Nonlinear dynamics of planetary gears using analyt

JOURNALOFSOUNDANDVIBRATIONJournalofSoundandVibration302(2007)577–595NonlineardynamicsofplanetarygearsusinganalyticalandfiniteelementmodelsVijayaKumarAmbarisha,RobertG.ParkerDepartmentofMechanicalEngineering,OhioStateUniversity,201W.19thAvenue,Columbus,OH43210,USAReceived9August2004;receivedinrevisedform19October2006;accepted27November2006Availableonline31January2007AbstractVibration-inducedgearnoiseanddynamicloadsremainkeyconcernsinmanytransmissionapplicationsthatuseplanetarygears.Toothseparationsatlargevibrationsintroducenonlinearityingearedsystems.Thepresentworkexaminesthecomplex,nonlineardynamicbehaviorofspurplanetarygearsusingtwomodels:(i)alumped-parametermodel,and(ii)afiniteelementmodel.Thetwo-dimensional(2D)lumped-parametermodelrepresentsthegearsaslumpedinertias,thegearmeshesasnonlinearspringswithtoothcontactlossandperiodicallyvaryingstiffnessduetochangingtoothcontactconditions,andthesupportsaslinearsprings.The2Dfiniteelementmodelisdevelopedfromauniquefiniteelement-contactanalysissolverspecializedforgeardynamics.Meshstiffnessvariationexcitation,cornercontact,andgeartoothcontactlossareallintrinsicallyconsideredinthefiniteelementanalysis.Thedynamicsofplanetarygearsshowarichspectrumofnonlinearphenomena.Nonlinearjumps,chaoticmotions,andperiod-doublingbifurcationsoccurwhenthemeshfrequencyoranyofitshigherharmonicsarenearanaturalfrequencyofthesystem.Responsesfromthedynamicanalysisusinganalyticalandfiniteelementmodelsaresuccessfullycomparedqualitativelyandquantitatively.Thesecomparisonsvalidatetheeffectivenessofthelumped-parametermodeltosimulatethedynamicsofplanetarygears.Meshphasingrulestosuppressrotationalandtranslationalvibrationsinplanetarygearsarevalidevenwhennonlinearityfromtoothcontactlossoccurs.Thesemeshphasingrules,however,arenotvalidinthechaoticandperiod-doublingregions.r2007ElsevierLtd.Allrightsreserved.1.IntroductionPlanetarygearsareeffectivepowertransmissionelementswherehightorquetoweightratios,largespeedreductionsincompactvolumes,co-axialshaftarrangements,highreliabilityandsuperiorefficiencyarerequired.Exampleapplicationsareautomotivetransmissions,tractors,windturbines,helicopters,andaircraftengines.Gearvibrationsareprimaryconcernsinmostplanetarygeartransmissionapplications,wherethemanifestproblemmaybenoiseordynamicforces.Noiselevelsexceeding110dBobservedinahelicoptercabinareattributedlargelytovibrationoftheplanetarygear.Largedynamicforcesincreasetheriskofgeartoothorbearingfailure.KahramanandBlankenship[1,2]performedexperimentsonaspurgearpairandobservedvariousnonlinearphenomenaincludinggeartoothcontactloss,period-doublingandchaos.ARTICLEINPRESS:10.1016/j.jsv.2006.11.028Correspondingauthor.Tel.:[phone];fax:[phone].E-mailaddress:[e-mail](R.G.Parker).Toothseparationsatlargevibrations,whicharecommoninspur–gearpairs,occureveninplanetarygearsasevidentfromtheexperimentsbyBotman[3].Planetarygearresearchershavedevelopedlumped-parametermodelsanddeformablegearmodelstoanalyzegeardynamics.Theliteraturemainlyaddressesstaticanalysis,naturalfrequenciesandvibrationmodes,modelingtoestimatedynamicforcesandresponses,andcancellationofmeshforcesusingtheplanetarygearsymmetrythroughmeshphasing.StudiesbyCunliffeetal.[4],Botman[5],HidakaandTerauchi[6],Hidakaetal.[7,8],andKahraman[9–11]involveplanetarygearmodelstoestimatenaturalfrequencies,vibrationmodesanddynamicforces.LinandParker[12,13]presenta2Drotational–translationaldegreeoffreedomspurgearmodelandmathematicallyshowtheuniquemodalpropertiesofequallyspacedanddiametricallyopposedplanetsystems.Allmodescanbeclassifiedasoneofrotational,translational,orplanetmodes.ThesensitivityofnaturalfrequenciesandmodestooperatingspeedsandvariousdesignparametersarestudiedbyLinandParker[14],whoalsoexaminenaturalfrequencyveeringphenomena[15].Meshstiffness-inducedparametricinstabilityisstudiedbyLinandParker[16].AhelicalplanetarygearmodelisformulatedandtheeffectofmeshphasingonthedynamicsofequallyspacedplanetsystemsisinvestigatedbyKahraman[9]andKahramanandBlankenship[17].Parker[18]showedtheeffectivenessofmeshphasinginsuppressingcertainharmonicsofplanetarygearvibrationmodesbasedonself-equilibrationofthedynamicmeshforcesatsun–planetandring–planetmeshes.AmbarishaandParker[19]extendedthisworktoderivedesignrulestosuppressplanetmoderesonances.Athoroughdescriptionoftherelativemeshphasingbetweenthesun–planetandring–planetmeshesinaplanetarygearsystemisgivenbyParkerandLin[20].AnonlineardynamicplanetarygearmodelisintroducedbyKahraman[11]andtheeffectsofvariousdesignparametersonthedynamicloadsharingoftheplanetsareexamined.VelexandFlamand[21]studiedtheplanetarygeardynamicsusinglumped-parametermodel.Inrecentyears,someresearchershaveuseddeformablegearbodydynamicmodels.Auniquefiniteelement-contactanalysisprogramisusedbyParkeretal.[22]tomodelnonlinearspurgeardynamics.Thefiniteelementresultscomparefavorablywithexperiments.Parkeretal.[23]usedthesamefiniteelementmethodtoexamineplanetarygeardynamics.KahramanandVi