determination of mapping relation between wheel

Research ArticleDetermination of Mapping Relation between WheelPolygonalisation and WheelRail Contact Force forRailway Freight Wagon Using Dynamic Simulation

Jian Mu Jing Zeng Qunsheng Wang and Hutang Sang

State Key Laboratory of Traction Power Southwest Jiaotong University Chengdu 610031 China

Correspondence should be addressed to Jing Zeng zengswjtueducn

Received 8 April 2021 Accepted 12 September 2021 Published 7 October 2021

Academic Editor Davood Younesian

Copyright copy 2021 JianMu et alis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

e polygonal wear around the wheel circumference could pose highly adverse influences on the wheelrail interactions andthereby the performance of the vehicle system In this study the effects of wheel polygonalisation on the dynamic responses of afreight wagon are investigated through development and simulations of a comprehensive coupled vehicle-track dynamic modele model integrates flexible ballasted track and wheelsets subsystem models so as to account for elastic deformations caused byimpact loads induced by the wheel polygonalisation Subsequently the vehicles with low-order polygonal wear whether in emptyor loaded conditions are simulated at different speeds considering different amplitudes and harmonic orders of the wheelpolygonalisation and thus the mapping relation between wheelrail impact force and wheel polygonalisation is obtained eresults reveal that the low-order wheel polygonalisation except 1st order and 3rd order can give rise to high-frequency impact loadsat the wheelrail interface and excite 1st-bend modes of the wheelset and ldquoP2 resonancerdquo leading to high-magnitude wheelrailcontact force at the corresponding speed

1 Introduction

Wheel polygonalisation is a form of wheel circumferentialuneven wear also referred to as wheel polygonalisation orwheel corrugation which generally exists in vehicles [1]e periodic circumferential wear has been associatedwith high magnitudes of high-frequency impact loads atthe wheelrail interface which contribute to undesireddynamic responses and reduced fatigue lives of the ve-hicle-track substructures especially with higher axle loadsand high-speed operations [2]

Owing to highly adverse effects of wheel polygonali-sation considerable efforts have been made to understandthe mechanisms leading to periodic circumferential defectsin railway wheels Wheel polygonalisation was initiallyobserved in European railways [3] e studies suggestedthat wheel polygonalisation is strongly related to bendingvibration of the wheelset track properties and dynamicunbalances in wheels Nielsen et al [4 5] presented a

comprehensive review of different types of wheel defectstogether with associated potential wear mechanisms andthe resulting influences on dynamic responses of the ve-hicle e study also proposed the wheel removal criterionin the presence of polygonalisation deformities e in-fluences of wheel out-of-roundness were also reviewed byBarke and Chiu [6] including the wheelrail impacts railfatigue rail joint deterioration sleeper degradation andnoise generation Liu and Zhai [7] investigated verticaldynamic wheelrail interaction resulting from two types ofout-of-round wheels at high speeds by a vertical vehicle-track coupled dynamics modele results demonstrate theinfluence of the out-of-round wheel on vehicle system ismainly related to the wheelset vibration and derivative ofwheel radial deviations can effectively reflect dynamicwheelrail contact force

A few studies have shown that high-magnitude impactloads could excite different bending and torsional defor-mation modes of the wheelset [8] which could cause higher

HindawiShock and VibrationVolume 2021 Article ID 4677851 13 pageshttpsdoiorg10115520214677851

lateral slippage of the wheel and wheelrail material ex-cavation and thereby enlargement of out-of-round (OOR)deformities Jin and his research team [9 10] experi-mentally investigated the effects and growth of polygonalwear of subway wheels and concluded that the ninth-orderpolygonal wear observed in wheel deformations was due tothe first bending mode of the wheelset e effects of wheelpolygonalisation on the dynamic responses of a high-speedrail vehicle are investigated through development andsimulations of a comprehensive coupled vehicle-trackdynamic model by Wu [11] e results suggested that thehigh-order wheel polygonalisation could give rise to high-frequency impact loads at the wheelrail interface andexcited some of the vibration modes of the wheelset leadingto high-magnitude axle box acceleration and dynamicstress in the wheelset axle e effects of the wheel poly-gonization on the fatigue damage of the gearbox housing ofa high-speed train are investigated through a multibodysystem (MBS) railway vehicle model by Wu [12] e re-sults stated that the fatigue damage with a 20th-orderpolygonal wear is 63 larger than that without the po-lygonal wear on the wheel Moreover the torsional vi-bration of the gear transmission system [13] and wheelrailnoise [14] are also affected by wheel polygonizationHowever almost all of the research focuses on single orderwheel polygonalisation obtained in high-speed vehicles orsubways

Wheel polygonalisation is common in freight wagon andits wear depth changes exponentially with time [15] It isimportant to establish the mapping relationship detectingwheel polygonalisation by the trackside detection for freightwagon condition repair In this study a comprehensivecoupled vehicle-track dynamic model of freight wagon isestablished so as to investigate the effects of low-order wheelpolygonalisation on wheelrail contact force e modelintegrates flexible ballasted track and wheelset subsystemmodels so as to account for elastic deformations that mayoccur in the presence of repetitive impact loads induced bythe wheel polygonalisation at the wheelrail interface eeffects of amplitude and harmonic order of wheel polygonalwear are subsequently investigated and the correspondingmapping relations are obtained to detect polygonal wheel bytrackside detection

2 Establishment of Vehicle-Track Model

A comprehensive model of a top open wagon vehicle-tracksystem is formulated through integration of models offlexible wheelsets a ballasted track and the freight wagonFigure 1 illustrates the simulation scheme comprising finiteelement (FE) models of the flexible wheelset and the bal-lasted track coupled with the open wagon model in theSIMPACK platform [11] e FE model of the wheelset isused to determine its modal properties via the eigen analysise modal vectors are subsequently integrated to the wagonmodel using the finite element multibody systems (FEMBS)interface available in SIMPACK [11] e resulting dynamicwheelrail contact forces the displacement of contact spotand speed are obtained to serve as excitations to the ballasted

track model to evaluate the deflection response of the tracke rail deflection response is subsequently integrated to thevehicle model using the SIMAT (SIMPACK-MATLAB)cosimulation interface as shown in Figure 1 to study itsdynamic responses in the presence of a wheel polygonalwear In simulation the track deflection is solved by a newfast numerical integration method [16] with the step size of00001s e component modelsrsquo formulations are describedin the following subsections

21 Flexible Wheelset Model e FE model of wheelset isdiscretized into 22066 Solid 185 elements as shown inFigure 1 e modal characteristics of wheelset are obtainedthrough eigen analysis e substructure method is intro-duced to deal with the wheelset FE model to speed up thedynamic calculation and the model DOF is transformedinto the structural model represented by the master DOFe nodes DOF where the force is applied or output and theboundary conditions are calculated is defined as the masterDOF and the other defined as the slave DOF e sub-structure method is based on Guyan reduction methodcondensing a set of elements into a superelement by matrixtransformation [17] e dynamic equation of the FE modelcan be expressed as

Meurou + D _u + Ku F (1)

where M K and D are mass matrix stiffness matrix anddamping matrix respectively u is displacement matrix andF is load matrix Assuming that there aremmaster DOF ands slave DOF the matrix form of (1) can be expressed as

Mmm Mms

Msm Mss1113890 1113891

euroumm

eurouss1113890 1113891 +

Dmm Dms

Dsm Dss1113890 1113891

_umm

_uss1113890 1113891

+Kmm Kms

Ksm Kss1113890 1113891

umm

uss1113890 1113891

Fmm

01113890 1113891

(2)

where Kmm Kms Ksm and Kss are mtimesm mtimesm stimesm andstimes s stiffness matrices respectively M and D are the cor-responding mass and damping matrices and Fmm is the loadvector acting on the master DOF Considering that Guyanreduction theory is based on the static transformation ofKu F [18] uss can be assumed as

uss minusKminus1s Ksmumm (3)

us

u umm

uss1113890 1113891

I

minusKminus1ss Ksm

1113890 1113891umm (4)

e DOF conversion matrix can be defined as

T I

minusKminus1ss Ksm

1113890 1113891 (5)

Substituting (4) and (5) into equation (1) the motionequation of the finite element model with reduced degrees offreedom is

2 Shock and Vibration

Mlowastmm euroumm + Dlowastmm _umm + Klowastmmu Flowast (6)

where

Μlowastmm TTMT

Dlowastmm TTDT

Klowastmm TTKT

Flowastmm TTF

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(7)

e selected master DOF must retain the structuralfeatures of the components as well as the master nodessuch as hinge points and force action points For thispurpose a total of 132 master nodes located on 13 crosssections of the wheelset along the axle direction are se-lected Each of the wheel and axle end has 2 cross-sectionsAdditional 5 cross-sections are selected on the axle be-tween the wheels Considering that the wheelrail impactforce induced by wheel polygonal wear can excite somemodes of the wheelset a total of 3 vibration modes withfrequency up to 229Hz are considered to determine thewheelset response and 3 modes of the wheelset are shownin Figure 2

22 e Ballasted Track Model Figure 3 illustrates theballasted track is composed of rail fasteners sleepers andballast e rail is represented by a finite length continuousTimoshenko beam discretely supported on ballast which isrepresented by mass block connected by spring-dampingelement via fasteners e model is formulated for the 100mrail with the sleeper bay of 0625m which is considered torepresent characteristics of the infinite length rail withreasonable accuracy and computational efficiency [19]

Both ends of the rail are assumed to be fixed while eachrail is subject to moving loads attributed to the lateral andvertical wheelrail contact forces Fwryj(t) and Fwrzj(t)(j 1sim4) e lateral and vertical forces due to discretesupports are represented as Fsyi(t) Fszi(t) (i 1simNs) whereNs is the number of supports Figure 3(c) also illustrates themoments caused by the wheelrail contact forces MGj(t)(j 1sim4) and the support forces Msi(t) (i 1simNs) which aretaken about the center of the rail xwj (j 1sim4) andxsi(i 1simNs) define the position of each wheel and the ithdiscrete support along the rail respectively

e governing equations describing the vertical z andbendingΨy deflections of the rail subjected to vertical wheelrail contact forces Fwrzj and support forces Fszi are obtainedas [20]

ρAr

z2z(x t)

zt2 + κzGAr

zψy(x y)

zxminus

z2z(x t)

zx21113890 1113891 minus 1113944

Ns

i1Fszi(t)δ x minus xsi( 1113857 + 1113944

Nw

j1Fwrzj(t)δ x minus xwj1113872 1113873

ρIy

z2ψy(x t)

zt2 + κzGAr ψy(x t) minus

zz(x t)

zx1113890 1113891 minus EIy

z2ψy(x t)

zx2 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)

where ρ is the mass density Ar is the rail cross-sectionalarea G is the shear modulus E is Youngrsquos modulus and Iyis the second moment of area of the rail cross-sectionabout the y-axis In the above equations κz is the verticalshear coefficient κz 05329 δ(x) is the Dirac delta

function and Nw is the number of wheelset considered inthe model

Similarly the governing equations describing the lateraly and yawΨz deflections of the rail subjected to lateral wheelrail contact forces Fwryj and support forces Fsyi are

Flexible Wheelset

FEMBSSIMAT

Wheelrail force

Ballasted trackspeedforcedisplacement

displacementv

Figure 1 Vehicle-track dynamic system model FEMBS finite element multibody systems

Shock and Vibration 3

ρAr

z2y(x t)

zt2 + κyGAr

zψz(x t)

δxminus

z2y(x t)

zx21113890 1113891 minus 1113944

Ns

i1Fsyi(t)δ x minus xsi( 1113857 + 1113944

Nw

j1Fwryj(t)δ x minus xwj1113872 1113873

ρIz

z2ψz(x t)

zt2 + κyGAr ψz(x t) minus

zy(x t)

zx1113890 1113891 minus EIz

z2ψz(x t)

zx2 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(9)

Kfz

Kr

Cr

Cfz

Ksbz

Kbz

Csbz

Cbz

Fastening

Rail

Ballast

Sleeper

+infin-infin

(a)

Kfy Kfy

KsbyKfz

Ksbz

Kbz Cbz

Csbz Csby

KfzCfz CfzCfy Cfy

2dB

Sleeper ys

zs

CrKr BallastBallast

Φs

zbL zbRCbzKbz

(b)

Fwrzj e

a

b bi

Fwryj

F1sy

F1szi F2szi

F2syi

MGj

Msi

hr

(c)

Figure 3 Schematic of ballasted track model (a) main view and (b) side view

Wheel incline samedirection mode 229 Hz

1st-bending mode115 Hz

Torsion mode100 Hz

Figure 2 Selected vibration modes of the wheelset


where Iz is the second moment of area of the rail crosssection about the z-axis and κy is the lateral shear coefficientκy 04507

e equation of motion describing the torsional de-flection Ψx of the rail subjected to the wheelrail forces andsupport forces is formulated as

ρI0z2ϕ(x t)

zt2 minus GK

z2ϕ(x t)

zt2 minus 1113944

Ns

i1Msi(t)δ x minus xsi( 1113857

+ 1113944Nw

j1MGj(t)δ x minus xwj1113872 1113873

(10)

where I0 is the polar moment of inertia of the rail crosssection and GK is the torsional stiffness coefficient

e above governing partial differential equations(7)sim(9) describing the deflections of the rail can be convertedto a series of ordinary differential equations using thevariable separation method in terms of generalized coor-dinates as

z(x t) 1113944

Nz

k1Zk(x)qzk(t)

ψz(x t) 1113944

Nψz

k1ψzk(x)wzk(t)

y(x t) 1113944

Ny

k1Yk(x)qyk(t)

ψy(x t) 1113944

Nψy

k1ψyk(x)wyk(t)

ϕ(x t) 1113944

Nϕ

k1ϕk(x)qTk(t)

(11)

where qyk(t) qzk(t) qTk(t) wzk(t) and wyk(t) are thevectors of corresponding modal coordinates N in theabove formulation relates to the number of modes con-sidered for each coordinate and Zk(x) Ψzk(x) Yk(x)Ψyk(x) and Φk(x) are the normalized shape functions ofthe rail Using the condition that the rail is simply sup-ported at two ends the normalized shape functions aregiven as

Zk(x)

2

ρIzl

1113971

sinkπl

x1113888 1113889

ψzk(x)

2

ρIyl

1113971

coskπl

x1113888 1113889

Yk(x)

2

ρIyl

1113971

sinkπl

x1113888 1113889

ψyk(x)

2

ρIzl

1113971

coskπl

x1113888 1113889

ϕk(x)

2

ρI0l

1113971

sinkπl

x1113888 1113889

(12)

Upon substituting the above relations in the governingequations of motion equations (7)sim(9) and applying thenormalized mode shape functions of the rails the partialdifferential equations of the rail are converted to a set ofordinary differential equations such that

eurogzk(t) +κzGAr

m

kπl

1113888 1113889

2

qzk(t) minus κzGAr

kπl

1

mρIy

1113971

wyk(t) minus 1113944

Ns

i1Fszi(t)Zk xsi( 1113857 + 1113944

Nw

j1Fwrzj(t)Zk xwj1113872 1113873

eurowyk(t) +κzGAr

ρIy

+EIy

ρIy

kπl

1113888 1113889

2⎡⎣ ⎤⎦wyk(t) minus κzGAr

kπl

1

mρIy

1113971

qzk(t) 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(13)

eurogyk(t) +κyGAr

m

kπl

1113888 1113889

2

qyk(t) minus κyGAr

kπl

1

mρIz

1113971

wzk(t) minus 1113944

Ns

i1Fsyi(t)Yk xsi( 1113857 + 1113944

Nw

j1Fwryj(t)Yk xwj1113872 1113873

eurowzk(t) +κyGAr

ρIz

+EIz

ρIz

kπl

1113888 1113889

2⎡⎣ ⎤⎦wzk(t) minus κyGAr

kπl

1

mρIz

1113971

qyk(t) 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(14)

eurogTk(t) +GK

ρI0

kπl

1113888 1113889

2

qTk(t) minus 1113944Ns

i1Msi(t)ϕk xsi( 1113857 + 1113944

Nw

j1MGj(t)ϕk xwj1113872 1113873 (15)


e solutions of equations (13)sim (15) yield deflectionresponses of the rail which are applied to the vehicle modelto determine the normal contact forces Fwrzi and Fwryi(i 1sim4) using the Hertzian contact model available in theSIMPACK platform

e high-magnitude impact forces induced by the wheeldefects are transmitted to the ballast through the fastenerswhich causes vibration of the ballast Meanwhile the vi-bration of the ballast directly contributes to the rail de-flections and thus the wheelrail contact forces As Figure 3shows the fastening is treated as the spring-damper elementwith vertical stiffness Kfz vertical damping coefficient Cfzlateral stiffness Kfy and lateral damping coefficient Cfz esleeper is modeled as a rigid body with mass ms and inertialmoment Isx e motions of the sleeper are described byvertical displacement zs and lateral displacement ys of themass center of the sleeper and rotation angle Φs e ballastis modeled as the equivalent mass block with mass mb andonly the vertical motion zbL (zbR) is considered Besides theequivalent shear spring-damping elements KrndashCr are setbetween track bed blocks to consider the shear interactionbetween left and right front and rear adjacent track bedsParameters of ballasted track model are listed in Table 1

23 Vehicle Model e mathematical model coupling thelateral and vertical motions is considered in order to ac-curately simulate the operation performance of the Chineseopen top wagon C80 e model consists of a rigid car bodysupported on two rigid bolsters via the center plates and sidebearings and four flexible wheelsets coupled to four sideframes through the primary rubber suspensions Meanwhiletwo side frames and one bolster are connected by coil springsand friction wedge dampers Each wheelset is considered as arotating flexible body whose deflection responses are ob-tained from modal superposition shown in Section 21

e freight wagon has friction elements such as centerplate and side bearing which will provide rotary frictiontorques between car body and bogies to ensure the vehiclehunting stability e rotary friction torque of the centerplate can be calculated by the following

Tc 1113946Rc

02πr

2μc

Pc

Ac

dr

23μcPcRc

(16)

where Pc is the load on the center plateAc and Rc are the areaand radius of the center plate and μc is the friction coefficientof the center plate surface e side bearing here is a doubleacting elastic side bearing which can effectively prevent theexcessive increase of the rotary friction torque as the gapreached and improve the curving performance of the freightwagon e rotary friction torque of the side bearing can becalculated as follows

Ts Ps + kzΔz( 1113857μsds ΔzleΔc

Ps + kzΔc( 1113857μsds ΔzgtΔc1113896 (17)

where Ps is side bearing preload kz is the vertical stiffness ofthe side bearing ∆c is side bearing gap with ∆c 6mm μs isthe friction coefficient of the side bearing surface and ds isthe distance between the side bearing and the bogie centerIn addition the Chinese type LM wheel profile and CHN60rail profile are adopted for the simulatione structural andsuspension parameters of freight wagon model are setaccording to Table 2

24 Wheel Polygonalisation Wheel polygonalisation is de-scribed by variations in the wheel radius as a function ofangular position of the wheel ϕh Figure 4(a) illustrates theOOR deformities measured on a wheel which suggestsnearly periodic variations or wheel polygonalisation emeasured wear profile is idealised by a periodic waveformsuperimposed on the wheel circumference as seen inFigure 4(b) e coordinates of a point on the circumferenceor the contact point can thus be expressed as

x Rw + Ap sin Npϕh1113872 1113873sin ϕh

y Rw + Ap sin Npϕh1113872 1113873cos ϕh

⎧⎪⎨

⎪⎩ϕh isin [0 2π] (18)

where x and y are the coordinates of a point on the wheelcircumference Ap denotes the amplitude of the polygonalwear Np is the order of harmonic of the polygonal wear andRw is the nominal wheel radius e radius of the wheel at agiven point on the wheel circumference can be given byRp Rw + Ap sin NpΦh

rough measurements of circumferential profiles of 99different wheels employed in high-speed freight and com-muter trains Johansson found that the order of polygo-nalisation in freight wagon is mostly below 10 [21] and thesimilar phenomenon is also reported in reference [22]erefore low-order wheel polygonalisation is simulated toestablish the mapping relationship between wheelrail forceand wheel polygonalisation in this paper

25 Model Validation e wheel polygonal wear inducedwheelrail contact forces have been widely investigatedexperimentally and analytically Most of these studieshowever focus on single order polygonalisatione validityof the coupled vehicle-track dynamic model in the presenceof wheel polygonal wear is thus examined by comparing thesimulation results with the reported measured data [5] esimulation results are obtained for the vehicle-track modelconsidering the reported experimental conditions 0125mamplitude of 10th-order polygonalisation while the wheelpolygonal wear is limited to the leading wheelset Simulationresults in terms of peak wheelrail contact force deviationsfrom the static load are compared with the reported mea-sured data in the 0ndash100 kmh speed range in Figure 5 ecomparisons suggest reasonably good agreements with theexperimental results in saturation force and variation ten-dency Both the simulation and experimental results showsaturation of the peak force in the 50ndash60 kmh speed rangeHowever the saturation forces appear at different speedswhich is caused by the structure of the track


Table 1 Parameters of the ballasted track

Variable Notation ValueE Youngrsquos modulus of rail 206times1011Nm2

G Shear modulus of rail 792times1010Nm2

ρ Density of rail 7850 kgm2

A Cross-section area of rail 7745times10minus3m2

Iy Second moments of area of rail cross section about the y-axis 5240times10minus6m2

Iz Second moments of area of rail cross section about the z-axis 3217times10minus5m4

I0 Solar moment of inertia of rail cross section 3741times10minus5m4

e Distance from load point to the central line of rail cross section 0023ma Distance from bottle surface to the torsional center of rail cross section 008147mb Distance from bottle surface leftright side to the central line of rail cross section 0075mhR Distance from top surface to the torsional center of rail cross section 0095mKfz Equivalent vertical stiffness of fastening 28times107NmKfy Equivalent lateral stiffness of fastening 30107NmCfz Equivalent vertical damping coefficient of fastening 3625times104NmCfy Equivalent lateral damping coefficient of fastening 30times104Nmms Mass of sleeper 349 kgIsx Moment of inertia of sleeper about the x-axis 199 kgm2

mb Mass of ballast equivalent mass block 1398 kgKfz Equivalent vertical stiffness of fastening 28times107NmKfy Equivalent lateral stiffness of fastening 30times107NmCfz Equivalent vertical damping coefficient of fastening 36times104NmiddotsmCfy Equivalent lateral damping coefficient of fastening 30times104NmiddotsmKsby Lateral equivalent stiffness between sleeper and ballast 70times107NmCsby Lateral equivalent damping coefficient between sleeper and ballast 60times104NmiddotsmKsbz Vertical equivalent stiffness between sleeper and ballast 70times107NmCsbz Vertical equivalent damping coefficient between sleeper and ballast 60times104NmiddotsmKbz Vertical equivalent stiffness between ballast and ground 67times107NmKbz Vertical equivalent damping coefficient between ballast and ground 31times 104NmiddotsmKr Vertical equivalent stiffness between ballasts 78times107NmCr Vertical equivalent damping coefficient between ballasts 80times104NmiddotsmdB Distance between mass centers of ballast and sleeper 06m

Table 2 Freight wagon model parameters

Parameters ValuesCar body mass 10829 kgSide frame mass 497 kgBolster mass 745 kgWheelset mass 1171 kgCar body roll moment of inertia around x-axis 6932times104 kgmiddotm2

Car body pitch moment of inertia around y-axis 1837times105 kgmiddotm2

Car body yaw moment of inertia around z-axis 2127times105 kgmiddotm2

Side frame roll moment of inertia around x-axis 20735 kgmiddotm2

Side frame pitch moment of inertia around y-axis 1882 kgmiddotm2

Side frame yaw moment of inertia around z-axis 1725 kgmiddotm2

Bolster roll moment of inertia around x-axis 2597 kgmiddotm2

Bolster pitch moment of inertia around y-axis 16158 kgmiddotm2

Bolster yaw moment of inertia around z-axis 25808 kgmiddotm2

Wheelset roll moment of inertia around x-axis 700 kgmiddotm2

Wheelset pitch moment of inertia around y-axis 72 kgmiddotm2

Wheelset yaw moment of inertia around z-axis 700 kgmiddotm2

Primary suspension vertical stiffness along z-axis 160MNmSecondary suspension vertical stiffness along z-axis 2233MNme distance between the side bearing and the bogie center 076mRadius of friction moment of center plate 0125mSide bearing preload 198 kN


3 Results and Discussions

In this study the measured polygonal wear is idealised by aharmonic variation in the wheel radius as described insection 24 e vehicle-track model simulations are sub-sequently performed to evaluate the dynamic responses ofthe vehicle system under excitations caused by polygonalwear e simulation results in each case are obtained underexcitations due to polygonal wear of both the left and rightwheels of the leading wheelset of the front bogie whileassuming negligible phase difference between the twowheelsrsquo deformities

31 Influences of Vehicle Speed and Wear AmplitudeFigure 6 illustrates the effect of forward speed on theresulting maximum and minimum wheelrail contact forcemagnitudes both empty and loaded vehicles consideringfrom 005mm to 03mm amplitude 6th-order polygonalwear on both the wheels of the leading wheelset Take po-lygonal wear with amplitude of 01mm as an example the

maximum contact forces of empty vehicle increase withvehicle speed in the 20ndash90 kmh speed range and decrease atspeeds exceeding 90 kmh while the maximum contact forcesof loaded vehicle reach saturation at speed of 100 kmhCorresponding to maximum of wheelrail contact force theminimum of wheelrail contact force of loaded vehicles andthemaximumwheel-rail force are approximately symmetricalwith the static load Meanwhile the minimum of wheelrailcontact force of empty vehicle decreases with the speed untilreducing to 0 at 90 kmh which indicates wheelrailseparation

Apart from the forward speed the magnitude of thewheelrail contact force is strongly dependent on the am-plitude of the polygonal wear Figure 7 illustrates the effect ofwear amplitude on maxima and minima of the wheelrailcontact force of empty and loaded vehicles respectivelycaused by the 6th-order polygonal wear at speed of 90 kmhand 100 kmh e results of empty clearly show a nearlylinear increase and decrease in maximum and minimumcontact force respectively with increasing wear amplitude atthe speed of 90 kmh Different from the variation at speed of90 kmh there is a rapid increase in the maximum contactforce while there is a decrease to 0 in the minimum between005mm amplitude and 01mm amplitude that the wheelrail separation contributes to a rapid increase in the max-imum contact force It is necessary to limit the polygonalwear amplitude preventing the separation of wheel and raile phenomenon in Figure 7(b) revealed by maximum andminimum contact force is similar to that in Figure 7(a)except for polygonal wear amplitude which is caused by thedifferent axel load between them

Figures 8(a) and 8(b) illustrate the frequency spectra ofwheelrail contact force deviations from the static load ofempty vehicle induced by 01mm amplitude and 02mmamplitude respectively of 6th-order polygonal wear in the20ndash120 kmh speed range e results suggest that thepassing frequency of wheel polygonalisation (foor) increaseswith the vehicle speed and variation tendency of force peaksalong the passing frequency is the same as that of maximumwheelrail contact force in Figure 6(a) which reveals that

01

00

-01

-02

180

210

240270

300

330

0

30

6090

150

120

Radi

us d

evia

tion

(mm

)

-03

-02

-01

00

01

(a)

ϕhx

y

Ap

Rp Rw

01

00

-01

-02

180

210

240270

300

330

0

30

6090

150

120

Radi

us d

evia

tion

(mm

)

-03

-02

-01

00

01

(b)

Figure 4 (a) Measured circumferential polygonal wear and (b) polygonal wear idealised by a harmonic waveform

80

Whe

elr

ail c

onta

ct fo

rce (

kN)

60

40

20

0

0 20 40 60Speed (kmh)

80 100 120

Johansson and NielsenSimulation results

Figure 5 Comparisons of the peak wheelrail contact force de-viations from the static load with vehicle speed obtained fromcoupled vehicle-track model with the reported measured data [5]


additional wheelrail contact force is induced by polygonalwear Besides the spectra also revealed force peaks corre-sponding to multiples of foor namely 2foor and 3foorappearing the same as that of the minimum of wheelrailforce reduced to 0 in Figure 6(a) ese are likely due towheelrail separation induced elastic deformations of thewheelset at some speeds Figures 8(c) and 8(d) illustrate thefrequency spectra of wheelrail contact force of loaded ve-hicle with 01mm amplitude and 02mm amplitude re-spectively of 6th-order polygonal wear in the 20ndash120 kmhspeed range Unlike empty vehicles there are no corre-sponding to multiples of foor which is due to no wheel-railseparation shown in Figure 6(b)

32 Influence of Order of PolygonalWear Figure 9 illustratesthe influence of order of polygonal wear on the maxima ofthe wheelrail contact force deviations from the static load

for 01mm wear amplitude in the 60ndash130 kmh speed rangee results reveal that 1st-order and 3rd-order wheel pol-ygonalisation whether empty or loaded vehicles havelimited influence on wheelrail contact forces Furthermorethe maxima contact forces for empty vehicles induced by 5th-order 7th-order and 9th-order polygons are saturated at110 kmh 80 kmh and 60 kmh respectively By contrastthe saturation forces induced by wheel polygonalisationappear at 115 kmh 85 kmh and 65 kmh is is likelycaused by the wheelrail coupled vertical vibration modenear 60Hz as reported in [23 24] is relatively low-fre-quency mode is also referred to as ldquoP2 forcerdquo or ldquoP2 reso-nancerdquo [25] and is related to effective masses of the wheelsetand the rail which cause the different passing frequencybetween empty and loaded vehicles and stiffness of theprimary suspension and the rail supports Meanwhile thepassing frequency induced by polygonal wear resonates withP2 which intensifies the wheelrail contact force resulting in

Maxima of wheelrailcontact force

Minima of wheelrailcontact force

Wheelrail separationSpeed (kmh)

120

Whe

elr

ail c

onta

ct fo

rce (

kN)

90

60

30

0

20 40 60 80 100 120

005 mm025 mm01 mm

015 mm

02 mm

03 mm

(a)



Wheelrail separation

300

250

Whe

elr

ail c

onta

ct fo

rce (

kN)

200

150

100

50

0

Speed (kmh)20 40 60 80 100 120

025 mm005 mm01 mm015 mm

02 mm

03 mm

(b)

Figure 6 Influence of vehicle speed on the maximum andminimumwheelrail contact force under the 6th-order polygonal wear (a) emptyvehicle and (b) loaded vehicle


Frequency (Hz)

Spee

d (km

h)

Amplitude = 01mm

Am

plitu

de (k

N)

15

10

5

0120

10080

6040

20250200150100

foor2foor 3foor

500

(a)

Frequency (Hz)

Spee

d (km

h)

Amplitude = 02mm

Am

plitu

de (k

N)

30

20

10

0120

10080

6040

20250200150100

foor2foor 3foor

500

(b)

Frequency (Hz)

Amplitude = 01mm

Spee

d (k

mh

)A

mpl

itude

(kN

)

30

20

10

0120

10080

6040

20250200150100

foor

500

(c)

Frequency (Hz)

Amplitude = 02mm

Spee

d (k

mh

)A

mpl

itude

(kN

)

60

40

20

0120

10080

6040

20250200150100

foor

500

(d)

Figure 8 Frequency spectra of wheelrail contact force in the 20ndash120 kmh speed range (a) empty vehicle (amplitude 01mm) (b) emptyvehicle (amplitude 02mm) (c) loaded vehicle (amplitude 01mm) (d) loaded vehicle (amplitude 02mm)

empty vehicle

90kmh100kmh

100W

heel

rai

l con

tact

forc

e (kN

) 80

60

40

20

0

005 010 015 020Polygonal wear amplitude (mm)

025 030

(a)

loaded vehicle

90kmh100kmh

300

Whe

elr

ail c

onta

ct fo

rce (

kN) 250

200

150

100

50

0


025 0300 3

(b)

Figure 7 Influence of polygonal wear amplitude on the maxima and minima of the wheelrail contact forces (a) empty vehicle and (b)loaded vehicle


the increase of polygon wear exponentially with time Ad-ditionally another peak value (113Hz) induced by 9th-orderpolygons whether empty or loaded vehicles appears near120 kmh which is close to the 1st-bending mode (115Hz) ofthe wheelset as Figure 2 shows

33 Mapping Relation As the limited influence is made by1st-order and 3rd-order wheel polygonalisation only themapping relations between induced wheelrail force and 5th-order 7th-order and 9th-order polygonalisation are

established in this section Figure 10 illustrates the mappingrelations between wheelrail forces induced by the polygo-nalisation with 01mm 02mm and 03mm amplitude eresults reveal that amplitude variation of the polygonalisa-tion only affects the amplitude of the saturation force andthe saturation forces of empty and loaded vehicles stillappear when the passing frequency is 58Hz and 113Hz62Hz and 113Hz respectively which is primarily affectedby P2 force and 1st-bend mode of wheelset Besides thedifference between saturated wheelrail forces induced bydifferent polygonal order with the same amplitude is little

1st bending mode115Hz

130120

110100Speed (kmh)

Number of harm

onic (order)

Whe

elr

ail c

onta

ct fo

rce (

kN)

9080

7060

(a) (b)

130120

110100Speed (kmh)

9080

7060

13

57

9

Number of harm

onic (order)

13

57

90

10

20

30

40

50

Whe

elr

ail c

onta

ct fo

rce (

kN)

0

10

20

30

40

50

Figure 9 Effect of order of polygonal wear on maxima of the wheelrail contact force deviations from the static load in the 60ndash120 kmhspeed range (a) empty vehicle and (b) loaded vehicle (amplitude 01mm)

Passing frequency58Hz

200

Whe

elr

ail c

onta

ct fo

rce (

kN) 160

120

80

40

060 70 80 90 100

Speed (kmh)110 120 130

5th-order polygonisation7th-order polygonisation9th-order polygonisation

(a)


250

Whe

elr

ail c

onta

ct fo

rce (

kN) 200

150

100

50

060 70 80 90 100

Speed (kmh)110 120 130


(b)

Figure 10 Mapping relation between the wheelrail contact force deviations from the static load and polygonal wear order with differenceamplitude


and the induced wheelrail forces are almost proportional tothe polygonal amplitude except when the wheel-rail sepa-ration occurs

e passing frequency induced by polygonal wear res-onates with P2 force which intensifies the wheelrail contactforce resulting in the increase of polygon wear exponentiallywith time When the wheel operation time is over a specifiedmaintenance period polygonal wear develops quickly

us it is very necessary to establish the mapping re-lationship between wheelrail force and wheel polygonali-sation to detect the state of passing vehicle wheelset bytrackside detection so as to find out and repair in time

4 Conclusion

A comprehensive coupled vehicle-track dynamic model ofopen top wagon C80 is established so as to investigate theeffects of low-order wheel polygonalisation on wheelrailcontact force e model integrates flexible ballasted trackand wheelset subsystem models so as to account for elasticdeformations that may occur in the presence of repetitiveimpact loads induced by the wheel polygonalisation eeffects of amplitude and harmonic order of wheel polygonalwear whether empty and loaded vehicles are subsequentlyinvestigated and the corresponding mapping relations areobtained to detect polygonal wheel by trackside detectionen the conclusions can be drawn as follows

(1) e induced wheelrail force is almost proportionalto the polygon amplitude except increasing rapidlywhen the wheelrail separation occurs

(2) 1st-order and 3rd-order polygons whether emptyvehicle and loaded vehicle have limited influence onwheelrail contact forces in the 20ndash130 kmh speedrange

(3) e saturation forces of empty and loaded vehicleappear when the passing frequency is 58Hz and113Hz and 62Hz and 113Hz respectively which isprimarily affected by P2 force and 1st-bend mode ofwheelset

Data Availability

e validity of the coupled vehicle-track dynamic model inthe presence of wheel polygonal wear is thus examined bycomparing the simulation results with the reported mea-sured data in [5]

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (Grant numbers U2034210 and52102441) and Independent Research and DevelopmentProject of the State Key Laboratory of Traction Power (Grantnumber 2019TPL-T18)

References

[1] X-s Jin ldquoKey problems faced in high-speed train operationrdquoJournal of Zhejiang University Science A vol 15 no 12pp 936ndash945 2014

[2] B Peng S Iwnicki P Shackleton and Y Song ldquoGeneralconditions for railway wheel polygonal wear to evolverdquo Ve-hicle System Dynamics vol 59 no 4 pp 568ndash587 2021

[3] Y Wu X Du H J Zhang Z F Wen and X-S Jin ldquoEx-perimental analysis of the mechanism of high-order polygonalwear of wheels of a high-speed trainrdquo Journal of ZhejiangUniversity - Science vol 18 no 8 pp 579ndash592 2017

[4] J C O Nielsen and A Johansson ldquoOut-of-round railwaywheels-a literature surveyrdquo Proceedings of the Institution ofMechanical Engineers - Part F Journal of Rail and RapidTransit vol 214 no 2 pp 79ndash91 2000

[5] A Johansson and J C O Nielsen ldquoOut-of-round railwaywheels-wheel-rail contact forces and track response derivedfrom field tests and numerical simulationsrdquo Proceedings of theInstitution of Mechanical Engineers - Part F Journal of Railand Rapid Transit vol 217 no 2 pp 135ndash146 2003

[6] D W Barke and W K Chiu ldquoA review of the effects of out-of-round wheels on track and vehicle componentsrdquo Pro-ceedings of the Institution of Mechanical Engineers - Part FJournal of Rail and Rapid Transit vol 219 no 3 pp 151ndash1752005

[7] X Y Liu and W M Zhai ldquoAnalysis of vertical dynamicwheelrail interaction caused by polygonal wheels on high-speed trainsrdquo Wear vol 314 no 2 pp 282ndash290 2014

[8] X W Wu S Rakheja and A K W Ahmed ldquoInfluence of aflexible wheelset on the dynamic responses of a high-speedrailway car due to a wheel flatrdquo Proceedings of the Institutionof Mechanical Engineers - Part F Journal of Rail and RapidTransit vol 232 no 4 pp 1033ndash1048 2018

[9] X S Jin L Wu J Y Fang S Q Zhong and L Ling ldquoAninvestigation into the mechanism of the polygonal wear ofmetro train wheels and its effect on the dynamic behaviour ofa wheelrail systemrdquo Vehicle System Dynamics vol 50 no 12pp 1817ndash1834 2012

[10] G Q Tao Z F Wen X R Liang D Ren and X Jin ldquoAninvestigation into the mechanism of the out-of-round wheelsof metro train and its mitigation measuresrdquo Vehicle SystemDynamics vol 57 no 1 pp 1ndash16 2019

[11] X W Wu S Rakheja S Qu P wu J Zeng andA K W Ahmed ldquoDynamic responses of a high-speed railwaycar due to wheel polygonalisationrdquo Vehicle System Dynamicsvol 56 no 12 pp 1817ndash1837 2018

[12] H Wu P Wu F S Li H Shi and K Xu ldquoFatigue analysis ofthe gearbox housing in high-speed trains under wheel pol-ygonization using a multibody dynamics algorithmrdquo Engi-neering Failure Analysis vol 100 pp 351ndash364 2019

[13] Z W Wang Y Cheng and G M Mei ldquoTorsional vibrationanalysis of the gear transmission system of high-speed trainswith wheel defectsrdquo Proceedings of the Institution of Me-chanical Engineers - Part F Journal of Rail and Rapid Transitvol 234 no 2 pp 123ndash133 2020

[14] J Zhang G X Han X B Xiao W Ruiqing Y Zhao andX-S Jin ldquoInfluence of wheel polygonal wear on interior noiseof high-speed trainsrdquo Journal of Zhejiang University - Sciencevol 15 no 12 pp 1002ndash1018 2014

[15] X W Wu S Rakheja W B Cai M Chi A K W Ahmadand S Qu ldquoA study of formation of high order wheel pol-ygonalizationrdquo Wear vol 424 pp 1ndash14 2019


[16] W M Zhai ldquoTwo simple fast integration methods for large-scale dynamic problems in eng-ineeringrdquo InternationalJournal for Numerical Methods in Engineering vol 39 no 24pp 4199ndash4214 1996

[17] R J Guyan ldquoReduction of stiffness and mass matricesrdquo AIAAJournal vol 3 no 2 p 380 1965

[18] Z S Ren G Yang S S Wang and S G Sun ldquoAnalysis ofvibration and frequency transmission of high speed EMUwith flexible modelrdquo Acta Mechanica Sinica vol 30 no 6pp 876ndash883 2014

[19] W M Zhai C B Cai Q C Wang and W Lu ldquoDynamiceffects of vehicles on tracks in the case of raising train speedsrdquoProceedings of the Institution of Mechanical Engineers - Part FJournal of Rail and Rapid Transit vol 215 no 2 pp 125ndash1352001

[20] C J Yang Y Xu W D Zhu W Fan W H Zhang andG M Mei ldquoA three-dimensional modal theory-based Tim-oshenko finite length beam model for train-track dynamicanalysisrdquo Journal of Sound and Vibration vol 479 Article ID115363 2020

[21] A Johansson ldquoOut-of-round railway wheels-assessment ofwheel tread irregularities in train trafficrdquo Journal of Sound andVibration vol 293 no 3-5 pp 795ndash806 2006

[22] Y G Ye D C Shi P Krause Q Tian and T Hecht ldquoWheelflat can cause or exacerbate wheel polygonizationrdquo VehicleSystem Dynamics vol 58 no 10 pp 1ndash30 2019

[23] X W Wu W B Cai M R Chi L Wei H Shi and M ZhuldquoInvestigation of the effects of sleeper-passing impacts on thehigh-speed trainrdquo Vehicle System Dynamics vol 53 no 12pp 1902ndash1917 2015

[24] R Dong Vertical Dynamics of Railway VehiclendashTrack SystemConcordia University Montreal Canada 1994

[25] K L Knothe and S L Grassie ldquoModelling of railway track andvehicletrack interaction at high frequenciesrdquo Vehicle SystemDynamics vol 22 no 3-4 pp 209ndash262 1993


lateral slippage of the wheel and wheelrail material ex-cavation and thereby enlargement of out-of-round (OOR)deformities Jin and his research team [9 10] experi-mentally investigated the effects and growth of polygonalwear of subway wheels and concluded that the ninth-orderpolygonal wear observed in wheel deformations was due tothe first bending mode of the wheelset e effects of wheelpolygonalisation on the dynamic responses of a high-speedrail vehicle are investigated through development andsimulations of a comprehensive coupled vehicle-trackdynamic model by Wu [11] e results suggested that thehigh-order wheel polygonalisation could give rise to high-frequency impact loads at the wheelrail interface andexcited some of the vibration modes of the wheelset leadingto high-magnitude axle box acceleration and dynamicstress in the wheelset axle e effects of the wheel poly-gonization on the fatigue damage of the gearbox housing ofa high-speed train are investigated through a multibodysystem (MBS) railway vehicle model by Wu [12] e re-sults stated that the fatigue damage with a 20th-orderpolygonal wear is 63 larger than that without the po-lygonal wear on the wheel Moreover the torsional vi-bration of the gear transmission system [13] and wheelrailnoise [14] are also affected by wheel polygonizationHowever almost all of the research focuses on single orderwheel polygonalisation obtained in high-speed vehicles orsubways

Wheel polygonalisation is common in freight wagon andits wear depth changes exponentially with time [15] It isimportant to establish the mapping relationship detectingwheel polygonalisation by the trackside detection for freightwagon condition repair In this study a comprehensivecoupled vehicle-track dynamic model of freight wagon isestablished so as to investigate the effects of low-order wheelpolygonalisation on wheelrail contact force e modelintegrates flexible ballasted track and wheelset subsystemmodels so as to account for elastic deformations that mayoccur in the presence of repetitive impact loads induced bythe wheel polygonalisation at the wheelrail interface eeffects of amplitude and harmonic order of wheel polygonalwear are subsequently investigated and the correspondingmapping relations are obtained to detect polygonal wheel bytrackside detection

2 Establishment of Vehicle-Track Model

A comprehensive model of a top open wagon vehicle-tracksystem is formulated through integration of models offlexible wheelsets a ballasted track and the freight wagonFigure 1 illustrates the simulation scheme comprising finiteelement (FE) models of the flexible wheelset and the bal-lasted track coupled with the open wagon model in theSIMPACK platform [11] e FE model of the wheelset isused to determine its modal properties via the eigen analysise modal vectors are subsequently integrated to the wagonmodel using the finite element multibody systems (FEMBS)interface available in SIMPACK [11] e resulting dynamicwheelrail contact forces the displacement of contact spotand speed are obtained to serve as excitations to the ballasted

track model to evaluate the deflection response of the tracke rail deflection response is subsequently integrated to thevehicle model using the SIMAT (SIMPACK-MATLAB)cosimulation interface as shown in Figure 1 to study itsdynamic responses in the presence of a wheel polygonalwear In simulation the track deflection is solved by a newfast numerical integration method [16] with the step size of00001s e component modelsrsquo formulations are describedin the following subsections

21 Flexible Wheelset Model e FE model of wheelset isdiscretized into 22066 Solid 185 elements as shown inFigure 1 e modal characteristics of wheelset are obtainedthrough eigen analysis e substructure method is intro-duced to deal with the wheelset FE model to speed up thedynamic calculation and the model DOF is transformedinto the structural model represented by the master DOFe nodes DOF where the force is applied or output and theboundary conditions are calculated is defined as the masterDOF and the other defined as the slave DOF e sub-structure method is based on Guyan reduction methodcondensing a set of elements into a superelement by matrixtransformation [17] e dynamic equation of the FE modelcan be expressed as

Meurou + D _u + Ku F (1)

where M K and D are mass matrix stiffness matrix anddamping matrix respectively u is displacement matrix andF is load matrix Assuming that there aremmaster DOF ands slave DOF the matrix form of (1) can be expressed as

Mmm Mms

Msm Mss1113890 1113891

euroumm

eurouss1113890 1113891 +

Dmm Dms

Dsm Dss1113890 1113891

_umm

_uss1113890 1113891

+Kmm Kms

Ksm Kss1113890 1113891

umm

uss1113890 1113891

Fmm

01113890 1113891

(2)

where Kmm Kms Ksm and Kss are mtimesm mtimesm stimesm andstimes s stiffness matrices respectively M and D are the cor-responding mass and damping matrices and Fmm is the loadvector acting on the master DOF Considering that Guyanreduction theory is based on the static transformation ofKu F [18] uss can be assumed as

uss minusKminus1s Ksmumm (3)

us

u umm

uss1113890 1113891

I

minusKminus1ss Ksm

1113890 1113891umm (4)

e DOF conversion matrix can be defined as

T I

minusKminus1ss Ksm

1113890 1113891 (5)

Substituting (4) and (5) into equation (1) the motionequation of the finite element model with reduced degrees offreedom is



where

Μlowastmm TTMT

Dlowastmm TTDT

Klowastmm TTKT

Flowastmm TTF

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(7)





ρAr

z2z(x t)

zt2 + κzGAr

zψy(x y)

zxminus

z2z(x t)

zx21113890 1113891 minus 1113944

Ns


Nw


ρIy

z2ψy(x t)


zz(x t)

zx1113890 1113891 minus EIy

z2ψy(x t)

zx2 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)




Flexible Wheelset

FEMBSSIMAT

Wheelrail force


displacementv



ρAr

z2y(x t)

zt2 + κyGAr

zψz(x t)

δxminus

z2y(x t)

zx21113890 1113891 minus 1113944

Ns


Nw


ρIz

z2ψz(x t)


zy(x t)

zx1113890 1113891 minus EIz

z2ψz(x t)

zx2 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(9)

Kfz

Kr

Cr

Cfz

Ksbz

Kbz

Csbz

Cbz

Fastening

Rail

Ballast

Sleeper

+infin-infin

(a)

Kfy Kfy

KsbyKfz

Ksbz

Kbz Cbz

Csbz Csby

KfzCfz CfzCfy Cfy

2dB

Sleeper ys

zs

CrKr BallastBallast

Φs

zbL zbRCbzKbz

(b)

Fwrzj e

a

b bi

Fwryj

F1sy

F1szi F2szi

F2syi

MGj

Msi

hr

(c)




Torsion mode100 Hz





ρI0z2ϕ(x t)

zt2 minus GK

z2ϕ(x t)

zt2 minus 1113944

Ns


+ 1113944Nw


(10)



z(x t) 1113944

Nz

k1Zk(x)qzk(t)

ψz(x t) 1113944

Nψz

k1ψzk(x)wzk(t)

y(x t) 1113944

Ny

k1Yk(x)qyk(t)

ψy(x t) 1113944

Nψy

k1ψyk(x)wyk(t)

ϕ(x t) 1113944

Nϕ

k1ϕk(x)qTk(t)

(11)


Zk(x)

2

ρIzl

1113971

sinkπl

x1113888 1113889

ψzk(x)

2

ρIyl

1113971

coskπl

x1113888 1113889

Yk(x)

2

ρIyl

1113971

sinkπl

x1113888 1113889

ψyk(x)

2

ρIzl

1113971

coskπl

x1113888 1113889

ϕk(x)

2

ρI0l

1113971

sinkπl

x1113888 1113889

(12)


eurogzk(t) +κzGAr

m

kπl

1113888 1113889

2

qzk(t) minus κzGAr

kπl

1

mρIy

1113971


Ns

i1Fszi(t)Zk xsi( 1113857 + 1113944

Nw

j1Fwrzj(t)Zk xwj1113872 1113873

eurowyk(t) +κzGAr

ρIy

+EIy

ρIy

kπl

1113888 1113889


kπl

1

mρIy

1113971

qzk(t) 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(13)

eurogyk(t) +κyGAr

m

kπl

1113888 1113889

2

qyk(t) minus κyGAr

kπl

1

mρIz

1113971


Ns

i1Fsyi(t)Yk xsi( 1113857 + 1113944

Nw

j1Fwryj(t)Yk xwj1113872 1113873

eurowzk(t) +κyGAr

ρIz

+EIz

ρIz

kπl

1113888 1113889


kπl

1

mρIz

1113971

qyk(t) 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(14)

eurogTk(t) +GK

ρI0

kπl

1113888 1113889

2


i1Msi(t)ϕk xsi( 1113857 + 1113944

Nw

j1MGj(t)ϕk xwj1113872 1113873 (15)






Tc 1113946Rc

02πr

2μc

Pc

Ac

dr

23μcPcRc

(16)








⎧⎪⎨

⎪⎩ϕh isin [0 2π] (18)




































01

00

-01

-02

180

210

240270

300

330

0

30

6090

150

120

Radi

us d

evia

tion

(mm

)

-03

-02

-01

00

01

(a)

ϕhx

y

Ap

Rp Rw

01

00

-01

-02

180

210

240270

300

330

0

30

6090

150

120

Radi

us d

evia

tion

(mm

)

-03

-02

-01

00

01

(b)


80

Whe

elr

ail c

onta

ct fo

rce (

kN)

60

40

20

0

0 20 40 60Speed (kmh)

80 100 120










120

Whe

elr

ail c

onta

ct fo

rce (

kN)

90

60

30

0

20 40 60 80 100 120

005 mm025 mm01 mm

015 mm

02 mm

03 mm

(a)




300

250

Whe

elr

ail c

onta

ct fo

rce (

kN)

200

150

100

50

0

Speed (kmh)20 40 60 80 100 120

025 mm005 mm01 mm015 mm

02 mm

03 mm

(b)



Frequency (Hz)

Spee

d (km

h)

Amplitude = 01mm

Am

plitu

de (k

N)

15

10

5

0120

10080

6040

20250200150100

foor2foor 3foor

500

(a)

Frequency (Hz)

Spee

d (km

h)

Amplitude = 02mm

Am

plitu

de (k

N)

30

20

10

0120

10080

6040

20250200150100

foor2foor 3foor

500

(b)

Frequency (Hz)

Amplitude = 01mm

Spee

d (k

mh

)A

mpl

itude

(kN

)

30

20

10

0120

10080

6040

20250200150100

foor

500

(c)

Frequency (Hz)

Amplitude = 02mm

Spee

d (k

mh

)A

mpl

itude

(kN

)

60

40

20

0120

10080

6040

20250200150100

foor

500

(d)


empty vehicle

90kmh100kmh

100W

heel

rai

l con

tact

forc

e (kN

) 80

60

40

20

0


025 030

(a)

loaded vehicle

90kmh100kmh

300

Whe

elr

ail c

onta

ct fo

rce (

kN) 250

200

150

100

50

0


025 0300 3

(b)







130120

110100Speed (kmh)

Number of harm

onic (order)

Whe

elr

ail c

onta

ct fo

rce (

kN)

9080

7060

(a) (b)

130120

110100Speed (kmh)

9080

7060

13

57

9

Number of harm

onic (order)

13

57

90

10

20

30

40

50

Whe

elr

ail c

onta

ct fo

rce (

kN)

0

10

20

30

40

50



200

Whe

elr

ail c

onta

ct fo

rce (

kN) 160

120

80

40

060 70 80 90 100

Speed (kmh)110 120 130


(a)


250

Whe

elr

ail c

onta

ct fo

rce (

kN) 200

150

100

50

060 70 80 90 100

Speed (kmh)110 120 130


(b)






4 Conclusion





Data Availability




Acknowledgments


References





























where

Μlowastmm TTMT

Dlowastmm TTDT

Klowastmm TTKT

Flowastmm TTF

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(7)





ρAr

z2z(x t)

zt2 + κzGAr

zψy(x y)

zxminus

z2z(x t)

zx21113890 1113891 minus 1113944

Ns


Nw


ρIy

z2ψy(x t)


zz(x t)

zx1113890 1113891 minus EIy

z2ψy(x t)

zx2 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)




Flexible Wheelset

FEMBSSIMAT

Wheelrail force


displacementv



ρAr

z2y(x t)

zt2 + κyGAr

zψz(x t)

δxminus

z2y(x t)

zx21113890 1113891 minus 1113944

Ns


Nw


ρIz

z2ψz(x t)


zy(x t)

zx1113890 1113891 minus EIz

z2ψz(x t)

zx2 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(9)

Kfz

Kr

Cr

Cfz

Ksbz

Kbz

Csbz

Cbz

Fastening

Rail

Ballast

Sleeper

+infin-infin

(a)

Kfy Kfy

KsbyKfz

Ksbz

Kbz Cbz

Csbz Csby

KfzCfz CfzCfy Cfy

2dB

Sleeper ys

zs

CrKr BallastBallast

Φs

zbL zbRCbzKbz

(b)

Fwrzj e

a

b bi

Fwryj

F1sy

F1szi F2szi

F2syi

MGj

Msi

hr

(c)




Torsion mode100 Hz





ρI0z2ϕ(x t)

zt2 minus GK

z2ϕ(x t)

zt2 minus 1113944

Ns


+ 1113944Nw


(10)



z(x t) 1113944

Nz

k1Zk(x)qzk(t)

ψz(x t) 1113944

Nψz

k1ψzk(x)wzk(t)

y(x t) 1113944

Ny

k1Yk(x)qyk(t)

ψy(x t) 1113944

Nψy

k1ψyk(x)wyk(t)

ϕ(x t) 1113944

Nϕ

k1ϕk(x)qTk(t)

(11)


Zk(x)

2

ρIzl

1113971

sinkπl

x1113888 1113889

ψzk(x)

2

ρIyl

1113971

coskπl

x1113888 1113889

Yk(x)

2

ρIyl

1113971

sinkπl

x1113888 1113889

ψyk(x)

2

ρIzl

1113971

coskπl

x1113888 1113889

ϕk(x)

2

ρI0l

1113971

sinkπl

x1113888 1113889

(12)


eurogzk(t) +κzGAr

m

kπl

1113888 1113889

2

qzk(t) minus κzGAr

kπl

1

mρIy

1113971


Ns

i1Fszi(t)Zk xsi( 1113857 + 1113944

Nw

j1Fwrzj(t)Zk xwj1113872 1113873

eurowyk(t) +κzGAr

ρIy

+EIy

ρIy

kπl

1113888 1113889


kπl

1

mρIy

1113971

qzk(t) 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(13)

eurogyk(t) +κyGAr

m

kπl

1113888 1113889

2

qyk(t) minus κyGAr

kπl

1

mρIz

1113971


Ns

i1Fsyi(t)Yk xsi( 1113857 + 1113944

Nw

j1Fwryj(t)Yk xwj1113872 1113873

eurowzk(t) +κyGAr

ρIz

+EIz

ρIz

kπl

1113888 1113889


kπl

1

mρIz

1113971

qyk(t) 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(14)

eurogTk(t) +GK

ρI0

kπl

1113888 1113889

2


i1Msi(t)ϕk xsi( 1113857 + 1113944

Nw

j1MGj(t)ϕk xwj1113872 1113873 (15)






Tc 1113946Rc

02πr

2μc

Pc

Ac

dr

23μcPcRc

(16)








⎧⎪⎨

⎪⎩ϕh isin [0 2π] (18)




































01

00

-01

-02

180

210

240270

300

330

0

30

6090

150

120

Radi

us d

evia

tion

(mm

)

-03

-02

-01

00

01

(a)

ϕhx

y

Ap

Rp Rw

01

00

-01

-02

180

210

240270

300

330

0

30

6090

150

120

Radi

us d

evia

tion

(mm

)

-03

-02

-01

00

01

(b)


80

Whe

elr

ail c

onta

ct fo

rce (

kN)

60

40

20

0

0 20 40 60Speed (kmh)

80 100 120










120

Whe

elr

ail c

onta

ct fo

rce (

kN)

90

60

30

0

20 40 60 80 100 120

005 mm025 mm01 mm

015 mm

02 mm

03 mm

(a)




300

250

Whe

elr

ail c

onta

ct fo

rce (

kN)

200

150

100

50

0

Speed (kmh)20 40 60 80 100 120

025 mm005 mm01 mm015 mm

02 mm

03 mm

(b)



Frequency (Hz)

Spee

d (km

h)

Amplitude = 01mm

Am

plitu

de (k

N)

15

10

5

0120

10080

6040

20250200150100

foor2foor 3foor

500

(a)

Frequency (Hz)

Spee

d (km

h)

Amplitude = 02mm

Am

plitu

de (k

N)

30

20

10

0120

10080

6040

20250200150100

foor2foor 3foor

500

(b)

Frequency (Hz)

Amplitude = 01mm

Spee

d (k

mh

)A

mpl

itude

(kN

)

30

20

10

0120

10080

6040

20250200150100

foor

500

(c)

Frequency (Hz)

Amplitude = 02mm

Spee

d (k

mh

)A

mpl

itude

(kN

)

60

40

20

0120

10080

6040

20250200150100

foor

500

(d)


empty vehicle

90kmh100kmh

100W

heel

rai

l con

tact

forc

e (kN

) 80

60

40

20

0


025 030

(a)

loaded vehicle

90kmh100kmh

300

Whe

elr

ail c

onta

ct fo

rce (

kN) 250

200

150

100

50

0


025 0300 3

(b)







130120

110100Speed (kmh)

Number of harm

onic (order)

Whe

elr

ail c

onta

ct fo

rce (

kN)

9080

7060

(a) (b)

130120

110100Speed (kmh)

9080

7060

13

57

9

Number of harm

onic (order)

13

57

90

10

20

30

40

50

Whe

elr

ail c

onta

ct fo

rce (

kN)

0

10

20

30

40

50



200

Whe

elr

ail c

onta

ct fo

rce (

kN) 160

120

80

40

060 70 80 90 100

Speed (kmh)110 120 130


(a)


250

Whe

elr

ail c

onta

ct fo

rce (

kN) 200

150

100

50

060 70 80 90 100

Speed (kmh)110 120 130


(b)






4 Conclusion





Data Availability




Acknowledgments


References




























ρAr

z2y(x t)

zt2 + κyGAr

zψz(x t)

δxminus

z2y(x t)

zx21113890 1113891 minus 1113944

Ns


Nw


ρIz

z2ψz(x t)


zy(x t)

zx1113890 1113891 minus EIz

z2ψz(x t)

zx2 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(9)

Kfz

Kr

Cr

Cfz

Ksbz

Kbz

Csbz

Cbz

Fastening

Rail

Ballast

Sleeper

+infin-infin

(a)

Kfy Kfy

KsbyKfz

Ksbz

Kbz Cbz

Csbz Csby

KfzCfz CfzCfy Cfy

2dB

Sleeper ys

zs

CrKr BallastBallast

Φs

zbL zbRCbzKbz

(b)

Fwrzj e

a

b bi

Fwryj

F1sy

F1szi F2szi

F2syi

MGj

Msi

hr

(c)




Torsion mode100 Hz





ρI0z2ϕ(x t)

zt2 minus GK

z2ϕ(x t)

zt2 minus 1113944

Ns


+ 1113944Nw


(10)



z(x t) 1113944

Nz

k1Zk(x)qzk(t)

ψz(x t) 1113944

Nψz

k1ψzk(x)wzk(t)

y(x t) 1113944

Ny

k1Yk(x)qyk(t)

ψy(x t) 1113944

Nψy

k1ψyk(x)wyk(t)

ϕ(x t) 1113944

Nϕ

k1ϕk(x)qTk(t)

(11)


Zk(x)

2

ρIzl

1113971

sinkπl

x1113888 1113889

ψzk(x)

2

ρIyl

1113971

coskπl

x1113888 1113889

Yk(x)

2

ρIyl

1113971

sinkπl

x1113888 1113889

ψyk(x)

2

ρIzl

1113971

coskπl

x1113888 1113889

ϕk(x)

2

ρI0l

1113971

sinkπl

x1113888 1113889

(12)


eurogzk(t) +κzGAr

m

kπl

1113888 1113889

2

qzk(t) minus κzGAr

kπl

1

mρIy

1113971


Ns

i1Fszi(t)Zk xsi( 1113857 + 1113944

Nw

j1Fwrzj(t)Zk xwj1113872 1113873

eurowyk(t) +κzGAr

ρIy

+EIy

ρIy

kπl

1113888 1113889


kπl

1

mρIy

1113971

qzk(t) 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(13)

eurogyk(t) +κyGAr

m

kπl

1113888 1113889

2

qyk(t) minus κyGAr

kπl

1

mρIz

1113971


Ns

i1Fsyi(t)Yk xsi( 1113857 + 1113944

Nw

j1Fwryj(t)Yk xwj1113872 1113873

eurowzk(t) +κyGAr

ρIz

+EIz

ρIz

kπl

1113888 1113889


kπl

1

mρIz

1113971

qyk(t) 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(14)

eurogTk(t) +GK

ρI0

kπl

1113888 1113889

2


i1Msi(t)ϕk xsi( 1113857 + 1113944

Nw

j1MGj(t)ϕk xwj1113872 1113873 (15)






Tc 1113946Rc

02πr

2μc

Pc

Ac

dr

23μcPcRc

(16)








⎧⎪⎨

⎪⎩ϕh isin [0 2π] (18)




































01

00

-01

-02

180

210

240270

300

330

0

30

6090

150

120

Radi

us d

evia

tion

(mm

)

-03

-02

-01

00

01

(a)

ϕhx

y

Ap

Rp Rw

01

00

-01

-02

180

210

240270

300

330

0

30

6090

150

120

Radi

us d

evia

tion

(mm

)

-03

-02

-01

00

01

(b)


80

Whe

elr

ail c

onta

ct fo

rce (

kN)

60

40

20

0

0 20 40 60Speed (kmh)

80 100 120










120

Whe

elr

ail c

onta

ct fo

rce (

kN)

90

60

30

0

20 40 60 80 100 120

005 mm025 mm01 mm

015 mm

02 mm

03 mm

(a)




300

250

Whe

elr

ail c

onta

ct fo

rce (

kN)

200

150

100

50

0

Speed (kmh)20 40 60 80 100 120

025 mm005 mm01 mm015 mm

02 mm

03 mm

(b)



Frequency (Hz)

Spee

d (km

h)

Amplitude = 01mm

Am

plitu

de (k

N)

15

10

5

0120

10080

6040

20250200150100

foor2foor 3foor

500

(a)

Frequency (Hz)

Spee

d (km

h)

Amplitude = 02mm

Am

plitu

de (k

N)

30

20

10

0120

10080

6040

20250200150100

foor2foor 3foor

500

(b)

Frequency (Hz)

Amplitude = 01mm

Spee

d (k

mh

)A

mpl

itude

(kN

)

30

20

10

0120

10080

6040

20250200150100

foor

500

(c)

Frequency (Hz)

Amplitude = 02mm

Spee

d (k

mh

)A

mpl

itude

(kN

)

60

40

20

0120

10080

6040

20250200150100

foor

500

(d)


empty vehicle

90kmh100kmh

100W

heel

rai

l con

tact

forc

e (kN

) 80

60

40

20

0


025 030

(a)

loaded vehicle

90kmh100kmh

300

Whe

elr

ail c

onta

ct fo

rce (

kN) 250

200

150

100

50

0


025 0300 3

(b)







130120

110100Speed (kmh)

Number of harm

onic (order)

Whe

elr

ail c

onta

ct fo

rce (

kN)

9080

7060

(a) (b)

130120

110100Speed (kmh)

9080

7060

13

57

9

Number of harm

onic (order)

13

57

90

10

20

30

40

50

Whe

elr

ail c

onta

ct fo

rce (

kN)

0

10

20

30

40

50



200

Whe

elr

ail c

onta

ct fo

rce (

kN) 160

120

80

40

060 70 80 90 100

Speed (kmh)110 120 130


(a)


250

Whe

elr

ail c

onta

ct fo

rce (

kN) 200

150

100

50

060 70 80 90 100

Speed (kmh)110 120 130


(b)






4 Conclusion





Data Availability




Acknowledgments


References






























ρI0z2ϕ(x t)

zt2 minus GK

z2ϕ(x t)

zt2 minus 1113944

Ns


+ 1113944Nw


(10)



z(x t) 1113944

Nz

k1Zk(x)qzk(t)

ψz(x t) 1113944

Nψz

k1ψzk(x)wzk(t)

y(x t) 1113944

Ny

k1Yk(x)qyk(t)

ψy(x t) 1113944

Nψy

k1ψyk(x)wyk(t)

ϕ(x t) 1113944

Nϕ

k1ϕk(x)qTk(t)

(11)


Zk(x)

2

ρIzl

1113971

sinkπl

x1113888 1113889

ψzk(x)

2

ρIyl

1113971

coskπl

x1113888 1113889

Yk(x)

2

ρIyl

1113971

sinkπl

x1113888 1113889

ψyk(x)

2

ρIzl

1113971

coskπl

x1113888 1113889

ϕk(x)

2

ρI0l

1113971

sinkπl

x1113888 1113889

(12)


eurogzk(t) +κzGAr

m

kπl

1113888 1113889

2

qzk(t) minus κzGAr

kπl

1

mρIy

1113971


Ns

i1Fszi(t)Zk xsi( 1113857 + 1113944

Nw

j1Fwrzj(t)Zk xwj1113872 1113873

eurowyk(t) +κzGAr

ρIy

+EIy

ρIy

kπl

1113888 1113889


kπl

1

mρIy

1113971

qzk(t) 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(13)

eurogyk(t) +κyGAr

m

kπl

1113888 1113889

2

qyk(t) minus κyGAr

kπl

1

mρIz

1113971


Ns

i1Fsyi(t)Yk xsi( 1113857 + 1113944

Nw

j1Fwryj(t)Yk xwj1113872 1113873

eurowzk(t) +κyGAr

ρIz

+EIz

ρIz

kπl

1113888 1113889


kπl

1

mρIz

1113971

qyk(t) 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(14)

eurogTk(t) +GK

ρI0

kπl

1113888 1113889

2


i1Msi(t)ϕk xsi( 1113857 + 1113944

Nw

j1MGj(t)ϕk xwj1113872 1113873 (15)






Tc 1113946Rc

02πr

2μc

Pc

Ac

dr

23μcPcRc

(16)








⎧⎪⎨

⎪⎩ϕh isin [0 2π] (18)




































01

00

-01

-02

180

210

240270

300

330

0

30

6090

150

120

Radi

us d

evia

tion

(mm

)

-03

-02

-01

00

01

(a)

ϕhx

y

Ap

Rp Rw

01

00

-01

-02

180

210

240270

300

330

0

30

6090

150

120

Radi

us d

evia

tion

(mm

)

-03

-02

-01

00

01

(b)


80

Whe

elr

ail c

onta

ct fo

rce (

kN)

60

40

20

0

0 20 40 60Speed (kmh)

80 100 120










120

Whe

elr

ail c

onta

ct fo

rce (

kN)

90

60

30

0

20 40 60 80 100 120

005 mm025 mm01 mm

015 mm

02 mm

03 mm

(a)




300

250

Whe

elr

ail c

onta

ct fo

rce (

kN)

200

150

100

50

0

Speed (kmh)20 40 60 80 100 120

025 mm005 mm01 mm015 mm

02 mm

03 mm

(b)



Frequency (Hz)

Spee

d (km

h)

Amplitude = 01mm

Am

plitu

de (k

N)

15

10

5

0120

10080

6040

20250200150100

foor2foor 3foor

500

(a)

Frequency (Hz)

Spee

d (km

h)

Amplitude = 02mm

Am

plitu

de (k

N)

30

20

10

0120

10080

6040

20250200150100

foor2foor 3foor

500

(b)

Frequency (Hz)

Amplitude = 01mm

Spee

d (k

mh

)A

mpl

itude

(kN

)

30

20

10

0120

10080

6040

20250200150100

foor

500

(c)

Frequency (Hz)

Amplitude = 02mm

Spee

d (k

mh

)A

mpl

itude

(kN

)

60

40

20

0120

10080

6040

20250200150100

foor

500

(d)


empty vehicle

90kmh100kmh

100W

heel

rai

l con

tact

forc

e (kN

) 80

60

40

20

0


025 030

(a)

loaded vehicle

90kmh100kmh

300

Whe

elr

ail c

onta

ct fo

rce (

kN) 250

200

150

100

50

0


025 0300 3

(b)







130120

110100Speed (kmh)

Number of harm

onic (order)

Whe

elr

ail c

onta

ct fo

rce (

kN)

9080

7060

(a) (b)

130120

110100Speed (kmh)

9080

7060

13

57

9

Number of harm

onic (order)

13

57

90

10

20

30

40

50

Whe

elr

ail c

onta

ct fo

rce (

kN)

0

10

20

30

40

50



200

Whe

elr

ail c

onta

ct fo

rce (

kN) 160

120

80

40

060 70 80 90 100

Speed (kmh)110 120 130


(a)


250

Whe

elr

ail c

onta

ct fo

rce (

kN) 200

150

100

50

060 70 80 90 100

Speed (kmh)110 120 130


(b)






4 Conclusion





Data Availability




Acknowledgments


References
































Tc 1113946Rc

02πr

2μc

Pc

Ac

dr

23μcPcRc

(16)








⎧⎪⎨

⎪⎩ϕh isin [0 2π] (18)




































01

00

-01

-02

180

210

240270

300

330

0

30

6090

150

120

Radi

us d

evia

tion

(mm

)

-03

-02

-01

00

01

(a)

ϕhx

y

Ap

Rp Rw

01

00

-01

-02

180

210

240270

300

330

0

30

6090

150

120

Radi

us d

evia

tion

(mm

)

-03

-02

-01

00

01

(b)


80

Whe

elr

ail c

onta

ct fo

rce (

kN)

60

40

20

0

0 20 40 60Speed (kmh)

80 100 120










120

Whe

elr

ail c

onta

ct fo

rce (

kN)

90

60

30

0

20 40 60 80 100 120

005 mm025 mm01 mm

015 mm

02 mm

03 mm

(a)




300

250

Whe

elr

ail c

onta

ct fo

rce (

kN)

200

150

100

50

0

Speed (kmh)20 40 60 80 100 120

025 mm005 mm01 mm015 mm

02 mm

03 mm

(b)



Frequency (Hz)

Spee

d (km

h)

Amplitude = 01mm

Am

plitu

de (k

N)

15

10

5

0120

10080

6040

20250200150100

foor2foor 3foor

500

(a)

Frequency (Hz)

Spee

d (km

h)

Amplitude = 02mm

Am

plitu

de (k

N)

30

20

10

0120

10080

6040

20250200150100

foor2foor 3foor

500

(b)

Frequency (Hz)

Amplitude = 01mm

Spee

d (k

mh

)A

mpl

itude

(kN

)

30

20

10

0120

10080

6040

20250200150100

foor

500

(c)

Frequency (Hz)

Amplitude = 02mm

Spee

d (k

mh

)A

mpl

itude

(kN

)

60

40

20

0120

10080

6040

20250200150100

foor

500

(d)


empty vehicle

90kmh100kmh

100W

heel

rai

l con

tact

forc

e (kN

) 80

60

40

20

0


025 030

(a)

loaded vehicle

90kmh100kmh

300

Whe

elr

ail c

onta

ct fo

rce (

kN) 250

200

150

100

50

0


025 0300 3

(b)







130120

110100Speed (kmh)

Number of harm

onic (order)

Whe

elr

ail c

onta

ct fo

rce (

kN)

9080

7060

(a) (b)

130120

110100Speed (kmh)

9080

7060

13

57

9

Number of harm

onic (order)

13

57

90

10

20

30

40

50

Whe

elr

ail c

onta

ct fo

rce (

kN)

0

10

20

30

40

50



200

Whe

elr

ail c

onta

ct fo

rce (

kN) 160

120

80

40

060 70 80 90 100

Speed (kmh)110 120 130


(a)


250

Whe

elr

ail c

onta

ct fo

rce (

kN) 200

150

100

50

060 70 80 90 100

Speed (kmh)110 120 130


(b)






4 Conclusion





Data Availability




Acknowledgments


References



























































01

00

-01

-02

180

210

240270

300

330

0

30

6090

150

120

Radi

us d

evia

tion

(mm

)

-03

-02

-01

00

01

(a)

ϕhx

y

Ap

Rp Rw

01

00

-01

-02

180

210

240270

300

330

0

30

6090

150

120

Radi

us d

evia

tion

(mm

)

-03

-02

-01

00

01

(b)


80

Whe

elr

ail c

onta

ct fo

rce (

kN)

60

40

20

0

0 20 40 60Speed (kmh)

80 100 120










120

Whe

elr

ail c

onta

ct fo

rce (

kN)

90

60

30

0

20 40 60 80 100 120

005 mm025 mm01 mm

015 mm

02 mm

03 mm

(a)




300

250

Whe

elr

ail c

onta

ct fo

rce (

kN)

200

150

100

50

0

Speed (kmh)20 40 60 80 100 120

025 mm005 mm01 mm015 mm

02 mm

03 mm

(b)



Frequency (Hz)

Spee

d (km

h)

Amplitude = 01mm

Am

plitu

de (k

N)

15

10

5

0120

10080

6040

20250200150100

foor2foor 3foor

500

(a)

Frequency (Hz)

Spee

d (km

h)

Amplitude = 02mm

Am

plitu

de (k

N)

30

20

10

0120

10080

6040

20250200150100

foor2foor 3foor

500

(b)

Frequency (Hz)

Amplitude = 01mm

Spee

d (k

mh

)A

mpl

itude

(kN

)

30

20

10

0120

10080

6040

20250200150100

foor

500

(c)

Frequency (Hz)

Amplitude = 02mm

Spee

d (k

mh

)A

mpl

itude

(kN

)

60

40

20

0120

10080

6040

20250200150100

foor

500

(d)


empty vehicle

90kmh100kmh

100W

heel

rai

l con

tact

forc

e (kN

) 80

60

40

20

0


025 030

(a)

loaded vehicle

90kmh100kmh

300

Whe

elr

ail c

onta

ct fo

rce (

kN) 250

200

150

100

50

0


025 0300 3

(b)







130120

110100Speed (kmh)

Number of harm

onic (order)

Whe

elr

ail c

onta

ct fo

rce (

kN)

9080

7060

(a) (b)

130120

110100Speed (kmh)

9080

7060

13

57

9

Number of harm

onic (order)

13

57

90

10

20

30

40

50

Whe

elr

ail c

onta

ct fo

rce (

kN)

0

10

20

30

40

50



200

Whe

elr

ail c

onta

ct fo

rce (

kN) 160

120

80

40

060 70 80 90 100

Speed (kmh)110 120 130


(a)


250

Whe

elr

ail c

onta

ct fo

rce (

kN) 200

150

100

50

060 70 80 90 100

Speed (kmh)110 120 130


(b)






4 Conclusion





Data Availability




Acknowledgments


References


































120

Whe

elr

ail c

onta

ct fo

rce (

kN)

90

60

30

0

20 40 60 80 100 120

005 mm025 mm01 mm

015 mm

02 mm

03 mm

(a)




300

250

Whe

elr

ail c

onta

ct fo

rce (

kN)

200

150

100

50

0

Speed (kmh)20 40 60 80 100 120

025 mm005 mm01 mm015 mm

02 mm

03 mm

(b)



Frequency (Hz)

Spee

d (km

h)

Amplitude = 01mm

Am

plitu

de (k

N)

15

10

5

0120

10080

6040

20250200150100

foor2foor 3foor

500

(a)

Frequency (Hz)

Spee

d (km

h)

Amplitude = 02mm

Am

plitu

de (k

N)

30

20

10

0120

10080

6040

20250200150100

foor2foor 3foor

500

(b)

Frequency (Hz)

Amplitude = 01mm

Spee

d (k

mh

)A

mpl

itude

(kN

)

30

20

10

0120

10080

6040

20250200150100

foor

500

(c)

Frequency (Hz)

Amplitude = 02mm

Spee

d (k

mh

)A

mpl

itude

(kN

)

60

40

20

0120

10080

6040

20250200150100

foor

500

(d)


empty vehicle

90kmh100kmh

100W

heel

rai

l con

tact

forc

e (kN

) 80

60

40

20

0


025 030

(a)

loaded vehicle

90kmh100kmh

300

Whe

elr

ail c

onta

ct fo

rce (

kN) 250

200

150

100

50

0


025 0300 3

(b)







130120

110100Speed (kmh)

Number of harm

onic (order)

Whe

elr

ail c

onta

ct fo

rce (

kN)

9080

7060

(a) (b)

130120

110100Speed (kmh)

9080

7060

13

57

9

Number of harm

onic (order)

13

57

90

10

20

30

40

50

Whe

elr

ail c

onta

ct fo

rce (

kN)

0

10

20

30

40

50



200

Whe

elr

ail c

onta

ct fo

rce (

kN) 160

120

80

40

060 70 80 90 100

Speed (kmh)110 120 130


(a)


250

Whe

elr

ail c

onta

ct fo

rce (

kN) 200

150

100

50

060 70 80 90 100

Speed (kmh)110 120 130


(b)






4 Conclusion





Data Availability




Acknowledgments


References




























Frequency (Hz)

Spee

d (km

h)

Amplitude = 01mm

Am

plitu

de (k

N)

15

10

5

0120

10080

6040

20250200150100

foor2foor 3foor

500

(a)

Frequency (Hz)

Spee

d (km

h)

Amplitude = 02mm

Am

plitu

de (k

N)

30

20

10

0120

10080

6040

20250200150100

foor2foor 3foor

500

(b)

Frequency (Hz)

Amplitude = 01mm

Spee

d (k

mh

)A

mpl

itude

(kN

)

30

20

10

0120

10080

6040

20250200150100

foor

500

(c)

Frequency (Hz)

Amplitude = 02mm

Spee

d (k

mh

)A

mpl

itude

(kN

)

60

40

20

0120

10080

6040

20250200150100

foor

500

(d)


empty vehicle

90kmh100kmh

100W

heel

rai

l con

tact

forc

e (kN

) 80

60

40

20

0


025 030

(a)

loaded vehicle

90kmh100kmh

300

Whe

elr

ail c

onta

ct fo

rce (

kN) 250

200

150

100

50

0


025 0300 3

(b)







130120

110100Speed (kmh)

Number of harm

onic (order)

Whe

elr

ail c

onta

ct fo

rce (

kN)

9080

7060

(a) (b)

130120

110100Speed (kmh)

9080

7060

13

57

9

Number of harm

onic (order)

13

57

90

10

20

30

40

50

Whe

elr

ail c

onta

ct fo

rce (

kN)

0

10

20

30

40

50



200

Whe

elr

ail c

onta

ct fo

rce (

kN) 160

120

80

40

060 70 80 90 100

Speed (kmh)110 120 130


(a)


250

Whe

elr

ail c

onta

ct fo

rce (

kN) 200

150

100

50

060 70 80 90 100

Speed (kmh)110 120 130


(b)






4 Conclusion





Data Availability




Acknowledgments


References
































130120

110100Speed (kmh)

Number of harm

onic (order)

Whe

elr

ail c

onta

ct fo

rce (

kN)

9080

7060

(a) (b)

130120

110100Speed (kmh)

9080

7060

13

57

9

Number of harm

onic (order)

13

57

90

10

20

30

40

50

Whe

elr

ail c

onta

ct fo

rce (

kN)

0

10

20

30

40

50



200

Whe

elr

ail c

onta

ct fo

rce (

kN) 160

120

80

40

060 70 80 90 100

Speed (kmh)110 120 130


(a)


250

Whe

elr

ail c

onta

ct fo

rce (

kN) 200

150

100

50

060 70 80 90 100

Speed (kmh)110 120 130


(b)






4 Conclusion





Data Availability




Acknowledgments


References































4 Conclusion





Data Availability




Acknowledgments


References




























determination of mapping relation between wheel

Documents