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

syonalMode-couplinginstabilityForentemsmatrixsystetorsionmodeofthedisc.&2009ElsevierLtd.Allrightsreserved.atedinformationthrouerviewpublishedofautmajoranalysis.cularly,Huangetal.6usedtheeigenvalueperturbationmethodsolutionformode-mergingbetweendiscdoubletmodepair.Duethecopice.uealbrake,forexample,thein-planemodeandhatmodeofthedisc.ARTICLEINPRESSContentslistsavailableofInternationalJournalofMechanicalSciences51(2009)284294describingthecontactkinematicsundertheundeformedconfig-tothestationarydiscassumption,thefiniteelement(FE)methodhasbeeneasilyimplementedasreferredtothereviewarticle2.Alternately,Caoetal.13studiedthemovingloadeffectfromaFEdiscbrakemodelwithmovingpads,wherethediscwasInthispaper,themethodologyofconstructingarotatingFEdiscbrakemodelisdeveloped.Consequently,itenablesustoexaminethesquealmechanismsinthephysicalFEbrakemodelsubjecttorotationeffects.Theglobalcontactmodel10urationisutilizedtodevelopcontactmodelingbetweentherotatingdiscandtwostationarypads.FromtheassumedmodeE-mailaddress:jkangkongju.ac.kr0020-7403/$doi:10.1todevelopthenecessaryconditionformode-mergingwithoutthedirecteigensolutions.Kangetal.7derivedtheclosed-formmechanismssincetheannularplateapproximationdoesnotrepresentallofmodalbehaviorsexistingonthephysicaldiscbrakesystem912,14.Thestabilityanalysisatthestaticsteady-slidingequilibriumofthestationarydiscandpadsprovidesthesquealmechanismasmode-mergingcharacterinthefrictionfrequencydomain.Parti-annularsectorplates.Thecomprehensiveanalysisexplainedstabilitycharacterinfluencedbymode-couplingandgyroseffect,andprovidedthephysicalbackgroundontheapproxima-tionsandmechanismsusedintheprevioussquealliteraturHowever,itstillcontainslimitationsonexaminingbrakesqequationsofmotion,therealpartsofeigenvalueshavebeencalculatedfordeterminingtheequilibriumstability.Intheliterature,therearetwomajordirectionsonthelinearsquealanalysis:thecomplexeigenvalueanalysisofthestaticsteady-slidingequilibrium38andthestabilityanalysisofrotatingmodeling,however,arotatingFEdiscbrakemodelhasnotbeendevelopedyet.Recently,Kangetal.14developedatheoreticaldiscbrakemodelinthecomprehensivemanner.Thediscbrakemodelconsistsofarotatingannularplateincontactwithtwostationary1.IntroductionDiscbrakesquealhasbeeninvestigforseveraldecades.MuchvaluablemechanismshasbeenaccumulatedKinkaidetal.1presentedtheovbrakesquealstudies.Ouyangetal.2focusedonthenumericalanalysissqueal.Theyhaveshownthatonesquealstudyisthelinearstability-seefrontmatter&2009ElsevierLtd.All016/j.ijmecsci.2009.02.003bymanyresearchersonsquealghouttheresearch.onthevariousdiscthereviewarticleomotivediscbrakeapproachonbrakeFromthelinearizedstationary,andtherefore,thegyroscopiceffectswereneglected.Gianninietal.15,16validatedthemode-mergingbehaviorassquealonsetbyusingtheexperimentalsquealfrequencies.Ontheotherhand,thestabilityofarotatingdiscbrakehasbeeninvestigatedintheanalyticalmanner.Therotatingdiscbrakesystemhasbeenmodeledasaring10andanannularplate12inpointcontactwithtwopads,andanannularplatesubjecttodistributedfrictionaltraction9.Withinclusionofgyroscopiceffect,therealpartsofeigenvalueshavebeensolvedwithrespecttosystemparameters.DuetothecomplexityoftherotatingdiscSquealanalysisofgyroscopicdiscbrakeonfiniteelementmethodJaeyoungKangDivisionofMechanicalandAutomotiveEngineering,CollegeofEngineering,KongjuNatiarticleinfoArticlehistory:Received9October2008Receivedinrevisedform30January2009Accepted12February2009Availableonline9March2009Keywords:GyroscopicDiscbrakeBrakesquealabstractInthispaper,thedynamicstationarypadsisstudied.structurebythefiniteelemandmovingcoordinatesystcorrespondinggyroscopicmethod.Thedynamicinstabpredictedwithrespecttospeeddependsonthevibrationthenegativeslopeoffrictionjournalhomepage:www.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.Thediscrotationwithconstantspeed(O)generatesfrictionstressesoverthecontactwithtwostationarypadsloadedbypre-normalload(N0).Thefrictionmaterialofthepadismodeledastheuniformcontactstiffness(kc),wherecontactstressesaredefinedontheglobalcontactmodel.Centrifugalforceisneglectedduetotheslowrotationinthebrakesquealproblem.Inordertodescribethecontactkinematics,thedisplacementvectorsofthediscandtoppadareexpressedinthereferencecoordinates(Fig.2),respectively,suchthatuðr；y；z；tÞ¼uðr；y；z；tÞerþvðr；y；z；tÞehþwðr；y；z；tÞez(1)up1ðr；y；z；tÞ¼up1ðr；y；z；tÞerþvp1ðr；y；z；tÞehþwp1ðr；y；z；tÞez(2)ARTICLEINPRESSConnectedwithcontactstiffnessoNoNConnectedwithcontactstiffnessClampedataninnerZrotatingshaftFig.1.Hat-discbrakesystem.tXeZreckNeutralsurfaceRotorpartFig.2.Coordinatesystemoftherotatingdisc,reference(y)andlocal(c)coordinates.BottomsurfaceofthetopPad:contactarea(Ac)Pw1pRw1pRv1pRuPuPvZPRJ.Kang/InternationalJournalofMechanicalSciences51(2009)284294285SegmentofundeformedtopsurfaceofthediscFig.3.ContactkinematicsatacontactpointP(orP0)intheglobalcontactmodel:(a)contactcontactwithPofthedisc.wherethesuperscripts,p1andp2denotethetopandbottompads,respectively,andthediscdisplacementisalsodefinedinthelocalcoordinates(Fig.2):uðr；c；z；tÞ¼uðr；c；z；tÞerþvðr；c；z；tÞehþwðr；c；z；tÞez(3)AsshowninFig.3,thecontactpointP0offrictionmaterialofthetoppadisassumedtobeincontactwithPofthediscandlaterallyfixedwithRofthetoppad,whichresultsinup1P0ðr；y；tÞ¼up1Rðr；y；tÞerþvp1Rðr；y；tÞehþwPðr；y；tÞez(4)Thevelocityvectorsofthediscandtoppadareobtainedfromthefollowingtime-derivatives.First,thepositionvectorsofthediscareexpressedinthelocalcoordinatesasr¼ðrþuÞerþvehþðzþwÞez(5)rP¼rjz¼h=2(6)Fordescribingthedirectionvectoroffrictionforce,thecontactvelocityvectorofthediscisderivedbytakingthetime-derivative1ppo+kc(wPwR)ZDisc1Nze1Nzeck1F1FPRPdisplacements；(b)contactforces.P0offrictionmaterialofthetoppadisinARTICLEINPRESSinEq.(6)inthereferencecoordinates:VP¼DrPDt(7)wherethecoordinatetransformationisgivenbythedifferentia-tioninthelocalcoordinatessuchthatDuðr；c；z；tÞDt¼uðr；y；z；tÞtþOuðr；y；z；tÞy(8)Dvðr；c；z；tÞDt¼vðr；y；z；tÞtþOvðr；y；z；tÞy(9)Dwðr；c；z；tÞDt¼wðr；y；z；tÞtþOwðr；y；z；tÞy(10)Sincethebrakepadisstationary,thecontactvelocityvectoratP0ofthetoppadissimplythepartialtime-derivativeofEq.(4):Vp1P0¼up1Rterþvp1RtehþwPtez(11)FromCoulombslawoffriction,contactfrictionforceisexpressedasF1¼C0m1C1N1VreljVrelj(12)wherethenormalloadisthesumofpre-stress(p0¼N0/Ac)andthenormalloadvariation:N1¼p0þkcðwPC0wp1RÞ(13)andtherelativevelocityattopcontactisgivenbyVrel¼VPC0Vp1P0(14)Inordertocapturethenegativeslopeeffect,thecontinuousfrictioncurve14isusedsuchthatm1ðtÞ¼fmkþðmsC0mkÞeC0ajVreljgr¼rctr(15)wherems,mkandaarethecontrolparametersdeterminingthemagnitudeandtheslopeofthefrictioncoefficient,andthefrictioncoefficientisassumedtobeuniformandcalculatedatthecentroidofthecontactarea(rctr).ThetransversevibrationsofthediscandpadcomponentsareexpressedinthemodalexpansionformofN¼ðNdþ2NpÞtruncatedmodesusingtheassumedmodemethod:wp1ðx；tÞXNpn¼1jp1z；nðxÞqp1nðtÞ(16)wðx；tÞXNdn¼1jz；nðxÞqnðtÞ(17)wp2ðx；tÞXNpn¼1jp2z；nðxÞqp2nðtÞ(18)whereNdandNparethenumbersofthetruncatedmodesofthediscandthepad,respectively,andwhereqp1¼fqp11qp12.qp1Npg(19)q¼fq1q2.qNdg(20)qp2¼fqp21qp22.qp2Npg(21)jp1z；nðxÞ,jz；nðxÞandjp2z；nðxÞarethenthtransversemodeshapeJ.Kang/InternationalJournalofMechanical286functionsobtainedfromtheeigenfunctionsofthetoppad,discandbottompadcomponents,respectively.Theradialandtangentialvibrations,ðup1；u；up2Þandðvp1；v；vp2Þcanbewritteninthemodalexpansionformassociatedwiththecorrespondingmodeshapefunctionsfjp1r；nðxÞ；jr；nðxÞ；jp2r；nðxÞg,fjp1y；nðxÞ；jy；nðxÞ；jp2y；nðxÞgaswell.Themodalcoordinatesarerearrangedinthevectorformforthefollowingdiscretization:fag¼qp1qqp28><>:9>=>；¼fa1a2.aNgT(22)FromthediscretizationofLagrangeequationbymodalcoordinates,thefriction-coupledequationsofmotionaregivenbyddtL_amC20C21C0Lam¼XNn¼1QmnðanÞ；m¼1；.；N；n¼1；.；N(23)L¼TC0ðUþUcÞ(24)dWC17XNm¼1XNn¼1QmnðanÞdam(25)whereUisthetotalstrainenergyoftheuncoupledcomponentdiscandtwopads,andT¼Tp1þTdþTp2(26)Td¼rZVdDrDtC1DrDtC18C19dV(27)Tp1¼rpZVpup1tC1up1tC18C19dV(28)Tp2¼rpZVpup2tC1up2tC18C19dV(29)Uc¼kc2ZAcðwPC0wp1RÞ2dAþUc；bottom(30)dW¼ZAcfðC0N1C0F1ÞC1dup1P0þðN1þF1ÞC1duPgdAþdWbottom(31)HereVdandVparethevolumesofthediscandpad,respectively.Inthesimilarmannerofobtainingthevirtualworkandcontactstrainenergyatthetopcontact,dWbottomandUc,bottomonthebottomcontactcanbederivedaswell.ThedirectionvectoroffrictionforceatthetopcontactislinearizedbyTaylorexpansionatthesteadyslidingequilibriumsuchthatVreljVrelj¼1rOðuP=tC0up1R=tÞþ1rðuP=yC0vPÞC26C27erþehþ1rwPyezþh:o:t.(32)whereh:o:tdenotesthehigherorderterms.HerewP=yisassociatedwithfrictionalfollowerforceasexplainedin11andneglectedinthesubsequentanalysisduetotheinsignificanceofthefrictionalfollowerforceasreferredto5,10,11and14.Usingthefiniteelementmethod,thetransversemodeshapefunctionsarediscretizedinthematrixformup1zC2C3¼up1z；1up1z；2C1C1C1up1z；Nphi¼jp1z；jðxiÞhi(33)½uzC138¼½uz；1uz；2.uz；NdC138¼½jz；jðxiÞC138(34)up2zC2C3¼up2z；1up2z；2C1C1C1up2z；Nphi¼jp2z；jðxiÞhi(35)Sciences51(2009)284294wherethelengthsoftheircolumnscorrespondtothenumbersofnodesinthecomponentFEmodel.Theradialandtangentialmodefunctionsarealsodenotedasf½up1rC138；½urC138；½up2rC138gandf½up1yC138；½uyC138；½up2yC138g.Fromthemass-normalizationandthelinearizationatthesteady-slidingequilibriumofEq.(23),thehomogeneouspartofthelinearizedequationsofmotiontakesthe(NC2N)matrixformsuchthataþð½GC138þ½CC138þ½RdC138þ½NsC138Þ_aþð½o2C138þ½AC138þ½BC138þ½FC138Þa¼0(36)wherethesystemmatricesaredescribedinEq.(37)andEqs.(A.1)(A.7)ofAppendixA.SubstitutingaðtÞ¼aoeltintoEq.(36)andsolvingRe(l)andIm(l)ofthecharacteristicequationresultinthedeterminationofthemodalstabilityandfrequency.HerethephysicalmeaningofeachsystemmatrixofEq.(36)isprovidedinthefollowing.½GC138ð¼C0½GC138TÞisthegyroscopicmatrixtobedescribedinEq.(37),Cisthestructuralmodaldampingmatrix,and½NsC138ð¼½NsC138TÞisthenegativeslopematrix.Thenegativefriction-sloperadialindiscmatrix.theEq.Kangetal.14,wherethefrictionalfollowerforceeffectswerewhere½GdC138¼OrZVd½urC138TuryC20C21C0½uyC138C18C19C0uryC20C21C0½uyC138C18C19T½urC138(þ½uyC138TuyyC20C21þ½urC138C18C19C0uyyC20C21þ½urC138C18C19T½uyC138þ½uzC138TuzyC20C21C0uzyC20C21T½uzC138)dV(38)Inordertoresolvetheabove,thehat-discandeachpadshouldbeuniformlymeshedinthecylindricalcoordinatesbyANSYS(oranyotherpre-processingFEsoftware).Ingeneral,thistaskistrickyandnotrecommendedforthepracticalpurpose.Alternately,theuniformdiscretizationwillbeachievedbyinterpolatingthemodalvectorsofirregularmeshesontothoseofuniformmeshes.Theonlypre-requisiteforthistaskistodiscretizethediscgeometryintheaxialdirection(asFig.1)generatingtheplanarmeshonARTICLEINPRESSznJ.Kang/InternationalJournalofMechanicalSciences51(2009)284294287showntobemarginalduetothedominantroleofBinthenumericalandanalyticalmanners.Inthefiniteelementapproach,severaltechnicaldifficultiesareencounteredincalculatingEqs.(27)(31)numericallyandsummarizedas:C15Themeshofthecontactareabetweenthediscandpadshouldbeidenticalinordertoconnectthefinitecontactforceelementsonthesamecontactpositionsofthematingparts.C15Tdrequiresthenumericaly-derivativesofmodalvectors.Particularly,thegyroscopicmatrixisgivenby½GC138¼½0C138½0C138½0C138½0C138½GdC138½0C138½0C138½0C138½0C138264375(37),(,)znlmkxyz(,)xyzitsincorporFig.uniformfollowerforceassociatedwith1=rðuP=yC0vPÞin(32),butneglectedinthesubsequentanalysisduetoinsignificance14aswell.Thelocalcontactmodel10atedwiththefrictionalfollowerforcescanbereferredtobyfrictionaleffectcanbereferredto17.½RdC138ð¼½RdC138TÞisthedissipativematrixstemmingfrom1=rOðuP=tC0up1R=tÞEq.(32).Also,o2isthenaturalfrequencymatrixoftheandpadcomponents,½AC138ð¼½AC138TÞisthecontactstiffnessOfthenon-symmetricstiffnessmatrix,½BC138ða½BC138TÞisnon-symmetricnon-conservativeworkmatrixproducedfriction-couple.½FC138ða½FC138TÞisderivedfromthein-planelmk4.Transversemodalvectoratz¼zkinterpolatedbytheuniformplanar-meshmethod:planarmeshinthepolarcoordinates.eachlayerperpendiculartotheaxis,wheretheplanarmeshesarenotyetuniform.Then,themodalvectorsassignedtotheplanarmeshofeachlayerareinterpolatedontothoseoftheuniformmeshinpolarcoordinatesbyMATLAB,whichwillbereferredtotheuniformplanar-meshmethod.Fig.4illustrateshowthemodalvectorontheirregularplanarmeshisinterpolatedontothatoftheuniformplanarmesh.Forthemodeshapeshownontheirregularmesh(Fig.5a),theinterpolatedmodalvectorontheuniformplanarmeshesassignedtothetopsurfacesoftherotorandhatpartsisdemonstratedasinFigs.5bandc.Fromtheuniformplanarmeshinthecylindricalcoordinates,thenumericaly-derivativesofthenthmodevectorcanbecalculatedatðri；yj；zkÞ,forexample,jz；nðri；yj；zkÞy¼jz；nðri；yjþ1；zkÞC0jz；nðri；yj；zkÞyjþ1C0yj(39)wherei¼1；.；Mr,j¼1；.；My,k¼1；.；Mz,andMr,My,Mzarethenumbersofthenodesofthehat-disc,respectively,inthecylindricalcoordinates(r,y,z).Fig.6demonstratestheseveraly-derivativemodalvectorsofthehat-discatagivenzk.Inordertoassignthefinitecontactforceelementtoeachfiniteelementofthediscandpadcontactsatthesamelocation,theplanarmeshtakeninthedisccontactsurfaceisdefinedonthepadcontactsurfaceaswell.Moreover,themodalvectorsonthepadcontactareinterpolatedontothoseofthedefinedplanarmesh.ConnectingthefinitecontactforceelementbetweenthediscandpadisreferredtoFig.7and18.Asaresult,the(,)ijkrz,(,)znijkrz(),(,)kxyzx(a)modalvectorontheirregularmesh；(b)modalvectorinterpolatedonthe