Linear eigenvalue analysis of the disc-brake squea

INTERNATIONALJOURNALFORNUMERICALMETHODSINENGINEERINGInt.J.Numer.Meth.Engng2004;61:1546–1563Publishedonline28September2004inWileyInterScience().DOI:10.1002/nme.1127Lineareigenvalueanalysisofthedisc-brakesquealproblemQ.Cao1,H.Ouyang1,∗,†,M.I.Friswell2andJ.E.Mottershead11DepartmentofEngineering,UniversityofLiverpool,LiverpoolL693GH,U.K.2DepartmentofAerospaceEngineering,UniversityofBristol,BristolBS8,U.K.SUMMARYThispaperpresentsanumericalmethodtocalculatetheunstablefrequenciesofacardiscbrakeandsuggestsasuitableanalysisprocedure.Thestationarycomponentsofthediscbrakearemodelledusingfiniteelementsandthediscasathinplate.Theseparatetreatmentsofthestationarycomponentsandtherotatingdiscfacilitatethemodellingofthediscbrakesquealasamovingloadproblem.Someuncertainsystemparametersofthestationarycomponentsandthediscaretunedtofitexperimentalresults.Alinear,complex-valued,asymmetriceigenvalueformulationisderivedforthefriction-inducedvibrationofthediscbrake.Predictedunstablefrequenciesarecomparedwithexperimentallyestablishedsquealfrequenciesofarealcardiscbrake.Copyright
2004JohnWiley&Sons,Ltd.KEYWORDS:discbrake;squeal;vibration;movingloads;friction;numericalmethod1.INTRODUCTIONFriction-inducedvibrationandnoiseemanatingfromcardiscbrakesisasourceofconsiderablediscomfortandleadstocustomerdissatisfaction.Thehighfrequencynoiseabove1kHz,knownassqueal,ismostannoyingandisverydifficulttoeliminate.Akay[1]recentlyreviewedfriction-inducednoise,includingcardiscbrakesqueal.Hequotedfromanindustrialsourceanestimateofthewarrantycostduetonoise,harshnessandvibration(togetherknownastheNVHproblem),includingdiscbrakesqueal,asUS$1billionayeartotheautomotiveindustryinNorthAmericaalone.North[2]observedthatthefirstconcertedefforttostudycardiscbrakesquealwasmadeattheMotorIndustrialResearchAssociation(MIRA)intheU.K.inthe1950s.Therenowexistsanextensiveliteratureontheoreticalandexperimentalinvestigationsofdiscbrakesqueal.Toappreciatethisverycomplexproblem,reviews[2–5]conductedatdifferenttimesarerecom-mended.Kinkaidetal.’srecentthoroughsurvey[6]deservesaspecialmention.Todate,the∗Correspondenceto:H.Ouyang,DepartmentofEngineering,UniversityofLiverpool,LiverpoolL693GH,U.K.†E-mail:[e-mail]/grantsponsor:EPSRC;contract/grantnumber:GR/L91061Contract/grantsponsor:TRWAutomotiveReceived4October2003Revised8January2004Copyright
2004JohnWiley&Sons,Ltd.Accepted20February2004LINEAREIGENVALUEANALYSISOFTHEDISC-BRAKESQUEALPROBLEM1547rotatingdisccarrierbracketcarrierbracketguidepinguidepincalliper(piston)padaventFigure1.Acardiscbrakeoffloatingcalliperdesign.discbrakevibrationandsquealproblemhasnotbeenwellsolved,eventhoughdiscbrakeshavebecomequieterovertheyears.Acardiscbrakesystemconsistsofarotatingdiscandstationary(non-rotating)pads,carrierbracket,calliperand(mounting)guidepins.Thepadsarelooselyhousedinthecalliperandlocatedbythecarrierbracket.Thecalliperitselfisallowedtoslidefairlyfreelyalongthetwomountingguidepinsinafloatingcalliperdesign.Atypicalfloating-typeventeddiscbrakesystemisshowninFigure1.Thediscisboltedtothecarwheelandthusrotatesatthesamespeedasthewheel.Whenthediscbrakeisapplied,thetwopadsarebroughtintocontactwiththediscsurfaces.Mostofthekineticenergyofthetravellingcarisconvertedtoheatthroughfriction.Butasmallpartofitconvertsintosoundenergyandgeneratesnoise.Asquealingbrakeisdifficultandexpensivetocorrect.Preferablythenoiseissueshouldberesolvedatthedesignstage.Thereareanumberofnamesforbrakenoisesindifferentfrequencyranges.Amongthese,highfrequencynoise,orsqueal,isthemostdifficultonetodealwith.Inparallelwithexperimentalstudy,modellingofdiscbrakesandsimulatingtheirdynamicsandacousticbehaviourisanefficientwayofunderstandingthesquealmechanismanddesigningquietdiscbrakes.Thestandardtechnologyinthecarmanufacturingindustryistouselargefiniteelementmodels.Byusingalargenumberoffiniteelements,anadequatestructuralmodelcanbeestablished.Theremainingissuesinmodellingarethefrictionmodels,thesquealmechanisms,thecontactmodelsandthedynamicsmodels.Mostanalysesofdiscbrakesquealadoptsimplefrictionlawssinceabrakesystemisverycomplicated.Varioussquealmechanismshavebeenused,includingthenegativegradi-entoffrictioncoefficientagainstrelativespeedandthestick-slipvibrationstudiedbyMills[7,8],thesprag–slipmodelputforwardbySpurr[9],North’sfollowerforceandfrictioncouple[2],therotatingfollowerforceadvancedbyMottersheadandco-workers[10,11],andsoon.Somerecentfiniteelementanalysesalsocontainastaticcontactanalysisofthecontactareaandthepressuredistributionofthedisc/padsinterface[12–14].Someevenembarkedonadynamiccontactanalysistoconsiderthemakingandbreakingofcontactbetweenthediscandthepads[15].Copyright
2004JohnWiley&Sons,Ltd.Int.J.Numer.Meth.Engng2004;61:1546–15631548Q.CAOETAL.Theauthorsthinkthatthedynamicsmodelsforthediscbrakesquealneedfurtherimprove-ment.Inadiscbrakesystem,thediscrotatespastthestationarypadshousedbythecarrierbracketandthecalliper.Asaresult,thepadsmatewithdifferentspatialareasasthediscrotatesandvibrates.Thisisamovingloadproblem.Becausediscbrakestendtosqu