Development of a wear model for the prediction of wheel and rail profile evolution in railway systems

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Wear

j o ur n a l ho me p a g e :w w w . e l s e v i e r . c o m / l o c a t e / w e a r

Development

of

a

wear

model

for

the

prediction

of

wheel

and

rail

proﬁle

evolution

in

railway

systems

M. Ignesti

a

_,

_M.

_Malvezzi

b

_,

_L.

_Marini

a

_,

_E.

_Meli

a,∗

_,

_A.

_Rindi

a a_Department_of_Energy_Engineering,_University_of_Florence,_Via_S._Marta_n._3,₅₀₁₃₉_Firenze,_Italy b_Department_of_Information_Engineering,_University_of_Siena,_Via_Roma_n._56,₅₃₁₀₀_Siena,_Italy

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received25May2011 Receivedinrevisedform 22December2011 Accepted3January2012 Available online 24 January 2012 Keywords:

Multibodymodeling Wheel–railcontact Wheel–railwear

a

b

s

t

r

a

c

t

Thepredictionofthewearatthewheel–railinterfaceisafundamentalproblemintherailwayfield, mainlycorrelatedtotheplanningofmaintenanceinterventions,vehiclestabilityandthepossibilityof researchingspecificstrategiesforthewheelandrailprofileoptimization.InthisworktheAuthorspresent amodelspecificallydevelopedfortheevaluationofthewheelandrailprofileevolutionduetowear, whoselayoutismadeupoftwomutuallyinteractivebutseparateunits:avehiclemodelforthedynamic analysisandamodelforthewearestimation.Thefirstoneismadeupoftwopartsthatinteractonline duringthedynamicsimulations:a3DmultibodymodeloftherailwayvehicleimplementedinSimpack Rail(acommercialsoftwarefortheanalysisofmultibodysystems)andaninnovative3Dglobalcontact model(developedbytheAuthorsinpreviousworks)forthedetectionofthecontactpointsbetweenwheel andrailandforthecalculationoftheforcesinthecontactpatches(implementedinC/C++environment). Thewearmodel,implementedintheMatlabenvironment,ismainlybasedonexperimentalrelationships foundinliteraturebetweentheremovedmaterialandtheenergydissipatedbyfrictionatthecontact. Itstartsfromtheoutputsofthedynamicsimulations(positionofcontactpoints,contactforcesand globalcreepages)andcalculatesthepressuresinsidethecontactpatchesthroughalocalcontactmodel (FASTSIMalgorithm);thenthematerialremovedduetowearisevaluatedandthewornprofilesofwheel andrailareobtained.Thisapproachallowstheevaluationofboththequantityofremovedmaterialand itsdistributionalongthewheelandrailprofilesinordertoanalyzethedevelopmentoftheprofilesshape duringtheirlifetime.

Thewholemodelisbasedonadiscreteprocess:eachdiscretestepconsistsinonedynamicsimulation andoneprofileupdatebymeansofthewearmodelwhile,withinthediscretestep,theprofilesare supposedtobeconstant.Thechoiceofanappropriatestepisfundamentalintermsofprecisionand computationalload.Moreoverthedifferenttimescalescharacterizingthewheelandrailwearevolution requirethedevelopmentofasuitablestrategyfortheprofileupdate:thestrategyproposedbytheAuthors isbasedbothonthetotaldistancetraveledbytheconsideredvehicleandonthetotaltonnageburdenon thetrack.TheentiremodelhasbeendevelopedandvalidatedincollaborationwithTrenitaliaS.p.A.and ReteFerroviariaItaliana(RFI),whichhaveprovidedthetechnicaldocumentationandtheexperimental resultsrelatingtosometestsperformedwiththevehicleDMUAln501MinuettoontheAosta-PreSaint Didierline.

1. Introduction

Thewearatthewheel–railinterfaceisanimportantproblem

intherailwayﬁeld.Theevolutionoftheproﬁleshapeduetowear

hasadeepeffectonthevehicledynamicsandonitsrunning

stabil-ity,leadingtoperformancevariationsbothinnegotiatingcurves

and instraight track. Thereforethe originalproﬁles have tobe

∗ Correspondingauthor.

E-mailaddresses:ignesti@mapp1.de.uniﬁ.it(M.Ignesti),malvezzi@dii.unisi.it

(M.Malvezzi),marini@mapp1.de.uniﬁ.it(L.Marini),meli@mapp1.de.uniﬁ.it

(E.Meli),rindi@mapp1.de.uniﬁ.it(A.Rindi).

periodicallyre-establishedbymeansofturning:particularly,from

asafetyviewpoint,thearisingofacontactgeometrywhichmay

compromisethevehiclestabilityorincreasethederailmentriskhas

tobeavoided.Areliablewearmodelcanalsobeusedtooptimize

theoriginalproﬁlesofwheelandrailandtoobtainamoreuniform

wear.Inthiswaytheoverallamountofremovedmaterialcanbe

reducedinordertoincreasethemeantimebetweentwo

mainte-nanceintervalsand,atthesametime,thedynamicalperformance

ofthewheel–railpaircanbekeptapproximatelyconstantbetween

twoturnings.

Becauseofallthesereasons,thedevelopmentofamathematical

modelforthepredictionofthewearatthewheel–railinterface

representsapowerfultool.

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Fig.1.Generalarchitectureofthemodel.

InthisworktheAuthorswillpresentaproceduretoestimatethe

evolutionofthewheelandrailproﬁleduetowearbasedonamodel

thatcombinesmultibodyandwearmodeling.More speciﬁcally,

thegenerallayoutofthemodelconsistsoftwo mutually

inter-activeparts:thevehiclemodel(multibodymodeland3Dglobal

contactmodel)andthewearmodel(localcontactmodelandwear

evaluation and proﬁles update). Concerning thevehicle model,

themultibodymodel,implementedintheSimpackRail

environ-ment,accuratelyreproducesthedynamicsofthevehicle,taking

intoaccountallthesigniﬁcantdegreesoffreedom.The3Dglobal

contactmodel,developedbytheAuthorsinpreviousworks[1,2],

detectsthewheel–railcontactpointsbymeansofaninnovative

algorithmbasedonsuitablesemi-analyticprocedures andthen,

foreachcontactpoint,calculatesthecontactforcesthroughHertz’s

andKalker’stheory[3–5].Thankstothenumericalefﬁciencyofthe

newcontactmodel,thetwomodelsinteractdirectlyonlineduring

thesimulationofthevehicledynamics.

Asregardsthewearestimation,themodelisbasedonalocal

contactmodel(inthiscasetheKalker’sFASTSIMalgorithm)and

onanexperimentalrelationshipforthecalculationoftheremoved

material[6,7].Thewearmodel,startingfromtheoutputsofthe

vehiclemodel(contactpoints,contactforcesandglobalcreepages),

calculatesthetotalamountofremovedmaterialduetowearand

itsdistributionalongthewheelandrailproﬁles.Theremovalofthe

materialiscarriedoutconsideringthethreedimensionalstructure

ofthecontactbodiesandthedifferenttimescalescharacterizing

thewearevolutiononthewheelandontherail.

Oneofthemostcriticalaspectsinthedevelopmentofawear

modelistheavailabilityofexperimentaldataforthevalidationof

themodel,becausethewearisalong-termphenomenonwhich

requiresseveralmonthsofmonitoringtocollectthedata.Ifonline

experimentalmeasurementcannotbecarriedout,theproblemcan

beovercomeusingtoolsprovidedbysoftware[8]orcarryingout

experimentalproofsonascaledtestrig[6].

Inthisworktheentiremodelhasbeenvalidatedbymeansof

theexperimentaldataprovided byTrenitaliaS.p.A.andRFI;the

dataconcerntheAosta-PreSaintDidierrailwaylineandthe

vehi-cleALSTOMDMUAln501Minuettowhich,inthisscenery,exhibits

seriousproblemsintermsofwear.

2. Generalarchitectureofthemodel

Thewholemodelconsistsoftwodifferentparts:thevehicle

modelandthewearmodel.Thegeneralarchitectureisshownin

theblockdiagraminFig.1.

Thevehiclemodelrepresentsthedynamicanalysisblockandis

composedbythemultibodymodelof thestudiedrailway

vehi-cle(inthisworktheALSTOMDMUAln501Minuetto)andthe3D

globalcontactmodelthat,duringthedynamicsimulation,interact

directlyonlinecreatingaloop.Ateachtimeintegrationstepthe

ﬁrstoneevaluatesthekinematicvariables (position,orientation

andtheirderivatives)relativetothewheelsetandconsequentlyto

eachwheel-railcontactpair.Startingfromthesequantities,the

secondonecalculatestheglobalcontactvariables(contactpoints

and contactforces, contactareasand globalcreepages).The 3D

globalcontactmodelisbasedbothonaninnovativealgorithmfor

thedetectionofthecontactpoints(developedbytheAuthorsin

previousworks[1,2])andonHertz’sandKalker’sglobaltheories

fortheevaluationofthecontactforces[3].Theglobalcontact

vari-ablesarethenpassedtothemultibodymodelinordertocarryon

thesimulationofthevehicledynamics.

The main inputs of the dynamic analysis block are the

rail-waytrackandthemultibodymodeloftheconsideredvehicle:in

thisresearchactivity,accordingtothespeciﬁcationsrequiredby

Trenitalia,thetrackisrepresentedbyastatisticalanalysisofthe

Aosta-PreSaintDidierlinebymeansofanensembleofNc

curvi-lineartracks,eachoflengthequaltoltrack.Thestatisticalanalysis

hasbeencarriedoutsplittingtheconsideredtrackinradiusclasses

andsuperelevationclasses.Thestatisticalapproachhasbeen

intro-ducedbecauseofthecomplexityandthelengthoftheAosta-Pre

SaintDidiertrack:forthesereasonsthesimulationoftheentireline

wouldhavecauseddifﬁcultybothintermsofmultibodymodeling

andintermsofnumericalefﬁciency(computationalloadand

mem-oryconsumption).Theoutputsofthevehiclemodelaretheglobal

contactvariablesevaluatedduringalltheNcdynamicsimulations

andrepresenttheinputsofthewearmodel.

ThedynamicsimulationshavebeenperformedinSimpackRail.

Morespeciﬁcally,themultibodymodelhasbeenbuiltusingdirectly

the SimpackRail environmentwhile the globalcontact model,

implementedinC/C++,hasbeencustomizedbymeansofthe

Sim-packUserRoutinemodule(implementedinFORTRANenvironment)

thatallowstohandletheinteractionbetweenSimpackandroutines

deﬁnedbytheuser.Thewearmodelismadeupofthreedistinct

phases:thelocalcontactmodel,thewearevaluationandtheproﬁle

update.Initiallythelocalcontactmodel(basedbothonHertz’slocal

theoryandonsimpliﬁedKalker’salgorithmFASTSIM),startingfrom

theglobalcontactvariables,evaluatesthelocalcontactvariables

(contactpressuresand localcreepages) and dividesthecontact

patchintoadhesionareaandcreeparea.Then,thedistributionof

removedmaterial(hypothesizingthecontactindryconditionsas

requiredbyTrenitaliaandRFI)iscalculatedbothonthewheeland

ontherailsurfaceonlywithinthecreepareausingan

experimen-tallawbetweentheremovalmaterialandtheenergydissipatedby

frictionatthecontactinterface[6,7].Finallytheproﬁlesofwheel

andrailareupdated:thenewproﬁlesaretheoutputsofone

dis-cretestepofthewholemodelloopandhavetobepassedbackto

thevehiclemodelinordertocontinuethewearcycledescribedin

Fig.1andtosimulatethevehicledynamicswithupdatedproﬁles.

Theevolutionofthewheelandrailproﬁlesisthereforeadiscrete

process.Inthisresearchthechoiceofthediscretestepsisoneof

themainissuesandhastoconsiderthedifferenceoftimescales

betweenthewheelandrailwearevolutionrate(aswillbeclariﬁed

inthefollowing).Forthewheelwearthefollowingconsiderations

arevalid:

1. thetotalmileagekmtottraveledbytheconsideredvehiclehas

beensubdividedinconstantstepsoflengthequaltokmstep;

2.withineachdiscretestepofthewholemodel(correspondingto

kmstepkilometerstraveledbythevehicle)thewheelproﬁleis

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Table1

Inertiapropertiesofthemultibodymodel.

MBSbody Mass(kg) Rollinertia

(kgm2₎ Pitchinertia (kgm2₎ Yawinertia (kgm2₎ CoachM 31,568 66,700 764,000 743,000 CoachT 14,496 30,600 245,000 236,000 Bogiem 3306 1578 2772 4200 Bogiet 3122 1674 3453 5011 Wheelsetm 2091 1073 120 1073 Wheelsett 1462 1027 120 1027

Thedepthoftherailweardoesnotdependonthedistance trav-eledbyvehiclebutonthenumberofvehiclesmovingonthetrack. Thereforeadifferentapproachforevaluatingthediscretestepfor therail,basedonthetotaltonnageburdenonthetrackMtot,is

needed:

1.dividingthetotaltonnageMtotbythevehiclemassMv,the

cor-respondingvehiclenumberNtothasbeencalculated;thenNtot

hasbeensubdividedinconstantstepsequaltoNstep;

2.withineachdiscretestep(correspondingtoNstepvehicles

mov-ingonthetrack)therailproﬁleissupposedtobeconstant. Finallythefollowingconsiderationsholdbothforthewheeland theraildiscretizationsteps:

• thenumberofdiscretizationstepsaffectsthemodelprecisionand thecomputationalload.Moreparticularly,increasingthestep number,themodelprecisionincreasesbut,atthesametime,the computationalloadincreasestoo:agoodcompromisemustbe researched;

• varioustypesofproﬁleupdatestrategiesareavailablein litera-ture[9,10]:theconstantstepandtheadaptivestepstrategiesare

themainones.Intheﬁrstoneaconstantupdatestepisdeﬁned,

whilethesecondoneisbasedonthedeﬁnitionofathreshold

valuethatimposesthemaximummaterialquantitytoremoveat

eachupdateoftheproﬁles.Thetwomethodshavebeencompared

andtheﬁrstonehasbeenchosenduetothefollowingreasons:

1.thephysicalphenomenonofthewearevolutionhasusuallyan

almostlinearcharacteristicandthusiswellsuitedtoaconstant

updatestep;

2.thetwomethodsleadtoverysimilarsystemevolutions(both

qualitativelyandquantitatively)buttheﬁrstoneis

computa-tionallymoreefﬁcient.

ThewearmodelhasbeenfullyimplementedintheMatlab

envi-ronment.

3. Thevehiclemodel

Inthissectionabriefdescriptionofthevehiclemodel

(com-posedbythemultibodymodelandtheglobalcontactmodel)is

given. In particular the global contact model willallow to the

detectionofthecontactpointsbetweenthewheelandrailand,

subsequently,ofthecontactforcesandtheglobalcreepagesinthe

contactpatch.

3.1. Themultibodymodel

TheDMUAln501Minuetto,apassengertransportunitwidely

usedontheItalianRailways,hasbeenchosenasbenchmark

vehi-cleforthisresearch;thephysicalandgeometricalcharacteristicsof

thevehiclecanbefoundinliterature[11,12].InTable1theinertia

propertiesofthevehicleareshown:motorsandgearboxeshave

notbeenmodeledandtheirinertiapropertieshavebeenincluded

inthemotorbogieandinthemotorwheelset(indicatedinTable1

Fig.2.Globalviewofthemultibodymodel.

withBogiemandWheelsetmrespectively)inordertotakeinto

accounttheirdifferentinﬂuenceontheunsprungandsprungmass.

ThemultibodymodelhasbeenrealizedintheSimpackRail

envi-ronment(seeFig.2)andconsistsofthirty-onerigidbodies:

• threecoaches;

• fourbogies:theintermediateones,interposedbetweentwo

suc-cessivecoaches,aretrailerbogieswhiletheotheronesaremotor

bogies;

• eightwheelsets:twoforeachbogie;

• sixteenaxleboxes:twoforeachwheelset.

Therigidbodiesareconnectedbymeansofappropriate

elas-tic and damping elements;particularly thevehicleis equipped

withtwosuspensionstages.Theprimarysuspensionsconnectthe

wheelsetstothebogies(seeFig.3)andcomprisetwospringsand

two verticaldampers,whilethesecondarysuspensionsconnect

thebogiestothecoaches(seeFig.4)andcomprisethefollowing

elements:

• twoairsprings;

• sixdampers(lateral,verticalandanti-yawdampers);

• onetractionrod;

• therollbar(notvisibleintheﬁgure);

• twolateralbumpstops.

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Table2

Mainlinearstiffnesspropertiesofthesuspensions.

MBSelement Longitudinalstiffness

(N/m) Lateralstiffness (N/m) Verticalstiffness (N/m) Rollstiffness (Nm/rad) Pitchstiffness (Nm/rad) Yawstiffness (Nm/rad) Primary 1,259,600 1,259,600 901,100 10,800 10,800 1000 Suspension Spring Secondary 120,000 120,000 398,000 – – – Suspension Airspring Secondary – – 2,600,000 – – – Suspension Rollbar

Fig.4.Bogieandsecondarysuspensions.

Boththestagesofsuspensionshavebeenmodelledbymeansof three-dimensionalviscoelasticforceelementstakingintoaccount allthemechanicalnonlinearities(bumpstopclearance,dampers androdbehavior).Themainlinearcharacteristicsofthe suspen-sionsareshowninTable2whilethenonlinearcharacteristicsare

imposedasafunctionofdisplacementandvelocityforthesprings

andthedampersrespectively(seeFigs.5and6).

Fig.5.Exampleofnonlinearcharacteristic:verticaldampingoftheprimary sus-pension.

3.2. Theglobalcontactmodel

Dynamicsimulationsofrailwayvehiclesneedareliableand

efﬁ-cientmethodtoevaluatethecontactpointsbetweenwheeland

rail,becausetheirpositionhasa considerableinﬂuencebothon

thedirectionandonthemagnitudeofthecontactforces.Inthis

workaspeciﬁccontactmodelhasbeenconsideredinsteadofthat

implementedinSimpackRailinordertoachievebetterreliability

andaccuracy[1,2].Theproposedcontactmodelisdividedintwo

parts:intheﬁrstonethecontactpointsaredetectedbymeansofan

innovativealgorithmdevelopedbytheAuthorsinpreviousworks

[1,2],whileinthesecondonetheglobalcontactforcesactingatthe

wheel–railinterfaceareevaluatedbymeansofHertz’sandKalker’s

globaltheories[3–5].

Thealgorithmforthecontactpointsdetectionstartsfromthe

standardideathatthecontactpointsmakestationarythedistance

betweenthewheelandrailsurfaces(seeFig.7(a));inmoredetails

thedistancehasapointofrelativeminimumifthereisno

pen-etrationbetweentheconsideredsurfaces,whileithasarelative

maximumintheothercase(seeSection3.2.1fortheanalytic

for-mulation of the problem).The main features ofthe innovative

adoptedalgorithmarethefollowing:

• itisafully3Dmodelthattakesintoaccountallthesixrelative

degreesoffreedom(DOF)betweenwheelandrail;

• itisabletosupportgenericrailwaytracksandgenericwheeland

railproﬁles;

• itassuresageneralandaccuratetreatmentofthemultiple

con-tactwithoutintroducingsimplifyingassumptionsontheproblem

geometryandlimitsonthenumberofcontactpointsdetected;

Fig.6. Exampleofnonlinearcharacteristic:lateralbumpstopstiffnessofthe sec-ondarysuspension.

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Fig.7.Distancemethod.

• itassureshighlynumericalefﬁciencymakingpossibletheonline

implementation within the commercial multibody software

(Simpack-Rail,Adams-Rail);inthiswayalsothenumerical

per-formanceofthecommercialmultibodysoftwareareimproved.

Twospeciﬁcreferencesystemshavetobeintroducedinorderto

simplifythemodel’sequations:theauxiliaryreferencesystemand

thelocalreferencesystem.TheauxiliarysystemOrxryrzrisdeﬁned

ontheplaneoftherailsandfollowsthemotionofthewheelset

duringthedynamicsimulations:thexraxisistangenttothecenter

lineofthetrackintheoriginOr,whosepositionisdeﬁnedsothat

theyrzrplanecontainsthecenterofmassGwofthewheelset,and

thezraxisisperpendiculartoplaneoftherails.Thelocalsystem

Owxwywzwisrigidlyconnectedtothewheelsetexceptforthe

rota-tionarounditsaxisandthexwaxisisparalleltothexryrplane(see

Fig.7(b)).Inthefollowing,forthesakeofsimplicity,thevariables

referredtothelocalsystemwillbemarkedwiththeapexw,while

thosereferredtotheauxiliarysystemwiththeapexr;the

vari-ablesbelongingtothewheelandtotherailwillbeindicatedwith

thesubscriptswandrrespectively.

Thankstothesereferencesystems,thedeﬁnitionofthewheel

andrailgeometricalsurfacesis easy.Inthelocalreference

sys-temthewheelsetcanbeconsideredasasimplerevolutionsurface

arounditsaxisyw(seeEq.(2));thegeneratingfunction,indicated

byw(yw),issupposedtobeknown(inthiswork,theproﬁleORES

1002forasinglewheelhasbeenchosen(seeFig.8)).Similarlythe

trackcanbelocallydescribedintheauxiliaryreferencesystemas

anestrusionsurfacealongthexraxis(seeEq.(3));thegenerating

function,indicatedbyr(yr),isknown(theproﬁleUIC60,with

lay-ingangle˛pequalto1/20rad,hasbeenchosenforasinglerail(see

Fig.8)).

WithreferencetoFig.1,theglobalcontactmodelcanbethought

ofasablackboxhavingthefollowinginputsandoutputs:

• Inputs: the kinematic variables evaluated by the multibody

model,i.e.thepositionOrw,theorientationmatrixRrw,theabsolute

velocity ˙Orwandtheabsoluteangularvelocityωrwofthewheel

sys-temandtheanalogousquantitiesoftherailsystemOr_r₌0,Rr

r=I,

˙Orandωrr;

• Outputs:theglobalcontactvariablesrelativetothewheel–rail

interface,likethepositionsPrwandPrrofthecontactpoints,the

contactforces(thenormalcomponentNr_and_the_tangential

com-ponentsTr

x andTyr),theglobalcreepagesεx,εyandεspandthe

contactpatch’sdimensionsa,b.

3.2.1. Thedistancemethodalgorithm

In this subsection thealgorithmusedfor detectingthe

con-tactpointswillbedescribed.Thedistancemethodalgorithm(see

Fig.7(a))isbasedonaclassicalformulationofthecontactproblem

inmultibodyﬁeld:

nr_r(Pr_r)∧nr_w(Pr_w)=n_rr(Pr_r)∧Rr_wnw_w(Pw_w)=0 (1a)

nr_r(Pr_r)∧dr=0 (1b)

where:

• Pw

wandPrrarethepositionsofthegenericpointonthewheel

sur-faceandontherailsurfaceexpressedintheirreferencesystems:

Pww(xw,yw)=

xw yw −

w(yw)2−x2w

T (2) Pr_r(xr,yr)=(xr yr r(yr))T (3) • nw

wandnrraretheoutgoingnormalunitvectorstothewheeland

railsurfacerespectivelyandaredeﬁnedasfollows:

nw w(Pww)= −

∂

Pww

∂

xw ∧

∂

Pw_w

∂

yw

∂

Pww

∂

xw ∧

∂

Pww

∂

yw

, n r r(Prr)=

∂

Pr_r

∂

xr ∧

∂

Pr_r

∂

yr

∂

Prr

∂

xr ∧

∂

Prr

∂

yr

(4) • Rr

wistherotationmatrixthatlinksthelocalreferencesystemto

theauxiliaryone;

• dr_is_the_distance_vector_between_two_generic_points_on_the_wheel

surface andontherailsurface (bothreferred totheauxiliary

system)anditisequalto:

dr(xw,yw,xr,yr)=Pwr(xw,yw)−Prr(xr,yr) (5)

wherePr_wisthepositionofthegenericpointofthewheelsurface

expressedintheauxiliarysystem:

Prw(xw,yw)=Orw+RrwPww(xw,yw). (6)

Theﬁrstcondition(Eq.(1a))ofthesystem(1)imposesthe

paral-lelismbetweenthenormalunitvectors,whilethesecondone(Eq.

(1b))requirestheparallelismbetweenthenormalunitvectorto

therailsurfaceandthedistancevector.

Alternatively,otherclassicalformulationsofthecontact

prob-lemcanbeused,byexampleimposingtheorthogonalitybetween

thedr_vector_and_the_tangential_planes_to_the_wheel_and_rail_surfaces

(respectivelyw andr);howeverthis approachleadstomore

complexandlessmanageablecalculationsandforthisreasonhas

notbeadopted.

The system (1) consists of six nonlinear equations in the

unknowns(xw,yw,xr,yr)(onlyfourequationsareindependentand

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Fig.8. Wheelandrailproﬁles.

ofthefourvariables(inthiscase(xw,xr,yr))asafunctionofyw,

reducingtheoriginal4Dproblemtoasimple1Dscalarequation.

Thereductionoftheproblemdimensionusingappropriate

ana-lyticalproceduresisthemostinnovativeaspectoftheproposed

method.ThesecondcomponentofEq.(1a)leadstothefollowing

equation:

r13

w(yw)2−xw2 =r11xw−r12w(yw)w(yw) (7)

where rij are the elements of the Rrw matrix. Calling A=r13,

B=w(yw),C=r11andD=r12w(yw)w(yw),thepreviousequation

becomes:

A

B2₋_x2

w=Cxw−D. (8)

Hence,removingtheradicalandsolvingforxw:

xw1,2(yw)=

CD±

C2_D2₋_(A2₊_C2_)(D2₋_A2_B2₎

A2₊_C2 ; (9)

ascanbeseen,therearetwopossiblevaluesofxwforeachyw.

FromtheﬁrstcomponentofEq.(1a)thefollowingrelationfor

r(yr)holds: r(yr)1,2 = r21xr1,2(yw)−r22w(yw)w(yw)−r23

w(yw)2−xw1,2(yw)2 r32w(yw)w(yw)+r33

w(yw)2−xw1,2(yw)2 . (10)

Ifr(yr)1,2isadecreasingmonotonousfunction(considering

sep-aratelythesidesofthetrack),Eq.(10)isnumericallyinvertible

andasinglepairyr 1,2(yw)existsforeachywvalue;otherwisethe

numericalinversionisstillpossiblebutwillproduceafurther

mul-tiplicationofthesolutionnumber.

BythesecondcomponentofEq.(1b)theexpressionofxr1,2(yw)

canbeobtained:

xr1,2(yw)=r11xw1,2(yw)+r12yw−r13

w(yw)2−xw1,2(yw)2. (11)

Finally,replacingthevariablesxw1,2(yw),xr1,2(yw)andyr1,2(yw)in

theﬁrstcomponentofEq.(1b),thefollowing1Dscalarequation

canbewritten: F1,2(yw) = −r

Gmz+r32yw−r33

w2₋_x w1,22−b

+ −

Gmy+r21xw1,2+r22yw−r23

w2₋_x w1,22 −yr1,2

=0 (12)

whereGwx,Gwy,Gwzarethecoordinatesofthewheelsetcenterof

massintheauxiliarysystem.Theexpression(12)consistsoftwo

scalarequationsinthevariableywthatcanbeeasilysolvedwith

appropriatenumericalalgorithms.

Theadvantagesofthisapproachbasedonthereductionofthe

algebraicproblemdimensionaremanyandcanbesummarizedas

follows:

• thereductionofthealgebraicproblemdimensionfrom4Dto1D

allowstoobtainanhighnumericalefﬁciencythatmakes

pos-sibletheonlineimplementationofthenewmethodwithinthe

multibodyvehiclemodels;

• theanalyticalapproachassuresanhighdegreeofaccuracyand

generality;

• the1Dproblemassuresaneasiermanagementofthemultiple

solutionsfromanalgebraicandanumericalpointofview;

• in1Dproblemalsoparticularlyelementarynumericalalgorithms

likethegridmethodarequiteefﬁcient.

Thus, onceobtainedthegenericsolution(indicatedwiththe

subscript i) ywi of Eq. (12),the complete solution (xwi,ywi, xri,

yri)ofthesystem(1)andconsequentlythecontactpointsPrwi=

Prw(xwi,ywi)andPrri=Prr(xri,yri)canbefoundbysubstitution.

However,sinceEqs.(1a)and(1b)containirrationalterms,the

genericsolution(xwi,ywi,xri,yri)mustsatisfythefollowing

analyt-icalconditions:

• thesolutionmustbereal;

• thesolutiondoesnothavetogeneratecomplexterms(thatcould

becausedbytheradicalsintheequations);

• thesolutionmustbeaneffectivesolutionofthesystem(1)(check

necessarybecauseoftheradicalremovalbysquaring).

Furthermore,fromaphysicalpointofview,alsothenextchecks

(8)

Fig.9. Convexityconditions.

• themultiplesolutionsobtainedfromtheanalyticalresolutionof

Eq.(12)mustbeindividuatedanderasedbecausetheyhaveno

physicalmeaning;

• thefollowingconvexityconditionsmustbesatisﬁedsothatthe

contactisphysicallypossible:

k1,wi+k1,ri>0

k2,wi+k2,ri>0

(13)

wherek1,wi,k2,wiarethenormalcurvatureofthewheelsurfacein

longitudinalandlateraldirection(referredtotheauxiliarysystem

andevaluatedintheithcontactpoint(xwi,ywi,xri,yri))whilek1,ri,

k2,riaretheanalogousquantitiesfortherailsurface.Becauseof

theproblemgeometry,theﬁrstoneofEq.(13)isalwayssatisﬁed

andthusonlythesecondonemustbeveriﬁed(seeFig.9).

• thegenericsolutionofthesystem(1)canbeaneffectivecontact

pointonlyifthenormalpenetration ˜pnbetweenthesurfacesof

wheelandrailisnegative(accordingtotheadoptedconvention),

i.e.theremustbeeffectivepenetrationbetweenthebodies:

˜pni=dri·nrr(Prri)=−d

r

i·nrw(Prwi)<0. (14)

3.2.2. Thecontactforces

Then,foreachcontactpoint,thenormalandtangentialcontact

forcesandtheglobalcreepagesonthecontactpatcharedetermined

(seeFig.10).

ThenormalforcesNr_(expressed_in_the_auxiliary_system)_are

calculatedbymeansofHertz’stheory[5,13]:

Nr=

−kh

˜pn

+kv|

v

n| sign (

v

n)−1 2

_sign(˜p n)−1 2 (15) where:

• ˜pnisthenormalpenetrationdeﬁnedbyEq.(14);

• istheHertz’sexponentequalto3/2;

• kvisthecontactdampingconstant(kv= 105Ns/m);

•

v

n=V·nrristhenormalpenetrationvelocity(Visthevelocityof

thecontactpointrigidlyconnectedtothewheelset);

Fig.10. Globalforcesactingatwheelandrailinterface.

• khisthehertzianconstant,functionbothofthematerial

prop-ertiesandofthegeometryofthecontactbodies(curvaturesand

semiaxesofthecontactpatch)[14,2].

Theglobalcreepagesε(longitudinalεx,lateralεyandspin

creep-ageεsp)arecalculatedasfollows:

εx= V· ir

˙Or_w

, εy= V· tr r

Prr

˙Or_w

, εsp= ωr w· nrr

Prr

˙Or_w

(16)

whereVisthevelocityofcontactpointrigidlyconnectedtothe

wheelset, ˙Orwisthewheelsetcenterofmassvelocity(takenasthe

referencevelocityforthecalculationoftheglobalcreepages),ωr

w

istheangularvelocityofwheelsetexpressedinauxiliarysystem,ir

istheunitvectorinlongitudinaldirectionoftheauxiliarysystem

andtrristhetangentialunitvectortotherailproﬁle.

The tangentialcontact forces ˜Tr

x, ˜Tyr and thespin torqueMspr

(expressedintheauxiliarysystem)arecalculatedbymeansofthe

Kalker’sglobaltheory:

˜

Tr

x=−f11εx, T˜yr=−f22εy−f23εsp

(17)

M_spr =f23εy−f33εsp (18)

wherethecoefﬁcientsfijarefunctionbothofthematerialsandof

thesemiaxesofthecontactpatch:

f11=abGC11, f22=abGC22

f23=(ab)3/2GC23, f33=(ab)2GC33 (19)

inwhichGisthewheelandrailcombinedshearmodulusandCijare

theKalker’scoefﬁcientsthatcanbefoundtabulatedinliterature

[3].Atthispoint,itisnecessarytointroduceasaturationonthe

tangentialcontactforces ˜Tr=[ ˜Tr

x T˜yr]T,inordertoconsiderthe

adhesionlimit(nottakeninaccountbythelinearKalker’stheory):

Tr

≤cNr (20)

wherecisthekineticfrictioncoefﬁcient.Consequentlythe

satu-ratedtangentialforcesTr_will_have_the_following_expression:

Tr=

˜Tr (21)

inwhichthesaturationcoefﬁcient

canbeevaluatedasfollows

[15,16]: =

⎧

⎨

⎩

cNr ˜ Tr

_˜ Tr cNr

−1 3

_˜ Tr cNr

2 + 1 27

_˜ Tr cNr

3

if ˜Tr_≤₃_c_Nr cNr ˜ Tr if ˜T r_>₃ cNr (22) where ˜Tr=

˜T r

.

4. Thewearmodel

Inthissectionthethreephases,inwhichthewearmodelhas

beendivided,willbedescribedindetails:thelocalcontactmodel,

theevaluationoftheamountofremovedmaterial(assumingdry

contactconditions)andthewheelandrailproﬁleupdate.

4.1. Thelocalcontactmodel

Thepurposeofthelocalcontactmodelisthecalculationofthe

localcontactvariables(normalandtangentialcontactstressespn,

ptandlocalcreeps,allevaluatedwithinthecontactpatch)starting

fromthecorrespondingglobalvariables(contactpointsPr_w,Pr_r,

con-tactforcesNr_,_Tr

x,Tyr,globalcreepageεandsemiaxesofthecontact

(9)

Fig.11.Contactpatchdiscretization.

ThismodelisbasedontheKalker’slocaltheoryinthe

simpli-ﬁedversionimplementedinthealgorithmFASTSIM;thisalgorithm

contains an extremely efﬁcient version (although necessarily

approximate)oftheKalkertheoryandthereforeiswidelyusedin

railwayﬁeld[17].

Forthelocalanalysisanewreferencesystemisdeﬁnedatthe

wheel–railinterfaceonthecontactplane(i.e.thecommontangent

planebetweenthewheelandrailsurfaces):thexandyaxesare

thelongitudinalandthetransversaldirectionofthecontactplane

respectively(seeFigs.11and13).Thealgorithmisbasedonthe

pro-portionalityhypothesisbetweenthetangentialcontactpressurept

andtheelasticdisplacementsu,bothevaluatedwithinthecontact

patch:

u(x,y)=Lpt(x,y), L=L(ε,a,b,G,) (23)

wheretheﬂexibilityL(functionoftheglobalcreepagesε,the

semi-axesofthecontactpatcha,b,thewheelandrailcombinedshear

modulusGandthewheelandrailcombinedPoisson’sratio)can

becalculatedasfollows:

L=|εx| L1+

εy

L2+c

εsp

L3

(ε2

x+ε2y+c2ε2sp)

1/2 (24)

withL1=8a/(3GC11),L2=8a/(3GC22),L3=a2/(4GcC23)andc=

√ ab

(theconstantsCij,functionsbothofthePoisson’sratioandofthe

ratioa/b,aretheKalker’sparametersandcanbefoundinliterature

[17]).

Thelocalcreepagesscanbecalculatedbyderivationconsidering

boththeelasticcreepagesandtherigidones:

s(x,y)=u(x,• y)+V

εx εy

(25)

whereV=

˙Or_w

isthelongitudinalvehiclespeed.Atthispoint

itisnecessarytodiscretizetheellipticalcontactpatchinagridof

pointsinwhichthequantitiespn,ptandswillbeevaluated.Initially

thetransversalaxis(withrespecttothemotiondirection)ofthe

contactellipsehasbeendividedinny−1equalpartsofmagnitude

y=2b/(ny−1)bymeansofnyequidistantnodes.Thenthe

longitu-dinalsectionsofthepatch(long2a(y)= 2a

1−y/b2)havebeen

dividedinnx−1equalpartsofmagnitude x(y)=2a(y)/(nx−1)by

meansofnxequidistantnodes(seeFig.11).Duetothisstrategythe

longitudinalgridresolutionisnotconstantbutincreasesnearthe

lateraledgesoftheellipse,wherethelengthsa(y)aresmaller.This

procedureprovidesmoreaccurateresultsrightnexttotheedgesof

theellipse,whereaconstantresolutiongridwouldgenerate

exces-sivenumericalnoise.Thevaluesofthenxandnyparametershave

toassuretherightbalancebetweenprecisionandcomputational

load;goodvaluesofcompromiseareintherange[2550].

Oncethecontactpatchisdiscretized,theFASTSIMalgorithm

allowstheiterativeevaluationofboththecontactpressuresvalue

pn,ptandthelocalcreepagesinordertodividethecontactpatch

inadhesionandslipzone.Indicatingthegenericpointofthegrid

with(xi,yj),1≤i≤nx1≤j≤ny,thenormalcontactpressurecanbe

expressedas: pn(xi,yj)= 3 2 Nr ab

1−x 2 i a2− y2 j b2 (26)

whereNr_is_the_normal_contact_force,_while_the_limit_adhesion

pres-surepAis: pA(xi,yj)=pt(xi−1,yj)−

εx εy

x(yj) L ; (27)

thus,knowingthevariablevaluesinthepoint(x_i−1,yj),itispossible

topasstothepoint(xi,yj)asfollows:

if

pA(xi,yj)

≤pn(xi,yj) arrow

pt(xi,yj)=pA(xi,yj) s(xi,yj)=0 (28a) if

pA(xi,yj)

>pn(xi,yj) arrow

pt(xi,yj)=pn(xi,yj)pA(xi,yj)/pA(xi,yj) s(xi,yj)= LV x(yj) (pt(xi,yj)−pA(xi,yj)) (28b)

whereisthestaticfrictioncoefﬁcient;Eqs.(28a)and(28b)hold

respectivelyintheadhesionandslipzone.Iteratingtheprocedure

for2≤i≤nxandsuccessivelyfor1≤j≤nyandassumingas

bound-aryconditionspt(x1,yj)=0,s(x1,yj)=0for1≤j≤ny(i.e.stressesand

creepageszerooutofthecontactpatch),thedesireddistributionof

pn(xi,yj),pt(xi,yj)ands(xi,yj)canbedetermined.

4.2. Thewearevaluation

Toevaluatethedistributionofremovedmaterialonwheeland

railduetowear(assumingdrycontactconditions)an

experimen-talrelationshipbetweenthevolumeofremovedmaterialandthe

frictionalwork[6,7]hasbeenused.Particularlytherelationshipis

abletodirectlyevaluatethespeciﬁcvolumeofremovedmaterial

ı

P_wijk(t)(x,y)andıP_rijk(t)(x,y)relatedtotheithcontactpointsP

jk wi(t)

andP_rijk(t)onthejthwheelandrailpairduringthekthoftheNc

dynamicsimulations.

Thecalculationofı

Pjk_i(t)(x,y)requiresﬁrstofalltheevaluation

ofthefrictionpowerdevelopedbythetangentialcontactstresses;

tothispurposethewearindexIW(expressedinN/mm2)isdeﬁned

asfollows:

IW=

p_t·s

V . (29)

Thisindex, bymeansof appropriateexperimentaltests, canbe

correlated with the wear rate KW (expressed in ␮g/(mmm2))

which representsthemassof removedmaterialforunit of

dis-tance traveled by the vehicle (expressed in m) and for unit

of surface (expressed in mm2_). _Wear _tests _carried _out _in _the

caseofmetal–metalcontactwithdrysurfacesusingatwindisc

(10)

Fig.12.TrendofthewearrateKW.

relationship between KW and IW adopted for the wear model

describedinthisworkisthefollowing(seeFig.12):

KW(IW)=

5.3∗IW IW<10.4 55.0 10.4≤IW≤77.2 61.9∗IW IW>77.2. (30)

OncethewearrateKW(IW)isknown(the samebothfor the

wheelandfortherail),thespeciﬁcvolumeofremovedmaterialon

thewheelandontherail(forunitofdistancetraveledbythe

vehi-cleandforunitofsurface)canbecalculatedasfollows(expressed

inmm3_/(m_mm2_)): ı P_wijk(t)(x,y)=KW(IW) 1 (31) ı P_rijk(t)(x,y)=KW(IW) 1 (32)

whereisthematerialdensity(expressedinkg/m3_).

4.3. Proﬁleupdate

Theproﬁleupdatestrategyisthesetofnumericalprocedures

thatallowsthecalculationofthenewproﬁles ofwheelwn(yw)

andrailrn(yr)(theproﬁlesatthenextstep),startingfromtheold

proﬁlesofwheelwo(yw)andrailro(yr)(i.e.theproﬁlesatthe

cur-rentstep)andallthedistributionsofremovedmaterialı

P_wijk(t)(x,y)

andı

P_rijk(t)(x,y).Theupdatestrategy,besidesevaluatingthenew

proﬁles,isnecessaryfortwoadditionalreasons:

1.thenecessitytoremovethenumericalnoisethatcharacterizes

thedistributionsı

Pjk_i(t)(x,y)andthat,duetononphysical

alter-ationsof thenew proﬁles,can causeproblems totheglobal

contactmodel;

2.theneed to mediatethe distributions ı

P_ijk(t)(x,y) in order to

obtainasingleproﬁlebothforthewheelandtherailasoutput

ofthewearmodel(asrequiredbythespeciﬁcationsofTrenitalia

andRFI).

Thefollowingmainstepscanbedistinguished:

• Longitudinalintegration: 1 2w(yjk_wi)

+a(y) −a(y) ı_Pjk wi(t) (x,y)dx=ıtot Pjk_wi(t)(y) (33) 1 ltrack

_+a(y) −a(y) ı P_rijk(t)(x,y)dx=ı tot P_rijk(t)(y) (34)

wherew(y_wijk)isthewheelradiusevaluatedinyjk_wiandltrackisthe

lengthofthesimulatedtrack.Thisﬁrstintegrationsums,inthe

longitudinaldirection,allthewearcontributesinsidethe

con-tactpathandaveragesthisquantityoverthewholelongitudinal

Fig.13. Normalabscissaforthewheelandrailproﬁle.

developmentofthewheelandoftherail(bymeansofthe

fac-tors1/2w(yjk_wi)and1/ltrack);inotherwordsitprovidesthemean

value ofremoved material(expressed inmm3_/(m_mm2_)). _The

differencebetweentheterms1/ltrackand1/2w(yjk_wi)(thetrack

lengthismuchgreaterthanthewheelcircumferencelength)is

themaincausethatleadsthewheeltowearmuchfasterthanthe

railandconsequentlytoadifferentscaleofmagnitudeofthetwo

investigatedphenomena.Thisreﬂectsthephysicalphenomena

thatthelifeoftherailismuchgreaterthanthatofthewheel.

Forthisreason,aswillbebetterexplainedinthefollowing,it

isnecessarytodevelopadifferentstrategyfortheupdateofthe

wheelandrailproﬁlerespectively.Inthisresearchthefollowing

strategieshavebeenadopted:

1.forthewheelupdatethemileagetraveledbyvehicleis

consid-ered.Thetotalmileagekmtot(derivedfromtheexperimental

dataprovidedbyTrenitaliaandRFI)issubdividedintoconstant

stepsoflengthequaltokmstep;

2.fortherailupdate,thetotaltonnageburdenonthetrack[18]is

considered.ThevehiclenumberNtotcorrespondingtothetotal

consideredtonnageissubdividedintoconstantstepsequalto

Nstep. • Trackintegration:

Tend Tin ıtot P_wijk(t)(y)V (t)dt≈

Tend Tin ıtot P_wijk(t)(sw−s cjk wi(t))V (t)dt= P_wijk(sw) (35)

Tend Tin ıtot Pjk ri(t) (y)V (t)dt≈

Tend Tin ıtot Pjk ri(t) (sr−scjk_ri (t))V (t)dt= _Pjk ri (sr); (36)

thetrackintegrationsumsallthewearcontributescomingfrom

thedynamicsimulationtoobtainthedepthofremovedmaterial

forwheel

P_wijk(sw)andrail P_rijk(sr)expressedinmm=mm

3_/mm2_.

Inordertohaveabetteraccuracyinthecalculationoftheworn

proﬁles,thenaturalabscissasswandsrofthecurvesw(yw)and

r(yr)havebeenintroduced.Inparticularthefollowingrelations

locallyhold(seeFig.13):

y≈sw−scjk_wi(t) y≈sr−scjk_ri (t) (37)

w(yw)=w(yw(sw))= ˜w(sw) r(yr)=r(yr(sr))= ˜r(sr) (38)

wherethenaturalabscissasofthecontactpointsscjk_wi andscjk_wi can

beevaluatedfromtheirpositionsP_wijkandP_rijk.

• Sumonthecontactpoints:

NPDC

i=1 Pjk_wi(sw)= w jk(sw) (39) NPDC

i=1 _Pjk ri (sr)= rjk(sr) (40)

(11)

Fig.14.Discretizationofthetotalmileage.

whereNPDCisthemaximumnumberofcontactpointsofeach

singlewheel(andrespectivelyofeachsinglerail);sincethe

num-berofcontactpointsonthewheel–railpairisusuallylessthan

NPDCandchangesinthetimeduringthedynamicsimulation,it

hasbeenassumedthatthewearcontributionassociatedtothe

ﬁctitiouspointsiszero.

• Averageonthevehiclewheelsandonthedynamicsimulations:

Nc

k=1 pk 1 Nw Nw

j=1 w jk(sw)= w (sw) (41) Nc

k=1 pk 1 Nw Nw

j=1 r jk(sr)= r (sr) (42)

whereNwisthenumberofvehiclewheelswhilethepk,1≤k≤Nc,

NC

k=1pk=1arethestatisticalweightsassociatedtothevarious

dynamic simulationsderived fromthestatisticalanalysis.The

averageonthenumberofwheel–railpairshastobeevaluated

inordertoobtainasoutputofthewearmodelasingleaverage

proﬁlebothforthewheelandfortherail(asrequiredbyTrenitalia

andRFI).

• Scaling:

theaimofthescalingprocedureistoamplifythesmallquantity

ofmaterialremovedduringtheNcdynamicsimulationsand,at

thesametime,tolimitthecomputationalload;usingthealmost

linearityofthewearmodelwiththetraveleddistance,itis

possi-bletoamplifytheremovedmaterialbymeansofascalingfactor

whichincreasesthedistancetraveledbythevehicle.

Thealmostlinearityofthewearmodelinsidethediscretesteps

kmstepin whichthetotalmileage traveledkmtot issubdivided

isaworkinghypothesiscomingfromthediscreteapproachof

themodel.Itisbasedontheideathatthewearrateinsidethe

simulateddistance(kmprove)remainsthesamealsoinsidethe

dis-cretestepkmstep,sincetheconsideredvehiclealwayscoversthe

sametracksofthestatisticalanalysisbothduringthesimulated

distance(kmprove)andduringthediscretestep(kmstep).

Inthisworkaconstantdiscretestephasbeenchosentoupdate

thewheelandrailproﬁles(seeFig.14):infactthismethodwell

adaptstothealmostlinearcharacteristicofthewearevolution.

Furthermoreitrequireslimitedcomputationalloadwithout

los-ingaccuracyifcomparedwithdifferentsuitablestrategiesasthe

adaptivestep[7].

Theevaluationofthediscretestep,withtheconsequentscaling

of w(sw)and

r

(sr),representsthemajordifferencebetween

wheelupdateandrailupdate:

1. theremovedmaterialonthewheelduetowearisproportional

tothedistancetraveledbythevehicle;infactapointofthe

wheelisfrequentlyincontactwiththerailinanumberoftimes

proportionaltothedistance.Thefollowingnomenclaturecan

beenintroduced(seeFig.14):

–kmtotisthetotalmileagetraveledbytheconsidered

vehi-cle(kmtotcanbechosendependingonthepurposeofthe

simulations,forexampleequaltothere-proﬁlingintervals);

– kmstepisthelengthofthediscretestepinwhichthetotal

mileagekmtotissubdivided;

–kmprove=ltrackistheoverallmileagetraveledbythevehicle

duringtheNcdynamicsimulations;thenecessityof

accept-ablecomputationaltimeforthemultibodysimulationsleads

toadoptsmallvalueofthekmprovelengthandforthis

rea-sontherelativeremovedmaterialhastobescaledwitha

multiplicativefactor.

Finallythematerialremovedonthewheelhastobescaled

accordingtothefollowinglaw:

w(sw)

kmstep

kmprove = w sc

(sw). (43)

Thechoiceofthespatialstepmustbeagoodcompromise

betweennumericalefﬁciencyandtheaccuracyrequired by

thewearmodel.Akmsteptoosmallcomparedtokmtotwould

provideaccurateresultsbutexcessivecalculationtimes;the

contraryhappenswithkmsteptoobigcomparedtokmtot.

2.thedepthofrailwearisnotproportionaltothedistance

trav-eledbythevehicle;infacttherailtendstowearoutonlyin

thezonewhereitiscrossedbythevehicleand,increasingthe

traveleddistance,thedepthofremovedmaterialremainsthe

same.Ontheotherhandtherailwearisproportionaltothe

totaltonnageMtotburdenontherailandthustothetotal

vehi-clenumberNtotmovingonthetrack.Therefore,ifNstepisthe

vehiclenumbermovinginadiscretestep,thequantityofrail

removedmaterialateachstepwillbe:

r(sr)∗Nstep=

r sc

(sr) (44)

whereNstepiscalculatedsubdividinginconstantstepthe

vehi-clenumberNtotcorrespondingtothetotaltonnagethathasto

besimulated;Ntotcanbeobtainedstartingfromthevehicle

massMv:

Ntot=Mtot

Mv . (45)

• Smoothingoftheremovedmaterial:

I

w sc(sw)

= w sc_sm(sw) (46) I

r sc(sr)

= r sc_sm(sr); (47)

thesmoothingoftheremovedmaterialfunctionisnecessaryto

removethenumericalnoisethataffectsthisquantityandthat

wouldbepassedtothenewproﬁles ˜wn(sw)and ˜rn(sr)ofwheel

andrailcausingproblemstotheglobalcontactmodel.Tothis

end,adiscreteﬁlter(i.e.amovingaverageﬁlterwithwindow

sizeequalto1%_÷5%ofthetotalnumberofpointsinwhichthe

proﬁlesarediscretized) hasbeenused;obviously thediscrete

ﬁlterhastoconservethemass.

• Proﬁleupdate:

yw(sw) ˜ wo(sw)

− w sc_sm(sw)nrw re−parameterization −→

yw(sw) ˜ wn(sw)

yr(sr) ˜ro(sr)

− r sc_sm(sr)nrr re−parameterization −→

yr(sr) ˜rn(sr)

; (48)

thelaststepconsistsin theupdateoftheoldproﬁles ˜wo(s)=

wo(y)and ˜ro(sr)=ro(yr)toobtainthenewproﬁles ˜wn(s)=wn(y)

and ˜rn(sr)=rn(yr);sincetheremovalofmaterialoccursinthe

normaldirectiontotheproﬁles(nr

wandnrraretheoutgoingunit

vectorforthewheelandrailproﬁlerespectively),onceremoved

thequantities w scsm(sw)and

r sc

sm(sr),are-parameterizationof

theproﬁlesisneededinordertoobtainagaincurves

(12)

5. Wearmodelvalidation

Inthis sectionthewearmodelvalidation phasewillbe

pre-sented. Initially, the set of Nc curvilinear tracks,on which the

dynamicsimulationsof theDMU Aln501Minuettovehiclehave

beenperformed,willbeintroduced(tracksextractedstartingfrom

thestatisticalanalysisoftheAosta-PreSaintDidiertrack,thedata

of which has been provided by RFI); moreover the wear

con-trolparametersforthewheelandrailwillbedeﬁned(theﬂange

heightFH,theﬂangethicknessFT,theﬂangesteepnessQRand

thequotaQMfortherail).Thentheexperimentaldata(provided

byTrenitalia)measuredontheAosta-PreSaintDidiertrackand

theirprocessingwillbeintroduced.Finally,thesimulation

strat-egyusedtoanalyzethewearbothonthewheelandontherailwill

bedescribedandtheresultsobtainedwiththewearmodelwillbe

analyzedandcomparedwiththeexperimentaldata.

5.1. StatisticalanalysisoftheAosta-PreSaintDidiertrack

StartingfromthedataofthewholeAosta-PreSaintDidiertrack

(providedbyRFI),thestatisticalanalysishasbeenperformedby

dividingthelinebothinradiusclasses(determinedbyRmin and

Rmax)andinsuperelevationclasses(determinedbyhminandhmax)

[11].Moreparticularlyﬁvesuperelevationsubclassesaredeﬁned

foreachradiusclass.Thesubclassesthatdonotincludecurvehave

notbeentakenintoaccountinthedeﬁnitionofthesetofNctracks.

AlltheNccurvedtracksareshowninTable3.

ThesetconsistsinNc=18distinctelements(17realcurvesand

thestraightline)characterizedbytheradiusvalueRc,the

superel-evationvalueH,thetravelingspeedVandthestatisticalweightpk

(with1≤k≤Nc)thatrepresentsthefrequency withwhich each

curveappearsonthe consideredrailwaytrack (Aosta-PreSaint

Didierline).TheradiiRcarecalculatedbymeansoftheweighted

meanonallthecurveradiiincludedinthecorresponding

superel-evationsubclass(theweightedfactoristhelengthofthecurvesin

therealtrack).Foreachsubclass,thevalueHisthemostfrequent

superelevationvalueamongthestandardvaluesthatcharacterize

thecurvesoftheconsideredsuperelevationsubclass.Thetraveling

speedsVarecalculatedimposingathresholdvalueonthe

uncom-pensatedaccelerationalim

nc=0.6m/s2: ˜ V2 Rc − H sg=a lim nc (49)

wheresistherailwaygaugeandgisthegravityacceleration.The

estimatedspeed ˜Vhasbeenthencomparedwiththemaximum

velocityVmaxonthelinetogetthedesiredtravelingspeed V=

min( ˜V ,Vmax).

5.2. Wearcontrolparameters

ThereferencequotasFH,FTandQRareintroducedinorderto

estimatethewheelproﬁleevolutionduetothewearwithout

nec-essarilyknowingthewholeproﬁleshape(seeFig.15).According

tothesequotastheuserwillbeablebothtoestablishedwhenthe

wornwheelproﬁlewillhavetobere-proﬁledandtodetectifthe

wearcompromisesthedynamicalstabilityofthevehicle[19].

Theproceduretodeﬁnethereferencequotasisthefollowing:

1.firstofallthepointP0isdefinedontheprofile,at70mmfrom

theinternalverticalfaceofthewheel;

2.thenthepointP1isintroducedontheproﬁle,2mmunderthe

ﬂangevertex;

3.ﬁnallythepointP2isdeterminedontheproﬁle,10mmunder

thepointP0;

4. thewear control parameters are then calculated as follows:

theﬂangethicknessFTisthehorizontaldistancebetweenthe

Fig.15.Deﬁnitionofthewheelwearcontrolparameters.

internalverticalfaceandthepointP2;theﬂangesteepnessQR

isthehorizontaldistancebetweenthepointsP1andP2,while

theﬂangeheightFHistheverticaldistancebetweenP0andthe

ﬂangevertex(allthedistancesareconsideredpositive).

Anadditionalcontrolparameteristhenintroducedtoevaluate

theevolutionofrailwear.ParticularlytheQMquotaisdeﬁnedas

therailheadheightinthepointyr=760mmwithrespecttothe

centerlineofthetrack:thisyrvaluedependsontherailwaygauge

(equalto1435mmintheAosta-PreSaintDidierline)andonthe

layingangle˛pofthetrack(equalto1/20rad).PhysicallytheQM

quotagivesinformationontherailheadwear(seeFig.16).

5.3. Experimentaldataandtheirprocessing

TheexperimentaldataprovidedbyTrenitaliaandRFIarerelated

onlytothewheelwearandconsistsinthewearcontrolparameters

measuredasafunctionofthetotaldistancetraveledbythe

con-sideredvehicleDMUAln501Minuetto;particularly,thedatahave

beenmeasuredonthreedifferentvehiclesoperatingonthesame

trackthatareconventionallycalledDM061,DM068,DM082.

AscanbeseenbyexampleinTable4forthevehicleMD061,

thereferencequotavalueshavebeenmeasuredforallthevehicle

wheels(eachvehiclehaseightwheelsetsasspeciﬁedinSection3.1).

Howeverthefollowingdataprocessinghasbeennecessaryinorder

toobtainasinglewheelproﬁlethatcouldbeeffectivelycompared

withtheproﬁleextractedfromthenumericalsimulationandto

reducethemeasurementerrors:

1.initiallythearithmeticmeanonallthesixteenvehiclewheelshas

beenevaluated;themeanisnecessarytoobtainasinglewheel

proﬁleand,atthesametime,toreducethemeasurementerrors

affectingtheexperimentaldata;

2.thenascalingofthequotavalueshasbeencarriedoutinorder

todeletetheoffsetontheinitialvalueoftheconsidered

quanti-ties:thisprocedureimposesthatallthewearcontrolparameters

startfromtheirnominalvalues(thestandardvaluesfortheORE

S1002proﬁlehavebeenused)inordertoremovetheinitial

differencesamongthevehiclesduetomeasurementerrors;

(13)

Table3

Dataofthecurvilineartracksofthestatisticalanalysis.

Rmin(m) Rmax(m) Superelevationhmin−hmax(mm) Rc(m) H(mm) V(km/h) pk(%)

147.1 156.3 0 – 10–40 – 60–80 – 90–120 150 120 55 0.77 130–160 – 156.3 166.7 0 – 10–40 – 60–80 – 90–120 160 110 55 0.48 130–160 165 140 55 0.56 166.7 178.6 0 – 10–40 – 60–80 – 90–120 170 110 55 0.82 130–160 175 130 55 1.55 178.6 192.3 0 – 10–40 – 60–80 – 90–120 190 100 55 8.37 130–160 180 130 55 0.45 192.3 208.3 0 – 10–40 – 60–80 – 90–120 200 90 55 20.64 130–160 200 130 60 4.00 208.3 227.3 0 – 10–40 – 60–80 220 80 55 0.70 90–120 220 100 55 3.76 130–160 – 227.3 250.0 0 – 10–40 – 60–80 240 80 55 7.26 90–120 240 110 60 5.28 130–160 – 250.0 312.5 0 – 10–40 – 60–80 270 70 55 3.91 90–120 270 90 60 5.29 130–160 – 312.5 416.7 0 – 10–40 – 60–80 370 60 55 2.26 90–120 345 100 70 1.63 130–160 – 416.7 ∞ 0 ∞ 0 70 32.27

3.the arithmetic mean on the three vehicle MD061, MD068, MD082hasnotbeencarriedout,inordertomaintaina disper-sionrangefortheexperimentaldata.

Theexperimentaldata,properlyprocessed,aresummarizedin Table5.Ascanbeseen,theﬂangeheightFHremainsapproximately

constantbecauseofthelowmileagetraveledbythevehicles,while

theﬂangethicknessFTandtheﬂangesteepnessQRdecreasealmost

linearlyandhighlight,accordingtothecharacteristicsofthetrack,

thewearconcentrationinthewheelﬂange.

5.4. Simulationstrategy

Inthissectionthesimulationcampaignperformedtostudythe

wearonwheelandrailisdescribed.AsexplainedinSection4.3,

thetwophenomenaevolveaccordingtodifferenttimescales

(sev-eralordersof magnitude)and afullysimulation ofsuchevents

wouldrequirea tooheavycomputationaleffort.For thisreason

thefollowingspeciﬁcalgorithmhasbeenadoptedforupdatingthe

proﬁles:

1.both for the wheel and for the rail ﬁve discrete stepshave

beenchosen,nsw=5andnsr=5,sotohaveagoodcompromise

betweencalculationtimesandresultaccuracy:

(a)thechoiceof thewheelkmstep (seeSection4.3)hasbeen

made considering the whole distance traveled equal to

kmtot≈3500km(gotfromexperimentaldata);thusthe

sin-glesteplengthwillbe:

kmstep=

kmtot

nsw ≈

700(km). (50)

(b) toestimatethevehiclenumberNtot(seeSection4.3)a

crite-rionpresentinliterature(basedonthetotaltonnageburden

onthetrack)hasbeenused[18].Particularlythereisa

pro-portionalityrelationshipbetweentonnageandwear:arail

wearof 1mmontherailheadheightevery100Mt

(14)

Table 4 Experimental data of the DMU Aln 501 Minuetto DM061. km quotas 1r 1l 2r 2l 3r 3l 4r 4l 5r 5l 6r 6l 7r 7l 8r 8l Wheel diameter Wheel diameter Wheel diameter Wheel diameter Wheel diameter Wheel diameter Wheel diameter Wheel diameter 816 mm 815 mm 824 mm 823 mm 823 mm 823 mm 819 mm 820 mm 0 F T 30.953 30.944 30.983 30.784 31.099 30.957 30.938 31.076 30.401 30.367 30.830 30.987 30.437 30.717 30.852 30.933 FH 27.970 27.894 28.141 28.043 27.969 28.187 28.030 28.271 28.245 27.918 28.141 27.982 28.013 27.937 28.333 27.883 QR 10.208 10.140 10.424 10.457 10.220 10.306 10.279 10.833 10.332 10.445 10.364 10.219 10.421 10.500 10.338 10.396 1426 FT 29.855 28.977 30.283 29.317 30.118 29.383 30.152 29.450 29.796 29.799 30.288 29.483 29.802 29.085 30.267 29.316 FH 28.010 27.923 28.104 28.108 28.000 28.249 28.095 28.278 28.248 28.284 28.247 28.030 28.997 28.003 30.383 27.919 QR 9.297 8.226 9.822 8.956 9.344 8.749 9.551 9.072 9.635 9.767 9.773 8.763 9.593 8.883 9.675 8.762 2001 FT 29.056 28.498 29.722 28.878 29.441 28.667 29.629 28.717 29.153 28.101 29.739 28.841 29.066 28.447 29.625 28.777 FH 27.990 27.880 28.161 28.080 29.998 28.248 28.128 28.283 28.290 27.994 28.273 28.022 28.027 28.014 28.362 27.957 QR 8.404 7.558 9.233 8.637 8.702 7.950 8.873 8.436 9.144 8.141 9.235 8.086 9.038 8.152 9.248 8.373 2575 FT 28.259 27.096 29.333 28.045 28.972 28.385 29.029 28.124 29.053 27.600 29.095 28.505 28.553 27.866 29.205 28.473 FH 28.009 27.089 28.173 28.020 28.063 28.243 28.090 28.241 28.285 27.963 28.244 28.085 28.030 28.018 28.352 27.968 QR 7.198 7.024 8.853 8.163 8.123 7.598 8.438 7.791 8.868 7.395 8.559 7.840 8.372 7.340 8.777 7.900

appreciablerailwear,amaximumvalueofremovedmaterial

depthof2mmontherailheadheighthasbeenhypothesized

(naturallythisvaluecanbechangedaccordingtothe

require-mentsofthesimulation).Thenumberofvehicles,ofknown

massMv(seeTable1),whichshouldevolveonthetrackto

reachthe200Mt,hasbeenthereforecalculated:

Ntot= Mtot Mv ≈ 2,000,000 (51) andthen: Nstep= Ntot nsr ≈ 400,000. (52)

2.the wear evolution on wheel and rail has been decoupled

becauseofthedifferentscalesofmagnitude:

(a)whilethewheel wearevolves,therail issupposed tobe

constant:infact,inthetimescaleconsidered,therailwear

variationisnegligible.

(b) becauseofthetimescalecharacteristicoftherailwear,each

discrete railproﬁlecomes incontact, withthesame

fre-quency,witheachpossiblewheelproﬁle.Forthisreason,for

eachrailproﬁle,thewholewheelwearevolution(fromthe

originalprofiletothefinalprofile)hasbeensimulated.

Basedonthetwoprevioushypotheses,thesimulationshave

beencarriedoutaccordingtothefollowingstrategy:

Wheelproﬁleevolutionatﬁrstrailstep:w0

i p1,1

⎧

⎨

⎩

w0 0 r0

→

w0 1 r0

→···→

w0 4 r0

→w0 5

Averageontherailsr(i₁+1)forthecalculationofthesecondrail

step:r1 p1,2

⎧

⎪

⎨

⎪

⎩

⎛

⎜

⎝

w0 0 r0 w0 1 r0 . . . ... w0 4 r0

⎞

⎟

⎠

→

⎛

⎜

⎝

r₁(1) r₁(2) . . . r₁(5)

⎞

⎟

⎠

→r1 . . .

Wheelproﬁleevolutionatfourthrailstep:w4

i p5,1

⎧

⎨

⎩

w4 0 r4

→

w4 1 r4

→···→

w4 4 r4

→w4 5

Averageontherailsr(i₅+1)forthecalculationoftheﬁfthrail

step:r5 p5,2

⎧

⎪

⎨

⎪

⎩

⎛

⎜

⎝

w4 0 r4 w4 1 r4 . . . ... w4 4 r4

⎞

⎟

⎠

→

⎛

⎜

⎝

r₅(1) r₅(2) . . . r₅(5)

⎞

⎟

⎠

→r5 (53)

wherewj_iindicatestheithstepofthewheelproﬁlethatevolves

(15)

Table5

Experimentaldataprocessed.

Vehicle Distancetraveled(km) FH(mm) FT(mm) QR(mm)

DM061 0 28.0 32.5 10.8 1426 28.2 31.5 9.8 2001 28.1 30.8 9.1 2575 28.0 30.2 8.6 DM068 0 28.0 32.5 10.8 1050 28.0 31.8 10.0 2253 28.0 30.2 8.5 2576 28.0 30.0 8.4 DM082 0 28.0 32.5 10.8 852 28.0 32.3 10.6 1800 28.0 31.3 9.6 2802 28.0 30.3 8.7 3537 27.6 30.0 8.3

thesameforeachjandcorrespondtotheunwornwheelproﬁle (ORES1002).

Initially the wheel (starting from the unworn proﬁle w0 0)

evolvesontheunwornrailproﬁler0inordertoproducethe

dis-cretewheelproﬁlesw0

0,w01,...,w50(stepp1,1).Thenthevirtual

railproﬁlesr(i+1)₁ ,obtainedbymeansofthesimulations(w0 i,r0)

with0≤i≤4,arearithmeticallyaveragedsoastogettheupdate railproﬁler1(stepp1,2).Thisprocedurecanberepeatednsrtimes

inordertoperformalltheraildiscretesteps(uptothestepp5,2).

Thecomputationaleffortrequiredbythesimulationstrategy isthefollowing:

(a)inthewheelwearstudy,foreachupdateoftherailproﬁle rj,thewholewheelwearloopw_ijwith0≤i≤4(nswstepsof

simulation)issimulated.Thecomputationaleffortresultsof nsw×nsr=25stepsbothforthedynamicanalysis(in

Sim-packRail) andfor thewearmodel necessarytocalculate theremovedmaterialonthewheel(inMatlab).Sothetotal numberofsimulationstepsare2(nsw×nsr)=50.

(b)intherailwearstudythedynamicanalysesarethesame asthepreviouscasebecauseforeach railstepthewheel proﬁlesw_ij(0≤i≤4)aresimulatedonrjinordertoobtain

r_j(i+1)andthustheupdatedrailproﬁlerj+1bymeansofan

arithmeticmean.Therefore,noadditionaldynamical anal-yses are needed. In this caseonly the wearmodel steps mustbesimulatedsoastogettheremovedmaterialonthe rail.Consequentlythetotalnumberofsimulationstepsis nsw×nsr=25.

Thecharacteristicsoftheprocessorusedinthesimulationsand themeancomputationaltimesrelativetoeachdiscretestepofthe wholemodelloop(dynamicalsimulationandwearsimulation)are brieﬂysummarizedinTable6.

5.5. Evolutionofwearcontrolparameters

In this section the evolutionof thewheel reference quotas

numericallyevaluatedbymeansofthewearmodel(ﬂange

thick-nessFT,ﬂangeheightFHandﬂangesteepnessQR)willbecompared

withtheexperimentaldataconcerningthethree DMUsAln 501

Table6

Computationaltime.

Processor Computationaltime

Dynamicsimulation

(SimpackRail)

Wearsimulation

(Matlab)

INTELXeonCPUE5430

2.66GHz8GBRAM

2h2s 31s

Fig.17.FTdimensionprogress.

Fig.18.FHdimensionprogress.

Minuettovehicles.FurthermoretherailreferencequotaQM evo-lutionwillbeshownandcomparedwiththecriterionpresentin literaturebasedonthetotaltonnageburdenonthetrack[18].

TheprogressofFTdimension,forthensrdiscretestepoftherail,

isshowninFig.17asafunctionofthemileage;asitcanbeseen,

thedecreaseofthedimensionisalmostlinearwiththetraveled

distanceexceptintheﬁrstphases,wheretheproﬁlesarestillnot

conformalenough.TheFHquotaprogressisrepresentedinFig.18

andshowsthat,duetothepresenceofmanysharpcurvesinthe

statisticalanalysisofthetrackandtothefewkilometerstraveled,

thewheelwearismainlylocalizedontheﬂangeratherthanonthe

treadandtheﬂangeheightremainsnearconstantinagreement

withexperimentaldata.TheQRtrendisshowninFig.19:alsothe

ﬂangesteepnessdecreasesalmostlinearly,leadingtoanincreaseof

(16)

Fig.20.QMdimensionprogress.

theconicityoftheﬂange.Finally,theevolutionofthewheelcontrol

parametersremainsquantitativelyandqualitativelysimilarasthe

railwearraises.

Althoughthesimulatedmileageisquiteshortconsideringthe

meantraveleddistancebetweentwoturningsofthewheelsina

standardscenery(infacttheFHquotaremainsalmostconstant),

thevariationsoftheFTandQRdimensionsareremarkableandit

highlightsthewearproblemsaffectingthevehicleDMUAln501

Minuettorunningalongtherailwaylinetakenintoaccount.

Inconclusion,thecomparisonsshowthattheoutputsof the

wearmodelareconsistentwiththeexperimentaldata,bothfor

theﬂangedimension(FH,FT)andfortheconicity(QR);theslightly

steeperdevelopmentoftheexperimentaldatathanthesimulation

canbeexplainedwiththedispersionoftheexperimentaldataand

withwearmechanisms,likeplasticandpittingwear,not

consid-eredinthedevelopedwearmodel.

Finally,theQMevolutionfortheanalysisoftherailwearis

pre-sentedinFig.20andshowsthealmostlineardependencebetween

therailwearandthetotaltonnageburdenonthetrack:theamount

ofremovedmaterialontherailproﬁleisinagreementwiththe

cri-terionpresentin literature(1mmontherailheadheightevery

100Mtofaccumulatedtonnage).

5.6. Evolutionofthewheelandrailproﬁles

Thewearevolutiononthewheelproﬁlesevolvingoneachrail

rj(with 0≤j≤nsrand nsr=5)is presentedin thefollowing(see

Figs.21and23–27).Asstatedpreviously,thewheelproﬁle

evolu-tionisdescribedbymeansofnsw=5stepsandthespatialstepkmstep

hasbeenchosenequalto700km,sincethetotalmileagekmtotis

3500km.Fig.22showsthecumulativedistributionsofremoved

materialinverticaldirectionzwonthewheelproﬁleatﬁrstrailstep

wKw(yw)=

Kw

i=1iw(yw)asafunctionofyw(1≤Kw≤nsw),where

w

i (yw)istheremovedmaterialbetweentwosubsequentdiscrete

stepsofthewheelproﬁleevolution.

Thequitelimiteddistancetraveledbythevehiclejustiﬁesthe

lowwearonthewheeltreadandentailsasmallreductionofthe

rollingradius.Howeverthehightortuosityoftheconsideredtrack

leadstoappreciablewearonthewheelﬂange.InFigs.21and23–27,

focusingontheﬂangezone,thehigherwearrateduringtheﬁrst

Fig.21.Evolutionofthewheelproﬁleonther0rail.

Fig.22. Cumulativedistributionsoftheremovedwheelmaterial.

Fig.23.Evolutionofthewheelproﬁleonther0railintheﬂangezone.

stepscanbeobservedbecauseoftheinitialnon-conformalcontact

thatcharacterizesthecouplingbetweentheORES1002wheel

pro-ﬁleandtheUIC60railproﬁlewithaninclinationof˛p=1/20rad;

thentheratedecreasesbecomingmoreregularandconstantinthe

laststeps,whenthecontactismoreandmoreconformal.

Alsoasregardsthewheelproﬁleevolution(asforthereference

quotas)thetrendremainsquantitativelyandqualitativelythesame

astherailwearraises(seeFigs.24–27).

In Figs.28 and 30theevolutionoftherailproﬁleis shown,

described bymeansofnsr=5discrete stepand withNstepequal

to400,000(the vehiclenumber Ntot,correspondingtothetotal

studied tonnage Mtot, is 2,000,000). In Fig. 29 the cumulative

distribution of the removed material on the rail proﬂie in zr

(17)

Fig.28.Evolutionoftherailproﬁle.

Fig.29.Cumulativedistributionsoftheremovedrailmaterial.

directionrKr(yr)=

Kr

i=1ri(yr)expressedasafunctionofyr(with

1≤Kr≤nsr) is shown, where _ir(yr) is the removed material

betweentwosubsequentdiscretestepsoftherailproﬁle

evolu-tion.Thevalueoftotaltonnagetakenintoaccount(Mtot=200Mt)

causesanappreciablewearontherailhead,whileitisnotsufﬁcient

toproduceanhighwearalsoontherailshouder(Fig.30).

Fig.30. Evolutionoftherailproﬁleinheadzone.

6. Conclusions

InthisworktheAuthorspresentedacompletemodelforthe

wheelandrailwearpredictioninrailwayapplication,developed

thanks tothe collaboration withTrenitalia S.p.A and Rete

Fer-roviariaItaliana(RFI),whichprovidedthenecessarytechnicaland

experimentaldataforthemodelvalidation. Thewholemodelis

madeupoftwomutuallyinteractiveparts.Theﬁrstoneevaluates

thevehicledynamicsandcomprisesboththemultibodymodelof

thevehicleimplementedinSimpackRailandaglobalwheel–rail

contactmodel(developedbytheAuthorsinpreviousworks)for

thecalculationofthecontactpointsandofthecontactforces.The

secondoneisthewearmodelwhich,startingfromtheoutputsof

themultibodysimulations,evaluatestheamountofmaterialtobe

removedduetowear.Theinteractionbetweenthetwopartsisnota

continuoustimeprocessbutoccursatdiscretesteps;consequently

theevolutionofthewheelandrailgeometryisdescribedthrough

severalintermediateproﬁles.

Inparticularasuitableupdatealgorithmhasbeendevelopedin

ordertoconsiderthedifferenttimescalecharacterizingthewheel

andrailwearevolution:thewheelwearhasbeenstudiedbasing

onthedistancetraveledbyvehicle,whiletherailwearhasbeen

evaluatedbasingonthetotaltonnageburdenontherailwaytrack.

The whole model has been validated on a critical scenario

in termsof wearinItalian railways:theALSTOM DMU Aln501

MinuettocirculatingontheAosta-PreSaintDidierrailwayline.A

statisticalapproachtodescribethetrackhasbeenusedtoreduce

thetotalcomputationaleffort.

As regards the wheel wear, the results obtained from the

dynamic simulationshavebeencompared withthe

experimen-taldataprovidedbyTrenitaliawhile,concerningtherailwear,the

comparisonwithexperimentalcriteriabasedonthetotaltonnage

burdenontherailwaytrackhavebeenconsidered.Thedeveloped

modelreproducesquitegoodtheevolutionofalltheproﬁle

charac-teristicdimensionsdescribinginsatisfyingwaythewearprogress

bothonthewheelandontherail.

Theresultsobtainedforthewheelproﬁleevolutionhighlights

how,intheparticularoperatingconditionsofthestudiedrailway

line,thewearisquitesevereandstronglylocalizedonthewheel

ﬂange,leadingtofrequentmaintenanceinterventions.Asregards

therailproﬁleevolution,thewearismainlyfocusedontherail

headduetotheinitiallowconformityofthecouplingORES

1002-UIC60with˛p=1/20radwhilethetotaltonnageconsideredisnot

sufﬁcienttoproduceanhighwearalsoontheshouderoftherail.

Futuredevelopmentswillbebasedonfurtherexperimentaldata

(relative tootherrailway trackwitha higher mileagethan the

Aosta-PreSaintDidierline)alwaysprovidedbyTrenitaliaandRFI

and referredtoadvancedwearonthewheel(especiallyonthe

wheeltread)and ontherail. Inthis wayotheranalysiswillbe

carriedoutinordertofurthervalidatethewholemodel.

Moreoverotherwearmechanismsinadditiontoabrasivewear

(plasticwear,pitting wearetc.),thatmaybecomequite

impor-tantespeciallyinconditionsofadvancedwear(inparticularonthe

wheelﬂange),willbeconsideredwithinthewearmodel.Finally,

acodeoptimization,concerningthevehiclemodel(inparticular

theglobalcontactmodel),thewearmodelandthewholeloop,is

scheduledforthefutureinordertoreducethesimulationtimes.

Acknowledgements

AuthorswouldliketothankEngg.R.CheliandG.Grandeof

Tren-italiaS.p.A.forprovidingandgivingthepermissiontoeditthedata

relativebothtothevehicleDMUAln501Minuettoandtothewheel

Development of a wear model for the prediction of wheel and rail profile evolution in railway systems

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Wear

Development

of

a

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model

for

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prediction

of

wheel

and

rail

proﬁle

evolution

in

railway

systems

M.

Ignesti

,

M.

Malvezzi

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L.

Marini

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E.

Meli

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A.

Rindi

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