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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-planeatScienceD/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,suchthatur;y;z;tur;y;z;tervr;y;z;tehwr;y;z;tez(1)up1r;y;z;tup1r;y;z;tervp1r;y;z;tehwp1r;y;z;tez(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):ur;c;z;tur;c;z;tervr;c;z;tehwr;c;z;tez(3)AsshowninFig.3,thecontactpointP0offrictionmaterialofthetoppadisassumedtobeincontactwithPofthediscandlaterallyfixedwithRofthetoppad,whichresultsinup1P0r;y;tup1Rr;y;tervp1Rr;y;tehwPr;y;tez(4)Thevelocityvectorsofthediscandtoppadareobtainedfromthefollowingtime-derivatives.First,thepositionvectorsofthediscareexpressedinthelocalcoordinatesasrruervehzwez(5)rPrjzh=2(6)Fordescribingthedirectionvectoroffrictionforce,thecontactvelocityvectorofthediscisderivedbytakingthetime-derivative1ppo+kc(wPwR)ZDisc1Nze1Nzeck1F1FPRPdisplacements;(b)contactforces.P0offrictionmaterialofthetoppadisinARTICLEINPRESSinEq.(6)inthereferencecoordinates:VPDrPDt(7)wherethecoordinatetransformationisgivenbythedifferentia-tioninthelocalcoordinatessuchthatDur;c;z;tDtur;y;z;ttOur;y;z;ty(8)Dvr;c;z;tDtvr;y;z;ttOvr;y;z;ty(9)Dwr;c;z;tDtwr;y;z;ttOwr;y;z;ty(10)Sincethebrakepadisstationary,thecontactvelocityvectoratP0ofthetoppadissimplythepartialtime-derivativeofEq.(4):Vp1P0up1Rtervp1RtehwPtez(11)FromCoulombslawoffriction,contactfrictionforceisexpressedasF1C0m1C1N1VreljVrelj(12)wherethenormalloadisthesumofpre-stress(p0N0/Ac)andthenormalloadvariation:N1p0kcwPC0wp1R(13)andtherelativevelocityattopcontactisgivenbyVrelVPC0Vp1P0(14)Inordertocapturethenegativeslopeeffect,thecontinuousfrictioncurve14isusedsuchthatm1tfmkmsC0mkeC0ajVreljgrrctr(15)wherems,mkandaarethecontrolparametersdeterminingthemagnitudeandtheslopeofthefrictioncoefficient,andthefrictioncoefficientisassumedtobeuniformandcalculatedatthecentroidofthecontactarea(rctr).ThetransversevibrationsofthediscandpadcomponentsareexpressedinthemodalexpansionformofNNd2Nptruncatedmodesusingtheassumedmodemethod:wp1x;tXNpn1jp1z;nxqp1nt(16)wx;tXNdn1jz;nxqnt(17)wp2x;tXNpn1jp2z;nxqp2nt(18)whereNdandNparethenumbersofthetruncatedmodesofthediscandthepad,respectively,andwhereqp1fqp11qp12.qp1Npg(19)qfq1q2.qNdg(20)qp2fqp21qp22.qp2Npg(21)jp1z;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.Themodalcoordinatesarerearrangedinthevectorformforthefollowingdiscretization:fagqp1qqp28:9=;fa1a2.aNgT(22)FromthediscretizationofLagrangeequationbymodalcoordinates,thefriction-coupledequationsofmotionaregivenbyddtL_amC20C21C0LamXNn1Qmnan;m1;.;N;n1;.;N(23)LTC0UUc(24)dWC17XNm1XNn1Qmnandam(25)whereUisthetotalstrainenergyoftheuncoupledcomponentdiscandtwopads,andTTp1TdTp2(26)TdrZVdDrDtC1DrDtC18C19dV(27)Tp1rpZVpup1tC1up1tC18C19dV(28)Tp2rpZVpup2tC1up2tC18C19dV(29)Uckc2ZAcwPC0wp1R2dAUc;bottom(30)dWZAcfC0N1C0F1C1dup1P0N1F1C1duPgdAdWbottom(31)HereVdandVparethevolumesofthediscandpad,respectively.Inthesimilarmannerofobtainingthevirtualworkandcontactstrainenergyatthetopcontact,dWbottomandUc,bottomonthebottomcontactcanbederivedaswell.ThedirectionvectoroffrictionforceatthetopcontactislinearizedbyTaylorexpansionatthesteadyslidingequilibriumsuchthatVreljVrelj1rOuP=tC0up1R=t1ruP=yC0vPC26C27ereh1rwPyezh:o:t.(32)whereh:o:tdenotesthehigherorderterms.HerewP=yisassociatedwithfrictionalfollowerforceasexplainedin11andneglectedinthesubsequentanalysisduetotheinsignificanceofthefrictionalfollowerforceasreferredto5,10,11and14.Usingthefiniteelementmethod,thetransversemodeshapefunctionsarediscretizedinthematrixformup1zC2C3up1z;1up1z;2C1C1C1up1z;Nphijp1z;jxihi(33)uzC138uz;1uz;2.uz;NdC138jz;jxiC138(34)up2zC2C3up2z;1up2z;2C1C1C1up2z;Nphijp2z;jxihi(35)Sciences51(2009)284294wherethelengthsoftheircolumnscorrespondtothenumbersofnodesinthecomponentFEmodel.Theradialandtangentialmodefunctionsarealsodenotedasfup1rC138;urC138;up2rC138gandfup1yC138;uyC138;up2yC138g.Fromthemass-normalizationandthelinearizationatthesteady-slidingequilibriumofEq.(23),thehomogeneouspartofthelinearizedequationsofmotiontakesthe(NC2N)matrixformsuchthataGC138CC138RdC138NsC138_ao2C138AC138BC138FC138a0(36)wherethesystemmatricesaredescribedinEq.(37)andEqs.(A.1)(A.7)ofAppendixA.SubstitutingataoeltintoEq.(36)andsolvingRe(l)andIm(l)ofthecharacteristicequationresultinthedeterminationofthemodalstabilityandfrequency.HerethephysicalmeaningofeachsystemmatrixofEq.(36)isprovidedinthefollowing.GC138C0GC138TisthegyroscopicmatrixtobedescribedinEq.(37),Cisthestructuralmodaldampingmatrix,andNsC138NsC138Tisthenegativeslopematrix.Thenegativefriction-sloperadialindiscmatrix.theEq.Kangetal.14,wherethefrictionalfollowerforceeffectswerewhereGdC138OrZVdurC138TuryC20C21C0uyC138C18C19C0uryC20C21C0uyC138C18C19TurC138(uyC138TuyyC20C21urC138C18C19C0uyyC20C21urC138C18C19TuyC138uzC138TuzyC20C21C0uzyC20C21TuzC138)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,thegyroscopicmatrixisgivenbyGC1380C1380C1380C1380C138GdC1380C1380C1380C1380C138264375(37),(,)znlmkxyz(,)xyzitsincorporFig.uniformfollowerforceassociatedwith1=ruP=yC0vPin(32),butneglectedinthesubsequentanalysisduetoinsignificance14aswell.Thelocalcontactmodel10atedwiththefrictionalfollowerforcescanbereferredtobyfrictionaleffectcanbereferredto17.RdC138RdC138Tisthedissipativematrixstemmingfrom1=rOuP=tC0up1R=tEq.(32).Also,o2isthenaturalfrequencymatrixoftheandpadcomponents,AC138AC138TisthecontactstiffnessOfthenon-symmetricstiffnessmatrix,BC138aBC138Tisnon-symmetricnon-conservativeworkmatrixproducedfriction-couple.FC138aFC138Tisderivedfromthein-planelmk4.Transversemodalvectoratzzkinterpolatedbytheuniformplanar-meshmethod:planarm
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