机械毕业设计英文外文翻译63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63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63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
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