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1附录A中英文翻译DynamictoothloadsofplanetarygearsetshavingtoothprofilewearC.Yuksela,A.Kahramanb,aTheUniversityofToledo,Toledo,OH43606,USAbDepartmentofMechanicalEngineering,TheOhioStateUniversity,Room255,650AckermanRoad,Columbus,OH43202,USAReceived25June2003receivedinrevisedform12January2004accepted10February2004AbstractAcomputationalmodelofaplanetarygearsetisemployedtostudytheinfluenceofsurfacewearonthedynamicbehaviorofatypicalplanetarygearset.Theoverallcomputationalschemecombinesawearmodelthatdefinesgeometricdescriptionofcontactinggeartoothsurfaceshavingwearandadeformablebodydynamicmodelofaplanetarygearset.ThewearmodelemploysaquasistaticgearcontactmodeltocomputecontactpressuresandArchardswearmodeltodeterminetheweardepthdistributions.Thewornsurfacesareinputintothedynamicmodeltoquantifytheimpactofwearongeartoothandmeshdynamicforces.Theresultsonaplanetarygearsethavingafixedplanetcarrierindicatesthatthedynamicbehaviorisnonlinearduetotoothseparationsinitsresonanceregions.Theresultsforworngearsurfacesindicatethatsurfacewearhasasignificantinfluenceinoffresonancespeedrangeswhileitsinfluencediminishesnearresonancepeaksprimarilyduetotoothseparations.Ó2004ElsevierLtd.Allrightsreserved.KeywordsPlanetarygearsetsGeardynamicsGearwear1.Introduction2Planetarygearsets,alsoknownasEpicyclicgeardrives,arecommonlyusedinalargenumberofautomotive,aerospaceandindustrialapplications.Theypossesnumerousadvantagesoverparallelaxisgeartrainsincludingcompactnessofdesign,availabilityofmultiplespeedreductionratios,andlessdemandingbearingrequirements.Mostcommonexamplesofplanetarygearsetscanbefoundinautomatictransmissions,gasturbines,jetengines,andhelicopterdrivetrains.Atypicalsimpleplanetarygearsetconsistsofasungear,aringgearandanumberofidenticalplanetgearstypically3–6meshingbothwiththesunandringgears.Acommoncarrierholdstheplanetsinplace.Dynamicanalysisofplanetarygearsisessentialforeliminatingnoiseandvibrationproblemsoftheproductstheyareusedin.Thedynamicforcesatthesunplanetandringplanetmeshesarethemainsourcesofsuchproblems.Althoughplanetarygearsetshavegenerallymorefavorablenoiseandvibrationcharacteristicscomparedtoparallelaxisgearsystems,planetarygearsetnoisestillremainstobeamajorproblem.Thedynamicgearmeshloadsthataremuchlargerthanthestaticloadsaretransmittedtothesupportingstructures,inmostcases,increasinggearnoise.Largerdynamicloadsalsoshortenthefatiguelifeofthecomponentsoftheplanetarygearsetincludinggearsandbearings.Surfacewearisconsideredoneofthemajorfailuremodesingearsystems.Incaseofplanetarygearsets,experimentaldatahasshownthatespeciallythesungearmeshesmightexperiencesignificantsurfacewearwhenrunundertypicaloperatingconditions1.Whilewearisafunctionofalargenumberofparameters,slidingdistanceandcontactpressurewereshowntobemostsignificantparametersinfluencinggearwear1.Wearoftoothprofilesresultsinauniquesurfacegeometrythataltersthegearmeshexcitationsintheformofkinematicmotionerrors,enhancingthedynamiceffects.Modelingofplanetarygearsetdynamicsreceivedsignificantattentionforthelast30years.Anumberofstudiesproposedlumpedparametermodelstopredictfreeandforcedvibrationcharacteristicsofplanetarygearsets2–9.Thesemodelsassumedrigidgear3wheels,connectedtoeachotherbyspringsrepresentingtheflexibilityofthemeshingteeth.Inthesestudiesexcept6,nonlineareffectsduetogearbacklashandtimevaryingparametersduetogearmeshstiffnessfluctuationswereneglected.ThecorrespondingEigenvaluesolutionofthelinearequationsofmotionresultedinnaturalmodes.Modalsummationtechniquewastypicallyusedtofindtheforcedresponseduetoexternalgearmeshdisplacementexcitationsdefinedtorepresentmotiontransmissionerrors.Theselumpedparametermodelsvaryindegreesoffreedomincluded,frompurelytorsionalmodels2–4,9totwo5,6,8orthreedimensionaltransversetorsionalmodels7.Whilethesemodelsservedwellindescribingthedynamicbehaviorofplanetarygearsetsqualitatively,theylackedcertaincriticalfeatures.First,thegearmeshmodelswerequitesimplisticwithacriticalassumptionthatcomplexgearmeshcontactinteractioncanberepresentedbyasimplemodelformedbyalinearspringandadamper.Thesemodelsdemandthatthevaluesofthegearmeshstiffnessanddamping,aswellasthekinematicmotiontransmissionerrorexcitation,mustbeknowninadvance.Itisalsoassumedthattheseparametervaluesdeterminedquasistaticallyremainunchangedunderdynamicconditions.Inaddition,gearrimdeflectionsandHertziancontactdeformationsarealsoneglected.Anothergroupofrecentmodels10,11usedmoresophisticatedfiniteelementbasedgearcontactmechanicsmodels.Thesecomputationalmodelsaddressalloftheshortcomingsofthelumpedparametermodelssincethegearmeshconditionsaremodeledasindividualnonlinearcontactproblems.Theneedforexternallydefinedgearmeshparametersiseliminatedwiththesemodels.Inaddition,rimdeflectionandsplinesupportconditionsaremodeledaccurately12.Thesemodelsarealsocapableofincludingtheinfluenceofthetoothprofilevariationsintheformofintentionalprofilemodifications,manufacturingerrorsorwearonthedynamicbehaviorofthesystem.Thestudyofwearofgearcontactisbecomingoneoftheemergingareasingeartechnology.Anumberofrecentgearwearmodelingefforts1,13–16formasolidfoundationformoreaccurate,largersystemanalyses.AllofthesemodelsuseArchardswearmodel17inconjunctionwithagearcontactmodelandrelativeslidingcalculations.Thesestudies4focusedonpredictionofwearofeitherspur13,14orhelical1,15,16gearpairsinaparallelaxisconfiguration.ThetoothcontactpressureswerecomputedinthesemodelsusingeithersimplifiedHertziancontact14–16orboundaryelement1formulationsunderquasistaticconditions.SlidingdistancecalculationswerecarriedoutkinematicallybyusingtheinvolutegeometryandArchardswearmodelwasusedwithanempiricalwearcoefficienttocomputethesurfaceweardepthdistribution.Anumberofstudiesinvestigatedtheinfluenceofwearongeardynamicsresponse18–20.Amongthem,KuangandLin18simulatedthetoothprofilewearprocessbythemodelproposedbyRef.14,andpredictedthevariationsofthedynamicloadsandthecorrespondingfrequencyspectraasafunctionofwearforasinglespurgearpair.WojnarowskiandOnishchenko20performedanalyticalandexperimentalinvestigationsoftheinfluenceofthetoothdeformationandwearonspurgeardynamics.Theystatedthatthechangeintheprofilesoftheteethduetowearmustbetakenintoaccountwhendealingwiththedurabilityofthegeartransmissionsaswell.Thesepreviousmodelsconsideredsurfaceweareffectsforonlyasinglespurgearpair,avoidingmultimeshgearsystemssuchastheplanetarygearsets.Theyfocusedononlyexternalgearsandusedlumpedparameterdynamicmodelsexcludingnonlinearandtimevaryingeffects.1.1.ObjectivesandscopeAsnoneofthepreviousstudiesonplanetarygearsetdynamicstookintoaccounttheeffectofwear,thisstudyisintendedtodescribetotheinfluenceofgeartoothsurfacewearondynamicbehaviorofplanetarygears.Adeformablebodydynamicmodelsimilartotheoneproposedearlier11willbeusedtoinvestigatetheinfluenceoftoothsurfacewearonthedynamicbehaviorofplanetarygearsets.Themainobjectivehereistoquantifytheinfluenceofsurfacewearondynamicbehaviorofplanetarygearsets.Aplanetarygearsetformedbyspurgearswillbeconsidered.Awearpredictionmodelwillbeproposedtopredictthegearsurfaceweardistributionunderquasistaticconditions.Differentamountsofweardepthswillthenbeintroducedinthedynamicmodeltoquantifythedifferencesindynamicbehaviorfromthebaselinebehaviorrepresentingagearsethavingnowear.Severalcomplexdynamic
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