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Equations for Spur Gears

by

Dr. Robert Rizza Associate Professor

Department of Mechanical Engineering Milwaukee School of Engineering

1025 N. Broadway Milwaukee, WI 53202

(414) 277-7377 Fax:(414) 277-2222

Email: rizza@msoe.edu http://people.msoe.edu/~rizza

The equations are believed to be correct. But, if you find any errors please let me know by writing to rizza@msoe.edu.

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Nomenclature Symbol* Meaning

a Addendum length b Dedendum length c Clearance C Gear and Pinion center distance

Cf (or ZR) Surface condition factor CH Hardness factor Cma Mesh alignment factor Cmc Load correction factor

CP (or ZE) Elastic coefficient Cpf Pinion proportion factor Cpm Pinion proportion modifier

d Pitch diameter E Young’s modulus hu Working depth

ht=a+b Whole depth F Face width Fn Transmitted load Fr Radial load Ft Tangential load I Surface Geometry factor φ Pressure angle m Module J Bending Strength Geometry Factor

KB Rim thickness factor KM (CM) Load distribution factor

Ko Overload factor KR Reliability factor KS Size factor KT Temperature factor KV Dynamic Factor MG Gear ratio (never less than 1) mB Backup ratio Mp Contact ratio Mv Angular velocity ratio n Number of revolutions per minute N Number of teeth ν Poisson ratio

ω, ωarm Angular velocity, Angular velocity of the arm Pb Base Pitch

Pc Circular Pitch Pd Diametral Pitch Qv Quality index r Pitch circle radii t Tooth thickness ρ Radii of curvature TF Oil film temperature Sb Allowable bending stress Sc Allowable surface contact stress Sfb Bending fatigue strength Sfc Surface fatigue contact strength T Torque tR Rim thickness Vt Velocity along the pitch circle YN Stress cycle factor for bending z Length of action

ZN Stress cycle for pitting resistance * Subscripts p and g are used to indicate the pinion and g for some of these parameters.

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KINEMATICS

Fundamental Relationships

out in inV

in out out

r Nmr N

ω= = =

ω out

Ain V

T 1mT m

= =

cdP

Nπ

= dNPd

= b cP P cos= φ

dmN

=

( ) ( ) ( ) ( )2 2 2 2p p p g g gz r a r cos r a r cos Csin= + − φ + + − φ − φ

dP

P zMcos

=π φ

MG= Mv or MA depending on which is greater than 1

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Standard Gears

Tables 1, 2, 3 (Norton tables 11-1, 11-2 and 11-3, or Shigley 13-1 to 13-3) Table 1 (Norton Table 11-1, Shigley Table 13-1): Standard Gear Parameters

Parameter Coarse Pitch (Pd < 20) Fine Pitch (Pd > 20) φ 20 or 25 degrees 20 degrees a 1.000/ Pd 1.000/ Pd b 1.250/ Pd 1.250/ Pd hu 2.000/ Pd 2.000/ Pd ht 2.250/ Pd 2.200/ Pd +0.002 in t 1.571/ Pd 1.571/ Pd

Fillet radius 0.3000/ Pd Not standardized Minimum basic clearance 0.250/ Pd 0.200/ Pd +0.002 in

Minimum width of top land

0.250/ Pd Not standardized

c 0.350/ Pd 0.350/ Pd +0.002 in Table 2 (Norton Table 11-2, Shigley Table 13-2): Standard Diametral Pitches

Coarse Pitch (Pd < 20) Fine Pitch (Pd > 20) 1.25 20 1.5 24 1.75 32

2 48 2.5 64 3 72 4 80 5 96 6 120 8 10 12 14 16 18

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Table 3 (Norton Table 11-3, Shigley 13-3): Standard Metric Modules

m (mm) Equivalent Pd (1/in) 0.3 84.67 0.4 63.50 0.5 50.80 0.8 31.75 1 25.40

1.25 20.32 1.5 16.93 2 12.70 3 8.47 4 6.35 5 5.08 6 4.23 8 3.18 10 2.54 12 2.12 16 1.59 20 1.27 25 1.02

Table 4 Minimum Number of Teeth Required for Avoiding Interference in a 20° Full-Depth Pinion and Full-Depth Gear (Norton Table 11-5). Minimum Pinion Teeth Maximum Gear Teeth

17 1309 16 101 15 45 14 26 13 16

Table 5 Minimum Number of Teeth Required for Avoiding Interference in a Full-Depth Pinion and a Full-Depth Rack (Norton Table 11-4).

Pressure Angle (φ) Minimum Number of Teeth. 14.5 32 20 18 25 12

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Gear Trains 1. Simple fixed Gears

out

in

product of the number of teeth of input gearsproduct of the number of teeth of output gears

ω=

ω

2. Planetary Gears

L arm

F arm

product of the number of teeth of input gears product of the number of teeth of output gears

ω − ω=

ω − ω

Stresses Transmitted Loads

t n

r n t

F F cosF F sin F tan

= φ= φ = φ

Velocity of the Pitch Circle

( )( )

d

t

dn Nn (in FPM) 12 12P

V dn in in/s n in rpsmNn in m/s, n in rps

π π⎧ =⎪⎪⎪= π⎨⎪π⎪⎪⎩

(See also Norton Table 11-7)

Bending Stress

t db o m v S B

F PK K K K K

FJσ = English Units

tb o m v S B

F K K K K KFmJ

σ = SI units

Bending Strength Geometry Factor, J For values of J see Norton Tables 11-8 to 11-15 or Shigley Figure 14-6 below.

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Mott Figure 9-17 (Shigley Figure 14-6).

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Dynamic Factor

tv t

v tt

v t

50 VK (V in ft/min)

50For Q 5 (V 2,500 ft/min (13m/s))50 200V

K (V in m/s)50

⎧ +=⎪

⎪≤ ≤ ⎨+⎪ =⎪⎩

( )

( )2

3

Bt

v t

Bt

v tv

v

A VK (V in ft/min)

A

A 200VK (V in m/s)For 6 Q 11 A

A=50+56 1-B

12 QB

4

⎧ ⎛ ⎞+⎪ = ⎜ ⎟⎪ ⎜ ⎟⎝ ⎠⎪

⎪ ⎛ ⎞+⎪⎪ = ⎜ ⎟≤ ≤ ⎨ ⎜ ⎟⎝ ⎠⎪

⎪⎪⎪ −⎪ =⎪⎩

Experimental results are limited by

( ) ( )

( )( )

2t max v

2v

t max

V A Q 3 in ft/min

A Q 3V in m/s

200

⎡ ⎤= + −⎣ ⎦

⎡ ⎤+ −⎣ ⎦=

Shigley Figure 14-9 Kv versus Vt.

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Load distribution factor, KM In general keep 8/Pd < F < 16/Pd. Km may be calculated by using Shigley equation 14-30: ( )m mc pf pm ma eK 1 C C C C C= + +

mc1 for uncrowned teeth

C0.8 for crowned teeth

⎧= ⎨

⎩

pf

2

F 0.025 F 1 in10d

FC 0.0375 0.0125F 1 <F 17 in10d

F 0.1109 0.0207F 0.000228F 17 <F 40 in10d

⎧ − ≤⎪⎪⎪= − + ≤⎨⎪⎪ − + − ≤⎪⎩

pf

7 2

F 0.025 F 25 mm10d

FC 0.0375 0.000492F 25 <F 432 mm10d

F 0.1109 0.000815F (3.53 x 10 )F 432 mm <F 1020 mm10d

−

⎧ − ≤⎪⎪⎪= − + ≤⎨⎪⎪ − + − ≤⎪⎩

Note for values F/(10d) < 0.05, F/(10d)=0.05 is used.

1pm

1

1 for straddle mounted pinion with S / S 0.175C

1.1 for straddle mounted pinion with S / S 0.175<⎧

= ⎨ ≥⎩

Shigley Figure 14-10. Definitions of S1 and S.

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2

maC A BF CF= + + (Shigley Table 14-9 or Table 5 for values of A, B and C) (Shigley Figure 14-11 may also be used) Table 5 (Shigley Table 14-9) Values of A, B and C Condition A B C Open gearing 0.247 0.0167 -0.765(10-4) Commercial, enclosed units

0.127 0.0158 -0.093(10-4)

Precision, enclosed units

0.0675 0.0128 -0.926(10-4)

Extraprecision enclosed gear units

0.0030 0.0102 -0.822(10-4)

e0.8 for gearing adjusted at assembly or compatability is improved by lapping or both

C1 for all other conditions

⎧= ⎨

⎩

Shigley Figure 14-11. Variation of Cma with F. Size correction factor, KS Most applications KS=1, see Table 6 (Mott, Table 9-6)

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Table 6: Suggested size Factors Diametral pitch, Pd Metric module, m Size factor, Ks

≥ 5 ≤ 5 1.00 4 6 1.05 3 8 1.15 2 12 1.25

1.25 20 4.00 Rim thickness factor, KB

B

BRB B

B t

2.2421.6 ln m 1.2mt

m , K1.0 m 1.2h1.0 solid gear

⎧ ≤⎪⎪= = ⎨ >⎪⎪⎩

See also Shigley Figure 14-16

Shigley Figure 14-11. KB as a function of backup ratio. Overload factor Ko Table 7 (Ko Norton Table 11-17 not in Shigley).

Driven Machine Driving Machine Uniform Moderate Shock Heavy Shock

Uniform (electric motor, turbine)

1.00 1.25 1.75 or higher

Light Shock (Multicylinder

engine)

1.25 1.50 2.00 or higher

Medium shock (Single-cylinder

engine)

1.50 1.75 2.25 or higher

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Surface Stress

m fP t o v s

cH R

E t o v sI

K CC F K K K US units

dF IK Z

Z F K K K SI unitsdF Z

⎧⎪⎪σ = ⎨⎪⎪⎩

For the value of d use the pitch diameter of the smaller of the two gears in contact. Surface Geometry factor, I (or ZI)

G

G

G

G

Mcos sin external gears2 M 1

IMcos sin internal gears

2 M 1

φ φ⎧⎪ +⎪= ⎨

φ φ⎪⎪ −⎩

Elastic Coefficient, Cp (or ZE) (in (lbf/in2)½ or (N/mm2)½)

p 2 2p g

p g

1C1 1

E E

=⎡ ⎤⎛ ⎞ ⎛ ⎞− ν − ν⎢ ⎥⎜ ⎟ ⎜ ⎟π +⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

See also Norton Table 11-18 (Shigley Table 14-8) for ν=0.3. Surface condition factor, Cf (or ZR) Standard values have yet to be established by AGMA.

Cf=1 for conventional manufactured gears, higher for rougher finishes. Fatigue Strength

Nfb t

T R

YS S

K K=

For values of St, see Tables 8 and 9 (Shigley tables 14-3 and 14-4).

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Table 8 (Shigley Table 14-3) St at 107 cycles and 0.99 Reliability for Steel Gears Allowable Bending Stress Number St

Psi Material

Designation Heat Treatment Minimum

Surface Hardness Grade 1 Grade 2 Grade 3

Through-hardened

Fig 14-2 Fig 14-2 Fig 14-2 -

Flame or induction

hardened with type A pattern

AGMA Table 8 45,000 55,000 -

Flame or induction

hardened with type B pattern

AGMA Table 8 22,000 22,000 -

Carburized and hardened

AGAM Table 9 55,000 65,000 or 70,000 (bainite and

cracks limited to grade 3 levels)

75,000

Steel

Nitrided (through

hardened steels)

83.5 HR15N Fig 14-3 Fig 14-3 -

Nitralloy 135M Nitralloy N,

and 2.5% chrome (no aluminum)

Nitrided 87.5 HR15N Fig 14-4 Fig 14-3 Fig 14-4

Shigley Figure 14-2

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Shigley Figure 14-3

Shigley Figure 14-4

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Table 9 (Shigley Table 14-4) Applied Bending Strength St for Iron and Bronze Gears at 107 Cycles and 0.99 Reliability

Material Material Designation

Heat Treatment Typical Minimum

Surface Hardness

Allowable Bending Stress

Number, St (psi) Class 20 As cast - 5000 Class 30 As cast 174 HB 8500

ASTM A48 gray

Class 40 As cast 201 HB 13,000 Grade 60-40-18 Annealed 140 HB 22,000-33,000 Grade 80-55-06 Quenched and

Tempered 179 HB 22,000-33,000

Grade 100-70-03 Quenched and Tempered

229 HB 27,000-40,000

ASTM A536 ductile (modular)

Iron

Grade 120-90-02 Quenched and Tempered

269 HB 31,000-44,000

Bronze Sand cast Min. Tensile strength 40,000

psi

5700

ASTM B-148 Alloy 954

Heat treated Min. Tensile strength 90,000

psi

23,600

Stress cycle factor for bending, YN

Shigley Figure 14-14 Values of YN.

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Temperature factor, KT

o

FT oF

F

1 T 250 FK 460 T T 250 F

620

⎧ ≤⎪= ⎨ +>⎪

⎩

From Norton not Shigley

Reliability factor, KR

R0.658 0.0759ln(1 R) 0.5 < R < 0.99

K0.50 - 0.109ln(1-R) 0.99 R 0.9999

− −⎧= ⎨ ≤ ≤⎩

Surface Fatigue Strength

N Hfc c

T R

Z CS S

K K=

Hardness factor, CH

a. Through hardness ( )H GC 1 A M 1= + −

p

g

p p

g g

p

g

HB0 1.2

HB

HB HBA 0.00898 0.00829 1.2 1.7

HB HB

HB0.00698 1.7

HB

⎧<⎪

⎪⎪⎪= − ≤ ≤⎨⎪⎪⎪ >⎪⎩

b. Surface hardened Pinion (> 48 HRC)

( )q

q

H g

0.0112R

0.052R

C 1 B 450 HB

0.00075e (english)B

0.00075e (SI)

−

−

= + −

⎧⎪= ⎨⎪⎩

Stress cycle for pitting resistance, ZN

See Shigley Figure 14-15 below.

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Shigley Figure 14-15. Allowable Surface Contact Stress Sc See Table 9 and 10 (Shigley Tables 14-6 and 14-7) Table 9 (Shigley Table 14-6) Repeatedly Applied Contact Strength Sc at 107 and 0.99 Reliability for Steel Gears

Allowable Contact Stres Number Sc, Psi Material Designation

Heat Treatment

Minimum Surface

Hardness Grade 1 Grade 2 Grade 3

Through hardened

Fig 14-5 Fig 14-5 Fig 14-5 -

50 HRC 170,000 190,000 - Flame or Induction hardened

54 HRC 175,000 195,000 -

Carburized and

hardened

AGMA Table 9

180,000 225,000 275,000

83.5 HR1 5N 150,000 163,000 175,000

Steel

Nitrided (through hardened

steels)

84.5 HR1 5N 155,000 168,000 180,000

2.5% Chrome (no aluminum)

Nitrided 87.5 HR1 5N 155,000 172,000 189,000

Nitralloy 135M Nitrided 90 HR1 5N 170,000 183,000 195,000 Nitralloy N Nitrided 90 HR1 5N 172,000 188,000 205,000

2.5% Chrome (no aluminum)

Nitrided 90 HR1 5N 176,000 196,000 216,000

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Table 10 (Shigley Table 14-7) Repeatedly Applied Contact Strength Sc at 107 and 0.99 Reliability for Iron and Bronze Gears

Material Material Designation

Heat Treatment Typical Minimum

Surface Hardness

Allowable Bending Stress

Number, St (psi) Class 20 As cast - 50,000-60,000 Class 30 As cast 174 HB 65,000-75,000

ASTM A48 gray

Class 40 As cast 201 HB 75,000-85,000 Grade 60-40-18 Annealed 140 HB 77,000-92,000 Grade 80-55-06 Quenched and

Tempered 179 HB 77,000-92,000

Grade 100-70-03 Quenched and Tempered

229 HB 92,000-112,000

ASTM A536 ductile (modular)

Iron

Grade 120-90-02 Quenched and Tempered

269 HB 103,000-126,000

Bronze Sand cast Min. Tensile strength 40,000

psi

30,000

ASTM B-148 Alloy 954

Heat treated Min. Tensile strength 90,000

psi

65,000

Shigley Figure 14-5