effect of shaft misalignment on the stresses distribution of spur gears- hani aziz ameen

8/6/2019 Effect of Shaft Misalignment on the Stresses Distribution of Spur Gears- Hani Aziz Ameen

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Eng. & Tech. Journal, Vol. 28, No. 7, 2010

* Pumps Engineering Department, Technical College, Al-Musaib /Babylon1321

Effect of Shaft Misalignment on The Stresses Distributionof Spur Gears

Dr. Hani Aziz Ameen*Received on :17/8/2009Accepted on:7/1/2010

AbstractShaft misalignment is considered as one of the common repeated

problems in most rotating machineries , which leads to generate vibrations andextra dynamic loads on transmitting gears teeth, also leads to non- uniformity indistribution of applied load along the meshing tooth face by being concentrated onone side of tooth face. The present work concentrated on the analysis of stressesgenerated on transmitting gear tooth, also studied the effect of misalignment angleon stress distribution and its concentration. This is important for the gear designand those who works in gear maintenance , because fracture is expected to initiateand propagate at locations of stress concentration . ANSYS program using finiteelement technique had been used, as this program is efficient and accurate tool instress analysis, especially for complicated shapes. Gear tooth model had beenanalyzed using finite element method in three dimensions. After calculatingtransmitted load and dynamic load, misalignment angle had been changed from(0,0.2 ,0.3 ,0.4 ,0.5 ) then its effect on distribution of applied load had beencalculated. The finite element program (ANSYS) had been executed for cases of misalignment angle (0 ,0.2 ,0.3 ,0.4 ,0.5 ). The results showed clearly, that thestresses distribution and its concentration on tooth changed with misalignmentangle and the equivalent stress is direct proportional with the misalignment angle.According to the values of generated stresses, the tooth fracture can be predicted.

Keywords : Machine Design , Gear Design , Finite Element Method, StressesAnalysis, ANSYS software, Misalignment

.

.

. .

ANSYS .

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Eng. & Tech. Journal, Vol. 28, No. 7, 2010 Effect of Shaft Misalignment on TheStresses Distribution of Spur Gears

1322

(0

0.2

0.3

0.4

0.5

) . ANSYS )0.50.40.30.20(

.

NomenclatureT applied torqueP transmitted power angular velocity

br ... radius of base circler radius of pitch circle pressure anglem moduleZ number of teeth

iF load at contact pointd pitch circle diameter (P.C.D.) ... total deflection at contactpoint

B bending deflection S shearing deflection

G deflection due to theinclination of foundationunder load

P compressive deflection12 , radii of curvature for the

B width of gearSm the location of applied loadE Young modulus possion s rationK tooth stiffness at contactpointFdyn dynamic loadF

t tangential component of

applied loadV linear velocityc parameter depending on the

material of gearing teeth andpressure angle

e error in the shape of tooth

a constant depending onpressure angle

gp E,E Young modulus for the

pinion and gear

B active gear width of contactzC spring constant

RWf directional error angle of misalignmentIntroduction

The reason of increasing thestresses concentration on gear teeth isthe misalignment of the shafts thattransmitting the movement, thismisalignment made non uniformdistribution of transmitted loads alonggear face, which is caused generation

of the stresses concentration inspecific region. Several researchesand studies have been done and useddifferent ways for the generatedstresses analysis on the movementtransmitting gear teeth. LewisEquation which is treated the spurgear as a cantilever beam and it is themain method for calculating thestresses in the root of the gear. Alsoseveral studies have been done usingPhotoelasticity Technique todetermine the stresses concentration.The widely used of finite elementmethod let the researchers to appliedit on stress analysis of spur gear . TaeH. Chang and A. Kubo [1] studies thestresses distribution in spur gear , andit is depended on the finite element



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method to analyze the stresses anddeformations in the gear and specifythe expected failure locations. V.Simon [2] studies the loads andstresses distributions on the teeth forspur and helical gears. M.Q. Abdullah[3] studies the analysis of stresses anddeformations on the gear teeth usingfinite element method, in this study acomplete program is built to generatethe symmetry and unsymmetry spurgear and analyze the stressesdistribution on it. D.G. Lewicki and

R. Ballarini [4] studies the stressesdistributions on the spur gear usingfinite element methods.

In this paper the effect of misalignment of shaft on the stressesdistribution and its concentration onthe spur gear using finite elementmethod during ANSYS program isinvestigated. Also the effect of shaftmisalignment angle on the loadsdistribution and on the stressesdistribution and its concentration onthe movement transmitted gear teeth

is studied.Transmitted Load by Meshing of TeethThe load distribution on the gearwidth [5 ] is determined as thefollowing steps:

sec) / rad()P(Power

)T(Torque=

. (1)

The values of transmitted power andrunning speed are specified in orderto calculate the torque, in this studythe values of P=400 kW and N=5000rpmSo, from Eq.(1) , T = 764.331 N.mThe vertical load (F i) which istransmitted by teeth meshing, can becalculated by:

bi r / TF = . (2))cos(.rrb = . (3)It is required to specify the basicradius ( r b ), Pressure angle( ),module(m) and number of teeth (Z) ,in order to calculate F i , so in thisstudy the following data are used :Pressure angle ( ) = 20 , Module( m ) = 4 mm , No. of teeth ( Z ) = 50teeth , (P.C.D) d=m.Z = 200 mm , r=d/2 = 100 mm, hence, from Eq.(3)r b = 93.969 mm , substituted into

Eq.(2) , the value of normal load canbe getFi=8133.843 NNow to calculate the share of normalload at any contact point as follows:Total Deflection ( ) at ContactPointThe total deflection at contact pointcan be calculated as follows[6]:

PGSB +++= ... (4)where,

3F

32i

m

2

m2

3F

2i

B

SBE

)z-(w )(cosF 6

]zS3z

S[SBE

z )(cosF 12

+

+ =

3)zw()Sw(

log2

)zw()Sw(

4)zw()Sw(

me

mm

..(5)

+

+=

me

F

2i

S

Swzw

log)zw(z

SBE )(cosF )1(2

(6)



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KFKF

SSzSnS

w = . (7)

F

2m

2i

G SBE )(cos S F24

= . (8)

+

+=

21

22

iP BE

)1( F 4 (9)

where :

B = Bending deflection S = Shearing deflection G = Deflection due to the

inclination of foundation underload

P = Compressive deflection12 , = Radii of curvature for

the two teeth gearing at contactpoint

Fig.(1) demonstrated theterms of teeth which are used incalculated the deflection . Thefollowing data are used in thisstudy:Sm = 6.338 mm , S F = 8.891 mm ,SK = 4.93 mm, n = 9 mm , z = 1mmw = 18.957 mm , B = 40 mm ,

=20 .Also , the Youngmodulus (E) and Possion s ratio( ) of the material of tooth is :E = 207000 N/mm 2 , 3.0=

Tooth Stiffness at Contact PointTo Calculated the tooth stiffness atcontact point as follows[6]:

= iFK .. (10)

If the applied load equal to thenormal load , hence the load atcontact point ( c ) equal to : F c = F i =8133.843 N so, it can be explainedthe stiffness of gear at point ( c )

using the material of tooth and its sizeas follows:

9

i

B 10x635.181F

= mm/N ,

9

i

S 10x745.228F

= mm/N

9

i

G 10x788.3682F

= mm/N ,

9

i

P 10x848.83F

= mm/N

mm/N10x018.4177 FFFFF

9

i

P

i

G

i

S

i

B

i

tot

=

+++=

itot F / 1

K=

= 239405.2 N/mm

= K cwhereKc = stiffness at point ( c ) .

Dynamic LoadIn order to demonstrate the

effect of the dynamic load ,Buckingham equation [7] is used,according to this equation, thedynamic load is depending on thelinear velocity (V), tangent load ( tF ),tooth width (B) and the materialproperties of the gear .It can beshown the Buckingham equation asfollows[8]:

t

ttdyn

FBceV21

)FBce(V21FF

++++= ...... (11)

)cos(FF it = ....... (12)where :e: error in the shape of tooth

It can be found from tolerancetable in Refs[7,8] corresponding tom= 4 mmthat m102e 6=



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c: Parameter depending on thematerial of gearing teeth and thepressure

angle [8],

)EE(

)E(Ea c

gp

gp

+=. (13)

a: constant depending on pressureangle , when =20 ,

0.111)20(a == gp E,E : Young modulus for the

pinion and gear ,2N/m10207EE 9gp ==

Substituting the values of gp E,E,a in

Eq.(13), the value of c can beobtained : 2N/mm5.11488c = Hence , the dynamic load (Eq.(11)will be : N94.16204Fdyn = i.e that the applied load on theanalyzed gear increased by twice thanthe transmitted load after the dynamiceffect is calculated.Active Gear Width of Contact

When the teeth is meshing in

correct way the load is distributeduniformly along the gear width asshown in Fig.(2), and when themisalignment is happen betweenshafts, so the meshing of teeth will benot correct and the contact will besubside in one side from the gear facewithout the other side asdemonstrated in Fig.(3).

The alteration of the applieddistribution load that concentrated onone side and decreasing gradually tothe other side and it can beapproximated the applied distributionload as a cubic parabolic distribution.Fig.(4) explained this distribution onthe active gear width of contact )B( and can be described in the followingequation[9] :

])B(x/ .[1F)x(F3

max = . (14)It can be defined the tangential loadon the gear width as follows [9] :

===B

0maxm FB4

3 F(x).dx.BFF (15)

Hence,

BF

Fm = (16)

B3F4

Fmax = (17)

It is found experimentally that )F( max can be calculated as [9]:

BB

f .CF RWzmax = . (18)where :

zC : Spring constant )mm / N( andcan be defined as [9]:

3z 10B

KC = . (19)K : gear stiffness at the location of applied load .)mm / N(

RWf : Directional error )( and canbe defined as [9]:

.B.10f 3

RW = . (20) : angle of misalignment (Radian)From Eq.(17) and Eq.(18) , it can bededuced that :

zRW .Cf 3BF4

B = . (21)

This equation represent the activegear width of contact, when themisalignment of gears is occurred .

Load Distribution on Gear in Caseof Shaft Misalignment [10-14]

The load distribution on the gearin case of misalignment can bespecified by putting a thin layer of soft material, which is explained thepressure zone on gear face during themeshing of teeth, and when the load



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three Dimensional model of gear isdone by drawing and meshing twodimension gear plane with elementplan42 and then dragging by solid45to form three dimensional gear.Fig.(8) shows the gear model inANSYS. Fig.(9) shows the loaddistribution when the misalignmentangle ( ) is zero, i.e. there isalignment of shafts, and it can be seenthat the distribution of load isuniform. The load distribution isapplied in ANSYS according to

equations (22) and (23) and its valuedepend on the misalignment angle(), these equations are programmedin ANSYS software by AnsysParametric Design Language (APDL)as follows :

/prep7! Data for Force FP=400*1000 ! power = 400 kWN=5000 ! rpmomg=5000*(2*3.14/60)T=P/omg*afun,deg : Psi=20

m=4 ! mm : Z=50dpcd=m*Z ! mm : r=dpcd/2 ! mmrb=r*cos(psi) ! mm : Fi=T/(rb/1000)!Date for stiffness KSm=6.338 : Sf=8.891Sk=4.93 : nn=9zz=1 : w=18.957B=40 : E=207000 :posi=0.3! DeltaBA1=12*Fi*zz*(cos(psi))**2A2=E*B*Sf**3A3=Sm**2+((zz**2)/3)-(Sm*zz)A4=6*Fi*(cos(psi))**2*(w-zz)**3

A5=(w-Sm)/(w-zz)DeltB=((A1/A2)*A3)+(A4/A2)*(A5*(4-A5)-2*log(A5)-3)DB=DeltB/Fi! DeltaSB1=2*(1+posi)*Fi*(cos(psi))**2

B2=E*B*Sf B3=(w-zz)/(w-Sm)DeltS=(B1/B2)*(zz+(w-zz)*log(B3))DS=DeltS/Fi! DeltaGC1=24*Fi*Sm**2*(cos(psi))**2C2=3.14*E*B*Sf DeltG=C1/C2DG= DeltG/Fi! DeltaPD1=4*Fi*(1+posi**2)D2=3.14*E*BD3= rb/(rb+rb)

DeltP=(D1/D2)*D3DP= DeltP/FiDtot=DB+DS+DG+DPKc=1/Dtot! Data for dynamic force FdynFt=Fi*cos(psi)ee=2e-3 : aa=0.111: Ep=E:Eg=Ecc= (aa*Ep*Eg)/(Ep+Eg)V= omg*rbE1=21*V*((B*cc*ee)+Ft)E2=21*V+sqrt((B*cc*ee)+Ft)Fdyn=Ft+(E1/E2)! Bdash

Alf=0.00349: fRW=1000*Alf*BCz= (Kc/B)*1e-3: F1=4*Fdyn*BF2=3*fRW*Cz : Bdash=sqrt(F1/F2)Fmax=(4*Fdyn)/(3*Bdash)! the function Fmax*(1-(x/Bdash)**3)X1=0: X2=1: N=1 ! no. of interval*Dim,Xs,,10 : *Dim,Ys,,10K1=0: K1=K1+1N1=N+K1-1 : L=(X2-X1)/N1H=L: A1=0! LOOP*DO,I,1,N1 : *DO,J,1,2

Xs(J)=X1+(I-1)*L+(J-1)*HXso=Xs(J) :! FUNCTIONYso=Fmax*(1-(Xso/Bdash)**3)Ys(J)=Yso*ENDDO



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A1=A1+H/2*(Ys(1)+Ys(2))*ENDDOFtag=A1Fnor=Ftag*tan(psi)FinishTables(1),(2),(3)&(4) are containedthe values of loads with ( = 0.2 ,0.3 , 0.4 & 0.5 ). Fig.(10) showsthe load distribution according to theangles of misalignment. This figureshows that the loads become lessstrong gradually in only one side of the gear face, while it is concentrated

on the other side.Results and DiscussionsMisalignment of shaft causes

changes in the teeth gearing .Thischanges the load distribution on theteeth. Manufacturing errors causesalternating loads on the gear. Localdeformation at the point of applyingthe load and misalignment of theshaft causes stress concentration (i.e.the stresses are concentrated on oneside of the teeth while graduallydecreasing on the other side) .

Consequently, sudden fracture in thetooth occurs. Generally, there appearsat the side of stress concentration,wearing or pitting and consequentdamage. The contact region betweenthe teeth of a sound gear may berepresented by straight line parallel tothe gear s axes of rotation, i.e. loaddistribution along the gear width isuniform (Fig.(9)). Misalignment isdue to manufacturing errors , bendingin the involutes curve of the gearetc, with consequent changes in the

teeth gearingmaking concentrated load on one sideof the teeth (Fig.(10)) .Fig.(11) explain the stress distributionon the gear width when = 0 i.e.shaft alignment, and it can be seen

that the stress distribution is uniformalong the gear face. Fig.(12) explainthe stress distribution on the gearwidth when = 0.2 , Fig.(12-a)shows thedistribution of stress is not uniformalong the gear face, and Fig(12-b)shows the deformation occurs in spurgear during misalignment. Fig.(13)demonstrates that when = 0.3 ,stress distribution on the gear widthoccurs; the non-uniformity of stressdistribution along the gear face is

demonstrated by Fig.(13-a), whileoccurrence of deformation in spurgear during misalignment isdemonstrated by Fig(13-b).

Fig.(14) demonstrates thatwhen = 0.4 , stress distribution onthe gear width occurs; the non-uniformity of stress distribution alongthe gear face is demonstrated byFig.(14-a), while occurrence of deformation in spur gear duringmisalignment is demonstrated byFig(14-b).

Fig.(15) demonstrates thatwhen = 0.5 , stress distribution onthe gear width occurs; the non-uniformity of stress distribution alongthe gear face is demonstrated byFig.(15-a), while occurrence of deformation in spur gear duringmisalignment is demonstrated byFig(15-b)

Fig.(16) shows therelationship between themisalignment angle and theequivalent stress at contact point on

spur gear, it can be shown that whenthe misalignment angle increases theequivalent stresses increase directly.Fig.(17) shows the relationshipbetween the misalignment angle andmax. deflection at contact point on



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spur gear , it can be seen that thedeflection is directly proportional tothe misalignment angle.Fig.(18) shows the comparisonbetween the present results and theresult from Ref. [3] for the tooththickness and shear stress at zeroangle of misalignment i.e. alignmentshafts. It can be shown that there aredifferences in the results, that due tothe mesh used in Ref.[3] is coarsemesh and due to thattheir results were far from the

ANSYS results.ConclusionsFrom this study the followingconclusion can be get :1-The values of equivalent stressesand its distribution is changed withchanging the misalignment angle,where the stresses concentration isincreased in the contact region and inthe root of the tooth with increasingthe misalignment angle, this isoccurred in the side of subside theload and decreasing in the other side

of the gear face.2- increasing the deformation of gearwith increasing misalignment angle.3- increasing the probability of fracturing the gear in the root withincreasing the misalignment angle.References [1] Chang T.H. and Kubo A., Simple Stress Formulae for a ThinRimmed Spur Gear , Journal of Mechanisms, Transmissions, andAutomation in Design, Vol.107,No.3,pp. 406-411, 1985.

[2] Simon V., Load and StressDistributions in Spur and HelicalGears , Journal of Mechanisms,Transmissions, and Automation inDesign , Vol.110,No. 2, pp. 197-202,1988.

[3] Abdullah M.Q, Analysis of Gear Tooth Stresses using FiniteElement Technique , M.Sc. thesis,University of Al-Nahreen,1994.[4] Lewicki D.G., and Ballarini R.,

Effect of Rim Thickness on GearCrack Propagation Path , 7 th InternationalPower Transmission and GearingConference, U.S.A., California, 1996.[5] Maitra G.M., Hand Book of Gear Design , Tata McGraw-HillCompany, India, New Delhi , 1997.

[6] Elkholy A.H., Tooth LoadSharing in High Contact Ratio SpurGears ,Journal of Mechanisms,Transmissions, and Automation inDesign , Vol. 107, No. 1, pp.11-16,1985.[7] Buckingham E., Analytical

Mechanics of Gears , McGraw-Hill,U.S.A., New York, 1949.[8] Pandya N.C., and Shah C.S.,

Elements of Machine Design ,Charotar Publishing House,

India,1986.[9] Niemann G., MachineElements , Allied Publishers Private,Ltd., India, New Delhi, 1980 .[10] Al-Musawi A.K., Vibration

Analysis and the Allowable Limits inRotating Machinery , M.Sc. thesis ,University of Baghdad , 1993.[11] Shaft Aligment is a Profitable

Form of Preventative Maintenance ,SKF Bearing MaintenanceHandbook, Hane Training, Inc., 2001.[12] Wowk V., Specifying Shaft

Alignment , McGraw-Hill Company,U.S.A., New York, 2000.[13] Shigley J.E., and Mischke C.R., Mechanical Engineering Design ,McGraw-Hill Company , U.S.A. ,New York, 1989.



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[14] Polak S., Gear box & GearSystem Problems , GearsConference, U.K., London, 1999.[15] Saeed Mouveni Finite Elementanalysis theory and applicationWith ANSYA, 1999.[16] Tim Langlais "ANSYS Short

course ", 1999, from internet.

[17] User's manual of FEA/ANSYS/Version 5.4/1997.[18] Basic structural, Training

Manual, (1996, ANSYS Inc.,(ANSYS on-line-help).

Table (1) when 32B, 2.0 == mmPoint 1 2 3 4 5 6 7 8 9 10 11 12 13Ft 671 671 671 670 669 668 666 663 659 654 648 640 632Fn 244 244 244 244 243 243 242 241 239 238 235 233 230

Point 14 15 16 17 18 19 20 21 22 23 24 25 26Ft 621 610 596 580 563 543 521 497 470 441 409 374 336Fn 226 222 217 211 205 197 189 181 171 160 149 136 122

Point 27 28 29 30 31 32Ft 295 251 203 152 40 21Fn 107 91 74 55 14 7

Table(2) when 62B, 3.0 == mmPoint 1 2 3 4 5 6 7 8 9 10 11 12 13Ft 822 822 821 820 818 814 809 803 794 783 769 753 733Fn 299 299 299 298 297 296 294 292 289 285 280 274 266

Point 14 15 16 17 18 19 20 21 22 23 24 25 26Ft 710 683 652 618 578 534 485 431 370 305 233 154 69Fn 258 248 237 224 210 194 176 156 135 111 84 56 25

Table (3) when 22B, 4.0 == mmPoint 1 2 3 4 5 6 7 8 9 10 11 12 13Ft 949 949 947 945 941 935 926 914 899 879 855 826 791Fn 345 345 345 344 342 340 337 333 327 320 311 300 288

Point 14 15 16 17 18 19 20 21 22Ft 750 702 648 586 516 438 351 254 147Fn 273 255 236 213 188 159 127 92 53



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Table(4) when 02B, 5.0 == mmPoint 1 2 3 4 5 6 7 8 9 10 11Ft 1061 1061 1059 1056 1049 1040 1026 1008 983 952 914Fn 386 386 385 384 382 378 373 366 358 346 333

Point 12 13 14 15 16 17 18 19 20Ft 869 814 750 676 590 494 384 261 125Fn 316 296 273 246 215 179 139 95 45

- a - - b -Figure (1) Tooth terms [6]

Figure (2) Uniform distributed of load Figure (3) Subside of the contactin along the gear width

one side of the gear



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Figure (4) Load distribution along the active gear width in contact whenmisalignment is occur

Figure (5) Configurations of load the regions on the tooth face duringmisalignment



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Figure (6) Configurations of the contact line between the gearing tooth

Figure (7) The components of equivalent forces for misalignment gear



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Figure (8) Gear Model in ANSYS

Figure (9) The load distributions at misalignment angle equal zero

= 0.2



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= 0.3

= 0.4

= 0.5 Figure (10) The load distributions at different misalignment angles



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(a) (b)Figure (11) Explain the stress distribution on the gear width when = 0

(a) (b)Figure (12) Explain the stress distribution on the gear width when = 0.2



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0.0 0.1 0.2 0.3 0.4 0.5misaliagnment angle

0

2000

4000

6000

8000

E q u

i v a

l e n

t s

t r e s s

[ M P a

]


Figure (16) The relationship between the misalignment angleand the equivalent stress on spur gear



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0.0 0.1 0.2 0.3 0.4 0.5misaliagnment angle

0.00

0.02

0.04

0.06

0.08

D e f l e c t i o n [ m m ]

Figure (17) The relationship between the misalignmentangle and the deflection on spur gear

Figure (18) The relationship between the tooth thicknessand shear stress when misalignment angle = 0

effect of shaft misalignment on the stresses distribution of spur gears- hani aziz ameen

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