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are involved. Since
X
b
(
ω
,
Φ
b
(
ω
, and
Θ
b
( )
be taken into account. Besides, the installed loca-
tion of the MTMD is also an essential issue.
According to the study by Wang and Lin (2005),
two parameters govern the TC effect of a building,
i.e., eccentricity,
e
, and uncoupled torsion/trans-
lation stiffness ratio,
k
ω
are independent and the ratio between them is
uncertain, it is not proper to use
η ω
j
( )
to measure
the performance of MTMD. In turn, the reduction
of area of transfer function can be a useful option
because earthquake usually has wide-band fre-
quency content. Therefore, the performance index
can be defined as
θ
/
. In addition, the
optimal locations for MTMD controlling the first
mode and the second mode can be obtained by
k
x
ϕ
ϕ
ϕ
ϕ
2
∞
∫
(
d
)
= −
12
for 1st mode
H
( )
ω
d
ω
s
opt
η
x
0
0
j
i
R
MTMD
=
(17)
(19)
22
j
2
∞
∫
H
( )
ω
d
ω
(
d
)
11
for 2nd mode
= −
η
x
s
opt
0
j
i
NOMTMD
0
21
where
x
i
=
x
b
,
ϕ
b
, or
θ
b
. In Equation (17), the
selection of
H
j
Figure 9 are plots of the optimal MTMD loca-
tion,
(
( )
is dependent upon the MTMD
control goal. Moreover,
R
j
is recognized as a
function of structural parameters:
ξ
1
,
ξ
2
, and
Φ
,
which should be known
in priori
, and the MTMD
parameters:
ρ
s
k
,
ξ
s
k
,
r
f
k
(the frequency ratio of
the
k
th unit to the controlled mode of the structure)
and
d
s
k
. The MTMD mass ratio is assigned based
on both considerations of construction cost and
structural capacity before the MTMD design.
Considering the same stiffness and damping coef-
ficients condition, which the physical parameters
of the MTMD unit can be obtained by Equation
(7), the MTMD parameters to be optimized be-
come
ω
d
s
0
opt
, for a square floor TC building under
two eccentricity cases,
e
)
η
x
i
=
0 3
(
r
= radius of gyration of the floor plan) and three
different uncoupled torsion/translation stiffness
ratios,
k / k
x
/
=
0
and
e
/
r
.
= 4
(white arrow),
k / k
x
=1
¸
¸
(gray arrow), and
k / k
¸
=
0 25
(black arrow).
It is seen that when
k / k
¸
= 4
, which means
that the torsional stiffness is large, only one MTMD
for the first structural mode is needed. The optimal
location is basically near the C.M. of the floor
when
e
.
<
0 3
. It is reasonable because the
first mode is dominated by floor translation. When
k / k
¸
=
0 2.
, which means the building motion
is dominated by floor torsion, only one MTMD
for the first mode is needed. The corresponding
optimal location,
(
/
r
.
d
)
1 45
.
r
opt
=
, is near the
positive edge of the floor. In this situation, the
MTMD is used to control the torsional motion,
so MTMD far away from the C.M. can generate
larger torque to the floor. When
k / k
¸
=1
, both
the first and the second structural modes are im-
portant. Two MTMDs respectively located at
opposite sides of the floor are required. The op-
timal locations of both MTMDs are between the
C.M. and the floor side.
(
r
)
, (
r
)
,
...,
(
r
)
s
0
f
opt
f
opt
f
opt
1
2
p
(
ξ
)
(18)
s
opt
0
d
s
0
where
d
s
0
is the distance from center of MTMD
to the C.M of the building floor. Comparing with
Equation (9), it is found that with the TC effect
of the building, multiple structural modes should
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