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Computational Structural Engineering InstituteAutumn Conference 2002Oct. 18 - 19, 2002
VIBRATION CONTROL OF BRIDGE FOR SERVICEABILITY
Jun-Sik Ha1), Ji-Seong Jo2), Sun-Kyu Park3), In-Won Lee4)
1) Graduate Student, Department of Civil and Environmental Engineering, KAIST
2) Ph.D. Candidate, Department of Civil and Environmental Engineering, KAIST
3) Professor, Department of Civil Engineering, SungKyunKwan Univ.
4) Professor, Department of Civil and Environmental Engineering, KAIST
Jun-Sik Ha1), Ji-Seong Jo2), Sun-Kyu Park3), In-Won Lee4)
1) Graduate Student, Department of Civil and Environmental Engineering, KAIST
2) Ph.D. Candidate, Department of Civil and Environmental Engineering, KAIST
3) Professor, Department of Civil Engineering, SungKyunKwan Univ.
4) Professor, Department of Civil and Environmental Engineering, KAIST
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 22
CONTENTS
INTRODUCTION
FORMULATION OF MATHEMATICAL MODEL
NUMERICAL EXAMPLE
CONCLUSIONS
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 33
INTRODUCTION
Bridges, which have lightweight, are more
vulnerable to heavy weight vehicle.
The vibration induced by moving loads makes
passengers uncomfortable.
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 44
Objective of Study
Propose passive control device for the improvement
of serviceability.
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 55
FORMULATION OF MATHEMATICAL MODEL
Modeling
H
),(vb tx
dC
dv
dK ,T C
L
tv tv tv tv tv tv
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 66
Equation of Motion
Control Device
SD PP ,T C
)vv( db m
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 77
(1)
(2)
2
b
cos2
)]vv([
CC
dSD
CC
CC AE
HmPP
AE
TL
3
bb cos2
)]vv([
cos)(v(t)v
CC
dSDCd AE
HmPPt
(3)
0)](v(t)[vcos2
)(v)(v)](v),2/(v[
db
3
b
tH
AE
tKtCttLm
CC
ddddd
SD PP ,T C
)vv( db m
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 88
(4)
(5)
(6)
Bridge
),(vv
t
v4
b
4
b
2
b
2
txfx
EIt
CA
L
xxtqxtx i
iii
sin)( where )()(),(vb
),()()()( txfqEIqCqAi
iii
iii
ii
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 99
(7)
(8)
si
iisi
iisi
iis txfqEIqCqA ),()()()(
Multiplying eq.(3) by s
dxtxfdxqEI
dxqCdxqA
L
s
L
iiis
L
iiis
L
iiis
00
00
),()(
)()(
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 1010
(9)
Applying orthogonal condition
dxtxfdxqmqdxCqdxAL
s
L
ssss
L
ss
L
s 00
22
0
2
0
2 ),(
L
L
nnnnnn
dxL
xnA
dxL
xntxf
tqtqtq
0
2
02
)(sin
]sin),([)()(2)(
Substituting mode shape function of beam
(10)
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 1111
(11)
ddddw
L
SDw
L
KCL
vtnP
dxL
xnLxPPvtxP
dxL
xntxf
vvsin
]sin)}2()()([{
]sin),([
0
0
(12)
L
vtnPtKtC
tqtqtqAL
wdddd
nnnnnn
sin)(v)(v
)]()(2)([2
2
2)(sin
0
2 ALdx
L
xnA
L where
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 1212
Optimization of Device Parameters
Using Pareto Optimization(“Engineering Optimization”, Singiresu S. Rao)
max
max
max
max )1(
uncon
con
uncon
con
a
a
d
dJ (13)
When J is minimized, the damping coefficient and spring constant in control device are optimum.
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 1313
NUMERICAL EXAMPLES
Composite Steel Plate Girder Bridge “Generalized of Design for Short Span Steel Bridges Using Rolled Beam”,Magazine of the Korean Society of Steel Construction, vol.14, No.1, pp. 77~82.
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 1414
Geometry and Material Properties
Geometry
4
2
00102.0
0708.0
)(18
mI
mA
mL
b
2003.0
m 1H
mAC
Bridge
Control device
Bridge
006.0
/14182
/1006.23
211
mkg
mNE
b
b
3
211
C
/6200
/101.2E
mkg
mN
C
Material Properties
Control device
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 1515
Vehicle velocity
Number of modes : 3
km/h70m/s4.91v
Coefficient of Pareto optimization
4.0
Parameters
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 1616
Optimization of Device Parameters The Normed Displacement of Mid-span
)/(101~101
)/(101~10161
61
msNC
mNK
d
d
As the damping coefficient and spring constant are increased, the normed displacement are decreased. Max : 46 % reduction
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 1717
The Normed Acceleration of Mid-span
The optimal damping
constant exists. Max : 36 % reduction
)/(101~101
)/(101~10161
61
msNC
mNK
d
d
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 1818
J of Mid-span
1
2
3
4
5
6
12
34
56
0.7
0.71
0.72
0.73
0.74
0.75
0.76
C(Ns/m)
K(N/m)
J
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 61
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
K(N/m)
C(Ns/m)
0.71031 0.71031 0.71031
0.71031 0.71031 0.71031
0.71341 0.71341 0.71341
0.71341 0.71341 0.71341
0.71652
0.71652 0.71652 0.716520.71962 0.71962 0.71962
0.72273 0.72273 0.722730.72584 0.72584 0.72584
0.72894 0.72894 0.728940.73205 0.73205 0.73205
0.73516 0.73516 0.735160.73826 0.73826 0.73826
0.74137 0.74137 0.741370.74448 0.74448 0.74448
0.74758 0.74758 0.747580.75069 0.75069 0.75069
0.75379 0.75379 0.75379
)/(101~101
)/(101~10161
61
msNC
mNK
d
d
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 1919
Optimal Damping Coefficient and Spring Constant
)/(100.1
)/(105.85
5
msNC
mNK
d
d
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 2020
Simulation Results Displacement of Mid-span
The maximum reduction
is 22%.
0 2 4 6 8
TIM E (s)
-0 .2
-0.1
0
DIS
PL
. a
t ce
nte
r (m
)
U ncontro lledC ontro lled
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 2121
Acceleration of Mid-span
The maximum reduction
is 21.1%.
0 2 4 6 8
TIM E (s)
- 8
- 4
0
4
8
AC
C. a
t ce
nte
r (m
/s^2
)
U ncontro lledC ontro lled
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 2222
mid-span responses
Uncontrolled Case
Controlled Case
Reduction (%)
Displacement
of mid-span(m)0.2241 0.1749 22.0
Acceleration
of mid-span(m/ )9.8963 7.8059 21.1
Table 1. Performances of proposed control device
2s
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 2323
CONCLUSIONS
Proposed Passive Control Device Proposed Passive Control Device
can control both displacement and acceleration simultaneously.
can decay the steady-state responses much faster.
Therefore, proposed passive control device could be effectively used for vibration control of bridges.
Therefore, proposed passive control device could be effectively used for vibration control of bridges.