Optimal Vibration Control of Adjacent Building Structures ... · Optimal Vibration Control of Adjacent Building Structures Interconnected by Viscoelastic Dampers 472 Table 1 Parameters

Journal of Civil Engineering and Architecture 11 (2017) 468-476 doi: 10.17265/1934-7359/2017.05.007

Optimal Vibration Control of Adjacent Building

Structures Interconnected by Viscoelastic Dampers

Ming-Yi Liu, An-Pei Wang and Ming-Yi Chiu

Department of Civil Engineering, Chung Yuan Christian University, Chungli District, Taoyuan City 32023, Taiwan

Abstract: The objective of this paper is to investigate the dynamic characteristics of two adjacent building structures interconnected by viscoelastic dampers under seismic excitations. The computational procedure for an analytical model including the system model formulation, complex modal analysis and seismic time history analysis is presented for this purpose. A numerical example is also provided to illustrate the analytical model. The complex modal analysis is conducted to determine the optimal damping ratio, the optimal damper stiffness and the optimal damper damping of the viscoelastic dampers for each mode of the system. For the damper stiffness and damping with optimal values, the responses can be categorized into underdamped and critically damped vibrations. Furthermore, compared to the viscous dampers with only the energy dissipation mechanism, the viscoelastic dampers with both the energy dissipation and redistribution mechanisms are more effective for increasing the damping ratio of the system. The seismic time history analysis is conducted to assess the effectiveness of the viscoelastic dampers for vibration control. Based on the optimal damping ratio, the optimal damper stiffness, the optimal damper damping of the viscoelastic dampers for a certain mode of the system, and the viscoelastic dampers can be used to effectively suppress the root-mean-square responses as well as the peak responses of the two adjacent buildings. Key words: Adjacent building structure, viscoelastic damper, complex modal analysis, seismic time history analysis.

1. Introduction

The impact of closely neighboring building

structures during earthquakes has become a problem in

modern cities. A variety of damping devices, such as

viscous dampers [1-4], friction dampers [5-7],

hysteretic dampers [8-11] and viscoelastic dampers

[12-17], interconnecting adjacent buildings have been

widely studied to prevent the pounding for

engineering applications. Among these types of

damping devices, the viscoelastic dampers can be used

to dissipate energy at all deformation levels.

Consequently, they have advantages in seismic

protection during various sizes of earthquakes [18].

Since the dynamic properties of viscoelastic dampers

between adjacent structures are complex, it is

necessary to fully understand the mechanism of the

system, which provides the essential information to

Corresponding author: Ming-Yi Liu, Ph.D./Associate Professor; research fields: wind engineering, structural dynamics, cable dynamics, random vibrations, and wind tunnel experiments. E-mail: [email protected].

accurately calculate the vibrations of the adjacent

structures under seismic excitations as well as to

reasonably assess the effectiveness of the viscoelastic

dampers for vibration control.

The objective of this paper is to investigate the

dynamic characteristics of two adjacent building

structures interconnected by viscoelastic dampers

under seismic excitations. The computational

procedure for an analytical model is presented for this

purpose. A numerical example is also provided to

illustrate the analytical model.

2. Analytical Model

The computational procedure for an analytical

model, including the system model formulation,

complex modal analysis and seismic time history

analysis, is presented in Fig. 1, which will be discussed

in this section.

The system model formulation is the basis of both

the complex modal analysis and seismic time history

D DAVID PUBLISHING

Optimal Vibration Control of Adjacent Building Structures Interconnected by Viscoelastic Dampers

469

Fig. 1 Computational procedure in this study.


470

)t(XG

)t(X LN−

)t(X1

)t(X2

)t(X L1N −−

)t(Xi

1K

2K

iK

L1NK −−

LNK −

1M

2M

LNM −

LNC −

2C

1C)t(XG

)t(XN

)t(X 1L+

)t(X 2L+

)t(X 1N−

)t(X iL+

1LK +

2LK +

iLK +

1NK −

NK

iLM +

1LM +

2LM +

1NM −

NM

NC

1NC −

iLC +

2LC +

1LC +

LM

LC

iM

iC

L1NM −−

L1NC −−

1LM −

1LC −

LK

1LK −

)t(X 1L−

)t(XL

1dC

1dK

2dC

2dK

diC

diK

)L1N(dC −−

)L1N(dK −−

)LN(dC −

)LN(dK −

Fig. 2 Analytical model in this study.

analysis. Fig. 2 illustrates the analytical model in this

study. Two adjacent structures with a total of N degrees

of freedom are simulated by an L-story and

(N − L)-story shear building models in the left and right

sides, respectively. N = 2L, N < 2L and N > 2L

individually represent the two buildings with equal

height, the left building with taller height and the right

building with taller height. Mi, Ki, Ci and ( )tX i

( )[ tX i , ( )]tX i

are the mass, stiffness, damping

and horizontal displacement (velocity, acceleration)

time history with respect to the ground motion of the

floor with the ith degree of freedom (i = 1, 2,…, N) for

the system, respectively, and t is the time. The

elevation of each floor in the left building and that of

the corresponding one in the right building are assumed

to be equal, and the neighboring floors at the same

elevation is interconnected by a viscoelastic damper.

Such damper simulated by the Kelvin-Voigt model

consists of a spring and a dashpot in parallel. Kdi and


471

Cdi are the stiffness and damping of the viscoelastic

damper between the two floors with the ith and (L +

i)th degrees of freedom, where either i = 1, 2,…, N − L

for N < 2L or i = 1, 2,…, L for N ≥ 2L. The seismic

excitation of the left building is assumed to be identical

to that of the right building, which can be simulated by

the horizontal displacement (acceleration) time history

of the ground motion ( )tXG· ( )[ ]tXG

[19]. Based on

the above mentioned parameters, one can derive the

equations of motion for the two adjacent building


under seismic excitations [12, 13].

The complex modal analysis is conducted for the

optimization of the parameters, placement and number

of viscoelastic dampers. Under the condition that both

the damper placement and number have been decided,

the determination of the optimal damper parameters is

emphasized in this study. The optimal damper

placement and number can also be estimated by the

same method. For this purpose, the analytical model

including viscoelastic dampers in certain locations of

two adjacent buildings is taken as an example to

illustrate the computational procedure. Based on the

complex eigenvalue equations derived from the

equations of motion, the natural frequency njf ,

damped frequency djf , damping ratio jζ and mode

shape jφ for the jth mode (j = 1, 2,…, N) of the

system can be calculated using the complex modal

analysis [20]. By varying both the damper stiffness and

damping with the other parameters of the analytical

model constant, the plot of damping ratio versus

damper stiffness and damping for each mode of the

system can be obtained. Under the objective function to

maximize the modal damping ratio, the maximum

applicate and the corresponding abscissa and ordinate

of the mentioned plot can be individually determined to

be the optimal damping ratio, the optimal damper

stiffness and the optimal damper damping of the

viscoelastic dampers for each mode of the system.

The seismic time history analysis is conducted to

assess the effectiveness of viscoelastic dampers for

vibration control. Based on the optimal parameters of

the viscoelastic dampers obtained by the complex

modal analysis, ( )tX i, ( )tX i and ( )tX i

(i = 1,

2,…, N), and the corresponding root-mean-square and

peak values for two types of systems: with and without

dampers, can be calculated according to the equations

of motion for these systems under seismic excitations

[19]. The effectiveness of the viscoelastic dampers can

be assessed by comparing the response reduction rates

between the mentioned two types of systems.

3. Numerical Example

A numerical example is provided to illustrate the

computational procedure for the analytical model

presented in Section 2. For this purpose, the system

including a viscoelastic damper between each floor of

each of the two five-story adjacent buildings with a

total of ten degrees of freedom (N = 10, L = 5) under

seismic excitations is developed with a modification

of the previous literature [21]. Table 1 lists the mass

Mi, stiffness Ki and damping Ci (i = 1, 2,…, 10) for

each degree of freedom of the two adjacent buildings.

Ci (i = 1, 2,…, 10) is set to zero due to the fact that the

damping ratio for each mode of the system is

contributed entirely from the viscoelastic dampers and

the contribution from the two adjacent buildings can

be neglected. Under these conditions, one can

accurately evaluate the effectiveness of the

viscoelastic dampers. The viscoelastic dampers at

each floor are assumed to have identical parameters,

i.e., the stiffness Kd (= Kd1 = Kd2,…, = Kd5) and

damping Cd (= Cd1 = Cd2,…, = Cd5). The former varies

from 0 to 8.0 × 107 N/m with an increment of 4.0 ×

105 N/m, while the latter varies from 0 to 8.0 × 105

N-s/m with an increment of 4.0 × 103 N-s/m. Fig. 3

shows the earthquake accelerogram based on white

noise excitations, which is selected as the horizontal

acceleration time history of the ground motion

( )tX G .


472

Table 1 Parameters of the two adjacent buildings.

Degree of freedom i Floor in the left building

Floor in the right building

Mass Mi (kg) Stiffness Ki (N/m) Damping Ci (N-s/m)

1 1

3.2 × 104 1.88 × 107 0.0

2 2 3.2 × 104 1.88 × 107 0.0

3 3 3.2 × 104 1.88 × 107 0.0

4 4 3.2 × 104 1.88 × 107 0.0

5 5 3.2 × 104 3.76 × 107 0.0

6

1 3.2 × 104 3.76 × 107 0.0

7 2 3.2 × 104 3.76 × 107 0.0

8 3 3.2 × 104 3.76 × 107 0.0

9 4 3.2 × 104 3.76 × 107 0.0

10 5 3.2 × 104 3.76 × 107 0.0

Fig. 3 Earthquake accelerogram based on white noise excitations.

The complex modal analysis is conducted to

determine the optimal damping ratio, the optimal

damper stiffness and the optimal damper damping of

the viscoelastic dampers for each mode of the system,

and the results are shown in both Fig. 4 and Table 2.

Figs. 4a and 4b depict the relationship among the

damping ratio jζ ( )2,1=j , dK and dC for the

first two modes of the system, respectively. For the

plot of each mode, the natural frequency njf , damped

frequency djf , jζ ( )1,2, ,10j = , dK and

dC for the case of maximum applicate are listed in

Table 2, in which the maximum applicate jζ

( )1,2, ,10j = and the corresponding abscissa

dK and ordinate dC are determined to be the

optimal parameters of the viscoelastic dampers. Under

these conditions, the responses can be categorized into

underdamped and critically damped vibrations. The

modes 1, 3 to 10 are dominated by the underdamped

vibration, in which jζ ( )10,,3,1 =j is lower

than unity. The mode 2 has the substantial

contribution from the critically damped vibration with

2ζ approaching unity. Figs. 4a and 4b also illustrate

that jζ ( )2,1=j of each mode for dK with

optimal value is higher than that for dK with zero value.

0.0 5.0 10.0 15.0 20.0 25.0

Time (s)

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

Acc

eler

atio

n (m

/s^

2)


473

Table 2 Optimal parameters of the viscoelastic dampers.

Mode j Natural frequency

njf (Hz) Damped frequency

djf (Hz) Damping ratio

jζ Damper stiffness

dK (N/m) Damper damping

dC (N-s/m)

1 1.1057 0.9581 0.4991 1.00 × 107 7.88 × 105 2 1.5528 0.1535 0.9951 7.60 × 106 7.44 × 105 3 3.3544 3.0425 0.4211 3.60 × 106 7.92 × 105 4 4.5326 1.7849 0.9192 7.60 × 106 6.88 × 105 5 5.4555 4.9660 0.4140 2.64 × 107 7.88 × 105 6 7.0441 6.4800 0.3921 1.12 × 107 7.96 × 105 7 7.1452 4.8442 0.7351 1.32 × 107 7.00 ×105 8 8.4759 7.8921 0.3647 7.20 × 106 7.84 × 105 9 9.1790 6.4313 0.7135 8.40 × 106 7.80 × 105 10 10.4691 8.8501 0.5342 3.20 × 105 7.56 × 105

(a)

(b)

Fig. 4 Plots of damping ratio versus damper stiffness and damping of the system: (a) the first mode; (b) the second mode.


474

Table 3 Dynamic responses of the fifth floor in the left building under seismic excitations.

System Root-mean-square response Peak response

Displacement (m)

Velocity (m/s)

Acceleration (m/s2)

Displacement (m)

Velocity (m/s)

Acceleration (m/s2)

Without damper 0.0866 0.5706 5.1383 0.2084 1.6080 17.9666

With damper 0.0245 0.2089 2.1514 0.0852 0.7951 7.8141 Response reduction rate (%)

71.7090 63.3894 58.1301 59.1171 50.5535 56.5076

Table 4 Dynamic responses of the fifth floor in the right building under seismic excitations.

System Root-mean-square response Peak response

Displacement (m)

Velocity (m/s)

Acceleration (m/s2)

Displacement (m)

Velocity (m/s)

Acceleration (m/s2)

Without damper 0.0394 0.4110 5.5819 0.1393 1.4697 25.6703

With damper 0.0235 0.2014 2.1066 0.0814 0.7713 8.0025 Response reduction rate (%)

40.3553 50.9976 62.2602 41.5650 47.5199 68.8258

(a) (b)

(c) (d)


475

(e) (f)

Fig. 5 Response time histories of the fifth floor in the two adjacent buildings under seismic excitations: (a) displacement in the left building; (b) displacement in the right building; (c) velocity in the left building; (d) velocity in the right building; (e) acceleration in the left building; (f) acceleration in the right building.

These results suggest that compared to the viscous

dampers ( dK with zero value) with only the energy

dissipation mechanism, the viscoelastic dampers ( dK

with optimal value) with both the energy dissipation

and redistribution mechanisms are more effective for

increasing the damping ratio of the system.

Based on the optimal 1ζ , dK and dC , the

seismic time history analysis is conducted to assess the

effectiveness of the viscoelastic dampers for vibration

control, and the results are shown in Fig. 5, Table 3

and 4. Figs. 5a-5f individually depict the horizontal

displacement, velocity and acceleration time histories

with respect to the ground motion of the fifth floor in

the left building ( )[ tX 5, ( )tX 5 , ( )]tX 5

, and those

in the right building ( )[ tX10, ( )tX10 , ( )]tX10

under seismic excitations for two types of systems:

with and without dampers. The root-mean-square

values, peak values and the corresponding response

reduction rates between the mentioned two types of

systems of ( )tX 5, ( )tX5 and ( )tX 5

in the left

building, and those of ( )tX10, ( )tX10 and ( )tX10

in the right building are shown in Tables 3 and 4,

respectively. It is revealed that the response reduction

rate of the root-mean-square value and that of the peak

value in the left building are individually greater than

58% and 50%, while the ones in the right building are

greater than 40% and 41%, respectively. Consequently,

the viscoelastic dampers can be used to effectively

suppress the root-mean-square responses as well as the

peak responses of the two adjacent buildings.

4. Conclusions

The objective of this paper is to investigate the

dynamic characteristics of two adjacent building


under seismic excitations. The computational

procedure for an analytical model including the system

model formulation, complex modal analysis and

seismic time history analysis is presented for this

purpose. A numerical example is also provided to

illustrate the analytical model.

The complex modal analysis is conducted to

determine the optimal damping ratio, the optimal

damper stiffness and the optimal damper damping of

the viscoelastic dampers for each mode of the system.

For the damper stiffness and damping with optimal

values, the responses can be categorized into

underdamped and critically damped vibrations. For

each of the modes dominated by the underdamped

vibration, the modal damping ratio is lower than unity.

While for each of the modes dominated by the critically

damped vibration, the modal damping ratio approaches


476

unity. Furthermore, compared to the viscous dampers

with only the energy dissipation mechanism, the

viscoelastic dampers with both the energy dissipation

and redistribution mechanisms are more effective for

increasing the damping ratio of the system.

The seismic time history analysis is conducted to

assess the effectiveness of the viscoelastic dampers for

vibration control. Based on the optimal damping ratio,

the optimal damper stiffness, the optimal damper

damping of the viscoelastic dampers for a certain mode

of the system, and the viscoelastic dampers can be used

to effectively suppress the root-mean-square responses

as well as the peak responses of the two adjacent

buildings.

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Optimal Vibration Control of Adjacent Building Structures ... · Optimal Vibration Control of Adjacent Building Structures Interconnected by Viscoelastic Dampers 472 Table 1 Parameters