[4]-Contribution-of-Gear-Body-to-Tooth-Deflections

ContributionofGearBodytoToothDeflections—ANewBidimensionalAnalyticalFormulaP.SainsotandP.VelexContactandSolidMechanicsLaboratory,UMRCNRS5514INSAdeLyon,20AvenueAlbertEinstein,69621VilleurbanneCedex,Francee-mail:[e-mail]ˆleMachinesetCommandes,CETIMSenlis,FranceThemagnitudeandvariationoftoothpaircomplianceaffectstoothloadingandgeardynamicssignificantly.Thispaperpresentsanimprovedfillet/foundationcomplianceanalysisbasedonthetheoryofMuskhelishviliappliedtocircularelasticrings.Assum-inglinearandconstantstressvariationsatrootcircle,theabovetheorymakesitpossibletoderiveananalyticalformulaforgearbody-inducedtoothdeflectionswhichcanbedirectlyintegratedintogearcomputercodes.ThecorrespondingresultsareinverygoodagreementwiththosefromfiniteelementmodelsandtheformulaisprovedtobesuperiortoWeber’swidelyusedequation,especiallyforlargegears.@DOI:10.1115/1.1758252#1IntroductionIn1949,Weber@1#,andlaterWeberandBanascheck@2#pro-posedamethodofcalculatingtoothdeflectionsofspurgearsbysuperimposingdisplacementswhicharisefromi!thecontactbe-tweentheteeth,ii!thetoothitselfconsideredasabeamand,iii!thegearbodyregardedasasemi-infiniteelasticplane.ExtensionsandvariantsofthemethodologywereintroducedbyO’Donnell@3–4#withregardtothefoundationeffectsandbyAttia@5#,Cor-nell@6#amongothers,concerninganalyticaldevelopments.WeberanalyzedthecontactcompliancebyusingtheHertzian2Dtheoryforcylindersincontactwiththedatumfordisplace-mentstakenatthetoothcenterline,andgavethenormalapproachbetweenthepartsincontactundertheform:dc54Fb12n2pEFln2Ak1k2a2n2~12n!G(1)withF:forceonthetoothb:toothfacewidthk1,k2:thedistancesonthepinion,onthegearbetweenthepointofcontactandthetoothcenterline~Fig.1!a5A8F/br1r2/r11r212n2/pE:halfwidthofcontactzone~intheprofiledirection!E,n:Young’smodulus,Poisson’sratio.r1,r2:radiiofcurvature~Fig.1!.Theotherwidely-usedformulasfortoothcontactdeflectioncomprisetheanalyticalformulaofLundberg@7#,theapproximatehertzianapproachoriginallyusedatHamiltonStandard@6#andthesemi-empiricalformuladevelopedbyPalmgrenforrollers@6#.Toothbendingdeformationswerederivedbyconsideringthetoothasacantileverofvariablecross-sectionandequatingthestrainenergytotheworkoftheexternalforce.ThecorrespondingWeberequationforbendingdisplacementsreads:db5Fb1Ecos2auF10.92E0uw~uw2y!2d~y!3dy13.1~110.294tg2au!E0uw1d~y!dyG(2)withau:pressureangleuwandd(y)aredefinedinFig.1Forthecontributionofgearbody,thetoothissupposedtoberigidandthewheelbodyismodeledasanelastichalfplaneonwhichthenormalandtangentialforcesaswellasthebendingmomentareapplied.Assumingalineardistributionofnormalstressandaconstantshearstressatthetoothroot,anestimateofthedisplacementinthedirectionofthetoothloadisgivenundertheform:dfw5Fb1Ecos2auFLSuwSfwD21MSuwSfwD1P~11Qtg2au!G(3)withSfw:tooththicknessatthecriticalsectionaccordingtoWeber~Fig.1!L,M,PandQ:constantswhichslightlydifferdependingontheauthorsasindicatedinTable1.Althoughtheequationsabovearebasedonasimplifiedbidi-mensionalapproach,theyarestillwidelyusedingeardesign.Forexample,themeshstiffnessformulasintheISOstandard6336@8#stemfrom~1!–~3!,whichweremodifiedtobringthevaluesincloseragreementwiththeexperimentalresults.Theobjectiveofthistechnicalbriefistoextendthepresenttheoryaboutthecontributionsofgearbodybydevelopinganoriginalsemi-analyticalformulawhichisnotbasedonthehalfplanehypothesis,butwhichaccountsforactualsoliddiskwheels.Thevalidityandthepracticalinterestofthisformulaareillus-tratedbycomparisonswithsomeresultsfromtheoriginalEq.~3!and2Dfiniteelementcalculations.2Theory2.1SolutionforElasticRings.Muskhelishvili@9#pre-sentedageneralbidimensionalsolutionforcircularringsbasedonacomplexpowerseriesrepresentationofdisplacements,stressesandexternalloads.Theparticularcaseofinterestingearingcor-respondstothemixedproblemofcalculatingradial(ur(u))andtangential(ut(u))displacementsattheouterboundaryofanelas-ticringintermsofthenormalandtangentialstresses,i.e.,srr(u)andsrt(u)appliedtothesameboundaryattheouterradiusRf~Fig.2!.AssumingthatthedisplacementsattheinnerradiusRaarenil,oneobtains:ur~u!1iut~u!5Rf2m(kPZUkeiku5Rf2mFU01(k51NUkeiku1U2ke2ikuG(4)]urRf]u1i]utRf]u5i2mF(k51Nk~Ukeiku2U2ke2iku!G(5)srr~u!2isrt~u!5s01(k51Nskeiku1s2ke2iku(6)whereiistheimaginarynumber(i2521)andmisaLame´’scoefficient.Foreveryharmonick,thestress-displacementrelationsforanelasticringcanbewrittenas:HUkU¯2kJ5FUA,kUB,kUB,2kUA,2kGHsks¯2kJ(7)ContributedbythePowerTransmissionandGearingCommitteeforpublicationintheJOURNALOFMECHANICALDESIGN.ManuscriptreceivedApril2003;revisedNovember2003.AssociateEditor:A.Kahraman.748ÕVol.126,JULY2004Copyright©2004byASMETransactionsoftheASMEDownloadedFrom::~11x!~h221!h2~12k2!~h221!22~x1h212k!~x1h222k!UB,k52x~h212k21!~h222k1x!~11k!1~12k!~h221!2~12k2!~h221!22~x1h212k!~x1h222k!x5324nforplanestrainx532n11nforplanestressh5Rf/RaU¯2kands¯2karetheconjugatesofU2kands2krespectively2.2ExternalLoadandStressCoefficients.Equations~4!to~7!makeitpossibletobuildthedi