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

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syonalMode-couplinginstabilityForentemsmatrixsystetorsionmodeofthedisc.2009ElsevierLtd.Allrightsreserved.atedinformationthrouerviewpublishedofautmajoranalysis.cularly,Huangetal.[6]usedtheeigenvalueperturbationmethodsolutionformode-mergingbetweendiscdoubletmodepair.Duethecopice.uealbrake,forexample,thein-planemodeandhatmodeofthedisc.ARTICLEINPRESSContentslistsavailableofInternationalJournalofMechanicalSciences512009284–294describingthecontactkinematicsundertheundeformedconfig-tothestationarydiscassumption,thefiniteelementFEmethodhasbeeneasilyimplementedasreferredtothereviewarticle[2].Alternately,Caoetal.[13]studiedthemovingloadeffectfromaFEdiscbrakemodelwithmovingpads,wherethediscwasInthispaper,themethodologyofconstructingarotatingFEdiscbrakemodelisdeveloped.Consequently,itenablesustoexaminethesquealmechanismsinthephysicalFEbrakemodelsubjecttorotationeffects.Theglobalcontactmodel[10]urationisutilizedtodevelopcontactmodelingbetweentherotatingdiscandtwostationarypads.FromtheassumedmodeE-mailaddressjkangkongju.ac.kr0020-7403/doi10.1todevelopthenecessaryconditionformode-mergingwithoutthedirecteigensolutions.Kangetal.[7]derivedtheclosed-formmechanismssincetheannularplateapproximationdoesnotrepresentallofmodalbehaviorsexistingonthephysicaldiscbrakesystem[9–12,14].Thestabilityanalysisatthestaticsteady-slidingequilibriumofthestationarydiscandpadsprovidesthesquealmechanismasmode-mergingcharacterinthefriction–frequencydomain.Parti-annularsectorplates.Thecomprehensiveanalysisexplainedstabilitycharacterinfluencedbymode-couplingandgyroseffect,andprovidedthephysicalbackgroundontheapproxima-tionsandmechanismsusedintheprevioussquealliteraturHowever,itstillcontainslimitationsonexaminingbrakesqequationsofmotion,therealpartsofeigenvalueshavebeencalculatedfordeterminingtheequilibriumstability.Intheliterature,therearetwomajordirectionsonthelinearsquealanalysisthecomplexeigenvalueanalysisofthestaticsteady-slidingequilibrium[3–8]andthestabilityanalysisofrotatingmodeling,however,arotatingFEdiscbrakemodelhasnotbeendevelopedyet.Recently,Kangetal.[14]developedatheoreticaldiscbrakemodelinthecomprehensivemanner.Thediscbrakemodelconsistsofarotatingannularplateincontactwithtwostationary1.IntroductionDiscbrakesquealhasbeeninvestigforseveraldecades.MuchvaluablemechanismshasbeenaccumulatedKinkaidetal.[1]presentedtheovbrakesquealstudies.Ouyangetal.[2]focusedonthenumericalanalysissqueal.Theyhaveshownthatonesquealstudyisthelinearstability-seefrontmatter2009ElsevierLtd.All016/j.ijmecsci.2009.02.003bymanyresearchersonsquealghouttheresearch.onthevariousdiscthereviewarticleomotivediscbrakeapproachonbrakeFromthelinearizedstationary,andtherefore,thegyroscopiceffectswereneglected.Gianninietal.[15,16]validatedthemode-mergingbehaviorassquealonsetbyusingtheexperimentalsquealfrequencies.Ontheotherhand,thestabilityofarotatingdiscbrakehasbeeninvestigatedintheanalyticalmanner.Therotatingdiscbrakesystemhasbeenmodeledasaring[10]andanannularplate[12]inpointcontactwithtwopads,andanannularplatesubjecttodistributedfrictionaltraction[9].Withinclusionofgyroscopiceffect,therealpartsofeigenvalueshavebeensolvedwithrespecttosystemparameters.DuetothecomplexityoftherotatingdiscSquealanalysisofgyroscopicdiscbrakeonfiniteelementmethodJaeyoungKangDivisionofMechanicalandAutomotiveEngineering,CollegeofEngineering,KongjuNatiarticleinfoArticlehistoryReceived9October2008Receivedinrevisedform30January2009Accepted12February2009Availableonline9March2009KeywordsGyroscopicDiscbrakeBrakesquealabstractInthispaper,thedynamicstationarypadsisstudied.structurebythefiniteelemandmovingcoordinatesystcorrespondinggyroscopicmethod.Thedynamicinstabpredictedwithrespecttospeeddependsonthevibrationthenegativeslopeoffrictionjournalhomepagewww.elsevierInternationalJournalrightsreserved.stembasedUniversity,Cheonan-Si,RepublicofKoreaofacarbrakesystemwitharotatingdiscincontactwithtwoactualgeometricapproximation,thediscismodeledasahat-discshapemethod.Fromacoordinatetransformationbetweenthereference,thecontactkinematicsbetweenthediscandpadsisdescribed.Theofthediscisconstructedbyintroducingtheuniformplanar-meshilityofagyroscopicnon-conservativebrakesystemisnumericallymparameters.Theresultsshowthatthesquealpropensityforrotationmodesparticipatinginsquealmodes.Moreover,itishighlightedthatcoefficienttakesanimportantroleingeneratingsquealinthein-planeatScienceDirect.com/locate/ijmecsciMechanicalSciencesmethod,theequationsofmotionofthefriction-engagedbrakesystemarederived.Thenumericalresultsdemonstrateseveralsquealmodesandexplainthecorrespondingsquealmechanisms.2.DerivationofequationsofmotionThediscpartofabrakesystemismodeledasahat-discshapestructureasshowninFig.1.Thehat-discissubjecttotheclampedboundaryconditionattheinnerrotatingshaftandthefreeboundaryconditionattheouterradius.Owingtothecomplexityofthegeometry,thefiniteelementmethodisutilizedformodalanalysis.ThediscrotationwithconstantspeedOgeneratesfrictionstressesoverthecontactwithtwostationarypadsloadedbypre-normalloadN0.Thefrictionmaterialofthepadismodeledastheuniformcontactstiffnesskc,wherecontactstressesaredefinedontheglobalcontactmodel.Centrifugalforceisneglectedduetotheslowrotationinthebrakesquealproblem.Inordertodescribethecontactkinematics,thedisplacementvectorsofthediscandtoppadareexpressedinthereferencecoordinatesFig.2,respectively,suchthatur;y;z;tur;y;z;tervr;y;z;tehwr;y;z;tez1up1r;y;z;tup1r;y;z;tervp1r;y;z;tehwp1r;y;z;tez2ARTICLEINPRESSConnectedwithcontactstiffnessoNoNConnectedwithcontactstiffnessΩClampedataninnerZrotatingshaftFig.1.Hat-discbrakesystem.tΩ⋅ψXθeZθreckΩNeutralsurfaceRotorpartFig.2.Coordinatesystemoftherotatingdisc,referenceyandlocalccoordinates.BottomsurfaceofthetopPadcontactareaAcPw1pRw1pRv1pRuPuPvZPRJ.Kang/InternationalJournalofMechanicalSciences512009284–294285SegmentofundeformedtopsurfaceofthediscFig.3.ContactkinematicsatacontactpointPorP0intheglobalcontactmodelacontactcontactwithPofthedisc.wherethesuperscripts,p1andp2denotethetopandbottompads,respectively,andthediscdisplacementisalsodefinedinthelocalcoordinatesFig.2ur;c;z;tur;c;z;tervr;c;z;tehwr;c;z;tez3AsshowninFig.3,thecontactpointP0offrictionmaterialofthetoppadisassumedtobeincontactwithPofthediscandlaterallyfixedwithRofthetoppad,whichresultsinup1P0r;y;tup1Rr;y;tervp1Rr;y;tehwPr;y;tez4Thevelocityvectorsofthediscandtoppadareobtainedfromthefollowingtime-derivatives.First,thepositionvectorsofthediscareexpressedinthelocalcoordinatesasrruervehzwez5rPrjzh26Fordescribingthedirectionvectoroffrictionforce,thecontactvelocityvectorofthediscisderivedbytakingthetime-derivative1ppokcwP–wRZDisc1N−ze1Nzeck1−F1FPR Pdisplacements;bcontactforces.P0offrictionmaterialofthetoppadisinARTICLEINPRESSinEq.6inthereferencecoordinatesVPDrPDt7wherethecoordinatetransformationisgivenbythedifferentia-tioninthelocalcoordinatessuchthatDur;c;z;tDtur;y;z;ttOur;y;z;ty8Dvr;c;z;tDtvr;y;z;ttOvr;y;z;ty9Dwr;c;z;tDtwr;y;z;ttOwr;y;z;ty10Sincethebrakepadisstationary,thecontactvelocityvectoratP0ofthetoppadissimplythepartialtime-derivativeofEq.4Vp1P0up1Rtervp1RtehwPtez11FromCoulomb’slawoffriction,contactfrictionforceisexpressedasF1C0m1C1N1VreljVrelj12wherethenormalloadisthesumofpre-stressp0N0/AcandthenormalloadvariationN1p0kcwPC0wp1R13andtherelativevelocityattopcontactisgivenbyVrelVPC0Vp1P014Inordertocapturethenegativeslopeeffect,thecontinuousfrictioncurve[14]isusedsuchthatm1tfmkmsC0mkeC0ajVreljgrrctr15wherems,mkandaarethecontrolparametersdeterminingthemagnitudeandtheslopeofthefrictioncoefficient,andthefrictioncoefficientisassumedtobeuniformandcalculatedatthecentroidofthecontactarearctr.ThetransversevibrationsofthediscandpadcomponentsareexpressedinthemodalexpansionformofNNd2Nptruncatedmodesusingtheassumedmodemethodwp1x;tffiXNpn1jp1z;nxqp1nt16wx;tffiXNdn1jz;nxqnt17wp2x;tffiXNpn1jp2z;nxqp2nt18whereNdandNparethenumbersofthetruncatedmodesofthediscandthepad,respectively,andwhereqp1fqp11qp12...qp1Npg19qfq1q2...qNdg20qp2fqp21qp22...qp2Npg21jp1z;nx,jz;nxandjp2z;nxarethenthtransversemodeshapeJ.Kang/InternationalJournalofMechanical286functionsobtainedfromtheeigenfunctionsofthetoppad,discandbottompadcomponents,respectively.Theradialandtangentialvibrations,up1;u;up2andvp1;v;vp2canbewritteninthemodalexpansionformassociatedwiththecorrespondingmodeshapefunctionsfjp1r;nx;jr;nx;jp2r;nxg,fjp1y;nx;jy;nx;jp2y;nxgaswell.Themodalcoordinatesarerearrangedinthevectorformforthefollowingdiscretizationfagqp1qqp289;fa1a2...aNgT22FromthediscretizationofLagrangeequationbymodalcoordinates,thefriction-coupledequationsofmotionaregivenbyddtL_amC20C21C0LamXNn1Qmnan;m1;...;N;n1;...;N23LTC0UUc24dWC17XNm1XNn1Qmnandam25whereUisthetotalstrainenergyoftheuncoupledcomponentdiscandtwopads,andTTp1TdTp226TdrZVdDrDtC1DrDtC18C19dV27Tp1rpZVpup1tC1up1tC18C19dV28Tp2rpZVpup2tC1up2tC18C19dV29Uckc2ZAcwPC0wp1R2dAUc;bottom30dWZAcfC0N1C0F1C1dup1P0N1F1C1duPgdAdWbottom31HereVdandVparethevolumesofthediscandpad,respectively.Inthesimilarmannerofobtainingthevirtualworkandcontactstrainenergyatthetopcontact,dWbottomandUc,bottomonthebottomcontactcanbederivedaswell.ThedirectionvectoroffrictionforceatthetopcontactislinearizedbyTaylorexpansionatthesteadyslidingequilibriumsuchthatVreljVrelj1rOuPtC0up1Rt1ruPyC0vPC26C27ereh1rwPyezhot.32wherehotdenotesthehigherorderterms.HerewPyisassociatedwithfrictionalfollowerforceasexplainedin[11]andneglectedinthesubsequentanalysisduetotheinsignificanceofthefrictionalfollowerforceasreferredto[5],[10,11]and[14].Usingthefiniteelementmethod,thetransversemodeshapefunctionsarediscretizedinthematrixformup1zC2C3up1z;1up1z;2C1C1C1up1z;Nphijp1z;jxihi33uzC138uz;1uz;2...uz;NdC138jz;jxiC13834up2zC2C3up2z;1up2z;2C1C1C1up2z;Nphijp2z;jxihi35Sciences512009284–294wherethelengthsoftheircolumnscorrespondtothenumbersofnodesinthecomponentFEmodel.Theradialandtangentialmodefunctionsarealsodenotedasfup1rC138;urC138;up2rC138gandfup1yC138;uyC138;up2yC138g.Fromthemass-normalizationandthelinearizationatthesteady-slidingequilibriumofEq.23,thehomogeneouspartofthelinearizedequationsofmotiontakestheNC2Nmatrixformsuchthat€aGC138CC138RdC138NsC138_ao2C138AC138BC138FC138a036wherethesystemmatricesaredescribedinEq.37andEqs.A.1–A.7ofAppendixA.SubstitutingataoeltintoEq.36andsolvingRelandImlofthecharacteristicequationresultinthedeterminationofthemodalstabilityandfrequency.HerethephysicalmeaningofeachsystemmatrixofEq.36isprovidedinthefollowing.GC138C0GC138TisthegyroscopicmatrixtobedescribedinEq.37,[C]isthestructuralmodaldampingmatrix,andNsC138NsC138Tisthenegativeslopematrix.Thenegativefriction-sloperadialindiscmatrix.theEq.Kangetal.[14],wherethefrictionalfollowerforceeffectswerewhereGdC138OrZVdurC138TuryC20C21C0uyC138C18C19C0uryC20C21C0uyC138C18C19TurC138uyC138TuyyC20C21urC138C18C19C0uyyC20C21urC138C18C19TuyC138uzC138TuzyC20C21C0uzyC20C21TuzC138dV38Inordertoresolvetheabove,thehat-discandeachpadshouldbeuniformlymeshedinthecylindricalcoordinatesbyANSYSoranyotherpre-processingFEsoftware.Ingeneral,thistaskistrickyandnotrecommendedforthepracticalpurpose.Alternately,theuniformdiscretizationwillbeachievedbyinterpolatingthemodalvectorsofirregularmeshesontothoseofuniformmeshes.Theonlypre-requisiteforthistaskistodiscretizethediscgeometryintheaxialdirectionasFig.1generatingtheplanarmeshonARTICLEINPRESSznϕJ.Kang/InternationalJournalofMechanicalSciences512009284–294287showntobemarginalduetothedominantroleof[B]inthenumericalandanalyticalmanners.Inthefiniteelementapproach,severaltechnicaldifficultiesareencounteredincalculatingEqs.27–31numericallyandsummarizedasC15Themeshofthecontactareabetweenthediscandpadshouldbeidenticalinordertoconnectthefinitecontactforceelementsonthesamecontactpositionsofthematingparts.C15Tdrequiresthenumericaly-derivativesofmodalvectors.Particularly,thegyroscopicmatrixisgivenbyGC1380C1380C1380C1380C138GdC1380C1380C1380C1380C13826437537,,,znlmkxyzϕ,,xyzitsincorporFig.uniformfollowerforceassociatedwith1ruPyC0vPin32,butneglectedinthesubsequentanalysisduetoinsignificance[14]aswell.Thelocalcontactmodel[10]atedwiththefrictionalfollowerforcescanbereferredtobyfrictionaleffectcanbereferredto[17].RdC138RdC138Tisthedissipativematrixstemmingfrom1rOuPtC0up1RtEq.32.Also,[o2]isthenaturalfrequencymatrixoftheandpadcomponents,AC138AC138TisthecontactstiffnessOfthenon-symmetricstiffnessmatrix,BC138aBC138Tisnon-symmetricnon-conservativeworkmatrixproducedfriction-couple.FC138aFC138Tisderivedfromthein-planelmk4.Transversemodalvectoratzzkinterpolatedbytheuniformplanar-meshmethodplanarmeshinthepolarcoordinates.eachlayerperpendiculartotheaxis,wheretheplanarmeshesarenotyetuniform.Then,themodalvectorsassignedtotheplanarmeshofeachlayerareinterpolatedontothoseoftheuniformmeshinpolarcoordinatesbyMATLAB,whichwillbereferredtotheuniformplanar-meshmethod.Fig.4illustrateshowthemodalvectorontheirregularplanarmeshisinterpolatedontothatoftheuniformplanarmesh.ForthemodeshapeshownontheirregularmeshFig.5a,theinterpolatedmodalvectorontheuniformplanarmeshesassignedtothetopsurfacesoftherotorandhatpartsisdemonstratedasinFigs.5bandc.Fromtheuniformplanarmeshinthecylindricalcoordinates,thenumericaly-derivativesofthenthmodevectorcanbecalculatedatri;yj;zk,forexample,jz;nri;yj;zkyjz;nri;yj1;zkC0jz;nri;yj;zkyj1C0yj39wherei1;...;Mr,j1;...;My,k1;...;Mz,andMr,My,Mzarethenumbersofthenodesofthehat-disc,respectively,inthecylindricalcoordinatesr,y,z.Fig.6demonstratestheseveraly-derivativemodalvectorsofthehat-discatagivenzk.Inordertoassignthefinitecontactforceelementtoeachfiniteelementofthediscandpadcontactsatthesamelocation,theplanarmeshtakeninthedisccontactsurfaceisdefinedonthepadcontactsurfaceaswell.Moreover,themodalvectorsonthepadcontactareinterpolatedontothoseofthedefinedplanarmesh.ConnectingthefinitecontactforceelementbetweenthediscandpadisreferredtoFig.7and[18].Asaresult,the,,ijkrzθ,,,znijkrzϕθ,,,kxyzxamodalvectorontheirregularmesh;bmodalvectorinterpolatedonthe
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