Dynamic models of railway track taking into
account of cross-section deformation at high
frequency
Jiannan Yang1, David Thompson1 and Atul Bhaskar2
1
Institute of Sound and Vibration Research, University of Southampton, Southampton SO17
1BJ, UK
Tel: +44 (0) 23 8059 2294, Fax: +44 (0) 23 8059 3190, E-mail: jny1e08@soton.ac.uk
2
School of Engineering Sciences, University of Southampton, Southampton, SO17 1BJ, UK
Summary
Track dynamic behaviour is important for the prediction of the rolling noise,
corrugation growth and track damage. Various models have been developed in the
literature but problems still remain. On the one hand, analytical models become
insufficient because of the need to include cross-section deformation at high
frequencies. On the other hand, FE models are straightforward but the truncation
of the infinite length is unavoidable. A new tapered plate rail model is developed
in this paper. This model takes into account all the main motions required for
frequencies below 7 kHz. The rail head is represented by a rectangular beam, the
web by a plate of constant thickness and the foot by a plate of variable thickness.
The out-of-plane and in-plane motions of the plates are approximated using cubic
and linear functions respectively based on the relevant wave speeds. Freely
propagating waves in the rail are studied by means of Hamilton’s principle.
Comparing the results in terms of the dispersion relations, the tapered plate rail
model shows good agreement with an FE model. Comparison with simpler beam
models confirms the improvements at high frequencies due to the taper of the foot.
1
Introduction
Railway rolling noise induced by the roughness of the rail and wheel surfaces
forms the principal source of noise from railway operation [1]. These surface
irregularities produce dynamic interaction forces between the wheel and rail. The
resulting high frequency vibrations are transmitted into both wheel and track and
then the sound is radiated from the vibration of the structures.
Dynamic models are required to predict the noise emitted from the track and
frequencies up to at least 5 kHz should be considered [2]. Below about 1500 Hz, a
single Timoshenko beam is sufficient to represent the rail dynamic behaviour [3].
However, it is more difficult to study the dynamic behaviour of railway track at
high frequencies due to cross-section deformation of the rail as found by
2
Thompson [4]. The vertical vibration is dominated by the rail foot flapping and the
lateral motion is even more complicated.
Most of the models taking into account of cross-section deformation are based
on finite element methods (FEM). Thompson [4] developed an FE model of a
finite length of rail using beam elements for the head and plates for the remaining
parts. Since such a model only allows frequencies to be predicted for a given
wavenumber, this is not enough to study the frequency response of the track. To
improve the application of FE methods to an infinite rail, the finite strip method
(FSM) and some other derivatives of FEM have been developed, e.g. [5]. The
deformations of regularly discretized elements in the infinite rail are connected, by
considering harmonic waves in the longitudinal direction. Gavric [6] and Gry [7]
used an alternative approach based on factorization of the function describing the
displacement field. A general shape function is used to describe the deformation
of the cross-section, while along the rail axis wave propagation is assumed.
To avoid the large number of degrees of freedom of FE methods, simplified
beam models were developed by Wu and Thompson [8-9], where the rail head and
foot were represented by infinite Timoshenko beams. The web was modelled as a
spring in the vertical direction or by an array of beams for the lateral direction.
The vertical and lateral vibrations of the rail were studied separately based on the
assumption of symmetry of the cross-section.
Foot deformation at high frequency is significant and an analytical model
which took account of the variable thickness of the rail foot was developed by
Bhaskar et al. [10]. However, little information is given for this model. In
addition, in-plane motions of the plates were approximated by simple beam
bending. This requires improvement because they are quite deep beams.
Moreover, the stretching of the web in the direction normal to the wave
propagation was neglected to simplify the model, which could result in
considerable discrepancies.
Therefore, a tapered plate rail model will be implemented in this paper. It will
be compared with both a 3D FE model and the simple beam models from [8-9].
2
Tapered plate model
The cross-section of the model (UIC60 rail) is shown in Fig. 1. This model takes
into account all the main motions required for frequencies below 7 kHz. The rail
head is represented by a rectangular beam, the web by a plate of constant
thickness and the foot by plates of variable thickness. The motion of the rail head
can be characterized by beam bending in two directions, torsion and compression
in the longitudinal direction. The deformation of the web and foot can be
described by plate out-of-plane bending and twisting, and in-plane stretching and
shear. In the rail axis (z direction), harmonic waves are assumed of the form e jkz
at frequency  (assumed time dependence e jt ).
The variational principle is employed to determine unknown deflections in
order to avoid solving the differential equations of plate theory. The out-of-plane
3
and in-plane motions of the plates are approximated using cubic and linear
functions respectively in the x–y plane. A total of 17 degrees of freedom are used
to represent the cross-section. The components are assembled with the stiffness
matrix formulated in terms of the wavenumber in the axial direction. The resulting
eigenvalue problem is solved to find the dispersion characteristics and the
propagation modes for the freely propagating waves in the rail.
Fig. 1. Mechanical idealization of the theoretical rail model. A, B, C and D are the four nodal
points which are located in the middle plane of each plate. Point E represents the centre of the
beam and the dashed lines represent the actual shape of the beam and plates.
2.1
Foot modelling using a tapered plate and web with constant thickness
The foot element AB is shown in Fig. 2 where the cross-section is defined in the
x–y plane. The local coordinate 1 denotes the distance from point B. The
deflections in the x, y and z directions are written as u, v and w respectively. The
variable thickness can be expressed as t f 1   S f 1  t f , where S f is the slope
B
B
and t f
A
tf
is the thickness at point B. The thickness at the point A is thus
B
 S f h f  t f where h f is the foot length.
The vibration energies of plate AB consist of both out-of-plane and in-plane
motion, and can be written as
U ABi
U ABo 
 3

1  Et f (1 ) 


  
 x x  z z  ( x z  x z G xz
 xz d 1dz ,
2  (1 2 )


1
TABi    t uu  ww d 1dz ,
2






1





 D v,11 v,11 v, zz v, zz 21 v,1z v,1z  (v,11 v, zz v,11 v, zz ) d1dz ,
2
1
TABo    tvvd1dz ,
2
(2.1)
(2.2)
(2.3)
(2.4)
4
with U for potential energy, T for kinetic energy and subscripts i, o represent
in-plane and out-of-plane respectively. Here E is Young’s modulus, G is the shear
modulus, ν is Poisson’s ratio,  x 
 xz 
w
u
and  z 
are normal strains and
z
1
Et 3f (1 )
u w
is the shear strain. D 

z 1
12 1 2


is the plate bending stiffness,
and the subscripts z and 1 denote the derivatives with respect to z and 1 .
z, w
1
y, v
t f  1 
A
t Bf
B
hf
x, u
Fig. 2. Schematic diagram of foot AB showing the relevant dimensions and the co-ordinate axes.
The highest derivatives appearing in the energy expressions are second and first
respectively for plate out-of-plane and in-plane motion. At each node, it is thus
necessary to take displacement and rotation as degrees of freedom for the bending
motion. Since only the first cantilever mode is of interest, the plate out-of-plane
bending deformation can be approximated by a polynomial having four constants,
that is, a cubic function. For the in-plane motion, the minimum requirement is a
linear function. The shape functions are therefore
 4
 j t  kz 
v AB (1 , z , t )    ai1i 1  e
,


 i 1

j t kz 
u AB (1 , z , t )   d1 d21  e
,
wAB (1 , z, t )   e1e21  e
j t kz 
.
(2.5)
(2.6)
(2.7)
For the rail web, it is quite convenient to put the plate slope equal to zero.
Although the web is quite long compared with the foot, for a steel plate with
constant thickness, at 5000 Hz the wavelengths of longitudinal and shear waves
are about 1.1 m and 0.63 m respectively. Therefore, one element of length 0.11 m
is still sufficient to satisfy the requirement of at least 6 elements per wavelength.
2.2
Rail head represented by a beam
The rail head can be modelled with a rigid cross-section because of its high
stiffness. Its dimensions are small compared with the wavelength of shear or
5
longitudinal waves in the frequency range of interest. At around 5000 Hz, the
minimum vertical and lateral bending wavelengths are about 0.35 m and 0.27 m
from an FE analysis (see [11]), which are about 9 and 4 times the corresponding
dimension of rail head. Although the rail behaviour actually becomes mainly
dominated by foot vibration, shear deformation and rotational inertia effects
become important for the lateral motion at high frequency. Therefore, it is
sufficient to use the elementary beam theory for the head for the vertical motion
but Timoshenko beam theory has to be employed for the lateral motion.
1
1
1
*

*
 EI yuE , zz uE , zz dz   EI xE , zE , z dz    AG vE ,Z E   vE ,Z E dz
2
2
2
1
1
  GJ E , z E , z dz   EAwE , z wE , z dz ,
2
2
1
1
1
1
*


Th    AuE uE dz    AvE*vE dz    I z EE dz    AwE wE dz
2
2
2
2
1

   J E  E dz ,
2
U
h

(2.8)
(2.9)
where A is the area of the cross-section of rail head, J is the torsion constant of the
cross-section, θE is the rotation of the rail head, Ix is the second moments of area of
the cross-section about x-axis the rail, but Iy is the second moments of area of the
head base axis (the longitudinal axis where the point D located).  is called the
Timoshenko shear coefficient and  5/ 6 is used here. E describes the
rotation of the head cross-section about the vertical.
2.3
Freely propagating waves in the rail
By applying the compatibility at points B and D, the total number of degrees of
freedom is 17. The following equation is obtained by applying Hamilton’s
principle based on a unit length of rail in the integral:
(  jk )4 K 4 (  jk )2 K 2 (  jk )K1 K 0  2M  {q}  {0}
(2.10)


The solutions to this equation represent waves in the free rail without support.
To solve this generalized polynomial eigenvalue problem in k for a given
frequency, it is convenient to convert it into state space form
where
0
I


0
0
A= 

0
0

2
- ( K 0  M ) -K 1
q0 
 
q
 A jkB 1 {0} ,
q 2 
q 3 
0
0
I
0
I
0 
B= 
,
0
I
0


-K 2 0 
0
(2.11)
0 
0 
0 

0 0 K4 
0 0
I 0
0 I
are both 68  68
matrices (I is the identity matrix and 0 is a matrix of zeros), and {qi }  jk i {q},
i=0, 1, 2, 3.
6
3
Results
The symmetric and anti-symmetric waves of a free rail are shown separately in
Figs 3 and 4 since they do not couple with each other due to the symmetry of the
cross-section. A short length of free UIC60 rail has been modelled using ANSYS
finite element package and the results are used as a reference. The dispersion
relations from simplified beam models [8-9] are also presented as comparison.
Fig. 3. Dispersion relation for vertical/longitudinal motion of free rail,  , tapered plate model;
, simplified beam model;
  , the original FE model.
From these two figures, it can be seen that both analytical models give good
results compared to FE analysis. However, the improvements brought by the
tapered plate model are still significant especially at high frequencies.
For the vertical motion, the foot flapping mode (iii) which cuts on about 5 kHz
is well represented by both plate and beam models, but the tapered plate model
gives much better representation for the vertical bending wave (i) for frequency
higher than 4 kHz. It is found from the mode shapes (not shown) that the vertical
bending wave is dominated by the deformation of the rail foot at high frequency.
Although the beam model takes account of the foot flapping by modelling the foot
as a separate beam, it cannot represent the deformation of the foot itself. In
addition, the double beam model does not give the longitudinal waves (ii) at all
because there is no degree of freedom in the rail direction assigned to this model.
It should be noted that the wave that cuts on at about 5 kHz from simple beam
model is actually the warping behaviour of one of the beams, but not the
second-order longitudinal wave (iv) where the head and the foot move anti-phase.
7
Fig. 4. Dispersion relation for lateral motion of free rail.  , tapered plate model;
simplified beam model;
  , the original FE model.
,
For the lateral motion, the torsion (II) and simple web bending (III) modes are
well represented by both analytical models but for the lateral bending wave (I),
these two models deviate from about 2 kHz. As found from the mode shapes from
the FE analysis, above 2 kHz the lateral bending wave is dominated by the rocking
of the foot. In the beam model, a constant thickness beam is used for the rail foot.
As found by Yang [11] this assumption makes the lateral bending motion much
less stiff than the real structure at high frequency. However, the tapered plate
model makes a big improvement on this by including the taper of the rail foot. The
remaining small difference from the FE model is due to the fact that the real
structure has a more complicated tapered foot. The discrepancy for the double web
bending wave (IV) is also due to the foot modelling.
4
Conclusions
A new analytical model of a rail with a tapered foot is presented. The rail head,
web and foot are constructed using a beam, a plate with constant thickness and a
tapered plate respectively. Cubic shape functions are used to approximate the plate
out-of-plane motions and linear shape functions are used for the in-plane motions.
The head beam is represented by an Euler beam vertically but a Timoshenko beam
laterally. The potential and kinetic energy of each part are found first using
stress-strain relations. Then the whole structure is assembled and the dispersion
relations of the free rail are found using Hamilton’s principle. It can be seen that
this tapered plate model gives good agreement with an FE model.
8
In addition, it is clear from the comparison with the simplified beam models
that big improvement on this is achieved at high frequency by including the taper
of the rail foot. First, it gives more complete modelling of the rail by considering
all possible motions up to 7 kHz. Second, at high frequency, the simplified beam
models tend to underestimate the vertical response and to overestimate the lateral
response. As the tapered plate model considers more details of the rail, more
parameters can be varied to optimize the rail design for noise reduction purposes.
In order to use the model to predict noise it is necessary to include the support
structure (ballast, sleepers and railpads) and to couple the model to a prediction of
acoustic radiation. This is beyond the scope of the paper. However, it can be
expected that the present model will give improved results, particularly for the
decay rate of vibration along the rail.
References
[1]. Thompson, D.J., Railway Noise and Vibration. Mechanisms, Modelling and
Means of Control. 2008: Oxford, Elsevier Science. 506pp.
[2]. Thompson, D.J., Wheel-rail noise: theoretical modelling of the generation of
vibrations, PhD thesis. 1990, University of Southampton.
[3]. Grassie, S.L., Gregory, R.W., Harrison, D., Johnson, K.L., Dynamic response
of railway track to high frequency vertical excitation. Journal of Mechanical
Engineering Science, 1982. 24(2): p. 77-90.
[4]. Thompson, D.J., Wheel-rail noise generation, Part III: Rail vibration. Journal
of Sound and Vibration, 1993. 161(3): p. 421-446.
[5]. Knothe, K., Strzyzakowski, Z., Willner, K., Rail vibrations in the high
frequency range. Journal of Sound and Vibration, 1994. 169(1): p. 111-123.
[6]. Gavric, L., Computation of propagative waves in free rail using a finite
element technique. Journal of Sound and Vibration, 1995. 185(3): p. 531-543.
[7]. Gry, L., Dynamic modelling of railway track based on wave propagation.
Journal of Sound and Vibration, 1996. 195(3): p. 477-505.
[8]. Wu, T.X., Thompson, D.J., A double Timoshenko beam for vertical vibration
analysis of railway track at high frequencies. Journal of Sound and Vibration,
1999. 224(2): p. 329-348.
[9]. Wu, T.X., Thompson, D.J., Analysis of lateral vibration behavior of railway
track at high frequencies using a continuously supported multiple beam
model. Journal of the Acoustical Society of America, 1999. 106(3 I): p.
1369-1376.
[10].Bhaskar, A., Johnson, K.L, Wood, G.D, Woodhouse, J., Wheel-rail dynamics
with closely conformal contact. Part 1: Dynamic modelling and stability
analysis. Proceedings of the Institution of Mechanical Engineers, Part F:
Journal of Rail and Rapid Transit, 1997. 211(1): p. 11-24.
[11].Yang, J., Dynamic models of railway track at high frequency. MSc Thesis,
ISVR, 2009, University of Southampton.