Civil Engineering Reference
In-Depth Information
The Laser Doppler Vibrometer (LDV) is a non-contact measuring device that measures displacement and velocity of a
remote point in structural members. The system is composed of three parts: the helium neon Class II laser head (Fig.
13.3c
),
the decoder unit (Fig.
13.3c
), and the reflective target (Fig.
13.3d
) attached to the structure. The laser head is mounted to a
tripod that is positioned underneath the target.
13.3
2-D Dimentional Dynamic Model and Model Validation
Figure
13.4
shows a two-dimensional (2D) model which was developed to simulate the dynamic behavior of the bridge-train
interaction The moving train composed of several vehicles, including locomotive car and freight cars or passenger cars, runs
over a simple supported bridge at the constant speed
v
. It is assumed that the wheel sets of each vehicle are kept in full
contact with the bridge at all times. This assumption is made to ensure that the dynamic responses of the bridge and vehicle
are linearly coupled, which can be computed using conventional time integration methods without iterations.
As shown in Fig.
13.4
, each vehicle is composed of one car body, two identical bogies, and four identical wheel sets. The
wheel sets and bogies are connected by primary suspensions, and the bogies and the carbody are connected by secondary
suspensions, which are modeled as a linear spring-dashpot units [
6
]. The car body and two bogies are each assigned two
DOFs, which are vertical displacement and rotation about the center point. Using rigid-body dynamics method, the
equations of motions for vehicle components can be given as a series of second-order ordinary differential equations in
the time domain.
In this study, the simple bridge is modeled as a linear elastic Bernoulli-Euler beam with identical sections. Using modal
superposition method, the equation of motion for the bridge subjected to moving train can be written as a series of second-
order ordinary differential equations with generalized displacements.
By combining vehicle equations and bridge equations together, the equations of motion for train-bridge system can be
presented in a matrix form as
½U
þ
½ U
þ
M
C
½
K
fg¼
U
fg
F
(13.3)
are the vectors of displacement,
velocity, and acceleration, respectively; and {F} represents the vector of exciting forces applied to the dynamic system.
To compute both the dynamic responses of the simple bridge and moving vehicles, the generalized matrix equation of
motion given in (
13.3
) will be solved using Newmark-
fg; U
; U
where [M], [C], [K] denote the mass, damping and stiffness matrices;
U
β
method. In this study,
β ¼
¼ and
γ ¼
½ are selected, which implies
a constant acceleration with unconditional numerical stability.
To validate the 2-D dynamic model described earlier, the field testing results including deflection, velocity, and strain
data are compared with the dynamic model results. As shown in Fig.
13.5
, the deflection and velocity results for the selected
bridge model show good agreement (within 10 %) with the field testing data.
Nth
l
v
2
jth
1
st
l
v
2
l
v
2
v
l
t
l
t
l
t
q
vj
m
v
Z
vj
I
v
q
vN
m
v
I
v
q
v
1
m
v
Z
v
1
I
v
Z
vN
c
3
c
3
...
k
s
k
s
k
s
c
s
q
tN2
q
tN1
q
tj2
m
t
I
t
Z
tj
q
tj1
q
t
12
Z
t
1
q
t
11
m
t
I
t
m
t
I
t
m
t
I
t
m
t
I
t
m
t
I
t
Z
tN
2
Z
tN
1
Z
tj
2
Z
t
12
k
p
c
p
k
p
c
p
k
p
c
p
k
p
c
p
k
p
c
p
c
p
k
p
l
w
l
w
l
w
l
w
l
w
2
m
w
m
w
m
w
m
w
l
w
m
w
2
2
2
2
m
w
2
r
(
x
)
Bridge
Z
b
L
Z
Fig. 13.4
2D train-bridge system model
