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1998/
1
黃震興
國立台灣科技大學營建系國家地震工程研究中心
加裝黏性阻尼器之結構減震設計
The difference of the pressure between each side of the piston head results in the damping force.
Longitudinal Cross Section of A Fluid Damper
The damping constant of the damper can be determined by adjusting the configuration of the orifice of the piston head.
1998/
2
Mechanical Properties of Fluid Viscous Dampers
Force-Velocity Relationship of Viscous Dampers
( )uuCFD && sgnα=
1=α
10 <<α
Linear Viscous Dampers
Nonlinear Viscous Dampers
Velocity, V
Dam
per F
orce
, FD
Line 2
Line 1: FD=CN1Vα, Nonlinear Damper with α<1
Line 3
Line 2: FD=CLV, Linear DamperLine 3: FD=CN2V
α, Nonlinear Damper with α>1
Line 1
Mechanical Properties of Fluid Viscous Dampers
Hysteresis Loops of Viscous Dampers
Force
Displacement
(Sinusoidal Motion)
Linear Damper
Nonlinear Damper
No storage stiffness!Viscous dampers won’t change the natural frequency of the primary structure.
1998/
3
H
DD
1θ
4θ
2θ
1θ 3θ
2θ
(a) (b)
(c) (d)
θ
D
H
黏性阻尼器安裝型式
Tai-Shin Bank
1998/
4
Tai-Shin BankHeadquarter
1998/
5
1998/
6
Site
御之苑
中間層鋼板連接上部抗彎構架
外部鋼板連接下部抗彎構架
基本構造圖
實績照片
剖面圖
填充粘性體
1998/
7
新店慈濟醫院
1998/
8
The Equivalent Damping Ratio of Structures with
Linear Viscous Dampers
1998/
9
Linear Viscous Damping Ratio
Sinusoidal Motionk
Viscous Damper
M
tuu ωsin0=)sin(0 δω += tPP
4 s
Dd W
Wπ
ξ =
Force
Displacement
DWsW
MDOF System withLinear Viscous Dampers
MDOF System with Linear Viscous Dampers
θdeff ξξξ += 0
inherent damping ratio
damping ratio attributed to added dampers
k
iFi
jVj
eff W4
WW
π
++β=β
∑∑
1998/
10
MDOF System with Linear Viscous DampersMDOF System with Linear Viscous Dampers
θdeff ξξξ += 0
SDOFs
Dd W
Wπ
ξ4
=
MDOFS
dampersallD
d Wπ4
W∑=ξ
MDOF System withLinear Viscous Dampers
MDOF System with Linear Viscous Dampers
Energy Dissipated by Linear Viscous Dampers( in one cycle of sinusoidal vibration )
tuu ωsin0=
( ) uCtPP &=+= δωsin0
P(t)
U(t)ωπ
ωωωπ
ωπ
20
2
0
2220
2
0
2
)(cos
uC
tdtuC
dtuCduuCduFW DD
=
=
===
∫
∫∫∫ &&
2rjj
2
Vj CT
2W δπ
=
1998/
11
MDOF System with Linear Viscous Dampers
S
jj
d Wπ
W
2
∑=ξ
Considering the First Mode of Vibration
jrjj
jj
jjj
j CT
uCW θφπωπ 222
0
2 cos2∑∑∑ ==
[ ] [ ]
∑∑ ==
ΦΦ=ΦΦ=
iii
iii
TT
s
mT
m
mKW
2
2
222
1
2
111
4 φπφω
ω
jφ
rjφjθ
∑=i
iik uF21W
Effective Damping Ratio of A Structure with Linear Viscous Dampers (FEMA273)
∑
∑
∑
∑
+=
+=
iii
jrjj
j
iii
jrjj
j
eff
m
CT
mT
CT
2
22
0
2
2
2
222
0
4
cos
42
cos2
φπ
θφξ
φππ
θφπ
ξξ
∑
∑ ∑++=
iii
j iFijrjj
eff uFT
WTuC πθπββ
2cos22
1998/
12
The Effective Damping Ratio of Structures with
Nonlinear Viscous Dampers
SDOF System with Nonlinear Viscous Dampers
Sinusoidal Motion
αuCFD &=Damper Force
tuu ωω sin0=&Velocity Time History
Nonlinear viscous damper
K
with damping constant, C
M
SDOF System with Nonlinear Viscous Dampers
1998/
13
Damping Ratio Contributed by Nonlinear Viscous Dampers
gd XMKXXCXCXM &&&&&& −=+++ α0
geq XMKXXCCXM &&&&& −=+++ )( 0
Nonlinear
Equivalent Linear
Approximated by
keff Wπ
ξξ40 +=∑j
jdW )(
work done by nonlinear damper
Equal Energy
Damping Ratio Contributed by Nonlinear Viscous Dampers
1998/
14
SDOF System with Nonlinear Viscous Dampers
Energy Dissipated by Nonlinear Viscous Dampers
( ) dttuC
dtuC
dtuFduFW DDD
∫=
∫=
∫=∫=
++
+
ωπ αα
ωπ α
ωπ
ωω 20
110
20
1
20
sin
&
& αuCFD &=
tuu ωω sin0=&
Let θω 2=t θωddt 2
=
( )
( )( )α
αω
θθθω
θθω
ω
ααα
π ααααα
π αα
+Γ+Γ
=
∫=
∫=
++
++++
++
2212
cossin22
2sin2
21
02
20
1110
2
011
0
uC
duC
duCWD ααωλ += 10uCWD
( )( )α
αλ α
+Γ+Γ
= +
2212
22
)()()(),(
cossin2)2
1,2
1(2/
0
bababaB
dnmnmB
FunctionBeta
+ΓΓΓ
=
=++ ∫ θθθ
π
3.13.141
3.23.170.95
3.23.200.9
3.23.240.85
3.33.270.8
3.33.300.75
3.33.340.7
3.43.380.65
3.43.420.6
3.53.460.55
3.53.500.5
3.53.540.45
3.63.580.4
3.63.630.35
3.73.670.3
3.73.720.25
3.83.770.2
3.83.830.15
3.93.880.1
FEMACalculatedExponent α
Values of Parameter λ
1998/
15
MDOF System with Linear Viscous Dampers
S
jj
d Wπ
W
2
∑=ξ
Considering the First Mode of Vibration (Linear Damper)
jrjj
jj
jjj
j CT
uCW θφπωπ 222
0
2 cos2∑∑∑ ==
[ ] [ ]
∑∑ ==
ΦΦ=ΦΦ=
iii
iii
TT
s
mT
m
mKW
2
2
222
1
2
111
4 φπφω
ω
jφ
rjφjθ
MDOF System with Nonlinear Viscous Dampers
Effective Damping Ratio of A Structurewith Nonlinear Viscous Dampers
θ ∑
∑−−
++
+=
iii
jrjj
j
eff mA
C
221
11
0 2
cos
φωπ
θφλξξ αα
αα
Displacement Dependent
A
1998/
16
非線性黏性阻尼器之位移設計法
於耐震設計地震反應譜作用下,阻尼比ξeff之位移反應
1
20 . 2 5 ( )1 . 5 0 . 5
4 0 1
d
d D
De f f
A S
S Z I C C T m
Cξ
= Γ
=
= ++
BISaD /
MDOF System with Nonlinear Viscous Dampers
Effective Damping Ratio of A Structurewith Nonlinear Viscous Dampers
θ ∑
∑−−
++
+=
iii
jrjj
j
eff mA
C
221
11
0 2
cos
φωπ
θφλξξ αα
αα
Displacement Dependent
1998/
17
The loading combination factorsCF1 and CF2
Responses at the instant of maximum acceleration
Responses at the instant of
maximum displacement
Responses at the instant of
maximum velocity
1CF × 2CF ×+||
F
u
A
CB
u0δcosu0
K
-u0
Loading Combination Factors
1998/
18
u
u.
u0
tt*
T 2t
-u0
δ/ωn
n/ωπ
nω
-u0
Time at max. acc.
F
u
A
CB
u0δcosu0
K
-u0
FEMA 273 規範建議:
最大加速度反應 =
( )[ ]effCF ξ2tancos 11
−=
上式僅適用於含線性阻尼器之構架
( )[ ]effCF ξ2tansin 12
−=
最大位移反應 x (CF1) + 最大速度反應 x (CF2)
1998/
19
A Design Example of A Structure with Linear Viscous Dampers
A Design Example
Properties of the Designed Structure
Mass
9378 kg
9378 kg
8155 kg
First ModeTθcos
0.494
0.805
1
0.83
0.87
0.87
0.33sec
∑
∑=
iii
jrjj
j
d m
CT
2
22
4
cos
φπ
θφξ
Assume all dampers have the same damping constant
0.494
0.311
0.195
1φ 1rφ
( )( )222
222222
494.09378805.09378181554494.083.0311.087.0195.087.033.0218.0
×+×+××+×+×××
==π
ξ Cd ( )mskNC −= 210
θ
Each floor contains two linear dampers
Assume the primary frame of the structure remains elastic
damping ratio attributed toviscous damper %18=dξ
inherent damping ratio of the structure %20 =ξ
Design of Linear Viscous Damping System
1998/
20
黃震興、曾義軒. “黏性阻尼器於晶圓廠耐震補強設計研究.”國家地震工程研究中心報告, NCREE-05-xxx, 2005
黃震興、曾義軒. “黏性阻尼器於橋樑隔減震設計應用研究.”國家地震工程研究中心報告, NCREE-05-xxx, 2005
黃震興、蔡俊祥. “黏性阻尼器於RC結構地震力反應試驗研究.”國家地震工程研究中心報告, NCREE-04-010, 2004.
黃震興、何松晏.“使用黏性阻尼器減震結構設計公式修正(II).”國家地震工程研究中心報告, NCREE-04-009, 2004.
黃震興、黃尹男、易序良. “使用黏性阻尼器減震結構設計公式修正(I).”國家地震工程研究中心報告, NCREE-03-12, 2003.
黃震興、黃尹男、李昭逸. “含黏性阻尼器減震結構之非彈性地震反應試驗與分析.”國家地震工程研究中心報告, NCREE-03-011, 2003.
黃震興、黃尹男、陳偉松.“LRB與FVD隔震系統於近斷層地震之設計.”國家地震工程研究中心報告, NCREE-02-013, 2003
黃震興、黃尹男、洪雅惠. “含非線性黏性阻尼器結構之減震試驗與分析.”國家地震工程研究中心報告, NCREE-02-020, 2002
黃震興、黃尹男,”使用線性黏性阻尼器結構之耐震試驗與分析.”國家地震工程研究中心報告,NCREE01-022, 2001。
參考文獻
FEMA 273/274 (1997)
FEMA 356 (2000)
FEMA 368/369 (2001)
MCEER-92-0032 (1992)
MCEER-97-0004 (1997)
MCEER-00-0010 (2000)
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