Geology Reference
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3
essentially similar to passive fluid damper, except
that the semi-active fluid damper has an external
valve which connects two sides of the cylinder and
modulates the output force. In this kind of damper,
adjustable damping property makes them capable
to generate wide range of damping force. Since
a small power or source just used for closing or
opening external valve, it can produce very large
damping force without need of large input energy
and can therefore operate on batteries (Pourzeynali
& Mousanejad 2010).
∑
ϕ
1
{ ( )}
X t
=
{ }
y t
( )
(16)
i
i
i
=
where
{
ϕ
i
is the
i
th
column of the modal matrix,
Φ
, and
y t
i
( )
is the
i
th
generalized modal coordinate
of the structure. Therefore, the equation of the
motion for the three first modes can be written as
follows:
M y t
{ ( )}
+
C y t
{ ( )}
+
K y t
{ ( )}
=
(17)
T
−
{ }
L x
+
[
ϕ
3
] { }(
l c x
+
k
r
x
)
g
d
rd
d
Structural Modeling of the Building
and
For a structure with a TMD and n degrees of free-
dom is subjected to earthquakes, the equations of
motion can be given as (Cao & Li, 2004)
M
=
[ ]
ϕ ϕ
T
M
[ ] ,
C
=
[ ]
ϕ ϕ
T
C
[ ] ,
3
3
3
3
(18)
{ }
=
{ }
K
=
[ ]
ϕ ϕ
T
K
[ ] ,
L
[ ]
ϕ
T
M R
3
3
3
M X
{ }
+
C X
{ }
+
K X
{ }
=
(14)
−
M R x
{ }
+
{ }(
l c x
+
k x
)
where
[
ϕ
3
is a matrix obtained from combination
of three first mode shapes of the building. In order
to optimally design of the TMD parameters, the
mass of TMD is assumed as a part of total mass
of the building
(
g
d
rd
d
rd
m x
+
c x
+
k x
= −
m l
T
X
−
m x
(15)
{ } { }
d
rd
d
rd
d
rd
d
d
g
t
m
Building
which can be expressed
)
as
where
x
rd
is relative displacement of the TMD
with respect to the top floor.
m
d
,
k
d
and
c
d
are
the mass, stiffness and damping of TMD, respec-
tively; and
{
l
is the
n
×1
location vector of the
control device.
The response of the building is depended on
its mode shapes and natural frequencies and can
be simulated by dominant modes. According
to the reference (Zuo & Nayfeh, 2003) the first
mode shape is dominant in earthquake excitation
if modal frequencies are well-separated. But in
this study, three first frequencies of the example
building are very close, thus, three first modes
of the main structure are considered for accurate
modeling of the building; therefore, the displace-
ment vector can be expressed as
=
′
×
0
m
m m
t
(19)
d
Building
It should be noted that to design a proper TMD
that be to absorb the entrance energy, its fre-
quency should be tuned close to the considered
frequencies of the building. In this study three
first building frequencies are very close to each
other, and therefore, the frequency of TMD,
ω
d
,
can be designed based on the ratio
( )
of the first
mode frequency of the building,
(
1
ω
d
= ×
β ω
(20)
The damping coefficient of the TMD can be
expressed as
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