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Frictionallyexcitedthermoelasticinstabilityindiscbrakes—TransientprobleminthefullcontactregimeAbstractExceedingthecriticalslidingvelocityindiscbrakescancauseunwantedformingofhotspots,non-uniformdistributionofcontactpressure,vibration,andalso,inmanycases,permanentdamageofthedisc.Consequently,inthelastdecade,agreatdealofconsiderationhasbeengiventomodelingmethodsofthermoelasticinstability(TEI),whichleadstotheseeffects.Modelsbasedonthefiniteelementmethodarealsobeingdevelopedinadditiontotheanalyticalapproach.TheanalyticalmodelofTEIdevelopmentdescribedinthepaperbyLeeandBarber[Frictionallyexcitedthermoelasticinstabilityinautomotivediskbrakes.ASMEJournalofTribology1993；115:607–14]hasbeenexpandedinthepresentedwork.Specificattentionwasgiventothemodificationoftheirmodel,tocatchthefactthatthearclengthofpadsislessthanthecircumferenceofthedisc,andtothedevelopmentoftemperatureperturbationamplitudeintheearlystageofbreaking,whenpadsareinthefullcontactwiththedisc.Awayisproposedhowtotakeintoaccountbothoftheinitialnon-flatnessofthediscfrictionsurfaceandchangeoftheperturbationshapeinsidethediscinthecourseofbraking.Keywords:Thermoelasticinstability；TEI；Discbrake；Hotspots1.IntroductionFormationofhotspotsaswellasnon-uniformdistributionofthecontactpressureisanunwantedeffectemergingindiscbrakesinthecourseofbrakingorduringengagementofatransmissionclutch.Iftheslidingvelocityishighenough,thiseffectcanbecomeunstableandcanresultindiscmaterialdamage,frictionalvibration,wear,etc.Therefore,alotofexperimentaleffortisbeingspenttounderstandbetterthiseffect(cf.Refs.)ortomodelitinthemostfeasiblefashion.Barberdescribedthethermoelasticinstability(TEI)asthecauseofthephenomenon.LaterDowandBurtonandBroducedamathematicalmodeltoestablishcriticalslidingvelocityforinstability,wheretwothermoelastichalf-planesareconsideredincontactalongtheircommoninterface.ItisinaworkbyLeeandBarberthattheeffectofthethicknesswasconsideredandthatamodelapplicablefordiscbrakeswasproposed.LeeandBarber’smodelismadeupwithametalliclayerslidingbetweentwohalf-planesoffrictionalmaterial.OnlyrecentlyaparametricanalysisofTEIindiscbrakeswasmadeorTEIinmulti-discclutchesandbrakeswasmodeled.TheevolutionofhotspotsamplitudeshasbeenaddressedinRefs.Usinganalyticalapproachortheeffectofintermittentcontactwasconsidered.Finally,thefiniteelementmethodwasalsoappliedtorendertheonsetofTEI(seeRef.).Theanalysisofnonlineartransientbehaviorinthemode,whenseparatedcontactregionsoccur,isevenaccomplishedinRef.Asinthecaseofotherengineeringproblemsofinstability,itturnsoutthatamoreaccuratepredictionbymathematicalmodelingisoftenquestionable.Thisismainlyimpartedbyneglectingvariousimperfectionsandrandomfluctuationsorbytheimpossibilitytodescribeallpossibleinfluencesappropriately.Therefore,someeffortarousedtointerpretresultsofcertainexperimentsinadditiontoclassicalTEI(see,e.g.Ref).ThispaperisrelatedtotheworkbyLeeandBarber[7].Usingananalyticalapproach,ittreatstheinceptionofTEIandthedevelopmentofhotspotsduringthefullcontactregimeinthediscbrakes.ThemodelproposedinSection2enablestocoverfinitethicknessofbothfrictionpadsandtheribbedportionofthedisc.Section3isdevotedtotheproblemsofmodelingofpartialdiscsurfacecontactwiththepads.Section4introducesthetermof‘‘thermalcapacityofperturbation’’emphasizingitsassociationwiththevalueofgrowthrate,ortheslidingvelocitymagnitude.Ananalysisofthediscfrictionsurfacesnon-flatnessanditsinfluenceoninitialamplitudeofperturbationsisputforwardintheSection5.Finally,theSection6offersamodeloftemperatureperturbationdevelopmentinitiatedbythementionedinitialdiscnon-flatnessinthecourseofbraking.Themodelbeinginuseherecomesfromadifferentialequationthatcoversthevariationofthe‘‘thermalcapacity’’duringthefullcontactregimeofthebraking.2.ElaborationofLeeandBarbermodelThebrakediscisrepresentedbythreelayers.Themiddleoneofthickness2a3standsfortheribbedportionofthediscwithfullsidewallsofthicknessa2connectedtoit.Thepadsarerepresentedbylayersofthicknessa1,whichareimmovableandpressedtoeachotherbyauniformpressurep.ThebrakediscslipsinbetweenthesepadsataconstantvelocityV.WewillinvestigatetheconditionsunderwhichaspatiallysinusoidalperturbationinthetemperatureandstressfieldscangrowexponentiallywithrespecttothetimeinasimilarmannertothatadoptedbyLeeandBarber.Itisevidencedintheirwork[7]thatitissufficienttohandleonlytheantisymmetricproblem.TheperturbationsthataresymmetricwithrespecttothemidplaneofthedisccangrowatavelocitywellabovetheslidingvelocityVthusbeingmadeuninteresting.Letusintroduceacoordinatesystem（x1；y1）fixedtooneofthepads(seeFig.1)thepointsofcontactsurfacebetweenthepadanddischavingy1=0.Furthermore,letacoordinatesystem（x2；y2）befixedtothediscwithy2=0forthepointsofthemidplane.Wesupposetheperturbationtohavearelativevelocityciwithrespecttothelayeri,andthecoordinatesystem（x；y）tomovetogetherwiththeperturbatedfield.ThenwecanwriteV=c1-c2；c2=c3；x=x1-c1t=x2-c2t,x2=x3；y=y2=y3=y1+a2+a3.Wewillsearchtheperturbationoftheuniformtemperaturefieldintheformandtheperturbationofthecontactpressureintheformwheretisthetime,bdenotesagrowthrate,subscriptIreferstoalayerinthemodel,andj=-1½istheimaginaryunit.Theparameterm=m（n）=2pin/cir=2pi/L,wherenisthenumberofhotspotsonthecircumferenceofthedisccirandLiswavelengthofperturbations.ThesymbolsT0mandp0mintheaboveformulaedenotetheamplitudesofinitialnon-uniformities(e.g.fluctuations).Bothperturbations(2)and(3)willbesearchedascomplexfunctionstheirrealpartdescribingtheactualperturbationoftemperatureorpressurefield.Obviously,ifthegrowthrateb<0,theinitialfluctuationsaredamped.Ontheotherhand,instabilitydevelopsifB〉0.2.1.TemperaturefieldperturbationHeatfluxinthedirectionofthex-axisiszerowhentheribbedportionofthediscisconsidered.Next,letusdenoteki=Ki/Qicpicoefficientofthelayeritemperaturediffusion.ParametersKi,Qi,cpiare,respectively,thethermalconductivity,densityandspecificheatofthematerialfori=1,2.Theyhavebeenre-calculatedtotheentirevolumeofthelayer(i=3)whentheribbedportionofthediscisconsidered.TheperturbationofthetemperaturefieldisthesolutionoftheequationsWithanditwillmeetthefollowingconditions:1，Thelayers1and2willhavethesametemperatureatthecontactsurface2，Thelayers2and3willreachthesametemperatureandthesameheatfluxinthedirectiony,3，AntisymmetricconditionatthemidplaneTheperturbationswillbezeroattheexternalsurfaceofafrictionpad(If,instead,zeroheatfluxthroughexternalsurfacehasbeenspecified,weobtainpracticallyidenticalnumericalsolutionforcurrentpads).Ifwewritethetemperaturedevelopmentinindividuallayersinasuitableformweobtainwhereand2.2.ThermoelasticstressesanddisplacementsForthesakeofsimplicity,letusconsidertheribbedportionofthedisctobeisotropicenvironmentwithcorrectedmodulusofelasticitythough,actually,thestiffnessofthislayerinthedirectionxdiffersfromthatinthedirectiony.Suchsimplificationis,however,admissibleastheyieldingcentrallayer3practicallydoesnottakeeffectonthediscflexuralrigidityunlikefullsidewalls(layer2).Givenathermalfieldperturbation,wecanexpressthestressstateanddisplacementscausedbythisperturbationforanylayer.Thethermoelasticproblemcanbesolvedbysuperimposingaparticularsolutiononthegeneralisothermalsolution.Welookfor

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