A numerical study on waves induced by wheel-rail contact

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Delft University of Technology

A numerical study on waves induced by wheel-rail contact

Yang, Zhen; Li, Zili

DOI

10.1016/j.ijmecsci.2019.105069

Publication date

2019

Document Version

Final published version

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International Journal of Mechanical Sciences

Citation (APA)

Yang, Z., & Li, Z. (2019). A numerical study on waves induced by wheel-rail contact. International Journal of

Mechanical Sciences, 161-162, [105069]. https://doi.org/10.1016/j.ijmecsci.2019.105069

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InternationalJournalofMechanicalSciences161–162(2019)105069

ContentslistsavailableatScienceDirect

International

Journal

of

Mechanical

Sciences

journalhomepage:www.elsevier.com/locate/ijmecsci

A

numerical

study

on

waves

induced

by

wheel-rail

contact

Zhen

Yang,

Zili

Li

∗

Delft University of Technology, Section of Railway Engineering, Stevinweg 1, 2628 CN, Delft, the Netherlands

a

r

t

i

c

l

e

i

n

f

o

Keywords: Wheel-rail contact Wave Rayleigh wave Crack Explicit FEM

a

b

s

t

r

a

c

t

Recentfiniteelement(FE)simulationshaverevealedthegenerationandpropagationofwavesinrailsurfaces inducedbywheel-railfrictionalrolling.Thesewaveshaverarelybeenaddressedintheliterature.Thispaper presentsanin-depthanalysisofthesewaves,aimingtogivenewinsightsintothecontactmechanics,aresearch area inwhichwaveshavegenerallybeenignored.Thestudyfirstcategorisesthesimulatedcontact-induced wavesaccordingtotheirgenerationmechanismsasimpact-induced,creepage-inducedandperturbation-induced waves.Thelinkbetweenthegenerationofperturbation-inducedwavesandthestick-slipcontactmechanism isthenexplored.Next,byexaminingtherailsurfacenodalmotionthatformsthewave,thecreepage-induced waveisdemonstratedtobeaRayleighwave;thisresultalsoshowsthattheexplicitFEmethodcaneffectively simulatephysicalcontact-inducedwavesandprovidereliabledynamiccontactsolutions.Finally,FEmodelling ispresentedtoinvestigatetheeffectsofsurfacecracksonthewaves,whichmaycontributetowave-basedcrack detection.

1. Introduction

The contact-induced wave phenomenon [1] has rarely been ad-dressedinthestudyofwheel-railrolling.Onepossiblereasonisthat thewavesinitiatedindynamicfrictionalrollingcontactandinﬂuenced bytheentirevibratingstructurescannotbereproducedbythebroadly usedcontacttheoriesbasedontheassumptionsofasteadystateanda halfspace,e.g.,Hertzcontacttheory[2]andKalker’stheories[3].The fundamentaldiﬃcultyliesisthecombinationofthestronglynon-linear frictionlawandthedynamicsofthesolids.

Toourknowledge,wheel-railcontact-inducedwaveswerefirst men-tionedin[4],inwhichwheel-railfrictionalrollingcontactwassolved withan explicit finiteelement method (FEM). Goodagreement was achievedwhencomparingtheobtainedexplicitfiniteelement(FE) con-tactsolutionwithHertzcontacttheoryandKalker’sboundaryelement contactsolution,butasmallpressurefluctuationexistedintheFE re-sults.Thisfluctuationwasconsideredtobecausedbyhigh-frequency vibrationandwavepropagationinthewheelandrailcontinuabecause theFEcontactsolutionsintrinsicallyincludealltherelevantvibration modesofthestructuresandcontinuaandtheassociatedwave propaga-tions[5].

The explicit FEM has since been increasingly employed for the simulation of wheel-rail dynamic contact involving, for example, impact[6–15],ﬂanging[16–18]andfriction-inducedinstability[19]. Theoverallpictureofthecontact-inducedwavepatternwasobserved whentheauthorsofthispapersimulatedthenon-steady-statetransition

∗_{Corresponding}_author.

E-mailaddresses:z.yang-1@tudelft.nl(Z.Yang),z.li@tudelft.nl(Z.Li).

ofwheel-railcontactfromasinglepointtotwopoints[18].Thewave wasfoundtobeinitiatednexttothejunctureoftheadhesionandslip regionsinthecontactpatch,wherethemaximumsurfaceshearstress islocated.Afterthisstudy,thewavesgeneratedbytheimpactsatan insulated rail joint [11] anda crossing [13] andthe waves caused bywheel-raillateralcreepage[19]werereproducedwithexplicitFE contact models. The explicit integration algorithm is considered to becomputationallyattractiveandnaturallysuitableforanalysingthe contact-inducedwavepropagationbecausethetotaldynamicresponse timethatmustbemodelledisonlyafewordersofmagnitudelonger thanthestabilitycriticaltimestep[20]andthecontactconditionsare updatedwithinasmalltimeinterval,whichfacilitatestheanalysisof high-frequencywavepropagation[21].

The aforementioned studies, however, only presented the wave phenomenaobservedinexplicitFEcontactsimulations.Thegeneration mechanismsandphysicalcharacteristicsofthesimulatedwaveshave not been examined. This study, in this context, ﬁrst categorises the waves observed in theprevious explicit FE wheel-rail contact simu-lations accordingtotheirgenerationmechanismsasimpact-induced, creepage-induced andperturbation-induced waves.The possible link between thegenerationofperturbation-inducedwaves andthe stick-slipcontactbehaviouristhendiscussed.Afterthat,thestudyanalyses thephysicalcharacteristicsofthecreepage-inducedwaveobservedin

[19],conﬁrmingthatthesimulatedwaveisaRayleighwave.Although theRayleighwavehasbeenextensivelyproposedtoenabledetection of thepresence of rail cracking [22,23], its practical application to

https://doi.org/10.1016/j.ijmecsci.2019.105069 Received15March2019;Accepted6August2019 Availableonline07August2019

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Z. Yang and Z. Li International Journal of Mechanical Sciences 161–162 (2019) 105069 Fig.1. Impact-inducedwavepatterns(seealso theanimationsin[25]and[26]).

fielddetectionisstillunderdevelopment.This studyfinallypresents anexplicitFEwheel-railcontactmodelwithacracktoinvestigatethe influenceofcracksonthewaves,whichmaycontributetowave-based crackdetection.

2. Categorisationandgenerationmechanismsofthesimulated waves

The contact-induced waves discovered in the previous explicit FE wheel-rail contact studies may be categorised according to the generation mechanisms as impact-induced, creepage-induced and perturbation-inducedwaves.Thegenerationmechanismsoftheformer twoappeartobeevident:thesigniﬁcantdynamiceﬀectorkinetic en-ergy[24]inducedbythewheel-railimpactorlargecreepageresultsin largeoscillationsof thewheel/rail surfaceparticlesinthevicinityof thecontactpatch.Thelargelocaloscillationsthenpropagateandform regularwavepatterns.Thegenerationmechanismofthe perturbation-inducedwaveishoweverlessapparentbecausetheperturbationarises duringseeminglysteady-staterolling.

2.1. Impact-inducedwaves

Thepropagationofelasticwavesinevitablyoccursuponimpact[24]. ThecontourgraphsofFig.1presenttwoexamplesofwheel-rail impact-inducedwaves.Becausetheimpactexcitationisnormaltothe wheel-railcontactsurface,thenormal(out-of-plane)nodalvibrationvelocities playmuchmoreimportantrolesthanthetangential(in-plane)onesin theformationoftheimpact-inducedwave[11].Themagnitudeofthe normalnodalvelocityontherailsurfaces isindicatedbythecolour depthofthecontourgraphs;theleadingandtrailingedgesofthecontact patchcanthusbeidentiﬁedbytheblueandredcolours,respectively.

Fig.1(a)showsanimpact-inducedwaveproducedbysimulatingthe impactofawheelonaninsulatedrailjoint(IRJ);detailsofthemodelling werepresentedin[11].WhenthewheelrollsovertheIRJandhitsthe railendontheothersideofthejoint,animpact-inducedwaveoccurs attheleadingedgeofthecontactpatchandpropagatesforwardalong thewheelrollingdirection.Fig.1(b)showsanothercaseofan impact-inducedwaveproducedbythewheel-railtwo-pointcontacttransition

discussedin[18].Theimpact-inducedwavearisesatthesecondcontact patchontherailgaugecornerimmediatelyafterthewheelﬂangecomes into contactwith, orhits,therailgauge corner.Thegenerationand propagationoftheimpact-inducedwavesshowninFigs.1(a)and(b) canbemoreclearlyseenintheanimations[25]and[26],respectively.

2.2. Creepage-inducedwaves

When large wheel-rail creepage occurs, wave patterns embody-ing thealternationof thecompression intensiﬁcationandrelaxation

[18]maybegenerated.Twoexamplesofcreepage-inducedwavesare presentedinthecontour/vectordiagramsinFig.2.Fig.2(a)showsa creepage-inducedwavepatterncalculatedwiththeexplicitFE squeal-excitingcontactmodelpresentedin[19],inwhichthewheel-railrolling contactwithalargelateralmotionofthewheelissimulated;Fig.2(b) showsanothercreepage-inducedwavepatternobservedinthe simula-tionofthewheel-railtwo-pointcontacttransition[18],inwhichlarge creepageoccursattheﬁrstcontactpatchontherailtopwhentherolling wheelnegotiateswiththerailviaﬂangecontact.Theanimations corre-spondingtoFig.2(a)and(b)displayingthegenerationandpropagation ofthecreepage-inducedwavecanbefoundin[27]and[28], respec-tively.

In thecontour/vector diagrams in Fig. 2, themagnitudes of the normalrelativevelocitiesofthewheel/railnodesareindicatedbythe colourdepthwithinthecontactpatches,andthetangentialrelative ve-locitiesbetweenthewheelnodesandtherailnodes,i.e.,themicro-slip, areindicatedbythebluearrows.Thearrowspointinthedirectionof themicro-slip,andtheirlengthsareproportionaltothemagnitude.Both thenormalandtangentialnodalvelocitiescontributetotheformation ofthecreepage-inducedwave.Thecreepage-inducedwaveappearsto propagateparalleltothemicro-slipvectors,andthemicro-slipinthe compressionrelaxationarea(darkercolour)islargerthanthatinthe adjacentcompressionintensiﬁcationarea(lightercolour).

2.3. Perturbation-inducedwaves

Perturbation-inducedwaveshavealsobeenobservedinthe previ-ousexplicitFEwheel-railcontactsimulations:perturbationofthenodal

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Z. Yang and Z. Li International Journal of Mechanical Sciences 161–162 (2019) 105069 Fig.2. Creepage-inducedwavepatterns(see alsotheanimationsin[27]and[28]).

velocitysuddenlyoccurswithinthecontactpatchandspreadsradially, consequentlydevelopingintoawavepattern.Themaindifferenceinthe perturbation-inducedwavecomparedwiththeformertwowavetypes isthattheperturbation-inducedwave,ortheperturbationtobe pre-cise,maybeinitiatedduringseeminglysteady-staterollingwithoutthe involvementofsignificantdynamiceffectscausedbyeitherimpactsor largecreepage.

Perturbation-inducedwavesarefoundtobeinitiatedeithercloseto theleadingedgeofthecontactpatch[11]orclosetothejunctureof theadhesion-slipregions[18].Twoanimationsofthegenerationand propagationofperturbation-inducedwavesarepresentedin[29]and

[30].Theperturbationinitiatesclosetotheleadingedgeofthecontact patchin[29]andatthejunctureoftheadhesion-slipregionsin[30]. TheanimationswereobtainedbyexplicitFEsimulationofwheel-rail frictionalrollingcontactwithoutwheellateralmotion(simulationcase 1in[19]).

Eachanimationconsistsoftwowindows:theupperwindowdisplays thesimulatedevolutionofthewheel-railrelativevelocitieswithinthe contactpatch,andthelowerwindowdisplaysthesimulatedevolution ofthenormalandtangentialvelocitiesoftherailsurfacenodeswithin theentiresolutionzone(aregionontherailtopsurfaceforwhichthe wheel-railcontactsolutionisoutput).Thewheel-railrelativevelocities (intheupperwindow)canclearlyindicatetherangeofthecontactpatch andtheadhesion-slipregionsandthusenabletheinitiationpositionof theperturbation tobelocated,whiletherailsurfacenodalvelocities (inthelowerwindow)canmoreclearlyshow thewavepropagation.

Figs.3(a)and(b),extractedfromtheupperwindowsoftheanimations in[29]and[30],respectively,showthegenerationprocessesofthetwo typicalperturbation-inducedwaves.Therangeofthecontactpatchis indicatedbythedashedblackovals,andtheinitialadhesionandslip regionsareroughlydividedbythedashedcurveswithinthecontact patch.TheintervalbetweeneachpairofconsecutivegraphsinFig.3as wellasthetimestepoftheanimationsin[29]and[30]is1μs.Notethat thetimestepsusedinalltheanimationsofthispaperaremorethan10 timeslargerthanthecomputationaltimestepusedinthesimulations. InFig.3,theperturbationoccursatinstantt2andgraduallyspreadsat instantst3∼t6.

Thedivisionoftheadhesionandslipregionscanbedetermined ei-therbythepresenceofmicro-slip,i.e.,micro-slipexistsonlyintheslip region,orbycomparingthewheel-railsurfaceshearstresswiththe trac-tionbound(theproductofthecontactpressureandcoeﬃcientof fric-tion),i.e.,anelementisintheslipregionifitssurfaceshearstressequals thetractionbound,asindicatedinFig.4.Notethatattheinitiation loca-tionsoftheperturbation-inducedwaves,i.e.,closetotheleadingedgeof thecontactpatchandclosetothejunctureoftheadhesion-slipregions, thesurfaceshearstressisclose,butnotequal,tothetractionbound, asindicatedbythetwogreencirclesinFig.4;therefore,thecontact nodes/elementsoriginallyinadhesionattheselocationsaremorelikely toslipthanthoseelsewherewithanincreaseofthesurfaceshearstress oradecreaseofthetractionbound(oradecreaseofthepressurewhen

thecoefficientoffrictionisconstant).Thisphenomenonmayoccur dur-ingthedynamicfrictionalrollingcontactinwhichthecontactstresses varyperiodically[19].AsshowninFig.5anditscorresponding anima-tionin[31],ineachperiod,amovinglocalpeakofthesurfaceshear stress,indicatedbythegreenarrowinFig.5,startsattheleadingedge ofthecontactpatch,movestowardsthetrailingedge,andultimately ex-itstheadhesionregionatthejunctureoftheadhesion-slipregion.This localpeakorincreaseofthesurfaceshearstressappearscapableof caus-ingsuddenslipintheoriginaladhesionareaeitherclosetotheleading edgeofthecontactpatchorclosetothejunctureoftheadhesion-slip regions,actingasaperturbationwithinthewheel-railcontactpatchand thendevelopingintoawave.Becausethevariationinthestress distri-butioniscausedbyvibration[32],theperturbation-inducedwavescan beconsideredtobeintrinsicallygeneratedbydynamiceffectssimilar totheimpact-inducedwaveandthecreepage-inducedwave.However, thedynamiceffectsofthewheel-railfrictionalrollingthattriggerthe perturbation-inducedwavemaybemuchlesssignificantthanthe dy-namiceffectsthatinitiatetheothertwotypesofwaves;therefore,the perturbation-induced waveobservedintheexplicit FEsimulationsis weakerandlastsforashortertime.

The initiation location of the perturbation-induced wave can be inﬂuenced bythe simulatedtraction condition of the rollingwheel.

Figs.6(a)and(b)extractedfromtheanimationsin[33]and[34]show theperturbation-inducedwavessimulatedbyabrakingwheelrolling model andatractive wheel rollingmodel, respectively. Thebraking rolling is simulated by applyingan opposite-direction torque tothe wheelaxle.Thedirectionsofthemicro-slipvectorswithintheslip re-gionsshowninFig.6(a)arethusoppositetothewheelrolling direc-tion.Althoughthetimestepsusedintheanimations[33]and[34](30 μs)arenotsuﬃcientlysmalltocapturetheentireprocessofwave gen-erationandpropagation,theanimationsandFig.6stillindicatethat the perturbation-induced waves generally occur at the leading edge of thecontactpatchinthesimulationofwheel braking,whereas the wavesoccurmoreoftenatthejunctureoftheadhesion-slipregionsin thesimulationofwheeltraction.Thecorrespondencebetweenthe trac-tion/brakingconditionandtheinitiationpositionofthewavemight sup-porttheactualexistenceofaperturbation-inducedwaveduring wheel-railfrictionalrollingcontactbecauseturbulencecausedbynumerical errorsoccursmorerandomly.

Perturbationsoccurregularlyintheanimationsin[33]and[34]with a periodof approximately 0.6ms (20timesteps in theanimations), correspondingwelltotheperiodofthesurfaceshearstressoscillation shown in Fig. 5 and [31]. This result supports the aforementioned ﬁndingthattheperturbationisgeneratedwhenthelocalpeakofthe surfaceshearstressperiodicallypassestheboundaryoftheadhesion regionateithertheleadingedgeorthejunctureoftheadhesion-slip regions. Moreover, Fig. 3shows that theadhesion region decreases graduallywiththespreadoftheperturbation.Theadhesionregioncan beexpectedtovanishundercertaincontactconditions(e.g.,by apply-ingacertaincoeﬃcientoffrictionorwhentheappliedtraction/braking

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Z. Yang and Z. Li International Journal of Mechanical Sciences 161–162 (2019) 105069 Fig. 3. Generation processes of two typical perturbation-inducedwaves(seealsothe ani-mationsin[29]and[30]).

Fig.4. Contactstressdistributionwithinthecontactpatch(thegreencircles indicatetheinitiationlocationsoftheperturbation:closetotheleadingedge andthejunctureoftheadhesion-slipregions).

forceexceedsathresholdvalue),andthestick-slipcontactbehaviour, characterised by sudden, periodic shear stress drops and markedly inﬂuencedbyvibrationandwaves[35],mayconsequentlyoccur.This study,therefore,seemstolinkthegenerationofperturbation-induced

waveswiththewheel-railfriction-inducedinstability– therootcauseof squeal[36]andpossiblyofcorrugation[37].Becausetheinitiationof theperturbationisinﬂuencedbythewheeltraction/brakingcondition asanalysedabove,squealandcorrugationcanpossiblybemitigatedby optimisingthetractionandbrakingcontrolstrategiesbasedonabetter understandingofperturbation-inducedwaves.

3. Physicalcharacteristicsofcreepage-inducedwaves

Thephysicalcharacteristicsofthewheel-railcontact-inducedwave canbeobtainedbyanalysingthenodalmotiononthecontactsurface thatformsthewave.Asafirstattempttoexaminethephysical char-acteristics of wheel-rail contact-inducedwaves, this studylimits the analysisofthewavecharacteristicstocreepage-inducedwavesbecause moredifficultiesarisewhenanalysingthesurfacenodalmotionforming impact-induced andperturbation-induced waves: theimpact-induced wavesshowninFig.1areinterferedwitheitherbythereflectedwave duetotherailjoint(see[38])orbythewavesgeneratedatthe con-tactpatchontherailtop(see[28]),andtheperturbation-inducedwave

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Z. Yang and Z. Li International Journal of Mechanical Sciences 161–162 (2019) 105069

Fig.5. Periodicsurfacestressdistributions(seealsotheanimationin[31].bluecurve:tractionbound;redcurve:surfaceshearstress;A:adhesionregion;S:slip region;thegreenarrowsindicatethepositionofthemovinglocalpeak)[19].

Fig.6. Perturbationinitiationssimulatedunderwheel brak-ing and traction conditions (see also the animations in [33]and[34]).

generatedbythemuchlesspronounceddynamiceﬀectisoftooshorta durationtoanalyse.

3.1. Preliminaryinferenceofthewavetype

Atypicalcreepage-inducedwavepatternis presented inthe con-tour/vectordiagramsinFig.7(seealso[39]forthecorresponding an-imationwithatimestepof0.3μs).Thewaveisobtainedby simulat-ingwheel-railfrictionalrollingcontactwithlargelateralmotionofthe wheel (simulationcase 4in[19]).Thewavephenomenacanbe ob-servedfromthedistributionofthewheel-railrelativevelocitieswithin thecontactpatch (Fig.7(a))andthedistribution of therailsurface nodalvelocitieswithintheentiresolutionzone(Fig.7(b)).Theregions ofthecontactpatchareindicatedbythedashedblackovalsinFig.7.In thecontour/vectordiagrams,themagnitudesanddirectionsofthe nor-mal(relative)velocitiesareindicatedbythecolourdepth,andthoseof thetangential(relative)velocitiesareindicatedbytheredarrows.The creepage-inducedwaveshowninFig.7(a)embodiesthealternationof thecompressionintensiﬁcationandrelaxationwithinthecontactpatch (seealsoFig.2),whereasthewaveinFig.7(b)reﬂectsthevibration velocitiesoftherailsurfacenodes.ThewavelengthsshowninFig.7are approximately6mm.

Assumingthat thetimestepused intheanimationin [39]– 0.3 μs– issuﬃcientlysmalltocapturethephysical characteristicsofthe wave,e.g.,wavespeedandtravellingdirection,andthustoidentifythe wavetype,threelinesofevidencelinkthesimulatedcreepage-induced

wavetoaRayleighwave.First,theobservedwaveis asurfacewave formedbythesimulatedrailsurfacenodalvelocities,ashasbeenshown. Second,boththenormalandtangentialnodalmotionscontributetothe formationofthewave,andthedirectionofthetangentialnodalmotion isroughlyparallelandoppositetothewavepropagationdirection(see

[39]),aswillbeshownindetailinSection3.3.Third,thewavespeed estimatedbydividingthewavetravellingdistanceof1mmineachtime step bythetimestep sizeof0.3μsapproximates theRayleigh wave speedinsteel(approximately3km/s);thisresultwillbefurtherveriﬁed inSection3.3.

3.2. Rayleighsurfacewave

ARayleighwaveisacommonlyknowntypeofsurfacewaveformed by retrograde elliptical particle motion on thesurface, as shown in

Fig.8:thesurfaceparticlesmoveinboththetangential(paralleland oppositetothewavepropagationdirection)andtransverse(normalto thesurface)directions,andthephasediﬀerencebetweenthe tangen-tialmotionandthetransversemotionis𝜋/2.Thetravellingspeedofa Rayleighwaveinsteelisapproximately3km/s.

3.3. Nodalmotionformingcreepage-inducedwaves

Section3.1 infersthatthe simulatedcreepage-inducedwaveis a Rayleighwavebasedontheassumptionthatthetimestepusedinthe animationin[39]issuﬃcientlysmall.Tovalidatethisassumptionand

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Fig.7. Creepage-inducedwavesimulatedbythewheellateralmotionmodel(seealsotheanimationin[39]).

Fig.8. RetrogradeellipticalparticlemotionofaRayleighwave.

theinferencemadeinSection3.1,thissectionanalysesthesimulated nodalmotionontherailsurfacethatformsthecreepage-inducedwave. ThenormalandtangentialvelocitiesofthreerailsurfacenodesN1∼N3, denotedbythesolidbluepointsinFig.7,areanalysed.Thenodesare locatedapproximatelyonthecentrallaterallineofthecontactpatch. N1andN2arewithinthecontactpatch,whileN3isoutsidethe con-tactpatch.Theselectednodeshavethesamelongitudinalcoordinateof 510mmandlateralcoordinatesof0mm,−4mmand−10mm.

Fig.9(a)plotsthetimehistoriesofthesimulatednormaland tangen-tialvelocitiesofthethreeselectednodesduringtheperiodof18.18ms ∼18.27ms(theanimation[39]displaysthewaveinthesameperiod). Thetangentialnodalvelocitieshavetrendcomponentsbecauseofthe wheellateralmotionprescribedinthesimulation.Thepowerspectrum densitiesofthetimehistoriesarecorrespondinglyshowninFig.9(b), whichindicatethatallthenodalvelocitiesarenarrow-bandsignalsand haveadominantfrequencyofapproximately0.495MHz.Thetravelling speedofthesimulatedwavemaythusbecalculatedbymultiplyingthe

wavelength𝜆w=6mmandthefrequencyfw=0.495MHzas:

𝑣𝑤=𝜆𝑤×𝑓𝑤≈ 3km∕s (1)

ThegoodagreementofthewavespeedcalculatedwithEq.(1)and that estimatedinSection3.1indicatesthat thetimestep ofthe ani-mation [39]is sufficientlysmalltoshow themaincharacteristicsof thewave.Thebandpass-filteredtimehistoriesaroundthedominant fre-quency(between0.48and0.51MHz)areplottedinFig.9(c),and close-upviewsareplottedinFig.9(d),whichclearlyshowthatboththe nor-malandtangentialnodalvelocitiescontributetotheformationofthe creepage-inducedwavewithcomparablemagnitudes.Fig.9(e)indicates thatthephasedifferencesbetweenthetangentialandnormalmotionsof N1andN2insidethecontactpatchareapproximately𝜋/2, correspond-ingtoaRayleighwave,whereasthatofN3outsidethecontactpatch isapproximately0.65𝜋,whichispossiblycausedbythesuperposition ofotherwaves.Fig.9(f)showsthatthecoherenceofthetangentialand normalmotionsis0.98forN1andN3andgreaterthan0.8forN2in thefrequencyrangeofinterest.Fig.9(g)plotstheretrogradeelliptical nodalmotionofN1∼N3indisplacement,whichstronglyindicatesthe presenceofaRayleighwave(seealso[40]fortheanimationsof the correspondingretrogradeellipticalnodalmotion).Theellipticaltrails ofeachnodefordifferentcyclesdonotexactlyoverlapduetothe dy-namiceffectsofwheel-railfrictionalrolling.

4. Wavesgeneratedbyacrack

Experimentalattempts[22,23]havesuggestedthatRayleighwaves can be usedtodetectrail cracks.However,their ﬁeldapplicationis stillunderdevelopment.AstheexplicitFEMcaneﬀectivelycapturethe contact-inducedRayleighwave,asanalysedinSection3,thissection presentsexplicitFEmodellingofwheel-raildynamicfrictionalrolling contactforarailtopsurfacewithacracktoinvestigatethecrack ef-fectsonthecontact-inducedwaves.Themodelwasdevelopedfromthe

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Fig. 9. Simulated surfacenodal motion(left, middle andright graphsarefortherailsurfacenodesN1,N2andN3,respectively).

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Fig.10. Wheel-railcontactmodelwithacrackontherailtopsurface.

Fig.11. Contact-inducedwavesinﬂuencedby acrack(seealsotheanimationsin[41]and [42]).

wheel-railcontactmodelwithoutwheellateralmotion(simulationcase 1in [19])byaddinga‘seam’in therailtop,asshowninFig.10(a).

Fig.10(b)showsthesizeandpositionoftheaddedcrack.Contact be-tweenthesurfacesofthecrackisincluded,withastaticcoeﬃcientof frictionof0.35.

Thecontour/vectordiagrams inFigs.11(a)and(b)showthe dis-tributionsoftherailsurfacenodalvelocitieswithintheentiresolution zonewhenthesimulatedwheelapproachesandrollsoverthecrack, re-spectively.Thecontactpatchisindicatedbythedashedblackoval;the crackisdenotedbytheboldblackline.Wavepatternscanbeobservedin

Fig.11,whicharegeneratedatthelocationofthecrackandpropagate radially.Thewavegeneratedwhenthewheelrollsoverthecrack(in

Fig.11(b))ismuchstrongerthanthewavegeneratedwhenthewheel approachesthecrack(inFig.11(a)).See[41]and[42]forthe corre-spondinganimationswithatimestepof1μs.Thecharacteristicsofthe wavesgeneratedbyrailcrackscanbeinvestigatedinfuturestudiesand, togetherwithexperimentalvalidation,beusedforthedevelopmentof wave-baseddetectionofrailsurfacecracks.

5. Conclusionsandfutureresearch

Thispaperpresentedananalysisofthecontact-inducedwaves sim-ulatedbyexplicitFEwheel-rail frictionalrollingcontactmodels. Ac-cordingtothegenerationmechanismsofthewaves,this study cate-gorisedthesimulatedwavesasimpact-induced,creepage-inducedand

perturbation-inducedwaves.Allthreetypesofwavesshouldbe intrinsi-callygeneratedbythedynamiceffectsofthewheel-railfrictionalrolling contact; thedynamic effect triggeringperturbation-induced waves is howevermuchlesssignificantthanthosecausingimpact-inducedand creepage-inducedwaves.

Thisstudyalsodiscussedapossiblelinkbetweenthegenerationof perturbation-inducedwavesandthestick-slipcontactmechanismand foundthattheinitiationlocationoftheperturbation-inducedwavecan beinﬂuencedbythetraction/brakingconditionoftherailwaywheel: the perturbation generallyoccursat theleadingedge of thecontact patchduringwheelbraking,whereasitoccursmoreoftenatthe junc-ture oftheadhesion-slipregionsduringwheeltraction.Thecauseof periodicperturbation-inducedwavesshouldbefurtherstudied.A bet-terunderstandingofperturbation-inducedwavesmaycontributetothe mitigationofrailcorrugationandsquealasconsequencesofstick-slip contactbyenablingoptimisationof thetractionandbraking control strategies.

Thisstudydemonstratedthatthesimulatedcreepage-inducedwave is a Rayleigh waveby comparing thephysical characteristics of the waves.ThereproductionoftheRayleighwaveconfirmedthatthe ex-plicitFEMisaneffectivetoolfortheanalysisofwheel-raildynamic con-tactandtheassociatedwaves.AnexplicitFEwheel-railcontactmodel withacrackintherailtopsurfacewasfinallypresentedtoinvestigate theeffectsofthecrackontherailsurfacewaves.Thegenerationand propagationofrailsurfacewaveswereobservedinsimulationsinwhich

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thewheelapproachedandrolledoverthecrack.Furtherinvestigationof thephysicalcharacteristicsofthecrack-generatedwavesandtheir ex-perimentalvalidationmayfacilitatethedevelopmentofawave-based crackdetectionmethod.

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