Geology Reference
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vector. By substituting
x
p
= Φ
η
into Equa-
tion (3) and premultiplying two sides of the build-
ing part by
Φ
T
to transform the system coordinates
from physical domain to modal domain, the equa-
tions of motion become
t
t
to the
j
th modal mass of the building,
µ
s
k
,
,
damp-
ing ratio of the
k
th MTMD unit,
ξ
s
k
,
and fre-
quency ratio of the
k
th MTMD unit to the
j
th
modal frequency of the building,
r
f
( )
( )
j
=
(
)
/
where
k
=1, 2, …,
p
. With the prior knowledge
of structural parameters,
ω
j
,
ξ
j
,
and
ϕ
ij
.
Considering the most economical MTMD
layout, identical stiffness coefficient,
k
s
0
,
and
damping coefficient,
c
s
0
,
are given. It can be ex-
pressed as:
ω
ω
s
j
k
k
*
*
*
M 0
M M v
η
( )
( )
t
t
C C
0
η
( )
( )
t
t
p
p
ps
+
+
*
*
T
C
*
v
sp
s
s
s
s
K K
0
*
*
η
( )
( )
t
t
Γ
Γ
p
ps
p
=
x t
( )
T
K
*
v
g
s
s
s
(4)
m
where the modal parameters of the primary build-
ing and the MTMD are involved.
k
=
st
(7a)
s
p
1
2 2
0
∑
ω
r
k
=
1
Conventional MTMD
Parameter Design
j
f
k
ξ
s
c
k
=
(7b)
k
The
j
th modal displacement of the building and the
stroke of the
k
th MTMD unit can be represented
in frequency domain as
s
r
ω
s
0
0
f
j
k
k
s
m
=
(7c)
0
η ω
=
H
ω
X
ω
(5a)
( )
( )
( )
s
r
2
ω
2
j
η
g
k
j
X
g
f
j
k
v
ω
=
H
ω
X
ω
(5b)
( )
( )
( )
p
s
v X
g
∑
1
k
s
k
g
where
m
=
m
is the total mass of MTMD
st
s
k
k
=
units. Based on Equations (7a), (7b), and (7c), the
damping ratio of each MTMD unit,
ξ
Statistically, the MTMD control effectiveness
can be evaluated by the performance index,
R
j
,
defined as
s s
1
,
,
,
2
and
ξ
s
p
can be expressed as
ξ
where
ξ
s
0
is a constant. Moreover, with the given MTMD
mass ratio,
µ
=
ξ
r
s
s
f
k
0
k
2
E
E
[
η
η
]
*
=
the modal mass
ratio of the
k
th MTMD unit can be calculated by
ϕ
m m
j
with MTMD
R
=
(6)
st j
,
ij
st
j
j
[
2
]
j
w/o MTMD
1
/
r
2
that is, the ratio of mean square response of dis-
placement or acceleration of the
j
th structural
mode with MTMD to that without MTMD.
R
j
is
a function of
j
th modal damping ratio of the pri-
mary building,
ξ
j
,
j
th mode shape at
i
ith floor of
the building,
ϕ
ij
,
ratio of the
k
th MTMD unit mass
f
µ
=
µ
(8)
k
s
,
j
st j
,
p
k
∑
2
1
r
f
l
l
=
1
Without any restriction on the frequency dis-
tribution of MTMD units, the optimization of
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