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Vehicle System Dynamics

Vol. 47, No. 11, November 2009, 13491376

Fundamentals of vehicletrack coupled dynamics

Wanming Zhai*, Kaiyun Wang and Chengbiao Cai

Train and Track Research Institute, Traction Power State Key Laboratory,Southwest Jiaotong University, Chengdu, P.R. China

(Received 14 A pril 2008; final version received 11 November 2008)

This paper presents a framework to investigate the dynamics of overall vehicletrack systems withemphasis on theoretical modelling, numerical simulation and experimental validation. A three-dimensional vehicletrack coupled dynamics model is developed in which a typical railway passengervehicle is modelled as a 35-degree-of-freedom multi-body system. A traditional ballasted track ismodelled as two parallel continuous beams supported by a discrete-elastic foundation of three layerswith sleepers and ballasts included. The non-ballasted slab track is modelled as two parallel continu-ous beams supported by a series of elastic rectangle plates on a viscoelastic foundation. The vehiclesubsystem and the track subsystem are coupled through a wheelrail spatial coupling model that con-siders rail vibrations in vertical, lateral and torsional directions. Random track irregularities expressedby track spectra are considered as system excitations by means of a timefrequency transformationtechnique. A fast explicit integration method is applied to solve the large nonlinear equations of motionof the system in the time domain. A computer program named TTISIM is developed to predict the

vertical and lateral dynamic responses of the vehicletrack coupled system. The theoretical modelis validated by full-scale field experiments, including the speed-up test on the BeijingQinhuangdaoline and the high-speed running test on the QinhuangdaoShenyang line. Differences in the dynamicresponses analysed by the vehicletrack coupled dynamics and by the classical vehicle dynamics areascertained in the case of vehicles passing through curved tracks.

Keywords: traintrack interaction; vehicle dynamics; vehicletrack coupled dynamics; curvingperformance; experimental validation

1. Introduction

The classical theory of railway vehicle dynamics [1,2] usually focuses on the railway vehicle

itself as the analysis object without consideration of the dynamic behaviour of the track system

that supports the vehicle, i.e. the track structure is assumed to be a rigid base. In fact, the railway

track is a typical elastic structure with damping. Vibrations of the vehicle can be transmitted to

the track via the wheelrail contact and excite vibrations of the elastic track structure, which

can in reverse influence the vibrations of the vehicle not only in the vertical direction but in

the lateral direction as well. Therefore, the vibrations of a vehicle and a track are essentially

coupled with each other.

*Corresponding author. Email: [email protected]

ISSN 0042-3114 print/ISSN 1744-5159 online 2009 Taylor & FrancisDOI: 10.1080/00423110802621561http://www.informaworld.com


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1350 W.M. Zhaiet al.

With the increase in train speed and vehicle axle load, effects of the dynamic interaction

between the vehicle and the track obviously intensify. Dynamic effects of the vehicle on the

track structure increase accordingly, especially in the case of increasing the train speed on

existing railway track structures such as in Chinese Railways (CR) [3]. Furthermore, a great

interest of modern railway designers as well as maintenance engineers is motivated by the

desire to improve ride comfort, to prevent vehicle hunting, to reduce deformations of the track,

to minimise damage and wear of both vehicle and track components and, most important of

all, to ensure running safety. All above aspects relate to the dynamic behaviour of the entire

vehicletrack system. Well understanding the dynamic characteristics of the entire system

provides the possibility to optimise the design parameters of both the vehicle and the track

components that would minimise the dynamic interactions between the vehicle and the track.

Therefore, it is necessary to systematically investigate the dynamics of a vehicle and a track

from the entire vehicletrack system. This is the basic idea of the vehicletrack coupled

dynamics, which was already adopted by the author in the early 1990s [46].

In fact, it seems to be a trend to consider the vehicle and the track as an integral sys-tem in recent studies on track dynamics and on wheelrail dynamic interactions [719]. The

rapid development of computation technique makes it easy to analyse a large complicate

vehicletrack coupled system. However, most of the research work deals with vertical dynamic

problems. Very few can be found on the lateral system dynamics [8,11,16,19]. It is important

to analyse the lateral dynamic behaviour of a vehicle running on a viscoelastic track structure

instead of a rigid track. Our field experiment results [20] showed that the lateral displacement

of the top of the rail could reach 45 mm for the wooden sleeper track and 23 mm for the

concrete sleeper track when a freight train is passing through a 287 m radius curve. The max-

imum values of the gauge enlargement corresponding to these two cases were measured to be

about 9 mm and about 3 mm, respectively. So large lateral motions of the rails will definitelycause great changes in the wheelrail contact geometry and will thereby eventually influence

the lateral and vertical wheelrail dynamic forces. It is not clear how big the difference is

between the lateral dynamic indices predicted by the models with and without consideration

of track vibration, for example the difference between the lateral wheelrail forces on a curved

track with a rigid track model and with an elastic track model. It is also not very clear which

parameters of the track system have a significant influence on the lateral dynamic performance

of the vehicle as well as on the lateral vehicletrack interaction.

This paper intends to systematically introduce the framework of the vehicletrack coupled

dynamics with emphasis on new advances made by the authors in recent years. A three-

dimensional model of a typical passenger coach on a ballasted track will be described in

detail. A dynamic model of a non-ballasted track is also developed in consideration of the

wide application of this type of track on high-speed railways. Random track irregularities

varying with the rail longitudinal direction are simulated based on the track spectra and used

as excitation input to the vehicletrack coupled system. The models will be validated using

field measurement data obtained from the speed-up test on the BeijingQinhuangdao line and

from the high-speed test on the QinhuangdaoShenyang line. Special attention will be paid to

the differences of the curving performances analysed with the vehicletrack coupled dynamics

and with the classical vehicle dynamics without consideration of elastic track structures.

2. Theoretical model

A vehicletrack coupled model is the theoretical foundation for analysing the vehicletrack

interaction. During the last decade, a series of vehicletrack system models have been

established for various research purposes. A vertical and lateral coupling model for a freight


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Vehicle System Dynamics 1351

car with three-piece bogies and a ballasted track was published in an early paper by the author

in 1996 [11]. A similar model for the vertical and lateral dynamics of the wagontrack system

was presented by Sunet al.in 2003 [16]. This paper will present a three-dimensional coupled

model for typical passenger coaches and tracks including the non-ballasted slab track.

2.1. Three-dimensional vehicletrack coupled model

Figures 13 illustrate the various components of the three-dimensional vehicletrack model in

which subsystems describing the vehicle and the track are spatially coupled by the wheelrail

interface.

In the vehicle submodel,the carbody is supportedon twodouble-axle bogies at each end. The

bogie frames are linked with the wheelsets through the primary suspensions and linked with the

car body through the secondary suspensions. Three-dimensional springdamper elements are

used to representthe primary andthe secondary suspensions.Yaw dampers andanti-roll springs

are considered in the secondary suspensions. Furthermore, the lateral clearances between

the car body and the stop-blocks on the bogie frames are also considered in the secondary

suspensions. The vehicle is assumed to move along the track with a constant travelling speed.

Each component of the vehicle has five degrees of freedom (DOFs): the vertical displacement

Z, the lateral displacementY, the roll angle , the yaw angle and the pitch angle with

respect to its centre of mass (the pitch for a wheelset corresponds to the variation of rotation

around its mean rotational speed). All angles are assumed to be small, which simplifies the

kinematics and the equations of motion. As a result, the total DOFs of the passenger vehicle

submodel are 35.

Figure 1. Three-dimensional vehicletrack coupled model (elevation).


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Figure 2. Three-dimensional vehicletrack coupled model (planform).

Figure 3. Three-dimensional vehicletrack coupled model (end view).

The track submodel shown in Figures 1 and 3 represents the typical ballasted track structure,

consisting of rails, rail pads, sleepers, the ballast and the subgrade. Both the left and the

right rails are treated as continuous BernoulliEuler beams which are discretely supported at

railsleeper junctions by three layers of springs and dampers, representing the elasticity and


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damping of the rail pad, the ballast and the subgrade, respectively. The model allows the input

of track parameters varying along the rails longitudinal direction, such as sleeper spacing or

track supporting stiffness. Three kinds of vibrations of the rails are considered: vertical Zr,

lateralYr and torsional r. The sleeper is assumed to be a rigid body with three DOFs of

the vertical displacementZs, the lateral displacement Ys and the roll angle displacement s.

Lateral springs and dampers are considered to represent the lateral dynamic properties in the

fastener system. Similarly, the lateral springs and dampers are used to represent the elasticity

and damping property between the sleeper and the ballast in the lateral direction.

For modelling the ballast, a five-parameter model of ballast under each rail-supporting point

is adopted [6,21], which is based upon the hypothesis that load transmission from a sleeper to

the ballast approximately coincides with cone distribution [22]. Only the vertical motionZbof

the ballast mass is taken into account. In order to account for the continuity and the coupling

effects of the interlocking ballast granules, a couple of shear stiffness (Kw) and shear damping

(Cw) is introduced between adjacent ballast masses in the ballast model. The ballast model

was recently modified [21] to be able to cover the practical situation in which an overlappingof adjacent ballast masses may occur, see Figure 4. In this case, the vibrating mass of ballast

under a rail-support point could be defined as the shadowed region as shown in Figure 4, where

h0 is the height of the overlapping regions,hb is the depth of ballast,ls is the sleeper spacing,

lb is the width of sleeper underside and is the ballast stress distribution angle.

If the analysed track structure is a non-ballasted track such as the slab track, the above track

dynamic model should be changed correspondingly. Figures 5 and 6 show the established

three-dimensional slab track model in which the track slabs are described as elastic rectangle

plates supported on viscoelastic foundation. In Figure 6, Ksvand Csv are the vertical stiffness

and damping of the CA layer, and Kshand Cshare the lateral stiffness and damping of the CA

layer, respectively. Because the lateral bending stiffness of the slab is very large, it is sufficientto consider the rigid mode of the slab vibration in the lateral direction.

Figure 4. Modified model of the ballast.

Figure 5. Three-dimensional slab track model (elevation).


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Figure 6. Three-dimensional slab track model (end view).

2.2. Equations of motion

2.2.1. Equations of motion of vehicle subsystem

By using the system of coordinates moving along the track with vehicle speed, the equationsof motion of the vehicle subsystem can be easily derived according to DAlemberts principle,

which can be described in the form of second-order differential equations in the time domain:

MVAV+ CV(VV)VV+ KV(XV)XV=FV(XV, VV, XT, VT) + FEXT, (1)

where XV, VV andAV are the vectors of displacements, velocities and accelerations of the

vehicle subsystem, respectively; MV is the mass matrix of the vehicle; CV(VV) and KV(XV)

are the damping and the stiffness matrices which can depend on the current state of the vehicle

subsystem to describe nonlinearities within the suspension; XT and VT are the vectors of

displacements and velocities of the track subsystem; FV(XV,VV,XT,VT) is the system loadvector representing the nonlinear wheelrail contact forces determined by the wheelrail

coupling model, which depend on the motionsXVand VVof the vehicle andXTand VTof the

track; andFEXT describes external forces including gravitational forces and forces resulting

from the centripetal acceleration when the vehicle is running through a curve [23].

There are several nonlinear factors existing in the suspension systems of passenger cars,

e.g. the nonlinear yaw dampers and the clearances between the car body and the stop-block

on the bogie frames. Generally, the yaw damper has a saturation nonlinearity, as shown in

Figure 7. The damping force of a yaw damper can be described by

Fxs=

Fmaxvxct

v0 if|vxct|< v0,Fmaxsign(vxct) if|vxct| v0,

(2)

Figure 7. Nonlinear characteristic of a yaw damper.


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where Fmaxis the saturation force of the damper; vxctis the relative velocity between two ends

of the damper connecting the car body and the bogie frame in the longitudinal direction; and

v0 is the unloading velocity of the damper.

Figure 8 shows a picture of a stop-block on the bogie frame used in the secondary suspension.

It can be clearly seen there is a lateral clearance between the car body and the stop-block on

the bogie frame. If the lateral clearance disappears, the car body will contact the stop-block

and a large contact stiffness will contribute to the secondary stiffness, as shown in Figure 9.

Therefore, the lateral force of a secondary suspension can be given by

Fys=

KsyYcb if|Ycb| sc,KsyYcb+ Ksc(|Ycb| sc)sign(Ycb) if|Ycb|> sc,

(3)

where sc is the lateral clearance between the car body and the stop-block; Ksy is the lateralstiffness of the secondary spring; Ksc is the contact stiffness in the lateral direction if the

car body is in contact with the stop-block; Ycb is the relative displacement between two

ends of the secondary spring connecting the car body and the bogie frame in the lateral

direction.

Figure 8. Lateral clearance between the car body and the stop-block on the bogie frame.

Figure 9. Nonlinear characteristic of lateral stiffness in the secondary suspension.


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2.2.2. Equations of motion of track subsystem

The equations of motion of the track subsystem are more complex than those of the vehicle

subsystem due to the spatial continuity of rails and slabs. The equations of motion of rails are

given in form of the fourth-order partial differential equations:Vertical vibration:

EIy4Zr(x,t)

x4 + mr

2Zr(x,t)

t2 =

Ni=1

Fsvi (t)(x xi ) +4

j=1Pj(t)(x xwj) (4)

Lateral vibration:

EIz4Yr(x,t)

x4 + mr

2Yr(x,t)

t2 =

Ni=1

Fsli (t)(x xi ) +4

j=1Qj(t)(x xwj) (5)

Torsional vibration:

GK2r(x,t)

x2 + I0

2r(x,t)

t2 =

Ni=1

Msi (t)(x xi ) +4

j=1Mwj(t)(x xwj) (6)

In Equations (4)(6), Zr(x, t),Yr(x, t) and r(x, t) are the vertical, lateral and torsional dis-

placements of the rail, respectively; mr is the rail mass per unit length; is the rail density;

EIy andEIz are the rail bending stiffness to the Y-axle and to the Z-axle, respectively; I0 is

the torsional inertia of the rail; GK is the rail torsional stiffness; Fsvi(t) and Fsli(t) are the

vertical and lateral dynamic forces at theith rail-supporting point;Pj(t) andQj(t) are thejth

wheelrail vertical and lateral forces; Msi(t) and Mwj(t) are the moments acting on the rails

due to the forcesFsvi(t) andFsli(t) and due to the forces Pj(t) andQj(t), respectively; and (x)is the Dirac delta function.

The moments Msi(t) and Mwj(t) can be determined based on the forces acting on the rail

cross-section, as shown in Figure 10:Msi (t )= [Fsv2i (t ) Fsv1i (t)]b Fsli (t)a,Mwj(t)=Qj(t)hr Pj(t)e,

(7)

where a is the vertical distance between the rail torsional centre and the lateral force from

the fastening system;b is half of the distance between two vertical forces from the fastening

Figure 10. Forces acting on the cross-section of the rail.


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system;hris the vertical distance from the rail torsional centre to the lateral wheelrail force;

ande is the lateral distance from the rail torsional centre to the vertical wheelrail force.

By describing the slab as a plate supported by a viscoelastic foundation, the vertical

vibrations of each slab are given by:

4w(x,y,t)

x4 + 2

4w(x,y,t)

x2y 2 +

4w(x,y,t)

y 4 + Cs

Ds

w(x,y,t)

t+ shs

Ds

2w(x,y,t)

t2

= 1Ds

Np

i=1Fsvi (t)(x xpi )(y ypi )

Nbj=1

FCAvj(t)(x xbj)(y ybj) . (8)

For the lateral motions, the slab is considered as a rigid body. Thus, the lateral vibrations are

described by:

sLsWshsys(t)=Np

i=1Fsli (t )

Nbj=1

FCAlj(t), (9)

where

Ds=Esh

3s

12(1 2s )(10)

in Equations (8)(10),w(x,y,t) andys are the vertical and lateral displacements of the slab;

Cs is the damping of the slab; Ds is the vertical bending stiffness of the slab; s is the slab

density;Ls,Ws andhs are the length, width and thickness of the slab, respectively; FCAvj (t)

andFCAlj(t) are the vertical and lateral dynamic forces at the jth supporting point under the

slab;Es and s are Youngs elastic modulus and Poissons ratio of the slab;Np and Nb are the

total number of rail fasteners on one slab and the total number of discrete supporting points

under one slab used in the calculation.

In order to solve the track subsystem equations with the time-stepping integration method,

the fourth-order partial differential equations of rails and slabs must be transformed into a

series of second-order ordinary differential equations in terms of the generalised coordinates,

which could be done by means of Ritzs method [6,11]. The transformation results for the rail

equations could be found in [11]. It should be pointed out that sufficient numbers of rail modesare necessary to describe the vibrations of the track system. Numerical trial results show that

0.5L/ls is enough to be chosen as the rail mode number to study the present vehicletrack

coupled dynamics, whereL is the effective length of the rail used in the calculation.

For the slab equation (8), the solution can be assumed as

w(x,y,t)=Nx

m=1

Nyn=1

Xm(x)Yn(y)Tmn(t), (11)

whereXm(x) andYn(y) are the mode functions of the slab withxandy coordinates respectively;Tmn(t) is the generalised coordinate in terms of time t; mand nare the mode numbers ofXm(x)

and Yn(y), respectively; andNxandNyare the total mode numbers of the slab mode functions

Xm(x) and Yn(y) selected for the calculation.


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For the rectangle plate supported on the viscoelastic foundation with free boundaries, the

mode functionsXm(x) and Yn(y) are given by:

X1(x)

=1,

X2(x)= 3

1 2xLs

,

Xm(x)=cosh(mx) + cos(mx) m[sinh(mx) + sin(mx)] (m >2),(12a)

Y1(y)=1,Y2(y)=

3

1 2y

Ws

,

Yn(y)=cosh(ny) + cos(ny) n[sinh(ny) + sin(ny)] (n >2),(12b)

where mand nare the frequency coefficients corresponding toXm(x) and Yn(y), and mand

n are the mode coefficients corresponding to Xm(x) and Yn(y), respectively.

3=4.73002

Ls,

m=2m 3

2Ls (m4),

(13a)

3=4.73002

Ws,

n=2n 3

2Ws (n4),

(13b)

3=0.982502,

m=cosh(mLs) cos(mLs)sinh(mLs) sin(mLs)

(m4),(14a)

3=0.982502,

n=cosh(nWs) cos(nWs)sinh(nWs) sin(nWs)

(n4).(14b)

Substituting Equation (11) into Equation (8) yields

Nxm=1

Nyn=1

Xm (x)Yn(y)Tmn(t) + 2Nx

m=1

Nyn=1

Xm(x)Y

n (y)Tmn(t)

+Nx

m=1

Nyn=1

Xm(x)Y

n (y)Tmn(t )+Cs

Ds

Nxm=1

Nyn=1

Xm(x)Yn(y)Tmn(t )

+ shsDs

Nx

m=1Ny

n=1Xm(x)Yn(y)Tmn(t)=

1

Ds

Np

i=1Fsvi (t)(x xpi )(y ypi )

1Ds

Nbj=1

FCAvj(t)(x xbj)(y ybj). (15)

Multiplying Equation (15) by Xk(x)Yh(y) and then applying integral in the slab area yields

the second-order ordinary differential equations of the slab vertical vibration in terms of the


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generalised coordinateTmn(t) as follows:

Tmn(t) +Cs

shsTmn(t) +

Ds

s hs

B2B3+ 2B4B5+ B1B6B1B2

Tmn(t)

= 1shsB1B2

Np

i=1Fsvi (t)Xm(xpi )Yn(ypi )

Nbj=1

FCAvj(t)Xm(xbj)Yn(ybj)

, (16)

wherem = 1,2, . . . ,Nx, n = 1,2, . . . ,Ny. Due to the orthogonality of the mode functions, theintegrals vanish fork= mand h= n.

B1=Ls

0 X2m(x)dx,

B2=

Ws

0

Y2n (y)dy,

B3=Ls

0 Xm (x)Xm(x)dx,

B4=Ls

0 Xm(x)Xm(x)dx,

B5=Ws

0 Yn (y)Yn(y)dy,

B6=Ws

0 Yn (y)Yn(y)dy.

(17)

Therefore, the final equations of the track subsystem, either for the ballasted track structure

including the equations of the sleepers and ballast blocks which are modelled as individual

rigid masses or for the non-ballasted track structure could be expressed in terms of the standardmatrix form as

MTAT+ CTVT+ KTXT=FT(XV, VV, XT, VT), (18)

where MT is the mass matrix of the track structure; CT and KT are the damping and the

stiffness matrices of the track subsystem; AV is the vector of accelerations of the system;

and FT(XV,VV,XT,VT) is the load vector of the track subsystem representing the nonlinear

wheelrail forces obtained by the wheelrail coupling model.

2.3. Wheelrail coupling model

The wheelrail coupling model is the essential element that couples the vehicle subsystem

with the track subsystem at the wheelrail interfaces.

Unlike the early wheelrail contact model used in the classical vehicle dynamics, in which

the rails are assumed to be fixed without any movement, the dynamic wheelrail coupling

model for the analysis of the three-dimensional vehicletrack coupled dynamics should con-

sider three kinds of rail motions in vertical, lateral and torsional directions, as shown in

Figure 11. In the model, CLcand CRcare the left and the right wheelrail contact points and Land Rare the left and the right wheelrail contact angles, respectively. The nonlinear Hertzian

elastic contact theory is used to calculate the wheelrail normal contact forces according to the

elastic compression deformation of wheels and rails at contact points in the normal directions,which depends on the displacements of wheels and rails. The tangential wheelrail creep

forces are calculated first by the use of Kalkers linear creep theory and then modified by the

ShenHedrickElkins nonlinear model [24]. Details of the new wheelrail coupling model

could be found in a recent paper by Chen and Zhai [25].


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Figure 11. Wheelrail coupling model.

The new spatial wheelrail coupling model eliminates the assumption that the wheel and the

rail are in contact all the time as used in the classical vehicle dynamics. Thus the current model

is capable of dealing with the situation that the wheel loses its contact with the rail existing in

practical railway operations, which may provide a new path to simulate the dynamic process

of derailment.

3. System excitation

As we know, the vehicletrack coupled vibrations are mainly excited by irregularities on the

surfaces of wheels and rails. Generally, there are two types of geometric irregularities existing

at the wheelrail system. One type is specific irregularities such as the wheel flat, the out-

of-round wheel, the dipped rail-joint and the rail corrugation, etc., which has been widely

discussed in the literature. The other type is random irregularities such as the roughness on

the surfaces of wheels and rails, which mainly influence the wheelrail rolling noise, and the

random deviation of the track geometry. Random track irregularities exist in railway lines

everywhere and influence the dynamic performance of vehicles moving on tracks. This paper

will focus on the excitation by the random irregularity of the track.

The random track irregularity is usually expressed by a one-sided power spectrum density

(PSD) function, namely track spectrum. The PSD function of the track spectrum measured on

CR mainlines is expressed as a unified formula

S(f)= A(f2 + Bf+ C)

f4 + Df3 + Ef2 + Ff+ G , (19)

where the unit ofS(f) is mm2/(1/m); f= 1/ is the spatial frequency in cycle/m ( is thewavelength, > 1m);A to G are the specific parameters which are different for vertical and

lateral rail irregularities. Table 1 gives the values of these parameters for the CR speed-raised

railway lines such as the BeijingShanghai line and the BeijingGuangzhou line. Figure 12compares the CR track spectrum with the AAR track spectra. It can be seen from Figure 12a

that the rail vertical profile irregularities of the CR track spectrum are usually located between

those of the AAR 5th class spectrum and 6th class spectrum. Figure 12b shows that the rail

alignment irregularities of the CR track spectrum are larger than those of the AAR 5th class


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Table 1. Values of parameters of the CR mainline track spectrum.

Parameter A B C D E F G

Left rail vertical 1.1029

1.4709 0.5941 0.8480 3.8016

0.2500 0.0112

Right rail vertical 0.8581 1.4607 0.5848 0.0407 2.8428 0.1989 0.0094Left rail alignment 0.2244 1.5746 0.6683 2.1466 1.7665 0.1506 0.0052Right rail alignment 0.3743 1.5894 0.7265 0.4353 0.9101 0.0270 0.0031Cross-level 0.1214 2.1603 2.0214 4.5089 2.2227 0.0396 0.0073

Figure 12. Comparison of the Chinese track spectrum withAAR track spectra: (a) rail vertical profile irregularities;(b) rail alignment irregularities.

spectrum if 1 m < < 7 m, smaller than those of the AAR 5th class spectrum if > 15 m, and

approximately the same at the wavelength range from 7 to 15 m.

The wavelength in the above track spectra is > 1 m. In order to excite high-frequency

wheelrail interactions, however, the irregularities with wavelength shorter than 1 m areneeded. A measurement result of this type of rail vertical irregularity on the Chinese

ShijiazhuangTaiyuan line is given here [26]

S(f)=0.036f3.15, (20)

where the range of the wavelength is 0.01 m 1 m.

In order to input the track spectrum intothe vehicletrack system, the PSDfunction shouldbe

transformed into vertical and lateral changes of the rail geometry varying with the tracks lon-

gitudinal distance (or in time domain) by means of a timefrequency transformation technique[27]. Figure 13 displays samples of the vertical and lateral irregularities of the left and right

rails representing the CR mainline condition, which are transformed from Equation (19) with

parameters listed in Table 1, superposed with vertical short wavelength irregularities on rails

in the form of Equation (20).


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Figure 13. Samples of the vertical and lateral rail irregularities obtained from the CR track spectra: (a) verticalirregularity of the left rail; (b) lateral irregularity of the left rail; (c) vertical irregularity of the right rail; (d) lateralirregularity of the right rail.


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Figure 14. Stability of the integration method for vehicletrack coupled dynamics: variation of the calculatedwheelrail forcesP1 andP2 with the time step t.

Numerical trial results show that the critical time step of the integration method is

tcr = 1.5 104 s when used for the solution of the present vehicletrack dynamic sys-tem, as shown in Figure 14. In order to get a high calculation accuracy, however, the actually

adopted effective time step is suggested to be te = 1.0 104 s.

5. Computer simulation

Computer simulation is a key technique to obtain the detailed responses of such a large

dynamic system. Two computer simulation programs were developed using the above fast

integration method to analyse the vehicletrack coupled dynamics. One is named VICT, which

was developed in the early 1990s and is used for analysing the vertical dynamic interactions

between vehicles and tracks. The basic features and applications of the VICT program could

be found in previously published articles; see, for example, [6].

Another program, named TTISIM (Train-Track Interaction SIMulation), was developed

by the authors in recent years at Southwest Jiaotong University. It is based on the three-

dimensional vehicletrack coupled model involving the new wheelrail coupling model. The

flow chart of the complete simulation process is illustrated in Figure 15.

Unlike the VICT, which is mainly used to analyse vertical dynamic effects of vehicles

on tracks, TTISIM can be used to investigate the dynamic behaviour of vehicles running on

flexible track structures, including the lateral stability, ride comfort, curving performance, etc.The major difference between TTISIM and other popularly used vehicle system dynamics

software might be that the track structure components are modelled in detail in TTISIM.

Actual profiles of wheels and rails measured, for example, by the MiniProf device, from

existing railways can be directly input to TTISIM as initial data. TTISIM can produce complete

responses of the vehicletrack system including the vertical and lateral wheelrail forces, rail

sleeper forces, vertical and lateral accelerations of car body, bogies and wheelsets, vertical

and lateral displacements and accelerations of rails, sleepers and ballasts, etc.

6. Field experimental validation

The three-dimensional vehicletrack coupled models and the simulation program TTISIM

have been validated by a lot of field measurements conducted with a variety of vehicles and

track conditions representing those of Chinese main trains and mainline tracks. Full-scale field


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Figure 15. Flow chart of the TTISIM simulation system.

tests include the speed-up test on the BeijingQinhuangdao line, the high-speed running test

on the QinhuangdaoShenyang line and the wheelrail dynamic interaction experiment on

curved tracks of the ChengduChongqing line, etc. In this paper, the main results from the

first two tests will be reported and compared with the calculated results obtained from the

three-dimensional vehicletrack coupled model.

The speed-up test on the BeijingQinhuangdao line was carried out by China Academy

of Railway Sciences and Beijing Railway Bureau in December 2000. The tested vehicle is adouble-deck passenger coach with two double-axle bogies. The track structure on the tested

section is a typical ballasted track used in CR with 60 kg/m rails and general concrete sleepers.

The main parameters of the tested vehicle and the track used in the simulation are listed in

Tables 2 and 3, respectively.


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Table 2. Main parameters of railway vehicles used in the simulation.

Double-deck Chinese Star

Notation Parameter passenger coach power car

Mc Car body mass (kg) 29,600 59,364.2Mt Bogie mass (kg) 1700 5630.8

Mw Wheelset mass (kg) 1900 1843.5

Icx Mass moment of inertia of car body aboutXaxis (kg m2) 5.802 104 1.305 105

Icy Mass moment of inertia of car body about Yaxis (kg m2) 2.139 106 1.723 106

Icz Mass moment of inertia of car body aboutZaxis (kg m2) 2.139 106 1.796 106

Itx Mass moment of inertia of bogie aboutXaxis (kg m2) 1600 2202

Ity Mass moment of inertia of bogie about Yaxis (kg m2) 1700 9487

Itz Mass moment of inertia of bogie aboutZaxis (kg m2) 1700 11,233

Iwx Mass moment of inertia of wheelset aboutXaxis (kg m2) 1067 1263

Iwy Mass moment of inertia of wheelset about Yaxis (kg m2) 140 219

Iwz Mass moment of inertia of wheelset aboutZaxis (kg m2) 1067 1285

Kpx Stiffness coefficient of primary suspension alongXaxis (MN/m) 24 30.8

Kpy Stiffness coefficient of primary suspension along Yaxis (MN/m) 5.1 4.878Kpz Stiffness coefficient of primary suspension alongZaxis (MN/m) 0.873 2.399 6

Ksx Stiffness coefficient of secondary suspension alongXaxis (MN/m) 1.2 0.315 6

Ksy Stiffness coefficient of secondary suspension along Y axis (MN/m) 0.3 0.315 6

Ksz Stiffness coefficient of secondary suspension alongZaxis (MN/m) 0.41 0.885 8

Cpz Damping coefficient of primary suspension alongZaxis (kN s/m) 30 30

Csy Damping coefficient of secondary suspension along Y axis (kN s/m) 25 50

Csz Damping coefficient of secondary suspension alongZaxis (kN s/m) 108.7 45

lc Semi-longitudinal distance between bogies (m) 9.0 5.73

lt Semi-longitudinal distance between wheelsets in bogie (m) 1.2 1.5

R0 Wheel radius (m) 0.4575 0.525

Table 3. Main parameters of the ballasted track used in the simulation.

Notation Parameter Value (per rail seat)

E Elastic modulus of rail (N/m2) 2.059 1011 Density of rail (kg/m3) 7.86 103

I0 Torsional inertia of rail (m4) 3.741 105

Iy Rail second moment of area about Yaxis (m4) 3.217 105

Iz Rail second moment of area aboutZaxis (m4) 5.24 106

GK Rail torsional stiffness (N m/rad) 1.9587 105mr Rail mass per unit length (kg/m) 60.64

Ms Sleeper mass (half) (kg) 125.5Kpv Fastener stiffness in vertical direction (N/m) 6.5 107Kph Fastener stiffness in lateral direction (N/m) 2.0 107Cpv Fastener damping in vertical direction (N s /m) 7.5 104Cph Fastener damping in lateral direction (N s /m) 5.0 104ls Sleeper spacing (m) 0.545

le Effective support length of half sleeper (m) 0.95

lb Sleeper width (m) 0.273

b Ballast density (kg/m3) 1.8 103

Eb Elastic modulus of ballast (Pa) 1.1 108Cb Ballast damping (N s/m) 5.88 104Kw Ballast shear stiffness (N/m) 7.84 107Cw Ballast shear damping (N s/m) 8.0

104

Ballast stress distribution angle () 35hb Ballast thickness (m) 0.45

Ef SubgradeK30 modulus (Pa/m) 9.0 107Cf Subgrade damping (N s/m) 3.115 104


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In the high-speed running test on the QinhuangdaoShenyang line, dynamic responses of

a vehicle running on a slab track was measured in 2002. The tested vehicle is a high-speed

power car, named Chinese Star, whose parameters are also given in Table 2. The tested track

is a non-ballasted slab track on a curve with a radius of 4000 m and with a superelevation of

115 mm. The length, width and thickness of the slab are 4.93, 2.4 and 0.19 m, respectively.

The slab mass per unit length is 1115.62 kg/m. The elastic modulus of the CA layer under the

slab is 300 MPa.

6.1. Comparison of dynamic wheelrail forces

In the speed-up test on the BeijingQinhuangdao line, the instrumented wheelsets were

installed in the tested vehicle to measure the wheelrail forces, which provided a possibility

for the comparison of time histories of the vertical and lateral wheelrail forces during

vehicle movement.Figures 16a and 17a give the measured lateral and vertical wheelrail forces, respectively,

when the tested vehicle is running with a velocity of 160 km/h through a curve section with

a radius of 1200 m and with a superelevation of 100 mm. The lengths of the circular curve

and the transition curve are 142.25 and 100 m, respectively. Figures 16b and 17b show the

Figure 16. Comparison of measured and calculated lateral wheelrail forces on a curved track.


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Figure 17. Comparison of measured and calculated vertical wheelrail forces on a curved track.

calculated time responses of the lateral and vertical wheelrail forces (data given with the same

time interval as the measured one). It can be seen from these figures that the time responses of

both the lateral and vertical wheelrail forces predicted by the current vehicletrack coupled

model are in reasonable agreement with the field measured responses.

6.2. Comparison of dynamic responses of vehicle system

For the comparison of vehicle dynamic responses, ride comfort indices are chosen. In con-

sideration of the CR standards, Sperling index is adopted here to evaluate the ride comfort of

vehicles, which could be calculated with Sperlings method based on the car body accelerations

and weighted by frequencies [29].

Figures 18 and 19 compare the measured and the calculated lateral and vertical Sper-

ling indices of the car body of the Chinese Star power car running on the tested slabtrack section in the QinhuangdaoShenyang line. The measured track irregularities on this

high-speed test section are used here as the system excitation. There are good correlations

between the calculated results and the measurement data in the tested speed range, as shown

in Figures 18 and 19.


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Figure 18. Comparison of measured and calculated lateral Sperling indices of the car body.

Figure 19. Comparison of measured and calculated vertical Sperling indices of the car body.

Figure 20. Accelerometers on the rail and on the slab.

6.3. Comparison of dynamic responses of track system

In the high-speed running test on the QinhuangdaoShenyang line, the vibrations of the slab

track structure were measured by the authors. Figure 20 shows a picture of the position of


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Figure 21. Comparison of measured and calculated vertical rail accelerations of the slab track.

Figure 22. Comparison of measured and calculated vertical slab accelerations of the slab track.

the accelerometers for measuring the vertical accelerations of the rail and the slab in the

experiment.

The measured results of peak values of the vertical accelerations of the rail and the slab are

given in Figures 21 and 22, respectively, in which the effective frequency band is 01500 Hz

for the measurement of track accelerations. The calculated results obtained with the currentmodel are also given in these figures, which are close to the measured data. It is shown from

Figure 22 that the vertical acceleration of the slab increases tardily with the addition of the

train speed in the tested speed range.

7. Case study

In order to ascertain the difference between the computational results obtained with the

vehicletrack coupled dynamics model and those obtained with a classical vehicle dynamicsmodel, vehicle curving performances on an elastic track and a rigid track are analysed, respec-

tively. Both results are compared for two cases: general vehicle curving on a very small-radius

curve with low speed (case I) and high-speed vehicle curving on a very large-radius curve

with high speed (case II).


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7.1. Case I: curving on very small-radius curve with low speed

In this case, a very small-radius curve is chosen so that severe lateral dynamic interaction

may occur. The radius of the curve is 287 m. The superelevation of the outer rail is 125 mm.

The track consists of rails with a mass of 60 kg/m, concrete sleepers and ordinary macadamballast, which belongs to the Chinese typical ballasted track structure. The vehicle chosen is

the Chinese general passenger car, which is the same as that used in the speed-up test on the

BeijingQinhuangdao line (Section 6). The parameters of the vehicle and the track used here

are the same as those given in Tables 2 and 3, respectively. The Chinese mainline track spectra,

superposed with the rail short wavelength irregularities in form of Equation (20), are used in

the calculation. The curving speed of the vehicle is 65 km/h.

Figure 23a and b shows the calculated lateral wheelrail forces of vehicle curving on an

elastic track model (using the vehicletrack coupled dynamics) and on a rigid track model

(using the classical vehicle dynamics), respectively. It can be seen that the lateral wheelrail

forces calculated with the vehicletrack coupled dynamics are generally smaller than thoseobtained with the classical vehicle dynamics. The deviation of both the maximum values is

16.3%. A comparison of the calculated vertical wheelrail forces for vehicle curving on the

elastic track model and on the rigid track model is shown in Figure 24. It can also be seen that

Figure 23. Calculated lateral wheelrail forces for vehicle curving on (a) the elastic track model and (b) the rigidtrack model.


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Figure 24. Calculated vertical wheelrail forces for vehicle curving on (a) the elastic track model and (b) the rigidtrack model.

modelling the track as an elastic structure leads to smaller dynamic wheelrail vertical forces

than assuming the track to be a rigid base. The maximum deviation of both results is 9.7%.

In Figures 25 and 26, the PSDs of the lateral and vertical wheelrail forces obtained with the

two models are compared. Figure 25 shows a significant difference for the lateral wheelrail

forces in the frequency range above 20 Hz, whereas little difference can be found at lower

frequencies. For the vertical wheelrail forces, a significant difference is observed for the

frequencies > 40 Hz, as shown in Figure 26. It could be concluded from this calculation that

the high-frequency vibrations of the track lead to differences in the spectral amplitudes. The

viscoelastic track structure may absorb some energy relating to the high-frequency vibrations.

Of course, the wheelset deformability also has some influence on the high-frequency wheel

rail forces, especially on the vertical force, which needs further research.

The main reason for the changes in the wheelrail forces when using the elastic track model

is the effect of the rail vibrations actually existing in the curving process. Figures 2729

depict the main results reflecting the rail lateral dynamic behaviour when the vehicle passes

through the curve. It can be seen from Figure 27 that the outer rail is shoved outwards by the

lateral wheelrail forces in the process of curving and the maximum lateral displacement ofthe rail reaches 1.9 mm. The outer rail rolls over with a maximum rolling angle of 0.38, as

shown in Figure 28. The combination of lateral displacements and rolling of the outer and the

inner rails results in a dynamic enlargement of the gauge (Figure 29). The maximum dynamic

enlargement of this gauge in the case is 2.1 mm.


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Figure 25. Comparison of power spectrum densities of lateral wheelrail forces obtained from vehicletrackcoupled dynamics (elastic track model) and from classical vehicle dynamics (rigid track model).

Figure 26. Comparison of power spectrum densities of vertical wheelrail forces obtained from vehicletrackcoupled dynamics (elastic track model) and from classical vehicle dynamics (rigid track model).

Figure 27. Time history of rail lateral displacement during curving.

Compared with the case of fixed rigid rails assumed in the classical theory of vehicledynamics, where the rail lateral displacement and the rail rolling angle are all equal to zero,

the elastic track modelling shows remarkable rail lateral motions during the vehicle curving.

This rail lateral motion causes obvious change in positions of the wheelrail contact points

and definitely results in the change of the lateral and vertical wheelrail dynamic forces.


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Figure 28. Time history of rail rolling angle during curving.

Figure 29. Dynamic gauge enlargement during curving.

Table 4. Comparison of high-speed curving performance indices calculated with the classical vehicledynamics and the vehicletrack coupled dynamics.

Classical Vehicletrack Relative

Dynamic index vehicle dynamics coupled dynamics deviation (%)

Wheelrail lateral force (kN) 28.03 24.38 13.02

Wheelrail vertical force (kN) 180.79 140.89 22.06

Wheelset lateral force (kN) 18.18 15.77 13.26

Derailment coefficient 0.36 0.29 19.44Reduction ratio of wheel-load 0.46 0.31 32.61

Dynamic gauge enlargement (mm) 0 0.51

7.2. Case II: curving on very large-radius curve with high speed

The Chinese Star high-speed power car is chosen as an example in this calculation, whose

parameters are given in Table 2. The curve has a radius of 6000 m and a maximum superele-

vation of 70 mm. The length of the transition curve is 180 m. The curving speed of the vehicle

is 250 km/h.

Table 4 summarises the main results representing the curving performances calculated withthe classical vehicle dynamics model assuming a rigid track and with the vehicletrack coupled

dynamics model using an elastic track model. The relative deviations between both results are

also given for each dynamic index in Table 4. It is clear from Table 4 that the elastic track

model produces much smaller values of dynamic indices than the rigid track model does. The


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reduction ratios are different for different indices. For example, the reduction ratios of the

lateral wheelrail dynamic force, the vertical wheelrail dynamic force and the derailment

coefficient are 13.02%, 22.06% and 19.44%, respectively.

8. Conclusions

A framework has been systematically presented in this paper for investigating the overall

dynamics of the vehicletrack system. A three-dimensional vehicletrack coupled model has

been established in which the vehicle subsystem and the track subsystem are coupled through

a spatial wheelrail coupling model that considers the rail vibrations in vertical, lateral and

torsional directions. Random track irregularities varying with the rails longitudinal direction

have been simulated based on track spectra to excite vibrations of the vehicletrack system.

Rail irregularities with short wavelengths are also considered in order to be able to excite

the high-frequency wheelrail interactions. The dynamic responses are solved numerically

with a fast integration method in the time domain. The vehicletrack coupled model has been

validated by full-scale field experiments, showing good correlation between theoretical and

experimental results.

Significant differences between the curving performances obtained from the vehicletrack

coupled dynamics and from the classical vehicle dynamics have been found for the cases

of a general vehicle curving on a very small-radius curve and a high-speed vehicle curving

on a large-radius curve. The deviation is usually located within the range of 1020%. The

vehicletrack coupled dynamics gives smaller values of the curving performance indices than

the classical vehicle dynamics modelling does.

It is found that vehicle curving can cause obvious lateral vibrations of rails and can make therails roll over under the action of lateral wheelrail forces, resulting in dynamic enlargement of

the gauge. The rails lateral motion can lead to changes in the position of the wheelrail contact

points, which influences the wheelrail forces and thereby eventually the curving performance

of the vehicle. Therefore, it is necessary to consider the vibration of the track structure when

evaluating the curving performance.

The vehicletrack coupled dynamics, including the validated models and the simulation

software reported in this paper, provides a method to investigate the dynamic behaviour of

the entire vehicletrack system, including the wheelrail interfaces, and also provides an

efficient way to systematically optimise the design parameters of both the vehicle and the

track components.A lot of future investigation is possible, for example on dynamic interactionsbetween vehicles and non-ballasted track structures, on the effect of track lateral properties

on vehicle lateral running behaviours and on the influence of curve characteristics on ride

comfort of high-speed vehicles, etc.

Acknowledgements

This research was supported by National Natural Science Foundation of China (NSFC) under grants 50838006,50823004 and 50521503 and by National Basic Research Program of China (973 Program) under grant2007CB714706. The authors would like to thank the reviewers of this paper for their valuable suggestions.

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