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)