TS31-03...(TVMD), using a rotational viscous mass damper by connecting it to the primary structure with a soft spring. The basic concept of the TVMD is the same as a tuned mass damper

Title: Seismic Control Design of Tall Buildings Using Tuned Viscous MassDampers

Authors: Kohju Ikago, Associate Professor, Tohoku UniversityYoshifumi Sugimura, Researcher, NTT Power and Building FacilitiesKenji Saito, Engineer, NTT Power and Building FacilitiesNorio Inoue, Graduate Student, Tohoku University

Subject: Structural Engineering

Keywords: DampingSeismicTuned Mass Damper

Publication Date: 2011

Original Publication: CTBUH 2011 Seoul Conference

Paper Type: 1. Book chapter/Part chapter2. Journal paper3. Conference proceeding4. Unpublished conference paper5. Magazine article6. Unpublished

© Council on Tall Buildings and Urban Habitat / Kohju Ikago; Yoshifumi Sugimura; Kenji Saito; Norio Inoue

ctbuh.org/papers

http://ctbuh.org/papers

1/24/2011

TS31-03

Seismic control design of tall buildings using tuned viscous mass dampers

Kohju Ikago1, Yoshifumi Sugimura2, Kenji Saito3 and Norio Inoue4

Graduate School of Engineering, Tohoku University, Sendai, Japan, [email protected]

Research & Development Headquarters, NTT Facilities, Inc., Tokyo, Japan, [email protected] Building Engineering Headquarters, NTT Facilities, Inc., Tokyo, Japan, [email protected]

Graduate School of Engineering, Tohoku University, Sendai, Japan, [email protected]

Kohju Ikago Biography Dr. Ikago is currently an associate professor of graduate school of engineering, Tohoku University. He graduated from Kyoto University, Japan, where he majored in structural engineering and studied structural optimization. He had been with an architectural design firm, NIKKEN SEKKEI Ltd. for fifteen years before he moved to Tohoku University. His experiences as a structural engineer include structural design of many base-isolated buildings, high-rise buildings and spatial frames. He now uses his previous experience to advantage in his academic activities. His current research topics are, seismic control design using a newly developed rotational viscous mass damper, use of structural optimization in structural design in practice, and inverse energy variational principle in anti-seismic design.

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Abstract Saito et al. developed a new seismic control system, which is denoted as the tuned viscous mass damper (TVMD), using a rotational viscous mass damper by connecting it to the primary structure with a soft spring. The basic concept of the TVMD is the same as a tuned mass damper (TMD) or a dynamic vibration absorber (DVA), and its optimum design is also obtained by using fixed points of its resonance curve as proposed by Den Hartog. Whereas a TMD for a building is effective against wind-induced vibrations, it is not necessarily effective against earthquake-induced vibrations if the effective mass ratio is relatively small. Hence, a large supplemental mass, larger than several percent of the effective mass of the primary structure, is required to achieve reduction in seismic vibrations, and installing such a large mass to the building may cause many problems. However, a large apparent mass, larger than several percent of the effective mass of the primary structure, can be easily obtained by a mass amplifying mechanism using a ball-screw and cylindrical flywheel with a small actual mass in the TVMD. Whereas a design method for a multiple-degree-of-freedom (MDOF) TVMD seismic control system based on a numerical optimization is presented by the authors, more simple design methods that are suitable for design in practice have not yet been presented. At the preliminary design stage, it is essential for structural designers to grasp the seismic response characteristics of the structure in terms of modal responses. Although complex eigenvalue analysis is required as a seismic control system with TVMDs is non-proportionally damped, commonly used in practice are not the complex modes but un-damped real modes. In this paper, a design example illustrates a simple seismic response estimation method that is useful for the seismic control design of a structure incorporated with TVMDs using the square-root-of-the-sum-of-square (SRSS) of the maximum modal responses derived from the un-damped real eigenvalue analysis. Keywords: Seismic control, Tuned Mass Damper, Fixed point method, Optimum design, Complex eigenvalue analysis

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Introduction

Saito et al. developed a new seismic control system using a rotational viscous mass damper (Fig. 1) by connecting it with a soft spring (Saito et al., 2008). The present device is denoted as the tuned viscous mass damper (TVMD). The basic concept of the TVMD is the same as a tuned mass damper (TMD) or a dynamic vibration absorber (DVA), and its optimum design is also obtained by using fixed points of its resonance curve (Den Hartog, 1956). Whereas a TMD for a building is effective against wind-induced vibrations (McNamara, 1979), it is not necessarily effective against earthquake-induced vibrations if the effective mass ratio is relatively small (Kaynia et al., 1981).

Figure 1. Schematic representation of the rotational viscous mass damper

Hence, a large supplemental mass, larger than several percent of the effective mass of the primary structure, is required to achieve reduction in seismic vibrations, and installing such a large mass to the building may cause many problems. However, a large apparent mass, larger than several percent of the effective mass of the primary structure, can be easily obtained by a mass amplifying mechanism using a ball-screw and cylindrical flywheel with a small actual mass in the TVMD. Whereas a design method for a multiple-degree-of-freedom (MDOF) TVMD seismic control system based on a numerical optimization is presented by the authors (Ikago et al., 2010), more simple design methods that are suitable for design in practice have not yet been presented. At the preliminary design stage, it is essential for structural designers to grasp the seismic response characteristics of the structure in terms of modal responses. Although complex eigenvalue analysis is required as a seismic control system with TVMDs is non-proportionally damped, commonly used in practice are not the complex modes but un-damped real modes. In this paper, a design example illustrates a simple seismic response estimation method, which is useful for the seismic control design of a structure incorporated with TVMDs, using the square-root-of-the-sum-of-square (SRSS) of the maximum modal responses derived from the un-damped real eigenvalue analysis.

Analysis model

In this study, we use a 10 story benchmark structure provided by the Japan Society for Seismic Isolation (JSSI) as a seismic control design example. Schematics and characteristics for the analytical model are shown in Fig. 2 and Tables 1 and 2. The benchmark structure is controlled by a TVMD system whose additional mass ratio is 0.06, which corresponds to the equivalent additional damping ratio of 10%.

viscous material inner cylinderouter cylinder

radial bearing

thrust bearingball nutball bearings

thrust bearingradial bearing

rotational motion

linear motion

sealcylindrical flywheel

ball screw

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Equatio

As shown inthe i th stothe TVMD ifreedom strdescribed a

on of mot

n Fig. 2 and ory, respectivinstalled in tructure inco

as follows:

tion of M

Table 1, ,i

mvely.

, ,,

r i d im c

the i th storyrporated wit

Tab

sto

1

Table 2.

Table

sto

19876

MDOF sei

,i

c and ik re

iand

,b ik rep

y, respectiveth one TVM

+Mx Cx&&

Figure

ble 1. Specific

ory mass[to10 8759 6498 6567 6606 6675 6704 6763 6802 6821 700

Un-dampedmode

period (sangular freq

e 3. Specific

ory ,r im

0 979 118 137 156 17

ismic co

epresent the resent the m

ely. Thus, theMD in each

+ = -Cx Kx M&

2. Analytica

cation of the primary stru

on] stiffness[k15855180112202524479291893061632826383023835527996

d fundamentae 1ssec) 2.0quency 3.1

cation for th

i[ton]

,bik [

75 1008 114

355 140507 156796 186

ntrol sys

mass, dampmass, dampine equation ostory, i.e. n

0xMr&&

l model

analytical mucture kN/m] height[50 410 450 490 490 460 460 420 450 460 6

al period of thst 2nd 3r01 0.76 0.413 8.27 13

he TVMD sy

kN/m],d i

c [k

120 95496 10059 13625 14631 17

stem with

ping coefficieng coefficien

of motion of TVMDs in

model

[m]

he model rd46.7

ystem

Ns/m]

57 087 330 478 762

h TVMD

ent and sheant and springa damped n

the whole

ar stiffness ofg stiffness ofn -degree-of-structure, is

(1)

f f -s

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5 1884 19542 1848 4 2020 20953 1982 3 2357 24448 2312 2 2360 24482 2316 1 1723 17870 1690

in which x is a 2n vector and consists of an n displacement vector of the primary system relative to the ground and an n deformation vector of the each damper in each story; 1 0n = in this study.

1 2 1 2{ }T

n d d dnx x x x x x= , , , , , , ,x L L (2)

where the superscript T denotes a transposed matrix. The lower half of the influence coefficient vector r is the n zero vector, whereas its upper half is the n unit vector, because the TVMDs are not activated by the ground motion but by the relative displacement in each story.

{11 100 0}T= , , , , , , ,r L L (3)

,M C and K denote the 2n dimension mass, damping and stiffness matrices, respectively.

p

r

殞油= 油油薏

M OM

O M (4)

p

d

殞油= 油油薏

C OC

O C (5)

11 12

21 22

p b b

b b

殞 +油= 油油薏

K K KK

K K (6)

where,

1

2

0 0

0

0

0 0

p

n

m

m

m

殞油油油油油油油油油油油油油油油油油油薏

=M

L

M

M O

L

(7)

1

2

0 0

0

0

0 0

r

r

r

rn

m

m

m


=M

L

M

M O

L

(8)

1 2 2

2 2 3 3

1 1

0

0 0

0

p

n n n n

n n

c c c

c c c c

c c c c

c c

殞油油油油油油油油油油油油油油 - -油油油油薏

+ -

- + -

=

- + -

-

C

L

M

O

M

L

(9)

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1

2

0 0

0

0

0 0

d

d

d

dn

c

c

c


=C

L

M

M O

L

(10)

1 2 2

2 2 3 3

1 1

0

0 0

0

p

n n n n

n n

k k k

k k k k

k k k k

k k

殞油油油油油油油油油油油油油油 - -油油油油薏

+ -

- + -

=

- + -

-

K

L

M

O

M

L

(11)

1 2 2

2 2 3 3

11

1 1

0

0 0

0

b b b

b b b b

b

b n b n bn bn

bn bn

k k k

k k k k

k k k k

k k

殞油油油油油油油油油油油油油油 , - , -油油油油薏

+ -

- + -

=

- + -

-

K

L

M

O

M

L

(12)

1 2

2 3

12

1

0

0

0

0 0

b b

b b

b

b n bn

bn

k k

k k

k k

k

殞油油油油油油油油油油油油油油 , -油油油油薏

-

-

=

-

-

K

L

M

O O

M

L

(13)

21 12

T

b b=K K (14)

1

2

22

0 0

0

0

0 0

b

b

b

bn

k

k

k


=K

L

M

M O

L

(15)

pC in Eq. (9) is the inherent damping matrix for the primary structure. If we assume that it is proportional to

the stiffness matrix and the damping ratio for the 1st mode of the primary structure 1x equals 0.02, it can

be rewritten as follows:

1

1 1

2 0.04p p p

p p

x

w w= =C K K (16)

where 1 pw is the lowest fundamental angular frequency of the un-damped primary structure.

Eigenvalue analyses

The equation of motion for the uncontrolled primary structure is,

0p p p p p p px+ + = -M x C x K x M 1&& & && (17)

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where, p

x is the displacement vector of the primary system relative to the ground and {1,1,1, ,1}T=1 L is the influence coefficient vector. The eigenvalue analysis problem for the un-controlled primary structure is described as follows: 0

p p p- W =K M (18)

Let

j pW and

j pu denote the j th eigenvalue and eigenvector derived from Eq. (18) respectively, the j th

fundamental angular frequency j pw

for the un-damped primary system

equals the square root of the j th

eigenvalue.

j p j pw = W (19)

Since we assume that the damping matrix pC is proportional to the stiffness matrix, pre-multiplying Eq. (17)

by T

iφ yields n modal equations by using the transformation =x uΦ , where 1 2

[ ]n

Lφ φ φΦ = and u is

the modal coordinate. 22 ( )

i i i i i i i gu u u x tx w w+ + = - G&& & && (20)

in which ( ) /T

i i p iMG= M 1φ is the modal participation factor, 1 1

( ) / 2 /T

i i p i i i iMx w x w w= =Cφ φ is the

damping ratio for the mode i and iM is the i th modal mass.

The solution of the second-order equation, Eq. (20) is given by,

0

( ) ( )exp[ ( )]sin ( ) ( ; )ti

i g i i i D i i i

i D

u t x t t d h tt x w t w t t xw

G= - - - - = - Gò &&

(21)

where ( ; )

i ih t x is referred to as the Duhamel integral

0

1( ; ) ( )exp[ ( )]sin ( )

t

i i g i i i D

i D

h t x t t dx t x w t w t tw

= - - -ò &&

(22)

and 21

i D i iw x w= - .

Thus the i th modal displacement response ( )i p

tx is obtained as follows: ( ) ( ) ( ; )

i p i i i i i it u t h t x= = - Gx φ φ

(23)

Since the TVMD system is activated by inter-story displacements of the primary structure, the effective modal mass of the additional vibration system tuned to the first mode is expressed as follows:

2 2

,1 1 1 , 1 1 12

( )n

r r r k k kk

M m mf f f-

=

= + -å

(24)

Thus, the effective mass ratio 1

/r

M Mm= . To give 8% to 10% critical damping to the structure, an

additional mass ratio of 0.06 is specified in this study (See Fig.4). The distribution of the additional masses is arranged in such a manner that it is proportional to that of story stiffness.

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,r i im ka=

(25)

Substituting Eq. (25) into Eq. (24) yields,

2 2

1 1 1 1 1 1 1 12

( )n

T

r j j j pj

M k ka f f f a-

=

殞油= + - =油油薏

å Kφ φ

(26)

Thus,

1 1

11 1 1 1

T

pr

T Tpp p

M m ma

×= = =

W

M

K K

φ φ

φ φ φ φ (27)

The optimal spring stiffness ,bik and damping coefficient ,d i

c is obtained as follows by using the fixed point method (Ikago et al. 2011). ( )

2

, ,, 2opt opt opt

b i r r d i d r rk m c mw z w= =

(28)

where,

1 1 3,

2 21

popt opt

r d

w mw z

mm= =

-- (29)

The calculated values are shown in Table 3. On the other hand, Eq. (1) is converted into 4n first order matrix equation to obtain complex modes,

( )g

x t+ = -Ay By Aw& &&

(30)

where,

,殞殞賃賃程程- 程程程程油油= , = = , =醬醬油油程程油油程程程程薏薏澱澱

O M M O r xA B w y

M C O K O x

&

. (31)

The eigenvalue problem of Eq. (30) can be expressed as follows: 늿

i i il= -B Aφ φ (32)

If we assume that this seismic control system is underdamped, the eigenvalues and eigenvectors given by Eq. (32) are 2n pairs of complex conjugate.

Here we express the j th pair of eigenvalues and eigenvectors as 2 1 2j jl l

-, and

2 1 2늿

j j-,φ φ respectively.

From the definition of y , it is obvious that ˆiφ is the form

ˆ i i

ii

l賃程程程= 醬程程程澱

φφ

φ (33)

The j th fundamental angular frequency j cw and corresponding damping ratio j c

x can be obtained as follows: 2 1 2j c j j

w l l-

=| |=| | (34)

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2 1 2

2 1 2

Re[ ] Re[ ]j j

j c

j j

l lx

l l-

-

= - = -| | | |

(35)

Eq. (30) can be reduced to 4n decoupled modal equations by using the modal coordinates z obtained from the transformation ˆ=y zΦ where 1 2 4

ˆ [ ]n

= LΦ φ φ φ .

The i th equation of motion is, ˆ ( )T

i i i i i gA z B z x t+ = Aw& &&φ (36)

or alternatively, ( )

i i i i gz z rx tl- =& && (37)

where, 늿T

i i iA = Aφ φ (38)

늿Ti i i i iB Al= =Bφ φ − (39)

ˆ / /T T

i i i i ir A A= = -Aw Mrφ φ (40)

Fig. 3 shows the comparison of participation mode vectors of the uncontrolled primary system and the controlled system with the TVMDs. By adding the supplemental vibration system to the primary system, the uncontrolled 1st mode is split into the 1st to 11th complex conjugate pairs in the controlled system. As shown in Fig. 3(a), combination of the split modes, the 1st and 11th pairs, is almost identical to the 1st mode of the uncontrolled primary system. The 12th and 13th conjugate pairs of modes of the TVMD system correspond to the 2nd and 3rd modes of the uncontrolled primary system, respectively. In Fig. 3(b), only the 1st and 11th conjugate pairs of participation vectors are significant, whereas the other conjugate pairs are insignificant because of their small components. Table 4 shows the comparison of fundamental angular frequencies and corresponding damping ratios of the uncontrolled primary system and the TVMD seismic control system. The values of fundamental angular frequencies of the 1st to 11th modes of the TVMD system are close to each other and they are also close to that of the 1st fundamental angular frequency of the uncontrolled primary system. Whereas the damping ratios of the 2nd and 3rd modes of the uncontrolled primary system are almost unchanged by adding the TVMDs, those of the 1st to 11th modes of the TVMD system is substantially increased. This means that the TVMD seismic control system can increase the damping ratio of the specified mode, and it almost never changes those of the other modes. This is one of the advantageous features of the TVMD seismic control system. Fig. 4 shows the relationship between the mass ratio and damping ratio obtained by the complex eigenvalue analysis of the equivalent SDOF system incorporated with a TVMD. The estimated damping ratio for the MDOF TVMD system can readily be obtained from Fig. 4. The damping ratios of the 1st and 11th conjugate pair modes are well approximated by those of 1st and 2nd conjugate pair modes of the equivalent SDOF system, respectively (Fig. 4). The damping ratios for the 2nd to 10th conjugate pairs of modes are almost

identical to the local damping ratio of the device 1 3

0.1522 2

opt

d

mz

m= =

-.

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Figure 3. Comparison of participation mode vectors of uncontrolled and controlled systems

Table 4. Comparison of angular frequencies and damping ratios mode uncontrolled mode TVMD system

angular frequency

damping ratio

angular frequency

Damping ratio

1 3.12 0.02 1 2.85 0.101 2 3.11 0.158 3 3.12 0.158 4 3.12 0.158 5 3.12 0.158 6 3.12 0.158 7 3.12 0.158 8 3.12 0.158 9 3.12 0.158 10 3.12 0.158 11 3.52 0.075

2 8.29 0.053 12 8.59 0.051 3 13.6 0.087 13 14.1 0.084

The solution of the first-order equation, Eq. (37), is the integral,

0

( )exp[ ( )]t

i i g iz r x t dt l t t= -ò && (41)

Combining i th conjugate pairs of responses obtains i th modal displacement response ( )i tx .

-2 0 20

2

4

6

8

101st mode

-0.5 0 0.50

2

4

6

8

102nd mode

-0.5 0 0.50

2

4

6

8

103rd mode

-0.5 0 0.50

2

4

6

8

101st pair

-5 0 5x 10-3

0

2

4

6

8

102nd pair

-1 0 1x 10-3

0

2

4

6

8

103rd pair

-2 0 2x 10-4

0

2

4

6

8

104th pair

-2 0 2x 10-4

0

2

4

6

8

105th pair

-5 0 5x 10-5

0

2

4

6

8

106th pair

-5 0 5x 10-5

0

2

4

6

8

107th pair

-2 0 2x 10-5

0

2

4

6

8

108th pair

-2 0 2x 10-5

0

2

4

6

8

109th pair

-1 0 1x 10-5

0

2

4

6

8

1010th pair

-0.5 0 0.50

2

4

6

8

1011th pair

-0.2 0 0.20

2

4

6

8

1012th pair

-0.2 0 0.20

2

4

6

8

1013th pair

real part imaginary part

undamped mode of the primary systemcombination of 1st and 11th pairsof modes of the controlled system

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ (a) participation mode vectors of the undamped primary system

(b) participation mode vectors of the controlled system with TVMD

stor

y

stor

y

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{2 1 2 1 2 20 0

2 1 2 1 0

2 1 2 1 0

( ) ( )exp[( )( )] ( )exp[( )( )]

2 Im( ) ( )exp[ ( )]sin ( )

Re( ) ( )exp[ ( )]cos

t t

i i i g i c i c i Dc i i g i c i c i Dc

t

i i g i c i c i Dc

t

i i g i c i c

t r x t i t d r x t i t d

r x t t t d

r x t t

x w w t t x w w t t

x w t w t t

x w t

- -

- -

- -

= - + - + - - -

= - - - -

+ - -

窒

ò

ò

x && &&

&&

&&

φ φ

φ

φ }( )i Dc

t dw t t-

(42)

where, 1i Dc i c i cw x w= - .

Whereas the first integral in Eq. (42) can readily be expressed in terms of the Duhamel integral ( ; )

i i ch t x , the

second integral in Eq. (42) with a cosine term can be obtained in terms of the derivative of the Duhamel integral.

0

( ; ) ( )exp[ ( )]cos ( ) ( ; )t

i i c g i c i c i Dc i c i c i i ch t x t t d h tx t x w t w t t x w x= - - - -ò& &&

(43)

Upon substituting Eq. (43) into Eq. (42) we obtain,

2 1 2 1 2 1 2 1( ) 2 Im( ) ( ; ) Re( ){ ( ; ) ( ; )}

i i i i Dc i i c i i i i c i c i c i i ct r h t r h t h tw x x x w x

- - - -

殞= - + +油薏x &φ φ (44)

From Eqs. (34) and (35), 2 1il

- and 2il

can be rewritten as follows: 2 1 2

,i i c i c i Dc i i c i c i Dc

i il x w w l x w w-

= - + = - - (45)

Using above relationship, Eq. (44) reduces to: ( ) ( ; ) ( ; )

i i i i c i i i ct h t h tx x= +x a b & (46)

where, 2 1 2 1 2 1 2 1 2 1

2Re( ), 2Re( )i i i i i i i

r rl- - - - -

= - = -a bφ φ (47)

Figure 4. Relationship between mass ratio and damping ratio

0 0.05 0.1 0.15 0.2

0.05

0.1

0.15

0.2

mass ratio

dam

ping

ratio

1st conjugate pair

2nd conjugate pair

μ

0.06

0.101

0.075

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Figure 5. Accelerogram of the input ground motion and time histories of inter-story drifts

( ; )

i i ch t x and ( ; )

i i ch t x& are readily evaluated by using a numerical method such as the Newmark’s b

method. In this study we employ a input ground motion recorded at Tohoku University in the 2011 Tohoku earthquake (M=9.0, PGA=3.33m/s/s), which occurred on 11 March 2011 and was the most powerful known earthquake to have hit Japan. The accelerogram of the ground motion is shown at the top of Fig. 5. Since the 1st mode of the uncontrolled primary system is split into the 11 conjugate pairs of modes in the TVMD system, a response obtained by combining the 1st to 11th modal responses can be regarded as the 1st modal response 1

ˆ( )tx .

{ }11

11

ˆ ( ) ( ) ( )i i i i

i

t h t h t=

= +åx a b & (48)

The 1st modal inter-story responses at the 10th, 5th, 1st stories obtained from Eq. (48) is shown in Fig. 5. They are compared with the approximated responses obtained from Eq. (23) by substituting 1

x with 0.075, which

is the lower damping ratio obtained from the 2nd mode of the equivalent SDOF system incorporated with the TVMD. As can be seen in Fig. 5, the 1st modal responses obtained by the 1st participation mode vector of uncontrolled primary system and the approximated damping ratio for the 1st mode obtained from the equivalent SDOF system with the TVMD (Fig. 4) gives good approximation to the complex valued analysis. For the higher modes, their time histories are readily approximated by those of the uncontrolled primary system since the TVMD system does not change the mode shapes and modal damping ratios.

0 10 20 30 40 50 60 70 80 90 100-4

-2

0

2

4

time [s]

acce

lera

tion

[m/s

/s]

input ground motion

0 10 20 30 40 50 60 70 80 90 100-20

0

20

time [s]

rela

tive

disp

.[m

m] inter-story drift of 10th story

0 10 20 30 40 50 60 70 80 90 100-40

-20

0

20

40

time [s]

rela

tive

disp

.[m

m] inter-story drift of 5th story

0 10 20 30 40 50 60 70 80 90 100-40

-20

0

20

40

time [s]

rela

tive

disp

.[m

m] inter-story drift of 1st story

exactapproximated

13.7(exact)13.1(approximated)

35.9(approximated)38.6(exact)

26.4(approximated)27.9(exact)

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Since the equivalent mass ratio and damping ratio for the tuned mode is obtained from the eigenvalue analysis of uncontrolled primary system, complex value analysis for the MDOF system is no longer required to approximate the modal responses. This suggests that the maximum response can be well estimated by the square-root-of-the-sum-of-square (SRSS).

Figure 6. Response spectra of the input ground motion

Figure 7. Comaprison of response estimation

Fig. 7 shows comparison of exact maximum responses obtained from the time history analysis and the SRSS approximation. As shown in Fig. 7, the SRSS estimation gives good approximation in the maximum response estimation in practical terms.

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Period [s]

Dis

plac

emen

t[m

]Displacement Spectra

h=0.087(3rd mode)

h=0.053(2nd mode)

h=0.075(1st mode)

1st mode

2nd mode

3rd modeTohoku Univ. record11 March, 2011The Tohoku earthquake

0 10 20 30 40 501

2

3

4

5

6

7

8

9

10

R

inter-story drift[mm]

stor

y

exactSRSS

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Conclusions

In this paper, the formulation of the equation of motion of the seismic control system incorporated with the new devices, the tuned viscous mass dampers, is shown. One of the advantageous features of the present device is that it can specify the modal damping ratio of the tuning mode. The modal damping added to the tuning mode can readily be obtained from the complex eigenvalue analysis of the equivalent SDOF system incorporated with the TVMD. Comparison of the modal responses derived from the complex valued modal analysis of the TVMD seismic control system and the real modal analysis using uncontrolled primary system illustrated that the latter estimation gives good approximation of exact maximum modal response in practical terms. Thus, a simple and easy seismic response estimation method that is suitable for preliminary design stage using the SRSS method is proposed.

Acknowledgements

This work was supported by Grant-in-Aid for Scientific Research (B) No. 21360260 provided by the Japan Society for the Promotion of Science (JSPS) and the research grant provided by the Japan Iron and Steel Federation (JISF).

References

Den Hartog J. P., (1956) Mechanical Vibrations (4th edn), Dover: New York. Ikago K., Sugimura Y., Saito K., Inoue N. (2010) Optimum Seismic Response Control of Multiple Degree of Freedom Structures using Tuned Viscous Mass Dampers, Proceedings of the Tenth International Conference on Computational Structures Technology, Valencia, Spain: Paper 164. DOI:10.4203/ccp.93.164. Ikago K., Saito K., Inoue N. (2011) Seismic control of single-degree-of-freedom structure using tuned viscous mass damper, Earthquake Engineering & Structural Dynamics, Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/eqe.1138. Kaynia A. M., Veneziano D., Biggs J. (1981) Seismic effectiveness of tuned mass dampers, Journal of the Structural Division ASCE, Vol. 107, No. 9, pp. 1465-1484. McNamara RJ. (1979) Tuned mass dampers for buildings, Journal of Structural Engineering ASCE, Vol. 103, No. 9, pp. 1785–1798. Saito K., Sugimura Y., Nakaminami S., Kida H. and Inoue N. (2008) Vibration Tests of 1-story Response Control System Using Inertial Mass and Optimized Soft Spring and Viscous Element, Proceedings of the 14th World Conference on Earthquake Engineering, Beijing, China, Paper ID 12-01-0128.

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Documents

TS31-03...(TVMD), using a rotational viscous mass damper by connecting it to the primary structure with a soft spring. The basic concept of the TVMD is the same as a tuned mass damper