Analysis of Spur and Helical Gears

prepared by Wayne Book

based on Norton, Machine Design and

Mischke and Shigley

Mechanical Engineering Design

The Gnashing of Teeth

• Simple model for loaded gears

• Beam for bending stress

• Cylinders in contact for surface contact stress

Idealized Shape of a Tooth for Stress Analysis

• Simple model: cantilever beam with applied force W

• Tooth thickness t• Length l• Face width F• Max stress at root (a)

lWt

Ft

a23

6

)12/(

2/)(

Ft

lW

Ft

tlW

I

Mc tt

Consider the Shape of a Tooth

• Uncertainties include:– point of load application l– point of maximum stress– appropriate load component– beam thickness

• Depends on pitch P, number of teeth N and pressure angle

• Conservative assumptions are made• Y = Lewis form factor

Introduce Lewis Shape Factor

t

l

Wr

Wt

W

x

3

26

42/

2/

ianglessimilar trBy 2

126

2

2

22

xPY

FY

PW

lPt

F

PWl

tx

t

l

x

t

Pt

l

F

PW

Ft

lW

tt

tt

•Rather than calculate Y(P,, N), create a table, e.g. 14-2

•Lewis equation has been improved by AGMA

Velocity Effect(Its Barth not Barf)

• Purely empirical adjustment for non-zero velocity

• Barth’s equation (1800’s) has been modified to account for current practice and accuracy

• V is velocity in ft/sec at the pitch line

• Kv= 1200/(1200+V) (Modified Barth)

• Metric form Kv= 6.1/(6.1+V), V in m/sec

• Compare to endurance strength (reversing) or use Goodman diagram (one direction)

• Apply notch sensitivity, Marin factors…. the works

FYK

PW

v

t

Surface Durability: Contact Stress

• Analyzed as two cylinders of length l in rolling contact with specified force

• Cylinder radii r1 and r2 vary with contact point

• Depends on elastic material properties and radii of cylinders

• Translate into gear nomenclature as shown on right

factorvelocity C

pinion andgear refer to subscripts PG,

modulus sYoung' E

ratio sPoisson'

11

1

11

cos

v

2/1

22

2/1

21

G

G

p

p

p

V

tpc

EE

C

rrFC

WC

AGMA Approach

• AGMA formula calculates stress for– Bending– Contact

• Stress is compared to an “allowable stress” (also called strength by Norton) based on strength and conditions

Bending Stress

• Many terms are similar to the Lewis equation

• Additional terms account for the application, load sharing and size

factorgeometry tooth

factor backup rim

factoron distributi load

factor size

widthface

pitch diametral

Lewisin asfactor velocity

factorn applicatio

factoridler

load l tangentia

J

K

K

K

F

P

K

K

K

W

J

KKK

F

P

K

KKW

B

m

s

d

v

a

I

t

Bmsd

v

aIt

loading

gear geometry

tooth form

J factor sample tableTip loading (low precision)

Distributed loading (higher precision)

Kv Velocity Factor (similar to Barth)(also provided in equations 11.16 – 11.19)

Load Distribution Factor

• Loads are less evenly distributed for wide face teeth

• Keep F (face width)

8/pd < F < 16/pd

• Nominally F = 12/pd

Application Factor

• Created to account for known but unquantified shock in load

• Electric motors are smooth while single cylinder engines have shock

• Centrifugal pumps are smooth loads while rock crushers have shock

Other FactorsSize, Rim Thickness, Idler

• Size– Fatigue tests are done on small specimins and

indications are that size results in weaker parts– Very large teeth might warrant Ks=1.25 to 1.5– Material properties created directly for gears

account for this

• Rim thickness– In large diameter gears, the centers are

connected to a rim by spokes.– KB reflects failures across the radius

• Idler: use KI = 1.42

Allowable Bending Stress

• Incorporate material strength St specific to gear materials

• St based on Brinell hardness of material

• Environmental and application factors– KL = life factor

– KT = temperature factor

– KR = reliability factorRT

Ltall KK

KS

Life Factor KL

(a specialized S-N curve)

BendingTemperature and Reliability Factors

• Strength data is based on 99% reliability. Adjust up or down.

• Temperatures up to 250 deg F use KT = 1

– Adjust for higher temperatures

620

460 FT

TK

AGMA Bending Fatigue Strengths (uncorrected)

Contact Stress

• Based on rolling cylinder model

• Added terms for size, load distribution, surface condition

factorgeometry tooth

factorcondition surface

factoron distributi load

diameterpitch

factor size

factor (dynamic) velocity

factorn applicatio

tcoefficien elastic

2/1

I

C

C

d

C

C

C

C

I

CC

Fd

C

C

CWC

f

m

s

v

a

P

fms

v

atPc

loading

gear geometry

tooth condition & geometry

material

Surface Geometry Factor I

h)depth teet fullfor (0

elongation addendumfraction

gearfor curvature of radius

pinionfor curvature of radius

angle pressure

pinion of radiuspitch

pinion ofpitch diametrial

sin

coscos1

11

cos

2

2

p

g

p

p

d

pg

dp

d

ppp

pgp

x

r

p

C

pr

p

xr

d

I

AGMA Elastic Coefficient(also from basic material properties and (11.23))

Other Surface Stress Factors

• Cf = 1 for standard manufacturing methods

• Ca, Cm, Cv, Cs are equal to corresponding K values from bending

Allowable Contact Stress(Norton calls Strength)

• Material strength SC is the basis, specific to gear materials

• Sc based on Brinell hardness of material or on tables in Norton

• Adjust for conditions– CL = life factor

– CH = hardness-ratio factor (pinion rel to gear)

– CT = temperature factor

– CR = reliability factor

RT

HLfcfc CC

CCSS

'

Surface Fatigue Strengths

Surface Fatigue Life Factor

Hardness Ratio Factor• Only applied to the gear material (not pinion)• Accounts for work hardening of the gear during

run-in• Depends on previous hardening (through hardened

vs surface hardened)

gear pinion, of hardness Brinnel,

00698.07.1

00829.000898.07.12.1

02.1

ratiogear

)1(1

gp

g

p

g

p

g

p

g

p

G

GH

HBHB

AthenHB

HB

HB

HBAthen

HB

HB

AthenHB

HB

m

mAC

microinchin roughness surface rms

.).(00075.0

.).(00075.0

)450(1

052.0

0112.0

q

R

R

gH

R

ISeB

SUeB

HBBC

q

q

Both through hardened Pinion surface hardened

Helical Gears – Brief Overview

• The treatment of tooth stresses for helical gears is very similar to spur gears

• Bending and Surface stresses must be analyzed

• AGMA formulas are analagous

• Tables also consider helix angle in range of 10 to 30 degrees

• For this class, be able to perform force analysis but we will not cover tooth stresses

Forces, Helical(Equations 12.3 in Norton)

t

n

t

n

tan

tancos

anglehelix

involute)(for angle pressurecircular

angle pressure normal

force radial sin

force axial sincos

force dtransmittecoscos

force total

nr

na

nt

WW

WW

WW

W

Bevel Gears

• Treat force analysis of intersecting, straight tooth, bevel gears

• Equations (12.8 in Norton)

Forces, Bevel Gears(Shigley, Fig 13-34)

•Assume forces concentrated at average radius

•Net force surface

•Decompose into transmitted, radial and axial forces

sintan

costan

tan;/

angle-half conepitch

angle pressure

ta

tr

tavet

WW

WW

WWrTW

Bending Strength from Hardness(Fig 14-2 Shigley)

Contact Strength from Hardness(Fig 14-3 Shigley)

"I've had a wonderful time, but this wasn't it."

- Groucho Marx (1895-1977)