Vibration Suppression of Vehicle-Bridge-Interaction System using Multiple Tuned Mass Dampers

International Journal of Modern Research in Engineering and Technology (IJMRET)
www.ijmret.org Volume 2 Issue 7 ǁ November 2017
w w w . i j m r e t . o r g I S S N : 2 4 5 6 - 5 6 2 8 Page 1
Vibration Suppression of Vehicle-Bridge-Interaction System using
Multiple Tuned Mass Dampers
Mehmet Akif KOÇ1
, İsmail ESEN2
, Yusuf ÇAY3
, Ömer ÇERLEK3
, Muhammed Asım
KESERCİOĞLU3
, RecepKILIÇ3
, Hüseyin DAL3
, Mustafa EROĞLU3
1
(Mechatronics Engineering, Sakarya University, Turkey)
2
(Mechanical Engineering, Karabuk University, Turkey)
3
(Mechanical Engineering, Sakarya University, Turkey)
ABSTRACT :In this study, vibration suppression of vehicle-bridge interaction system has been studied in
terms of bridge dynamics. In vibration suppression, some configurations of tuned passive mass dampers have
been considered. Motion equations of vehicle and absorbers are coupled with the motion equation of the bridge
beam using first four mode of beam and Lagrange equations. Then the coupled equation of vehicle-bridge-
absorbers (VBA) has been solved using the fourth order Raunge-Kutta algorithm. It is proved that the
configuration of the absorbers that are placed on the anti-nodes of first and second mode shapes of the bridge
beam is the most effective one of all others.
KEYWORDS -Anti-nodes,Tuned mass damper, Vehicle bridge interaction,Vibration suppression
I. INTRODUCTION
Vibrations of bridges under moving loads
are vital in engineering and have been studied over a
century. Early studies of the moving load problem
have been carried out by structural engineers who
accepted the moving load as a moving point force.
Later, with the implementation of the increased
transport speed, the subject has also been examined
by the automotive and railway engineers. Now the
subject, as vehicle-bridge-interaction (VBI), has
been studied in terms of both vehicle and bridge
dynamics with the new direction such as vibration
suppression of the both systems. In this field, some
of analytical studies[1,2] are valuable for the
dynamics of the bridge beams under different
loading cases and end support conditions. When
higher transportation speeds are implemented, the
dynamic amplification factor (DAF) is considered as
important due to the inertial effects of the moving
loads in rail and other transportation systems.
Vibration suppression and control of the bridge
beams have been studied by [3–6]. It has been
reported that using Passive Tuned Mass Dampers
(PTMD), the vibration of a high-speed train bridge
at 300 km/h had been reduced by 21 percent.
Another application of PTMD has been reported by
[4] for a thin bridge under successive moving loads.
Under a periodic excitation [10] has been studied
vibration reduction of a thin beam by PTMD. It has
been reported in[6] that it is impossible to suppress
all the vibration due to the VBI, since the excitation
frequency of the moving load is not constant. For a
constant velocity of the load [7], for vibration
absorbers [8] , for Timoshenko beams [9], for
optimized damped-absorbers [10] are the other
valuable studies in this field. Plate structures under
moving loads are also research interests and some
FEM of such system can be found in [11]. Resonant
response of a beam plate of a high-speed a railway
bridge has been optimized by [12] using passive
viscous dampers. Vibration suppression of a plate
structure under random excitation using PTMD and
its H∞ and H2 optimizations have been given in [13]
and it is the early study of PTMD. It has reported
that the most effective use of PTMD is to place it at
a point where the mode shape is maximum at an
anti-node. One of the other implementation field of
the PTDM is defence science applications such as
projectile barrel interaction in various type of heavy
and light gun systems. For example, studies [14,15]
have reported that the muzzle vibrations of barrels
can be decreased at 50 percent using proper
optimized PTMD.
One of the biggest challenges in PTDM
use, the influence frequency of the moving vehicle is
not constant and when the vehicle velocity is altered,
it is changed. Secondly, there can be more than one
vehicle on the bridge system with different

characteristics. Thirdly, the moving mass case,
taking into account the effects of mass inertia, the
natural frequencies of the bridge beam are changed
depending on the position of the moving load.
Further, a VBI system is coupled and its
linearization is difficult.
In order to suppress vibrations of the
vehicle and bridge beam, in this study, the idea of
the usage of the anti-nodes for the location of the
PTDM in [13] has been extended, and developed
such that at least the anti-nodes of the first and
second mode shapes may help more effectively.
Since a general vibration, response of a vibrating
system can be approximated by superimposing of
the effect of each vibration mode. Using this idea, in
this study the vibration response of the beam has
been approximated including the effect of the first
four modes. Then tuned mass dampers are
considered for the anti-node of the first mode at the
mid-point of the bridge beam, and the anti-nodes of
the second mode that are the 0.25 and 0.75 of the
bridge length L. The tuning of the absorber at the
mid-point is carried out using the fundamental
frequency while the tunings of the absorbers at the
0.25 and 0.75 locations, the second mode frequency
is used. Except this, the effects of one undamped
absorber and damped absorber are also studied as a
comparison. Moreover, the effect of the other
position of the PTDM on beam has been analysed
widely. From this perspective, this study can help
both structural and automotive engineers in order to
extend the service life of the bridges and predict the
dynamic forces applied to the vehicle from VBI, and
to achieve desired ride-safety and passenger
comfort.
II. MATHEMATICAL MODELLING
In order to reduce vibrations of the vehicle
and bridge due to vehicle-bridge interaction, a
special system shown in Fig. 1 has been studied.
The system consists of a single degree of freedom
vehicle, a simply supported Euler- Bernoulli beam
on which the vehicle moves with constant velocity v,
and pasive vibration absorbers that are placed at
0.25, 0.5 and 0.75 of the length of the beam. Where
m is the mass of the vehicle, k and c spring constant
and the damping coefficient of the suspension
system, while y is the vertical displacement of the
vehicle body; and parameters ki and mi(i = 1, 2, 3),
respectively, are the stiffness and mass of the
absorbers at the given locations i=1, 2, and 3, as
shown in Fig. 1. Symbol ycis the displacement of
the wheel at the contact point of the beam and wb(x,
t) is the displacement of the beam at point x and time
t. The displacements of the absorbers are
represented by y1, y2, y3from left to rigth.
In the formulation for the vehicle-bridge-
absorbers (VBA) analysis following assumptions
will be adopted:
 The bridge is modelled as a simple
supported beam based on Euler-Bernoulli
theory.
 The vehicle is modelled as a single DOF
lumped parameter system.
 Only one vehicle is accepted moving on the
bridge with constant velocity v.
 The wheel is always in contact with the
bridge.
 The effect of road roughness upon vehicle
and bridge dynamic is not considered
during analysis in this study.

Figure 1. Model of Euler-Bernoulli Bridge beam with attached PTMDs and subjected to moving vehicle.
With all these assumptions, for the system
of vehicle-bridge- absorbers (VBA) shown in Fig.
1 the kinetic and potential energy are expressed as
follows, respectively:
2 2
0
2 2 2
1 1 2 2 3 3
( , ) ( )1
,
2
( ) ( ) ( )

 
       
 
  
  
  
L
b
k
w x t dx my t
E
m y t m y t m y t
(1a) (1a)
 
   
 
22
0
2 2
1 1 2 2
2
3 3
( , ) ( ) ( ( ), ) +
1
( ) ( / 4, ) ( ) ( / 2, ) ( , ( )),
2
( ) (3 / 4, )


 
     
 
 
    
 
  
 
 

L
b b
p b b
b
EI w x t dx k y t w t t
E k y t w L t k y t w L t H x t
k y t w L t
(1b)
whereμ and EI, refer to the mass of the
unit length and flexural rigitidy of the bridge
girder, respectively.
For the system shown in Fig.1, in order to
obtain the equations of motion one can use the
virtual work principle, Hamilton's principle and
D'Alembert's principle. In this study, motion
equation of the VBA integrated system is optained
using Langrange’s equations and mode expansion
method. For any point x on the beam at time tthe
deflection function wb (x, t) can be aproximated
using the Galerkin’s method:
1
1
1
( , ) ( ) ( ),
( , ) ( ) ( ),
( , ) ( ) ( ),
2
( ) sin , 1,2,..., .
 
 
 







 
 
  
 




n
b i bi
i
n
b i bi
i
n
b i bi
i
i
w x t x t
w x t x t
w x t x t
i x
x i n
L L
(2)
whereηbi is the i-th time dependent
generalized nodal coordinate,φi is i-th mode shape
function. The applied static axle load of the
vehicle at the contact point of the wheel can be
determined as:
( , ) ( ( ( ))),  cf x t mgH x t (3)
whereH(x-ξ(t))is the Heaviside function.
For the vehicle- bridge system the Rayleigh’s
dissipation function can be expressed as below:
 2 21
( , ) [ ( ) ( ( ), )] ( ( )) ,
2
      eq b bD c w x t c y t w t t H x t (4)
In Eq. (4) ceq is the equivalent damping
coefficient of bridge girder. In addition, for the
given system, the Lagrangian (L = Ek- Ep) is equal
to the difference of the kinetic and potential
energies. In such a case, for the five independent
coordinates, the Lagrange equations can be as
follows:
x
y
EI,c ,
vt
L
0.5L
ck
k2k1 k3
0.75L
t
0.25L
m1 m2 m3
m
v=const.
F
w(tt)b
F
y(t)
y (t)1
y (t)2
y (t)3
eq

d
0 , 1,
d ( ) ( ) ( )
   
    
    k k k
L L R
k
t p t p t p t
(5a)
d
, i 1,2,3,4,
d ( ) ( ) ( )  
   
    
     i
i i i
L L D
Q
t t t t
(5b)
With the state variables that are:
   1 2 3 4( ) ( ) , (t) (t) (t) (t) (t) ,     
T T
p t y t (6)
And the corresponding generalized forces
are:
0
( ) ( , ) , 1,2,3,4, 
L
i i cQ x f x t dx i (7)
Using the orthogonality conditions and
Galerkin approach for displacements of the beam,
(Eq. (2)) the equation of motion is obtained for
VBA interaction model given in Fig. 1. In this
case, the equation for the acceleration of the
vehicle body and absorbers are written as follows,
respectively:
 
1
- [ ( ) - ( , )]- [ ( ) - ( , )] ,  b by c y t w x t k y t w x t
m
(8a) (
 1 1
1
1
- [ ( ) - ( 4, )] , by k y t w L t
m
(8b) (
 2 2
2
1
- [ ( ) - ( 2, )] , by k y t w L t
m
(8c) (
 3 3
3
1
- [ ( ) - (3 4, )] , by k y t w L t
m
(8d)
Dynamic equations of the bridge as the n
second order ordinary differential equations can be
expressed as follows:
 
 
 
   
 
2
1
1
1 1 2 2
3 3
( ) ( ) ( ) ( )
( ), - ( )
( )
( ), - ( )
( 4, ) - ( ) ( 2, ) - ( )
(3 4, ) - ( ) 0 1,2,3,4.
   

 

  
      
  
      
 
  
 
 
i i eq i i i i
c b
i
b b
b
N t c x t t
f c w t t y t
t
k w t t y t
k w L t y t k w L t y t
k w L t y t i
(
(9)
whereΛ is:
11, 0
0 ,
for t t
elsewhere
ì £ <ïïL = í
ïïî
(10)
Where t1is the time when the vehicle
leaves the bridge. The coefficient Λis time
dependent and is used with Eigen function
φi(ξ(t))in order to determine motion equation.
In Eqs. (8a), (8b-c) and (9), shows the
total eight second order differential equations for
the vehicle, bridge and absorbers. The first four
equations are for vehicle and the absorbers and
they are transferred into first order equations by
[16]. In addition, the bridge dynamic is expressed
by using the second order differential equations in
Eq. (9). In this study, the bridge dynamic is
calculated by considering the first four-vibration
mode and it is represented with four differential
equations of second order. These four differential
equations are transferred to eight equations by [16].
In order to solve this equation system, comprising
of sixteen equations, a fourth order Runge-Kutta
method has been used[16].
III. NUMERICAL ANALYSIS
In this section, using four different VBA
models (Model A, B, C, D) the effects of the
absorbers on the vehicle and bridge dynamics have
been studied extensively. System equations of
motion Eq. (8a-d) for the VBA is solved by using
Runge-Kutta algorithm of the fourth order, with a
special m.file prepared MATLAB © environment.
The integration time step size for numerical
analysis during the study Δt = 0.01 s, the solution
final time is taken to be t = L / v. In addition, the
parameters used in this study for the systems VBA
are presented in Table 1.
When the vehicle moves over the bridge,
it forces the bridge to vibrate. At certain vehicle
speeds, the bridge enters into resonance and the
amount of oscillation increases significantly. The
velocity of the vehicle that causes resonance is
called the critical velocity vcr and it is calculated
using the circular natural frequencies of the beam.
For a simply supported beam the natural
frequencies of jth
vibration mode is expressed
as[17]:
4 4
2
4
( / )j
j EI
rad s
L
p
w
m
= (11)
When Eq. (11) and the excitation
frequency of a moving vehicle on the bridge are
rearranged, the following is obtained[16]:
1/2
2
,
2j j cr
vL v
f j EI v
w w m
a
w p p
æ ö÷ç= = = =÷ç ÷÷çè ø
(12)
Using Eq. (12), Fig. 2 shows the first three
modes of vibration of the bridge beam. Frequency
values of the first three modes of vibration of the

bridge beam are, respectively, f1=0.2108 Hz,
f2=0.8432 Hz, and f3=1.8972 Hz. Using these
values and Eq. (22), the critical speeds for vehicles
moving on the bridge are calculated, respectively,
as vcr1=42.15 m/s, vcr2=168.62 m/s, and vcr3=252.82
m/s.
Figure2.Mode shapes of the bridge.
In order to analyze the effects of the
absorbers on the dynamics of vehicles and bridges
four different models of the VBA shown in Fig. 3.
In the model A as shown in Fig. 3a, an undamped
mass spring system is suspended from the bridge at
the mid-point. The main vibration displacement of
the bridge system is affected mostly from the
fundemantal mode of the beam, for such reason the
absorber is placed at the mid point where the
maximum deflection occurs at the midpoint when
the beam vibrates with only the first mode. The
first natural frequency of the bridge is f1 = 0.2108
Hz. Therefore, the absorber is tuned to resonate at
the first natural frequency
1
1 1 1(2 ) 0.2108f k mp -
= = . Thus, it is expected
that this absorber can reduce the vibration energy
of the beam caused from the first vibration mode of
the bridge. In the second model, model B, a
viscous damper added to the spring mass system.
The damping coefficient of the absorber was
chosen as c2=10000 Nsm-1
, not on purpose but to
show the effect of any damping in the absorber.
The critical damping value
2 2 2 22 2 26476.46cr nc m k mw= = = Nsm-1
, and
the damping ratio of the damper is
( )2 2 2 2 2 2 22 (2 ) 0.377cr nc c c m c k mz w= = = =
obtained using the parameters listed in Table 1.
When this damper is added, the natural frequency
of the absorber has reduced from 0.2108 Hz to
0.195 Hz that is the damped natural frequency,
where 2
1 1.226 / 0.195 Hz.d n rad sw w z= - = =
it is expected that adding damping may not be
useful in order to reduce the vibration energy of the
first mode. However, it should be known that in
some case the damping could be considered in
terms of design criteria. These issues are examined
in detail in the following sections.
Figure 3. Model of a vehicle-bridge-absorber interaction system (a) an undamped absorber at middle
of the bridge; (b) A damped absorber at middle of the bridge; (c) three undamped absorbers at 0.25L, 0.5L, and
0.75L of bridge length; (d) without absorber.
The Models A and B can reduce oscillations originating from the first mode of the
0 20 40 60 80 100
-1
-0.5
0
0.5
1
x 10
-3
Length of the bridge (m)
Mode 1 ( 0.2108 Hz. )
Mode 2 ( 0.8432 Hz. )
Mode 3 (1.8972 Hz. )
x
y
x=vt
L
L/2
ck
k c2 2
x
y
x=vt
L
L/2
ck
x
y
x=vt
L
ck
x
y
x=vt
L
L/2
ck
k2
k2k1 k3
Model A Model B
Model C Model D
3L/4
L/4
m1 m2 m3
v=const.
m2
m2
m m
m m
mode 1 mode 1
mode 1
mode 1
mode 2
v=const.
v=const.
v=const.
EI,c ,eq
EI,c ,eq
EI,c ,eq
EI,c ,eq
a-) b-)
c-) d-)

bridge during vibration. Because these absorbers
prepared at a middle point of maximum amplitude
of the first mode of the bridge and their resonance
frequency is set to the first oscillation frequency of
the bridge. However, structures are not only
affected from the first mode of vibration the
contribution of the higher modes are also involved.
Therefore, in addition to the first vibration mode
for the second vibration mode of the bridge a
second and a third absorber are placed at the
locations 0.25 L and 0.75 L of the bridge; and they
are tuned to the second natural frequency of the
bridge beam. As seen from Figure 3 the peaks of
the second mode shape of the beam are at these
locations. In order to compare with the effects of
the three previous models, the Model D has no
absorber. The parameters of the bridge and the
vehicle used in the study are presented in Table 1.
Table 1.Prpoperties of the vehicle and bridge.
Bridge Vehicle parameters
L (m)
E
(Gpa)
I (m4
)
μ
(kg/m)
ceq
(Ns/m)
100
207
0.174
20000
1750
m (kg)
k (kg)
c (kg)
k1 ( N/m
)
k2 ( N/m
)
2172
85439.6
2219.6
280401.
59
17525.0
8
c2 ( Ns/m)
m1( Ns/m)
m2( Ns/m)
m3( Ns/m)
k3 ( N/m )
10000
10000
10000
10000
280401.
59
Based on the four different VBA models
in Figure 3, Figure 4, displays the midpoint
displacements wb (x=L/2, t) of the bridge beam for
a vehicle speed of v=90 km/h = 25 m/s that this
speed is chosen so that it is applicable in
transportation.
For Model D model that there is no
absorber, the maximum midpoint displacement is
20.34 mm when the vehicle is at about 0.75 L,
while for Model A, the maximum displacement is
observed as 15.8 mm, which means a decrease of
22.32 percent when compared with no absorber
case. After a damped absorber in Model B is used,
the maximum midpoint displacement is 17.6 mm
when the vehicle has reached to 81.2 percent of the
bridge length. When a damping added to the
absorber, the occurrence time of the maximum
bridge midpoint displacement shifted to forward,
because of the added phase difference from
damping. Damping in the passive absorbers could
not reduce the displacements much when compared
to the undamped absorbers. For Model C, a special
case when three absorbers are used, the maximum
displacement of the midpoint of the bridge is 15
mm and; that means the displacement has been
reduced by 26.25%. All the results are presented in
Table 2.
Table 2.Comparation of the four different models according to
bridge midpoint maximum displacement (m).
Model Max. Mid-point
disp.(mm)
Location of
the vehicle
(%)
According to
the Model D,
the reduction
rate (%)
A 15.8 73 22.32
B 17.6 81.2 13.47
C 15 87.5 26.25
D 20.34 75 ----
Figure 4. The effect of the absorber upon bridge midpoint dynamic response for vehicle constant velocity v=25
m/s and time step size ∆t=0.01 s (a) Bridge midpoint displacement; (b) Absorber displacement.
In traditional highway or railroad bridge
0 20 40 60 80 100
-20
-15
-10
-5
0
x 10
-3
Position of the vehicle on the bridge ( m )
Midpointdisplacementofthebridge(m)
model A
model B
model C
model D
0 20 40 60 80 100
-0.03
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
Position of the vehicle on the bridge ( m )
Displacementoftheabsorbers(m)
Model A
Model B
Model C 0.25L
Model C 0.5L
Model C 0.75L

design, the Dynamic Amplification Factor (DAF) is
considered. DAF means the ratio of the maximum
dynamic displacement to the static displacement
when the load is located at the midpoint of the
bridge; and can be expressed as given below:
2 2
   
    
   
d s
L L
DAF R R (13)
The DAF of any bridge system can be
affected by many parameters such as vehicle
velocity, mass of the vehicle, natural frequency of
the bridge, and road roughness, etc. considering the
Models A, B, C and D, the DAFs of the different
velocities of the vhecile are presented in Figure 5.
As shown in Figure 5, for the Model D, that is no
absorber on the bridge the maximum value of the
DAF is 1.65 for the vehicle speed v = 26 m/s, while
for the Model B, the maximum value of the DAF is
1.47 for the vehicle speed v = 18.5 m / s; and for
the Model C the maximum value of the DAF is
1.45 for the vehicle speed v = 13.5 m / s; and for
the model A the DAF=1.37 at v=12.35 m/s. Up to
a speed of 31 m/s, the Model A showed the best
performance, but after this speed, the Model C was
the best in reducing the vibrations. Considering
this situation it can genarally be accepted that
velocity of the vehicle determine the behaviour of
the absorbers. This is because; the forcing
frequency of the VBI system is determined by the
velocity of the vehicle.
Figure 5. DAF response of the bridge mid-point for
four different vehicle-bridge-absorber interaction
system.
IV. CONCLUSION
In this study, in order to reduce the
vibration of the vehicle that is caused by vehicle –
bridge- interaction (VBI), the effects of various
dynamic vibration absorbers on the bridge
dynamics have been investigated. Due to the
complexity of the loading in moving load problem,
that the interaction is a non-linear coupled system
of the two multiple degree of freedom subsystems.
In the vibration of any beam system one can
consider the general forced vibration response of
the beam can be calculated by super positioning the
mode shapes of consecutive modes at least
including the first-four modes. However, in this
assumption the effect of the first fundamental mode
and the second mode is compromises the 80
percent of the response even in very flexible
systems. In such a case, the anti-node points of the
two first nodes can be used as the location of the
absorbers. The anti-node is the midpoint in the
first mode, and they are at 0.25L and 0.75 L in the
second mode shape. From the results of the
analysis, the Model C in this study, represents this
situation where three absorbers have been placed
on the beam at the location of the anti-nodes of the
first and second modes, is the best design for the
reduction of the vibrations of the bridge.
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[1] L. Fryba, Vibration solids and structures under
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0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
0.4
0.6
0.8
1
1.2
1.4
1.6
Vehicle velocity (m/s)
DAF
1=2:
p
K a2=M a2 = 0:2108H z:
Model A
Model B
Model C
Model D
max. DAF
value 1.65
Vehicle
velocity
18.5
m/s,
Vehicle velocity 26
m/s,
max DAF
value 1.47
Vehicle velocity 12.35
m/s, max DAF 1.37
Vehicle velocity 13.5
m/s, max DAF 1.45
Vehicle velocity
31
m/s,DAF value
1.25

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017-0913-y.
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Kren Kirişlerinin Dinamik (Mukavemet)
Analizi, Istanbul Technical University, 2009.

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