Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Porto, Portugal, 30 June - 2 July 2014
A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)
ISSN: 2311-9020; ISBN: 978-972-752-165-4
An efficient numerical method for the vehicle-bridge nonlinear dynamic interaction
1
Zhibin Jin1, Xiaozhen Li1, Dejun Liu1, Shiling Pei2
Department of Bridge Eng., Southwest Jiaotong University, 610031, Chengdu, P.R.China
2
Department of Civil and Enviromental Eng., Colorado School of Mines, Golden, USA
email: [email protected], [email protected]
ABSTRACT: This study aims at providing an efficient and accurate method for the time history simulation of the vehicle-bridge
dynamic interaction. With the geometric nonlinearity in wheel-rail contact fully considered, a simplified assumption was
adopted that vertical wheel-load fluctuation has no influence on the lateral wheel-rail interaction. The wheel-rail creep forces are
computed using Kalker’s theory, and the normal forces are solved from wheel-set vertical and torsional equilibrium equations.
The method limited the number of nonlinear parameters for each wheel-set to five. In addition, mode superposition method is
adopted for bridge responses to minimize the overall system DOFs. The virtual work principle method is used to derive the
system matrix. Numerical examples show that the present method gives results comparable in terms of accuracy to that using a
more complex model. But it is at least 50 times faster than the detailed model. The efficiency and accuracy of the proposed
method will make it suitable in the reliability-based vehicle-bridge interaction analyses.
KEY WORDS: Vehicle-bridge; Dynamic interaction; Wheel-rail contact; Mode superposition.
1
INTRODUCTION
The rail way vehicles passing across a bridge arise vibrations
both to the bridge and vehicles. These vibrations are of great
concerns in bridge and vehicle design, and have been the topic
of research for many years. In the earlier stage, the vehicle is
simply treated as moving loads along the bridge[1], but
significant difference to the effect of a real vehicle has been
observed[2]. Another limitation of moving load model is that
it cannot account for the lateral vibrations of the bridge. To
take account the effect of vehicle vibration, various dynamic
interaction models are developed later [3-12].
All these interaction models are solved numerically based
on different wheel-rail interaction assumptions. For example,
Yang and Yau et al.[3] assumed the wheel-set motion is
identical to the rail beneath; Xia, Zhang and Xu et al. [4-5]
took a set sine waves as the lateral gap between the wheel and
rail; Li, Qiang et al.[6-7] allows the wheel has independent
yaw and lateral motion; Zhai and Xia et al. [8-10] taken all 5
DOFs of the wheel-set are independent. The solution scheme
for vehicle-bridge interaction varies in existing research
works, including direct time history integration[4-5], iterative
approach [6-7], the VBI element method [3,13], and explicit
integration method[8-10]. Through combination of wheel-rail
model assumption and integration scheme, a family of
solution procedures was developed. But it is very challenging
to increase the computational efficiency of these traditional
methods. A simulation can often take a long time to complete
even on state-of-the-art computers. This has become a major
barrier for studying the vehicle-bridge system from a
probabilistic perspective.
The intent of this paper is to develop an efficient but still
realistic model for vehicle-bridge random simulations. The
wheel-rail interaction in this model is modified from [14-15].
The main modification here is the consideration of the
rotational motion of rail, which is of significant for vehicles
on the bridge. This model fully considers the wheel-rail
geometric nonlinearity, which maybe the most significant
nonlinear factor in vehicle bridge interaction. The model also
adopts the mode superposition method for the bridge and
results a hybrid model in which the vehicle are represented by
physical DOFs, while the bridge by generalized DOFs in the
frequency domain. The stiffness formulation was conducted
through principle of virtual work using Matlab. The accuracy
and efficiency of this method are proved in an example by
comparing with that of detailed wheel-rail model.
2
THE VEHICLE AND BRIDGE AND EXCITATION
MODELS
The interaction model consists of two sub systems, namely,
the vehicle and the bridge. There are a variety of vehicle
models in literatures, here we use the model with 23 DOFs
proposed by Li and Qiang[6] as depicted in Figure 1 and 2.
A passage vehicle is composed of one car body, two trucks,
and four wheel-sets. These components are all treated as rigid
bodies. For each rigid body, 5 DOFs are considered, omitting
the longitudinal motion. The 5 DOFs of the car-body is
, thoses of the the i-th truck(i=1,2) is:
. Symbols
denotes the
lateral, vertical, rolling, pitching and yawing motion
respectively; suberscripts “c”and “t” stand for “car-body” and
“truck”.
ψ
1 Carbody
z
θ
y
2lc
ϕ
x
3
Rear truch
12
4
11
7
Bridge deck
1
3
Front truch 7
9
10
Left rail
6 Wheel-set
Right rail
8
2
2
5
6
5
2lt
4
er
Section
controid
hr
Figure 1. Vehicle-bridge interaction model.
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Carbody
Secondary
suspension
2b2
h1
Truck
h2
h3
Primary
suspension
Wheel-set
2b1
Figure 2. Front view of the vehicle.
For the wheel-set, since the wheels are assumed not to lift
from the rail, the vertical
and rolling
DOFs are
not free ones but dependent on the track motion and wheelset’s lateral motion. Thus, only yaw and lateral motions are
free DOFs. Displacements of the j-th(j=1,2,3,4) wheel-sets is
represented by
. Totally, a passage
vehicle’s motion are described by 23 DOFs as
.
To minimize the number of the bridge’s DOFs, we use the
mode superposition method to represent the bridge. The mode
superposition method[16] is a classical technique using
generalized DOFs to decrease the physical DOFs of a system.
The DOFs of bridge is the former N modal displacements ,
n=1~N. The n-th mode shapes is denoted as
, which is
mass normalized. Other parameters related to the bridge
includes: the n-th natural frequency , n-th damping ratio .
For a given bridge, all these parameters can be obtained from
an eigenvalue analysis using any structure software.
Rail irregularities are random excitations to the vehiclebridge system. Several nations have established rail
irregularities spectrums based on field measurements [17]. We
consider three components of rail irregularities, namely, the
vertical, lateral and torsional irregularity. The rail
irregularities to the j-th wheel-set are depicted below. In
Figure 3,
is the vertical rail irregularity at where the j-th
wheel-set locates longitudinally,
and
are the lateral and
torsional irregularities respectively.
3
VIRTUAL WORK FORMULATION
When a systems is at equilibrium, the virtual work sum done
by the forces
(both internal and external, either linear or
nonlinear) on any permissible virtual displacement δxi is
zero, that is
(1)
If the inertial forces are treated as external forces, the
formula above can be applied to dynamic problems. Knowing
the virtual work of a system, the mass, stiffness, damping and
external force matrix can be easily derived [16].
The virtual work considered in the study including virtual
work done by vehicle suspension spring and damping forces
and the work by bridge structure inertia and internal forces.
We write these forces in terms of vehicle motions firstly, then
calculate the virtual works done by these forces.
We give each spring and damper set an index number (from
1 to 12 in the boxes of figure 1), also an index to each rigid
body of the vehicle (from 1 to 7), as depicted in figure 1.
For the i-th spring set, the two ends of the spring are
connected to rigid body “ ” and “ ”, and the connection
1342
point in the local coordinate of body “ ” is
and
]in “ ”.
The stretches in x, y and z directions of the i-th spring are:
(2)
where
(3)
and
are the displacements of the “s” and “e” body.
Note that the longitudinal motion is not included.
It’s easy to know, the virtual work done by this spring is
(4)
For damping forces, the virtual work can be expressed in a
similar way.
For bridge structure, take the n-th mode of bridge as a
vibrator with mass as one, stiffness as , and damping
coefficient as
, the virtual work done by the stiffness
and damping force of this vibrator can be written as
,
4
(5)
THE WHEEL-RAIL INTERACTION
The contact geometry is the essential non-linear factor in the
vehicle-bridge interaction. The key contact geometric
parameters are show in figure 4. The nonlinear geometric
parameters include: contact angles, and , at the left and
right rail; rolling radius and ; and rotational angle of the
wheel-set
. For a real flanged wheel profile, these
parameters are all non-linear functions of wheel-rail relative
motion in lateral direction .
In this study, the Chinese LM wheel tread and UIC60 rail
profile are used.
z
Right
Left
y
o
rR r0
ϕw
rL r0
Rail center
yw
δR
δL
Rail
center
Figure 4. Wheel-rail contact geometry.
In the stationary track case in reference[15], the lateral
movement of wheel in the track frame
is identical that in
the inertial frame
. While in our moving track case (see
Fig.5), the lateral movement at wheel in the inertial frame and
the track frame are related by
(6)
LA
z
o
wheelset
z'
rail
NR
ϕw
δR
ϕw
ϕt y
o'
y'
ϕw δL NL
track
bridge deck
Figure 5. Movement of Wheel-set on the Bridge.
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
All nonlinear parameters in wheel rail contact depend on the
lateral relative displacement of the wheel-set to the track.
Note that this relative displacement should be calculated in
frame o’x’y’z’.
There are two type forces between the wheel and the rail:
contact normal forces and the creep forces. The normal forces
are due to the normal compress between the wheel and rail
materials. And the creep forces are due to the slip and
adhesion between the wheel and the rail along their contact
path. Figure 6 depicts the wheel-rail forces, where
and
are normal forces at the left and right wheel-rail contact point,
and
are the longitudinal creep forces at the left and
right wheel-rail contact point respectively,
and
are the
lateral creep forces,
and
are the creep moment along
the vertical direction.
Since the wheel-rail contact forces on stationary tracks
haven been well established in [15], we derive the formulas of
these forces on the moving track on the bridge in a parallel
way.
The total wheel-rail lateral force
and yawing moment
can be expressed in matrix form as
In equation 9,
=-
(9)
′
is the lateral displacement between
the wheel-set and the rail,
and
are the stiffness and
damping matrix of wheel-rail contact respectively as
(10.a)
(10.b)
In equation 10 , , ,
and
are lateral, lateral/spin,
spin
and
longitudinal
creep
coefficients,
respectively;
and are nonlinear terms whose
formulas can be found in [15].
Knowing the wheel-rail forces, the virtual work by the
wheel rail can be obtained as
rL r0
rR r0
FLy
FLz
FRy
NL
NR
FRz
ML
MR
wheelset
o
MRz
FRy
rail FRx
MLz
FLy
y
ψw
x
FLx
rail
Figure.6. Wheel-rail forces.
In derivation of the wheel-rail normal forces on the
stationary track, the normal contact forces at left
and right
wheel
are solved from two equilibrium equations. This is
possible because the wheel and rail are assumed to be in
constant contact, and neither vertical nor rolling motion is an
independent freedom. Instead they are determined by lateral
motion of the wheel-set.
In our moving track case, component of normal forces along
y direction is
(7)
where is the vertical wheel-set load, and the nonlinear term
is used to calculate the gravitational stiffness can be found
in reference [15]. Here an additional term
appears in
contrast to the stationary track case.
The gravitational stiffness moment can be approximated as
(8)
where is the contact angle at wheel-set’s centered position,
a is half of the wheel-set gauge and
is the yaw angle of the
wheel-set. The gravitational stiffness moment is the same as
on the unrolled track.
Creep forces are calculated from the creepages between
wheel and rail using the Kalker’s linear theory. The creep
forces on the moving track can be expressed in the similar
way to that of the stationary track in [15]. The only difference
lies in that Eq.6 is used to calculate the creepage.
(11)
where system equations and solving method
The total virtual work of the vehicle-bridge system is the sum
of all virtual work from vehicle suspensions, wheel rail forces,
bridge modal forces and inertial forces. Then differentiate the
total virtual work with first respect to the virtual
displacements then to the displacements, velocities or rail
irregularities. The differentiated results are members of the
stiffness, damping and load distributing matrixes.
The resulting vehicle-bridge system equation derived using
the virtual work method is
(12)
where M is the diagonal mass matrix; and are the damping
and stiffness matrix both dependent of time and the wheelrail lateral displacement
; X is the displacement vector;
is the load distribute matrix, and the irregularity vector.
The nonlinear equation 12 is solved iteratively at discretized
time points using the Newmark’s integration method. Here
only one loop of iteration for wheel-rail nonlinear geometry is
required.
5
5.1
NUMERICAL EXAMPLES
Bridge parameters
The bridge in study is a continuous rigid frame with span of
88+168+88m, and two piers with height 120m. The bridge is
made of reinforced concrete, whose modulus is 3.4 × 104
MPa, density 2550 kg/m3. Damping ratio is 0.05 for the first
and second frequencies. The properties of the girder and pier
sections are listed in Table 2. In table 2, A is the area of the
sections, J the torsional moment of inertial and Iy, Iz the
bending moment about horizontal and vertical directions
respectively. The first lateral and vertical mode shapes of the
bridge are plotted in figure 7.
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Table 2. Section properties of the bridge sections.
2
4
4
5.3
4
A/m
J/m
Iy/m
Iz/m
Girder
17.12
151.06
93.62
187.20
Pier
48.28
871.31
564.62
522.49
Lateral /mm
Vertical /mm
-3
Torsional /10 rad
10
Irregularities
Secion
Rail irregularities
The rail irregularities are simulated from the USA spectrum,
class 6. The wave length of the irregularities ranges from 1 to
100 m. The simulated results are plotted in figure 8.
5
0
-5
(a) First lateral mode(plane view, f=2.14 rad/sec)
-10
0
100
200
300
400
500
Distance/m
Figure 8. Simulated rail irregularities
5.4
5.2
Vehicle parameters
A typical railway vehicle is selected in this analysis. The
parameters of the vehicle are listed in table 3. The train is
composed of 8 identical vehicles. The train runs at 100 km/h
across the bridge.
Table 3. Vehicle parameters.
Parameter
Mw
Iwz
Mt
Itx
Ity
Itz
Mc
Icx
Icy
Icz
k1x
k1y
k1z
k2x
k2y
k2z
c1x
c1y
Value
2100 kg
1030 kg.m2
2600 kg
2100 kg.m2
1420 kg.m2
2600 kg.m2
3.41×104 kg
1.10×105 kg.m2
1.70×106 kg.m2
1.70×106 kg.m2
1.57×107 N/m
7.48×106 N/m
1.18×106 N/m
1.78×105 N/m
1.78×105 N/m
2.21×105 N/m
0 N.s/m
0 N.s/m
Parameter
c1z
c2x
c2y
c2z
L
lc
lt
b1
b2
h1
h2
h3
a
δ0
f11
f12
f22
f33
Value
1.96×104 N.s/m
1.10×106 N.s/m
2.18×104 N.s/m
9.80×103 N.s/m
26.0 m
8.75 m
1.25 m
1.00 m
1.23 m
0.50 m
0.20 m
0.065 m
0.75 m
0.05 rad
8.28×106 N
2.22×104 N.m
11.3 N.m2
8.13×106 N
Most parameters in table 3 have been defined early in this
paper except for the inertial parameters. For the inertial
parameters in table 3, M denotes mass, I denotes mass
moment; the subscribers, ‘w’ stands for wheel-set, ‘c’ for carbody and t for truck; ‘x’, ‘y’ and ‘z’ represent directions.
1344
Detailed Model
Present Model
0.10
0.05
0.00
-0.05
-0.10
0
5
10
15
Time / s
20
25
30
Figure 9. Lateral displacement at the middle main-span
Deflection at midspan / mm
Figure 7. Mode shapes of the bridge
The dynamic responses of the bridge, the vehicle and their
interaction are simulated using two methods. One method
used the detailed wheel-rail model and iteration solving
scheme, the other is the present method.
Figure 9 and 10 are the lateral displacement and deflection
at middle of the main-span of the bridge. It is clear, that the
amplitude and phase of the bridge responses from the two
methods are pretty close.
Lateral displacement
at midspan / mm
(b) First vertical mode(front view, f=5.94 rad/sec)
Numerical results
0
-5
-10
Detailed model
Present model
-15
0
5
10
15
Time / s
20
25
30
Figure 10. Deflection at the middle main-span
The time histories of vehicle-bridge interaction forces are
shown in figure 11. From the figure, the coincidence of the
two methods can be observed, only small difference occurs in
high frequencies components. This difference may be aroused
from neglecting the effect of the vertical force fluctuations on
the lateral forces in the present method.
Wheel-rail lateral force / kN
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
40
Detailed model
Present model
20
0
-20
-40
4
6
8
10
Time / s
12
14
Figure 11. Wheel-rail lateral force
The lateral and vertical accelerations at the centroid of the
car-body are shown in figure 12 and 13 respectively. The
results from the two method fit fairly well from the point view
of engineering. Especially, the vertical acceleration fit much
better than the lateral one. This can be understood since the
vertical wheel-rail models are identical, and the lateral motion
of the vehicle has minor effects on the vertical ones.
Lateral acceleration of
2
the carbody / m/s
0.4
achieved by 2 approaches: (1) a simplified wheel-rail nonlinear interaction model which cancels two loops of iteration
of the more detailed model; (2) The mode superposition
method is used to decrease the bridge DOFs significantly.
The accuracy of this model is still maintained by retain the
most important non-linearity of vehicle-bridge system, the
nonlinear wheel-rail contact geometry.
Also, we introduced the virtual work method to derive the
sophisticated motion equations of vehicle-bridge system to
ease the task of manually operations. This approach is easy to
expand to vehicles of other topology.
Numerical results show that vehicle-bridge responses using
present method are of adequate accuracy compared to those
from a more detail model, while the present method is about
60 times faster than the detailed model. With these merits, the
present model may find its future application in the random
simulations of vehicle-bridge vibrations.
ACKNOWLEDGMENTS
This optional section contains acknowledgments.
This study is supported by the National Natural Science
Foundation of China (Grant No. 51008250), Southwest
Jiaotong University Foundation for Young Researchers (Grant
No. 2007Q110, SWJTU12BR037).
Detailed model
Present model
0.2
0.0
REFERENCES
[1]
-0.2
[2]
-0.4
3
6
9
Time / s
12
15
[3]
Figure 12. Lateral acceleration at the centroid of the car-body
[4]
[5]
Detailed model
Present model
Vertical acceleration of
2
the carbody / m/s
0.2
[6]
0.0
[7]
-0.2
[8]
3
6
9
Time / s
12
15
[9]
Figure 13. Vertical acceleration at the centroid of the car-body
Although the present method loses minor accuracy in the
lateral behavior of the vehicle-bridge system, the tradeoff for
efficiency is significant. For such a large span bridge
interaction with 8 vehicles, the present method only token 317
second to simulate on a laptop using Matlab program. While
the detailed method programmed in C++ language used 302
minutes. Or the present method is about 60 times efficient as
the detailed model.
[10]
[11]
[12]
[13]
6
CONCLUSION
This paper presented a new model for the vehicle-bridge
dynamic interaction analysis. This model intends to get a
balance between the efficiency and accuracy. The efficiency is
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