机械毕业设计英文外文翻译64超高速行星齿轮组合中内部齿轮的有限元分析.docx

机械毕业设计英文外文翻译64超高速行星齿轮组合中内部齿轮的有限元分析.docx

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机械毕业设计英文外文翻译64超高速行星齿轮组合中内部齿轮的有限元分析

翻译部分

英文原文

FiniteElementAnalysisofinternalGearinHigh-SpeedPlanetaryGearUnits

Abstract:

Thestressandtheelasticdeflectionofinternalringgearinhigh-speedspurplanetarygearunitsareinvestigated.Arimthicknessparameterisdefinedastheflexibilityofinternalringgearandthegearcase.ThefiniteelementmodelofthewholeinternalringgearisestablishedbymeansofPro/EandANSYS.Theloadsonmeshingteethofinternalringgearareappliedaccordingtothecontactratioandtheload-sharingcoefficient.Withthefiniteelementanalysis（FEA）,theinfluencesofflexibilityandfittingstatusonthestressandelasticdeflectionofinternalringgeararepredicted.Thesimulationrevealsthattheprincipalstressanddeflectionincreasewiththedecreaseofrimthicknessofinternalringgear.Moreover,largerspringstiffnesshelpstoreducethestressanddeflectionofinternalringgear.Therefore,theflexibilityofinternalringgearmustbeconsideredduringthedesignofhigh-speedplanetarygeartransmissions.

Keywords:

planetarygeartransmissions;internalringgear;finiteelementmethod

High-speedplanetarygeartransmissionsarewidelyusedinaerospaceandautomotiveengineeringduetotheadvantagesoflargereductionratio,highloadcapacity,compactnessandstability.Greatattentionhasbeenpaidtothedynamicpredictionofgearunitsforthepurposeofvibrationreductionandnoisecontrolinthepastdecades（1-8）.asoneofthekeyparts,internalgearmustbedesignedcarefullysinceitsflexibilityhasastronginfluenceonthegeartrain’sperformance.studieshaveshownthattheflexibilityofinternalgearsignificantlyaffectsthedynamicbehaviorsofplanetarygeartrains（9）.inordertogetstressesanddeflectionsofringgear,severalfiniteelementanalysismodelswereproposed（10-14）.however,mostofthemodelsdealtwithonlyasegmentoftheinternalringgearwithathinrim.thegearsegmentwasconstrainedwithcorrespondingboundaryconditionsandappointloadwasexertedonasingletoothalongthelineofactionwithoutconsideringthechangeoverbetweenthesingleanddoublecontactzoneinacompletemeshcycleofagiventooth.Afiniteelement/semi-analyticalnonlinearcontractmodelwaspresentedtoinvestigatetheeffectofinternalgearflexibilityonthequasi-staticbehaviorofaplanetarygearset（15）.Byconsideringthedeflectionsofallgearsandsupportconditionsofsplines,thestressesanddeflectionswerequantifiedasafunctionofrimthickness.Comparedwiththepreviouswork,thismodelconsideredthewholetransmissionsystem.However,themethoddescribedinRef.（15）requiresahighlevelofexpertisebeforeitcanevenbesuccessful.

Thepurposeofthispaperistoinvestigatetheeffectsofrimthicknessandsupportconditionsonthestressandthedeflectionofinternalgearinahigh-speedspurplanetarygeartransmission.Firstly,afiniteelementmodelforacompleteinternalgearfixedtogearcasewithstraightsplinesiscreatedbymeansofPro/EandANSYS.Then,properboundaryconditionsareappliedtosimulatingtheactualsupportconditions.Meanwhilethecontactratioandloadsharingareconsideredtoapplysuitableloadsonmeshingteeth.Finally,withthecommercialfiniteelementcodeofAPDLinANSYS,theinfluencesofrimthicknessandsupportconditiononinternalringgearstressanddeflectionareanalyzed.

1finiteelementmodel

1.1examplesystem

Athree-planetplanetarygearset（quenchedandtemperedsteel5140）definedinTab.1istakenasanexampletostudytheinfluenceofrimthicknessandsupportconditions.

AsshowninFig.1,threeplanetsareequallyspacedaroundthesungearwith120·apartfromeachother.Here,allthegearsinthegearunitarestandardinvolutespurgears.Thesungearischosenastheinputmemberwhilethecarrier,whichisnotindicatedinFig.1forthesakeofclarity,ischosenastheoutputmember.Theinternalringgearissetstationarybyusing6splinesevenlyspacedroundtheoutercircletoconstraintherigidbodymotionofringgear.

Adimensionlessinternalgearrimthicknessparameter

isdefinedastheratioofrimthicknesstothetoothheightasfollows:

（1）

Wherer0,rf,raaretheouter,dedendumandaddendumradiusofinternalgear,respectively.

Asmaller

indicatesamoreflexibleringgearandviceversa.internalgearswithdifferentvaluesof

=1.0,1.5,2.0,2.5areinvestigatedinthispaper.Inallthesecases,thewidthsofringgearare44mm,andtheconnectingsplinesare34mminlengthand14mminwidth,whiletheheightsofsplinesineachcaseare5mm,6mm,7mmand8mm,respectively.

Afiniteelementmodelfortheinternalgearwith

=1.5isshowninFig.2,whichcontains69813elementsand112527nodes.

Fig.2Finiteelementmodelofinternalringgear

1.2loadsandboundaryconditions

Theinternalgearisfixedtogearcasethroughsplinesandmesheswithplanetgears.Assumingthattheloadisevenlydistributedtoeachplanetandallfrictionsarenegligible,themeshingforcebetweeneachplanetandtheringisasfollows:

WhereTcistheoveralloutputtorque;iscistheoverallreductionratio;rsistheradiusofsungear;npdenotesthenumberofplanets;

isthepressureangle.

Inaddition,byconsideringthecontactratioandloadsharingfactors,wecanfinallydeterminethemeshpositionsandtheproportionsoftheloadcarriedbyeachtoothofthering.TheloadstateoftheringisshowninFig.3.

Here,thephaseanglebetweeneachplanetis120。

andFi（1,….,6）isthenormalmeshingforceactingontheteethofinternalgear.Forclaritypurpose,onlytheteethinmeshareplottedinFig.3.afterobtainingthemeshingforcesactingoninternalgear,wecanapplythemtothefiniteelementmodel.Tobespecific,themeshingforcesareevenlydistributedtothecorrespondingnodesalongthelineofengagement.

Assupportconditionscanbeverycomplicatedifconsideringthecontactproblems,specialsubstitutemustbemadetomodeltheactualcontactsatthesplines.Inthispaper,thesplinesarecoupledwiththeringbytheoverlappednodesandsixspringsequallyspacedbetweentheoutersurfaceoftheringandthehousingsurfaceareappliedtosimulatingthesupportconditions.Thesupportconditionbetweentheringandthehousingisindicatedthroughthestiffnessofthesesprings.Theprocesscanbedetailedasfollows.Asinglenodeneedstobedefinedforeachspline-to-housingconnection.ThisisachievedsuingCOMBINE14elementsateachsplineposition,whichconnectthesplinestothepointsatthehousingsurfacewithaninfinitestiffness.Alldegreeoffreedoms（DOFs）ofthesepredefinednodesareconstrained.AttheotherendofeachspringelementisacommonnodeconnectedwithsplinewhoseDOFsexceptinradialdirectionareallconstrained.Inaddition,thenodesontheloadedsurfaceofeachsplineareconstrainedincircumferentialDOF.AndtheaxialDOFoftheringisconstrained.

ThesupportconditionsimulatedwithspringsisshownasFig.4

2FEAresults

Byapplyingproperloadsandboundaryconditions,afiniteelementanalysiscanbeconductedtofigureouttheeffectsofrimthicknessandsupportconditionsoninternalgearstressanddeflection.Astotheexamplesystem,thestressanddeflectionarepredictedat24discreteangularpositionswithanincrementof5。

whichspana120..rotationofthecarrier.thisensuresthatanytoothofinternalgeargoesthroughacompletemeshingcyclebecausethenumberofplanetsis3.

2.1effectofrimthicknessoninternalgearstressanddeflection

InFig.5,themaximumprincipalstress（Misesstress）oftheringateachdiscretepositionisplottedagainstthecarrierrotationangleforfourdifferentringrimthickness（

=1.0,1.5,2.0,2.5）.here,thespringstiffnessis33N/mm.

FromFig.5,wecanseethatwiththedecreaseof

themaximumstressintheringincreases.hence,therimthicknessoftheringcannotbetoosmallforthesakeofgeardurability.Andfurtherinvestigationsrevealthatthecriticalpointatwhichthemaximumstressoccursmovesfromthefilletregiontotherootoftoothwhen

decrease.

Fig.6showsthedeflectionshapesofringswithdifferentrimthickness.Theringdeflectionsfor

=1.0and

=2.0aredemonstratedinFig.7withthesamedeflectionmagnificationfactorof50.

Obviously,when

increases,thedeflectionofringdecreases.Theamountofradialdeflectionoftheringinbothoutwardandinwarddirectionisplottedasafunctionof

inFig.7.here,thepositiveamountsdenotetheoutwarddeflectionswhilethenegativeonesdenotetheinwarddeflection.When

=1.0,themaximumout-wardandinwardradialdeflectionsarepredictedtobe0.139and0.122mm,respectively.Iftheringsipermittedtodeflectsomuch,thosemanufacturingerrorsassociatedwiththeinternalgearsuchastheroundnesserrorandrun-outerrorcanbetoleratedaslongastheirmagnitudesarelesstheamountofdeflection.

2.2effectofspringstiffnessoninternalgearstressanddeflection

ThemaximumprincipalstressoftheringwithvariedspringstiffnesskisshowninFig.8.here,theunitofstiffnessisN/mm.obviously,themaximumprincipalstressoftheringwith

=1.0ismuchmoresensitivetothesupportstiffnessthanthatoftheringwith

=2.5.andforaringwithagiven

themaximumprincipalstressincreaseswiththedecreaseofspringstiffness.

Fig.9demonstratestheinfluenceofspringstiffnessonthemaximumradialdeflectionofthering.Similarlythemaximumradialdeflectionsoftheringwith

=1.0ismuchmoresensitivetothesupportstiffnessthanthatofthering