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

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

SYONALMODECOUPLINGINSTABILITYFORENTEMSMATRIXSYSTETORSIONMODEOFTHEDISC2009ELSEVIERLTDALLRIGHTSRESERVEDATEDINFORMATIONTHROUERVIEWPUBLISHEDOFAUTMAJORANALYSISCULARLY,HUANGETAL6USEDTHEEIGENVALUEPERTURBATIONMETHODSOLUTIONFORMODEMERGINGBETWEENDISCDOUBLETMODEPAIRDUETHECOPICEUEALBRAKE,FOREXAMPLE,THEINPLANEMODEANDHATMODEOFTHEDISCARTICLEINPRESSCONTENTSLISTSAVAILABLEOFINTERNATIONALJOURNALOFMECHANICALSCIENCES512009284–294DESCRIBINGTHECONTACTKINEMATICSUNDERTHEUNDEFORMEDCONFIGTOTHESTATIONARYDISCASSUMPTION,THEFINITEELEMENTFEMETHODHASBEENEASILYIMPLEMENTEDASREFERREDTOTHEREVIEWARTICLE2ALTERNATELY,CAOETAL13STUDIEDTHEMOVINGLOADEFFECTFROMAFEDISCBRAKEMODELWITHMOVINGPADS,WHERETHEDISCWASINTHISPAPER,THEMETHODOLOGYOFCONSTRUCTINGAROTATINGFEDISCBRAKEMODELISDEVELOPEDCONSEQUENTLY,ITENABLESUSTOEXAMINETHESQUEALMECHANISMSINTHEPHYSICALFEBRAKEMODELSUBJECTTOROTATIONEFFECTSTHEGLOBALCONTACTMODEL10URATIONISUTILIZEDTODEVELOPCONTACTMODELINGBETWEENTHEROTATINGDISCANDTWOSTATIONARYPADSFROMTHEASSUMEDMODEEMAILADDRESSJKANGKONGJUACKR00207403/DOI101TODEVELOPTHENECESSARYCONDITIONFORMODEMERGINGWITHOUTTHEDIRECTEIGENSOLUTIONSKANGETAL7DERIVEDTHECLOSEDFORMMECHANISMSSINCETHEANNULARPLATEAPPROXIMATIONDOESNOTREPRESENTALLOFMODALBEHAVIORSEXISTINGONTHEPHYSICALDISCBRAKESYSTEM9–12,14THESTABILITYANALYSISATTHESTATICSTEADYSLIDINGEQUILIBRIUMOFTHESTATIONARYDISCANDPADSPROVIDESTHESQUEALMECHANISMASMODEMERGINGCHARACTERINTHEFRICTION–FREQUENCYDOMAINPARTIANNULARSECTORPLATESTHECOMPREHENSIVEANALYSISEXPLAINEDSTABILITYCHARACTERINFLUENCEDBYMODECOUPLINGANDGYROSEFFECT,ANDPROVIDEDTHEPHYSICALBACKGROUNDONTHEAPPROXIMATIONSANDMECHANISMSUSEDINTHEPREVIOUSSQUEALLITERATURHOWEVER,ITSTILLCONTAINSLIMITATIONSONEXAMININGBRAKESQEQUATIONSOFMOTION,THEREALPARTSOFEIGENVALUESHAVEBEENCALCULATEDFORDETERMININGTHEEQUILIBRIUMSTABILITYINTHELITERATURE,THEREARETWOMAJORDIRECTIONSONTHELINEARSQUEALANALYSISTHECOMPLEXEIGENVALUEANALYSISOFTHESTATICSTEADYSLIDINGEQUILIBRIUM3–8ANDTHESTABILITYANALYSISOFROTATINGMODELING,HOWEVER,AROTATINGFEDISCBRAKEMODELHASNOTBEENDEVELOPEDYETRECENTLY,KANGETAL14DEVELOPEDATHEORETICALDISCBRAKEMODELINTHECOMPREHENSIVEMANNERTHEDISCBRAKEMODELCONSISTSOFAROTATINGANNULARPLATEINCONTACTWITHTWOSTATIONARY1INTRODUCTIONDISCBRAKESQUEALHASBEENINVESTIGFORSEVERALDECADESMUCHVALUABLEMECHANISMSHASBEENACCUMULATEDKINKAIDETAL1PRESENTEDTHEOVBRAKESQUEALSTUDIESOUYANGETAL2FOCUSEDONTHENUMERICALANALYSISSQUEALTHEYHAVESHOWNTHATONESQUEALSTUDYISTHELINEARSTABILITYSEEFRONTMATTER2009ELSEVIERLTDALL016/JIJMECSCI200902003BYMANYRESEARCHERSONSQUEALGHOUTTHERESEARCHONTHEVARIOUSDISCTHEREVIEWARTICLEOMOTIVEDISCBRAKEAPPROACHONBRAKEFROMTHELINEARIZEDSTATIONARY,ANDTHEREFORE,THEGYROSCOPICEFFECTSWERENEGLECTEDGIANNINIETAL15,16VALIDATEDTHEMODEMERGINGBEHAVIORASSQUEALONSETBYUSINGTHEEXPERIMENTALSQUEALFREQUENCIESONTHEOTHERHAND,THESTABILITYOFAROTATINGDISCBRAKEHASBEENINVESTIGATEDINTHEANALYTICALMANNERTHEROTATINGDISCBRAKESYSTEMHASBEENMODELEDASARING10ANDANANNULARPLATE12INPOINTCONTACTWITHTWOPADS,ANDANANNULARPLATESUBJECTTODISTRIBUTEDFRICTIONALTRACTION9WITHINCLUSIONOFGYROSCOPICEFFECT,THEREALPARTSOFEIGENVALUESHAVEBEENSOLVEDWITHRESPECTTOSYSTEMPARAMETERSDUETOTHECOMPLEXITYOFTHEROTATINGDISCSQUEALANALYSISOFGYROSCOPICDISCBRAKEONFINITEELEMENTMETHODJAEYOUNGKANGDIVISIONOFMECHANICALANDAUTOMOTIVEENGINEERING,COLLEGEOFENGINEERING,KONGJUNATIARTICLEINFOARTICLEHISTORYRECEIVED9OCTOBER2008RECEIVEDINREVISEDFORM30JANUARY2009ACCEPTED12FEBRUARY2009AVAILABLEONLINE9MARCH2009KEYWORDSGYROSCOPICDISCBRAKEBRAKESQUEALABSTRACTINTHISPAPER,THEDYNAMICSTATIONARYPADSISSTUDIEDSTRUCTUREBYTHEFINITEELEMANDMOVINGCOORDINATESYSTCORRESPONDINGGYROSCOPICMETHODTHEDYNAMICINSTABPREDICTEDWITHRESPECTTOSPEEDDEPENDSONTHEVIBRATIONTHENEGATIVESLOPEOFFRICTIONJOURNALHOMEPAGEWWWELSEVIERINTERNATIONALJOURNALRIGHTSRESERVEDSTEMBASEDUNIVERSITY,CHEONANSI,REPUBLICOFKOREAOFACARBRAKESYSTEMWITHAROTATINGDISCINCONTACTWITHTWOACTUALGEOMETRICAPPROXIMATION,THEDISCISMODELEDASAHATDISCSHAPEMETHODFROMACOORDINATETRANSFORMATIONBETWEENTHEREFERENCE,THECONTACTKINEMATICSBETWEENTHEDISCANDPADSISDESCRIBEDTHEOFTHEDISCISCONSTRUCTEDBYINTRODUCINGTHEUNIFORMPLANARMESHILITYOFAGYROSCOPICNONCONSERVATIVEBRAKESYSTEMISNUMERICALLYMPARAMETERSTHERESULTSSHOWTHATTHESQUEALPROPENSITYFORROTATIONMODESPARTICIPATINGINSQUEALMODESMOREOVER,ITISHIGHLIGHTEDTHATCOEFFICIENTTAKESANIMPORTANTROLEINGENERATINGSQUEALINTHEINPLANEATSCIENCEDIRECTCOM/LOCATE/IJMECSCIMECHANICALSCIENCESMETHOD,THEEQUATIONSOFMOTIONOFTHEFRICTIONENGAGEDBRAKESYSTEMAREDERIVEDTHENUMERICALRESULTSDEMONSTRATESEVERALSQUEALMODESANDEXPLAINTHECORRESPONDINGSQUEALMECHANISMS2DERIVATIONOFEQUATIONSOFMOTIONTHEDISCPARTOFABRAKESYSTEMISMODELEDASAHATDISCSHAPESTRUCTUREASSHOWNINFIG1THEHATDISCISSUBJECTTOTHECLAMPEDBOUNDARYCONDITIONATTHEINNERROTATINGSHAFTANDTHEFREEBOUNDARYCONDITIONATTHEOUTERRADIUSOWINGTOTHECOMPLEXITYOFTHEGEOMETRY,THEFINITEELEMENTMETHODISUTILIZEDFORMODALANALYSISTHEDISCROTATIONWITHCONSTANTSPEEDOGENERATESFRICTIONSTRESSESOVERTHECONTACTWITHTWOSTATIONARYPADSLOADEDBYPRENORMALLOADN0THEFRICTIONMATERIALOFTHEPADISMODELEDASTHEUNIFORMCONTACTSTIFFNESSKC,WHERECONTACTSTRESSESAREDEFINEDONTHEGLOBALCONTACTMODELCENTRIFUGALFORCEISNEGLECTEDDUETOTHESLOWROTATIONINTHEBRAKESQUEALPROBLEMINORDERTODESCRIBETHECONTACTKINEMATICS,THEDISPLACEMENTVECTORSOFTHEDISCANDTOPPADAREEXPRESSEDINTHEREFERENCECOORDINATESFIG2,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;TEZ2ARTICLEINPRESSCONNECTEDWITHCONTACTSTIFFNESSONONCONNECTEDWITHCONTACTSTIFFNESSΩCLAMPEDATANINNERZROTATINGSHAFTFIG1HATDISCBRAKESYSTEMTΩ⋅ΨXΘEZΘRECKΩNEUTRALSURFACEROTORPARTFIG2COORDINATESYSTEMOFTHEROTATINGDISC,REFERENCEYANDLOCALCCOORDINATESBOTTOMSURFACEOFTHETOPPADCONTACTAREAACPW1PRW1PRV1PRUPUPVZPRJKANG/INTERNATIONALJOURNALOFMECHANICALSCIENCES512009284–294285SEGMENTOFUNDEFORMEDTOPSURFACEOFTHEDISCFIG3CONTACTKINEMATICSATACONTACTPOINTPORP0INTHEGLOBALCONTACTMODELACONTACTCONTACTWITHPOFTHEDISCWHERETHESUPERSCRIPTS,P1ANDP2DENOTETHETOPANDBOTTOMPADS,RESPECTIVELY,ANDTHEDISCDISPLACEMENTISALSODEFINEDINTHELOCALCOORDINATESFIG2UR;C;Z;TUR;C;Z;TERVR;C;Z;TEHWR;C;Z;TEZ3ASSHOWNINFIG3,THECONTACTPOINTP0OFFRICTIONMATERIALOFTHETOPPADISASSUMEDTOBEINCONTACTWITHPOFTHEDISCANDLATERALLYFIXEDWITHROFTHETOPPAD,WHICHRESULTSINUP1P0R;Y;TUP1RR;Y;TERVP1RR;Y;TEHWPR;Y;TEZ4THEVELOCITYVECTORSOFTHEDISCANDTOPPADAREOBTAINEDFROMTHEFOLLOWINGTIMEDERIVATIVESFIRST,THEPOSITIONVECTORSOFTHEDISCAREEXPRESSEDINTHELOCALCOORDINATESASRRUERVEHZWEZ5RPRJZH26FORDESCRIBINGTHEDIRECTIONVECTOROFFRICTIONFORCE,THECONTACTVELOCITYVECTOROFTHEDISCISDERIVEDBYTAKINGTHETIMEDERIVATIVE1PPOKCWP–WRZDISC1N−ZE1NZECK1−F1FPRPDISPLACEMENTS;BCONTACTFORCESP0OFFRICTIONMATERIALOFTHETOPPADISINARTICLEINPRESSINEQ6INTHEREFERENCECOORDINATESVPDRPDT7WHERETHECOORDINATETRANSFORMATIONISGIVENBYTHEDIFFERENTIATIONINTHELOCALCOORDINATESSUCHTHATDUR;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,THECONTACTVELOCITYVECTORATP0OFTHETOPPADISSIMPLYTHEPARTIALTIMEDERIVATIVEOFEQ4VP1P0UP1RTERVP1RTEHWPTEZ11FROMCOULOMB’SLAWOFFRICTION,CONTACTFRICTIONFORCEISEXPRESSEDASF1C0M1C1N1VRELJVRELJ12WHERETHENORMALLOADISTHESUMOFPRESTRESSP0N0/ACANDTHENORMALLOADVARIATIONN1P0KCWPC0WP1R13ANDTHERELATIVEVELOCITYATTOPCONTACTISGIVENBYVRELVPC0VP1P014INORDERTOCAPTURETHENEGATIVESLOPEEFFECT,THECONTINUOUSFRICTIONCURVE14ISUSEDSUCHTHATM1TFMKMSC0MKEC0AJVRELJGRRCTR15WHEREMS,MKANDAARETHECONTROLPARAMETERSDETERMININGTHEMAGNITUDEANDTHESLOPEOFTHEFRICTIONCOEFFICIENT,ANDTHEFRICTIONCOEFFICIENTISASSUMEDTOBEUNIFORMANDCALCULATEDATTHECENTROIDOFTHECONTACTAREARCTRTHETRANSVERSEVIBRATIONSOFTHEDISCANDPADCOMPONENTSAREEXPRESSEDINTHEMODALEXPANSIONFORMOFNND2NPTRUNCATEDMODESUSINGTHEASSUMEDMODEMETHODWP1X;TffiXNPN1JP1Z;NXQP1NT16WX;TffiXNDN1JZ;NXQNT17WP2X;TffiXNPN1JP2Z;NXQP2NT18WHERENDANDNPARETHENUMBERSOFTHETRUNCATEDMODESOFTHEDISCANDTHEPAD,RESPECTIVELY,ANDWHEREQP1FQP11QP12QP1NPG19QFQ1Q2QNDG20QP2FQP21QP22QP2NPG21JP1Z;NX,JZ;NXANDJP2Z;NXARETHENTHTRANSVERSEMODESHAPEJKANG/INTERNATIONALJOURNALOFMECHANICAL286FUNCTIONSOBTAINEDFROMTHEEIGENFUNCTIONSOFTHETOPPAD,DISCANDBOTTOMPADCOMPONENTS,RESPECTIVELYTHERADIALANDTANGENTIALVIBRATIONS,UP1;U;UP2ANDVP1;V;VP2CANBEWRITTENINTHEMODALEXPANSIONFORMASSOCIATEDWITHTHECORRESPONDINGMODESHAPEFUNCTIONSFJP1R;NX;JR;NX;JP2R;NXG,FJP1Y;NX;JY;NX;JP2Y;NXGASWELLTHEMODALCOORDINATESAREREARRANGEDINTHEVECTORFORMFORTHEFOLLOWINGDISCRETIZATIONFAGQP1QQP289;FA1A2ANGT22FROMTHEDISCRETIZATIONOFLAGRANGEEQUATIONBYMODALCOORDINATES,THEFRICTIONCOUPLEDEQUATIONSOFMOTIONAREGIVENBYDDTL_AMC20C21C0LAMXNN1QMNAN;M1;;N;N1;;N23LTC0UUC24DWC17XNM1XNN1QMNANDAM25WHEREUISTHETOTALSTRAINENERGYOFTHEUNCOUPLEDCOMPONENTDISCANDTWOPADS,ANDTTP1TDTP226TDRZVDDRDTC1DRDTC18C19DV27TP1RPZVPUP1TC1UP1TC18C19DV28TP2RPZVPUP2TC1UP2TC18C19DV29UCKC2ZACWPC0WP1R2DAUC;BOTTOM30DWZACFC0N1C0F1C1DUP1P0N1F1C1DUPGDADWBOTTOM31HEREVDANDVPARETHEVOLUMESOFTHEDISCANDPAD,RESPECTIVELYINTHESIMILARMANNEROFOBTAININGTHEVIRTUALWORKANDCONTACTSTRAINENERGYATTHETOPCONTACT,DWBOTTOMANDUC,BOTTOMONTHEBOTTOMCONTACTCANBEDERIVEDASWELLTHEDIRECTIONVECTOROFFRICTIONFORCEATTHETOPCONTACTISLINEARIZEDBYTAYLOREXPANSIONATTHESTEADYSLIDINGEQUILIBRIUMSUCHTHATVRELJVRELJ1ROUPTC0UP1RT1RUPYC0VPC26C27EREH1RWPYEZHOT32WHEREHOTDENOTESTHEHIGHERORDERTERMSHEREWPYISASSOCIATEDWITHFRICTIONALFOLLOWERFORCEASEXPLAINEDIN11ANDNEGLECTEDINTHESUBSEQUENTANALYSISDUETOTHEINSIGNIFICANCEOFTHEFRICTIONALFOLLOWERFORCEASREFERREDTO5,10,11AND14USINGTHEFINITEELEMENTMETHOD,THETRANSVERSEMODESHAPEFUNCTIONSAREDISCRETIZEDINTHEMATRIXFORMUP1ZC2C3UP1Z;1UP1Z;2C1C1C1UP1Z;NPHIJP1Z;JXIHI33UZC138UZ;1UZ;2UZ;NDC138JZ;JXIC13834UP2ZC2C3UP2Z;1UP2Z;2C1C1C1UP2Z;NPHIJP2Z;JXIHI35SCIENCES512009284–294WHERETHELENGTHSOFTHEIRCOLUMNSCORRESPONDTOTHENUMBERSOFNODESINTHECOMPONENTFEMODELTHERADIALANDTANGENTIALMODEFUNCTIONSAREALSODENOTEDASFUP1RC138;URC138;UP2RC138GANDFUP1YC138;UYC138;UP2YC138GFROMTHEMASSNORMALIZATIONANDTHELINEARIZATIONATTHESTEADYSLIDINGEQUILIBRIUMOFEQ23,THEHOMOGENEOUSPARTOFTHELINEARIZEDEQUATIONSOFMOTIONTAKESTHENC2NMATRIXFORMSUCHTHAT€AGC138CC138RDC138NSC138_AO2C138AC138BC138FC138A036WHERETHESYSTEMMATRICESAREDESCRIBEDINEQ37ANDEQSA1–A7OFAPPENDIXASUBSTITUTINGATAOELTINTOEQ36ANDSOLVINGRELANDIMLOFTHECHARACTERISTICEQUATIONRESULTINTHEDETERMINATIONOFTHEMODALSTABILITYANDFREQUENCYHERETHEPHYSICALMEANINGOFE

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