14 Spur and Helical Gears Chapter Outline 14-1The Lewis Bending Equation 14-2Surface Durability 14-3AGMA Stress Equations 14-4AGMA Strength Equations 14-5Geometry Factors I and J ( Z I and Y J ) 14-6The Elastic Coefficient Cp ( Z E ) 14-7Dynamic Factor K v 14-8Overload Factor K o 14-9Surface Condition Factor C f ( Z R ) 14-10Size Factor K s 14-11Load-Distribution Factor K m ( K H ) 14-12Hardness-Ratio Factor C H ( Z W ) 14-13Stress Cycle Life Factors Y N and Z N 14-14Reliability Factor KR ( Y Z ) 14-15Temperature Factor KT ( Y θ ) 14-16Rim-Thickness Factor K B 14-17Safety Factors S F and S H 14-18Analysis 14-19Design 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 K v 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
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 ν 1, ν 2, E 1, and E 2 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 r 1 and r 2 are the radii of curvature on the pinion- and gear-tooth profiles at the point of contact. Using an elastic coefficient C p And a velocity factor K v where the sign is negative because σ C is a compressive stress.
6 AGMA Stress Equation The fundamental equations for bending resistance are where for U.S. customary units (SI units), W t is the tangential transmitted load, lbf (N) K o is the overload factor K v is the dynamic factor K s is the size factor P d is the transverse diameteral pitch F (b) is the face width of the narrower member, in (mm) K m (KH) is the load-distribution factor K B is the rim-thickness factor J (Y J ) is the geometry factor for bending strength (which includes root fillet stress- concentration factor Kf ) (m t ) is the transverse metric module The fundamental equation for pitting resistance is C p (Z E ) is an elastic coefficient, lbf/in2 (N/mm2) C f (Z R ) is the surface condition factor d P (d w1 ) is the pitch diameter of the pinion, in (mm) I (Z I ) is the geometry factor for pitting resistance
7 AGMA Strength Equation The equation for the allowable bending stress is where for U.S. customary units (SI units), S t is the allowable bending stress, lbf/in2 (N/mm2) Y N is the stress cycle factor for bending stress K T (Y θ ) are the temperature factors K R (Y Z ) are the reliability factors S F is the AGMA factor of safety, a stress ratio The equation for the allowable contact stress σ c,all is where the upper equation is in U.S. customary units and the lower equation is in SI units. Also, S c is the allowable contact stress, lbf/in2 (N/mm2) Z N is the stress cycle life factor C H (Z W ) are the hardness ratio factors for pitting resistance K T (Y θ ) are the temperature factors K R (Y Z ) are the reliability factors S H 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 m F. This is defined as where p x 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 m N. 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 m G 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 m N, we obtain a factor valid for both spur and helical gears. where m N = 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 K o 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 C f or Z R 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 C f 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 K s in equation is less than 1, use K s = 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, C mf, where
16 Hardness-Ratio Factor The hardness-ratio factor C H is used only for the gear. The values of C H 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 10 7 load cycles applied. The purpose of the load cycle factors Y N and Z N is to modify the gear strength for lives other than 10 7 cycles.
18 Reliability Factor The reliability factor accounts for the effect of the statistical distributions of material fatigue failures. The gear strengths S t and S c are based on a reliability of 99 percent. A least-squares regression fit is
19 Rim-Thickness Factor The rim-thickness factor K B, adjusts the estimated bending stress for the thin-rimmed gear. It is a function of the backup ratio m B where t R = rim thickness below the tooth, in, and h t = the tooth height. The rim-thickness factor K B is given by
20 Safety Factor The ANSI/AGMA standards contain a safety factor S F guarding against bending fatigue failure and safety factor S H guarding against pitting failure. The role of the overload factor K o 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 K o.
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, K o Unquantifiable risk: design factor n d Tooth system: φ, ψ, addendum, dedendum, root fillet radius Gear ratio m G, N p, N G Quality number Q v Diametral pitch P d Face width F Pinion material, core hardness, case hardness Gear material, core hardness, case hardness a priori decisions design decisions