2 14 Spur and Helical Gears Chapter Outline 14-1The Lewis Bending Equation14-2Surface Durability14-3AGMA Stress Equations14-4AGMA Strength Equations14-5Geometry Factors I and J (ZI and YJ)14-6The Elastic Coefficient Cp (ZE)14-7Dynamic Factor Kv14-8Overload Factor Ko14-9Surface Condition Factor Cf (ZR)14-10Size Factor Ks14-11Load-Distribution Factor Km (KH)14-12Hardness-Ratio Factor CH (ZW)14-13Stress Cycle Life Factors YN and ZN14-14Reliability Factor KR (YZ)14-15Temperature Factor KT (Yθ)14-16Rim-Thickness Factor KB14-17Safety Factors SF and SH14-18Analysis14-19Design of a Gear Mesh1-1Design1-2Mechanical Engineering Design
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 isAssume that the maximum stress in a gear tooth occurs at point a. By similar trianglesLetting y = 2x/3p, we haveThis completes the development of the original Lewis equation.The factor y is called the Lewis form factor.
4 Dynamic EffectsWhen 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 EquationThe Barth equation is often modified ,for cut or milled teeth.Introducing the velocity factor gives
5 Surface DurabilityThe 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 equationThe Hertz contact stress between two cylinders isr1 and r2 are the radii of curvature on the pinion- and gear-tooth profiles at the point of contact.Using an elastic coefficient CpwhereAnd a velocity factor Kvν1, 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.
6 AGMA Stress EquationThe fundamental equations for bending resistance are(mt ) is the transverse metric moduleThe fundamental equation for pitting resistance iswhere 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 factorKo is the overload factordP (dw1) is the pitch diameter of the pinion, in (mm)Kv is the dynamic factorKs is the size factorI (ZI ) is the geometry factor for pittingPd is the transverse diameteral pitchresistanceF (b) is the face width of the narrower member, in (mm)Km (KH) is the load-distribution factorKB is the rim-thickness factorJ (YJ ) is the geometry factor for bending strength (which includes root fillet stress-concentration factor Kf )
7 AGMA Strength Equation The equation for the allowable bending stress iswhere 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 stressKT (Yθ ) are the temperature factorsKR (YZ ) are the reliability factorsSF is the AGMA factor of safety, a stress ratioThe equation for the allowable contact stress σc ,all iswhere 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 factorCH (ZW) are the hardness ratio factors for pitting resistanceSH is the AGMA factor of safety, a stress ratio
8 Geometry Factor JThe determination of I and J depends upon the face- contact ratio mF . This is defined aswhere 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 IThe factor I is also called the pitting-resistance geometry factor by AGMA.Define speed ratio mG asThe 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.
11 Dynamic FactorDynamic 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 Qvwhere
12 Overloading FactorThe 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 onSurface finish as affected by, but not limited to, cutting, shaving, lapping, grinding, shotpeeningResidual stressPlastic 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 FactorThe 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 equationWhen 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 FactorsThe 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 FactorThe 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 FactorThe rim-thickness factor KB, adjusts the estimated bending stress for the thin-rimmed gear. It is a function of the backup ratio mBwhere tR = rim thickness below the tooth, in, and ht = the tooth height.The rim-thickness factor KB is given by
20 Safety FactorThe 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.