Wt

A

B

C

The above photo-elastic analysis of two gear teeth in contact shows that there are two types of high stress on the teeth. At A and B we see the tensile and compressive stresses due to bending of the tooth. Note that the compressive stress has a greater magnitude due to the radially inward component of the tooth force Wt. The bending stress is cyclic as it occurs once per revolution of the gear and will, thus, lead to a potential fatigue failure.

At C we have a contact stress situation as the two, approximately cylindrical surfaces roll and slide on each other during tooth contact. This stress may lead to a surface fatigue of the tooth.

Lewis Equation for Gear Strength.Historically, the first equation used for the bending stress was the Lewis equation. This is derived by treating the tooth as a simple cantilever and with tooth contact occurring at the tip as shown above. Only the tangent component (Wt) is considered. It is also assumed that only one pair of teeth is in contact. Stress concentrations at the tooth root fillet are ignored. It can be shown that the maximum bending stress occurs at the tangent points on the parabola shown above. Use of the standard equation for bending stress (σ = Mc/I) leads to:

and, letting y = 2x/3p we get the original Lewis

equation: where p = circular pitch

Replacing p with the diametral pitch P ( = π/p ) gives the more usual form:

3/2FxppWt=σ

FpyWt=σ

σ = WtP/FY, where

Wt = tangential tooth load

P = diametral pitch

F = face width of tooth

σ = bending stress in gear tooth

Y = Lewis form factor

The Lewis form factor is a function of the number of teeth N and the pressure angle φ and is given in the following chart. Note that the old 14.5 degree form factor is not generally used today.

Dynamic effects:

It is found that, if a pair of gears is run under load, the safe load on them decreases as the running speed increases. Specifically, it is a function of the pitch line velocity V = πdN/12 ft/min where d is the pitch diameter in inches and N is the rotation speed in rpm.

This is accounted for by the use of a velocity factor Kv.

Values of Kv derived by Barth are used with the Lewis equation:

Kv = (600 + V)/600 for crude gears (typically cast metal)

Kv = (1200 +V)/1200 for cut or milled teeth

Hence we get the gear stress as

We will not use the Lewis equation or the Barth velocity factors in this course as they have been superseded by the AGMA equations.

FYPWK tv=σ

AGMA Gear Stress EquationThis equation modifies the Lewis equation to take into account:

1. the effect of the radial forceWr

2. the root stress concentrations

3. the effects of having multiple pairs of teeth in contact

σ = WtPKvKoKmKsKB/FJJ = AGMA gear geometry factor

Kv = velocity factor

Ko = overload factor

Km = mounting factor

Ks = size factor

KB = rim thickness factor

One important difference is that the AGMA geometry factor J is a function of the numbers of teeth on both gears.

All of the equations and data in this document applies to design of spur gears using standard US units (inches, pounds, etc.)

Dynamic factor Kv:

AGMA has defined quality control numbers Qv. Classes 3 to 7 cover most commercial quality gears and classes 8 to 12 cover high precision gears. Values of Kv as a function of Qv and pitch line velocity (in ft/min) are given in the chart above. These are derived from the following equations:

where

The maximum velocity for any given Qv value is:

The following chart and equations give older values for Kv. These are used in the gear.exe program discussed elsewhere in the ME356 web site notes..

B

v AVAK

+= ( ) ( ) 3/21225.015650 vQBandBA −=−+=

( )[ ] min/3 2max ftQAV v −+=

The overload and load distribution factors are highly qualitative in nature as you will see from sections 14-8 and 14-11 in your text. For the purposes of this course you should use the values given in the following two tables.

Overload factor KO

Load distribution factor Km

Size factor Ks: use the following table

1.010121.1005

1.029101.1244

1.05281.1563

1.06571.1762.5

1.08161.2022

Factor KsPitch PFactor KsPitch P

The rim thickness factor is only needed for large gears with a spoked hub – see secton 14-16 in your test.

Gear surface stresses:Surface failure of gear teeth has two causes – (a) fatigue failure leading to pitting or spalling and (b) wear due to the sliding contact that occurs between involute tooth surface during tooth contact (except at the pitch point). These failures are related to the high, very local contact stresses that occur. The compressive stress at the contact point between two cylinders is given by:

where F = force pressing the cylinders together

l = length of the cylinders

b = half width of the contact surface

Where ν1, E1, ν2, E2, d1 and d2 are the elastic constants and diameters of the cylinders.

For two gear teeth we can replace F by Wt/cos φ, pmax by σC, d by 2r and l by the face width F. This gives the surface contact compressive stress as:

blFp

π2

max =

( )[ ] ( )[ ]( ) ( )

2/1

21

2221

21

/1/1/1/12

+−+−

=dd

EElFb ννπ

( ) ( )( )[ ] ( )[ ]2

221

21

212

/1/1/1/1

cos EErr

FWt

C ννφπσ

−+−+

=

Where r1 and r2 are the radii of curvature of the tooth surfaces at the contact point. As the first evidence of tooth wear is seen at the pitch point, the curvatures at the pitch point are used:

Where dP and dG are the pitch diameters of the pinion and gear respectively.

Note that the denominator in the above contains only the elastic constants for the materials of the gear and the pinion. We thus define the elastic coefficient Cp:

2sin

2sin

21φφ GP dranddr ==

( )[ ] ( )[ ]2/1

22 /1/11

−+−

=PPGG

p EEC

ννπ

This is tabulated below for various common combinations of gear and pinion materials.

The expression in the contact stress equation that contains only values that are dependent on the geometry of the gear and pinion can be written as:

We get the surface geometry factor for spur gears only.

All of this leads to:

P

G

P

GG

GP dd

NNmratiospeedtheifand

ddrrI ==

+=+= ,,11

sin211

21 φ

12sincos

+=

G

G

mmI φφ

AGMA Surface Stress Equation:

Wt, Ko, Kv, Ks, Km and F are the same quantities that were defined above for the bending stress equation. The elastic coefficient Cp and the geometry factor I are defined above. dP is the pinion diameter.

The surface condition factor Cf has not yet been evaluated by AGMA so its value is always 1.0.

IC

FdKKKKWC f

P

msvotpC =σ

Strength of gear materials:

AGMA uses its own data for strengths. These values should only be used in gear tooth strength calculations.

The allowable bending strength numbers (St) are given in your text in Figures 14-2, 14-3, 14-4 and Tables 14-3, 14-4 (pages 735-737).

These values are modified by a number of factors to arrive at the allowable bending stress σall.

Where: YN is the stress cycle factor (Figure 14-14, page 751)

KT is the temperature factor =1.0 for temp ≤ 250 F.

KR is the reliability factor (Table 14-10, page 752)

SF is the AGMA factor of safety

RT

N

F

tall KK

YSS

=σ

The allowable contact strength numbers are given in Figure 14-5 (page 738) and Tables 16-6, 14-7 (pages 739-740) in your text.

These values are modified by a number of factors to arrive at the allowable bending stress σc,all.

where KT, KR are defined above and:

ZN is the stress cycle life factor (Figure 14-15, page 751)

SH is the AGMA factor of safety

CH is the hardness ratio factor. This is discussed in section 14-12 in your text. For the purposes of this course, use CH = 1.0.

RTH KKSHNC

allcCZS

=,σ

Design equations for spur gears:

These are simply two separate equations that must both be used in a gear set design.

1. Bending strength - σ = σall

2. Surface strength - σC = σc,all

There are two variables to be found in the design of a gear set consisting of a gear and a pinion, the face width F and the diametral pitch P. The pitch P must be one of the preferred standard values (see Table 13-2 in your text). Note that you can choose any of these pitch values and then solve for F. You should aim for tooth proportions such that the face width lies in the range 3p to 5p where p is the circular pitch p = π/P inch. If you must violate this rule, do so with values less than 3p.

You should use the bending strength equation separately for the gear and pinion if they are made of different materials or have different heat treatment. Otherwise, you only have to design for the pinion as the larger gear will have a larger geometry factor J.

The surface strength calculation is a completely different case and must always be done in addition to the bending strength calculation. You only have to do this for the gear set member with the lower contact strength as the calculation already incorporates the elastic properties and geometry of both gears.