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外文资料--Squeal analysis of gyroscopic disc brake system based on finite element method.pdf外文资料--Squeal analysis of gyroscopic disc brake system based on finite element method.pdf -- 1 元

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syonalModecouplinginstabilityForentemsmatrixsystetorsionmodeofthedisc.2009ElsevierLtd.Allrightsreserved.atedinformationthrouerviewpublishedofautmajoranalysis.cularly,Huangetal.6usedtheeigenvalueperturbationmethodsolutionformodemergingbetweendiscdoubletmodepair.Duethecopice.uealbrake,forexample,theinplanemodeandhatmodeofthedisc.ARTICLEINPRESSContentslistsavailableofInternationalJournalofMechanicalSciences512009284–294describingthecontactkinematicsundertheundeformedconfigtothestationarydiscassumption,thefiniteelementFEmethodhasbeeneasilyimplementedasreferredtothereviewarticle2.Alternately,Caoetal.13studiedthemovingloadeffectfromaFEdiscbrakemodelwithmovingpads,wherethediscwasInthispaper,themethodologyofconstructingarotatingFEdiscbrakemodelisdeveloped.Consequently,itenablesustoexaminethesquealmechanismsinthephysicalFEbrakemodelsubjecttorotationeffects.Theglobalcontactmodel10urationisutilizedtodevelopcontactmodelingbetweentherotatingdiscandtwostationarypads.FromtheassumedmodeEmailaddressjkangkongju.ac.kr00207403/doi10.1todevelopthenecessaryconditionformodemergingwithoutthedirecteigensolutions.Kangetal.7derivedtheclosedformmechanismssincetheannularplateapproximationdoesnotrepresentallofmodalbehaviorsexistingonthephysicaldiscbrakesystem9–12,14.Thestabilityanalysisatthestaticsteadyslidingequilibriumofthestationarydiscandpadsprovidesthesquealmechanismasmodemergingcharacterinthefriction–frequencydomain.Partiannularsectorplates.Thecomprehensiveanalysisexplainedstabilitycharacterinfluencedbymodecouplingandgyroseffect,andprovidedthephysicalbackgroundontheapproximationsandmechanismsusedintheprevioussquealliteraturHowever,itstillcontainslimitationsonexaminingbrakesqequationsofmotion,therealpartsofeigenvalueshavebeencalculatedfordeterminingtheequilibriumstability.Intheliterature,therearetwomajordirectionsonthelinearsquealanalysisthecomplexeigenvalueanalysisofthestaticsteadyslidingequilibrium3–8andthestabilityanalysisofrotatingmodeling,however,arotatingFEdiscbrakemodelhasnotbeendevelopedyet.Recently,Kangetal.14developedatheoreticaldiscbrakemodelinthecomprehensivemanner.Thediscbrakemodelconsistsofarotatingannularplateincontactwithtwostationary1.IntroductionDiscbrakesquealhasbeeninvestigforseveraldecades.MuchvaluablemechanismshasbeenaccumulatedKinkaidetal.1presentedtheovbrakesquealstudies.Ouyangetal.2focusedonthenumericalanalysissqueal.Theyhaveshownthatonesquealstudyisthelinearstabilityseefrontmatter2009ElsevierLtd.All016/j.ijmecsci.2009.02.003bymanyresearchersonsquealghouttheresearch.onthevariousdiscthereviewarticleomotivediscbrakeapproachonbrakeFromthelinearizedstationary,andtherefore,thegyroscopiceffectswereneglected.Gianninietal.15,16validatedthemodemergingbehaviorassquealonsetbyusingtheexperimentalsquealfrequencies.Ontheotherhand,thestabilityofarotatingdiscbrakehasbeeninvestigatedintheanalyticalmanner.Therotatingdiscbrakesystemhasbeenmodeledasaring10andanannularplate12inpointcontactwithtwopads,andanannularplatesubjecttodistributedfrictionaltraction9.Withinclusionofgyroscopiceffect,therealpartsofeigenvalueshavebeensolvedwithrespecttosystemparameters.DuetothecomplexityoftherotatingdiscSquealanalysisofgyroscopicdiscbrakeonfiniteelementmethodJaeyoungKangDivisionofMechanicalandAutomotiveEngineering,CollegeofEngineering,KongjuNatiarticleinfoArticlehistoryReceived9October2008Receivedinrevisedform30January2009Accepted12February2009Availableonline9March2009KeywordsGyroscopicDiscbrakeBrakesquealabstractInthispaper,thedynamicstationarypadsisstudied.structurebythefiniteelemandmovingcoordinatesystcorrespondinggyroscopicmethod.Thedynamicinstabpredictedwithrespecttospeeddependsonthevibrationthenegativeslopeoffrictionjournalhomepagewww.elsevierInternationalJournalrightsreserved.stembasedUniversity,CheonanSi,RepublicofKoreaofacarbrakesystemwitharotatingdiscincontactwithtwoactualgeometricapproximation,thediscismodeledasahatdiscshapemethod.Fromacoordinatetransformationbetweenthereference,thecontactkinematicsbetweenthediscandpadsisdescribed.Theofthediscisconstructedbyintroducingtheuniformplanarmeshilityofagyroscopicnonconservativebrakesystemisnumericallymparameters.Theresultsshowthatthesquealpropensityforrotationmodesparticipatinginsquealmodes.Moreover,itishighlightedthatcoefficienttakesanimportantroleingeneratingsquealintheinplaneatScienceDirect.com/locate/ijmecsciMechanicalSciencesmethod,theequationsofmotionofthefrictionengagedbrakesystemarederived.Thenumericalresultsdemonstrateseveralsquealmodesandexplainthecorrespondingsquealmechanisms.2.DerivationofequationsofmotionThediscpartofabrakesystemismodeledasahatdiscshapestructureasshowninFig.1.Thehatdiscissubjecttotheclampedboundaryconditionattheinnerrotatingshaftandthefreeboundaryconditionattheouterradius.Owingtothecomplexityofthegeometry,thefiniteelementmethodisutilizedformodalanalysis.ThediscrotationwithconstantspeedOgeneratesfrictionstressesoverthecontactwithtwostationarypadsloadedbyprenormalloadN0.Thefrictionmaterialofthepadismodeledastheuniformcontactstiffnesskc,wherecontactstressesaredefinedontheglobalcontactmodel.Centrifugalforceisneglectedduetotheslowrotationinthebrakesquealproblem.Inordertodescribethecontactkinematics,thedisplacementvectorsofthediscandtoppadareexpressedinthereferencecoordinatesFig.2,respectively,suchthatuðryztÞ¼uðryztÞerþvðryztÞehþwðryztÞez1up1ðryztÞ¼up1ðryztÞerþvp1ðryztÞehþwp1ðryztÞez2ARTICLEINPRESSConnectedwithcontactstiffnessoNoNConnectedwithcontactstiffnessΩClampedataninnerZrotatingshaftFig.1.Hatdiscbrakesystem.tΩ⋅ψXθeZθreckΩNeutralsurfaceRotorpartFig.2.Coordinatesystemoftherotatingdisc,referenceyandlocalccoordinates.BottomsurfaceofthetopPadcontactareaAcPw1pRw1pRv1pRuPuPvZPRJ.Kang/InternationalJournalofMechanicalSciences512009284–294285SegmentofundeformedtopsurfaceofthediscFig.3.ContactkinematicsatacontactpointPorP0intheglobalcontactmodelacontactcontactwithPofthedisc.wherethesuperscripts,p1andp2denotethetopandbottompads,respectively,andthediscdisplacementisalsodefinedinthelocalcoordinatesFig.2uðrcztÞ¼uðrcztÞerþvðrcztÞehþwðrcztÞez3AsshowninFig.3,thecontactpointP0offrictionmaterialofthetoppadisassumedtobeincontactwithPofthediscandlaterallyfixedwithRofthetoppad,whichresultsinup1P0ðrytÞ¼up1RðrytÞerþvp1RðrytÞehþwPðrytÞez4Thevelocityvectorsofthediscandtoppadareobtainedfromthefollowingtimederivatives.First,thepositionvectorsofthediscareexpressedinthelocalcoordinatesasr¼ðrþuÞerþvehþðzþwÞez5rP¼rjz¼h26Fordescribingthedirectionvectoroffrictionforce,thecontactvelocityvectorofthediscisderivedbytakingthetimederivative1ppokcwP–wRZDisc1N−ze1Nzeck1−F1FPRPdisplacementsbcontactforces.P0offrictionmaterialofthetoppadisinARTICLEINPRESSinEq.6inthereferencecoordinatesVP¼DrPDt7wherethecoordinatetransformationisgivenbythedifferentiationinthelocalcoordinatessuchthatDuðrcztÞDt¼uðryztÞtþOuðryztÞy8DvðrcztÞDt¼vðryztÞtþOvðryztÞy9DwðrcztÞDt¼wðryztÞtþOwðryztÞy10Sincethebrakepadisstationary,thecontactvelocityvectoratP0ofthetoppadissimplythepartialtimederivativeofEq.4Vp1P0¼up1Rterþvp1RtehþwPtez11FromCoulombslawoffriction,contactfrictionforceisexpressedasF1¼C0m1C1N1VreljVrelj12wherethenormalloadisthesumofprestressp0¼N0/AcandthenormalloadvariationN1¼p0þkcðwPC0wp1RÞ13andtherelativevelocityattopcontactisgivenbyVrel¼VPC0Vp1P014Inordertocapturethenegativeslopeeffect,thecontinuousfrictioncurve14isusedsuchthatm1ðtÞ¼fmkþðmsC0mkÞeC0ajVreljgr¼rctr15wherems,mkandaarethecontrolparametersdeterminingthemagnitudeandtheslopeofthefrictioncoefficient,andthefrictioncoefficientisassumedtobeuniformandcalculatedatthecentroidofthecontactarearctr.ThetransversevibrationsofthediscandpadcomponentsareexpressedinthemodalexpansionformofN¼ðNdþ2NpÞtruncatedmodesusingtheassumedmodemethodwp1ðxtÞffiXNpn¼1jp1znðxÞqp1nðtÞ16wðxtÞffiXNdn¼1jznðxÞqnðtÞ17wp2ðxtÞffiXNpn¼1jp2znðxÞqp2nðtÞ18whereNdandNparethenumbersofthetruncatedmodesofthediscandthepad,respectively,andwhereqp1¼fqp11qp12...qp1Npg19q¼fq1q2...qNdg20qp2¼fqp21qp22...qp2Npg21jp1znðxÞ,jznðxÞandjp2znðxÞarethenthtransversemodeshapeJ.Kang/InternationalJournalofMechanical286functionsobtainedfromtheeigenfunctionsofthetoppad,discandbottompadcomponents,respectively.Theradialandtangentialvibrations,ðup1uup2Þandðvp1vvp2Þcanbewritteninthemodalexpansionformassociatedwiththecorrespondingmodeshapefunctionsfjp1rnðxÞjrnðxÞjp2rnðxÞg,fjp1ynðxÞjynðxÞjp2ynðxÞgaswell.Themodalcoordinatesarerearrangedinthevectorformforthefollowingdiscretizationfag¼qp1qqp289¼fa1a2...aNgT22FromthediscretizationofLagrangeequationbymodalcoordinates,thefrictioncoupledequationsofmotionaregivenbyddtL_amC20C21C0Lam¼XNn¼1QmnðanÞm¼1...Nn¼1...N23L¼TC0ðUþUcÞ24dWC17XNm¼1XNn¼1QmnðanÞdam25whereUisthetotalstrainenergyoftheuncoupledcomponentdiscandtwopads,andT¼Tp1þTdþTp226Td¼rZVdDrDtC1DrDtC18C19dV27Tp1¼rpZVpup1tC1up1tC18C19dV28Tp2¼rpZVpup2tC1up2tC18C19dV29Uc¼kc2ZAcðwPC0wp1RÞ2dAþUcbottom30dW¼ZAcfðC0N1C0F1ÞC1dup1P0þðN1þF1ÞC1duPgdAþdWbottom31HereVdandVparethevolumesofthediscandpad,respectively.Inthesimilarmannerofobtainingthevirtualworkandcontactstrainenergyatthetopcontact,dWbottomandUc,bottomonthebottomcontactcanbederivedaswell.ThedirectionvectoroffrictionforceatthetopcontactislinearizedbyTaylorexpansionatthesteadyslidingequilibriumsuchthatVreljVrelj¼1rOðuPtC0up1RtÞþ1rðuPyC0vPÞC26C27erþehþ1rwPyezþhot.32wherehotdenotesthehigherorderterms.HerewPyisassociatedwithfrictionalfollowerforceasexplainedin11andneglectedinthesubsequentanalysisduetotheinsignificanceofthefrictionalfollowerforceasreferredto5,10,11and14.Usingthefiniteelementmethod,thetransversemodeshapefunctionsarediscretizedinthematrixformup1zC2C3¼up1z1up1z2C1C1C1up1zNphi¼jp1zjðxiÞhi33½uzC138¼½uz1uz2...uzNdC138¼½jzjðxiÞC13834up2zC2C3¼up2z1up2z2C1C1C1up2zNphi¼jp2zjðxiÞhi35Sciences512009284–294wherethelengthsoftheircolumnscorrespondtothenumbersofnodesinthecomponentFEmodel.Theradialandtangentialmodefunctionsarealsodenotedasf½up1rC138½urC138½up2rC138gandf½up1yC138½uyC138½up2yC138g.FromthemassnormalizationandthelinearizationatthesteadyslidingequilibriumofEq.23,thehomogeneouspartofthelinearizedequationsofmotiontakestheNC2Nmatrixformsuchthat€aþð½GC138þ½CC138þ½RdC138þ½NsC138Þ_aþð½o2C138þ½AC138þ½BC138þ½FC138Þa¼036wherethesystemmatricesaredescribedinEq.37andEqs.A.1–A.7ofAppendixA.SubstitutingaðtÞ¼aoeltintoEq.36andsolvingRelandImlofthecharacteristicequationresultinthedeterminationofthemodalstabilityandfrequency.HerethephysicalmeaningofeachsystemmatrixofEq.36isprovidedinthefollowing.½GC138ð¼C0½GC138TÞisthegyroscopicmatrixtobedescribedinEq.37,Cisthestructuralmodaldampingmatrix,and½NsC138ð¼½NsC138TÞisthenegativeslopematrix.Thenegativefrictionsloperadialindiscmatrix.theEq.Kangetal.14,wherethefrictionalfollowerforceeffectswerewhere½GdC138¼OrZVd½urC138TuryC20C21C0½uyC138C18C19C0uryC20C21C0½uyC138C18C19T½urC138þ½uyC138TuyyC20C21þ½urC138C18C19C0uyyC20C21þ½urC138C18C19T½uyC138þ½uzC138TuzyC20C21C0uzyC20C21T½uzC138dV38Inordertoresolvetheabove,thehatdiscandeachpadshouldbeuniformlymeshedinthecylindricalcoordinatesbyANSYSoranyotherpreprocessingFEsoftware.Ingeneral,thistaskistrickyandnotrecommendedforthepracticalpurpose.Alternately,theuniformdiscretizationwillbeachievedbyinterpolatingthemodalvectorsofirregularmeshesontothoseofuniformmeshes.TheonlyprerequisiteforthistaskistodiscretizethediscgeometryintheaxialdirectionasFig.1generatingtheplanarmeshonARTICLEINPRESSznϕJ.Kang/InternationalJournalofMechanicalSciences512009284–294287showntobemarginalduetothedominantroleofBinthenumericalandanalyticalmanners.Inthefiniteelementapproach,severaltechnicaldifficultiesareencounteredincalculatingEqs.27–31numericallyandsummarizedasC15Themeshofthecontactareabetweenthediscandpadshouldbeidenticalinordertoconnectthefinitecontactforceelementsonthesamecontactpositionsofthematingparts.C15Tdrequiresthenumericalyderivativesofmodalvectors.Particularly,thegyroscopicmatrixisgivenby½GC138¼½0C138½0C138½0C138½0C138½GdC138½0C138½0C138½0C138½0C13826437537,,,znlmkxyzϕ,,xyzitsincorporFig.uniformfollowerforceassociatedwith1rðuPyC0vPÞin32,butneglectedinthesubsequentanalysisduetoinsignificance14aswell.Thelocalcontactmodel10atedwiththefrictionalfollowerforcescanbereferredtobyfrictionaleffectcanbereferredto17.½RdC138ð¼½RdC138TÞisthedissipativematrixstemmingfrom1rOðuPtC0up1RtÞEq.32.Also,o2isthenaturalfrequencymatrixoftheandpadcomponents,½AC138ð¼½AC138TÞisthecontactstiffnessOfthenonsymmetricstiffnessmatrix,½BC138ða½BC138TÞisnonsymmetricnonconservativeworkmatrixproducedfrictioncouple.½FC138ða½FC138TÞisderivedfromtheinplanelmk4.Transversemodalvectoratz¼zkinterpolatedbytheuniformplanarmeshmethodplanarmeshinthepolarcoordinates.eachlayerperpendiculartotheaxis,wheretheplanarmeshesarenotyetuniform.Then,themodalvectorsassignedtotheplanarmeshofeachlayerareinterpolatedontothoseoftheuniformmeshinpolarcoordinatesbyMATLAB,whichwillbereferredtotheuniformplanarmeshmethod.Fig.4illustrateshowthemodalvectorontheirregularplanarmeshisinterpolatedontothatoftheuniformplanarmesh.ForthemodeshapeshownontheirregularmeshFig.5a,theinterpolatedmodalvectorontheuniformplanarmeshesassignedtothetopsurfacesoftherotorandhatpartsisdemonstratedasinFigs.5bandc.Fromtheuniformplanarmeshinthecylindricalcoordinates,thenumericalyderivativesofthenthmodevectorcanbecalculatedatðriyjzkÞ,forexample,jznðriyjzkÞy¼jznðriyjþ1zkÞC0jznðriyjzkÞyjþ1C0yj39wherei¼1...Mr,j¼1...My,k¼1...Mz,andMr,My,Mzarethenumbersofthenodesofthehatdisc,respectively,inthecylindricalcoordinatesr,y,z.Fig.6demonstratestheseveralyderivativemodalvectorsofthehatdiscatagivenzk.Inordertoassignthefinitecontactforceelementtoeachfiniteelementofthediscandpadcontactsatthesamelocation,theplanarmeshtakeninthedisccontactsurfaceisdefinedonthepadcontactsurfaceaswell.Moreover,themodalvectorsonthepadcontactareinterpolatedontothoseofthedefinedplanarmesh.ConnectingthefinitecontactforceelementbetweenthediscandpadisreferredtoFig.7and18.Asaresult,the,,ijkrzθ,,,znijkrzϕθ,,,kxyzxamodalvectorontheirregularmeshbmodalvectorinterpolatedonthe
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