooth heightfacewidth The maximum of h1b1 and h2b2 is to be used but not higher than 13 For double helical gears use only the facewidth of one helix

If the tooth root facewidth (b1 or b2) is considerably wider than b the value of KFβ(1or2) is to be specially considered as it may even exceed KHβ

Eg in pinion-rack lifting systems for jack up rigs where b = b2 asymp mn and b1 asymp 3 mn the typical KHβ asymp KFβ2 asymp 1 and KFβ1 asymp 13

192 Measurement of face load factors Primarily

KFβ may be determined by a number of strain gauges distrib-uted over the facewidth Such strain gauges must be put in

Classification Notes- No 412 11 May 2003

DET NORSKE VERITAS

exactly the same position relative to the root fillet Relations in 191 apply for conversion to KHβ

Secondarily

KHβ may be evaluated by observed contact patterns on vari-ous defined load levels It is imperative that the various test loads are well defined Usually it is also necessary to evalu-ate the elastic deflections Some teeth at each 90 degrees are to be painted with a suitable lacquer Always consider the poorest of the contact patterns

After having run the gear for a suitable time at test load 1 (the lowest) observe the contact pattern with respect to ex-tension over the facewidth Evaluate that KHβ by means of the methods mentioned in this section Proceed in the same way for the next higher test load etc until there is a full face contact pattern From these data the initial mesh misalign-ment (ie without elastic deflections) can be found by ex-trapolation and then also the KHβ at design load can be found by calculation and extrapolation See example

Figure 11 Example of experimental determination of KHβ

It must be considered that inaccurate gears may accumulate a larger observed contact pattern than the actual single mesh to mesh contact patterns This is particularly important for lapped bevel gears Ground or hard metal hobbed bevel gears are assumed to present an accumulated contact pattern that is practically equal the actual single mesh to mesh con-tact patterns As a rough guidance the (observed) accumu-lated contact pattern of lapped bevel gears may be reduced by 10 in order to assess the single mesh to mesh contact pattern which is used in 199

193 Theoretical determination of KHβ The methods described in 193 to 198 may be used for cy-lindrical gears The principles may to some extent also be used for bevel gears but a more practical approach is given in 199

General For gears where the tooth contact pattern cannot be verified during assembly or under load all assumptions are to be well on the safe side

KHβ is to be determined in the plane of contact

The influence parameters considered in this method are

bull mean mesh stiffness cγ (see 111) (if necessary also variable stiffness over b)

bull mean unit load Fmb = Fbt KA Kγ Kvb (for double helical gears see 17 for use of Kγ)

bull misalignment fsh due to elastic deflections of shafts and gear bodies (both pinion and wheel)

bull misalignment fdefl due to elastic deflections of and work-ing positions in bearings

bull misalignment fbe due to bearing clearance tolerances bull misalignment fma due to manufacturing tolerances bull helix modifications as crowning end relief helix correc-

tion bull running in amount yβ (see 112)

In practice several other parameters such as centrifugal ex-pansion thermal expansion housing deflection etc contrib-ute to KHβ However these parameters are not taken into account unless in special cases when being considered as particularly important

When all or most of the am parameters are to be considered the most practical way to determine KHβ is by means of a graphical approach described in 1931

If cγ can be considered constant over the facewidth and no helix modifications apply KHβ can be determined analyti-cally as described in 1932

1931 Graphical method The graphical method utilises the superposition principle and is as follows

bull Calculate the mean mesh deflection Mδ as a function of

γm c and bF see 111

bull Draw a base line with length b and draw up a rectangu-lar with height δM (The area δM b is proportional to the transmitted force)

bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the areas above and below this zero line are equal

Figure 12 fsh balanced around zero line

bull Superimpose these ordinates of the fsh curve to the previ-ous load distribution curve (The area under this new load distribution curve is still δM b)

bull Calculate the bearing deflections andor working posi-tions in the bearings and evaluate the influence fdefl in the plane of contact This is a straight line and is bal-anced around a zero line as indicated in Fig 14 but with one distinct direction Superimpose these ordinates to the previous load distribution curve

12 Classification Notes- No 412

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DET NORSKE VERITAS

bull The amount of crowning end relief or helix correction (defined in the plane of contact) is to be balanced around a zero line similarly to fsh

Figure 13 Crowning Cc balanced aound zero line

bull Superimpose these ordinates to the previous load distri-bution curve In case of high crowning etc as eg often applied to bevel gears the new load distribution curve may cross the base line (the real zero line) The result is areas with negative load that is not real as the load in those areas should be zero Thus corrective actions must be made but for practical reasons it may be postponed to after next operation

bull The amount of initial mesh misalignment bema ff + (defined in the plane of contact) is to be bal-

anced around a zero line If the direction of bema ff + is known (due to initial contact check) or if the direction of fbe is known due to design (eg overhang bevel pin-ion) this should be taken into account If direction un-known the influence of bema ff + in both directions as well as equal zero should be considered

Figure 14 fma+fbe in both directions balanced around zero line

Superimpose these ordinates to the previous load distri-bution curve This results in up to 3 different curves of which the one with the highest peak is to be chosen for further evaluation

bull If the chosen load distribution curve crosses the base line (ie mathematically negative load) the curve is to be corrected by adding the negative areas and dividing this with the active facewidth The (constant) ordinates of this rectangular correction area are to be subtracted from the positive part of the load distribution curve It is advisable to check that the area covered under this new load distribution curve is still equal δM b

bull If cγ cannot be considered as constant over b then cor-rect the ordinates of the load distribution curve with the local (on various positions over the facewidth) ratio between local mesh stiffness and average mesh stiffness cγ (average over the active facewidth only) Note that the result is to be a curve that covers the same area δM b as before

bull The influence of running in yβ is to be determined as in 112 whereby the value for Fβx is to be taken as twice the distance between the peak of the load distribution curve and δM

bull Determine

KHβ = M

β

δycurveofpeak minus

1932 Simplified analytical method for cylindrical gears The analytical approach is similar to 1931 but has a more limited application as cγ is assumed constant over the face-width and no helix modification applies

bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the area above and below this zero line are equal see Fig 12 The max positive ordinate is frac12∆fsh

bull Calculate the initial mesh alignment as Fβx= deflbemash ffff plusmnplusmnplusmn∆ The negative signs may only be used if this is justified andor verified by a contact pattern test Otherwise al-ways use positive signs If a negative sign is justified the value of Fβx is not to be taken less than the largest of each of these elements

bull Calculate the effective mesh misalignment as Fβy = Fβx - yβ (yβ see 112)

bull Determine

2KforF2

bFc1K βH

m

βγγH le+=β

or

2KforF

bFc2K Hβ

m

βγγH gt=β

where cγ as used here is the effective mesh stiffness see 111

194 Determination of fsh fsh is the mesh misalignment due to elastic deflections Usu-ally it is sufficient to consider the combined mesh deflection of the pinion body and shaft and the wheel shaft The calcu-lation is to be made in the plane of contact (of the considered gear mesh) and to consider all forces (incl axial) acting on the shafts Forces from other meshes can be parted into components parallel respectively vertical to the considered plane of contact Forces vertical to this plane of contact have no influence on fsh

It is advised to use following diameters for toothed elements

d + 2 x mn for bending and shear deflection

d + 2 mn (x ndash ha0 + 02) for torsional deflection

Usually fsh is calculated on basis of an evenly distributed load If the analysis of KHβ shows a considerable maldis-

Classification Notes- No 412 13 May 2003

DET NORSKE VERITAS

tribution in term of hard end contact or if it is known by other reasons that there exists a hard end contact the load should be correspondingly distributed when calculating fsh In fact the whole KHβ procedure can be used iteratively 2-3 iterations will be enough even for almost triangular load distributions

195 Determination of fdefl fdefl is the mesh misalignment in the plane of contact due to bearing deflections and working positions (housing deflec-tion may be included if determined)

First the journal working positions in the bearings are to be determined The influence of external moments and forces must be considered This is of special importance for twin pinion single output gears with all 3 shafts in one plane

For rolling bearings fdefl is further determined on basis of the elastic deflection of the bearings An elastic bearing deflec-tion depends on the bearing load and size and number of rolling elements Note that the bearing clearance tolerances are not included here

For fluid film bearings fdefl is further determined on basis of the lift and angular shift of the shafts due to lubrication oil film thickness Note that fbe takes into account the influence of the bearing clearance tolerance

When working positions bearing deflections and oil film lift are combined for all bearings the angular misalignment as projected into the plane of the contact is to be determined fdefl is this angular misalignment (radians) times the face-width

196 Determination of fbe fbe is the mesh misalignment in the plane of contact due to tolerances in bearing clearances In principle fbe and fdefl could be combined But as fdefl can be determined by analy-sis and has a distinct direction and fbe is dependent on toler-ances and in most cases has no distinct direction (ie + toler-ance) it is practicable to separate these two influences

Due to different bearing clearance tolerances in both pinion and wheel shafts the two shaft axis will have an angular mis-alignment in the plane of contact that is superimposed to the working positions determined in 195 fbe is the facewidth times this angular misalignment Note that fbe may have a distinct direction or be given as a + tolerance or a combina-tion of both For combination of + tolerance it is adviced to use

fbe= 22be

21be ff ++plusmn

fbe is particularly important for overhang designs for gears with widely different kinds of bearings on each side and when the bearings have wide tolerances on clearances In general it shall be possible to replace standard bearings with-out causing the real load distribution to exceed the design premises For slow speed gears with journal bearings the expected wear should also be considered

197 Determination of fma fma is the mesh misalignment due to manufacturing toler-ances (helix slope deviation) of pinion fHβ1 wheel fHβ2 and housing bore

For gear without specifically approved requirements to as-sembly control the value of fma is to be determined as

fma= 22Hβ

21Hβ ff +

For gears with specially approved assembly control the value of fma will depend on those specific requirements

198 Comments to various gear types For double helical gears KHβ is to be determined for both helices Usually an even load share between the helices can be assumed If not the calculation is to be made as de-scribed in 171

For planetary gears the free floating sun pinion suffers only twist no bending It must be noted that the total twist is the sum of the twist due to each mesh If the value of 1K γ ne this must be taken into account when calculating the total sun pinion twist (ie twist calculated with the force per mesh without Kγ and multiplied with the number of planets)

When planets are mounted on spherical bearings the mesh misalignments sun-planet respectively planet-annulus will be balanced Ie the misalignment will be the average between the two theoretical individual misalignments The faceload distribution on the flanks of the planets can take full advan-tage of this However as the sun and annulus mesh with several planets with possibly different lead errors the sun and annulus cannot obtain the above mentioned advantage to the full extent

199 Determination of KHβ for bevel gears If a theoretical approach similar to 193-198 is not docu-mented the following may be used

testeff

H Kb

b851851K sdot

minussdot=β

beff b represents the relative active facewidth (regarding lapped gears see 192 last part)

Higher values than beff b = 090 are normally not to be used in the formula

For dual directional gears it may be difficult to obtain a high beff b in both directions In that case the smaller beff b is to be used

Ktest represents the influence of the bearing arrangement shaft stiffness bearing stiffness housing stiffness etc on the faceload distribution and the verification thereof Expected variations in length- and height-wise tooth profile is also accounted for to some extent

a) Ktest = 1 For ground or hard metal hobbed gears with the specified contact pattern verified at full rating or at full torque slow turning at a condition representative for the thermal expansion at normal operation

14 Classification Notes- No 412

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DET NORSKE VERITAS

It also applies when the bearing arrangementsupport has insignificant elastic deflections and thermal axial expansion However each initial mesh contact must be verified to be within acceptance criteria that are cali-brated against a type test at full load Reproduction of the gear tooth length- and height-wise profile must also be verified This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon)

b) Ktest = 1 + 04middot(beffbndash06) For designs with possible influence of thermal expansion in the axial direction of the pinion The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load

c) Ktest = 12 if mesh is only checked by toolmakerrsquos blue or by spin test contact For gears in this category beffb gt 085 is not to be used in the calculation

110 Transversal Load Distribution Factors KHα and KFα The transverse load distribution factors KHα for contact stresses and for scuffing KFα for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairs of teeth in mesh

The following relations may be used

Cylindrical gears

( )

minus+==

tH

αptγγHαFα F

byfc0409

2ε

KK

valid for 2εγ le

( ) ( )tH

αptγ

γ

γHαFα F

byfcε

1ε20409KK

minusminus+==

valid for 2ε γ gt

where

FtH = Ft KA Kγ Kv KHβ

cγ = See 111

γα = See 112

fpt = Maximum single pitch deviation (microm) of pinion or wheel or maximum total profile form devia-tion Fα of pinion or wheel if this is larger than the maximum single pitch deviation

Note In case of adequate equivalent tip relief adapted to the load half of the above mentioned fpt can be introduced A tip relief is considered adequate when the average of Ca1 and Ca2 is within plusmn40 of the value of Ceff in 432

Limitations of KHα and KFα

If the calculated values for

KFα = KHα lt 1 use KFα = KHα = 10

If the calculated value of KHα gt 2εα

γ

Zε

ε use KHα = 2

εα

γ

Zε

ε

If the calculated value of KFα gt εα

γ

Yεε

use KFα = εα

γ

Yεε

where Yε = αnε

075025+ (for εαn see 331c)

Bevel gears

For ground or hard metal hobbed gears KFα = KHα = 1

For lapped gears KFα = KHα = 11

111 Tooth Stiffness Constants cacute and cγ The tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having 1 mm facewidth by an amount of 1 microm in the plane of contact cacute is the maximum stiffness of a single pair of teeth cγ is the mean value of the mesh stiffness in a transverse plane (brief term mesh stiffness)

Both valid for high unit load (Unit load = Ft middot KA middot Kγb)

Cylindrical gears The real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bending stiff-nesses For high unit loads the Hertzian stiffness has little importance and can be disregarded This approach is on the safe side for determination of KHβ and KHα However for moderate or low loads Kv may be underestimated due to determination of a too high resonance speed

The linear approach is described in A

An optional approach for inclusion of the non-linear stiffness is described in B

A The linear approach

BRCCq

βcos08acutec =

and

( )025ε075cc αγ +prime=

where

( )[ ]n02a01a

B α2000212

hh12051C minusminus

+minus+=

12n1n

x000635z

025791z

015551004723q minus++=

12

2n

22

1n

1 x005290z

x024188x000193z

x011654+minusminusminus

+ 000182 x22

Classification Notes- No 412 15 May 2003

DET NORSKE VERITAS

(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as

( )( )nR m5s

sR e5

bbln1C +=

where

bs = thickness of a central web

sR = average thickness of rim (net value from tooth root to inside of rim)

The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered

Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ

B The non-linear approach

In the following an example is given on how to consider the non-linearity

The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm

With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as

( )10δKbF

minus= for 500bFgt

minus=

500Fb10δK

bF for 500

bFlt

with γAt KK

bF

bF

sdotsdot= etc (Nmm) ie unit load incorporat-

ing the relevant factors as

KA middot Kγ for determination of Kv

KA middot Kγ middot Kv for determination of KHβ

KA middot Kγ middot Kv middot KHβ for determination of KHα

δ = mesh deflection (microm)

K = applicable stiffness (c or cγ)

Use of stiffnesses for KV KHβ and KHα

For calculation of Kv and KHα the stiffness is calculated as follows

When Fb lt 500

the stiffness is determined as δ∆

∆ bF

where the increment is chosen as eg ∆ Fb = 10 and thus

50010Fb10

K10Fb∆δ +

++

=

When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly

or an equivalent stiffness determined as δsdotb

F

Bevel gears

In lack of more detailed relationship between stiffness and geometry the following may be used

b085

b13cacute eff=

b085b

16c effγ =

beff not to be used in excess of 085 b in these formulae

Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)

112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements

yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload

Cay is defined as the running-in amount that compensates for lack of tip relief

The following relations may be used

For not surface hardened steel

ptHlim

α fσ160y =

yβ = βxlimH

f320σ

with the following upper limits

V lt 5 ms 5-10 ms gt 10 ms

yα max none

limH

12800σ limH

6400σ

yβ max none

limH

25600σ limH

12800σ

For surface hardened steel

yα = 0075 fpt but not more than 3 for any speed

yβ = 015 Fβx but not more than 6 for any speed

16 Classification Notes- No 412

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DET NORSKE VERITAS

For all kinds of steel

5145189718

1C2

limHay +

minusσ

=

When pinion and wheel material differ the following ap-plies

bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv

bull Use ( )2β1ββ yy21y += in the calculation of KHβ

bull Use ( )2ay1aya CC21C += in the calculation of Kv

bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing

calculation if no design tip relief is foreseen

Classification Notes- No 412 17 May 2003

DET NORSKE VERITAS

2 Calculation of Surface Durability

21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213

Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted

Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213

For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength

For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied

22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater

Calculation of surface durability for helical gears is based on the contact stress at the pitch point

For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies

Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact

Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses

The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses

The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel

221 Contact stress Cylindrical gears

( )HαHβvγA

1

tβεEHDBH KKKKK

bud1uF

ZZZZZσ+

=

where

ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)

ZH = Zone factor for pitch point (see 231)

ZE = Elasticity factor (see 24)

Zε = Contact ratio factor (see 25)

Zβ = Helix angle factor (see 26)

Ft KA Kγ Kv KHβ KHα see 15 ndash 110

d1 b u see 12 ndash 15

Bevel gears

( )HαHβvγA

v1v

vmtKEMH KKKKK

bud1uF

ZZZ105σ+

sdot=

where

105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)

ZE KA etc see above

ZM = mid-zone factor see 233

ZK = bevel gear factor see 27

Fmt dv1 uv see 12 ndash 15

It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks

222 Permissible contact stress

XWRvLH

NHlimHP ZZZZZ

SZσ

σ =

where

σH lim = Endurance limit for contact stresses (see 28)

ZN = Life factor for contact stresses (see 29)

SH = Required safety factor according to the rules

ZLZvZR = Oil film influence factors (see 210)

ZW = Work hardening factor (see 211)

ZX = Size factor (see 212)

18 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

23 Zone Factors ZH ZBD and ZM

231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder

wtt2

wtbH sinααcos

cosαcosβ2Z =

232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel

For εβ ge 1 ZBD = 1

For internal gears ZD = 1

For εβ = 0 (spur gears)

( )

π

minusεminusminus

π

minusminus

α=

α2

2

2b

2a

1

2

1b

1a

wtB

z211

dd

z21

dd

tanZ

( )

π

minusεminusminus

π

minusminus

α=

α1

2

1b

1a

2

2

2b

2a

wtD

z211

dd

z21

dd

tanZ

If ZB lt 1 use ZB = 1

If ZD lt 1 use ZD = 1

For 0 lt εβ lt 1

ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)

233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears

εminusminus

εminusminus

αβ=

αα btm2

2vb2

2vabtm2

1vb2val

2v1vvtbmM

pddpdd

ddtancos2Z

This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint

234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel

24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses

For steel against steel ZE = 1898

25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses

αε1Z =ε for 1εβ ge

( )α

ββ

αε ε

εε1

3ε4

Z +minusminus

= for εβ lt 1

26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability

cosβZβ =

27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply

The following may be used ZK = 080

28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ

σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting

For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply

For this purpose pitting is defined by

bull for not surface hardened gears pitted area ge 2 of total active flank area

bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given

material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies

The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as

Classification Notes- No 412 19 May 2003

DET NORSKE VERITAS

the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99

σHlim σH105 σH10

3

Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade

1650 1500

2500 2400

3100 3100

Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)

1000

13 σHlim

13 σHlim

Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV

Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV

Carbon steel 15 HV + 250 16 σHlim 16 σHlim

These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15

29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply

If this is not documented by approved fatigue tests the fol-lowing method may be used

For all steels except nitrided

7L 105N sdotge ZN = 1 or

01570

L

7

N N105Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 5middot107 510N

logZ037

L

7

N N105Z

sdot=

NL = 105 WXRVLHlim

WstX10H1010NN ZZZZZσ

ZZσZZ

55

5=

103 lt NL lt 105 )Z(Zlog05

L

5

10NN

5N103N10

5N10ZZ

==

3L 10N le

WXRVLlimH

Wst10X10H10NN ZZZZZ

ZZZZ

33

3σ

σ==

(but not less than ZN105)

For nitrided steels

6L 102N sdotge ZN = 1 or

00980

L

6

N N102Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 2middot106 510N

Zlog76860

L

6

N N102Z

sdot=

5L 10N le

XWRVL

10XWst10NN ZZZZZ

ZZ13ZZ

55 ==

Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles

210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity

The following methods may be applied in connection with the endurance limit

20 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Surface hardened steels Not surface hardened steels

ZL ( )24013421

360910ν+

+

( )24013421

680830ν+

+

ZV ( )v3280

140930+

+ ( )v3280300850

++

ZR

080

ZrelR3

150

ZrelR3

where

ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits

For values of ν40 gt 500 use ν40 = 500

RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm

RZrel

= ( ) 31

cZ2Z1 ρ

10RR50

+

RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)

For 5

L 10N le ZL ZV ZR = 10

211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface

The following approximation may be used for the endurance limit

Surface hardened steel against not surface hardened steel 150

ZeqW R

31700

130HB21Z

minus

minus=

where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130

RZeq = equivalent roughness

033

c40

066

ZS

ZHZHZEQ ρvν

15000RRRR

=

If RZeq gt 16 then use RZeq = 16

If RZeq lt 15 then use RZeq = 15

where

RZH = surface roughness of the hard member before run in

RZS = surface roughness of the soft member before run in

ν40 = see 210

If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem

Through hardened pinion against softer wheel

( )

minussdotminus+= 000829

HBHB0008981u1Z

2

1W

For 21HBHB

2

1 le use ZW = 1

For 71HBHB

2

1 gt use 17HBHB

2

1 =

For u gt 20 use u = 20

For static strength (lt 105 cycles)

Surface hardened against not surface hardened

ZWst = 105

Through hardened pinion against softer wheel

ZWst = 1

212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality

ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213

213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented

The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent

Classification Notes- No 412 21 May 2003

DET NORSKE VERITAS

stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically

The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence

The real Hertzian stresses σHR are determined as

For helical gears with εβ gt 1

σHR = σH

For helical gears with εβ lt 1 and spur gears

ε

α

ββ ε

ε+εminus

sdotσ=σZ

1

HHR

For bevel gears

KHHR Z

1σσ sdot=

The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)

The coordinates tz and HV are to be compared with the de-sign specification such as

bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400

and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)

In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods

For high cycle fatigue (gt3 106 cycles) the following applies

sdot+

minussdotsdotsdot= o90

05azt

05azt

cosSσ04HV

H

HHHR

applicable to 05a

zt

Hge

For 05at

H

z lt the value for 05at

H

z = applies

56300ρSσ12a cHHR

Hsdotsdot

sdot=

Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength

If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows

reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue

For static strength (lt103 cycles) the following applies

sdot+

minussdotsdotsdot= o90

07at

06a

zt

cosSσ019HV

Hst

z

HstHHR

applicable to 06at

Hst

z ge

56300ρSσa cHHR

Hstsdotsdot

=

In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies

For limited life fatigue (103 lt cycles lt 3106 )

For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance

The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram

minussdotminus

= sdot logN3477

logSlogSlogS

310H6103HHN

36 10H103H Slog86281Slog86280 sdot+sdot sdot

22 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3 Calculation of Tooth Strength

31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding

For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36

For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning

Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα

In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314

A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired

It should be noted that this part 3 does not cover fractures caused by

bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining

Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears

Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied

32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level

321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance

Normally the stresses for pinion and wheel are calculated as

Cylindrical gears

FαFβvγAβSFn

tF K K K K K Y Y Y

m bFσ =

where

YF = Tooth form factor (see 33)

YS = Stress correction factor (see 34)

Yβ = Helix angle factor (see 36)

Ft KA Kγ Kv KFβ KFα see 15 ndash 110

b see 13

Bevel gears

FαFβvγASaFamn

mtF K K K K K Y Y Y

m bFσ ε=

where

YFa = Tooth form factor see 33

YSa = Stress correction factor see 34

Yε = Contact ratio factor see 35

Fmt KA etc see 15 ndash 110

b see 13

322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is

CXRrelTrelTF

NMFEFP YYYY

SYY

δσ

=σ

Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN

where

σFE = Local tooth root bending endurance limit of reference test gear (see 37)

YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)

YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)

SF = Required safety factor according to the rules

YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)

YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)

Classification Notes- No 412 23 May 2003

DET NORSKE VERITAS

YX = Size factor (see 312)

YC = Case depth factor (see 313)

33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress

YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears

YFa applies to load application at the tooth tip and is used for bevel gears

Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears

Figure 31 External tooth in normal section

Figure 32 Internal tooth in normal section

Definitions

n

2

n

Fn

enFn

Fe

F

α cosms

α cosmh6

Y

=

n

2

n

Fn

anFn

Fa

Fa

α cosms

α cosmh6

Y

=

In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth

YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)

The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing

mn with mnm

zn with zvn

αt with αvt

β with βm

with undercut without undercut

Fig 33 Dimensions and basic rack profile of the teeth (finished profile)

Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless

331 Determination of parameters

( )n

n

prnfPnfP m

α cossαsin 1ρ

αtan h4πE

minusminusminusminus=

For external gears fPfP ρρ =

For internal gears ( )0z

195fPfP0

fPfP 10363156ρhxρρ

sdotminus+

+=

where

z0 = number of teeth of pinion cutter

x0 = addendum modification coefficient of pinion cutter

hfP = addendum of pinion cutter

ρfP = tip radius of pinion cutter

xhρG fpfp +minus=

24 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

τmE

2π

z2H

nnminus

minus=

with

3πτ = for external gears

6πτ = for internal gears

Htan zG2

nminusϑ=ϑ

(to be solved iteratively suitable start value6π

=ϑ for exter-

nal gears and 3π for internal gears)

a) Tooth root chord sFn For external gears

minus

ϑ+

ϑminus= ρ

cosG3

3πsinz

ms

fpnn

Fn

For bevel gears with a tooth thickness modification

xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn

For internal gears

minus

ϑ+

ϑminus= ρ

cosG

6πsinz

ms

fPnn

Fn

b) Root fillet radius ρF at 30ordm tangent

( )G2coszcosG2ρ

mρ

2n

2

fpn

F

minusϑϑ+=

c) Determination of bending moment arm hF dn = zn mn

b2α

αn βcosε

ε =

dan = dn + 2 ha

pbn = π mn cos αn

dbn = dn cos αn

( )4

d1εpzz

2dd

zz2d

2bn

2

αnbn

2bn

2an

en +

minusminus

minus=

en

bnen d

dcos arcα =

ennnn

e α invα invα x tan 22π

z1

minus+

+=γ

αFen = αen ndash γe

For external gears

( )

minus=

n

enFenee

n

Fe

mdαtan sin γ γcos

21

mh

]ρcos

G3πcosz fpn +

ϑminus

ϑminusminus

For internal gears

( ) minus

sdotsdotminus=

n

enFenee

n

Fe

mdα tanγ sinγ cos

21

mh

minus

ϑminus

ϑminussdot ρ

cosG3

6πcosz fPn

332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as

250ε205for 0666ε2366Y αnαnDT leleminus=

205εfor 10Y αnDT lt=

34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section

Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm

YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip

YS can be determined as follows

( )

+

+=L23121

1

sS qL01312Y

where Fe

Fn

hsL = and

F

Fns ρ2

sq = (see 33)

YSa can be calculated by replacing hFe with hFa in the above formulae

Note a) Range of validity 1 lt qs lt 8

In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered

b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with

Classification Notes- No 412 25 May 2003

DET NORSKE VERITAS

g

g

ρt

0613

13

minus

where

tg = depth of the grinding notch

ρg = radius of the grinding notch

c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles

35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears

The following may be used

Yε = 0625

36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank

The following may be used (β input in degrees)

Yβ = 1 ndash εβ β120

When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula

However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)

37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM

σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision

If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules

σFE

Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)

bull of specially approved high grade

1050

bull of normal grade

minus CrNiMo steels with approved process

1000

minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850

Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

840

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

720

Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)

07 HV + 300

Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)

025 σB + 125

Alloyed quenched and tempered steel 04 σB + 200

Carbon steel 025 σB + 250

Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40

1) These values are valid for a root radius

bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)

bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)

bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for

6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked

2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)

26 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears

YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0

YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst

For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316

The following method may be used within a stress ratio ndash12 lt R lt 05

381 For idlers planets and PTO with ice class

M1M1R1

1Yor Y MstM

+minus

minus=

where

R = stress ratio = min stress divided by max stress

For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12

For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as

branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus

For a power take off (PTO) with ice class see 161 c

M considers the mean stress influence on the endurance (or static) strength amplitudes

M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress

Following M values may be used

Endurance limit

Static strength

Case hardened 08 ndash 015 Ys 1) 07

If shot peened 04 06

Nitrided 03 03

Induction or flame hardened

04 06

Not surface hardened steel 03 05

Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M

The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches

382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106

For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers

For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used

383 For gears with shrinkage stresses and unidirectional load For endurance strength

FE

fitM σ

σM1

M21Y+

minus=

σFE is the endurance limit for R = 0

For static strength YMst = 1 and σfit accounted for in 39b

σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor

n

Ffit m

ρ215scf minus=

384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is

( ) ( ) FE

fitM σ

σR1M1

M2

M1M1R1

1Yminussdot+

minus

+minus

minus=

Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12

For static strength

M1M1R1

1YMst

+minus

minus=

The effect of σfit is accounted for in 39 b

Classification Notes- No 412 27 May 2003

DET NORSKE VERITAS

385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed

F

y

Sσ225

for not surface hardened fillets

FSHV5 for surface hardened fillets

39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles

Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN

If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used

Determination of the σ ndash N ndash curve

a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF

b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ

FstS1

FPstσ minussdotsdotsdot=

where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability

σFst

Alloyed case hardened steel 1) 2300

Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

1250

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

1050

Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)

18 HV + 800

Steel with not surface hardened fillets the smaller value of 2)

18 σB or 225 σy

1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet

c) Calculate YN as

NL gt 3middot106

YN = 1 or 001

L

6

N N103Y

sdot= ie Yn = 092 for 1010

The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2

103ltNLlt3middot106exp

L

6

N N103Y

sdot=

cycles6103forσcycles310forσlog02876exp

FP

FPst

sdot=

NL lt 103 cycles6103forσcycles310forσ

NYFP

FPst

sdot=

or simply use σFPst as mentioned in b) directly

Guidance on number of load cycles NL for various applica-tions

bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)

bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied

28 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage

YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit

The following method may be used

For endurance limit

for not surface hardened fillets

( )

024

s024

relT σ103133q21σ1012201351

Ysdotsdotminus

+sdotsdotminus+= minus

minus

δ

for all surface hardened fillets except nitrided

( )

106q21002451

Y srelT

++=δ

for nitrided fillets

( )

1347q2101421

Y srelT

++=δ

For static strength

for not surface hardened fillets1)

( )( )( )025

02

02502s

relTst σ3000821σ300 1Y0821Y

+

minus+=δ

for surface hardened fillets except nitrided

YδrelTst = 044 YS + 012

for nitrided fillets

YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not

exceed the yield point and thereby alter the residual stress level See also 39b footnote 2

311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness

YRrelT differs for endurance limit and static strength

The following method may be used

For endurance limit

YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels

YδrelT = 43 ndash 326 (Ry + 1)0005

for nitrided steels

For static strength

YRrelTst = 1 for all Ry and all materials

For a fillet without any longitudinal machining trace Ry asymp Rz

312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength

The following may be used

For endurance limit

YX = 1 for mn le 5 generally

YX = 103 ndash 0006 mn for 5 lt mn lt 30

YX = 085 for mn ge 30

for not surface hard-ened steels

YX = 105 ndash 001 mn for 5 lt mn ge 25

YX = 08 for mn ge 25

for surface hardened steels

For static strength

YXst = 1 for all mn and all materials

313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength

YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength

In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength

The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses

The following simplified method for YC may be used

YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)

YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10

For endurance limit

+

+=nFFE

C m02ρt31

σconstY

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

Classification Notes- No 412 5 May 2003

DET NORSKE VERITAS

x = addendum modification coefficient z = number of teeth zn = virtual number of spur teeth αn = normal pressure angle at ref cylinder αt = transverse pressure angle at ref cylinder αa = transverse pressure angle at tip cylinder αwt = transverse pressure angle at pitch cylinder β = helix angle at ref cylinder βb = helix angle at base cylinder βa = helix angle at tip cylinder εα = transverse contact ratio εβ = overlap ratio εγ = total contact ratio ρa0 = tip radius of tool ref to mn ρfp = root radius of basic rack ref to mn ( = ρa0) ρC = effective radius (mm) of curvature at pitch point ρF = root fillet radius (mm) in the critical section σB = ultimate tensile strength (Nmm2) σy = yield strength resp 02 proof stress (Nmm2)

Index 1 refers to the pinion 2 to the wheel

Index n refers to normal section or virtual spur gear of a heli-cal gear

Index w refers to pitch point

Special additional symbols for bevel gears are as follows

Σ = angle between intersection axis Kϑ = angle modification (Klingelnberg)

m0 = tool module (Klingelnberg) δ = pitch cone angle xsm = tooth thickness modification coefficient (mid-

face) R = pitch cone distance (mm)

Index v refers to the virtual (equivalent) helical cylindrical gear

Index m refers to the midsection of the bevel gear

13 Geometrical Definitions For internal gearing z2 a da2 dw2 d2 and db2 are negative x2 is positive if da2 is increased ie the numeric value is de-creased

The pinion has the smaller number of teeth ie

11

2 ge=zz

u

For calculation of surface durability b is the common face-width on pitch diameter

For tooth strength calculations b1 or b2 are facewidths at the respective tooth roots If b1 or b2 differ much from b above they are not to be taken more than 1 module on either side of b

Cylindrical gears

tan αt = tan αn cos β tan βb = tan β cos αt

tan βa = tan β da d

cos αa = dbda

d = z mn cos β mt = mn cos β db = d cos αt = dw cos αwt

a = 05 (dw1 + dw2)

dw1dw2 = z1 z2

inv α = tan α - α (radians)

inv αwt = inv αt + 2 tan αn (x1 + x2)(z1 + z2)

zn = z (cos2 βb cos β)

1

aw1fw1α T

ξξε

+=

where ξfw1 is to be taken as the smaller of

bull wtfw1 αtanξ =

bull soi1

b1wtfw1 d

dacostan -tanαξ =

bull 1

2wt

a2

b2fw1 z

ztanα

d

dacostan ξ

minus=

and

2

1fw2aw1 z

zξξ = where ξfw2 is calculated as ξfw1

substituting the values for the wheel by the values for the pinion and visa versa

11 z

2πT =

( ) +

sdot+minusminussdot= minus

2sinαρρxhm

2d2d nfpfp1fpnsoi1

21

t

nfpfplfpn2

tanα)sinαρρx(hm

sdot+minusminus

nmsinbπ

β=εβ

(for double helix b is to be taken as the width of one helix)

εy = βα εε +

ρC = ( )2

b

wt

u1βcos

αsinua

+

v = 311 10dn

60π minus

6 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

pbt = βcos

αcosmπ tn

sat =

minus++

at

ninvααinv

z

αtanx22

π

d a

san = aβcossat

14 Bevel Gear Conversion Formulae and Specific Formulae Conversion of bevel gears to virtual equivalent helical cylin-drical gears is based on the bevel gear midsection The con-version formulae are

Number of teeth

zv12 = z12 cos δ12

(δ1 + δ2 = Σ)

Gear ratio

uv = 1v

2vzz

tan αvt = tan αn cos βm

tan βbm = tan βm cos αvt

Base pitch

pbtm = m

vtnmβcos

αcosmπ

Reference pitch diameters

dv12 = 21

21m

cosdδ

Centre distance

av = 05 (dv1 + dv2)

Tip diameters

dva 12 = dv 12 + 2 ham 12

Addenda

for gears with constant addenda (Klingelnberg)

ham 12 = mmn (1 + xm 12)

for gears with variable addenda (Gleason)

ham 12 = ha 12 ndash b2 tan (δa 12 ndash δ12)

(when ha is addendum at outer end and δa is the outer cone angle)

Addendum modification coefficients

xm 12 = mn

12am21am

m2

hh minus

Base circle

dvb 12 = dv 12 cos αvt

Transverse contact ratio)

εα = btmP

αsinadd05dd05 vtv2

2vb2

2va2

1vb2

1va minusminus+minus

Overlap ratio) (theoretical value for bevel gears with no crowning but used as approximations in the calculation pro-cedures)

εβ = nm

mmπβsinb

Total contact ratio)

εγ = 2β

2α εε +

( Note that index laquovraquo is left out in order to combine formu-lae for cylindrical and bevel gears)

Tangential speed at midsection

vmt = 3m11 10dn

60π minus

Effective radius of curvature (normal section)

ρvc = ( )2vbm

vtvv

u1βcosαsinua

+

Length of line of contact

lb = ( )( )( )

1εifε

ε1ε2ε

βcosεb

β2γ

2βα

2γ

bm

α ltminusminusminus

lb = 1εifβcosε

εbβ

bmγ

α ge

15 Nominal Tangential Load Ft Fbt Fmt and Fmbt The nominal tangential load (tangential to the reference cyl-inder with diameter d and perpendicular to an axial plane) is calculated from the nominal (rated) torque T transmitted by the gear set

Cylindrical gears

dT2000

Ft = t

tbt αcos

FF =

Bevel gears

mmt d

T2000F =

vt

mtmbt αcos

FF =

Classification Notes- No 412 7 May 2003

DET NORSKE VERITAS

16 Application Factors KA and KAP The application factor KA accounts for dynamic overloads from sources external to the gearing

It is distinguished between the influence of repetitive cyclic torques KA (161) and the influence of temporary occasional peak torques KAP (162)

Calculations are always to be made with KA In certain cases additional calculations with KAP may be necessary

For gears with a defined load spectrum the calculation with a KA may be replaced by a fatigue damage calculation as given in Appendix A

161 KA For gears designed for long or infinite life at nominal rated torque KA is defined as the ratio between the maximum re-petitive cyclic torque applied to the gear set and nominal rated torque

This definition is suitable for main propulsion gears and most of the auxiliary gears

KA can be determined by measurements or system analysis or may be ruled by conventions (ice classes) (For the pur-pose of a preliminary (but not binding) calculation before KA is determined it is advised to apply either the max values mentioned below or values known from similar plants)

a) For main propulsion gears KA can be taken from the (mandatory) torsional vibration analysis thereby con-sidering all permissible driving conditions) Unless specially agreed the rules do not allow KA in excess of 135 for diesel propulsion) With turbine or electric propulsion KA would normally not exceed 12 However special attention should be given to thrusters that are arranged in such a way that heavy vessel movements andor manoeuvring can cause severe load fluctuations This means eg thrusters positioned far from the rolling axis of vessels that could be susceptible to rolling If leading to propeller air suction the condi-tions may be even worse The above mentioned movements or manoeuvring will result in increased propeller excitation If the thruster is driven by a diesel engine the engine mean torque is limited to 100 However thrusters driven by electric motors can suffer temporary mean torque much above 100 unless a suitable load control system (limiting available e-motor torque) is provided

b) For main propulsion gears with ice class notation (see Rules Pt5 Ch1 Sec J500) KA ice has to be taken as the higher value of the applicable (rule defined) ice shock torque referred to nominal rated torque and the value under a) The Baltic ice class notations refer to a few millions ice shock loads Thus the life factors may be put YN=1 and ZN=12 (except for nitrided gears where ZN = 1 applies) Additionally the calculations with the normal KA (no ice class) are to fulfil the normal requirements For polar ice class notations KA ice applies to all criteria and for long or infinite life

c) For a power take off (PTO) branch from a main propul-sion gear with ice class ice shocks result in negative torques It is assumed that the PTO branch is unloaded when the ice shock load occurs The influence of these reverse shock loads may be taken into account as follows The negative torque (reversed load) expressed by means of an application factor based on rated forward load (T or Ft) is KAreverse = KA ice ndash1 (the minus 1 be-cause no mean torque assumed) KAice to be calculated as in the ice class rules This KAreverse should be used for back flank considera-tions such as pitting and scuffing The influence on tooth bending strength (forward di-rection) may be simplified by using the factor YM = 1 minus 03 middot KAreverse KA

d) For diesel driven auxiliaries KA can be taken from the torsional vibration analysis if available For units where no vibration analysis is required (lt 200 kW) or available it is advised to apply KA as the upper allow-able value 135)

e) For turbine or electro driven auxiliaries the same as for c) applies however the practical upper value is 12

) For diesel driven gears more information on KA for mis-firing and normal driving is given in Appendix B

162 KAP The peak overload factor KAP is defined as the ratio between the temporary occasional peak overload torque and the nominal rated torque

For plants where high temporary occasional peak torques can occur (ie in excess of the above mentioned KA) the gearing (if nitrided) has to be checked with regard to static strength Unless otherwise specified the same safety factors as for infinite life apply

The scuffing safety is to be specially considered whereby the KA applies in connection with the bulk temperature and the KAP applies for the flash temperature calculation and should replace KA in the formulae in 431 432 and 441

KAP can be evaluated from the torsional impact vibration calculation (as required by the rules)

If the overloads have a duration corresponding to several revolutions of the shafts the scuffing safety has to be consid-ered on basis of this overload both with respect to bulk and flash temperature This applies to plants with ice class nota-tions (Baltic and polar) and to plants with prime movers which have high temporary overload capacity such as eg electric motors (provided the driven member can have a con-siderable increase in demand torque as eg azimuth thrusters during manoeuvring)

For plants without additional ice class notation KAP should normally not exceed 15

8 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

163 Frequent overloads For plants where high overloads or shock loads occur regu-larly the influence of this is to be considered by means of cumulative fatigue (see Appendix A)

17 Load Sharing Factor Kγ The load sharing factor Kγ accounts for the maldistribution of load in multiple-path transmissions (dual tandem epicyclic double helix etc) Kγ is defined as the ratio between the max load through an actual path and the evenly shared load

171 General method Kγ mainly depends on accuracy and flexibility of the branches (eg quill shaft planet support external forces etc) and should be considered on basis of measurements or of relevant analysis as eg

Kγ = δ+δ f

δ = total compliance of a branch under full load (assuming even load share) referred to gear mesh

f = minusminusminusminus+++ 23

22

21 fff where f1 f2 etc are the

main individual errors that may contribute to a maldis-tribution between the branches Eg tooth pitch errors planet carrier pitch errors bearing clearance influences etc Compensating effects should also be considered

For double helical gears

An external axial force Fext applied from sources outside the actual gearing (eg thrust via or from a tooth coupling) will cause a maldistribution of forces between the two helices Expressed by a load sharing factor the

βsdot

plusmn=γ tanFF1K

t

ext

If the direction of Fext is known the calculation should be carried out separately for each helix and with the tangential force corrected with the pertinent Kγ If the direction of Fext is unknown both combinations are to be calculated and the higher σH or σF to be used

172 Simplified method If no relevant analysis is available the following may apply

For epicyclic gears

Kγ = 32501 minus+ pln

where npl = number of planets ( gt3 )

For multistage gears with locked paths and gear stages sepa-rated by quill shafts (see figure below)

Figure 10 Locked paths gear

Kγ = ( )φ201+

where φ = quill shaft twist (degrees) under full load

18 Dynamic Factor Kv The dynamic factor Kv accounts for the internally generated dynamic loads

Kv is defined as the ratio between the maximum load that dynamically acts on the tooth flanks and the maximum ex-ternally applied load Ft KA Kγ

In the following 2 different methods (181 and 182) are described In case of controversy between the methods the next following is decisive ie the methods are listed with increasing priority

It is important to observe the limitations for the method in 181 In particular the influence of lateral stiffness of shafts is often underestimated and resonances occur at considerably lower speed than determined in 1811

However for low speed gears with vmiddotz1 lt 300 calculations may be omitted and the dynamic factor simplified to Kv=105

181 Single resonance method For a single stage gear Kv may be determined on basis of the relative proximity (or resonance ratio) N between actual speed n1 and the lowest resonance speed nE1

N = 1E

1

nn

Note that for epicyclic gears n is the relative speed ie the speed that multiplied with z gives the mesh frequency

1811 Determination of critical speed It is not advised to apply this method for multimesh gears for N gt 085 as the influence of higher modes has to be consid-ered see 182 In case of significant lateral shaft flexibility (eg overhung mounted bevel gears) the influence of cou-pled bending and torsional vibrations between pinion and wheel should be considered if N ge 075 see 182

nE1 = redmγc

1zπ

31030 sdot

where

cγ is the actual mesh stiffness per unit facewidth see 111

Classification Notes- No 412 9 May 2003

DET NORSKE VERITAS

For gears with inactive ends of the facewidth as eg due to high crowning or end relief such as often applied for bevel gears the use of cγ in connection with determination of natu-ral frequencies may need correction cγ is defined as stiffness per unit facewidth but when used in connection with the total mesh stiffness it is not as simple as cγmiddotb as only a part of the facewidth is active Such corrections are given in 111

mred is the reduced mass of the gear pair per unit facewidth and referred to the plane of contact

For a single gear stage where no significant inertias are closely connected to neither pinion nor wheel mred is calcu-lated as

mred = 21

21mm

mm+

The individual masses per unit facewidth are calculated as

m12 = 221b

21

)2d(b

I

where I is the polar moment of inertia (kgmm2)

The inertia of bevel gears may be approximated as discs with diameter equal the midface pitch diameter and width equal to b However if the shape of the pinion or wheel body differs much from this idealised cylinder the inertia should be cor-rected accordingly

For all kind of gears if a significant inertia (eg a clutch) is very rigidly connected to the pinion or wheel it should be added to that particular inertia (pinion or wheel) If there is a shaft piece between these inertias the torsional shaft stiffness alters the system into a 3-mass (or more) system This can be calculated as in 182 but also simplified as a 2-mass system calculated with only pinion and wheel masses

1812 Factors used for determination of Kv Non-dimensional gear accuracy dependent parameters

( )bKKF

yfcB

At

pptp

γ

minus=

( )bKKF

yFcBAt

ff

γ

α minus=

Non-dimensional tip relief parameter

bKKFcC

1BγAt

ak sdotsdot

sdotminus=

For gears of quality grade (ISO 1328) Q = 7 or coarser Bk = 1

For gears with Q le 6 and excessive tip relief Bk is limited to max 1

For gears (all quality grades) with tip relief of more than 2middotCeff (see 432) the reduction of αε has to be considered (see 443)

where

fpt = the single pitch deviation (ISO 1328) max of pinion or wheel

Fα = the total profile form deviation (ISO 1328) max of pinion or wheel (Note Fα is pt not available for bevel gears thus use Fα = fpt)

yp and yf = the respective running-in allowances and may be calculated similarly to yα in 112 ie the value of fpt is replaced by Fα for yf

cacute = the single tooth stiffness see 111

Ca = the amount of tip relief see 433 In case of different tip relief on pinion and wheel the value that results in the greater value of Bk is to be used If Ca is zero by design the value of running-in tip relief Cay (see 112) may be used in the above formula

1813 Kv in the subcritical range Cylindrical gears N le 085

Bevel gears N le 075

Kv = 1 + N K

K = Cv1 Bp + Cv2 Bf + Cv3 Bk

Cv1 accounts for the pitch error influence

Cv1 = 032

Cv2 accounts for profile error influence

Cv2 = 034 for γε le 2

Cv2 = 03ε

057

γ minus for γε gt 2

Cv3 accounts for the cyclic mesh stiffness variation

Cv3 = 023 for γε le 2

Cv3 = 156ε

0096

γ minus for γε gt 2

1814 Kv in the main resonance range Cylindrical gears 085 lt N le 115

Bevel gears 075 lt N le 125

Running in this range should preferably be avoided and is only allowed for high precision gears

Kv = 1 + Cv1 Bp + Cv2 Bf + Cv4 Bk

Cv4 accounts for the resonance condition with the cyclic mesh stiffness variation

Cv4 = 090 for γε le 2

Cv4 = 144ε

ε005057

γ

γ

minus

minus for γε gt 2

10 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

1815 Kv in the supercritical range Cylindrical gears N ge 15

Bevel gears N ge 15

Special care should be taken as to influence of higher vibra-tion modes andor influence of coupled bending (ie lateral shaft vibrations) and torsional vibrations between pinion and wheel These influences are not covered by the following approach

Kv = Cv5 Bp + Cv6 Bf + Cv7

Cv5 accounts for the pitch error influence

Cv5 = 047

Cv6 accounts for the profile error influence

Cv6 = 047 for εγ le 2

Cv6 = 174ε

012

γ minus for εγ gt 2

Cv7 relates the maximum externally applied tooth loading to the maximum tooth loading of ideal accurate gears operating in the supercritical speed sector when the circumferential vibration becomes very soft

Cv7 = 075 for εγ le 15

Cv7 = [ ] 8750)2(sin 0125 +minusβεπ for 15 lt εγ le 25

Cv7 = 10 for εγ gt 25

1816 Kv in the intermediate range Cylindrical gears 115 lt N lt 15

Bevel gears 125 lt N lt 15

Comments raised in 1814 and 1815 should be observed

Kv is determined by linear interpolation between Kv for N = 115 respectively 125 and N = 15 as

Cylindrical gears

( ) ( ) ( )[ ]51Nv151Nv51Nvv KK350

N51KK === minussdot

minus

+=

Bevel gears

( ) ( ) ( )[ ]51Nv251Nv51Nvv KK250

N51KK === minussdot

minus

+=

182 Multi-resonance method For high speed gear (vgt40 ms) for multimesh medium speed gears for gears with significant lateral shaft flexibility etc it is advised to determine Kv on basis of relevant dy-namic analysis

Incorporating lateral shaft compliance requires transforma-tion of even a simple pinion-wheel system into a lumped multi-mass system It is advised to incorporate all relevant inertias and torsional shaft stiffnesses into an equivalent (to pinion speed) system Thereby the mesh stiffness appears as an equivalent torsional stiffness

cγ b (db12)2 (Nmrad)

The natural frequencies are found by solving the set of dif-ferential equations (one equation per inertia) Note that for a gear put on a laterally flexible shaft the coupling bending-torsionals is arranged by introducing the gear mass and the lateral stiffness with its relation to the torsional displacement and torque in that shaft

Only the natural frequency (ies) having high relative dis-placement and relative torque through the actual pinion-wheel flexible element need(s) to be considered as critical frequency (ies)

Kv may be determined by means of the method mentioned in 181 thereby using N as the least favourable ratio (in case of more than one pinion-wheel dominated natural frequency) Ie the N-ratio that results in the highest Kv has to be consid-ered

The level of the dynamic factor may also be determined on basis of simulation technique using numeric time integration with relevant tooth stiffness variation and pitchprofile er-rors

19 Face Load Factors KHβ and KFβ The face load factors KHβ for contact stresses and for scuff-ing KFβ for tooth root stresses account for non-uniform load distribution across the facewidth

KHβ is defined as the ratio between the maximum load per unit facewidth and the mean load per unit facewidth

KFβ is defined as the ratio between the maximum tooth root stress per unit facewidth and the mean tooth root stress per unit facewidth The mean tooth root stress relates to the con-sidered facewidth b1 respectively b2

Note that facewidth in this context is the design facewidth b even if the ends are unloaded as often applies to eg bevel gears

The plane of contact is considered

191 Relations between KHβ and KFβ

KFβ = expβHK 2(hb)hb1

1exp++

=

where hb is the ratio tooth heightfacewidth The maximum of h1b1 and h2b2 is to be used but not higher than 13 For double helical gears use only the facewidth of one helix

If the tooth root facewidth (b1 or b2) is considerably wider than b the value of KFβ(1or2) is to be specially considered as it may even exceed KHβ

Eg in pinion-rack lifting systems for jack up rigs where b = b2 asymp mn and b1 asymp 3 mn the typical KHβ asymp KFβ2 asymp 1 and KFβ1 asymp 13

192 Measurement of face load factors Primarily

KFβ may be determined by a number of strain gauges distrib-uted over the facewidth Such strain gauges must be put in

Classification Notes- No 412 11 May 2003

DET NORSKE VERITAS

exactly the same position relative to the root fillet Relations in 191 apply for conversion to KHβ

Secondarily

KHβ may be evaluated by observed contact patterns on vari-ous defined load levels It is imperative that the various test loads are well defined Usually it is also necessary to evalu-ate the elastic deflections Some teeth at each 90 degrees are to be painted with a suitable lacquer Always consider the poorest of the contact patterns

After having run the gear for a suitable time at test load 1 (the lowest) observe the contact pattern with respect to ex-tension over the facewidth Evaluate that KHβ by means of the methods mentioned in this section Proceed in the same way for the next higher test load etc until there is a full face contact pattern From these data the initial mesh misalign-ment (ie without elastic deflections) can be found by ex-trapolation and then also the KHβ at design load can be found by calculation and extrapolation See example

Figure 11 Example of experimental determination of KHβ

It must be considered that inaccurate gears may accumulate a larger observed contact pattern than the actual single mesh to mesh contact patterns This is particularly important for lapped bevel gears Ground or hard metal hobbed bevel gears are assumed to present an accumulated contact pattern that is practically equal the actual single mesh to mesh con-tact patterns As a rough guidance the (observed) accumu-lated contact pattern of lapped bevel gears may be reduced by 10 in order to assess the single mesh to mesh contact pattern which is used in 199

193 Theoretical determination of KHβ The methods described in 193 to 198 may be used for cy-lindrical gears The principles may to some extent also be used for bevel gears but a more practical approach is given in 199

General For gears where the tooth contact pattern cannot be verified during assembly or under load all assumptions are to be well on the safe side

KHβ is to be determined in the plane of contact

The influence parameters considered in this method are

bull mean mesh stiffness cγ (see 111) (if necessary also variable stiffness over b)

bull mean unit load Fmb = Fbt KA Kγ Kvb (for double helical gears see 17 for use of Kγ)

bull misalignment fsh due to elastic deflections of shafts and gear bodies (both pinion and wheel)

bull misalignment fdefl due to elastic deflections of and work-ing positions in bearings

bull misalignment fbe due to bearing clearance tolerances bull misalignment fma due to manufacturing tolerances bull helix modifications as crowning end relief helix correc-

tion bull running in amount yβ (see 112)

In practice several other parameters such as centrifugal ex-pansion thermal expansion housing deflection etc contrib-ute to KHβ However these parameters are not taken into account unless in special cases when being considered as particularly important

When all or most of the am parameters are to be considered the most practical way to determine KHβ is by means of a graphical approach described in 1931

If cγ can be considered constant over the facewidth and no helix modifications apply KHβ can be determined analyti-cally as described in 1932

1931 Graphical method The graphical method utilises the superposition principle and is as follows

bull Calculate the mean mesh deflection Mδ as a function of

γm c and bF see 111

bull Draw a base line with length b and draw up a rectangu-lar with height δM (The area δM b is proportional to the transmitted force)

bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the areas above and below this zero line are equal

Figure 12 fsh balanced around zero line

bull Superimpose these ordinates of the fsh curve to the previ-ous load distribution curve (The area under this new load distribution curve is still δM b)

bull Calculate the bearing deflections andor working posi-tions in the bearings and evaluate the influence fdefl in the plane of contact This is a straight line and is bal-anced around a zero line as indicated in Fig 14 but with one distinct direction Superimpose these ordinates to the previous load distribution curve

12 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

bull The amount of crowning end relief or helix correction (defined in the plane of contact) is to be balanced around a zero line similarly to fsh

Figure 13 Crowning Cc balanced aound zero line

bull Superimpose these ordinates to the previous load distri-bution curve In case of high crowning etc as eg often applied to bevel gears the new load distribution curve may cross the base line (the real zero line) The result is areas with negative load that is not real as the load in those areas should be zero Thus corrective actions must be made but for practical reasons it may be postponed to after next operation

bull The amount of initial mesh misalignment bema ff + (defined in the plane of contact) is to be bal-

anced around a zero line If the direction of bema ff + is known (due to initial contact check) or if the direction of fbe is known due to design (eg overhang bevel pin-ion) this should be taken into account If direction un-known the influence of bema ff + in both directions as well as equal zero should be considered

Figure 14 fma+fbe in both directions balanced around zero line

Superimpose these ordinates to the previous load distri-bution curve This results in up to 3 different curves of which the one with the highest peak is to be chosen for further evaluation

bull If the chosen load distribution curve crosses the base line (ie mathematically negative load) the curve is to be corrected by adding the negative areas and dividing this with the active facewidth The (constant) ordinates of this rectangular correction area are to be subtracted from the positive part of the load distribution curve It is advisable to check that the area covered under this new load distribution curve is still equal δM b

bull If cγ cannot be considered as constant over b then cor-rect the ordinates of the load distribution curve with the local (on various positions over the facewidth) ratio between local mesh stiffness and average mesh stiffness cγ (average over the active facewidth only) Note that the result is to be a curve that covers the same area δM b as before

bull The influence of running in yβ is to be determined as in 112 whereby the value for Fβx is to be taken as twice the distance between the peak of the load distribution curve and δM

bull Determine

KHβ = M

β

δycurveofpeak minus

1932 Simplified analytical method for cylindrical gears The analytical approach is similar to 1931 but has a more limited application as cγ is assumed constant over the face-width and no helix modification applies

bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the area above and below this zero line are equal see Fig 12 The max positive ordinate is frac12∆fsh

bull Calculate the initial mesh alignment as Fβx= deflbemash ffff plusmnplusmnplusmn∆ The negative signs may only be used if this is justified andor verified by a contact pattern test Otherwise al-ways use positive signs If a negative sign is justified the value of Fβx is not to be taken less than the largest of each of these elements

bull Calculate the effective mesh misalignment as Fβy = Fβx - yβ (yβ see 112)

bull Determine

2KforF2

bFc1K βH

m

βγγH le+=β

or

2KforF

bFc2K Hβ

m

βγγH gt=β

where cγ as used here is the effective mesh stiffness see 111

194 Determination of fsh fsh is the mesh misalignment due to elastic deflections Usu-ally it is sufficient to consider the combined mesh deflection of the pinion body and shaft and the wheel shaft The calcu-lation is to be made in the plane of contact (of the considered gear mesh) and to consider all forces (incl axial) acting on the shafts Forces from other meshes can be parted into components parallel respectively vertical to the considered plane of contact Forces vertical to this plane of contact have no influence on fsh

It is advised to use following diameters for toothed elements

d + 2 x mn for bending and shear deflection

d + 2 mn (x ndash ha0 + 02) for torsional deflection

Usually fsh is calculated on basis of an evenly distributed load If the analysis of KHβ shows a considerable maldis-

Classification Notes- No 412 13 May 2003

DET NORSKE VERITAS

tribution in term of hard end contact or if it is known by other reasons that there exists a hard end contact the load should be correspondingly distributed when calculating fsh In fact the whole KHβ procedure can be used iteratively 2-3 iterations will be enough even for almost triangular load distributions

195 Determination of fdefl fdefl is the mesh misalignment in the plane of contact due to bearing deflections and working positions (housing deflec-tion may be included if determined)

First the journal working positions in the bearings are to be determined The influence of external moments and forces must be considered This is of special importance for twin pinion single output gears with all 3 shafts in one plane

For rolling bearings fdefl is further determined on basis of the elastic deflection of the bearings An elastic bearing deflec-tion depends on the bearing load and size and number of rolling elements Note that the bearing clearance tolerances are not included here

For fluid film bearings fdefl is further determined on basis of the lift and angular shift of the shafts due to lubrication oil film thickness Note that fbe takes into account the influence of the bearing clearance tolerance

When working positions bearing deflections and oil film lift are combined for all bearings the angular misalignment as projected into the plane of the contact is to be determined fdefl is this angular misalignment (radians) times the face-width

196 Determination of fbe fbe is the mesh misalignment in the plane of contact due to tolerances in bearing clearances In principle fbe and fdefl could be combined But as fdefl can be determined by analy-sis and has a distinct direction and fbe is dependent on toler-ances and in most cases has no distinct direction (ie + toler-ance) it is practicable to separate these two influences

Due to different bearing clearance tolerances in both pinion and wheel shafts the two shaft axis will have an angular mis-alignment in the plane of contact that is superimposed to the working positions determined in 195 fbe is the facewidth times this angular misalignment Note that fbe may have a distinct direction or be given as a + tolerance or a combina-tion of both For combination of + tolerance it is adviced to use

fbe= 22be

21be ff ++plusmn

fbe is particularly important for overhang designs for gears with widely different kinds of bearings on each side and when the bearings have wide tolerances on clearances In general it shall be possible to replace standard bearings with-out causing the real load distribution to exceed the design premises For slow speed gears with journal bearings the expected wear should also be considered

197 Determination of fma fma is the mesh misalignment due to manufacturing toler-ances (helix slope deviation) of pinion fHβ1 wheel fHβ2 and housing bore

For gear without specifically approved requirements to as-sembly control the value of fma is to be determined as

fma= 22Hβ

21Hβ ff +

For gears with specially approved assembly control the value of fma will depend on those specific requirements

198 Comments to various gear types For double helical gears KHβ is to be determined for both helices Usually an even load share between the helices can be assumed If not the calculation is to be made as de-scribed in 171

For planetary gears the free floating sun pinion suffers only twist no bending It must be noted that the total twist is the sum of the twist due to each mesh If the value of 1K γ ne this must be taken into account when calculating the total sun pinion twist (ie twist calculated with the force per mesh without Kγ and multiplied with the number of planets)

When planets are mounted on spherical bearings the mesh misalignments sun-planet respectively planet-annulus will be balanced Ie the misalignment will be the average between the two theoretical individual misalignments The faceload distribution on the flanks of the planets can take full advan-tage of this However as the sun and annulus mesh with several planets with possibly different lead errors the sun and annulus cannot obtain the above mentioned advantage to the full extent

199 Determination of KHβ for bevel gears If a theoretical approach similar to 193-198 is not docu-mented the following may be used

testeff

H Kb

b851851K sdot

minussdot=β

beff b represents the relative active facewidth (regarding lapped gears see 192 last part)

Higher values than beff b = 090 are normally not to be used in the formula

For dual directional gears it may be difficult to obtain a high beff b in both directions In that case the smaller beff b is to be used

Ktest represents the influence of the bearing arrangement shaft stiffness bearing stiffness housing stiffness etc on the faceload distribution and the verification thereof Expected variations in length- and height-wise tooth profile is also accounted for to some extent

a) Ktest = 1 For ground or hard metal hobbed gears with the specified contact pattern verified at full rating or at full torque slow turning at a condition representative for the thermal expansion at normal operation

14 Classification Notes- No 412

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DET NORSKE VERITAS

It also applies when the bearing arrangementsupport has insignificant elastic deflections and thermal axial expansion However each initial mesh contact must be verified to be within acceptance criteria that are cali-brated against a type test at full load Reproduction of the gear tooth length- and height-wise profile must also be verified This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon)

b) Ktest = 1 + 04middot(beffbndash06) For designs with possible influence of thermal expansion in the axial direction of the pinion The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load

c) Ktest = 12 if mesh is only checked by toolmakerrsquos blue or by spin test contact For gears in this category beffb gt 085 is not to be used in the calculation

110 Transversal Load Distribution Factors KHα and KFα The transverse load distribution factors KHα for contact stresses and for scuffing KFα for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairs of teeth in mesh

The following relations may be used

Cylindrical gears

( )

minus+==

tH

αptγγHαFα F

byfc0409

2ε

KK

valid for 2εγ le

( ) ( )tH

αptγ

γ

γHαFα F

byfcε

1ε20409KK

minusminus+==

valid for 2ε γ gt

where

FtH = Ft KA Kγ Kv KHβ

cγ = See 111

γα = See 112

fpt = Maximum single pitch deviation (microm) of pinion or wheel or maximum total profile form devia-tion Fα of pinion or wheel if this is larger than the maximum single pitch deviation

Note In case of adequate equivalent tip relief adapted to the load half of the above mentioned fpt can be introduced A tip relief is considered adequate when the average of Ca1 and Ca2 is within plusmn40 of the value of Ceff in 432

Limitations of KHα and KFα

If the calculated values for

KFα = KHα lt 1 use KFα = KHα = 10

If the calculated value of KHα gt 2εα

γ

Zε

ε use KHα = 2

εα

γ

Zε

ε

If the calculated value of KFα gt εα

γ

Yεε

use KFα = εα

γ

Yεε

where Yε = αnε

075025+ (for εαn see 331c)

Bevel gears

For ground or hard metal hobbed gears KFα = KHα = 1

For lapped gears KFα = KHα = 11

111 Tooth Stiffness Constants cacute and cγ The tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having 1 mm facewidth by an amount of 1 microm in the plane of contact cacute is the maximum stiffness of a single pair of teeth cγ is the mean value of the mesh stiffness in a transverse plane (brief term mesh stiffness)

Both valid for high unit load (Unit load = Ft middot KA middot Kγb)

Cylindrical gears The real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bending stiff-nesses For high unit loads the Hertzian stiffness has little importance and can be disregarded This approach is on the safe side for determination of KHβ and KHα However for moderate or low loads Kv may be underestimated due to determination of a too high resonance speed

The linear approach is described in A

An optional approach for inclusion of the non-linear stiffness is described in B

A The linear approach

BRCCq

βcos08acutec =

and

( )025ε075cc αγ +prime=

where

( )[ ]n02a01a

B α2000212

hh12051C minusminus

+minus+=

12n1n

x000635z

025791z

015551004723q minus++=

12

2n

22

1n

1 x005290z

x024188x000193z

x011654+minusminusminus

+ 000182 x22

Classification Notes- No 412 15 May 2003

DET NORSKE VERITAS

(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as

( )( )nR m5s

sR e5

bbln1C +=

where

bs = thickness of a central web

sR = average thickness of rim (net value from tooth root to inside of rim)

The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered

Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ

B The non-linear approach

In the following an example is given on how to consider the non-linearity

The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm

With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as

( )10δKbF

minus= for 500bFgt

minus=

500Fb10δK

bF for 500

bFlt

with γAt KK

bF

bF

sdotsdot= etc (Nmm) ie unit load incorporat-

ing the relevant factors as

KA middot Kγ for determination of Kv

KA middot Kγ middot Kv for determination of KHβ

KA middot Kγ middot Kv middot KHβ for determination of KHα

δ = mesh deflection (microm)

K = applicable stiffness (c or cγ)

Use of stiffnesses for KV KHβ and KHα

For calculation of Kv and KHα the stiffness is calculated as follows

When Fb lt 500

the stiffness is determined as δ∆

∆ bF

where the increment is chosen as eg ∆ Fb = 10 and thus

50010Fb10

K10Fb∆δ +

++

=

When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly

or an equivalent stiffness determined as δsdotb

F

Bevel gears

In lack of more detailed relationship between stiffness and geometry the following may be used

b085

b13cacute eff=

b085b

16c effγ =

beff not to be used in excess of 085 b in these formulae

Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)

112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements

yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload

Cay is defined as the running-in amount that compensates for lack of tip relief

The following relations may be used

For not surface hardened steel

ptHlim

α fσ160y =

yβ = βxlimH

f320σ

with the following upper limits

V lt 5 ms 5-10 ms gt 10 ms

yα max none

limH

12800σ limH

6400σ

yβ max none

limH

25600σ limH

12800σ

For surface hardened steel

yα = 0075 fpt but not more than 3 for any speed

yβ = 015 Fβx but not more than 6 for any speed

16 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

For all kinds of steel

5145189718

1C2

limHay +

minusσ

=

When pinion and wheel material differ the following ap-plies

bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv

bull Use ( )2β1ββ yy21y += in the calculation of KHβ

bull Use ( )2ay1aya CC21C += in the calculation of Kv

bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing

calculation if no design tip relief is foreseen

Classification Notes- No 412 17 May 2003

DET NORSKE VERITAS

2 Calculation of Surface Durability

21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213

Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted

Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213

For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength

For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied

22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater

Calculation of surface durability for helical gears is based on the contact stress at the pitch point

For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies

Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact

Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses

The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses

The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel

221 Contact stress Cylindrical gears

( )HαHβvγA

1

tβεEHDBH KKKKK

bud1uF

ZZZZZσ+

=

where

ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)

ZH = Zone factor for pitch point (see 231)

ZE = Elasticity factor (see 24)

Zε = Contact ratio factor (see 25)

Zβ = Helix angle factor (see 26)

Ft KA Kγ Kv KHβ KHα see 15 ndash 110

d1 b u see 12 ndash 15

Bevel gears

( )HαHβvγA

v1v

vmtKEMH KKKKK

bud1uF

ZZZ105σ+

sdot=

where

105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)

ZE KA etc see above

ZM = mid-zone factor see 233

ZK = bevel gear factor see 27

Fmt dv1 uv see 12 ndash 15

It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks

222 Permissible contact stress

XWRvLH

NHlimHP ZZZZZ

SZσ

σ =

where

σH lim = Endurance limit for contact stresses (see 28)

ZN = Life factor for contact stresses (see 29)

SH = Required safety factor according to the rules

ZLZvZR = Oil film influence factors (see 210)

ZW = Work hardening factor (see 211)

ZX = Size factor (see 212)

18 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

23 Zone Factors ZH ZBD and ZM

231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder

wtt2

wtbH sinααcos

cosαcosβ2Z =

232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel

For εβ ge 1 ZBD = 1

For internal gears ZD = 1

For εβ = 0 (spur gears)

( )

π

minusεminusminus

π

minusminus

α=

α2

2

2b

2a

1

2

1b

1a

wtB

z211

dd

z21

dd

tanZ

( )

π

minusεminusminus

π

minusminus

α=

α1

2

1b

1a

2

2

2b

2a

wtD

z211

dd

z21

dd

tanZ

If ZB lt 1 use ZB = 1

If ZD lt 1 use ZD = 1

For 0 lt εβ lt 1

ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)

233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears

εminusminus

εminusminus

αβ=

αα btm2

2vb2

2vabtm2

1vb2val

2v1vvtbmM

pddpdd

ddtancos2Z

This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint

234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel

24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses

For steel against steel ZE = 1898

25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses

αε1Z =ε for 1εβ ge

( )α

ββ

αε ε

εε1

3ε4

Z +minusminus

= for εβ lt 1

26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability

cosβZβ =

27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply

The following may be used ZK = 080

28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ

σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting

For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply

For this purpose pitting is defined by

bull for not surface hardened gears pitted area ge 2 of total active flank area

bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given

material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies

The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as

Classification Notes- No 412 19 May 2003

DET NORSKE VERITAS

the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99

σHlim σH105 σH10

3

Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade

1650 1500

2500 2400

3100 3100

Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)

1000

13 σHlim

13 σHlim

Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV

Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV

Carbon steel 15 HV + 250 16 σHlim 16 σHlim

These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15

29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply

If this is not documented by approved fatigue tests the fol-lowing method may be used

For all steels except nitrided

7L 105N sdotge ZN = 1 or

01570

L

7

N N105Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 5middot107 510N

logZ037

L

7

N N105Z

sdot=

NL = 105 WXRVLHlim

WstX10H1010NN ZZZZZσ

ZZσZZ

55

5=

103 lt NL lt 105 )Z(Zlog05

L

5

10NN

5N103N10

5N10ZZ

==

3L 10N le

WXRVLlimH

Wst10X10H10NN ZZZZZ

ZZZZ

33

3σ

σ==

(but not less than ZN105)

For nitrided steels

6L 102N sdotge ZN = 1 or

00980

L

6

N N102Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 2middot106 510N

Zlog76860

L

6

N N102Z

sdot=

5L 10N le

XWRVL

10XWst10NN ZZZZZ

ZZ13ZZ

55 ==

Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles

210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity

The following methods may be applied in connection with the endurance limit

20 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Surface hardened steels Not surface hardened steels

ZL ( )24013421

360910ν+

+

( )24013421

680830ν+

+

ZV ( )v3280

140930+

+ ( )v3280300850

++

ZR

080

ZrelR3

150

ZrelR3

where

ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits

For values of ν40 gt 500 use ν40 = 500

RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm

RZrel

= ( ) 31

cZ2Z1 ρ

10RR50

+

RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)

For 5

L 10N le ZL ZV ZR = 10

211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface

The following approximation may be used for the endurance limit

Surface hardened steel against not surface hardened steel 150

ZeqW R

31700

130HB21Z

minus

minus=

where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130

RZeq = equivalent roughness

033

c40

066

ZS

ZHZHZEQ ρvν

15000RRRR

=

If RZeq gt 16 then use RZeq = 16

If RZeq lt 15 then use RZeq = 15

where

RZH = surface roughness of the hard member before run in

RZS = surface roughness of the soft member before run in

ν40 = see 210

If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem

Through hardened pinion against softer wheel

( )

minussdotminus+= 000829

HBHB0008981u1Z

2

1W

For 21HBHB

2

1 le use ZW = 1

For 71HBHB

2

1 gt use 17HBHB

2

1 =

For u gt 20 use u = 20

For static strength (lt 105 cycles)

Surface hardened against not surface hardened

ZWst = 105

Through hardened pinion against softer wheel

ZWst = 1

212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality

ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213

213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented

The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent

Classification Notes- No 412 21 May 2003

DET NORSKE VERITAS

stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically

The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence

The real Hertzian stresses σHR are determined as

For helical gears with εβ gt 1

σHR = σH

For helical gears with εβ lt 1 and spur gears

ε

α

ββ ε

ε+εminus

sdotσ=σZ

1

HHR

For bevel gears

KHHR Z

1σσ sdot=

The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)

The coordinates tz and HV are to be compared with the de-sign specification such as

bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400

and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)

In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods

For high cycle fatigue (gt3 106 cycles) the following applies

sdot+

minussdotsdotsdot= o90

05azt

05azt

cosSσ04HV

H

HHHR

applicable to 05a

zt

Hge

For 05at

H

z lt the value for 05at

H

z = applies

56300ρSσ12a cHHR

Hsdotsdot

sdot=

Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength

If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows

reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue

For static strength (lt103 cycles) the following applies

sdot+

minussdotsdotsdot= o90

07at

06a

zt

cosSσ019HV

Hst

z

HstHHR

applicable to 06at

Hst

z ge

56300ρSσa cHHR

Hstsdotsdot

=

In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies

For limited life fatigue (103 lt cycles lt 3106 )

For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance

The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram

minussdotminus

= sdot logN3477

logSlogSlogS

310H6103HHN

36 10H103H Slog86281Slog86280 sdot+sdot sdot

22 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3 Calculation of Tooth Strength

31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding

For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36

For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning

Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα

In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314

A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired

It should be noted that this part 3 does not cover fractures caused by

bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining

Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears

Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied

32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level

321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance

Normally the stresses for pinion and wheel are calculated as

Cylindrical gears

FαFβvγAβSFn

tF K K K K K Y Y Y

m bFσ =

where

YF = Tooth form factor (see 33)

YS = Stress correction factor (see 34)

Yβ = Helix angle factor (see 36)

Ft KA Kγ Kv KFβ KFα see 15 ndash 110

b see 13

Bevel gears

FαFβvγASaFamn

mtF K K K K K Y Y Y

m bFσ ε=

where

YFa = Tooth form factor see 33

YSa = Stress correction factor see 34

Yε = Contact ratio factor see 35

Fmt KA etc see 15 ndash 110

b see 13

322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is

CXRrelTrelTF

NMFEFP YYYY

SYY

δσ

=σ

Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN

where

σFE = Local tooth root bending endurance limit of reference test gear (see 37)

YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)

YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)

SF = Required safety factor according to the rules

YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)

YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)

Classification Notes- No 412 23 May 2003

DET NORSKE VERITAS

YX = Size factor (see 312)

YC = Case depth factor (see 313)

33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress

YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears

YFa applies to load application at the tooth tip and is used for bevel gears

Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears

Figure 31 External tooth in normal section

Figure 32 Internal tooth in normal section

Definitions

n

2

n

Fn

enFn

Fe

F

α cosms

α cosmh6

Y

=

n

2

n

Fn

anFn

Fa

Fa

α cosms

α cosmh6

Y

=

In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth

YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)

The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing

mn with mnm

zn with zvn

αt with αvt

β with βm

with undercut without undercut

Fig 33 Dimensions and basic rack profile of the teeth (finished profile)

Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless

331 Determination of parameters

( )n

n

prnfPnfP m

α cossαsin 1ρ

αtan h4πE

minusminusminusminus=

For external gears fPfP ρρ =

For internal gears ( )0z

195fPfP0

fPfP 10363156ρhxρρ

sdotminus+

+=

where

z0 = number of teeth of pinion cutter

x0 = addendum modification coefficient of pinion cutter

hfP = addendum of pinion cutter

ρfP = tip radius of pinion cutter

xhρG fpfp +minus=

24 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

τmE

2π

z2H

nnminus

minus=

with

3πτ = for external gears

6πτ = for internal gears

Htan zG2

nminusϑ=ϑ

(to be solved iteratively suitable start value6π

=ϑ for exter-

nal gears and 3π for internal gears)

a) Tooth root chord sFn For external gears

minus

ϑ+

ϑminus= ρ

cosG3

3πsinz

ms

fpnn

Fn

For bevel gears with a tooth thickness modification

xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn

For internal gears

minus

ϑ+

ϑminus= ρ

cosG

6πsinz

ms

fPnn

Fn

b) Root fillet radius ρF at 30ordm tangent

( )G2coszcosG2ρ

mρ

2n

2

fpn

F

minusϑϑ+=

c) Determination of bending moment arm hF dn = zn mn

b2α

αn βcosε

ε =

dan = dn + 2 ha

pbn = π mn cos αn

dbn = dn cos αn

( )4

d1εpzz

2dd

zz2d

2bn

2

αnbn

2bn

2an

en +

minusminus

minus=

en

bnen d

dcos arcα =

ennnn

e α invα invα x tan 22π

z1

minus+

+=γ

αFen = αen ndash γe

For external gears

( )

minus=

n

enFenee

n

Fe

mdαtan sin γ γcos

21

mh

]ρcos

G3πcosz fpn +

ϑminus

ϑminusminus

For internal gears

( ) minus

sdotsdotminus=

n

enFenee

n

Fe

mdα tanγ sinγ cos

21

mh

minus

ϑminus

ϑminussdot ρ

cosG3

6πcosz fPn

332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as

250ε205for 0666ε2366Y αnαnDT leleminus=

205εfor 10Y αnDT lt=

34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section

Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm

YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip

YS can be determined as follows

( )

+

+=L23121

1

sS qL01312Y

where Fe

Fn

hsL = and

F

Fns ρ2

sq = (see 33)

YSa can be calculated by replacing hFe with hFa in the above formulae

Note a) Range of validity 1 lt qs lt 8

In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered

b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with

Classification Notes- No 412 25 May 2003

DET NORSKE VERITAS

g

g

ρt

0613

13

minus

where

tg = depth of the grinding notch

ρg = radius of the grinding notch

c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles

35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears

The following may be used

Yε = 0625

36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank

The following may be used (β input in degrees)

Yβ = 1 ndash εβ β120

When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula

However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)

37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM

σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision

If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules

σFE

Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)

bull of specially approved high grade

1050

bull of normal grade

minus CrNiMo steels with approved process

1000

minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850

Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

840

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

720

Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)

07 HV + 300

Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)

025 σB + 125

Alloyed quenched and tempered steel 04 σB + 200

Carbon steel 025 σB + 250

Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40

1) These values are valid for a root radius

bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)

bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)

bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for

6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked

2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)

26 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears

YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0

YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst

For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316

The following method may be used within a stress ratio ndash12 lt R lt 05

381 For idlers planets and PTO with ice class

M1M1R1

1Yor Y MstM

+minus

minus=

where

R = stress ratio = min stress divided by max stress

For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12

For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as

branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus

For a power take off (PTO) with ice class see 161 c

M considers the mean stress influence on the endurance (or static) strength amplitudes

M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress

Following M values may be used

Endurance limit

Static strength

Case hardened 08 ndash 015 Ys 1) 07

If shot peened 04 06

Nitrided 03 03

Induction or flame hardened

04 06

Not surface hardened steel 03 05

Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M

The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches

382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106

For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers

For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used

383 For gears with shrinkage stresses and unidirectional load For endurance strength

FE

fitM σ

σM1

M21Y+

minus=

σFE is the endurance limit for R = 0

For static strength YMst = 1 and σfit accounted for in 39b

σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor

n

Ffit m

ρ215scf minus=

384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is

( ) ( ) FE

fitM σ

σR1M1

M2

M1M1R1

1Yminussdot+

minus

+minus

minus=

Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12

For static strength

M1M1R1

1YMst

+minus

minus=

The effect of σfit is accounted for in 39 b

Classification Notes- No 412 27 May 2003

DET NORSKE VERITAS

385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed

F

y

Sσ225

for not surface hardened fillets

FSHV5 for surface hardened fillets

39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles

Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN

If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used

Determination of the σ ndash N ndash curve

a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF

b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ

FstS1

FPstσ minussdotsdotsdot=

where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability

σFst

Alloyed case hardened steel 1) 2300

Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

1250

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

1050

Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)

18 HV + 800

Steel with not surface hardened fillets the smaller value of 2)

18 σB or 225 σy

1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet

c) Calculate YN as

NL gt 3middot106

YN = 1 or 001

L

6

N N103Y

sdot= ie Yn = 092 for 1010

The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2

103ltNLlt3middot106exp

L

6

N N103Y

sdot=

cycles6103forσcycles310forσlog02876exp

FP

FPst

sdot=

NL lt 103 cycles6103forσcycles310forσ

NYFP

FPst

sdot=

or simply use σFPst as mentioned in b) directly

Guidance on number of load cycles NL for various applica-tions

bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)

bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied

28 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage

YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit

The following method may be used

For endurance limit

for not surface hardened fillets

( )

024

s024

relT σ103133q21σ1012201351

Ysdotsdotminus

+sdotsdotminus+= minus

minus

δ

for all surface hardened fillets except nitrided

( )

106q21002451

Y srelT

++=δ

for nitrided fillets

( )

1347q2101421

Y srelT

++=δ

For static strength

for not surface hardened fillets1)

( )( )( )025

02

02502s

relTst σ3000821σ300 1Y0821Y

+

minus+=δ

for surface hardened fillets except nitrided

YδrelTst = 044 YS + 012

for nitrided fillets

YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not

exceed the yield point and thereby alter the residual stress level See also 39b footnote 2

311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness

YRrelT differs for endurance limit and static strength

The following method may be used

For endurance limit

YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels

YδrelT = 43 ndash 326 (Ry + 1)0005

for nitrided steels

For static strength

YRrelTst = 1 for all Ry and all materials

For a fillet without any longitudinal machining trace Ry asymp Rz

312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength

The following may be used

For endurance limit

YX = 1 for mn le 5 generally

YX = 103 ndash 0006 mn for 5 lt mn lt 30

YX = 085 for mn ge 30

for not surface hard-ened steels

YX = 105 ndash 001 mn for 5 lt mn ge 25

YX = 08 for mn ge 25

for surface hardened steels

For static strength

YXst = 1 for all mn and all materials

313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength

YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength

In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength

The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses

The following simplified method for YC may be used

YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)

YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10

For endurance limit

+

+=nFFE

C m02ρt31

σconstY

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

6 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

pbt = βcos

αcosmπ tn

sat =

minus++

at

ninvααinv

z

αtanx22

π

d a

san = aβcossat

14 Bevel Gear Conversion Formulae and Specific Formulae Conversion of bevel gears to virtual equivalent helical cylin-drical gears is based on the bevel gear midsection The con-version formulae are

Number of teeth

zv12 = z12 cos δ12

(δ1 + δ2 = Σ)

Gear ratio

uv = 1v

2vzz

tan αvt = tan αn cos βm

tan βbm = tan βm cos αvt

Base pitch

pbtm = m

vtnmβcos

αcosmπ

Reference pitch diameters

dv12 = 21

21m

cosdδ

Centre distance

av = 05 (dv1 + dv2)

Tip diameters

dva 12 = dv 12 + 2 ham 12

Addenda

for gears with constant addenda (Klingelnberg)

ham 12 = mmn (1 + xm 12)

for gears with variable addenda (Gleason)

ham 12 = ha 12 ndash b2 tan (δa 12 ndash δ12)

(when ha is addendum at outer end and δa is the outer cone angle)

Addendum modification coefficients

xm 12 = mn

12am21am

m2

hh minus

Base circle

dvb 12 = dv 12 cos αvt

Transverse contact ratio)

εα = btmP

αsinadd05dd05 vtv2

2vb2

2va2

1vb2

1va minusminus+minus

Overlap ratio) (theoretical value for bevel gears with no crowning but used as approximations in the calculation pro-cedures)

εβ = nm

mmπβsinb

Total contact ratio)

εγ = 2β

2α εε +

( Note that index laquovraquo is left out in order to combine formu-lae for cylindrical and bevel gears)

Tangential speed at midsection

vmt = 3m11 10dn

60π minus

Effective radius of curvature (normal section)

ρvc = ( )2vbm

vtvv

u1βcosαsinua

+

Length of line of contact

lb = ( )( )( )

1εifε

ε1ε2ε

βcosεb

β2γ

2βα

2γ

bm

α ltminusminusminus

lb = 1εifβcosε

εbβ

bmγ

α ge

15 Nominal Tangential Load Ft Fbt Fmt and Fmbt The nominal tangential load (tangential to the reference cyl-inder with diameter d and perpendicular to an axial plane) is calculated from the nominal (rated) torque T transmitted by the gear set

Cylindrical gears

dT2000

Ft = t

tbt αcos

FF =

Bevel gears

mmt d

T2000F =

vt

mtmbt αcos

FF =

Classification Notes- No 412 7 May 2003

DET NORSKE VERITAS

16 Application Factors KA and KAP The application factor KA accounts for dynamic overloads from sources external to the gearing

It is distinguished between the influence of repetitive cyclic torques KA (161) and the influence of temporary occasional peak torques KAP (162)

Calculations are always to be made with KA In certain cases additional calculations with KAP may be necessary

For gears with a defined load spectrum the calculation with a KA may be replaced by a fatigue damage calculation as given in Appendix A

161 KA For gears designed for long or infinite life at nominal rated torque KA is defined as the ratio between the maximum re-petitive cyclic torque applied to the gear set and nominal rated torque

This definition is suitable for main propulsion gears and most of the auxiliary gears

KA can be determined by measurements or system analysis or may be ruled by conventions (ice classes) (For the pur-pose of a preliminary (but not binding) calculation before KA is determined it is advised to apply either the max values mentioned below or values known from similar plants)

a) For main propulsion gears KA can be taken from the (mandatory) torsional vibration analysis thereby con-sidering all permissible driving conditions) Unless specially agreed the rules do not allow KA in excess of 135 for diesel propulsion) With turbine or electric propulsion KA would normally not exceed 12 However special attention should be given to thrusters that are arranged in such a way that heavy vessel movements andor manoeuvring can cause severe load fluctuations This means eg thrusters positioned far from the rolling axis of vessels that could be susceptible to rolling If leading to propeller air suction the condi-tions may be even worse The above mentioned movements or manoeuvring will result in increased propeller excitation If the thruster is driven by a diesel engine the engine mean torque is limited to 100 However thrusters driven by electric motors can suffer temporary mean torque much above 100 unless a suitable load control system (limiting available e-motor torque) is provided

b) For main propulsion gears with ice class notation (see Rules Pt5 Ch1 Sec J500) KA ice has to be taken as the higher value of the applicable (rule defined) ice shock torque referred to nominal rated torque and the value under a) The Baltic ice class notations refer to a few millions ice shock loads Thus the life factors may be put YN=1 and ZN=12 (except for nitrided gears where ZN = 1 applies) Additionally the calculations with the normal KA (no ice class) are to fulfil the normal requirements For polar ice class notations KA ice applies to all criteria and for long or infinite life

c) For a power take off (PTO) branch from a main propul-sion gear with ice class ice shocks result in negative torques It is assumed that the PTO branch is unloaded when the ice shock load occurs The influence of these reverse shock loads may be taken into account as follows The negative torque (reversed load) expressed by means of an application factor based on rated forward load (T or Ft) is KAreverse = KA ice ndash1 (the minus 1 be-cause no mean torque assumed) KAice to be calculated as in the ice class rules This KAreverse should be used for back flank considera-tions such as pitting and scuffing The influence on tooth bending strength (forward di-rection) may be simplified by using the factor YM = 1 minus 03 middot KAreverse KA

d) For diesel driven auxiliaries KA can be taken from the torsional vibration analysis if available For units where no vibration analysis is required (lt 200 kW) or available it is advised to apply KA as the upper allow-able value 135)

e) For turbine or electro driven auxiliaries the same as for c) applies however the practical upper value is 12

) For diesel driven gears more information on KA for mis-firing and normal driving is given in Appendix B

162 KAP The peak overload factor KAP is defined as the ratio between the temporary occasional peak overload torque and the nominal rated torque

For plants where high temporary occasional peak torques can occur (ie in excess of the above mentioned KA) the gearing (if nitrided) has to be checked with regard to static strength Unless otherwise specified the same safety factors as for infinite life apply

The scuffing safety is to be specially considered whereby the KA applies in connection with the bulk temperature and the KAP applies for the flash temperature calculation and should replace KA in the formulae in 431 432 and 441

KAP can be evaluated from the torsional impact vibration calculation (as required by the rules)

If the overloads have a duration corresponding to several revolutions of the shafts the scuffing safety has to be consid-ered on basis of this overload both with respect to bulk and flash temperature This applies to plants with ice class nota-tions (Baltic and polar) and to plants with prime movers which have high temporary overload capacity such as eg electric motors (provided the driven member can have a con-siderable increase in demand torque as eg azimuth thrusters during manoeuvring)

For plants without additional ice class notation KAP should normally not exceed 15

8 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

163 Frequent overloads For plants where high overloads or shock loads occur regu-larly the influence of this is to be considered by means of cumulative fatigue (see Appendix A)

17 Load Sharing Factor Kγ The load sharing factor Kγ accounts for the maldistribution of load in multiple-path transmissions (dual tandem epicyclic double helix etc) Kγ is defined as the ratio between the max load through an actual path and the evenly shared load

171 General method Kγ mainly depends on accuracy and flexibility of the branches (eg quill shaft planet support external forces etc) and should be considered on basis of measurements or of relevant analysis as eg

Kγ = δ+δ f

δ = total compliance of a branch under full load (assuming even load share) referred to gear mesh

f = minusminusminusminus+++ 23

22

21 fff where f1 f2 etc are the

main individual errors that may contribute to a maldis-tribution between the branches Eg tooth pitch errors planet carrier pitch errors bearing clearance influences etc Compensating effects should also be considered

For double helical gears

An external axial force Fext applied from sources outside the actual gearing (eg thrust via or from a tooth coupling) will cause a maldistribution of forces between the two helices Expressed by a load sharing factor the

βsdot

plusmn=γ tanFF1K

t

ext

If the direction of Fext is known the calculation should be carried out separately for each helix and with the tangential force corrected with the pertinent Kγ If the direction of Fext is unknown both combinations are to be calculated and the higher σH or σF to be used

172 Simplified method If no relevant analysis is available the following may apply

For epicyclic gears

Kγ = 32501 minus+ pln

where npl = number of planets ( gt3 )

For multistage gears with locked paths and gear stages sepa-rated by quill shafts (see figure below)

Figure 10 Locked paths gear

Kγ = ( )φ201+

where φ = quill shaft twist (degrees) under full load

18 Dynamic Factor Kv The dynamic factor Kv accounts for the internally generated dynamic loads

Kv is defined as the ratio between the maximum load that dynamically acts on the tooth flanks and the maximum ex-ternally applied load Ft KA Kγ

In the following 2 different methods (181 and 182) are described In case of controversy between the methods the next following is decisive ie the methods are listed with increasing priority

It is important to observe the limitations for the method in 181 In particular the influence of lateral stiffness of shafts is often underestimated and resonances occur at considerably lower speed than determined in 1811

However for low speed gears with vmiddotz1 lt 300 calculations may be omitted and the dynamic factor simplified to Kv=105

181 Single resonance method For a single stage gear Kv may be determined on basis of the relative proximity (or resonance ratio) N between actual speed n1 and the lowest resonance speed nE1

N = 1E

1

nn

Note that for epicyclic gears n is the relative speed ie the speed that multiplied with z gives the mesh frequency

1811 Determination of critical speed It is not advised to apply this method for multimesh gears for N gt 085 as the influence of higher modes has to be consid-ered see 182 In case of significant lateral shaft flexibility (eg overhung mounted bevel gears) the influence of cou-pled bending and torsional vibrations between pinion and wheel should be considered if N ge 075 see 182

nE1 = redmγc

1zπ

31030 sdot

where

cγ is the actual mesh stiffness per unit facewidth see 111

Classification Notes- No 412 9 May 2003

DET NORSKE VERITAS

For gears with inactive ends of the facewidth as eg due to high crowning or end relief such as often applied for bevel gears the use of cγ in connection with determination of natu-ral frequencies may need correction cγ is defined as stiffness per unit facewidth but when used in connection with the total mesh stiffness it is not as simple as cγmiddotb as only a part of the facewidth is active Such corrections are given in 111

mred is the reduced mass of the gear pair per unit facewidth and referred to the plane of contact

For a single gear stage where no significant inertias are closely connected to neither pinion nor wheel mred is calcu-lated as

mred = 21

21mm

mm+

The individual masses per unit facewidth are calculated as

m12 = 221b

21

)2d(b

I

where I is the polar moment of inertia (kgmm2)

The inertia of bevel gears may be approximated as discs with diameter equal the midface pitch diameter and width equal to b However if the shape of the pinion or wheel body differs much from this idealised cylinder the inertia should be cor-rected accordingly

For all kind of gears if a significant inertia (eg a clutch) is very rigidly connected to the pinion or wheel it should be added to that particular inertia (pinion or wheel) If there is a shaft piece between these inertias the torsional shaft stiffness alters the system into a 3-mass (or more) system This can be calculated as in 182 but also simplified as a 2-mass system calculated with only pinion and wheel masses

1812 Factors used for determination of Kv Non-dimensional gear accuracy dependent parameters

( )bKKF

yfcB

At

pptp

γ

minus=

( )bKKF

yFcBAt

ff

γ

α minus=

Non-dimensional tip relief parameter

bKKFcC

1BγAt

ak sdotsdot

sdotminus=

For gears of quality grade (ISO 1328) Q = 7 or coarser Bk = 1

For gears with Q le 6 and excessive tip relief Bk is limited to max 1

For gears (all quality grades) with tip relief of more than 2middotCeff (see 432) the reduction of αε has to be considered (see 443)

where

fpt = the single pitch deviation (ISO 1328) max of pinion or wheel

Fα = the total profile form deviation (ISO 1328) max of pinion or wheel (Note Fα is pt not available for bevel gears thus use Fα = fpt)

yp and yf = the respective running-in allowances and may be calculated similarly to yα in 112 ie the value of fpt is replaced by Fα for yf

cacute = the single tooth stiffness see 111

Ca = the amount of tip relief see 433 In case of different tip relief on pinion and wheel the value that results in the greater value of Bk is to be used If Ca is zero by design the value of running-in tip relief Cay (see 112) may be used in the above formula

1813 Kv in the subcritical range Cylindrical gears N le 085

Bevel gears N le 075

Kv = 1 + N K

K = Cv1 Bp + Cv2 Bf + Cv3 Bk

Cv1 accounts for the pitch error influence

Cv1 = 032

Cv2 accounts for profile error influence

Cv2 = 034 for γε le 2

Cv2 = 03ε

057

γ minus for γε gt 2

Cv3 accounts for the cyclic mesh stiffness variation

Cv3 = 023 for γε le 2

Cv3 = 156ε

0096

γ minus for γε gt 2

1814 Kv in the main resonance range Cylindrical gears 085 lt N le 115

Bevel gears 075 lt N le 125

Running in this range should preferably be avoided and is only allowed for high precision gears

Kv = 1 + Cv1 Bp + Cv2 Bf + Cv4 Bk

Cv4 accounts for the resonance condition with the cyclic mesh stiffness variation

Cv4 = 090 for γε le 2

Cv4 = 144ε

ε005057

γ

γ

minus

minus for γε gt 2

10 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

1815 Kv in the supercritical range Cylindrical gears N ge 15

Bevel gears N ge 15

Special care should be taken as to influence of higher vibra-tion modes andor influence of coupled bending (ie lateral shaft vibrations) and torsional vibrations between pinion and wheel These influences are not covered by the following approach

Kv = Cv5 Bp + Cv6 Bf + Cv7

Cv5 accounts for the pitch error influence

Cv5 = 047

Cv6 accounts for the profile error influence

Cv6 = 047 for εγ le 2

Cv6 = 174ε

012

γ minus for εγ gt 2

Cv7 relates the maximum externally applied tooth loading to the maximum tooth loading of ideal accurate gears operating in the supercritical speed sector when the circumferential vibration becomes very soft

Cv7 = 075 for εγ le 15

Cv7 = [ ] 8750)2(sin 0125 +minusβεπ for 15 lt εγ le 25

Cv7 = 10 for εγ gt 25

1816 Kv in the intermediate range Cylindrical gears 115 lt N lt 15

Bevel gears 125 lt N lt 15

Comments raised in 1814 and 1815 should be observed

Kv is determined by linear interpolation between Kv for N = 115 respectively 125 and N = 15 as

Cylindrical gears

( ) ( ) ( )[ ]51Nv151Nv51Nvv KK350

N51KK === minussdot

minus

+=

Bevel gears

( ) ( ) ( )[ ]51Nv251Nv51Nvv KK250

N51KK === minussdot

minus

+=

182 Multi-resonance method For high speed gear (vgt40 ms) for multimesh medium speed gears for gears with significant lateral shaft flexibility etc it is advised to determine Kv on basis of relevant dy-namic analysis

Incorporating lateral shaft compliance requires transforma-tion of even a simple pinion-wheel system into a lumped multi-mass system It is advised to incorporate all relevant inertias and torsional shaft stiffnesses into an equivalent (to pinion speed) system Thereby the mesh stiffness appears as an equivalent torsional stiffness

cγ b (db12)2 (Nmrad)

The natural frequencies are found by solving the set of dif-ferential equations (one equation per inertia) Note that for a gear put on a laterally flexible shaft the coupling bending-torsionals is arranged by introducing the gear mass and the lateral stiffness with its relation to the torsional displacement and torque in that shaft

Only the natural frequency (ies) having high relative dis-placement and relative torque through the actual pinion-wheel flexible element need(s) to be considered as critical frequency (ies)

Kv may be determined by means of the method mentioned in 181 thereby using N as the least favourable ratio (in case of more than one pinion-wheel dominated natural frequency) Ie the N-ratio that results in the highest Kv has to be consid-ered

The level of the dynamic factor may also be determined on basis of simulation technique using numeric time integration with relevant tooth stiffness variation and pitchprofile er-rors

19 Face Load Factors KHβ and KFβ The face load factors KHβ for contact stresses and for scuff-ing KFβ for tooth root stresses account for non-uniform load distribution across the facewidth

KHβ is defined as the ratio between the maximum load per unit facewidth and the mean load per unit facewidth

KFβ is defined as the ratio between the maximum tooth root stress per unit facewidth and the mean tooth root stress per unit facewidth The mean tooth root stress relates to the con-sidered facewidth b1 respectively b2

Note that facewidth in this context is the design facewidth b even if the ends are unloaded as often applies to eg bevel gears

The plane of contact is considered

191 Relations between KHβ and KFβ

KFβ = expβHK 2(hb)hb1

1exp++

=

where hb is the ratio tooth heightfacewidth The maximum of h1b1 and h2b2 is to be used but not higher than 13 For double helical gears use only the facewidth of one helix

If the tooth root facewidth (b1 or b2) is considerably wider than b the value of KFβ(1or2) is to be specially considered as it may even exceed KHβ

Eg in pinion-rack lifting systems for jack up rigs where b = b2 asymp mn and b1 asymp 3 mn the typical KHβ asymp KFβ2 asymp 1 and KFβ1 asymp 13

192 Measurement of face load factors Primarily

KFβ may be determined by a number of strain gauges distrib-uted over the facewidth Such strain gauges must be put in

Classification Notes- No 412 11 May 2003

DET NORSKE VERITAS

exactly the same position relative to the root fillet Relations in 191 apply for conversion to KHβ

Secondarily

KHβ may be evaluated by observed contact patterns on vari-ous defined load levels It is imperative that the various test loads are well defined Usually it is also necessary to evalu-ate the elastic deflections Some teeth at each 90 degrees are to be painted with a suitable lacquer Always consider the poorest of the contact patterns

After having run the gear for a suitable time at test load 1 (the lowest) observe the contact pattern with respect to ex-tension over the facewidth Evaluate that KHβ by means of the methods mentioned in this section Proceed in the same way for the next higher test load etc until there is a full face contact pattern From these data the initial mesh misalign-ment (ie without elastic deflections) can be found by ex-trapolation and then also the KHβ at design load can be found by calculation and extrapolation See example

Figure 11 Example of experimental determination of KHβ

It must be considered that inaccurate gears may accumulate a larger observed contact pattern than the actual single mesh to mesh contact patterns This is particularly important for lapped bevel gears Ground or hard metal hobbed bevel gears are assumed to present an accumulated contact pattern that is practically equal the actual single mesh to mesh con-tact patterns As a rough guidance the (observed) accumu-lated contact pattern of lapped bevel gears may be reduced by 10 in order to assess the single mesh to mesh contact pattern which is used in 199

193 Theoretical determination of KHβ The methods described in 193 to 198 may be used for cy-lindrical gears The principles may to some extent also be used for bevel gears but a more practical approach is given in 199

General For gears where the tooth contact pattern cannot be verified during assembly or under load all assumptions are to be well on the safe side

KHβ is to be determined in the plane of contact

The influence parameters considered in this method are

bull mean mesh stiffness cγ (see 111) (if necessary also variable stiffness over b)

bull mean unit load Fmb = Fbt KA Kγ Kvb (for double helical gears see 17 for use of Kγ)

bull misalignment fsh due to elastic deflections of shafts and gear bodies (both pinion and wheel)

bull misalignment fdefl due to elastic deflections of and work-ing positions in bearings

bull misalignment fbe due to bearing clearance tolerances bull misalignment fma due to manufacturing tolerances bull helix modifications as crowning end relief helix correc-

tion bull running in amount yβ (see 112)

In practice several other parameters such as centrifugal ex-pansion thermal expansion housing deflection etc contrib-ute to KHβ However these parameters are not taken into account unless in special cases when being considered as particularly important

When all or most of the am parameters are to be considered the most practical way to determine KHβ is by means of a graphical approach described in 1931

If cγ can be considered constant over the facewidth and no helix modifications apply KHβ can be determined analyti-cally as described in 1932

1931 Graphical method The graphical method utilises the superposition principle and is as follows

bull Calculate the mean mesh deflection Mδ as a function of

γm c and bF see 111

bull Draw a base line with length b and draw up a rectangu-lar with height δM (The area δM b is proportional to the transmitted force)

bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the areas above and below this zero line are equal

Figure 12 fsh balanced around zero line

bull Superimpose these ordinates of the fsh curve to the previ-ous load distribution curve (The area under this new load distribution curve is still δM b)

bull Calculate the bearing deflections andor working posi-tions in the bearings and evaluate the influence fdefl in the plane of contact This is a straight line and is bal-anced around a zero line as indicated in Fig 14 but with one distinct direction Superimpose these ordinates to the previous load distribution curve

12 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

bull The amount of crowning end relief or helix correction (defined in the plane of contact) is to be balanced around a zero line similarly to fsh

Figure 13 Crowning Cc balanced aound zero line

bull Superimpose these ordinates to the previous load distri-bution curve In case of high crowning etc as eg often applied to bevel gears the new load distribution curve may cross the base line (the real zero line) The result is areas with negative load that is not real as the load in those areas should be zero Thus corrective actions must be made but for practical reasons it may be postponed to after next operation

bull The amount of initial mesh misalignment bema ff + (defined in the plane of contact) is to be bal-

anced around a zero line If the direction of bema ff + is known (due to initial contact check) or if the direction of fbe is known due to design (eg overhang bevel pin-ion) this should be taken into account If direction un-known the influence of bema ff + in both directions as well as equal zero should be considered

Figure 14 fma+fbe in both directions balanced around zero line

Superimpose these ordinates to the previous load distri-bution curve This results in up to 3 different curves of which the one with the highest peak is to be chosen for further evaluation

bull If the chosen load distribution curve crosses the base line (ie mathematically negative load) the curve is to be corrected by adding the negative areas and dividing this with the active facewidth The (constant) ordinates of this rectangular correction area are to be subtracted from the positive part of the load distribution curve It is advisable to check that the area covered under this new load distribution curve is still equal δM b

bull If cγ cannot be considered as constant over b then cor-rect the ordinates of the load distribution curve with the local (on various positions over the facewidth) ratio between local mesh stiffness and average mesh stiffness cγ (average over the active facewidth only) Note that the result is to be a curve that covers the same area δM b as before

bull The influence of running in yβ is to be determined as in 112 whereby the value for Fβx is to be taken as twice the distance between the peak of the load distribution curve and δM

bull Determine

KHβ = M

β

δycurveofpeak minus

1932 Simplified analytical method for cylindrical gears The analytical approach is similar to 1931 but has a more limited application as cγ is assumed constant over the face-width and no helix modification applies

bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the area above and below this zero line are equal see Fig 12 The max positive ordinate is frac12∆fsh

bull Calculate the initial mesh alignment as Fβx= deflbemash ffff plusmnplusmnplusmn∆ The negative signs may only be used if this is justified andor verified by a contact pattern test Otherwise al-ways use positive signs If a negative sign is justified the value of Fβx is not to be taken less than the largest of each of these elements

bull Calculate the effective mesh misalignment as Fβy = Fβx - yβ (yβ see 112)

bull Determine

2KforF2

bFc1K βH

m

βγγH le+=β

or

2KforF

bFc2K Hβ

m

βγγH gt=β

where cγ as used here is the effective mesh stiffness see 111

194 Determination of fsh fsh is the mesh misalignment due to elastic deflections Usu-ally it is sufficient to consider the combined mesh deflection of the pinion body and shaft and the wheel shaft The calcu-lation is to be made in the plane of contact (of the considered gear mesh) and to consider all forces (incl axial) acting on the shafts Forces from other meshes can be parted into components parallel respectively vertical to the considered plane of contact Forces vertical to this plane of contact have no influence on fsh

It is advised to use following diameters for toothed elements

d + 2 x mn for bending and shear deflection

d + 2 mn (x ndash ha0 + 02) for torsional deflection

Usually fsh is calculated on basis of an evenly distributed load If the analysis of KHβ shows a considerable maldis-

Classification Notes- No 412 13 May 2003

DET NORSKE VERITAS

tribution in term of hard end contact or if it is known by other reasons that there exists a hard end contact the load should be correspondingly distributed when calculating fsh In fact the whole KHβ procedure can be used iteratively 2-3 iterations will be enough even for almost triangular load distributions

195 Determination of fdefl fdefl is the mesh misalignment in the plane of contact due to bearing deflections and working positions (housing deflec-tion may be included if determined)

First the journal working positions in the bearings are to be determined The influence of external moments and forces must be considered This is of special importance for twin pinion single output gears with all 3 shafts in one plane

For rolling bearings fdefl is further determined on basis of the elastic deflection of the bearings An elastic bearing deflec-tion depends on the bearing load and size and number of rolling elements Note that the bearing clearance tolerances are not included here

For fluid film bearings fdefl is further determined on basis of the lift and angular shift of the shafts due to lubrication oil film thickness Note that fbe takes into account the influence of the bearing clearance tolerance

When working positions bearing deflections and oil film lift are combined for all bearings the angular misalignment as projected into the plane of the contact is to be determined fdefl is this angular misalignment (radians) times the face-width

196 Determination of fbe fbe is the mesh misalignment in the plane of contact due to tolerances in bearing clearances In principle fbe and fdefl could be combined But as fdefl can be determined by analy-sis and has a distinct direction and fbe is dependent on toler-ances and in most cases has no distinct direction (ie + toler-ance) it is practicable to separate these two influences

Due to different bearing clearance tolerances in both pinion and wheel shafts the two shaft axis will have an angular mis-alignment in the plane of contact that is superimposed to the working positions determined in 195 fbe is the facewidth times this angular misalignment Note that fbe may have a distinct direction or be given as a + tolerance or a combina-tion of both For combination of + tolerance it is adviced to use

fbe= 22be

21be ff ++plusmn

fbe is particularly important for overhang designs for gears with widely different kinds of bearings on each side and when the bearings have wide tolerances on clearances In general it shall be possible to replace standard bearings with-out causing the real load distribution to exceed the design premises For slow speed gears with journal bearings the expected wear should also be considered

197 Determination of fma fma is the mesh misalignment due to manufacturing toler-ances (helix slope deviation) of pinion fHβ1 wheel fHβ2 and housing bore

For gear without specifically approved requirements to as-sembly control the value of fma is to be determined as

fma= 22Hβ

21Hβ ff +

For gears with specially approved assembly control the value of fma will depend on those specific requirements

198 Comments to various gear types For double helical gears KHβ is to be determined for both helices Usually an even load share between the helices can be assumed If not the calculation is to be made as de-scribed in 171

For planetary gears the free floating sun pinion suffers only twist no bending It must be noted that the total twist is the sum of the twist due to each mesh If the value of 1K γ ne this must be taken into account when calculating the total sun pinion twist (ie twist calculated with the force per mesh without Kγ and multiplied with the number of planets)

When planets are mounted on spherical bearings the mesh misalignments sun-planet respectively planet-annulus will be balanced Ie the misalignment will be the average between the two theoretical individual misalignments The faceload distribution on the flanks of the planets can take full advan-tage of this However as the sun and annulus mesh with several planets with possibly different lead errors the sun and annulus cannot obtain the above mentioned advantage to the full extent

199 Determination of KHβ for bevel gears If a theoretical approach similar to 193-198 is not docu-mented the following may be used

testeff

H Kb

b851851K sdot

minussdot=β

beff b represents the relative active facewidth (regarding lapped gears see 192 last part)

Higher values than beff b = 090 are normally not to be used in the formula

For dual directional gears it may be difficult to obtain a high beff b in both directions In that case the smaller beff b is to be used

Ktest represents the influence of the bearing arrangement shaft stiffness bearing stiffness housing stiffness etc on the faceload distribution and the verification thereof Expected variations in length- and height-wise tooth profile is also accounted for to some extent

a) Ktest = 1 For ground or hard metal hobbed gears with the specified contact pattern verified at full rating or at full torque slow turning at a condition representative for the thermal expansion at normal operation

14 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

It also applies when the bearing arrangementsupport has insignificant elastic deflections and thermal axial expansion However each initial mesh contact must be verified to be within acceptance criteria that are cali-brated against a type test at full load Reproduction of the gear tooth length- and height-wise profile must also be verified This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon)

b) Ktest = 1 + 04middot(beffbndash06) For designs with possible influence of thermal expansion in the axial direction of the pinion The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load

c) Ktest = 12 if mesh is only checked by toolmakerrsquos blue or by spin test contact For gears in this category beffb gt 085 is not to be used in the calculation

110 Transversal Load Distribution Factors KHα and KFα The transverse load distribution factors KHα for contact stresses and for scuffing KFα for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairs of teeth in mesh

The following relations may be used

Cylindrical gears

( )

minus+==

tH

αptγγHαFα F

byfc0409

2ε

KK

valid for 2εγ le

( ) ( )tH

αptγ

γ

γHαFα F

byfcε

1ε20409KK

minusminus+==

valid for 2ε γ gt

where

FtH = Ft KA Kγ Kv KHβ

cγ = See 111

γα = See 112

fpt = Maximum single pitch deviation (microm) of pinion or wheel or maximum total profile form devia-tion Fα of pinion or wheel if this is larger than the maximum single pitch deviation

Note In case of adequate equivalent tip relief adapted to the load half of the above mentioned fpt can be introduced A tip relief is considered adequate when the average of Ca1 and Ca2 is within plusmn40 of the value of Ceff in 432

Limitations of KHα and KFα

If the calculated values for

KFα = KHα lt 1 use KFα = KHα = 10

If the calculated value of KHα gt 2εα

γ

Zε

ε use KHα = 2

εα

γ

Zε

ε

If the calculated value of KFα gt εα

γ

Yεε

use KFα = εα

γ

Yεε

where Yε = αnε

075025+ (for εαn see 331c)

Bevel gears

For ground or hard metal hobbed gears KFα = KHα = 1

For lapped gears KFα = KHα = 11

111 Tooth Stiffness Constants cacute and cγ The tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having 1 mm facewidth by an amount of 1 microm in the plane of contact cacute is the maximum stiffness of a single pair of teeth cγ is the mean value of the mesh stiffness in a transverse plane (brief term mesh stiffness)

Both valid for high unit load (Unit load = Ft middot KA middot Kγb)

Cylindrical gears The real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bending stiff-nesses For high unit loads the Hertzian stiffness has little importance and can be disregarded This approach is on the safe side for determination of KHβ and KHα However for moderate or low loads Kv may be underestimated due to determination of a too high resonance speed

The linear approach is described in A

An optional approach for inclusion of the non-linear stiffness is described in B

A The linear approach

BRCCq

βcos08acutec =

and

( )025ε075cc αγ +prime=

where

( )[ ]n02a01a

B α2000212

hh12051C minusminus

+minus+=

12n1n

x000635z

025791z

015551004723q minus++=

12

2n

22

1n

1 x005290z

x024188x000193z

x011654+minusminusminus

+ 000182 x22

Classification Notes- No 412 15 May 2003

DET NORSKE VERITAS

(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as

( )( )nR m5s

sR e5

bbln1C +=

where

bs = thickness of a central web

sR = average thickness of rim (net value from tooth root to inside of rim)

The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered

Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ

B The non-linear approach

In the following an example is given on how to consider the non-linearity

The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm

With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as

( )10δKbF

minus= for 500bFgt

minus=

500Fb10δK

bF for 500

bFlt

with γAt KK

bF

bF

sdotsdot= etc (Nmm) ie unit load incorporat-

ing the relevant factors as

KA middot Kγ for determination of Kv

KA middot Kγ middot Kv for determination of KHβ

KA middot Kγ middot Kv middot KHβ for determination of KHα

δ = mesh deflection (microm)

K = applicable stiffness (c or cγ)

Use of stiffnesses for KV KHβ and KHα

For calculation of Kv and KHα the stiffness is calculated as follows

When Fb lt 500

the stiffness is determined as δ∆

∆ bF

where the increment is chosen as eg ∆ Fb = 10 and thus

50010Fb10

K10Fb∆δ +

++

=

When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly

or an equivalent stiffness determined as δsdotb

F

Bevel gears

In lack of more detailed relationship between stiffness and geometry the following may be used

b085

b13cacute eff=

b085b

16c effγ =

beff not to be used in excess of 085 b in these formulae

Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)

112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements

yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload

Cay is defined as the running-in amount that compensates for lack of tip relief

The following relations may be used

For not surface hardened steel

ptHlim

α fσ160y =

yβ = βxlimH

f320σ

with the following upper limits

V lt 5 ms 5-10 ms gt 10 ms

yα max none

limH

12800σ limH

6400σ

yβ max none

limH

25600σ limH

12800σ

For surface hardened steel

yα = 0075 fpt but not more than 3 for any speed

yβ = 015 Fβx but not more than 6 for any speed

16 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

For all kinds of steel

5145189718

1C2

limHay +

minusσ

=

When pinion and wheel material differ the following ap-plies

bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv

bull Use ( )2β1ββ yy21y += in the calculation of KHβ

bull Use ( )2ay1aya CC21C += in the calculation of Kv

bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing

calculation if no design tip relief is foreseen

Classification Notes- No 412 17 May 2003

DET NORSKE VERITAS

2 Calculation of Surface Durability

21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213

Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted

Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213

For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength

For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied

22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater

Calculation of surface durability for helical gears is based on the contact stress at the pitch point

For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies

Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact

Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses

The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses

The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel

221 Contact stress Cylindrical gears

( )HαHβvγA

1

tβεEHDBH KKKKK

bud1uF

ZZZZZσ+

=

where

ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)

ZH = Zone factor for pitch point (see 231)

ZE = Elasticity factor (see 24)

Zε = Contact ratio factor (see 25)

Zβ = Helix angle factor (see 26)

Ft KA Kγ Kv KHβ KHα see 15 ndash 110

d1 b u see 12 ndash 15

Bevel gears

( )HαHβvγA

v1v

vmtKEMH KKKKK

bud1uF

ZZZ105σ+

sdot=

where

105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)

ZE KA etc see above

ZM = mid-zone factor see 233

ZK = bevel gear factor see 27

Fmt dv1 uv see 12 ndash 15

It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks

222 Permissible contact stress

XWRvLH

NHlimHP ZZZZZ

SZσ

σ =

where

σH lim = Endurance limit for contact stresses (see 28)

ZN = Life factor for contact stresses (see 29)

SH = Required safety factor according to the rules

ZLZvZR = Oil film influence factors (see 210)

ZW = Work hardening factor (see 211)

ZX = Size factor (see 212)

18 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

23 Zone Factors ZH ZBD and ZM

231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder

wtt2

wtbH sinααcos

cosαcosβ2Z =

232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel

For εβ ge 1 ZBD = 1

For internal gears ZD = 1

For εβ = 0 (spur gears)

( )

π

minusεminusminus

π

minusminus

α=

α2

2

2b

2a

1

2

1b

1a

wtB

z211

dd

z21

dd

tanZ

( )

π

minusεminusminus

π

minusminus

α=

α1

2

1b

1a

2

2

2b

2a

wtD

z211

dd

z21

dd

tanZ

If ZB lt 1 use ZB = 1

If ZD lt 1 use ZD = 1

For 0 lt εβ lt 1

ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)

233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears

εminusminus

εminusminus

αβ=

αα btm2

2vb2

2vabtm2

1vb2val

2v1vvtbmM

pddpdd

ddtancos2Z

This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint

234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel

24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses

For steel against steel ZE = 1898

25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses

αε1Z =ε for 1εβ ge

( )α

ββ

αε ε

εε1

3ε4

Z +minusminus

= for εβ lt 1

26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability

cosβZβ =

27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply

The following may be used ZK = 080

28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ

σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting

For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply

For this purpose pitting is defined by

bull for not surface hardened gears pitted area ge 2 of total active flank area

bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given

material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies

The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as

Classification Notes- No 412 19 May 2003

DET NORSKE VERITAS

the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99

σHlim σH105 σH10

3

Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade

1650 1500

2500 2400

3100 3100

Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)

1000

13 σHlim

13 σHlim

Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV

Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV

Carbon steel 15 HV + 250 16 σHlim 16 σHlim

These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15

29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply

If this is not documented by approved fatigue tests the fol-lowing method may be used

For all steels except nitrided

7L 105N sdotge ZN = 1 or

01570

L

7

N N105Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 5middot107 510N

logZ037

L

7

N N105Z

sdot=

NL = 105 WXRVLHlim

WstX10H1010NN ZZZZZσ

ZZσZZ

55

5=

103 lt NL lt 105 )Z(Zlog05

L

5

10NN

5N103N10

5N10ZZ

==

3L 10N le

WXRVLlimH

Wst10X10H10NN ZZZZZ

ZZZZ

33

3σ

σ==

(but not less than ZN105)

For nitrided steels

6L 102N sdotge ZN = 1 or

00980

L

6

N N102Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 2middot106 510N

Zlog76860

L

6

N N102Z

sdot=

5L 10N le

XWRVL

10XWst10NN ZZZZZ

ZZ13ZZ

55 ==

Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles

210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity

The following methods may be applied in connection with the endurance limit

20 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Surface hardened steels Not surface hardened steels

ZL ( )24013421

360910ν+

+

( )24013421

680830ν+

+

ZV ( )v3280

140930+

+ ( )v3280300850

++

ZR

080

ZrelR3

150

ZrelR3

where

ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits

For values of ν40 gt 500 use ν40 = 500

RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm

RZrel

= ( ) 31

cZ2Z1 ρ

10RR50

+

RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)

For 5

L 10N le ZL ZV ZR = 10

211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface

The following approximation may be used for the endurance limit

Surface hardened steel against not surface hardened steel 150

ZeqW R

31700

130HB21Z

minus

minus=

where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130

RZeq = equivalent roughness

033

c40

066

ZS

ZHZHZEQ ρvν

15000RRRR

=

If RZeq gt 16 then use RZeq = 16

If RZeq lt 15 then use RZeq = 15

where

RZH = surface roughness of the hard member before run in

RZS = surface roughness of the soft member before run in

ν40 = see 210

If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem

Through hardened pinion against softer wheel

( )

minussdotminus+= 000829

HBHB0008981u1Z

2

1W

For 21HBHB

2

1 le use ZW = 1

For 71HBHB

2

1 gt use 17HBHB

2

1 =

For u gt 20 use u = 20

For static strength (lt 105 cycles)

Surface hardened against not surface hardened

ZWst = 105

Through hardened pinion against softer wheel

ZWst = 1

212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality

ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213

213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented

The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent

Classification Notes- No 412 21 May 2003

DET NORSKE VERITAS

stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically

The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence

The real Hertzian stresses σHR are determined as

For helical gears with εβ gt 1

σHR = σH

For helical gears with εβ lt 1 and spur gears

ε

α

ββ ε

ε+εminus

sdotσ=σZ

1

HHR

For bevel gears

KHHR Z

1σσ sdot=

The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)

The coordinates tz and HV are to be compared with the de-sign specification such as

bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400

and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)

In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods

For high cycle fatigue (gt3 106 cycles) the following applies

sdot+

minussdotsdotsdot= o90

05azt

05azt

cosSσ04HV

H

HHHR

applicable to 05a

zt

Hge

For 05at

H

z lt the value for 05at

H

z = applies

56300ρSσ12a cHHR

Hsdotsdot

sdot=

Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength

If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows

reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue

For static strength (lt103 cycles) the following applies

sdot+

minussdotsdotsdot= o90

07at

06a

zt

cosSσ019HV

Hst

z

HstHHR

applicable to 06at

Hst

z ge

56300ρSσa cHHR

Hstsdotsdot

=

In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies

For limited life fatigue (103 lt cycles lt 3106 )

For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance

The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram

minussdotminus

= sdot logN3477

logSlogSlogS

310H6103HHN

36 10H103H Slog86281Slog86280 sdot+sdot sdot

22 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3 Calculation of Tooth Strength

31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding

For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36

For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning

Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα

In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314

A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired

It should be noted that this part 3 does not cover fractures caused by

bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining

Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears

Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied

32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level

321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance

Normally the stresses for pinion and wheel are calculated as

Cylindrical gears

FαFβvγAβSFn

tF K K K K K Y Y Y

m bFσ =

where

YF = Tooth form factor (see 33)

YS = Stress correction factor (see 34)

Yβ = Helix angle factor (see 36)

Ft KA Kγ Kv KFβ KFα see 15 ndash 110

b see 13

Bevel gears

FαFβvγASaFamn

mtF K K K K K Y Y Y

m bFσ ε=

where

YFa = Tooth form factor see 33

YSa = Stress correction factor see 34

Yε = Contact ratio factor see 35

Fmt KA etc see 15 ndash 110

b see 13

322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is

CXRrelTrelTF

NMFEFP YYYY

SYY

δσ

=σ

Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN

where

σFE = Local tooth root bending endurance limit of reference test gear (see 37)

YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)

YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)

SF = Required safety factor according to the rules

YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)

YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)

Classification Notes- No 412 23 May 2003

DET NORSKE VERITAS

YX = Size factor (see 312)

YC = Case depth factor (see 313)

33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress

YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears

YFa applies to load application at the tooth tip and is used for bevel gears

Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears

Figure 31 External tooth in normal section

Figure 32 Internal tooth in normal section

Definitions

n

2

n

Fn

enFn

Fe

F

α cosms

α cosmh6

Y

=

n

2

n

Fn

anFn

Fa

Fa

α cosms

α cosmh6

Y

=

In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth

YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)

The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing

mn with mnm

zn with zvn

αt with αvt

β with βm

with undercut without undercut

Fig 33 Dimensions and basic rack profile of the teeth (finished profile)

Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless

331 Determination of parameters

( )n

n

prnfPnfP m

α cossαsin 1ρ

αtan h4πE

minusminusminusminus=

For external gears fPfP ρρ =

For internal gears ( )0z

195fPfP0

fPfP 10363156ρhxρρ

sdotminus+

+=

where

z0 = number of teeth of pinion cutter

x0 = addendum modification coefficient of pinion cutter

hfP = addendum of pinion cutter

ρfP = tip radius of pinion cutter

xhρG fpfp +minus=

24 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

τmE

2π

z2H

nnminus

minus=

with

3πτ = for external gears

6πτ = for internal gears

Htan zG2

nminusϑ=ϑ

(to be solved iteratively suitable start value6π

=ϑ for exter-

nal gears and 3π for internal gears)

a) Tooth root chord sFn For external gears

minus

ϑ+

ϑminus= ρ

cosG3

3πsinz

ms

fpnn

Fn

For bevel gears with a tooth thickness modification

xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn

For internal gears

minus

ϑ+

ϑminus= ρ

cosG

6πsinz

ms

fPnn

Fn

b) Root fillet radius ρF at 30ordm tangent

( )G2coszcosG2ρ

mρ

2n

2

fpn

F

minusϑϑ+=

c) Determination of bending moment arm hF dn = zn mn

b2α

αn βcosε

ε =

dan = dn + 2 ha

pbn = π mn cos αn

dbn = dn cos αn

( )4

d1εpzz

2dd

zz2d

2bn

2

αnbn

2bn

2an

en +

minusminus

minus=

en

bnen d

dcos arcα =

ennnn

e α invα invα x tan 22π

z1

minus+

+=γ

αFen = αen ndash γe

For external gears

( )

minus=

n

enFenee

n

Fe

mdαtan sin γ γcos

21

mh

]ρcos

G3πcosz fpn +

ϑminus

ϑminusminus

For internal gears

( ) minus

sdotsdotminus=

n

enFenee

n

Fe

mdα tanγ sinγ cos

21

mh

minus

ϑminus

ϑminussdot ρ

cosG3

6πcosz fPn

332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as

250ε205for 0666ε2366Y αnαnDT leleminus=

205εfor 10Y αnDT lt=

34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section

Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm

YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip

YS can be determined as follows

( )

+

+=L23121

1

sS qL01312Y

where Fe

Fn

hsL = and

F

Fns ρ2

sq = (see 33)

YSa can be calculated by replacing hFe with hFa in the above formulae

Note a) Range of validity 1 lt qs lt 8

In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered

b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with

Classification Notes- No 412 25 May 2003

DET NORSKE VERITAS

g

g

ρt

0613

13

minus

where

tg = depth of the grinding notch

ρg = radius of the grinding notch

c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles

35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears

The following may be used

Yε = 0625

36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank

The following may be used (β input in degrees)

Yβ = 1 ndash εβ β120

When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula

However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)

37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM

σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision

If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules

σFE

Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)

bull of specially approved high grade

1050

bull of normal grade

minus CrNiMo steels with approved process

1000

minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850

Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

840

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

720

Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)

07 HV + 300

Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)

025 σB + 125

Alloyed quenched and tempered steel 04 σB + 200

Carbon steel 025 σB + 250

Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40

1) These values are valid for a root radius

bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)

bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)

bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for

6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked

2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)

26 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears

YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0

YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst

For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316

The following method may be used within a stress ratio ndash12 lt R lt 05

381 For idlers planets and PTO with ice class

M1M1R1

1Yor Y MstM

+minus

minus=

where

R = stress ratio = min stress divided by max stress

For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12

For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as

branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus

For a power take off (PTO) with ice class see 161 c

M considers the mean stress influence on the endurance (or static) strength amplitudes

M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress

Following M values may be used

Endurance limit

Static strength

Case hardened 08 ndash 015 Ys 1) 07

If shot peened 04 06

Nitrided 03 03

Induction or flame hardened

04 06

Not surface hardened steel 03 05

Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M

The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches

382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106

For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers

For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used

383 For gears with shrinkage stresses and unidirectional load For endurance strength

FE

fitM σ

σM1

M21Y+

minus=

σFE is the endurance limit for R = 0

For static strength YMst = 1 and σfit accounted for in 39b

σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor

n

Ffit m

ρ215scf minus=

384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is

( ) ( ) FE

fitM σ

σR1M1

M2

M1M1R1

1Yminussdot+

minus

+minus

minus=

Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12

For static strength

M1M1R1

1YMst

+minus

minus=

The effect of σfit is accounted for in 39 b

Classification Notes- No 412 27 May 2003

DET NORSKE VERITAS

385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed

F

y

Sσ225

for not surface hardened fillets

FSHV5 for surface hardened fillets

39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles

Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN

If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used

Determination of the σ ndash N ndash curve

a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF

b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ

FstS1

FPstσ minussdotsdotsdot=

where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability

σFst

Alloyed case hardened steel 1) 2300

Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

1250

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

1050

Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)

18 HV + 800

Steel with not surface hardened fillets the smaller value of 2)

18 σB or 225 σy

1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet

c) Calculate YN as

NL gt 3middot106

YN = 1 or 001

L

6

N N103Y

sdot= ie Yn = 092 for 1010

The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2

103ltNLlt3middot106exp

L

6

N N103Y

sdot=

cycles6103forσcycles310forσlog02876exp

FP

FPst

sdot=

NL lt 103 cycles6103forσcycles310forσ

NYFP

FPst

sdot=

or simply use σFPst as mentioned in b) directly

Guidance on number of load cycles NL for various applica-tions

bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)

bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied

28 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage

YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit

The following method may be used

For endurance limit

for not surface hardened fillets

( )

024

s024

relT σ103133q21σ1012201351

Ysdotsdotminus

+sdotsdotminus+= minus

minus

δ

for all surface hardened fillets except nitrided

( )

106q21002451

Y srelT

++=δ

for nitrided fillets

( )

1347q2101421

Y srelT

++=δ

For static strength

for not surface hardened fillets1)

( )( )( )025

02

02502s

relTst σ3000821σ300 1Y0821Y

+

minus+=δ

for surface hardened fillets except nitrided

YδrelTst = 044 YS + 012

for nitrided fillets

YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not

exceed the yield point and thereby alter the residual stress level See also 39b footnote 2

311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness

YRrelT differs for endurance limit and static strength

The following method may be used

For endurance limit

YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels

YδrelT = 43 ndash 326 (Ry + 1)0005

for nitrided steels

For static strength

YRrelTst = 1 for all Ry and all materials

For a fillet without any longitudinal machining trace Ry asymp Rz

312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength

The following may be used

For endurance limit

YX = 1 for mn le 5 generally

YX = 103 ndash 0006 mn for 5 lt mn lt 30

YX = 085 for mn ge 30

for not surface hard-ened steels

YX = 105 ndash 001 mn for 5 lt mn ge 25

YX = 08 for mn ge 25

for surface hardened steels

For static strength

YXst = 1 for all mn and all materials

313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength

YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength

In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength

The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses

The following simplified method for YC may be used

YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)

YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10

For endurance limit

+

+=nFFE

C m02ρt31

σconstY

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

Classification Notes- No 412 7 May 2003

DET NORSKE VERITAS

16 Application Factors KA and KAP The application factor KA accounts for dynamic overloads from sources external to the gearing

It is distinguished between the influence of repetitive cyclic torques KA (161) and the influence of temporary occasional peak torques KAP (162)

Calculations are always to be made with KA In certain cases additional calculations with KAP may be necessary

For gears with a defined load spectrum the calculation with a KA may be replaced by a fatigue damage calculation as given in Appendix A

161 KA For gears designed for long or infinite life at nominal rated torque KA is defined as the ratio between the maximum re-petitive cyclic torque applied to the gear set and nominal rated torque

This definition is suitable for main propulsion gears and most of the auxiliary gears

KA can be determined by measurements or system analysis or may be ruled by conventions (ice classes) (For the pur-pose of a preliminary (but not binding) calculation before KA is determined it is advised to apply either the max values mentioned below or values known from similar plants)

a) For main propulsion gears KA can be taken from the (mandatory) torsional vibration analysis thereby con-sidering all permissible driving conditions) Unless specially agreed the rules do not allow KA in excess of 135 for diesel propulsion) With turbine or electric propulsion KA would normally not exceed 12 However special attention should be given to thrusters that are arranged in such a way that heavy vessel movements andor manoeuvring can cause severe load fluctuations This means eg thrusters positioned far from the rolling axis of vessels that could be susceptible to rolling If leading to propeller air suction the condi-tions may be even worse The above mentioned movements or manoeuvring will result in increased propeller excitation If the thruster is driven by a diesel engine the engine mean torque is limited to 100 However thrusters driven by electric motors can suffer temporary mean torque much above 100 unless a suitable load control system (limiting available e-motor torque) is provided

b) For main propulsion gears with ice class notation (see Rules Pt5 Ch1 Sec J500) KA ice has to be taken as the higher value of the applicable (rule defined) ice shock torque referred to nominal rated torque and the value under a) The Baltic ice class notations refer to a few millions ice shock loads Thus the life factors may be put YN=1 and ZN=12 (except for nitrided gears where ZN = 1 applies) Additionally the calculations with the normal KA (no ice class) are to fulfil the normal requirements For polar ice class notations KA ice applies to all criteria and for long or infinite life

c) For a power take off (PTO) branch from a main propul-sion gear with ice class ice shocks result in negative torques It is assumed that the PTO branch is unloaded when the ice shock load occurs The influence of these reverse shock loads may be taken into account as follows The negative torque (reversed load) expressed by means of an application factor based on rated forward load (T or Ft) is KAreverse = KA ice ndash1 (the minus 1 be-cause no mean torque assumed) KAice to be calculated as in the ice class rules This KAreverse should be used for back flank considera-tions such as pitting and scuffing The influence on tooth bending strength (forward di-rection) may be simplified by using the factor YM = 1 minus 03 middot KAreverse KA

d) For diesel driven auxiliaries KA can be taken from the torsional vibration analysis if available For units where no vibration analysis is required (lt 200 kW) or available it is advised to apply KA as the upper allow-able value 135)

e) For turbine or electro driven auxiliaries the same as for c) applies however the practical upper value is 12

) For diesel driven gears more information on KA for mis-firing and normal driving is given in Appendix B

162 KAP The peak overload factor KAP is defined as the ratio between the temporary occasional peak overload torque and the nominal rated torque

For plants where high temporary occasional peak torques can occur (ie in excess of the above mentioned KA) the gearing (if nitrided) has to be checked with regard to static strength Unless otherwise specified the same safety factors as for infinite life apply

The scuffing safety is to be specially considered whereby the KA applies in connection with the bulk temperature and the KAP applies for the flash temperature calculation and should replace KA in the formulae in 431 432 and 441

KAP can be evaluated from the torsional impact vibration calculation (as required by the rules)

If the overloads have a duration corresponding to several revolutions of the shafts the scuffing safety has to be consid-ered on basis of this overload both with respect to bulk and flash temperature This applies to plants with ice class nota-tions (Baltic and polar) and to plants with prime movers which have high temporary overload capacity such as eg electric motors (provided the driven member can have a con-siderable increase in demand torque as eg azimuth thrusters during manoeuvring)

For plants without additional ice class notation KAP should normally not exceed 15

8 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

163 Frequent overloads For plants where high overloads or shock loads occur regu-larly the influence of this is to be considered by means of cumulative fatigue (see Appendix A)

17 Load Sharing Factor Kγ The load sharing factor Kγ accounts for the maldistribution of load in multiple-path transmissions (dual tandem epicyclic double helix etc) Kγ is defined as the ratio between the max load through an actual path and the evenly shared load

171 General method Kγ mainly depends on accuracy and flexibility of the branches (eg quill shaft planet support external forces etc) and should be considered on basis of measurements or of relevant analysis as eg

Kγ = δ+δ f

δ = total compliance of a branch under full load (assuming even load share) referred to gear mesh

f = minusminusminusminus+++ 23

22

21 fff where f1 f2 etc are the

main individual errors that may contribute to a maldis-tribution between the branches Eg tooth pitch errors planet carrier pitch errors bearing clearance influences etc Compensating effects should also be considered

For double helical gears

An external axial force Fext applied from sources outside the actual gearing (eg thrust via or from a tooth coupling) will cause a maldistribution of forces between the two helices Expressed by a load sharing factor the

βsdot

plusmn=γ tanFF1K

t

ext

If the direction of Fext is known the calculation should be carried out separately for each helix and with the tangential force corrected with the pertinent Kγ If the direction of Fext is unknown both combinations are to be calculated and the higher σH or σF to be used

172 Simplified method If no relevant analysis is available the following may apply

For epicyclic gears

Kγ = 32501 minus+ pln

where npl = number of planets ( gt3 )

For multistage gears with locked paths and gear stages sepa-rated by quill shafts (see figure below)

Figure 10 Locked paths gear

Kγ = ( )φ201+

where φ = quill shaft twist (degrees) under full load

18 Dynamic Factor Kv The dynamic factor Kv accounts for the internally generated dynamic loads

Kv is defined as the ratio between the maximum load that dynamically acts on the tooth flanks and the maximum ex-ternally applied load Ft KA Kγ

In the following 2 different methods (181 and 182) are described In case of controversy between the methods the next following is decisive ie the methods are listed with increasing priority

It is important to observe the limitations for the method in 181 In particular the influence of lateral stiffness of shafts is often underestimated and resonances occur at considerably lower speed than determined in 1811

However for low speed gears with vmiddotz1 lt 300 calculations may be omitted and the dynamic factor simplified to Kv=105

181 Single resonance method For a single stage gear Kv may be determined on basis of the relative proximity (or resonance ratio) N between actual speed n1 and the lowest resonance speed nE1

N = 1E

1

nn

Note that for epicyclic gears n is the relative speed ie the speed that multiplied with z gives the mesh frequency

1811 Determination of critical speed It is not advised to apply this method for multimesh gears for N gt 085 as the influence of higher modes has to be consid-ered see 182 In case of significant lateral shaft flexibility (eg overhung mounted bevel gears) the influence of cou-pled bending and torsional vibrations between pinion and wheel should be considered if N ge 075 see 182

nE1 = redmγc

1zπ

31030 sdot

where

cγ is the actual mesh stiffness per unit facewidth see 111

Classification Notes- No 412 9 May 2003

DET NORSKE VERITAS

For gears with inactive ends of the facewidth as eg due to high crowning or end relief such as often applied for bevel gears the use of cγ in connection with determination of natu-ral frequencies may need correction cγ is defined as stiffness per unit facewidth but when used in connection with the total mesh stiffness it is not as simple as cγmiddotb as only a part of the facewidth is active Such corrections are given in 111

mred is the reduced mass of the gear pair per unit facewidth and referred to the plane of contact

For a single gear stage where no significant inertias are closely connected to neither pinion nor wheel mred is calcu-lated as

mred = 21

21mm

mm+

The individual masses per unit facewidth are calculated as

m12 = 221b

21

)2d(b

I

where I is the polar moment of inertia (kgmm2)

The inertia of bevel gears may be approximated as discs with diameter equal the midface pitch diameter and width equal to b However if the shape of the pinion or wheel body differs much from this idealised cylinder the inertia should be cor-rected accordingly

For all kind of gears if a significant inertia (eg a clutch) is very rigidly connected to the pinion or wheel it should be added to that particular inertia (pinion or wheel) If there is a shaft piece between these inertias the torsional shaft stiffness alters the system into a 3-mass (or more) system This can be calculated as in 182 but also simplified as a 2-mass system calculated with only pinion and wheel masses

1812 Factors used for determination of Kv Non-dimensional gear accuracy dependent parameters

( )bKKF

yfcB

At

pptp

γ

minus=

( )bKKF

yFcBAt

ff

γ

α minus=

Non-dimensional tip relief parameter

bKKFcC

1BγAt

ak sdotsdot

sdotminus=

For gears of quality grade (ISO 1328) Q = 7 or coarser Bk = 1

For gears with Q le 6 and excessive tip relief Bk is limited to max 1

For gears (all quality grades) with tip relief of more than 2middotCeff (see 432) the reduction of αε has to be considered (see 443)

where

fpt = the single pitch deviation (ISO 1328) max of pinion or wheel

Fα = the total profile form deviation (ISO 1328) max of pinion or wheel (Note Fα is pt not available for bevel gears thus use Fα = fpt)

yp and yf = the respective running-in allowances and may be calculated similarly to yα in 112 ie the value of fpt is replaced by Fα for yf

cacute = the single tooth stiffness see 111

Ca = the amount of tip relief see 433 In case of different tip relief on pinion and wheel the value that results in the greater value of Bk is to be used If Ca is zero by design the value of running-in tip relief Cay (see 112) may be used in the above formula

1813 Kv in the subcritical range Cylindrical gears N le 085

Bevel gears N le 075

Kv = 1 + N K

K = Cv1 Bp + Cv2 Bf + Cv3 Bk

Cv1 accounts for the pitch error influence

Cv1 = 032

Cv2 accounts for profile error influence

Cv2 = 034 for γε le 2

Cv2 = 03ε

057

γ minus for γε gt 2

Cv3 accounts for the cyclic mesh stiffness variation

Cv3 = 023 for γε le 2

Cv3 = 156ε

0096

γ minus for γε gt 2

1814 Kv in the main resonance range Cylindrical gears 085 lt N le 115

Bevel gears 075 lt N le 125

Running in this range should preferably be avoided and is only allowed for high precision gears

Kv = 1 + Cv1 Bp + Cv2 Bf + Cv4 Bk

Cv4 accounts for the resonance condition with the cyclic mesh stiffness variation

Cv4 = 090 for γε le 2

Cv4 = 144ε

ε005057

γ

γ

minus

minus for γε gt 2

10 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

1815 Kv in the supercritical range Cylindrical gears N ge 15

Bevel gears N ge 15

Special care should be taken as to influence of higher vibra-tion modes andor influence of coupled bending (ie lateral shaft vibrations) and torsional vibrations between pinion and wheel These influences are not covered by the following approach

Kv = Cv5 Bp + Cv6 Bf + Cv7

Cv5 accounts for the pitch error influence

Cv5 = 047

Cv6 accounts for the profile error influence

Cv6 = 047 for εγ le 2

Cv6 = 174ε

012

γ minus for εγ gt 2

Cv7 relates the maximum externally applied tooth loading to the maximum tooth loading of ideal accurate gears operating in the supercritical speed sector when the circumferential vibration becomes very soft

Cv7 = 075 for εγ le 15

Cv7 = [ ] 8750)2(sin 0125 +minusβεπ for 15 lt εγ le 25

Cv7 = 10 for εγ gt 25

1816 Kv in the intermediate range Cylindrical gears 115 lt N lt 15

Bevel gears 125 lt N lt 15

Comments raised in 1814 and 1815 should be observed

Kv is determined by linear interpolation between Kv for N = 115 respectively 125 and N = 15 as

Cylindrical gears

( ) ( ) ( )[ ]51Nv151Nv51Nvv KK350

N51KK === minussdot

minus

+=

Bevel gears

( ) ( ) ( )[ ]51Nv251Nv51Nvv KK250

N51KK === minussdot

minus

+=

182 Multi-resonance method For high speed gear (vgt40 ms) for multimesh medium speed gears for gears with significant lateral shaft flexibility etc it is advised to determine Kv on basis of relevant dy-namic analysis

Incorporating lateral shaft compliance requires transforma-tion of even a simple pinion-wheel system into a lumped multi-mass system It is advised to incorporate all relevant inertias and torsional shaft stiffnesses into an equivalent (to pinion speed) system Thereby the mesh stiffness appears as an equivalent torsional stiffness

cγ b (db12)2 (Nmrad)

The natural frequencies are found by solving the set of dif-ferential equations (one equation per inertia) Note that for a gear put on a laterally flexible shaft the coupling bending-torsionals is arranged by introducing the gear mass and the lateral stiffness with its relation to the torsional displacement and torque in that shaft

Only the natural frequency (ies) having high relative dis-placement and relative torque through the actual pinion-wheel flexible element need(s) to be considered as critical frequency (ies)

Kv may be determined by means of the method mentioned in 181 thereby using N as the least favourable ratio (in case of more than one pinion-wheel dominated natural frequency) Ie the N-ratio that results in the highest Kv has to be consid-ered

The level of the dynamic factor may also be determined on basis of simulation technique using numeric time integration with relevant tooth stiffness variation and pitchprofile er-rors

19 Face Load Factors KHβ and KFβ The face load factors KHβ for contact stresses and for scuff-ing KFβ for tooth root stresses account for non-uniform load distribution across the facewidth

KHβ is defined as the ratio between the maximum load per unit facewidth and the mean load per unit facewidth

KFβ is defined as the ratio between the maximum tooth root stress per unit facewidth and the mean tooth root stress per unit facewidth The mean tooth root stress relates to the con-sidered facewidth b1 respectively b2

Note that facewidth in this context is the design facewidth b even if the ends are unloaded as often applies to eg bevel gears

The plane of contact is considered

191 Relations between KHβ and KFβ

KFβ = expβHK 2(hb)hb1

1exp++

=

where hb is the ratio tooth heightfacewidth The maximum of h1b1 and h2b2 is to be used but not higher than 13 For double helical gears use only the facewidth of one helix

If the tooth root facewidth (b1 or b2) is considerably wider than b the value of KFβ(1or2) is to be specially considered as it may even exceed KHβ

Eg in pinion-rack lifting systems for jack up rigs where b = b2 asymp mn and b1 asymp 3 mn the typical KHβ asymp KFβ2 asymp 1 and KFβ1 asymp 13

192 Measurement of face load factors Primarily

KFβ may be determined by a number of strain gauges distrib-uted over the facewidth Such strain gauges must be put in

Classification Notes- No 412 11 May 2003

DET NORSKE VERITAS

exactly the same position relative to the root fillet Relations in 191 apply for conversion to KHβ

Secondarily

KHβ may be evaluated by observed contact patterns on vari-ous defined load levels It is imperative that the various test loads are well defined Usually it is also necessary to evalu-ate the elastic deflections Some teeth at each 90 degrees are to be painted with a suitable lacquer Always consider the poorest of the contact patterns

After having run the gear for a suitable time at test load 1 (the lowest) observe the contact pattern with respect to ex-tension over the facewidth Evaluate that KHβ by means of the methods mentioned in this section Proceed in the same way for the next higher test load etc until there is a full face contact pattern From these data the initial mesh misalign-ment (ie without elastic deflections) can be found by ex-trapolation and then also the KHβ at design load can be found by calculation and extrapolation See example

Figure 11 Example of experimental determination of KHβ

It must be considered that inaccurate gears may accumulate a larger observed contact pattern than the actual single mesh to mesh contact patterns This is particularly important for lapped bevel gears Ground or hard metal hobbed bevel gears are assumed to present an accumulated contact pattern that is practically equal the actual single mesh to mesh con-tact patterns As a rough guidance the (observed) accumu-lated contact pattern of lapped bevel gears may be reduced by 10 in order to assess the single mesh to mesh contact pattern which is used in 199

193 Theoretical determination of KHβ The methods described in 193 to 198 may be used for cy-lindrical gears The principles may to some extent also be used for bevel gears but a more practical approach is given in 199

General For gears where the tooth contact pattern cannot be verified during assembly or under load all assumptions are to be well on the safe side

KHβ is to be determined in the plane of contact

The influence parameters considered in this method are

bull mean mesh stiffness cγ (see 111) (if necessary also variable stiffness over b)

bull mean unit load Fmb = Fbt KA Kγ Kvb (for double helical gears see 17 for use of Kγ)

bull misalignment fsh due to elastic deflections of shafts and gear bodies (both pinion and wheel)

bull misalignment fdefl due to elastic deflections of and work-ing positions in bearings

bull misalignment fbe due to bearing clearance tolerances bull misalignment fma due to manufacturing tolerances bull helix modifications as crowning end relief helix correc-

tion bull running in amount yβ (see 112)

In practice several other parameters such as centrifugal ex-pansion thermal expansion housing deflection etc contrib-ute to KHβ However these parameters are not taken into account unless in special cases when being considered as particularly important

When all or most of the am parameters are to be considered the most practical way to determine KHβ is by means of a graphical approach described in 1931

If cγ can be considered constant over the facewidth and no helix modifications apply KHβ can be determined analyti-cally as described in 1932

1931 Graphical method The graphical method utilises the superposition principle and is as follows

bull Calculate the mean mesh deflection Mδ as a function of

γm c and bF see 111

bull Draw a base line with length b and draw up a rectangu-lar with height δM (The area δM b is proportional to the transmitted force)

bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the areas above and below this zero line are equal

Figure 12 fsh balanced around zero line

bull Superimpose these ordinates of the fsh curve to the previ-ous load distribution curve (The area under this new load distribution curve is still δM b)

bull Calculate the bearing deflections andor working posi-tions in the bearings and evaluate the influence fdefl in the plane of contact This is a straight line and is bal-anced around a zero line as indicated in Fig 14 but with one distinct direction Superimpose these ordinates to the previous load distribution curve

12 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

bull The amount of crowning end relief or helix correction (defined in the plane of contact) is to be balanced around a zero line similarly to fsh

Figure 13 Crowning Cc balanced aound zero line

bull Superimpose these ordinates to the previous load distri-bution curve In case of high crowning etc as eg often applied to bevel gears the new load distribution curve may cross the base line (the real zero line) The result is areas with negative load that is not real as the load in those areas should be zero Thus corrective actions must be made but for practical reasons it may be postponed to after next operation

bull The amount of initial mesh misalignment bema ff + (defined in the plane of contact) is to be bal-

anced around a zero line If the direction of bema ff + is known (due to initial contact check) or if the direction of fbe is known due to design (eg overhang bevel pin-ion) this should be taken into account If direction un-known the influence of bema ff + in both directions as well as equal zero should be considered

Figure 14 fma+fbe in both directions balanced around zero line

Superimpose these ordinates to the previous load distri-bution curve This results in up to 3 different curves of which the one with the highest peak is to be chosen for further evaluation

bull If the chosen load distribution curve crosses the base line (ie mathematically negative load) the curve is to be corrected by adding the negative areas and dividing this with the active facewidth The (constant) ordinates of this rectangular correction area are to be subtracted from the positive part of the load distribution curve It is advisable to check that the area covered under this new load distribution curve is still equal δM b

bull If cγ cannot be considered as constant over b then cor-rect the ordinates of the load distribution curve with the local (on various positions over the facewidth) ratio between local mesh stiffness and average mesh stiffness cγ (average over the active facewidth only) Note that the result is to be a curve that covers the same area δM b as before

bull The influence of running in yβ is to be determined as in 112 whereby the value for Fβx is to be taken as twice the distance between the peak of the load distribution curve and δM

bull Determine

KHβ = M

β

δycurveofpeak minus

1932 Simplified analytical method for cylindrical gears The analytical approach is similar to 1931 but has a more limited application as cγ is assumed constant over the face-width and no helix modification applies

bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the area above and below this zero line are equal see Fig 12 The max positive ordinate is frac12∆fsh

bull Calculate the initial mesh alignment as Fβx= deflbemash ffff plusmnplusmnplusmn∆ The negative signs may only be used if this is justified andor verified by a contact pattern test Otherwise al-ways use positive signs If a negative sign is justified the value of Fβx is not to be taken less than the largest of each of these elements

bull Calculate the effective mesh misalignment as Fβy = Fβx - yβ (yβ see 112)

bull Determine

2KforF2

bFc1K βH

m

βγγH le+=β

or

2KforF

bFc2K Hβ

m

βγγH gt=β

where cγ as used here is the effective mesh stiffness see 111

194 Determination of fsh fsh is the mesh misalignment due to elastic deflections Usu-ally it is sufficient to consider the combined mesh deflection of the pinion body and shaft and the wheel shaft The calcu-lation is to be made in the plane of contact (of the considered gear mesh) and to consider all forces (incl axial) acting on the shafts Forces from other meshes can be parted into components parallel respectively vertical to the considered plane of contact Forces vertical to this plane of contact have no influence on fsh

It is advised to use following diameters for toothed elements

d + 2 x mn for bending and shear deflection

d + 2 mn (x ndash ha0 + 02) for torsional deflection

Usually fsh is calculated on basis of an evenly distributed load If the analysis of KHβ shows a considerable maldis-

Classification Notes- No 412 13 May 2003

DET NORSKE VERITAS

tribution in term of hard end contact or if it is known by other reasons that there exists a hard end contact the load should be correspondingly distributed when calculating fsh In fact the whole KHβ procedure can be used iteratively 2-3 iterations will be enough even for almost triangular load distributions

195 Determination of fdefl fdefl is the mesh misalignment in the plane of contact due to bearing deflections and working positions (housing deflec-tion may be included if determined)

First the journal working positions in the bearings are to be determined The influence of external moments and forces must be considered This is of special importance for twin pinion single output gears with all 3 shafts in one plane

For rolling bearings fdefl is further determined on basis of the elastic deflection of the bearings An elastic bearing deflec-tion depends on the bearing load and size and number of rolling elements Note that the bearing clearance tolerances are not included here

For fluid film bearings fdefl is further determined on basis of the lift and angular shift of the shafts due to lubrication oil film thickness Note that fbe takes into account the influence of the bearing clearance tolerance

When working positions bearing deflections and oil film lift are combined for all bearings the angular misalignment as projected into the plane of the contact is to be determined fdefl is this angular misalignment (radians) times the face-width

196 Determination of fbe fbe is the mesh misalignment in the plane of contact due to tolerances in bearing clearances In principle fbe and fdefl could be combined But as fdefl can be determined by analy-sis and has a distinct direction and fbe is dependent on toler-ances and in most cases has no distinct direction (ie + toler-ance) it is practicable to separate these two influences

Due to different bearing clearance tolerances in both pinion and wheel shafts the two shaft axis will have an angular mis-alignment in the plane of contact that is superimposed to the working positions determined in 195 fbe is the facewidth times this angular misalignment Note that fbe may have a distinct direction or be given as a + tolerance or a combina-tion of both For combination of + tolerance it is adviced to use

fbe= 22be

21be ff ++plusmn

fbe is particularly important for overhang designs for gears with widely different kinds of bearings on each side and when the bearings have wide tolerances on clearances In general it shall be possible to replace standard bearings with-out causing the real load distribution to exceed the design premises For slow speed gears with journal bearings the expected wear should also be considered

197 Determination of fma fma is the mesh misalignment due to manufacturing toler-ances (helix slope deviation) of pinion fHβ1 wheel fHβ2 and housing bore

For gear without specifically approved requirements to as-sembly control the value of fma is to be determined as

fma= 22Hβ

21Hβ ff +

For gears with specially approved assembly control the value of fma will depend on those specific requirements

198 Comments to various gear types For double helical gears KHβ is to be determined for both helices Usually an even load share between the helices can be assumed If not the calculation is to be made as de-scribed in 171

For planetary gears the free floating sun pinion suffers only twist no bending It must be noted that the total twist is the sum of the twist due to each mesh If the value of 1K γ ne this must be taken into account when calculating the total sun pinion twist (ie twist calculated with the force per mesh without Kγ and multiplied with the number of planets)

When planets are mounted on spherical bearings the mesh misalignments sun-planet respectively planet-annulus will be balanced Ie the misalignment will be the average between the two theoretical individual misalignments The faceload distribution on the flanks of the planets can take full advan-tage of this However as the sun and annulus mesh with several planets with possibly different lead errors the sun and annulus cannot obtain the above mentioned advantage to the full extent

199 Determination of KHβ for bevel gears If a theoretical approach similar to 193-198 is not docu-mented the following may be used

testeff

H Kb

b851851K sdot

minussdot=β

beff b represents the relative active facewidth (regarding lapped gears see 192 last part)

Higher values than beff b = 090 are normally not to be used in the formula

For dual directional gears it may be difficult to obtain a high beff b in both directions In that case the smaller beff b is to be used

Ktest represents the influence of the bearing arrangement shaft stiffness bearing stiffness housing stiffness etc on the faceload distribution and the verification thereof Expected variations in length- and height-wise tooth profile is also accounted for to some extent

a) Ktest = 1 For ground or hard metal hobbed gears with the specified contact pattern verified at full rating or at full torque slow turning at a condition representative for the thermal expansion at normal operation

14 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

It also applies when the bearing arrangementsupport has insignificant elastic deflections and thermal axial expansion However each initial mesh contact must be verified to be within acceptance criteria that are cali-brated against a type test at full load Reproduction of the gear tooth length- and height-wise profile must also be verified This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon)

b) Ktest = 1 + 04middot(beffbndash06) For designs with possible influence of thermal expansion in the axial direction of the pinion The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load

c) Ktest = 12 if mesh is only checked by toolmakerrsquos blue or by spin test contact For gears in this category beffb gt 085 is not to be used in the calculation

110 Transversal Load Distribution Factors KHα and KFα The transverse load distribution factors KHα for contact stresses and for scuffing KFα for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairs of teeth in mesh

The following relations may be used

Cylindrical gears

( )

minus+==

tH

αptγγHαFα F

byfc0409

2ε

KK

valid for 2εγ le

( ) ( )tH

αptγ

γ

γHαFα F

byfcε

1ε20409KK

minusminus+==

valid for 2ε γ gt

where

FtH = Ft KA Kγ Kv KHβ

cγ = See 111

γα = See 112

fpt = Maximum single pitch deviation (microm) of pinion or wheel or maximum total profile form devia-tion Fα of pinion or wheel if this is larger than the maximum single pitch deviation

Note In case of adequate equivalent tip relief adapted to the load half of the above mentioned fpt can be introduced A tip relief is considered adequate when the average of Ca1 and Ca2 is within plusmn40 of the value of Ceff in 432

Limitations of KHα and KFα

If the calculated values for

KFα = KHα lt 1 use KFα = KHα = 10

If the calculated value of KHα gt 2εα

γ

Zε

ε use KHα = 2

εα

γ

Zε

ε

If the calculated value of KFα gt εα

γ

Yεε

use KFα = εα

γ

Yεε

where Yε = αnε

075025+ (for εαn see 331c)

Bevel gears

For ground or hard metal hobbed gears KFα = KHα = 1

For lapped gears KFα = KHα = 11

111 Tooth Stiffness Constants cacute and cγ The tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having 1 mm facewidth by an amount of 1 microm in the plane of contact cacute is the maximum stiffness of a single pair of teeth cγ is the mean value of the mesh stiffness in a transverse plane (brief term mesh stiffness)

Both valid for high unit load (Unit load = Ft middot KA middot Kγb)

Cylindrical gears The real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bending stiff-nesses For high unit loads the Hertzian stiffness has little importance and can be disregarded This approach is on the safe side for determination of KHβ and KHα However for moderate or low loads Kv may be underestimated due to determination of a too high resonance speed

The linear approach is described in A

An optional approach for inclusion of the non-linear stiffness is described in B

A The linear approach

BRCCq

βcos08acutec =

and

( )025ε075cc αγ +prime=

where

( )[ ]n02a01a

B α2000212

hh12051C minusminus

+minus+=

12n1n

x000635z

025791z

015551004723q minus++=

12

2n

22

1n

1 x005290z

x024188x000193z

x011654+minusminusminus

+ 000182 x22

Classification Notes- No 412 15 May 2003

DET NORSKE VERITAS

(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as

( )( )nR m5s

sR e5

bbln1C +=

where

bs = thickness of a central web

sR = average thickness of rim (net value from tooth root to inside of rim)

The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered

Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ

B The non-linear approach

In the following an example is given on how to consider the non-linearity

The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm

With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as

( )10δKbF

minus= for 500bFgt

minus=

500Fb10δK

bF for 500

bFlt

with γAt KK

bF

bF

sdotsdot= etc (Nmm) ie unit load incorporat-

ing the relevant factors as

KA middot Kγ for determination of Kv

KA middot Kγ middot Kv for determination of KHβ

KA middot Kγ middot Kv middot KHβ for determination of KHα

δ = mesh deflection (microm)

K = applicable stiffness (c or cγ)

Use of stiffnesses for KV KHβ and KHα

For calculation of Kv and KHα the stiffness is calculated as follows

When Fb lt 500

the stiffness is determined as δ∆

∆ bF

where the increment is chosen as eg ∆ Fb = 10 and thus

50010Fb10

K10Fb∆δ +

++

=

When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly

or an equivalent stiffness determined as δsdotb

F

Bevel gears

In lack of more detailed relationship between stiffness and geometry the following may be used

b085

b13cacute eff=

b085b

16c effγ =

beff not to be used in excess of 085 b in these formulae

Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)

112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements

yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload

Cay is defined as the running-in amount that compensates for lack of tip relief

The following relations may be used

For not surface hardened steel

ptHlim

α fσ160y =

yβ = βxlimH

f320σ

with the following upper limits

V lt 5 ms 5-10 ms gt 10 ms

yα max none

limH

12800σ limH

6400σ

yβ max none

limH

25600σ limH

12800σ

For surface hardened steel

yα = 0075 fpt but not more than 3 for any speed

yβ = 015 Fβx but not more than 6 for any speed

16 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

For all kinds of steel

5145189718

1C2

limHay +

minusσ

=

When pinion and wheel material differ the following ap-plies

bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv

bull Use ( )2β1ββ yy21y += in the calculation of KHβ

bull Use ( )2ay1aya CC21C += in the calculation of Kv

bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing

calculation if no design tip relief is foreseen

Classification Notes- No 412 17 May 2003

DET NORSKE VERITAS

2 Calculation of Surface Durability

21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213

Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted

Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213

For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength

For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied

22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater

Calculation of surface durability for helical gears is based on the contact stress at the pitch point

For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies

Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact

Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses

The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses

The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel

221 Contact stress Cylindrical gears

( )HαHβvγA

1

tβεEHDBH KKKKK

bud1uF

ZZZZZσ+

=

where

ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)

ZH = Zone factor for pitch point (see 231)

ZE = Elasticity factor (see 24)

Zε = Contact ratio factor (see 25)

Zβ = Helix angle factor (see 26)

Ft KA Kγ Kv KHβ KHα see 15 ndash 110

d1 b u see 12 ndash 15

Bevel gears

( )HαHβvγA

v1v

vmtKEMH KKKKK

bud1uF

ZZZ105σ+

sdot=

where

105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)

ZE KA etc see above

ZM = mid-zone factor see 233

ZK = bevel gear factor see 27

Fmt dv1 uv see 12 ndash 15

It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks

222 Permissible contact stress

XWRvLH

NHlimHP ZZZZZ

SZσ

σ =

where

σH lim = Endurance limit for contact stresses (see 28)

ZN = Life factor for contact stresses (see 29)

SH = Required safety factor according to the rules

ZLZvZR = Oil film influence factors (see 210)

ZW = Work hardening factor (see 211)

ZX = Size factor (see 212)

18 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

23 Zone Factors ZH ZBD and ZM

231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder

wtt2

wtbH sinααcos

cosαcosβ2Z =

232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel

For εβ ge 1 ZBD = 1

For internal gears ZD = 1

For εβ = 0 (spur gears)

( )

π

minusεminusminus

π

minusminus

α=

α2

2

2b

2a

1

2

1b

1a

wtB

z211

dd

z21

dd

tanZ

( )

π

minusεminusminus

π

minusminus

α=

α1

2

1b

1a

2

2

2b

2a

wtD

z211

dd

z21

dd

tanZ

If ZB lt 1 use ZB = 1

If ZD lt 1 use ZD = 1

For 0 lt εβ lt 1

ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)

233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears

εminusminus

εminusminus

αβ=

αα btm2

2vb2

2vabtm2

1vb2val

2v1vvtbmM

pddpdd

ddtancos2Z

This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint

234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel

24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses

For steel against steel ZE = 1898

25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses

αε1Z =ε for 1εβ ge

( )α

ββ

αε ε

εε1

3ε4

Z +minusminus

= for εβ lt 1

26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability

cosβZβ =

27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply

The following may be used ZK = 080

28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ

σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting

For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply

For this purpose pitting is defined by

bull for not surface hardened gears pitted area ge 2 of total active flank area

bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given

material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies

The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as

Classification Notes- No 412 19 May 2003

DET NORSKE VERITAS

the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99

σHlim σH105 σH10

3

Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade

1650 1500

2500 2400

3100 3100

Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)

1000

13 σHlim

13 σHlim

Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV

Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV

Carbon steel 15 HV + 250 16 σHlim 16 σHlim

These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15

29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply

If this is not documented by approved fatigue tests the fol-lowing method may be used

For all steels except nitrided

7L 105N sdotge ZN = 1 or

01570

L

7

N N105Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 5middot107 510N

logZ037

L

7

N N105Z

sdot=

NL = 105 WXRVLHlim

WstX10H1010NN ZZZZZσ

ZZσZZ

55

5=

103 lt NL lt 105 )Z(Zlog05

L

5

10NN

5N103N10

5N10ZZ

==

3L 10N le

WXRVLlimH

Wst10X10H10NN ZZZZZ

ZZZZ

33

3σ

σ==

(but not less than ZN105)

For nitrided steels

6L 102N sdotge ZN = 1 or

00980

L

6

N N102Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 2middot106 510N

Zlog76860

L

6

N N102Z

sdot=

5L 10N le

XWRVL

10XWst10NN ZZZZZ

ZZ13ZZ

55 ==

Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles

210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity

The following methods may be applied in connection with the endurance limit

20 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Surface hardened steels Not surface hardened steels

ZL ( )24013421

360910ν+

+

( )24013421

680830ν+

+

ZV ( )v3280

140930+

+ ( )v3280300850

++

ZR

080

ZrelR3

150

ZrelR3

where

ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits

For values of ν40 gt 500 use ν40 = 500

RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm

RZrel

= ( ) 31

cZ2Z1 ρ

10RR50

+

RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)

For 5

L 10N le ZL ZV ZR = 10

211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface

The following approximation may be used for the endurance limit

Surface hardened steel against not surface hardened steel 150

ZeqW R

31700

130HB21Z

minus

minus=

where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130

RZeq = equivalent roughness

033

c40

066

ZS

ZHZHZEQ ρvν

15000RRRR

=

If RZeq gt 16 then use RZeq = 16

If RZeq lt 15 then use RZeq = 15

where

RZH = surface roughness of the hard member before run in

RZS = surface roughness of the soft member before run in

ν40 = see 210

If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem

Through hardened pinion against softer wheel

( )

minussdotminus+= 000829

HBHB0008981u1Z

2

1W

For 21HBHB

2

1 le use ZW = 1

For 71HBHB

2

1 gt use 17HBHB

2

1 =

For u gt 20 use u = 20

For static strength (lt 105 cycles)

Surface hardened against not surface hardened

ZWst = 105

Through hardened pinion against softer wheel

ZWst = 1

212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality

ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213

213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented

The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent

Classification Notes- No 412 21 May 2003

DET NORSKE VERITAS

stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically

The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence

The real Hertzian stresses σHR are determined as

For helical gears with εβ gt 1

σHR = σH

For helical gears with εβ lt 1 and spur gears

ε

α

ββ ε

ε+εminus

sdotσ=σZ

1

HHR

For bevel gears

KHHR Z

1σσ sdot=

The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)

The coordinates tz and HV are to be compared with the de-sign specification such as

bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400

and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)

In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods

For high cycle fatigue (gt3 106 cycles) the following applies

sdot+

minussdotsdotsdot= o90

05azt

05azt

cosSσ04HV

H

HHHR

applicable to 05a

zt

Hge

For 05at

H

z lt the value for 05at

H

z = applies

56300ρSσ12a cHHR

Hsdotsdot

sdot=

Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength

If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows

reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue

For static strength (lt103 cycles) the following applies

sdot+

minussdotsdotsdot= o90

07at

06a

zt

cosSσ019HV

Hst

z

HstHHR

applicable to 06at

Hst

z ge

56300ρSσa cHHR

Hstsdotsdot

=

In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies

For limited life fatigue (103 lt cycles lt 3106 )

For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance

The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram

minussdotminus

= sdot logN3477

logSlogSlogS

310H6103HHN

36 10H103H Slog86281Slog86280 sdot+sdot sdot

22 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3 Calculation of Tooth Strength

31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding

For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36

For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning

Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα

In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314

A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired

It should be noted that this part 3 does not cover fractures caused by

bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining

Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears

Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied

32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level

321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance

Normally the stresses for pinion and wheel are calculated as

Cylindrical gears

FαFβvγAβSFn

tF K K K K K Y Y Y

m bFσ =

where

YF = Tooth form factor (see 33)

YS = Stress correction factor (see 34)

Yβ = Helix angle factor (see 36)

Ft KA Kγ Kv KFβ KFα see 15 ndash 110

b see 13

Bevel gears

FαFβvγASaFamn

mtF K K K K K Y Y Y

m bFσ ε=

where

YFa = Tooth form factor see 33

YSa = Stress correction factor see 34

Yε = Contact ratio factor see 35

Fmt KA etc see 15 ndash 110

b see 13

322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is

CXRrelTrelTF

NMFEFP YYYY

SYY

δσ

=σ

Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN

where

σFE = Local tooth root bending endurance limit of reference test gear (see 37)

YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)

YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)

SF = Required safety factor according to the rules

YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)

YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)

Classification Notes- No 412 23 May 2003

DET NORSKE VERITAS

YX = Size factor (see 312)

YC = Case depth factor (see 313)

33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress

YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears

YFa applies to load application at the tooth tip and is used for bevel gears

Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears

Figure 31 External tooth in normal section

Figure 32 Internal tooth in normal section

Definitions

n

2

n

Fn

enFn

Fe

F

α cosms

α cosmh6

Y

=

n

2

n

Fn

anFn

Fa

Fa

α cosms

α cosmh6

Y

=

In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth

YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)

The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing

mn with mnm

zn with zvn

αt with αvt

β with βm

with undercut without undercut

Fig 33 Dimensions and basic rack profile of the teeth (finished profile)

Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless

331 Determination of parameters

( )n

n

prnfPnfP m

α cossαsin 1ρ

αtan h4πE

minusminusminusminus=

For external gears fPfP ρρ =

For internal gears ( )0z

195fPfP0

fPfP 10363156ρhxρρ

sdotminus+

+=

where

z0 = number of teeth of pinion cutter

x0 = addendum modification coefficient of pinion cutter

hfP = addendum of pinion cutter

ρfP = tip radius of pinion cutter

xhρG fpfp +minus=

24 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

τmE

2π

z2H

nnminus

minus=

with

3πτ = for external gears

6πτ = for internal gears

Htan zG2

nminusϑ=ϑ

(to be solved iteratively suitable start value6π

=ϑ for exter-

nal gears and 3π for internal gears)

a) Tooth root chord sFn For external gears

minus

ϑ+

ϑminus= ρ

cosG3

3πsinz

ms

fpnn

Fn

For bevel gears with a tooth thickness modification

xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn

For internal gears

minus

ϑ+

ϑminus= ρ

cosG

6πsinz

ms

fPnn

Fn

b) Root fillet radius ρF at 30ordm tangent

( )G2coszcosG2ρ

mρ

2n

2

fpn

F

minusϑϑ+=

c) Determination of bending moment arm hF dn = zn mn

b2α

αn βcosε

ε =

dan = dn + 2 ha

pbn = π mn cos αn

dbn = dn cos αn

( )4

d1εpzz

2dd

zz2d

2bn

2

αnbn

2bn

2an

en +

minusminus

minus=

en

bnen d

dcos arcα =

ennnn

e α invα invα x tan 22π

z1

minus+

+=γ

αFen = αen ndash γe

For external gears

( )

minus=

n

enFenee

n

Fe

mdαtan sin γ γcos

21

mh

]ρcos

G3πcosz fpn +

ϑminus

ϑminusminus

For internal gears

( ) minus

sdotsdotminus=

n

enFenee

n

Fe

mdα tanγ sinγ cos

21

mh

minus

ϑminus

ϑminussdot ρ

cosG3

6πcosz fPn

332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as

250ε205for 0666ε2366Y αnαnDT leleminus=

205εfor 10Y αnDT lt=

34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section

Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm

YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip

YS can be determined as follows

( )

+

+=L23121

1

sS qL01312Y

where Fe

Fn

hsL = and

F

Fns ρ2

sq = (see 33)

YSa can be calculated by replacing hFe with hFa in the above formulae

Note a) Range of validity 1 lt qs lt 8

In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered

b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with

Classification Notes- No 412 25 May 2003

DET NORSKE VERITAS

g

g

ρt

0613

13

minus

where

tg = depth of the grinding notch

ρg = radius of the grinding notch

c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles

35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears

The following may be used

Yε = 0625

36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank

The following may be used (β input in degrees)

Yβ = 1 ndash εβ β120

When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula

However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)

37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM

σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision

If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules

σFE

Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)

bull of specially approved high grade

1050

bull of normal grade

minus CrNiMo steels with approved process

1000

minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850

Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

840

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

720

Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)

07 HV + 300

Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)

025 σB + 125

Alloyed quenched and tempered steel 04 σB + 200

Carbon steel 025 σB + 250

Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40

1) These values are valid for a root radius

bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)

bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)

bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for

6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked

2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)

26 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears

YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0

YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst

For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316

The following method may be used within a stress ratio ndash12 lt R lt 05

381 For idlers planets and PTO with ice class

M1M1R1

1Yor Y MstM

+minus

minus=

where

R = stress ratio = min stress divided by max stress

For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12

For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as

branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus

For a power take off (PTO) with ice class see 161 c

M considers the mean stress influence on the endurance (or static) strength amplitudes

M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress

Following M values may be used

Endurance limit

Static strength

Case hardened 08 ndash 015 Ys 1) 07

If shot peened 04 06

Nitrided 03 03

Induction or flame hardened

04 06

Not surface hardened steel 03 05

Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M

The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches

382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106

For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers

For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used

383 For gears with shrinkage stresses and unidirectional load For endurance strength

FE

fitM σ

σM1

M21Y+

minus=

σFE is the endurance limit for R = 0

For static strength YMst = 1 and σfit accounted for in 39b

σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor

n

Ffit m

ρ215scf minus=

384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is

( ) ( ) FE

fitM σ

σR1M1

M2

M1M1R1

1Yminussdot+

minus

+minus

minus=

Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12

For static strength

M1M1R1

1YMst

+minus

minus=

The effect of σfit is accounted for in 39 b

Classification Notes- No 412 27 May 2003

DET NORSKE VERITAS

385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed

F

y

Sσ225

for not surface hardened fillets

FSHV5 for surface hardened fillets

39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles

Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN

If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used

Determination of the σ ndash N ndash curve

a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF

b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ

FstS1

FPstσ minussdotsdotsdot=

where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability

σFst

Alloyed case hardened steel 1) 2300

Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

1250

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

1050

Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)

18 HV + 800

Steel with not surface hardened fillets the smaller value of 2)

18 σB or 225 σy

1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet

c) Calculate YN as

NL gt 3middot106

YN = 1 or 001

L

6

N N103Y

sdot= ie Yn = 092 for 1010

The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2

103ltNLlt3middot106exp

L

6

N N103Y

sdot=

cycles6103forσcycles310forσlog02876exp

FP

FPst

sdot=

NL lt 103 cycles6103forσcycles310forσ

NYFP

FPst

sdot=

or simply use σFPst as mentioned in b) directly

Guidance on number of load cycles NL for various applica-tions

bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)

bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied

28 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage

YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit

The following method may be used

For endurance limit

for not surface hardened fillets

( )

024

s024

relT σ103133q21σ1012201351

Ysdotsdotminus

+sdotsdotminus+= minus

minus

δ

for all surface hardened fillets except nitrided

( )

106q21002451

Y srelT

++=δ

for nitrided fillets

( )

1347q2101421

Y srelT

++=δ

For static strength

for not surface hardened fillets1)

( )( )( )025

02

02502s

relTst σ3000821σ300 1Y0821Y

+

minus+=δ

for surface hardened fillets except nitrided

YδrelTst = 044 YS + 012

for nitrided fillets

YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not

exceed the yield point and thereby alter the residual stress level See also 39b footnote 2

311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness

YRrelT differs for endurance limit and static strength

The following method may be used

For endurance limit

YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels

YδrelT = 43 ndash 326 (Ry + 1)0005

for nitrided steels

For static strength

YRrelTst = 1 for all Ry and all materials

For a fillet without any longitudinal machining trace Ry asymp Rz

312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength

The following may be used

For endurance limit

YX = 1 for mn le 5 generally

YX = 103 ndash 0006 mn for 5 lt mn lt 30

YX = 085 for mn ge 30

for not surface hard-ened steels

YX = 105 ndash 001 mn for 5 lt mn ge 25

YX = 08 for mn ge 25

for surface hardened steels

For static strength

YXst = 1 for all mn and all materials

313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength

YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength

In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength

The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses

The following simplified method for YC may be used

YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)

YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10

For endurance limit

+

+=nFFE

C m02ρt31

σconstY

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

8 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

163 Frequent overloads For plants where high overloads or shock loads occur regu-larly the influence of this is to be considered by means of cumulative fatigue (see Appendix A)

17 Load Sharing Factor Kγ The load sharing factor Kγ accounts for the maldistribution of load in multiple-path transmissions (dual tandem epicyclic double helix etc) Kγ is defined as the ratio between the max load through an actual path and the evenly shared load

171 General method Kγ mainly depends on accuracy and flexibility of the branches (eg quill shaft planet support external forces etc) and should be considered on basis of measurements or of relevant analysis as eg

Kγ = δ+δ f

δ = total compliance of a branch under full load (assuming even load share) referred to gear mesh

f = minusminusminusminus+++ 23

22

21 fff where f1 f2 etc are the

main individual errors that may contribute to a maldis-tribution between the branches Eg tooth pitch errors planet carrier pitch errors bearing clearance influences etc Compensating effects should also be considered

For double helical gears

An external axial force Fext applied from sources outside the actual gearing (eg thrust via or from a tooth coupling) will cause a maldistribution of forces between the two helices Expressed by a load sharing factor the

βsdot

plusmn=γ tanFF1K

t

ext

If the direction of Fext is known the calculation should be carried out separately for each helix and with the tangential force corrected with the pertinent Kγ If the direction of Fext is unknown both combinations are to be calculated and the higher σH or σF to be used

172 Simplified method If no relevant analysis is available the following may apply

For epicyclic gears

Kγ = 32501 minus+ pln

where npl = number of planets ( gt3 )

For multistage gears with locked paths and gear stages sepa-rated by quill shafts (see figure below)

Figure 10 Locked paths gear

Kγ = ( )φ201+

where φ = quill shaft twist (degrees) under full load

18 Dynamic Factor Kv The dynamic factor Kv accounts for the internally generated dynamic loads

Kv is defined as the ratio between the maximum load that dynamically acts on the tooth flanks and the maximum ex-ternally applied load Ft KA Kγ

In the following 2 different methods (181 and 182) are described In case of controversy between the methods the next following is decisive ie the methods are listed with increasing priority

It is important to observe the limitations for the method in 181 In particular the influence of lateral stiffness of shafts is often underestimated and resonances occur at considerably lower speed than determined in 1811

However for low speed gears with vmiddotz1 lt 300 calculations may be omitted and the dynamic factor simplified to Kv=105

181 Single resonance method For a single stage gear Kv may be determined on basis of the relative proximity (or resonance ratio) N between actual speed n1 and the lowest resonance speed nE1

N = 1E

1

nn

Note that for epicyclic gears n is the relative speed ie the speed that multiplied with z gives the mesh frequency

1811 Determination of critical speed It is not advised to apply this method for multimesh gears for N gt 085 as the influence of higher modes has to be consid-ered see 182 In case of significant lateral shaft flexibility (eg overhung mounted bevel gears) the influence of cou-pled bending and torsional vibrations between pinion and wheel should be considered if N ge 075 see 182

nE1 = redmγc

1zπ

31030 sdot

where

cγ is the actual mesh stiffness per unit facewidth see 111

Classification Notes- No 412 9 May 2003

DET NORSKE VERITAS

For gears with inactive ends of the facewidth as eg due to high crowning or end relief such as often applied for bevel gears the use of cγ in connection with determination of natu-ral frequencies may need correction cγ is defined as stiffness per unit facewidth but when used in connection with the total mesh stiffness it is not as simple as cγmiddotb as only a part of the facewidth is active Such corrections are given in 111

mred is the reduced mass of the gear pair per unit facewidth and referred to the plane of contact

For a single gear stage where no significant inertias are closely connected to neither pinion nor wheel mred is calcu-lated as

mred = 21

21mm

mm+

The individual masses per unit facewidth are calculated as

m12 = 221b

21

)2d(b

I

where I is the polar moment of inertia (kgmm2)

The inertia of bevel gears may be approximated as discs with diameter equal the midface pitch diameter and width equal to b However if the shape of the pinion or wheel body differs much from this idealised cylinder the inertia should be cor-rected accordingly

For all kind of gears if a significant inertia (eg a clutch) is very rigidly connected to the pinion or wheel it should be added to that particular inertia (pinion or wheel) If there is a shaft piece between these inertias the torsional shaft stiffness alters the system into a 3-mass (or more) system This can be calculated as in 182 but also simplified as a 2-mass system calculated with only pinion and wheel masses

1812 Factors used for determination of Kv Non-dimensional gear accuracy dependent parameters

( )bKKF

yfcB

At

pptp

γ

minus=

( )bKKF

yFcBAt

ff

γ

α minus=

Non-dimensional tip relief parameter

bKKFcC

1BγAt

ak sdotsdot

sdotminus=

For gears of quality grade (ISO 1328) Q = 7 or coarser Bk = 1

For gears with Q le 6 and excessive tip relief Bk is limited to max 1

For gears (all quality grades) with tip relief of more than 2middotCeff (see 432) the reduction of αε has to be considered (see 443)

where

fpt = the single pitch deviation (ISO 1328) max of pinion or wheel

Fα = the total profile form deviation (ISO 1328) max of pinion or wheel (Note Fα is pt not available for bevel gears thus use Fα = fpt)

yp and yf = the respective running-in allowances and may be calculated similarly to yα in 112 ie the value of fpt is replaced by Fα for yf

cacute = the single tooth stiffness see 111

Ca = the amount of tip relief see 433 In case of different tip relief on pinion and wheel the value that results in the greater value of Bk is to be used If Ca is zero by design the value of running-in tip relief Cay (see 112) may be used in the above formula

1813 Kv in the subcritical range Cylindrical gears N le 085

Bevel gears N le 075

Kv = 1 + N K

K = Cv1 Bp + Cv2 Bf + Cv3 Bk

Cv1 accounts for the pitch error influence

Cv1 = 032

Cv2 accounts for profile error influence

Cv2 = 034 for γε le 2

Cv2 = 03ε

057

γ minus for γε gt 2

Cv3 accounts for the cyclic mesh stiffness variation

Cv3 = 023 for γε le 2

Cv3 = 156ε

0096

γ minus for γε gt 2

1814 Kv in the main resonance range Cylindrical gears 085 lt N le 115

Bevel gears 075 lt N le 125

Running in this range should preferably be avoided and is only allowed for high precision gears

Kv = 1 + Cv1 Bp + Cv2 Bf + Cv4 Bk

Cv4 accounts for the resonance condition with the cyclic mesh stiffness variation

Cv4 = 090 for γε le 2

Cv4 = 144ε

ε005057

γ

γ

minus

minus for γε gt 2

10 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

1815 Kv in the supercritical range Cylindrical gears N ge 15

Bevel gears N ge 15

Special care should be taken as to influence of higher vibra-tion modes andor influence of coupled bending (ie lateral shaft vibrations) and torsional vibrations between pinion and wheel These influences are not covered by the following approach

Kv = Cv5 Bp + Cv6 Bf + Cv7

Cv5 accounts for the pitch error influence

Cv5 = 047

Cv6 accounts for the profile error influence

Cv6 = 047 for εγ le 2

Cv6 = 174ε

012

γ minus for εγ gt 2

Cv7 relates the maximum externally applied tooth loading to the maximum tooth loading of ideal accurate gears operating in the supercritical speed sector when the circumferential vibration becomes very soft

Cv7 = 075 for εγ le 15

Cv7 = [ ] 8750)2(sin 0125 +minusβεπ for 15 lt εγ le 25

Cv7 = 10 for εγ gt 25

1816 Kv in the intermediate range Cylindrical gears 115 lt N lt 15

Bevel gears 125 lt N lt 15

Comments raised in 1814 and 1815 should be observed

Kv is determined by linear interpolation between Kv for N = 115 respectively 125 and N = 15 as

Cylindrical gears

( ) ( ) ( )[ ]51Nv151Nv51Nvv KK350

N51KK === minussdot

minus

+=

Bevel gears

( ) ( ) ( )[ ]51Nv251Nv51Nvv KK250

N51KK === minussdot

minus

+=

182 Multi-resonance method For high speed gear (vgt40 ms) for multimesh medium speed gears for gears with significant lateral shaft flexibility etc it is advised to determine Kv on basis of relevant dy-namic analysis

Incorporating lateral shaft compliance requires transforma-tion of even a simple pinion-wheel system into a lumped multi-mass system It is advised to incorporate all relevant inertias and torsional shaft stiffnesses into an equivalent (to pinion speed) system Thereby the mesh stiffness appears as an equivalent torsional stiffness

cγ b (db12)2 (Nmrad)

The natural frequencies are found by solving the set of dif-ferential equations (one equation per inertia) Note that for a gear put on a laterally flexible shaft the coupling bending-torsionals is arranged by introducing the gear mass and the lateral stiffness with its relation to the torsional displacement and torque in that shaft

Only the natural frequency (ies) having high relative dis-placement and relative torque through the actual pinion-wheel flexible element need(s) to be considered as critical frequency (ies)

Kv may be determined by means of the method mentioned in 181 thereby using N as the least favourable ratio (in case of more than one pinion-wheel dominated natural frequency) Ie the N-ratio that results in the highest Kv has to be consid-ered

The level of the dynamic factor may also be determined on basis of simulation technique using numeric time integration with relevant tooth stiffness variation and pitchprofile er-rors

19 Face Load Factors KHβ and KFβ The face load factors KHβ for contact stresses and for scuff-ing KFβ for tooth root stresses account for non-uniform load distribution across the facewidth

KHβ is defined as the ratio between the maximum load per unit facewidth and the mean load per unit facewidth

KFβ is defined as the ratio between the maximum tooth root stress per unit facewidth and the mean tooth root stress per unit facewidth The mean tooth root stress relates to the con-sidered facewidth b1 respectively b2

Note that facewidth in this context is the design facewidth b even if the ends are unloaded as often applies to eg bevel gears

The plane of contact is considered

191 Relations between KHβ and KFβ

KFβ = expβHK 2(hb)hb1

1exp++

=

where hb is the ratio tooth heightfacewidth The maximum of h1b1 and h2b2 is to be used but not higher than 13 For double helical gears use only the facewidth of one helix

If the tooth root facewidth (b1 or b2) is considerably wider than b the value of KFβ(1or2) is to be specially considered as it may even exceed KHβ

Eg in pinion-rack lifting systems for jack up rigs where b = b2 asymp mn and b1 asymp 3 mn the typical KHβ asymp KFβ2 asymp 1 and KFβ1 asymp 13

192 Measurement of face load factors Primarily

KFβ may be determined by a number of strain gauges distrib-uted over the facewidth Such strain gauges must be put in

Classification Notes- No 412 11 May 2003

DET NORSKE VERITAS

exactly the same position relative to the root fillet Relations in 191 apply for conversion to KHβ

Secondarily

KHβ may be evaluated by observed contact patterns on vari-ous defined load levels It is imperative that the various test loads are well defined Usually it is also necessary to evalu-ate the elastic deflections Some teeth at each 90 degrees are to be painted with a suitable lacquer Always consider the poorest of the contact patterns

After having run the gear for a suitable time at test load 1 (the lowest) observe the contact pattern with respect to ex-tension over the facewidth Evaluate that KHβ by means of the methods mentioned in this section Proceed in the same way for the next higher test load etc until there is a full face contact pattern From these data the initial mesh misalign-ment (ie without elastic deflections) can be found by ex-trapolation and then also the KHβ at design load can be found by calculation and extrapolation See example

Figure 11 Example of experimental determination of KHβ

It must be considered that inaccurate gears may accumulate a larger observed contact pattern than the actual single mesh to mesh contact patterns This is particularly important for lapped bevel gears Ground or hard metal hobbed bevel gears are assumed to present an accumulated contact pattern that is practically equal the actual single mesh to mesh con-tact patterns As a rough guidance the (observed) accumu-lated contact pattern of lapped bevel gears may be reduced by 10 in order to assess the single mesh to mesh contact pattern which is used in 199

193 Theoretical determination of KHβ The methods described in 193 to 198 may be used for cy-lindrical gears The principles may to some extent also be used for bevel gears but a more practical approach is given in 199

General For gears where the tooth contact pattern cannot be verified during assembly or under load all assumptions are to be well on the safe side

KHβ is to be determined in the plane of contact

The influence parameters considered in this method are

bull mean mesh stiffness cγ (see 111) (if necessary also variable stiffness over b)

bull mean unit load Fmb = Fbt KA Kγ Kvb (for double helical gears see 17 for use of Kγ)

bull misalignment fsh due to elastic deflections of shafts and gear bodies (both pinion and wheel)

bull misalignment fdefl due to elastic deflections of and work-ing positions in bearings

bull misalignment fbe due to bearing clearance tolerances bull misalignment fma due to manufacturing tolerances bull helix modifications as crowning end relief helix correc-

tion bull running in amount yβ (see 112)

In practice several other parameters such as centrifugal ex-pansion thermal expansion housing deflection etc contrib-ute to KHβ However these parameters are not taken into account unless in special cases when being considered as particularly important

When all or most of the am parameters are to be considered the most practical way to determine KHβ is by means of a graphical approach described in 1931

If cγ can be considered constant over the facewidth and no helix modifications apply KHβ can be determined analyti-cally as described in 1932

1931 Graphical method The graphical method utilises the superposition principle and is as follows

bull Calculate the mean mesh deflection Mδ as a function of

γm c and bF see 111

bull Draw a base line with length b and draw up a rectangu-lar with height δM (The area δM b is proportional to the transmitted force)

bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the areas above and below this zero line are equal

Figure 12 fsh balanced around zero line

bull Superimpose these ordinates of the fsh curve to the previ-ous load distribution curve (The area under this new load distribution curve is still δM b)

bull Calculate the bearing deflections andor working posi-tions in the bearings and evaluate the influence fdefl in the plane of contact This is a straight line and is bal-anced around a zero line as indicated in Fig 14 but with one distinct direction Superimpose these ordinates to the previous load distribution curve

12 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

bull The amount of crowning end relief or helix correction (defined in the plane of contact) is to be balanced around a zero line similarly to fsh

Figure 13 Crowning Cc balanced aound zero line

bull Superimpose these ordinates to the previous load distri-bution curve In case of high crowning etc as eg often applied to bevel gears the new load distribution curve may cross the base line (the real zero line) The result is areas with negative load that is not real as the load in those areas should be zero Thus corrective actions must be made but for practical reasons it may be postponed to after next operation

bull The amount of initial mesh misalignment bema ff + (defined in the plane of contact) is to be bal-

anced around a zero line If the direction of bema ff + is known (due to initial contact check) or if the direction of fbe is known due to design (eg overhang bevel pin-ion) this should be taken into account If direction un-known the influence of bema ff + in both directions as well as equal zero should be considered

Figure 14 fma+fbe in both directions balanced around zero line

Superimpose these ordinates to the previous load distri-bution curve This results in up to 3 different curves of which the one with the highest peak is to be chosen for further evaluation

bull If the chosen load distribution curve crosses the base line (ie mathematically negative load) the curve is to be corrected by adding the negative areas and dividing this with the active facewidth The (constant) ordinates of this rectangular correction area are to be subtracted from the positive part of the load distribution curve It is advisable to check that the area covered under this new load distribution curve is still equal δM b

bull If cγ cannot be considered as constant over b then cor-rect the ordinates of the load distribution curve with the local (on various positions over the facewidth) ratio between local mesh stiffness and average mesh stiffness cγ (average over the active facewidth only) Note that the result is to be a curve that covers the same area δM b as before

bull The influence of running in yβ is to be determined as in 112 whereby the value for Fβx is to be taken as twice the distance between the peak of the load distribution curve and δM

bull Determine

KHβ = M

β

δycurveofpeak minus

1932 Simplified analytical method for cylindrical gears The analytical approach is similar to 1931 but has a more limited application as cγ is assumed constant over the face-width and no helix modification applies

bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the area above and below this zero line are equal see Fig 12 The max positive ordinate is frac12∆fsh

bull Calculate the initial mesh alignment as Fβx= deflbemash ffff plusmnplusmnplusmn∆ The negative signs may only be used if this is justified andor verified by a contact pattern test Otherwise al-ways use positive signs If a negative sign is justified the value of Fβx is not to be taken less than the largest of each of these elements

bull Calculate the effective mesh misalignment as Fβy = Fβx - yβ (yβ see 112)

bull Determine

2KforF2

bFc1K βH

m

βγγH le+=β

or

2KforF

bFc2K Hβ

m

βγγH gt=β

where cγ as used here is the effective mesh stiffness see 111

194 Determination of fsh fsh is the mesh misalignment due to elastic deflections Usu-ally it is sufficient to consider the combined mesh deflection of the pinion body and shaft and the wheel shaft The calcu-lation is to be made in the plane of contact (of the considered gear mesh) and to consider all forces (incl axial) acting on the shafts Forces from other meshes can be parted into components parallel respectively vertical to the considered plane of contact Forces vertical to this plane of contact have no influence on fsh

It is advised to use following diameters for toothed elements

d + 2 x mn for bending and shear deflection

d + 2 mn (x ndash ha0 + 02) for torsional deflection

Usually fsh is calculated on basis of an evenly distributed load If the analysis of KHβ shows a considerable maldis-

Classification Notes- No 412 13 May 2003

DET NORSKE VERITAS

tribution in term of hard end contact or if it is known by other reasons that there exists a hard end contact the load should be correspondingly distributed when calculating fsh In fact the whole KHβ procedure can be used iteratively 2-3 iterations will be enough even for almost triangular load distributions

195 Determination of fdefl fdefl is the mesh misalignment in the plane of contact due to bearing deflections and working positions (housing deflec-tion may be included if determined)

First the journal working positions in the bearings are to be determined The influence of external moments and forces must be considered This is of special importance for twin pinion single output gears with all 3 shafts in one plane

For rolling bearings fdefl is further determined on basis of the elastic deflection of the bearings An elastic bearing deflec-tion depends on the bearing load and size and number of rolling elements Note that the bearing clearance tolerances are not included here

For fluid film bearings fdefl is further determined on basis of the lift and angular shift of the shafts due to lubrication oil film thickness Note that fbe takes into account the influence of the bearing clearance tolerance

When working positions bearing deflections and oil film lift are combined for all bearings the angular misalignment as projected into the plane of the contact is to be determined fdefl is this angular misalignment (radians) times the face-width

196 Determination of fbe fbe is the mesh misalignment in the plane of contact due to tolerances in bearing clearances In principle fbe and fdefl could be combined But as fdefl can be determined by analy-sis and has a distinct direction and fbe is dependent on toler-ances and in most cases has no distinct direction (ie + toler-ance) it is practicable to separate these two influences

Due to different bearing clearance tolerances in both pinion and wheel shafts the two shaft axis will have an angular mis-alignment in the plane of contact that is superimposed to the working positions determined in 195 fbe is the facewidth times this angular misalignment Note that fbe may have a distinct direction or be given as a + tolerance or a combina-tion of both For combination of + tolerance it is adviced to use

fbe= 22be

21be ff ++plusmn

fbe is particularly important for overhang designs for gears with widely different kinds of bearings on each side and when the bearings have wide tolerances on clearances In general it shall be possible to replace standard bearings with-out causing the real load distribution to exceed the design premises For slow speed gears with journal bearings the expected wear should also be considered

197 Determination of fma fma is the mesh misalignment due to manufacturing toler-ances (helix slope deviation) of pinion fHβ1 wheel fHβ2 and housing bore

For gear without specifically approved requirements to as-sembly control the value of fma is to be determined as

fma= 22Hβ

21Hβ ff +

For gears with specially approved assembly control the value of fma will depend on those specific requirements

198 Comments to various gear types For double helical gears KHβ is to be determined for both helices Usually an even load share between the helices can be assumed If not the calculation is to be made as de-scribed in 171

For planetary gears the free floating sun pinion suffers only twist no bending It must be noted that the total twist is the sum of the twist due to each mesh If the value of 1K γ ne this must be taken into account when calculating the total sun pinion twist (ie twist calculated with the force per mesh without Kγ and multiplied with the number of planets)

When planets are mounted on spherical bearings the mesh misalignments sun-planet respectively planet-annulus will be balanced Ie the misalignment will be the average between the two theoretical individual misalignments The faceload distribution on the flanks of the planets can take full advan-tage of this However as the sun and annulus mesh with several planets with possibly different lead errors the sun and annulus cannot obtain the above mentioned advantage to the full extent

199 Determination of KHβ for bevel gears If a theoretical approach similar to 193-198 is not docu-mented the following may be used

testeff

H Kb

b851851K sdot

minussdot=β

beff b represents the relative active facewidth (regarding lapped gears see 192 last part)

Higher values than beff b = 090 are normally not to be used in the formula

For dual directional gears it may be difficult to obtain a high beff b in both directions In that case the smaller beff b is to be used

Ktest represents the influence of the bearing arrangement shaft stiffness bearing stiffness housing stiffness etc on the faceload distribution and the verification thereof Expected variations in length- and height-wise tooth profile is also accounted for to some extent

a) Ktest = 1 For ground or hard metal hobbed gears with the specified contact pattern verified at full rating or at full torque slow turning at a condition representative for the thermal expansion at normal operation

14 Classification Notes- No 412

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DET NORSKE VERITAS

It also applies when the bearing arrangementsupport has insignificant elastic deflections and thermal axial expansion However each initial mesh contact must be verified to be within acceptance criteria that are cali-brated against a type test at full load Reproduction of the gear tooth length- and height-wise profile must also be verified This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon)

b) Ktest = 1 + 04middot(beffbndash06) For designs with possible influence of thermal expansion in the axial direction of the pinion The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load

c) Ktest = 12 if mesh is only checked by toolmakerrsquos blue or by spin test contact For gears in this category beffb gt 085 is not to be used in the calculation

110 Transversal Load Distribution Factors KHα and KFα The transverse load distribution factors KHα for contact stresses and for scuffing KFα for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairs of teeth in mesh

The following relations may be used

Cylindrical gears

( )

minus+==

tH

αptγγHαFα F

byfc0409

2ε

KK

valid for 2εγ le

( ) ( )tH

αptγ

γ

γHαFα F

byfcε

1ε20409KK

minusminus+==

valid for 2ε γ gt

where

FtH = Ft KA Kγ Kv KHβ

cγ = See 111

γα = See 112

fpt = Maximum single pitch deviation (microm) of pinion or wheel or maximum total profile form devia-tion Fα of pinion or wheel if this is larger than the maximum single pitch deviation

Note In case of adequate equivalent tip relief adapted to the load half of the above mentioned fpt can be introduced A tip relief is considered adequate when the average of Ca1 and Ca2 is within plusmn40 of the value of Ceff in 432

Limitations of KHα and KFα

If the calculated values for

KFα = KHα lt 1 use KFα = KHα = 10

If the calculated value of KHα gt 2εα

γ

Zε

ε use KHα = 2

εα

γ

Zε

ε

If the calculated value of KFα gt εα

γ

Yεε

use KFα = εα

γ

Yεε

where Yε = αnε

075025+ (for εαn see 331c)

Bevel gears

For ground or hard metal hobbed gears KFα = KHα = 1

For lapped gears KFα = KHα = 11

111 Tooth Stiffness Constants cacute and cγ The tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having 1 mm facewidth by an amount of 1 microm in the plane of contact cacute is the maximum stiffness of a single pair of teeth cγ is the mean value of the mesh stiffness in a transverse plane (brief term mesh stiffness)

Both valid for high unit load (Unit load = Ft middot KA middot Kγb)

Cylindrical gears The real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bending stiff-nesses For high unit loads the Hertzian stiffness has little importance and can be disregarded This approach is on the safe side for determination of KHβ and KHα However for moderate or low loads Kv may be underestimated due to determination of a too high resonance speed

The linear approach is described in A

An optional approach for inclusion of the non-linear stiffness is described in B

A The linear approach

BRCCq

βcos08acutec =

and

( )025ε075cc αγ +prime=

where

( )[ ]n02a01a

B α2000212

hh12051C minusminus

+minus+=

12n1n

x000635z

025791z

015551004723q minus++=

12

2n

22

1n

1 x005290z

x024188x000193z

x011654+minusminusminus

+ 000182 x22

Classification Notes- No 412 15 May 2003

DET NORSKE VERITAS

(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as

( )( )nR m5s

sR e5

bbln1C +=

where

bs = thickness of a central web

sR = average thickness of rim (net value from tooth root to inside of rim)

The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered

Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ

B The non-linear approach

In the following an example is given on how to consider the non-linearity

The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm

With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as

( )10δKbF

minus= for 500bFgt

minus=

500Fb10δK

bF for 500

bFlt

with γAt KK

bF

bF

sdotsdot= etc (Nmm) ie unit load incorporat-

ing the relevant factors as

KA middot Kγ for determination of Kv

KA middot Kγ middot Kv for determination of KHβ

KA middot Kγ middot Kv middot KHβ for determination of KHα

δ = mesh deflection (microm)

K = applicable stiffness (c or cγ)

Use of stiffnesses for KV KHβ and KHα

For calculation of Kv and KHα the stiffness is calculated as follows

When Fb lt 500

the stiffness is determined as δ∆

∆ bF

where the increment is chosen as eg ∆ Fb = 10 and thus

50010Fb10

K10Fb∆δ +

++

=

When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly

or an equivalent stiffness determined as δsdotb

F

Bevel gears

In lack of more detailed relationship between stiffness and geometry the following may be used

b085

b13cacute eff=

b085b

16c effγ =

beff not to be used in excess of 085 b in these formulae

Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)

112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements

yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload

Cay is defined as the running-in amount that compensates for lack of tip relief

The following relations may be used

For not surface hardened steel

ptHlim

α fσ160y =

yβ = βxlimH

f320σ

with the following upper limits

V lt 5 ms 5-10 ms gt 10 ms

yα max none

limH

12800σ limH

6400σ

yβ max none

limH

25600σ limH

12800σ

For surface hardened steel

yα = 0075 fpt but not more than 3 for any speed

yβ = 015 Fβx but not more than 6 for any speed

16 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

For all kinds of steel

5145189718

1C2

limHay +

minusσ

=

When pinion and wheel material differ the following ap-plies

bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv

bull Use ( )2β1ββ yy21y += in the calculation of KHβ

bull Use ( )2ay1aya CC21C += in the calculation of Kv

bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing

calculation if no design tip relief is foreseen

Classification Notes- No 412 17 May 2003

DET NORSKE VERITAS

2 Calculation of Surface Durability

21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213

Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted

Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213

For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength

For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied

22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater

Calculation of surface durability for helical gears is based on the contact stress at the pitch point

For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies

Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact

Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses

The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses

The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel

221 Contact stress Cylindrical gears

( )HαHβvγA

1

tβεEHDBH KKKKK

bud1uF

ZZZZZσ+

=

where

ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)

ZH = Zone factor for pitch point (see 231)

ZE = Elasticity factor (see 24)

Zε = Contact ratio factor (see 25)

Zβ = Helix angle factor (see 26)

Ft KA Kγ Kv KHβ KHα see 15 ndash 110

d1 b u see 12 ndash 15

Bevel gears

( )HαHβvγA

v1v

vmtKEMH KKKKK

bud1uF

ZZZ105σ+

sdot=

where

105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)

ZE KA etc see above

ZM = mid-zone factor see 233

ZK = bevel gear factor see 27

Fmt dv1 uv see 12 ndash 15

It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks

222 Permissible contact stress

XWRvLH

NHlimHP ZZZZZ

SZσ

σ =

where

σH lim = Endurance limit for contact stresses (see 28)

ZN = Life factor for contact stresses (see 29)

SH = Required safety factor according to the rules

ZLZvZR = Oil film influence factors (see 210)

ZW = Work hardening factor (see 211)

ZX = Size factor (see 212)

18 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

23 Zone Factors ZH ZBD and ZM

231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder

wtt2

wtbH sinααcos

cosαcosβ2Z =

232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel

For εβ ge 1 ZBD = 1

For internal gears ZD = 1

For εβ = 0 (spur gears)

( )

π

minusεminusminus

π

minusminus

α=

α2

2

2b

2a

1

2

1b

1a

wtB

z211

dd

z21

dd

tanZ

( )

π

minusεminusminus

π

minusminus

α=

α1

2

1b

1a

2

2

2b

2a

wtD

z211

dd

z21

dd

tanZ

If ZB lt 1 use ZB = 1

If ZD lt 1 use ZD = 1

For 0 lt εβ lt 1

ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)

233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears

εminusminus

εminusminus

αβ=

αα btm2

2vb2

2vabtm2

1vb2val

2v1vvtbmM

pddpdd

ddtancos2Z

This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint

234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel

24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses

For steel against steel ZE = 1898

25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses

αε1Z =ε for 1εβ ge

( )α

ββ

αε ε

εε1

3ε4

Z +minusminus

= for εβ lt 1

26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability

cosβZβ =

27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply

The following may be used ZK = 080

28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ

σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting

For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply

For this purpose pitting is defined by

bull for not surface hardened gears pitted area ge 2 of total active flank area

bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given

material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies

The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as

Classification Notes- No 412 19 May 2003

DET NORSKE VERITAS

the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99

σHlim σH105 σH10

3

Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade

1650 1500

2500 2400

3100 3100

Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)

1000

13 σHlim

13 σHlim

Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV

Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV

Carbon steel 15 HV + 250 16 σHlim 16 σHlim

These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15

29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply

If this is not documented by approved fatigue tests the fol-lowing method may be used

For all steels except nitrided

7L 105N sdotge ZN = 1 or

01570

L

7

N N105Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 5middot107 510N

logZ037

L

7

N N105Z

sdot=

NL = 105 WXRVLHlim

WstX10H1010NN ZZZZZσ

ZZσZZ

55

5=

103 lt NL lt 105 )Z(Zlog05

L

5

10NN

5N103N10

5N10ZZ

==

3L 10N le

WXRVLlimH

Wst10X10H10NN ZZZZZ

ZZZZ

33

3σ

σ==

(but not less than ZN105)

For nitrided steels

6L 102N sdotge ZN = 1 or

00980

L

6

N N102Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 2middot106 510N

Zlog76860

L

6

N N102Z

sdot=

5L 10N le

XWRVL

10XWst10NN ZZZZZ

ZZ13ZZ

55 ==

Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles

210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity

The following methods may be applied in connection with the endurance limit

20 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Surface hardened steels Not surface hardened steels

ZL ( )24013421

360910ν+

+

( )24013421

680830ν+

+

ZV ( )v3280

140930+

+ ( )v3280300850

++

ZR

080

ZrelR3

150

ZrelR3

where

ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits

For values of ν40 gt 500 use ν40 = 500

RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm

RZrel

= ( ) 31

cZ2Z1 ρ

10RR50

+

RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)

For 5

L 10N le ZL ZV ZR = 10

211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface

The following approximation may be used for the endurance limit

Surface hardened steel against not surface hardened steel 150

ZeqW R

31700

130HB21Z

minus

minus=

where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130

RZeq = equivalent roughness

033

c40

066

ZS

ZHZHZEQ ρvν

15000RRRR

=

If RZeq gt 16 then use RZeq = 16

If RZeq lt 15 then use RZeq = 15

where

RZH = surface roughness of the hard member before run in

RZS = surface roughness of the soft member before run in

ν40 = see 210

If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem

Through hardened pinion against softer wheel

( )

minussdotminus+= 000829

HBHB0008981u1Z

2

1W

For 21HBHB

2

1 le use ZW = 1

For 71HBHB

2

1 gt use 17HBHB

2

1 =

For u gt 20 use u = 20

For static strength (lt 105 cycles)

Surface hardened against not surface hardened

ZWst = 105

Through hardened pinion against softer wheel

ZWst = 1

212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality

ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213

213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented

The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent

Classification Notes- No 412 21 May 2003

DET NORSKE VERITAS

stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically

The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence

The real Hertzian stresses σHR are determined as

For helical gears with εβ gt 1

σHR = σH

For helical gears with εβ lt 1 and spur gears

ε

α

ββ ε

ε+εminus

sdotσ=σZ

1

HHR

For bevel gears

KHHR Z

1σσ sdot=

The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)

The coordinates tz and HV are to be compared with the de-sign specification such as

bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400

and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)

In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods

For high cycle fatigue (gt3 106 cycles) the following applies

sdot+

minussdotsdotsdot= o90

05azt

05azt

cosSσ04HV

H

HHHR

applicable to 05a

zt

Hge

For 05at

H

z lt the value for 05at

H

z = applies

56300ρSσ12a cHHR

Hsdotsdot

sdot=

Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength

If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows

reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue

For static strength (lt103 cycles) the following applies

sdot+

minussdotsdotsdot= o90

07at

06a

zt

cosSσ019HV

Hst

z

HstHHR

applicable to 06at

Hst

z ge

56300ρSσa cHHR

Hstsdotsdot

=

In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies

For limited life fatigue (103 lt cycles lt 3106 )

For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance

The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram

minussdotminus

= sdot logN3477

logSlogSlogS

310H6103HHN

36 10H103H Slog86281Slog86280 sdot+sdot sdot

22 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3 Calculation of Tooth Strength

31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding

For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36

For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning

Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα

In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314

A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired

It should be noted that this part 3 does not cover fractures caused by

bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining

Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears

Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied

32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level

321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance

Normally the stresses for pinion and wheel are calculated as

Cylindrical gears

FαFβvγAβSFn

tF K K K K K Y Y Y

m bFσ =

where

YF = Tooth form factor (see 33)

YS = Stress correction factor (see 34)

Yβ = Helix angle factor (see 36)

Ft KA Kγ Kv KFβ KFα see 15 ndash 110

b see 13

Bevel gears

FαFβvγASaFamn

mtF K K K K K Y Y Y

m bFσ ε=

where

YFa = Tooth form factor see 33

YSa = Stress correction factor see 34

Yε = Contact ratio factor see 35

Fmt KA etc see 15 ndash 110

b see 13

322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is

CXRrelTrelTF

NMFEFP YYYY

SYY

δσ

=σ

Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN

where

σFE = Local tooth root bending endurance limit of reference test gear (see 37)

YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)

YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)

SF = Required safety factor according to the rules

YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)

YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)

Classification Notes- No 412 23 May 2003

DET NORSKE VERITAS

YX = Size factor (see 312)

YC = Case depth factor (see 313)

33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress

YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears

YFa applies to load application at the tooth tip and is used for bevel gears

Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears

Figure 31 External tooth in normal section

Figure 32 Internal tooth in normal section

Definitions

n

2

n

Fn

enFn

Fe

F

α cosms

α cosmh6

Y

=

n

2

n

Fn

anFn

Fa

Fa

α cosms

α cosmh6

Y

=

In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth

YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)

The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing

mn with mnm

zn with zvn

αt with αvt

β with βm

with undercut without undercut

Fig 33 Dimensions and basic rack profile of the teeth (finished profile)

Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless

331 Determination of parameters

( )n

n

prnfPnfP m

α cossαsin 1ρ

αtan h4πE

minusminusminusminus=

For external gears fPfP ρρ =

For internal gears ( )0z

195fPfP0

fPfP 10363156ρhxρρ

sdotminus+

+=

where

z0 = number of teeth of pinion cutter

x0 = addendum modification coefficient of pinion cutter

hfP = addendum of pinion cutter

ρfP = tip radius of pinion cutter

xhρG fpfp +minus=

24 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

τmE

2π

z2H

nnminus

minus=

with

3πτ = for external gears

6πτ = for internal gears

Htan zG2

nminusϑ=ϑ

(to be solved iteratively suitable start value6π

=ϑ for exter-

nal gears and 3π for internal gears)

a) Tooth root chord sFn For external gears

minus

ϑ+

ϑminus= ρ

cosG3

3πsinz

ms

fpnn

Fn

For bevel gears with a tooth thickness modification

xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn

For internal gears

minus

ϑ+

ϑminus= ρ

cosG

6πsinz

ms

fPnn

Fn

b) Root fillet radius ρF at 30ordm tangent

( )G2coszcosG2ρ

mρ

2n

2

fpn

F

minusϑϑ+=

c) Determination of bending moment arm hF dn = zn mn

b2α

αn βcosε

ε =

dan = dn + 2 ha

pbn = π mn cos αn

dbn = dn cos αn

( )4

d1εpzz

2dd

zz2d

2bn

2

αnbn

2bn

2an

en +

minusminus

minus=

en

bnen d

dcos arcα =

ennnn

e α invα invα x tan 22π

z1

minus+

+=γ

αFen = αen ndash γe

For external gears

( )

minus=

n

enFenee

n

Fe

mdαtan sin γ γcos

21

mh

]ρcos

G3πcosz fpn +

ϑminus

ϑminusminus

For internal gears

( ) minus

sdotsdotminus=

n

enFenee

n

Fe

mdα tanγ sinγ cos

21

mh

minus

ϑminus

ϑminussdot ρ

cosG3

6πcosz fPn

332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as

250ε205for 0666ε2366Y αnαnDT leleminus=

205εfor 10Y αnDT lt=

34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section

Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm

YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip

YS can be determined as follows

( )

+

+=L23121

1

sS qL01312Y

where Fe

Fn

hsL = and

F

Fns ρ2

sq = (see 33)

YSa can be calculated by replacing hFe with hFa in the above formulae

Note a) Range of validity 1 lt qs lt 8

In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered

b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with

Classification Notes- No 412 25 May 2003

DET NORSKE VERITAS

g

g

ρt

0613

13

minus

where

tg = depth of the grinding notch

ρg = radius of the grinding notch

c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles

35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears

The following may be used

Yε = 0625

36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank

The following may be used (β input in degrees)

Yβ = 1 ndash εβ β120

When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula

However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)

37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM

σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision

If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules

σFE

Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)

bull of specially approved high grade

1050

bull of normal grade

minus CrNiMo steels with approved process

1000

minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850

Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

840

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

720

Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)

07 HV + 300

Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)

025 σB + 125

Alloyed quenched and tempered steel 04 σB + 200

Carbon steel 025 σB + 250

Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40

1) These values are valid for a root radius

bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)

bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)

bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for

6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked

2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)

26 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears

YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0

YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst

For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316

The following method may be used within a stress ratio ndash12 lt R lt 05

381 For idlers planets and PTO with ice class

M1M1R1

1Yor Y MstM

+minus

minus=

where

R = stress ratio = min stress divided by max stress

For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12

For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as

branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus

For a power take off (PTO) with ice class see 161 c

M considers the mean stress influence on the endurance (or static) strength amplitudes

M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress

Following M values may be used

Endurance limit

Static strength

Case hardened 08 ndash 015 Ys 1) 07

If shot peened 04 06

Nitrided 03 03

Induction or flame hardened

04 06

Not surface hardened steel 03 05

Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M

The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches

382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106

For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers

For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used

383 For gears with shrinkage stresses and unidirectional load For endurance strength

FE

fitM σ

σM1

M21Y+

minus=

σFE is the endurance limit for R = 0

For static strength YMst = 1 and σfit accounted for in 39b

σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor

n

Ffit m

ρ215scf minus=

384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is

( ) ( ) FE

fitM σ

σR1M1

M2

M1M1R1

1Yminussdot+

minus

+minus

minus=

Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12

For static strength

M1M1R1

1YMst

+minus

minus=

The effect of σfit is accounted for in 39 b

Classification Notes- No 412 27 May 2003

DET NORSKE VERITAS

385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed

F

y

Sσ225

for not surface hardened fillets

FSHV5 for surface hardened fillets

39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles

Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN

If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used

Determination of the σ ndash N ndash curve

a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF

b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ

FstS1

FPstσ minussdotsdotsdot=

where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability

σFst

Alloyed case hardened steel 1) 2300

Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

1250

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

1050

Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)

18 HV + 800

Steel with not surface hardened fillets the smaller value of 2)

18 σB or 225 σy

1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet

c) Calculate YN as

NL gt 3middot106

YN = 1 or 001

L

6

N N103Y

sdot= ie Yn = 092 for 1010

The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2

103ltNLlt3middot106exp

L

6

N N103Y

sdot=

cycles6103forσcycles310forσlog02876exp

FP

FPst

sdot=

NL lt 103 cycles6103forσcycles310forσ

NYFP

FPst

sdot=

or simply use σFPst as mentioned in b) directly

Guidance on number of load cycles NL for various applica-tions

bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)

bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied

28 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage

YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit

The following method may be used

For endurance limit

for not surface hardened fillets

( )

024

s024

relT σ103133q21σ1012201351

Ysdotsdotminus

+sdotsdotminus+= minus

minus

δ

for all surface hardened fillets except nitrided

( )

106q21002451

Y srelT

++=δ

for nitrided fillets

( )

1347q2101421

Y srelT

++=δ

For static strength

for not surface hardened fillets1)

( )( )( )025

02

02502s

relTst σ3000821σ300 1Y0821Y

+

minus+=δ

for surface hardened fillets except nitrided

YδrelTst = 044 YS + 012

for nitrided fillets

YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not

exceed the yield point and thereby alter the residual stress level See also 39b footnote 2

311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness

YRrelT differs for endurance limit and static strength

The following method may be used

For endurance limit

YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels

YδrelT = 43 ndash 326 (Ry + 1)0005

for nitrided steels

For static strength

YRrelTst = 1 for all Ry and all materials

For a fillet without any longitudinal machining trace Ry asymp Rz

312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength

The following may be used

For endurance limit

YX = 1 for mn le 5 generally

YX = 103 ndash 0006 mn for 5 lt mn lt 30

YX = 085 for mn ge 30

for not surface hard-ened steels

YX = 105 ndash 001 mn for 5 lt mn ge 25

YX = 08 for mn ge 25

for surface hardened steels

For static strength

YXst = 1 for all mn and all materials

313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength

YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength

In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength

The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses

The following simplified method for YC may be used

YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)

YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10

For endurance limit

+

+=nFFE

C m02ρt31

σconstY

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

Classification Notes- No 412 9 May 2003

DET NORSKE VERITAS

For gears with inactive ends of the facewidth as eg due to high crowning or end relief such as often applied for bevel gears the use of cγ in connection with determination of natu-ral frequencies may need correction cγ is defined as stiffness per unit facewidth but when used in connection with the total mesh stiffness it is not as simple as cγmiddotb as only a part of the facewidth is active Such corrections are given in 111

mred is the reduced mass of the gear pair per unit facewidth and referred to the plane of contact

For a single gear stage where no significant inertias are closely connected to neither pinion nor wheel mred is calcu-lated as

mred = 21

21mm

mm+

The individual masses per unit facewidth are calculated as

m12 = 221b

21

)2d(b

I

where I is the polar moment of inertia (kgmm2)

The inertia of bevel gears may be approximated as discs with diameter equal the midface pitch diameter and width equal to b However if the shape of the pinion or wheel body differs much from this idealised cylinder the inertia should be cor-rected accordingly

For all kind of gears if a significant inertia (eg a clutch) is very rigidly connected to the pinion or wheel it should be added to that particular inertia (pinion or wheel) If there is a shaft piece between these inertias the torsional shaft stiffness alters the system into a 3-mass (or more) system This can be calculated as in 182 but also simplified as a 2-mass system calculated with only pinion and wheel masses

1812 Factors used for determination of Kv Non-dimensional gear accuracy dependent parameters

( )bKKF

yfcB

At

pptp

γ

minus=

( )bKKF

yFcBAt

ff

γ

α minus=

Non-dimensional tip relief parameter

bKKFcC

1BγAt

ak sdotsdot

sdotminus=

For gears of quality grade (ISO 1328) Q = 7 or coarser Bk = 1

For gears with Q le 6 and excessive tip relief Bk is limited to max 1

For gears (all quality grades) with tip relief of more than 2middotCeff (see 432) the reduction of αε has to be considered (see 443)

where

fpt = the single pitch deviation (ISO 1328) max of pinion or wheel

Fα = the total profile form deviation (ISO 1328) max of pinion or wheel (Note Fα is pt not available for bevel gears thus use Fα = fpt)

yp and yf = the respective running-in allowances and may be calculated similarly to yα in 112 ie the value of fpt is replaced by Fα for yf

cacute = the single tooth stiffness see 111

Ca = the amount of tip relief see 433 In case of different tip relief on pinion and wheel the value that results in the greater value of Bk is to be used If Ca is zero by design the value of running-in tip relief Cay (see 112) may be used in the above formula

1813 Kv in the subcritical range Cylindrical gears N le 085

Bevel gears N le 075

Kv = 1 + N K

K = Cv1 Bp + Cv2 Bf + Cv3 Bk

Cv1 accounts for the pitch error influence

Cv1 = 032

Cv2 accounts for profile error influence

Cv2 = 034 for γε le 2

Cv2 = 03ε

057

γ minus for γε gt 2

Cv3 accounts for the cyclic mesh stiffness variation

Cv3 = 023 for γε le 2

Cv3 = 156ε

0096

γ minus for γε gt 2

1814 Kv in the main resonance range Cylindrical gears 085 lt N le 115

Bevel gears 075 lt N le 125

Running in this range should preferably be avoided and is only allowed for high precision gears

Kv = 1 + Cv1 Bp + Cv2 Bf + Cv4 Bk

Cv4 accounts for the resonance condition with the cyclic mesh stiffness variation

Cv4 = 090 for γε le 2

Cv4 = 144ε

ε005057

γ

γ

minus

minus for γε gt 2

10 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

1815 Kv in the supercritical range Cylindrical gears N ge 15

Bevel gears N ge 15

Special care should be taken as to influence of higher vibra-tion modes andor influence of coupled bending (ie lateral shaft vibrations) and torsional vibrations between pinion and wheel These influences are not covered by the following approach

Kv = Cv5 Bp + Cv6 Bf + Cv7

Cv5 accounts for the pitch error influence

Cv5 = 047

Cv6 accounts for the profile error influence

Cv6 = 047 for εγ le 2

Cv6 = 174ε

012

γ minus for εγ gt 2

Cv7 relates the maximum externally applied tooth loading to the maximum tooth loading of ideal accurate gears operating in the supercritical speed sector when the circumferential vibration becomes very soft

Cv7 = 075 for εγ le 15

Cv7 = [ ] 8750)2(sin 0125 +minusβεπ for 15 lt εγ le 25

Cv7 = 10 for εγ gt 25

1816 Kv in the intermediate range Cylindrical gears 115 lt N lt 15

Bevel gears 125 lt N lt 15

Comments raised in 1814 and 1815 should be observed

Kv is determined by linear interpolation between Kv for N = 115 respectively 125 and N = 15 as

Cylindrical gears

( ) ( ) ( )[ ]51Nv151Nv51Nvv KK350

N51KK === minussdot

minus

+=

Bevel gears

( ) ( ) ( )[ ]51Nv251Nv51Nvv KK250

N51KK === minussdot

minus

+=

182 Multi-resonance method For high speed gear (vgt40 ms) for multimesh medium speed gears for gears with significant lateral shaft flexibility etc it is advised to determine Kv on basis of relevant dy-namic analysis

Incorporating lateral shaft compliance requires transforma-tion of even a simple pinion-wheel system into a lumped multi-mass system It is advised to incorporate all relevant inertias and torsional shaft stiffnesses into an equivalent (to pinion speed) system Thereby the mesh stiffness appears as an equivalent torsional stiffness

cγ b (db12)2 (Nmrad)

The natural frequencies are found by solving the set of dif-ferential equations (one equation per inertia) Note that for a gear put on a laterally flexible shaft the coupling bending-torsionals is arranged by introducing the gear mass and the lateral stiffness with its relation to the torsional displacement and torque in that shaft

Only the natural frequency (ies) having high relative dis-placement and relative torque through the actual pinion-wheel flexible element need(s) to be considered as critical frequency (ies)

Kv may be determined by means of the method mentioned in 181 thereby using N as the least favourable ratio (in case of more than one pinion-wheel dominated natural frequency) Ie the N-ratio that results in the highest Kv has to be consid-ered

The level of the dynamic factor may also be determined on basis of simulation technique using numeric time integration with relevant tooth stiffness variation and pitchprofile er-rors

19 Face Load Factors KHβ and KFβ The face load factors KHβ for contact stresses and for scuff-ing KFβ for tooth root stresses account for non-uniform load distribution across the facewidth

KHβ is defined as the ratio between the maximum load per unit facewidth and the mean load per unit facewidth

KFβ is defined as the ratio between the maximum tooth root stress per unit facewidth and the mean tooth root stress per unit facewidth The mean tooth root stress relates to the con-sidered facewidth b1 respectively b2

Note that facewidth in this context is the design facewidth b even if the ends are unloaded as often applies to eg bevel gears

The plane of contact is considered

191 Relations between KHβ and KFβ

KFβ = expβHK 2(hb)hb1

1exp++

=

where hb is the ratio tooth heightfacewidth The maximum of h1b1 and h2b2 is to be used but not higher than 13 For double helical gears use only the facewidth of one helix

If the tooth root facewidth (b1 or b2) is considerably wider than b the value of KFβ(1or2) is to be specially considered as it may even exceed KHβ

Eg in pinion-rack lifting systems for jack up rigs where b = b2 asymp mn and b1 asymp 3 mn the typical KHβ asymp KFβ2 asymp 1 and KFβ1 asymp 13

192 Measurement of face load factors Primarily

KFβ may be determined by a number of strain gauges distrib-uted over the facewidth Such strain gauges must be put in

Classification Notes- No 412 11 May 2003

DET NORSKE VERITAS

exactly the same position relative to the root fillet Relations in 191 apply for conversion to KHβ

Secondarily

KHβ may be evaluated by observed contact patterns on vari-ous defined load levels It is imperative that the various test loads are well defined Usually it is also necessary to evalu-ate the elastic deflections Some teeth at each 90 degrees are to be painted with a suitable lacquer Always consider the poorest of the contact patterns

After having run the gear for a suitable time at test load 1 (the lowest) observe the contact pattern with respect to ex-tension over the facewidth Evaluate that KHβ by means of the methods mentioned in this section Proceed in the same way for the next higher test load etc until there is a full face contact pattern From these data the initial mesh misalign-ment (ie without elastic deflections) can be found by ex-trapolation and then also the KHβ at design load can be found by calculation and extrapolation See example

Figure 11 Example of experimental determination of KHβ

It must be considered that inaccurate gears may accumulate a larger observed contact pattern than the actual single mesh to mesh contact patterns This is particularly important for lapped bevel gears Ground or hard metal hobbed bevel gears are assumed to present an accumulated contact pattern that is practically equal the actual single mesh to mesh con-tact patterns As a rough guidance the (observed) accumu-lated contact pattern of lapped bevel gears may be reduced by 10 in order to assess the single mesh to mesh contact pattern which is used in 199

193 Theoretical determination of KHβ The methods described in 193 to 198 may be used for cy-lindrical gears The principles may to some extent also be used for bevel gears but a more practical approach is given in 199

General For gears where the tooth contact pattern cannot be verified during assembly or under load all assumptions are to be well on the safe side

KHβ is to be determined in the plane of contact

The influence parameters considered in this method are

bull mean mesh stiffness cγ (see 111) (if necessary also variable stiffness over b)

bull mean unit load Fmb = Fbt KA Kγ Kvb (for double helical gears see 17 for use of Kγ)

bull misalignment fsh due to elastic deflections of shafts and gear bodies (both pinion and wheel)

bull misalignment fdefl due to elastic deflections of and work-ing positions in bearings

bull misalignment fbe due to bearing clearance tolerances bull misalignment fma due to manufacturing tolerances bull helix modifications as crowning end relief helix correc-

tion bull running in amount yβ (see 112)

In practice several other parameters such as centrifugal ex-pansion thermal expansion housing deflection etc contrib-ute to KHβ However these parameters are not taken into account unless in special cases when being considered as particularly important

When all or most of the am parameters are to be considered the most practical way to determine KHβ is by means of a graphical approach described in 1931

If cγ can be considered constant over the facewidth and no helix modifications apply KHβ can be determined analyti-cally as described in 1932

1931 Graphical method The graphical method utilises the superposition principle and is as follows

bull Calculate the mean mesh deflection Mδ as a function of

γm c and bF see 111

bull Draw a base line with length b and draw up a rectangu-lar with height δM (The area δM b is proportional to the transmitted force)

bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the areas above and below this zero line are equal

Figure 12 fsh balanced around zero line

bull Superimpose these ordinates of the fsh curve to the previ-ous load distribution curve (The area under this new load distribution curve is still δM b)

bull Calculate the bearing deflections andor working posi-tions in the bearings and evaluate the influence fdefl in the plane of contact This is a straight line and is bal-anced around a zero line as indicated in Fig 14 but with one distinct direction Superimpose these ordinates to the previous load distribution curve

12 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

bull The amount of crowning end relief or helix correction (defined in the plane of contact) is to be balanced around a zero line similarly to fsh

Figure 13 Crowning Cc balanced aound zero line

bull Superimpose these ordinates to the previous load distri-bution curve In case of high crowning etc as eg often applied to bevel gears the new load distribution curve may cross the base line (the real zero line) The result is areas with negative load that is not real as the load in those areas should be zero Thus corrective actions must be made but for practical reasons it may be postponed to after next operation

bull The amount of initial mesh misalignment bema ff + (defined in the plane of contact) is to be bal-

anced around a zero line If the direction of bema ff + is known (due to initial contact check) or if the direction of fbe is known due to design (eg overhang bevel pin-ion) this should be taken into account If direction un-known the influence of bema ff + in both directions as well as equal zero should be considered

Figure 14 fma+fbe in both directions balanced around zero line

Superimpose these ordinates to the previous load distri-bution curve This results in up to 3 different curves of which the one with the highest peak is to be chosen for further evaluation

bull If the chosen load distribution curve crosses the base line (ie mathematically negative load) the curve is to be corrected by adding the negative areas and dividing this with the active facewidth The (constant) ordinates of this rectangular correction area are to be subtracted from the positive part of the load distribution curve It is advisable to check that the area covered under this new load distribution curve is still equal δM b

bull If cγ cannot be considered as constant over b then cor-rect the ordinates of the load distribution curve with the local (on various positions over the facewidth) ratio between local mesh stiffness and average mesh stiffness cγ (average over the active facewidth only) Note that the result is to be a curve that covers the same area δM b as before

bull The influence of running in yβ is to be determined as in 112 whereby the value for Fβx is to be taken as twice the distance between the peak of the load distribution curve and δM

bull Determine

KHβ = M

β

δycurveofpeak minus

1932 Simplified analytical method for cylindrical gears The analytical approach is similar to 1931 but has a more limited application as cγ is assumed constant over the face-width and no helix modification applies

bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the area above and below this zero line are equal see Fig 12 The max positive ordinate is frac12∆fsh

bull Calculate the initial mesh alignment as Fβx= deflbemash ffff plusmnplusmnplusmn∆ The negative signs may only be used if this is justified andor verified by a contact pattern test Otherwise al-ways use positive signs If a negative sign is justified the value of Fβx is not to be taken less than the largest of each of these elements

bull Calculate the effective mesh misalignment as Fβy = Fβx - yβ (yβ see 112)

bull Determine

2KforF2

bFc1K βH

m

βγγH le+=β

or

2KforF

bFc2K Hβ

m

βγγH gt=β

where cγ as used here is the effective mesh stiffness see 111

194 Determination of fsh fsh is the mesh misalignment due to elastic deflections Usu-ally it is sufficient to consider the combined mesh deflection of the pinion body and shaft and the wheel shaft The calcu-lation is to be made in the plane of contact (of the considered gear mesh) and to consider all forces (incl axial) acting on the shafts Forces from other meshes can be parted into components parallel respectively vertical to the considered plane of contact Forces vertical to this plane of contact have no influence on fsh

It is advised to use following diameters for toothed elements

d + 2 x mn for bending and shear deflection

d + 2 mn (x ndash ha0 + 02) for torsional deflection

Usually fsh is calculated on basis of an evenly distributed load If the analysis of KHβ shows a considerable maldis-

Classification Notes- No 412 13 May 2003

DET NORSKE VERITAS

tribution in term of hard end contact or if it is known by other reasons that there exists a hard end contact the load should be correspondingly distributed when calculating fsh In fact the whole KHβ procedure can be used iteratively 2-3 iterations will be enough even for almost triangular load distributions

195 Determination of fdefl fdefl is the mesh misalignment in the plane of contact due to bearing deflections and working positions (housing deflec-tion may be included if determined)

First the journal working positions in the bearings are to be determined The influence of external moments and forces must be considered This is of special importance for twin pinion single output gears with all 3 shafts in one plane

For rolling bearings fdefl is further determined on basis of the elastic deflection of the bearings An elastic bearing deflec-tion depends on the bearing load and size and number of rolling elements Note that the bearing clearance tolerances are not included here

For fluid film bearings fdefl is further determined on basis of the lift and angular shift of the shafts due to lubrication oil film thickness Note that fbe takes into account the influence of the bearing clearance tolerance

When working positions bearing deflections and oil film lift are combined for all bearings the angular misalignment as projected into the plane of the contact is to be determined fdefl is this angular misalignment (radians) times the face-width

196 Determination of fbe fbe is the mesh misalignment in the plane of contact due to tolerances in bearing clearances In principle fbe and fdefl could be combined But as fdefl can be determined by analy-sis and has a distinct direction and fbe is dependent on toler-ances and in most cases has no distinct direction (ie + toler-ance) it is practicable to separate these two influences

Due to different bearing clearance tolerances in both pinion and wheel shafts the two shaft axis will have an angular mis-alignment in the plane of contact that is superimposed to the working positions determined in 195 fbe is the facewidth times this angular misalignment Note that fbe may have a distinct direction or be given as a + tolerance or a combina-tion of both For combination of + tolerance it is adviced to use

fbe= 22be

21be ff ++plusmn

fbe is particularly important for overhang designs for gears with widely different kinds of bearings on each side and when the bearings have wide tolerances on clearances In general it shall be possible to replace standard bearings with-out causing the real load distribution to exceed the design premises For slow speed gears with journal bearings the expected wear should also be considered

197 Determination of fma fma is the mesh misalignment due to manufacturing toler-ances (helix slope deviation) of pinion fHβ1 wheel fHβ2 and housing bore

For gear without specifically approved requirements to as-sembly control the value of fma is to be determined as

fma= 22Hβ

21Hβ ff +

For gears with specially approved assembly control the value of fma will depend on those specific requirements

198 Comments to various gear types For double helical gears KHβ is to be determined for both helices Usually an even load share between the helices can be assumed If not the calculation is to be made as de-scribed in 171

For planetary gears the free floating sun pinion suffers only twist no bending It must be noted that the total twist is the sum of the twist due to each mesh If the value of 1K γ ne this must be taken into account when calculating the total sun pinion twist (ie twist calculated with the force per mesh without Kγ and multiplied with the number of planets)

When planets are mounted on spherical bearings the mesh misalignments sun-planet respectively planet-annulus will be balanced Ie the misalignment will be the average between the two theoretical individual misalignments The faceload distribution on the flanks of the planets can take full advan-tage of this However as the sun and annulus mesh with several planets with possibly different lead errors the sun and annulus cannot obtain the above mentioned advantage to the full extent

199 Determination of KHβ for bevel gears If a theoretical approach similar to 193-198 is not docu-mented the following may be used

testeff

H Kb

b851851K sdot

minussdot=β

beff b represents the relative active facewidth (regarding lapped gears see 192 last part)

Higher values than beff b = 090 are normally not to be used in the formula

For dual directional gears it may be difficult to obtain a high beff b in both directions In that case the smaller beff b is to be used

Ktest represents the influence of the bearing arrangement shaft stiffness bearing stiffness housing stiffness etc on the faceload distribution and the verification thereof Expected variations in length- and height-wise tooth profile is also accounted for to some extent

a) Ktest = 1 For ground or hard metal hobbed gears with the specified contact pattern verified at full rating or at full torque slow turning at a condition representative for the thermal expansion at normal operation

14 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

It also applies when the bearing arrangementsupport has insignificant elastic deflections and thermal axial expansion However each initial mesh contact must be verified to be within acceptance criteria that are cali-brated against a type test at full load Reproduction of the gear tooth length- and height-wise profile must also be verified This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon)

b) Ktest = 1 + 04middot(beffbndash06) For designs with possible influence of thermal expansion in the axial direction of the pinion The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load

c) Ktest = 12 if mesh is only checked by toolmakerrsquos blue or by spin test contact For gears in this category beffb gt 085 is not to be used in the calculation

110 Transversal Load Distribution Factors KHα and KFα The transverse load distribution factors KHα for contact stresses and for scuffing KFα for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairs of teeth in mesh

The following relations may be used

Cylindrical gears

( )

minus+==

tH

αptγγHαFα F

byfc0409

2ε

KK

valid for 2εγ le

( ) ( )tH

αptγ

γ

γHαFα F

byfcε

1ε20409KK

minusminus+==

valid for 2ε γ gt

where

FtH = Ft KA Kγ Kv KHβ

cγ = See 111

γα = See 112

fpt = Maximum single pitch deviation (microm) of pinion or wheel or maximum total profile form devia-tion Fα of pinion or wheel if this is larger than the maximum single pitch deviation

Note In case of adequate equivalent tip relief adapted to the load half of the above mentioned fpt can be introduced A tip relief is considered adequate when the average of Ca1 and Ca2 is within plusmn40 of the value of Ceff in 432

Limitations of KHα and KFα

If the calculated values for

KFα = KHα lt 1 use KFα = KHα = 10

If the calculated value of KHα gt 2εα

γ

Zε

ε use KHα = 2

εα

γ

Zε

ε

If the calculated value of KFα gt εα

γ

Yεε

use KFα = εα

γ

Yεε

where Yε = αnε

075025+ (for εαn see 331c)

Bevel gears

For ground or hard metal hobbed gears KFα = KHα = 1

For lapped gears KFα = KHα = 11

111 Tooth Stiffness Constants cacute and cγ The tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having 1 mm facewidth by an amount of 1 microm in the plane of contact cacute is the maximum stiffness of a single pair of teeth cγ is the mean value of the mesh stiffness in a transverse plane (brief term mesh stiffness)

Both valid for high unit load (Unit load = Ft middot KA middot Kγb)

Cylindrical gears The real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bending stiff-nesses For high unit loads the Hertzian stiffness has little importance and can be disregarded This approach is on the safe side for determination of KHβ and KHα However for moderate or low loads Kv may be underestimated due to determination of a too high resonance speed

The linear approach is described in A

An optional approach for inclusion of the non-linear stiffness is described in B

A The linear approach

BRCCq

βcos08acutec =

and

( )025ε075cc αγ +prime=

where

( )[ ]n02a01a

B α2000212

hh12051C minusminus

+minus+=

12n1n

x000635z

025791z

015551004723q minus++=

12

2n

22

1n

1 x005290z

x024188x000193z

x011654+minusminusminus

+ 000182 x22

Classification Notes- No 412 15 May 2003

DET NORSKE VERITAS

(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as

( )( )nR m5s

sR e5

bbln1C +=

where

bs = thickness of a central web

sR = average thickness of rim (net value from tooth root to inside of rim)

The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered

Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ

B The non-linear approach

In the following an example is given on how to consider the non-linearity

The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm

With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as

( )10δKbF

minus= for 500bFgt

minus=

500Fb10δK

bF for 500

bFlt

with γAt KK

bF

bF

sdotsdot= etc (Nmm) ie unit load incorporat-

ing the relevant factors as

KA middot Kγ for determination of Kv

KA middot Kγ middot Kv for determination of KHβ

KA middot Kγ middot Kv middot KHβ for determination of KHα

δ = mesh deflection (microm)

K = applicable stiffness (c or cγ)

Use of stiffnesses for KV KHβ and KHα

For calculation of Kv and KHα the stiffness is calculated as follows

When Fb lt 500

the stiffness is determined as δ∆

∆ bF

where the increment is chosen as eg ∆ Fb = 10 and thus

50010Fb10

K10Fb∆δ +

++

=

When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly

or an equivalent stiffness determined as δsdotb

F

Bevel gears

In lack of more detailed relationship between stiffness and geometry the following may be used

b085

b13cacute eff=

b085b

16c effγ =

beff not to be used in excess of 085 b in these formulae

Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)

112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements

yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload

Cay is defined as the running-in amount that compensates for lack of tip relief

The following relations may be used

For not surface hardened steel

ptHlim

α fσ160y =

yβ = βxlimH

f320σ

with the following upper limits

V lt 5 ms 5-10 ms gt 10 ms

yα max none

limH

12800σ limH

6400σ

yβ max none

limH

25600σ limH

12800σ

For surface hardened steel

yα = 0075 fpt but not more than 3 for any speed

yβ = 015 Fβx but not more than 6 for any speed

16 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

For all kinds of steel

5145189718

1C2

limHay +

minusσ

=

When pinion and wheel material differ the following ap-plies

bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv

bull Use ( )2β1ββ yy21y += in the calculation of KHβ

bull Use ( )2ay1aya CC21C += in the calculation of Kv

bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing

calculation if no design tip relief is foreseen

Classification Notes- No 412 17 May 2003

DET NORSKE VERITAS

2 Calculation of Surface Durability

21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213

Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted

Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213

For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength

For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied

22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater

Calculation of surface durability for helical gears is based on the contact stress at the pitch point

For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies

Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact

Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses

The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses

The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel

221 Contact stress Cylindrical gears

( )HαHβvγA

1

tβεEHDBH KKKKK

bud1uF

ZZZZZσ+

=

where

ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)

ZH = Zone factor for pitch point (see 231)

ZE = Elasticity factor (see 24)

Zε = Contact ratio factor (see 25)

Zβ = Helix angle factor (see 26)

Ft KA Kγ Kv KHβ KHα see 15 ndash 110

d1 b u see 12 ndash 15

Bevel gears

( )HαHβvγA

v1v

vmtKEMH KKKKK

bud1uF

ZZZ105σ+

sdot=

where

105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)

ZE KA etc see above

ZM = mid-zone factor see 233

ZK = bevel gear factor see 27

Fmt dv1 uv see 12 ndash 15

It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks

222 Permissible contact stress

XWRvLH

NHlimHP ZZZZZ

SZσ

σ =

where

σH lim = Endurance limit for contact stresses (see 28)

ZN = Life factor for contact stresses (see 29)

SH = Required safety factor according to the rules

ZLZvZR = Oil film influence factors (see 210)

ZW = Work hardening factor (see 211)

ZX = Size factor (see 212)

18 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

23 Zone Factors ZH ZBD and ZM

231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder

wtt2

wtbH sinααcos

cosαcosβ2Z =

232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel

For εβ ge 1 ZBD = 1

For internal gears ZD = 1

For εβ = 0 (spur gears)

( )

π

minusεminusminus

π

minusminus

α=

α2

2

2b

2a

1

2

1b

1a

wtB

z211

dd

z21

dd

tanZ

( )

π

minusεminusminus

π

minusminus

α=

α1

2

1b

1a

2

2

2b

2a

wtD

z211

dd

z21

dd

tanZ

If ZB lt 1 use ZB = 1

If ZD lt 1 use ZD = 1

For 0 lt εβ lt 1

ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)

233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears

εminusminus

εminusminus

αβ=

αα btm2

2vb2

2vabtm2

1vb2val

2v1vvtbmM

pddpdd

ddtancos2Z

This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint

234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel

24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses

For steel against steel ZE = 1898

25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses

αε1Z =ε for 1εβ ge

( )α

ββ

αε ε

εε1

3ε4

Z +minusminus

= for εβ lt 1

26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability

cosβZβ =

27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply

The following may be used ZK = 080

28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ

σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting

For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply

For this purpose pitting is defined by

bull for not surface hardened gears pitted area ge 2 of total active flank area

bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given

material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies

The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as

Classification Notes- No 412 19 May 2003

DET NORSKE VERITAS

the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99

σHlim σH105 σH10

3

Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade

1650 1500

2500 2400

3100 3100

Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)

1000

13 σHlim

13 σHlim

Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV

Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV

Carbon steel 15 HV + 250 16 σHlim 16 σHlim

These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15

29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply

If this is not documented by approved fatigue tests the fol-lowing method may be used

For all steels except nitrided

7L 105N sdotge ZN = 1 or

01570

L

7

N N105Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 5middot107 510N

logZ037

L

7

N N105Z

sdot=

NL = 105 WXRVLHlim

WstX10H1010NN ZZZZZσ

ZZσZZ

55

5=

103 lt NL lt 105 )Z(Zlog05

L

5

10NN

5N103N10

5N10ZZ

==

3L 10N le

WXRVLlimH

Wst10X10H10NN ZZZZZ

ZZZZ

33

3σ

σ==

(but not less than ZN105)

For nitrided steels

6L 102N sdotge ZN = 1 or

00980

L

6

N N102Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 2middot106 510N

Zlog76860

L

6

N N102Z

sdot=

5L 10N le

XWRVL

10XWst10NN ZZZZZ

ZZ13ZZ

55 ==

Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles

210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity

The following methods may be applied in connection with the endurance limit

20 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Surface hardened steels Not surface hardened steels

ZL ( )24013421

360910ν+

+

( )24013421

680830ν+

+

ZV ( )v3280

140930+

+ ( )v3280300850

++

ZR

080

ZrelR3

150

ZrelR3

where

ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits

For values of ν40 gt 500 use ν40 = 500

RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm

RZrel

= ( ) 31

cZ2Z1 ρ

10RR50

+

RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)

For 5

L 10N le ZL ZV ZR = 10

211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface

The following approximation may be used for the endurance limit

Surface hardened steel against not surface hardened steel 150

ZeqW R

31700

130HB21Z

minus

minus=

where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130

RZeq = equivalent roughness

033

c40

066

ZS

ZHZHZEQ ρvν

15000RRRR

=

If RZeq gt 16 then use RZeq = 16

If RZeq lt 15 then use RZeq = 15

where

RZH = surface roughness of the hard member before run in

RZS = surface roughness of the soft member before run in

ν40 = see 210

If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem

Through hardened pinion against softer wheel

( )

minussdotminus+= 000829

HBHB0008981u1Z

2

1W

For 21HBHB

2

1 le use ZW = 1

For 71HBHB

2

1 gt use 17HBHB

2

1 =

For u gt 20 use u = 20

For static strength (lt 105 cycles)

Surface hardened against not surface hardened

ZWst = 105

Through hardened pinion against softer wheel

ZWst = 1

212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality

ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213

213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented

The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent

Classification Notes- No 412 21 May 2003

DET NORSKE VERITAS

stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically

The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence

The real Hertzian stresses σHR are determined as

For helical gears with εβ gt 1

σHR = σH

For helical gears with εβ lt 1 and spur gears

ε

α

ββ ε

ε+εminus

sdotσ=σZ

1

HHR

For bevel gears

KHHR Z

1σσ sdot=

The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)

The coordinates tz and HV are to be compared with the de-sign specification such as

bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400

and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)

In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods

For high cycle fatigue (gt3 106 cycles) the following applies

sdot+

minussdotsdotsdot= o90

05azt

05azt

cosSσ04HV

H

HHHR

applicable to 05a

zt

Hge

For 05at

H

z lt the value for 05at

H

z = applies

56300ρSσ12a cHHR

Hsdotsdot

sdot=

Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength

If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows

reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue

For static strength (lt103 cycles) the following applies

sdot+

minussdotsdotsdot= o90

07at

06a

zt

cosSσ019HV

Hst

z

HstHHR

applicable to 06at

Hst

z ge

56300ρSσa cHHR

Hstsdotsdot

=

In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies

For limited life fatigue (103 lt cycles lt 3106 )

For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance

The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram

minussdotminus

= sdot logN3477

logSlogSlogS

310H6103HHN

36 10H103H Slog86281Slog86280 sdot+sdot sdot

22 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3 Calculation of Tooth Strength

31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding

For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36

For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning

Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα

In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314

A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired

It should be noted that this part 3 does not cover fractures caused by

bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining

Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears

Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied

32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level

321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance

Normally the stresses for pinion and wheel are calculated as

Cylindrical gears

FαFβvγAβSFn

tF K K K K K Y Y Y

m bFσ =

where

YF = Tooth form factor (see 33)

YS = Stress correction factor (see 34)

Yβ = Helix angle factor (see 36)

Ft KA Kγ Kv KFβ KFα see 15 ndash 110

b see 13

Bevel gears

FαFβvγASaFamn

mtF K K K K K Y Y Y

m bFσ ε=

where

YFa = Tooth form factor see 33

YSa = Stress correction factor see 34

Yε = Contact ratio factor see 35

Fmt KA etc see 15 ndash 110

b see 13

322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is

CXRrelTrelTF

NMFEFP YYYY

SYY

δσ

=σ

Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN

where

σFE = Local tooth root bending endurance limit of reference test gear (see 37)

YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)

YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)

SF = Required safety factor according to the rules

YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)

YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)

Classification Notes- No 412 23 May 2003

DET NORSKE VERITAS

YX = Size factor (see 312)

YC = Case depth factor (see 313)

33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress

YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears

YFa applies to load application at the tooth tip and is used for bevel gears

Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears

Figure 31 External tooth in normal section

Figure 32 Internal tooth in normal section

Definitions

n

2

n

Fn

enFn

Fe

F

α cosms

α cosmh6

Y

=

n

2

n

Fn

anFn

Fa

Fa

α cosms

α cosmh6

Y

=

In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth

YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)

The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing

mn with mnm

zn with zvn

αt with αvt

β with βm

with undercut without undercut

Fig 33 Dimensions and basic rack profile of the teeth (finished profile)

Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless

331 Determination of parameters

( )n

n

prnfPnfP m

α cossαsin 1ρ

αtan h4πE

minusminusminusminus=

For external gears fPfP ρρ =

For internal gears ( )0z

195fPfP0

fPfP 10363156ρhxρρ

sdotminus+

+=

where

z0 = number of teeth of pinion cutter

x0 = addendum modification coefficient of pinion cutter

hfP = addendum of pinion cutter

ρfP = tip radius of pinion cutter

xhρG fpfp +minus=

24 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

τmE

2π

z2H

nnminus

minus=

with

3πτ = for external gears

6πτ = for internal gears

Htan zG2

nminusϑ=ϑ

(to be solved iteratively suitable start value6π

=ϑ for exter-

nal gears and 3π for internal gears)

a) Tooth root chord sFn For external gears

minus

ϑ+

ϑminus= ρ

cosG3

3πsinz

ms

fpnn

Fn

For bevel gears with a tooth thickness modification

xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn

For internal gears

minus

ϑ+

ϑminus= ρ

cosG

6πsinz

ms

fPnn

Fn

b) Root fillet radius ρF at 30ordm tangent

( )G2coszcosG2ρ

mρ

2n

2

fpn

F

minusϑϑ+=

c) Determination of bending moment arm hF dn = zn mn

b2α

αn βcosε

ε =

dan = dn + 2 ha

pbn = π mn cos αn

dbn = dn cos αn

( )4

d1εpzz

2dd

zz2d

2bn

2

αnbn

2bn

2an

en +

minusminus

minus=

en

bnen d

dcos arcα =

ennnn

e α invα invα x tan 22π

z1

minus+

+=γ

αFen = αen ndash γe

For external gears

( )

minus=

n

enFenee

n

Fe

mdαtan sin γ γcos

21

mh

]ρcos

G3πcosz fpn +

ϑminus

ϑminusminus

For internal gears

( ) minus

sdotsdotminus=

n

enFenee

n

Fe

mdα tanγ sinγ cos

21

mh

minus

ϑminus

ϑminussdot ρ

cosG3

6πcosz fPn

332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as

250ε205for 0666ε2366Y αnαnDT leleminus=

205εfor 10Y αnDT lt=

34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section

Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm

YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip

YS can be determined as follows

( )

+

+=L23121

1

sS qL01312Y

where Fe

Fn

hsL = and

F

Fns ρ2

sq = (see 33)

YSa can be calculated by replacing hFe with hFa in the above formulae

Note a) Range of validity 1 lt qs lt 8

In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered

b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with

Classification Notes- No 412 25 May 2003

DET NORSKE VERITAS

g

g

ρt

0613

13

minus

where

tg = depth of the grinding notch

ρg = radius of the grinding notch

c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles

35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears

The following may be used

Yε = 0625

36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank

The following may be used (β input in degrees)

Yβ = 1 ndash εβ β120

When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula

However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)

37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM

σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision

If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules

σFE

Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)

bull of specially approved high grade

1050

bull of normal grade

minus CrNiMo steels with approved process

1000

minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850

Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

840

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

720

Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)

07 HV + 300

Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)

025 σB + 125

Alloyed quenched and tempered steel 04 σB + 200

Carbon steel 025 σB + 250

Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40

1) These values are valid for a root radius

bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)

bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)

bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for

6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked

2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)

26 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears

YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0

YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst

For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316

The following method may be used within a stress ratio ndash12 lt R lt 05

381 For idlers planets and PTO with ice class

M1M1R1

1Yor Y MstM

+minus

minus=

where

R = stress ratio = min stress divided by max stress

For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12

For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as

branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus

For a power take off (PTO) with ice class see 161 c

M considers the mean stress influence on the endurance (or static) strength amplitudes

M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress

Following M values may be used

Endurance limit

Static strength

Case hardened 08 ndash 015 Ys 1) 07

If shot peened 04 06

Nitrided 03 03

Induction or flame hardened

04 06

Not surface hardened steel 03 05

Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M

The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches

382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106

For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers

For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used

383 For gears with shrinkage stresses and unidirectional load For endurance strength

FE

fitM σ

σM1

M21Y+

minus=

σFE is the endurance limit for R = 0

For static strength YMst = 1 and σfit accounted for in 39b

σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor

n

Ffit m

ρ215scf minus=

384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is

( ) ( ) FE

fitM σ

σR1M1

M2

M1M1R1

1Yminussdot+

minus

+minus

minus=

Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12

For static strength

M1M1R1

1YMst

+minus

minus=

The effect of σfit is accounted for in 39 b

Classification Notes- No 412 27 May 2003

DET NORSKE VERITAS

385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed

F

y

Sσ225

for not surface hardened fillets

FSHV5 for surface hardened fillets

39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles

Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN

If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used

Determination of the σ ndash N ndash curve

a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF

b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ

FstS1

FPstσ minussdotsdotsdot=

where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability

σFst

Alloyed case hardened steel 1) 2300

Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

1250

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

1050

Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)

18 HV + 800

Steel with not surface hardened fillets the smaller value of 2)

18 σB or 225 σy

1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet

c) Calculate YN as

NL gt 3middot106

YN = 1 or 001

L

6

N N103Y

sdot= ie Yn = 092 for 1010

The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2

103ltNLlt3middot106exp

L

6

N N103Y

sdot=

cycles6103forσcycles310forσlog02876exp

FP

FPst

sdot=

NL lt 103 cycles6103forσcycles310forσ

NYFP

FPst

sdot=

or simply use σFPst as mentioned in b) directly

Guidance on number of load cycles NL for various applica-tions

bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)

bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied

28 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage

YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit

The following method may be used

For endurance limit

for not surface hardened fillets

( )

024

s024

relT σ103133q21σ1012201351

Ysdotsdotminus

+sdotsdotminus+= minus

minus

δ

for all surface hardened fillets except nitrided

( )

106q21002451

Y srelT

++=δ

for nitrided fillets

( )

1347q2101421

Y srelT

++=δ

For static strength

for not surface hardened fillets1)

( )( )( )025

02

02502s

relTst σ3000821σ300 1Y0821Y

+

minus+=δ

for surface hardened fillets except nitrided

YδrelTst = 044 YS + 012

for nitrided fillets

YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not

exceed the yield point and thereby alter the residual stress level See also 39b footnote 2

311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness

YRrelT differs for endurance limit and static strength

The following method may be used

For endurance limit

YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels

YδrelT = 43 ndash 326 (Ry + 1)0005

for nitrided steels

For static strength

YRrelTst = 1 for all Ry and all materials

For a fillet without any longitudinal machining trace Ry asymp Rz

312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength

The following may be used

For endurance limit

YX = 1 for mn le 5 generally

YX = 103 ndash 0006 mn for 5 lt mn lt 30

YX = 085 for mn ge 30

for not surface hard-ened steels

YX = 105 ndash 001 mn for 5 lt mn ge 25

YX = 08 for mn ge 25

for surface hardened steels

For static strength

YXst = 1 for all mn and all materials

313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength

YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength

In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength

The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses

The following simplified method for YC may be used

YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)

YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10

For endurance limit

+

+=nFFE

C m02ρt31

σconstY

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

10 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

1815 Kv in the supercritical range Cylindrical gears N ge 15

Bevel gears N ge 15

Special care should be taken as to influence of higher vibra-tion modes andor influence of coupled bending (ie lateral shaft vibrations) and torsional vibrations between pinion and wheel These influences are not covered by the following approach

Kv = Cv5 Bp + Cv6 Bf + Cv7

Cv5 accounts for the pitch error influence

Cv5 = 047

Cv6 accounts for the profile error influence

Cv6 = 047 for εγ le 2

Cv6 = 174ε

012

γ minus for εγ gt 2

Cv7 relates the maximum externally applied tooth loading to the maximum tooth loading of ideal accurate gears operating in the supercritical speed sector when the circumferential vibration becomes very soft

Cv7 = 075 for εγ le 15

Cv7 = [ ] 8750)2(sin 0125 +minusβεπ for 15 lt εγ le 25

Cv7 = 10 for εγ gt 25

1816 Kv in the intermediate range Cylindrical gears 115 lt N lt 15

Bevel gears 125 lt N lt 15

Comments raised in 1814 and 1815 should be observed

Kv is determined by linear interpolation between Kv for N = 115 respectively 125 and N = 15 as

Cylindrical gears

( ) ( ) ( )[ ]51Nv151Nv51Nvv KK350

N51KK === minussdot

minus

+=

Bevel gears

( ) ( ) ( )[ ]51Nv251Nv51Nvv KK250

N51KK === minussdot

minus

+=

182 Multi-resonance method For high speed gear (vgt40 ms) for multimesh medium speed gears for gears with significant lateral shaft flexibility etc it is advised to determine Kv on basis of relevant dy-namic analysis

Incorporating lateral shaft compliance requires transforma-tion of even a simple pinion-wheel system into a lumped multi-mass system It is advised to incorporate all relevant inertias and torsional shaft stiffnesses into an equivalent (to pinion speed) system Thereby the mesh stiffness appears as an equivalent torsional stiffness

cγ b (db12)2 (Nmrad)

The natural frequencies are found by solving the set of dif-ferential equations (one equation per inertia) Note that for a gear put on a laterally flexible shaft the coupling bending-torsionals is arranged by introducing the gear mass and the lateral stiffness with its relation to the torsional displacement and torque in that shaft

Only the natural frequency (ies) having high relative dis-placement and relative torque through the actual pinion-wheel flexible element need(s) to be considered as critical frequency (ies)

Kv may be determined by means of the method mentioned in 181 thereby using N as the least favourable ratio (in case of more than one pinion-wheel dominated natural frequency) Ie the N-ratio that results in the highest Kv has to be consid-ered

The level of the dynamic factor may also be determined on basis of simulation technique using numeric time integration with relevant tooth stiffness variation and pitchprofile er-rors

19 Face Load Factors KHβ and KFβ The face load factors KHβ for contact stresses and for scuff-ing KFβ for tooth root stresses account for non-uniform load distribution across the facewidth

KHβ is defined as the ratio between the maximum load per unit facewidth and the mean load per unit facewidth

KFβ is defined as the ratio between the maximum tooth root stress per unit facewidth and the mean tooth root stress per unit facewidth The mean tooth root stress relates to the con-sidered facewidth b1 respectively b2

Note that facewidth in this context is the design facewidth b even if the ends are unloaded as often applies to eg bevel gears

The plane of contact is considered

191 Relations between KHβ and KFβ

KFβ = expβHK 2(hb)hb1

1exp++

=

where hb is the ratio tooth heightfacewidth The maximum of h1b1 and h2b2 is to be used but not higher than 13 For double helical gears use only the facewidth of one helix

If the tooth root facewidth (b1 or b2) is considerably wider than b the value of KFβ(1or2) is to be specially considered as it may even exceed KHβ

Eg in pinion-rack lifting systems for jack up rigs where b = b2 asymp mn and b1 asymp 3 mn the typical KHβ asymp KFβ2 asymp 1 and KFβ1 asymp 13

192 Measurement of face load factors Primarily

KFβ may be determined by a number of strain gauges distrib-uted over the facewidth Such strain gauges must be put in

Classification Notes- No 412 11 May 2003

DET NORSKE VERITAS

exactly the same position relative to the root fillet Relations in 191 apply for conversion to KHβ

Secondarily

KHβ may be evaluated by observed contact patterns on vari-ous defined load levels It is imperative that the various test loads are well defined Usually it is also necessary to evalu-ate the elastic deflections Some teeth at each 90 degrees are to be painted with a suitable lacquer Always consider the poorest of the contact patterns

After having run the gear for a suitable time at test load 1 (the lowest) observe the contact pattern with respect to ex-tension over the facewidth Evaluate that KHβ by means of the methods mentioned in this section Proceed in the same way for the next higher test load etc until there is a full face contact pattern From these data the initial mesh misalign-ment (ie without elastic deflections) can be found by ex-trapolation and then also the KHβ at design load can be found by calculation and extrapolation See example

Figure 11 Example of experimental determination of KHβ

It must be considered that inaccurate gears may accumulate a larger observed contact pattern than the actual single mesh to mesh contact patterns This is particularly important for lapped bevel gears Ground or hard metal hobbed bevel gears are assumed to present an accumulated contact pattern that is practically equal the actual single mesh to mesh con-tact patterns As a rough guidance the (observed) accumu-lated contact pattern of lapped bevel gears may be reduced by 10 in order to assess the single mesh to mesh contact pattern which is used in 199

193 Theoretical determination of KHβ The methods described in 193 to 198 may be used for cy-lindrical gears The principles may to some extent also be used for bevel gears but a more practical approach is given in 199

General For gears where the tooth contact pattern cannot be verified during assembly or under load all assumptions are to be well on the safe side

KHβ is to be determined in the plane of contact

The influence parameters considered in this method are

bull mean mesh stiffness cγ (see 111) (if necessary also variable stiffness over b)

bull mean unit load Fmb = Fbt KA Kγ Kvb (for double helical gears see 17 for use of Kγ)

bull misalignment fsh due to elastic deflections of shafts and gear bodies (both pinion and wheel)

bull misalignment fdefl due to elastic deflections of and work-ing positions in bearings

bull misalignment fbe due to bearing clearance tolerances bull misalignment fma due to manufacturing tolerances bull helix modifications as crowning end relief helix correc-

tion bull running in amount yβ (see 112)

In practice several other parameters such as centrifugal ex-pansion thermal expansion housing deflection etc contrib-ute to KHβ However these parameters are not taken into account unless in special cases when being considered as particularly important

When all or most of the am parameters are to be considered the most practical way to determine KHβ is by means of a graphical approach described in 1931

If cγ can be considered constant over the facewidth and no helix modifications apply KHβ can be determined analyti-cally as described in 1932

1931 Graphical method The graphical method utilises the superposition principle and is as follows

bull Calculate the mean mesh deflection Mδ as a function of

γm c and bF see 111

bull Draw a base line with length b and draw up a rectangu-lar with height δM (The area δM b is proportional to the transmitted force)

bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the areas above and below this zero line are equal

Figure 12 fsh balanced around zero line

bull Superimpose these ordinates of the fsh curve to the previ-ous load distribution curve (The area under this new load distribution curve is still δM b)

bull Calculate the bearing deflections andor working posi-tions in the bearings and evaluate the influence fdefl in the plane of contact This is a straight line and is bal-anced around a zero line as indicated in Fig 14 but with one distinct direction Superimpose these ordinates to the previous load distribution curve

12 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

bull The amount of crowning end relief or helix correction (defined in the plane of contact) is to be balanced around a zero line similarly to fsh

Figure 13 Crowning Cc balanced aound zero line

bull Superimpose these ordinates to the previous load distri-bution curve In case of high crowning etc as eg often applied to bevel gears the new load distribution curve may cross the base line (the real zero line) The result is areas with negative load that is not real as the load in those areas should be zero Thus corrective actions must be made but for practical reasons it may be postponed to after next operation

bull The amount of initial mesh misalignment bema ff + (defined in the plane of contact) is to be bal-

anced around a zero line If the direction of bema ff + is known (due to initial contact check) or if the direction of fbe is known due to design (eg overhang bevel pin-ion) this should be taken into account If direction un-known the influence of bema ff + in both directions as well as equal zero should be considered

Figure 14 fma+fbe in both directions balanced around zero line

Superimpose these ordinates to the previous load distri-bution curve This results in up to 3 different curves of which the one with the highest peak is to be chosen for further evaluation

bull If the chosen load distribution curve crosses the base line (ie mathematically negative load) the curve is to be corrected by adding the negative areas and dividing this with the active facewidth The (constant) ordinates of this rectangular correction area are to be subtracted from the positive part of the load distribution curve It is advisable to check that the area covered under this new load distribution curve is still equal δM b

bull If cγ cannot be considered as constant over b then cor-rect the ordinates of the load distribution curve with the local (on various positions over the facewidth) ratio between local mesh stiffness and average mesh stiffness cγ (average over the active facewidth only) Note that the result is to be a curve that covers the same area δM b as before

bull The influence of running in yβ is to be determined as in 112 whereby the value for Fβx is to be taken as twice the distance between the peak of the load distribution curve and δM

bull Determine

KHβ = M

β

δycurveofpeak minus

1932 Simplified analytical method for cylindrical gears The analytical approach is similar to 1931 but has a more limited application as cγ is assumed constant over the face-width and no helix modification applies

bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the area above and below this zero line are equal see Fig 12 The max positive ordinate is frac12∆fsh

bull Calculate the initial mesh alignment as Fβx= deflbemash ffff plusmnplusmnplusmn∆ The negative signs may only be used if this is justified andor verified by a contact pattern test Otherwise al-ways use positive signs If a negative sign is justified the value of Fβx is not to be taken less than the largest of each of these elements

bull Calculate the effective mesh misalignment as Fβy = Fβx - yβ (yβ see 112)

bull Determine

2KforF2

bFc1K βH

m

βγγH le+=β

or

2KforF

bFc2K Hβ

m

βγγH gt=β

where cγ as used here is the effective mesh stiffness see 111

194 Determination of fsh fsh is the mesh misalignment due to elastic deflections Usu-ally it is sufficient to consider the combined mesh deflection of the pinion body and shaft and the wheel shaft The calcu-lation is to be made in the plane of contact (of the considered gear mesh) and to consider all forces (incl axial) acting on the shafts Forces from other meshes can be parted into components parallel respectively vertical to the considered plane of contact Forces vertical to this plane of contact have no influence on fsh

It is advised to use following diameters for toothed elements

d + 2 x mn for bending and shear deflection

d + 2 mn (x ndash ha0 + 02) for torsional deflection

Usually fsh is calculated on basis of an evenly distributed load If the analysis of KHβ shows a considerable maldis-

Classification Notes- No 412 13 May 2003

DET NORSKE VERITAS

tribution in term of hard end contact or if it is known by other reasons that there exists a hard end contact the load should be correspondingly distributed when calculating fsh In fact the whole KHβ procedure can be used iteratively 2-3 iterations will be enough even for almost triangular load distributions

195 Determination of fdefl fdefl is the mesh misalignment in the plane of contact due to bearing deflections and working positions (housing deflec-tion may be included if determined)

First the journal working positions in the bearings are to be determined The influence of external moments and forces must be considered This is of special importance for twin pinion single output gears with all 3 shafts in one plane

For rolling bearings fdefl is further determined on basis of the elastic deflection of the bearings An elastic bearing deflec-tion depends on the bearing load and size and number of rolling elements Note that the bearing clearance tolerances are not included here

For fluid film bearings fdefl is further determined on basis of the lift and angular shift of the shafts due to lubrication oil film thickness Note that fbe takes into account the influence of the bearing clearance tolerance

When working positions bearing deflections and oil film lift are combined for all bearings the angular misalignment as projected into the plane of the contact is to be determined fdefl is this angular misalignment (radians) times the face-width

196 Determination of fbe fbe is the mesh misalignment in the plane of contact due to tolerances in bearing clearances In principle fbe and fdefl could be combined But as fdefl can be determined by analy-sis and has a distinct direction and fbe is dependent on toler-ances and in most cases has no distinct direction (ie + toler-ance) it is practicable to separate these two influences

Due to different bearing clearance tolerances in both pinion and wheel shafts the two shaft axis will have an angular mis-alignment in the plane of contact that is superimposed to the working positions determined in 195 fbe is the facewidth times this angular misalignment Note that fbe may have a distinct direction or be given as a + tolerance or a combina-tion of both For combination of + tolerance it is adviced to use

fbe= 22be

21be ff ++plusmn

fbe is particularly important for overhang designs for gears with widely different kinds of bearings on each side and when the bearings have wide tolerances on clearances In general it shall be possible to replace standard bearings with-out causing the real load distribution to exceed the design premises For slow speed gears with journal bearings the expected wear should also be considered

197 Determination of fma fma is the mesh misalignment due to manufacturing toler-ances (helix slope deviation) of pinion fHβ1 wheel fHβ2 and housing bore

For gear without specifically approved requirements to as-sembly control the value of fma is to be determined as

fma= 22Hβ

21Hβ ff +

For gears with specially approved assembly control the value of fma will depend on those specific requirements

198 Comments to various gear types For double helical gears KHβ is to be determined for both helices Usually an even load share between the helices can be assumed If not the calculation is to be made as de-scribed in 171

For planetary gears the free floating sun pinion suffers only twist no bending It must be noted that the total twist is the sum of the twist due to each mesh If the value of 1K γ ne this must be taken into account when calculating the total sun pinion twist (ie twist calculated with the force per mesh without Kγ and multiplied with the number of planets)

When planets are mounted on spherical bearings the mesh misalignments sun-planet respectively planet-annulus will be balanced Ie the misalignment will be the average between the two theoretical individual misalignments The faceload distribution on the flanks of the planets can take full advan-tage of this However as the sun and annulus mesh with several planets with possibly different lead errors the sun and annulus cannot obtain the above mentioned advantage to the full extent

199 Determination of KHβ for bevel gears If a theoretical approach similar to 193-198 is not docu-mented the following may be used

testeff

H Kb

b851851K sdot

minussdot=β

beff b represents the relative active facewidth (regarding lapped gears see 192 last part)

Higher values than beff b = 090 are normally not to be used in the formula

For dual directional gears it may be difficult to obtain a high beff b in both directions In that case the smaller beff b is to be used

Ktest represents the influence of the bearing arrangement shaft stiffness bearing stiffness housing stiffness etc on the faceload distribution and the verification thereof Expected variations in length- and height-wise tooth profile is also accounted for to some extent

a) Ktest = 1 For ground or hard metal hobbed gears with the specified contact pattern verified at full rating or at full torque slow turning at a condition representative for the thermal expansion at normal operation

14 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

It also applies when the bearing arrangementsupport has insignificant elastic deflections and thermal axial expansion However each initial mesh contact must be verified to be within acceptance criteria that are cali-brated against a type test at full load Reproduction of the gear tooth length- and height-wise profile must also be verified This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon)

b) Ktest = 1 + 04middot(beffbndash06) For designs with possible influence of thermal expansion in the axial direction of the pinion The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load

c) Ktest = 12 if mesh is only checked by toolmakerrsquos blue or by spin test contact For gears in this category beffb gt 085 is not to be used in the calculation

110 Transversal Load Distribution Factors KHα and KFα The transverse load distribution factors KHα for contact stresses and for scuffing KFα for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairs of teeth in mesh

The following relations may be used

Cylindrical gears

( )

minus+==

tH

αptγγHαFα F

byfc0409

2ε

KK

valid for 2εγ le

( ) ( )tH

αptγ

γ

γHαFα F

byfcε

1ε20409KK

minusminus+==

valid for 2ε γ gt

where

FtH = Ft KA Kγ Kv KHβ

cγ = See 111

γα = See 112

fpt = Maximum single pitch deviation (microm) of pinion or wheel or maximum total profile form devia-tion Fα of pinion or wheel if this is larger than the maximum single pitch deviation

Note In case of adequate equivalent tip relief adapted to the load half of the above mentioned fpt can be introduced A tip relief is considered adequate when the average of Ca1 and Ca2 is within plusmn40 of the value of Ceff in 432

Limitations of KHα and KFα

If the calculated values for

KFα = KHα lt 1 use KFα = KHα = 10

If the calculated value of KHα gt 2εα

γ

Zε

ε use KHα = 2

εα

γ

Zε

ε

If the calculated value of KFα gt εα

γ

Yεε

use KFα = εα

γ

Yεε

where Yε = αnε

075025+ (for εαn see 331c)

Bevel gears

For ground or hard metal hobbed gears KFα = KHα = 1

For lapped gears KFα = KHα = 11

111 Tooth Stiffness Constants cacute and cγ The tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having 1 mm facewidth by an amount of 1 microm in the plane of contact cacute is the maximum stiffness of a single pair of teeth cγ is the mean value of the mesh stiffness in a transverse plane (brief term mesh stiffness)

Both valid for high unit load (Unit load = Ft middot KA middot Kγb)

Cylindrical gears The real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bending stiff-nesses For high unit loads the Hertzian stiffness has little importance and can be disregarded This approach is on the safe side for determination of KHβ and KHα However for moderate or low loads Kv may be underestimated due to determination of a too high resonance speed

The linear approach is described in A

An optional approach for inclusion of the non-linear stiffness is described in B

A The linear approach

BRCCq

βcos08acutec =

and

( )025ε075cc αγ +prime=

where

( )[ ]n02a01a

B α2000212

hh12051C minusminus

+minus+=

12n1n

x000635z

025791z

015551004723q minus++=

12

2n

22

1n

1 x005290z

x024188x000193z

x011654+minusminusminus

+ 000182 x22

Classification Notes- No 412 15 May 2003

DET NORSKE VERITAS

(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as

( )( )nR m5s

sR e5

bbln1C +=

where

bs = thickness of a central web

sR = average thickness of rim (net value from tooth root to inside of rim)

The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered

Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ

B The non-linear approach

In the following an example is given on how to consider the non-linearity

The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm

With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as

( )10δKbF

minus= for 500bFgt

minus=

500Fb10δK

bF for 500

bFlt

with γAt KK

bF

bF

sdotsdot= etc (Nmm) ie unit load incorporat-

ing the relevant factors as

KA middot Kγ for determination of Kv

KA middot Kγ middot Kv for determination of KHβ

KA middot Kγ middot Kv middot KHβ for determination of KHα

δ = mesh deflection (microm)

K = applicable stiffness (c or cγ)

Use of stiffnesses for KV KHβ and KHα

For calculation of Kv and KHα the stiffness is calculated as follows

When Fb lt 500

the stiffness is determined as δ∆

∆ bF

where the increment is chosen as eg ∆ Fb = 10 and thus

50010Fb10

K10Fb∆δ +

++

=

When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly

or an equivalent stiffness determined as δsdotb

F

Bevel gears

In lack of more detailed relationship between stiffness and geometry the following may be used

b085

b13cacute eff=

b085b

16c effγ =

beff not to be used in excess of 085 b in these formulae

Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)

112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements

yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload

Cay is defined as the running-in amount that compensates for lack of tip relief

The following relations may be used

For not surface hardened steel

ptHlim

α fσ160y =

yβ = βxlimH

f320σ

with the following upper limits

V lt 5 ms 5-10 ms gt 10 ms

yα max none

limH

12800σ limH

6400σ

yβ max none

limH

25600σ limH

12800σ

For surface hardened steel

yα = 0075 fpt but not more than 3 for any speed

yβ = 015 Fβx but not more than 6 for any speed

16 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

For all kinds of steel

5145189718

1C2

limHay +

minusσ

=

When pinion and wheel material differ the following ap-plies

bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv

bull Use ( )2β1ββ yy21y += in the calculation of KHβ

bull Use ( )2ay1aya CC21C += in the calculation of Kv

bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing

calculation if no design tip relief is foreseen

Classification Notes- No 412 17 May 2003

DET NORSKE VERITAS

2 Calculation of Surface Durability

21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213

Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted

Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213

For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength

For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied

22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater

Calculation of surface durability for helical gears is based on the contact stress at the pitch point

For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies

Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact

Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses

The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses

The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel

221 Contact stress Cylindrical gears

( )HαHβvγA

1

tβεEHDBH KKKKK

bud1uF

ZZZZZσ+

=

where

ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)

ZH = Zone factor for pitch point (see 231)

ZE = Elasticity factor (see 24)

Zε = Contact ratio factor (see 25)

Zβ = Helix angle factor (see 26)

Ft KA Kγ Kv KHβ KHα see 15 ndash 110

d1 b u see 12 ndash 15

Bevel gears

( )HαHβvγA

v1v

vmtKEMH KKKKK

bud1uF

ZZZ105σ+

sdot=

where

105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)

ZE KA etc see above

ZM = mid-zone factor see 233

ZK = bevel gear factor see 27

Fmt dv1 uv see 12 ndash 15

It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks

222 Permissible contact stress

XWRvLH

NHlimHP ZZZZZ

SZσ

σ =

where

σH lim = Endurance limit for contact stresses (see 28)

ZN = Life factor for contact stresses (see 29)

SH = Required safety factor according to the rules

ZLZvZR = Oil film influence factors (see 210)

ZW = Work hardening factor (see 211)

ZX = Size factor (see 212)

18 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

23 Zone Factors ZH ZBD and ZM

231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder

wtt2

wtbH sinααcos

cosαcosβ2Z =

232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel

For εβ ge 1 ZBD = 1

For internal gears ZD = 1

For εβ = 0 (spur gears)

( )

π

minusεminusminus

π

minusminus

α=

α2

2

2b

2a

1

2

1b

1a

wtB

z211

dd

z21

dd

tanZ

( )

π

minusεminusminus

π

minusminus

α=

α1

2

1b

1a

2

2

2b

2a

wtD

z211

dd

z21

dd

tanZ

If ZB lt 1 use ZB = 1

If ZD lt 1 use ZD = 1

For 0 lt εβ lt 1

ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)

233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears

εminusminus

εminusminus

αβ=

αα btm2

2vb2

2vabtm2

1vb2val

2v1vvtbmM

pddpdd

ddtancos2Z

This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint

234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel

24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses

For steel against steel ZE = 1898

25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses

αε1Z =ε for 1εβ ge

( )α

ββ

αε ε

εε1

3ε4

Z +minusminus

= for εβ lt 1

26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability

cosβZβ =

27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply

The following may be used ZK = 080

28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ

σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting

For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply

For this purpose pitting is defined by

bull for not surface hardened gears pitted area ge 2 of total active flank area

bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given

material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies

The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as

Classification Notes- No 412 19 May 2003

DET NORSKE VERITAS

the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99

σHlim σH105 σH10

3

Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade

1650 1500

2500 2400

3100 3100

Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)

1000

13 σHlim

13 σHlim

Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV

Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV

Carbon steel 15 HV + 250 16 σHlim 16 σHlim

These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15

29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply

If this is not documented by approved fatigue tests the fol-lowing method may be used

For all steels except nitrided

7L 105N sdotge ZN = 1 or

01570

L

7

N N105Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 5middot107 510N

logZ037

L

7

N N105Z

sdot=

NL = 105 WXRVLHlim

WstX10H1010NN ZZZZZσ

ZZσZZ

55

5=

103 lt NL lt 105 )Z(Zlog05

L

5

10NN

5N103N10

5N10ZZ

==

3L 10N le

WXRVLlimH

Wst10X10H10NN ZZZZZ

ZZZZ

33

3σ

σ==

(but not less than ZN105)

For nitrided steels

6L 102N sdotge ZN = 1 or

00980

L

6

N N102Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 2middot106 510N

Zlog76860

L

6

N N102Z

sdot=

5L 10N le

XWRVL

10XWst10NN ZZZZZ

ZZ13ZZ

55 ==

Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles

210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity

The following methods may be applied in connection with the endurance limit

20 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Surface hardened steels Not surface hardened steels

ZL ( )24013421

360910ν+

+

( )24013421

680830ν+

+

ZV ( )v3280

140930+

+ ( )v3280300850

++

ZR

080

ZrelR3

150

ZrelR3

where

ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits

For values of ν40 gt 500 use ν40 = 500

RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm

RZrel

= ( ) 31

cZ2Z1 ρ

10RR50

+

RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)

For 5

L 10N le ZL ZV ZR = 10

211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface

The following approximation may be used for the endurance limit

Surface hardened steel against not surface hardened steel 150

ZeqW R

31700

130HB21Z

minus

minus=

where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130

RZeq = equivalent roughness

033

c40

066

ZS

ZHZHZEQ ρvν

15000RRRR

=

If RZeq gt 16 then use RZeq = 16

If RZeq lt 15 then use RZeq = 15

where

RZH = surface roughness of the hard member before run in

RZS = surface roughness of the soft member before run in

ν40 = see 210

If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem

Through hardened pinion against softer wheel

( )

minussdotminus+= 000829

HBHB0008981u1Z

2

1W

For 21HBHB

2

1 le use ZW = 1

For 71HBHB

2

1 gt use 17HBHB

2

1 =

For u gt 20 use u = 20

For static strength (lt 105 cycles)

Surface hardened against not surface hardened

ZWst = 105

Through hardened pinion against softer wheel

ZWst = 1

212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality

ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213

213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented

The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent

Classification Notes- No 412 21 May 2003

DET NORSKE VERITAS

stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically

The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence

The real Hertzian stresses σHR are determined as

For helical gears with εβ gt 1

σHR = σH

For helical gears with εβ lt 1 and spur gears

ε

α

ββ ε

ε+εminus

sdotσ=σZ

1

HHR

For bevel gears

KHHR Z

1σσ sdot=

The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)

The coordinates tz and HV are to be compared with the de-sign specification such as

bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400

and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)

In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods

For high cycle fatigue (gt3 106 cycles) the following applies

sdot+

minussdotsdotsdot= o90

05azt

05azt

cosSσ04HV

H

HHHR

applicable to 05a

zt

Hge

For 05at

H

z lt the value for 05at

H

z = applies

56300ρSσ12a cHHR

Hsdotsdot

sdot=

Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength

If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows

reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue

For static strength (lt103 cycles) the following applies

sdot+

minussdotsdotsdot= o90

07at

06a

zt

cosSσ019HV

Hst

z

HstHHR

applicable to 06at

Hst

z ge

56300ρSσa cHHR

Hstsdotsdot

=

In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies

For limited life fatigue (103 lt cycles lt 3106 )

For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance

The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram

minussdotminus

= sdot logN3477

logSlogSlogS

310H6103HHN

36 10H103H Slog86281Slog86280 sdot+sdot sdot

22 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3 Calculation of Tooth Strength

31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding

For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36

For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning

Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα

In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314

A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired

It should be noted that this part 3 does not cover fractures caused by

bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining

Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears

Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied

32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level

321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance

Normally the stresses for pinion and wheel are calculated as

Cylindrical gears

FαFβvγAβSFn

tF K K K K K Y Y Y

m bFσ =

where

YF = Tooth form factor (see 33)

YS = Stress correction factor (see 34)

Yβ = Helix angle factor (see 36)

Ft KA Kγ Kv KFβ KFα see 15 ndash 110

b see 13

Bevel gears

FαFβvγASaFamn

mtF K K K K K Y Y Y

m bFσ ε=

where

YFa = Tooth form factor see 33

YSa = Stress correction factor see 34

Yε = Contact ratio factor see 35

Fmt KA etc see 15 ndash 110

b see 13

322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is

CXRrelTrelTF

NMFEFP YYYY

SYY

δσ

=σ

Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN

where

σFE = Local tooth root bending endurance limit of reference test gear (see 37)

YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)

YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)

SF = Required safety factor according to the rules

YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)

YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)

Classification Notes- No 412 23 May 2003

DET NORSKE VERITAS

YX = Size factor (see 312)

YC = Case depth factor (see 313)

33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress

YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears

YFa applies to load application at the tooth tip and is used for bevel gears

Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears

Figure 31 External tooth in normal section

Figure 32 Internal tooth in normal section

Definitions

n

2

n

Fn

enFn

Fe

F

α cosms

α cosmh6

Y

=

n

2

n

Fn

anFn

Fa

Fa

α cosms

α cosmh6

Y

=

In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth

YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)

The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing

mn with mnm

zn with zvn

αt with αvt

β with βm

with undercut without undercut

Fig 33 Dimensions and basic rack profile of the teeth (finished profile)

Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless

331 Determination of parameters

( )n

n

prnfPnfP m

α cossαsin 1ρ

αtan h4πE

minusminusminusminus=

For external gears fPfP ρρ =

For internal gears ( )0z

195fPfP0

fPfP 10363156ρhxρρ

sdotminus+

+=

where

z0 = number of teeth of pinion cutter

x0 = addendum modification coefficient of pinion cutter

hfP = addendum of pinion cutter

ρfP = tip radius of pinion cutter

xhρG fpfp +minus=

24 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

τmE

2π

z2H

nnminus

minus=

with

3πτ = for external gears

6πτ = for internal gears

Htan zG2

nminusϑ=ϑ

(to be solved iteratively suitable start value6π

=ϑ for exter-

nal gears and 3π for internal gears)

a) Tooth root chord sFn For external gears

minus

ϑ+

ϑminus= ρ

cosG3

3πsinz

ms

fpnn

Fn

For bevel gears with a tooth thickness modification

xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn

For internal gears

minus

ϑ+

ϑminus= ρ

cosG

6πsinz

ms

fPnn

Fn

b) Root fillet radius ρF at 30ordm tangent

( )G2coszcosG2ρ

mρ

2n

2

fpn

F

minusϑϑ+=

c) Determination of bending moment arm hF dn = zn mn

b2α

αn βcosε

ε =

dan = dn + 2 ha

pbn = π mn cos αn

dbn = dn cos αn

( )4

d1εpzz

2dd

zz2d

2bn

2

αnbn

2bn

2an

en +

minusminus

minus=

en

bnen d

dcos arcα =

ennnn

e α invα invα x tan 22π

z1

minus+

+=γ

αFen = αen ndash γe

For external gears

( )

minus=

n

enFenee

n

Fe

mdαtan sin γ γcos

21

mh

]ρcos

G3πcosz fpn +

ϑminus

ϑminusminus

For internal gears

( ) minus

sdotsdotminus=

n

enFenee

n

Fe

mdα tanγ sinγ cos

21

mh

minus

ϑminus

ϑminussdot ρ

cosG3

6πcosz fPn

332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as

250ε205for 0666ε2366Y αnαnDT leleminus=

205εfor 10Y αnDT lt=

34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section

Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm

YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip

YS can be determined as follows

( )

+

+=L23121

1

sS qL01312Y

where Fe

Fn

hsL = and

F

Fns ρ2

sq = (see 33)

YSa can be calculated by replacing hFe with hFa in the above formulae

Note a) Range of validity 1 lt qs lt 8

In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered

b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with

Classification Notes- No 412 25 May 2003

DET NORSKE VERITAS

g

g

ρt

0613

13

minus

where

tg = depth of the grinding notch

ρg = radius of the grinding notch

c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles

35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears

The following may be used

Yε = 0625

36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank

The following may be used (β input in degrees)

Yβ = 1 ndash εβ β120

When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula

However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)

37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM

σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision

If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules

σFE

Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)

bull of specially approved high grade

1050

bull of normal grade

minus CrNiMo steels with approved process

1000

minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850

Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

840

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

720

Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)

07 HV + 300

Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)

025 σB + 125

Alloyed quenched and tempered steel 04 σB + 200

Carbon steel 025 σB + 250

Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40

1) These values are valid for a root radius

bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)

bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)

bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for

6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked

2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)

26 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears

YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0

YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst

For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316

The following method may be used within a stress ratio ndash12 lt R lt 05

381 For idlers planets and PTO with ice class

M1M1R1

1Yor Y MstM

+minus

minus=

where

R = stress ratio = min stress divided by max stress

For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12

For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as

branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus

For a power take off (PTO) with ice class see 161 c

M considers the mean stress influence on the endurance (or static) strength amplitudes

M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress

Following M values may be used

Endurance limit

Static strength

Case hardened 08 ndash 015 Ys 1) 07

If shot peened 04 06

Nitrided 03 03

Induction or flame hardened

04 06

Not surface hardened steel 03 05

Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M

The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches

382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106

For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers

For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used

383 For gears with shrinkage stresses and unidirectional load For endurance strength

FE

fitM σ

σM1

M21Y+

minus=

σFE is the endurance limit for R = 0

For static strength YMst = 1 and σfit accounted for in 39b

σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor

n

Ffit m

ρ215scf minus=

384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is

( ) ( ) FE

fitM σ

σR1M1

M2

M1M1R1

1Yminussdot+

minus

+minus

minus=

Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12

For static strength

M1M1R1

1YMst

+minus

minus=

The effect of σfit is accounted for in 39 b

Classification Notes- No 412 27 May 2003

DET NORSKE VERITAS

385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed

F

y

Sσ225

for not surface hardened fillets

FSHV5 for surface hardened fillets

39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles

Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN

If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used

Determination of the σ ndash N ndash curve

a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF

b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ

FstS1

FPstσ minussdotsdotsdot=

where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability

σFst

Alloyed case hardened steel 1) 2300

Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

1250

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

1050

Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)

18 HV + 800

Steel with not surface hardened fillets the smaller value of 2)

18 σB or 225 σy

1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet

c) Calculate YN as

NL gt 3middot106

YN = 1 or 001

L

6

N N103Y

sdot= ie Yn = 092 for 1010

The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2

103ltNLlt3middot106exp

L

6

N N103Y

sdot=

cycles6103forσcycles310forσlog02876exp

FP

FPst

sdot=

NL lt 103 cycles6103forσcycles310forσ

NYFP

FPst

sdot=

or simply use σFPst as mentioned in b) directly

Guidance on number of load cycles NL for various applica-tions

bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)

bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied

28 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage

YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit

The following method may be used

For endurance limit

for not surface hardened fillets

( )

024

s024

relT σ103133q21σ1012201351

Ysdotsdotminus

+sdotsdotminus+= minus

minus

δ

for all surface hardened fillets except nitrided

( )

106q21002451

Y srelT

++=δ

for nitrided fillets

( )

1347q2101421

Y srelT

++=δ

For static strength

for not surface hardened fillets1)

( )( )( )025

02

02502s

relTst σ3000821σ300 1Y0821Y

+

minus+=δ

for surface hardened fillets except nitrided

YδrelTst = 044 YS + 012

for nitrided fillets

YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not

exceed the yield point and thereby alter the residual stress level See also 39b footnote 2

311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness

YRrelT differs for endurance limit and static strength

The following method may be used

For endurance limit

YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels

YδrelT = 43 ndash 326 (Ry + 1)0005

for nitrided steels

For static strength

YRrelTst = 1 for all Ry and all materials

For a fillet without any longitudinal machining trace Ry asymp Rz

312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength

The following may be used

For endurance limit

YX = 1 for mn le 5 generally

YX = 103 ndash 0006 mn for 5 lt mn lt 30

YX = 085 for mn ge 30

for not surface hard-ened steels

YX = 105 ndash 001 mn for 5 lt mn ge 25

YX = 08 for mn ge 25

for surface hardened steels

For static strength

YXst = 1 for all mn and all materials

313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength

YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength

In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength

The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses

The following simplified method for YC may be used

YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)

YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10

For endurance limit

+

+=nFFE

C m02ρt31

σconstY

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

Classification Notes- No 412 11 May 2003

DET NORSKE VERITAS

exactly the same position relative to the root fillet Relations in 191 apply for conversion to KHβ

Secondarily

KHβ may be evaluated by observed contact patterns on vari-ous defined load levels It is imperative that the various test loads are well defined Usually it is also necessary to evalu-ate the elastic deflections Some teeth at each 90 degrees are to be painted with a suitable lacquer Always consider the poorest of the contact patterns

After having run the gear for a suitable time at test load 1 (the lowest) observe the contact pattern with respect to ex-tension over the facewidth Evaluate that KHβ by means of the methods mentioned in this section Proceed in the same way for the next higher test load etc until there is a full face contact pattern From these data the initial mesh misalign-ment (ie without elastic deflections) can be found by ex-trapolation and then also the KHβ at design load can be found by calculation and extrapolation See example

Figure 11 Example of experimental determination of KHβ

It must be considered that inaccurate gears may accumulate a larger observed contact pattern than the actual single mesh to mesh contact patterns This is particularly important for lapped bevel gears Ground or hard metal hobbed bevel gears are assumed to present an accumulated contact pattern that is practically equal the actual single mesh to mesh con-tact patterns As a rough guidance the (observed) accumu-lated contact pattern of lapped bevel gears may be reduced by 10 in order to assess the single mesh to mesh contact pattern which is used in 199

193 Theoretical determination of KHβ The methods described in 193 to 198 may be used for cy-lindrical gears The principles may to some extent also be used for bevel gears but a more practical approach is given in 199

General For gears where the tooth contact pattern cannot be verified during assembly or under load all assumptions are to be well on the safe side

KHβ is to be determined in the plane of contact

The influence parameters considered in this method are

bull mean mesh stiffness cγ (see 111) (if necessary also variable stiffness over b)

bull mean unit load Fmb = Fbt KA Kγ Kvb (for double helical gears see 17 for use of Kγ)

bull misalignment fsh due to elastic deflections of shafts and gear bodies (both pinion and wheel)

bull misalignment fdefl due to elastic deflections of and work-ing positions in bearings

bull misalignment fbe due to bearing clearance tolerances bull misalignment fma due to manufacturing tolerances bull helix modifications as crowning end relief helix correc-

tion bull running in amount yβ (see 112)

In practice several other parameters such as centrifugal ex-pansion thermal expansion housing deflection etc contrib-ute to KHβ However these parameters are not taken into account unless in special cases when being considered as particularly important

When all or most of the am parameters are to be considered the most practical way to determine KHβ is by means of a graphical approach described in 1931

If cγ can be considered constant over the facewidth and no helix modifications apply KHβ can be determined analyti-cally as described in 1932

1931 Graphical method The graphical method utilises the superposition principle and is as follows

bull Calculate the mean mesh deflection Mδ as a function of

γm c and bF see 111

bull Draw a base line with length b and draw up a rectangu-lar with height δM (The area δM b is proportional to the transmitted force)

bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the areas above and below this zero line are equal

Figure 12 fsh balanced around zero line

bull Superimpose these ordinates of the fsh curve to the previ-ous load distribution curve (The area under this new load distribution curve is still δM b)

bull Calculate the bearing deflections andor working posi-tions in the bearings and evaluate the influence fdefl in the plane of contact This is a straight line and is bal-anced around a zero line as indicated in Fig 14 but with one distinct direction Superimpose these ordinates to the previous load distribution curve

12 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

bull The amount of crowning end relief or helix correction (defined in the plane of contact) is to be balanced around a zero line similarly to fsh

Figure 13 Crowning Cc balanced aound zero line

bull Superimpose these ordinates to the previous load distri-bution curve In case of high crowning etc as eg often applied to bevel gears the new load distribution curve may cross the base line (the real zero line) The result is areas with negative load that is not real as the load in those areas should be zero Thus corrective actions must be made but for practical reasons it may be postponed to after next operation

bull The amount of initial mesh misalignment bema ff + (defined in the plane of contact) is to be bal-

anced around a zero line If the direction of bema ff + is known (due to initial contact check) or if the direction of fbe is known due to design (eg overhang bevel pin-ion) this should be taken into account If direction un-known the influence of bema ff + in both directions as well as equal zero should be considered

Figure 14 fma+fbe in both directions balanced around zero line

Superimpose these ordinates to the previous load distri-bution curve This results in up to 3 different curves of which the one with the highest peak is to be chosen for further evaluation

bull If the chosen load distribution curve crosses the base line (ie mathematically negative load) the curve is to be corrected by adding the negative areas and dividing this with the active facewidth The (constant) ordinates of this rectangular correction area are to be subtracted from the positive part of the load distribution curve It is advisable to check that the area covered under this new load distribution curve is still equal δM b

bull If cγ cannot be considered as constant over b then cor-rect the ordinates of the load distribution curve with the local (on various positions over the facewidth) ratio between local mesh stiffness and average mesh stiffness cγ (average over the active facewidth only) Note that the result is to be a curve that covers the same area δM b as before

bull The influence of running in yβ is to be determined as in 112 whereby the value for Fβx is to be taken as twice the distance between the peak of the load distribution curve and δM

bull Determine

KHβ = M

β

δycurveofpeak minus

1932 Simplified analytical method for cylindrical gears The analytical approach is similar to 1931 but has a more limited application as cγ is assumed constant over the face-width and no helix modification applies

bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the area above and below this zero line are equal see Fig 12 The max positive ordinate is frac12∆fsh

bull Calculate the initial mesh alignment as Fβx= deflbemash ffff plusmnplusmnplusmn∆ The negative signs may only be used if this is justified andor verified by a contact pattern test Otherwise al-ways use positive signs If a negative sign is justified the value of Fβx is not to be taken less than the largest of each of these elements

bull Calculate the effective mesh misalignment as Fβy = Fβx - yβ (yβ see 112)

bull Determine

2KforF2

bFc1K βH

m

βγγH le+=β

or

2KforF

bFc2K Hβ

m

βγγH gt=β

where cγ as used here is the effective mesh stiffness see 111

194 Determination of fsh fsh is the mesh misalignment due to elastic deflections Usu-ally it is sufficient to consider the combined mesh deflection of the pinion body and shaft and the wheel shaft The calcu-lation is to be made in the plane of contact (of the considered gear mesh) and to consider all forces (incl axial) acting on the shafts Forces from other meshes can be parted into components parallel respectively vertical to the considered plane of contact Forces vertical to this plane of contact have no influence on fsh

It is advised to use following diameters for toothed elements

d + 2 x mn for bending and shear deflection

d + 2 mn (x ndash ha0 + 02) for torsional deflection

Usually fsh is calculated on basis of an evenly distributed load If the analysis of KHβ shows a considerable maldis-

Classification Notes- No 412 13 May 2003

DET NORSKE VERITAS

tribution in term of hard end contact or if it is known by other reasons that there exists a hard end contact the load should be correspondingly distributed when calculating fsh In fact the whole KHβ procedure can be used iteratively 2-3 iterations will be enough even for almost triangular load distributions

195 Determination of fdefl fdefl is the mesh misalignment in the plane of contact due to bearing deflections and working positions (housing deflec-tion may be included if determined)

First the journal working positions in the bearings are to be determined The influence of external moments and forces must be considered This is of special importance for twin pinion single output gears with all 3 shafts in one plane

For rolling bearings fdefl is further determined on basis of the elastic deflection of the bearings An elastic bearing deflec-tion depends on the bearing load and size and number of rolling elements Note that the bearing clearance tolerances are not included here

For fluid film bearings fdefl is further determined on basis of the lift and angular shift of the shafts due to lubrication oil film thickness Note that fbe takes into account the influence of the bearing clearance tolerance

When working positions bearing deflections and oil film lift are combined for all bearings the angular misalignment as projected into the plane of the contact is to be determined fdefl is this angular misalignment (radians) times the face-width

196 Determination of fbe fbe is the mesh misalignment in the plane of contact due to tolerances in bearing clearances In principle fbe and fdefl could be combined But as fdefl can be determined by analy-sis and has a distinct direction and fbe is dependent on toler-ances and in most cases has no distinct direction (ie + toler-ance) it is practicable to separate these two influences

Due to different bearing clearance tolerances in both pinion and wheel shafts the two shaft axis will have an angular mis-alignment in the plane of contact that is superimposed to the working positions determined in 195 fbe is the facewidth times this angular misalignment Note that fbe may have a distinct direction or be given as a + tolerance or a combina-tion of both For combination of + tolerance it is adviced to use

fbe= 22be

21be ff ++plusmn

fbe is particularly important for overhang designs for gears with widely different kinds of bearings on each side and when the bearings have wide tolerances on clearances In general it shall be possible to replace standard bearings with-out causing the real load distribution to exceed the design premises For slow speed gears with journal bearings the expected wear should also be considered

197 Determination of fma fma is the mesh misalignment due to manufacturing toler-ances (helix slope deviation) of pinion fHβ1 wheel fHβ2 and housing bore

For gear without specifically approved requirements to as-sembly control the value of fma is to be determined as

fma= 22Hβ

21Hβ ff +

For gears with specially approved assembly control the value of fma will depend on those specific requirements

198 Comments to various gear types For double helical gears KHβ is to be determined for both helices Usually an even load share between the helices can be assumed If not the calculation is to be made as de-scribed in 171

For planetary gears the free floating sun pinion suffers only twist no bending It must be noted that the total twist is the sum of the twist due to each mesh If the value of 1K γ ne this must be taken into account when calculating the total sun pinion twist (ie twist calculated with the force per mesh without Kγ and multiplied with the number of planets)

When planets are mounted on spherical bearings the mesh misalignments sun-planet respectively planet-annulus will be balanced Ie the misalignment will be the average between the two theoretical individual misalignments The faceload distribution on the flanks of the planets can take full advan-tage of this However as the sun and annulus mesh with several planets with possibly different lead errors the sun and annulus cannot obtain the above mentioned advantage to the full extent

199 Determination of KHβ for bevel gears If a theoretical approach similar to 193-198 is not docu-mented the following may be used

testeff

H Kb

b851851K sdot

minussdot=β

beff b represents the relative active facewidth (regarding lapped gears see 192 last part)

Higher values than beff b = 090 are normally not to be used in the formula

For dual directional gears it may be difficult to obtain a high beff b in both directions In that case the smaller beff b is to be used

Ktest represents the influence of the bearing arrangement shaft stiffness bearing stiffness housing stiffness etc on the faceload distribution and the verification thereof Expected variations in length- and height-wise tooth profile is also accounted for to some extent

a) Ktest = 1 For ground or hard metal hobbed gears with the specified contact pattern verified at full rating or at full torque slow turning at a condition representative for the thermal expansion at normal operation

14 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

It also applies when the bearing arrangementsupport has insignificant elastic deflections and thermal axial expansion However each initial mesh contact must be verified to be within acceptance criteria that are cali-brated against a type test at full load Reproduction of the gear tooth length- and height-wise profile must also be verified This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon)

b) Ktest = 1 + 04middot(beffbndash06) For designs with possible influence of thermal expansion in the axial direction of the pinion The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load

c) Ktest = 12 if mesh is only checked by toolmakerrsquos blue or by spin test contact For gears in this category beffb gt 085 is not to be used in the calculation

110 Transversal Load Distribution Factors KHα and KFα The transverse load distribution factors KHα for contact stresses and for scuffing KFα for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairs of teeth in mesh

The following relations may be used

Cylindrical gears

( )

minus+==

tH

αptγγHαFα F

byfc0409

2ε

KK

valid for 2εγ le

( ) ( )tH

αptγ

γ

γHαFα F

byfcε

1ε20409KK

minusminus+==

valid for 2ε γ gt

where

FtH = Ft KA Kγ Kv KHβ

cγ = See 111

γα = See 112

fpt = Maximum single pitch deviation (microm) of pinion or wheel or maximum total profile form devia-tion Fα of pinion or wheel if this is larger than the maximum single pitch deviation

Note In case of adequate equivalent tip relief adapted to the load half of the above mentioned fpt can be introduced A tip relief is considered adequate when the average of Ca1 and Ca2 is within plusmn40 of the value of Ceff in 432

Limitations of KHα and KFα

If the calculated values for

KFα = KHα lt 1 use KFα = KHα = 10

If the calculated value of KHα gt 2εα

γ

Zε

ε use KHα = 2

εα

γ

Zε

ε

If the calculated value of KFα gt εα

γ

Yεε

use KFα = εα

γ

Yεε

where Yε = αnε

075025+ (for εαn see 331c)

Bevel gears

For ground or hard metal hobbed gears KFα = KHα = 1

For lapped gears KFα = KHα = 11

111 Tooth Stiffness Constants cacute and cγ The tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having 1 mm facewidth by an amount of 1 microm in the plane of contact cacute is the maximum stiffness of a single pair of teeth cγ is the mean value of the mesh stiffness in a transverse plane (brief term mesh stiffness)

Both valid for high unit load (Unit load = Ft middot KA middot Kγb)

Cylindrical gears The real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bending stiff-nesses For high unit loads the Hertzian stiffness has little importance and can be disregarded This approach is on the safe side for determination of KHβ and KHα However for moderate or low loads Kv may be underestimated due to determination of a too high resonance speed

The linear approach is described in A

An optional approach for inclusion of the non-linear stiffness is described in B

A The linear approach

BRCCq

βcos08acutec =

and

( )025ε075cc αγ +prime=

where

( )[ ]n02a01a

B α2000212

hh12051C minusminus

+minus+=

12n1n

x000635z

025791z

015551004723q minus++=

12

2n

22

1n

1 x005290z

x024188x000193z

x011654+minusminusminus

+ 000182 x22

Classification Notes- No 412 15 May 2003

DET NORSKE VERITAS

(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as

( )( )nR m5s

sR e5

bbln1C +=

where

bs = thickness of a central web

sR = average thickness of rim (net value from tooth root to inside of rim)

The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered

Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ

B The non-linear approach

In the following an example is given on how to consider the non-linearity

The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm

With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as

( )10δKbF

minus= for 500bFgt

minus=

500Fb10δK

bF for 500

bFlt

with γAt KK

bF

bF

sdotsdot= etc (Nmm) ie unit load incorporat-

ing the relevant factors as

KA middot Kγ for determination of Kv

KA middot Kγ middot Kv for determination of KHβ

KA middot Kγ middot Kv middot KHβ for determination of KHα

δ = mesh deflection (microm)

K = applicable stiffness (c or cγ)

Use of stiffnesses for KV KHβ and KHα

For calculation of Kv and KHα the stiffness is calculated as follows

When Fb lt 500

the stiffness is determined as δ∆

∆ bF

where the increment is chosen as eg ∆ Fb = 10 and thus

50010Fb10

K10Fb∆δ +

++

=

When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly

or an equivalent stiffness determined as δsdotb

F

Bevel gears

In lack of more detailed relationship between stiffness and geometry the following may be used

b085

b13cacute eff=

b085b

16c effγ =

beff not to be used in excess of 085 b in these formulae

Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)

112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements

yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload

Cay is defined as the running-in amount that compensates for lack of tip relief

The following relations may be used

For not surface hardened steel

ptHlim

α fσ160y =

yβ = βxlimH

f320σ

with the following upper limits

V lt 5 ms 5-10 ms gt 10 ms

yα max none

limH

12800σ limH

6400σ

yβ max none

limH

25600σ limH

12800σ

For surface hardened steel

yα = 0075 fpt but not more than 3 for any speed

yβ = 015 Fβx but not more than 6 for any speed

16 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

For all kinds of steel

5145189718

1C2

limHay +

minusσ

=

When pinion and wheel material differ the following ap-plies

bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv

bull Use ( )2β1ββ yy21y += in the calculation of KHβ

bull Use ( )2ay1aya CC21C += in the calculation of Kv

bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing

calculation if no design tip relief is foreseen

Classification Notes- No 412 17 May 2003

DET NORSKE VERITAS

2 Calculation of Surface Durability

21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213

Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted

Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213

For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength

For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied

22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater

Calculation of surface durability for helical gears is based on the contact stress at the pitch point

For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies

Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact

Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses

The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses

The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel

221 Contact stress Cylindrical gears

( )HαHβvγA

1

tβεEHDBH KKKKK

bud1uF

ZZZZZσ+

=

where

ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)

ZH = Zone factor for pitch point (see 231)

ZE = Elasticity factor (see 24)

Zε = Contact ratio factor (see 25)

Zβ = Helix angle factor (see 26)

Ft KA Kγ Kv KHβ KHα see 15 ndash 110

d1 b u see 12 ndash 15

Bevel gears

( )HαHβvγA

v1v

vmtKEMH KKKKK

bud1uF

ZZZ105σ+

sdot=

where

105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)

ZE KA etc see above

ZM = mid-zone factor see 233

ZK = bevel gear factor see 27

Fmt dv1 uv see 12 ndash 15

It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks

222 Permissible contact stress

XWRvLH

NHlimHP ZZZZZ

SZσ

σ =

where

σH lim = Endurance limit for contact stresses (see 28)

ZN = Life factor for contact stresses (see 29)

SH = Required safety factor according to the rules

ZLZvZR = Oil film influence factors (see 210)

ZW = Work hardening factor (see 211)

ZX = Size factor (see 212)

18 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

23 Zone Factors ZH ZBD and ZM

231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder

wtt2

wtbH sinααcos

cosαcosβ2Z =

232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel

For εβ ge 1 ZBD = 1

For internal gears ZD = 1

For εβ = 0 (spur gears)

( )

π

minusεminusminus

π

minusminus

α=

α2

2

2b

2a

1

2

1b

1a

wtB

z211

dd

z21

dd

tanZ

( )

π

minusεminusminus

π

minusminus

α=

α1

2

1b

1a

2

2

2b

2a

wtD

z211

dd

z21

dd

tanZ

If ZB lt 1 use ZB = 1

If ZD lt 1 use ZD = 1

For 0 lt εβ lt 1

ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)

233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears

εminusminus

εminusminus

αβ=

αα btm2

2vb2

2vabtm2

1vb2val

2v1vvtbmM

pddpdd

ddtancos2Z

This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint

234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel

24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses

For steel against steel ZE = 1898

25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses

αε1Z =ε for 1εβ ge

( )α

ββ

αε ε

εε1

3ε4

Z +minusminus

= for εβ lt 1

26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability

cosβZβ =

27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply

The following may be used ZK = 080

28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ

σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting

For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply

For this purpose pitting is defined by

bull for not surface hardened gears pitted area ge 2 of total active flank area

bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given

material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies

The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as

Classification Notes- No 412 19 May 2003

DET NORSKE VERITAS

the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99

σHlim σH105 σH10

3

Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade

1650 1500

2500 2400

3100 3100

Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)

1000

13 σHlim

13 σHlim

Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV

Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV

Carbon steel 15 HV + 250 16 σHlim 16 σHlim

These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15

29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply

If this is not documented by approved fatigue tests the fol-lowing method may be used

For all steels except nitrided

7L 105N sdotge ZN = 1 or

01570

L

7

N N105Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 5middot107 510N

logZ037

L

7

N N105Z

sdot=

NL = 105 WXRVLHlim

WstX10H1010NN ZZZZZσ

ZZσZZ

55

5=

103 lt NL lt 105 )Z(Zlog05

L

5

10NN

5N103N10

5N10ZZ

==

3L 10N le

WXRVLlimH

Wst10X10H10NN ZZZZZ

ZZZZ

33

3σ

σ==

(but not less than ZN105)

For nitrided steels

6L 102N sdotge ZN = 1 or

00980

L

6

N N102Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 2middot106 510N

Zlog76860

L

6

N N102Z

sdot=

5L 10N le

XWRVL

10XWst10NN ZZZZZ

ZZ13ZZ

55 ==

Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles

210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity

The following methods may be applied in connection with the endurance limit

20 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Surface hardened steels Not surface hardened steels

ZL ( )24013421

360910ν+

+

( )24013421

680830ν+

+

ZV ( )v3280

140930+

+ ( )v3280300850

++

ZR

080

ZrelR3

150

ZrelR3

where

ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits

For values of ν40 gt 500 use ν40 = 500

RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm

RZrel

= ( ) 31

cZ2Z1 ρ

10RR50

+

RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)

For 5

L 10N le ZL ZV ZR = 10

211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface

The following approximation may be used for the endurance limit

Surface hardened steel against not surface hardened steel 150

ZeqW R

31700

130HB21Z

minus

minus=

where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130

RZeq = equivalent roughness

033

c40

066

ZS

ZHZHZEQ ρvν

15000RRRR

=

If RZeq gt 16 then use RZeq = 16

If RZeq lt 15 then use RZeq = 15

where

RZH = surface roughness of the hard member before run in

RZS = surface roughness of the soft member before run in

ν40 = see 210

If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem

Through hardened pinion against softer wheel

( )

minussdotminus+= 000829

HBHB0008981u1Z

2

1W

For 21HBHB

2

1 le use ZW = 1

For 71HBHB

2

1 gt use 17HBHB

2

1 =

For u gt 20 use u = 20

For static strength (lt 105 cycles)

Surface hardened against not surface hardened

ZWst = 105

Through hardened pinion against softer wheel

ZWst = 1

212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality

ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213

213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented

The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent

Classification Notes- No 412 21 May 2003

DET NORSKE VERITAS

stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically

The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence

The real Hertzian stresses σHR are determined as

For helical gears with εβ gt 1

σHR = σH

For helical gears with εβ lt 1 and spur gears

ε

α

ββ ε

ε+εminus

sdotσ=σZ

1

HHR

For bevel gears

KHHR Z

1σσ sdot=

The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)

The coordinates tz and HV are to be compared with the de-sign specification such as

bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400

and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)

In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods

For high cycle fatigue (gt3 106 cycles) the following applies

sdot+

minussdotsdotsdot= o90

05azt

05azt

cosSσ04HV

H

HHHR

applicable to 05a

zt

Hge

For 05at

H

z lt the value for 05at

H

z = applies

56300ρSσ12a cHHR

Hsdotsdot

sdot=

Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength

If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows

reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue

For static strength (lt103 cycles) the following applies

sdot+

minussdotsdotsdot= o90

07at

06a

zt

cosSσ019HV

Hst

z

HstHHR

applicable to 06at

Hst

z ge

56300ρSσa cHHR

Hstsdotsdot

=

In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies

For limited life fatigue (103 lt cycles lt 3106 )

For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance

The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram

minussdotminus

= sdot logN3477

logSlogSlogS

310H6103HHN

36 10H103H Slog86281Slog86280 sdot+sdot sdot

22 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3 Calculation of Tooth Strength

31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding

For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36

For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning

Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα

In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314

A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired

It should be noted that this part 3 does not cover fractures caused by

bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining

Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears

Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied

32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level

321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance

Normally the stresses for pinion and wheel are calculated as

Cylindrical gears

FαFβvγAβSFn

tF K K K K K Y Y Y

m bFσ =

where

YF = Tooth form factor (see 33)

YS = Stress correction factor (see 34)

Yβ = Helix angle factor (see 36)

Ft KA Kγ Kv KFβ KFα see 15 ndash 110

b see 13

Bevel gears

FαFβvγASaFamn

mtF K K K K K Y Y Y

m bFσ ε=

where

YFa = Tooth form factor see 33

YSa = Stress correction factor see 34

Yε = Contact ratio factor see 35

Fmt KA etc see 15 ndash 110

b see 13

322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is

CXRrelTrelTF

NMFEFP YYYY

SYY

δσ

=σ

Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN

where

σFE = Local tooth root bending endurance limit of reference test gear (see 37)

YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)

YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)

SF = Required safety factor according to the rules

YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)

YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)

Classification Notes- No 412 23 May 2003

DET NORSKE VERITAS

YX = Size factor (see 312)

YC = Case depth factor (see 313)

33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress

YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears

YFa applies to load application at the tooth tip and is used for bevel gears

Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears

Figure 31 External tooth in normal section

Figure 32 Internal tooth in normal section

Definitions

n

2

n

Fn

enFn

Fe

F

α cosms

α cosmh6

Y

=

n

2

n

Fn

anFn

Fa

Fa

α cosms

α cosmh6

Y

=

In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth

YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)

The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing

mn with mnm

zn with zvn

αt with αvt

β with βm

with undercut without undercut

Fig 33 Dimensions and basic rack profile of the teeth (finished profile)

Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless

331 Determination of parameters

( )n

n

prnfPnfP m

α cossαsin 1ρ

αtan h4πE

minusminusminusminus=

For external gears fPfP ρρ =

For internal gears ( )0z

195fPfP0

fPfP 10363156ρhxρρ

sdotminus+

+=

where

z0 = number of teeth of pinion cutter

x0 = addendum modification coefficient of pinion cutter

hfP = addendum of pinion cutter

ρfP = tip radius of pinion cutter

xhρG fpfp +minus=

24 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

τmE

2π

z2H

nnminus

minus=

with

3πτ = for external gears

6πτ = for internal gears

Htan zG2

nminusϑ=ϑ

(to be solved iteratively suitable start value6π

=ϑ for exter-

nal gears and 3π for internal gears)

a) Tooth root chord sFn For external gears

minus

ϑ+

ϑminus= ρ

cosG3

3πsinz

ms

fpnn

Fn

For bevel gears with a tooth thickness modification

xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn

For internal gears

minus

ϑ+

ϑminus= ρ

cosG

6πsinz

ms

fPnn

Fn

b) Root fillet radius ρF at 30ordm tangent

( )G2coszcosG2ρ

mρ

2n

2

fpn

F

minusϑϑ+=

c) Determination of bending moment arm hF dn = zn mn

b2α

αn βcosε

ε =

dan = dn + 2 ha

pbn = π mn cos αn

dbn = dn cos αn

( )4

d1εpzz

2dd

zz2d

2bn

2

αnbn

2bn

2an

en +

minusminus

minus=

en

bnen d

dcos arcα =

ennnn

e α invα invα x tan 22π

z1

minus+

+=γ

αFen = αen ndash γe

For external gears

( )

minus=

n

enFenee

n

Fe

mdαtan sin γ γcos

21

mh

]ρcos

G3πcosz fpn +

ϑminus

ϑminusminus

For internal gears

( ) minus

sdotsdotminus=

n

enFenee

n

Fe

mdα tanγ sinγ cos

21

mh

minus

ϑminus

ϑminussdot ρ

cosG3

6πcosz fPn

332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as

250ε205for 0666ε2366Y αnαnDT leleminus=

205εfor 10Y αnDT lt=

34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section

Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm

YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip

YS can be determined as follows

( )

+

+=L23121

1

sS qL01312Y

where Fe

Fn

hsL = and

F

Fns ρ2

sq = (see 33)

YSa can be calculated by replacing hFe with hFa in the above formulae

Note a) Range of validity 1 lt qs lt 8

In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered

b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with

Classification Notes- No 412 25 May 2003

DET NORSKE VERITAS

g

g

ρt

0613

13

minus

where

tg = depth of the grinding notch

ρg = radius of the grinding notch

c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles

35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears

The following may be used

Yε = 0625

36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank

The following may be used (β input in degrees)

Yβ = 1 ndash εβ β120

When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula

However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)

37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM

σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision

If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules

σFE

Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)

bull of specially approved high grade

1050

bull of normal grade

minus CrNiMo steels with approved process

1000

minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850

Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

840

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

720

Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)

07 HV + 300

Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)

025 σB + 125

Alloyed quenched and tempered steel 04 σB + 200

Carbon steel 025 σB + 250

Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40

1) These values are valid for a root radius

bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)

bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)

bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for

6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked

2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)

26 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears

YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0

YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst

For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316

The following method may be used within a stress ratio ndash12 lt R lt 05

381 For idlers planets and PTO with ice class

M1M1R1

1Yor Y MstM

+minus

minus=

where

R = stress ratio = min stress divided by max stress

For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12

For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as

branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus

For a power take off (PTO) with ice class see 161 c

M considers the mean stress influence on the endurance (or static) strength amplitudes

M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress

Following M values may be used

Endurance limit

Static strength

Case hardened 08 ndash 015 Ys 1) 07

If shot peened 04 06

Nitrided 03 03

Induction or flame hardened

04 06

Not surface hardened steel 03 05

Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M

The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches

382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106

For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers

For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used

383 For gears with shrinkage stresses and unidirectional load For endurance strength

FE

fitM σ

σM1

M21Y+

minus=

σFE is the endurance limit for R = 0

For static strength YMst = 1 and σfit accounted for in 39b

σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor

n

Ffit m

ρ215scf minus=

384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is

( ) ( ) FE

fitM σ

σR1M1

M2

M1M1R1

1Yminussdot+

minus

+minus

minus=

Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12

For static strength

M1M1R1

1YMst

+minus

minus=

The effect of σfit is accounted for in 39 b

Classification Notes- No 412 27 May 2003

DET NORSKE VERITAS

385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed

F

y

Sσ225

for not surface hardened fillets

FSHV5 for surface hardened fillets

39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles

Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN

If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used

Determination of the σ ndash N ndash curve

a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF

b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ

FstS1

FPstσ minussdotsdotsdot=

where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability

σFst

Alloyed case hardened steel 1) 2300

Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

1250

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

1050

Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)

18 HV + 800

Steel with not surface hardened fillets the smaller value of 2)

18 σB or 225 σy

1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet

c) Calculate YN as

NL gt 3middot106

YN = 1 or 001

L

6

N N103Y

sdot= ie Yn = 092 for 1010

The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2

103ltNLlt3middot106exp

L

6

N N103Y

sdot=

cycles6103forσcycles310forσlog02876exp

FP

FPst

sdot=

NL lt 103 cycles6103forσcycles310forσ

NYFP

FPst

sdot=

or simply use σFPst as mentioned in b) directly

Guidance on number of load cycles NL for various applica-tions

bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)

bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied

28 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage

YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit

The following method may be used

For endurance limit

for not surface hardened fillets

( )

024

s024

relT σ103133q21σ1012201351

Ysdotsdotminus

+sdotsdotminus+= minus

minus

δ

for all surface hardened fillets except nitrided

( )

106q21002451

Y srelT

++=δ

for nitrided fillets

( )

1347q2101421

Y srelT

++=δ

For static strength

for not surface hardened fillets1)

( )( )( )025

02

02502s

relTst σ3000821σ300 1Y0821Y

+

minus+=δ

for surface hardened fillets except nitrided

YδrelTst = 044 YS + 012

for nitrided fillets

YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not

exceed the yield point and thereby alter the residual stress level See also 39b footnote 2

311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness

YRrelT differs for endurance limit and static strength

The following method may be used

For endurance limit

YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels

YδrelT = 43 ndash 326 (Ry + 1)0005

for nitrided steels

For static strength

YRrelTst = 1 for all Ry and all materials

For a fillet without any longitudinal machining trace Ry asymp Rz

312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength

The following may be used

For endurance limit

YX = 1 for mn le 5 generally

YX = 103 ndash 0006 mn for 5 lt mn lt 30

YX = 085 for mn ge 30

for not surface hard-ened steels

YX = 105 ndash 001 mn for 5 lt mn ge 25

YX = 08 for mn ge 25

for surface hardened steels

For static strength

YXst = 1 for all mn and all materials

313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength

YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength

In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength

The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses

The following simplified method for YC may be used

YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)

YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10

For endurance limit

+

+=nFFE

C m02ρt31

σconstY

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

12 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

bull The amount of crowning end relief or helix correction (defined in the plane of contact) is to be balanced around a zero line similarly to fsh

Figure 13 Crowning Cc balanced aound zero line

bull Superimpose these ordinates to the previous load distri-bution curve In case of high crowning etc as eg often applied to bevel gears the new load distribution curve may cross the base line (the real zero line) The result is areas with negative load that is not real as the load in those areas should be zero Thus corrective actions must be made but for practical reasons it may be postponed to after next operation

bull The amount of initial mesh misalignment bema ff + (defined in the plane of contact) is to be bal-

anced around a zero line If the direction of bema ff + is known (due to initial contact check) or if the direction of fbe is known due to design (eg overhang bevel pin-ion) this should be taken into account If direction un-known the influence of bema ff + in both directions as well as equal zero should be considered

Figure 14 fma+fbe in both directions balanced around zero line

Superimpose these ordinates to the previous load distri-bution curve This results in up to 3 different curves of which the one with the highest peak is to be chosen for further evaluation

bull If the chosen load distribution curve crosses the base line (ie mathematically negative load) the curve is to be corrected by adding the negative areas and dividing this with the active facewidth The (constant) ordinates of this rectangular correction area are to be subtracted from the positive part of the load distribution curve It is advisable to check that the area covered under this new load distribution curve is still equal δM b

bull If cγ cannot be considered as constant over b then cor-rect the ordinates of the load distribution curve with the local (on various positions over the facewidth) ratio between local mesh stiffness and average mesh stiffness cγ (average over the active facewidth only) Note that the result is to be a curve that covers the same area δM b as before

bull The influence of running in yβ is to be determined as in 112 whereby the value for Fβx is to be taken as twice the distance between the peak of the load distribution curve and δM

bull Determine

KHβ = M

β

δycurveofpeak minus

1932 Simplified analytical method for cylindrical gears The analytical approach is similar to 1931 but has a more limited application as cγ is assumed constant over the face-width and no helix modification applies

bull Calculate the elastic deflection fsh in the plane of contact Balance this deflection curve around a zero line so that the area above and below this zero line are equal see Fig 12 The max positive ordinate is frac12∆fsh

bull Calculate the initial mesh alignment as Fβx= deflbemash ffff plusmnplusmnplusmn∆ The negative signs may only be used if this is justified andor verified by a contact pattern test Otherwise al-ways use positive signs If a negative sign is justified the value of Fβx is not to be taken less than the largest of each of these elements

bull Calculate the effective mesh misalignment as Fβy = Fβx - yβ (yβ see 112)

bull Determine

2KforF2

bFc1K βH

m

βγγH le+=β

or

2KforF

bFc2K Hβ

m

βγγH gt=β

where cγ as used here is the effective mesh stiffness see 111

194 Determination of fsh fsh is the mesh misalignment due to elastic deflections Usu-ally it is sufficient to consider the combined mesh deflection of the pinion body and shaft and the wheel shaft The calcu-lation is to be made in the plane of contact (of the considered gear mesh) and to consider all forces (incl axial) acting on the shafts Forces from other meshes can be parted into components parallel respectively vertical to the considered plane of contact Forces vertical to this plane of contact have no influence on fsh

It is advised to use following diameters for toothed elements

d + 2 x mn for bending and shear deflection

d + 2 mn (x ndash ha0 + 02) for torsional deflection

Usually fsh is calculated on basis of an evenly distributed load If the analysis of KHβ shows a considerable maldis-

Classification Notes- No 412 13 May 2003

DET NORSKE VERITAS

tribution in term of hard end contact or if it is known by other reasons that there exists a hard end contact the load should be correspondingly distributed when calculating fsh In fact the whole KHβ procedure can be used iteratively 2-3 iterations will be enough even for almost triangular load distributions

195 Determination of fdefl fdefl is the mesh misalignment in the plane of contact due to bearing deflections and working positions (housing deflec-tion may be included if determined)

First the journal working positions in the bearings are to be determined The influence of external moments and forces must be considered This is of special importance for twin pinion single output gears with all 3 shafts in one plane

For rolling bearings fdefl is further determined on basis of the elastic deflection of the bearings An elastic bearing deflec-tion depends on the bearing load and size and number of rolling elements Note that the bearing clearance tolerances are not included here

For fluid film bearings fdefl is further determined on basis of the lift and angular shift of the shafts due to lubrication oil film thickness Note that fbe takes into account the influence of the bearing clearance tolerance

When working positions bearing deflections and oil film lift are combined for all bearings the angular misalignment as projected into the plane of the contact is to be determined fdefl is this angular misalignment (radians) times the face-width

196 Determination of fbe fbe is the mesh misalignment in the plane of contact due to tolerances in bearing clearances In principle fbe and fdefl could be combined But as fdefl can be determined by analy-sis and has a distinct direction and fbe is dependent on toler-ances and in most cases has no distinct direction (ie + toler-ance) it is practicable to separate these two influences

Due to different bearing clearance tolerances in both pinion and wheel shafts the two shaft axis will have an angular mis-alignment in the plane of contact that is superimposed to the working positions determined in 195 fbe is the facewidth times this angular misalignment Note that fbe may have a distinct direction or be given as a + tolerance or a combina-tion of both For combination of + tolerance it is adviced to use

fbe= 22be

21be ff ++plusmn

fbe is particularly important for overhang designs for gears with widely different kinds of bearings on each side and when the bearings have wide tolerances on clearances In general it shall be possible to replace standard bearings with-out causing the real load distribution to exceed the design premises For slow speed gears with journal bearings the expected wear should also be considered

197 Determination of fma fma is the mesh misalignment due to manufacturing toler-ances (helix slope deviation) of pinion fHβ1 wheel fHβ2 and housing bore

For gear without specifically approved requirements to as-sembly control the value of fma is to be determined as

fma= 22Hβ

21Hβ ff +

For gears with specially approved assembly control the value of fma will depend on those specific requirements

198 Comments to various gear types For double helical gears KHβ is to be determined for both helices Usually an even load share between the helices can be assumed If not the calculation is to be made as de-scribed in 171

For planetary gears the free floating sun pinion suffers only twist no bending It must be noted that the total twist is the sum of the twist due to each mesh If the value of 1K γ ne this must be taken into account when calculating the total sun pinion twist (ie twist calculated with the force per mesh without Kγ and multiplied with the number of planets)

When planets are mounted on spherical bearings the mesh misalignments sun-planet respectively planet-annulus will be balanced Ie the misalignment will be the average between the two theoretical individual misalignments The faceload distribution on the flanks of the planets can take full advan-tage of this However as the sun and annulus mesh with several planets with possibly different lead errors the sun and annulus cannot obtain the above mentioned advantage to the full extent

199 Determination of KHβ for bevel gears If a theoretical approach similar to 193-198 is not docu-mented the following may be used

testeff

H Kb

b851851K sdot

minussdot=β

beff b represents the relative active facewidth (regarding lapped gears see 192 last part)

Higher values than beff b = 090 are normally not to be used in the formula

For dual directional gears it may be difficult to obtain a high beff b in both directions In that case the smaller beff b is to be used

Ktest represents the influence of the bearing arrangement shaft stiffness bearing stiffness housing stiffness etc on the faceload distribution and the verification thereof Expected variations in length- and height-wise tooth profile is also accounted for to some extent

a) Ktest = 1 For ground or hard metal hobbed gears with the specified contact pattern verified at full rating or at full torque slow turning at a condition representative for the thermal expansion at normal operation

14 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

It also applies when the bearing arrangementsupport has insignificant elastic deflections and thermal axial expansion However each initial mesh contact must be verified to be within acceptance criteria that are cali-brated against a type test at full load Reproduction of the gear tooth length- and height-wise profile must also be verified This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon)

b) Ktest = 1 + 04middot(beffbndash06) For designs with possible influence of thermal expansion in the axial direction of the pinion The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load

c) Ktest = 12 if mesh is only checked by toolmakerrsquos blue or by spin test contact For gears in this category beffb gt 085 is not to be used in the calculation

110 Transversal Load Distribution Factors KHα and KFα The transverse load distribution factors KHα for contact stresses and for scuffing KFα for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairs of teeth in mesh

The following relations may be used

Cylindrical gears

( )

minus+==

tH

αptγγHαFα F

byfc0409

2ε

KK

valid for 2εγ le

( ) ( )tH

αptγ

γ

γHαFα F

byfcε

1ε20409KK

minusminus+==

valid for 2ε γ gt

where

FtH = Ft KA Kγ Kv KHβ

cγ = See 111

γα = See 112

fpt = Maximum single pitch deviation (microm) of pinion or wheel or maximum total profile form devia-tion Fα of pinion or wheel if this is larger than the maximum single pitch deviation

Note In case of adequate equivalent tip relief adapted to the load half of the above mentioned fpt can be introduced A tip relief is considered adequate when the average of Ca1 and Ca2 is within plusmn40 of the value of Ceff in 432

Limitations of KHα and KFα

If the calculated values for

KFα = KHα lt 1 use KFα = KHα = 10

If the calculated value of KHα gt 2εα

γ

Zε

ε use KHα = 2

εα

γ

Zε

ε

If the calculated value of KFα gt εα

γ

Yεε

use KFα = εα

γ

Yεε

where Yε = αnε

075025+ (for εαn see 331c)

Bevel gears

For ground or hard metal hobbed gears KFα = KHα = 1

For lapped gears KFα = KHα = 11

111 Tooth Stiffness Constants cacute and cγ The tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having 1 mm facewidth by an amount of 1 microm in the plane of contact cacute is the maximum stiffness of a single pair of teeth cγ is the mean value of the mesh stiffness in a transverse plane (brief term mesh stiffness)

Both valid for high unit load (Unit load = Ft middot KA middot Kγb)

Cylindrical gears The real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bending stiff-nesses For high unit loads the Hertzian stiffness has little importance and can be disregarded This approach is on the safe side for determination of KHβ and KHα However for moderate or low loads Kv may be underestimated due to determination of a too high resonance speed

The linear approach is described in A

An optional approach for inclusion of the non-linear stiffness is described in B

A The linear approach

BRCCq

βcos08acutec =

and

( )025ε075cc αγ +prime=

where

( )[ ]n02a01a

B α2000212

hh12051C minusminus

+minus+=

12n1n

x000635z

025791z

015551004723q minus++=

12

2n

22

1n

1 x005290z

x024188x000193z

x011654+minusminusminus

+ 000182 x22

Classification Notes- No 412 15 May 2003

DET NORSKE VERITAS

(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as

( )( )nR m5s

sR e5

bbln1C +=

where

bs = thickness of a central web

sR = average thickness of rim (net value from tooth root to inside of rim)

The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered

Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ

B The non-linear approach

In the following an example is given on how to consider the non-linearity

The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm

With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as

( )10δKbF

minus= for 500bFgt

minus=

500Fb10δK

bF for 500

bFlt

with γAt KK

bF

bF

sdotsdot= etc (Nmm) ie unit load incorporat-

ing the relevant factors as

KA middot Kγ for determination of Kv

KA middot Kγ middot Kv for determination of KHβ

KA middot Kγ middot Kv middot KHβ for determination of KHα

δ = mesh deflection (microm)

K = applicable stiffness (c or cγ)

Use of stiffnesses for KV KHβ and KHα

For calculation of Kv and KHα the stiffness is calculated as follows

When Fb lt 500

the stiffness is determined as δ∆

∆ bF

where the increment is chosen as eg ∆ Fb = 10 and thus

50010Fb10

K10Fb∆δ +

++

=

When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly

or an equivalent stiffness determined as δsdotb

F

Bevel gears

In lack of more detailed relationship between stiffness and geometry the following may be used

b085

b13cacute eff=

b085b

16c effγ =

beff not to be used in excess of 085 b in these formulae

Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)

112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements

yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload

Cay is defined as the running-in amount that compensates for lack of tip relief

The following relations may be used

For not surface hardened steel

ptHlim

α fσ160y =

yβ = βxlimH

f320σ

with the following upper limits

V lt 5 ms 5-10 ms gt 10 ms

yα max none

limH

12800σ limH

6400σ

yβ max none

limH

25600σ limH

12800σ

For surface hardened steel

yα = 0075 fpt but not more than 3 for any speed

yβ = 015 Fβx but not more than 6 for any speed

16 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

For all kinds of steel

5145189718

1C2

limHay +

minusσ

=

When pinion and wheel material differ the following ap-plies

bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv

bull Use ( )2β1ββ yy21y += in the calculation of KHβ

bull Use ( )2ay1aya CC21C += in the calculation of Kv

bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing

calculation if no design tip relief is foreseen

Classification Notes- No 412 17 May 2003

DET NORSKE VERITAS

2 Calculation of Surface Durability

21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213

Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted

Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213

For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength

For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied

22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater

Calculation of surface durability for helical gears is based on the contact stress at the pitch point

For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies

Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact

Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses

The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses

The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel

221 Contact stress Cylindrical gears

( )HαHβvγA

1

tβεEHDBH KKKKK

bud1uF

ZZZZZσ+

=

where

ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)

ZH = Zone factor for pitch point (see 231)

ZE = Elasticity factor (see 24)

Zε = Contact ratio factor (see 25)

Zβ = Helix angle factor (see 26)

Ft KA Kγ Kv KHβ KHα see 15 ndash 110

d1 b u see 12 ndash 15

Bevel gears

( )HαHβvγA

v1v

vmtKEMH KKKKK

bud1uF

ZZZ105σ+

sdot=

where

105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)

ZE KA etc see above

ZM = mid-zone factor see 233

ZK = bevel gear factor see 27

Fmt dv1 uv see 12 ndash 15

It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks

222 Permissible contact stress

XWRvLH

NHlimHP ZZZZZ

SZσ

σ =

where

σH lim = Endurance limit for contact stresses (see 28)

ZN = Life factor for contact stresses (see 29)

SH = Required safety factor according to the rules

ZLZvZR = Oil film influence factors (see 210)

ZW = Work hardening factor (see 211)

ZX = Size factor (see 212)

18 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

23 Zone Factors ZH ZBD and ZM

231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder

wtt2

wtbH sinααcos

cosαcosβ2Z =

232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel

For εβ ge 1 ZBD = 1

For internal gears ZD = 1

For εβ = 0 (spur gears)

( )

π

minusεminusminus

π

minusminus

α=

α2

2

2b

2a

1

2

1b

1a

wtB

z211

dd

z21

dd

tanZ

( )

π

minusεminusminus

π

minusminus

α=

α1

2

1b

1a

2

2

2b

2a

wtD

z211

dd

z21

dd

tanZ

If ZB lt 1 use ZB = 1

If ZD lt 1 use ZD = 1

For 0 lt εβ lt 1

ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)

233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears

εminusminus

εminusminus

αβ=

αα btm2

2vb2

2vabtm2

1vb2val

2v1vvtbmM

pddpdd

ddtancos2Z

This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint

234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel

24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses

For steel against steel ZE = 1898

25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses

αε1Z =ε for 1εβ ge

( )α

ββ

αε ε

εε1

3ε4

Z +minusminus

= for εβ lt 1

26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability

cosβZβ =

27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply

The following may be used ZK = 080

28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ

σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting

For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply

For this purpose pitting is defined by

bull for not surface hardened gears pitted area ge 2 of total active flank area

bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given

material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies

The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as

Classification Notes- No 412 19 May 2003

DET NORSKE VERITAS

the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99

σHlim σH105 σH10

3

Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade

1650 1500

2500 2400

3100 3100

Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)

1000

13 σHlim

13 σHlim

Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV

Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV

Carbon steel 15 HV + 250 16 σHlim 16 σHlim

These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15

29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply

If this is not documented by approved fatigue tests the fol-lowing method may be used

For all steels except nitrided

7L 105N sdotge ZN = 1 or

01570

L

7

N N105Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 5middot107 510N

logZ037

L

7

N N105Z

sdot=

NL = 105 WXRVLHlim

WstX10H1010NN ZZZZZσ

ZZσZZ

55

5=

103 lt NL lt 105 )Z(Zlog05

L

5

10NN

5N103N10

5N10ZZ

==

3L 10N le

WXRVLlimH

Wst10X10H10NN ZZZZZ

ZZZZ

33

3σ

σ==

(but not less than ZN105)

For nitrided steels

6L 102N sdotge ZN = 1 or

00980

L

6

N N102Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 2middot106 510N

Zlog76860

L

6

N N102Z

sdot=

5L 10N le

XWRVL

10XWst10NN ZZZZZ

ZZ13ZZ

55 ==

Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles

210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity

The following methods may be applied in connection with the endurance limit

20 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Surface hardened steels Not surface hardened steels

ZL ( )24013421

360910ν+

+

( )24013421

680830ν+

+

ZV ( )v3280

140930+

+ ( )v3280300850

++

ZR

080

ZrelR3

150

ZrelR3

where

ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits

For values of ν40 gt 500 use ν40 = 500

RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm

RZrel

= ( ) 31

cZ2Z1 ρ

10RR50

+

RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)

For 5

L 10N le ZL ZV ZR = 10

211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface

The following approximation may be used for the endurance limit

Surface hardened steel against not surface hardened steel 150

ZeqW R

31700

130HB21Z

minus

minus=

where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130

RZeq = equivalent roughness

033

c40

066

ZS

ZHZHZEQ ρvν

15000RRRR

=

If RZeq gt 16 then use RZeq = 16

If RZeq lt 15 then use RZeq = 15

where

RZH = surface roughness of the hard member before run in

RZS = surface roughness of the soft member before run in

ν40 = see 210

If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem

Through hardened pinion against softer wheel

( )

minussdotminus+= 000829

HBHB0008981u1Z

2

1W

For 21HBHB

2

1 le use ZW = 1

For 71HBHB

2

1 gt use 17HBHB

2

1 =

For u gt 20 use u = 20

For static strength (lt 105 cycles)

Surface hardened against not surface hardened

ZWst = 105

Through hardened pinion against softer wheel

ZWst = 1

212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality

ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213

213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented

The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent

Classification Notes- No 412 21 May 2003

DET NORSKE VERITAS

stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically

The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence

The real Hertzian stresses σHR are determined as

For helical gears with εβ gt 1

σHR = σH

For helical gears with εβ lt 1 and spur gears

ε

α

ββ ε

ε+εminus

sdotσ=σZ

1

HHR

For bevel gears

KHHR Z

1σσ sdot=

The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)

The coordinates tz and HV are to be compared with the de-sign specification such as

bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400

and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)

In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods

For high cycle fatigue (gt3 106 cycles) the following applies

sdot+

minussdotsdotsdot= o90

05azt

05azt

cosSσ04HV

H

HHHR

applicable to 05a

zt

Hge

For 05at

H

z lt the value for 05at

H

z = applies

56300ρSσ12a cHHR

Hsdotsdot

sdot=

Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength

If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows

reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue

For static strength (lt103 cycles) the following applies

sdot+

minussdotsdotsdot= o90

07at

06a

zt

cosSσ019HV

Hst

z

HstHHR

applicable to 06at

Hst

z ge

56300ρSσa cHHR

Hstsdotsdot

=

In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies

For limited life fatigue (103 lt cycles lt 3106 )

For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance

The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram

minussdotminus

= sdot logN3477

logSlogSlogS

310H6103HHN

36 10H103H Slog86281Slog86280 sdot+sdot sdot

22 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3 Calculation of Tooth Strength

31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding

For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36

For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning

Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα

In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314

A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired

It should be noted that this part 3 does not cover fractures caused by

bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining

Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears

Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied

32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level

321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance

Normally the stresses for pinion and wheel are calculated as

Cylindrical gears

FαFβvγAβSFn

tF K K K K K Y Y Y

m bFσ =

where

YF = Tooth form factor (see 33)

YS = Stress correction factor (see 34)

Yβ = Helix angle factor (see 36)

Ft KA Kγ Kv KFβ KFα see 15 ndash 110

b see 13

Bevel gears

FαFβvγASaFamn

mtF K K K K K Y Y Y

m bFσ ε=

where

YFa = Tooth form factor see 33

YSa = Stress correction factor see 34

Yε = Contact ratio factor see 35

Fmt KA etc see 15 ndash 110

b see 13

322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is

CXRrelTrelTF

NMFEFP YYYY

SYY

δσ

=σ

Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN

where

σFE = Local tooth root bending endurance limit of reference test gear (see 37)

YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)

YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)

SF = Required safety factor according to the rules

YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)

YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)

Classification Notes- No 412 23 May 2003

DET NORSKE VERITAS

YX = Size factor (see 312)

YC = Case depth factor (see 313)

33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress

YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears

YFa applies to load application at the tooth tip and is used for bevel gears

Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears

Figure 31 External tooth in normal section

Figure 32 Internal tooth in normal section

Definitions

n

2

n

Fn

enFn

Fe

F

α cosms

α cosmh6

Y

=

n

2

n

Fn

anFn

Fa

Fa

α cosms

α cosmh6

Y

=

In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth

YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)

The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing

mn with mnm

zn with zvn

αt with αvt

β with βm

with undercut without undercut

Fig 33 Dimensions and basic rack profile of the teeth (finished profile)

Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless

331 Determination of parameters

( )n

n

prnfPnfP m

α cossαsin 1ρ

αtan h4πE

minusminusminusminus=

For external gears fPfP ρρ =

For internal gears ( )0z

195fPfP0

fPfP 10363156ρhxρρ

sdotminus+

+=

where

z0 = number of teeth of pinion cutter

x0 = addendum modification coefficient of pinion cutter

hfP = addendum of pinion cutter

ρfP = tip radius of pinion cutter

xhρG fpfp +minus=

24 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

τmE

2π

z2H

nnminus

minus=

with

3πτ = for external gears

6πτ = for internal gears

Htan zG2

nminusϑ=ϑ

(to be solved iteratively suitable start value6π

=ϑ for exter-

nal gears and 3π for internal gears)

a) Tooth root chord sFn For external gears

minus

ϑ+

ϑminus= ρ

cosG3

3πsinz

ms

fpnn

Fn

For bevel gears with a tooth thickness modification

xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn

For internal gears

minus

ϑ+

ϑminus= ρ

cosG

6πsinz

ms

fPnn

Fn

b) Root fillet radius ρF at 30ordm tangent

( )G2coszcosG2ρ

mρ

2n

2

fpn

F

minusϑϑ+=

c) Determination of bending moment arm hF dn = zn mn

b2α

αn βcosε

ε =

dan = dn + 2 ha

pbn = π mn cos αn

dbn = dn cos αn

( )4

d1εpzz

2dd

zz2d

2bn

2

αnbn

2bn

2an

en +

minusminus

minus=

en

bnen d

dcos arcα =

ennnn

e α invα invα x tan 22π

z1

minus+

+=γ

αFen = αen ndash γe

For external gears

( )

minus=

n

enFenee

n

Fe

mdαtan sin γ γcos

21

mh

]ρcos

G3πcosz fpn +

ϑminus

ϑminusminus

For internal gears

( ) minus

sdotsdotminus=

n

enFenee

n

Fe

mdα tanγ sinγ cos

21

mh

minus

ϑminus

ϑminussdot ρ

cosG3

6πcosz fPn

332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as

250ε205for 0666ε2366Y αnαnDT leleminus=

205εfor 10Y αnDT lt=

34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section

Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm

YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip

YS can be determined as follows

( )

+

+=L23121

1

sS qL01312Y

where Fe

Fn

hsL = and

F

Fns ρ2

sq = (see 33)

YSa can be calculated by replacing hFe with hFa in the above formulae

Note a) Range of validity 1 lt qs lt 8

In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered

b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with

Classification Notes- No 412 25 May 2003

DET NORSKE VERITAS

g

g

ρt

0613

13

minus

where

tg = depth of the grinding notch

ρg = radius of the grinding notch

c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles

35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears

The following may be used

Yε = 0625

36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank

The following may be used (β input in degrees)

Yβ = 1 ndash εβ β120

When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula

However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)

37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM

σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision

If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules

σFE

Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)

bull of specially approved high grade

1050

bull of normal grade

minus CrNiMo steels with approved process

1000

minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850

Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

840

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

720

Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)

07 HV + 300

Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)

025 σB + 125

Alloyed quenched and tempered steel 04 σB + 200

Carbon steel 025 σB + 250

Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40

1) These values are valid for a root radius

bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)

bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)

bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for

6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked

2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)

26 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears

YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0

YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst

For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316

The following method may be used within a stress ratio ndash12 lt R lt 05

381 For idlers planets and PTO with ice class

M1M1R1

1Yor Y MstM

+minus

minus=

where

R = stress ratio = min stress divided by max stress

For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12

For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as

branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus

For a power take off (PTO) with ice class see 161 c

M considers the mean stress influence on the endurance (or static) strength amplitudes

M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress

Following M values may be used

Endurance limit

Static strength

Case hardened 08 ndash 015 Ys 1) 07

If shot peened 04 06

Nitrided 03 03

Induction or flame hardened

04 06

Not surface hardened steel 03 05

Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M

The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches

382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106

For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers

For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used

383 For gears with shrinkage stresses and unidirectional load For endurance strength

FE

fitM σ

σM1

M21Y+

minus=

σFE is the endurance limit for R = 0

For static strength YMst = 1 and σfit accounted for in 39b

σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor

n

Ffit m

ρ215scf minus=

384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is

( ) ( ) FE

fitM σ

σR1M1

M2

M1M1R1

1Yminussdot+

minus

+minus

minus=

Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12

For static strength

M1M1R1

1YMst

+minus

minus=

The effect of σfit is accounted for in 39 b

Classification Notes- No 412 27 May 2003

DET NORSKE VERITAS

385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed

F

y

Sσ225

for not surface hardened fillets

FSHV5 for surface hardened fillets

39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles

Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN

If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used

Determination of the σ ndash N ndash curve

a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF

b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ

FstS1

FPstσ minussdotsdotsdot=

where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability

σFst

Alloyed case hardened steel 1) 2300

Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

1250

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

1050

Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)

18 HV + 800

Steel with not surface hardened fillets the smaller value of 2)

18 σB or 225 σy

1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet

c) Calculate YN as

NL gt 3middot106

YN = 1 or 001

L

6

N N103Y

sdot= ie Yn = 092 for 1010

The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2

103ltNLlt3middot106exp

L

6

N N103Y

sdot=

cycles6103forσcycles310forσlog02876exp

FP

FPst

sdot=

NL lt 103 cycles6103forσcycles310forσ

NYFP

FPst

sdot=

or simply use σFPst as mentioned in b) directly

Guidance on number of load cycles NL for various applica-tions

bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)

bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied

28 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage

YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit

The following method may be used

For endurance limit

for not surface hardened fillets

( )

024

s024

relT σ103133q21σ1012201351

Ysdotsdotminus

+sdotsdotminus+= minus

minus

δ

for all surface hardened fillets except nitrided

( )

106q21002451

Y srelT

++=δ

for nitrided fillets

( )

1347q2101421

Y srelT

++=δ

For static strength

for not surface hardened fillets1)

( )( )( )025

02

02502s

relTst σ3000821σ300 1Y0821Y

+

minus+=δ

for surface hardened fillets except nitrided

YδrelTst = 044 YS + 012

for nitrided fillets

YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not

exceed the yield point and thereby alter the residual stress level See also 39b footnote 2

311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness

YRrelT differs for endurance limit and static strength

The following method may be used

For endurance limit

YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels

YδrelT = 43 ndash 326 (Ry + 1)0005

for nitrided steels

For static strength

YRrelTst = 1 for all Ry and all materials

For a fillet without any longitudinal machining trace Ry asymp Rz

312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength

The following may be used

For endurance limit

YX = 1 for mn le 5 generally

YX = 103 ndash 0006 mn for 5 lt mn lt 30

YX = 085 for mn ge 30

for not surface hard-ened steels

YX = 105 ndash 001 mn for 5 lt mn ge 25

YX = 08 for mn ge 25

for surface hardened steels

For static strength

YXst = 1 for all mn and all materials

313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength

YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength

In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength

The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses

The following simplified method for YC may be used

YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)

YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10

For endurance limit

+

+=nFFE

C m02ρt31

σconstY

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

Classification Notes- No 412 13 May 2003

DET NORSKE VERITAS

tribution in term of hard end contact or if it is known by other reasons that there exists a hard end contact the load should be correspondingly distributed when calculating fsh In fact the whole KHβ procedure can be used iteratively 2-3 iterations will be enough even for almost triangular load distributions

195 Determination of fdefl fdefl is the mesh misalignment in the plane of contact due to bearing deflections and working positions (housing deflec-tion may be included if determined)

First the journal working positions in the bearings are to be determined The influence of external moments and forces must be considered This is of special importance for twin pinion single output gears with all 3 shafts in one plane

For rolling bearings fdefl is further determined on basis of the elastic deflection of the bearings An elastic bearing deflec-tion depends on the bearing load and size and number of rolling elements Note that the bearing clearance tolerances are not included here

For fluid film bearings fdefl is further determined on basis of the lift and angular shift of the shafts due to lubrication oil film thickness Note that fbe takes into account the influence of the bearing clearance tolerance

When working positions bearing deflections and oil film lift are combined for all bearings the angular misalignment as projected into the plane of the contact is to be determined fdefl is this angular misalignment (radians) times the face-width

196 Determination of fbe fbe is the mesh misalignment in the plane of contact due to tolerances in bearing clearances In principle fbe and fdefl could be combined But as fdefl can be determined by analy-sis and has a distinct direction and fbe is dependent on toler-ances and in most cases has no distinct direction (ie + toler-ance) it is practicable to separate these two influences

Due to different bearing clearance tolerances in both pinion and wheel shafts the two shaft axis will have an angular mis-alignment in the plane of contact that is superimposed to the working positions determined in 195 fbe is the facewidth times this angular misalignment Note that fbe may have a distinct direction or be given as a + tolerance or a combina-tion of both For combination of + tolerance it is adviced to use

fbe= 22be

21be ff ++plusmn

fbe is particularly important for overhang designs for gears with widely different kinds of bearings on each side and when the bearings have wide tolerances on clearances In general it shall be possible to replace standard bearings with-out causing the real load distribution to exceed the design premises For slow speed gears with journal bearings the expected wear should also be considered

197 Determination of fma fma is the mesh misalignment due to manufacturing toler-ances (helix slope deviation) of pinion fHβ1 wheel fHβ2 and housing bore

For gear without specifically approved requirements to as-sembly control the value of fma is to be determined as

fma= 22Hβ

21Hβ ff +

For gears with specially approved assembly control the value of fma will depend on those specific requirements

198 Comments to various gear types For double helical gears KHβ is to be determined for both helices Usually an even load share between the helices can be assumed If not the calculation is to be made as de-scribed in 171

For planetary gears the free floating sun pinion suffers only twist no bending It must be noted that the total twist is the sum of the twist due to each mesh If the value of 1K γ ne this must be taken into account when calculating the total sun pinion twist (ie twist calculated with the force per mesh without Kγ and multiplied with the number of planets)

When planets are mounted on spherical bearings the mesh misalignments sun-planet respectively planet-annulus will be balanced Ie the misalignment will be the average between the two theoretical individual misalignments The faceload distribution on the flanks of the planets can take full advan-tage of this However as the sun and annulus mesh with several planets with possibly different lead errors the sun and annulus cannot obtain the above mentioned advantage to the full extent

199 Determination of KHβ for bevel gears If a theoretical approach similar to 193-198 is not docu-mented the following may be used

testeff

H Kb

b851851K sdot

minussdot=β

beff b represents the relative active facewidth (regarding lapped gears see 192 last part)

Higher values than beff b = 090 are normally not to be used in the formula

For dual directional gears it may be difficult to obtain a high beff b in both directions In that case the smaller beff b is to be used

Ktest represents the influence of the bearing arrangement shaft stiffness bearing stiffness housing stiffness etc on the faceload distribution and the verification thereof Expected variations in length- and height-wise tooth profile is also accounted for to some extent

a) Ktest = 1 For ground or hard metal hobbed gears with the specified contact pattern verified at full rating or at full torque slow turning at a condition representative for the thermal expansion at normal operation

14 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

It also applies when the bearing arrangementsupport has insignificant elastic deflections and thermal axial expansion However each initial mesh contact must be verified to be within acceptance criteria that are cali-brated against a type test at full load Reproduction of the gear tooth length- and height-wise profile must also be verified This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon)

b) Ktest = 1 + 04middot(beffbndash06) For designs with possible influence of thermal expansion in the axial direction of the pinion The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load

c) Ktest = 12 if mesh is only checked by toolmakerrsquos blue or by spin test contact For gears in this category beffb gt 085 is not to be used in the calculation

110 Transversal Load Distribution Factors KHα and KFα The transverse load distribution factors KHα for contact stresses and for scuffing KFα for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairs of teeth in mesh

The following relations may be used

Cylindrical gears

( )

minus+==

tH

αptγγHαFα F

byfc0409

2ε

KK

valid for 2εγ le

( ) ( )tH

αptγ

γ

γHαFα F

byfcε

1ε20409KK

minusminus+==

valid for 2ε γ gt

where

FtH = Ft KA Kγ Kv KHβ

cγ = See 111

γα = See 112

fpt = Maximum single pitch deviation (microm) of pinion or wheel or maximum total profile form devia-tion Fα of pinion or wheel if this is larger than the maximum single pitch deviation

Note In case of adequate equivalent tip relief adapted to the load half of the above mentioned fpt can be introduced A tip relief is considered adequate when the average of Ca1 and Ca2 is within plusmn40 of the value of Ceff in 432

Limitations of KHα and KFα

If the calculated values for

KFα = KHα lt 1 use KFα = KHα = 10

If the calculated value of KHα gt 2εα

γ

Zε

ε use KHα = 2

εα

γ

Zε

ε

If the calculated value of KFα gt εα

γ

Yεε

use KFα = εα

γ

Yεε

where Yε = αnε

075025+ (for εαn see 331c)

Bevel gears

For ground or hard metal hobbed gears KFα = KHα = 1

For lapped gears KFα = KHα = 11

111 Tooth Stiffness Constants cacute and cγ The tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having 1 mm facewidth by an amount of 1 microm in the plane of contact cacute is the maximum stiffness of a single pair of teeth cγ is the mean value of the mesh stiffness in a transverse plane (brief term mesh stiffness)

Both valid for high unit load (Unit load = Ft middot KA middot Kγb)

Cylindrical gears The real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bending stiff-nesses For high unit loads the Hertzian stiffness has little importance and can be disregarded This approach is on the safe side for determination of KHβ and KHα However for moderate or low loads Kv may be underestimated due to determination of a too high resonance speed

The linear approach is described in A

An optional approach for inclusion of the non-linear stiffness is described in B

A The linear approach

BRCCq

βcos08acutec =

and

( )025ε075cc αγ +prime=

where

( )[ ]n02a01a

B α2000212

hh12051C minusminus

+minus+=

12n1n

x000635z

025791z

015551004723q minus++=

12

2n

22

1n

1 x005290z

x024188x000193z

x011654+minusminusminus

+ 000182 x22

Classification Notes- No 412 15 May 2003

DET NORSKE VERITAS

(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as

( )( )nR m5s

sR e5

bbln1C +=

where

bs = thickness of a central web

sR = average thickness of rim (net value from tooth root to inside of rim)

The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered

Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ

B The non-linear approach

In the following an example is given on how to consider the non-linearity

The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm

With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as

( )10δKbF

minus= for 500bFgt

minus=

500Fb10δK

bF for 500

bFlt

with γAt KK

bF

bF

sdotsdot= etc (Nmm) ie unit load incorporat-

ing the relevant factors as

KA middot Kγ for determination of Kv

KA middot Kγ middot Kv for determination of KHβ

KA middot Kγ middot Kv middot KHβ for determination of KHα

δ = mesh deflection (microm)

K = applicable stiffness (c or cγ)

Use of stiffnesses for KV KHβ and KHα

For calculation of Kv and KHα the stiffness is calculated as follows

When Fb lt 500

the stiffness is determined as δ∆

∆ bF

where the increment is chosen as eg ∆ Fb = 10 and thus

50010Fb10

K10Fb∆δ +

++

=

When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly

or an equivalent stiffness determined as δsdotb

F

Bevel gears

In lack of more detailed relationship between stiffness and geometry the following may be used

b085

b13cacute eff=

b085b

16c effγ =

beff not to be used in excess of 085 b in these formulae

Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)

112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements

yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload

Cay is defined as the running-in amount that compensates for lack of tip relief

The following relations may be used

For not surface hardened steel

ptHlim

α fσ160y =

yβ = βxlimH

f320σ

with the following upper limits

V lt 5 ms 5-10 ms gt 10 ms

yα max none

limH

12800σ limH

6400σ

yβ max none

limH

25600σ limH

12800σ

For surface hardened steel

yα = 0075 fpt but not more than 3 for any speed

yβ = 015 Fβx but not more than 6 for any speed

16 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

For all kinds of steel

5145189718

1C2

limHay +

minusσ

=

When pinion and wheel material differ the following ap-plies

bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv

bull Use ( )2β1ββ yy21y += in the calculation of KHβ

bull Use ( )2ay1aya CC21C += in the calculation of Kv

bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing

calculation if no design tip relief is foreseen

Classification Notes- No 412 17 May 2003

DET NORSKE VERITAS

2 Calculation of Surface Durability

21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213

Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted

Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213

For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength

For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied

22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater

Calculation of surface durability for helical gears is based on the contact stress at the pitch point

For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies

Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact

Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses

The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses

The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel

221 Contact stress Cylindrical gears

( )HαHβvγA

1

tβεEHDBH KKKKK

bud1uF

ZZZZZσ+

=

where

ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)

ZH = Zone factor for pitch point (see 231)

ZE = Elasticity factor (see 24)

Zε = Contact ratio factor (see 25)

Zβ = Helix angle factor (see 26)

Ft KA Kγ Kv KHβ KHα see 15 ndash 110

d1 b u see 12 ndash 15

Bevel gears

( )HαHβvγA

v1v

vmtKEMH KKKKK

bud1uF

ZZZ105σ+

sdot=

where

105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)

ZE KA etc see above

ZM = mid-zone factor see 233

ZK = bevel gear factor see 27

Fmt dv1 uv see 12 ndash 15

It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks

222 Permissible contact stress

XWRvLH

NHlimHP ZZZZZ

SZσ

σ =

where

σH lim = Endurance limit for contact stresses (see 28)

ZN = Life factor for contact stresses (see 29)

SH = Required safety factor according to the rules

ZLZvZR = Oil film influence factors (see 210)

ZW = Work hardening factor (see 211)

ZX = Size factor (see 212)

18 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

23 Zone Factors ZH ZBD and ZM

231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder

wtt2

wtbH sinααcos

cosαcosβ2Z =

232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel

For εβ ge 1 ZBD = 1

For internal gears ZD = 1

For εβ = 0 (spur gears)

( )

π

minusεminusminus

π

minusminus

α=

α2

2

2b

2a

1

2

1b

1a

wtB

z211

dd

z21

dd

tanZ

( )

π

minusεminusminus

π

minusminus

α=

α1

2

1b

1a

2

2

2b

2a

wtD

z211

dd

z21

dd

tanZ

If ZB lt 1 use ZB = 1

If ZD lt 1 use ZD = 1

For 0 lt εβ lt 1

ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)

233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears

εminusminus

εminusminus

αβ=

αα btm2

2vb2

2vabtm2

1vb2val

2v1vvtbmM

pddpdd

ddtancos2Z

This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint

234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel

24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses

For steel against steel ZE = 1898

25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses

αε1Z =ε for 1εβ ge

( )α

ββ

αε ε

εε1

3ε4

Z +minusminus

= for εβ lt 1

26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability

cosβZβ =

27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply

The following may be used ZK = 080

28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ

σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting

For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply

For this purpose pitting is defined by

bull for not surface hardened gears pitted area ge 2 of total active flank area

bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given

material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies

The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as

Classification Notes- No 412 19 May 2003

DET NORSKE VERITAS

the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99

σHlim σH105 σH10

3

Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade

1650 1500

2500 2400

3100 3100

Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)

1000

13 σHlim

13 σHlim

Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV

Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV

Carbon steel 15 HV + 250 16 σHlim 16 σHlim

These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15

29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply

If this is not documented by approved fatigue tests the fol-lowing method may be used

For all steels except nitrided

7L 105N sdotge ZN = 1 or

01570

L

7

N N105Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 5middot107 510N

logZ037

L

7

N N105Z

sdot=

NL = 105 WXRVLHlim

WstX10H1010NN ZZZZZσ

ZZσZZ

55

5=

103 lt NL lt 105 )Z(Zlog05

L

5

10NN

5N103N10

5N10ZZ

==

3L 10N le

WXRVLlimH

Wst10X10H10NN ZZZZZ

ZZZZ

33

3σ

σ==

(but not less than ZN105)

For nitrided steels

6L 102N sdotge ZN = 1 or

00980

L

6

N N102Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 2middot106 510N

Zlog76860

L

6

N N102Z

sdot=

5L 10N le

XWRVL

10XWst10NN ZZZZZ

ZZ13ZZ

55 ==

Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles

210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity

The following methods may be applied in connection with the endurance limit

20 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Surface hardened steels Not surface hardened steels

ZL ( )24013421

360910ν+

+

( )24013421

680830ν+

+

ZV ( )v3280

140930+

+ ( )v3280300850

++

ZR

080

ZrelR3

150

ZrelR3

where

ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits

For values of ν40 gt 500 use ν40 = 500

RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm

RZrel

= ( ) 31

cZ2Z1 ρ

10RR50

+

RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)

For 5

L 10N le ZL ZV ZR = 10

211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface

The following approximation may be used for the endurance limit

Surface hardened steel against not surface hardened steel 150

ZeqW R

31700

130HB21Z

minus

minus=

where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130

RZeq = equivalent roughness

033

c40

066

ZS

ZHZHZEQ ρvν

15000RRRR

=

If RZeq gt 16 then use RZeq = 16

If RZeq lt 15 then use RZeq = 15

where

RZH = surface roughness of the hard member before run in

RZS = surface roughness of the soft member before run in

ν40 = see 210

If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem

Through hardened pinion against softer wheel

( )

minussdotminus+= 000829

HBHB0008981u1Z

2

1W

For 21HBHB

2

1 le use ZW = 1

For 71HBHB

2

1 gt use 17HBHB

2

1 =

For u gt 20 use u = 20

For static strength (lt 105 cycles)

Surface hardened against not surface hardened

ZWst = 105

Through hardened pinion against softer wheel

ZWst = 1

212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality

ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213

213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented

The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent

Classification Notes- No 412 21 May 2003

DET NORSKE VERITAS

stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically

The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence

The real Hertzian stresses σHR are determined as

For helical gears with εβ gt 1

σHR = σH

For helical gears with εβ lt 1 and spur gears

ε

α

ββ ε

ε+εminus

sdotσ=σZ

1

HHR

For bevel gears

KHHR Z

1σσ sdot=

The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)

The coordinates tz and HV are to be compared with the de-sign specification such as

bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400

and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)

In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods

For high cycle fatigue (gt3 106 cycles) the following applies

sdot+

minussdotsdotsdot= o90

05azt

05azt

cosSσ04HV

H

HHHR

applicable to 05a

zt

Hge

For 05at

H

z lt the value for 05at

H

z = applies

56300ρSσ12a cHHR

Hsdotsdot

sdot=

Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength

If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows

reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue

For static strength (lt103 cycles) the following applies

sdot+

minussdotsdotsdot= o90

07at

06a

zt

cosSσ019HV

Hst

z

HstHHR

applicable to 06at

Hst

z ge

56300ρSσa cHHR

Hstsdotsdot

=

In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies

For limited life fatigue (103 lt cycles lt 3106 )

For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance

The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram

minussdotminus

= sdot logN3477

logSlogSlogS

310H6103HHN

36 10H103H Slog86281Slog86280 sdot+sdot sdot

22 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3 Calculation of Tooth Strength

31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding

For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36

For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning

Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα

In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314

A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired

It should be noted that this part 3 does not cover fractures caused by

bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining

Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears

Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied

32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level

321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance

Normally the stresses for pinion and wheel are calculated as

Cylindrical gears

FαFβvγAβSFn

tF K K K K K Y Y Y

m bFσ =

where

YF = Tooth form factor (see 33)

YS = Stress correction factor (see 34)

Yβ = Helix angle factor (see 36)

Ft KA Kγ Kv KFβ KFα see 15 ndash 110

b see 13

Bevel gears

FαFβvγASaFamn

mtF K K K K K Y Y Y

m bFσ ε=

where

YFa = Tooth form factor see 33

YSa = Stress correction factor see 34

Yε = Contact ratio factor see 35

Fmt KA etc see 15 ndash 110

b see 13

322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is

CXRrelTrelTF

NMFEFP YYYY

SYY

δσ

=σ

Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN

where

σFE = Local tooth root bending endurance limit of reference test gear (see 37)

YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)

YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)

SF = Required safety factor according to the rules

YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)

YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)

Classification Notes- No 412 23 May 2003

DET NORSKE VERITAS

YX = Size factor (see 312)

YC = Case depth factor (see 313)

33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress

YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears

YFa applies to load application at the tooth tip and is used for bevel gears

Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears

Figure 31 External tooth in normal section

Figure 32 Internal tooth in normal section

Definitions

n

2

n

Fn

enFn

Fe

F

α cosms

α cosmh6

Y

=

n

2

n

Fn

anFn

Fa

Fa

α cosms

α cosmh6

Y

=

In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth

YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)

The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing

mn with mnm

zn with zvn

αt with αvt

β with βm

with undercut without undercut

Fig 33 Dimensions and basic rack profile of the teeth (finished profile)

Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless

331 Determination of parameters

( )n

n

prnfPnfP m

α cossαsin 1ρ

αtan h4πE

minusminusminusminus=

For external gears fPfP ρρ =

For internal gears ( )0z

195fPfP0

fPfP 10363156ρhxρρ

sdotminus+

+=

where

z0 = number of teeth of pinion cutter

x0 = addendum modification coefficient of pinion cutter

hfP = addendum of pinion cutter

ρfP = tip radius of pinion cutter

xhρG fpfp +minus=

24 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

τmE

2π

z2H

nnminus

minus=

with

3πτ = for external gears

6πτ = for internal gears

Htan zG2

nminusϑ=ϑ

(to be solved iteratively suitable start value6π

=ϑ for exter-

nal gears and 3π for internal gears)

a) Tooth root chord sFn For external gears

minus

ϑ+

ϑminus= ρ

cosG3

3πsinz

ms

fpnn

Fn

For bevel gears with a tooth thickness modification

xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn

For internal gears

minus

ϑ+

ϑminus= ρ

cosG

6πsinz

ms

fPnn

Fn

b) Root fillet radius ρF at 30ordm tangent

( )G2coszcosG2ρ

mρ

2n

2

fpn

F

minusϑϑ+=

c) Determination of bending moment arm hF dn = zn mn

b2α

αn βcosε

ε =

dan = dn + 2 ha

pbn = π mn cos αn

dbn = dn cos αn

( )4

d1εpzz

2dd

zz2d

2bn

2

αnbn

2bn

2an

en +

minusminus

minus=

en

bnen d

dcos arcα =

ennnn

e α invα invα x tan 22π

z1

minus+

+=γ

αFen = αen ndash γe

For external gears

( )

minus=

n

enFenee

n

Fe

mdαtan sin γ γcos

21

mh

]ρcos

G3πcosz fpn +

ϑminus

ϑminusminus

For internal gears

( ) minus

sdotsdotminus=

n

enFenee

n

Fe

mdα tanγ sinγ cos

21

mh

minus

ϑminus

ϑminussdot ρ

cosG3

6πcosz fPn

332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as

250ε205for 0666ε2366Y αnαnDT leleminus=

205εfor 10Y αnDT lt=

34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section

Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm

YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip

YS can be determined as follows

( )

+

+=L23121

1

sS qL01312Y

where Fe

Fn

hsL = and

F

Fns ρ2

sq = (see 33)

YSa can be calculated by replacing hFe with hFa in the above formulae

Note a) Range of validity 1 lt qs lt 8

In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered

b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with

Classification Notes- No 412 25 May 2003

DET NORSKE VERITAS

g

g

ρt

0613

13

minus

where

tg = depth of the grinding notch

ρg = radius of the grinding notch

c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles

35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears

The following may be used

Yε = 0625

36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank

The following may be used (β input in degrees)

Yβ = 1 ndash εβ β120

When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula

However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)

37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM

σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision

If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules

σFE

Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)

bull of specially approved high grade

1050

bull of normal grade

minus CrNiMo steels with approved process

1000

minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850

Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

840

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

720

Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)

07 HV + 300

Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)

025 σB + 125

Alloyed quenched and tempered steel 04 σB + 200

Carbon steel 025 σB + 250

Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40

1) These values are valid for a root radius

bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)

bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)

bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for

6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked

2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)

26 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears

YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0

YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst

For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316

The following method may be used within a stress ratio ndash12 lt R lt 05

381 For idlers planets and PTO with ice class

M1M1R1

1Yor Y MstM

+minus

minus=

where

R = stress ratio = min stress divided by max stress

For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12

For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as

branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus

For a power take off (PTO) with ice class see 161 c

M considers the mean stress influence on the endurance (or static) strength amplitudes

M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress

Following M values may be used

Endurance limit

Static strength

Case hardened 08 ndash 015 Ys 1) 07

If shot peened 04 06

Nitrided 03 03

Induction or flame hardened

04 06

Not surface hardened steel 03 05

Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M

The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches

382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106

For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers

For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used

383 For gears with shrinkage stresses and unidirectional load For endurance strength

FE

fitM σ

σM1

M21Y+

minus=

σFE is the endurance limit for R = 0

For static strength YMst = 1 and σfit accounted for in 39b

σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor

n

Ffit m

ρ215scf minus=

384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is

( ) ( ) FE

fitM σ

σR1M1

M2

M1M1R1

1Yminussdot+

minus

+minus

minus=

Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12

For static strength

M1M1R1

1YMst

+minus

minus=

The effect of σfit is accounted for in 39 b

Classification Notes- No 412 27 May 2003

DET NORSKE VERITAS

385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed

F

y

Sσ225

for not surface hardened fillets

FSHV5 for surface hardened fillets

39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles

Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN

If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used

Determination of the σ ndash N ndash curve

a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF

b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ

FstS1

FPstσ minussdotsdotsdot=

where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability

σFst

Alloyed case hardened steel 1) 2300

Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

1250

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

1050

Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)

18 HV + 800

Steel with not surface hardened fillets the smaller value of 2)

18 σB or 225 σy

1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet

c) Calculate YN as

NL gt 3middot106

YN = 1 or 001

L

6

N N103Y

sdot= ie Yn = 092 for 1010

The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2

103ltNLlt3middot106exp

L

6

N N103Y

sdot=

cycles6103forσcycles310forσlog02876exp

FP

FPst

sdot=

NL lt 103 cycles6103forσcycles310forσ

NYFP

FPst

sdot=

or simply use σFPst as mentioned in b) directly

Guidance on number of load cycles NL for various applica-tions

bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)

bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied

28 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage

YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit

The following method may be used

For endurance limit

for not surface hardened fillets

( )

024

s024

relT σ103133q21σ1012201351

Ysdotsdotminus

+sdotsdotminus+= minus

minus

δ

for all surface hardened fillets except nitrided

( )

106q21002451

Y srelT

++=δ

for nitrided fillets

( )

1347q2101421

Y srelT

++=δ

For static strength

for not surface hardened fillets1)

( )( )( )025

02

02502s

relTst σ3000821σ300 1Y0821Y

+

minus+=δ

for surface hardened fillets except nitrided

YδrelTst = 044 YS + 012

for nitrided fillets

YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not

exceed the yield point and thereby alter the residual stress level See also 39b footnote 2

311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness

YRrelT differs for endurance limit and static strength

The following method may be used

For endurance limit

YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels

YδrelT = 43 ndash 326 (Ry + 1)0005

for nitrided steels

For static strength

YRrelTst = 1 for all Ry and all materials

For a fillet without any longitudinal machining trace Ry asymp Rz

312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength

The following may be used

For endurance limit

YX = 1 for mn le 5 generally

YX = 103 ndash 0006 mn for 5 lt mn lt 30

YX = 085 for mn ge 30

for not surface hard-ened steels

YX = 105 ndash 001 mn for 5 lt mn ge 25

YX = 08 for mn ge 25

for surface hardened steels

For static strength

YXst = 1 for all mn and all materials

313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength

YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength

In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength

The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses

The following simplified method for YC may be used

YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)

YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10

For endurance limit

+

+=nFFE

C m02ρt31

σconstY

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

14 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

It also applies when the bearing arrangementsupport has insignificant elastic deflections and thermal axial expansion However each initial mesh contact must be verified to be within acceptance criteria that are cali-brated against a type test at full load Reproduction of the gear tooth length- and height-wise profile must also be verified This can be made through 3D measurements or by initial contact movements caused by defined axial offsets of the pinion (tolerances to be agreed upon)

b) Ktest = 1 + 04middot(beffbndash06) For designs with possible influence of thermal expansion in the axial direction of the pinion The initial mesh contact verified with low load or spin test where the acceptance criteria are calibrated against a type test at full load

c) Ktest = 12 if mesh is only checked by toolmakerrsquos blue or by spin test contact For gears in this category beffb gt 085 is not to be used in the calculation

110 Transversal Load Distribution Factors KHα and KFα The transverse load distribution factors KHα for contact stresses and for scuffing KFα for tooth root stresses account for the effects of pitch and profile errors on the transversal load distribution between 2 or more pairs of teeth in mesh

The following relations may be used

Cylindrical gears

( )

minus+==

tH

αptγγHαFα F

byfc0409

2ε

KK

valid for 2εγ le

( ) ( )tH

αptγ

γ

γHαFα F

byfcε

1ε20409KK

minusminus+==

valid for 2ε γ gt

where

FtH = Ft KA Kγ Kv KHβ

cγ = See 111

γα = See 112

fpt = Maximum single pitch deviation (microm) of pinion or wheel or maximum total profile form devia-tion Fα of pinion or wheel if this is larger than the maximum single pitch deviation

Note In case of adequate equivalent tip relief adapted to the load half of the above mentioned fpt can be introduced A tip relief is considered adequate when the average of Ca1 and Ca2 is within plusmn40 of the value of Ceff in 432

Limitations of KHα and KFα

If the calculated values for

KFα = KHα lt 1 use KFα = KHα = 10

If the calculated value of KHα gt 2εα

γ

Zε

ε use KHα = 2

εα

γ

Zε

ε

If the calculated value of KFα gt εα

γ

Yεε

use KFα = εα

γ

Yεε

where Yε = αnε

075025+ (for εαn see 331c)

Bevel gears

For ground or hard metal hobbed gears KFα = KHα = 1

For lapped gears KFα = KHα = 11

111 Tooth Stiffness Constants cacute and cγ The tooth stiffness is defined as the load which is necessary to deform one or several meshing gear teeth having 1 mm facewidth by an amount of 1 microm in the plane of contact cacute is the maximum stiffness of a single pair of teeth cγ is the mean value of the mesh stiffness in a transverse plane (brief term mesh stiffness)

Both valid for high unit load (Unit load = Ft middot KA middot Kγb)

Cylindrical gears The real stiffness is a combination of the progressive Hertzian contact stiffness and the linear tooth bending stiff-nesses For high unit loads the Hertzian stiffness has little importance and can be disregarded This approach is on the safe side for determination of KHβ and KHα However for moderate or low loads Kv may be underestimated due to determination of a too high resonance speed

The linear approach is described in A

An optional approach for inclusion of the non-linear stiffness is described in B

A The linear approach

BRCCq

βcos08acutec =

and

( )025ε075cc αγ +prime=

where

( )[ ]n02a01a

B α2000212

hh12051C minusminus

+minus+=

12n1n

x000635z

025791z

015551004723q minus++=

12

2n

22

1n

1 x005290z

x024188x000193z

x011654+minusminusminus

+ 000182 x22

Classification Notes- No 412 15 May 2003

DET NORSKE VERITAS

(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as

( )( )nR m5s

sR e5

bbln1C +=

where

bs = thickness of a central web

sR = average thickness of rim (net value from tooth root to inside of rim)

The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered

Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ

B The non-linear approach

In the following an example is given on how to consider the non-linearity

The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm

With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as

( )10δKbF

minus= for 500bFgt

minus=

500Fb10δK

bF for 500

bFlt

with γAt KK

bF

bF

sdotsdot= etc (Nmm) ie unit load incorporat-

ing the relevant factors as

KA middot Kγ for determination of Kv

KA middot Kγ middot Kv for determination of KHβ

KA middot Kγ middot Kv middot KHβ for determination of KHα

δ = mesh deflection (microm)

K = applicable stiffness (c or cγ)

Use of stiffnesses for KV KHβ and KHα

For calculation of Kv and KHα the stiffness is calculated as follows

When Fb lt 500

the stiffness is determined as δ∆

∆ bF

where the increment is chosen as eg ∆ Fb = 10 and thus

50010Fb10

K10Fb∆δ +

++

=

When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly

or an equivalent stiffness determined as δsdotb

F

Bevel gears

In lack of more detailed relationship between stiffness and geometry the following may be used

b085

b13cacute eff=

b085b

16c effγ =

beff not to be used in excess of 085 b in these formulae

Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)

112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements

yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload

Cay is defined as the running-in amount that compensates for lack of tip relief

The following relations may be used

For not surface hardened steel

ptHlim

α fσ160y =

yβ = βxlimH

f320σ

with the following upper limits

V lt 5 ms 5-10 ms gt 10 ms

yα max none

limH

12800σ limH

6400σ

yβ max none

limH

25600σ limH

12800σ

For surface hardened steel

yα = 0075 fpt but not more than 3 for any speed

yβ = 015 Fβx but not more than 6 for any speed

16 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

For all kinds of steel

5145189718

1C2

limHay +

minusσ

=

When pinion and wheel material differ the following ap-plies

bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv

bull Use ( )2β1ββ yy21y += in the calculation of KHβ

bull Use ( )2ay1aya CC21C += in the calculation of Kv

bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing

calculation if no design tip relief is foreseen

Classification Notes- No 412 17 May 2003

DET NORSKE VERITAS

2 Calculation of Surface Durability

21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213

Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted

Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213

For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength

For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied

22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater

Calculation of surface durability for helical gears is based on the contact stress at the pitch point

For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies

Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact

Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses

The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses

The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel

221 Contact stress Cylindrical gears

( )HαHβvγA

1

tβεEHDBH KKKKK

bud1uF

ZZZZZσ+

=

where

ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)

ZH = Zone factor for pitch point (see 231)

ZE = Elasticity factor (see 24)

Zε = Contact ratio factor (see 25)

Zβ = Helix angle factor (see 26)

Ft KA Kγ Kv KHβ KHα see 15 ndash 110

d1 b u see 12 ndash 15

Bevel gears

( )HαHβvγA

v1v

vmtKEMH KKKKK

bud1uF

ZZZ105σ+

sdot=

where

105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)

ZE KA etc see above

ZM = mid-zone factor see 233

ZK = bevel gear factor see 27

Fmt dv1 uv see 12 ndash 15

It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks

222 Permissible contact stress

XWRvLH

NHlimHP ZZZZZ

SZσ

σ =

where

σH lim = Endurance limit for contact stresses (see 28)

ZN = Life factor for contact stresses (see 29)

SH = Required safety factor according to the rules

ZLZvZR = Oil film influence factors (see 210)

ZW = Work hardening factor (see 211)

ZX = Size factor (see 212)

18 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

23 Zone Factors ZH ZBD and ZM

231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder

wtt2

wtbH sinααcos

cosαcosβ2Z =

232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel

For εβ ge 1 ZBD = 1

For internal gears ZD = 1

For εβ = 0 (spur gears)

( )

π

minusεminusminus

π

minusminus

α=

α2

2

2b

2a

1

2

1b

1a

wtB

z211

dd

z21

dd

tanZ

( )

π

minusεminusminus

π

minusminus

α=

α1

2

1b

1a

2

2

2b

2a

wtD

z211

dd

z21

dd

tanZ

If ZB lt 1 use ZB = 1

If ZD lt 1 use ZD = 1

For 0 lt εβ lt 1

ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)

233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears

εminusminus

εminusminus

αβ=

αα btm2

2vb2

2vabtm2

1vb2val

2v1vvtbmM

pddpdd

ddtancos2Z

This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint

234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel

24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses

For steel against steel ZE = 1898

25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses

αε1Z =ε for 1εβ ge

( )α

ββ

αε ε

εε1

3ε4

Z +minusminus

= for εβ lt 1

26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability

cosβZβ =

27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply

The following may be used ZK = 080

28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ

σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting

For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply

For this purpose pitting is defined by

bull for not surface hardened gears pitted area ge 2 of total active flank area

bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given

material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies

The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as

Classification Notes- No 412 19 May 2003

DET NORSKE VERITAS

the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99

σHlim σH105 σH10

3

Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade

1650 1500

2500 2400

3100 3100

Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)

1000

13 σHlim

13 σHlim

Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV

Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV

Carbon steel 15 HV + 250 16 σHlim 16 σHlim

These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15

29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply

If this is not documented by approved fatigue tests the fol-lowing method may be used

For all steels except nitrided

7L 105N sdotge ZN = 1 or

01570

L

7

N N105Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 5middot107 510N

logZ037

L

7

N N105Z

sdot=

NL = 105 WXRVLHlim

WstX10H1010NN ZZZZZσ

ZZσZZ

55

5=

103 lt NL lt 105 )Z(Zlog05

L

5

10NN

5N103N10

5N10ZZ

==

3L 10N le

WXRVLlimH

Wst10X10H10NN ZZZZZ

ZZZZ

33

3σ

σ==

(but not less than ZN105)

For nitrided steels

6L 102N sdotge ZN = 1 or

00980

L

6

N N102Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 2middot106 510N

Zlog76860

L

6

N N102Z

sdot=

5L 10N le

XWRVL

10XWst10NN ZZZZZ

ZZ13ZZ

55 ==

Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles

210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity

The following methods may be applied in connection with the endurance limit

20 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Surface hardened steels Not surface hardened steels

ZL ( )24013421

360910ν+

+

( )24013421

680830ν+

+

ZV ( )v3280

140930+

+ ( )v3280300850

++

ZR

080

ZrelR3

150

ZrelR3

where

ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits

For values of ν40 gt 500 use ν40 = 500

RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm

RZrel

= ( ) 31

cZ2Z1 ρ

10RR50

+

RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)

For 5

L 10N le ZL ZV ZR = 10

211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface

The following approximation may be used for the endurance limit

Surface hardened steel against not surface hardened steel 150

ZeqW R

31700

130HB21Z

minus

minus=

where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130

RZeq = equivalent roughness

033

c40

066

ZS

ZHZHZEQ ρvν

15000RRRR

=

If RZeq gt 16 then use RZeq = 16

If RZeq lt 15 then use RZeq = 15

where

RZH = surface roughness of the hard member before run in

RZS = surface roughness of the soft member before run in

ν40 = see 210

If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem

Through hardened pinion against softer wheel

( )

minussdotminus+= 000829

HBHB0008981u1Z

2

1W

For 21HBHB

2

1 le use ZW = 1

For 71HBHB

2

1 gt use 17HBHB

2

1 =

For u gt 20 use u = 20

For static strength (lt 105 cycles)

Surface hardened against not surface hardened

ZWst = 105

Through hardened pinion against softer wheel

ZWst = 1

212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality

ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213

213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented

The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent

Classification Notes- No 412 21 May 2003

DET NORSKE VERITAS

stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically

The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence

The real Hertzian stresses σHR are determined as

For helical gears with εβ gt 1

σHR = σH

For helical gears with εβ lt 1 and spur gears

ε

α

ββ ε

ε+εminus

sdotσ=σZ

1

HHR

For bevel gears

KHHR Z

1σσ sdot=

The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)

The coordinates tz and HV are to be compared with the de-sign specification such as

bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400

and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)

In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods

For high cycle fatigue (gt3 106 cycles) the following applies

sdot+

minussdotsdotsdot= o90

05azt

05azt

cosSσ04HV

H

HHHR

applicable to 05a

zt

Hge

For 05at

H

z lt the value for 05at

H

z = applies

56300ρSσ12a cHHR

Hsdotsdot

sdot=

Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength

If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows

reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue

For static strength (lt103 cycles) the following applies

sdot+

minussdotsdotsdot= o90

07at

06a

zt

cosSσ019HV

Hst

z

HstHHR

applicable to 06at

Hst

z ge

56300ρSσa cHHR

Hstsdotsdot

=

In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies

For limited life fatigue (103 lt cycles lt 3106 )

For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance

The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram

minussdotminus

= sdot logN3477

logSlogSlogS

310H6103HHN

36 10H103H Slog86281Slog86280 sdot+sdot sdot

22 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3 Calculation of Tooth Strength

31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding

For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36

For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning

Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα

In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314

A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired

It should be noted that this part 3 does not cover fractures caused by

bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining

Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears

Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied

32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level

321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance

Normally the stresses for pinion and wheel are calculated as

Cylindrical gears

FαFβvγAβSFn

tF K K K K K Y Y Y

m bFσ =

where

YF = Tooth form factor (see 33)

YS = Stress correction factor (see 34)

Yβ = Helix angle factor (see 36)

Ft KA Kγ Kv KFβ KFα see 15 ndash 110

b see 13

Bevel gears

FαFβvγASaFamn

mtF K K K K K Y Y Y

m bFσ ε=

where

YFa = Tooth form factor see 33

YSa = Stress correction factor see 34

Yε = Contact ratio factor see 35

Fmt KA etc see 15 ndash 110

b see 13

322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is

CXRrelTrelTF

NMFEFP YYYY

SYY

δσ

=σ

Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN

where

σFE = Local tooth root bending endurance limit of reference test gear (see 37)

YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)

YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)

SF = Required safety factor according to the rules

YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)

YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)

Classification Notes- No 412 23 May 2003

DET NORSKE VERITAS

YX = Size factor (see 312)

YC = Case depth factor (see 313)

33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress

YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears

YFa applies to load application at the tooth tip and is used for bevel gears

Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears

Figure 31 External tooth in normal section

Figure 32 Internal tooth in normal section

Definitions

n

2

n

Fn

enFn

Fe

F

α cosms

α cosmh6

Y

=

n

2

n

Fn

anFn

Fa

Fa

α cosms

α cosmh6

Y

=

In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth

YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)

The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing

mn with mnm

zn with zvn

αt with αvt

β with βm

with undercut without undercut

Fig 33 Dimensions and basic rack profile of the teeth (finished profile)

Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless

331 Determination of parameters

( )n

n

prnfPnfP m

α cossαsin 1ρ

αtan h4πE

minusminusminusminus=

For external gears fPfP ρρ =

For internal gears ( )0z

195fPfP0

fPfP 10363156ρhxρρ

sdotminus+

+=

where

z0 = number of teeth of pinion cutter

x0 = addendum modification coefficient of pinion cutter

hfP = addendum of pinion cutter

ρfP = tip radius of pinion cutter

xhρG fpfp +minus=

24 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

τmE

2π

z2H

nnminus

minus=

with

3πτ = for external gears

6πτ = for internal gears

Htan zG2

nminusϑ=ϑ

(to be solved iteratively suitable start value6π

=ϑ for exter-

nal gears and 3π for internal gears)

a) Tooth root chord sFn For external gears

minus

ϑ+

ϑminus= ρ

cosG3

3πsinz

ms

fpnn

Fn

For bevel gears with a tooth thickness modification

xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn

For internal gears

minus

ϑ+

ϑminus= ρ

cosG

6πsinz

ms

fPnn

Fn

b) Root fillet radius ρF at 30ordm tangent

( )G2coszcosG2ρ

mρ

2n

2

fpn

F

minusϑϑ+=

c) Determination of bending moment arm hF dn = zn mn

b2α

αn βcosε

ε =

dan = dn + 2 ha

pbn = π mn cos αn

dbn = dn cos αn

( )4

d1εpzz

2dd

zz2d

2bn

2

αnbn

2bn

2an

en +

minusminus

minus=

en

bnen d

dcos arcα =

ennnn

e α invα invα x tan 22π

z1

minus+

+=γ

αFen = αen ndash γe

For external gears

( )

minus=

n

enFenee

n

Fe

mdαtan sin γ γcos

21

mh

]ρcos

G3πcosz fpn +

ϑminus

ϑminusminus

For internal gears

( ) minus

sdotsdotminus=

n

enFenee

n

Fe

mdα tanγ sinγ cos

21

mh

minus

ϑminus

ϑminussdot ρ

cosG3

6πcosz fPn

332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as

250ε205for 0666ε2366Y αnαnDT leleminus=

205εfor 10Y αnDT lt=

34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section

Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm

YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip

YS can be determined as follows

( )

+

+=L23121

1

sS qL01312Y

where Fe

Fn

hsL = and

F

Fns ρ2

sq = (see 33)

YSa can be calculated by replacing hFe with hFa in the above formulae

Note a) Range of validity 1 lt qs lt 8

In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered

b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with

Classification Notes- No 412 25 May 2003

DET NORSKE VERITAS

g

g

ρt

0613

13

minus

where

tg = depth of the grinding notch

ρg = radius of the grinding notch

c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles

35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears

The following may be used

Yε = 0625

36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank

The following may be used (β input in degrees)

Yβ = 1 ndash εβ β120

When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula

However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)

37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM

σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision

If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules

σFE

Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)

bull of specially approved high grade

1050

bull of normal grade

minus CrNiMo steels with approved process

1000

minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850

Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

840

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

720

Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)

07 HV + 300

Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)

025 σB + 125

Alloyed quenched and tempered steel 04 σB + 200

Carbon steel 025 σB + 250

Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40

1) These values are valid for a root radius

bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)

bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)

bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for

6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked

2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)

26 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears

YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0

YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst

For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316

The following method may be used within a stress ratio ndash12 lt R lt 05

381 For idlers planets and PTO with ice class

M1M1R1

1Yor Y MstM

+minus

minus=

where

R = stress ratio = min stress divided by max stress

For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12

For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as

branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus

For a power take off (PTO) with ice class see 161 c

M considers the mean stress influence on the endurance (or static) strength amplitudes

M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress

Following M values may be used

Endurance limit

Static strength

Case hardened 08 ndash 015 Ys 1) 07

If shot peened 04 06

Nitrided 03 03

Induction or flame hardened

04 06

Not surface hardened steel 03 05

Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M

The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches

382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106

For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers

For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used

383 For gears with shrinkage stresses and unidirectional load For endurance strength

FE

fitM σ

σM1

M21Y+

minus=

σFE is the endurance limit for R = 0

For static strength YMst = 1 and σfit accounted for in 39b

σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor

n

Ffit m

ρ215scf minus=

384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is

( ) ( ) FE

fitM σ

σR1M1

M2

M1M1R1

1Yminussdot+

minus

+minus

minus=

Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12

For static strength

M1M1R1

1YMst

+minus

minus=

The effect of σfit is accounted for in 39 b

Classification Notes- No 412 27 May 2003

DET NORSKE VERITAS

385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed

F

y

Sσ225

for not surface hardened fillets

FSHV5 for surface hardened fillets

39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles

Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN

If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used

Determination of the σ ndash N ndash curve

a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF

b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ

FstS1

FPstσ minussdotsdotsdot=

where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability

σFst

Alloyed case hardened steel 1) 2300

Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

1250

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

1050

Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)

18 HV + 800

Steel with not surface hardened fillets the smaller value of 2)

18 σB or 225 σy

1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet

c) Calculate YN as

NL gt 3middot106

YN = 1 or 001

L

6

N N103Y

sdot= ie Yn = 092 for 1010

The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2

103ltNLlt3middot106exp

L

6

N N103Y

sdot=

cycles6103forσcycles310forσlog02876exp

FP

FPst

sdot=

NL lt 103 cycles6103forσcycles310forσ

NYFP

FPst

sdot=

or simply use σFPst as mentioned in b) directly

Guidance on number of load cycles NL for various applica-tions

bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)

bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied

28 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage

YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit

The following method may be used

For endurance limit

for not surface hardened fillets

( )

024

s024

relT σ103133q21σ1012201351

Ysdotsdotminus

+sdotsdotminus+= minus

minus

δ

for all surface hardened fillets except nitrided

( )

106q21002451

Y srelT

++=δ

for nitrided fillets

( )

1347q2101421

Y srelT

++=δ

For static strength

for not surface hardened fillets1)

( )( )( )025

02

02502s

relTst σ3000821σ300 1Y0821Y

+

minus+=δ

for surface hardened fillets except nitrided

YδrelTst = 044 YS + 012

for nitrided fillets

YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not

exceed the yield point and thereby alter the residual stress level See also 39b footnote 2

311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness

YRrelT differs for endurance limit and static strength

The following method may be used

For endurance limit

YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels

YδrelT = 43 ndash 326 (Ry + 1)0005

for nitrided steels

For static strength

YRrelTst = 1 for all Ry and all materials

For a fillet without any longitudinal machining trace Ry asymp Rz

312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength

The following may be used

For endurance limit

YX = 1 for mn le 5 generally

YX = 103 ndash 0006 mn for 5 lt mn lt 30

YX = 085 for mn ge 30

for not surface hard-ened steels

YX = 105 ndash 001 mn for 5 lt mn ge 25

YX = 08 for mn ge 25

for surface hardened steels

For static strength

YXst = 1 for all mn and all materials

313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength

YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength

In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength

The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses

The following simplified method for YC may be used

YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)

YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10

For endurance limit

+

+=nFFE

C m02ρt31

σconstY

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

Classification Notes- No 412 15 May 2003

DET NORSKE VERITAS

(for internal gears use zn2 equal infinite and x2 = 0) ha0 = hfp for all practical purposes CR considers the increased flexibility of the wheel teeth if the wheel is not a solid disc and may be calculated as

( )( )nR m5s

sR e5

bbln1C +=

where

bs = thickness of a central web

sR = average thickness of rim (net value from tooth root to inside of rim)

The formula is valid for bs b ge 02 and sRmn ge 1 Outside this range of validity and if the web is not centrally posi-tioned CR has to be specially considered

Note CR is the ratio between the average mesh stiffness over the facewidth and the mesh stiffness of a gear pair of solid discs The local mesh stiffness in way of the web corresponds to the mesh stiffness with CR = 1 The local mesh stiffness where there is no web support will be less than calculated with CR above Thus eg a centrally positioned web will have an effect corresponding to a longitudinal crowning of the teeth See also 1931 regarding KHβ

B The non-linear approach

In the following an example is given on how to consider the non-linearity

The relation between unit load Fb as a function of mesh deflection δ is assumed to be a progressive curve up to 500 Nmm and from there on a straight line This straight line when extended to the baseline is assumed to intersect at 10microm

With these assumptions the unit force Fb as a function of mesh deflection δ can be expressed as

( )10δKbF

minus= for 500bFgt

minus=

500Fb10δK

bF for 500

bFlt

with γAt KK

bF

bF

sdotsdot= etc (Nmm) ie unit load incorporat-

ing the relevant factors as

KA middot Kγ for determination of Kv

KA middot Kγ middot Kv for determination of KHβ

KA middot Kγ middot Kv middot KHβ for determination of KHα

δ = mesh deflection (microm)

K = applicable stiffness (c or cγ)

Use of stiffnesses for KV KHβ and KHα

For calculation of Kv and KHα the stiffness is calculated as follows

When Fb lt 500

the stiffness is determined as δ∆

∆ bF

where the increment is chosen as eg ∆ Fb = 10 and thus

50010Fb10

K10Fb∆δ +

++

=

When F b gt 500 the stiffness is c or cγ For calculation of KHβ the mesh deflection δ is used directly

or an equivalent stiffness determined as δsdotb

F

Bevel gears

In lack of more detailed relationship between stiffness and geometry the following may be used

b085

b13cacute eff=

b085b

16c effγ =

beff not to be used in excess of 085 b in these formulae

Bevel gears with heightwise and lengthwise crowning have progressive mesh stiffness The values mentioned above are only valid for high loads They should not be used for de-termination of Ceff (see 432) or KHβ (see 1931)

112 Running-in Allowances The running-in allowances account for the influence of run-ning-in wear on the various error elements

yα respectively yβ are the running-in amounts which reduce the influence of pitch and profile errors respectively influ-ence of localised faceload

Cay is defined as the running-in amount that compensates for lack of tip relief

The following relations may be used

For not surface hardened steel

ptHlim

α fσ160y =

yβ = βxlimH

f320σ

with the following upper limits

V lt 5 ms 5-10 ms gt 10 ms

yα max none

limH

12800σ limH

6400σ

yβ max none

limH

25600σ limH

12800σ

For surface hardened steel

yα = 0075 fpt but not more than 3 for any speed

yβ = 015 Fβx but not more than 6 for any speed

16 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

For all kinds of steel

5145189718

1C2

limHay +

minusσ

=

When pinion and wheel material differ the following ap-plies

bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv

bull Use ( )2β1ββ yy21y += in the calculation of KHβ

bull Use ( )2ay1aya CC21C += in the calculation of Kv

bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing

calculation if no design tip relief is foreseen

Classification Notes- No 412 17 May 2003

DET NORSKE VERITAS

2 Calculation of Surface Durability

21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213

Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted

Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213

For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength

For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied

22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater

Calculation of surface durability for helical gears is based on the contact stress at the pitch point

For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies

Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact

Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses

The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses

The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel

221 Contact stress Cylindrical gears

( )HαHβvγA

1

tβεEHDBH KKKKK

bud1uF

ZZZZZσ+

=

where

ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)

ZH = Zone factor for pitch point (see 231)

ZE = Elasticity factor (see 24)

Zε = Contact ratio factor (see 25)

Zβ = Helix angle factor (see 26)

Ft KA Kγ Kv KHβ KHα see 15 ndash 110

d1 b u see 12 ndash 15

Bevel gears

( )HαHβvγA

v1v

vmtKEMH KKKKK

bud1uF

ZZZ105σ+

sdot=

where

105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)

ZE KA etc see above

ZM = mid-zone factor see 233

ZK = bevel gear factor see 27

Fmt dv1 uv see 12 ndash 15

It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks

222 Permissible contact stress

XWRvLH

NHlimHP ZZZZZ

SZσ

σ =

where

σH lim = Endurance limit for contact stresses (see 28)

ZN = Life factor for contact stresses (see 29)

SH = Required safety factor according to the rules

ZLZvZR = Oil film influence factors (see 210)

ZW = Work hardening factor (see 211)

ZX = Size factor (see 212)

18 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

23 Zone Factors ZH ZBD and ZM

231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder

wtt2

wtbH sinααcos

cosαcosβ2Z =

232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel

For εβ ge 1 ZBD = 1

For internal gears ZD = 1

For εβ = 0 (spur gears)

( )

π

minusεminusminus

π

minusminus

α=

α2

2

2b

2a

1

2

1b

1a

wtB

z211

dd

z21

dd

tanZ

( )

π

minusεminusminus

π

minusminus

α=

α1

2

1b

1a

2

2

2b

2a

wtD

z211

dd

z21

dd

tanZ

If ZB lt 1 use ZB = 1

If ZD lt 1 use ZD = 1

For 0 lt εβ lt 1

ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)

233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears

εminusminus

εminusminus

αβ=

αα btm2

2vb2

2vabtm2

1vb2val

2v1vvtbmM

pddpdd

ddtancos2Z

This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint

234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel

24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses

For steel against steel ZE = 1898

25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses

αε1Z =ε for 1εβ ge

( )α

ββ

αε ε

εε1

3ε4

Z +minusminus

= for εβ lt 1

26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability

cosβZβ =

27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply

The following may be used ZK = 080

28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ

σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting

For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply

For this purpose pitting is defined by

bull for not surface hardened gears pitted area ge 2 of total active flank area

bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given

material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies

The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as

Classification Notes- No 412 19 May 2003

DET NORSKE VERITAS

the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99

σHlim σH105 σH10

3

Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade

1650 1500

2500 2400

3100 3100

Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)

1000

13 σHlim

13 σHlim

Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV

Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV

Carbon steel 15 HV + 250 16 σHlim 16 σHlim

These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15

29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply

If this is not documented by approved fatigue tests the fol-lowing method may be used

For all steels except nitrided

7L 105N sdotge ZN = 1 or

01570

L

7

N N105Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 5middot107 510N

logZ037

L

7

N N105Z

sdot=

NL = 105 WXRVLHlim

WstX10H1010NN ZZZZZσ

ZZσZZ

55

5=

103 lt NL lt 105 )Z(Zlog05

L

5

10NN

5N103N10

5N10ZZ

==

3L 10N le

WXRVLlimH

Wst10X10H10NN ZZZZZ

ZZZZ

33

3σ

σ==

(but not less than ZN105)

For nitrided steels

6L 102N sdotge ZN = 1 or

00980

L

6

N N102Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 2middot106 510N

Zlog76860

L

6

N N102Z

sdot=

5L 10N le

XWRVL

10XWst10NN ZZZZZ

ZZ13ZZ

55 ==

Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles

210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity

The following methods may be applied in connection with the endurance limit

20 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Surface hardened steels Not surface hardened steels

ZL ( )24013421

360910ν+

+

( )24013421

680830ν+

+

ZV ( )v3280

140930+

+ ( )v3280300850

++

ZR

080

ZrelR3

150

ZrelR3

where

ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits

For values of ν40 gt 500 use ν40 = 500

RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm

RZrel

= ( ) 31

cZ2Z1 ρ

10RR50

+

RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)

For 5

L 10N le ZL ZV ZR = 10

211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface

The following approximation may be used for the endurance limit

Surface hardened steel against not surface hardened steel 150

ZeqW R

31700

130HB21Z

minus

minus=

where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130

RZeq = equivalent roughness

033

c40

066

ZS

ZHZHZEQ ρvν

15000RRRR

=

If RZeq gt 16 then use RZeq = 16

If RZeq lt 15 then use RZeq = 15

where

RZH = surface roughness of the hard member before run in

RZS = surface roughness of the soft member before run in

ν40 = see 210

If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem

Through hardened pinion against softer wheel

( )

minussdotminus+= 000829

HBHB0008981u1Z

2

1W

For 21HBHB

2

1 le use ZW = 1

For 71HBHB

2

1 gt use 17HBHB

2

1 =

For u gt 20 use u = 20

For static strength (lt 105 cycles)

Surface hardened against not surface hardened

ZWst = 105

Through hardened pinion against softer wheel

ZWst = 1

212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality

ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213

213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented

The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent

Classification Notes- No 412 21 May 2003

DET NORSKE VERITAS

stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically

The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence

The real Hertzian stresses σHR are determined as

For helical gears with εβ gt 1

σHR = σH

For helical gears with εβ lt 1 and spur gears

ε

α

ββ ε

ε+εminus

sdotσ=σZ

1

HHR

For bevel gears

KHHR Z

1σσ sdot=

The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)

The coordinates tz and HV are to be compared with the de-sign specification such as

bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400

and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)

In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods

For high cycle fatigue (gt3 106 cycles) the following applies

sdot+

minussdotsdotsdot= o90

05azt

05azt

cosSσ04HV

H

HHHR

applicable to 05a

zt

Hge

For 05at

H

z lt the value for 05at

H

z = applies

56300ρSσ12a cHHR

Hsdotsdot

sdot=

Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength

If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows

reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue

For static strength (lt103 cycles) the following applies

sdot+

minussdotsdotsdot= o90

07at

06a

zt

cosSσ019HV

Hst

z

HstHHR

applicable to 06at

Hst

z ge

56300ρSσa cHHR

Hstsdotsdot

=

In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies

For limited life fatigue (103 lt cycles lt 3106 )

For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance

The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram

minussdotminus

= sdot logN3477

logSlogSlogS

310H6103HHN

36 10H103H Slog86281Slog86280 sdot+sdot sdot

22 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3 Calculation of Tooth Strength

31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding

For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36

For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning

Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα

In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314

A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired

It should be noted that this part 3 does not cover fractures caused by

bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining

Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears

Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied

32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level

321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance

Normally the stresses for pinion and wheel are calculated as

Cylindrical gears

FαFβvγAβSFn

tF K K K K K Y Y Y

m bFσ =

where

YF = Tooth form factor (see 33)

YS = Stress correction factor (see 34)

Yβ = Helix angle factor (see 36)

Ft KA Kγ Kv KFβ KFα see 15 ndash 110

b see 13

Bevel gears

FαFβvγASaFamn

mtF K K K K K Y Y Y

m bFσ ε=

where

YFa = Tooth form factor see 33

YSa = Stress correction factor see 34

Yε = Contact ratio factor see 35

Fmt KA etc see 15 ndash 110

b see 13

322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is

CXRrelTrelTF

NMFEFP YYYY

SYY

δσ

=σ

Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN

where

σFE = Local tooth root bending endurance limit of reference test gear (see 37)

YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)

YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)

SF = Required safety factor according to the rules

YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)

YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)

Classification Notes- No 412 23 May 2003

DET NORSKE VERITAS

YX = Size factor (see 312)

YC = Case depth factor (see 313)

33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress

YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears

YFa applies to load application at the tooth tip and is used for bevel gears

Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears

Figure 31 External tooth in normal section

Figure 32 Internal tooth in normal section

Definitions

n

2

n

Fn

enFn

Fe

F

α cosms

α cosmh6

Y

=

n

2

n

Fn

anFn

Fa

Fa

α cosms

α cosmh6

Y

=

In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth

YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)

The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing

mn with mnm

zn with zvn

αt with αvt

β with βm

with undercut without undercut

Fig 33 Dimensions and basic rack profile of the teeth (finished profile)

Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless

331 Determination of parameters

( )n

n

prnfPnfP m

α cossαsin 1ρ

αtan h4πE

minusminusminusminus=

For external gears fPfP ρρ =

For internal gears ( )0z

195fPfP0

fPfP 10363156ρhxρρ

sdotminus+

+=

where

z0 = number of teeth of pinion cutter

x0 = addendum modification coefficient of pinion cutter

hfP = addendum of pinion cutter

ρfP = tip radius of pinion cutter

xhρG fpfp +minus=

24 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

τmE

2π

z2H

nnminus

minus=

with

3πτ = for external gears

6πτ = for internal gears

Htan zG2

nminusϑ=ϑ

(to be solved iteratively suitable start value6π

=ϑ for exter-

nal gears and 3π for internal gears)

a) Tooth root chord sFn For external gears

minus

ϑ+

ϑminus= ρ

cosG3

3πsinz

ms

fpnn

Fn

For bevel gears with a tooth thickness modification

xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn

For internal gears

minus

ϑ+

ϑminus= ρ

cosG

6πsinz

ms

fPnn

Fn

b) Root fillet radius ρF at 30ordm tangent

( )G2coszcosG2ρ

mρ

2n

2

fpn

F

minusϑϑ+=

c) Determination of bending moment arm hF dn = zn mn

b2α

αn βcosε

ε =

dan = dn + 2 ha

pbn = π mn cos αn

dbn = dn cos αn

( )4

d1εpzz

2dd

zz2d

2bn

2

αnbn

2bn

2an

en +

minusminus

minus=

en

bnen d

dcos arcα =

ennnn

e α invα invα x tan 22π

z1

minus+

+=γ

αFen = αen ndash γe

For external gears

( )

minus=

n

enFenee

n

Fe

mdαtan sin γ γcos

21

mh

]ρcos

G3πcosz fpn +

ϑminus

ϑminusminus

For internal gears

( ) minus

sdotsdotminus=

n

enFenee

n

Fe

mdα tanγ sinγ cos

21

mh

minus

ϑminus

ϑminussdot ρ

cosG3

6πcosz fPn

332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as

250ε205for 0666ε2366Y αnαnDT leleminus=

205εfor 10Y αnDT lt=

34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section

Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm

YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip

YS can be determined as follows

( )

+

+=L23121

1

sS qL01312Y

where Fe

Fn

hsL = and

F

Fns ρ2

sq = (see 33)

YSa can be calculated by replacing hFe with hFa in the above formulae

Note a) Range of validity 1 lt qs lt 8

In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered

b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with

Classification Notes- No 412 25 May 2003

DET NORSKE VERITAS

g

g

ρt

0613

13

minus

where

tg = depth of the grinding notch

ρg = radius of the grinding notch

c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles

35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears

The following may be used

Yε = 0625

36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank

The following may be used (β input in degrees)

Yβ = 1 ndash εβ β120

When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula

However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)

37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM

σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision

If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules

σFE

Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)

bull of specially approved high grade

1050

bull of normal grade

minus CrNiMo steels with approved process

1000

minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850

Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

840

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

720

Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)

07 HV + 300

Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)

025 σB + 125

Alloyed quenched and tempered steel 04 σB + 200

Carbon steel 025 σB + 250

Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40

1) These values are valid for a root radius

bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)

bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)

bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for

6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked

2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)

26 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears

YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0

YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst

For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316

The following method may be used within a stress ratio ndash12 lt R lt 05

381 For idlers planets and PTO with ice class

M1M1R1

1Yor Y MstM

+minus

minus=

where

R = stress ratio = min stress divided by max stress

For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12

For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as

branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus

For a power take off (PTO) with ice class see 161 c

M considers the mean stress influence on the endurance (or static) strength amplitudes

M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress

Following M values may be used

Endurance limit

Static strength

Case hardened 08 ndash 015 Ys 1) 07

If shot peened 04 06

Nitrided 03 03

Induction or flame hardened

04 06

Not surface hardened steel 03 05

Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M

The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches

382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106

For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers

For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used

383 For gears with shrinkage stresses and unidirectional load For endurance strength

FE

fitM σ

σM1

M21Y+

minus=

σFE is the endurance limit for R = 0

For static strength YMst = 1 and σfit accounted for in 39b

σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor

n

Ffit m

ρ215scf minus=

384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is

( ) ( ) FE

fitM σ

σR1M1

M2

M1M1R1

1Yminussdot+

minus

+minus

minus=

Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12

For static strength

M1M1R1

1YMst

+minus

minus=

The effect of σfit is accounted for in 39 b

Classification Notes- No 412 27 May 2003

DET NORSKE VERITAS

385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed

F

y

Sσ225

for not surface hardened fillets

FSHV5 for surface hardened fillets

39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles

Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN

If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used

Determination of the σ ndash N ndash curve

a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF

b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ

FstS1

FPstσ minussdotsdotsdot=

where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability

σFst

Alloyed case hardened steel 1) 2300

Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

1250

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

1050

Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)

18 HV + 800

Steel with not surface hardened fillets the smaller value of 2)

18 σB or 225 σy

1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet

c) Calculate YN as

NL gt 3middot106

YN = 1 or 001

L

6

N N103Y

sdot= ie Yn = 092 for 1010

The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2

103ltNLlt3middot106exp

L

6

N N103Y

sdot=

cycles6103forσcycles310forσlog02876exp

FP

FPst

sdot=

NL lt 103 cycles6103forσcycles310forσ

NYFP

FPst

sdot=

or simply use σFPst as mentioned in b) directly

Guidance on number of load cycles NL for various applica-tions

bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)

bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied

28 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage

YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit

The following method may be used

For endurance limit

for not surface hardened fillets

( )

024

s024

relT σ103133q21σ1012201351

Ysdotsdotminus

+sdotsdotminus+= minus

minus

δ

for all surface hardened fillets except nitrided

( )

106q21002451

Y srelT

++=δ

for nitrided fillets

( )

1347q2101421

Y srelT

++=δ

For static strength

for not surface hardened fillets1)

( )( )( )025

02

02502s

relTst σ3000821σ300 1Y0821Y

+

minus+=δ

for surface hardened fillets except nitrided

YδrelTst = 044 YS + 012

for nitrided fillets

YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not

exceed the yield point and thereby alter the residual stress level See also 39b footnote 2

311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness

YRrelT differs for endurance limit and static strength

The following method may be used

For endurance limit

YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels

YδrelT = 43 ndash 326 (Ry + 1)0005

for nitrided steels

For static strength

YRrelTst = 1 for all Ry and all materials

For a fillet without any longitudinal machining trace Ry asymp Rz

312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength

The following may be used

For endurance limit

YX = 1 for mn le 5 generally

YX = 103 ndash 0006 mn for 5 lt mn lt 30

YX = 085 for mn ge 30

for not surface hard-ened steels

YX = 105 ndash 001 mn for 5 lt mn ge 25

YX = 08 for mn ge 25

for surface hardened steels

For static strength

YXst = 1 for all mn and all materials

313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength

YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength

In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength

The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses

The following simplified method for YC may be used

YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)

YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10

For endurance limit

+

+=nFFE

C m02ρt31

σconstY

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

16 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

For all kinds of steel

5145189718

1C2

limHay +

minusσ

=

When pinion and wheel material differ the following ap-plies

bull Use the larger of fpt1 - yα1 and fpt2 - yα2 to replace fpt - yα in the calculation of KHα and Kv

bull Use ( )2β1ββ yy21y += in the calculation of KHβ

bull Use ( )2ay1aya CC21C += in the calculation of Kv

bull Use ( )2ay1ay2a1a CC21CC +== in the scuffing

calculation if no design tip relief is foreseen

Classification Notes- No 412 17 May 2003

DET NORSKE VERITAS

2 Calculation of Surface Durability

21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213

Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted

Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213

For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength

For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied

22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater

Calculation of surface durability for helical gears is based on the contact stress at the pitch point

For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies

Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact

Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses

The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses

The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel

221 Contact stress Cylindrical gears

( )HαHβvγA

1

tβεEHDBH KKKKK

bud1uF

ZZZZZσ+

=

where

ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)

ZH = Zone factor for pitch point (see 231)

ZE = Elasticity factor (see 24)

Zε = Contact ratio factor (see 25)

Zβ = Helix angle factor (see 26)

Ft KA Kγ Kv KHβ KHα see 15 ndash 110

d1 b u see 12 ndash 15

Bevel gears

( )HαHβvγA

v1v

vmtKEMH KKKKK

bud1uF

ZZZ105σ+

sdot=

where

105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)

ZE KA etc see above

ZM = mid-zone factor see 233

ZK = bevel gear factor see 27

Fmt dv1 uv see 12 ndash 15

It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks

222 Permissible contact stress

XWRvLH

NHlimHP ZZZZZ

SZσ

σ =

where

σH lim = Endurance limit for contact stresses (see 28)

ZN = Life factor for contact stresses (see 29)

SH = Required safety factor according to the rules

ZLZvZR = Oil film influence factors (see 210)

ZW = Work hardening factor (see 211)

ZX = Size factor (see 212)

18 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

23 Zone Factors ZH ZBD and ZM

231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder

wtt2

wtbH sinααcos

cosαcosβ2Z =

232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel

For εβ ge 1 ZBD = 1

For internal gears ZD = 1

For εβ = 0 (spur gears)

( )

π

minusεminusminus

π

minusminus

α=

α2

2

2b

2a

1

2

1b

1a

wtB

z211

dd

z21

dd

tanZ

( )

π

minusεminusminus

π

minusminus

α=

α1

2

1b

1a

2

2

2b

2a

wtD

z211

dd

z21

dd

tanZ

If ZB lt 1 use ZB = 1

If ZD lt 1 use ZD = 1

For 0 lt εβ lt 1

ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)

233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears

εminusminus

εminusminus

αβ=

αα btm2

2vb2

2vabtm2

1vb2val

2v1vvtbmM

pddpdd

ddtancos2Z

This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint

234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel

24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses

For steel against steel ZE = 1898

25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses

αε1Z =ε for 1εβ ge

( )α

ββ

αε ε

εε1

3ε4

Z +minusminus

= for εβ lt 1

26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability

cosβZβ =

27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply

The following may be used ZK = 080

28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ

σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting

For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply

For this purpose pitting is defined by

bull for not surface hardened gears pitted area ge 2 of total active flank area

bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given

material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies

The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as

Classification Notes- No 412 19 May 2003

DET NORSKE VERITAS

the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99

σHlim σH105 σH10

3

Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade

1650 1500

2500 2400

3100 3100

Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)

1000

13 σHlim

13 σHlim

Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV

Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV

Carbon steel 15 HV + 250 16 σHlim 16 σHlim

These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15

29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply

If this is not documented by approved fatigue tests the fol-lowing method may be used

For all steels except nitrided

7L 105N sdotge ZN = 1 or

01570

L

7

N N105Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 5middot107 510N

logZ037

L

7

N N105Z

sdot=

NL = 105 WXRVLHlim

WstX10H1010NN ZZZZZσ

ZZσZZ

55

5=

103 lt NL lt 105 )Z(Zlog05

L

5

10NN

5N103N10

5N10ZZ

==

3L 10N le

WXRVLlimH

Wst10X10H10NN ZZZZZ

ZZZZ

33

3σ

σ==

(but not less than ZN105)

For nitrided steels

6L 102N sdotge ZN = 1 or

00980

L

6

N N102Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 2middot106 510N

Zlog76860

L

6

N N102Z

sdot=

5L 10N le

XWRVL

10XWst10NN ZZZZZ

ZZ13ZZ

55 ==

Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles

210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity

The following methods may be applied in connection with the endurance limit

20 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Surface hardened steels Not surface hardened steels

ZL ( )24013421

360910ν+

+

( )24013421

680830ν+

+

ZV ( )v3280

140930+

+ ( )v3280300850

++

ZR

080

ZrelR3

150

ZrelR3

where

ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits

For values of ν40 gt 500 use ν40 = 500

RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm

RZrel

= ( ) 31

cZ2Z1 ρ

10RR50

+

RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)

For 5

L 10N le ZL ZV ZR = 10

211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface

The following approximation may be used for the endurance limit

Surface hardened steel against not surface hardened steel 150

ZeqW R

31700

130HB21Z

minus

minus=

where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130

RZeq = equivalent roughness

033

c40

066

ZS

ZHZHZEQ ρvν

15000RRRR

=

If RZeq gt 16 then use RZeq = 16

If RZeq lt 15 then use RZeq = 15

where

RZH = surface roughness of the hard member before run in

RZS = surface roughness of the soft member before run in

ν40 = see 210

If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem

Through hardened pinion against softer wheel

( )

minussdotminus+= 000829

HBHB0008981u1Z

2

1W

For 21HBHB

2

1 le use ZW = 1

For 71HBHB

2

1 gt use 17HBHB

2

1 =

For u gt 20 use u = 20

For static strength (lt 105 cycles)

Surface hardened against not surface hardened

ZWst = 105

Through hardened pinion against softer wheel

ZWst = 1

212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality

ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213

213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented

The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent

Classification Notes- No 412 21 May 2003

DET NORSKE VERITAS

stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically

The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence

The real Hertzian stresses σHR are determined as

For helical gears with εβ gt 1

σHR = σH

For helical gears with εβ lt 1 and spur gears

ε

α

ββ ε

ε+εminus

sdotσ=σZ

1

HHR

For bevel gears

KHHR Z

1σσ sdot=

The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)

The coordinates tz and HV are to be compared with the de-sign specification such as

bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400

and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)

In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods

For high cycle fatigue (gt3 106 cycles) the following applies

sdot+

minussdotsdotsdot= o90

05azt

05azt

cosSσ04HV

H

HHHR

applicable to 05a

zt

Hge

For 05at

H

z lt the value for 05at

H

z = applies

56300ρSσ12a cHHR

Hsdotsdot

sdot=

Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength

If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows

reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue

For static strength (lt103 cycles) the following applies

sdot+

minussdotsdotsdot= o90

07at

06a

zt

cosSσ019HV

Hst

z

HstHHR

applicable to 06at

Hst

z ge

56300ρSσa cHHR

Hstsdotsdot

=

In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies

For limited life fatigue (103 lt cycles lt 3106 )

For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance

The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram

minussdotminus

= sdot logN3477

logSlogSlogS

310H6103HHN

36 10H103H Slog86281Slog86280 sdot+sdot sdot

22 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3 Calculation of Tooth Strength

31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding

For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36

For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning

Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα

In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314

A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired

It should be noted that this part 3 does not cover fractures caused by

bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining

Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears

Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied

32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level

321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance

Normally the stresses for pinion and wheel are calculated as

Cylindrical gears

FαFβvγAβSFn

tF K K K K K Y Y Y

m bFσ =

where

YF = Tooth form factor (see 33)

YS = Stress correction factor (see 34)

Yβ = Helix angle factor (see 36)

Ft KA Kγ Kv KFβ KFα see 15 ndash 110

b see 13

Bevel gears

FαFβvγASaFamn

mtF K K K K K Y Y Y

m bFσ ε=

where

YFa = Tooth form factor see 33

YSa = Stress correction factor see 34

Yε = Contact ratio factor see 35

Fmt KA etc see 15 ndash 110

b see 13

322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is

CXRrelTrelTF

NMFEFP YYYY

SYY

δσ

=σ

Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN

where

σFE = Local tooth root bending endurance limit of reference test gear (see 37)

YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)

YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)

SF = Required safety factor according to the rules

YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)

YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)

Classification Notes- No 412 23 May 2003

DET NORSKE VERITAS

YX = Size factor (see 312)

YC = Case depth factor (see 313)

33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress

YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears

YFa applies to load application at the tooth tip and is used for bevel gears

Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears

Figure 31 External tooth in normal section

Figure 32 Internal tooth in normal section

Definitions

n

2

n

Fn

enFn

Fe

F

α cosms

α cosmh6

Y

=

n

2

n

Fn

anFn

Fa

Fa

α cosms

α cosmh6

Y

=

In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth

YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)

The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing

mn with mnm

zn with zvn

αt with αvt

β with βm

with undercut without undercut

Fig 33 Dimensions and basic rack profile of the teeth (finished profile)

Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless

331 Determination of parameters

( )n

n

prnfPnfP m

α cossαsin 1ρ

αtan h4πE

minusminusminusminus=

For external gears fPfP ρρ =

For internal gears ( )0z

195fPfP0

fPfP 10363156ρhxρρ

sdotminus+

+=

where

z0 = number of teeth of pinion cutter

x0 = addendum modification coefficient of pinion cutter

hfP = addendum of pinion cutter

ρfP = tip radius of pinion cutter

xhρG fpfp +minus=

24 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

τmE

2π

z2H

nnminus

minus=

with

3πτ = for external gears

6πτ = for internal gears

Htan zG2

nminusϑ=ϑ

(to be solved iteratively suitable start value6π

=ϑ for exter-

nal gears and 3π for internal gears)

a) Tooth root chord sFn For external gears

minus

ϑ+

ϑminus= ρ

cosG3

3πsinz

ms

fpnn

Fn

For bevel gears with a tooth thickness modification

xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn

For internal gears

minus

ϑ+

ϑminus= ρ

cosG

6πsinz

ms

fPnn

Fn

b) Root fillet radius ρF at 30ordm tangent

( )G2coszcosG2ρ

mρ

2n

2

fpn

F

minusϑϑ+=

c) Determination of bending moment arm hF dn = zn mn

b2α

αn βcosε

ε =

dan = dn + 2 ha

pbn = π mn cos αn

dbn = dn cos αn

( )4

d1εpzz

2dd

zz2d

2bn

2

αnbn

2bn

2an

en +

minusminus

minus=

en

bnen d

dcos arcα =

ennnn

e α invα invα x tan 22π

z1

minus+

+=γ

αFen = αen ndash γe

For external gears

( )

minus=

n

enFenee

n

Fe

mdαtan sin γ γcos

21

mh

]ρcos

G3πcosz fpn +

ϑminus

ϑminusminus

For internal gears

( ) minus

sdotsdotminus=

n

enFenee

n

Fe

mdα tanγ sinγ cos

21

mh

minus

ϑminus

ϑminussdot ρ

cosG3

6πcosz fPn

332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as

250ε205for 0666ε2366Y αnαnDT leleminus=

205εfor 10Y αnDT lt=

34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section

Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm

YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip

YS can be determined as follows

( )

+

+=L23121

1

sS qL01312Y

where Fe

Fn

hsL = and

F

Fns ρ2

sq = (see 33)

YSa can be calculated by replacing hFe with hFa in the above formulae

Note a) Range of validity 1 lt qs lt 8

In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered

b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with

Classification Notes- No 412 25 May 2003

DET NORSKE VERITAS

g

g

ρt

0613

13

minus

where

tg = depth of the grinding notch

ρg = radius of the grinding notch

c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles

35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears

The following may be used

Yε = 0625

36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank

The following may be used (β input in degrees)

Yβ = 1 ndash εβ β120

When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula

However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)

37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM

σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision

If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules

σFE

Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)

bull of specially approved high grade

1050

bull of normal grade

minus CrNiMo steels with approved process

1000

minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850

Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

840

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

720

Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)

07 HV + 300

Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)

025 σB + 125

Alloyed quenched and tempered steel 04 σB + 200

Carbon steel 025 σB + 250

Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40

1) These values are valid for a root radius

bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)

bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)

bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for

6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked

2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)

26 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears

YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0

YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst

For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316

The following method may be used within a stress ratio ndash12 lt R lt 05

381 For idlers planets and PTO with ice class

M1M1R1

1Yor Y MstM

+minus

minus=

where

R = stress ratio = min stress divided by max stress

For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12

For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as

branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus

For a power take off (PTO) with ice class see 161 c

M considers the mean stress influence on the endurance (or static) strength amplitudes

M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress

Following M values may be used

Endurance limit

Static strength

Case hardened 08 ndash 015 Ys 1) 07

If shot peened 04 06

Nitrided 03 03

Induction or flame hardened

04 06

Not surface hardened steel 03 05

Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M

The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches

382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106

For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers

For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used

383 For gears with shrinkage stresses and unidirectional load For endurance strength

FE

fitM σ

σM1

M21Y+

minus=

σFE is the endurance limit for R = 0

For static strength YMst = 1 and σfit accounted for in 39b

σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor

n

Ffit m

ρ215scf minus=

384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is

( ) ( ) FE

fitM σ

σR1M1

M2

M1M1R1

1Yminussdot+

minus

+minus

minus=

Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12

For static strength

M1M1R1

1YMst

+minus

minus=

The effect of σfit is accounted for in 39 b

Classification Notes- No 412 27 May 2003

DET NORSKE VERITAS

385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed

F

y

Sσ225

for not surface hardened fillets

FSHV5 for surface hardened fillets

39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles

Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN

If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used

Determination of the σ ndash N ndash curve

a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF

b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ

FstS1

FPstσ minussdotsdotsdot=

where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability

σFst

Alloyed case hardened steel 1) 2300

Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

1250

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

1050

Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)

18 HV + 800

Steel with not surface hardened fillets the smaller value of 2)

18 σB or 225 σy

1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet

c) Calculate YN as

NL gt 3middot106

YN = 1 or 001

L

6

N N103Y

sdot= ie Yn = 092 for 1010

The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2

103ltNLlt3middot106exp

L

6

N N103Y

sdot=

cycles6103forσcycles310forσlog02876exp

FP

FPst

sdot=

NL lt 103 cycles6103forσcycles310forσ

NYFP

FPst

sdot=

or simply use σFPst as mentioned in b) directly

Guidance on number of load cycles NL for various applica-tions

bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)

bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied

28 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage

YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit

The following method may be used

For endurance limit

for not surface hardened fillets

( )

024

s024

relT σ103133q21σ1012201351

Ysdotsdotminus

+sdotsdotminus+= minus

minus

δ

for all surface hardened fillets except nitrided

( )

106q21002451

Y srelT

++=δ

for nitrided fillets

( )

1347q2101421

Y srelT

++=δ

For static strength

for not surface hardened fillets1)

( )( )( )025

02

02502s

relTst σ3000821σ300 1Y0821Y

+

minus+=δ

for surface hardened fillets except nitrided

YδrelTst = 044 YS + 012

for nitrided fillets

YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not

exceed the yield point and thereby alter the residual stress level See also 39b footnote 2

311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness

YRrelT differs for endurance limit and static strength

The following method may be used

For endurance limit

YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels

YδrelT = 43 ndash 326 (Ry + 1)0005

for nitrided steels

For static strength

YRrelTst = 1 for all Ry and all materials

For a fillet without any longitudinal machining trace Ry asymp Rz

312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength

The following may be used

For endurance limit

YX = 1 for mn le 5 generally

YX = 103 ndash 0006 mn for 5 lt mn lt 30

YX = 085 for mn ge 30

for not surface hard-ened steels

YX = 105 ndash 001 mn for 5 lt mn ge 25

YX = 08 for mn ge 25

for surface hardened steels

For static strength

YXst = 1 for all mn and all materials

313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength

YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength

In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength

The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses

The following simplified method for YC may be used

YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)

YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10

For endurance limit

+

+=nFFE

C m02ρt31

σconstY

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

Classification Notes- No 412 17 May 2003

DET NORSKE VERITAS

2 Calculation of Surface Durability

21 Scope and General Remarks Part 2 includes the calculations of flank surface durability as limited by pitting spalling case crushing and subsurface yielding Endurance and time limited flank surface fatigue is calculated by means of 22 ndash 212 In a way also tooth frac-tures starting from the flank due to subsurface fatigue is in-cluded through the criteria in 213

Pitting itself is not considered as a critical damage for slow speed gears However pits can create a severe notch effect that may result in tooth breakage This is particularly im-portant for surface hardened teeth but also for high strength through hardened teeth For high-speed gears pitting is not permitted

Spalling and case crushing are considered similar to pitting but may have a more severe effect on tooth breakage due to the larger material breakouts initiated below the surface Subsurface fatigue is considered in 213

For jacking gears (self-elevating offshore units) or similar slow speed gears designed for very limited life the max static (or very slow running) surface load for surface hard-ened flanks is limited by the subsurface yield strength

For case hardened gears operating with relatively thin lubri-cation oil films grey staining (micropitting) may be the lim-iting criterion for the gear rating Specific calculation meth-ods for this purpose are not given here but are under consid-eration for future revisions Thus depending on experience with similar gear designs limitations on surface durability rating other than those according to 22 - 213 may be ap-plied

22 Basic Equations Calculation of surface durability (pitting) for spur gears is based on the contact stress at the inner point of single pair contact or the contact at the pitch point whichever is greater

Calculation of surface durability for helical gears is based on the contact stress at the pitch point

For helical gears with 0 lt εβ lt 1 a linear interpolation be-tween the above mentioned applies

Calculation of surface durability for spiral bevel gears is based on the contact stress at the midpoint of the zone of contact

Alternatively for bevel gears the contact stress may be cal-culated with the program ldquoBECALrdquo In that case KA and Kv are to be included in the applied tooth force but not KHβ and KHα The calculated (real) Hertzian stresses are to be multi-plied with ZK in order to be comparable with the permissible contact stresses

The contact stresses calculated with the method in part 2 are based on the Hertzian theory but do not always represent the real Hertzian stresses

The corresponding permissible contact stresses σHP are to be calculated for both pinion and wheel

221 Contact stress Cylindrical gears

( )HαHβvγA

1

tβεEHDBH KKKKK

bud1uF

ZZZZZσ+

=

where

ZBD = Zone factor for inner point of single pair contact for pinion resp wheel (see 232)

ZH = Zone factor for pitch point (see 231)

ZE = Elasticity factor (see 24)

Zε = Contact ratio factor (see 25)

Zβ = Helix angle factor (see 26)

Ft KA Kγ Kv KHβ KHα see 15 ndash 110

d1 b u see 12 ndash 15

Bevel gears

( )HαHβvγA

v1v

vmtKEMH KKKKK

bud1uF

ZZZ105σ+

sdot=

where

105 is a correlation factor to reach real Hertzian stresses (when ZK = 1)

ZE KA etc see above

ZM = mid-zone factor see 233

ZK = bevel gear factor see 27

Fmt dv1 uv see 12 ndash 15

It is assumed that the heightwise crowning is chosen so as to result in the maximum contact stresses at or near the mid-point of the flanks

222 Permissible contact stress

XWRvLH

NHlimHP ZZZZZ

SZσ

σ =

where

σH lim = Endurance limit for contact stresses (see 28)

ZN = Life factor for contact stresses (see 29)

SH = Required safety factor according to the rules

ZLZvZR = Oil film influence factors (see 210)

ZW = Work hardening factor (see 211)

ZX = Size factor (see 212)

18 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

23 Zone Factors ZH ZBD and ZM

231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder

wtt2

wtbH sinααcos

cosαcosβ2Z =

232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel

For εβ ge 1 ZBD = 1

For internal gears ZD = 1

For εβ = 0 (spur gears)

( )

π

minusεminusminus

π

minusminus

α=

α2

2

2b

2a

1

2

1b

1a

wtB

z211

dd

z21

dd

tanZ

( )

π

minusεminusminus

π

minusminus

α=

α1

2

1b

1a

2

2

2b

2a

wtD

z211

dd

z21

dd

tanZ

If ZB lt 1 use ZB = 1

If ZD lt 1 use ZD = 1

For 0 lt εβ lt 1

ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)

233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears

εminusminus

εminusminus

αβ=

αα btm2

2vb2

2vabtm2

1vb2val

2v1vvtbmM

pddpdd

ddtancos2Z

This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint

234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel

24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses

For steel against steel ZE = 1898

25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses

αε1Z =ε for 1εβ ge

( )α

ββ

αε ε

εε1

3ε4

Z +minusminus

= for εβ lt 1

26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability

cosβZβ =

27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply

The following may be used ZK = 080

28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ

σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting

For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply

For this purpose pitting is defined by

bull for not surface hardened gears pitted area ge 2 of total active flank area

bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given

material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies

The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as

Classification Notes- No 412 19 May 2003

DET NORSKE VERITAS

the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99

σHlim σH105 σH10

3

Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade

1650 1500

2500 2400

3100 3100

Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)

1000

13 σHlim

13 σHlim

Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV

Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV

Carbon steel 15 HV + 250 16 σHlim 16 σHlim

These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15

29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply

If this is not documented by approved fatigue tests the fol-lowing method may be used

For all steels except nitrided

7L 105N sdotge ZN = 1 or

01570

L

7

N N105Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 5middot107 510N

logZ037

L

7

N N105Z

sdot=

NL = 105 WXRVLHlim

WstX10H1010NN ZZZZZσ

ZZσZZ

55

5=

103 lt NL lt 105 )Z(Zlog05

L

5

10NN

5N103N10

5N10ZZ

==

3L 10N le

WXRVLlimH

Wst10X10H10NN ZZZZZ

ZZZZ

33

3σ

σ==

(but not less than ZN105)

For nitrided steels

6L 102N sdotge ZN = 1 or

00980

L

6

N N102Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 2middot106 510N

Zlog76860

L

6

N N102Z

sdot=

5L 10N le

XWRVL

10XWst10NN ZZZZZ

ZZ13ZZ

55 ==

Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles

210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity

The following methods may be applied in connection with the endurance limit

20 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Surface hardened steels Not surface hardened steels

ZL ( )24013421

360910ν+

+

( )24013421

680830ν+

+

ZV ( )v3280

140930+

+ ( )v3280300850

++

ZR

080

ZrelR3

150

ZrelR3

where

ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits

For values of ν40 gt 500 use ν40 = 500

RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm

RZrel

= ( ) 31

cZ2Z1 ρ

10RR50

+

RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)

For 5

L 10N le ZL ZV ZR = 10

211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface

The following approximation may be used for the endurance limit

Surface hardened steel against not surface hardened steel 150

ZeqW R

31700

130HB21Z

minus

minus=

where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130

RZeq = equivalent roughness

033

c40

066

ZS

ZHZHZEQ ρvν

15000RRRR

=

If RZeq gt 16 then use RZeq = 16

If RZeq lt 15 then use RZeq = 15

where

RZH = surface roughness of the hard member before run in

RZS = surface roughness of the soft member before run in

ν40 = see 210

If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem

Through hardened pinion against softer wheel

( )

minussdotminus+= 000829

HBHB0008981u1Z

2

1W

For 21HBHB

2

1 le use ZW = 1

For 71HBHB

2

1 gt use 17HBHB

2

1 =

For u gt 20 use u = 20

For static strength (lt 105 cycles)

Surface hardened against not surface hardened

ZWst = 105

Through hardened pinion against softer wheel

ZWst = 1

212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality

ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213

213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented

The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent

Classification Notes- No 412 21 May 2003

DET NORSKE VERITAS

stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically

The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence

The real Hertzian stresses σHR are determined as

For helical gears with εβ gt 1

σHR = σH

For helical gears with εβ lt 1 and spur gears

ε

α

ββ ε

ε+εminus

sdotσ=σZ

1

HHR

For bevel gears

KHHR Z

1σσ sdot=

The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)

The coordinates tz and HV are to be compared with the de-sign specification such as

bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400

and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)

In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods

For high cycle fatigue (gt3 106 cycles) the following applies

sdot+

minussdotsdotsdot= o90

05azt

05azt

cosSσ04HV

H

HHHR

applicable to 05a

zt

Hge

For 05at

H

z lt the value for 05at

H

z = applies

56300ρSσ12a cHHR

Hsdotsdot

sdot=

Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength

If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows

reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue

For static strength (lt103 cycles) the following applies

sdot+

minussdotsdotsdot= o90

07at

06a

zt

cosSσ019HV

Hst

z

HstHHR

applicable to 06at

Hst

z ge

56300ρSσa cHHR

Hstsdotsdot

=

In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies

For limited life fatigue (103 lt cycles lt 3106 )

For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance

The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram

minussdotminus

= sdot logN3477

logSlogSlogS

310H6103HHN

36 10H103H Slog86281Slog86280 sdot+sdot sdot

22 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3 Calculation of Tooth Strength

31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding

For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36

For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning

Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα

In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314

A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired

It should be noted that this part 3 does not cover fractures caused by

bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining

Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears

Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied

32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level

321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance

Normally the stresses for pinion and wheel are calculated as

Cylindrical gears

FαFβvγAβSFn

tF K K K K K Y Y Y

m bFσ =

where

YF = Tooth form factor (see 33)

YS = Stress correction factor (see 34)

Yβ = Helix angle factor (see 36)

Ft KA Kγ Kv KFβ KFα see 15 ndash 110

b see 13

Bevel gears

FαFβvγASaFamn

mtF K K K K K Y Y Y

m bFσ ε=

where

YFa = Tooth form factor see 33

YSa = Stress correction factor see 34

Yε = Contact ratio factor see 35

Fmt KA etc see 15 ndash 110

b see 13

322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is

CXRrelTrelTF

NMFEFP YYYY

SYY

δσ

=σ

Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN

where

σFE = Local tooth root bending endurance limit of reference test gear (see 37)

YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)

YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)

SF = Required safety factor according to the rules

YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)

YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)

Classification Notes- No 412 23 May 2003

DET NORSKE VERITAS

YX = Size factor (see 312)

YC = Case depth factor (see 313)

33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress

YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears

YFa applies to load application at the tooth tip and is used for bevel gears

Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears

Figure 31 External tooth in normal section

Figure 32 Internal tooth in normal section

Definitions

n

2

n

Fn

enFn

Fe

F

α cosms

α cosmh6

Y

=

n

2

n

Fn

anFn

Fa

Fa

α cosms

α cosmh6

Y

=

In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth

YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)

The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing

mn with mnm

zn with zvn

αt with αvt

β with βm

with undercut without undercut

Fig 33 Dimensions and basic rack profile of the teeth (finished profile)

Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless

331 Determination of parameters

( )n

n

prnfPnfP m

α cossαsin 1ρ

αtan h4πE

minusminusminusminus=

For external gears fPfP ρρ =

For internal gears ( )0z

195fPfP0

fPfP 10363156ρhxρρ

sdotminus+

+=

where

z0 = number of teeth of pinion cutter

x0 = addendum modification coefficient of pinion cutter

hfP = addendum of pinion cutter

ρfP = tip radius of pinion cutter

xhρG fpfp +minus=

24 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

τmE

2π

z2H

nnminus

minus=

with

3πτ = for external gears

6πτ = for internal gears

Htan zG2

nminusϑ=ϑ

(to be solved iteratively suitable start value6π

=ϑ for exter-

nal gears and 3π for internal gears)

a) Tooth root chord sFn For external gears

minus

ϑ+

ϑminus= ρ

cosG3

3πsinz

ms

fpnn

Fn

For bevel gears with a tooth thickness modification

xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn

For internal gears

minus

ϑ+

ϑminus= ρ

cosG

6πsinz

ms

fPnn

Fn

b) Root fillet radius ρF at 30ordm tangent

( )G2coszcosG2ρ

mρ

2n

2

fpn

F

minusϑϑ+=

c) Determination of bending moment arm hF dn = zn mn

b2α

αn βcosε

ε =

dan = dn + 2 ha

pbn = π mn cos αn

dbn = dn cos αn

( )4

d1εpzz

2dd

zz2d

2bn

2

αnbn

2bn

2an

en +

minusminus

minus=

en

bnen d

dcos arcα =

ennnn

e α invα invα x tan 22π

z1

minus+

+=γ

αFen = αen ndash γe

For external gears

( )

minus=

n

enFenee

n

Fe

mdαtan sin γ γcos

21

mh

]ρcos

G3πcosz fpn +

ϑminus

ϑminusminus

For internal gears

( ) minus

sdotsdotminus=

n

enFenee

n

Fe

mdα tanγ sinγ cos

21

mh

minus

ϑminus

ϑminussdot ρ

cosG3

6πcosz fPn

332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as

250ε205for 0666ε2366Y αnαnDT leleminus=

205εfor 10Y αnDT lt=

34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section

Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm

YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip

YS can be determined as follows

( )

+

+=L23121

1

sS qL01312Y

where Fe

Fn

hsL = and

F

Fns ρ2

sq = (see 33)

YSa can be calculated by replacing hFe with hFa in the above formulae

Note a) Range of validity 1 lt qs lt 8

In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered

b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with

Classification Notes- No 412 25 May 2003

DET NORSKE VERITAS

g

g

ρt

0613

13

minus

where

tg = depth of the grinding notch

ρg = radius of the grinding notch

c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles

35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears

The following may be used

Yε = 0625

36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank

The following may be used (β input in degrees)

Yβ = 1 ndash εβ β120

When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula

However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)

37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM

σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision

If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules

σFE

Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)

bull of specially approved high grade

1050

bull of normal grade

minus CrNiMo steels with approved process

1000

minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850

Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

840

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

720

Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)

07 HV + 300

Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)

025 σB + 125

Alloyed quenched and tempered steel 04 σB + 200

Carbon steel 025 σB + 250

Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40

1) These values are valid for a root radius

bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)

bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)

bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for

6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked

2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)

26 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears

YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0

YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst

For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316

The following method may be used within a stress ratio ndash12 lt R lt 05

381 For idlers planets and PTO with ice class

M1M1R1

1Yor Y MstM

+minus

minus=

where

R = stress ratio = min stress divided by max stress

For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12

For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as

branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus

For a power take off (PTO) with ice class see 161 c

M considers the mean stress influence on the endurance (or static) strength amplitudes

M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress

Following M values may be used

Endurance limit

Static strength

Case hardened 08 ndash 015 Ys 1) 07

If shot peened 04 06

Nitrided 03 03

Induction or flame hardened

04 06

Not surface hardened steel 03 05

Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M

The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches

382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106

For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers

For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used

383 For gears with shrinkage stresses and unidirectional load For endurance strength

FE

fitM σ

σM1

M21Y+

minus=

σFE is the endurance limit for R = 0

For static strength YMst = 1 and σfit accounted for in 39b

σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor

n

Ffit m

ρ215scf minus=

384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is

( ) ( ) FE

fitM σ

σR1M1

M2

M1M1R1

1Yminussdot+

minus

+minus

minus=

Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12

For static strength

M1M1R1

1YMst

+minus

minus=

The effect of σfit is accounted for in 39 b

Classification Notes- No 412 27 May 2003

DET NORSKE VERITAS

385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed

F

y

Sσ225

for not surface hardened fillets

FSHV5 for surface hardened fillets

39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles

Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN

If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used

Determination of the σ ndash N ndash curve

a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF

b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ

FstS1

FPstσ minussdotsdotsdot=

where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability

σFst

Alloyed case hardened steel 1) 2300

Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

1250

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

1050

Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)

18 HV + 800

Steel with not surface hardened fillets the smaller value of 2)

18 σB or 225 σy

1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet

c) Calculate YN as

NL gt 3middot106

YN = 1 or 001

L

6

N N103Y

sdot= ie Yn = 092 for 1010

The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2

103ltNLlt3middot106exp

L

6

N N103Y

sdot=

cycles6103forσcycles310forσlog02876exp

FP

FPst

sdot=

NL lt 103 cycles6103forσcycles310forσ

NYFP

FPst

sdot=

or simply use σFPst as mentioned in b) directly

Guidance on number of load cycles NL for various applica-tions

bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)

bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied

28 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage

YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit

The following method may be used

For endurance limit

for not surface hardened fillets

( )

024

s024

relT σ103133q21σ1012201351

Ysdotsdotminus

+sdotsdotminus+= minus

minus

δ

for all surface hardened fillets except nitrided

( )

106q21002451

Y srelT

++=δ

for nitrided fillets

( )

1347q2101421

Y srelT

++=δ

For static strength

for not surface hardened fillets1)

( )( )( )025

02

02502s

relTst σ3000821σ300 1Y0821Y

+

minus+=δ

for surface hardened fillets except nitrided

YδrelTst = 044 YS + 012

for nitrided fillets

YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not

exceed the yield point and thereby alter the residual stress level See also 39b footnote 2

311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness

YRrelT differs for endurance limit and static strength

The following method may be used

For endurance limit

YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels

YδrelT = 43 ndash 326 (Ry + 1)0005

for nitrided steels

For static strength

YRrelTst = 1 for all Ry and all materials

For a fillet without any longitudinal machining trace Ry asymp Rz

312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength

The following may be used

For endurance limit

YX = 1 for mn le 5 generally

YX = 103 ndash 0006 mn for 5 lt mn lt 30

YX = 085 for mn ge 30

for not surface hard-ened steels

YX = 105 ndash 001 mn for 5 lt mn ge 25

YX = 08 for mn ge 25

for surface hardened steels

For static strength

YXst = 1 for all mn and all materials

313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength

YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength

In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength

The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses

The following simplified method for YC may be used

YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)

YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10

For endurance limit

+

+=nFFE

C m02ρt31

σconstY

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

18 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

23 Zone Factors ZH ZBD and ZM

231 Zone factor ZH The zone factor ZH accounts for the influence on contact stresses of the tooth flank curvature at the pitch point and converts the tangential force at the reference cylinder to the normal force at the pitch cylinder

wtt2

wtbH sinααcos

cosαcosβ2Z =

232 Zone factors ZBD The zone factors ZBD account for the influence on contact stresses of the tooth flank curvature at the inner point of sin-gle pair contact in relation to ZH Index B refers to pinion D to wheel

For εβ ge 1 ZBD = 1

For internal gears ZD = 1

For εβ = 0 (spur gears)

( )

π

minusεminusminus

π

minusminus

α=

α2

2

2b

2a

1

2

1b

1a

wtB

z211

dd

z21

dd

tanZ

( )

π

minusεminusminus

π

minusminus

α=

α1

2

1b

1a

2

2

2b

2a

wtD

z211

dd

z21

dd

tanZ

If ZB lt 1 use ZB = 1

If ZD lt 1 use ZD = 1

For 0 lt εβ lt 1

ZBD = ZBD (for spur gears) ndash εβ (ZBD (for spur gears) ndash 1)

233 Zone factor ZM The mid-zone factor ZM accounts for the influence of the contact stress at the mid point of the flank and applies to spi-ral bevel gears

εminusminus

εminusminus

αβ=

αα btm2

2vb2

2vabtm2

1vb2val

2v1vvtbmM

pddpdd

ddtancos2Z

This factor is the product of ZH and ZM-B in ISO 10300 with the condition that the heightwise crowning is sufficient to move the peak load towards the midpoint

234 Inner contact point For cylindrical or bevel gears with very low number of teeth the inner contact point (A) may be close to the base circle In order to avoid a wear edge near A it is required to have suitable tip relief on the wheel

24 Elasticity Factor ZE The elasticity factor ZE accounts for the influence of the material properties as modulus of elasticity and Poissonrsquos ratio on the contact stresses

For steel against steel ZE = 1898

25 Contact Ratio Factor Zε The contact ratio factor Zε accounts for the influence of the transverse contact ratio εα and the overlap ratio εβ on the contact stresses

αε1Z =ε for 1εβ ge

( )α

ββ

αε ε

εε1

3ε4

Z +minusminus

= for εβ lt 1

26 Helix Angle Factor Zβ The helix angle factor Zβ accounts for the influence of helix angle (independent of its influence on Zε) on the surface du-rability

cosβZβ =

27 Bevel Gear Factor ZK The bevel gear factor accounts for the difference between the real Hertzian stresses in spiral bevel gears and the contact stresses assumed responsible for surface fatigue (pitting) ZK adjusts the contact stresses in such a way that the same per-missible stresses as for cylindrical gears may apply

The following may be used ZK = 080

28 Values of Endurance Limit σHlim and Static Strength 5H10σ 3H10σ

σHlim is the limit of contact stress that may be sustained for 5middot107 cycles without the occurrence of progressive pitting

For most materials 5middot107 cycles are considered to be the be-ginning of the endurance strength range or lower knee of the σ-N curve (See also Life Factor ZN) However for nitrided steels 2middot106 apply

For this purpose pitting is defined by

bull for not surface hardened gears pitted area ge 2 of total active flank area

bull for surface hardened gears pitted area ge 05 of total active flank area or ge 4 of one particular tooth flank area 510Hσ and 310Hσ are the contact stresses which the given

material can withstand for 105 respectively 103 cycles without subsurface yielding or flank damages as pitting spalling or case crushing when adequate case depth applies

The following listed values for σHlim 510Hσ and 310Hσ may only be used for materials subjected to a quality control as

Classification Notes- No 412 19 May 2003

DET NORSKE VERITAS

the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99

σHlim σH105 σH10

3

Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade

1650 1500

2500 2400

3100 3100

Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)

1000

13 σHlim

13 σHlim

Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV

Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV

Carbon steel 15 HV + 250 16 σHlim 16 σHlim

These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15

29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply

If this is not documented by approved fatigue tests the fol-lowing method may be used

For all steels except nitrided

7L 105N sdotge ZN = 1 or

01570

L

7

N N105Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 5middot107 510N

logZ037

L

7

N N105Z

sdot=

NL = 105 WXRVLHlim

WstX10H1010NN ZZZZZσ

ZZσZZ

55

5=

103 lt NL lt 105 )Z(Zlog05

L

5

10NN

5N103N10

5N10ZZ

==

3L 10N le

WXRVLlimH

Wst10X10H10NN ZZZZZ

ZZZZ

33

3σ

σ==

(but not less than ZN105)

For nitrided steels

6L 102N sdotge ZN = 1 or

00980

L

6

N N102Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 2middot106 510N

Zlog76860

L

6

N N102Z

sdot=

5L 10N le

XWRVL

10XWst10NN ZZZZZ

ZZ13ZZ

55 ==

Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles

210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity

The following methods may be applied in connection with the endurance limit

20 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Surface hardened steels Not surface hardened steels

ZL ( )24013421

360910ν+

+

( )24013421

680830ν+

+

ZV ( )v3280

140930+

+ ( )v3280300850

++

ZR

080

ZrelR3

150

ZrelR3

where

ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits

For values of ν40 gt 500 use ν40 = 500

RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm

RZrel

= ( ) 31

cZ2Z1 ρ

10RR50

+

RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)

For 5

L 10N le ZL ZV ZR = 10

211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface

The following approximation may be used for the endurance limit

Surface hardened steel against not surface hardened steel 150

ZeqW R

31700

130HB21Z

minus

minus=

where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130

RZeq = equivalent roughness

033

c40

066

ZS

ZHZHZEQ ρvν

15000RRRR

=

If RZeq gt 16 then use RZeq = 16

If RZeq lt 15 then use RZeq = 15

where

RZH = surface roughness of the hard member before run in

RZS = surface roughness of the soft member before run in

ν40 = see 210

If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem

Through hardened pinion against softer wheel

( )

minussdotminus+= 000829

HBHB0008981u1Z

2

1W

For 21HBHB

2

1 le use ZW = 1

For 71HBHB

2

1 gt use 17HBHB

2

1 =

For u gt 20 use u = 20

For static strength (lt 105 cycles)

Surface hardened against not surface hardened

ZWst = 105

Through hardened pinion against softer wheel

ZWst = 1

212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality

ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213

213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented

The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent

Classification Notes- No 412 21 May 2003

DET NORSKE VERITAS

stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically

The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence

The real Hertzian stresses σHR are determined as

For helical gears with εβ gt 1

σHR = σH

For helical gears with εβ lt 1 and spur gears

ε

α

ββ ε

ε+εminus

sdotσ=σZ

1

HHR

For bevel gears

KHHR Z

1σσ sdot=

The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)

The coordinates tz and HV are to be compared with the de-sign specification such as

bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400

and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)

In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods

For high cycle fatigue (gt3 106 cycles) the following applies

sdot+

minussdotsdotsdot= o90

05azt

05azt

cosSσ04HV

H

HHHR

applicable to 05a

zt

Hge

For 05at

H

z lt the value for 05at

H

z = applies

56300ρSσ12a cHHR

Hsdotsdot

sdot=

Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength

If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows

reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue

For static strength (lt103 cycles) the following applies

sdot+

minussdotsdotsdot= o90

07at

06a

zt

cosSσ019HV

Hst

z

HstHHR

applicable to 06at

Hst

z ge

56300ρSσa cHHR

Hstsdotsdot

=

In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies

For limited life fatigue (103 lt cycles lt 3106 )

For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance

The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram

minussdotminus

= sdot logN3477

logSlogSlogS

310H6103HHN

36 10H103H Slog86281Slog86280 sdot+sdot sdot

22 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3 Calculation of Tooth Strength

31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding

For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36

For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning

Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα

In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314

A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired

It should be noted that this part 3 does not cover fractures caused by

bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining

Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears

Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied

32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level

321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance

Normally the stresses for pinion and wheel are calculated as

Cylindrical gears

FαFβvγAβSFn

tF K K K K K Y Y Y

m bFσ =

where

YF = Tooth form factor (see 33)

YS = Stress correction factor (see 34)

Yβ = Helix angle factor (see 36)

Ft KA Kγ Kv KFβ KFα see 15 ndash 110

b see 13

Bevel gears

FαFβvγASaFamn

mtF K K K K K Y Y Y

m bFσ ε=

where

YFa = Tooth form factor see 33

YSa = Stress correction factor see 34

Yε = Contact ratio factor see 35

Fmt KA etc see 15 ndash 110

b see 13

322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is

CXRrelTrelTF

NMFEFP YYYY

SYY

δσ

=σ

Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN

where

σFE = Local tooth root bending endurance limit of reference test gear (see 37)

YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)

YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)

SF = Required safety factor according to the rules

YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)

YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)

Classification Notes- No 412 23 May 2003

DET NORSKE VERITAS

YX = Size factor (see 312)

YC = Case depth factor (see 313)

33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress

YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears

YFa applies to load application at the tooth tip and is used for bevel gears

Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears

Figure 31 External tooth in normal section

Figure 32 Internal tooth in normal section

Definitions

n

2

n

Fn

enFn

Fe

F

α cosms

α cosmh6

Y

=

n

2

n

Fn

anFn

Fa

Fa

α cosms

α cosmh6

Y

=

In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth

YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)

The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing

mn with mnm

zn with zvn

αt with αvt

β with βm

with undercut without undercut

Fig 33 Dimensions and basic rack profile of the teeth (finished profile)

Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless

331 Determination of parameters

( )n

n

prnfPnfP m

α cossαsin 1ρ

αtan h4πE

minusminusminusminus=

For external gears fPfP ρρ =

For internal gears ( )0z

195fPfP0

fPfP 10363156ρhxρρ

sdotminus+

+=

where

z0 = number of teeth of pinion cutter

x0 = addendum modification coefficient of pinion cutter

hfP = addendum of pinion cutter

ρfP = tip radius of pinion cutter

xhρG fpfp +minus=

24 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

τmE

2π

z2H

nnminus

minus=

with

3πτ = for external gears

6πτ = for internal gears

Htan zG2

nminusϑ=ϑ

(to be solved iteratively suitable start value6π

=ϑ for exter-

nal gears and 3π for internal gears)

a) Tooth root chord sFn For external gears

minus

ϑ+

ϑminus= ρ

cosG3

3πsinz

ms

fpnn

Fn

For bevel gears with a tooth thickness modification

xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn

For internal gears

minus

ϑ+

ϑminus= ρ

cosG

6πsinz

ms

fPnn

Fn

b) Root fillet radius ρF at 30ordm tangent

( )G2coszcosG2ρ

mρ

2n

2

fpn

F

minusϑϑ+=

c) Determination of bending moment arm hF dn = zn mn

b2α

αn βcosε

ε =

dan = dn + 2 ha

pbn = π mn cos αn

dbn = dn cos αn

( )4

d1εpzz

2dd

zz2d

2bn

2

αnbn

2bn

2an

en +

minusminus

minus=

en

bnen d

dcos arcα =

ennnn

e α invα invα x tan 22π

z1

minus+

+=γ

αFen = αen ndash γe

For external gears

( )

minus=

n

enFenee

n

Fe

mdαtan sin γ γcos

21

mh

]ρcos

G3πcosz fpn +

ϑminus

ϑminusminus

For internal gears

( ) minus

sdotsdotminus=

n

enFenee

n

Fe

mdα tanγ sinγ cos

21

mh

minus

ϑminus

ϑminussdot ρ

cosG3

6πcosz fPn

332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as

250ε205for 0666ε2366Y αnαnDT leleminus=

205εfor 10Y αnDT lt=

34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section

Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm

YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip

YS can be determined as follows

( )

+

+=L23121

1

sS qL01312Y

where Fe

Fn

hsL = and

F

Fns ρ2

sq = (see 33)

YSa can be calculated by replacing hFe with hFa in the above formulae

Note a) Range of validity 1 lt qs lt 8

In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered

b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with

Classification Notes- No 412 25 May 2003

DET NORSKE VERITAS

g

g

ρt

0613

13

minus

where

tg = depth of the grinding notch

ρg = radius of the grinding notch

c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles

35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears

The following may be used

Yε = 0625

36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank

The following may be used (β input in degrees)

Yβ = 1 ndash εβ β120

When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula

However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)

37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM

σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision

If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules

σFE

Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)

bull of specially approved high grade

1050

bull of normal grade

minus CrNiMo steels with approved process

1000

minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850

Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

840

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

720

Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)

07 HV + 300

Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)

025 σB + 125

Alloyed quenched and tempered steel 04 σB + 200

Carbon steel 025 σB + 250

Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40

1) These values are valid for a root radius

bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)

bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)

bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for

6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked

2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)

26 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears

YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0

YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst

For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316

The following method may be used within a stress ratio ndash12 lt R lt 05

381 For idlers planets and PTO with ice class

M1M1R1

1Yor Y MstM

+minus

minus=

where

R = stress ratio = min stress divided by max stress

For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12

For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as

branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus

For a power take off (PTO) with ice class see 161 c

M considers the mean stress influence on the endurance (or static) strength amplitudes

M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress

Following M values may be used

Endurance limit

Static strength

Case hardened 08 ndash 015 Ys 1) 07

If shot peened 04 06

Nitrided 03 03

Induction or flame hardened

04 06

Not surface hardened steel 03 05

Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M

The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches

382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106

For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers

For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used

383 For gears with shrinkage stresses and unidirectional load For endurance strength

FE

fitM σ

σM1

M21Y+

minus=

σFE is the endurance limit for R = 0

For static strength YMst = 1 and σfit accounted for in 39b

σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor

n

Ffit m

ρ215scf minus=

384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is

( ) ( ) FE

fitM σ

σR1M1

M2

M1M1R1

1Yminussdot+

minus

+minus

minus=

Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12

For static strength

M1M1R1

1YMst

+minus

minus=

The effect of σfit is accounted for in 39 b

Classification Notes- No 412 27 May 2003

DET NORSKE VERITAS

385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed

F

y

Sσ225

for not surface hardened fillets

FSHV5 for surface hardened fillets

39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles

Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN

If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used

Determination of the σ ndash N ndash curve

a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF

b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ

FstS1

FPstσ minussdotsdotsdot=

where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability

σFst

Alloyed case hardened steel 1) 2300

Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

1250

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

1050

Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)

18 HV + 800

Steel with not surface hardened fillets the smaller value of 2)

18 σB or 225 σy

1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet

c) Calculate YN as

NL gt 3middot106

YN = 1 or 001

L

6

N N103Y

sdot= ie Yn = 092 for 1010

The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2

103ltNLlt3middot106exp

L

6

N N103Y

sdot=

cycles6103forσcycles310forσlog02876exp

FP

FPst

sdot=

NL lt 103 cycles6103forσcycles310forσ

NYFP

FPst

sdot=

or simply use σFPst as mentioned in b) directly

Guidance on number of load cycles NL for various applica-tions

bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)

bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied

28 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage

YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit

The following method may be used

For endurance limit

for not surface hardened fillets

( )

024

s024

relT σ103133q21σ1012201351

Ysdotsdotminus

+sdotsdotminus+= minus

minus

δ

for all surface hardened fillets except nitrided

( )

106q21002451

Y srelT

++=δ

for nitrided fillets

( )

1347q2101421

Y srelT

++=δ

For static strength

for not surface hardened fillets1)

( )( )( )025

02

02502s

relTst σ3000821σ300 1Y0821Y

+

minus+=δ

for surface hardened fillets except nitrided

YδrelTst = 044 YS + 012

for nitrided fillets

YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not

exceed the yield point and thereby alter the residual stress level See also 39b footnote 2

311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness

YRrelT differs for endurance limit and static strength

The following method may be used

For endurance limit

YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels

YδrelT = 43 ndash 326 (Ry + 1)0005

for nitrided steels

For static strength

YRrelTst = 1 for all Ry and all materials

For a fillet without any longitudinal machining trace Ry asymp Rz

312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength

The following may be used

For endurance limit

YX = 1 for mn le 5 generally

YX = 103 ndash 0006 mn for 5 lt mn lt 30

YX = 085 for mn ge 30

for not surface hard-ened steels

YX = 105 ndash 001 mn for 5 lt mn ge 25

YX = 08 for mn ge 25

for surface hardened steels

For static strength

YXst = 1 for all mn and all materials

313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength

YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength

In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength

The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses

The following simplified method for YC may be used

YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)

YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10

For endurance limit

+

+=nFFE

C m02ρt31

σconstY

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

Classification Notes- No 412 19 May 2003

DET NORSKE VERITAS

the one referred to in the rules Results of approved fatigue tests may also be used as the basis for establishing these values The defined survival probability is 99

σHlim σH105 σH10

3

Alloyed case hardened steels (surface hardness 58-63 HRC) - of specially approved high grade - of normal grade

1650 1500

2500 2400

3100 3100

Nitrided steel of approved grade gas nitrided (surface hardness 700-800 HV) 1250 13 σHlim 13 σHlim

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500-700 HV)

1000

13 σHlim

13 σHlim

Alloyed flame or induction hardened steel (surface hardness 500-650 HV) 075 HV + 750 16 σHlim 45 HV

Alloyed quenched and tempered steel 14 HV + 350 16 σHlim 45 HV

Carbon steel 15 HV + 250 16 σHlim 16 σHlim

These values refer to forged or hot rolled steel For cast steel the values for σHlim are to be reduced by 15

29 Life Factor ZN The life factor ZN takes account of a higher permissible contact stress if only limited life (number of cycles NL) is demanded or lower permissible contact stress if very high number of cycles apply

If this is not documented by approved fatigue tests the fol-lowing method may be used

For all steels except nitrided

7L 105N sdotge ZN = 1 or

01570

L

7

N N105Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 5middot107 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 5middot107 510N

logZ037

L

7

N N105Z

sdot=

NL = 105 WXRVLHlim

WstX10H1010NN ZZZZZσ

ZZσZZ

55

5=

103 lt NL lt 105 )Z(Zlog05

L

5

10NN

5N103N10

5N10ZZ

==

3L 10N le

WXRVLlimH

Wst10X10H10NN ZZZZZ

ZZZZ

33

3σ

σ==

(but not less than ZN105)

For nitrided steels

6L 102N sdotge ZN = 1 or

00980

L

6

N N102Z

sdot=

Ie ZN = 092 for 1010 cycles

The ZN = 1 from 2middot106 on may only be used when the mate-rial cleanliness is of approved high grade (see Rules Pt4 Ch2) and the lubrication is optimised by a specially approved filtering process

105 lt NL lt 2middot106 510N

Zlog76860

L

6

N N102Z

sdot=

5L 10N le

XWRVL

10XWst10NN ZZZZZ

ZZ13ZZ

55 ==

Note that when no index indicating number of cycles is used the factors are valid for 5middot107 (respectively 2middot106 for nitrid-ing) cycles

210 Influence Factors on Lubrication Film ZL ZV and ZR The lubricant factor ZL accounts for the influence of the type of lubricant and its viscosity the speed factor ZV ac-counts for the influence of the pitch line velocity and the roughness factor ZR accounts for influence of the surface roughness on the surface endurance capacity

The following methods may be applied in connection with the endurance limit

20 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Surface hardened steels Not surface hardened steels

ZL ( )24013421

360910ν+

+

( )24013421

680830ν+

+

ZV ( )v3280

140930+

+ ( )v3280300850

++

ZR

080

ZrelR3

150

ZrelR3

where

ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits

For values of ν40 gt 500 use ν40 = 500

RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm

RZrel

= ( ) 31

cZ2Z1 ρ

10RR50

+

RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)

For 5

L 10N le ZL ZV ZR = 10

211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface

The following approximation may be used for the endurance limit

Surface hardened steel against not surface hardened steel 150

ZeqW R

31700

130HB21Z

minus

minus=

where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130

RZeq = equivalent roughness

033

c40

066

ZS

ZHZHZEQ ρvν

15000RRRR

=

If RZeq gt 16 then use RZeq = 16

If RZeq lt 15 then use RZeq = 15

where

RZH = surface roughness of the hard member before run in

RZS = surface roughness of the soft member before run in

ν40 = see 210

If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem

Through hardened pinion against softer wheel

( )

minussdotminus+= 000829

HBHB0008981u1Z

2

1W

For 21HBHB

2

1 le use ZW = 1

For 71HBHB

2

1 gt use 17HBHB

2

1 =

For u gt 20 use u = 20

For static strength (lt 105 cycles)

Surface hardened against not surface hardened

ZWst = 105

Through hardened pinion against softer wheel

ZWst = 1

212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality

ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213

213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented

The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent

Classification Notes- No 412 21 May 2003

DET NORSKE VERITAS

stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically

The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence

The real Hertzian stresses σHR are determined as

For helical gears with εβ gt 1

σHR = σH

For helical gears with εβ lt 1 and spur gears

ε

α

ββ ε

ε+εminus

sdotσ=σZ

1

HHR

For bevel gears

KHHR Z

1σσ sdot=

The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)

The coordinates tz and HV are to be compared with the de-sign specification such as

bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400

and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)

In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods

For high cycle fatigue (gt3 106 cycles) the following applies

sdot+

minussdotsdotsdot= o90

05azt

05azt

cosSσ04HV

H

HHHR

applicable to 05a

zt

Hge

For 05at

H

z lt the value for 05at

H

z = applies

56300ρSσ12a cHHR

Hsdotsdot

sdot=

Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength

If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows

reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue

For static strength (lt103 cycles) the following applies

sdot+

minussdotsdotsdot= o90

07at

06a

zt

cosSσ019HV

Hst

z

HstHHR

applicable to 06at

Hst

z ge

56300ρSσa cHHR

Hstsdotsdot

=

In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies

For limited life fatigue (103 lt cycles lt 3106 )

For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance

The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram

minussdotminus

= sdot logN3477

logSlogSlogS

310H6103HHN

36 10H103H Slog86281Slog86280 sdot+sdot sdot

22 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3 Calculation of Tooth Strength

31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding

For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36

For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning

Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα

In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314

A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired

It should be noted that this part 3 does not cover fractures caused by

bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining

Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears

Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied

32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level

321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance

Normally the stresses for pinion and wheel are calculated as

Cylindrical gears

FαFβvγAβSFn

tF K K K K K Y Y Y

m bFσ =

where

YF = Tooth form factor (see 33)

YS = Stress correction factor (see 34)

Yβ = Helix angle factor (see 36)

Ft KA Kγ Kv KFβ KFα see 15 ndash 110

b see 13

Bevel gears

FαFβvγASaFamn

mtF K K K K K Y Y Y

m bFσ ε=

where

YFa = Tooth form factor see 33

YSa = Stress correction factor see 34

Yε = Contact ratio factor see 35

Fmt KA etc see 15 ndash 110

b see 13

322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is

CXRrelTrelTF

NMFEFP YYYY

SYY

δσ

=σ

Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN

where

σFE = Local tooth root bending endurance limit of reference test gear (see 37)

YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)

YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)

SF = Required safety factor according to the rules

YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)

YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)

Classification Notes- No 412 23 May 2003

DET NORSKE VERITAS

YX = Size factor (see 312)

YC = Case depth factor (see 313)

33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress

YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears

YFa applies to load application at the tooth tip and is used for bevel gears

Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears

Figure 31 External tooth in normal section

Figure 32 Internal tooth in normal section

Definitions

n

2

n

Fn

enFn

Fe

F

α cosms

α cosmh6

Y

=

n

2

n

Fn

anFn

Fa

Fa

α cosms

α cosmh6

Y

=

In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth

YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)

The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing

mn with mnm

zn with zvn

αt with αvt

β with βm

with undercut without undercut

Fig 33 Dimensions and basic rack profile of the teeth (finished profile)

Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless

331 Determination of parameters

( )n

n

prnfPnfP m

α cossαsin 1ρ

αtan h4πE

minusminusminusminus=

For external gears fPfP ρρ =

For internal gears ( )0z

195fPfP0

fPfP 10363156ρhxρρ

sdotminus+

+=

where

z0 = number of teeth of pinion cutter

x0 = addendum modification coefficient of pinion cutter

hfP = addendum of pinion cutter

ρfP = tip radius of pinion cutter

xhρG fpfp +minus=

24 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

τmE

2π

z2H

nnminus

minus=

with

3πτ = for external gears

6πτ = for internal gears

Htan zG2

nminusϑ=ϑ

(to be solved iteratively suitable start value6π

=ϑ for exter-

nal gears and 3π for internal gears)

a) Tooth root chord sFn For external gears

minus

ϑ+

ϑminus= ρ

cosG3

3πsinz

ms

fpnn

Fn

For bevel gears with a tooth thickness modification

xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn

For internal gears

minus

ϑ+

ϑminus= ρ

cosG

6πsinz

ms

fPnn

Fn

b) Root fillet radius ρF at 30ordm tangent

( )G2coszcosG2ρ

mρ

2n

2

fpn

F

minusϑϑ+=

c) Determination of bending moment arm hF dn = zn mn

b2α

αn βcosε

ε =

dan = dn + 2 ha

pbn = π mn cos αn

dbn = dn cos αn

( )4

d1εpzz

2dd

zz2d

2bn

2

αnbn

2bn

2an

en +

minusminus

minus=

en

bnen d

dcos arcα =

ennnn

e α invα invα x tan 22π

z1

minus+

+=γ

αFen = αen ndash γe

For external gears

( )

minus=

n

enFenee

n

Fe

mdαtan sin γ γcos

21

mh

]ρcos

G3πcosz fpn +

ϑminus

ϑminusminus

For internal gears

( ) minus

sdotsdotminus=

n

enFenee

n

Fe

mdα tanγ sinγ cos

21

mh

minus

ϑminus

ϑminussdot ρ

cosG3

6πcosz fPn

332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as

250ε205for 0666ε2366Y αnαnDT leleminus=

205εfor 10Y αnDT lt=

34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section

Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm

YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip

YS can be determined as follows

( )

+

+=L23121

1

sS qL01312Y

where Fe

Fn

hsL = and

F

Fns ρ2

sq = (see 33)

YSa can be calculated by replacing hFe with hFa in the above formulae

Note a) Range of validity 1 lt qs lt 8

In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered

b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with

Classification Notes- No 412 25 May 2003

DET NORSKE VERITAS

g

g

ρt

0613

13

minus

where

tg = depth of the grinding notch

ρg = radius of the grinding notch

c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles

35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears

The following may be used

Yε = 0625

36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank

The following may be used (β input in degrees)

Yβ = 1 ndash εβ β120

When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula

However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)

37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM

σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision

If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules

σFE

Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)

bull of specially approved high grade

1050

bull of normal grade

minus CrNiMo steels with approved process

1000

minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850

Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

840

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

720

Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)

07 HV + 300

Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)

025 σB + 125

Alloyed quenched and tempered steel 04 σB + 200

Carbon steel 025 σB + 250

Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40

1) These values are valid for a root radius

bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)

bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)

bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for

6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked

2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)

26 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears

YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0

YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst

For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316

The following method may be used within a stress ratio ndash12 lt R lt 05

381 For idlers planets and PTO with ice class

M1M1R1

1Yor Y MstM

+minus

minus=

where

R = stress ratio = min stress divided by max stress

For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12

For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as

branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus

For a power take off (PTO) with ice class see 161 c

M considers the mean stress influence on the endurance (or static) strength amplitudes

M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress

Following M values may be used

Endurance limit

Static strength

Case hardened 08 ndash 015 Ys 1) 07

If shot peened 04 06

Nitrided 03 03

Induction or flame hardened

04 06

Not surface hardened steel 03 05

Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M

The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches

382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106

For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers

For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used

383 For gears with shrinkage stresses and unidirectional load For endurance strength

FE

fitM σ

σM1

M21Y+

minus=

σFE is the endurance limit for R = 0

For static strength YMst = 1 and σfit accounted for in 39b

σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor

n

Ffit m

ρ215scf minus=

384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is

( ) ( ) FE

fitM σ

σR1M1

M2

M1M1R1

1Yminussdot+

minus

+minus

minus=

Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12

For static strength

M1M1R1

1YMst

+minus

minus=

The effect of σfit is accounted for in 39 b

Classification Notes- No 412 27 May 2003

DET NORSKE VERITAS

385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed

F

y

Sσ225

for not surface hardened fillets

FSHV5 for surface hardened fillets

39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles

Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN

If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used

Determination of the σ ndash N ndash curve

a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF

b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ

FstS1

FPstσ minussdotsdotsdot=

where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability

σFst

Alloyed case hardened steel 1) 2300

Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

1250

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

1050

Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)

18 HV + 800

Steel with not surface hardened fillets the smaller value of 2)

18 σB or 225 σy

1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet

c) Calculate YN as

NL gt 3middot106

YN = 1 or 001

L

6

N N103Y

sdot= ie Yn = 092 for 1010

The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2

103ltNLlt3middot106exp

L

6

N N103Y

sdot=

cycles6103forσcycles310forσlog02876exp

FP

FPst

sdot=

NL lt 103 cycles6103forσcycles310forσ

NYFP

FPst

sdot=

or simply use σFPst as mentioned in b) directly

Guidance on number of load cycles NL for various applica-tions

bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)

bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied

28 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage

YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit

The following method may be used

For endurance limit

for not surface hardened fillets

( )

024

s024

relT σ103133q21σ1012201351

Ysdotsdotminus

+sdotsdotminus+= minus

minus

δ

for all surface hardened fillets except nitrided

( )

106q21002451

Y srelT

++=δ

for nitrided fillets

( )

1347q2101421

Y srelT

++=δ

For static strength

for not surface hardened fillets1)

( )( )( )025

02

02502s

relTst σ3000821σ300 1Y0821Y

+

minus+=δ

for surface hardened fillets except nitrided

YδrelTst = 044 YS + 012

for nitrided fillets

YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not

exceed the yield point and thereby alter the residual stress level See also 39b footnote 2

311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness

YRrelT differs for endurance limit and static strength

The following method may be used

For endurance limit

YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels

YδrelT = 43 ndash 326 (Ry + 1)0005

for nitrided steels

For static strength

YRrelTst = 1 for all Ry and all materials

For a fillet without any longitudinal machining trace Ry asymp Rz

312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength

The following may be used

For endurance limit

YX = 1 for mn le 5 generally

YX = 103 ndash 0006 mn for 5 lt mn lt 30

YX = 085 for mn ge 30

for not surface hard-ened steels

YX = 105 ndash 001 mn for 5 lt mn ge 25

YX = 08 for mn ge 25

for surface hardened steels

For static strength

YXst = 1 for all mn and all materials

313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength

YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength

In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength

The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses

The following simplified method for YC may be used

YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)

YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10

For endurance limit

+

+=nFFE

C m02ρt31

σconstY

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

20 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Surface hardened steels Not surface hardened steels

ZL ( )24013421

360910ν+

+

( )24013421

680830ν+

+

ZV ( )v3280

140930+

+ ( )v3280300850

++

ZR

080

ZrelR3

150

ZrelR3

where

ν40 = Kinematic oil viscosity at 40ordmC (mm2s) For case hardened steels the influence of a high bulk temperature (see 4 Scuffing) should be con-sidered Eg bulk temperatures in excess of 120ordmC for long periods may cause reduced flank surface endurance limits

For values of ν40 gt 500 use ν40 = 500

RZrel = The mean roughness between pinion and wheel (after running in) relative to an equivalent radius of curvature at the pitch point ρc = 10mm

RZrel

= ( ) 31

cZ2Z1 ρ

10RR50

+

RZ = Mean peak to valley roughness (microm) (DIN defini-tion) (roughly RZ = 6 Ra)

For 5

L 10N le ZL ZV ZR = 10

211 Work Hardening Factor ZW The work hardening factor ZW accounts for the increase of surface durability of a soft steel gear when meshing the soft steel gear with a surface hardened or substantially harder gear with a smooth surface

The following approximation may be used for the endurance limit

Surface hardened steel against not surface hardened steel 150

ZeqW R

31700

130HB21Z

minus

minus=

where HB = the Brinell hardness of the soft member For HB gt 470 use HB = 470 For HB lt 130 use HB = 130

RZeq = equivalent roughness

033

c40

066

ZS

ZHZHZEQ ρvν

15000RRRR

=

If RZeq gt 16 then use RZeq = 16

If RZeq lt 15 then use RZeq = 15

where

RZH = surface roughness of the hard member before run in

RZS = surface roughness of the soft member before run in

ν40 = see 210

If values of ZW lt 1 are evaluated ZW = 1 should be used for flank endurance However the low value for ZW may indi-cate a potential wear problem

Through hardened pinion against softer wheel

( )

minussdotminus+= 000829

HBHB0008981u1Z

2

1W

For 21HBHB

2

1 le use ZW = 1

For 71HBHB

2

1 gt use 17HBHB

2

1 =

For u gt 20 use u = 20

For static strength (lt 105 cycles)

Surface hardened against not surface hardened

ZWst = 105

Through hardened pinion against softer wheel

ZWst = 1

212 Size Factor ZX The size factor accounts for statistics indicating that the stress levels at which fatigue damage occurs decrease with an increase of component size as a consequence of the influ-ence on subsurface defects combined with small stress gradi-ents and of the influence of size on material quality

ZX may be taken unity provided that subsurface fatigue for surface hardened pinions and wheels is considered eg as in the following subsection 213

213 Subsurface Fatigue This is only applicable to surface hardened pinions and wheels The main objective is to have a subsurface safety against fatigue (endurance limit) or deformation (static strength) which is at least as high as the safety SH required for the surface The following method may be used as an approximation unless otherwise documented

The high cycle fatigue (gt3106 cycles) is assumed to mainly depend on the orthogonal shear stresses Static strength (lt103 cycles) is assumed to depend mainly on equivalent

Classification Notes- No 412 21 May 2003

DET NORSKE VERITAS

stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically

The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence

The real Hertzian stresses σHR are determined as

For helical gears with εβ gt 1

σHR = σH

For helical gears with εβ lt 1 and spur gears

ε

α

ββ ε

ε+εminus

sdotσ=σZ

1

HHR

For bevel gears

KHHR Z

1σσ sdot=

The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)

The coordinates tz and HV are to be compared with the de-sign specification such as

bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400

and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)

In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods

For high cycle fatigue (gt3 106 cycles) the following applies

sdot+

minussdotsdotsdot= o90

05azt

05azt

cosSσ04HV

H

HHHR

applicable to 05a

zt

Hge

For 05at

H

z lt the value for 05at

H

z = applies

56300ρSσ12a cHHR

Hsdotsdot

sdot=

Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength

If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows

reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue

For static strength (lt103 cycles) the following applies

sdot+

minussdotsdotsdot= o90

07at

06a

zt

cosSσ019HV

Hst

z

HstHHR

applicable to 06at

Hst

z ge

56300ρSσa cHHR

Hstsdotsdot

=

In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies

For limited life fatigue (103 lt cycles lt 3106 )

For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance

The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram

minussdotminus

= sdot logN3477

logSlogSlogS

310H6103HHN

36 10H103H Slog86281Slog86280 sdot+sdot sdot

22 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3 Calculation of Tooth Strength

31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding

For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36

For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning

Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα

In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314

A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired

It should be noted that this part 3 does not cover fractures caused by

bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining

Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears

Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied

32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level

321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance

Normally the stresses for pinion and wheel are calculated as

Cylindrical gears

FαFβvγAβSFn

tF K K K K K Y Y Y

m bFσ =

where

YF = Tooth form factor (see 33)

YS = Stress correction factor (see 34)

Yβ = Helix angle factor (see 36)

Ft KA Kγ Kv KFβ KFα see 15 ndash 110

b see 13

Bevel gears

FαFβvγASaFamn

mtF K K K K K Y Y Y

m bFσ ε=

where

YFa = Tooth form factor see 33

YSa = Stress correction factor see 34

Yε = Contact ratio factor see 35

Fmt KA etc see 15 ndash 110

b see 13

322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is

CXRrelTrelTF

NMFEFP YYYY

SYY

δσ

=σ

Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN

where

σFE = Local tooth root bending endurance limit of reference test gear (see 37)

YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)

YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)

SF = Required safety factor according to the rules

YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)

YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)

Classification Notes- No 412 23 May 2003

DET NORSKE VERITAS

YX = Size factor (see 312)

YC = Case depth factor (see 313)

33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress

YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears

YFa applies to load application at the tooth tip and is used for bevel gears

Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears

Figure 31 External tooth in normal section

Figure 32 Internal tooth in normal section

Definitions

n

2

n

Fn

enFn

Fe

F

α cosms

α cosmh6

Y

=

n

2

n

Fn

anFn

Fa

Fa

α cosms

α cosmh6

Y

=

In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth

YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)

The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing

mn with mnm

zn with zvn

αt with αvt

β with βm

with undercut without undercut

Fig 33 Dimensions and basic rack profile of the teeth (finished profile)

Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless

331 Determination of parameters

( )n

n

prnfPnfP m

α cossαsin 1ρ

αtan h4πE

minusminusminusminus=

For external gears fPfP ρρ =

For internal gears ( )0z

195fPfP0

fPfP 10363156ρhxρρ

sdotminus+

+=

where

z0 = number of teeth of pinion cutter

x0 = addendum modification coefficient of pinion cutter

hfP = addendum of pinion cutter

ρfP = tip radius of pinion cutter

xhρG fpfp +minus=

24 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

τmE

2π

z2H

nnminus

minus=

with

3πτ = for external gears

6πτ = for internal gears

Htan zG2

nminusϑ=ϑ

(to be solved iteratively suitable start value6π

=ϑ for exter-

nal gears and 3π for internal gears)

a) Tooth root chord sFn For external gears

minus

ϑ+

ϑminus= ρ

cosG3

3πsinz

ms

fpnn

Fn

For bevel gears with a tooth thickness modification

xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn

For internal gears

minus

ϑ+

ϑminus= ρ

cosG

6πsinz

ms

fPnn

Fn

b) Root fillet radius ρF at 30ordm tangent

( )G2coszcosG2ρ

mρ

2n

2

fpn

F

minusϑϑ+=

c) Determination of bending moment arm hF dn = zn mn

b2α

αn βcosε

ε =

dan = dn + 2 ha

pbn = π mn cos αn

dbn = dn cos αn

( )4

d1εpzz

2dd

zz2d

2bn

2

αnbn

2bn

2an

en +

minusminus

minus=

en

bnen d

dcos arcα =

ennnn

e α invα invα x tan 22π

z1

minus+

+=γ

αFen = αen ndash γe

For external gears

( )

minus=

n

enFenee

n

Fe

mdαtan sin γ γcos

21

mh

]ρcos

G3πcosz fpn +

ϑminus

ϑminusminus

For internal gears

( ) minus

sdotsdotminus=

n

enFenee

n

Fe

mdα tanγ sinγ cos

21

mh

minus

ϑminus

ϑminussdot ρ

cosG3

6πcosz fPn

332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as

250ε205for 0666ε2366Y αnαnDT leleminus=

205εfor 10Y αnDT lt=

34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section

Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm

YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip

YS can be determined as follows

( )

+

+=L23121

1

sS qL01312Y

where Fe

Fn

hsL = and

F

Fns ρ2

sq = (see 33)

YSa can be calculated by replacing hFe with hFa in the above formulae

Note a) Range of validity 1 lt qs lt 8

In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered

b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with

Classification Notes- No 412 25 May 2003

DET NORSKE VERITAS

g

g

ρt

0613

13

minus

where

tg = depth of the grinding notch

ρg = radius of the grinding notch

c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles

35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears

The following may be used

Yε = 0625

36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank

The following may be used (β input in degrees)

Yβ = 1 ndash εβ β120

When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula

However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)

37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM

σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision

If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules

σFE

Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)

bull of specially approved high grade

1050

bull of normal grade

minus CrNiMo steels with approved process

1000

minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850

Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

840

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

720

Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)

07 HV + 300

Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)

025 σB + 125

Alloyed quenched and tempered steel 04 σB + 200

Carbon steel 025 σB + 250

Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40

1) These values are valid for a root radius

bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)

bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)

bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for

6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked

2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)

26 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears

YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0

YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst

For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316

The following method may be used within a stress ratio ndash12 lt R lt 05

381 For idlers planets and PTO with ice class

M1M1R1

1Yor Y MstM

+minus

minus=

where

R = stress ratio = min stress divided by max stress

For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12

For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as

branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus

For a power take off (PTO) with ice class see 161 c

M considers the mean stress influence on the endurance (or static) strength amplitudes

M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress

Following M values may be used

Endurance limit

Static strength

Case hardened 08 ndash 015 Ys 1) 07

If shot peened 04 06

Nitrided 03 03

Induction or flame hardened

04 06

Not surface hardened steel 03 05

Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M

The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches

382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106

For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers

For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used

383 For gears with shrinkage stresses and unidirectional load For endurance strength

FE

fitM σ

σM1

M21Y+

minus=

σFE is the endurance limit for R = 0

For static strength YMst = 1 and σfit accounted for in 39b

σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor

n

Ffit m

ρ215scf minus=

384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is

( ) ( ) FE

fitM σ

σR1M1

M2

M1M1R1

1Yminussdot+

minus

+minus

minus=

Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12

For static strength

M1M1R1

1YMst

+minus

minus=

The effect of σfit is accounted for in 39 b

Classification Notes- No 412 27 May 2003

DET NORSKE VERITAS

385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed

F

y

Sσ225

for not surface hardened fillets

FSHV5 for surface hardened fillets

39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles

Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN

If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used

Determination of the σ ndash N ndash curve

a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF

b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ

FstS1

FPstσ minussdotsdotsdot=

where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability

σFst

Alloyed case hardened steel 1) 2300

Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

1250

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

1050

Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)

18 HV + 800

Steel with not surface hardened fillets the smaller value of 2)

18 σB or 225 σy

1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet

c) Calculate YN as

NL gt 3middot106

YN = 1 or 001

L

6

N N103Y

sdot= ie Yn = 092 for 1010

The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2

103ltNLlt3middot106exp

L

6

N N103Y

sdot=

cycles6103forσcycles310forσlog02876exp

FP

FPst

sdot=

NL lt 103 cycles6103forσcycles310forσ

NYFP

FPst

sdot=

or simply use σFPst as mentioned in b) directly

Guidance on number of load cycles NL for various applica-tions

bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)

bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied

28 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage

YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit

The following method may be used

For endurance limit

for not surface hardened fillets

( )

024

s024

relT σ103133q21σ1012201351

Ysdotsdotminus

+sdotsdotminus+= minus

minus

δ

for all surface hardened fillets except nitrided

( )

106q21002451

Y srelT

++=δ

for nitrided fillets

( )

1347q2101421

Y srelT

++=δ

For static strength

for not surface hardened fillets1)

( )( )( )025

02

02502s

relTst σ3000821σ300 1Y0821Y

+

minus+=δ

for surface hardened fillets except nitrided

YδrelTst = 044 YS + 012

for nitrided fillets

YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not

exceed the yield point and thereby alter the residual stress level See also 39b footnote 2

311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness

YRrelT differs for endurance limit and static strength

The following method may be used

For endurance limit

YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels

YδrelT = 43 ndash 326 (Ry + 1)0005

for nitrided steels

For static strength

YRrelTst = 1 for all Ry and all materials

For a fillet without any longitudinal machining trace Ry asymp Rz

312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength

The following may be used

For endurance limit

YX = 1 for mn le 5 generally

YX = 103 ndash 0006 mn for 5 lt mn lt 30

YX = 085 for mn ge 30

for not surface hard-ened steels

YX = 105 ndash 001 mn for 5 lt mn ge 25

YX = 08 for mn ge 25

for surface hardened steels

For static strength

YXst = 1 for all mn and all materials

313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength

YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength

In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength

The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses

The following simplified method for YC may be used

YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)

YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10

For endurance limit

+

+=nFFE

C m02ρt31

σconstY

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

Classification Notes- No 412 21 May 2003

DET NORSKE VERITAS

stresses (von Mises) Both are influenced by residual stresses but this is only considered roughly and empirically

The subsurface working stresses at depths inside the peak of the orthogonal shear stresses respectively the equivalent stresses are only dependent on the (real) Hertzian stresses Surface related conditions as expressed by ZL ZV and ZR are assumed to have a negligible influence

The real Hertzian stresses σHR are determined as

For helical gears with εβ gt 1

σHR = σH

For helical gears with εβ lt 1 and spur gears

ε

α

ββ ε

ε+εminus

sdotσ=σZ

1

HHR

For bevel gears

KHHR Z

1σσ sdot=

The necessary hardness HV is given as a function of the net depth tz (net = after grinding or hard metal hobbing and per-pendicular to the flank)

The coordinates tz and HV are to be compared with the de-sign specification such as

bull for flame and induction hardening tHVmin HVmin bull for nitriding t400min HV = 400 bull for case hardening t550min HV = 550 t400min HV = 400

and t300min HV = 300 (the latter only if the core hardness lt 300 If the core hardness gt 400 the t400 is to be re-placed by a fictive t400 = 16 t550)

In addition the specified surface hardness is not to be less than the max necessary hardness (at tz = 05aH) This applies to all hardening methods

For high cycle fatigue (gt3 106 cycles) the following applies

sdot+

minussdotsdotsdot= o90

05azt

05azt

cosSσ04HV

H

HHHR

applicable to 05a

zt

Hge

For 05at

H

z lt the value for 05at

H

z = applies

56300ρSσ12a cHHR

Hsdotsdot

sdot=

Where aH is half the hertzian contact width multiplied by an empirical factor of 12 that takes into account the possible influence of reduced compressive residual stresses (or even tensile residual stresses) on the local fatigue strength

If any of the specified hardness depths including the surface hardness is below the curve described by HV = f (tz) the actual safety factor against subsurface fatigue is determined as follows

reduce SH stepwise in the formula for HV and aH until all specified hardness depths and surface hardness balance with the corrected curve The safety factor obtained through this method is the safety against subsurface fatigue

For static strength (lt103 cycles) the following applies

sdot+

minussdotsdotsdot= o90

07at

06a

zt

cosSσ019HV

Hst

z

HstHHR

applicable to 06at

Hst

z ge

56300ρSσa cHHR

Hstsdotsdot

=

In the case of insufficient specified hardness depths the same procedure for determination of the actual safety factor as above applies

For limited life fatigue (103 lt cycles lt 3106 )

For this purpose it is necessary to extend the correction of safety factors to include also higher values than required Ie in the case of more than sufficient hardness and depths the safety factor in the formulae for both high cycle fatigue and static strength are to be increased until necessary and speci-fied values balance

The actual safety factor for a given number of cycles N be-tween 103 and 3106 is found by linear interpolation in a dou-ble logarithmic diagram

minussdotminus

= sdot logN3477

logSlogSlogS

310H6103HHN

36 10H103H Slog86281Slog86280 sdot+sdot sdot

22 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3 Calculation of Tooth Strength

31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding

For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36

For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning

Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα

In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314

A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired

It should be noted that this part 3 does not cover fractures caused by

bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining

Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears

Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied

32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level

321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance

Normally the stresses for pinion and wheel are calculated as

Cylindrical gears

FαFβvγAβSFn

tF K K K K K Y Y Y

m bFσ =

where

YF = Tooth form factor (see 33)

YS = Stress correction factor (see 34)

Yβ = Helix angle factor (see 36)

Ft KA Kγ Kv KFβ KFα see 15 ndash 110

b see 13

Bevel gears

FαFβvγASaFamn

mtF K K K K K Y Y Y

m bFσ ε=

where

YFa = Tooth form factor see 33

YSa = Stress correction factor see 34

Yε = Contact ratio factor see 35

Fmt KA etc see 15 ndash 110

b see 13

322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is

CXRrelTrelTF

NMFEFP YYYY

SYY

δσ

=σ

Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN

where

σFE = Local tooth root bending endurance limit of reference test gear (see 37)

YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)

YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)

SF = Required safety factor according to the rules

YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)

YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)

Classification Notes- No 412 23 May 2003

DET NORSKE VERITAS

YX = Size factor (see 312)

YC = Case depth factor (see 313)

33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress

YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears

YFa applies to load application at the tooth tip and is used for bevel gears

Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears

Figure 31 External tooth in normal section

Figure 32 Internal tooth in normal section

Definitions

n

2

n

Fn

enFn

Fe

F

α cosms

α cosmh6

Y

=

n

2

n

Fn

anFn

Fa

Fa

α cosms

α cosmh6

Y

=

In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth

YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)

The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing

mn with mnm

zn with zvn

αt with αvt

β with βm

with undercut without undercut

Fig 33 Dimensions and basic rack profile of the teeth (finished profile)

Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless

331 Determination of parameters

( )n

n

prnfPnfP m

α cossαsin 1ρ

αtan h4πE

minusminusminusminus=

For external gears fPfP ρρ =

For internal gears ( )0z

195fPfP0

fPfP 10363156ρhxρρ

sdotminus+

+=

where

z0 = number of teeth of pinion cutter

x0 = addendum modification coefficient of pinion cutter

hfP = addendum of pinion cutter

ρfP = tip radius of pinion cutter

xhρG fpfp +minus=

24 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

τmE

2π

z2H

nnminus

minus=

with

3πτ = for external gears

6πτ = for internal gears

Htan zG2

nminusϑ=ϑ

(to be solved iteratively suitable start value6π

=ϑ for exter-

nal gears and 3π for internal gears)

a) Tooth root chord sFn For external gears

minus

ϑ+

ϑminus= ρ

cosG3

3πsinz

ms

fpnn

Fn

For bevel gears with a tooth thickness modification

xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn

For internal gears

minus

ϑ+

ϑminus= ρ

cosG

6πsinz

ms

fPnn

Fn

b) Root fillet radius ρF at 30ordm tangent

( )G2coszcosG2ρ

mρ

2n

2

fpn

F

minusϑϑ+=

c) Determination of bending moment arm hF dn = zn mn

b2α

αn βcosε

ε =

dan = dn + 2 ha

pbn = π mn cos αn

dbn = dn cos αn

( )4

d1εpzz

2dd

zz2d

2bn

2

αnbn

2bn

2an

en +

minusminus

minus=

en

bnen d

dcos arcα =

ennnn

e α invα invα x tan 22π

z1

minus+

+=γ

αFen = αen ndash γe

For external gears

( )

minus=

n

enFenee

n

Fe

mdαtan sin γ γcos

21

mh

]ρcos

G3πcosz fpn +

ϑminus

ϑminusminus

For internal gears

( ) minus

sdotsdotminus=

n

enFenee

n

Fe

mdα tanγ sinγ cos

21

mh

minus

ϑminus

ϑminussdot ρ

cosG3

6πcosz fPn

332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as

250ε205for 0666ε2366Y αnαnDT leleminus=

205εfor 10Y αnDT lt=

34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section

Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm

YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip

YS can be determined as follows

( )

+

+=L23121

1

sS qL01312Y

where Fe

Fn

hsL = and

F

Fns ρ2

sq = (see 33)

YSa can be calculated by replacing hFe with hFa in the above formulae

Note a) Range of validity 1 lt qs lt 8

In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered

b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with

Classification Notes- No 412 25 May 2003

DET NORSKE VERITAS

g

g

ρt

0613

13

minus

where

tg = depth of the grinding notch

ρg = radius of the grinding notch

c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles

35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears

The following may be used

Yε = 0625

36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank

The following may be used (β input in degrees)

Yβ = 1 ndash εβ β120

When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula

However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)

37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM

σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision

If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules

σFE

Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)

bull of specially approved high grade

1050

bull of normal grade

minus CrNiMo steels with approved process

1000

minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850

Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

840

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

720

Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)

07 HV + 300

Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)

025 σB + 125

Alloyed quenched and tempered steel 04 σB + 200

Carbon steel 025 σB + 250

Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40

1) These values are valid for a root radius

bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)

bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)

bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for

6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked

2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)

26 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears

YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0

YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst

For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316

The following method may be used within a stress ratio ndash12 lt R lt 05

381 For idlers planets and PTO with ice class

M1M1R1

1Yor Y MstM

+minus

minus=

where

R = stress ratio = min stress divided by max stress

For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12

For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as

branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus

For a power take off (PTO) with ice class see 161 c

M considers the mean stress influence on the endurance (or static) strength amplitudes

M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress

Following M values may be used

Endurance limit

Static strength

Case hardened 08 ndash 015 Ys 1) 07

If shot peened 04 06

Nitrided 03 03

Induction or flame hardened

04 06

Not surface hardened steel 03 05

Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M

The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches

382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106

For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers

For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used

383 For gears with shrinkage stresses and unidirectional load For endurance strength

FE

fitM σ

σM1

M21Y+

minus=

σFE is the endurance limit for R = 0

For static strength YMst = 1 and σfit accounted for in 39b

σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor

n

Ffit m

ρ215scf minus=

384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is

( ) ( ) FE

fitM σ

σR1M1

M2

M1M1R1

1Yminussdot+

minus

+minus

minus=

Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12

For static strength

M1M1R1

1YMst

+minus

minus=

The effect of σfit is accounted for in 39 b

Classification Notes- No 412 27 May 2003

DET NORSKE VERITAS

385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed

F

y

Sσ225

for not surface hardened fillets

FSHV5 for surface hardened fillets

39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles

Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN

If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used

Determination of the σ ndash N ndash curve

a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF

b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ

FstS1

FPstσ minussdotsdotsdot=

where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability

σFst

Alloyed case hardened steel 1) 2300

Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

1250

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

1050

Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)

18 HV + 800

Steel with not surface hardened fillets the smaller value of 2)

18 σB or 225 σy

1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet

c) Calculate YN as

NL gt 3middot106

YN = 1 or 001

L

6

N N103Y

sdot= ie Yn = 092 for 1010

The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2

103ltNLlt3middot106exp

L

6

N N103Y

sdot=

cycles6103forσcycles310forσlog02876exp

FP

FPst

sdot=

NL lt 103 cycles6103forσcycles310forσ

NYFP

FPst

sdot=

or simply use σFPst as mentioned in b) directly

Guidance on number of load cycles NL for various applica-tions

bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)

bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied

28 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage

YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit

The following method may be used

For endurance limit

for not surface hardened fillets

( )

024

s024

relT σ103133q21σ1012201351

Ysdotsdotminus

+sdotsdotminus+= minus

minus

δ

for all surface hardened fillets except nitrided

( )

106q21002451

Y srelT

++=δ

for nitrided fillets

( )

1347q2101421

Y srelT

++=δ

For static strength

for not surface hardened fillets1)

( )( )( )025

02

02502s

relTst σ3000821σ300 1Y0821Y

+

minus+=δ

for surface hardened fillets except nitrided

YδrelTst = 044 YS + 012

for nitrided fillets

YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not

exceed the yield point and thereby alter the residual stress level See also 39b footnote 2

311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness

YRrelT differs for endurance limit and static strength

The following method may be used

For endurance limit

YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels

YδrelT = 43 ndash 326 (Ry + 1)0005

for nitrided steels

For static strength

YRrelTst = 1 for all Ry and all materials

For a fillet without any longitudinal machining trace Ry asymp Rz

312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength

The following may be used

For endurance limit

YX = 1 for mn le 5 generally

YX = 103 ndash 0006 mn for 5 lt mn lt 30

YX = 085 for mn ge 30

for not surface hard-ened steels

YX = 105 ndash 001 mn for 5 lt mn ge 25

YX = 08 for mn ge 25

for surface hardened steels

For static strength

YXst = 1 for all mn and all materials

313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength

YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength

In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength

The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses

The following simplified method for YC may be used

YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)

YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10

For endurance limit

+

+=nFFE

C m02ρt31

σconstY

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

22 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3 Calculation of Tooth Strength

31 Scope and General Remarks Part 3 include the calculation of tooth root strength as limited by tooth root cracking (surface or subsurface initiated) and yielding

For rim thickness sR ge 35middotmn the strength is calculated by means of 32 ndash 313 For cylindrical gears the calculation is based on the assumption that the highest tooth root tensile stress arises by application of the force at the outer point of single tooth pair contact of the virtual spur gears The method has however a few limitations that are mentioned in 36

For bevel gears the calculation is based on force application at the tooth tip of the virtual cylindrical gear Subsequently the stress is converted to load application at the mid point of the flank due to the heightwise crowning

Bevel gears may also be calculated with the program BECAL In that case KA and Kv are to be included in the applied tooth force but not KFβ and KFα

In case of a thin annulus or a thin gear rim etc radial crack-ing can occur rather than tangential cracking (from root fillet to root fillet) Cracking can also start from the compression fillet rather than the tension fillet For rim thickness sR lt 35middotmn a special calculation procedure is given in 315 and 316 and a simplified procedure in 314

A tooth breakage is often the end of the life of a gear trans-mission Therefore a high safety SF against breakage is re-quired

It should be noted that this part 3 does not cover fractures caused by

bull oil holes in the tooth root space bull wear steps on the flank bull flank surface distress such as pits spalls or grey staining

Especially the latter is known to cause oblique fractures starting from the active flank predominately in spiral bevel gears but also sometimes in cylindrical gears

Specific calculation methods for these purposes are not given here but are under consideration for future revisions Thus depending on experience with similar gear designs limita-tions other than those outlined in part 3 may be applied

32 Tooth Root Stresses The local tooth root stress is defined as the max principal stress in the tooth root caused by application of the tooth force Ie the stress ratio R = 0 Other stress ratios such as for eg idler gears (R asymp -12) shrunk on gear rims (R gt 0) etc are considered by correcting the permissible stress level

321 Local tooth root stress The local tooth root stress for pinion and wheel may be as-sessed by strain gauge measurements or FE calculations or similar For both measurements and calculations all details are to be agreed in advance

Normally the stresses for pinion and wheel are calculated as

Cylindrical gears

FαFβvγAβSFn

tF K K K K K Y Y Y

m bFσ =

where

YF = Tooth form factor (see 33)

YS = Stress correction factor (see 34)

Yβ = Helix angle factor (see 36)

Ft KA Kγ Kv KFβ KFα see 15 ndash 110

b see 13

Bevel gears

FαFβvγASaFamn

mtF K K K K K Y Y Y

m bFσ ε=

where

YFa = Tooth form factor see 33

YSa = Stress correction factor see 34

Yε = Contact ratio factor see 35

Fmt KA etc see 15 ndash 110

b see 13

322 Permissible tooth root stress The permissible local tooth root stress for pinion respectively wheel for a given number of cycles N is

CXRrelTrelTF

NMFEFP YYYY

SYY

δσ

=σ

Note that all these factors YM etc are applicable to 3middot106 cycles when used in this formula for σFP The influence of other number of cycles on these factors is covered by the calculation of YN

where

σFE = Local tooth root bending endurance limit of reference test gear (see 37)

YM = Mean stress influence factor which accounts for other loads than constant load direction eg idler gears temporary change of load di-rection pre-stress due to shrinkage etc (see 38)

YN = Life factor for tooth root stresses related to reference test gear dimensions (see 39)

SF = Required safety factor according to the rules

YδrelT = Relative notch sensitivity factor of the gear to be determined related to the reference test gear (see 310)

YRrelT = Relative (root fillet) surface condition factor of the gear to be determined related to the reference test gear (see 311)

Classification Notes- No 412 23 May 2003

DET NORSKE VERITAS

YX = Size factor (see 312)

YC = Case depth factor (see 313)

33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress

YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears

YFa applies to load application at the tooth tip and is used for bevel gears

Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears

Figure 31 External tooth in normal section

Figure 32 Internal tooth in normal section

Definitions

n

2

n

Fn

enFn

Fe

F

α cosms

α cosmh6

Y

=

n

2

n

Fn

anFn

Fa

Fa

α cosms

α cosmh6

Y

=

In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth

YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)

The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing

mn with mnm

zn with zvn

αt with αvt

β with βm

with undercut without undercut

Fig 33 Dimensions and basic rack profile of the teeth (finished profile)

Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless

331 Determination of parameters

( )n

n

prnfPnfP m

α cossαsin 1ρ

αtan h4πE

minusminusminusminus=

For external gears fPfP ρρ =

For internal gears ( )0z

195fPfP0

fPfP 10363156ρhxρρ

sdotminus+

+=

where

z0 = number of teeth of pinion cutter

x0 = addendum modification coefficient of pinion cutter

hfP = addendum of pinion cutter

ρfP = tip radius of pinion cutter

xhρG fpfp +minus=

24 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

τmE

2π

z2H

nnminus

minus=

with

3πτ = for external gears

6πτ = for internal gears

Htan zG2

nminusϑ=ϑ

(to be solved iteratively suitable start value6π

=ϑ for exter-

nal gears and 3π for internal gears)

a) Tooth root chord sFn For external gears

minus

ϑ+

ϑminus= ρ

cosG3

3πsinz

ms

fpnn

Fn

For bevel gears with a tooth thickness modification

xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn

For internal gears

minus

ϑ+

ϑminus= ρ

cosG

6πsinz

ms

fPnn

Fn

b) Root fillet radius ρF at 30ordm tangent

( )G2coszcosG2ρ

mρ

2n

2

fpn

F

minusϑϑ+=

c) Determination of bending moment arm hF dn = zn mn

b2α

αn βcosε

ε =

dan = dn + 2 ha

pbn = π mn cos αn

dbn = dn cos αn

( )4

d1εpzz

2dd

zz2d

2bn

2

αnbn

2bn

2an

en +

minusminus

minus=

en

bnen d

dcos arcα =

ennnn

e α invα invα x tan 22π

z1

minus+

+=γ

αFen = αen ndash γe

For external gears

( )

minus=

n

enFenee

n

Fe

mdαtan sin γ γcos

21

mh

]ρcos

G3πcosz fpn +

ϑminus

ϑminusminus

For internal gears

( ) minus

sdotsdotminus=

n

enFenee

n

Fe

mdα tanγ sinγ cos

21

mh

minus

ϑminus

ϑminussdot ρ

cosG3

6πcosz fPn

332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as

250ε205for 0666ε2366Y αnαnDT leleminus=

205εfor 10Y αnDT lt=

34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section

Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm

YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip

YS can be determined as follows

( )

+

+=L23121

1

sS qL01312Y

where Fe

Fn

hsL = and

F

Fns ρ2

sq = (see 33)

YSa can be calculated by replacing hFe with hFa in the above formulae

Note a) Range of validity 1 lt qs lt 8

In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered

b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with

Classification Notes- No 412 25 May 2003

DET NORSKE VERITAS

g

g

ρt

0613

13

minus

where

tg = depth of the grinding notch

ρg = radius of the grinding notch

c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles

35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears

The following may be used

Yε = 0625

36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank

The following may be used (β input in degrees)

Yβ = 1 ndash εβ β120

When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula

However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)

37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM

σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision

If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules

σFE

Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)

bull of specially approved high grade

1050

bull of normal grade

minus CrNiMo steels with approved process

1000

minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850

Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

840

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

720

Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)

07 HV + 300

Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)

025 σB + 125

Alloyed quenched and tempered steel 04 σB + 200

Carbon steel 025 σB + 250

Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40

1) These values are valid for a root radius

bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)

bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)

bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for

6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked

2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)

26 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears

YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0

YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst

For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316

The following method may be used within a stress ratio ndash12 lt R lt 05

381 For idlers planets and PTO with ice class

M1M1R1

1Yor Y MstM

+minus

minus=

where

R = stress ratio = min stress divided by max stress

For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12

For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as

branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus

For a power take off (PTO) with ice class see 161 c

M considers the mean stress influence on the endurance (or static) strength amplitudes

M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress

Following M values may be used

Endurance limit

Static strength

Case hardened 08 ndash 015 Ys 1) 07

If shot peened 04 06

Nitrided 03 03

Induction or flame hardened

04 06

Not surface hardened steel 03 05

Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M

The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches

382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106

For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers

For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used

383 For gears with shrinkage stresses and unidirectional load For endurance strength

FE

fitM σ

σM1

M21Y+

minus=

σFE is the endurance limit for R = 0

For static strength YMst = 1 and σfit accounted for in 39b

σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor

n

Ffit m

ρ215scf minus=

384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is

( ) ( ) FE

fitM σ

σR1M1

M2

M1M1R1

1Yminussdot+

minus

+minus

minus=

Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12

For static strength

M1M1R1

1YMst

+minus

minus=

The effect of σfit is accounted for in 39 b

Classification Notes- No 412 27 May 2003

DET NORSKE VERITAS

385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed

F

y

Sσ225

for not surface hardened fillets

FSHV5 for surface hardened fillets

39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles

Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN

If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used

Determination of the σ ndash N ndash curve

a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF

b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ

FstS1

FPstσ minussdotsdotsdot=

where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability

σFst

Alloyed case hardened steel 1) 2300

Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

1250

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

1050

Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)

18 HV + 800

Steel with not surface hardened fillets the smaller value of 2)

18 σB or 225 σy

1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet

c) Calculate YN as

NL gt 3middot106

YN = 1 or 001

L

6

N N103Y

sdot= ie Yn = 092 for 1010

The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2

103ltNLlt3middot106exp

L

6

N N103Y

sdot=

cycles6103forσcycles310forσlog02876exp

FP

FPst

sdot=

NL lt 103 cycles6103forσcycles310forσ

NYFP

FPst

sdot=

or simply use σFPst as mentioned in b) directly

Guidance on number of load cycles NL for various applica-tions

bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)

bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied

28 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage

YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit

The following method may be used

For endurance limit

for not surface hardened fillets

( )

024

s024

relT σ103133q21σ1012201351

Ysdotsdotminus

+sdotsdotminus+= minus

minus

δ

for all surface hardened fillets except nitrided

( )

106q21002451

Y srelT

++=δ

for nitrided fillets

( )

1347q2101421

Y srelT

++=δ

For static strength

for not surface hardened fillets1)

( )( )( )025

02

02502s

relTst σ3000821σ300 1Y0821Y

+

minus+=δ

for surface hardened fillets except nitrided

YδrelTst = 044 YS + 012

for nitrided fillets

YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not

exceed the yield point and thereby alter the residual stress level See also 39b footnote 2

311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness

YRrelT differs for endurance limit and static strength

The following method may be used

For endurance limit

YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels

YδrelT = 43 ndash 326 (Ry + 1)0005

for nitrided steels

For static strength

YRrelTst = 1 for all Ry and all materials

For a fillet without any longitudinal machining trace Ry asymp Rz

312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength

The following may be used

For endurance limit

YX = 1 for mn le 5 generally

YX = 103 ndash 0006 mn for 5 lt mn lt 30

YX = 085 for mn ge 30

for not surface hard-ened steels

YX = 105 ndash 001 mn for 5 lt mn ge 25

YX = 08 for mn ge 25

for surface hardened steels

For static strength

YXst = 1 for all mn and all materials

313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength

YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength

In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength

The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses

The following simplified method for YC may be used

YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)

YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10

For endurance limit

+

+=nFFE

C m02ρt31

σconstY

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

Classification Notes- No 412 23 May 2003

DET NORSKE VERITAS

YX = Size factor (see 312)

YC = Case depth factor (see 313)

33 Tooth Form Factors YF YFa The tooth form factors YF and YFa take into account the in-fluence of the tooth form on the nominal bending stress

YF applies to load application at the outer point of single tooth pair contact of the virtual spur gear pair and is used for cylindrical gears

YFa applies to load application at the tooth tip and is used for bevel gears

Both YF and YFa are based on the distance between the con-tact points of the 30˚-tangents at the root fillet of the tooth profile for external gears respectively 60˚ tangents for inter-nal gears

Figure 31 External tooth in normal section

Figure 32 Internal tooth in normal section

Definitions

n

2

n

Fn

enFn

Fe

F

α cosms

α cosmh6

Y

=

n

2

n

Fn

anFn

Fa

Fa

α cosms

α cosmh6

Y

=

In the case of helical gears YF and YFa are determined in the normal section ie for a virtual number of teeth

YFa differs from YF by the bending moment arm hFa and αFan and can be determined by the same procedure as YF with exception of hFe and αFan For hFa and αFan all indices e will change to a (tip)

The following formulae apply to cylindrical gears but may also be used for bevel gears when replacing

mn with mnm

zn with zvn

αt with αvt

β with βm

with undercut without undercut

Fig 33 Dimensions and basic rack profile of the teeth (finished profile)

Tool and basic rack data such as hfP ρfp and spr etc are re-ferred to mn ie dimensionless

331 Determination of parameters

( )n

n

prnfPnfP m

α cossαsin 1ρ

αtan h4πE

minusminusminusminus=

For external gears fPfP ρρ =

For internal gears ( )0z

195fPfP0

fPfP 10363156ρhxρρ

sdotminus+

+=

where

z0 = number of teeth of pinion cutter

x0 = addendum modification coefficient of pinion cutter

hfP = addendum of pinion cutter

ρfP = tip radius of pinion cutter

xhρG fpfp +minus=

24 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

τmE

2π

z2H

nnminus

minus=

with

3πτ = for external gears

6πτ = for internal gears

Htan zG2

nminusϑ=ϑ

(to be solved iteratively suitable start value6π

=ϑ for exter-

nal gears and 3π for internal gears)

a) Tooth root chord sFn For external gears

minus

ϑ+

ϑminus= ρ

cosG3

3πsinz

ms

fpnn

Fn

For bevel gears with a tooth thickness modification

xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn

For internal gears

minus

ϑ+

ϑminus= ρ

cosG

6πsinz

ms

fPnn

Fn

b) Root fillet radius ρF at 30ordm tangent

( )G2coszcosG2ρ

mρ

2n

2

fpn

F

minusϑϑ+=

c) Determination of bending moment arm hF dn = zn mn

b2α

αn βcosε

ε =

dan = dn + 2 ha

pbn = π mn cos αn

dbn = dn cos αn

( )4

d1εpzz

2dd

zz2d

2bn

2

αnbn

2bn

2an

en +

minusminus

minus=

en

bnen d

dcos arcα =

ennnn

e α invα invα x tan 22π

z1

minus+

+=γ

αFen = αen ndash γe

For external gears

( )

minus=

n

enFenee

n

Fe

mdαtan sin γ γcos

21

mh

]ρcos

G3πcosz fpn +

ϑminus

ϑminusminus

For internal gears

( ) minus

sdotsdotminus=

n

enFenee

n

Fe

mdα tanγ sinγ cos

21

mh

minus

ϑminus

ϑminussdot ρ

cosG3

6πcosz fPn

332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as

250ε205for 0666ε2366Y αnαnDT leleminus=

205εfor 10Y αnDT lt=

34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section

Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm

YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip

YS can be determined as follows

( )

+

+=L23121

1

sS qL01312Y

where Fe

Fn

hsL = and

F

Fns ρ2

sq = (see 33)

YSa can be calculated by replacing hFe with hFa in the above formulae

Note a) Range of validity 1 lt qs lt 8

In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered

b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with

Classification Notes- No 412 25 May 2003

DET NORSKE VERITAS

g

g

ρt

0613

13

minus

where

tg = depth of the grinding notch

ρg = radius of the grinding notch

c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles

35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears

The following may be used

Yε = 0625

36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank

The following may be used (β input in degrees)

Yβ = 1 ndash εβ β120

When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula

However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)

37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM

σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision

If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules

σFE

Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)

bull of specially approved high grade

1050

bull of normal grade

minus CrNiMo steels with approved process

1000

minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850

Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

840

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

720

Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)

07 HV + 300

Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)

025 σB + 125

Alloyed quenched and tempered steel 04 σB + 200

Carbon steel 025 σB + 250

Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40

1) These values are valid for a root radius

bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)

bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)

bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for

6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked

2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)

26 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears

YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0

YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst

For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316

The following method may be used within a stress ratio ndash12 lt R lt 05

381 For idlers planets and PTO with ice class

M1M1R1

1Yor Y MstM

+minus

minus=

where

R = stress ratio = min stress divided by max stress

For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12

For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as

branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus

For a power take off (PTO) with ice class see 161 c

M considers the mean stress influence on the endurance (or static) strength amplitudes

M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress

Following M values may be used

Endurance limit

Static strength

Case hardened 08 ndash 015 Ys 1) 07

If shot peened 04 06

Nitrided 03 03

Induction or flame hardened

04 06

Not surface hardened steel 03 05

Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M

The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches

382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106

For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers

For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used

383 For gears with shrinkage stresses and unidirectional load For endurance strength

FE

fitM σ

σM1

M21Y+

minus=

σFE is the endurance limit for R = 0

For static strength YMst = 1 and σfit accounted for in 39b

σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor

n

Ffit m

ρ215scf minus=

384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is

( ) ( ) FE

fitM σ

σR1M1

M2

M1M1R1

1Yminussdot+

minus

+minus

minus=

Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12

For static strength

M1M1R1

1YMst

+minus

minus=

The effect of σfit is accounted for in 39 b

Classification Notes- No 412 27 May 2003

DET NORSKE VERITAS

385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed

F

y

Sσ225

for not surface hardened fillets

FSHV5 for surface hardened fillets

39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles

Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN

If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used

Determination of the σ ndash N ndash curve

a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF

b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ

FstS1

FPstσ minussdotsdotsdot=

where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability

σFst

Alloyed case hardened steel 1) 2300

Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

1250

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

1050

Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)

18 HV + 800

Steel with not surface hardened fillets the smaller value of 2)

18 σB or 225 σy

1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet

c) Calculate YN as

NL gt 3middot106

YN = 1 or 001

L

6

N N103Y

sdot= ie Yn = 092 for 1010

The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2

103ltNLlt3middot106exp

L

6

N N103Y

sdot=

cycles6103forσcycles310forσlog02876exp

FP

FPst

sdot=

NL lt 103 cycles6103forσcycles310forσ

NYFP

FPst

sdot=

or simply use σFPst as mentioned in b) directly

Guidance on number of load cycles NL for various applica-tions

bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)

bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied

28 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage

YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit

The following method may be used

For endurance limit

for not surface hardened fillets

( )

024

s024

relT σ103133q21σ1012201351

Ysdotsdotminus

+sdotsdotminus+= minus

minus

δ

for all surface hardened fillets except nitrided

( )

106q21002451

Y srelT

++=δ

for nitrided fillets

( )

1347q2101421

Y srelT

++=δ

For static strength

for not surface hardened fillets1)

( )( )( )025

02

02502s

relTst σ3000821σ300 1Y0821Y

+

minus+=δ

for surface hardened fillets except nitrided

YδrelTst = 044 YS + 012

for nitrided fillets

YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not

exceed the yield point and thereby alter the residual stress level See also 39b footnote 2

311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness

YRrelT differs for endurance limit and static strength

The following method may be used

For endurance limit

YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels

YδrelT = 43 ndash 326 (Ry + 1)0005

for nitrided steels

For static strength

YRrelTst = 1 for all Ry and all materials

For a fillet without any longitudinal machining trace Ry asymp Rz

312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength

The following may be used

For endurance limit

YX = 1 for mn le 5 generally

YX = 103 ndash 0006 mn for 5 lt mn lt 30

YX = 085 for mn ge 30

for not surface hard-ened steels

YX = 105 ndash 001 mn for 5 lt mn ge 25

YX = 08 for mn ge 25

for surface hardened steels

For static strength

YXst = 1 for all mn and all materials

313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength

YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength

In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength

The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses

The following simplified method for YC may be used

YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)

YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10

For endurance limit

+

+=nFFE

C m02ρt31

σconstY

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

24 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

τmE

2π

z2H

nnminus

minus=

with

3πτ = for external gears

6πτ = for internal gears

Htan zG2

nminusϑ=ϑ

(to be solved iteratively suitable start value6π

=ϑ for exter-

nal gears and 3π for internal gears)

a) Tooth root chord sFn For external gears

minus

ϑ+

ϑminus= ρ

cosG3

3πsinz

ms

fpnn

Fn

For bevel gears with a tooth thickness modification

xsm affects mainly sFn but also hFe and αFen The total influence of xsm on YFa Ysa can be approximated by only adding 2 xsm to sFn mn

For internal gears

minus

ϑ+

ϑminus= ρ

cosG

6πsinz

ms

fPnn

Fn

b) Root fillet radius ρF at 30ordm tangent

( )G2coszcosG2ρ

mρ

2n

2

fpn

F

minusϑϑ+=

c) Determination of bending moment arm hF dn = zn mn

b2α

αn βcosε

ε =

dan = dn + 2 ha

pbn = π mn cos αn

dbn = dn cos αn

( )4

d1εpzz

2dd

zz2d

2bn

2

αnbn

2bn

2an

en +

minusminus

minus=

en

bnen d

dcos arcα =

ennnn

e α invα invα x tan 22π

z1

minus+

+=γ

αFen = αen ndash γe

For external gears

( )

minus=

n

enFenee

n

Fe

mdαtan sin γ γcos

21

mh

]ρcos

G3πcosz fpn +

ϑminus

ϑminusminus

For internal gears

( ) minus

sdotsdotminus=

n

enFenee

n

Fe

mdα tanγ sinγ cos

21

mh

minus

ϑminus

ϑminussdot ρ

cosG3

6πcosz fPn

332 Gearing with εαn gt 2 For deep tooth form gearing ( )25ε2 αn lele produced with a verified grade of accuracy of 4 or better and with applied profile modification to obtain a trapezoidal load distribution along the path of contact the YF may be corrected by the factor YDT as

250ε205for 0666ε2366Y αnαnDT leleminus=

205εfor 10Y αnDT lt=

34 Stress Correction Factors YS YSa The stress correction factors YS and YSa take into account the conversion of the nominal bending stress to the local tooth root stress Thereby YS and YSa cover the stress increasing effect of the notch (fillet) and the fact that not only bending stresses arise at the root A part of the local stress is inde-pendent of the bending moment arm This part increases the more the decisive point of load application approaches the critical tooth root section

Therefore in addition to its dependence on the notch radius the stress correction is also dependent on the position of the load application ie the size of the bending moment arm

YS applies to the load application at the outer point of single tooth pair contact YSa to the load application at tooth tip

YS can be determined as follows

( )

+

+=L23121

1

sS qL01312Y

where Fe

Fn

hsL = and

F

Fns ρ2

sq = (see 33)

YSa can be calculated by replacing hFe with hFa in the above formulae

Note a) Range of validity 1 lt qs lt 8

In case of sharper root radii (ie produced with tools having too sharp tip radii) YS resp YSa must be specially considered

b) In case of grinding notches (due to insufficient protuberance of the hob) YS resp YSa can rise considerably and must be multiplied with

Classification Notes- No 412 25 May 2003

DET NORSKE VERITAS

g

g

ρt

0613

13

minus

where

tg = depth of the grinding notch

ρg = radius of the grinding notch

c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles

35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears

The following may be used

Yε = 0625

36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank

The following may be used (β input in degrees)

Yβ = 1 ndash εβ β120

When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula

However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)

37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM

σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision

If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules

σFE

Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)

bull of specially approved high grade

1050

bull of normal grade

minus CrNiMo steels with approved process

1000

minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850

Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

840

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

720

Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)

07 HV + 300

Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)

025 σB + 125

Alloyed quenched and tempered steel 04 σB + 200

Carbon steel 025 σB + 250

Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40

1) These values are valid for a root radius

bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)

bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)

bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for

6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked

2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)

26 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears

YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0

YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst

For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316

The following method may be used within a stress ratio ndash12 lt R lt 05

381 For idlers planets and PTO with ice class

M1M1R1

1Yor Y MstM

+minus

minus=

where

R = stress ratio = min stress divided by max stress

For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12

For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as

branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus

For a power take off (PTO) with ice class see 161 c

M considers the mean stress influence on the endurance (or static) strength amplitudes

M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress

Following M values may be used

Endurance limit

Static strength

Case hardened 08 ndash 015 Ys 1) 07

If shot peened 04 06

Nitrided 03 03

Induction or flame hardened

04 06

Not surface hardened steel 03 05

Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M

The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches

382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106

For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers

For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used

383 For gears with shrinkage stresses and unidirectional load For endurance strength

FE

fitM σ

σM1

M21Y+

minus=

σFE is the endurance limit for R = 0

For static strength YMst = 1 and σfit accounted for in 39b

σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor

n

Ffit m

ρ215scf minus=

384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is

( ) ( ) FE

fitM σ

σR1M1

M2

M1M1R1

1Yminussdot+

minus

+minus

minus=

Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12

For static strength

M1M1R1

1YMst

+minus

minus=

The effect of σfit is accounted for in 39 b

Classification Notes- No 412 27 May 2003

DET NORSKE VERITAS

385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed

F

y

Sσ225

for not surface hardened fillets

FSHV5 for surface hardened fillets

39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles

Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN

If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used

Determination of the σ ndash N ndash curve

a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF

b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ

FstS1

FPstσ minussdotsdotsdot=

where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability

σFst

Alloyed case hardened steel 1) 2300

Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

1250

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

1050

Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)

18 HV + 800

Steel with not surface hardened fillets the smaller value of 2)

18 σB or 225 σy

1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet

c) Calculate YN as

NL gt 3middot106

YN = 1 or 001

L

6

N N103Y

sdot= ie Yn = 092 for 1010

The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2

103ltNLlt3middot106exp

L

6

N N103Y

sdot=

cycles6103forσcycles310forσlog02876exp

FP

FPst

sdot=

NL lt 103 cycles6103forσcycles310forσ

NYFP

FPst

sdot=

or simply use σFPst as mentioned in b) directly

Guidance on number of load cycles NL for various applica-tions

bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)

bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied

28 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage

YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit

The following method may be used

For endurance limit

for not surface hardened fillets

( )

024

s024

relT σ103133q21σ1012201351

Ysdotsdotminus

+sdotsdotminus+= minus

minus

δ

for all surface hardened fillets except nitrided

( )

106q21002451

Y srelT

++=δ

for nitrided fillets

( )

1347q2101421

Y srelT

++=δ

For static strength

for not surface hardened fillets1)

( )( )( )025

02

02502s

relTst σ3000821σ300 1Y0821Y

+

minus+=δ

for surface hardened fillets except nitrided

YδrelTst = 044 YS + 012

for nitrided fillets

YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not

exceed the yield point and thereby alter the residual stress level See also 39b footnote 2

311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness

YRrelT differs for endurance limit and static strength

The following method may be used

For endurance limit

YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels

YδrelT = 43 ndash 326 (Ry + 1)0005

for nitrided steels

For static strength

YRrelTst = 1 for all Ry and all materials

For a fillet without any longitudinal machining trace Ry asymp Rz

312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength

The following may be used

For endurance limit

YX = 1 for mn le 5 generally

YX = 103 ndash 0006 mn for 5 lt mn lt 30

YX = 085 for mn ge 30

for not surface hard-ened steels

YX = 105 ndash 001 mn for 5 lt mn ge 25

YX = 08 for mn ge 25

for surface hardened steels

For static strength

YXst = 1 for all mn and all materials

313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength

YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength

In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength

The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses

The following simplified method for YC may be used

YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)

YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10

For endurance limit

+

+=nFFE

C m02ρt31

σconstY

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

Classification Notes- No 412 25 May 2003

DET NORSKE VERITAS

g

g

ρt

0613

13

minus

where

tg = depth of the grinding notch

ρg = radius of the grinding notch

c) The formulae for YS resp YSa are only valid for αn = 20˚ However the same formulae can be used as a safe approximation for other pressure angles

35 Contact Ratio Factor Yε The contact ratio factor Yε covers the conversion from load application at the tooth tip to the load application at the mid point of the flank (heightwise) for bevel gears

The following may be used

Yε = 0625

36 Helix Angle Factor Yβ The helix angle factor Yβ takes into account the difference between the helical gear and the virtual spur gear in the nor-mal section on which the calculation is based in the first step In this way it is accounted for that the conditions for tooth root stresses are more favourable because the lines of contact are sloping over the flank

The following may be used (β input in degrees)

Yβ = 1 ndash εβ β120

When εβ gt 1 use εβ = 1 and when β gt 30deg use β = 30deg in the formula

However the above equation for Yβ may only be used for gears with β gt 25deg if adequate tip relief is applied to both pinion and wheel (adequate = at least 05 middot Ceff see 432)

37 Values of Endurance Limit σFE σFE is the local tooth root stress (max principal) which the material can endure permanently with 99 survival prob-ability 3106 load cycles is regarded as the beginning of the endurance limit or the lower knee of the σ ndash N curve σFE is defined as the unidirectional pulsating stress with a minimum stress of zero (disregarding residual stresses due to heat treatment) Other stress conditions such as alternating or pre-stressed etc are covered by the conversion factor YM

σFE can be found by pulsating tests or gear running tests for any material in any condition If the approval of the gear is to be based on the results of such tests all details on the testing conditions have to be approved by the Society Fur-ther the tests may have to be made under the Societys su-pervision

If no fatigue tests are available the following listed values for σFE may be used for materials subjected to a quality con-trol as the one referred to in the rules

σFE

Alloyed case hardened steels 1) (fillet surface hardness 58 ndash 63 HRC)

bull of specially approved high grade

1050

bull of normal grade

minus CrNiMo steels with approved process

1000

minus CrNi and CrNiMo steels generally 920 minus MnCr steels generally 850

Nitriding steel of approved grade quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

840

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

720

Alloyed quenched and tempered steel flame or induction hardened 2) (incl entire root fillet) (fillet surface hardness 500 ndash 650 HV)

07 HV + 300

Alloyed quenched and tempered steel flame or induction hardened (excl entire root fillet) (σB = uts of base material)

025 σB + 125

Alloyed quenched and tempered steel 04 σB + 200

Carbon steel 025 σB + 250

Note All numbers given above are valid for separate forgings and for blanks cut from bars forged according to a qualified procedure see Pt 4 Ch 2 Sec 3 For rolled steel the values are to be reduced with 10 For blanks cut from forged bars that are not qualified as mentioned above the values are to be reduced with 20 For cast steel reduce with 40

1) These values are valid for a root radius

bull being unground If however any grinding is made in the root fillet area in such a way that the residual stresses may be affected σFE is to be reduced by 20 (If the grinding also leaves a notch see 34)

bull with fillet surface hardness 58 ndash 63 HRC In case of lower surface hardness than 58 HRC σFE is to be reduced with 20(58 ndash HRC) where HRC is the detected hardness (This may lead to a permissible tooth root stress that varies along the facewidth If so the actual tooth root stresses may also be considered along facewidth)

bull not being shot peened In case of approved shot peening σFE may be increased by 200 for gears where σFE is reduced by 20 due to root grinding Otherwise σFE may be increased by 100 for

6nm le and 100 ndash 5 (mn - 6) for mn gt 6 However the possible adverse influence on the flanks regarding grey staining should be considered and if necessary the flanks should be masked

2) The fillet is not to be ground after surface hardening Regarding possible root grinding see 1)

26 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears

YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0

YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst

For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316

The following method may be used within a stress ratio ndash12 lt R lt 05

381 For idlers planets and PTO with ice class

M1M1R1

1Yor Y MstM

+minus

minus=

where

R = stress ratio = min stress divided by max stress

For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12

For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as

branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus

For a power take off (PTO) with ice class see 161 c

M considers the mean stress influence on the endurance (or static) strength amplitudes

M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress

Following M values may be used

Endurance limit

Static strength

Case hardened 08 ndash 015 Ys 1) 07

If shot peened 04 06

Nitrided 03 03

Induction or flame hardened

04 06

Not surface hardened steel 03 05

Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M

The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches

382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106

For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers

For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used

383 For gears with shrinkage stresses and unidirectional load For endurance strength

FE

fitM σ

σM1

M21Y+

minus=

σFE is the endurance limit for R = 0

For static strength YMst = 1 and σfit accounted for in 39b

σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor

n

Ffit m

ρ215scf minus=

384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is

( ) ( ) FE

fitM σ

σR1M1

M2

M1M1R1

1Yminussdot+

minus

+minus

minus=

Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12

For static strength

M1M1R1

1YMst

+minus

minus=

The effect of σfit is accounted for in 39 b

Classification Notes- No 412 27 May 2003

DET NORSKE VERITAS

385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed

F

y

Sσ225

for not surface hardened fillets

FSHV5 for surface hardened fillets

39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles

Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN

If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used

Determination of the σ ndash N ndash curve

a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF

b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ

FstS1

FPstσ minussdotsdotsdot=

where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability

σFst

Alloyed case hardened steel 1) 2300

Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

1250

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

1050

Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)

18 HV + 800

Steel with not surface hardened fillets the smaller value of 2)

18 σB or 225 σy

1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet

c) Calculate YN as

NL gt 3middot106

YN = 1 or 001

L

6

N N103Y

sdot= ie Yn = 092 for 1010

The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2

103ltNLlt3middot106exp

L

6

N N103Y

sdot=

cycles6103forσcycles310forσlog02876exp

FP

FPst

sdot=

NL lt 103 cycles6103forσcycles310forσ

NYFP

FPst

sdot=

or simply use σFPst as mentioned in b) directly

Guidance on number of load cycles NL for various applica-tions

bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)

bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied

28 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage

YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit

The following method may be used

For endurance limit

for not surface hardened fillets

( )

024

s024

relT σ103133q21σ1012201351

Ysdotsdotminus

+sdotsdotminus+= minus

minus

δ

for all surface hardened fillets except nitrided

( )

106q21002451

Y srelT

++=δ

for nitrided fillets

( )

1347q2101421

Y srelT

++=δ

For static strength

for not surface hardened fillets1)

( )( )( )025

02

02502s

relTst σ3000821σ300 1Y0821Y

+

minus+=δ

for surface hardened fillets except nitrided

YδrelTst = 044 YS + 012

for nitrided fillets

YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not

exceed the yield point and thereby alter the residual stress level See also 39b footnote 2

311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness

YRrelT differs for endurance limit and static strength

The following method may be used

For endurance limit

YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels

YδrelT = 43 ndash 326 (Ry + 1)0005

for nitrided steels

For static strength

YRrelTst = 1 for all Ry and all materials

For a fillet without any longitudinal machining trace Ry asymp Rz

312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength

The following may be used

For endurance limit

YX = 1 for mn le 5 generally

YX = 103 ndash 0006 mn for 5 lt mn lt 30

YX = 085 for mn ge 30

for not surface hard-ened steels

YX = 105 ndash 001 mn for 5 lt mn ge 25

YX = 08 for mn ge 25

for surface hardened steels

For static strength

YXst = 1 for all mn and all materials

313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength

YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength

In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength

The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses

The following simplified method for YC may be used

YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)

YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10

For endurance limit

+

+=nFFE

C m02ρt31

σconstY

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

26 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

38 Mean stress influence Factor YM The mean stress influence factor YM takes into account the influence of other working stress conditions than pure pulsa-tions (R = 0) such as eg load reversals idler gears planets and shrink-fitted gears

YM (YMst) are defined as the ratio between the endurance (or static) strength with a stress ratio R ne 0 and the endurance (or static) strength with R = 0

YM and YMst apply only to a calculation method that assesses the positive (tensile) stresses and is therefore suitable for comparison between the calculated (positive) working stress σF and the permissible stress σFP calculated with YM or YMst

For thin rings (annulus) in epicyclic gears where the com-pression fillet may be decisive special considerations apply see 316

The following method may be used within a stress ratio ndash12 lt R lt 05

381 For idlers planets and PTO with ice class

M1M1R1

1Yor Y MstM

+minus

minus=

where

R = stress ratio = min stress divided by max stress

For designs with the same force applied on both forward- and back-flank R may be assumed to ndash 12

For designs with considerably different forces on forward- and back-flank such as eg a marine propulsion wheel with a power take off pinion R may be assessed as

branchmain theoffacewidth unit per forcepto offacewidt unit per force21minus

For a power take off (PTO) with ice class see 161 c

M considers the mean stress influence on the endurance (or static) strength amplitudes

M is defined as the reduction of the endurance strength am-plitude for a certain increase of the mean stress divided by that increase of the mean stress

Following M values may be used

Endurance limit

Static strength

Case hardened 08 ndash 015 Ys 1) 07

If shot peened 04 06

Nitrided 03 03

Induction or flame hardened

04 06

Not surface hardened steel 03 05

Cast steels 04 06 1)For bevel gears use Ys=2 for determination of M

The listed M values for the endurance limit are independent of the fillet shape (Ys) except for case hardening In princi-ple there is a dependency but wide variations usually only occur for case hardening eg smooth semicircular fillets versus grinding notches

382 For gears with periodical change of rotational direction For case hardened gears with full load applied periodically in both directions such as side thrusters the same formula for YM as for idlers (with R = ndash 12) may be used together with the M values for endurance limit This simplified approach is valid when the number of changes of direction exceeds 100 and the total number of load cycles exceeds 3middot106

For gears of other materials YM will normally be higher than for a pure idler provided the number of changes of direction is below 3middot106 A linear interpolation in a diagram with loga-rithmic number of changes of direction may be used ie from YM = 09 with one change to YM (idler) for 3middot106 changes This is applicable to YM for endurance limit For static strength use YM as for idlers

For gears with occasional full load in reversed direction such as the main wheel in a reversing gear box YM = 09 may be used

383 For gears with shrinkage stresses and unidirectional load For endurance strength

FE

fitM σ

σM1

M21Y+

minus=

σFE is the endurance limit for R = 0

For static strength YMst = 1 and σfit accounted for in 39b

σfit is the shrinkage stress in the fillet (30˚ tangent) and may be found by multiplying the nominal tangential (hoop) stress with a stress concentration factor

n

Ffit m

ρ215scf minus=

384 For shrink-fitted idlers and planets When combined conditions apply such as idlers with shrink-age stresses the design factor for endurance strength is

( ) ( ) FE

fitM σ

σR1M1

M2

M1M1R1

1Yminussdot+

minus

+minus

minus=

Symbols as above but note that the stress ratio R in this par-ticular connection should disregard the influence of σfit ie R normally equal ndash 12

For static strength

M1M1R1

1YMst

+minus

minus=

The effect of σfit is accounted for in 39 b

Classification Notes- No 412 27 May 2003

DET NORSKE VERITAS

385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed

F

y

Sσ225

for not surface hardened fillets

FSHV5 for surface hardened fillets

39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles

Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN

If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used

Determination of the σ ndash N ndash curve

a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF

b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ

FstS1

FPstσ minussdotsdotsdot=

where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability

σFst

Alloyed case hardened steel 1) 2300

Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

1250

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

1050

Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)

18 HV + 800

Steel with not surface hardened fillets the smaller value of 2)

18 σB or 225 σy

1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet

c) Calculate YN as

NL gt 3middot106

YN = 1 or 001

L

6

N N103Y

sdot= ie Yn = 092 for 1010

The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2

103ltNLlt3middot106exp

L

6

N N103Y

sdot=

cycles6103forσcycles310forσlog02876exp

FP

FPst

sdot=

NL lt 103 cycles6103forσcycles310forσ

NYFP

FPst

sdot=

or simply use σFPst as mentioned in b) directly

Guidance on number of load cycles NL for various applica-tions

bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)

bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied

28 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage

YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit

The following method may be used

For endurance limit

for not surface hardened fillets

( )

024

s024

relT σ103133q21σ1012201351

Ysdotsdotminus

+sdotsdotminus+= minus

minus

δ

for all surface hardened fillets except nitrided

( )

106q21002451

Y srelT

++=δ

for nitrided fillets

( )

1347q2101421

Y srelT

++=δ

For static strength

for not surface hardened fillets1)

( )( )( )025

02

02502s

relTst σ3000821σ300 1Y0821Y

+

minus+=δ

for surface hardened fillets except nitrided

YδrelTst = 044 YS + 012

for nitrided fillets

YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not

exceed the yield point and thereby alter the residual stress level See also 39b footnote 2

311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness

YRrelT differs for endurance limit and static strength

The following method may be used

For endurance limit

YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels

YδrelT = 43 ndash 326 (Ry + 1)0005

for nitrided steels

For static strength

YRrelTst = 1 for all Ry and all materials

For a fillet without any longitudinal machining trace Ry asymp Rz

312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength

The following may be used

For endurance limit

YX = 1 for mn le 5 generally

YX = 103 ndash 0006 mn for 5 lt mn lt 30

YX = 085 for mn ge 30

for not surface hard-ened steels

YX = 105 ndash 001 mn for 5 lt mn ge 25

YX = 08 for mn ge 25

for surface hardened steels

For static strength

YXst = 1 for all mn and all materials

313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength

YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength

In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength

The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses

The following simplified method for YC may be used

YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)

YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10

For endurance limit

+

+=nFFE

C m02ρt31

σconstY

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

Classification Notes- No 412 27 May 2003

DET NORSKE VERITAS

385 Additional requirements for peak loads The total stress range (σmax ndash σmin) in a tooth root fillet is not to exceed

F

y

Sσ225

for not surface hardened fillets

FSHV5 for surface hardened fillets

39 Life Factor YN The life factor YN takes into account that in the case of limited life (number of cycles) a higher tooth root stress can be permitted and that lower stresses may apply for very high number of cycles

Decisive for the strength at limited life is the σ ndash N ndash curve of the respective material for given hardening module fillet radius roughness in the tooth root etc Ie the factors YδrelT YRelT YX and YM have an influence on YN

If no σ ndash N ndash curve for the actual material and hardening etc is available the following method may be used

Determination of the σ ndash N ndash curve

a) Calculate the permissible stress σFP for the beginning of the endurance limit (3middot106 cycles) including the influ-ence of all relevant factors as SF YδrelT YRelT YX YM and YC ie σFP = σFE middotYM middotYδrelT middotYRelT middotYX middotYC SF

b) Calculate the permissible laquostaticraquo stress (le 103 load cy-cles) including the influence of all relevant factors as SFst YδrelTst YMst and YCst ( )fitσCstYδrelTstYMstYFstσ

FstS1

FPstσ minussdotsdotsdot=

where σFst is the local tooth root stress which the material can resist without cracking (surface hardened materials) or unacceptable deformation (not surface hardened materials) with 99 survival probability

σFst

Alloyed case hardened steel 1) 2300

Nitriding steel quenched tempered and gas nitrided (surface hardness 700 ndash 800 HV)

1250

Alloyed quenched and tempered steel bath or gas nitrided (surface hardness 500 ndash 700 HV)

1050

Alloyed quenched and tempered steel flame or induction hardened (fillet surface hardness 500 ndash 650 HV)

18 HV + 800

Steel with not surface hardened fillets the smaller value of 2)

18 σB or 225 σy

1) This is valid for a fillet surface hardness of 58 ndash 63 HRC In case of lower fillet surface hardness than 58 HRC σFst is to be reduced with 30(58 ndash HRC) where HRC is the actual hardness Shot peening or grinding notches are not considered to have any significant influence on σFst 2) Actual stresses exceeding the yield point (σy or σ02) will alter the residual stresses locally in the ldquotensionrdquo fillet re-spectively ldquocompressionrdquo fillet This is only to be utilised for gears that are not later loaded with a high number of cycles at lower loads that could cause fatigue in the ldquocompressionrdquo fillet

c) Calculate YN as

NL gt 3middot106

YN = 1 or 001

L

6

N N103Y

sdot= ie Yn = 092 for 1010

The YN = 1 from 3middot106 on may only be used when special material cleanness applies see rules Pt4 Ch2

103ltNLlt3middot106exp

L

6

N N103Y

sdot=

cycles6103forσcycles310forσlog02876exp

FP

FPst

sdot=

NL lt 103 cycles6103forσcycles310forσ

NYFP

FPst

sdot=

or simply use σFPst as mentioned in b) directly

Guidance on number of load cycles NL for various applica-tions

bull For propulsion purpose normally NL = 1010 at full load (yachts etc may have lower values)

bull For auxiliary gears driving generators that normally operate with 70-90 of rated power NL = 108 with rated power may be applied

28 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage

YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit

The following method may be used

For endurance limit

for not surface hardened fillets

( )

024

s024

relT σ103133q21σ1012201351

Ysdotsdotminus

+sdotsdotminus+= minus

minus

δ

for all surface hardened fillets except nitrided

( )

106q21002451

Y srelT

++=δ

for nitrided fillets

( )

1347q2101421

Y srelT

++=δ

For static strength

for not surface hardened fillets1)

( )( )( )025

02

02502s

relTst σ3000821σ300 1Y0821Y

+

minus+=δ

for surface hardened fillets except nitrided

YδrelTst = 044 YS + 012

for nitrided fillets

YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not

exceed the yield point and thereby alter the residual stress level See also 39b footnote 2

311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness

YRrelT differs for endurance limit and static strength

The following method may be used

For endurance limit

YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels

YδrelT = 43 ndash 326 (Ry + 1)0005

for nitrided steels

For static strength

YRrelTst = 1 for all Ry and all materials

For a fillet without any longitudinal machining trace Ry asymp Rz

312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength

The following may be used

For endurance limit

YX = 1 for mn le 5 generally

YX = 103 ndash 0006 mn for 5 lt mn lt 30

YX = 085 for mn ge 30

for not surface hard-ened steels

YX = 105 ndash 001 mn for 5 lt mn ge 25

YX = 08 for mn ge 25

for surface hardened steels

For static strength

YXst = 1 for all mn and all materials

313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength

YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength

In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength

The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses

The following simplified method for YC may be used

YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)

YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10

For endurance limit

+

+=nFFE

C m02ρt31

σconstY

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

28 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

310 Relative Notch Sensitivity Factor YδrelT The dynamic (respectively static) relative notch sensitivity factor YδrelT (YδrelTst) indicate to which extent the theoreti-cally concentrated stress lies above the endurance limits (re-spectively static strengths) in the case of fatigue (respectively overload) breakage

YδrelT is a function of the material and the relative stress gra-dient It differs for static strength and endurance limit

The following method may be used

For endurance limit

for not surface hardened fillets

( )

024

s024

relT σ103133q21σ1012201351

Ysdotsdotminus

+sdotsdotminus+= minus

minus

δ

for all surface hardened fillets except nitrided

( )

106q21002451

Y srelT

++=δ

for nitrided fillets

( )

1347q2101421

Y srelT

++=δ

For static strength

for not surface hardened fillets1)

( )( )( )025

02

02502s

relTst σ3000821σ300 1Y0821Y

+

minus+=δ

for surface hardened fillets except nitrided

YδrelTst = 044 YS + 012

for nitrided fillets

YδrelTst = 06 + 02 YS 1) These values are only valid if the local stresses do not

exceed the yield point and thereby alter the residual stress level See also 39b footnote 2

311 Relative Surface Condition Factor YRrelT The relative surface condition factor YRrelT takes into ac-count the dependence of the tooth root strength on the sur-face condition in the tooth root fillet mainly the dependence on the peak to valley surface roughness

YRrelT differs for endurance limit and static strength

The following method may be used

For endurance limit

YRrelT = 1675 ndash 053 (Ry + 1)01 for surface hardened steels and alloyed quenched and tempered steels except nitrided

YRrelT = 53 ndash 42 (Ry + 1)001 for carbon steels

YδrelT = 43 ndash 326 (Ry + 1)0005

for nitrided steels

For static strength

YRrelTst = 1 for all Ry and all materials

For a fillet without any longitudinal machining trace Ry asymp Rz

312 Size Factor YX The size factor YX takes into account the decrease of the strength with increasing size YX differs for endurance limit and static strength

The following may be used

For endurance limit

YX = 1 for mn le 5 generally

YX = 103 ndash 0006 mn for 5 lt mn lt 30

YX = 085 for mn ge 30

for not surface hard-ened steels

YX = 105 ndash 001 mn for 5 lt mn ge 25

YX = 08 for mn ge 25

for surface hardened steels

For static strength

YXst = 1 for all mn and all materials

313 Case Depth Factor YC The case depth factor YC takes into account the influence of hardening depth on tooth root strength

YC applies only to surface hardened tooth roots and is dif-ferent for endurance limit and static strength

In case of insufficient hardening depth fatigue cracks can develop in the transition zone between the hardened layer and the core For static strength yielding shall not occur in the transition zone as this would alter the surface residual stresses and therewith also the fatigue strength

The major parameters are case depth stress gradient permis-sible surface respectively subsurface stresses and subsurface residual stresses

The following simplified method for YC may be used

YC consists of a ratio between permissible subsurface stress (incl influence of expected residual stresses) and permissible surface stress This ratio is multiplied with a bracket con-taining the influence of case depth and stress gradient (The empirical numbers in the bracket are based on a high number of teeth and are somewhat on the safe side for low number of teeth)

YC and YCst may be calculated as given below but calculated values above 10 are to be put equal 10

For endurance limit

+

+=nFFE

C m02ρt31

σconstY

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

Classification Notes- No 412 29 May 2003

DET NORSKE VERITAS

For static strength

+

+=nFFst

Cst m02ρt31

σconstY

where const and t are connected as

Hardening process

t = endurance limit const =

static strength const =

t550 640 1900

t400 500 1200 Case hard-ening

t300 380 800

Nitriding t400 500 1200

Induction- or flame hardening

tHVmin 11 HVmin 25 HVmin

For symbols see 213

In addition to these requirements to minimum case depths for endurance limit some upper limitations apply to case hard-ened gears

The max depth to 550 HV should not exceed

1) 13 of the top land thickness san unless adequate tip relief is applied (see 110)

2) 025 mn If this is exceeded the following applies addi-tionally in connection with endurance limit

minusminus= 025

mt

1Yn

max550C

314 Thin rim factor YB Where the rim thickness is not sufficient to provide full sup-port for the tooth root the location of a bending failure may be through the gear rim rather than from root fillet to root fillet

YB is not a factor used to convert calculated root stresses at the 30deg tangent to actual stresses in a thin rim tension fillet Actually the compression fillet can be more susceptible to fatigue

YB is a simplified empirical factor used to de-rate thin rim gears (external as well as internal) when no detailed calcula-tion of stresses in both tension and compression fillets are available

Figure 34 Examples on thin rims

YB is applicable in the range 175 lt sRmn lt 35

YB = 115 middot ln (8324 middot mnsR)

(for sRmn ge 35 YB = 1)

(for sRmn le 175 use 315)

σF as calculated in 321 is to be multiplied with YB when sRmn lt 35 Thus YB is used for both high and low cycle fatigue

Note This method is considered to be on the safe side for external gear rims However for internal gear rims without any flange or web stiffeners the method may not be on the safe side and it is advised to check with the method in 315316

315 Stresses in Thin Rims For rim thickness sR lt 35 mn the safety against rim cracking has to be checked

The following method may be used

3151 General The stresses in the standardised 30ordm tangent section tension side are slightly reduced due to decreasing stress correction factor with decreasing relative rim thickness sRmn On the other hand during the complete stress cycle of that fillet a certain amount of compression stresses are also introduced The complete stress range remains approximately constant Therefore the standardised calculation of stresses at the 30ordm tangent may be retained for thin rims as one of the necessary criteria

The maximum stress range for thin rims usually occurs at the 60ordm ndash 80ordm tangents both for laquotensionraquo and laquocompressionraquo side The following method assumes the 75ordm tangent to be the decisive Therefore in addition to the am criterion ap-plied at the 30ordm tangent it is necessary to evaluate the max and min stresses at the 75ordm tangent for both laquotensionraquo (loaded flank) fillet and laquocompressionraquo (back-flank) fillet For this purpose the whole stress cycle of each fillet should be considered but usually the following simplification is justified

Figure 35 Nomenclature of fillets

Index laquoTraquo means laquotensileraquo fillet laquoCraquo means laquocompressionraquo fillet

σFTmin and σFCmax are determined on basis of the nominal rim stresses times the stress concentration factor Y75

σFCmin and σFTmax are determined on basis of superposition of nominal rim stresses times Y75 plus the tooth bending stresses at 75ordm tangent

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

30 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

3152 Stress concentration factors at the 75ordm tangents The nominal rim stress consists of bending stresses due to local bending moments tangential stresses due to the tan-gential force Ft and radial shear stresses due to Fr

The major influence is given by the bending stresses The influence of the tangential stresses is minor and even though its stress concentration factor is slightly higher than for bending it is considered to be safe enough when the sum of these nominal stresses are combined with the stress concen-tration factor for bending The influence of the radial shear stress is neglected

The stress concentration factor relating nominal rim stresses to local fillet stresses at the 75ordm tangent may be calculated as

75n

R75

n

R

75

ρmsρ185

ms3

Y+

sdot=

where ρ75 is the root radius at the 75ordm tangent ref to mn Usually ρ75 is closed to the tool radius ρao and

ρ75 = ρao

is a safe approximation compensating for the am simplifi-cations to the laquounsaferaquo side

The tooth root stresses of the loaded tooth are decreasing with decreasing relative rim thickness approximately with the empirical factor

15

R

ncorr s

m311Y

minus=

3153 Nominal rim stresses The bending moment applied to the rim consists of a part of the tooth tilt moment Ft (hF + 05 sR) and the bending caused by the radial force Fr

The sectional modulus (first moment of area) which is used for determination of the nominal bending stresses is not nec-essarily the same for the 2 am bending moments If flanges webs etc outside the toothed section contribute to stiffening the rim against various deflections the influence of these stiffeners should be considered Eg an end flange will have an almost negligible influence on the effective sectional modulus for the stresses due to tooth tilt as the deflection caused by the tilt moment is rather small and would not much involve the flange On the other hand the radial forces as for instance from the meshes in an annulus would cause considerable radial deflections that the flange might restrict to a substantial amount When considering the stiff-ening of such flanges or webs on basis of simplified models it is advised to use an effective rim thickness sR = sR + 02 mn for the first moment of area of the rim (toothed part) cross section

For a high number of rim teeth it may be assumed that the rim bending moments in the fillets adjacent to the loaded tooth are of the same magnitude as right under the applied

force This assumption is reasonable for an annulus but rather much on the safe side for a hollow pinion

The influence of Ft on the nominal tangential stress is simpli-fied by half of it for compressive stresses (σ1) and the other half for tensile stresses (σ2) Applying these assumptions the nominal rim stresses adjacent to the loaded tooth are

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt1 minus

ϑminus

+minus=σ

( ) ( )A2

FW

f RFW

s05hF05 t

R

r

T

RFt2 +

ϑminus

+=σ

where σ1 σ2 see Fig 35

A = minimum area of cross section (usually bR sR)

WT = the sectional modulus of rim with respect to tooth tilt moment (usually 6sb 2

rR )

WR = the sectional modulus of the rim including the influence of stiffeners as flanges webs etc (WR ge WT)

bR = the width of the rim

R = the radius of the neutral axis in the rim ie from wheel centre to midpoint of rim

( )ϑf = a function for bending moment distribution around the rim For a rim (pinion) with one mesh only the ( )ϑf at the position of load application is 024 For an annulus with 3 or more meshes ( )ϑf at the position of each load application is approx 3 planets ( )ϑf = 019 4 planets ( )ϑf = 014 5 planets ( )ϑf = 011 6 planets ( )ϑf = 009

It must be checked if the max (tensile) stress in the compres-sion fillet really occurs when the fillet is adjacent to the loaded tooth In principle the stress variation through a complete rotation should be considered and the max value used The max value is usually never less than 0 For an annulus eg the tilt moment is zero in the mid position be-tween the planet meshes whilst the bending moment due to the radial forces is half of that at the mesh but with opposite sign

If these formulae are applied to idler gears as eg planets the influence of nominal tangential stresses must be cor-rected by deleting Ft(2 A) for σ1 and using FtA for σ2 Further the influence of Fr on the nominal bending stresses is usually negligible due to the planet bearing support

3154 Root fillet stresses Determination of min and max stresses in the laquotensionraquo fillet

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

Classification Notes- No 412 31 May 2003

DET NORSKE VERITAS

Minimum stress

751FTmin YσK σ =

Maximum stress

corrF752 Yσ03Yσ KσFTmax +=

where

03 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the tension stresses at the 75ordm tangent which add to the rim related stresses (03 also takes into account that full superposition of nominal stresses times stress concentration factors from both laquosides of the corner filletraquo would result in too high stresses)

αβγ sdotsdotsdotsdot= FFvA KKKKKK

Determination of min and max stresses in the laquocompres-sionraquo fillet

Minimum stress

corrF751FCmin Yσ036Yσ Kσ minus=

Maximum stress

752FCmax Yσ Kσ =

where

036 is an empirical factor relating the tension stresses (σF) at the 30ordm tangent to the part of the compression stresses at the 75ordm tangent which add to the rim related stresses

For gears with reversed loads as idler gears and planets there is no distinct laquotensionraquo or laquocompressionraquo fillet The mini-mum stress σFmin is the minimum of σFTmin and σFCmin (usually the latter is decisive) The maximum stress σFTmax is the maximum of σFTmax and σFCmax (usually the former is deci-sive)

316 Permissible Stresses in Thin Rims

3161 General The safety against fatigue fracture respectively overload fracture is to be at least at the same level as for solid gears The laquoordinaryraquo criteria at the 30ordm tangent apply as given in 31 through 313

Additionally the following criteria at the 75ordm tangent may apply

3162 For gt3middot106 cycles The permissible stresses for the laquotensionraquo fillets and for the laquocompressionraquo fillets are determined by means of a relevant fatigue diagram

If the actual tooth root stress (tensile or compressive) ex-ceeds the yield strength to the material the induced residual stresses are to be taken into account

For determination of permissible stresses the following is defined

R = stress ratio ie FTmax

FTminσσ respectively

FCmax

FCmin

σσ

∆σ = stress range ie σFTmax ndash σFTmin resp σFCmax ndash σFCmin

(For idler gears and planets Fmax

FminσσR =

and ∆σ = σFmax minus σFmin)

The permissible stress range ∆σρ for the laquotensionraquo respec-tively laquocompressionraquo fillets can be calculated as

For R gt minus1 FPp σ

R1R1031

13∆σ

minus+

+=

For minus infin lt R lt minus1 FPp σ

R1R10151

13∆σ

minus+

+=

where

σFP = see 32 determined for unidirectional stresses (YM = 1)

If the yield strength σy is exceeded in either tension or com-pression residual stresses are induced This may be consid-ered by correcting the stress ratio R for the respective fillets (tension or compression)

Eg if yFCmin σσ gt (ie exceeded in compression) the

difference yFCmin σσ∆ minus= affects the stress ratio as

∆σ

σ∆σ∆σR

FCmax

y

FCmax

FCmin

+

minus=

++

=

Similarly the stress ratio in the tension fillet may require cor-rection

If the yield strength is exceeded in tension σFTmax gt σy the difference ∆ = σFTmax ndash σy affects the stress ratio as

y

FTmin

FTmax

FTmin

σ∆σ

∆σ∆σR minus=

minusminus

=

Checking for possible exceeding of the yield strength has to be made with the highest torque and with the KAP if this ex-ceeds KA

3163 Forle 103 cycles The permissible stress range ∆σp is not to exceed

For R gt minus1 FPstpst σ

R1R1051

15∆σ

minus+

+=

For ndash infin lt R lt minus1 FPstpst σ

R1R10251

15∆σ

minus+

+=

For all values of R ∆σpst is limited by

not surface hardened F

y

Sσ225

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

32 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

surface hardened CF

YSHV5

Definition of ∆σ and R see 3162 with particular attention to possible correction of R if the yield strength is exceeded

σFPst see 32 determined for unidirectional stresses (YM = 1) and lt 103 cycles

3164 For 103 lt cycles lt 3106 ∆σp is to be determined by linear interpolation a log-log dia-gram

∆σp at NL load cycles is

6L 103p

exp

L

6

N p ∆σN103∆σ sdot

sdot=

6

3

103p

10p

∆σ

∆σlog28760exp

sdot

=

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

Classification Notes- No 412 33 May 2003

DET NORSKE VERITAS

4 Calculation of Scuffing Load Capacity

41 Introduction High surface temperatures due to high loads and sliding ve-locities can cause lubricant films to break down This sei-zure or welding together of areas of tooth surface is termed scuffing

In contrast to pitting and fatigue breakage which show a dis-tinct incubation period a single short overloading can lead to a scuffing failure In the ISO-TR13989 two criteria are mentioned The method used in this Classification Note is based on the principles of the flash temperature criterion

Note Bulk temperature in excess of 120ordmC for long periods may have an adverse effect on the surface durability see 211

42 General Criteria In no point along the path of contact the local contact tem-perature may exceed the permissible temperature ie

oilS

oilSB S

ϑ+ϑminusϑ

leϑ

50SB minusϑleϑ

where

Bϑ = max contact temperature along the path of contact

maxflaMBB ϑ+ϑ=ϑ

MBϑ = bulk temperature see 434

maxflaϑ = max flash temperature along the path of con-tact see 44

Sϑ = scuffing temperature see below

oilϑ = oil temperature before it reaches the mesh (max applicable for the actual load case to be used ie normally alarm temperature except for ice classes where a max expected tem-perature applies)

SS = required safety factor according to the Rules

The scuffing temperature Sϑ may be calculated as

L2

wrelT

002

40S XFZGX

ν100112085780

sdot

sdot++=ϑ

where

XwrelT = relative welding factor

XwrelT

Through hardened steel 10

Phosphated steel 125

Copper-plated steel 150

Nitrided steel 150

Less than 10 retained austenite

115

10 ndash 20 retained austenite 10

Casehardened steel

20 ndash 30 retained austenite 085

Austenitic steel 045

FZG = load stage according to FZG-Test A8390 (Note This is the load stage where scuffing oc-curs However due to scatter in test results cal-culations are to be made with one load stage less than the specification)

XL = lubricant factor

= 10 for mineral oils

= 08 for polyalfaolefins

= 07 for non-water-soluble polyglycols

= 06 for water-soluble polyglycols

= 15 for traction fluids

= 13 for phosphate esters

ν40 = kinematic oil viscosity at 40˚C (mm2s)

Application of other test methods such as the Ryder the FZG-Ryder R46574 and the FZG L-42 Test 141195110 may be specially considered

For high speed gears with very short time of contact Sϑ may be increased as follows provided use of EP-oils

Addition to the calculated scuffing temperature Sϑ

If micros 18tc ge no addition

If micros 18tc lt add ( )cwrelT t18X18 minussdotsdot

where

ct = contact time ( micros ) which is the time needed to cross the Hertzian contact width

( ) [ ]microsu1cosβn

uσ340tb1

Hc +sdotsdot

sdot=

σH as calculated in 221

For bevel gears use vu in stead of u

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

34 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

43 Influence Factors

431 Coefficient of friction The following coefficient of friction may apply

L025a

005oil

02

redC ΣC

Bt X R ηρv

w0048micro minus

=

where

wBt = specific tooth load (Nmm)

vΣC = sum of tangential velocities at pitch point At pitch line velocities gt 50 ms the limiting value of vΣC at v = 50 ms is to be used

ρredC = relative radius of curvature (transversal plane) at the pitch point

Cylindrical gears

HαHβvγAbt

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

twΣC αsin v2v =

bCredC β cos ρρ =

Bevel gears

HαHβvγAtmb

Bt KKKKKb

Fw sdotsdotsdotsdotsdot= (see 1)

vtmtΣC αsin v2v =

bmvCredC β cos ρρ =

ηoil = dynamic viscosity (mPa s) at oilϑ calculated as

1000

ρ νη oiloil =

where ρ in kgm3 approximated as

( ) 70 15oil15 minusϑminusρ=ρ and νoil is kinematic viscosity at oilϑ and may be calculated by means of the following equation

( ) ( )++=+ 08νloglog08νloglog 100oil ( )

sdotminus

ϑ+minus313log373log

273log373log oil

( ) ( )( )08νloglog08νloglog 10040 +minus+

Ra = composite arithmetic mean roughness (micron) of pinion an wheel calculated as ( )2a1aa RR 05R +=This is defined as the roughness on the new flanks ie as manufactured

XL = see 42

432 Effective tip relief Ceff Ceff is the effective tip relief that amount of tip relief which compensates for the elastic deformation of the gear mesh ie

zero load at the tooth tip It is assumed (simplified) to be equal for both pinion and wheel

Cylindrical gears

for helical

γ

γ=cb

KKFC Abt

eff (see 1)

For spur

cbKKF

C Abteff

γ= (see 1)

Alternatively for spur and helical gears the non-linear ap-proach in 111 may be used (taking Ceff=δ)

Bevel gears

bcKFC Ambt

effγ

= (see 1)

where

γ

α

ε2ε44c

+sdot

=γ

433 Tip relief and extension Cylindrical gears

The extension of the tip relief is not to result in an effective contact ratio 1ltεα when the gear is unloaded (exceptions to this may only apply for applications where the gear is not to run at light loads) This means that the unrelieved part of the path of contact is to be minimum pbt It is further assumed that this unrelieved part is placed centrally on the path of contact

If root relief applies it has to be calculated as equivalent tip relief Ie pinion root relief (at mesh position A) is added to Ca2 and wheel root relief (at mesh position E) is added to Ca1 If no design tip relief or root relief on the mating gear is specified (ie if Ca1+Croot2 = 0 and visa versa) use the run-ning in amount see 112

Bevel gears

Bevel gears are to have heightwise crowning ie no distinct relievedunrelieved area This may be treated as tip and root relief For calculation purposes the root relief is combined with the tip relief of the mating member into an equivalent tip relief If no resulting tip relieves are specified the equivalent tip relives may be calculated as an approxima-tion based on the tool crowning Ca tool (per mille of tool module m0) as follows

2root1aeq1a CCC +=

1root2aeq2a CCC +=

where

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

Classification Notes- No 412 35 May 2003

DET NORSKE VERITAS

2

0

n0101atoolal m

)m2(mA1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb1

2vb1

21vavt1n1 αtan dddαsin 05x1mA

2

0

n020atool22a m

)mm(2A1middotmCC

minus+minussdot=

( )

sdotminusminussdotsdotminus+= vtvb2

2vb2

2va2vt2n2 αtan dddαsin 05x1mA

2

0

20atool1root1 m

A1mCC

minussdotsdot=

2

0

10atool2root2 m

A1mCC

minussdotsdot=

If the pressure angles of the cutter blades are modified (and verified) to balance the tip relieves the following may be assumed

( )eq2aeq1aeq2aeq1a C andC calculated of sum50CC sdot==

Throughout the following Ca1 and Ca2 mean the equivalent tip relieves Ca1eq and Ca2eq

434 Bulk temperature The bulk temperature may be calculated as

flaaveragempsoilMB X X 50 ϑ+ϑ=ϑ

where

Xs = lubrication factor

= 12 for spray lubrication

= 10 for dip lubrication (provided both pinion and wheel are dip lubricated and tip speed lt 5 ms)

= 10 for spray lubrication with additional cooling spray (spray on both pinion and wheel or spray on pinion and dip lubrication of wheel)

= 02 for meshes fully submerged in oil

Xmp = contact factor ( )pmp nX += 150

np = number of mesh contact on the pinion (for small gear ratii the number of wheel meshes should be used if higher)

flaaverageϑ

= average of the integrated flash temperature (see 44) along the path of contact

( )

AE

E

Ayyyfla

flaaverage

d

ΓminusΓ

ΓΓϑ=ϑint =

For high speed gears (v gt 50 ms) it may be necessary to assess the bulk temperature on the basis of a thermal rating of the entire gear transmission

44 The Flash Temperature flaϑ

441 Basic formula The local flash temperature flaϑ may be calculated as

( ) 41redy

y2ly

211

43Btcorrfla

unXwX3250

y ρ

ρminusρ

micro=ϑ Γ

(For bevel gears replace u with uv)

and is to be calculated stepwise along the path of contact from A to E

where

micro = coefficient of friction see 431

Xcorr = correction factor taking empirically into account the increased scuffing risk in the beginning of the approach path due to mesh starting without any previously built up oil film and possible shuffling away oil before meshing if insufficient tip relief

( )

3

AD

yaeffcorr 50

CC1X

ΓminusΓε

Γminus+=

α

Ca = tip relief of driven member Xcorr is only applicable in the approach path and if Ca lt Ceff otherwise 10

wBt = unit load see 431

yXΓ = load sharing factor see 443

n1 = pinion rpm

Γy ρly etc see 442

442 Geometrical relations The various radii of flank curvature (transversal plane) are

ρ1y = pinion flank radius at mesh point y

ρ2y = wheel flank radius at mesh point y

ρredy = equivalent radius of curvature at mesh point y

y2y1

y2y1redy

ρ+ρ

ρρ=ρ

Cylindrical gears

twy

1y αsin au1Γ1

ρ+

+=

twy

2y αsin au1Γu

ρ+

minus=

Note that for internal gears a and u are negative

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

36 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Bevel gears

vtvv

y1y αsin a

u1Γ1

ρ+

+=

vtvv

yv2y αsin a

u1Γu

ρ+

minus=

Γ is the parameter on the path of contact and y is any point between A end E

At the respective ends Γ has the following values

Root piniontip wheel

Cylindrical gears

( )

minus

minusminus= 1

αtan 1dd

zzΓ

tw

2b2a2

1

2A

Bevel gears

( )

minus

minusminus= 1

αtan 1dd

uΓvt

2vb2va2

vA

Tip pinionroot wheel

Cylindrical gears

( )

1αtan

1ddΓ

tw

2b1a1

E minusminus

=

Bevel gears

( )

1αtan

1ddΓ

vt

2vb1va1

E minusminus

=

At inner point of single pair contact

Cylindrical gears

tw1

EB α tan zπ2ΓΓ minus=

Bevel gears

vtv1

EB α tan zπ2ΓΓ minus=

At outer point of single pair contact

Cylindrical gears

tw1AD α tan z

π2ΓΓ +=

Bevel gears

vt v1

AD αtan z π2ΓΓ +=

At pitch point ΓC = 0

The points F and G (only applicable to cylindrical gears) limiting the extension of tip relief (so as to maintain a mini-mum contact ratio of unity for unloaded gears) are at

2

BAF

Γ+Γ=Γ

2

EDG

Γ+Γ=Γ

443 Load sharing factor XΓ The load sharing factor XΓ accounts for the load sharing be-tween the various pairs of teeth in mesh along the path of contact

XΓ is to be calculated stepwise from A to E using the pa-rameter Γy

4431 Cylindrical gears with β = 0 and no tip relief

Figure 41

ByAAB

Ay for31

31X

yΓltΓleΓ

ΓminusΓ

ΓminusΓ+=Γ

DyB for1Xy

ΓltΓleΓ=Γ

EyDDE

yE for31

31X

yΓleΓltΓ

ΓminusΓ

ΓminusΓ+=Γ

4432 Cylindrical gears with β = 0 and tip relief Tip relief on the pinion reduces XΓ in the range G ndash E and increases correspondingly XΓ in the range F ndash B

Tip relief on the wheel reduces XΓ in the range A ndash F and increases correspondingly XΓ in the range D ndash G

Following remains generally

DyB for1Xy

ΓleΓleΓ=Γ

21XXGF== ΓΓ

In the following it must be distinguished between Ca lt Ceff respectively Ca gt Ceff This is shown by an example below where Ca1 lt Ceff and Ca2 gt Ceff

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

Classification Notes- No 412 37 May 2003

DET NORSKE VERITAS

Figure 42

Note When Ca gt Ceff the path of contact is shortened by A ndash A respectively E ndash E The single pair contact path is extended into B respectively D If this shift is significant it is necessary to consider the negative effect on surface durability (B) and bending stresses (B and D)

Range A - F

For effa2 CC le

minus=Γ

eff

2a

CC1

31X

A

FyAeff

2a

AF

Ay for C3

C61XX

AyΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+= ΓΓ

For effa2 CC ge

AyAfor0Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

AFAAminus

minusΓminusΓ+Γ=Γ

FyAeff

2a

AF

Ay

eff

2a for 21

CC

CC1X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+minus=Γ

Range F - B

For effa1 CC le

ByFeff

1a

FB

Fy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓ+=Γ

For effa1 CC ge

ByFeff

1a

FB

Fy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓ+=Γ

ByB for1Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

FBFB

minus

ΓminusΓ+Γ=Γ

Range D ndash G

For effa2 CC le

GyDeff

2a

DG

Dy

eff

2a for C 3C

61

C 3C

32X

yΓleΓleΓ

+

ΓminusΓ

ΓminusΓminus+=Γ F

or effa2 CC ge

DyD for1Xy

ΓleΓleΓ=Γ

with ( )21

CC

1CC

eff

2a

eff

2a

DGDDminus

minusΓminusΓ+Γ=Γ

GyDeff

2a

DG

Dy

eff

2a for21

CC

CCX ΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

Range G - E

For effa1 CC le

minus=Γ

eff

1a

CC1

31X

E

EyGeff

1a

GE

Gy forC3

C61

21X

yΓleΓleΓ

sdot

+ΓminusΓ

ΓminusΓminus=Γ

For effa1 CC gt

EyGeff

1a

GE

Gy for21

CC

21X

yΓleΓleΓ

minus

ΓminusΓ

ΓminusΓminus=Γ

EyE for0Xy

ΓleΓleΓ=Γ

with 1

CC2

eff

1a

GEGE

minus

ΓminusΓ+Γ=Γ

4433 Gears with β gt 0 buttressing Due to oblique contact lines over the flanks a certain but-tressing may occur near A and E

This applies to both cylindrical and bevel gears with tip relief lt Ceff The buttressing Xbutt is simplified as a linear function within the ranges A ndash H respectively I ndash E

Figure 43

1εwhen13X βbutt EAge=

1εwhenε031X ββbutt EAlt+=

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

38 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Cylindrical gears

bIEAH βsin 02ΓΓΓΓ =minus=minus

Bevel gears

bmIEAH βsin 02ΓΓΓΓ =minus=minus

4434 Cylindrical gears with 2εγ le and no tip relief

yXΓ is obtained by multiplication of

yXΓ in 4431 with

Xbutt in 4433

4435 Gears with 2εγ gt and no tip relief

Applicable to both cylindrical and bevel gears

IyH for1Xy

ΓleΓleΓε

=α

Γ

IyHybutt and forX1Xy

ΓgtΓΓltΓε

=α

Γ

Figure 44

4436 Cylindrical gears with 2εγ le and tip relief

yXΓ is obtained by multiplication of

yXΓ in 4432 with Xbutt

in 4433

4437 Cylindrical gears with 2εγ gt and tip relief

Tip relief on the pinion (respectively wheel) reduces XГ in the range G ndash E (respectively A ndash F) and increases XГ in the range F ndash G

yXΓ is obtained by multiplication of

yXΓ as described below

with Xbutt in 4433

In the XГ example below the influence of tip relief is shown (without the influence of Xbutt) by means of

eff2aeff1a CC and CC ltgt

Tip relief gt Ceff causes new end points A respectively E of the path of contact

Figure 45

Range A ndash F

( ) ( )( ) eff

a2αa1α

AF

Ay

eff

2aeff

C 12C 13εC 1ε

CCCX

y +εε++minus

ΓminusΓ

ΓminusΓ+

εminus

=ααα

Γ

eff2aFyA CC if for leΓleΓleΓ

eff2aFyA CifCfor and geΓleΓleΓ

eff2aAyA CC iffor0Xy

gtΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) a2αa1α

αeffa2AFAA C 1ε 3C 1ε

1ε 2 CC with++minus+minus

ΓminusΓ+Γ=Γ

Range F ndash G

( )( )( ) GyF

eff

2a1a forC 1 2CC 11X

yΓleΓleΓ

+εε+minusε

+ε

=αα

α

αΓ

Range G ndash E

( ) ( )( ) eff

2a1a

GE

Gy

C 1 2C 1C 1 3XX

GFy +εεminusε++ε

ΓminusΓ

ΓminusΓminus=

αα

ααΓΓ minus

effa1EyG CC if for leΓleΓleΓ

effa1EyG CC if Γfor and geΓleΓle

eff1aEyE CC if for0Xy

geΓleΓleΓ=Γ

( ) ( ) ( )( ) ( ) 2a1a

eff1aGEEE C 1C 1 3

1 2 CC withminusε++ε+εminus

ΓminusΓminusΓ=Γαα

α

4438 Bevel gears with εγ more than approx 18 and heightwise crowning For Ca1 = Ca2 = Ceff the following applies

Figure 46

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

Classification Notes- No 412 39 May 2003

DET NORSKE VERITAS

( )AEM 50 Γ+Γ=Γ

( )( )2AD

3

2My651X

y ΓminusΓε

ΓminusΓminus

ε=

ααΓ

For tip relief lt Ceff yXΓ is found by linear interpolation be-

tween ( )effay CCX =Γ and ( )0CayX =Γ as in 4435

The interpolation is to be made stepwise from A to M with the influence of Ca2 and from M to E with the influence of Ca1 (For 2a1a CC ne there is a discontinuity at M)

Eg with Ca1 = 04 Ceff and Ca2 = 055 Ceff then

Range A ndash M

)CC()0C( eff2ay2ayyX 550X 450X =Γ=ΓΓ +=

Range M ndash E

)CC()0C( eff1ay1ayyX 40X 06X =Γ=ΓΓ +=

For tip relief gt Ceff the new end points A and E are found as

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

2aADAA

( )

minusΓminusΓ

ε+Γ=Γ α 1

CC

6 eff

1aADEE

Range A ndash A

0Xy=Γ

Range A ndash M

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

MA

2My

eff

2a1

CC4

351Xy

Range M ndash E

( )( )

ΓminusΓ

ΓminusΓminus

minusε=

αΓ 2

ME

2My

eff

1a1

CC4

351Xy

Range E ndash E

0Xy=Γ

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

40 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix A Fatigue Damage Accumulation

The Palmgren-Miner cumulative damage calculation princi-ple is used The procedure may be applied as follows

A1 Stress Spectrum From the individual torque classes the torques (Ti) at the peak values of class intervals and the associated number of cycles (NLi) for both pinion and wheel are to be listed from the highest to the lowest torque

(In case of a cyclic torque variation within the torque classes it is advised to use the peak torque If the cyclic variation is such that the same teeth will repeatedly suffer the peak torque this is a must)

The stress spectra for tooth roots and flanks (σFi σHi) with all relevant factors (except KA) are to be calculated on the basis of the torque spectrum The load dependent K-factors are to be determined for each torque class

A2 σminusN-curve The stress versus load cycle curves for tooth roots and flanks (both pinion and wheel) are to be drawn on the basis of per-missible stresses (ie including the demanded minimum safety factors) as determined in 2 respectively 3 If different safety levels for high cycle fatigue and low cycle fatigue are desired this may be expressed by different demand safety factors applied at the endurance limit respectively at static strength

A3 Damage accumulation The individual damage ratio Di at ith stress level is defined as

Fi

Lii N

ND =

where

NLi = The number of applied cycles at ith stress

NFi = The number of cycles to failure at ith stress

Basically stresses σi below the permissible stress level for infinite life (if a constant ZN or YN is accepted) do not con-tribute to the damage sum However calculating the actual safety factor Sact as described below all the σi for which the product S σi is bigger than or equal to the permissible stress level for infinite life contribute to the damage sum and thus to the determination of Sact The final value of S is decisive

(NFi can be found mathematically by putting the permissible stress σpi equal the actual stress σi thereby finding the actual life factor This life factor can be solved with regard to load cycles ie NFi)

The damage sum ΣDi is not to exceed unity

If ΣDi ne 1 the safety against cumulative fatigue damage is different from the applied demand safety factor For deter-mination of this theoretical safety factor an iteration proce-dure is required as described in the following flowchart

S is correction factor with which the actual safety factor Sact can be found

Sact is the demand safety factor (used in determination of the permissible stresses in the σ ndash N ndash curve) times the correc-tion factor S

The full procedure is to be applied for pinion and wheel tooth roots and flanks

Note If alternating stresses occur in a spectrum of mainly pulsating stresses the alternating stresses may be replaced by equiva-lent pulsating stresses ie by means of division with the ac-tual mean stress influence factor YM

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

Classification Notes- No 412 41 May 2003

DET NORSKE VERITAS

Appendix B Application Factors for Diesel Driven Gears

For diesel driven gears the application factor KA depends on torsional vibrations Both normal operation and misfiring conditions have to be considered

Normally these two running conditions can be covered by only one calculation

B1 Definitions

Normal operation KAnorm = 0

normv0

TTT +

where

T0 = rated nominal torque

Tv norm = vibratory torque amplitude for normal opera-tion (see rules Pt4 Ch3 Sec1 G301 for defi-nition of ldquonormalrdquo irregularity)

Misfiring operation KA misf = 0

misfv

TTT +

where

T = remaining nominal torque when one cylinder out of action

Tv misf = vibratory torque amplitude in misfiring con-dition This refers to a permissible misfiring condition ie a condition that does not require automatic or immediate corrective actions as speed or pitch reduction

The normal operation is assumed to last for a very high num-ber of cycles such as 1010

The misfiring operation is assumed to last for a limited dura-tion such as 107 cycles

B2 Determination of decisive load Assuming life factor at 1010 cycles as YN = ZN = 092 which usually is relevant the calculation may be performed only once with the combination having the highest value of appli-cation factorlife factor

For bending stresses and scuffing the higher value of

092

K normA and 098

K misfA

For contact stresses the higher value of

092K normA and

113K misfA (but

097K misfA for nitrided gears)

B3 Simplified procedure Note that this is only a guidance and is not a binding con-vention

T = oTZ

1Zsdot

minus where Z = number of cylinders

( ) ideal Videal Vmisf Vnorm V TTT24ZT +minus=

where TV ideal = vibratory torque with all cylinders perfectly equal See also rules Pt4 Ch3 Sec1 G300

When using trends from torsional vibration analysis and measurements the following may be used

TV ideal To is close to zero for engines with few cylinders and using a suitable elastic coupling and increases with relative coupling stiffness and number of cylinders

TV misf To may be high for engines with few cylinders and decreases with number of cylinders

This can be indicated as

200Z

TT

ο

idealV asymp and 80Z04

TT

o

misfV minusasymp

Inserting this into the formulae for the two application fac-tors the following guidance can be given

118112K misfA minusasymp

115110K normA minusasymp

Since KA norm is to be combined with the lower life factors the decisive load condition will be the normal one and a KA of 115 will cover most relevant cases when a suitable elas-tic coupling is chosen

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width

42 Classification Notes- No 412

May 2003

DET NORSKE VERITAS

Appendix C Calculation of Pinion-Rack

Pinion-racks used for elevating of mobile offshore units are open gears that are subjected to wear and tear With normal specifications such as only a few hundred total operation cycles (site to site) the tooth bending stresses for static strength will be decisive for the lay out however with ex-ception of surface hardened pinions where case crushing has to be considered

In the following the use of part 1 and 3 for pinion-racks is shown including relevant simplifications

C1 Pinion tooth root stresses Since the load spectrum normally is dominated by high torques and few load cycles (in the range up to 1-2000) the static strength is decisive

The actual stress is calculated as

β1FSaFan1

tF1 KYY

mbFσ sdotsdotsdotsdot

=

b1 is limited to b2 + 2middotmn

YFa and YSa replace YF and YS because load application at tooth tip has to be assumed for such inaccurate gears

Pinions often use a non-involute profile in the dedendum part of the flank eg a constant radius equal the radius of curva-ture at reference circle For such pinions sFn and hFa are to be measured directly on a sectional drawing of the pinion tooth

Due to high loads and narrow facewidths it may be assumed that KFβ2 = KHβ = 10 However when b1 gt b2 then KFβ1 gt10

If no detailed documentation of KFβ1 is available the fol-lowing may be used

KFβ1 = 1 + 015middot(b1b2 ndash 1)

The permissible stress (not surface hardened) is calculated as

δrelTstF

Fst1FP1 Y

Sσσ sdot=

The mean stress influence due to leg lifting may be disre-garded

The actual and permissible stresses should be calculated for the relevant loads as given in the rules

C2 Rack tooth root stresses The actual stress is calculated as

SaFan2

tF2 YY

mbFσ sdotsdotsdot

=

See C1 for details

The permissible stress is calculated as

δrelTstF

Fst2FP2 Y

Sσσ sdot=

For alloyed steels (Ni Cr Mo) with high toughness and duc-tility the value of YδrelTst may be put equal to YSa

C3 Surface hardened pinions For surface hardened pinions the maximum load is not to cause crushing of the hardened layer of the flank

In principle the calculation described in 213 may be used but when the theoretical Hertzian stress exceeds the ap-proximately 18 times the yield strength of the rack material plastic deformation will occur This will limit the peak Hertzian stress but increases the contact width and thus the penetration of stresses into the depth

An approximation may be based on an assessment of contact width determined by means of equal areas under the theoreti-cal (elastic range) Hertzian contact and the elasto-plastic contact stress (limited to 18 middot σy) with the unknown width