a dynamic model for the vertical interaction of the rail

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A dynamic model for the vertical interaction of the rail

track and wagon system

Y.Q. Sun a, M. Dhanasekar b,*

a Centre for Railway Engineering, Central Queensland University, Rockhampton 4702, Australiab Faculty of Engineering and Physical Systems, Central Queensland University, Rockhampton 4702, Australia

Received 10 December 2000

Abstract

With the advent of high-speed trains, there is a renewed interest in the rail track–vehicle interaction studies. As part

of an ongoing investigation of the track system optimisation and fatigue of the track components, a dynamic model is

developed to examine the vertical interaction of the rail track and the wagon system. Wagon with four wheelsets

representing two bogies is modelled as a 10 degree of freedom subsystem, the track is modelled as a four-layer sub-

system and the two subsystems are coupled together via the non-linear Hertz contact mechanism. The current model is

validated using several field test data and other numerical models reported in the literature by other researchers. Ó 2002

Elsevier Science Ltd. All rights reserved.

Keywords: Rail track; Wagon; Timoshenko beam; Hertz contact; Wheel/rail irregularities; Steady-state responses

1. Introduction

As different divisions within railway departments traditionally have been managing the wagon and track

separately, the dynamics of the rail track and wagon are often studied as two relatively independent

problems. For example, the wagon dynamics is examined by assuming the track as either a rigid support or

as an elastic foundation using commercial software such as NUCARS (Blader et al., 1989). The longitudinal,

the lateral and the vertical dynamics of a wagon or a train in motion are examined using the software.

The track dynamics, on the other hand, is investigated either by simplified beams on elastic foundation

approach (Jenkins et al., 1974; Grassie and Cox, 1982; Duffy, 1990; Sato, 1977; Newton and Clark, 1979;Clark et al., 1982; Tunna, 1988; Ilias and M}uuller, 1993; Cai and Raymond, 1994; Ishida et al., 1997) or by

the finite element model of rail track system (Lin and Trerhewey, 1988; Thompson, 1991; Dong et al., 1994;

Luo et al., 1996). Usually these track models have been assumed to be excited either by a single wheel or by

a single bogie with two wheelsets rolling on the rail.

Knothe and Grassie (1993) presented a state-of-the-art review on the modelling of railway track and

vehicle–track interaction. It is generally found that the Euler or Timoshenko beams representing the rail

International Journal of Solids and Structures 39 (2002) 1337–1359

www.elsevier.com/locate/ijsolstr

*Corresponding author. Tel.: +61-7-4930-9677; fax: +61-7-4930-9382.

E-mail address: [email protected] (M. Dhanasekar).

0020-7683/02/$ - see front matterÓ

2002 Elsevier Science Ltd. All rights reserved.P II: S0 0 2 0 -7 6 8 3 (0 1 )0 0 2 2 4 -4



resting on an elastic foundation provides only a limited insight into the dynamic response of various track

components. An improvement to such models is achieved by accounting the discrete spacing of the sleepers.

The discrete support models and the finite element model allow improved prediction of the rail response

and offer the potential for refinement by including all conceivable track components as ‘‘layers’’.

In the track–vehicle interaction research, several field experiments have also been reported in the liter-

ature. Two of them were conducted by the British Railways (Jenkins et al., 1974; Newton and Clark, 1979).

Other experimental investigations include those carried out by the Swedish railways (Fermeer and Nielsen,

1995) and the South African railways (Fr€oohling et al., 1997).

Jenkins et al. (1974) varied the train speed up to 160 km/h. Defect was induced via dip at the fishplate

joints of the rail. Two peak forces due to impact of the wheel and the rail were detected. The first peak with

very high frequency was classed as P1 force and the second peak with relatively low frequency was classed

as P2 force. Newton and Clark (1979) conducted further field experiment and measured the impact contact

forces when the trains passed over a purpose made indentation on the top surface of the rail. A range of

train speed from 27 to 117 km/h was considered in the investigation. The results of these experiments are

used as a source of validating several numerical models reported in the literature. The model presented in

this paper also has been validated using these results in addition to other results.Fermeer and Nielsen (1995) reported a full-scale experiment carried out on the West Coast line in Sweden

using a wagon equipped with instrumented wheelsets at speeds up to 275 km/h. Five consecutive sleepers and

one rail instrumented with accelerometers and strain gauges were used. The influence of wagon speed and

axle load on dynamic responses was studied. It was concluded that the pad stiffness and the axle load largely

affected the contact forces due to wheel flats. This test data are also used to validate the model in this paper.

Fr€oohling et al. (1997) reported a more detailed controlled field test conducted in South Africa. The

purpose of the test was to understand the possible detrimental effects caused by the low frequency contact

forces. It was concluded that the track dynamic responses were affected by the vehicle load, the vehicle

speed, the track geometry, the track stiffness, and the accumulating traffic.

With the increase in the axle load and wagon speed, the cost of damage to track components and de-

railment risks increase substantially. This leads to widespread interests in the investigation of the dynamicinteractions of the rail track and the wagon. More refined analytical models of the rail track and wagon

system have, therefore, started emerging in the literature with the potential to optimise the design pa-

rameters of both the rail track and wagon components that would reduce the dynamic interactions. The

model reported in this paper is classed to this category of investigation. As part of an ongoing research at

the centre for railway engineering (CRE) to optimise the track and wagon components, a model containing

all components of the track and the wagon subsystems has been developed. Although an extensive field-

testing is desirable to fully validate such a detailed model, limited validation would be possible by com-

paring the wheel–rail interface forces predicted by the model with the results reported in the literature. Since

the second approach is much cheaper, as a preliminary phase of the ongoing research, we have validated the

detailed model via the results reported in the literature by other researchers. Should the objective be only to

evaluate the wheel–rail interface forces, the detailed model presented in this paper would not have been

necessary. The detailed model, on the other hand, provides an insight into the dissipation of the interface

forces into the components of the track and the wagon. Furthermore the detailed track–wagon model

allows the investigation of the effect of unequal axle loads (anticipated in freight wagons).

This paper describes a track submodel developed using the discrete beam concept. All track components

are assembled exactly as per the conventional ballasted track structure used in the heavy haul railway

network. This submodel comprises of the rails, the fasteners, the pads, the sleepers, the ballast, the sub-

ballast and the subgrade arranged in four layers. The wagon submodel comprises of a rigid car body, two

two-piece bogies and four wheelsets that are connected by secondary and primary suspensions. The wheel–

rail contact patch submodel is developed according to the non-linear Hertz theory of contact (Johnson,

1985). Although more sophisticated models are available for the definition of contact (for example, in-

1338 Y.Q. Sun, M. Dhanasekar / International Journal of Solids and Structures 39 (2002) 1337–1359



cluding friction), from the perspective of the overall dynamics of the track–wagon system, it was decided to

use the simple Hertz theory.

The model reported in this paper could, therefore, be described as a full wagon-four-layer track model

capable of predicting the vertical dynamic interaction when the wagon runs on the track at a steady-state

velocity in the longitudinal direction. The model could simulate the interaction of the wheel and the rail

with and without defects or irregularities (as excitation sources). Two types of excitation sources, namely,

periodic and impulse, are considered in the simulation. The model is capable of predicting the distribution

of the dynamic responses at the wheel–rail interface in the downward direction to the track subgrade and in

the upward direction to the car body. The characteristics of the steady-state responses, and impact re-

sponses are presented in this paper.

2. The vertical dynamic model for the rail track and wagon interaction

Prior to presenting our model, we discuss some models that are recently reported in the literature. Cai

and Raymond (1992) reported a track dynamic interaction model consisting of one bogie. The wheelsetmodel included two unsprung masses, side frame mass and pitch inertia, and primary suspensions. The

track was modelled as a 40-sleeper long discretely supported system of elastic beams representing the rails

and the sleepers. This model was used to examine the dynamic response due to various wheel and rail

defects. It was found that the wheel and rail impact behaviour depends highly on the train speed. It was also

found that a wheel with an irregular profile causes not only a high impact force on itself, but also greatly

increase the impact force on the adjacent wheel.

Dahlberg (1995) reported a theoretical model similar to that of Cai and Raymond (1992) with one bogie

and track with a view to modelling the field experiments. This model was used to investigate the sensitivity

of the parameters such as the wagon speed, the axle load, the wheel base of a bogie, the defects in rail and

wheel on the dynamic behaviour of the track and wagon components.

Zhai and Sun (1993) presented a detailed model that represented the wagon as two bogies multi-bodysystem and the track as an infinite Euler beam supported on a discrete–continuous elastic foundation

consisting of three layers of rail, sleeper, and ballast. The significance of mutual dynamic influence of the

neighbouring wheelsets via the rail and the bogie was determined in the paper.

Ripke and Knothe (1995) developed a model similar to that of Zhai and Sun (1993) but used the

Timoshenko beam formulation to model the rail and sleepers instead of the Euler formulation adopted by

Zhai and Sun (1993). This model was used to investigate the effects of the local defects of the track on the

contact forces.

Our model is pictorially represented in Fig. 1. The track submodel consists of the rail, the pads, the

fasteners, the sleepers, the ballast, the subballast and the subgrade. The first layer of model consists of

the rail represented as a continuous Timoshenko beam that is discretely supported on the fasteners and the

pads represented by the linear spring and damping elements. The enclosed dash-line box signifies that the

mass of the pad and the fasteners are disregarded. The sleeper is represented in the second layer with its

mass and viscoelastic properties (spring and damper enclosed within solid-line box that signifies the mass

being included). The third and the fourth layers of the model consist of the ballast and the subballast

respectively. The ballast and the subballast are considered as pyramids for calculating the effective mass,

stiffness and damping coefficients. The viscoelastic springs and dampers connecting one pyramid to the

other represent the continuity of the ballast and subballast in the longitudinal direction. The subgrade is

modelled as the viscoelastic elements without mass connecting the subballast to the ground.

The wagon submodel consists of the wagon body, the two two-piece bogies and the four wheelsets as

rigid bodies. The spring and the damping elements representing the secondary suspension connect the

wagon body with the two bogies. Similarly, the spring and the damping elements representing the primary

Y.Q. Sun, M. Dhanasekar / International Journal of Solids and Structures 39 (2002) 1337–1359 1339



suspension connect the bogies with the wheelsets. While the car body and bogies are allowed vertical

displacement and in-plane rotation (bounce and pitch motions respectively) the wheelsets are allowed

vertical displacement (bounce motion) only as shown in Fig. 1. The wagon is thus represented by a 10

degree of freedom (DOF) system.

2.1. Rail

Rail is modelled as an infinitely continuous (with its vertical deformation and rotation vanishing at both

ends long enough to be considered as infinity) Timoshenko beam shown in Fig. 2.

The Timoshenko beam theory (Dym, 1973) expresses the equations for the vertical deflection and ro-

tation of the rail at any point under the action of forces as shown in Eq. (1):

Fig. 1. The dynamic model of rail track and wagon system.




q A o2wR

ot 2À GAk ðo2wR

o x2 À o/R

o xÞ ¼ ÀP N s

i¼1

F RSidð x À xiÞ þP4

j¼1

P WR jdð x À x jÞ

q I o2/R

ot 2

ÀGAk

ðowR

o x

À/R

Þ ÀEI o

2/R

o x2

¼0

8><

>:ð1Þ

where wR is the vertical deflection of the rail, /R is the rotation of the rail, q is the rail density, A is the area

of the rail cross-section, G is the shear modulus of the rail, E is the Young’s modulus of the rail material, I is

the second moment of area of the rail section, k is the Timoshenko shear coefficient, FRSi is the reaction

force between the rail and the ith sleeper, P WR j is the contact force between the jth wheel and the rail, dð xÞ is

the Dirac delta function, xi is the position of the ith sleeper, x j is the position of the jth wheel, and N s is the

number of sleepers considered. The subscript i is used for the sleeper count and j for the wheel count.

The vertical deflection wR and rotation /R of the rail are obtained using modal superposition as given in

Eq. (2):

wR ¼P N c

h¼1

N wðh; xÞW hðt Þ

/R ¼ P N c

h¼1

N /ðh; xÞUhðt Þ

8>><>>: ð

2Þ

where N wðh; xÞ and N /ðh; xÞ are the hth mode shape functions of the vertical deflection and rotation re-

spectively of the rail, W hðt Þ and Uhðt Þ are the hth mode time coefficients of the rail vertical deflection and

rotation respectively of the rail, N c is the number of modes considered and x represents the linear coordinate

along the length of the rail beam.

By substituting Eq. (2) into Eq. (1), we modify the partial differential equation (1) into ordinary dif-

ferential equation shown in Eq. (3) below. This transformation facilitated the application of the Newmark-

b method to solve the equations.

d2W hdt 2

þGk q

Ph LÀ Á2

W h

À ffiffi A I q Gk

qPh LÀ ÁUh

¼ ÀP N s

i¼1

FRSi N w

ðh; xi

Þ þ P4

j¼1

P WR j N w

ðh; x j

Þ ðh

¼1; 2; . . . ; N c

Þd2Uh

dt 2þ GAk

q I þ E

qPh L

À Á2

Uh À ffiffi

A I

q Gk q

Ph L

À ÁW h ¼ 0

8><>: ð3Þ

in which L is the length of the rail considered, and the reaction force between the rail and the ith sleeper F RSi

is expressed as in Eq. (4):

F RSi ¼ ðC pi þ C f iÞX N c

m¼1

N wðm; xiÞ _W W m þ ð K pi þ K f iÞX N c

m¼1

N wðm; xiÞW m À ðC pi þ C f iÞ _wwsi À ð K pi þ K f iÞwsi ð4Þ

where wsi is the vertical displacement of the ith sleeper. C pi, K pi and C f i, K f i are the damping and stiffness

coefficients of the ith pad and the ith fastener respectively.

Fig. 2. Timoshenko beam model of the rail.




In Eq. (3), the contact force P WR j between the jth wheel and the rail is determined by the non-linear Hertz

contact theory and is given in Eq. (5),

P WR jðt Þ ¼C H ww j

ðt

Þ Àwr

ð xp j; t

Þ Àwd

ðt

ÞÈ É3=2if ww j

ðt

Þ Àwr

ð xp j; t

Þ Àwd

ðt

Þ> 0

0 if ww jðt Þ À wrð xp j; t Þ À wdðt Þ < 0& ð5Þ

where ww jðt Þ and wrð xp j; t Þ are the displacements of the wheel and the rail at the jth contact point, wdðt Þ is

the wheel and/or the rail irregularity function (for example, an out-of-round wheel, rail corrugation or rail

surface geometric irregularity), C H is the Hertz contact coefficient that can be deduced from Johnson (1985)

as follows:

C H ¼ 4G wr

ffiffiffiffiffi Re

p 3ð1 À mwrÞ ð6Þ

in which G wr is the shear modulus, mwr is the Poisson’s ratio, and Re ¼ ffiffiffiffiffirR

p (r is the rolling radius of wheel,

R ¼ qw Rt=ðqw À r Þ (qw and Rt are the wheel profile radius and the rail profile radius respectively)).

In Eq. (5) the contact force P WR j is calculated based on the relative displacement between the wheel andthe rail at the point of contact xp j. This point is easily determined by keeping the angle between the vertical

diameter of the wheel and the axis of the rail as 90° for the non-defect wheels and rails. However, where

defect (in the wheel or the rail) is encountered, the angle between the diameter of the wheel drawn through

the point of contact and the axis of the rail varies from 90°. The exact point of contact is determined in such

cases by dividing the contact length obtained from static Hertz analysis into smaller segments and checking

each segment for potential contact. A similar approach has been reported by Dong et al. (1994).

2.2. Pads, fasteners and sleepers

Rubber or high-density polyethylene mats that are used as a bearing layer between the rails and the

concrete sleepers are commonly known as pads. Rail fastener connects the rail and the sleeper together. Theelasticity of the fastener is measured by the spring rate, which is the amount of deflection proportional to

the clamping force. In the model, both the pads and the fasteners are modelled as the linear springs and

dampers without mass.

Sleeper is the track component that ties the two rails together thereby providing monolithic action to the

track. Sleepers are positioned between the rails and the ballast and are represented in the model by their

mass, stiffness and damping properties. The stiffness of sleepers is calculated using the influence coefficient

approach by considering the sleepers as beams on elastic foundation proposed by Profillidis (2000). The

track structure has been considered as medium quality for the evaluation of the sleeper stiffness. The

damping coefficient is then determined based on the values of stiffness and mass.

2.3. Ballast and subballast

The ballast ensures damping of the vibrations and distributes the load evenly to the subgrade. The

subballast protects the top surface of the subgrade from penetration of the ballast stone particles, in ad-

dition to, further distributing the load. Ahlbeck et al. (1975) developed the ballast pyramid model based on

the theory of elasticity. The ballast–subballast pyramid model shown in Fig. 3 assumes that the loading and

pressure distribution is uniform throughout the depth. In Fig. 3 the model is divided into the upper and

lower sections, which reflects the actual transmission of the loading. Zhai and Sun (1993) defined the vi-

bration of the ballast as a single block based on the observation that the accelerations of the individual

particles in both upper and lower surfaces of the ballast block do not vary significantly even though such a

conclusion is not universal. The oscillating mass of each ballast block is calculated by multiplying the




volume of ballast block with the ballast density. According to Ahlbeck et al. (1975), the stiffness of the ith

ballast block K bl is:

K bl ¼ 2 tan hbð Ls À BsÞ E b

lnLsð2tanhb H bþ BsÞ

Bsð2tanhb H bþ LsÞ

h i ð7Þ

in which Ls and Bs are the effective length and width of the support area of the rail seat, E b is the modulus of

elasticity of the ballast (in N/m2), hb is the internal friction angle of ballast (20° is chosen for ballast as

Ahlbeck et al. (1975) suggested), and H b is the height of the ballast.

Similarly, the stiffness of the ith subballast K sb is:

K sb ¼ 2tan hsbð Ls À BsÞ E sb

ln

ð2tanhb H bþ LsÞð2tan hsb H sbþ2tan hb H bþ BsÞð2tanhb H bþ BsÞð2tanhsb H sbþ2tanhb H bþ LsÞh i

ð8Þ

in which E sb is the modulus of elasticity of the subballast (in N/m2), hsb is the internal friction angle of the

subballast (35° is chosen for subballast), H sb is the height of the subballast.

The damping coefficients of the ballast and the subballast are determined as 40% of their critical

damping coefficients. This damping ratio (40%) is considered realistic for earth structures and is found that

these values are within the range given by Grassie et al. (1982) (for example, the post-tamping and pre-

tamping tracks were 30 and 82 kN s/m respectively).

The oscillating masses of each ballast block M bl and subballast block M sb are:

M bl ¼ qb½ Ls Bs þ H b tan hbð Ls þ BsÞ þ 43 H 2b tan2 hb ð9Þ

M sb ¼ qsb½ð Ls þ 2tan hbÞð Bs þ 2tan hbÞ þ H sb tanhsbð Ls þ Bs þ 4tan hbÞ þ43 H

2

sb tan2

hsb ð10ÞThe subgrade stiffness K sg is:

K sg ¼ E sgð2tan hsb H sb þ 2tan hb H b þ LsÞð2tan hsb H sb þ 2tan hb H b þ BsÞ ð11Þin which E sg is the modulus of the subgrade expressed in N/m3.

In the longitudinal direction the continuity of the ballast and the subballast are ensured by including

viscoelastic elements (without mass) connecting the blocks of ballast and subballast in their respective

layers. The coefficients of these longitudinal springs and dampers were calculated by multiplying the re-

spective vertical stiffness and damping coefficients by a factor of 0.3. This factor is not sensitive to the

dynamic responses on the interface between the wagon and the track.

Fig. 3. The ballast and subballast pyramid model.




3. Equations of motion

In the simulation model presented here, the wagon is assumed to be under a steady state motion in the

longitudinal direction. The dynamic equations of motions in the vertical direction for the track and wagon

subsystems and the interface are presented in this section.

3.1. The track dynamics

The equations of motion for the ith sleeper, ballast and subballast blocks are established from the basic

dynamic equilibrium concept.

For the ith sleeper:

M s€wwsi þ C pi

þ C f i þ C sliC bli

C sli þ C bli

_wwsi þ K pi

þ K f i þ K sli K bli

K sli þ K bli

wsi À ðC pi þ C f iÞ

ÂX N c

m¼1

N wðm; xiÞ _W W m À ð K pi þ K f iÞX N c

m¼1

N wðm; xiÞW m À C sliC bli

C sli þ C bli

_wwbli À K sli K bli

K sli þ K bli

wbli ¼ 0 ð12Þ

for the ith ballast block:

M bl€wwbli þ C sliC bli

C sli þ C bli

þ C sbi þ 2C jbi

_wwbli þ K sli K bli

K sli þ K bli

þ K sbi þ 2 K jbi

wbli À C sliC bli

C sli þ C bli

_wwsi

À K sli K bli

K sli þ K bli

wsi À C sbi _wwsbi À K sbiwsbi À C jbi _wwblðiÀ1Þ À K jbiwblðiÀ1Þ À C jbi _wwblðiþ1Þ À K jbiwblðiþ1Þ ¼ 0 ð13Þ

for the ith subballast block:

M sb€ww

sbi þ ðC

sbi þC

sgi þ2C

jsbiÞ_ww

sbi þ ð K

sbi þK

sgi þ2 K

jsbiÞw

sbi ÀC

sbi_ww

bli ÀK

sbiw

bli ÀC

jsbi_ww

sbðiÀ1ÞÀ K jsbiwsbðiÀ1Þ À C jsbi _wwblðiþ1Þ À K jbiwblðiþ1Þ ¼ 0 ð14Þin Eqs. (12)–(14), M s, M bl, M sb are the masses of the sleeper, the ballast block and the subballast block

respectively; C sli, K sli, C bli, K bli and C sbi, K sbi are the damping and stiffness coefficients of the ith sleeper, the

ith ballast block and the ith subballast block respectively; C jbi, K jbi and C jsbi, K jsbi are the damping and the

stiffness coefficients between the ith ballast block and its adjacent ballast blocks and between the ith

subballast block and its adjacent subballast blocks respectively, C sgi, K sgi are the damping and stiffness

coefficients of the subgrade; wbli, wsbi are the vertical displacements of the ith ballast and subballast blocks;

wblðiÀ1Þ, wblðiþ1Þ, wsbðiÀ1Þ, wsbðiþ1Þ are the vertical displacements of the adjacent ballast and subballast blocks of

the ith ballast and subballast blocks.

3.2. The wagon dynamics

The wagon body is connected with the two bolsters, which rests on the secondary suspension. As ex-

plained before, the wagon body presents bounce and pitch motions in the vertical plane. Similarly, the side

frames that are the connection structures between the primary and the secondary suspensions have bounce

and pitch motions. The wheelsets have only bounce motion. According to the model shown in Fig. 1, the

equations of motion of the wagon components are deduced as follows:

for the bounce of wagon body:

M c€wwc þ 2C sc _wwc þ 2 K scwc À C sc _wwb1 À K scwb1 À C sc _wwb2 À K scwb2 ¼ 0 ð15Þ




for the pitch of wagon body:

J c €//c þ 2C sc L2c_//c þ 2 K sc L

2c/c À C sc Lc _wwb1 À K sc Lcwb1 þ C sc Lc _wwb2 þ K sc Lcwb2 ¼ 0 ð16Þ

for the bounce of the rear side frame:

M b1€wwb1 þ ðC sc þ 2C prÞ _wwb1 þ ð K sc þ 2 K prÞwb1 À C scð _wwc þ Lc_//cÞ À K scðwc þ Lc/Þ À C pr _www1

À K prww1 À C pr _www2 À K prww2 ¼ 0 ð17Þfor the pitch of the rear side frame:

J b1€//b1 þ 2C pr L

2w_//b1 þ 2 K pr L

2w/b1 À C pr Lw _www1 À K pr Lwww1 þ C pr Lw _www2 þ K pr Lwww2 ¼ 0 ð18Þ

for the bounce of the front side frame:

M b2€wwb2 þ ðC sc þ 2C prÞ _wwb2 þ ð K sc þ 2 K prÞwb2 À C scð _wwc À Lc_//cÞ À K scðwc À Lc/Þ À C pr _www3

À K prww3 À C pr_

www4 À K prww4 ¼ 0 ð19Þfor the pitch of the front side frame:

J b2€//b2 þ 2C pr L

2w_//b2 þ 2 K pr L

2w/b2 À C pr Lw _www3 À K pr Lwww3 þ C pr Lw _www4 þ K pr Lwww4 ¼ 0 ð20Þ

for the bounce of rear wheelset:

M w€www1 þ C pr _www1 þ K prww1 À C prð _wwb1 þ Lw_//b1Þ À K prðwb1 þ Lw/b1Þ þ P WR1 ¼ 0 ð21Þ

for the bounce of the third wheelset:

M w€www2 þ C pr _www2 þ K prww2 À C prð _wwb1 À Lw_//b1Þ À K prðwb1 À Lw/b1Þ þ P WR2 ¼ 0 ð22Þ

for the bounce of the second wheelset:

M w€www3 þ C pr _www3 þ K prww3 À C prð _wwb2 þ Lw_//b2Þ À K prðwb2 þ Lw/b2Þ þ P WR3 ¼ 0 ð23Þ

for the bounce of the leading wheelset:

M w€www4 þ C pr _www4 þ K prww4 À C prð _wwb2 À Lw_//b2Þ À K prðwb2 À Lw/b2Þ þ P WR4 ¼ 0 ð24Þ

in the Eqs. (15)–(24), M c, M b1, M b2, M w are the masses of the wagon body, front bogie, rear bogie and the

four wheelsets respectively; K sc, C sc, K pr, C pr are the stiffness and damping coefficients of the secondary and

the primary suspensions; wc, /c are the vertical displacement and rotation of the wagon body; wb1, /b1, wb2,

/b2 are the vertical displacement and rotation of the front bogie and the rear bogie and ww1, ww2, ww3, ww4

are the vertical displacements of the four wheelsets.

3.3. The dynamics of the interface

The equations of the complete system are obtained by assembling the above equations in a matrix form

as shown in Eqs. (25a) and (25b) for the wagon and rail track respectively.

M W½ f€wwWg þ C W½ f _wwWg þ K W½ wWf g ¼ F Wf g ð25aÞ

M T½ f€wwTg þ C T½ f _wwTg þ K T½ wTf g ¼ F Tf g ð25bÞwhere

fwWg ¼ fwc /c wb1 /b1 wb2 /b2 ww1 ww2 ww3 ww4gT




f F Wg ¼ f0 0 0 0 0 0 P WR1 P WR2 P WR3 P WR4gTis the force vector that is consisted of the contact forces

between the wheels and the rail;

fwT

g ¼ fW 1 . . . W N cU1 . . .U N c ws1 . . . ws N s wbl1 . . . wbl N s wsb1 . . . wsb N s

gT

f F Tg ¼ f ~ F F 1 . . . ~ F F h . . . ~ F F N c 0 . . . 0gTis the force vector that includes the reaction forces between the rail and the

sleepers and the contact forces between the wheels and the rail, in which

~ F F h ¼ ÀX N s

i¼1

F RSi N wðh; xiÞ þX4

j¼1

P WR j N wðh; x jÞ ðh ¼ 1; 2; . . . ; N cÞ:

3.4. Wheel/rail irregularities

Irregularities in wheel and/or rail generate sharp peak responses in the track–wagon system. Some ir-

regularities cause periodic excitation whilst others cause non-periodic or localised excitation defined asimpulse excitation in this paper. The periodic irregularities include the rail corrugations, the out-of-round

wheels or the rounded flat wheels, and the non-periodic irregularities include the indentation on the rail-

head due to the spalling or the defect of welded joint and the dipped-joint.

The periodic irregularities are represented by cosine functions. Table 1 shows some cases of the periodic

excitation sources and the corresponding expression wdðt Þ. When the excitation source is non-periodic and

Ld6 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2ra À a2p

, (where Ld is the wavelength of irregularity, r is the rolling radius of wheel, a is the wave

depth of the irregularity), the wheel and the rail will not be in contact with the trough of the irregularity.

For small Ld, when the flat wheel runs on the perfect rail or the perfect wheel runs at the defective rail, the

instantaneous rotating centre of the wheel suddenly moves down or up, inducing a vertical impact velocity.

In this situation, we call these excitation sources as the impulse excitation sources. Various impact velocities

have been deduced to simulate small wheel flats, the short length of rail shelling and spalling, and the

defects from rail welded joints. These impact velocities are shown in Table 2.

3.5. Solution technique

Eqs. (25a) and (25b) is solved using the Newmark-b method; A similar numerical integration method has

also been used by Zhai (1996). For the case without the excitation, the term wdðt Þ in the expression of

Table 1

Harmonic excitation sources

Name and geometry Expression

Out-of-round wheel W dðt Þ ¼ að1 À cosXt Þ=2X ¼ 2pV

Ldð06 t 6 Ld=V Þ

Indentation on rail surface a –– wave length of irregularity

Ld –– wave depth of irregularity

(for out-of-round wheel, Ld is the length of arc)




contact force shown in Eq. (5) is disregarded. Some typical functions for wdðt Þ have been discussed in the

above section. The flow chart for the solution technique is schematically presented in Fig. 4.

4. Validation of the model

4.1. Dynamic response without defects

Example 1: To validate the model illustrated in this paper its predictions of the responses are compared

with the responses reported by Dong et al. (1994) for the case where there is no excitation source (in otherwords perfect wheels running on the defect free surface of the rail). The purpose of investigating the steady-

state responses without defects is to obtain the characteristics of the responses, and more importantly to

determine the natural frequencies of the coupled wagon and track system.

Dong et al. (1994) modelled a problem of one perfect wheel travelling at a constant speed (148 km/h)

over a defect-free rail surface by just one wheel rolling on a track with just two layers. To predict the

responses reported by Dong et al. (1994), it was required to simplify the model reported in this paper.

Accordingly, Eqs. (7)–(11), (13) and (14) were removed from the track subsystem and Eqs. (15)–(20) and

(22)–(24) were removed from the wagon subsystem. Eq. (21) was simplified as the rolling wheel was as-

signed only the static wheel load. This has substantially reduced the overall complexity of the model

presented in this paper. The resulting one wheel-two layer track system contained 341 equations of motion.

This corresponds to 100 sleeper long track. The parameters obtained from Dong et al. (1994) used in the

execution of the simplified current model is shown in Table 3.

The rail and wheel displacements at the contact point and the contact force factor (the contact force

divided by the static wheel load) evaluated by the current model and that of Dong et al. (1994) are shown in

Fig. 5(a) and (b) respectively. Both figures show that the prediction of the responses by the current model is

in good agreement to the responses reported by Dong et al. (1994). The period of vibration predicted by

both models is almost the same (0.0192 s), which reflects that the period is obtained by dividing the sleeper

spacing by the wagon speed. The difference between the displacement of the wheel and the rail at any

defined time interval is also almost exactly same (0.1 mm) as that of Dong et al. (1994). However, whilst the

trough displacement of both the rail and the wheel during the entire period of simulation compare very well

with that of Dong et al. (1994), the peak displacement vary. The situation seems to be just reverse for the

Table 2

Impulse excitation sources

Name and geometry Impulse velocity

Raise on welding joint V 0¼

V ð

2 H

r Þ1=2

r ––

wheel radius

V –– wagon spped

Dipped-joint V 0 ¼ ða1 þ a2ÞV

a1, a2 –– dip angles of joint




contact force factor shown in Fig. 5(b). The difference of the maximum amplitudes of the two results is

about 5.6% and 1.95% respectively for the displacement and contact force factor. It is found that the

amplitudes of the dynamic responses of both the wheel and rail displacements and the contact force at the

contact point are sensitive to the damping coefficients of both the pad and the ballast. From Table 3 it could

be seen that the damping coefficients of the pad and the ballast are ‘‘calculated’’ indirectly from the data

provided in the paper and perhaps these values differ to the actual value. This might partly explain thedifference.

4.2. Dynamic responses with defects

Example 2: Most wheel flats are created by wheel slide during the application of break. Newton and

Clark (1979) published a good set of data obtained from controlled field experiment for a long indention

(150 mm length and 2.15 mm depth) on the top surface of the rail to simulate a wheel flat.

We have used these field experimental data for validation of the dynamic model of wagon and track

interaction reported in this paper. All the equations from Eqs. (3) to (24) are therefore used in the simu-

Fig. 4. Flow chart for solution technique.




lation. The total number of equations of motion solved for the simulation was 550. Once again, as inExample 1, the track used was 100 sleeper long. Due to the limited parameters given by Newton and Clark

(1979), some parameters were extracted from a commonly known wagon and track system in Australia

(McClanachan, 1999; Zhang et al., 1998). All parameters used in the simulation are listed in Table 4.

Although our detailed model was used in the evaluation of the impact response of the track system, it was

of general interest to feel for the static mean stiffness of the track. For this purpose the static mean stiffness

of the track was calculated as 26.9 MN/m from the sleeper, ballast, the subballast and the subgrade stiffness

values (Table 4). The static mean stiffness of 26.9 MN/m is within an acceptable range of values provided in

Grassie (1992) and Knothe and Grassie (1993).

The dynamic response of the whole system was simulated when the wagon travelled at a constant speed

of 117 km/h and passed through the indentation on the rail top surface. Fig. 6(a) shows the comparison of

the wheel and rail contact force factor calculated using the current model with the field experimental data

obtained by Newton and Clark (1979). It can be seen that when the wheel touches the indentation, the

contact force reduces to zero. This really means that the wheel and the rail separates for a while; when they

meets again, a large peak force is induced between the wheel and the rail, and the impact is accompanied by

the obvious oscillation as the natural frequency of the wagon and track system. However, the oscillation

decays very fast. This phenomenon is predicted by both the experimental data and the simulation results.

For the large peak force, the magnitude of the current model has a certain deviation compared with the

measured value by Newton and Clark (1979) and the error is about 18.4%. However, the duration of this

peak force is almost the same for both results.

The contact force is sensitive to the damping coefficients of the pad, and the sleeper. As these values are

only ‘‘assumed’’ or ‘‘calculated’’ (as shown in Table 4), there is a possibility of them being different to that

Table 3

All parameters for Example 1: Dong et al. (1994)

Notation Parameter Value

Wagon subsystem

F s Static wheel loada 82 kN

M w Wheel massa 500 kg

Track subsystem

mr Rail mass per metera 56 kg/m

Ar Rail cross-section areaa 7:17 Â 10À3 m2

E r Elastic modulus of raila 2:07 Â 1011 N/m2

G r Shear modulus of raila 8:1 Â 1010 N/m2

I r Rail second moment of areaa 2:35 Â 10À5 m4

k Timoshenko shear coefficienta 0.34

C p Pad dampingb 21.8 kN s/m

K p Pad stiffnessa 200 MN/m

ms Sleeper massb 50 kg

S l Sleeper spacinga 0.79 m

C b Ballast dampingb 21.8 kN s/m

K b Ballast stiffnessa 31.6 MN/m

Interface subsystem

C H Hertz spring constanta 1:0 Â 1011 N/m3=2

aThe values given by Dong et al. (1994).bThe value is calculated based on the related parameters given by Dong et al. (1994). For example, the damping coefficients of both

pad and ballast are equally given by multiplying the foundation damping per unit length (27.6 kN s/m2) with sleeper spacing; the

sleeper mass is obtained by subtracting the rail mass per meter from the track mass per unit length (119 kg/m), and then multiplying the

sleeper spacing.




of the actual. Perhaps this explains the difference in the peak magnitude of the field test result and that of

the current model.

Fig. 6(b) shows the time histories of the displacements of the wheel and the rail at the contact point and

the rail irregularity. The rail profile is also shown in the figure. It can be seen that without the contact force,

the rail jumps up due to its elasticity while the wheel moves down due to gravity. When the wheel and the

rail meets again, a large contact force is induced. Fig. 6(c) shows the time histories of displacements of the

sleeper, the ballast and the subballast. It can be seen that the obvious dynamic responses occur with a time

lag due to their damping effect.

Fig. 5. (a) Comparison of the displacements of the wheel and the rail at the point of contact with Dong et al. (1994) and (b) comparison

of the contact force factor with Dong et al. (1994).




The time histories of the secondary and the primary suspension force factor (the force divided by the

static wheel load) are shown in Fig. 6(d). It can be seen that the secondary suspension force almost remains

unchanged during the wheel impact on the rail while the primary suspension force drops due to the sep-

aration of the wheel and the rail and peaks when the wheel and the rail meets again. It can also be seen that

the adjacent primary suspension force has a similar response. Such periodic loading due to a large wheel

Table 4

Basic parameters for Example 2: Newton and Clark (1979)


Wagon system

M c Wagon body mass (loaded)a 58400 kg

J c Wagon mass moment of inertia about Y axisa 617 282 kgm2

M b Bogie massa 3600 kg

J b Bogie mass moment of inertia about Y axisa 1801 kgm2

K sc Secondary suspension stiffnessa 2555 kN/m

C sc Secondary suspension dampinga 30 kNs/m

K pr Primary suspension stiffnessb 6500 kN/m

C pr Primary suspension dampingb 10 kNs/m

Lc Distance between two bogie Y -direction centrelinesa 10.36 m

Lw Wheelset basea 1.675 m

r Wheel radiusc 0.50 m

Track subsystem

mr Rail mass per meterc 56 kg/m

Ar Rail cross-section areac 7:17 Â 10À3 m2

E r Elastic modulus of railc 2:07 Â 1011 N/m2

G r Shear modulus of railc 8:1 Â 1010 N/m2

I r Rail second moment of areac 2:35 Â 10À5 m4

r Rail profile radius on topa 0.30 m

k Timoshenko shear coefficientc 0.34

m Poisson’s ratio of raila 0.27

C p Pad dampingb 70 kNs/m

K p Pad stiffnessc 200 MN/m

C f Fastener dampingb 0.4 kNs/m

K f Fastener stiffnessa 1 MN/m

ms Sleeper massd 50 kg

S l Sleeper spacingc 0.79 m

C s Sleeper dampingb

50 kNs/m K s Sleeper stiffnessb 79 MN/m

Rt Rail top profile radiusa 0.30 m

Ls Effective length of rail seat support areaa 0.18 m

Bs Effective width of rail seat support areaa 0.164 m

H b Ballast heighta 0.30 m

H sb Subballast heighta 0.15 m

E b Elastic modulus of ballasta 130 MN/m2

E sb Elastic modulus of subballasta 200 MN/m2

E s Elastic modulus of subgradea 65 MN/m3

Interface subsystem

C H Hertz spring constantc 1:0 Â 1011 N/m3=2

aAssumed values based on a wagon and track system in Australia (M.J. McClanachan, Y.J. Zhang). The wagon body mass is

adjusted to have a static wheel force of 82 kN.bAssumed value (V.A. Profillidis, M.J. McClanachan, Y.J. Zhang).c The values given by Newton and Clark (1979).dThe value is calculated based on the related parameters given by Newton and Clark (1979). Sleeper mass is obtained by subtracting

the rail mass per meter from the track mass per unit length (119 kg/m), and then multiplying the sleeper spacing.




flats would certainly cause fatigue failure of the primary suspension. Similar conclusion has been made by

Ahlbeck and Hadden (1985) with reference to bearing fatigue.

Example 3: Fermeer and Nielsen (1995) reported their results of the field testing on the study of vertical

interaction between the wagon and the track. The experiments were carried out with wheel flats on the

instrumented wheelset, which were artificially grounded with an initial circumferential length of 40 mm and

a depth of 0.35 mm. Such length is the limit value for the flat wheel service in the Swedish railways. The

main objective of the experiments was to examine the influence of the vehicle speed, the axle load, and the

rail pad stiffness on the dynamic responses of the vehicle and that of the track.

These field experimental data due to a small wheel flat (relative to the previous example) were used for

the validation of the current model. The same number of equations of motion (550), and the number of

sleepers (100) as in the previous example was used. Only a few parameters are reported by Fermeer and

Nielsen (1995), and most other parameters were extracted based on the commonly known wagon and track

system in Australia (McClanachan, 1999; Zhang et al., 1998). The parameters used in the simulation are

listed in Table 5. Similar to Example 2, the static mean stiffness of the track used in the current example was

Fig. 6. (a) Comparison of the contact force factor predicted by the current model for Newton and Clark (1979) experimental results, (b)

the displacements of the wheel and the rail at the contact point predicted by the current model for Newton and Clark (1979) ex-

periments, (c) the displacements of the sleeper, the ballast and the subballast predicted by the current model for Newton and Clark

(1979) experiments and (d) the secondary and primary suspension forces predicted by the current model for Newton and Clark (1979)

experiments.




also calculated from the sleeper, ballast, the subballast and the subgrade stiffness values (Table 5) as 28.8

MN/m. The static mean stiffness of 28.8 MN/m is within an acceptable range of values provided in Grassie

(1992) and Knothe and Grassie (1993).

The dynamic response of the whole system was simulated when the wagon travelled at a constant speed

of 70 km/h with the flat wheel.

Table 5

Basic parameters for Example 3: Fermeer and Nielsen (1995)


Wagon system



M b Bogie massa 3600 kg

J b Bogie mass moment of inertia about Y axisa 1801 kg m2

K sc Secondary suspension stiffnessa 2555 kN/m



C pr Primary suspension dampingb 10 kNs/m

Lc Distance between two bogie Y -direction centrelinesa 10.36 m

Lw Wheelset basea 1.675 m

r Wheel radiusc 0.475 m

Track subsystem

mr Rail mass per meterc 60 kg/m








C p Pad dampingc 45 kNs/m

K p Pad stiffnessc 140 MN/m

C f Fastener dampingb 0.4 kNs/m


ms Sleeper massa 270 kg

S l Sleeper spacinga 0.685 m

C s Sleeper dampingb

130 kN s/m K s Sleeper stiffnessb 98 MN/m








E s Modulus of subgradea 65 MN/m3

Interface subsystem

C H Hertz spring constantd 0:87 Â 1011 N/m3=2


adjusted to have a static wheel force of 106 kN.bAssumed values (V.A. Profillidis, M.J. McClanachan, Y.J. Zhang).c The values given by Fermeer and Nielsen (1995).dThe value is calculated using formula reported in this paper.




Fig. 7. (a1) Theoretical prediction of contact force by Fermeer and Nielsen (1995), (a2) experimental prediction of contact force by

Fermeer and Nielsen (1995), (a3) theoretical prediction of contact force by current model. (b) The displacements of the wheel and the

rail at the contact point predicted by the current model for Fermeer and Nielsen (1995) experiments.




Fig. 7(a) shows the comparison of the measured contact force from Fermeer and Nielsen (1995) and the

calculated contact forces using the current model. The contact forces calculated by Fermeer and Nielsen

(1995) is also provided in the figure. To avoid congestion and improve clarity, Fig. 7(a) is presented as three

graphs in Fig. 7(a1), Fig. 7(a2) and Fig. 7(a3) respectively. It can be seen from Fig. 7(a1)–(a3), that the

numerical results predicted by our model are in reasonable agreement with the measured responses although

the experimental and predicted frequencies differ especially in the non-peak region. However, it may be

stated that amongst the results of the two models provided in Fig. 7(a1) and (a3), the result of our model

compares better with the experimental result especially at the peak region of the response curve. Both nu-

merical models predict lower level of vibration in the non-peak region relative to the experimental results.

The first positive peak force of the calculated and the measured results is about 1.5 times of the static

wheel load. The (larger) difference in the second positive peak force may be attributed to the values of

damping coefficient used for several components in the absence of the actual data (Table 5).

The displacement time histories of the wheel and the rail at the contact point calculated in this paper is

presented in Fig. 7(b), which exhibits the continuous penetration of the wheel into the rail. During impact

the maximum magnitudes of the displacement increase by about 11% of the average displacement.

Example 4: This is an example from the impulse source, which is available from Zhai (1996) (written inChinese). Zhai (1996) reported the experimental data in order to verify a model developed by him. The

impulse excitation source was from a dipped-joint with the total angle a1 þ a2 ¼ 0:02 radian (a1, a2 are the

dip angles of joints).

The full model with 550 equations of motion (100 sleeper long track) was used in the simulation. Due to

incomplete parameters given by Zhai (1996), we assumed the missing data based on the commonly known

wagon and track system used in Australia (McClanachan, 1999; Zhang et al., 1998). The data used in the

simulation are presented in Table 6. Similar to Examples 2 and 3, the static mean stiffness of the track used

in the current example was also calculated from the sleeper, ballast, the subballast and the subgrade stiffness

values (Table 5) as 28.8 MN/m. The static mean stiffness of 28.8 MN/m is within an acceptable range of

values provided in Grassie (1992) and Knothe and Grassie (1993).

The dynamic response was calculated when the wagon travelled at a speed of 48.7 km/h and passedthrough this dipped-joint.

Fig. 8(a) shows the comparison of the contact forces from the experimental data and that calculated data

by Zhai (1996) with the results simulated in this paper. The displacements of the wheel and the rail at the

contact point are also reported in this paper. From Fig. 8(a), the magnitude of the peak force and its du-

ration are in good agreement with the experimental and calculated data by Zhai (1996), and the peak force is

about twice as the static wheel load in this situation. It should be noted that the frequencies calculated from

the current model is in better agreement with the experimental data than the calculated values of Zhai (1996)

especially in the non-peak region. Fig. 8(b) shows the displacement of the rail and the wheel at the contact

point. In this situation, the wheel and the rail are always in contact with each other without any separation.

5. Conclusions

A model for simulating the dynamic interaction of the rail track and wagon system for steady state travel

of the wagon has been presented. The wheels of the wagon and the rail may or may not have defects. The

current model has been validated using four sets of data reported in the literature by several authors.

The model reported in this paper is capable of predicting the dynamic responses of both the wagon and

the rail track components. The model is also capable of examining the influence of the properties of the rail

track and the wagon components on the impact forces and other dynamic responses of the rail track and

wagon system. It is possible to achieve design optimisation for the improved performance of wagon and

track using this model.




In addition, the model can be easily reduced to a one wheel–track model or one bogie-track model due to

its flexibility of removing a few equations as demonstrated in Example 1. This can provide with the means

to investigate the effectiveness of various models for the vertical interaction of rail track and wagon system.

Table 6

Basic parameters for Example 4: Zhai (1996)


Wagon system



M b Bogie massb 3823 kg

J b Bogie mass moment of inertia about Y axisa 1801 kg m2

K sc Secondary suspension stiffnessb 2300 kN/m



C pr Primary suspension dampingc 10 kNs/m

Lc Distance between two bogie Y -direction centrelinesb 9 m

Lw Wheelset baseb 2.9 m

r Wheel radiusb 0.625 m

Track subsystem

mr Rail mass per meterb 60 kg/m








C p Pad dampingc 70 kNs/m

K p Pad stiffnessb 44 MN/m

C f Fastener dampingc 0.4 kNs/m


ms Sleeper massb 251 kg

S l Sleeper spacingb 0.544 m

C s Sleeper dampingc

130 kN s/m K s Sleeper stiffnessc 98 MN/m








E s Modulus of subgradea 65 MN/m3

Interface subsystem

C H Hertz spring constantd 0:92 Â 1011 N/m3=2


adjusted to have a static wheel force of about 100 kN.bThe values given by Zhai (1996).c Assumed values (V.A. Profillidis, M.J. McClanachan, Y.J. Zhang).dThe value is calculated using formula reported in this paper.




From the examples reported in this paper it could be concluded that exact match between the field data

and model would be possible only when all data, in particular the damping coefficients of various com-

ponents, are precisely known. The examples also indicate that

(i) even though the periodic defect is more serious than the impulse excitation (due to the former’s po-

tential to cause fatigue failure), the impulse excitation produces much higher impact forces (as a percent-

age of static loading);

Fig. 8. (a) Comparison of the contact force predicted by the current model for Zhai (1996) experiments (results of Zhai (1996) model

are also shown) and (b) comparison of the contact force predicted by the current model for Zhai (1996) experiments (results of Zhai

(1996) model is also shown).




(ii) the secondary suspension is not affected by the wheel–rail impact forces and the effect of impact on

primary suspension is also marginal.

Acknowledgements

Central Queensland University and Queensland Rail financially supported this research via a scholarship

awarded to the first author. The support and encouragement of Professor Dudley Roach, Director of the

Centre for Railway Engineering and the assistance of Mr. Brain Hagaman, and Mr. Tim McSweeney of the

Queensland Rail are thankfully acknowledged.

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a dynamic model for the vertical interaction of the rail

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