1

CHAPTER 14

Spur and Helical Gears

In this chapter we are going to analyze and design Spur and

Helical gears to:

Resist bending failure of the teeth

Pitting failure of tooth surfaces.

Failure by bending will occur when the significant tooth

stress equals or exceeds either the yield strength or the

bending endurance strength.

A surface failure occurs when the significant contact stress

equals or exceeds the surface endurance strength.

2

In the first two sections a little of the history of the analyses

from which current methodology developed is going to be

covered.

The American Gear Manufacturers Association (AGMA)

has for many years been the responsible authority for the

distribution of knowledge pertaining to the design and

analysis of gearing. The methods this organization presents

are in general use when strength and wear are primary

considerations.

3

The general AGMA approach requires a great many charts and graphs.

Many of these charts and graphs can be omitted by choosing a single pressure angle and by using only full-depth teeth.

This simplification reduces the complexity and makes possible a better development of the fundamentals and hence should constitute an ideal introduction to the use of the general AGMA method.

4

14.1 The Lewis Bending Equation

Static Effect

An equation for estimating the bending stress in gear teeth in which the tooth form entered into the formulation was introduced by Wilfred Lewis.

To derive the basic Lewis equation (See Fig. 14-la) which shows a cantilever of cross-sectional dimensions F and t, having a length l and a load Wt, uniformly distributed across the face width F.

5

The section modulus I/c is Ft2/6, and therefore the bending

stress is:

(a)2

6

/ Ft

lW

cI

M t

6

The maximum stress in a gear tooth occurs at point a as shown

in figure 14-1a.

Using the similarity of triangles, we can write:

(B)

=l

x

t / 2

l

tx

l

t

t

x

4

2/

2/

2

7

From equation (a): we can rewrite it as the following:

(C)

Substitute (b) into (c) and multiply the numerator and

denominator by the circular pitch p, we find

(d)

Letting y = 2x/3p, we have

(14-1)

6/4

1

4/

1

6/

16222 ltF

W

ltF

W

Ft

lW ttt

xpF

pW t

3/2

Fpy

W t

8

Equation (14-1) is known as Lewis equation and y is called

the Lewis form factor.

Most engineers prefer to employ the diametral pitch in

determining the stresses. This is done by substituting P=π / p

and Y= π y in equation (14-1). This gives

(14-2)

Where

(14-3)

Equation (14-2) considers only the bending of the tooth. And

the effect of the radial load Wr is neglected.

FY

PW t

3

2xPY

9

The value of Y can be found

in table 14-2. (These values

are for a normal pressure

angle of 20o, Full-Depth

Teeth, and a Diametral Pitch

of Unity in the Plane of

Rotation)

Number of

TeethY

Number of

TeethY

12 0.245 28 0.353

13 0.261 30 0.359

14 0.277 34 0.371

15 0.290 38 0.384

16 0.296 43 0.397

17 0.303 50 0.409

18 0.309 60 0.422

19 0.314 75 0.435

20 0.322 100 0.447

21 0.328 150 0.460

22 0.331 300 0.472

24 0.337 400 0.480

26 0.346 Rack 0.485

10

Eq. (14-3) also implies that

the teeth do not share the load

the greatest force is exerted at the tip of the tooth.

the contact ratio should be somewhat greater than unity, say

about 1.5, to achieve a quality gearset. If, in fact, the gears are

cut with sufficient accuracy, the tip-load condition is not the

worst, because another pair of teeth will be in contact when

this condition occurs.

11

Examination of run-in teeth will show that the heaviest loads

occur near the middle of the tooth. Therefore the maximum

stress probably occurs while a single pair of teeth is carrying

the full load, at a point where another pair of teeth is just on

the verge of coming into contact.

12

Dynamic Effect

Dynamic effect occurs when a pair of gears is driven at

moderate or high speed and noise is generated

Barth first expressed the velocity factor in terms of AGMA

standards and can be represented as (SI units):

(14-6a)

Where V is the pitch-line velocity in m/s

profile)cast iron,(cast 05.3

05.3 VK

13

Barth equation is modified for cut or milled profile (see

equation 14-6b)

(14-6b)

Later AGMA added two velocity factors for shaped and

ground profiles and can be written as:

(14-6c)

(14-6d)

profile) milledor (cut 10.6

10.6 VK

profile) shapedor (hobbed 56.3

56.3 VK

profile) groundor (shaved 56.5

56.5 VK

14

V is in ft/min

15

By introducing the velocity factor into (14-2) gives

(14-7)

The metric version of this equation is

(14-8)

Where

FY

PWK t

FmY

WK t

(N) load theofcomponent tangential

(mm) modile them

(mm) width face the

tW

F

16

As a general rule, spur gears should have a face width F from

3 to 5 times the circular pitch p.

Equations (14-7) and (14-8) are important because they form

the basis for AGMA approach to the bending strength of gear

teeth. They are in general use for estimating the capacity of

gear drives when life and reliability are not important

considerations.

The equations can be useful in obtaining a preliminary

estimate of gear sizes needed for various applications.

17

Example 14-1

A stock spur gear is available having a diametral pitch of 8

teeth/in, a 1.50 in face, 16 teeth, and a pressure angle of 20°

with full-depth teeth. The material is AISI 1020 steel in as-

rolled condition. Use a design factor of nd = 3 to rate the

horsepower output of the gear corresponding to a speed of

1200 rev/m and moderate applications.

18

Solution

The term moderate applications seems to imply that the gear

can be rated by using the yield strength as a criterion of failure.

From Table A-20, we find:

Sut = 55 kpsi=379MPa and Sy = 30kpsi=207MPa

A design factor of 3 means that the allowable bending stress is

207/3 = 69 MPa

The pitch diameter is N/P = 16/8 = 2 in, so the pitch-line

velocity is:

smftdn

V /2028.3)628(0051.0min/62812

1200)2(

12

19

Using milled profile (eq. 14-6b) for the velocity factor

From table 14-2, for N =16 teeth Y = 0.296, thus

525.11.6

3028.9

1.6

1.6

VK

6.93hphp)746/1.5171(1.5171)10(60

16201200)4.252(

)10(60

35,-13equation From

1620525.1

)69)(296.0)(4.25125.0)(1.38(

33

WdnW

H

NK

FmYW

FmY

WK

t

allt

t

20

Example 14-2

Estimate the horsepower rating of the gear in the previous

example (14-1) based on obtaining an infinite life in bending.

21

Solution

The rotating-beam endurance limit is estimated from Eq. (7-8)

To obtain the surface finish Marin factor ka, for machine surface

MPa191)379(504.0504.0' ute SS

934.0)191(70.2

265.0,7.2

265.0

b

uta aSk

ba

22

For the size factor kb

948.03.0

495.025.0

0555.0)8(2

)296.0(3

2

3

281.08

25.1

8

125.11

,)/4(

5.1

)(808.0)(808.0

0107.

5.0

5.05.0

e

b

e

e

dk

indint

inP

Yx

inPP

l

xlt

inF

Fthbd

23

for load factor kc =1, for bending

Stress concentration factor Kf

For a 20o full-depth tooth the radius of the root fillet is

denoted rf, where rf = 0.3/P=0.3/8=0.0375 in

From Figure A-15-6

MPa12.169)191)(1)(1)(1)(1)(948.0)(934.0(

1

'

efedcbas

fed

SkkkkkkS

kkk

15.025.0

0375.0

t

r

d

r f

43.1)168.1(63.01)1(1

63.0 7.20, figure From

1.68 ,3 can take we,/ Since

tf

t

KqK

q

KD/ddD

24

These results should be accepted only as preliminary estimates

to alert you to the nature of bending in gear teeth.

hp96.3)746/2952(2952)10(60

58.925525.1

)42.39)(296.0)(4.25125.0(1.38

MPa42.39/

MPa373.563

12.169

3

WdnW

H

NK

FmYW

K

n

S

t

v

allt

foallall

d

e

oall

25

Dolan and Broghamer stress

concentration formula

(14-9)

Where

ML

fl

t

r

tHK

diameterpitch the dedendum, the radius,fillet angle, pressure

2/

4583662.0290.0

4583662.0316.0

4583662.034.0

2

dbr

rbd

rbr

M

L

H

f

f

f

26

Gear Teeth Contact

27

14.2 Surface Durability

In this section, the failure of the surfaces of gear teeth, which

is generally called wear is studied. To obtain an expression for

the surface-contact stress, the Hertz theory is employed.

It was shown that the contact stress between two cylinders

may be computed from the equation:

(a)

Wherebl

Fp

2max

cylinders oflength

togethercylinders two thepressing force

pressure surfacelargest max

l

F

p

28

(14-10)

2/1

21

2

2

21

2

1

/1/1

/1/12width-half

dd

EE

l

Fb

E1, ν1

E2, ν2

29

To adapt these relations to the notation used in gearing, we

replace F by Wt/cos(ϕ), d by 2r, and l by the face width F.

With these changes, we can substitute the value of b as given

by equation (14-10) in equation (a). Replacing pmax by σC, the

surface compressive stress (Hertzian stress) is found from the

equation:

(14-11)

Where r1 and r2 are the instantaneous values of the radii of

curvature on the pinion- and gear-tooth profiles, respectively,

at the point of contact.

2

2

21

2

1

212

C/1/1

/1/1

cos EE

rr

F

W t

30

By accounting for load sharing in the value of Wt used,

equation (14-11) can be solved for the Hertzian stress for any

or all points from the beginning to the end of tooth contact

Noted that the first evidence of wear occurs near the pitch line.

The radii of curvature of the tooth profiles at the pitch point

are:

(14-12)

where ϕ is the pressure angle and dP and dG are the pitch

diameters of the pinion and the gear, respectively.

In equation (14-11), that the denominator of the second group

of terms contains four elastic constants, two for the pinion and

two for the gear.

2

sin

2

sin21

GP dr

dr

31

As a simple means of combining and tabulating the results for

various combinations of pinion and gear materials, AGMA

defines an elastic coefficient CP by the equation

(14-13)

With this simplification, and the addition of a velocity factor

Kν, equation (14-11) can be written as:

(14-14)

Where the sign is negative because σC is a compressive stress.

2/1

22 11

1

G

G

P

P

P

EE

C

2/1

21

11

cos

rrF

WKC

t

PC

32

Example 14-3

The pinion of Example 14-1 and 14-2 is to be mated with a 50

tooth gear manufactured of ASTM No. 50 cost iron. Using the

tangential load of 382 lbf, estimate the factor of safety of the

drive based on the possibility of a surface fatigue failure.

33

Solution From Table A-5:

From example 14-1, the pinion pitch diameter is dP = 2in, and

for the gear dG = 50/8 =6.25in

Thus from equation (14-12):

1817

105.14

02111

1030

02921

1

211.0 Mpsi,5.14 ,292.0 Mpsi,30

2/1

6

2

6

2

P

GGPP

C

EE

in069.12

20sin25.6 in,342.0

2

20sin221

rr

34

The face width is given as F = 1.5 in and Kν =1.525 from

example 14-1. Substituting all these values in equation (14-14)

with pressure angle of 20o, thus the contact stress is

psi72400

069.1

1

342.0

1

20cos5.1

380525.11817

11

cos

2/1

2/1

21

rrF

WKC

t

PC

35

The surface endurance strength of cast iron can be estimated

from

For table A-24, HB = 262 for ASTM No. 50 cast iron, Thus

SC = 0.32(262) = 83.8 kpsi

Thus the factor of safety can be found as:

cycles, 10for kpsi, 32.0 8

BC HS

35.14.72

8.83

load imposed

loadfunction of loss22

n

36

14.3 AGMA Stress Equations

Two fundamental stress equations are used in the AGMA

methodology, these are:

The bending stress

The pitting resistance (contact stress).

In AGMA terminology, these are called stress numbers and

are designated by a lowercase letter s instead of the Greek

lower case σ. The fundamental equations are:

(14-15)

J

BH

t

so

t

Bmdso

t

Y

KK

bmKKKW

J

KK

F

PKKKW

1

units) (SI

units)customary (U.S.

37

Where for U.S. customary units (SI units),

)factor ion concentrat-stressfillet root includes(which

strengh bendingfor factor geometry the

factor thickness-rim the

factoron distributi-load the

(mm)in member,narrower theof width face the

pitch diameteral e transversthe

factor size the

factor dynamic the

factor overload the

(N) Ibf load, ed transmittl tangentiathe

f

J

B

Hm

d

s

o

t

K

YJ

K

KK

bF

P

K

K

K

W

mt: transverse metric module

38

Before you try to digest the meaning of all these terms in

equation (14-15), view them as advice from AGMA

concerning items the designer should consider whether you

follows the voluntary standard or not. These items include

issues such as:

Transmitted load magnitude

Overload

Dynamic augmentation of transmitted load

Size

Geometry: pitch and face width

Distribution of load across the teeth

Rim support of the tooth

Lewis form factor and root fillet stress concentration

39

The fundamental equation for pitting resistance (contact stress)

is

(14-16)

Where

The evaluation of all these factors is explained in the sections

that follow.

I

R

w

Hso

t

E

f

P

m

so

t

P

C

Z

Z

bd

KKKKWZ

I

C

Fd

KKKKWC

1

units) (SI

units)customary (U.S.

resistance pittingfor factor geometry the)(

(mm)in pinion, theofdiameter pitch the)(

factorcondition surface the)(

)N/mm(Ibf/in t,coefficien elastican )(

1

22

I

WP

Rf

Ep

ZI

dd

ZC

ZC

40

14.4 AGMA Strength Equations

41

Following this convention, values for AGMA bending strength, designated here as St, are to be found in Figs. 14-2, 14-3, and 14-4, and in Tables 14-3 and 14-4.

Since AGMA strengths are not identified with other strengths such as Sut, Se, or Sy as used elsewhere in this book, their use should be restricted to gear problems.

42

Figure 14-2: Allowable bending stress number for through-hardened steels. The SI equations are St , σFP =0.533HB + 88.3 MPa, grade 1, and σFP = 0.703HB+ 113 MPa, for grade 2.

43

Figure 14-3: Allowable bending stress number for nitrided

through-hardened steel gears (i.e., AISI4140, 4340, St. The SI

equations are σFP =0.568HB + 83.8 MPa, grade 1, and σFP =

0.749HB+ 110 MPa, for grade 2.

44

Figure 14-4: Allowable bending stress number for nitriding through-

hardened steel gears St. The SI equations are σFP =0.594HB + 87.76

MPa Nitralloy grade 1, σFP =0.784HB + 114.81 MPa Nitralloy grade

2, σFP =0.7255HB + 63.89 MPa 2.5% chrome, grade 1, σFP

=0.7255HB + 153.63 MPa 2.5% chrome, grade 2, and σFP =

0.7255HB+ 201.91 MPa 2.5% chrome, grade 3.

45

46

In the AGMA approach the strengths are modified by various

factors that produce limiting values of the bending stress and

the contact stress. Using similar notation to that in equation (1-

6) we shall term the resulting modifications the allowable

bending stress σall and the allowable contact stress σC, all . The

equation for the allowable bending stress is:

(14-17)

Where for the U.S. customary units (SI units),

Z

N

F

FP

RT

N

F

t

all

YY

Y

S

KK

Y

S

S

units) (SI

units)customary (U.S.

47

ratio stress a safety, offactor AGMA the

factorsy reliabilit the

factors re temperatuthe

stress bendingfor factor cycle stress the

)(N/mmlbf/in stress, bending allowable 22

F

ZR

T

N

FPt

S

YK

YK

Y

S

48

The equation for the allowable contact stress σC,all is

(14-18)

Where the upper equation is in U.S customary units and the

lower equation is in SI units, Also

Z

WN

H

HP

RT

HN

H

C

allC

YY

ZZ

S

KK

CZ

S

S

,

units) (SI

units)customary (U.S.

49

The values for the AGMA allowable stress numbers

(strengths) for bending and contact stress are for

Unidirectional loading

10 million stress cycles

99 percent reliability

ratio stress a safety, offactor AGMA the

factorsy reliabilit the

factors re temperatuthe

resistance pittingfor factors ratio hardness the

factor life cycle stress the

)(N/mmlbf/in stress,contact allowable 22

H

ZR

T

H

N

HPC

S

YK

YK

C

Z

S

50

Figure 14-5: Contact-fatigue strength SC at 107 cycles and

0.99 reliablity for through-hardened steel gears. The SI

equations are σHP =2.22HB + 200 MPa, grade 1, and σHP

=2.41HB + 237 MPa, grade 2

51