Journal of Transportation Technologies, 2012, 2, 150-157
http://dx.doi.org/10.4236/jtts.2012.22016 Published Online April 2012 (http://www.SciRP.org/journal/jtts)
Lateral Stability Analysis of Heavy-Haul Vehicle on
Curved Track Based on Wheel/Rail Coupled Dynamics
Kaiyun Wang, Pengfei Liu
Traction Power State Key Laboratory, Southwest Jiaotong University, Chengdu, China
Received December 16, 2011; revised February 7, 2012; accepted March 1, 2012
ABSTRACT
Being viewed from the standpoint of whole system, the hunting stability of a heavy-haul railway vehicle on a curved
track is investigated in this paper. First, a model to simulate dynamic performance of the heavy-haul vehicle on the
elastic track is developed. Secondly, the reason of the hunting motion is analyzed, and a bifurcation diagram for the
vehicle on the curved track is put forward to simulate the nonlinear critical speed. Results show that the hunting motion
of the heavy-haul vehicle will appear due to the larger conicity, the initial lateral shift and the wheelset angle of attack.
With the hunting motion appearing, the lateral shift and force of the wheelset are changed sharply and periodically with
a wave of circa 3.6 m. There is obvious difference in the bifurcation diagram between on a curved track and on a tan-
gent track. Relative to the centerline of the track, each vehicle body on the curved track has two stable cycles. As for the
curved track with a radius of 600 m and a superelevation of 55 mm, the nonlinear critical speed of the heavy-haul vehi-
cle is 76.4 km/h.
Keywords: Hunting; Stability; Curved Track; Heavy-Haul Railway; Coupled Dynamics
1. Introduction
Developing heavy-haul railways is one efficient measure
to increase the transport volume. However, being re-
stricted to the landform in the heavy-haul railway, there
are many sharp curved tracks with the radius ranging
from 500 m to 1000 m. When a heavy-haul vehicle is
negotiating these curved tracks with low speeds, the phe-
nomenon of the hunting motion, usually taking place on
the tangent track, will also appear. Due to the hunting
motion on the curved tracks, the interaction force be-
tween the wheel and the rail is seriously enhanced, the
performance of the negotiation is severely deteriorated,
the rail is sharply worn, the rail life is clearly shortened,
and even the vehicle will derail on the curved track. So,
much attention should be paid to the vehicle hunting mo-
tion on the curved track.
Studies on the running stability of the railway vehicle
have been performed for almost half of century. The first
bifurcation analysis of the free running wheelset was
carried out by Huilgol [1] and in this research a hopf bi-
furcation from the steady state was revealed. The first
observation of chaotic oscillations in models of railway
vehicles was carried out by True et al. [2] and Petersen
[3]. Dukkipati [4] developed the mathematical linear
models to determine the lateral stability or hunting of
North American standard three-piece freight truck on
track/roller stands, and the theoretical model results were
compared with field test data performed by the Associa-
tion of American Railroads (AAR). In the meanwhile,
using linear models, a comparative study on the dynamic
stability and steady state curving behavior of some un-
conventional railway truck designs was carried out by
Dukkipati [5]. Further works demonstrating the hunting
stability of high-speed vehicle were carried out by Lee
and his team [6,7], including the stability on the tangent
track [6] and the stability on the curved track [7]. Lee et
al. [8] modeled eight degrees of freedom (DOFs) for
truck system moving on curved tracks, and it was re-
ported that the critical hunting speeds evaluated by using
the eight-DOF system differed significantly from those
done by the six-DOF system.
Unfortunately, up to now, there is little study about the
lateral stability of the heavy-haul vehicle on the curved
track, let alone the stability research based on the cou-
pled model between the vehicle and the track.
Therefore, being viewed from the standpoint of the
coupled system presented by Zhai et al. [9,10], investiga-
tion efforts will be focused on the nonlinear hunting sta-
bility of the heavy-haul vehicle on the curved track in
this paper. Firstly, a model to simulate the dynamic per-
formance of the heavy-haul vehicle on the elastic track is
established. Secondly, the reason of the hunting motion
on the curved track is analyzed. Lastly, a bifurcation dia-
C
opyright © 2012 SciRes. JTTs

K. Y. WANG ET AL. 151
gram for the vehicle on curved track is put forward to
simulate the nonlinear critical speed.
2. Dynamic Model of Heavy-Haul Vehicle on
Curved Track
A curved track contains three main elements such as ra-
dius, cant, and gauge. When a heavy-haul vehicle with
full freight passes through the small-radius curved track,
the interaction between the wheel and the rail is very
serious, as causes the track vibrating obviously. Figure 1
indicates a measured result [11] of dynamic gauge en-
largements on a small-radius curve track. It can be seen
from the Figure 1 that the dynamic gauge enlargement is
very clear and the maximum value is close to 3.5 mm.
Therefore, for an analysis on the vehicle dynamic per-
formance, including the stability on the curved track, it is
very necessary to take the dynamic vibration of the track
into account.
2.1. Coordinate Systems
The descriptions of the configuration and the orientation
of the railway vehicle on the track are related to the defi-
nition of coordinate systems. In order to describe the
absolute motion of the heavy-haul vehicle on a curved
track, two coordinate systems are needed: the inertial
coordinate system and the body fixed coordinate system.
Only taking a wheelset on a curved track for an example,
the coordinate systems are shown in Figure 2, where the
inertial coordinate system (Oi, Xi, Yi, Zi) is located on the
center line of the track and can not move with the
wheelset, and the wheelset fixed coordinate system (Ow,
Xw, Yw, Zw) is located in the mass center of the wheelset
and can move with the wheelset. In addition, the positive
directions of y in both systems are toward the inner rail
of the curved track or the right rail of the tangent track.
Relative to the inertial system, any of the vehicle body
has six DOFs, including three DOFs of transfer motions
in longitudinal (X), lateral (Y), and vertical (Z), three
45 50 55 60 65 70
0
1
2
3
4
5
Dynamic gauge enlargement (mm)
Speed (km/h)
Figure 1. Measured result of dynamic gauge enlargements
on the small-radius curve track.
Figure 2. Coordinate systems on curved track.
DOFs of a pitch motion (
 ), a rolling motion (  ), and a
yaw motion (
 ). The absolute motion of the body of the
vehicle is the vector sum of the transfer and the rotation
motions. The relationship between the inertial system
and the body fixed system is shown in Figure 3 [12], in
which, p stands any point of a body, the vector rp (Oi, Xi,
Yi, Zi) in the inertial system stands for its static and spa-
tial position, the vector rp (OB, XB, YB, ZB) in the body
fixed system stands for its dynamic and spatial position
with respect to the inertial system,
 rc is the transfer
vector in the body fixed system with respect to the iner-
tial system, and
 ri is the absolute vector of a body.
So,
 ri is calculated by:
icBpp

 Arr rr (1)
in which, AB is the rotation matrix in the body fixed sys-
tem with respect to the inertial system and is formed as
following:
101
11
01 1
BB
BB
BB

B


 



 

 
 

 

 


 
A
where,
 is the angle difference of two coordinate sys-
tems caused by the superelevation (around X axis),
 is
the angle difference of two coordinate systems (around Z
axis), in addition, these parameter values are determined
by the line type of the curved track.
According to the Equation (1), all the absolute vectors
of vehicle bodies can be calculated. If a body A and a
body B are conjoined by a suspension, the relative dis-
placement of the suspension points between A and B is
given by:
ABB A

 rrr (2)
in which, the
 rA and  rB stand for the absolute vector of
body A and body B, respectively.
Therefore, the relative velocity of the suspension points

K. Y. WANG ET AL. 152
Figure 3. The vector of body with respect to the inertial
system.
between A and B is also given by:

A
d
d
B
AB t

 r v (3)
If the stiffness matrix and the damping matrix in this
suspension are K and C, respectively, the force of the
suspension points is calculated as following:
AB AB
 FKr Cv (4)
2.2. Model of Heavy-Haul Vehicle
Figure 4 shows a three-piece structure of a heavy-haul
vehicle which includes two side frames and one bolster.
Nowadays, the three-piece structure is widely applied in
the heavy-haul vehicle in CR. In order to reduce the un-
sprung mass, the elastic rubber is set in the axle-box, as
is the primary suspension. Apart from this, the compo-
nents of the structure are similar to those introduced by
Xia [13].
The vehicle is modeled as a multi-body dynamic sys-
tem, as is shown in Figure 5. In Figure 5, Z, Y and Φ
denote the DOFs of vertical, lateral and roll motions of a
component; subscripts c, t and w represent carbody, side
frame and wheelset; subscripts L and R denote the left
and right side respectively; Ksz and Csz are the vertical
stiffness and damping of the secondary suspension; Ksy
and Csy are the lateral stiffness and damping of the sec-
ondary suspension; Kpz and Cpz are the vertical stiffness
and damping of the primary suspension; and Kpy and Cpy
are the lateral stiffness and damping of the primary sus-
pension.
There are eleven principal components, including one
carbody, two bolsters, four side frames, and four wheel-
sets. It is shown that the carbody is connected to the bol-
ster via the center plates modeled as a spherical joint.
Because the radius of the spherical joint is small com-
pared with the other dimensions of the carbody, and the
effect of the friction produced by the spherical joint on
the roll rotation of the carbody and the bolster is ne-
glected. But the effect of the friction torque on the yaw
rotation is still considered. In the secondary suspension,
the bolsters are elastically connected to the side frames
by springs vertically, and a one-dimension model without
Centerplate
Whee
l
Axl e
Primary suspension
Secondary suspension
Side frame
Bolster
Figure 4. The construction of the he avy-haul vehicle in CR.
Φ
c
Φ
w
Yc
YtL
Yw
Z
Zc
ZtL
Ksy
C
syK
szC szYtR
ZtR
Cpy
w
K
py
Kpz
Cpz
Figure 5. Description of heavy-haul vehicle dynamics model
with end view.
stick mode is applied to simulate the dry friction in the
wedge damper system. The primary suspension is mod-
eled as the parameters of Kp and Cp in longitudinal, lat-
eral and vertical.
All of components are assumed to be rigid. The DOFs
of a heavy-haul vehicle are given in Table 1, where the
total DOFs are 47.
2.3. Model of Track
A five-parameter model [9,10] is adopted to model the
railway track, as is shown in Figure 6. The rail is mod-
eled as a Bernoulli-Euler beam discretely supported at
masses. The three layers of discrete springs and dampers
represent the elasticity and damping effects of the rail
fastening, the ballast, and the subgrade respectively.
2.4. Model of the Wheel-Rail Contact
The wheelsets provide the supports for the entire vehicle
and supply the contact forces that keep the vehicle sys-
tem on the track. A new model of three-dimensional
geometrical contact between wheel and rail [10,14] is

K. Y. WANG ET AL. 153
Table 1. DOFs of a heavy-haul vehicle.
Component Long. Lat. Vert. Roll Pitch Yaw
Carbody - Yc Zc 
c

c

c
Bolster
(i = 1, 2) - - - - -

Bi
Front side frame
(i = 1, 2) XtFi YtFi ZtFi -

tFi

tFi
Rear side frame
(i = 1, 2) XtRi YtRi ZtRi -

tRi

tRi
Wheelset
(i = 1 - 4) - Ywi Zwi

wi

wi

wi
Figure 6. Description of a five-parameter track model.
adopted. All the contact parameters are calculated online,
including the contact point and its curvature, the conicity,
and so on. The normal force of wheel-rail contact is de-
scribed by a non-linear Hertzian contact theory, and the
tangential force is calculated by a Shen-Hedrick-Elkins
formula [15].
3. Reason of Hunting Motion of Heavy-Haul
Vehicle on Curved Track
When a heavy-haul vehicle negotiates a curved track at a
low speed, some unbalanced centrifugal forces will ap-
pear in the vehicle system due to the multiple effects
such as speed, curve radius, superelevation, and so on.
Under the action of the unbalanced centrifugal forces,
wheelsets will deviate from the track centerline. Both the
theoretical and experimental results [11] show that the
lateral shifts of wheelsets are bigger than 4 mm usually.
Figure 7 illustrates the theoretical result of the conic-
ity for the wheel tread of LM and the rail profile of the
heavy-haul railway. During the curved negotiation, the
lateral shifts of wheelsets are bigger than 4 mm. So the
conicity value is more than 0.1, as can help to improve
the self-steering ability of the wheelset. It is worth point-
ing out that the conicity value of the curved track is big-
ger than the value of the tangent track. In addition, as for
the vehicle system on the curved track, the lateral shift is
a kind of outside excitation, and the wheelset centerline
is not under the radial situation, and the wheelset angle of
attack, i.e. the angle between the wheelset axis and the
radius of the curve, will appear on the wheelset.
Consequently, under the effects of the large conicity,
Figure 7. Conicity versus lateral shift of wheelset.
the excitation of the lateral shift, and the wheelset angle
of attack, the hunting motion appears easily for the
wheelsets of a heavy-haul vehicle. Especially for the
wagons assembled with the worn components and the
worsened suspensions, the hunting stability on curved
track is much worse.
Furthermore, a numerical example of the hunting mo-
tion of a heavy-haul vehicle with no freight is given, in
which, the radius is 600 m, the superelevation of outer
rail is 55 mm, the negotiation speed is 85 km/h, the tran-
sitions are made of parabolic curves, and other dynamic
parameters are evaluated [10,16]. The calculated results
of the lateral shift of the wheelset are presented in Figure
8, including the time and frequency domain results. It can
be seen from Figure 8(a) that the phenomenon of the
hunting motion appears at the point of spiral to curve, at
the point of curve to spiral, and in the whole circular line,
and there is lateral and periodic oscillation on the wheel-
set. Under this situation of the hunting motion, the
dominant frequency of the lateral shift is circa 6.5 Hz, as
is shown in Figure 8(b).
According to the following formula
v
f


(5)
in which,
 stands the wave length, v is the vehicle speed,
and f is the dominant frequency.
So, the hunting wave length is circa 3.6 m corre-
spondingly, which is quite close to the hunting wave
length on a tangent track [17].
Figure 9 shows the calculated results of the wheelset
lateral force under this condition of the hunting motion.
It is noticeable that the lateral interaction forces between
the wheel and the rail change severely and fluctuate pe-
riodically when the phenomenon of the hunting motion
of the wheelset appears. The main reason is that, due to
the effects of the lateral shift and wheelset angle of attack,
the severe impact contact may happen between the root
of the wheel flange and the rail side.

K. Y. WANG ET AL. 154
Lateral shift of wheelset (mm)
(a)
110
0.0
0.2
0.4
0.6
0.8
1.0
100
Amplitude
Frequency (Hz)
6.49Hz
(b)
Figure 8. Calculated results of lateral shift of wheelset on
curved track, (a) Response in time domain, (b) Response in
frequency domain.
0100 200 300400 500
-4
0
4
8
12
16
Wheelsets
Rail
Contact
point
Lateral force of wheelset (kN)
Distance (m)
Figure 9. Calculated results of wheelset lateral force.
Figure 10 shows the lateral displacement of outer rail
under this case. It can be seen that there is lateral and
periodic oscillation on the outer rail during the hunting
motion of the wheelset. However, due to the centrifugal
forces, the rail will not go back to its resting position.
4. Bifurcation Diagram for Heavy-Haul
Vehicle on Curved Track
The results [16,17] have proved that once the hunting
motion of a vehicle occurs on a tangent track, the wheel-
set will have a lateral and periodic motion, and there is a
stable limit cycle relative to the track centerline. The
lateral stability characteristic could be described as an
“S” shape curve, as is shown in Figure 11 [2,16,17],
Figure 10. Calculated results of lateral displacement of
outer rail.
Amplitude of limit cycle
Figure 11. Typical bifurcation diagram for nonlinear vehi-
cle system on tangent track.
where the solid line and the dashed line represent the
stable and the unstable limit cycle, respectively.
Whereas, according to the calculated results mentioned
above, when the hunting motion occurs on a curved track
for the wheelset, there is a lateral and periodic oscillation
deviating from the centerline of the track. Meanwhile,
due to the deviation of the wheelset from the track cen-
terline, two stable limit cycles relative to the centerline of
the track can be observed simultaneously, as is shown in
Figure 12. One stable limit is defined as “large ring”,
which represents the motion state of the wheelset with
the largest amplitude of the lateral shift. The other stable
limit is defined as “small ring”, which describes the mo-
tion state of the wheelset with the smallest amplitude of
the lateral shift. Besides, the dashed line in Figure 12
represents the boundary between the “large ring” and the
“small ring”.
So, on the basis of the typical bifurcation diagram on
tangent track (shown in Figure 11) and the motion char-
acteristics on a curved track (shown in Figure 12), the
bifurcation diagram of nonlinear vehicle system on a
curved track can be obtained, as is shown in Figure 13,
where the solid lines (CD and C′D′) and the dashed lines
(BD and BD′) also represent stable and unstable limit
cycle, respectively, and the horizontal line (AB) indicates
the equilibrium position of the system deviating from the

K. Y. WANG ET AL. 155
-11-10-9-8-7 -6
-0.10
-0.05
0.00
0.05
0.10
small ring
Wheelset lateral velocity (m/s)
Wheelset lateral shift (mm)
large ring
Figure 12. Wheelset lateral shift versus velocity with hunt-
ing motion.
D'
C
C'
VDVB
D
B
A
O
Amplitude of limit cycle
Train speed
Figure 13. Bifurcation diagram for nonlinear vehicle system
on curved track.
track centerline.
In Figure 13, the abscissa denotes the train speed; the
ordinate denotes the amplitude of limit cycle vibration
depending on the system disturbances, e.g., the amplitude
of the lateral displacement of the leading wheelset. When
the train speed V is less than VD, the system vibration will
stabilize to the equilibrium position for any external dis-
turbances, as is similar to the diagram on tangent track
(shown in Figure 11). When the train speed V is larger
than VB, two stable limit cycles will be observed no mat-
ter what amplitude of the external disturbance is. How-
ever, in the interval VD < V < VB, the situation of system
vibration will depend on the amplitude of the external
disturbance.
It also can be seen from Figure 13 that, given an ex-
ternal disturbance with a large disturbance and an initial
speed being lower than VD, if the vehicle speed is in-
creased little by little, two stable limit cycles will appear
simultaneously when the speed is VD, and the ordinate
values at points D and D′ are the amplitudes of the limit
cycles. On the contrary, given an external disturbance
with a large disturbance and an initial speed being higher
than VB, if the vehicle speed is reduced step by step, two
stable limit cycles will disappear immediately when the
speed is VD and the system vibration will stabilize to the
equilibrium position at points D and D′.
So, according to the bifurcation diagram for nonlinear
vehicle system on the curved track, the speed VD at
points D and D′ is called the nonlinear critical speed, and
the speed VB at point B is called the linear critical speed.
5. Nonlinear Critical Speed of Heavy-Haul
Vehicle on Curved Track
In order to calculate the nonlinear critical speed of a
heavy-haul vehicle on a curved track accurately, the points
D and D' in Figure 13 should be found out. Therefore,
the “Speed Reducing Method” (SRM) put forward in
paper [17,18] is adopted here. It is noticeable that the
value of the initial external disturbance should be more
than 8mm for a curved track.
Using the SRM, the nonlinear critical speed of a
heavy-haul vehicle on a curved track can be calculated.
Taking a typical heavy-haul vehicle running on a curved
track in CR for an example, the dynamic parameters are
the same as those referred above, the radius of the curved
track is 600 m and the superelevation is 55 mm. Figure
14 shows the dynamic response of the lateral shift of the
wheelset when the speed is reduced from 140 km/h to 42
km/h gradually. It can be seen from Figure 14 that, at
speeds ranging between 140 km/h and 76.4 km/h, there
are the lateral and periodic oscillation on the wheelset
and there are two limit cycles with amplitudes of about
10 mm and 7 mm respectively, and when the speed is
76.4 km/h, the lateral shift of the wheelset will stabilize
to equilibrium position suddenly. So, it can be deduced
that the nonlinear critical speed of the vehicle is 76.4
km/h under these given situations.
On the basis of the research experience and results,
Wang [19] has pointed out that the ratio of the nonlinear
critical speed to the largest operating speed should be
more than 1.2. Consequently, for a curved track with a
radius of 600 m and a superelevation of 55 mm, the
maximum negotiation speed of this heavy-haul freight
140 120 10080604
0
-12
-10
-8
-6
-4
-2
0
Stabilization point
Equilibrium position
Reduced speeds
Lateral shift of wheelset (mm)
S
p
eed
(
km/h
)
Periodical position
Figure 14. Wheelset lateral shift versus speeds of heavy-
haul vehicle on curved track.

K. Y. WANG ET AL. 156
wagon should be 61 km/h. However, according to the
code for design of railway line [20], it is prescribed that
the maximum negotiation speed of a freight truck on this
curved track is 65 km/h. Moreover, the practical opera-
tion experience also proves the fact that the negotiation
speed usually reaches 65 km/h on this curved track. Thus,
when a freight vehicle passes through this curved track
with a speed of 65 km/h, the nonlinear critical speed of
the vehicle should be more than 76.4 km/h. That is to say,
the lateral stability of this type vehicle cannot meet the
real requirement for the curved track, and the hunting
motion appears usually.
Furthermore, compared with the results in paper [17]
that the nonlinear critical speed of this heavy-haul vehi-
cle on a tangent track is 134 km/h, it can be concluded
that the nonlinear critical speed of a vehicle on a curved
track is lower than that on a tangent track. In other words,
the performance of the lateral stability on a curved track
is worse than that on a tangent track.
6. Conclusions
1) Due to effects of various factors such as the large
conicity, the excitation of the lateral shift and the wheel-
set angle of attack, the hunting motion appears easily
when a heavy-haul vehicle negotiates a curved track at a
low speed. Under this situation of the hunting motion,
there are two stable limit cycles relative to the track cen-
terline, as is different with the phenomenon on a tangent
track where there is one stable limit cycle.
2) The nonlinear critical speed of the heavy-haul vehi-
cle on the curved track can be calculated by the SRM. As
for the curved track with a radius of 600 m and a su-
perelevation of 55 mm, the nonlinear critical speed of the
heavy-haul vehicle is 76.4 km/h, which is lower than the
speed on the tangent track.
3) Finally, for the curved track, much attention should
be paid to the lateral stability, as well as the running
safety.
7. Acknowledgements
The authors wish to acknowledge the support and moti-
vation provided by National Natural Science Foundation
of China (51075340) and Tong Education Foundation for
Young Teachers in the Higher Education Institutions
(121075) and Program for Innovation Research Team in
University in China (No. IRT1178).
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