Identification of short-wavelength contact wire irregularities in electrified railway pantograph–catenary system

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Mechanism and Machine Theory

journalhomepage:www.elsevier.com/locate/mechmachtheory

Identification of short-wavelength contact wire irregularities in electrified railway pantograph–catenary system

Yang Song

^∗

, Anders Rønnquist, Tengjiao Jiang, Petter Nåvik

Department of Structural Engineering, Norwegian University of Science and Technology, Richard Birkelands vei 1A, Trondheim 7491, Norway

a rt i c l e i nf o

Article history:

Received 25 February 2021 Accepted 17 March 2021 Available online 29 March 2021 Keywords:

Railway

Pantograph–catenary interaction Contact force

Irregularity

Ensemble empirical mode

a b s t ra c t

The effect oftwo commontypes ofshort-wavelength irregularities(local imperfections and periodicshort-wavelengthirregularities)oftherailwaycatenaryonthepantograph–

catenaryinteractionperformancearestudiedandtheirpotentialidentificationapproaches areexploredinthispaper.Theanalysisoftheintrinsicmodefunctionsofpanheadaccel- eration indicates thattheeffectoflocalimperfectioncan bereflected inthe high-order deformation mode ofthe contact wire.The cut-off frequency issuggested to coverthe wavelength smallerthan1/8 ofthedropperto dropperdistance,whichcanbeused to identifythe localimperfection. TheinstantaneousenergyobtainedbytheHilbert trans- formisusedtolocalisethelocalimperfection.Theeffectofperiodicshort-wavelengthir- regularitiescanberecognisedastheintroductionofnon-Gaussianbehaviourinthecontact forceatspecificwavelengths.Thus, spectralkurtosisisutilised toidentifythedeviating wavelength.Theshort-wavelengthirregularitycanbelocalisedbythetime-frequencyanal- ysisoftheintrinsicmodefunctioncontainingtheidentifieddeviatingwavelength.Theex- ampleswithmeasurementdataindicatethevalidationofthepresentmethodswithsome improvementstothecurrentequipment.

© 2021 The Authors. Published by Elsevier Ltd.

ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Pantograph–catenarysystemsarewidelyusedinthemodernelectrifiedrailwaytopowertheelectrictrainviathesliding contact between the registration strip of the pantograph andthe contactwire of the catenary, as shown in Fig. 1. The contactqualityofpantograph–catenaryisofgreatimportanceasitdirectlydeterminesthereliabilityandsafetyofelectric transmission [1]. Generally,the catenary suffers long-termimpacts fromthe pantographs and various disturbances from environmental conditions.These factors, together with mounting imprecisions and inadequate maintenance, presentthe primary sourceofcontactwireirregularity(CWI),whichisthecommonearlyfaultinthecatenaryanddirectlyaffectsthe interactionperformancebetweenthepantographcollectorandthecontactwire.

∗Corresponding author.

E-mail addresses: [email protected] (A. Rønnquist), [email protected] (T. Jiang), [email protected] (P. Nåvik).

https://doi.org/10.1016/j.mechmachtheory.2021.104338

0094-114X/© 2021 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )

Fig. 1. Schematic of a pantograph-catenary system.

1.1. Problemdescription

Generally, CWIs can be divided into three types. The first is the geometrical distortion, as shown in Fig. 2(a), which is normally caused by the deformation of messenger, contactand dropper wires. The second one is the periodic short-wavelength CWI(PSW-CWI)causedbywearandmanufacturingdefectinthecontactwireasshowninFig.2(b).The last one isthelocal imperfectioncausedby thewirekinksasshowninFig.2(c). Thefirst type ofCWIhasa longwave- length andmainlyaffects thelow-frequencyperformance ofpantograph–catenaryinteraction.In contrast,thesecond and thirdonespresentthemainsourceofshort-wavelengthdisturbancetothepantograph–catenaryinteraction,whichdirectly causeshard-spotandcontactloss.TheCWIsaredesiredtoidentifyandcorrectbeforetheydevelopintosevere faultsand resultincatastrophicconsequences.The currentstandard [2]specifies thatmainstatistics(suchasthemeanvalue,maxi- mum valueandstandarddeviation)ofthecontactforceofpantograph-catenarycanbeusedto assessthecontactquality.

The previous researches[3]indicate thatthe contactforce filteredat0–20Hzcan reflectthe geometricaldeformation of the catenary. Apart fromthe conventional dynamicinspection, some railway operators also use diesel train to regularly monitorthevariationofthecontactwireheight.Buttheeffectoftheshort-wavelengthirregularitiesisnotdistinct inthe time-historyofcontactforceorthecontactwireheightandchallengingtobeidentifiedbytheregularinspection[4]. 1.2. Literaturereview

Thestudyofthepantograph–catenaryinteractionhasattractedever-increasinginterestfromboththescientificcommu- nity andthe industry,as it hasbeen recognisedas themost vulnerable partof the traction powersystem [5,6]. Due to theconsiderableexpenseoffieldexperiment, varioustypesofmathematicalmodelsofcatenaryweredevelopedbasedon the mode superposition[7],the finite elementmethod [8–10]andthe analytical expression[11,12].The pantograph was normally assumedasa lumped massmodel whichcould describe several criticalmodes ofreal one [13]. The modelling of3D pantograph hasbeenstate-of-the-art, asit canfullydescribe therealconfiguration,whichisessential fortheopti- misation ofparameters[14],theinclusionofspatialvehiclevibration[15]andthedevelopmentofcontrollers[16,17].The pantograph–catenary system worksunder complex serviceconditions resultant from the high-speedairflow, wind loads, temperature variations, snow andicecoating, electromagneticforces, vehicleexcitations, catenarydefects andanomalies.

The mathematical descriptions of thesefactors’effect on pantograph–catenaryinteraction havebeen developed by many researchers. The aerodynamicforcesapplied to the pantograph andtheir impact on the contactforce were evaluated by Pombo etal.[18]. Differentfromthetraditional vortex-inducedvibration [19,20],thewind loadnormallycausesthe buf- feting of the catenary and affects its interaction withpantographs, which was investigated in [21,22]. The aerodynamic instability of the catenaryinextremeconditions wasalso investigatedin [23]. The locomotiveexcitation responses eval- uated by vehicle-trackinteraction simulation[24] andexperimental test [25] were introduced to evaluatetheir influence onpantograph–catenaryinteractionperformance[26].Themultiplepantographinteractionwithahigh-speedcatenarywas

Fig. 2. Three types of contact wire irregularity.

analysed by Xu etal.[27] andLiuet al.[28], andtwo formulas fortheoptimal pantograph interval were proposed.The realanomaliesofthecatenary,includingthedefectivedropper[29]andtensionloss[30]wereaddedintheassessmentof the pantograph–catenaryinteraction.Fortheperspective ofaccidentprevention, someresearchers devotedtheir attention to theimprovementofmonitoringanddetectiontechniqueforthepantograph–catenarysystem. Exceptusing thecontact force, Nåvik etal.[31] proposedan acceleration-basedindextoidentifythe pantographfault. Jiangetal.[32] proposeda robust line-trackingphotogrammetrymethodformonitoringthe upliftofcontactwireincomplex backgrounds.Liuetal.

[33]developed a detection method for structure parameters of the catenary cantilever using 3D point cloud data.

The effectoflong-wavelengthCWI andits detectiontechniquehasbeenwidelystudied andreportedin exitinglitera- ture. Songetal.[34]proposedaTCUD(TargetConfigurationUnderDeadLoads)-basedmethodtomodelthecatenarywith realistic CWIs, andinvestigatedthestochastic effectofcontactwireheight variationonthe contactforce[3].Wangetal.

[4]presentedatime-frequencyrepresentation(TFR)toidentifyandlocaliseCWIsfromthecontactforcebasedontheZhao–

Atlas–Marksdistribution.Asfortheshort-wavelengthCWI,thefollowingworksdeservestobementioned.

ThefirstonewasperformedbyWangetal.[35].Therealcontactwirewearwasincludedinthenumericalsimulationto evaluatetheinteractionperformanceofthepantograph–catenary.Theresultsindicatedthatthestandarddeviationoflow- frequencycontactforcewasnotsignificantlychangedbytheslightwear,whichpointedoutthatthetraditionalassessment indicatorsmightbeinvalidtoevaluatethecontributionoftheshort-wavelengthCWI.Thesecond[36]proposedtheconcept of catenarystructural wavelength (CSW)to quantify the contribution ofCWIs to interaction performance. The CSW was definedaccordingtothecatenarygeometry(includingthespanlength,halfspanlengthanddropperinterval).Theeffectof CWIscould beextractedbyremoving theCSWfromthemeasured accelerationorcontactforceusingensembleempirical mode decomposition(EEMD).However, it isdifficultto definea CSWinactual operation forthefollowingthree reasons.

Firstly,thecatenarymodesaredependentonthegeometricalinformationandthedeformationofthetensionedwires[37]. It isinsufficient toonly includethegeometrical structuralwavelengths (namelythe dropperinterval andspanlength) in theCSWwithoutdeformationmodes.Secondly,thereisaninherentvariationofthecatenarygeometryinactualoperation, whichresultsinawide-bandstructuralwavelengthandposesachallengetodefinetheCSW.Thirdly,theCWImayhavethe samewavelengthasthecontactforce sincemostshort-wavelength CWIsarethedirectconsequenceofthecontactforce’s long-termimpact.ThecontributionofCWIscannotbeevaluatedbymerelyremovingtheCSWfromthemeasurementsignal.

In [38],the high-frequencyaccelerationof thepantograph headwasutilisedto identifythelocal imperfectionemploying RMS(RootMeanSquare).Inthiswork,theaccelerationsignalwasmeasuredatupto200Hzusingasensitiveaccelerometer.

Thissolutionmaynotbesuitableforallthecountries,asmostrailwayoperatorsonlymeasurethedynamicparameters(e.g.

stagger,contactforce,accelerationandheight)withasparsedistancestep(e.g.0.5mformostcountrieslikeChina,Norway andtheUK).Therefore,thecurrentissueiswhetheritispossibletomakethebestuseofthecurrentsparsemeasurement datatoidentifythelocalimperfection.AnotherproblemisthattheRMSisnotreliabletodetecttheimperfection,andmore sensitiveindicesshouldbeinvestigated.

1.3. Scopeandcontribution

Asshownintheaboveliteraturereview,thelocalimperfectioncanbeidentifiedthroughthepanheadaccelerationwith extremely highcut-off frequencies.Ifa highcut-off frequencycannot be reached,the effectivenessofthe existingidenti- fication methodforthe short-wavelength irregularityshould be re-evaluated.The mainobjectiveof thisworkisto study potential identificationmethods fortheshort-wavelength irregularities andthelocalimperfection fromthemeasurement contactforceandpanheadaccelerationrespectively.Thescopeandcontributioncanbesummarisedasfollows:

1) Explorethepossibilityto identifythelocalimperfectionfromthepanhead accelerationwithlimitedcut-off frequency, andinvestigateasensitiveindexbettertorepresentthelocalimperfectionfromthepantographheadacceleration.

2) EvaluatetheeffectofPSW-CWIonthecontactforce,andseekavalidindextorepresentthecontributionofCWIwitha specificwavelength.

Toachieve themaingoals,thispaperpresentsamathematicalmodelofthepantograph–catenaryinteractionbasedon the absolutecoordinate nodalformulation(ANCF), which hasbeen proven tobe an effectivemethod to describethe ge- ometrical nonlinearity in railwaydynamics [39]. The short-wavelength CWI is properly included in the catenary model.

EEMD isutilised to analysethe implicationofshort-wavelength CWI ineach IMF. Addressingthefeature ofeach type of short-wavelength CWI,thepotentialsolutions arediscussed.Forthelocalimperfection,theminimumcut-off frequencyof the pantograph head accelerationwhich can reflectthe effectof localimperfection is investigated.The Hilbertspectrum is utilised toidentify andlocalisethe localimperfection. ForthePSW-CWI, the spectral kurtosis [40] isadoptedto indi- catethenon-Gaussianbehaviourofcontactforce witha specificwavelengthcausedbythePSW-CWI. Thetime-frequency analysisisperformedtolocalisethePSW-CWIfromthecorresponding IMFcontainingtheidentifieddeviatingwavelength.

Using both numerical andmeasurement data, thevalidations of the proposed identificationapproaches fortwo types of short-wavelengthCWIareverified.

Fig. 3. Catenary model based on ANCF beam and cable elements.

1.4. Paperorganization

Theoutlineofthispaperisasfollows.AbackgrounddescriptionandliteraturereviewaregiveninSection1.Themath- ematicalmodelofthepantograph–catenarysystemtogeneratethedataforsubsequentanalysesisdescribedinSection 2. Section3discussesthepotentialsolutionsforthetwotypesofshort-wavelengthCWI.Theidentificationapproachforlocal imperfection isproposed inSection 4withapplicationstoboth numericalandmeasurementdata.Section 5presentsthe identificationapproachforPSW-CWI.Section6concludesthispaper.

2. Numericalmodellingofpantograph–catenary

Inthissection,amathematicalmodelofthepantograph–catenarysystemisbuilttogeneratenumericaldataforsubse- quentanalyses.ThecatenaryismodelledbyANCF.TheTCUDmethodpresentedin[41]isemployedtocalculatetheinitial configurationofthecatenary.Thepantographisconsideredasalumpedmassmodel.Thevalidationofthepresentmodel isverifiedaccordingtothebenchmark[42]andthemeasurementdataiscollectedfromtheNorwegianrailnetwork.

2.1. Finiteelementmodelofcatenary

TheANCFisanonlinearfiniteelementapproachtodescribethegeometricalnonlinearityoflargedeformation.Asshown inFig.3,theANCFbeamelementisutilisedtomodelthetensionedwires(includingcontactwire,messengerwireandstitch wire).TheANCFcableelementisadoptedtomodelthedropperwire.Thesteadyarmismodelledbythetrusselement.The clawsandclamps onthe wireareassumed aslumped masses. Foran ANCFbeamelement, thenodaldegree offreedom (DOF)vectorthatcontainsthedisplacementsandthegradientsaredefinedas:

e=

x_i y_i z_i ^∂_∂χ^{xi ∂}_∂χ^yi _∂χ^∂zi x_j y_j z_j ^∂_∂χ^{xj ∂}_∂χ^{yj ∂}_∂χ^zj

T

(1) inwhich,

χ

^{isthelocalcoordinateintheundeformed}configurationrangingfrom0totheelementlengthL₀.Theposition vectorinthedeformedconfigurationrisinterpolatedusingtheshapefunctionmatrixSas

r=Se (2)

Scanbedefinedasfollows:

S=

_S

1 S2 S3 S4

S1 S2 S3 S4

S₁ S₂ S₃ S₄

S₁

( ξ )

⁼¹⁻³

ξ

²⁺²

ξ

³

S₂

( ξ )

=l₀

ξ

+

ξ

³−2

ξ

²

S₃

( ξ )

=3

ξ

²−2

ξ

³

S₄

( ξ )

=l₀

ξ

³−

ξ

²

(3)

Fig. 4. Lumped mass model of pantograph with flexible collectors.

Thestrainenergyobtainedfromthecontributionofaxialandbendingdeformationisexpressedby U=1

2 L0

0

EA

ε

²l +EI

κ

²

d

χ

⁽⁴⁾

inwhich,EisYoung’s modulus,Aisthesectionarea,Iisthemomentinertialofthewire,

ε

l isthelongitudinalstrainand

κ

^{isthecurvature.The}generalisedelasticforcescanbedefinedas Q=

∂

^U

∂

^e

T

=Kee (5)

In this way,the element stiffnessmatrix Ke is obtained. In the shape-finding procedure,the tangent stiffness matrix is usedtocalculatethe incrementalnodal DOFvector ^{e andthe} incrementalunstrained length ^L0. Thecorresponding tangentstiffnessmatricesK_TandK_L canbeobtainedasfollows:

^F=

∂

^Q

∂

^e

^e+

∂

^Q

∂

^L0

^L0=KT

^e+KL

^{L0 (6)}

Similarly, the tangent stiffness matricesof the ANCF cable element can also be derived. It should be noted that the cableelement usedto modelthedropper canonlywithstandtensionbutnot compression.The axialstiffnesschangesto zerowhenthedropper worksincompression.Assemblingtheelementmatricesyields theglobalincremental equilibrium equationforthewholecatenaryasfollows:

^FG=K^G_T

^UC+K^G_L

^L0 (7)

where ^{FG isthe globalunbalancedforce vector. KG}T andK^G_L are the globalstiffnessmatricesrelatedto theincremental nodal displacementvector^UC andthe incrementalunstrained length vector^L0,respectively.It isseenthat [K^G_T K^G_L] is not a square matrix.The total numberof unknownsin Eq.(7)exceeds the total numberofequations, whichleads to undeterminedsolutions.Hence,additionalconstraintconditionsareprovidedtosuppressundesiredmovements, according tothedesignspecifications.Thus,thefollowingadditionalconstraintconditionsaredefined.

• Theverticalpositionsofthedropperpointinthecontactwirearerestrictedtodescribethereservedpre-sag.

• Thelongitudinaldirectionofeachnodeisrestrictedtosuppressthelongitudinalmovement.

• Thetensionsareappliedtostitchwiresandtheendpointsofmessengerandcontactwires.

Withthe help oftheabove threetypesof constraints,[K^G_T K^G_L] isreducedto asquare matrix.The strainedandun- strainedlengthsofalltheelementscanbecalculatedbysolvingEq.(7).Inthisway,theinitialconfigurationofthecatenary isdetermined.IntroducingaconsistentmassmatrixandRayleighdamping,theequationofmotionforthecatenarysystem iswrittenby

M^G_CU¨C

(

^t

)

^+CGCU˙C

(

^t

)

^+KGC

(

^t

)

^UC

(

^t

)

^=FGC

(

^t

)

⁽⁸⁾

2.2. Pantographmodel

The pantographis modelledasalumped massmodelwithflexiblecollectors asshowninFig. 4.Eachcollectorisdis- cretizedintoseveralEuler-Bernoullibeamelements.Themassofthepantographheadisevenlydistributedalongthecollec- tor.Generally,thecollectorismadeoftwotypesofmaterials,namelyGraphiteandAluminium.Thebendingstiffnessofthe collectoriscalculatedbasedonthemixturetheory[43].Apartfromthepantographaerodynamics[44,45],theflexibilityof theframeworkdoesnothaveasignificantcontributiontotheresponsewithinthefrequencyrangeofinterestinthiswork.

Therefore,thepantographframeworkistreatedasalumpedmasshere.Apenaltyfunctionmethodisutilisedtocouplethe twosystems.Basedontheassumptionoftherelativepenetrationgeneratedbetweenthetwocontactsurfaces,thecontact force fc canbecalculatedby:

fc=

ks

δ

0

i f

δ

^>0

i f

δ

≤0 (9)

inwhich,thepenetration

δ

^{canbeevaluatedby}

δ

^=zp−zc−zir (10)

Table 1

Static validation of the present model against benchmark.

Dropper No.

Pre-sag Elasticity

Benchmark (mm) Present (mm) Error (%) Benchmark (mm/N) Present (mm/N) Error (%)

Sup 0 0 0 0.206 0.19257 6.52

1 0 0 0 0.165 0.15647 5.17

2 24 24 0 0.273 0.26774 1.93

3 41 41 0 0.345 0.3268 5.28

4 52 52 0 0.388 0.36832 5.07

5 55 55 0 0.4 0.37509 6.23

6 52 52 0 0.388 0.36832 5.07

7 41 41 0 0.345 0.3268 5.28

8 24 24 0 0.273 0.26774 1.93

9 0 0 0 0.165 0.15647 5.17

Sup 0 0 0 0.206 0.19257 6.52

Table 2

Dynamic validation of the present model against benchmark.

Benchmark Present model Error

Mean [N] 169 169.15 0.09%

Std. (0–20 Hz) [N] 53.91 52.59 2.45%

Std. ed. (0–2 Hz) [N] 38.27 38.25 0.05%

tStd. d. (0–5 Hz) [N] 41.04 41.00 0.10%

(Std. 5–20 Hz) [N] 34.80 32.99 5.20%

Max. [N] 313.22 305.85 2.35%

Min . [N] 60.40 56.22 6.9%

where z_p,z_c andz_ir are thevertical displacements ofthe pantograph head,the contactwire and the irregularity.Using Eq.(9),theequationofmotionforthepantograph–catenarysystemcanbewrittenby

M^GU¨

(

^t

)

^+CGU˙

(

^t

)

^+KG

(

^t

)

^U

(

^t

)

^=FG

(

^t

)

⁽¹¹⁾

in whichM^G,C^G andK^G(^t)^{are themass,dampingandstiffnessmatricesforthewholesystem,} respectively. F^G(^t) ^isthe external force vector. A Newmarkintegration schemeis adopted to solve Eq.(11). The stiffnessmatrix K^G(^t) ^{is updated} accordingtothecatenarydeformationineachtimesteptofullydescribethegeometricalnonlinearityandtheslacknessof droppers.

2.3. Modelvalidation

Twonumericalsimulationsarepresentedtovalidatethepresentmodelofpantograph–catenaryinteraction.Thefirstone istousethereferencemodelinthebenchmark[42]andcomparetheresultswiththemeanvaluesoftheresultsobtained by ten mainstream software. The second one is to use the design data in the Norwegian network to model a realistic pantograph-catenarysystemandcomparetheresultswiththemeasurementdatacollectedbyaregularinspectionvehicle.

Thebenchmarkprovidestheresultsforbothstaticanddynamicvalidations[46].Inthestaticvalidation,thepre-sagand elasticityof thecontactwireobtainedby thepresentmodelare comparedwiththe benchmarkinTable 1.Itshowsthat thepre-sagobtainedby thepresentmethodisexactlythesameasthebenchmark.Themaximumerroroftheelasticityis just6.52%. Thedynamicvalidation isimplementedby comparing thekey contactforce statisticswiththe benchmark.As showninTable2,theresultsofthepresentmodelshowgoodconsistencywiththebenchmark.Themostsignificanterror onlyreaches6.9%fortheminimumcontactforce.

Anothernumericalexampleisimplementedbycomparisonwiththemeasurementdatafromthe“Gardermobanen” rail lineinNorway, goingfromOslotoEidsvoll.The propertyparameters ofthe catenaryfortheanalysissection ispresented inTable3.Theinspection vehicleregularlyrunsat160km/h.Thisspeed isadoptedinthisvalidationandthesubsequent analyses.ThepantographontheroofofaninspectiontrainisWBL85,ofwhichthelumped-massparameterscanbefound inTable4.Inthiscase,thepantographheadismodelledbytwodeformablecollectors.Thefirst-ordernaturalfrequencyof thecollectoriscalculatedas65Hz,whichisconsistentwiththeexperimentalresultsin[47].BasedontheTCUDmethod, theinitial configurationofthecatenaryiscalculatedandpresentedinFig.5.Thecomparisonwiththemeasurement data ispresentedinTable5.Itisseen thatthemaximumerrorofthesimulationresultagainst themeasurementdatareaches 12.18%,whichisstilllowerthantheacceptancethresholdof20%specifiedinthestandard[46].

3. Potentialsolutions

Inthissection,thepotentialsolutionsforidentifyingthetwoclassictypesofshort-wavelengthCWIsarediscussedbased on thesimulation results.Thelocalimperfection with2mmamplitudeis addedtothe positionof620m inthecontact

Table 3

Catenary property parameters.

Total Length 1.012 km

Contact Wire Tension 15 kN

Messenger Wire Tension 15 kN

Stitch Wire Tension 2.8 kN

Contact Wire Area 120 mm ²

Messenger Wire Area 65.8 mm ²

Stitch Wire Area 3.44 mm ²

Contact Wire Linear Density 1.07 kg/m

Messenger Wire Linear Density 0.596 kg/m

Number of spans in contact with pantograph 18

Table 4

Pantograph parameters.

Mass of panhead 5.2 kg

Mass of frame 15.2 kg

Contact spring 100,000 N/m

Spring between panhead and frame 5400 N/m

Spring between frame and train roof 0 N/m

Damping between panhead and frame 40 N s/m

Damping between frame and train roof up 63.5 N s/m

Damping between frame and train roof down 63.5 N s/m

Friction between frame and train roof 7 N

Static uplift force 50 N

Aerodynamic uplift force 0.0068 v ²

Fig. 5. Initial configuration of catenary.

wire.ThePSW-CWIisrepresentedbyasinusoidalfunctionwitha3.3mwavelengthand0.5mamplitude,whichisapplied totheeffectiverangefrom625.9mto643.46m.

3.1. Potentialsolutionforlocalimperfection

The numerical simulation is performed with the abovementioned classic local imperfection in the catenary. An ac- celerometer is placed onthe edge ofeach collector [38]. Inthis work, theacceleration ofthe front collectoris takenas theanalysisobject.Theresultingpanheadaccelerationwith200Hzcut-off frequencyispresentedinFig.6.Itisseenthat the panhead accelerationexperiences asignificant peakwhen thepantograph passesthelocal imperfection.Accordingto [38],theeffectoflocalimperfectioncanbefullydescribedbythepanheadaccelerationwhenthecut-off frequencyreaches 200Hz.However,mostrailwayoperatorsaroundtheworlddonotequiptheinstrumentedpantographwithhigh-sensitive accelerometers.Thissectioninvestigatesthepotentialofusinglow-frequencyaccelerationtoidentifythelocalimperfection.

EEMDisadata-drivenalgorithmwhichhasbeenusedinvariousindustrialbackgrounds[48,49]todecomposeasignalinto severalIMFs.Inthiswork,EEMDisusedtoinvestigatetheeffectoflocalimperfectiononeachIMFoftheaccelerationsignal.

Fig. 6. Acceleration of front collector with 200 Hz cut-off frequency.

The empirical mode decomposition (EMD) can decompose a given signal x(^t) ^{into a number N of IMFs d}j(^t)^{, j}= 1,2,· · ·,N anda residual r(^t) ^{through an iterative procedure. The sumof all the IMFs and theresidual precisely yields} the original signal. However, mode mixingis an urgent issue fortraditional EMD, which largelyrestricts the useof this method [50].Toresolve thisproblem, EEMDis proposed basedonthe dyadicproperty ofEMD whendealingwithwhite noise. EEMDutilises additionalwhitenoiseto ensurethefull physicalmeaningoftheIMFs. Fig.7showstheIMFs ofthe panhead accelerationandtheir spectrums overthewavelength.It isseenthat theeffect ofthelocalimperfectionis only observed from thefirst three IMFs butnot significant in other IMFs. The physical meaningof each IMF can be revealed by observing its spectrum. It isseen IMFs 6 and7 inFigs. 7 (h-i)are relatedto the span length ofthe catenary.IMF 5 in Fig. 7 (f-g) denotes the dropper to dropper (DD)interval. The firstfour IMFs represent the deformation mode of the contact wirerelatedto the 1/16, 1/8,1/4 and1/2 ofDD, respectively. Itis seen that the effectof thelocal imperfection is reflected fromthe mode which has a corresponding wavelength smaller than 1/4 of DD. If the cut-off frequencycan coverthewavelengthof1/8ofDD,thelocalimperfectioncanbesignificantlyreflectedinthepanheadacceleration.Forthe givencase,thedropperintervalsare8–12m,andtheminimum1/8ofDDisaround1m.Assumingthatthetrainspeedis 160km/h,thecut-off frequencyshouldbe improvedto45Hztofullycoverthewavelengthof1/8ofDD.Consideringthe intrinsicnonlinearityofcatenaryandthelossofinformationinthefilter,ahighercut-off frequencyisdesiredinpractice.

The identificationeffectivenessisdiscussed inthenext Section withdifferentcut-off frequencies.It shouldbe notedthat the Norwegian railwayoperator implements theregular inspection vehicles witha measurement distancestep of0.5 m, whichisclosetotheNyquistfrequencyandcannotfullycoverthecontributionof1/8ofDDintherealmeasurement.Butit mayhavethepotentialtobeimprovedtocoverapartofthe1/8ofDDwithoutasignificanteffort.Basedonthisidea,the EEMDcanbeusedtodecomposethesignalandextracttheIMFrelatedto1/8ofDD.Thewholeidentificationprocedureis giveninSection4withtheapplicationtothenumericalandmeasurementdata.

3.2. Potentialsolutionforpsw-cwi

ThenumericalsimulationsareperformedwithandwithouttheabovementionedPSW-CWI.Theresultingcontactforces filtered within0–20Hzare presentedinFig.8.It isseenthat theearly PSW-CWIonly causesaslightdisturbanceto the contactforce time-history.The contactforcestandard deviationincreasesby1.04%when thePSW-CWIisintroduced.The traditional indices specified inthe standard [2]are not allowed to quantifythe effect ofPSW-CWIon the contactforce.

ThroughthefrequencyanalysisinFig.9,itisseenthatthePSW-CWIwavelengthisthesameasoneoftheoriginalcontact force wavelengthcomponents.The PSW-CWIonlyleadstoaslightdifferenceintheenergyatawavelengthof3.3m.The

Table 5

Dynamic validation of present model against measurement data.

Measurement Present model Error

Mean [N] 92.14 92.34 0.22%

Std. (0–20 Hz) [N] 11.53 10.59 8.15%

Std. (0–5 Hz) [N] 9.01 8.86 1.66%

Std. (5–20 Hz) [N] 6.33 5.72 9.64%

Max . [N] 126.64 125.44 0.95%

Min. [N] 52.85 57.14 8.12%

Range of vertical position of the point of contact [mm] 78 68.5 12.18%

method proposed in[36] is not allowed toidentify thistype ofCWI. In thiswork, anew approachis proposed to solve thisproblem. ThePSW-CWIcanberegardedasaseriesoflocalsinusoidaldisturbanceswithaspecificwavelengthto the contactforce.ThepresenceofsuchdisturbancecanaffecttheGaussiandistributionofthesignalatspecificfrequencies.This featurecanbeutilisedtoevaluatetheeffectofPSW-CWIandidentifythedeviatingwavelength.Thecommonindicatorthat canindicatethenon-Gaussianbehaviourinthefrequencydomainisthespectralkurtosis(SK)[40].TheSKofagivensignal canbecalculatedaccordingtothefollowingmathematicalprocedure.

Fig. 7. IMFs of panhead acceleration and their frequency spectrums: a) original signal and its spectrum; b) IMF 1 and its spectrum; c) IMF 2 and its spectrum; d) IMF 3 and its spectrum; e) IMF 4 and its spectrum; f) IMF 5 and its spectrum; g) IMF 6 and its spectrum; h) IMF 7 and its spectrum; i) IMF 8 and its spectrum;

Fig. 7. Continued

Fig. 8. Contact force with 20 Hz cut-off frequency.

Fig. 9. Spectrum of contact force with and without PSW-CWI.

Foragivensignalx(^t)^{,theshort-timeFouriertransform(STFT)S}(^t,f)^{canbecalculatedby} S

(

^t,f

)

^{= +∞}

−∞ x

(

^t

)

^w

(

^t−

τ )

^{e−2πf tdt (12)}

inwhich,w(^t−

τ

)^{isthewindowfunctionusedinSTFT,and}

τ

^{isthewindowlength.TheSKcanbecalculatedas:}

K

(

^f

)

=

|

^S

(

^t,f

) |

⁴

|

^S

(

^t,f

) |

²

^{2−2, f=0 (13)}

where

·

^isthetime-averageoperator.

Through theabove procedure,the SKsofcontactforce withandwithoutPSW-CWIare presentedinFig.10.Itisseen thattheSKundergoesasignificantincreaseatthewavelengthof3.3m,whichindicatesthattheGaussianbehaviourofthe contactforce atthisspecific wavelengthdeteriorates.In thisway,thedeviating wavelengthcan be determined.However, it shouldbe notedthat anSK benchmarkto representthe healthystatues shouldbe providedforthe comparison.As for the measurement data, multipleinspections can be used to comparetheir SKs anddetermine the deviating wavelength.

Then EEMD can be used to extract the IMF containing thedeviating wavelength.The PSW-CWI can be localised by the time-frequency analysis. The whole identification procedure is givenin Section 5 with the applicationto numerical and measurementdata.

Fig. 10. Spectral kurtosis of contact force with and without PSW-CWI.

Fig. 11. Identification procedure for local impact.

4. Identificationoflocalimperfection

Through the analysisinSection 3.1, thissection proposesthe identificationprocedure forthelocalimperfection using theaccelerationwiththecut-off frequencycoveringa1/8ofDDwavelength.Thenthenumericalandmeasurementdataare adoptedtovalidatetheproposedmethod.

4.1. Identificationprocedure

The identification procedurefor thelocal imperfectionfrom thepanhead acceleration isproposed inFig. 11. The IMF relatedto1/8ofDDisextractedfromtheaccelerationsignalthroughtheEEMD.DifferenttothepreviousmobileRMS[38], a moresensitiveindicator,instantaneousenergyobtainedby theHilberttransformisusedheretoindicatethepositionof local imperfection.The localimperfectionis localisedby searchingthe extremely largepeak ofthe instantaneousenergy.

ThemathematicalprocedurefortheHilberttransformisgivenasfollows.

Foragiventimeseriesx(^t)^{,ananalyticalsignalz}(^t)^{canbeformulatedas}

z

(

^t

)

^=x

(

^t

)

^+iy

(

^t

)

⁽¹⁴⁾

inwhich,iistheimaginaryunit,y(^t)^{istheHilberttransformofx}(^t)^{,whichcanbecalculatedby} y

(

^t

)

= 1

π

_+∞

−∞

x

(

^s

)

t−sds (15)

Inthisway,theanalyticalsignalcanbere-writteninthefollowingform:

z

(

^t

)

^=a

(

^t

)

^{eiθ(t) (16)}

inwhich,a(^t)^istheinstantaneousamplitudeand

θ

(^t)^istheinstantaneousphase.Theinstantaneousenergycanbecalcu- latedby

|

^a(^t)

|

^2,andtheinstantaneousfrequencyisdefinedby ^dθ(_dt^t).

4.2. Applicationtonumericalresults

Thevalidationofthepresentidentificationprocedureisdemonstratedthroughseveralnumericalsimulations.Thetrain speed inthesenumericalexamplesissetassameastheinspectionvehiclespeed160km/h.The localimperfectionswith differentamplitudesandpositionsareincludedinthecatenarytoperformthenumericalsimulations.

1) Withdifferentcut-off frequencies

The obtainedsignal in Fig. 6 is adopted here to analyse the identification effectiveness and feasibility. The accelera- tions filtered with 40 Hz, 50 Hz, 60 Hz and 70 Hz cut-off frequencies are presented in Fig. 12. When the cut-off fre- quency moves up to 60Hz, the impact of localimperfection can be significantly observed fromthe acceleration, which is not difficult to be identified through a traditionalmobile RMS [38].When the cut-off frequency is justover the crit- ical value (45 Hz) to cover the wavelength of 1/8 of DD, the effect of local imperfection is not distinct in the accel- eration (see Fig. 12(b)), and the mobile RMS may lose its effectiveness. The EEMD is used to decompose the accel- eration, and the IMFs’ related to the 1/8 of DD with different cut-off frequencies are presented in Fig. 13. The cor- responding instantaneous energies are presented in Fig. 14. It is seen that the local imperfection cannot be identified when the cut-off frequency is 40 Hz, which is lower than the critical value (45 Hz) to cover the wavelength of 1/8 of

Fig. 12. Panhead accelerations with different cut-off frequencies: (a) 40 Hz; (b) 50 Hz; (c) 60 Hz; (d) 70 Hz.

Fig. 13. IMFs related to 1/8 of DD with different cut-off frequencies: (a) 40 Hz; (b) 50 Hz; (c) 60 Hz; (d) 70 Hz.

DD. Even though the local imperfection effect cannot be distinctly seen in the panhead acceleration time-history with 50 Hz cut-off frequency, it can still be accurately identified and localised by the present method. The increase of the cut-off frequency makes theidentificationresults moresignificant, whichprovides thefeasibilityto identifysmallerlocal imperfections.

Fig. 14. Instantaneous energies of the IMF related to 1/8 of DD with different cut-off frequencies: (a) 40 Hz; (b) 50 Hz; (c) 60 Hz; (d) 70 Hz.

Fig. 15. Identification of local imperfection at different positions using 50 Hz cut-off frequency: (a) Panhead acceleration; (b) Extracted IMF related to 1/8 of DD; (c) Instantaneous energy.

2) Withdifferentpositions

Threelocalimperfectionsareaddedtothepositionsat440mand520minthecontactwiretoperformthenumerical simulations.Usinga50Hzcut-off frequency,theresultingaccelerationsarepresentedinFig.15(a).Thecorresponding IMF foreachcaseisextractedandpresentedinFig.15(b),whichisrelatedtothe1/8ofDDwavelengthcomponent.Employing theHilberttransform, theinstantaneousenergies arepresentedinFig.15(c).It isseenthatsome significantpeakscanbe observed atthe positions oflocalimperfection. Apartfromthe onescausedby the localimperfection, some other peaks

Fig. 16. Identification of local imperfection at different positions using 60 Hz cut-off freqeuncy: (a) Panhead acceleration; (b) Extracted IMF related to 1/8 of DD; (c) Instantaneous energy.

can also be observedin theinstantaneous energy, whichmay be attributedto theimplication ofelasticitydiscontinuity, wave reflectionandlumpedmassinthecontactwire.The bestsolutionistoincreasethecut-off frequencytocovermore deformablemodes.Theidentificationresultswith60Hzcut-off frequencyarepresentedinFig.16.Itisseenthattheidenti- ficationapproachprovidesmorereliableresultswithahighercut-off frequency.Ifthecut-off frequencycannotbereached, somepotentialsolutionsarediscussedinSection6.

4.3. Applicationtomeasurementdata

Thepresentidentificationprocedureisimplementedwiththemeasurementpanheadaccelerationoftwotensilesections (from41.47to43.36km)inthe“Gardermobanen” railwayline.Thedistancestepforthismeasurementis0.5m.Thepanhead accelerationandits spectrumare presentedinFig.17(a)and(b),respectively. Itshould benoticed thatthemeasurement dataare filteredwith20Hzcut-off frequency accordingtothe assessmentstandard[2],whichcannotevenfullydescribe the1/4ofDDwavelengths.ThroughEEMD,thefirstfiveIMFsandtheirspectrumsarepresentedinFig.17(b–f).Itisseen that thefirst onerepresentstheintrinsicmode relatedtothewavelengthof1/4 ofDD.Throughthe Hilberttransformto the first IMF, theinstantaneous energy versus thedistance ispresented inFig. 18.It is seen that themaximum peak of instantaneous energy appears ataround 42.37 km,whichis the overlaptransition ofthe two tensilesections(according to thedesignspecification).Thevariation ofthecontactwireheightinthe transitionoftwo tensilesectionspresentsthe commonlocalimperfectionforthepantograph-catenaryinteraction.Thesmallerlocalimperfection,whichaffectsthehigher- order deformation mode of the contactwire cannot be identified by the low cut-off frequency. The cut-off frequencyis desiredto moveup tocoverthedoublesamplingstep,whichcan describewavelengthsdown to1/8ofDD andhavethe potentialtoidentifysmallerlocalimperfection.

5. IdentificationofPSW-CWI

AccordingtotheanalysisinSection3.2,thissectionproposestheidentificationprocedureforthePSW-CWI.Thenumer- icalandmeasurementdataareadoptedtovalidatetheproposedmethod.

5.1. Identificationprocedure

The identificationprocedureforthePSW-CWIfromthecontactforceis proposedinFig.19.TheSKofcontactforce is utilised toidentify thedeviating wavelength.Then EEMD isusedto extractthe IMFcontainingthedeviating wavelength.

Fig. 17. (a) Original measured panhead acceleration and its spectrum; (b) IMF 1 and its spectrum; (c) IMF 2 and its spectrum; (d) IMF 3 and its spectrum;

(e) IMF 4 and its spectrum; (f) IMF 5 and its spectrum.

Fig. 18. Instantaneous energy of the first IMF.

Fig. 19. Identification procedure for PSW-CWI.

Thetime-frequencyanalysisisperformedtolocalisethePSW-CWIbasedonthesmoothedpseudo-Wigner-Villedistribution (SPWD).ThegeneralisedquadraticTFRC(^t,

ω

;

φ

)^ofthecorrespondingIMFseries f(^t)^{withthekernel}

φ

(^t,

ν

)^isdefinedas [51]

C

(

^t,

ω

;

φ )

=_4π¹2

φ (

^t,

ν )

^{e−jωτe−jν(t−u)}

f

(

^u+^τ₂

)

^f∗

(

^u−^τ₂

)

^dud

τ

^d

ν

⁽¹⁸⁾

wherethe f^∗denotescomplexconjugationoff.tand

ω

^aretheinstantaneoustimeandangularfrequencyrespectively.u,

τ

andvaretherunningpositions,timeandfrequencyvariablesintheintegrationrespectively.ThequadraticTFRisobtained through a triple integral, including dual Fourier transform in time (t-u) and

τ

^{with the} autocorrelation of f(^t)^{. Mainly} C(^t,

ω

;

φ

) ^{describesthe signal energy}distribution inthe time-frequencydomain. When thekernel

φ

(^t,

ν

)=1,C(^t,

ω

;

φ

) isa WVD(Wigner-VilleDistribution),whichisthebasicformofthequadraticclass.Accordingto[4],thebasicformdoes not havean excellent performance todescribe thetime-frequencycharacteristics ofcontactforces,astheWVD’svariants sufferfromsevereinterferenceofcross-term.Inthispaper,thesmoothedpseudo-WVD(SPWD)isemployedtoreducethe interferenceofcross-term.ThekernelofthistypicalquadraticTFRisexpressedasfollows:

φ (

^t,

ν )

⁼

η τ

2

η

^∗

−

τ

2

G

( ν )

⁽¹⁹⁾

where

η

(

τ

) ^{isthetime window. G}(

ν

) ^{isafrequency window.Comparedtothe basicWVDkernel,theSPWD kernelacts} asafilteroftheWVDforthepurposetosuppressthecross-term.TheSPWDapplies windowfunctionsindependentlyon boththetimeandfrequencydomainsoftheWVD,whichstronglydependsonthepriorinformationofthetargetsignalto suppressthecross-termscorrectly.

5.2. Applicationtonumericalresults

Thevalidationofthepresentidentificationprocedureisdemonstratedthroughseveralnumericalsimulations.ThePSW- CWIswithdifferentamplitudesandwavelengthsareincludedinthecatenarytoperformthenumericalsimulations.

1) Withdifferentamplitudes

Three typesof PSW-CWIwithawavelength of3.3m andtheamplitudes of0.4mm,0.5mm and0.6mm are added to thecontactwire.The amplitudesofCWI areslight withrespect tothe contactwirediameter(which is6.15mm)and representtheearlywear.TheeffectiverangeofPSW-CWIissetfrom626mto643.5m.Theresultingcontactforcesfiltered with the20Hz cut-off frequency are presentedin Fig.20(a).The SKs forthesethree casesare calculated andpresented in Fig.20(b).Tofacilitatethecomparison,the SKswithout PSW-CWIarealso presentedasthebenchmark. Thedeviating wavelengthsforthesethreecasescanbeidentifiedbysearchingforthesignificantincreaseofSKagainstitsoriginalvalue.

Then EEMDisappliedtoextractthecorresponding IMFcontainingthedeviating wavelength.TheextractedIMFs andtheir spectrums areshowninFig.20(c).It isseenthatthe deviatingwavelengthof3.3mcanbe observedinthespectrumfor each case.The timehistoryoftheextractedIMFundergoesaswellintheeffectiverangeofthePSW-CWI. Employingthe SPWD,theTFRforeachcaseispresentedinFig.20(d).Lookingatthedeviatingspatialfrequencyof0.303m⁻¹,theenergy oftheextractedIMFshowsasuddenincreaseintheeffectiverangeofthePSW-CWI.

2) Withdifferentwavelengths

ThePSW-CWImayhavethesamewavelengthcomponentsasthecontactforce,whichposesthechallengetoidentifythe deviatingwavelengthfromthecontactforce.Inthissection,anotherwavelengthcomponentofthecontactforceisadopted to representthedominantwavelength ofPSW-CWI. Accordingto thecontactforce spectrumin Fig.8,the wavelengthof 3.6 m isadopted, andtheamplitude ischosen as0.5mm. The effectiverange isset asthe sameas theabove analysis.

Though the numerical simulation, theresulting contact forces are presented in Fig.21(a). The resulting SK asshown in Fig. 21(b)indicates anon-Gaussianbehaviour causedbythe PSW-CWIataroundthespecific wavelength.ThroughEEMD, thecorrespondingIMFcontainingthedeviatingwavelengthisextracted.TheTFRoftheIMFindicatesthataPSW-CWIwith aspatialfrequencyofabout0.28m⁻¹existsataround630minthecontactwire.Thenumericalexampledemonstratesthe validationofthepresentidentificationapproachforthePSW-CWIwithdifferentwavelengths.

Fig. 20. Identification of PSW-CWI with different am plitudes: a) Resulting contact forces; b) Spectral kurtosis; c) Extracted IMFs containing deviating wavelength and their spectrums; d) Time-frequency representations.

5.3. Applicationwithmeasurementdata

Thepresentidentificationprocedureisimplementedusingthemeasurementcontactforce.Throughtheaboveapplication examples withnumericaldata,itisseenthattheessential partoftheidentificationprocedureistoidentifythedeviating wavelength,whichrequiresabenchmarkofSKforcomparison.Inthenumericalsimulation,thecontactforcewithoutPSW- CWI canbetakenasthebenchmark.Inactual operation,themeasurementdatafrommultipleinspectionscanbeusedto

Fig. 21. Identification of PSW-CWI with 3.6 m wavelength: a) Resulting contact forces; b) Spectral kurtosis; c) Extracted IMFs containing deviating wave- length and their spectrums; d) Time-frequency representation.

comparetheirSKstoidentifythedeviatingwavelength.ThecontactforcesfromtwotimesofmeasurementinOslo-Eidsvoll railway arepresented inFig. 22(a).Due tothe longinterval betweentwo measurements, they can describethe catenary healthstatusfordifferentserviceperiods.ThroughthecomparisonoftheirSKsasshowninFig.22(b),itisseenthattheSK ofmeasurement2undergoesasignificantincreaseatthewavelengthof3.23m,whichdoesnotexistinmeasurement1.It canbeexpectedthatadeviatingwavelengthmayexistinthecontactforceofmeasurement2.ThecorrespondingIMFscon- tainingthedeviatingwavelength,togetherwiththeirspectrums,arepresentedinFig.22(c).Itisseenthatthespectrumof measurement2issignificantlyhigherthanmeasurement1ataroundthedeviatingwavelength.Tofacilitatethecomparison, theTFRsoftheIMFoftwomeasurementsatthedeviatingwavelengthof3.23marepresentedinFig.22(d).Itisseenthat severalTFR peaksofmeasurement2canbe observed.ThePSW-CWIwitha3.23mwavelengthisa commonwavelength ofweargeneratedinlong-termoperations.Themostserious PSW-CWIappearsataround42.9km,whichdeservesspecial attentioninthemaintenancefollowingmeasurement2.

6. Discussions

Accordingtotheaboveanalysisresults,somediscussionsaredrawnhere.

6.1. Localimperfection

The localimperfection contributes a transient impact to the acceleration. The previous researches mainly used high- frequency acceleration to identifyit. The numerical analysis resultsindicate that thecut-off frequency doesnot haveto reachanextremelyhighvalue.Itissuggestedtoincreasethecut-off frequencytocoverthe1/8ofDDwavelength,inwhich theeffectoflocalimperfectioncanbereflectedinthepanheadacceleration.However,thecurrentmeasurementacceleration

Fig. 22. Identification of PSW-CWI using measurement data: a) Contact forces; b) Spectral kurtosis; c) Extracted IMFs containing deviating wavelength and their spectrums; d) Comparison of TFRs at deviating wavelength.

obtainedfromtheNorwegian railwaynetworkcanonlyreflect apartofthe1/4ofDD wavelength.Onlysomesignificant imperfections, such as the overlap section, can be identified withthe currentmeasurement response. The difference of the identificationapproachinthiswork againstthe previousone [38] isthat theEEMDis usedtoextract thehigh-order deformation mode whichismoresensitive tothe transientimpact insteadofimplementing RMStotheoriginal signal. A shortfall of theidentification approach can be seen inFig. 15. The elasticitydiscontinuity, wave reflection, lumped mass inthe contactwire,andeventheerrorsinsignal processingcan alsointroducepeaksintheinstantaneous energy,which maydisturbtheidentificationresults.Thepresentmethodcanonlygivepotentialpositionsoflocalimperfectionwithalow cut-off frequency.Thebestsolutionistoimprovethecut-off frequency,whichallowsfortheidentificationofsmallerlocal imperfections.Anotherpotentialsolutionistocomparemultiplemeasurementresultsandtaketheincidentalpeaksasthe suspectlocalimperfections.Thephysicalmeaningofthesepeaksdeservestoberevealedinthefuture.

6.2. PSW-CWI

The time-frequency analysismethodhasbeen proven toeffectivelycapturethe PSW-CWIfromthecontactforce [35]. However,thePSW-CWImayhavethesamewavelengthasthecontactforce.Anindicatorisdesiredtoidentifythedeviating wavelengthbeforeimplementingthetime-frequencyanalysistolocalisethePSW-CWI.Theabovenumericalsimulationhas demonstratedthattheSKcanbeusedtoindicatethenon-GaussianbehaviourcausedbythePSW-CWI,whichisapotential indicator todetermine thedeviating wavelength.It canalsobe seenfromFig.22that themeasurement dataobtainedin differentserviceperiodsshowasignificantchangeofthecontactforceSKatsomecertainwavelengths.Thenextstepisto collaboratewiththemaintenancedepartmenttovalidatetheidentificationresults.

7. Conclusions

Thispaperpresentspotentialsolutionsforidentifyingtwocommontypesofshort-wavelengthirregularities.Thepresent identificationapproachesarevalidatedwithseveralnumericalexamples.Theapplicationexampleswithmeasurementdata indicate that presentmethodshave thepotential to beused inactual operation withsome improvementsinthe current monitoringtrains.Themainconclusionsaresummarisedasfollows:

1) The local imperfection can be detected fromthe panhead acceleration withhigher orders of deformable modes. The analysisresultsindicatethatcut-off frequencyshouldbeimprovedtocoverthe1/8ofDDwavelength.EEMDisutilised