Delft University of Technology
Numerical modeling of wheel-rail squeal-exciting contact
Yang, Zhen; Li, Zili
DOI
10.1016/j.ijmecsci.2019.02.012
Publication date
2019
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Final published version
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International Journal of Mechanical Sciences
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Yang, Z., & Li, Z. (2019). Numerical modeling of wheel-rail squeal-exciting contact. International Journal of
Mechanical Sciences, 153-154, 490-499. https://doi.org/10.1016/j.ijmecsci.2019.02.012
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InternationalJournalofMechanicalSciences153–154(2019)490–499
ContentslistsavailableatScienceDirect
International
Journal
of
Mechanical
Sciences
journalhomepage:www.elsevier.com/locate/ijmecsci
Numerical
modeling
of
wheel-rail
squeal-exciting
contact
Zhen Yang, Zili Li
∗Delft University of Technology, Section of Railway Engineering, Stevinweg 1, 2628 CN Delft, The Netherlands
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r
t
i
c
l
e
i
n
f
o
Keywords: Wheel-rail contact Dynamic interaction Squeal Numerical modeling Explicit FEMa
b
s
t
r
a
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t
Complexfrictionalrollingcontactandhigh-frequencywheeldynamicbehaviormakemodelingsquealgreatly challenging.Thefalling-frictioneffectandwheelmode-couplingbehaviorarebelievedtobethetwomain mech-anismsthatgenerateunstablewheelvibrationandtheresultingsquealnoise.Torigorouslyconsiderboth mech-anismsinonemodel,weproposeanexplicitfiniteelement(FE)modeltosimulatewheel-raildynamicfrictional rolling.Wheel-railsqueal-excitingcontactisinvestigatedwithconsiderationsofdynamiceffects,unsteady lat-eralcreepageandvelocity-dependentfriction.Withtheinclusionofthedynamiceffectsinthecontactsolution, large-creepage-inducedwaves,whichsharefeatureswithRayleighwaves,arediscovered.Thesolutionsofthe dynamiccontactcalculatedusingtheproposedmodelindicatethattheexplicitFEmethodisabletoreproduce thefalling-frictioneffect.Thetransientanalysesofwheel-railfrictionalrollingwithwheellateralcreepageshow thecouplingoftheaxialandradialdynamicsoftherollingwheelmodel,suggestingthattheexplicitFEmethod canalsoreproducethemode-couplingbehavior.Thisstudyimprovestheunderstandingandmodelingof squeal-excitingcontactfromtheperspectiveofwheel-raildynamicinteraction.
1. Introductiontofriction-inducedsqueal
Squeal, which is an intrinsically nonlinear and transient phe-nomenon,posesgreatchallengesinmodeling[1].Systematicstudieson railwaycurvesquealbeganinthe1970swhenRudd[2]describedthe wheel-railfrictionalcharacteristic(thefunctionaldependencebetween friction force and wheel-rail relative velocity) in terms of ‘negative damping’(generalizedasfull‘stick-slip’)or‘fallingfriction’,and associ-atedthischaracteristicwiththemechanismofwheelsqueal.Sincethen, variousincreasinglysophisticatedtheoreticalmodelshaveadoptedparts ofthismechanismbasedonRudd’sseminalmodel[3–8].
Analyticalresearch[9],however,revealedthatinstabilitycanalso occurwithconstantorevenpositivecreepforce characteristicswhen additionalmechanicaldegreesoffreedomareconsidered.Similarly, fi-niteelement(FE)analysesbythecomplexeigenvalueapproachrevealed thatinstabilityariseswhentwomodescoupleinthepresenceoffriction [10].Toconsidertheinfluenceoftheverticaldynamicsonfriction,the ‘mode-couplingdynamicinstability’mechanismhavebeenemployedin combinationwithfalling-frictiontheoriestopredictfriction-induced in-stability[8,11–13].Chiello,etal.[14]concludedthatthetwotypes ofdestabilizationcausedbythefalling-frictionmechanismandbythe mode-couplingmechanismmaybecombinedineachparticular situa-tionofsqueal;thus,bothmechanismsshouldbeexplored[15].
Thegenerationofsquealischaracterizedas‘enigmatic’[16]or ‘er-ratic’[17,18]inrailwayresearchbecausefieldobservationsofsqueal
∗Correspondingauthor.
E-mailaddresses:z.yang-1@tudelft.nl(Z.Yang),z.li@tudelft.nl(Z.Li).
phenomenaoftenfailtobeexplainedbytheexistingtheories.The con-tact modelingaccounting for thefrictional instability of a vibrating wheel is considered thecentralpart of thesqueal prediction model [19].Oneshortcomingoftheexistingsquealpredictionmodelsisthe incompleterepresentationofthecontactmodel.Thesimplificationsof thepointcontactmodelsandKalker’scontactmodelswithdiscretized contactsurface(widelyusedintheexistingsquealpredictionmodels) haveunknowninfluencesonthepredictionoffriction-inducedvibration [20].Asreportedbyarecentreviewofthesquealstudy[21],although Kalker’scontactmodelscantreatsteady-statecreepage,moredetailed contactmodelsthatincludetransienteffectsmaybeneededforacorrect representationofthesquealmechanisms.
ThisstudyproposesanexplicitFEdynamiccontactmodelto simu-latewheel-railsqueal-excitingcontactwithunsteadylateralcreepage. Unsteadylateralcreepage,particularlybetweentheleadinginnerwheel andthelowrail,isthoughttobethemaincauseofsqueal[22].The twocommonlyconsideredmechanismsleadingtosqueal—the falling-frictionandmode-couplingmechanisms—areaddressedinthisstudy. Incontrastwiththesteady-statecontactassumedinKalker’stheories, theproposedexplicitFEcontactmodelaccommodatesthedynamic,or transient,effectsinvolvedinwheel-railinteractionsbycouplingthe cal-culationofwheel-railfrictionalrollingcontactwiththecalculationof wheel/railstructuraldynamics(seeSection2.1).Here,thedynamic ef-fectsdenotethattheinertiaofmaterialelementsinfluencesthestress field when the elements ‘flow’ through the deforming region [23]. Withtheinclusionofthedynamiceffectsinthecontactsolution,the
https://doi.org/10.1016/j.ijmecsci.2019.02.012
Received5November2018;Receivedinrevisedform14January2019;Accepted11February2019 Availableonline19February2019
Fig.1.Thestructureandtheschematiclogicofthisstudy.
large-creepage-induced waves, which share features with Rayleigh waves,arediscovered.Thisstudyreferstowheel-railcontactwith struc-turaldynamiceffectsaswheel-raildynamicinteraction.Theaimofthis workistocontributetothebettermodelingandunderstandingof fric-tioninstabilityandsquealgenerationfromtheperspectiveofwheel-rail dynamicinteraction.
Fig.1shows thestructure andtheschematiclogicof this study. Afterreviewingstudiesonthetwowidelyacceptedgeneration mech-anismsinSection1, Section2presentstheexplicitFEwheel-rail dy-namicinteractionmodel.Section3thenanalyzesthesimulatedcontact forces(Section3.1),thedynamic contactsolutions (Section3.2)and thewheeldynamicbehavior(Section3.3).Theanalysessuggestthat theexplicitFEmodelcanreproducethefalling-frictioneffectandthe wheelmode-couplingbehaviorandthusconfirmthattheexplicitfinite elementmethod(FEM)issuitableformodelingsqueal-excitingcontact. 2. ExplicitFEwheel-raildynamicinteractionmodel
2.1. AlgorithmsoftheexplicitFEwheel-raildynamicinteractionanalysis
Thissubsectionsummarizesthecorealgorithmsemployedinthe ex-plicitFEwheel-railinteractionanalyseswithafocusonthe mathemati-calmodelandthenumericalsolutionprocedure.Theexplicitintegration schemeusesthecentraldifferencemethodtoapproximatethe accel-erationvector̈u.Theequilibriumequationforthewheel-raildynamic interactionproblemattimesteptcanbewrittenasEq.(1):
̈𝐮𝑡=𝐌−1(𝐟
ext𝑡−𝐟int𝑡+𝐟con𝑡 )
(1) whereM,fint andfextarethemassmatrix,internal forcevectorand externalforcevector,respectively,andfconisthecontactforcevector, whichcanbeincludedasacontributiontotheexternallyapplied trac-tions[24,25].Thevelocityanddisplacementvectorscanthenbe ob-tainedusingcentraldifferencetimeintegrationviaEqs.(2)and(3):
̇𝐮𝑡+1∕2= ̇𝐮𝑡−1∕2+Δ𝑡𝑡+1∕2̈𝐮𝑡 (2)
𝐮𝑡+1=𝐮𝑡+Δ𝑡𝑡+1̇𝐮𝑡+1∕2 (3)
whereΔtisthetimestepsize.TheCourant–Friedrichs–Lewystability condition[26]was used in thisstudy toguarantee theconvergence oftheexplicitintegration,whichrequiresthatasoundwavemaynot crossthesmallestelementwithinonetimestep.Thecriticaltimestep characterizedbythesmallestelementmayvaryinnonlineardynamic analysis[24]becauseofchangesinthematerialparametersand/or ge-ometry.Ascalefactorof0.9isthusemployedtocontrolthetimestep (Δt=0.9×criticaltimestep),sothattheconvergenceconditioncanbe satisfiedwithcertainty.Inthisstudy,thetimestepoftheexplicit inte-grationΔt≈86ns.
Table1outlinesthenumericalprocedurefortheexplicitFE wheel-raildynamicinteractionanalysis,whichiscomposedprimarilyof cal-culationsofthemassmatrixandforcevectorsinequilibriumEq.(1).A lumpedmassmatrixcanbeconstructedpriortotheiterationtopromote
Table1
NumericalprocedurefortheexplicitFEwheel-raildynamicinteractionanalysis.
Initialize algorithm: apply initial conditions; define contact pairs; construct the lumped mass matrix M; and set the termination time.
LOOP1 t = 0, 1, 2,…, n (time step number)
(I) Apply load conditions to construct the external force vector 𝐟 𝑡 ext . (II) Process elements to construct the internal force vector 𝐟 𝑡
int . (III) Construct the wheel-rail contact force vector 𝐟 𝑡
con using the penalty contact method.
LOOP2 N = 1, 2,…, m (slave wheel node number) i. Locate the master rail segment for the slave wheel node N .
ii. Locate the wheel-rail contact point (projection of the slave node on the master segment).
iii. Calculate the contact force. END LOOP2
(IV) Update the nodal accelerations ̈u 𝑡 , velocities ̇u 𝑡+1∕2 and displacements u t+1 . (V) Check for termination.
END LOOP1
Output: wheel/rail nodal force and nodal motion ( ̈u , ̇u and u)
theefficiencyandpracticalityoftheexplicitFEM.Fortheforce vec-tors,theexternalforcevectorfextcanbecalculateddirectlybythegiven loadconditions;theinternalforcevectorfintcontributedbystressescan becalculatedbytheconstitutiveandstrain-displacementformulations builtintheelementandmaterialmodels;andthecontactforcevector fconcanbecalculatedusingthepenaltycontactmethod[27].Whenthe penaltycontactalgorithmisapplied,thetimestepsizeΔtusedinthe ex-plicitintegrationshouldnotexceedthecontact-basedcriticaltimestep toavoidcontactinstability.Thecontact-basedcriticaltimestep deter-minedbythepenaltycontactalgorithmisproportionaltomin{√𝑚𝐽∕𝑘} wheremJ(J=1,2)isthemassattachedtothecontact"spring"andkis
thepenalty contactstiffness[28]. Inthisstudy,thepenalty contact-basedcriticaltimestepis118ns,biggerthanthetimestepofthe ex-plicitintegrationΔt(86ns);theconvergenceofcontactcalculationcan, therefore,beguaranteed.
ThenumericalsolutionprocedurepresentedinTable1containstwo loops.Theouterloopisconstructedmainlybyformulatingtheequation ofmotionandsolvingtheequationwiththecentraldifferencescheme. Incontrast,theinnerloopcalculatesthewheel-railcontact,whichis calledasasubroutineateachtimesteppriortoupdatingthestructural dynamicresponses.Thecalculationofwheel/raildynamicsandthe cal-culationof wheel-railcontactarethereforecoupledinthenumerical algorithm.
TheCoulombfrictionlawisappliedtosolvethewheel-railtangential contact[27].Atrialtangentialcontactnodalforcef∗ attimestept+1 canbedefinedasEq.(4):
𝐟∗=𝐟
cT𝑡−𝑘Δ𝐞 (4)
wherefcTtisthetangential contactforce calculatedattimestept,k isthepenaltycontactstiffness,andΔeistheincrementalmovementof theslavewheelnodealongtherailsurface.Thetractionboundfboundtat timesteptintheCoulombfrictionlawistheproductofthemagnitudeof thenormalforcefcNtatthesametimestepandthecoefficientoffriction
Z. Yang and Z. Li International Journal of Mechanical Sciences 153–154 (2019) 490–499
(COF)𝜇,asEq.(5):
𝑓bound𝑡=𝜇||𝐟cN𝑡|| (5)
Thetangentialcontactforceattimestept+1canthusbewrittenas Eq.(6): 𝐟cT𝑡+1= { 𝐟∗ 𝑖𝑓|𝐟∗| ≤𝑓 bound𝑡+1 𝑓bound𝑡+1𝐟∗∕|𝐟∗| 𝑖𝑓|𝐟∗| >𝑓bound𝑡+1 (6) TheCOF𝜇,whichisconsideredaconstantintheclassicalCoulomb’s law,canvarywithvariousfactorsinwheel-railcontact,suchassliding speed,contactpressure, surfacelubricationor contamination, rough-ness,temperatureandhumidity[29].BecausetheexplicitFEMcouples thecalculationofwheel/raildynamicresponseswiththecalculationof wheel-railcontactinthetimedomain,avelocity-dependentCoulomb’s lawwithafunctionalCOFcanbeimplementedintheexplicitFEM.The COFupdatedateachtimestepcanbeexpressedintermsofthestatic anddynamicfrictioncoefficients𝜇sand𝜇d,respectively,adecay
con-stantcandwheel-railrelativeslidingvelocities ̇𝐮relbetweentheslave nodesandmastersegmentsatthesametimestepusingEqs.(7)and(8):
𝜇(̇𝐮rel )
=𝜇𝑑+(𝜇𝑠−𝜇𝑑)e−𝑐|̇𝐮 rel| (7)
̇𝐮rel=Δ𝐞∕Δ𝑡 (8)
Thedecayconstantcdescribeshowquicklythestaticcoefficient ap-proachesthedynamiccoefficientandcanbedeterminedbyfittingthe measuredresults[29].
2.2. Wheel-raildynamicinteractionmodelwithwheellateralmotion
Fig.2(a)showstheemployedthree-dimensionalexplicitFE wheel-trackdynamic interactionmodelwithwheel lateral motion.A10-m lengthofhalf-trackandahalfwheelsetwithsprungmassofthecar bodyandthebogiewereconsidered.Thewheel,therailandthe sleep-ersweremodeledusing8-nodesolidelements.Toachievehighsolution accuracywithareasonablemodelsize,nonuniformmeshingwasused. Themeshsizearoundtheinitialpositionofthewheel-railcontactand
Fig.2. Wheel-raildynamicinteractionmodel.(a)3DFEmodel;(b)simulated frictionalrollingwithwheellateralmotion.
Table2
Prescribeddisplacementboundaryconditionsappliedtothesimulationcases.
Prescribed displacement boundary conditions
Simulation cases with a constant COF
Simulation case with a velocity-dependent COF No lateral wheel motion case 1
Small lateral wheel motion case 2
Medium lateral wheel motion case 3 case 5
Large lateral wheel motion case 4
the150-mmlengthofthesolutionzonewas1mm.Thelumpedmassof thecarbodyandbogiewasmodeledasmasselementsthatwere con-nectedtothewheelsetbytheprimarysuspensionofthevehiclewith parallellinearspringsandviscousdampers.Becausethesleepersand ballasthavelittleinfluenceonthehigh-frequencydynamicbehavior un-derstudy,eachsleepermodelcontainedonly12solidelementsandthe ballastwassimplifiedasverticalspringanddamperelementswiththe displacementconstrainedinthelateralandlongitudinaldirections.The sameconstrainttypewasusedatthetwoendsoftherailmodel.The pa-rametersinvolvedinthetrackmodelaretakenprimarilyfrom[30].The wheel-railcontactwasdefinedwithrealgeometryandwiththewheel flangeincluded.Thewheelgeometrycorrespondedtoapassengercar wheeloftheDutchrailwaywiththestandardprofileofS1002;therail was UIC54E1withaninclinationof1:40.Nogeometryirregularities wereconsideredonthesurfacesofthewheelorrail.
Thewheel-raildynamicinteractionwassimulatedbyemployingthe softwareANSYS&LS-DYNA:thestaticequilibriumofthesystemunder gravitywasfirstsolvedwithanimplicitsolverofANSYS;andanexplicit solverofLS-DYNA,whosecorealgorithmsispresentedinSection2.1, was thenemployedtosimulatewheel-raildynamic frictionalrolling. Thedynamicsimulationappliedthedisplacementfieldcalculatedwith theimplicitsolverforstressinitialization.Inthetransientdynamic sim-ulation,thewheelrolledfromitsinitialpositionattimet0=0stothe so-lutionzonewithaninitialspeedof100km/halongtherail,asshownin Fig.2(b).Thedynamicsarisingfromthewheel/railinitialkinematicand potentialenergyduetoimperfectstaticequilibrium[31]haverelaxed attimet1=15ms.Thewheellateralmotionwassubsequentlysimulated fromtimet2=16msbyapplyingtheprescribeddisplacementboundary conditionslistedinTable2tobothendsofthewheelhalf-axle(except forincase1).Notethat,inthisstudy,theangleofattackwas approx-imatedtozerobecause,inthesimulations,theinitialanglesofattack werezeroandbothendsofthewheelhalf-axlemodelappliedthesame displacementboundaryconditionduringtransientrolling.Fig.3(a)and (b)presenttheprescribeddisplacementboundaryconditionsappliedin thesimulationsandtheresultinglateralwheeldisplacements, respec-tively. Thetwo graphs sharethe sametrend, butthesimulated dis-placementsfluctuatedduetotheflexibilityofthewheelandaxle;the
Fig.3.Timehistoriesoftheprescribeddisplacementboundaryconditionsand theresultinglateralwheeldisplacements.(a)Prescribeddisplacementboundary conditionsappliedtothesimulationcases;(b)simulatedresultinglateralwheel displacements.
unsteadylateralcreepagewasthussimulated.Thewheelenteredthe so-lutionzoneattimet3=16.5msandexitedatt4=18.7ms.Thedynamic evolutionofthecontactsolutionswascapturedbetweent3andt4.
Adrivingwheelwasmodeledinthisstudybyapplyingatorqueon thewheelaxle.Thetorqueinitiallyincreasedgraduallyfromzerountil reachingitsmaximumvalueat6msandremainingconstant.A longi-tudinalcreepforce wasthus generated.TheexplicitFEMcanalsobe employedtosimulatefrictionalrollingcontactbetweenadrivenwheel andrail,e.g.,byapplyingalongitudinaltractionforceonthewheelaxle [32],which,however,producesalongitudinalcreepforceinthe oppo-sitedirection.Thecorrespondingtractioncoefficientequalsthequotient ofthelongitudinalcreepforceandthewheelloadandis considered tobeconstantinsteady-staterolling[31],whereasitvarieswithtime indynamicrollingbecauseofvibration.Thequasi-steady-statetraction coefficientwas0.27inthisstudy.AconstantCOF=0.45wasusedfor simulationcases1–4.
Knothe,etal.[33]reportedthatthefalling-friction characteristic canbereproducedonlybyassumingthatCOFdependsonthe wheel-railslidingvelocity;simulationcase5listedinTable2wastherefore conductedwitha velocity-dependentCOF. Simulation case 5shared thesameconfigurationascase3(exceptforCOF).Whena velocity-dependentCOFisapplied,theoverallCOFmaydifferfromthelocal COFatanode[29].
3. Analysesofwheel-railcontactanddynamicsresults
Thissectionfirstanalyzesthecontactforcescalculatedbythe ex-plicitFEmodelinSection3.1toprovideabroadoverviewofthe wheel-raildynamicinteractionwithwheellateralmotion.Toreproducethe twopotentialmechanismsofsquealintroducedinSection1(the falling-frictionandmode-couplingmechanisms),thesimulatedwheel-rail dy-namiccontactsolutionsandstructuraldynamicresponsesareanalyzed inSections3.2and3.3,respectively.
3.1. Contactforces
Fig.4(a)and(b)showthetimehistoriesofthecontactforces sim-ulatedwithoutwheellateralmotion(case1)andwithmediumlateral
motion(case3),respectively.Beforetimet1=15ms,thesimulated con-tactforcesfluctuatedmostlyduetovibrationsexcitedbyinitialkinetic andpotentialenergyintheunrelaxedsystem.Thetractionbound (de-notedbythecyancurve)fluctuated aroundthestaticvalue50kN.It couldbeassumedthataquasi-steadystatewasenteredafterthe oscilla-tionsweredampedtolessthan10%ofthestaticvaluesattimet1.The creepforceistheresultantforceofthelateralandlongitudinalcontact forces.InFig.4(a),thesimulatedcreepforce(denotedbytheredcurve) almostoverlapsthelongitudinalforce(denotedbythegreencurve) be-causethevalueofthelateralforce(denotedbythebluecurve)issmall. Bycontrast,asshowninFig.4(b),thelateralforcesimulatedbycase3 jumpedtolargevalueaftert2=16msduetotheenforcementofa pre-scribeddisplacementboundarycondition;consequently,thecreepforce reachedthetractionboundandfrictionsaturationoccurredat approx-imately19ms.Thedynamicevolutionofthecontactsolutionbetween timest3=16.5msandt4=18.7mswereoutputandareanalyzedinthe nextsubsection.
3.2. Contactsolutions
This subsectionanalyzesthedynamic contactsolutionscalculated usingtheproposedexplicitFEwheel-railinteractionmodel,including contactstressesandthedistributionsofmicro-slipandadhesion-slip re-gionswithinthecontactpatch.Thissubsectionalsodiscussesthe influ-encesoflateralcreepageandvelocity-dependentCOFonthedynamic contactsolutions.
3.2.1. Contactsolutionswithdynamiceffects
Fig.5(a)showsamagnifiedviewofthetimehistoriesofthe con-tactforcescalculatedbycase1betweentimest3=16.5andt4=18.7ms (withinthesolutionzone).Theclose-upviewshowsthatthesimulated tractionboundandcreepforceoscillatedperiodicallybutwerenot ex-actlyinphase.Eighttimepointswithafixedinterval(0.06ms)were selectedfromacertainperiodofthecreepforcefluctuation,designated 1–8inthemagnifiedviewinFig.5(a).Fig.5(b)plotsthedistributionsof thesimulatedsurfaceshearstress(redcurve)andtractionbound(blue curve)withinthecontactpatchattheseeightselectedtimepoints.The adhesionandslipregionsdeterminedbythecontactstressaredenoted Fig.4. Timehistoriesofthesimulatedcontactforces. (a)Contactforces simulatedbycase1; (b)contact forcessimulatedbycase3.
Z. Yang and Z. Li International Journal of Mechanical Sciences 153–154 (2019) 490–499
Fig.5.Periodicsurfacestressdistributionsfor simula-tioncase1.(a)Magnifiedview(withinthesolution zone)of thetimehistoriesof thecontactforcesfor simulationcase1;(b)stressdistributionscalculatedat correspondingtimepointsdenotedby1–8in(a)(blue curve:tractionbound;redcurve:surfaceshearstress; A:adhesionregion;andS:slipregion;thegreen ar-rowsindicatethepositionofthemovinglocalpeak). (Forinterpretationofthereferencestocolorinthis fig-urelegend,thereaderisreferredtothewebversionof thisarticle.)
‘A’and‘S’inthefigure,respectively.Acomparisonofthestress distri-butionscalculatedattheeighttimepointsinFig.5(b)indicatesthat thesurfaceshearstresswithintheadhesionregionincreasedgradually fromthe1sttimepointtothe4thtimepointanddecreasedfromthe5th timepointtothe8thtimepoint.Thistrendisinaccordancewiththe variationinthecreepforceshowninthemagnifiedviewofFig.5(a). Theanimationdisplayingthevariationinthesurfacestresswithtime [34]indicatesthatthesurfaceshearstressandthetractionboundvaried periodicallywiththecontactforces.Asreportedby[10],theinterface pressuredistributionvarieswithtimeduringvibration;thus,the peri-odicvariationinthesurfacenormalandshearstressesreproducedin thisnumericalexamplemayreflectthedynamiceffectsinvolvedinthe explicitFEcontactsolutions.
Inaddition,amovinglocalpeakoftheshearstress,indicatedbythe greenarrowsinFig.5(b),wasobservedwithintheadhesionregioninthe variationprocessofthecontactstress.Thepeakstartsattheleadingedge ofthecontactpatch,movestowardsthetrailingpart,andultimately exitstheadhesionregionatthejunctureoftheadhesion-slipregions. Becausetheshearstressis closetothetractionbound whencloseto theleadingedgeofthecontactpatchandwhenclosetothejunctureof theadhesion-slipregions,suddenfrictionsaturation,oraturbulenceof micro-slip[35],mayarisewhensuchalocalpeakofshearstressenters orexitstheadhesionregion.
3.2.2. Contactsolutionswithlateralcreepage
Contactstressandmicro-sliparesymmetricallydistributedwith re-specttothelongitudinalaxiswithinthecontactpatchwhencreepage existsonlyinthelongitudinaldirection.Whensimulatingwheellateral
motioninthis study,theresulting wheel-raillateralcreepagecaused asymmetricdistributionsofcontactstressandmicro-slipwithinthe con-tactpatch,asshowninFig.6.Tocomparecontactsolutionswithabroad rangeoflateralcreepage,Fig.6plotssixtypicalcontactsolutions ob-tainedforseveralsimulationcases:(a)and(b)wereobtainedwith sim-ulation case1at17.58msandwithcase2at17.58ms,respectively; (c)and(e)wereobtainedwithcase3at17.58msand18.54ms, re-spectively;and(d)and(f)wereobtainedwithcase4at17.07msand 17.94ms,respectively.Fromtoptobottom,thegraphsinFig.6display thesimulatedstressdistributionsalongthelongitudinalcenterlineofthe contactpatch(1strow),stressdistributions(2ndrow)andmicro-slip distributions(3rdrow)withinthecontactpatch,aswellasshearstress andadhesion-slipregiondistributionsobtainedwithKalker’sboundary elementmethodprogramCONTACT[36](4throw).The correspond-ingcreepagevalues(lateralcreepage𝜂 andlongitudinalcreepage𝜉)are presentedinTable3(exceptforFig.6(f)).
Traditionalrollingcontacttheoriesaregenerallybasedonthe half-space assumption;themainparts of contactbodies not close tothe contactpointcan,therefore,beconsideredasrigid.Thecreepagemay thenbeconvenientlycalculatedwiththe‘rigid’wheel-railrelative ve-locityandthewheelrollingvelocity.However,thisisnolongervalid forFEcontactmodels,whichdropthehalf-spaceassumptionandare flexibleeverywhere.Thetraditionalmethodofthecreepagecalculation isthus notapplicableforthepresentedexplicitFEwheel-railcontact model.ConsideringthattheexplicitFEMandCONTACTshouldprovide similarcreepage-creepforcerelationships,asreportedin[37],we esti-matedthecorrespondingcreepagevaluesoftheexplicitFEcontact solu-tionsusingCONTACTinthisstudy:thecreepforcescalculatedwiththe
Fig.6. Simulatedcontactsolutionswithlateralcreepage(1strow:stressdistributionsalongthelongitudinalcenterline;2ndrow:stressdistributionswithinthe contactpatch(colordepth:pressuremagnitude;bluearrows:directionsandmagnitudesoftheshearstress);3rdrow:micro-slipdistributions(colordepth:magnitude ofthenormalwheel-railrelativevelocity;redarrows:directionsandmagnitudesofthemicro-slips);and4throw:shearstressandadhesion-slipregiondistributions calculatedwithKalker’sCONTACT[36](blackarrows:directionsandmagnitudesoftheshearstress).(a)simulationcase1at17.58ms;(b)case2at17.58ms;(c) case3at17.58ms;(d)case4at17.07ms;(e)case3at18.54ms;and(f)case4at17.94ms.(Forinterpretationofthereferencestocolorinthisfigurelegend,the readerisreferredtothewebversionofthisarticle.)
Table3
InputforcesandoutputcreepageinthesimulationswithCONTACT.
Calculated forces and creepage corresponding to the
graphs in Fig. 6 (a) (b) (c) (d) (e) (f)
Normal load F n (kN) 126.68 125.61 123.87 102.60 116.19 106.46 Lateral creep force
Fty (kN) 2.19 9.30 20.27 18.41 36.83 47.04 Longitudinal creep force F tx (kN) 39.46 39.19 38.12 33.59 34.01 6.09 Lateral creepage 𝜂 (%) 0.019 0.080 0.185 0.199 0.440 – Longitudinal creepage 𝜉 (%) 0.304 0.308 0.317 0.331 0.326 –
explicitFEmodel(seeTable3)wereinputintothesimulationswith CONTACTastheprescribedtangentialforces,andtheotherinput pa-rametersusedintheCONTACTsimulationswerethesameasthose in-volvedintheexplicitFEsimulations.Contactsolutionsandthecreepage valueswerethenobtainedwithCONTACTandpresentedinthe4throw ofFig.6andinTable3,respectively.Notethatnocreepagevaluewas presentedforFig.6(f)becausewhenfrictionsaturationoccurs,a pre-scribedcreepforcecorrespondstononuniquecreepage.
Inthecontour/vectordiagramsshowingthestressdistributions(2nd row),thepressuremagnitudecorrespondstothedepthofthecolor;the bluearrowsindicatethedirectionsoftheshearstress,andtheirlengths areproportional totheshearstressmagnitude.Similarly,inthe con-tour/vectordiagramsshowing themicro-slipdistributions(3rdrow), thedepthof thecolorwithin thecontactpatchindicatesthe magni-tudeofthenormalwheel-railrelativevelocity;theredarrowsindicate thedirectionsof themicro-slip,andtheirlengthsareproportionalto themicro-slipmagnitude.Theasymmetryofthedistributionofthe con-tactpatchcanbecharacterizedbyanorientationangle𝜃,asshownin Fig.6(c).Theorientationangleincreaseswiththelateralcreepagefrom Fig.6(a)to(e).ForthesolutionscalculatedwithCONTACT(4throw),
theblackarrowsindicatethedirectionsoftheshearstress,andtheir lengthsareproportionaltotheshearstressmagnitude.Theorientation anglesobtainedwithCONTACTareconsistentwiththoseobtainedwith theexplicitFEM.
Thedistributionsoftheadhesion-slipregionsdeterminedbythe sim-ulatedexplicit FEcontactstressesin the1strowandthemicro-slips in the3rdrowareinlinewitheach other.They arealsoin reason-ableagreementwiththedistributionsoftheadhesion-slipregions de-terminedwithCONTACTinthe4throw.Inthegraphsofthemicro-slip distribution(3rdrow),theslipregionscoveredwithmicro-slipvectors arelocatedat thetrailingpartofthecontactpatch,whilethe adhe-sionregions‘A’indicatedbydashedovalsshrinkwithincreasinglateral creepagefromFig.6(a)to(e).Fig.6(f)showsfrictionsaturation.
Thecolordepthoutsidethecontactpatchcorrespondstoazero nor-malrelativevelocityin thegraphsof themicro-slipdistribution(3rd row).Thus,thecolordepthsattheleadingandtrailingedgesarelighter anddarker,respectively,indicatingapositiveandnegativenormal rela-tivevelocity.Wavephenomenaindicatedbythevariationincolordepth canbeobservedinthemicro-slipdistributionscalculatedwiththe ex-plicitFEMinFig.6(e)and(f).Becausethesurfaceelementsofthewheel andrailwereincontact,furthermovementtowardsorawayfromeach othercausedtransientintensificationorrelaxationof thecontact, re-spectively.Thewaveswerethereforeessentiallyembodiedinthe alter-nationofcompressionintensificationandrelaxation[35].Accordingto thevibrationfrequenciesoftherailsurfacenodesandthewavelengths observedinFig.6,thewavespeedcanbeestimatedasapproximately 3km/s,whichisinlinewiththespeedofRayleighwavetravelingin steel. Moreover,thephasedifferencebetween normalandtangential surfacenodalvibrationsisapproximately𝜋/2,whichalsocorresponds wellwiththeRayleighwave.
3.2.3. Contactsolutionswithavelocity-dependentCOF
Avelocity-dependentCOFdefinedinEq.(7)wasadoptedin simula-tioncase5.Thedecaycoefficientc,staticCOF𝜇sanddynamicCOF𝜇d
weresetto6,0.5and0.32,respectively,asusedin[29].Fig.7presents oneexample ofthecontactsolutionscalculatedwithsimulationcase
Z. Yang and Z. Li International Journal of Mechanical Sciences 153–154 (2019) 490–499
Fig.7. Oneexampleof thecontactsolutions calcu-latedusingavelocity-dependentCOF(simulationcase 5.S1,S2andS3:slipregions).(a)Adhesion-slip divi-siondeterminedbythemicro-slip(redarrows: micro-slips); (b) stress distribution (color depth: pressure magnitude;blue arrows: directions andmagnitudes oftheshearstress);(c)adhesion-slipdivisions deter-minedbythestressdistributionsextracted from(b) alongthefourlongitudinallinesofthecontactpatches indicatedbyblackarrows.(Forinterpretationofthe referencestocolorinthisfigurelegend,thereaderis referredtothewebversionofthisarticle.)
5at16.68ms.Althoughsimulationcase5sharedthesame configura-tionascase3(exceptforCOF),theorientationangleshowninFig.7(b) ismuchsmallerthanthoseshowninFig.6(c)and(e)becauselateral creepforceincreasedwhenthewheelpassedthesolutionzone(seethe bluecurvebetweentimet3andt4inFig.4(b))andthecontactsolution presentedinFig.7wascalculatedatanearliertimestepthanthoseof Fig.6(c)and(e)andthushadasmallerlateralcreepage.
Fig. 7(a) shows the simulated micro-slip distribution within the contactpatch.Compared totheadhesion-slipdistributionssimulated withaconstantCOFshowninFig.6,theadhesion-slipdistributionin Fig.7showsalessregularpattern:theslipregionsdenotedbytheblack ovalsS1,S2andS3arescatteredinthetrailingpartofthecontactpatch. However,thisresultisbelievedtobearealisticcontactcondition[29]. ThecontactpressuresandsurfaceshearstressesplottedinFig.7(c) wereextractedfromthestressdistributionshowninFig.7(b)alongthe fourlongitudinallinesofthecontactpatchesindicatedbyblackarrows. InFig.7(c),theredandbluecurvesindicatethesimulatedsurfaceshear stressFtandthetractionbound (theproductofthecontactpressure
Fnandthevelocity-dependentCOF𝜇(v)),respectively,thegreencurve
indicatestheproductofthecontactpressureandthestaticCOF𝜇s,and
theblackcurveindicatestheproductofthecontactpressureandthe dynamicCOF𝜇d.
AccordingtoCoulomb’slawofstatics,thebluecurveinFig.7(c) shouldcoincideeitherwiththegreencurveoftheadhesionregionsor withtheredcurveoftheslipregions.Discrepancieswere,however, ob-servedinthenumericalsolutionsattwodatapoints,asindicatedbythe tworedarrows:thebluecurvesatthesetwodatapointsshouldhave
overlappedwiththered curvesbecausethecalculatedmicro-slipsat theselocationsarenonzeroinFig.7(a),asindicatedbythetwogreen arrows.Thesediscrepanciesmaystemfromtwoaspects.First,theshear stressandtractionboundshowninFig.7(c)wereobtainedatdifferent timesteps.Becausethevelocity-dependentCOF𝜇(̇𝐮rel)usedto calcu-latethetractionbound(Eq.(5))andthetangentialforce(Eq.(6))at acertaintimestep reliesonthenodaldisplacementsobtained atthe previoustimestep(Table1step(d)),thetractionbounddisplayedin Fig.7(c)wasinfactcalculatedonetimesteplaterthantheshearstress. Consideringthatthetimestepusedin thesimulationwassmall (ap-proximately86ns),thesecondaspectisexpectedtodominate:the dis-crepancieswereduetothedifferentsensitivitiesoftheapproachesused todeterminetheslipregion.Themicro-slipshowninFig.7(a)andthe velocity-dependentCOF𝜇(v)usedforFig.7werecalculatedusingthe wheel-railrelativevelocities,whereastheshearstressFtwasobtained usingtherelativedisplacement(Eq.(4)).Becausethedisplacementwas obtainedbyintegratingvelocity(Eq.(3))andintegrationcanactasa filter,thetractionboundshowninFig.7(c)isexpectedtoreflect dy-namiceffectsmoresensitivelythanthesurfaceshearstresscalculated usingthedisplacement.
Theadhesion-slipdistributiondeterminedusingthecalculated con-tactstressesinFig.7(c)isinreasonableagreementwiththatdetermined usingthecalculatedmicro-slipinFig.7(a),indicatingthattheexplicit FEM can predictwheel-rail dynamic contactsolutionswith velocity-dependent COFs.Becausethefalling-frictioncharacteristiccan be re-producedonlybythevelocity-dependentCOF[33],theexplicitFEMis expectedtobecapableofreproducingthefalling-frictioncharacteristic.
Fig.8. Comparisonof thecalculatedandmeasuredwheel modalfrequencies(solidlines:measured;dottedlines:FE sim-ulated).
3.3. Wheeldynamicbehavior
Basedontheanalysisofthesimulatedwheel-raildynamiccontact solutionwithvelocity-dependentCOF,theprecedingsubsection indi-catedthattheexplicitFEMiscapableofaddressingthefalling-friction generationmechanismofwheelsqueal.Asreportedin[38],anexplicit FEmodelmayincludeinitssolutionallrelevantvibrationmodesand associatedwavepropagationsaslongastheelementsandtimesteps aresufficientlysmall.Thecomputationaltimestep(ΔtinEqs.(2)and (3))ofthetransientanalysesinthisstudyfluctuatedaround86ns;thus, thevibrationfrequencyupto5.8MHzcantheoreticallybepredicted. Thissubsectionanalyzesthewheeldynamicbehaviorandcomparesthe wheeldynamicresponsessimulatedwithandwithoutlateralmotionto addressthemode-couplingmechanismofsqueal.
3.3.1. Validationofwheeldynamicbehavior
Wheeldynamicbehaviorplaysamoreimportantroleinthe gener-ationofhigh-frequencysquealthantrackdynamicbehavior[22].Each tonalfrequencyofsquealisexpectedtorelatetoawheel mode[17, 39]. Toaccuratelysimulate wheeldynamic responsesduring rolling, thisstudyfirstvalidatedthedynamicbehavioroftheexplicitFEwheel submodelwithalaboratoryhammertest.AtypicalNS-intercitywheel usedintheDutchrailwaywasmeasured.Becausematerialdampingof awheelisgenerallyverylowandtheexactvalueofthewheelmodal dampingis notcriticalfornoiseprediction[22],thewheel dynamic behaviorcanbecharacterizedbythemodesandthecorresponding nat-uralfrequencies[21,30].Thewheelmodesaregenerallycharacterized bythenumbersofnodaldiametersandnodalcircles[22].Fig.8 com-paresthewheelvibrationmodesmeasuredbythehammertestandthose identifiedthroughtheFEmodalanalysiswithinthefrequencyrangeof squeal(upto10kHz).Thenaturalfrequenciesofthewheelmodesin var-iousdirectionsareplottedagainstthenumberofnodaldiameters. Mea-suredresultsareplottedusingsolidlinesandthecorrespondingresults fromtheFEmodalanalysisarepresentedusingdottedlines.Reasonable agreementwasreached.Allthesephysicalmodeswerethusincludedin thetransientdynamicsimulationbyvirtueofthefullFEmodelandthe smalltimestep[31].
3.3.2. Squeal-likewheelvibration
Afterthevalidationofwheeldynamicbehavior,wheeldynamic re-sponsesweresimulatedusingtheproposed explicitFEwheel-rail dy-namicinteractionmodel.Fig.9(a)and(b)showthetimehistoriesand thecorrespondingpowerspectrumdensities(PSDs),respectively,ofthe lateralvibrationoftherollingwheelcalculatedinsimulationcases1–4. Thesqueal-likevibrationsignalsrepresentedbylargeamplitude limit-cycleswereproducedbytheexplicitFEmodelwhenapplyingthewheel lateralmotion.Theamplitudesofthelimitcyclesofthetimehistories
Fig.9. TimehistoriesandPSDsofthewheellateralvibration simulatedby differentsimulationcases.(a)Timehistories;(b)PSDs.
increasedwithincreasingamplitudeofthelateralmotionappliedtothe wheelmodel.Thethreedominantfrequenciesofthesqueal-like vibra-tionsignalsshowninPSDsinFig.9(b)are163Hz,1172Hzand1921Hz. Comparingthesefrequenciestothewheelmodalfrequenciesidentified inSubsection3.3.1,wecandeterminethewheelmodesexcitedinthe simulationsofwheel-rail dynamicinteractionwithwheel lateral mo-tion:themodewithzeronodalcircleandzeronodaldiameter(0,0),the modewithzeronodalcircleandthreenodaldiameters(0,3),andthe modewithzeronodalcircleandfournodaldiameters(0,4),asshown inFig.9(b)andTable4.Theresultscorrespondwelltotheconclusion thatzero-nodal-circlemodestendtobeexcitedinsqueal[22].
3.3.3. Mode-couplingbehavior
Fig.10comparesthewheelvibrationscalculatedwithnolateral mo-tion(case1)andwithlargelateralmotion(case4).Thecomparisonof thesimulatedwheellateralvibrationsinFig.10(a)and(d)indicatesthat
Z. Yang and Z. Li International Journal of Mechanical Sciences 153–154 (2019) 490–499
Fig.10. WPSsofthesimulatedwheelvibrationwithout lat-eralwheelmotion(case1)andwithlargelateralwheelmotion (case4).(a)Lateralvibrationforcase1;(b)verticalvibration forcase1;(c)longitudinalvibrationforcase1;(d)lateral vi-brationforcase4;(e)verticalvibrationforcase4;(f) longitu-dinalvibrationforcase4.
Table4
Correspondingdominantfrequenciesofthesimulatedvibrationsandthewheel modes.
Dominant frequency in
PSDs Corresponding modal frequency Corresponding mode
163 Hz 183 Hz 0 nodal circle, 0 nodal diameter
1172 Hz 1075 Hz 0 nodal circle, 3 nodal diameters
1921 Hz 1934 Hz 0 nodal circle, 4 nodal diameters
theenforcementofthelateraldisplacementboundaryconditionattime
t2=16msexcitedtheout-of-planeaxialmodes(0,0),(0,3)and(0,4),as denotedinFig.10(d).Thecomparisonsofthesimulatedwheelvertical andlongitudinalvibrationsinFig.10(b)and(e)andinFig.10(c)and(f) indicatethattheprescribedwheelmotionintheaxledirectionmayalso excitethein-planeradialwheelmodeswithone nodaldiameterand twonodaldiameters,whicharedenotedbymodeRadial1andmode Radial2inFig.10(e)and(f),respectively. Thecouplingoftheaxial andradialdynamicsshowninFig.10suggeststhatthemode-coupling mechanismcan bereproducedbytheproposedexplicitFE wheel-rail dynamicinteractionmodel.
The dominant frequency component of approximately 150Hz in Fig.10(f)was excitedmainlybythelongitudinal creepforce,which generallyfluctuatedinantiphasewiththelateralcreepforceaftertime
t3(seeFig.4(b)).Thedominantfrequencyofthecreepforce fluctua-tionscanbeestimatedbyFig.4(b)asapproximately150Hz.However, becausetheaxialwheelmode((0,0):183Hz)withaclosemodal fre-quencywasexcited,theenergyamplitudeofthecalculatedwheel lat-eralvibrationconcentratedatapproximately150Hzwasmuchlarger thanthatofthelongitudinalvibration,asindicatedbythecolorbarsof Fig.10(d)and(f).
4. Conclusionsandfutureresearch
We have proposed an explicit FE wheel-rail dynamic interaction model with wheel lateral motion toaddress the falling-friction and mode-couplingmechanisms,whicharecommonlyconsideredto gener-atesqueal.TheexplicitFEMcouplesthecalculationoffrictionalrolling contactwiththecalculationofstructuraldynamicresponsesandis in-trinsicallysuitableformodelingfriction-inducedunstablevibration.The simulationresultsindicatedthattheproposedmodelcanreproduceboth mechanisms,thusconfirmingthattheexplicitFEMisreliablefor pre-dictingwheel-railsqueal-excitingcontactandispromisingforthe accu-ratepredictionofsqueal.
Wehaveanalyzedthedynamiccontactsolutionscalculatedbythe explicitFEwheel-railinteractionmodel,includingcontactstressesand thedistributionsofmicro-slipandadhesion-slipregionswithinthe con-tactpatch.ThecontactsolutionsobtainedwiththeexplicitFE arein reasonableagreementwiththoseobtainedwithKalker’sboundary el-ementprogramCONTACT. Wavephenomenacausedbylargelateral creepagewereobservedintheexplicitFEcontactsolutions.Amoving local peakofshearstresswas discoveredwithin theadhesion region inthevariationofcontactstress,whichwasconcludedtorelatetothe generationoftheturbulence-inducedwavesreportedin[35].
Wealsodiscussedtheinfluenceofthevelocity-dependentCOFon thedynamic contactsolutions.Applyingthevelocity-dependent COF resulted in a less regular adhesion-slip distribution pattern, which mayshowamorerealisticcontactcondition.Thedistributionsofthe adhesion-slipregionsdeterminedbythesimulatedcontactstressesand bythemicro-slipsweremutuallyconsistentwhentheconstantCOFwas used, whereas small discrepancieswere observedwhen the velocity-dependent COFwasused.Thediscrepancieswereconcludedtostem fromthedifferentsensitivitiesoftheapproachesusedtodeterminethe slipregion.
ThedynamicbehavioroftheFEwheelmodelwasvalidatedthrough alaboratoryhammertest.Byrelatingthedominantfrequenciesofthe simulatedsqueal-likevibrationtotheidentifiedwheelmodal frequen-cies,threezero-nodal-circlemodespronetobeexcitedinsquealwere determinedandfoundtocorrespondwelltopreviouslyreportedresults. Because‘enigmatic’squealissensitivetobothstructuraldynamic be-haviorandcontactcondition,thedynamiccontactalgorithmsshouldbe furtherstudiedandexperimentallyvalidated.Inaddition,the displace-mentboundaryconditionsappliedtotheproposedexplicitFE wheel modelmaynotbe sufficientlyaccurateforsimulatingwheelcurving behaviorandrollingcontactonthecurvetracks.Morerealistic kine-maticboundaryconditionsorafullwheelsetmodelmayberequiredto reliablycalculateunstablewheelvibrationandconsequentsqueal. Acknowledgments
ThisworkwassupportedbytheChinaScholarshipCouncilandthe DutchrailwayinfrastructuremanagerProRail.
Declarationsofinterest None.
Figuresreproducedincolorintheprintedversion:None.
Supplementarymaterials
Supplementarymaterialassociatedwiththisarticlecanbefound,in theonlineversion,atdoi:10.1016/j.ijmecsci.2019.02.012.
References
[1] Pieringer A. A numerical investigation of curve squeal in the case of constant wheel/rail friction. J Sound Vibration 2014;333:4295–313.
https://doi.org/10.1016/j.jsv.2014.04.024 .
[2] Rudd MJ. Wheel/rail noise —Part II: Wheel squeal. J Sound Vibration 1976;46:381– 94. https://doi.org/10.1016/0022-460x(76)90862-2 .
[3] Remington PJ. Wheel/rail squeal and impact noise: What do we know? What don’t we know? Where do we go from here? J Sound Vibration 1987;116:339–53.
https://doi.org/10.1016/s0022-460x(87)81306-8 .
[4] Fingberg U. A model of wheel-rail squealing noise. J Sound Vibration 1990;143:365– 77. https://doi.org/10.1016/0022-460x(90)90729-j .
[5] Périard FJ . Wheel-Rail Noise Generation: Curve Squealing by Trams Doctoral thesis. Delft University of Technology; 1998 .
[6] Heckl MA, Abrahams ID. Curve squeal of train wheels, Part 1: Mathe- matical model for its generation. J Sound Vibration 2000;229:669–93.
https://doi.org/10.1006/jsvi.1999.2510 .
[7] de Beer FG, Janssens MHA, Kooijman PP. Squeal noise of rail-bound vehicles influenced by lateral contact position. J Sound Vibration 2003;267:497–507.
https://doi.org/10.1016/s0022-460x(03)00710-7 .
[8] Huang ZY . Theoretical modelling of railway curve squeal. Doctoral thesis. University of Southampton; 2007 .
[9] Graf M, Ostermeyer GP. Friction-induced vibration and dynamic friction laws: Instability at positive friction–velocity-characteristic. Tribol Int 2015;92:255–8.
https://doi.org/10.1016/j.triboint.2015.06.019 .
[10] Ouyang H, Nack W, Yuan Y, Chen F. Numerical analysis of automotive disc brake squeal: a review. Int J Vehicle Noise Vibration 2005;1:207.
https://doi.org/10.1504/ijvnv.2005.007524 .
[11] Kang J, Krousgrill CM, Sadeghi F. Analytical formulation of mode-coupling instability in disc–pad coupled system. Int J Mech Sci 2009;51:52–63.
https://doi.org/10.1016/j.ijmecsci.2008.11.002 .
[12] Giner-Navarro J, Martínez-Casas J, Denia FD, Baeza L. Study of railway curve squeal in the time domain using a high-frequency vehicle/track interaction model. J Sound Vibration 2018;431:177–91. https://doi.org/10.1016/j.jsv.2018.06.004 .
[13] Li Z, Wang X, Zhang Q, Guan Z, Mo J, Ouyang H. Model reduction for friction- induced vibration of multi-degree-of-freedom systems and experimental validation. Int J Mech Sci 2018;145:106–19. https://doi.org/10.1016/j.ijmecsci.2018.06.039 .
[14] Chiello O, Ayasse JB, Vincent N, Koch JR. Curve squeal of urban rolling stock —Part 3: Theoretical model. J Sound Vibration 2006;293:710–27.
https://doi.org/10.1016/j.jsv.2005.12.010 .
[15] Ding B, Squicciarini G, Thompson D, Corradi R. An assessment of mode-coupling and falling-friction mechanisms in railway curve squeal through a simplified approach. J Sound Vibration 2018;423:126–40. https://doi.org/10.1016/j.jsv.2018.02.048 .
[16] Jiang J, Anderson DC, Dwight R. The Mechanisms of Curve Squeal. Noise Vibration Mitigation Rail Transport Syst 2015;126:587–94.
https://doi.org/10.1007/978-3-662-44832-8_69 .
[17] Vincent N, Koch JR, Chollet H, Guerder JY. Curve squeal of urban rolling stock —Part 1: State of the art and field measurements. J Sound Vibration 2006;293:691–700.
https://doi.org/10.1016/j.jsv.2005.12.008 .
[18] Glocker C, Cataldi-Spinola E, Leine RI. Curve squealing of trains: Mea- surement, modelling and simulation. J Sound Vibration 2009;324:365–86.
https://doi.org/10.1016/j.jsv.2009.01.048 .
[19] Zenzerovic I, Kropp W, Pieringer A. An engineering time-domain model for curve squeal: Tangential point-contact model and Green ׳s functions approach. J Sound Vibration 2016;376:149–65. https://doi.org/10.1016/j.jsv.2016.04.037 .
[20] Lai V-V, Chiello O, Brunel J-F, Dufrénoy P. Full finite element models and reduction strategies for the simulation of friction-induced vibrations of rolling contact systems. J Sound Vibration 2019;444:197–215. https://doi.org/10.1016/j.jsv.2018.12.024 .
[21] Thompson DJ, Squicciarini G, Ding B, Baeza L. A state-of-the-art re- view of curve squeal noise: Phenomena, mechanisms, modelling and mit- igation. Noise Vibration Mitigation Rail Transport Syst 2018;139:3–41.
https://doi.org/10.1007/978-3-319-73411-8_1 .
[22] Thompson DJ . Railway noise and vibration: Mechanisms, modelling and means of control. Elsevier; 2009 .
[23] Johnson KL . Contact mechanics. Cambridge University Press; 1985 .
[24] Bathe KJ . Finite element procedures. Prentice Hall; 1996 .
[25] Wu SR , Gu L . Introduction to the explicit finite element method for nonlinear tran- sient dynamics. John Wiley & Sons; 2012. Incorporated .
[26] Courant R, Friedrichs K, Lewy H. On the partial difference equations of mathematical physics. IBM J Res Dev 1967;11:215–34. https://doi.org/10.1147/rd.112.0215 .
[27] Hallquist JO, Goudreau GL, Benson DJ. Sliding interfaces with contact-impact in large-scale lagrangian computations. Comput Methods Appl Mech Eng 1985;51:107– 37. https://doi.org/10.1016/0045-7825(85)90030-1 .
[28] Hallquist J.O. LS-DYNA Theory Manual. 2006.
[29] Zhao X, Li Z. A solution of transient rolling contact with velocity dependent friction by the explicit finite element method. Eng Comput 2016;33:1033–50.
https://doi.org/10.1108/ec-09-2014-0180 .
[30] Yang Z, Boogaard A, Chen R, Dollevoet R, Li Z. Numerical and experimental study of wheel-rail impact vibration and noise generated at an insulated rail joint. Int J Impact Eng 2018;113:29–39. https://doi.org/10.1016/j.ijimpeng.2017.11.008 .
[31] Zhao X, Li ZL. The solution of frictional wheel-rail rolling contact with a 3D tran- sient finite element model: Validation and error analysis. Wear 2011;271:444–52.
https://doi.org/10.1016/j.wear.2010.10.007 .
[32] Wei ZL, Li ZL, Qian ZW, Chen R, Dollevoet R. 3D FE modelling and validation of frictional contact with partial slip in compression-shift-rolling evolution. Int J Rail Transport 2016;4:20–36. https://doi.org/10.1080/23248378.2015.1094753 .
[33] Knothe K, Wille R, Zastrau BW. Advanced contact mechanics–road and rail. Vehicle Syst. Dynamics 2001;35:361–407. https://doi.org/10.1076/vesd.35.4.361.2043 .
[34] Animation for Fig. 5. https://youtu.be/3jeei3NTP84 . [accessed 18.08.11]. [35] Yang Z, Li ZL, Dollevoet R. Modelling of non-steady-state transition from
single-point to two-point rolling contact. Tribol Int 2016;101:152–63.
https://doi.org/10.1016/j.triboint.2016.04.023 .
[36] Kalker JJ. Three-dimensional elastic bodies in rolling contact. Netherlands: Springer; 1990 https://doi.org/10.1007/978-94-015-7889-9 .
[37] Vo KD, Tieu AK, Zhu HT, Kosasih PB. A 3D dynamic model to investigate wheel–rail contact under high and low adhesion. Int J Mech Sci 2014;85:63–75.
https://doi.org/10.1016/j.ijmecsci.2014.05.007 .
[38] Zhao X, Li ZL. A three-dimensional finite element solution of frictional wheel- rail rolling contact in elasto-plasticity. Proc Inst Mech Eng Part J J Eng Tribol 2015;229:86–100. https://doi.org/10.1177/1350650114543717 .
[39] Squicciarini G, Usberti S, Thompson DJ, Corradi R, Barbera A. Curve squeal in the presence of two wheel/rail contact points. Noise Vibration Mitigation Rail Transport Syst 2015;126:603–10. https://doi.org/10.1007/978-3-662-44832-8_71 .
