Chapter 14Spur and Helical Gears

Copyright © 2011 by The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Shigley’s Mechanical Engineering Design 9th Edition in SI units

Richard G. Budynas and J. Keith Nisbett

Prepared by

Kuei-Yuan ChanAssociate Professor of Mechanical Engineering

National Cheng Kung University

14 Spur and Helical Gears

Chapter Outline

14-1 The Lewis Bending Equation

14-2 Surface Durability

14-3 AGMA Stress Equations

14-4 AGMA Strength Equations

14-5 Geometry Factors I and J (ZI and YJ)

14-6 The Elastic Coefficient Cp (ZE)

14-7 Dynamic Factor Kv

14-8 Overload Factor Ko

14-9 Surface Condition Factor Cf (ZR)

14-10 Size Factor Ks

14-11 Load-Distribution Factor Km (KH)

14-12 Hardness-Ratio Factor CH (ZW)

14-13 Stress Cycle Life Factors YN and ZN

14-14 Reliability Factor KR (YZ)

14-15 Temperature Factor KT (Yθ)

14-16 Rim-Thickness Factor KB

14-17 Safety Factors SF and SH

14-18 Analysis

14-19 Design of a Gear Mesh

3

The Lewis Bending Equation

• Wilfred Lewis introduced an equation for estimating the bending stress in gear teeth in which the tooth form entered into the formulation.

• A cantilever of cross-sectional dimensions F and t has a length l and a load W t, uniformly distributed across the face width F. Its bending stress is

• Assume that the maximum stress in a gear tooth occurs at point a. By similar triangles

• Letting y = 2x/3p, we have

This completes the development of the original Lewis equation.

• The factor y is called the Lewis form factor.

4

Dynamic Effects

• When a pair of gears is driven at moderate or high speed and noise is generated, it is certain that dynamic effects are present.

• AGMA standards ANSI/AGMA 2001-D04 and 2101-D04 contain this caution:“ Dynamic factor Kv has been redefined as the reciprocal of that used inprevious AGMA standards. It is now greater than 1.0. In earlier AGMAstandards it was less than 1.0. ”

• Barth Equation

• The Barth equation is often modified ,for cut or milled teeth.

• Introducing the velocity factor gives

5

Surface Durability

• The surfaces of gear teeth wear includes pitting, due to repetitions of high contact stresses; scoring, a lubrication failure; and abrasion, due to the presence of foreign material.

• The Hertz contact stress between two cylinders is

where

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

• Replacing F by W t/cos φ, d by 2r, and l by the face width F, the surface compressive stress (Hertzian stress) is found from the equation

r1 and r2 are the radii of curvature on the pinion- and gear-tooth profiles at the point of contact.

• Using an elastic coefficient Cp

And a velocity factor Kv

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

6

AGMA Stress Equation

• The fundamental equations for bending resistance are

where for U.S. customary units (SI units),Wt is the tangential transmitted load, lbf (N)Ko is the overload factorKv is the dynamic factorKs is the size factorPd is the transverse diameteral pitchF (b) is the face width of the narrower member, in (mm)Km (KH) is the load-distribution factorKB is the rim-thickness factor

J (YJ ) is the geometry factor for bending strength (which includes root fillet stress-concentration factor Kf )(mt ) is the transverse metric module• The fundamental equation

for pitting resistance is

Cp (ZE ) is an elastic coefficient, √lbf/in2 (√N/mm2)Cf (ZR) is the surface condition factordP (dw1) is the pitch diameter of the pinion, in (mm)I (ZI ) is the geometry factor for pittingresistance

7

AGMA Strength Equation• The equation for the allowable bending stress is

where for U.S. customary units (SI units),St is the allowable bending stress, lbf/in2 (N/mm2)

YN is the stress cycle factor for bending stress

KT (Yθ ) are the temperature factors

KR (YZ ) are the reliability factors

SF is the AGMA factor of safety, a stress ratio

• The equation for the allowable contact stress σc ,all is

where the upper equation is in U.S. customary units and the lower equation is in SI units. Also,

Sc is the allowable contact stress, lbf/in2 (N/mm2)

ZN is the stress cycle life factor

CH (ZW) are the hardness ratio factors for pitting resistance

KT (Yθ ) are the temperature factors

KR (YZ ) are the reliability factors

SH is the AGMA factor of safety, a stress ratio

8

Geometry Factor J

• The determination of I and J depends upon the face-contact ratio mF . This is defined as

where px is the axial pitch and F is the face width.

• Bending-Strength Geometry Factor J (YJ ) :The AGMA factor J employs a fatigue stress-concentration factor Kf ; and a tooth load-sharing ratio mN . The resulting equation for J for spur and helical gears is

9

Geometry Factor I

• The factor I is also called the pitting-resistance geometry factor by AGMA.

• Define speed ratio mG as

The geometry factor I for external spur and helical gears is the denominator of the second term in the brackets.

• By adding the load-sharing ratio mN , we obtain a factor valid for both spur and helical gears.

where mN = 1 for spur gears.

10

The Elastic Coefficient

11

Dynamic Factor

• Dynamic factors are used to account for inaccuracies in the manufacture and meshing of gear teeth in action.

• To account for these effects, AGMA has defined a set of quality numbers defining the tolerances for gears of various sizes manufactured to a specified accuracy.• Quality numbers 3 to 7 will include most commercial-quality gears. Quality numbers 8 to 12 are of precision quality.

• The dynamic factor based on Qv

where

12

Overloading Factor

• The overload factor Ko is intended to make allowance for all externally applied loads in excess of the nominal tangential load W t in a particular application.

13

Surface Condition Factor

• The surface condition factor Cf or ZR is used only in the pitting resistance equation.

• It depends on Surface finish as affected by, but not limited to, cutting,

shaving, lapping, grinding, shotpeening Residual stress Plastic effects (work hardening)

• Standard surface conditions for gear teeth have not yet been established. AGMA specifies a value of Cf greater than unity.

14

Size Factor

• The size factor reflects nonuniformity of material properties due to size.

• Standard size factors for gear teeth have not yet been established AGMA recommends a size factor greater than unity.

• If Ks in equation is less than 1, use Ks = 1.

15

Load-Distribution Factor• The load-distribution factor modified the stress equations to

reflect nonuniform distribution of load across the line of contact.

• The load-distribution factor under these conditions is currently given by the face load distribution factor, Cmf , where

16

Hardness-Ratio Factor

• The hardness-ratio factor CH is used only for the gear. The values of CH

are obtained from the equation

• When surface-hardened pinions with hardness of 48 Rockwell C scale (Rockwell C48) or harder are run with through-hardened gears (180–400 Brinell), a work hardening occurs.

17

Stress Cycle Factors

• The AGMA strengths are based on 107 load cycles applied. The purpose of the load cycle factors YN and ZN is to modify the gear strength for lives other than 107 cycles.

18

Reliability Factor

• The reliability factor accounts for the effect of the statistical distributions of material fatigue failures.

• The gear strengths St and Sc are based on a reliability of 99 percent.

• A least-squares regression fit is

19

Rim-Thickness Factor

• The rim-thickness factor KB, adjusts the estimated bending stress for the thin-rimmed gear. It is a function of the backup ratio mB

where tR = rim thickness below the tooth, in, and ht = the tooth height.

• The rim-thickness factor KB is given by

20

Safety Factor

• The ANSI/AGMA standards contain a safety factor SF guarding against bending fatigue failure and safety factor SH guarding against pitting failure.

•

•

• The role of the overload factor Ko is to include predictable excursions of load beyond W t based on experience. A safety factor is intended to account for unquantifiable elements in addition to Ko.

21

Analysis Example 1

22

Analysis Example 2

23

Design of a Gear Mesh

• A useful decision set for spur and helical gears includes Function: load, speed, reliability, life, Ko

Unquantifiable risk: design factor nd

Tooth system: φ, ψ, addendum, dedendum, root fillet radius Gear ratio mG, Np, NG

Quality number Qv

Diametral pitch Pd

Face width F Pinion material, core hardness, case hardness Gear material, core hardness, case hardness

a priori decisions

design decisions