Geology Reference
In-Depth Information
U
p
= Φ
η
(13)
K
ps
*
=
2
2
2
−
ω ρ ϕ
(
+
ϕ
d
)
−
ω ρ ϕ
(
+
ϕ
d
)
−
ω ρ ϕ
(
+
ϕ
d
)
where
s
s
11
21
s
s
s
11
21
s
s
s
11
21
s
1
1
1
2
2
2
p
p
p
2
2
2
−
ω ρ ϕ
(
+
ϕ
d
)
−
ω ρ ϕ
(
+
ϕ
d
)
−
ω ρ
(
ϕ
+
ϕ
d
s
p
)
s
s
12
22
s
s
s
12
22
s
s
s
12
22
1
1
1
2
2
2
p
p
ϕ
ϕ
η
η
11
12
1
Φ =
,
η
=
(14)
ϕ
ϕ
ϕ
ϕ
ϕ
Γ
b
= −
11
11
21
21
22
2
ϕ
ϕ
ϕ
12
12
22
According to the study by Ueng
et al.
(2008),
the building floor is equipped with an MTMD
system with
p
units along the y-axis, as shown in
Figure 8(b), and considering flexible foundation,
the equations of motion of the building-MTMD
system can be given in modal domain by
d
d
1 1
1 1
s
1
s
Γ
s
= −
2
d
p
1 1
s
M 0
M M
v
*
η
( )
( )
t
t
C C
0
*
*
η
( )
( )
t
t
x
p
p
ps
+
+
b
*
T
v
C
ϕ
θ
X
b
T
=
s
s
sp
s
s
b
K K
0
*
*
η
( )
( )
t
t
Γ
Γ
p
ps
=
p
X
( )
t
b
T
v
b
K
s
s
s
(15)
(
)
where
ξ
j
and
ω
j
j
=
1 ~
are the
j
th modal
damping ratio and modal frequency of the pri-
mary building;
ρ
s
k
,
ξ
s
k
, and
ω
s
k
are the mass
ratio, damping ratio and natural frequency of the
k
th MTMD unit;
d
s
k
is the distance between the
k
th MTMD unit and the C.M. of the floor.
It is seen that Equation (15) is similar to Equa-
tion (4) except multiple input excitations,
X
b
,
rather than single input excitation,
x
g
. Taking
Fourier transform on Equation (15), the modal
displacement of the
j
th mode of the building can
be expressed in terms of transfer functions as
where
1 0
0 1
2
ξ ω
0
*
*
1
1
M
=
,
C
=
,
p
p
0
2
ξ ω
2
2
2
ω
0
K
*
1
=
ω
2
p
0
M I C
=
,
=
diag
.(
2
ξ ω
)
,
K
=
diag
.(
ω
)
2
s
s
s
s
s
s
k
k
k
*
M
sp
(
)
=
T
ϕ
+
ϕ
d
ϕ
+
ϕ
d
ϕ
+
ϕ
d
η ω
=
H
ω
X
ω
+
( )
( )
( )
11
21
s
11
21
s
11
21
s
1
2
p
j
η
x
b
Θ
)
(16)
ϕ
+
ϕ
d
ϕ
+
ϕ
d
ϕ
+
ϕ
d
s
p
j b
H
ω
Φ
ω
+
H
ω
ω
( )
( )
( )
(
12
22
s
12
22
s
12
22
1
2
η ϕ
b
η θ
b
j
b
j b
where
X
b
(
ω
,
Φ
b
(
ω
, and
Θ
b
(
ω
are the Fourier
transform of
x t
C
ps
*
=
−
2
ξ ω ρ ϕ
(
+
ϕ
d
)
−
2
ξ ω ρ ϕ
(
+
ϕ
d
)
−
2
ξ ω
ρ ϕ
(
+
ϕ
d
)
s
s
s
11
21
s
s
s
s
11
21
s
s
s
s
11
21
s
b
( )
,
ϕ
b
( )
, and
θ
b
( )
, respec-
tively. Equation (16) takes the similar form to
Equation (5a) except that three transfer functions
1
1
1
1
2
2
2
2
p
p
p
p
−
2
ξ ω ρ ϕ
(
+
ϕ
d
)
−
2
ξ ω ρ ϕ
(
+
ϕ
d
)
−
2
ξ ω ρ ϕ
(
+
ϕ
d
)
s
s
s
12
22
s
s
s
s
12
22
s
s
s
s
12
22
s
1
1
1
1
2
2
2
2
p
p
p
p
Search WWH ::
Custom Search