Mass Distribution of Multiple Tuned Mass Dampers for Vibration Control of Structures Under Earthquake Load

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459,ISO 9001:2008 Certified Journal, Volume 3, Issue 8, August 2013)

198

Mass Distribution of Multiple Tuned Mass Dampers for

Vibration Control of Structures Under Earthquake Load

Rama Debbarma

1

, Sanjoy Hazari

2 1

Associate Professor, 2PG student,Department of Civil Engineering, National Institute of Technology, Agartala, INDIA

Abstract-- This paper deals the effectiveness of multiple tuned mass dampers (MTMD) to control the vibration of structures under earthquake load. For this, the stochastic earthquake response quantities with MTMD, STMD and without dampers are computed in random vibration framework using state space formulation to study the possible improvement of performance of MTMD system with STMD. The parametric study is conducted to delineate the influence of several parameters (mass ratio, damping ratio of structure, peak ground acceleration and time period of structures) on the effectiveness and robustness of MTMDs considering uniform mass and non uniform mass distribution in comparison with single tuned mass damper (STMD). It is demonstrated that the MTMDs configuration is more effective to controlling the seismic motion of the primary system. A numerical study is performed to study the effectiveness of MTMDs and safety of primary structures.

Keywords-- Earthquake load, mass distribution, multiple tuned mass dampers, parametric study, vibration control.

I. INTRODUCTION

Recently, due to shortage of land space, economic requirements and new developments of construction techniques have caused an increased presence of skyscrapers and other tall structures. Due to advanced construction techniques, these skyscrapers and tall structures have become relatively lighter, flexible and lightly damped by using high strength materials. The structural vibration caused by dynamic loadings such as earthquake and wind loadings. The occupants, especially in the upper floors of tall buildings are feeling discomfort due to structural vibration. Thus, mitigating the responses of such structures to external dynamic loads have gain

tremendous interest to the structural engineering

researchers. Now-a-days, various vibration control scheme are capable to reduce the structural vibration. Tuned mass damper is the oldest passive vibration control device. In dynamic vibration control of structures, the tuned mass damper (TMD) has been implemented as an effective passive control device to mitigate the structural vibration.

A TMD is a passive vibration control device consisting of a mass, damping, and a spring; it is attached to a main building structure for suppressing undesirable vibrations induced by earthquake loads. The natural frequency of the TMD is tuned in resonance with the fundamental mode of the building structure, so that the huge amount of the structural vibrating energy is transferred to the TMD and dissipated by the damping as the building structure is subjected to earthquake loads. Multiple tuned mass dampers (MTMD) can be proposed in a parallel or series configuration. These can be incorporated in a primary structural system at one location or distributed spatially. Bergman et al. (1989, 1991) investigated the performance of MTMDS, spatially distributed in a primary structure. The application of MTMD for single degree system have been studied by (Xu and Igusa, 1992, Igusa and Xu,1994). It has been demonstrated that MTMD with distributed natural frequencies are more effective than a single TMD. The effectiveness and robustness of MTMD under dynamic load were studied by Yamaguchi and Harnpornchai (1993), Abe and Fujino(1994), Kareem and Kline (1995), Jangid (1995). The present paper deals the effectiveness and robustness of MTMD to controlling the vibration of structure under random earthquake loads. A parametric study is conducted to investigate the performance of MTMD considering uniform and non uniform mass distribution system in comparison with single TMD using state space formulation. A numerical example is taken to evaluate the effectiveness of MTMD considering several parameters under random earthquake loadings.

II. THEORITICAL FORMULATION

A. Designing of MTMD:

The aim of designing MTMD is to tune damper parameters to the fundamental mode of vibration. It means that the natural damper frequency (or a group of dampers)

d



must be close to the natural frequency of fundamental

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199 Moreover, the damping coefficient of the damper must

be appropriately chosen by (Zuo and Nayfeh, 2005) and

c

_j

is obtained using equations developed by Den Hartog (1956) for the SDOF damper.

The optimum parameters of such a damper (or group of MTMD) can be obtained from the formulae given in a paper (Warburton 1982). The optimal frequency ratio is determined from:





2 d

2 2

s

2

2 1

  



 _{ } (1)

Where,

n

j _n

j 1 2 s 2 d

s d d j

j 1

s s d

m

K k

, , , m m

M M m





 



    



(2)

B. The Dynamic Equation of Motion of Structure and MTMD System:

The equation of motion of a sdof system attached with MTMD (as shown in Fig.1) can be expressed as,

b

z

   

MY CY KY Mr (3)

Where,Y[ , ,x x x_{s 1 2},...,x_n]T is the relative displacement vector, andr



0 I



T, where I is an nx1 unit vector. M, C and K represent the mass, damping and stiffness matrix of the combined system.

Fig.1 Structure-MTMD system

M Ms 0 0 m

 

  

  (4)

Where,

M

_s is the mass of the structure and

m

is the

matrix of dampers.



1 2 3 4 5 n



m



diag m ,

m ,

m ,

m ,

m

... m

The stiffness matrix

K

of the considered system can be

written in the block form below,

*

s d

*T

K k k

K

k k

  

  

  (5)

Where, K_s is the stiffness of structure.

n

d j

j 1

k k







,





*

1 2 3 4 5 n

k  k k k k k ... k



1 2 3 4 5 n



kdiag k , k , k , k , k , ... k

The damping matrix of the system

C

is in a form

similar to that of the stiffness matrix

K

. The specific blocks of this matrix are shown below:

*

s d

*T

C c c

C

c c

  

  

  (6)

Where,

C

_S is the damping of the structure.

Where,

n d j

j 1

c

c











*

1 2 3 4 5 n

c  c c c c c ... c





j j

j j j

c 3

8 1 2 m k

 

  ,

j j

j j

c

2m

 



,



1 2 3 4 5 n



cdiag c , c , c , c , c , ... c

Introducing the state space vector,

1 2 1 2

,

,

,....

,

,

...

(

)

T

s



x x x

s

x x x x

n s

x

n

Y

, Equation (3) can

be written as, 1

k

1

c k2

2

c ...

n

k cn

1

x x2 xn

1

m m2 mn

s

k cs

s

m

s

x

b

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200 Y_s Α Y_{s s}rz_b (7)

k c

0

where, _s    _

 

Α

Η Η

,

k -1 , c -1

H = M K H = M C

In which

r





0, ]

I

Twith I and 0 is the (n+1)x(n+1) unit and null matrices, respectively

C. Determination of Response Covariance:

The structure -MTMD system as shown in Fig.1 is

subjected to stochastic load due to the random seismic acceleration that excites the primary structure at base. A

widely adopted stationary model of

z

_b

 

t

is obtained by filtering a white noise process acting at the bed rock through a linear filter which represents the surface ground. This is the well-known Kanai– Tajimi stochastic process

[Tajimi 1960] which is able to characterize the input frequency content for a wide range of practical situations. The process of excitation at the base can be described as:

2

2

( ) 2 ( )

and ( ) ( ) ( ) 2

f f f f f f

b f f f f f f

x t x x t

z t x t t x x

   

   

   

    (8)

Where,



( )t is a stationary Gaussian zero mean white

noise process, representing the excitation at the bed rock,

f



is the base filter frequency and



_f is the filter or ground damping. Defining the global state space vector is

defined as:

(

, ,_{1 2},.... , , ,_{1 2},... ,

)

T

f n f

x x x x x x x x x

s n s

Z



x

,

Eqn. (7) and (8) leads to an algebraic matrix equation of order six i.e. the so called Lyapunov equation (Lutes and Sarkani 2001):

ΑR RΑ T Β 0

(9)

The details of the state space matrix A and Bin Eqn. (9) are as below:

 

A 0 I

H_k H_c

 

  

  (10)

Where,

2 2

2

2

2

2 .

. 2

0 0 0 0 2

0 0 ... 0

f

f f f

f f

f f f

f f f

 _ 

  _{   }

 _ 

 

 _  _{   }

  _

 

  _{ }

 _  _{ }

 

  _{ }

 

_{ }   

 

 _  _{    }

 

 

  _{ }

  _{ }

  _{ }

  _{ }

 

 

 

 

 

-1

-1

k c

Μ Κ

Μ C

H H

0 0

0 2 _So

 

 

 

 

  

 

 

 



 

 

Β ₍₁₁₎

The space state covariance matrix Ris obtained as the solution of the Lyapunov equation. The state space

covariance matrix is represented by the

sub-matrices

R

_zz

,

R

_zz

,

R

_zz

and

R

_zz. The root mean square (rms) displacement of the primary system can be then obtained as:

_xs  Rzz(1,1) (12)

III. NUMERICAL STUDY

A single degree of freedom primary system with an attached MTMD as shown in Fig.1 subjected to stochastic earthquake excitation is undertaken to study the effectiveness and robustness of the MTMD considering uniform and non uniform mass distribution system. The primary system has the following mass and stiffness values: ms=2.5×106 kg; k1=1.0×107N/m. Unless mentioned otherwise, following nominal values are assumed for

various parameters: structural damping,



_s



3%

, mass

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201 The PSD of the white noise process at bed rock, S0 is related to the standard deviation _z of ground acceleration

(Crandall and Mark) by:





2

0 2

2

1 4

b

f z

f f

S  

  



 . For

numerical study, the peak ground acceleration is taken as,

PGA=0.2g, where „g‟ is the acceleration due to gravity. It is

assumed that 3

b

z

PGA  . The mean value of the filter

frequency

(



_f

)

and damping

(



_f

)

are taken

as

9 rad / sec



and 0.4, respectively. The rms displacement without damper is 0.1246 m.

The variations of the rms displacement of the primary structure with mass ratio are shown in Figs.2 for considering STMD and MTMD with uniform and non uniform mass distribution system. It is seen that rms displacement reduces with increasing mass ratio. Also, it is observed that the reduction of rms displacement is more for considering MTMD with uniform mass (U) and non uniform mass (NU) distribution system with compared to STMD. Further, it seen that the rms displacement is decreases when Nos. of MTMD is more. But, rms displacement increases for non uniform mass distribution in case of 5 Nos of MTMD. It is also observed that the tuning ratio decreases with increasing mass ratio and damping ratio of damper increases with increasing mass ratio (results are not given here). The similar results are shown in Figs. 3 for varying damping ratio of primary structure. From Fig.4, it is found that the rms displacement increases with increasing value of peak ground acceleration for all cases. But, the rms displacement decreases for MTMD case with compared to STMD case. Similar observations are also shown in Fig.5.

0 2 4 6 8 10

0.075 0.080 0.085 0.090 0.095 0.100 0.105 0.110 0.115 0.120 0.125

r

m

s d

is

p

lace

m

e

n

t,



(m

)

Mass ratio,(%)

STMD MTMD=3 Nos.(U) MTMD=3 Nos.(NU) MTMD=5 Nos.(U) MTMD=5 Nos.(NU)

Fig.2: Variation of rms displacement of structures with uniform mass and non uniform mass (central= 0.05 Ms /2) distribution of MTMD

for s3%and pga=0.2g.

1 2 3 4 5

0.08 0.10 0.12 0.14 0.16 0.18

r

m

s d

isp

lace

m

e

n

t,



(m

)

Damping ratio ofstructures,_s(%) STMD MTMD=3 Nos.(U) MTMD=3 Nos.(NU) MTMD=5 Nos.(U) MTMD=5 Nos.(NU)

Fig.3: Variation of rms displacement of structures with uniform mass and non uniform mass (central= 0.05 Ms /2) distribution of MTMD

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202

0.1 0.2 0.3 0.4 0.5

0.05 0.10 0.15 0.20 0.25 0.30

r

m

s d

isp

lace

m

e

n

t,



(m

)

Peak ground acceleration,pga(c*g)

STMD MTMD=3 Nos.(U) MTMD=3 Nos.(NU) MTMD=5 Nos.(U) MTMD=5 Nos.(NU)

Fig.4: Variation of rms displacement of structures with uniform mass and non uniform mass (central= 0.05 Ms /2) distribution of MTMD

for  5%and s3%

1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.000

0.025 0.050 0.075 0.100 0.125 0.150 0.175

r

m

s d

is

p

lace

m

e

n

t,



(m

)

Time period of structures (sec)

STMD MTMD=3 Nos.(U) MTMD=3 Nos.(NU) MTMD=5 Nos.(U) MTMD=5 Nos.(U)

Fig.5: Variation of rms displacement of structures with uniform mass and non uniform mass (central= 0.05 Ms /2) distribution of MTMD

for  5%, s3%and pga=0.2g.

IV. CONCLUSIONS

The performance of MTMDs for controlling the response of a SDOF system is investigated in this paper. The parametric study is conducted to evaluate the influence of several parameters (mass ratio, damping ratio of structure, peak ground acceleration and time period of structure) on the effectiveness and robustness of MTMDs in comparison with STMD. It is observed that rms displacement reduces with increasing mass ratio for MTMD and STMD cases.

But, the effectiveness and robustness of MTMD is more in comparison with STMD. It is also seen that tuning ratio decreases with increasing mass ratio and damping ratio of damper increases with increasing the same. Further, it is observed that the rms displacement of primary structure reduces more when no. of MTMD is more in case uniform mass distribution system. The rms displacement increases in case of non uniform mass distribution system when nos. of MTMD is five. Similar observation can be found for different damping ratio of primary structure for fixed mass ratio of MTMD with uniform and non uniform mass distribution system in comparison with STMD. Similar trends also obtained for time period of structures and peak ground acceleration.

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