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Anyakwo, A., Pislaru, Crinela, Ball, Andrew and Fengshou, Gu
Modelling the Dynamic Behaviour of the Wheel-Rail Interface by Using a Novel 3D Wheel-Rail
Contact Model
Original Citation
Anyakwo, A., Pislaru, Crinela, Ball, Andrew and Fengshou, Gu (2011) Modelling the Dynamic
Behaviour of the Wheel-Rail Interface by Using a Novel 3D Wheel-Rail Contact Model. In: 5th IET
Conference on Railway Condition Monitoring and Non-Destructive Testing (RCM 2011) , 29-30
November 2011, Derby Conference Centre, UK.
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MODELING THE DYNAMIC BEHAVIOUR OF THE WHEEL
RAIL INTERFACE USING A NOVEL 3D WHEEL-RAIL
CONTACT MODEL
A. Anyakwo*, C. Pislaru, A. Ball, F. Gu
*Diagnostic Engineering Research Centre, University of Huddersfield, U.K, Email: arthur.anyakwo@hud.ac.uk
Keywords: multibody modelling, wheel-rail interface, wheel
profiles, vehicle dynamics, condition monitoring
Abstract
Methods for multibody modelling and simulation should
accurately replicate the dynamic behaviour of rail-wheel
interface including precise values for wheel-rail contact
positions. This paper studies the development of a novel 3-D
wheel-rail contact model which is used for dynamic
simulation of a suspended wheelset with parameters listed for
a typical Mark IV coach. The contact point locations on the
wheel and rail are determined by the minimum difference
method considering the lateral displacement, yaw angle and
the roll angle. The proposed new 3D wheel-rail contact model
can be applied in railway condition monitoring techniques to
estimate the wheel geometry parameters and thus to achieve
practical optimised wheel-rail interfaces.
1
Introduction
The dynamic behaviour of railway vehicle on the track is
influenced by wheel-rail interaction. Any slight deviation in
the shape of the wheel and rail profiles affects the movement
of the vehicle on the track. The implementation of a
comprehensive rail vehicle dynamic model in the multibody
simulation software package requires the location of all
contact points on the wheel-rail contact. Two dimensional
wheel-rail contact models are limited to the two dimensional
motion of the two surfaces and are thus not very suitable for
application in steep rail track curves where the yaw angles are
large. Wickers [1] applied 2D wheel-rail contact method to
calculate the wheel-rail contact coordinates considering the
lateral displacement and the roll angle as inputs. Also the 2D
wheel-rail contact method was used to design the wheel
profiles considering the contact angle function [2] or the
rolling radius difference function [3].
The lateral displacement, roll angle and the yaw angle are
used to depict the movement of the wheelset on the track in
three dimensions. Two methods used for 3-D wheel-rail
contact are rigid contact method and semi-elastic method. The
rigid contact method [4,5] comprises a set of algebraic
nonlinear differential equations used to describe the dynamics
of the wheel-rail contact in 3D. Indentation and lift are not
considered due to the fact that the wheel movement is made
up of five degrees of freedom (DOF) with respect to the rail.
The semi-elastic methods allow the management of multiple
contact points and are generally used for automotive and
railway applications [5, 6]. The wheel is assumed to have six
DOF with respect to the rail. The normal contact forces acting
on the wheel-rail contact are defined as a function of
indentation using Hertz theory for two surfaces in contact.
They require look-up tables with the values of the wheel-rail
co-ordinates depending on the lateral displacement, yaw and
roll angles. The number of simulated contact points is limited
so the management of multiple contact points is hard to
achieve.
Numerical iterative algorithms (such as Simplex and
Compass methods [7] can be applied to determine the
location of contact points by minimizing the difference
between rail and wheel surfaces. These algorithms allow
multiple contact points to be effectively managed because
there are no additional geometrical constraints applied for the
wheel-rail contact model. These methods could be used for
real-time applications but they need a starting point and end
point for simulations that are sometimes difficult to choose.
In this paper the minimum difference method (semi-analytic
method) is used to determine the location of wheel-rail
contact points based on the calculated local minima [8, 9].
This method reduces the problem to a one dimensional scalar
problem that can be easily solved numerically without
iterations. Also the two point contacts can be managed
effectively. These wheel-rail contact co-ordinate positions are
then used to determine the rolling radius difference function,
contact angle function, normal and tangential forces.
Dynamic simulations of the wheelset on the track are carried
out in MATLAB using numerical differential techniques to
plot the lateral excursion and the yaw angle of the wheelset
on the track. Figure 1 shows the stages of the development for
the proposed 3D wheel-rail contact model representing the
dynamic behaviour of the single wheelset on a straight track.
The proposed method could be employed by the real
multibody simulation software packages (such as SIMPACK,
VAMPIRE, VTSIM) for real time implementation and
condition monitoring purposes using hardware in the loop
techniques [7]. The condition monitoring systems used for
modern railways should include effective measurement
elements, robust post analysis and decision support and
estimation in real-time of the wheelset parameters [10].
2 Development of mathematical 3D wheel-rail
contact model
Generative functions for P8 wheel profiles and BS 113A rail
profile are applied as inputs. The contact position locations on
the wheelset and rail are obtained by using the piecewise
cubic interpolation polynomial and the calculated lateral
displacement, roll angle and yaw angle (their initial values are
assumed to be zero). The variation between the wheelset and
rail positions is applied to the minimum difference method
algorithm. The rolling radius difference function and the
contact angle function are determined when the indentation is
negative. The block ‘Normal contact problem’ (see Fig. 1)
represents the calculation of the normal contact forces and
contact patch dimensions by using Hertz theory. These values
and the rolling radii of left and right wheel functions are used
to evaluate the tangential contact forces. These calculated
values and the primary suspension parameters corresponding
to BR MK IV coach [11] are included in the differential
equations describing the wheelset dynamic behaviour. The
equations are solved using Runge-Kutta method.
Fig. 1 Development of 3D Wheel-rail contact model
2.1 Reference Frame Definitions for the Track
Three frames of reference are used to define wheel-rail
contact geometry. They include the fixed reference frame, the
auxiliary reference frame and the local reference frame. The
track reference frames are shown in Fig. 2.
. The fixed reference system (O
f
, X
f
, Y
f
, Z
f
) defines the track
as a three dimensional curve. The auxiliary reference system
(O
a
,X
a
,Y
a
,Z
a
)
follows the wheelset during program
simulations. It is defined on the rail tracks. The X
a
axis is
tangential to the track centreline in the longitudinal direction
of point O
a
. Y
a
is the lateral direction with respect to the rail
plane while Z
a
is in the normal direction with respect to the
plane of the rail.
The unit normal vectors obtained from the auxiliary system
can be defined as follows;
(1)
where A
cant
is the rotation matrix defined as a function of the
cant angle β. The cant angle for the rail is 1/20 radians
Fig. 2 Reference frames for the wheel-rail contact model
The unit normal vectors obtained from the auxiliary system
can be defined as follows [8, 9];
(1)
where A
cant
is the rotation matrix defined as a function of the
cant angle β. The cant angle for the rail is 1/20 radians
β
β
β
β
β
(2)
The local reference system in O
w
,X
w
,Y
w
,Z
w
is defined
whereby Y
w
is rigidly fixed to the wheelset axle. The origin of
the wheelset O
w
corresponds with the centre of gravity G of
the wheelset. Let
and
represent the position of a point
in the auxiliary and reference local frame respectively, then
the kinematic equation is generally expressed as follows:
(3)
where
is the wheelset centre coordinates of mass
expressed with respect to the auxiliary system and
is a
function of the yaw angle
ψ
and the roll angle
φ
.
! "
ψ#
φ
ψ#
φ
ψ#
φ
ψ#
φ
φ
φ$ (4)
%
&
%
'
(5)
where#%
&
and %
'
are the lateral and vertical displacement of
the wheelset respectively.
In the local reference system the wheelset function is derived
by a generative function that represents half of the wheelset
axle. The generative profile of the wheelset ()
with P8
wheel profiles on each wheel is shown in Fig. 3.
The position of a point on the axle local reference frame can
be represented as follows
*
+)
! *
)
,()
*
-
$ (6)
Fig. 3 Wheelset generative function
Similarly, the position of the same generic point on the
wheelset with respect to the auxiliary reference system is;
*
+)
.
*
+)
!/
0
1
$ (7)
The generative rail function is plotted in Fig. 4 with BS 113A
rail profiles on both rails. Also a zoomed in portion of the
wheel-rail profile is shown in Fig. 5.
Fig. 4 Rail generative function
Fig. 5 Zoom-in portions for right BS 113A and P8 profile
In the auxiliary system, the coordinates of the point on the rail
can be expressed as:
*
+)
*
)
2)
(8)
2.2 Minimum Difference Method
The minimum difference method is used in order to simplify
and further improve the computational burden associated with
the minimum distance method. The main motivation for using
this method is that the contact points in the wheel and rail
surfaces minimize the difference between the wheel-rail
contacts in the direction of the unit normal vector C
a
.
Fig. 6 Minimum difference (right wheel-rail contact)
The minimum difference method definition is illustrated in
Fig. 7 where C =
and D =
3*
+)
*
+)
*
+)
45
(9)
Where
*
+)
! /
0
60
$ (10)
For each wheel-rail contact the difference 3*
+)
is a
function of two variables *
and#)
. The contact points can
thus be found by solving synchronously for the two variables
using numerical optimization technique such as Simplex
method. For real-time simulations it is preferable to reduce
Equation (9) to one dimensional form in one variable )
which can easily be solved numerically. Introducing the
definitions of the contact positions for the wheel and the rail
in the respective frames in Equation (9) it yields [8,9];
3*
+)
%
'
7
8
4
*
+)
2%
&
7
4
*
+)
(11)
Where a
11
to a
33
is equivalent to the rotation matrix variables
defined in Equation (5) and 7
9
:
+7
:
and 7
8
:
are the transpose of
the row vectors of column [A
2
].
!7
9
:
7
:
7
8
:
$7
99
7
9
7
98
7
9
7
7
8
7
8
7
88
(12)
Taking the partial derivatives of Equation (11) with respect to
the variables*
+)
, that
;<
;=
>
and
;<
;?
>
and equating it to
zero we will have two different representative equations.
Carrying out further reductions of these equations leads to a
quadratic solution of the variable *
with two roots.
Substituting the roots of the quadratic equation *
as a
function of )
and substituting into the second component of
the partial derivative of the difference, we have that the
following expression
@
9+
)
;<=
>
9+
;?
>
(13)
Equation (13) has now been reduced to a simple equation
with variable )
ranging from (692 ≤ )
≤ 815) mm for the
right wheel-rail contact geometry part and from (-815 ≤ )
≤ -692) mm for the left wheel-rail contact geometry. The
equation has two real values of *
and )
corresponding to
the twenty-one specified numerical points. The solution of the
variables must satisfy Equation (13). The following
indentation condition must be considered:
A
B
3
B
4C
B
D (14)
Note that from Equation (11) the vertical displacement u
z
adds to the difference equation. Taking the partial derivatives
of Equation (11) eliminates u
z
. Hence the only parameters
used in determining minimum of the contact points depend on
three variables; φ roll angle, yaw angle ψ and lateral
displacement u
y
. (see Table 1)
Input
parameter Range Step
φ
(rad) -0.01 – 0.01 0.0005
ψ
(rad) -0.01 – 0.01 0.0005
u
y
(mm) -10 – 10 0.5
Table 1 Parameters for wheelset degree of freedom
The rolling radius difference function can be obtained from
the numerical simulations by substituting the values of
variables *
+)
into Equation (7). The constant u
z
from
the wheel-rail geometry description in Fig. 3 and Fig. 4 is
491mm. Extracting the third component of Equation (7) in
the vertical direction and subtracting the results obtained from
the nominal rolling radius given for the simulation as 460
mm, the rolling radius difference is realized in Fig. 8. For
simulation purposes, a look-up table showing the rolling
radius difference function and the contact angle function
which is a function is shown below. A distance of 0.5 mm
spacing was chosen and then piecewise cubic interpolation
was used to interpolate between the functions described.
From the simulations carried out for the BS 113A profile and
P8 wheel profile, 0.5 mm increments was sufficient enough to
carry out dynamic simulations. Due to the conformal nature
of the P8 profile the contact angle function [2] derived was
found by substituting values of the lateral co-ordinate
function into the derivative of the wheelset generative
function as follows
E7FGH7CI
J-
J?
>
K (15)
The rolling radius difference function and the contact angle
function is shown in Fig. 4 and Fig. 5. Note that since the
wheelset is symmetrical, hence the same calculation for the
contact angle occurs at the left wheel-rail contact where the
lateral co-ordinate )
is negative.
2.3 Normal contact problem
The normal contact forces [12] acting on the wheel-rail
contact patch depend on the axle load, wheelset mass and
contact angle. The wheelset is assumed to be on a straight
track with the axle load (W
load
) of 110kN. The mass (m) of
the wheelset is based on parameters for the British MK IV
vehicle [11]. The roll angle has very little effect on the
contact angle function since the values are really small and
they depend on the rolling radius difference function.
Fig. 7 Rolling radius difference function
Fig. 8 Contact angle function (Right wheel-rail contact)
Anyakwo et al [12] determined the normal contact forces
acting on the wheel-rail contact patch. These forces are used
to estimate the contact patch size dimensions based on Hertz
contact theory [4]. The simulated values for the wheel-rail
contact patch dimensions considering the wheelset in central
position is shown in Fig. 9.
Fig. 9 Wheel-rail contact patch at central position
In these cases it is necessary to calculate the Hertz radii of
curvature in order to determine the rolling radii. To minimize
the errors, the longitudinal radius of curvature is considered to
be a function of the contact angle and rolling radius difference
function as follows:
2
I
L
MNO#
θ
K (16)
Where R
x
is the longitudinal radius of curvature of the wheel
and R
is the rolling radius of the left and right wheel-rail
contact [14]. The normal contact pressure acting on the
contact patch is semi-ellipsoidal in shape with maximum
contact pressure occuring at the centre of the elliptical contact
patch. It can be calculated as thus;
P
8Q
R
SI
K
I
&
K
T
U4V
(17)
Where x and y are the co-ordinates of the wheel-rail contact
patch in the longitudinal and lateral directions respectively
and N is the normal contact force.
2.3 Tangential Contact Problem
The tangential contact problem involves calculating the creep
forces that are developed in the wheel-rail interface as a result
of braking, traction and acceleration. The creepages (lateral,
longitudinal and spin) are used to calculate the creep forces
acting on the contact patch using Kalker’s linear coefficient
tables [15]. The creepages for a dynamic wheel-rail contact
can be calculated as follows;
Lateral creepage (right/left wheel-rail contact)
W
J&
XJ
ψ
(18)
Longitudinal creepage
WYZ[BZ\
WJ
ψ
XJ
λ
&
L
]
(19)
WYZW^_
WJ
ψ
XJ
λ
&
L
]
(20)
Spin creepage
`aB[BZ\
λ
L
]
J
ψ
XJ
(21)
`aBW^_
λ
L
]
J
ψ
XJ
(22)
Where
λ
is the effective conicity of the wheel profile. For
new P8 wheel profile the equivalent conicity is non-linear due
to the wheel profile design. The effective conicity can be
expressed as a function of the lateral displacement and can be
calculated as follows;
λ
0
LL<
&
(23)
Where RRD is the rolling radius difference function expressed
as;
RRD(y) 2
[
02
W
0 (24)
R
r
and R
l
represent the rolling radius of the right wheel and
left wheel-rail contact respectively and y is the lateral
displacement. V is the forward velocity of the wheelset and R
0
is the nominal rolling radius difference.
The creep forces are determined using Kalker’s theory. For
small creepage values, the creep force/creepage ratio is linear.
This implies that the creep forces increase linearly as the
creepage increases. For large creepages, the creep forces must
be limited by applying Coulomb’s maximal saturation law. At
this region the creep/creepage ratio is highly non-linear. This
occurs at flange contact. Kalker’s linear theory calculation at
the saturation region is generally non-linear. Heuristic non-
linear creep force model can be used to calculate the limiting
values that occur at the wheel-rail contact patch in this case.
The saturation constant d [13] is used to limit the creep forces
calculated via Kalker’s contact theory:
bc
9
η
dI
η
9
8
η
9
e
η
8
Kf##########
η
Dg
9
η
+#################################################
η
hgi (25)
where
η
S
jk
l
mk&
l
n
]4o
pq
T (26)
where µ represents the coefficient of friction (0.3) and N is
the normal contact force acting on the contact patch. η is the
unlimited normalized creep force ratio while F
x
and F
y
are the
longitudinal and lateral creep forces developed at the wheel-
rail interface. Normalized re-calculated creep forces at the
saturation region can be defined as thus:
r
br
(27)
r
&
br
&
(28)
s
'
bs
'
(29)
where F
x
, F
y
represent the longitudinal, lateral creep forces.
M
z
is the spin creep moment. All the creep forces depend on
the Kalker’s linear creep coefficient, shear modulus of
rigidity and Kalker’s linear coefficients [12].
2.4 Wheelset dynamic behaviour
The wheelset dynamic behaviour of a straight track can be
investigated by summing all the creep forces and normal
contact forces generated at the wheel-rail contact patch. The
summation of the total creep forces plus the forces generated
as a result of longitudinal shift variation of the wheelset as it
moves on the track [13]. Details of calculation of the normal
contact vertical forces and moments as a result of the
longitudinal variation can be found in [12]. For simulation
purposes the following parameters based on British Mark IV
coach is displayed in [11], [13]. The suspended parameters
used for the simulation can be found in [13].
3 Numerical Simulation results
The two degree of freedom differential equations are solved
using numerical differentiation. Runge Kutta’s fourth other
method was used to solve the equations for initial value inputs
of the lateral displacement and the yaw angle. Initial input
parameter for the lateral displacement is given as y = 7.5mm
thats just at flange contact. The yaw angle input variable from
the wheelset geometry is given as 0.00125 radians.
To illustrate dynamic simulation of the wheelset on a straight
track, forward velocity V = 2.5m/s was used. The response of
the suspended wheelset on the track shows that the wheelset
stabilized after 15 second for both lateral displacement and
yaw angle as shown in Fig. 10 and Fig. 11. The wavy
sinusoidal response of the wheelset for a given lateral
displacement and yaw angle indicates that the conicity plays a
vital role in the critical stability of the wheelset.
Fig. 10 Simulated lateral displacement response for V =
2.5m/s
Fig. 11 Simulated yaw angle response for V = 2.5m/s
The non-linear conicity function in Equation (23) was used to
for dynamic analysis. For an improved dynamic response of
the single wheelset, the conicity function needs to be
linearized to obtain the equivalent conicity function. Several
methods exist in literature for linearizing the conicity
function. In this paper, the main motivation was towards
simulating the dynamic behaviour of the track. Future work
would concentrate on obtaining the equivalent conicity
function that would be used for railway vehicle dynamic
simulations and for condition monitoring applications where
the wheel profile parameters and the forces could be
estimated.
4 Conclusions
This paper studies the development of a novel 3-D wheel-rail
contact model which is used for dynamic simulation of a
suspended wheelset with parameters listed for a typical Mark
IV coach. The contact point locations on the wheel and rail
are determined by the minimum difference method
considering the lateral displacement, yaw angle and the roll
angle. The proposed new 3D wheel-rail contact model
accurately replicate the dynamic behaviour of rail-wheel
interface and can be employed in railway condition
monitoring techniques to estimate the wheel geometry
parameters and achieve optimised wheel-rail interfaces
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