Gear Analysis

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Analysis of Spur and Helical Gears prepared by Wayne Book based on Norton, Machine Design and Mischke and Shigley Mechanical Engineering Design The Gnashing of Teeth Simple model for loaded gears Beam for bending stress Cylinders in contact for surface contact stress Idealized Shape of a Tooth for Stress Analysis Simple model: cantilever beam with applied force W Tooth thickness t Length l Face width F Max stress at root (a) l Wt F t a 2 36) 12 / (2 / ) (Ftl WFtt l WIMct t= = = oConsider the Shape of a Tooth Uncertainties include: point of load application l point of maximum stress appropriate load component beam thickness Depends on pitch P, number of teeth N and pressure angle | Conservative assumptions are made Y = Lewis form factor Introduce Lewis Shape Factor t l Wr Wt W x 3264 2 /2 /iangles similar tr By 212 6222 2xPYFYP WlP tFP WltxtlxtPtlFP WFtl Wt tt t== == == =ooRather than calculate Y(P,, N), create a table, e.g. 14-2 Lewis equation has been improved by AGMA Velocity Effect (Its Barth not Barf) Purely empirical adjustment for non-zero velocity Barths equation (1800s) has been modified to account for current practice and accuracy V is velocity in ft/sec at the pitch line Kv= 1200/(1200+V)(Modified Barth) Metric form Kv= 6.1/(6.1+V), V in m/sec Compare to endurance strength (reversing) or use Goodman diagram (one direction) Apply notch sensitivity, Marin factors. the works FY KP Wvt= oSurface Durability:Contact Stress Analyzed as two cylinders of length l in rolling contact with specified force Cylinder radii r1 and r2 vary with contact point Depends on elastic material properties and radii of cylinders Translate into gear nomenclature as shown on right factor velocityCpinion and gearrefer to subscripts P G,modulus s Young' Eratio s Poisson'1111 1cosv2 / 1222 / 12 1===((((((

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\|+ =vvvt|oGGpppVtp cE ECr r F CWCAGMA Approach AGMA formula calculates stress for Bending Contact Stress is compared to an allowable stress (also called strength by Norton) based on strength and conditions Bending Stress Many terms are similar to the Lewis equation Additional terms account for the application, load sharing and size factor geometrytooth factor backup rimfactor ondistributi loadfactor size width facepitch diametralLewis inas factorvelocity factor napplicatiofactor idler load l tangentia===========JKKKFPKKKWJK K KFPKK KWBmsdvaItB m s dva Itoloading gear geometry tooth form J factor sample table Tip loading (low precision) Distributed loading (higher precision) Kv Velocity Factor (similar to Barth) (also provided in equations 11.16 11.19) Load Distribution Factor Loads are less evenly distributed for wide face teeth Keep F (face width) 8/pd < F < 16/pd

Nominally F = 12/pd

Application Factor Created to account for known but unquantified shock in load Electric motors are smooth while single cylinder engines have shock Centrifugal pumps are smooth loads while rock crushers have shock Other Factors Size, Rim Thickness, Idler Size Fatigue tests are done on small specimins and indications are that size results in weaker parts Very large teeth might warrant Ks=1.25 to 1.5 Material properties created directly for gears account for this Rim thickness In large diameter gears, the centers are connected to a rim by spokes. KB reflects failures across the radius Idler: use KI = 1.42 Allowable Bending Stress Incorporate material strength St specific to gear materials St based on Brinell hardness of material Environmental and application factors KL = life factor KT = temperature factor KR = reliability factor R TL tallK KK S= oLife Factor KL (a specialized S-N curve) BendingTemperature and Reliability Factors Strength data is based on 99% reliability.Adjust up or down. Temperatures up to 250 deg F use KT = 1 Adjust for higher temperatures 620460FTTK+=AGMA Bending Fatigue Strengths (uncorrected) Contact Stress Based on rolling cylinder model Added terms for size, load distribution, surface condition factor geometrytooth factor conditionsurfacefactor ondistributi loaddiameter pitch factor sizefactor (dynamic) velocity factor napplicatiot coefficien elastic2 / 1========((

=ICCdCCCCIC CFdCCC WCfmsvaPf msva tP coloading gear geometry tooth condition & geometry material Surface Geometry Factor I ( )h) depth teet full for(0elongation addendum fraction gear forcurvature of radiuspinion forcurvature of radiusangle pressurepinion of radius pitch pinion of pitchdiametrialsincos cos11 1cos22======= ||.|

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\|=pgppdp gdpdpp ppg pxrpCprpxrdI| | |t| |AGMA Elastic Coefficient (also from basic material properties and (11.23)) Other Surface Stress Factors Cf = 1 for standard manufacturing methods Ca, Cm, Cv, Cs are equal to corresponding K values from bending Allowable Contact Stress (Norton calls Strength) Material strength SC is the basis, specific to gear materials Sc based on Brinell hardness of material or on tables in Norton Adjust for conditions CL = life factor CH = hardness-ratio factor (pinion rel to gear) CT = temperature factor CR = reliability factor R TH L fcfcC CC C SS'=Surface Fatigue Strengths Surface Fatigue Life Factor Hardness Ratio Factor Only applied to the gear material (not pinion) Accounts for work hardening of the gear during run-in Depends on previous hardening (through hardened vs surface hardened) gear pinion, of hardness Brinnel ,00698 . 0 7 . 100829 . 0 00898 . 0 7 . 1 2 . 10 2 . 1ratio gear ) 1 ( 1== > =