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 DOI: 10.1177/0954409712458403published online 29 August 2012

2013 227: 203 originallyProceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid TransitJian Zhang, Yan Zhao, Ya-hui Zhang, Xue-song Jin, Wan-xie Zhong, Frederic W Williams and David Kennedy

based on the pseudo excitation methodslab track system using a parallel algorithm−Non-stationary random vibration of a coupled vehicle

  

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Original Article

Non-stationary random vibration ofa coupled vehicle–slab track system usinga parallel algorithm based on the pseudoexcitation method

Jian Zhang1, Yan Zhao1, Ya-hui Zhang1, Xue-song Jin2,Wan-xie Zhong1, Frederic W Williams3 and David Kennedy3

Abstract

A parallel algorithm based on the pseudo excitation method (PEM) is applied to investigate the non-stationary random

response of a vertically coupled vehicle–slab track system subjected to random excitation induced by track irregularity.

The vehicle is simplified as a multibody system with 10 degrees of freedom and the slab track is represented by a three

layer Bernoulli–Euler beam model which includes the rail, slab and roadbed. Linear wheel–rail contact provides inter-

action between the vehicle and slab track models. The track irregularity is assumed to be a Gaussian random process

with zero mean value and the non-stationary random response of the coupled vehicle–slab track system is solved using a

combination of PEM and step-by-step integration. A parallel algorithm based on PEM is used to accelerate the compu-

tation and the effectiveness of the proposed method is verified by the Monte Carlo method. The basic random per-

formance of the coupled vehicle–slab track system is analysed and the effects of the vehicle speed and slab track

parameters on the random responses of the vehicle and slab track are investigated. The results indicate that the vehicle

speed has a significant effect on the random response of the coupled vehicle–slab track system and that the rail pad

parameters are the most important of the slab track parameters discussed.

Keywords

Vehicle–slab track interaction, random vibration, pseudo excitation method

Date received: 20 December 2011; accepted: 18 July 2012

Introduction

The two main types of railway track structure areballasted track and slab track. Of these, the slabtrack has a lower maintenance cost, better stiffnessuniformity and higher running stability1–3 but it hasa higher construction cost. Slab track systems are usedfor high-speed lines in Europe, China and Japan.2,4

As the train speed increases the dynamic responses ofthe vehicle and track become more severe5 and thisadversely affects vehicle and track maintenance cost,ride comfort and vehicle running safety. The vibra-tions of the vehicle and track are mainly induced byrandom track irregularity6 and it is necessary to cal-culate the random responses of the vehicle and trackin order to understand their essential vibrationcharacteristics.

Xiang et al.7 presented a new track segment elem-ent to analyse three-dimensional coupled vehicle–slabtrack dynamics. The vertical displacement of the slabwas calculated simply by the lateral finite stripmethod, and the responses of the track displacement

and wheel–rail force were investigated. Lei andZhang2 proposed a new type of slab track elementto calculate the vibration of a vertical coupled vehi-cle–slab track system. The global mass, damping andstiffness matrices were efficiently assembled by usingthis slab track element and this led to an investigationof the effects of various slab track parameters on thedynamic responses of the vehicle and track. The vibra-tions of the vehicle and track were analysed in thetransition zone between the slab track and ballastedtrack. Multibody dynamics software was used with a

1State Key Laboratory of Structural Analysis for Industrial Equipment,

Dalian University of Technology, People’s Republic of China2State Key Laboratory of Traction Power, Southwest Jiaotong

University, People’s Republic of China3School of Engineering, Cardiff University, UK

Corresponding author:

Ya-hui Zhang, State Key Laboratory of Structural Analysis for Industrial

Equipment, Faculty of Vehicle Engineering and Mechanics, Dalian

University of Technology, Dalian 116024, People’s Republic of China.

Email: zhangyh@dlut.edu.cn

Proc IMechE Part F:

J Rail and Rapid Transit

227(3) 203–216

! IMechE 2012

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Timoshenko flexible track model to calculate thedynamic responses of two types of innovative slabtrack for comparison with a ballasted track8 withthe uneven stiffness of the slab track forming the exci-tation of the vehicle–track interaction. The dynamicresponse of a coupled vehicle–slab track system wassolved using Green’s functions in the frequency andtime domains,9 with the vehicle model simplified as atwo-mass oscillator (ignoring the bogie pitch) and theslab track was treated as an infinite two-layer beam. Apolygonal wheel and rail corrugation were consideredto analyse the interaction between vehicle and slabtrack. Bitzenbauer and Dinkel10 presented a semi-ana-lytical method based on Fourier transforms to calcu-late the dynamic response of a coupled vehicle–slabtrack system subjected to cosine irregularity, by rep-resenting the slab track as a continuously supportedmodel to obtain a closed-form solution in the Fourierdomain. Zhai et al.11 investigated the interactionbetween vehicle and slab track with a verticallycoupled vehicle–slab track model and with the railand slab simulated by a Bernoulli–Euler beam andwith irregularity at rail joints considered when calcu-lating the dynamic responses of the vehicle and slabtrack. Zhai et al.12 then developed a three-dimensionaldynamic model in which the slab was described as anelastic plane mounted on a viscoelastic foundation.Xiao et al.13 established a coupled vehicle–slab trackmodel to investigate the influence of vertical and lat-eral earthquake amplitudes and vehicle speed on therunning safety of a vehicle, with the slab model simu-lated by solid finite elements. Different coupled vehi-cle–track dynamic models were established toinvestigate the difference between ballasted trackand three types of slab track.3 The dynamic responsesof the vehicle and track were solved by usingNASTRAN to compute the eigenmodes of the railand slab, which were integrated into SIMPACK, acommercial multibody system software package.A spatial sample of the random track irregularitywas obtained from the power spectral density (PSD)function of the track irregularity. The track irregular-ity in the time domain was considered as the externalexcitation to calculate the responses of the wheel–railforce and rail pad force. Luo14 presented a verticallycoupled vehicle–slab track model in which the slabtrack was modelled by thin shell elements and thevehicle–slab track system was excited by a spatialsample of the random track irregularity in the timedomain. Hence, the dynamic responses of the vehicleand slab track were compared by changing the slabtrack parameters. Galvin et al.15 developed a three-dimensional coupled train–track–soil model to inves-tigate the dynamic interaction in the transition zonebetween ballasted track and slab track. The vehiclewas simplified as a multibody model in which theinteraction between two axles linked to the samebogie was ignored and the track and soil were mod-elled by a combination of the finite element method

and the boundary element method. The spatial sampleof the track irregularity was generated from the PSDto form the external excitation. Galvin et al.4 com-pared the dynamic response of slab and ballastedtracks using a similar method.

Although random track irregularity plays a signifi-cant role in inducing interaction between vehicle andslab track, at present few authors have investigatedthe random vibration of coupled vehicle–slab tracksystems. Monte Carlo methods have been used inwhich the spatial sample of the track irregularity isgenerated from the PSD as the external excita-tion.3,14,15 However, many spatial samples must begenerated for accurate calculation of the random per-formance of the coupled vehicle–ballasted tracksystem,16 resulting in high computational costs. Linand Zhang17 proposed the use of the pseudo excita-tion method (PEM) to efficiently and accuratelycalculate the stationary/non-stationary random vibra-tion of large-scale structures. This method has beensuccessfully applied to analyse the random vibrationof coupled vehicle–ballasted track16,18 and vehicle–bridge19 systems. However, the algorithm has nowbeen developed to remove the restrictions whichmeant that in Zhang et al.16 and Lu et al.18 it couldonly be applied to calculate the random response ofthe periodic track and in Zhang et al.19 it was notsuitable for computation of random responses oflarge-scale structures. Although PEM is well suitedfor parallel computation due to the independence ofcalculations at each frequency point, such parallelcomputation has rarely been applied to analyselarge-scale coupled vehicle–structure systems.

In this paper a vertically coupled vehicle–slab trackmodel is established to analyse the non-stationaryrandom vibration of the vehicle and slab track. Thevehicle is treated as a multibody model in which theelasticity of the car body, bogies and wheelsets isignored. The slab track is considered as flexible andthe rail, slab and roadbed are modeled by Bernoulli–Euler beam theory. The vehicle and slab track arecoupled through a linearized wheel–rail force andthe non-stationary random vibration of the coupledvehicle–slab track system is obtained by a combin-ation of PEM and step-by-step integration. A parallelalgorithm based on PEM is adopted to improve com-putational efficiency. This paper extensively investi-gates the influence of the vehicle speed and slabtrack parameters on the random responses of thevehicle and slab track.

Vertically coupled vehicle–slab trackdynamics model

A vertically coupled vehicle–slab track dynamicsmodel is established as shown in Figure 1. It isassumed that the vehicle runs along the slab track ata constant velocity. The vehicle model is simplified asa multi-rigid-body system which consists of one car

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body mounted on two bogies, four wheelsets, fourprimary suspensions and two secondary suspensions.The primary and secondary suspensions are repre-sented by parallel spring–damping elements. The carbody and bogies are connected through the secondarysuspensions. Each bogie is supported on two wheel-sets, which are linked to the bogie through the pri-mary suspensions. The car body and the two bogieseach undergo vertical displacement and pitch rota-tion, but only vertical displacement is considered inthe four wheelsets. Thus, there are 10 degrees of free-dom in the vehicle model. The bogies and wheelsetsare assumed to be numbered from right to leftin Figure 1.

The equation of motion for the vehicle model is

Mv €xv tð Þ þ Cv _xv tð Þ þ Kvxv tð Þ ¼ fv xv tð Þ,xt tð Þð Þ ð1Þ

where Mv, Cv and Kv are the mass, damping and stiff-ness matrices of the vehicle model, respectively, €xv tð Þ,_xv tð Þ and xv tð Þ are the acceleration, velocity and dis-placement vectors of the vehicle model, respectively,xt tð Þ denotes the displacement vector of the slab trackmodel and fv xv tð Þ, xt tð Þð Þ denotes the external loadvector of the vehicle model.

The slab track consists of the rail, slab, roadbed,rail pad, cement asphalt mortar (CAM) and founda-tion surface layer (FSL). The slab track is consideredas flexible, and is simplified using a three-layer beammodel. The rail, slab and roadbed are described byBernoulli–Euler beam theory. The rail is discretelysupported by the slab. The rail fastener, CAM andFSL are represented by parallel spring–dampingelements.

The equation of motion for the slab track is

Mt €xt tð Þ þ Ct _xt tð Þ þ Ktxt tð Þ ¼ ft xv tð Þ, xt tð Þð Þ ð2Þ

where Mt, Ct and Kt are the mass, damping and stiff-ness matrices of the slab track model, respectively,€xt tð Þ and _xt tð Þ are the acceleration and velocity vectors

of the slab track model, respectively and ft xv tð Þ, xt tð Þð Þ

denotes the external load vector of the slab trackmodel.

The vehicle model and the slab track model arecoupled by the linearized wheel–rail force, which isexpressed as

fwr ¼ kh xw � xr � xirrð Þ ð3Þ

where kh denotes the contact stiffness of the wheel andrail, xw denotes the wheel displacement, xr denotes therail displacement at the wheel position and xirrdenotes the track irregularity.

The contact stiffness of the wheel and rail isexpressed as18

kh ¼ 1:51

GP1=30 ð4Þ

where G denotes the Hertzian constant and P0

denotes the static wheel load.G is expressed for a worn wheel profile as20

G ¼ 3:86R�0:115 � 10�8m=N2=3 ð5Þ

where R is the wheel radius.According to equations (1), (2) and (3), the equa-

tion of motion for the coupled vehicle–slab tracksystem can be written as

Mvt €xvt tð Þ þ Cvt _xvt tð Þ þ Kvt tð Þxvt tð Þ ¼ fvt tð Þ ð6Þ

where Mvt, Cvt and Kvt tð Þ are the mass, damping andstiffness matrices of the coupled vehicle–slab trackmodel, respectively, €xvt tð Þ, _xvt tð Þ and xvt tð Þ are theacceleration, velocity and displacement vectors ofthe coupled vehicle–track model, respectively andfvt tð Þ is the external load vector which is determinedby the track irregularity.

In equation (6) the stiffness matrix is time-depen-dent and is determined by the contact location

z

xBogie

Car body

Wheelset

RailRail pad

Slab

CAM

Roadbed

FSL

Figure 1. Vertically coupled vehicle–slab track dynamics model.

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between wheel and rail. The stiffness matrix isupdated at each time step and the time-variant termof the stiffness matrix is transferred into the right sideof the equation (6) for convenient calculation of thesystem response.12,19,21,22 The coupled vehicle–slabtrack system is divided into two time-invariantsubsystems: the vehicle subsystem and the slab tracksubsystem. The interaction between these subsys-tems is accomplished through the wheel–railforce. The responses of the vehicle and slab trackcan be obtained independently using step-by-stepintegration.

PEM for linear time-variant systemof fully coherent multiple excitation

The equation of motion for the linear time-variantsystem is

M tð Þ €x tð Þ þ C tð Þ _x tð Þ þ K tð Þx tð Þ ¼ f tð Þ ð7Þ

where M tð Þ, C tð Þ and K tð Þ are the mass, damping andstiffness matrices of the linear time-variant system,respectively, €x tð Þ, _x tð Þ and x tð Þ are the acceleration,velocity and displacement vectors of the linear time-variant system, respectively and f tð Þ denotes therandom external force vector.

The fully coherent multiple excitation isexpressed as

r tð Þ ¼ r t� t1ð Þ, r t� t2ð Þ, . . . , r t� tnð Þ½ �T

ð8Þ

where the superscript T denotes transpose, the randomexcitation r tð Þ is a stationary random Gaussian pro-cess with zero mean value, ti denotes the time lag witht1 ¼ 0 and n is the number of excitations.

The random external force is expressed as

f tð Þ ¼ ! tð Þr tð Þ ð9Þ

where ! tð Þ denotes the index matrix representing theexternal force location, whose elements are either zeroor one.

Using Duhamel’s integral, an arbitrary responseinduced by the random external force can beexpressed as

x tð Þ ¼

Z t

0

H t� �, �ð Þf tð Þd� ð10Þ

where H t� �, �ð Þ denotes the impulse response func-tion of the linear time-variant system.

The cross-correlation matrix of two arbitraryresponses is expressed as

Rxkxl tð Þ ¼ E xk tkð Þxl tlð ÞT

� �ð11Þ

where E½ � denotes the expectation operator.

Substituting equations (9) and (10) into equa-tion (11) gives

Rxkxl tð Þ¼

Z tk

0

Z tl

0

H tk � �k, �kð Þ! �kð Þ

E r �kð ÞrT �lð Þ

� �!T �lð ÞH

T tl � �l, �lð Þd�kd�l

ð12Þ

Using the Wiener–Khintchine theorem gives

E r �kð ÞrT �lð Þ

� �¼

Z þ1�1

v�vTSf !ð Þei!�d! ð13Þ

where v ¼ e�i!t1 , e�i!t2 , . . . , e�i!tn� �T

, Sf !ð Þ is the PSDof the random excitation and the superscript � denotescomplex conjugate.

Substituting equation (13) into equation (12) gives

Rxkxl tð Þ ¼

Z þ1�1

I�k !, tkð ÞSf !ð ÞITl !, tlð Þd! ð14Þ

where

Ik !, tkð Þ ¼

Z tk

0

H tk � �k, �kð Þ! �kð Þvei!tkd�k,

Il !, tlð Þ ¼

Z tl

0

H tl � �l, �lð Þ! �lð Þvei!tld�l:

Using the Wiener–Khintchine theorem, the cross-PSD of xk tð Þ and xl tð Þ is

Sxkxl !, tð Þ ¼ I�k !, tð ÞSf !ð ÞITl !, tð Þ ð15Þ

By letting k ¼ l in equation (15), the auto-PSD ofthe arbitrary response xk is

Sxkxk !, tð Þ ¼ Sf !ð Þ

Z t

0

Z t

0

H t� �1, �1ð Þ

� ! �1ð Þv�vT!T �2ð ÞH

T t� �2, �2ð Þd�1d�2

ð16Þ

Substituting the pseudo excitation17

~f tð Þ ¼ ! tð ÞvffiffiffiffiffiffiffiffiffiffiffiSf !ð Þ

pei!t ð17Þ

into equation (10) gives the pseudo response

~x !, tð Þ ¼

Z t

0

H t� �, �ð Þ! tð ÞvffiffiffiffiffiffiffiffiffiffiffiSf !ð Þ

pei!td� ð18Þ

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The auto-PSD of the response is calculated as

Sxx !, tð Þ ¼ ~x� !, tð Þ~xT !, tð Þ

¼ Sf !ð Þ

Z t

0

Z t

0

H t� �1, �1ð Þ! �1ð Þv�vT!T �2ð Þ

�HT t� �2, �2ð Þd�1d�2 ð19Þ

i.e. the same as equation (16). For a linear time-var-iant system, the PSD of an arbitrary non-stationaryrandom response can be calculated by PEM.

PEM applied to the coupled vehicle–slabtrack system

The coupled vehicle–slab track system is excited bymultipoint random external forces from the sametrack irregularity. A time lag exits between two arbi-trary random external forces. The track irregularity isassumed to be a Gaussian random process with zeromean value.6 The random excitation of the trackirregularity is expressed as

r tð Þ ¼ r t� t1ð Þ, r t� t2ð Þ, r t� t3ð Þ, r t� t4ð Þ½ �T

ð20Þ

where t1 ¼ 0, t2 ¼ 2lt=v, t3 ¼ 2lc=v andt4 ¼ 2 lc þ ltð Þ=v; lc denotes half the distance betweenthe two bogie centres; lt denotes half the distancebetween the two axles of the same bogie and v is thevehicle speed.

The pseudo excitation of the track irregularity isintroduced as

~f tð Þ ¼ ! tð Þ e�i!t1 , e�i!t2 , e�i!t3 , e�i!t4� �T ffiffiffiffiffiffiffiffiffiffiffi

Sr !ð Þp

ei!t

ð21Þ

where Sr !ð Þ denotes the PSD of the track irregularityand ! is the circular frequency.

According to equations (10) and (21), an arbitrarynon-stationary pseudo response ~u !, tð Þ is obtained,and then the corresponding PSD is calculated as

Suu !, tð Þ ¼ ~u� !, tð Þ~uT !, tð Þ ð22Þ

For practical problems, the PSD in equation (22) iscalculated over a specified frequency range. Thestandard deviation (SD) of the arbitrary response iscalculated as

�u ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ !2

!1

Suu !, tð Þd!

sð23Þ

where !1 and !2 are the minimum and maximumcomputational frequencies, respectively.

Parallel algorithm for PEM

The specified frequency range is divided into asequence of equally spaced frequency points, at eachof which the PSD in equation (22) is calculated. For alarge-scale coupled vehicle–slab track system, thecomputational cost of the traditional PEM is highand increases with the number of frequency points.However, the random responses of the vehicle andslab track are calculated independently at each fre-quency point, with no exchange of data between dif-ferent frequency points. Thus, PEM is well suited forparallel computation. Generally, the number of pro-cessors is less than the number of frequency points, sothat a parallel algorithm is easily derived by lettingeach processor independently calculate the randomresponses of the vehicle and slab track at a specifiednumber of frequency points. This parallel computa-tion process is shown in Figure 2. First, the frequencypoints are distributed as equally as possible to eachprocessor. Second, the random responses of the vehi-cle and slab track are calculated by each processor atits specified frequency points. Finally, the results cal-culated by each processor are merged to give results atall frequency points.

The speed-up ratio is defined as the ratio of CPUtime of the serial computation to that of the parallelcomputation. The theoretical speed-up ratio is equalto the number of processors used. However, it is dif-ficult to obtain this theoretical value due to the add-itional cost of set-up and overhead calculations andbecause some processors may have one more

Figure 2. Flowchart for parallel computation.

Table 1. Speed-up ratio for different numbers of

the processors.

Processors Time (s) Ratio

1 2849.9 1.0

2 1679.4 1.7

4 841.4 3.4

6 532.3 5.4

8 399.0 7.1

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frequency point than others. One node with up toeight processors was used to investigate the speed-upratios for different numbers of processors, for thecoupled vehicle–slab track system for which the par-ameters are given in the section ‘Numerical resultsand discussion’. The number of frequency pointswas 40. Accurate speed-up ratios were obtained fordifferent numbers of processors by using the Windowsoperating system on a 2.67GHz CPU HP Z800 whichhad one node with eight processors. The results aregiven in Table 1, from which it is seen that the ratios

increase almost linearly with the number of proces-sors. If the number of processors exceeds thenumber of computational frequency points, a morecomplex parallel algorithm would need to be devel-oped for the step-by-step integration at each fre-quency point.

Numerical results and discussion

A 1.86GHz CPU Lenovo Shenteng 7000 with eightnodes each having eight processors was used to com-pute the random response of the coupled vehicle–slabtrack system. The vehicle starts at the left-hand end ofa 100m long slab track, with a speed of 200 km/h(except in the section ‘Effect of vehicle speed on therandom response of the coupled vehicle–slab tracksystem’) and the steady state response of the coupledvehicle–slab track system were obtained at the middleof the slab track thereby ensuring that the end condi-tions did not influence the obtained results. The vehi-cle and slab track parameters are listed in Tables 2and 3.11,14 The random vertical track irregularityPSD23 was adopted to give

S �ð Þ ¼Av�

2c

�2 þ�2r

� ��2 þ�2

c

� �m2= rad=mð Þ ð24Þ

where �c and �r are cutoff frequencies, Av is theroughness constant and � is the spatial circular fre-quency. The values used were �c ¼ 0:8246 rad=m,�r ¼ 0:0206 rad=m and Av ¼ 4:032� 10�7 m�rad,with a computational frequency range of between0.5 and 100Hz and 200 frequency points. Five alter-native sets of rail pad, CAM and FSL parameterswere used, see Table 4.

Verification by Monte Carlo method

The random response of the coupled vehicle–slabtrack system was calculated both by PEM and bythe Monte Carlo method with 10, 100 and 500 spatialsamples. The results are compared in Figure 3 whichgives the SDs of the wheel–rail force, bogie verticalacceleration and rail acceleration, from which it isseen that the Monte Carlo results become close tothose calculated by PEM as the number of samplesis increased. For 500 spatial samples the maximumSD differences are 6.8, 8.5 and 9.5%, for the wheel–rail force, bogie vertical acceleration and rail acceler-ation, respectively. Note that the wheel–rail and bogieresults are periodic due to the slab track stiffnesschanging periodically, with periodicity equal to theslab length, i.e. 5m.

Random response of the coupled vehicle–slab tracksystem

The wheel–rail force plays a key role for the coupledvehicle–slab track system and so Figure 4 shows the

Table 2. Vehicle parameters.

Parameter Value

Car body mass 39,600 kg

Car body pitch moment of inertia 1,940,400 kg �m2

Bogie mass 3200 kg

Bogie pitch moment of inertia 1752 kg �m2

Wheelset mass 2000 kg

Primary suspension stiffness 2352 kN/m

Primary suspension damping 39.2 kN � s/m

Secondary suspension stiffness 411 kN/m

Secondary suspension damping 19.6 kN � s/m

Distance between two bogie centres 17.5 m

Distance between two axles of same bogie 2.5 m

Wheel rolling radius 0.43 m

Table 3. Slab track parameters.

Parameter Value

Rail mass per unit length 60.64 kg/m

Area moment of the rail cross-section 3.217� 10�5 m4

Rail elastic modulus 2.059� 1011N/m2

Rail pad stiffness 6.0� 107 N/m

Rail pad damping 3.625� 104 N � s/m

Slab length 5 m

Slab width 2.4 m

Slab height 0.19 m

Slab elastic modulus 42,000 MPa

CAM length 5 m

CAM width 2.4 m

CAM height 0.05 m

CAM elastic modulus 100 MPa

CAM damping12 8.3� 104 N � s/m

Roadbed length 5 m

Roadbed width 2.8 m

Roadbed height 0.3 m

Roadbed elastic modulus 42,000 MPa

FSL length 100 m

FSL width 6.2 m

FSL height 0.4 m

FSL elastic modulus 220 MPa

FSL damping 8.3� 104 N � s/m

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wheel–rail force PSDs for the first two wheelsets,which also vary periodically with time due to the peri-odic variation of the slab track stiffness. Within eachsuch cycle, the wheel–rail force PSD varies quasi-per-iodically due to the rail being supported discretely bythe rail pads, so that its quasi-periodic length is equalto the rail pad spacing, i.e. 0.625m (a similar phenom-enon occurs in Figure 5(b) and (c)). The main peakPSD values for the first two wheel–rail forces differ

because of bogie pitch and occur at approximately50.5 and 51Hz, respectively, with their maximum fluc-tuations being 0.79� 106 and 0.27� 106N2/Hz,respectively. The wheel–rail force PSDs for the thirdand fourth wheelsets are similar to those for the firsttwo wheelsets, respectively. The wheel–rail forcePSD distribution is mainly affected by the vehiclespeed and the rail pad stiffness, see the next twosubsections.

Table 4. Alternative rail pad, CAM and FSL parameters.

Parameters

Case

1 2 3 4 5

Rail pad Stiffness (106 N/m) 20 40 60 80 100

Damping (104 N � s/m) 20 40 60 80 100

CAM Elastic modulus (MPa) 200 400 600 800 1 000

Damping (104 N � s/m) 4 6 8 10 12

FSL Elastic modulus (MPa) 160 180 200 220 240

Damping (104 N � s/m) 4 6 8 10 12

(a) (b)

(c)

Figure 3. Comparison between PEM and Monte Carlo method: (a) wheel–rail force SD; (b) bogie vertical acceleration SD; and

(c) rail acceleration SD.

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Figure 5(a) to (c) show the acceleration PSDs ofthe vehicle’s components. Their maximum ampli-tudes decrease dramatically from wheelset to carbody due to the effectiveness of the primary andsecondary suspensions, particular the former, sothat the vertical acceleration PSD of the car bodyvaries only slightly with time. Thus, the verticalacceleration PSD of the car body is hardly affectedby the slab track stiffness. Its maximum value occursat approximately 0.99Hz, see Figure 5(a), which isclose to its vertical natural frequency. Figure 5(b)shows that the four peak values for the first bogie

occur at approximately 6, 20, 47 and 61.5Hz,respectively, the first of which is close to the verticalnatural frequency of the bogie. The three other peakvalues are mainly related to vehicle speed, seethe next subsection. Compared with the firsttwo peak values, the third and fourth peak valuesclearly vary more with time. The second bogie verti-cal acceleration PSD is similar to that of the firstbogie and so is not shown. The first wheelset accel-eration PSD is shown in Figure 5(c) and the wheel–rail force PSD for the first wheelset behavessimilarly.

Figure 4. Wheel–rail force PSD for (a) the first wheelset and (b) the second wheelset.

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Figure 5(d) to (f) show the PSDs of the accel-erations of the slab track components which wereobtained at 61.875m from the left-hand end of theslab track. Their maximum amplitudes decreasefrom rail to roadbed with the rail pad clearly play-ing an important role in isolating the vibration.From Figure 5(d) the main frequency of the railacceleration PSD is seen to occur at approximately51.5Hz (which approximates the frequency of themaximum PSD of the wheel–rail force), and its twomain peak values are induced by the first twowheelsets passing over the rail, with the distance

between them being equal to the distance betweenthe axles of the bogie, i.e. 2.5m. The peak valueinduced by the first wheel–rail force exceeds thatfor the second wheel–rail force because the firstwheel–rail force exceeds the second one. Thereare also small peak values between 70 and100Hz. The PSD of the slab and roadbed acceler-ations are respectively shown in Figure 5(e) and (f)and their maximum peak values both occur at89.5Hz. This frequency is not induced by the fre-quency of the maximum PSD of the wheel—railforce, which is isolated by the rail pad.

Figure 5. PSD of accelerations (in m4/s2/Hz) for: (a) car body; (b) first bogie; (c) first wheelset; (d) rail; (e) slab; and (e) roadbed.

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Effect of vehicle speed on the random response ofthe coupled vehicle–slab track system

The vehicle speed was varied between 200 and350 km/h at intervals of 50 km/h to investigate theeffect of vehicle speed on the random response ofthe coupled vehicle–slab track system. Figure 6(a)shows variation with vehicle speed of the maximumPSD of the first two wheel–rail forces and the num-bers on it are the frequency of each maximum; thesenumbers vary only to a small extent except for the firstwheel–rail force at 250 km/h. The variation of themaximum PSD of the first wheel–rail force is clearlythe more severe of the two between 300 and 350 km/h.The variation of the wheelset acceleration PSD is notshown because it is similar to that of the wheel–railforce PSD.

In Figure 6(b) the regions A and B represent thevariation of the first two peak values of the car body’svertical acceleration PSD with vehicle speed. In regionA, the peak values increase with vehicle speed,although their frequencies are almost identicalbecause these peaks are mainly influenced by the ver-tical natural frequency of the car body. In contrast, inregion B the frequency of the peak value shifts

towards the right as the vehicle speed increases, withthe peak values being altered very little, since they arecaused by the phase difference of the excitation of thetrack irregularity, which depends on the vehicle speed.

In Figure 6(c) the peak values of the bogie verticalacceleration PSDs at 200, 250, 300 and 350 km/h aredenoted by A, B, C and D, respectively, and theirvalues increase as the vehicle speed increases. Thefrequencies of the first peak values are close to thevertical natural frequency of the bogie and so arelittle influenced by the vehicle speed and theybecome larger with increasing vehicle speed, so thatat higher speeds the frequency of the maximum ismainly precipitated by the natural frequency of thebogie. The frequencies and amplitudes of other peakvalues also change with vehicle speed, and are againcaused by the phase difference of the excitation of thetrack irregularity. Such complex phenomena werealso reported by Zhang et al.16

Figure 6(d) shows the variation of the vehicle speedof the maximum PSD of the accelerations of the slabtrack components. The peak value for the railincreases with vehicle speed, but the values for theslab and roadbed at 300 km/h are lower than at350 km/h. The considered maximum frequency ofthe excitation is only 100Hz due to lack of knowledge

(a) (b)

(c) (d)

Figure 6. Effect of vehicle speed on the PSD for (a) wheel–rail vertical force (N2/Hz); (b) car body acceleration (m4/s2/Hz); (c) bogie

acceleration (m4/s2/Hz); and (d) rail acceleration (m4/s2/Hz).

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about the track irregularity PSD for a high-speedline at high frequency, which may excite the higherfrequencies of the PSD of the slab and roadbed accel-erations at 350 km/h.

Effect of slab track parameters on the randomresponse of the coupled vehicle–slab track system

Figure 7(a) shows that the maximum PSD of the firsttwo wheel–rail forces increase with rail pad stiffnesswhereas Figure 7(b) shows that they decrease with railpad damping. The maximum PSDs of the first twowheel–rail forces increase by 469.1 and 359.2%,

respectively, when the rail pad stiffness is increasedfive-fold. The numbers printed on Figure 7(a) arethe frequencies of the maximum PSD and can beseen to increase with rail pad stiffness, whereasFigure 7(b) shows that the rail pad damping hardlyalters the frequency of the maximum.

The variation range of the maximum PSD of thecar body vertical acceleration was between 0.0245 and0.0246 m4/s2/Hz as rail pad stiffness increased and sothis maximum is unaffected by the rail padparameters.

Figure 7(c) shows that the variance of the firstpeak value of the bogie vertical acceleration PSD is

(a) (b)

(c) (d)

(e) (f)

Figure 7. Effect of rail pad parameters on the PSD (a) wheel–rail vertical force (N2/Hz); (b) car body acceleration (m4/s2/Hz);

(c) bogie acceleration (m4/s2/Hz); and (d) rail acceleration (m4/s2/Hz).

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relatively small due to it being mainly induced by thenatural frequency of the bogie, while the variance ofother peak values is complex. Figure 7(d) shows thatonly the third peak decreases as the rail pad dampingincreases.

Figure 7(e) shows that the maximum PSD of therail, slab and roadbed accelerations increase with thepad stiffness. Figure 7(f) shows that the frequencies ofthe maximum PSD of the rail and roadbed acceler-ations are not affected significantly by increasing therail pad damping and nor is the frequency of the max-imum PSD of the slab acceleration, except for the

first value. The rail pad damping affects the maximumrail acceleration PSD much more than the slab androadbed acceleration PSDs.

Figure 8 shows the effects of CAM and FSL stiff-ness and damping on vehicle and slab track compo-nents, showing whichever is appropriate ofacceleration or wheel–rail force PSD. It can be seenthat the FSL stiffness has much the greatest effectexcept for the rail, although the FSL damping canalso be significant. Frequency values are not printedon Figure 8; however, the frequency of the maximumPSD of the bogie vertical acceleration changed from

(a) (b)

(c) (d)

(e) (f)

Figure 8. Effect of CAM and FSL parameters on the maximum PSD for: (a) wheel–rail vertical force (N2/Hz); (b) wheel–rail vertical

force (N2/Hz); (c) bogie acceleration (m4/s2/Hz); (d) bogie acceleration (m4/s2/Hz); (e) rail acceleration (m4/s2/Hz); and (f) rail

acceleration (m4/s2/Hz).

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47 to 60Hz when the CAM stiffness changed from 200to 400MPa. In contrast, the frequencies of the max-imum PSD of the vehicle and slab track componentaccelerations are hardly affected by the other CAMand FSL parameters.

Conclusions

A coupled vehicle–slab track model has been devel-oped to investigate the non-stationary random vibra-tion of coupled vehicle–slab track systems. Thevehicle is treated as a multibody model, in which theelasticity of the car body, bogie and wheelset are neg-lected. The slab track is flexible with the rail, slab androadbed modeled by Bernoulli–Euler beam theory.The vehicle and slab track are coupled through thelinearized wheel–rail force, and the non-stationaryrandom vibration of the coupled vehicle–slab tracksystem is calculated by combining PEM with step-by-step integration. A parallel algorithm based onPEM was adopted to improve computational effi-ciency. The paper investigated how the randomresponses of the vehicle and slab track are influencedby the vehicle speed and the slab track parameters,leading to the following conclusions.

1. The parallel algorithm based on PEM is easilyimplemented and compared with serial computa-tion; it significantly improves computational effi-ciency when calculating the random response ofthe coupled vehicle–slab track system.

2. The random response of the car body’s verticalacceleration is time-invariant with the maximumvertical acceleration PSD of the car body beinginduced by its vertical natural frequency. ThePSD of the vertical accelerations of the bogieand wheelset vary periodically with time as alsodoes the wheel–rail force. The periodic lengths ofthese PSD are equal to the slab length. The mainfrequency of the rail acceleration PSD is close tothe frequency of the maximum PSD of the wheel–rail force.

3. The PSD of the vertical accelerations of the carbody, bogie and rail increase with vehicle speed, asalso does the first wheel–rail force. The vehiclespeed barely affects the frequency of the maximumvertical acceleration PSD of the car body, wheelsetand rail.

4. Compared with the CAM and FSL stiffness anddamping, the stiffness and damping of the rail padhas much more influence on the random responseof the coupled vehicle–slab track system. Thedamping of the rail pad evidently reduces the max-imum vertical acceleration PSD of the bogie,wheelset and rail. However, the corresponding fre-quencies of these maximum PSDs hardly vary.For the case of the CAM and FSL parameters,the maximum vertical acceleration PSDs of thebogie, slab and roadbed are evidently most

affected by the FSL stiffness compared with theother parameters.

5. Study of track irregularity for high-speed lines athigh frequency needs to be developed further tobetter understand the random responses.

Funding

This work was supported by the National Basic ResearchProgram of China under grant 2010CB832704, the National

Natural Science Foundation of China under grants10972048 and 11172056, the High PerformanceComputing Department of the Network and Information

Center, Dalian University of Technology and the CardiffAdvanced Chinese Engineering Centre.

Conflict of interest

There are no conflicting interests associated with the finan-

cial support of this paper.

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Appendix 1

Notation

Av roughness constantC damping matrix of the linear time-

variant systemCt, Cv, Cvt damping matrices of the slab track,

vehicle and coupled vehicle–slabtrack models

E½ � expectation operatorf random external force vector

ft, fv, fvt external load vectors of the slabtrack, vehicle and coupled vehicle–slab track models

G Hertzian constantH impulse response function of the

linear time-variant systemkh contact stiffness of the wheel and railK stiffness matrix of the linear time-

variant systemKt, Kv, Kvt stiffness matrices of the slab track,

vehicle and coupled vehicle–slabtrack models.

lc, lt half the distances between two bogiecentres and axles of the same bogie

M mass matrix of the linear time-var-iant system

Mt, Mv, Mvt mass matrices of the slab track,vehicle and coupled vehicle–slabtrack models.

n, N numbers of excitations and fre-quency points

P0 static wheel loadr random excitationR wheel radiusSf PSD of the random excitationSr PSD of the track irregularityti time lagv vehicle speedx track irregularity locationx, _x, €x displacement, velocity and accelera-

tion vectors of the linear time-variantsystem

xirr track irregularityxr, xw rail displacement at wheel position

and wheel displacementxt, xv, xvt displacement, velocity and accelera-

tion vectors of the slab track, vehicle_xt, _xv, _xvt andxt, xv, xvt coupled vehicle–slab track models

! index matrix�k random phase angles! circular frequency!1, !2 minimum and maximum computa-

tional frequencies!k circular frequency within the com-

putational frequency range�! circular frequency interval� spatial circular frequency�c, �r cutoff frequencies

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