Dynamic models of railway track taking into
account of cross

section deformation at
high
frequency
Jiannan Yang
1
,
David Thompson
1
and
Atul Bhaskar
2
1
Institute of Sound and
V
ibration 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 p
roblems 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 unavoid
able. 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
. T
he rail head is represented by a rectangular beam, the
web by a plate of constant thickness and the f
oot 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 Hamilto
n’s principle.
Comparing the results in terms of the dispersion relation
s
, the tapered plate rail
model shows good agreement with
an
FE model.
C
omparison
with simpler beam
models
confirms the improvements at high frequencies
due to
the taper of the foot.
1
I
ntroduction
Railway rolling noise induced by the roughness of the rail and wheel surface
s
forms the princip
al
source of noise from railway operation
[1]
.
T
hese surface
irregularities produce dynamic interaction forces between the wheel and rail.
T
he
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 requir
ed to predict the noise emitted from the track and
frequencies up to at least 5 kHz should be considered
[2]
.
B
elow about 1500 Hz, a
single Timoshen
ko beam is sufficient to represent the rail dynamic
behaviour
[3]
.
However, i
t
is
more
difficult to study the dynamic
behaviour
of railway track
at
high frequencies
due to cross

section deformation
of the rail
a
s found by
2
Thompson
[4]
.
T
he vertical vibration is dominated by the rail foot flapping and the
la
teral 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 a
n
FE model of
a
finite length of
rail
using
beam eleme
nts for the head and plates for the remaining
parts.
S
ince
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.
T
o
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]
. T
he
deformation
s
of regularly discretized elements in the infinite rail ar
e connected,
b
y
considering harmonic wave
s
in
the
longitudinal direction
.
Gavric
[6]
and
Gry
[7]
used an
alternative
approach based on factorization of t
he
function
describing the
displacement
field.
A g
eneral
shape
function
is used to
describ
e
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, simplifi
ed
beam model
s we
re developed by Wu and Thompson
[8

9]
,
where
the
rail head and
foot we
re represente
d by infinite Timoshenko beams
. T
he web
wa
s
modelled
as
a
spring in the vertical direction or by an array of beams
for the
lateral
direction
.
The
vertical and lateral vibration
s
of the rail
we
re
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. I
n
addition, in

plane
motions
of the plates were approximated by simple beam
bending.
T
his
requires
improve
ment
because they are quite deep beams.
M
oreover, the
s
tretching
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
. I
t will
be compared with b
oth
a
3D
F
E model
and
the
simple beam model
s
from
[8

9]
.
2
T
apered 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
. T
he rail
head is represented by a rectangular beam, the web by a p
late of constant
thickness and the foot by plate
s
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
d
escribed 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
jkz
e
at frequency
(assumed time
dep
endence
j t
e
)
.
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 cu
bic and linear
functions respectively
in the
x
–
y
plane
. A total of 1
7
degrees of freedom are used
to represent the cross

section. The components are assembled with the stiffness
matrix formulated in terms of the wavenum
ber in the axial direction. The
resul
ting
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.
P
oint E
represent
s the centre of the
beam and the dashed lines represent the actual shape of the beam and plates.
2.1
F
oot
modelling
using a tapered plate
and web with constant thic
kness
The foot element
AB
is shown in Fig
.
2
where the cross

section is defined in
the
x
–
y
plane.
T
he local
coordinate
1
denotes the distance from point B.
T
he
deflections in the
x, y
and
z
directions are written as
u, v
and
w
res
pectively. The
variable
thickness can be expressed
as
1
1
B
f f f
t S t
, where
f
S
is the slope
and
B
f
t
is the thickness at point B.
T
he thickness at the point A is thus
A B
f f f f
t S h t
where
f
h
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
3
1
1
( )
1
(,
2
2
(1 )
x x z z x z x z xz xz
ABi
Et
f
U G d dz
(
2
.
1
)
1
1
,
2
ABi
T t u u w w d dz
(
2
.
2
)
1 1 1 1 1 1 1 1 1 1
,,,,,,,,,,
1
1
2 1 ( ),
2
zz zz z z zz zz
ABo
U D v v v v v v v v v v d dz
(
2
.
3
)
1
1
,
2
ABo
T tv vd dz
(
2
.
4
)
4
with
U
for potential energy,
T
for kinetic energy and subscript
s
i, o
repre
sent
in

plane and out

of

plane respectively
.
Here
E
is Young
’
s
modulus
,
G
is the shear
modulus,
ν
is
Poisson’s
ratio
,
1
x
u
and
z
w
z
are normal strains
and
1
xz
u w
z
is the shear st
rain
.
3
1
2
( )
12 1
Et
f
D
is the plate bending stiffness,
and the
subscript
s
z
and
1
denote the derivatives with respect to
z
and
1
.
B
1
1
f
t
B
f
t
z,
w
A
x
,
u
y,
v
h
f
Fig.
2
.
S
chematic diagram of foot AB showing the relevant
dimensions
and the
co

ordinate
axes
.
T
he highest derivatives appearing in the energy expressions are second and first
respectively for plate out

of

plane
and in

plane motion.
A
t 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 cons
tants
,
that is, a cubic function.
F
or
the
in

plane motion, the minimum requirement is
a
linear function.
The shape functions are therefore
1
,
1
1
4
(,,)
1
j t kz
i
i
AB
v z t a e
i
(
2
.
5
)
1 2 1
1
(,,),
j t kz
AB
u z t d d e
(
2
.
6
)
1 2 1
1
(,,).
j t kz
AB
w z t e e e
(
2
.
7
)
For the rail web, it is quite convenient to put the plat
e slope equal
to
zero.
Although
the web is quite long compared with the foot, f
or 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
T
he rail head can be
modelled
with
a
rigid
cross

section
because of
its
high
stiffness. I
ts
dimensions
are small compared wi
th the wavelength of shear or
5
longitudinal waves in the frequency range of interest.
At around 5000 Hz, the
minimum
vertical and lateral bending wavelength
s
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
ma
inly
dominated by
foot
vibration
,
shear deformation and rotational inertia effect
s
become important for
the lateral motion
at
high frequency. Therefore, it is
s
ufficient
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
*
2 2 2
1 1
,
2 2
y E zz E zz E z E Z E E Z E
E z
E z E z E z E z
U EI u u dz EI dz AG v v dz
x
h
GJ dz EAw w dz
(
2
.
8
)
*
*
1 1 1 1
2 2 2 2
1
,
2
E E E
h E E E E E
E E
T Au u dz Av v dz I dz Aw w dz
z
J dz
(
2
.
9
)
where
A
is the area of the
cross

section of
rail head,
J
is the torsion const
ant of the
cross

section,
θ
E
is the rotation of the rail head,
I
x
is
the second moment
s
of area of
the cross

section about
x

axis
the
rail
,
but
I
y
is
the second moment
s
of area
of the
head base axis (the longitudinal axis where the point D located).
is c
alled 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.
T
he following equation
is
obtained
by
applying Hamilton
’
s
principle
based on
a
unit length
of rail
in the integral
:
4 2 2
4 2 1 0
( ) ( ) ( ) { } { }
jk jk jk
K K K K M q 0
(
2
.
10
)
T
he solutions to this equation represent waves in the free rail without support.
T
o solve this
generalized
polynomial eigenvalue problem
in
k
for a given
frequency
, it is
convenient
to convert it into state space form
0
{ },
jk
1
2
3
q
q
A B 0
q
q
(
2
.
11
)
where
2
( )
0 1 2
0 I 0 0
0 0 I 0
A=
0 0 0 I
 K M K K 0
,
4
I 0 0 0
0 I 0 0
B=
0 0 I 0
0 0 0 K
are both
68 68
matrices
(
I
is
the
identity
matrix
and
0
is a matrix of zeros)
,
and
{ } { },
i
i
jk
q q
i
=0, 1, 2, 3.
6
3
Re
sults
T
he symmetric and anti

symmetric
waves of a free rail
are shown
separately
in
Fig
s
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 elem
ent 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/l
ongitudinal motion of f
ree rail,
,
tapered plate mode
l
;
,
simplified beam model
;
,
the original FE model
.
From these two figures, it can be seen that both analytical models give good
results comp
ared 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 b
eam 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 r
ail foot at high frequency.
A
lthough the beam model takes account of the foot flapping by modelling the foot
as
a separate
beam, it can
not
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
beam
s, but not
the
second

o
rder
longitudinal wave
(iv) where
the head and the foot
move
anti

phase
.
7
Fig. 4
.
D
ispersion relation for lateral motion of free rail
.
,
tapered plate mode
l
;
,
simplified beam model
;
,
the original FE model
.
F
or
the
later
al
motion, the torsion (II) and simple web bending (III) modes are
well
represented by both analytical models
b
ut for the
lateral bending wave
(I)
,
these two models deviate from about 2 kHz.
A
s
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.
H
owever, the taper
ed plate
model makes
a
big improvement on this by
including the
taper
of
the
rail foot
. T
he
remaining
small difference from
the
FE model is due to
the fact th
at
the
real
structure has
a
more complicated tapered foot.
The discrepancy for the double web
bend
ing 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.
Cu
bic
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.
T
he potential and kinetic ene
rgy
of each part are found first using
stress

strain relations. Then the whole structure is assembled and t
h
e
dispersion
relation
s
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 m
odel.
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 consideri
ng
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
purpose
s
.
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 radiat
ion. 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.
R
eferences
[1].
Thompson, D.J.,
Railway Noise and V
ibration. 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., Greg
ory, 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.
J
ournal
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 wav
es 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 o
f railway track at high frequency. MSc Thesis
,
ISVR,
2009, University of Southampton.
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