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Vibration Control of Bridges using Visco-elastic Post and Tendon
Ji-Seong Jo
Ph. D. Candidate, Department of Civil and Environmental Engineering, Korea Advanced Institute of
Science and Technology, Taejon 305-701, Korea
Jun-Sik Ha
Graduate Student, Department of Civil and Environmental Engineering, Korea Advanced Institute of
Science and Technology, Taejon 305-701, Korea
In-Won Lee
Professor, Department of Civil and Environmental Engineering, Korea Advanced Institute of Science and
Technology, Taejon 305-701, Korea
ABSTRACT
This paper presents a passive vibration control system for reducing excessive traffic-induced vibration of
bridges. The proposed system is developed by combining merits of king-post and tuned mass damper
mechanism. The king-post mechanism that increases the stiffness of the bridge is used to reduce transient
response of bridges when vehicles are on the bridge and the tuned mass damper mechanism that absorbs
vibration energy is used to reduce steady-state response after vehicles cross the bridge. To verify the
performance of the proposed system, a numerical simulation conducted on the existing bridge undergoing
vibration problems due to moving vehicles. The simulation results show that the maximum displacement and
acceleration at mid-span are more efficiently diminished than king-post or tuned mass damper alone.
Therefore, the proposed system can be used to improve the serviceability and the structural safety of
bridges under serious traffic-induced vibration.
1
Key words : King-post mechanism; tuned mass damper; moving vehicle; serviceability; safety
1. INTRODUCTION
In recent years, while heavy and high speed vehicles are increasing in number due to increase in
domestic and foreign trade, the developments in design and construction technology in civil engineering
enable the construction of more light and slender structures, which causes structures to be vulnerable to
dynamic loads, especially moving loads. Large deflections and vibrations of bridges induced by the heavy
and high speed vehicles may significantly increase the maximum internal stresses and affect the safety and
serviceability of the bridges. Therefore, vibration control system is necessary to reduce the response of
bridges and thus to improve the serviceability and the structural safety of bridges under serious trafficinduced vibration.
For dynamic loadings such as wind and earthquake, there have been extensive researches on vibration
control systems of bridges. Various base isolation systems which decouple the structure and/or its contents
from potentially damaging earthquake induced ground or support motions are analyzed and compared by
Park et al [1]. Kawashima and Unjoh [2] proposed a variable damper to control the response of bridges
against earthquake and showed the effectiveness of the system by experimental and numerical results.
Symans and Kelly [3] proposed a fuzzy logic controller for bridges using semi-active seismic base isolation
systems. Erkus and Fujino [4] investigated semi-active control for seismic protection of elevated highway
bridges. Also, for cable–stayed bridges there have been proposed many vibration control systems (Yang
[5], Fujino et al [6], Jung et al [7]).
However, for vehicle-induced moving loads there have been quite a few researches on vibration control
systems of bridges. Soong [8-9] introduced a king-post beam to control vibration of bridges under moving
loads. Though the system was shown to be very efficient in reducing both vehicle-induced transient and
steady vibrations, it is an active type and it needs very large control force to reduce the vibration.
2
Therefore, it is very impractical to apply that for existing bridges. Kwon et al. [10] developed TMD (tuned
mass damper) to reduce the vibration of simple-span bridges under moving loads. The system is a passive
type and it needs no external power to control the vibration of bridges. However, this system has no
versatility of the active system and it was shown that the steady-state responses are reduced efficiently but
the transient responses are not. Kajikawa [11] tried full scale implementation of TMD for reducing traffic
induced bridge vibration. However it was found that TMD gave minor control effect. Pattern et al [12]
installed a semi-active hydraulic vibration absorbers system installed in the bridge and this system showed
good control performance.
In this paper, a passive vibration control system for reducing excessive traffic-induced vibration of
bridges is proposed. The proposed system consists of visco-elastic post and tendon. The tendon with
elastic part of the post increases the stiffness of the bridge and reduces transient response of bridges when
vehicles are on the bridge and the visco-elatic post with the mass of the system plays the role of the TMD
that absorbs vibration energy and reduces steady-state response after vehicles cross the bridge. To verify
the performance of the proposed system, a numerical simulation conducted on the existing bridge
undergoing serious vibration problems due to moving vehicles.
2. RESPONSE OF BRIDGES
The response of bridges under moving vehicles was obtained by various researchers such as
Hayashikawa and Watanabe [13], Biggs [14], Chen and Coffey [15] and Kwon [10]. It is well known that
the response of bridges are affected by many factors such as stiffness and damping of the suspension, mass
inertia, speed, wheel spacing of the vehicle, the roughness of the bridge deck surface and the centripetal
and Coriolis forces and so on (Savin [16], Michaltsos [17]). The objective of this study is not to obtain the
analytical or numerical solutions for the above cases but to develop a passive control system to reduce
3
vibrations, so the moving vehicles are modeled as moving loads and the simple span bridge is considered
for convenience. The Bernoulli-Euler beams having equivalent stiffness for idealizing the bridge is shown in
Fig. 1. For the formulation of equations, in this and next chapter the following assumptions are introduced:
(1) The bridge is idealized as having equivalent stiffness.
(2) Only vertical modes of the bridge are considered.
(3) The vehicle moves at a constant speed on the bridge.
2.1 Equations of motion [15]
The equation of motion of the bridge (Fig.1) can be expressed as:
EI
? 4y
?y
? 2y
?
C
?
?
A
? F ( x, t )
? x4
?t
? t2
(1)
In Eq. (1) EI, C and ? A are the flexural stiffness, the damping of the bridge and the mass per unit length of
the beam, y ( x, t ) is the vertical deflection of the beam in terms of the special coordinate x and the time
variable t. F ( x, t ) is the force acting on the beam.
2.2 Dynamic response [10]
To obtain dynamic response using eigenpairs calculated, the bridge response can be defined as the
combination of normal modes and generalized coordinates.
?
y ( x, t )? ? ? n ( x) qn (t )
(2)
n ?1
where ? n (x ) , q n (t ) and n are the nth mode shape, nth generalized coordinate and mode number.
The mode shape of the beam, ? (x ) can be represented as:
? ( x) ? C1 sinh ? x ? C2 cosh ? x ? C3 sin ? x ? C4 cos ? x
4
(3)
? A?
EI
?4 ?
2
(4)
where ? and ? are the eigenvalue and natural frequency of the bridge respectively and Ci (i ? 1,2,3,4)
are constants to be determined according to boundary conditions.
In the case of simple beam, the following boundary conditions must be satisfied.
? ( 0) ? 0
d 2?
dx 2
?0
x? 0
(5)
? (L ) ? 0
d?
dx 2
2
?0
x? L
Introduce the boundary conditions in the eq. (5) to the eq. (3) and then the following equations are
obtained:
C2 ? C4 ? 0
? 2 (C2 ? C4 ) ? 0
C1 sinh ? L ? C3 sin ? L ? 0
(6)
? 2 (C1 sinh ? L ? C3 sin ? L ) ? 0
From the eq. (6), the characteristic equation for ? is:
sinh ? L
sin ? L
?0
? 2 sinh ? L ? ? 2 sin ? L
(7)
From the nontrivial solution of eq. (7), the mode shape of the beam can be written as:
? n ( x) ? C sin ? n x
(8)
where n is the mode number and C is a constant.
Introduce eq. (2) and eq. (8) to eq. (1) and then the equation of motion for generalized coordinate
q n (t ) is:
q??n ? 2? n? nq?n ? ? n2 qn ?
1
M n2
5
L
? F ( x, t )?
0
n
( x) dx
(9)
where
L
M n2 ? ? ? A? n2 dx ?
0
? AL
2
(10)
where L is the total length of the bridge, ? n the nth modal damping ratio, ? n the nth natural frequency.
3. RESPONSE OF BRIDGES WITH VISCO-ELASTIC POST AND TENDON
To reduce the vibration of bridges under moving loads, a passive type control system with visco-elastic
post and tendon is developed as shown in Fig. 2. The one end of visco-elastic post is connected to the
main girder of the bridge at mid span and the other end is connected to the middle of the tendon. The
tendon is post-tensioned to be in tensioned state during the vibration. The jacking force of the tension can
also be adjusted to give an appropriate camber to the bridge.
The transient response of the bridge when the vehicle is on deck consists of both the static deflection by
the weight of the vehicle and the fluctuating deflection by the dynamic effect of the vehicle. The steady-state
response of the bridge is the free vibration with initial conditions at the time when vehicles cross the bridge.
Kwon et al [10] showed that TMD (tuned mass damper) can reduce the steady-state responses efficiently
but the transient responses are not reduced because the main portion of the transient responses is occupied by
the static defle ctions. The visco-elastic post with mass from the tendon and itself takes the roll of the TMD
and the post-stressed tendon gives additional stiffness to the bridge.
3.1 Equation of motion of the visco-elastic post and tendon
The free body diagram for visco-elastic post in Fig. 2 is shown in Fig. 3. In Fig. 2, yb and z are the
displacements corresponding to the bridge and the effective mass. yd is the relative displacement of the
effective mass; and v is the velocity of the vehicle; and md is the effective mass of the tendon; K d , Cd and
6
H are the spring constant, the damping coefficient and the height of the post; T and ? C are the tension
and expansion of the tendon; ? is the angle between post and tendon.
The damping force( PD ) and spring force( PS ) of visco-elastic post can be represented.
PD ? Cd ?y?d (t )
PS ? K d ?y d (t )
(11)
From the Fig. 3, we can obtain the following force equilibrium equation.
T??
PD ? PS ? m d ??y?d (t ) ? ?y?b ( L / 2, t ) ?
2 cos?
(12)
TLC
EC AC
(13)
?C ?
where LC , EC and AC are the length, Young’s modulus and the area of the tendon.
Substituting the eq. (12) to the eq. (13), the following equation is obtained.
?C ? ?
[ PD ? PS ? md ( ?y?d (t ) ? ?y?b ( L / 2, t ))] H
2 EC AC cos 2 ?
(14)
The response of the effective mass of the system can be written as:
yd (t ) ? yb ( L / 2, t ) ?
?C
cos ?
(15)
Introducing the eq. (14) to the eq. (15), the following equation is obtained.
yd (t ) ? yb ( L / 2, t ) ? ?
[ PD ? PS ? md ( ?y?d (t ) ? ?y?b ( L / 2, t ))] H
2 EC AC cos3 ?
(16)
Rewrite the eq. (16) as following:
md ( ?y?d ( t ) ? ?y?b (t )) ? PD ? PS ?
2 EC AC cos 3 ?
{ yd ( t ) ? y b ( L / 2, t )} ? 0
H
Substituting the eq. (11) to the eq. (17) and rearranging the eq. (17), the equation of motion of the
effective mass can be obtained as following:
7
(17)
md ?y?b ( L / 2, t ) ?
?
2 EC AC cos3 ?
2 EC AC cos 3 ?
yb ( L / 2, t ) ? md ?y?d (t ) ? Cd y?d (t ) ? ?? K d ?
H
H
?
?
?? y d (t ) ? 0
?
(18)
3.2 Equation of motion of the bridge with visco-elastic post and tendon
In this part the equations of motion for the bridge with the proposed system, when a moving load exist, is
obtained. The effect of the proposed system on bridge can be considered as the combined forces by
spring( PS ) and damper( PD ). So the force on bridge F ( x, t ) in the eq. (1) can be written as:
F ( x, t )? Pw ? ( x ? vt ) ? ?Cd y?d ( t ) ? K d yd (t )? ?? ( x ? L / 2)
(19)
where Pw is the force transmitted by vehicle and ? ??
? stands for Dirac’s delta function.
Using the above eq. (19) and mode shape function in eq. (8), the integral part of eq. (9) can be
evaluated as:
L
? F ( x, t)?
0
n
( x )dx ? Pw sin
n? vt
n?
? [ Cd y?d ( t ) ? K d yd (t )] ?sin
L
2
(20)
From the eq. (9) and eq. (20), the following equation of motion for the general coordinate is obtained.
q??i (t ) ? 2? i? iq?i (t ) ? ? i2 qi (t )
2Cd
i?
2K d
i?
2 Pw
i? vt
?
sin
y?d (t ) ?
sin
yd (t ) ?
sin
? AL
2
? AL
2
? AL
L
(i ? 1,2,? , n)
(21)
The matrix form of the eq. (18) and eq. (21) becomes
MX??? CX? ? KX ? R
? q1 ?
?q ?
?? 2 ??
X ? ???
, R?
?q ?
? n?
?? y d ?? (n? 1)?1
8
? vt ?
?
? sin L ?
? 2? vt ?
? sin
?
?
L ?
? ? ?
? n? vt ?
?sin
?
L ?
?
?? 0 ?? ( n ? 1) ?1
(22)
(23)
? 1
?
0
?
M?? 0
?
? ?
?md sin ?
?
2
?
?2? 1? 1
?
? 0
C ? ??
0
?
? ?
?
?? 0
?
? 12
?
?
?
0
?
K??
?
?
?
0
?
? 2 EC AC cos 3 ?
?
sin
?
?
H
2
0
?
0
1
0
?
0
?
0
?
0
1
?
md sin
md sin
2?
2
0
?
0
2? 2? 2
0
?
0
?
2? n?
?
?
?
0
?
0
0
0
?
?
?
?
?
?
?
?
?
?
??( n? 1)? ( n? 1)
?
?
?
2 EC AC cos 3 ?
2?
sin
H
2
? 2Cd
?
sin
? AL
2
? 2Cd
2?
sin
? AL
2
?
? 2Cd
n?
sin
? AL
2
Cd
?
2
2
0
n
n?
2
0?
?
?
?
??
?
0?
md ?
?( n? 1)? ( n? 1)
?
2
n
2 EC AC cos3 ?
n?
sin
H
2
? 2 Kd
?
sin
? AL
2
? 2 Kd
2?
sin
? AL
2
?
? 2K d
n?
sin
? AL
2
2 EC AC cos 3 ?
Kd ?
H
(24)
(25)
?
?
?
?
?
?
?
?
?
?
?
?( n? 1)? ( n? 1)
(26)
where M ,C and K are the mass, damping and stiffness matrices of order n+1, respectively, and R the
load vector of order n+1, vector X the total system degrees of freedom of order n+1. Eq. (22) can be
solved by using the conventional numerical methods.
4. NUMERICAL EXAMPLE
To verify the efficiency of the proposed system, the Kum-ho bridge located at Kyung-bu highway in
Korea, which is undergoing serious vibrations due to vehicle load is used for numerical example. The viscoelastic post and tendon is installed at midpoint of the bridge. The parameters of the system are determined
9
by minimizing a pre-defined object function considering the deflection and acceleration at the midpoint of
the bridge. The displacements and accelerations at the midpoint of the bridges without any vibration control
system are compared with those of the bridge with TMD, rigid-post and tendon, and the proposed viscoelastic post and tendon.
4.1 Dimensions of the bridge
The Kum-ho bridge is a simple span bridge of pre-stressed concrete beam type. The bridge is
simplified as a Bernoulli-Euler beam with equivalent stiffness regardless of its complex cross section shape.
Table 1 shows material and section properties of the bridge mentioned. The geometry of the bridge is
shown in Fig. 5.
4.2 Vehicles loads
The vehicle used in the analysis is truck loads (HS20) according to the provisions of AASHTO. The
truck has 3 wheels in the longitudinal direction and 2 wheels in the transverse direction as shown in Fig. 4.
The loads on one longitudinal beam determined by considering the lanes on the bridge and the influence line
along the transverse direction. The calculated wheel loads are shown in Table 2. The speed of the vehicle is
determined by varying the velocity of the vehicle from 60 km/h to 120 km/h and observing the maximum
displacements at the midpoint. Fig. 6 shows the midpoint vertical displacements of bridge versus vehicle
speeds. As shown in Fig. 6, the response of the bridge under the vehicle moving at a speed of 74 km/h is
larger than that of speed of 100 km/h which is limited by Korea traffic administration. The vehicle speed of
74 km/h is considered to be controlled and the parameters of the proposed system are determined using
this speed.
10
4.3 Tuning conditions of visco-elastic post and tendon
The proposed control system has three parameters to be determined beforehand, that is the damping
and spring coefficients of the visco-elastic post and the cross sectional area of the tendon. In this study, the
cross sectional area of the tendon is determined by considering commercial tendons used in practice and
resulting properties are shown in Table 3. The damping and spring coefficient of the visco-elastic post
cannot be determined using the tuning methods for TMD such as Den Hartog’s [18] because the tendon
gives additional stiffness to the system in series. The tuning method for TMD by Den Hartog’s is based on
minimizing the displacement of the main system, so it is not appropriate to minimize accelerations of the
system also. In some cases, reducing accelerations can be a very important factor to design vibration
control system of bridges, especially when people on the bridge feel uneasiness due to vibrations. To
include effects of both displacements and accelerations, an object function J is introduced as follows.
J ?? ?
dcon ? max
a
? (1 ? ? ) con ? max
d uncon? max
auncon? max
(27)
where d uncon? max and a uncon? max are the maximum displacement and acceleration at midpoint when no
vibration control devoice is installed respectively, d con? max and a con? max are the maximum displacement and
acceleration at midpoint when the vibration control devoice is installed respectively, ? stands for absolute
value and ? (0.0~1.0) is a weight parameter can be adjusted according to design purpose. When
displacements are to be controlled, the value of ? should be taken as 1.0 and when accelerations are to be
controlled, the value of ? should be taken as 0.0. The value of ? between 0.0 and 1.0 means
displacements and accelerations are to be controlled with weighting value of ? and 1 ? ? respectively. The
minimization of the object function when damping coefficient Cd is varied from 104 ( N ?s / m) to
107 ( N ?s / m) and stiffness coefficient K d is varied from 104 ( N / m) to 107 ( N / m) is shown in Fig. 7 for
? ? 0.5 . The damping coefficients Cd
and spring coefficients K d
11
when ?
=0.5 are
Cd ? 7.6823 ? 10 4 ( N ?s / m ) and K d ? 1.3130 ? 10 6 ( N / m ) . These tuned parameters will be used for
numerical simulation.
4.4 Response of the bridges
The constant-average-acceleration method, a kind of Newmark ? , is used to guarantee the numerical
stability unconditionally.
4.4.1 Displacements of the midpoint of the bridge
The vertical displacements at midpoint of the bridge under moving load with speed of 74km/h, when no
control system is installed, are shown in Fig. 8-(a). As shown in Fig. 8-(a), the most of the transient
response is static deflection by vehicle weight. Fig. 8-(b) presents the displacements when rigid-post with
tendon is installed. The transient response is reduced due to increased stiffness by the system but the steady
state response is not. Fig. 8-(c) presents the displacements when TMD is installed. The steady-state
response is reduced but transient response is not. Fig. 8-(d) presents the displacements when the proposed
system is installed. Both transient response and steady state response are reduced. The maximum
displacements for each system are summarized in Table 4. The maximum deflection is 0.0955m in
uncontrolled case, 0.0936m, 0.0687m and 0.0770m with TMD, rigid-post and proposed control system.
Suppressions of the maximum displacement are about 2 %, 28.1% and 19.4 % in each case comparing to
uncontrolled (uninstalled) case. These maximum values are occurred at transient responses and when the
proposed system is tuned with ? ? 1 , more reduced maximum displacement can be obtained.
4.4.2 Accelerations of the mid-span of the bridge
12
The vertical accelerations at midpoint of the bridge under moving load with speed of 74km/h are shown
in Fig. 9-(a). Fig. 9-(b) presents the accelerations when rigid-post with tendon is installed, the rigid-post is
not appropriate to reduce accelerations. Fig. 9-(c) presents the accelerations when TMD is installed and
Fig. 9-(d) presents the accelerations when the proposed system is installed. The maximum accelerations for
each system is installed are summarized in Table 4. The maximum acceleration is 1.7711 m/s2 in
uncontrolled case, 1.4996 m/s2, 1.9452 m/s2 and 0.9280 m/s2 with TMD, king-post mechanism and
proposed control system. Suppressions of maximum acceleration are 15.3 %, -9.8% and 47.6 % in each
case. The proposed control system is more efficient for the reductions of maximum acceleration than the
rigid-post with tendon and TMD.
5. Conclusions
A passive vibration control system for reducing excessive traffic-induced vibration of bridges is
proposed. The proposed system consists of visco-elastic post and tendon. The tendon and elastic part of
the system increases the stiffness of the bridge and reduces transient response when vehicles are on the
bridge like that of the king post mechanism. The visco-elastic post with the mass of the system absorbs
vibration energy and reduces steady-state response after vehicles cross the bridge like that of TMD. From
the numerical simulation conducted on the existing bridge undergoing vibration problems, it is shown that
the proposed control system reduces both transient and steady-state responses efficiently. Therefore, the
proposed system has merits of both king-post and TMD and can be used reduce deflection and
acceleration of the bridge due to moving loads simultaneously.
13
Acknowledgement
This research was supported by the National Research Laboratory (NRL) program (Grant No.: 2000N-NL-01-C-251) from the Ministry of Science and Technology in Korea. The financial support is
gratefully acknowledged.
References
[1] K. S. Park, H. J. Jung and I. W. Lee. A Comparative Study on Aseismic Performances of Base
Isolation Systems for Multi-span Continuous Bridge. Engineering Structures 2002; 24:1001-1013.
[2] K. Kawashima and S. Unjoh. Seismic Response Control of Bridges by Variable Dampers. Journal of
Structural Engineering 1994, ASCE; 120(9):2583-2601.
[3] M. D. Symans and S. W. Kelly. Fuzzy Logic Control of Bridge Structures using Intelligent Semi-active
Isolation Systems. Earthquake Engineering and Structural Dynamics 1999; 28:37-60.
[4] B. Erkus, M. Ab? and Y. Fujino. Investigation of Semi-active Control for Seismic Protection of
Elevated Highway Bridges. Engineering Structures 2002; 24:281-293.
[5] J. N. Yang. Active Control and Stability of Cable-stayed Bridge. Journal of Engineering Mechanics
Division 1979. ASCE; 105(4):677-694.
[6] Y. Fujino, T. Susumpow and P. Warnichai. Active Control of Cable and Cable-structure System.
JSME International Series C 1995; 38(2):260-266.
[7] H. J. Jung, B. F. Spencer, Jr. and I. W. Lee. Seismic Protection of Cable-Stayed Bridges Using
Magnetorheological Dampers. The Seventh U. S. National Conference on Earthquake Engineering
2002.
14
[8] T. T. Soong and G. F. Dargush. Passive Energy Dissipation Systems in Structural Engineering. John
Wiley & Sons, 1997.
[9] T. T. Soong, Active Structural Control : Theory and Practice. John Wiley & Sons. 1990.
[10] H. C. Kwon, M. C. Kim and I. W. Lee. Vibration Control of Bridges under Moving Loads.
Computers and structures 1998; 66(4):473-480.
[11] Y. Kajikawa. Controll of Traffic Vibration on Urban Viaduct with Tuned Mass Damper. Journal of
Structural Engineering 1989. JSCE; 35A:585-595.
[12] W. N. Patten, R. L. Sack and Q. He. Controlled Semi-active Hydraulic Vibration Absorbers for
Bridges. Journal of Structural Engineering 1996. ASCE; 122(4):187-192.
[13] T. Hayashikawa and N. Watanabe. Dynamic Behavior of Continuous Beams with Moving Loads.
Journal of Engineering Mechanics Division 1981. ASCE; 107(EM1):229-246.
[14] J. M. Biggs. Introduction to Structural Dynamics. McGraw-Hill, 1982.
[15] Y. Cai, S. S. Chen, D. M. Rote and H. T. Coffey. Vehicle/Guideway Interaction for High Speed
Vehicles on a Flexible Guideway. Journal of Sound and Vibration 1994; 175(5):625-646.
[16] E. Savin. Dynamic Amplication Factor and Response Spectrum for the Evaluation of Vibrations of
Beams under Successive Moving Loads. Journal of Sound and Vibration 2001; 248(2):267-288.
[17] G. T. Michaltsos. Dynamic Behaviour of a Single-span Beam Subjected to Loads Moving with
Variabl Speeds. Journal of Sound and Vibration 2002; 258(2):359-372.
[18] D. Hartog. Mechanical Vibrations. New York: Dover Publications, 1985.
15
List of Figures
Fig. 1. Idealized Bernoulli-Euller beam
Fig. 2. Bridge with visco-elastic post and tendon
Fig. 3. Free body diagram of the proposed control system
Fig. 4. Vehicle loads: HS20
Fig. 5. Geometry of the bridge
Fig. 6. Maximum displacement at midpoint of the bridge (m)
Fig. 7. Object function J ( ? ? 0.5 )
Fig. 8. The comparison of displacement at midpoint: (a) uncontrolled; (b) king-post; (c) TMD; (d)
proposed
Fig. 9. The comparison of acceleration at midpoint: (a) uncontrolled; (b) king-post; (c) TMD; (d) proposed
16
List of Tables
Table 1. Material and section properties of the bridge
Table 2. Moving load
Table 3. Material and section properties of the tendon
Table 4. Summary of numerical analysis results
17
F
v m/sec
L
Fig.1 Idealized Bernoulli-Euller beam
18
vt
x
yb(x,t)
K
E C , AC
Cd
d
H
?
LC
z(t)
md
y d(t) ? z(t) ? yb(L/ 2 ,t)
L
Fig.2 Bridge with visco-elastic post and tendon
19
y b ( L / 2, t )
T,?
C
PD ? PS
?
m d [ ?y?d ( t ) ? ?y?b ( L / 2, t )]
Fig. 3 Free body diagram of the proposed control system
20
19'? 4"
14'? 0"
W ? 156080 N
0.4W
0.4W
0.1W
6'
Fig. 4 Vehicle loads: HS20
21
42 m
914 mm
2150 mm
600 mm
Fig. 5 Geometry of the bridge
22
d i s p l a c e me n t ( m )
0.096
0.095
0.094
0.093
0.092
60
80
100
120
velocity(m/s)
Fig. 6 Maximum displacement at midpoint of the bridge (m)
23
J min ? 0.665
n x : step number for Cd
n y : step number for Kd
Fig. 7 Object function J ( ? ? 0.5 )
24
d i s p l a c e me n t ( m )
0.04
0
-0.04
Static + dynamic
Static
-0.08
-0.12
0
2
4
6
8
10
time(sec)
d i s p l ac emen t ( m)
(a)
0.04
0
-0.04
Uncontroll ed
King-Post
-0.08
-0.12
0
2
4
6
8
10
time(sec)
(b)
displ acement ( m)
0.04
0
-0.04
Uncontrolled
TMD
-0.08
-0.12
0
2
4
6
8
10
time(sec)
(c)
di spl ac ement ( m )
0.04
0
-0.04
Uncontrolled
Proposed
-0.08
-0.12
0
2
4
6
8
10
time(sec)
(d)
Fig. 8 The comparison of displacement at midpoint: (a) uncontrolled; (b) king-post; (c) TMD ;(d)
proposed
25
accel erat i on ( m/ s 2)
2
1
0
-1
-2
0
2
4
6
8
10
time(sec)
acce l er at i on( m/ s 2)
(a)
2
1
0
-1
Unc ontrol led
Ki ng -Post
-2
0
2
4
6
8
10
time(sec)
ac ce l er at i on( m/ s 2)
(b)
2
1
0
-1
Uncontrolled
TMD
-2
0
2
4
6
8
10
time(sec)
a c ce l er a t i on ( m/ s 2 )
(c)
2
1
0
-1
Uncontrolled
Proposed
-2
0
2
4
6
8
10
time(sec)
(d)
Fig. 9 The comparison of acceleration at midpoint: (a) uncontrolled; (b) king-post; (c) TMD; (d)
proposed
26
Table 1. Material and section properties of the bridge
Elastic modulus
Mass density
Damping ratio
Section area
Section inertia
(N/m2)
(kg/m3)
(%)
(m2)
(m4)
2.94? 1010
2,096
0.6
0.3162
0.1551
27
Table 2. Moving load
Front wheel
Middle and rear wheel
(N)
(N)
31216
124863
28
Table 3. Material and section properties of the tendon
Elastic modulus
Mass density
Section area
(N/m2)
(kg/m3)
(m2)
2.1? 1011
7500
0.0028
29
Table 4. Summary of numerical analysis results
Uncontrolled
Max. displacements
TMD
King-post
Proposed
0.0936
0.0687
0.0770
(2.0%)
(28.1%)
(19.4%)
0.0301
0.0221
0.0250
(0.7%)
(27.1%)
(17.5%)
1.4996
1.9452
0.9280
(15.3%)
(-9.8%)
(47.6%)
0.3561
0.9129
0.2865
(56.2%)
(-12.3%)
(64.8%)
0.0955
(m)
RMS displacements
0.0303
(m)
Max. accelerations
(m/sec2)
1.7711
RMS accelerations
(m/sec2)
0.8128
( ) : reduction ratio
30