Chapter 14 Spur and Helical Gears Shigley’s Mechanical Engineering Design 9th Edition in SI units Richard G. Budynas and J. Keith Nisbett Chapter 14 Spur and Helical Gears Prepared by Kuei-Yuan Chan National Cheng Kung University Prepared by Kuei-Yuan Chan Associate Professor of Mechanical Engineering National Cheng Kung University Copyright © 2011 by The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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 1-1Design1-2Mechanical Engineering Design

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.

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 in previous AGMA standards. It is now greater than 1.0. In earlier AGMA standards it was less than 1.0. ” Barth Equation The Barth equation is often modified ,for cut or milled teeth. Introducing the velocity factor gives

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. 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 The Hertz contact stress between two cylinders is 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 where And a velocity factor Kv ν1, ν2, E1, and E2 are the elastic constants and d1 and d2 are the diameters of the two contacting cylinders. where the sign is negative because σC is a compressive stress.

AGMA Stress Equation The fundamental equations for bending resistance are (mt ) is the transverse metric module The fundamental equation for pitting resistance is where for U.S. customary units (SI units), Cp (ZE ) is an elastic coefficient, √lbf/in2 (√N/mm2) Wt is the tangential transmitted load, lbf (N) Cf (ZR) is the surface condition factor Ko is the overload factor dP (dw1) is the pitch diameter of the pinion, in (mm) Kv is the dynamic factor Ks is the size factor I (ZI ) is the geometry factor for pitting Pd is the transverse diameteral pitch resistance F (b) is the face width of the narrower member, in (mm) Km (KH) is the load-distribution factor KB is the rim-thickness factor J (YJ ) is the geometry factor for bending strength (which includes root fillet stress-concentration factor Kf )

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 SH is the AGMA factor of safety, a stress ratio

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

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.

The Elastic Coefficient

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

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.

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.

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.

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

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.

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.

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

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

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.

Analysis Example 1

Analysis Example 2

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