Simulation Modelling of the Railway Vehicle Guidance Mechanism

SYSTEMS DIVISION RESEARCH 2013 1
Simulation Modelling of the Railway Vehicle
Guidance Mechanism
Priya Parthasarathy, Dr. Christopher Ward, Dr. Roger Dixon
Abstract—This paper presents the initial stage of work
carried out on the simulation of a simplified rail vehicle model.
The stability analysis of the wheelset with a change in the
lateral track position has also been discussed. It puts forward
the significance of yaw suspension stiffness on the dynamic
stability of the wheelset. The results of the simplified rail
vehicle modelling and a brief idea about the future work have
been analysed.
I. INTRODUCTION
The initial phase of the work was to develop a simple rail
vehicle model to understand the dynamics of a rail vehicle
system. This involved simulating a linearised vehicle model
with a single wheelset and a suspended mass. The wheelset
is modelled in the lateral and yaw direction and the
suspended mass in the lateral direction. The lateral position
of the wheelset and the mass suspension and the yaw angle
of the wheelset have been determined for a 10mm step
change in the lateral position of the track.
II. BACKGROUND
The low adhesion at the wheel rail contact causing huge
problems to the railway network is one important aspect to
be outlined. This project puts forward the real time
estimation of low adhesion conditions with the use of
advanced monitoring techniques.
The first step of this work is to develop simulation
models of a simple rail vehicle and increasing the
complexity levels from simple linear models to nonlinear
vehicle models as the project progresses.
III. INITIAL STAGE OF WORK
A linearised rail vehicle model comprising of a single
wheelset and a suspended mass as shown in Fig 1[1] have
been simulated to test the stability of the wheelset in
response to the track displacement. The wheelset equations
are a combination of suspension forces, creep forces
generated at the wheel-rail contact and gravitational stiffness
generated by the geometry of the wheel and rail.
Fig 1 shows the simplified rail vehicle model with a single
wheelset and a suspended mass. The suspension forces and
the yaw moments, the lateral displacement of the wheelset
and the mass are also shown in the figure below.
It can be seen that there is a 10mm step change in the track
position for which the lateral positions and the yaw angle
are observed.
10mm
Wheelset
fΨ
Ym
kΨ
y
ky
fy
Suspended
mass
Track
10mm
Fig 1 Diagram of the simplified rail vehicle model with a single wheelset
and suspended mass
The differential equations of the lateral position and yaw
angle of the wheelset and the mass lateral position are as
shown below.
y = 1 m − 2f !!
v
ψ = 1 I
−2f !!
y + r !λ
l
lλ
r !
− Wλ
l
y − r !λ
l
d − vψ − 2f !"
v
ψ
y − d) + F !" (1)
y − d + l! v ψ
+ 2f !"
v
− 2f !"
v
y − r !λ
l
y + r !λ
d − vψ
l
ψ + lWλψ − I !"vλ
lr !
y + I !"vλ
d
lr !
+ M !" (2)
y ! = !
! !
−F !" (3)
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SYSTEMS DIVISION RESEARCH 2013 2
The suspension forces and the yaw moments are given as
F !" = k ! y ! − y + f ! y ! − y (4)
M !" = −k ! ψ − f ! ψ (5)
A complete list of the parameters and states used in
equations 1-5 are given in Table 1.
Parameters Values Units
f 11
f 22
f 23
f 33
f y
fψ
k y
kψ
m
mm
I
Iwy
l
v
W
r0
F sy
M sψ
y
y m
ψ
λ
d
Longitudinal creep coefficient
Lateral creep coefficient
Lateral/spin creep coefficient
Spin creep coefficient
Lateral damper coefficient
Yaw damper coefficient
Lateral suspension stiffness
Yaw suspension stiffness
Wheelset mass
Suspended mass
Wheelset yaw inertia
Wheelset roll inertia
Half wheelset width
Velocity
Wheelset width
Nominal rolling radius
Suspension lateral force
Yaw moment
States
Wheelset lateral position
Suspended mass lateral position
Wheelset yaw angle
Conicity
Lateral track displacement
7.44e6
6.79e6
13.7e3
0
50e3
0
0.23e6
2.5e6
1250
8000
700
200
0.7452
20
12263
0.45
-
-
-
-
-
-
-
N
N
N
N
Ns/m
Ns/rad
N/m
N/rad
kg
kg
kg/m 2
kg/m 2
m
m/s
N
m
N
Nm/rad
m
m
rad
m
Fig 3 Pole Zero plot to show the stability of the wheelset for a step change
in the track position
It is clear that the wheelset shows a stable response for a
10mm lateral track displacement. This can be verified from
fig 3 as the poles are located on the left half of the s-plane.
The pole zero plot was obtained by state space analysis of
the rail vehicle model to determine the stability of the
system.
Additionally, the effect of primary yaw stiffness (k ψ) on the
stability of the wheelset has been studied. This was
performed by varying the value of the primary yaw stiffness
and determining how the stability level varies with the
change in the stiffness. Initially, the value of yaw suspension
stiffness was assigned as 2.5x10 6 Nm, which when the
system remained stable with a single oscillation damping
out quickly as shown in fig 2.
When the stiffness was reduced to a very low value of 2500
Nm, the response became unsteady and the wheel sets were
unstable. This can be made evident from fig 4 below.
Table 1 Parameter values and states used in the simulation model
Figure 2 shows the lateral position of the wheelset and the
mass and yaw angle for a step change of 10mm in the track
at a step time of 1 second. This shows that the position of
the wheelset and the suspended mass follows the step
change in the track position.
Fig 4 Lateral position of the wheelset and mass, yaw angle for the change
in primary yaw stiffness of 2.5x10 3 Nm
The above result can be further analysed using the pole zero
analysis, which is as shown in fig 5.
Fig 2 Lateral position of the wheelset and mass, yaw angle for the
corresponding track displacement
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SYSTEMS DIVISION RESEARCH 2013 3
be verified using the pole zero analysis as in fig 7.
It can be understood from the above results that the change
in the primary yaw stiffness can have influence in the
stability of the wheelsets.
Fig 5 Pole Zero plot shows the instability of the wheelset for the yaw
stiffness of 2.5x10 3 Nm
The response of the system was again observed by
increasing the value of primary yaw stiffness to 2.5x10 9 .
This has been carried out to examine the system’s stability
for a massive range of stiffness.
IV. FUTURE WORK
The simplified rail vehicle modelling has been followed by
the simulation of a linearised half vehicle model consisting
of two wheelsets, a bogie and a half vehicle body. The next
phase of the project will be increasing the complexity in the
vehicle suspension system and the contact mechanics in a
non-linear vehicle model. One of the several advanced
techniques will be applied for the condition monitoring of
the railway vehicle.
V. CONCLUSION
In order to understand the dynamic interaction between
wheel and rail, the initial stage of the project was to design a
simplified model of a rail vehicle. The results produced
from the single wheelset and suspended mass modelling
were as expected and the stability of the wheelset with the
step change in the track lateral position have also been
examined. An estimator model will be developed for the real
time estimation of low adhesion at the wheel rail interface.
Advanced filtering techniques will be analysed for the
estimation of rail vehicle dynamics.
ACKNOWLEDGEMENT
This work is being funded by Engineering and Physical
Sciences Research Council of UK (EPSRC).
Fig 6 Wheelset and suspended mass response for the varied primary yaw
stiffness of 2.5x10 9
REFERENCES
[1] Charles, G., Goodall, R., and Dixon, R. (2008b) “A Least Mean
Squared Approach to Wheel Rail Profile Estimation,” in Proceedings
of the 4 th International Conference on Railway Condition Monitoring.
[2] Charles, G., Goodall, R., Dixon, R., Model based condition
monitoring at the wheel rail interface Vehicle System Dynamics,
46(1), 415–430.
[3] Garg, V. and Dukkipati, R., Dynamics of Railway Vehicle System,
Academic Press, 1984.
[4] Iwnicki S. (Ed.) Handbook of railway vehicles dynamics, 2006, p. 535
(Tailor & Francis Group, Boca Raton, London, New York).
[5] Kalman, R.E, A new approach to linear filtering and prediction,
Tran.ASME.J.Basic Eng.82 (1960), pp.35-45.
[6] Ward, C., Goodall, R., and Dixon, R., Wheel rail profile condition
monitoring, in Proceedings of the UKACC control conference,
Coventry (2010a).
[7] Ward, C., Goodall, R., Dixon, R., and Charles, G.A., Adhesion
Estimation at the wheel rail interface using advanced model based
filtering Vehicle System Dynamics Vol. 00, No. 00, January 2012, 1–
23
Fig 7 Pole Zero analysis for the corresponding change in yaw stiffness
(2.5x10 9 )
The wheelset takes a very long time (say few minutes) to
follow the track’s lateral shift as shown in fig 6. The impact
was due to the increasing in primary yaw stiffness to a huge
range, making the system stiffer. However, the variation had
not affected the stability of the rail vehicle system. This can
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