螺旋齿轮接触建模与齿偏转-外文文献

TribologyInternational40(2007)ArtoDesign,4forfoundationplaysr2005ElsevierLtd.Allrightsreserved.Keywords:Helicalgear;Contact;Deformation;Loaddistribution;Modelingarealisticanalysisofhelicalgearcontactalsorequiresgivenoperatingconditions.Gearcontactratio12andequalloaddistributioninthecaseoftwoteethincontactfiniteelementmethodwithspurgears24.CoyandChaowith3Dfiniteelements8,9.ThisismainlybecauseinthiscaseFEMcontactmodelingiscomputationallyexpensiveandtime-consumingduetothesmallgridsizewhichisARTICLEINPRESSnecessaryonthegearflanksurface.Differentmethodshavebeenintroducedtoovercomethisproblem.Vedmar10separatedstructuralandcontactanalysisbycombiningthe0301-679X/$-seefrontmatterr2005ElsevierLtd.Allrightsreserved.doi:10.1016/j.triboint.2005.11.004C3Correspondingauthor.Tel.:+358331154442.E-mailaddresses:juha.hedlundtut.fi(J.Hedlund),arto.lehtovaaratut.fi(A.Lehtovaara).informationonstructuraldeformations,suchastoothdeflection.Themajorityofgearcontactanalyseswithintribologicalstudiesaremadeonspurgears.Typically,teethcontactthroughthelineofactionismodeledasaconstantlychangingrollercontact,whoseradius,speedandloadareapproximatedfromidealinvolutegeargeometryinthe5studiedfiniteelementgridsizedimensionstocovertheHertziancontact.Duetal.6andArafaetal.7laterenhancedcontactmodelingasapartofstructuralanalysisbyusinggapelementsforthecalculationofspurgeardeformation.Onlyafewhelicalgearcontactstudies,whichincludestructuraldeformationsofthegear,havebeenperformed1.IntroductionHelicalgearsaregenerallyusedinindustryandtheircontactbehaviordeservesmoreattentiontoestablisharealisticbaseforthedetailedstudyofgearfriction,wearandlife.Thegearcontactstressesderivedfromtoothcontactforcesandgeometryareveryimportantfordetermininggearpitting,i.e.lifeperformance.Toothcontactforcesalongthelineofactiondependessentiallyonloadsharingbetweenmeshingteeth,and,therefore,(halfofthesingle-toothload)areoftenassumed.Deforma-tionsarecalculatedaccordingtotheHertzlinecontacttheory,otherwiseassumingrigidtoothbehavior.Somestudiesaremadebyslicingthehelicalgeartoaseriesofspurgearsandtreatingtheseslicesasspurgears1.Finite-element-basedcalculationmodelsarewidelyacceptedforcalculatingstructuraldeformationsandstressesinspurandhelicalgearsinthecaseofconcentratedloads.Ingeartransmissionanddynamicanalyses,typicaldeformationstudieshaveusedthetwo-dimensional(2D)ModelingofhelicalgearcontactJuhaHedlundC3,TampereUniversityofTechnology,MachineAvailableonlineAbstractThemajorityofgeartribologicalstudiesaremadeonspurgears.contactbehaviordeservesmoreattentiontoestablisharealisticbasemodelingofhelicalgearcontactwithtoothdeflection.Acalculationsurfaceprofilesareconstructedfromgeartoolgeometrybysimulatingelementsforthecalculationoftoothdeflectionincludingtoothbending,contactanalysiswithstructuralanalysistoavoidlargemeshes.Toothloadsharingbetweenthemeshingteeth,whereascontactflexibilitycalculationmethodswasalsostudied.613619withtoothdeflectionLehtovaaraP.O.Box589,33101Tampere,FinlandJanuary2006However,helicalgearsaregenerallyusedinindustry,andtheirdetailedfriction,wearandlifestudies.Thisstudyfocusesonthemodelforhelicalgearcontactanalysisisintroduced.Helicalgearthehobbingprocess.Themodelusesthree-dimensionalfiniteshearingandtoothfoundationflexibility.Themodelcombinesflexibilitywasfoundtohaveanessentialroleincontactonlyaminorrole.T/locate/tribointARTICLEINNomenclatureacontactellipseradiibcontactellipseradiibggearwidthdcontactdeformationatcenterofcontactEelasticitymodulusE0reducedmodulusofelasticity2=E0121C0v21=E11C0v22=E2C138etijdifferencebetweencontactsurfaceprofilesFccalculatedtotalcontactforceFijforceatsurfacenodei,jh1distancefrompitchlinei,jindicatesgridnodesKijstiffnessvalueatgridpoint(i,j)KeqijreducedtotalstiffnessLeffectivelengthofthelineofactionMg1rotationmatrixM2gtranslationmatrixM21transformationmatrixM12transformationmatrixmnnormalmodulepcontactpressureJ.Hedlund,A.Lehtovaara/Tribology614finiteelementmethodandtheWeber&Banaschekdeformationformulatostudythecontactbehaviorofinvolutehelicalgears.Brauer11utilizedthegloballocalFEapproach,whichcombineslocaldenseandglobalcoarseelementmeshes,applyingittocalculatetransmis-sionerrorinanti-backlashconicalinvolutegears.PimsarnandKazerounian12avoidedtheuseofcontactelementsbyintroducingafastpseudo-interferencemethodforthecalculationofmeshstiffnessinthecaseofaspurgearpair.Thecalculationwasdonewith2Dplaneelements.Thismethodwasbasedonanelasticfoundationmodel.Inthispaper,aparameterizedcalculationmodelforhelicalgearcontactanalysisisintroduced.Themodelisbasedon3Dfiniteelements.Themodelcombinescontactanalysisandstructuralanalysistoavoidlargemeshes.TheeffectoftoothfoundationandHertziandeformationonloadsharingbetweenthemeshingteethisanalyzedwiththedevelopedmodel.Thecapabilityofpseudo-interference-basedmethodutilizing3Dfiniteelementsincontactcalculationsisalsostudied.2.GeometrymodelNumericalapproachisusedtocreatehelicalgeargeometrybysimulatingthehobbingprocess.Thismethodp0maximumcontactpressurer1x,r1yprincipalradiiofsolid1PRESSr2x,r2yprincipalradiiofsolid2rp1pitchradiussinitialseparationofsurfacesTpiniontorqueucontactnormaldeformationWappliedtotalforcex,x0coordinatesy,y0coordinateszdepthcoordinatez1,z2numberofteethbhelixangledcompositerigid-bodymovementegtotalcontactratiozijforcednodaldisplacementnpoissonratioxcoordinatealongthelineofactionrradiusfrotationangleSubscripts1referstobody12referstobody2International40(2007)613619isbasedonawidesetofnumericalcalculationpointsandtheirsynchronizing,whichallowsdeviationsfrominvolutegeometry.Twomovablecoordinatesystemsareused,whicharerigidlyconnectedtogear(x1,y1)andrack(x2,y2),asshowninFig.1.CoordinatetransformationmatrixM21consistsofarotationmatrixandatranslationmatrix.Aninversedtransformationmatrixgivesthegearprofileinastationarycoordinatesystem.TherotationmatrixisgivenasMg1cosfC0sinf00sinfcosf00001000012666437775(1)andthetranslationmatrixinthecaseofahelicalgearasM2g100rp1f010C0rp1h1001000012666437775,(2)M21M2gMg1,(3)caseappliedtestcaseTransformationmatrixM12describestoolrackrollingwithoutslidingaroundthepitchcircleofthegear,asinthecaseofhobbing.ThegearteethprofilecanbecreatedfromthesetofcurvesshowninFig.2.ThegearteethprofileisARTICLEINPRESSFig.1.Rotatingandtranslatingcoordinatesystems13.M12MC0121cosfsinf0sinfrp1sinfh1C0cosfrp1fC0sinfcosf0cosfrp1cosfh1sinfrp1f00cosf2sinf20000cosf2sinf226666643777775.4J.Hedlund,A.Lehtovaara/TribologyInternational40(2007)613619615Fig.2.Atoolrackrollingaroundthepitchcircle,thesetofcurves.extrudedalongtoahelicoidcurveto3Dsurfacegeometry.Contactlineisdeterminedbysearchingtheminimumdistancebetweentwoundeformedmatingsurfaces.Finally,thetoothpairsurfacegeometryandthecontactlinearenumericallyobtained.3.FiniteelementmodelEight-nodetrilinearhexahedralelementsareexploitedintheFEMmodel.Thedevelopedmodelhasitsownmesher,solverandpostprocessor.MatlabTMisusedasaprogram-mingtool.Thesurfacegeometrydataobtainedfromthegeometrymodelisutilizedinthefiniteelementmodel.Theparameterizedmeshercreates3Dsolidelementsfromthesurfacedataandaddsanotherparameterizedmeshbelowthetooth,whichcoversthetoothfoundation.Afterstiffnessandmassmatrixintegration,thesetwomeshesaresummedupinanassemblyprocess.Thestiffnessmatrixandmassmatrixassemblyprocessesareperformedbyfreedom-pointer-techniqueusinganelementfreedomtable.Contactloadvaluesaresharedtoequivalentnodalloads.Finally,themodelgivesnodalstiffness,displacementandstressvaluesasoutput.Thetotalstiffnessvaluesofgearmeshcanbecalculatedfromthesenodaldisplacementvalues.4.ContactformulationGearcontactgeometryisdescribedwithtwonon-conformalsmoothsurfaces,asshowninFig.3.UndertheactionofnormalloadW,thetwobodiesaredeformedandapproacheachotherbydistanced.Withintherealcontact,elasticdeformationu(x)addedwithinitialFig.3.Schematicviewofundeformedsmoothsurfaceprofilesincontact.contactparametersarecalculatedbyapproachingthecontactbodiesgraduallyuntilloadbalanceisachieved(Eq.(11).Theconditionofthecontactischeckedateachiterationstepatthenodalpoints,i.e.parameteretijisupdated.TotalcontactforceisdeterminedasFcXiXjKeqijetijifetij40;etij0.(11)TheFEM-basedcontactmodeltakesintoaccountstructureboundaries,i.e.nohalfspaceassumptionisneeded.5.Resultanddiscussion5.1.ContactmodeltestcaseTheFEM-basedcontactmodelwastestedagainstARTICLEINPRESSInternational40(2007)613619separations(x)mustbeequaltotherigid-bodymovementdandoutsidetherealcontactgreaterthandasfollows:uxsxd;px40,(5)uxsx4d;px0.(6)Inaddition,theresultingpressuredistributionmustsatisfytheforcebalanceinnormaldirectionwithtotalforceWappliedonthecontactingbodies.Itfollows:Z1C01Z1C01px;ydxdyW.(7)For3DelasticcontactproblemstheBoussinesqfor-mulationcanbeused.Thebasicequationforsurfacepressuredeformationofsurfacesinz-directionu(x)betweensemi-infinitesolidsis14ux2pE0Z1C01Z1C01px0;y0dx0dy0xC0x02yC0y02q.(8)Contactproblemswitharbitraryundeformedsurfaceprofilesneedtobesolvedbynumericalmethods.Thenumericalsolutionprocessistypicallyiterative,becausepressuredistributionandrealcontactareadistributionareunknown,whereastotalload,materialpropertiesandinitialcontactgeometryareknownparameters.Thewell-knownHertzsolutiontothecontactproblemisbasedoncases,wheretheundeformedgeometryofcontactingsolidscanberepresentedingeneraltermsbytwoellipsoids.Thesolutionrequiresthecalculationofellipticityparameterandcompleteellipticintegrals.AsimplifiedsolutionoftheclassicalHertztheoryofellipticalcontactsolutionispresentedin15.Thiscalculationmethodisnon-iterativeandfast.Thesolutionincludesanelastichalfspaceassumption.4.1.FEM-basedcontactmodelInitially,geometricaloverlapbetweenthecontactbodiesischosentoproduceacalculationdomaingreaterthanthefinalcontactarea.Theloadingvectoractinginthecalculationdomainisacombinationofnodaldisplace-ments(overlap)andzeroloads,producingnon-homoge-nousFEMboundaryconditions.Thisapproachisappliedtocalculateforcedistributionoverthedomain.ForcedistributionFij,whichislocatedatthesurface,isusedforthecalculationofcontactstiffnessvaluesKijineverynode(i,j)asfollows:KijFijBij.(9)Reducedstiffnessvaluesinthegridbetweenthecontactbodiesaredeterminedasfollows:KeqijKij1Kij2Kij1Kij2.(10)J.Hedlund,A.Lehtovaara/Tribology616Afterthereducedcontactstiffnessisestablishedandtheinitialseparationoftheundeformedsurfacesisknown,simplifiedHertzsformulas15inthecaseofcircularandellipticalcontact.ThetestcasedimensionsandloadconditionsareshowninTable1andtheresultsinTable2.Bothsurfaceshavethesamematerialproperties.Theellipticaltestcasewaschosentoevaluatethecrownedspurgearcontact.Meshsizewaslimitedto4500elementspercontactbodyinthetestcasecalculation.Onesurfacecalculationdomainconsistsof900nodalpoints.Thedeformedcontactsurfaceandcalculationgridareshowninthecaseofcircularcontact(Fig.4).TheresultsshowthattheFEM-basedcontactmodelgivesreasonableapproximationofcontactparameterstakingintoaccountthefairlycoarsegridsize.Theminorsemi-axisoftheellipseespeciallysuffersfromgriddimensions.Meshsizeandshapehaveacertaineffectontheresultsunlessthemeshsizeisfineenough.Itisobviousthattheaccuracyofresultswilldecreaseastheellipticityratioofthecontactincreases.Table2ComparisonresultsModela(mm)b(mm)d(mm)p0(GPa)Hertz/circular0.370.3718.13.56FEM/circular0.40.419.13.11Table1TestcasespecificationsCaseCircularEllipticalE(GPa)206n0.3r1x(mm)12.312.3r2x(mm)18.618.6r1y(mm)12.31000r2y(mm)18.61000W(N)10005000Hertz/elliptical0.274.1320.72.11FEM/elliptical0.44.020.22.36HelicalgearcontactwasstudiedwiththeFZGtestrigrelatedgeardatashowninTable3.Inthisexample,totalcontactratiowasover2,whichmeansthattherearealwaysatleasttwoteethpairsincontact.ThetestcaseelementmeshandthematingtoothsurfaceforpinionareshowninFig.5.Theelementmeshofthegearisequal.ThecalculatedforcedistributioncurvesofthetestcaseareshowninFig.6.Thelineofactionisdescribedwithnon-dimensionalparametercx/L,whereListheeffectivelengthofthelineofaction.Fig.6showsthatthegeneraltrendinforcedistributionremainsinthedifferentmodeltestcases.Theareabetweenthesharpedgesnearthemiddlerepresentsasituationwherethreeteethpairscarrythetotalload.Inthecaseofhelicalgears,thistransitionfromtwotothreeteethandviceversaoccursquitesmoothly.Contrarytospurgears,singletoothforceishigh,whenallthreeteethareincontact.Thisisbecauseflexibilityisloweratthetoothtipcornersthaninthemiddleandrootarea.Thedifferenttestcasesproducecleardifferencesincontactforcebehavior.Theflexibilityoftoothfoundationhasthemostcrucialeffectonthedistributionofcontactforcealongthelineofaction.ContactflexibilityhaslessARTICLEINPRESSInternational40(2007)613619617whichmakesthecontactcurvaturechangealongthecontactlineandthelineofaction.Also,loadsharingbetweenthegearteethiscomplicatedpartlybecausethetotalforceisoftensharedbetweenthreeteethpairs.Therealisticforceactingonasingletoothatanylocationalongthelineofactionisthebasicparameterintribologicalcontactstudies.Forcedistributionbetweenmeshingteethpairswasstudiedinthedevelopedmodel.Fourdifferenttestcaseswereestablishedwherethemodelallows:(1)toothandtoothfoundationdeformationswithrigidcontact;(2)toothandtoothfoundationdeformationswithcontactdeformation;Incase(3)(4)Thecalculninesolvedsingleteethpairvectorcalculsolved.Contactondistributradiusameanbyhelicalgearmesh,contactismorecomplexthanintheofspurgears.Thecontactareahasareal3Dnature,5.2.HelicalgearcontactcaseFig.4.Deformedcontactsurface.J.Hedlund,A.Lehtovaara/Tribologytoothdeformationwithrigidfoundationandrigidcontact;toothandcontactdeformationwithrigidfoundation.stiffnessvectorforatoothpairconsistsof30ationpointsalongthelineofaction.Twothousandhundredandtwenty-sixelementspertoothwereateverycalculationpoint.Thestiffnessvectorofagearpairwascopiedwithoffsettorepresentotherincontact.Thetotalmeshstiffnessvectorofagearwasobtainedbysummingupthesesinglestiffnesss.Finally,displacementalongthelineofactionwasatedandthecontactforceofasingletoothwasstiffnessalongthelineofactionwascalculatedtheHertzlinecontactformulabyusingtheforceionfromtestcase1.Thevaluesoftoothflankarecalculatedoverthecontactlineandestimatedasvalue.Theoverallreducedstiffnesswasobtainedtheiterativemethod,asinEq.(10).impact,butinterestingly,itshiftstheforcedistributioncurveslightlytotherightincertainareas.Thisisbecausethecombinedcontactradiusisasymmetricoverthepitchpoint.Theloaddistributionwasobservedtobesensitivetostiffnesspropertiesatthestartandendpointsofthelineofaction.Onecontactpoint(cC00.236),showninFig.6,waschosenforacloserstudy.ThiscontactpointwasestimatedTable3Testcasegeardatamn(mm)2.75bg(mm)20b(deg)12z126z239eg2.084T(Nm)143Fig.5.Elementmeshofpiniontoothusedinthetestcase.ARTICLEINPRESSforcealongthelineofaction.Table4ContactspecificationsE(GPa)220n0.3r1x(mm)8.7r2x(mm)23.2r1y(mm)7000r2y(mm)7000withtheHertzianellipticalcontactformula16andtheFEM-basedcontactmodel.Thiscontactpointrepresentsthesituationwheretwoteethpairscarrythetotalload.Thechosencontactsituationwascalculatedwithtwodifferentforcescorrespondingtothecalculatedtestcases2and4.Theloadsharingdiffersdependingonthemodelingofthetoothfoundation.TheradiusofsurfaceprofileswasapproximatedwiththecircumcirclemethodandthesevaluesareshowninTable4.SomecrowningwasincludedFig.6.SingletoothcontactJ.Hedlund,A.Lehtovaara/Tribology618inthecontactlinedirection.Atthestudiedcontactpoint,differentloadsharingbetweencases2and4hasonlyaminoreffectoncontactparameters(Table5).However,forcedifferencebetweenthedifferenttestcasesisgreateratsomeothercontactpoints,asshowninFig.6.Especiallyinthebeginningofgearpairengagement,forcedifferencebetweenthedifferenttestcasesmayberemarkable.Inthestudiedhelicalgearcontact,theestimatedellipticityratiobecomesveryhigh.Thisisthecaseevenwhenthestudiedgearwasrathernarrow.ThisisthemainreasonwhythecalculatedFEMresultsarelessaccuratethanintheearliercontactmodeltestcase.Howaccuratetheassumptionofellipticalcontactisinhelicalgearcontactisnotstudiedhere.Futurestudieswilldeterminethefinalcapabilityoftheusedcontactmodels.However,theFEM-basedcontactmodelhaspotentialespeciallyinthecalculationofedgecontacts,i.e.incaseswhicharenotfullycoveredbyanalyticalformulas.6.ConclusionsAcalculationmodelfortheanalysisofhelicalgearcontactisintroduced.HelicalgearsurfaceprofilesareconstructedfromgeartoolgeometrybysimulatingtheInternational40(2007)613619hobbingprocess.Thisprocedureallowsdeviationsfromidealinvolutegeometry.Thegearpaircontactlineisnumericallydefineddirectfromthegearsurfacegeometry.Themodeluses3Dfiniteelementsforthecalculationoftoothdeflectionincludingtoothbending,shearingandtoothfoundationflexibility.Themodelcombinescontactanalysisandstructuralanalysistoavoidlargemeshes.Theflexibilityoftoothfoundationwasfoundtohaveanessentialroleincontactloadsharingbetweenthemeshingteeth,whereascontactflexibilityplaysonlyaminorrole.Thisindicatesthatreasonabledistributionoftoothcontactforcealongthelineofactionmaybegeneratedbyusingflexibleteethandflexibletoothfoundation,butallowingrigidcontact.Table5CalculationresultsCaseCase4Case2HertzFEMHertzFEMWcase(N)15701723.5a(mm)4.5b(mm)20.2d(mm)5.64.946.15.27p0(GPa)1.151.281.181.37TheFEM-basedcontactmodelgivesareasonableapproximationofcontactparameterswhenthemeshsizeisfineenough.Contactshapes,suchasinhelicalgears,requiresmallelementsize,i.e.alargenumberofelementstoavoidelementdimensionaldistortion.However,theFEM-basedcontactmodelhaspotentialincalculatingedgecontacts.References1Flod