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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: myliu@cycu.edu.tw.
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.
Optimal Vibration Control of Adjacent Building Structures Interconnected by Viscoelastic Dampers
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
Optimal Vibration Control of Adjacent Building Structures Interconnected by Viscoelastic Dampers
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
structures interconnected by viscoelastic dampers
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 .
Optimal Vibration Control of Adjacent Building Structures Interconnected by Viscoelastic Dampers
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)
Optimal Vibration Control of Adjacent Building Structures Interconnected by Viscoelastic Dampers
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.
Optimal Vibration Control of Adjacent Building Structures Interconnected by Viscoelastic Dampers
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)
Optimal Vibration Control of Adjacent Building Structures Interconnected by Viscoelastic Dampers
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
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. 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
Optimal Vibration Control of Adjacent Building Structures Interconnected by Viscoelastic Dampers
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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