Geology Reference
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If modal vibrations are approximately isolated
for the displacements of a selected mode
j
at floor
number
i
,
v
where
µ
j
,
ζ
Sj
,
ζ
Aj
,
ω
Sj
and
ω
Aj
are the general-
ized mass ratio, the light modal structural damp-
ing, the TLCGD damping, the circular natural
frequency of the original structure and the circu-
lar natural frequency of the TLCGD, respec-
tively.
m
Si
,
m
j
and
L
j
denote the mass of floor
i
, the modal mass and the modified participation
factor, respectively.
v
Ai j
,
and
w
Ai j
,
denote the
modal displacements of the reference point
A
in
y
- and
z
-direction, respectively. Again the dead
fluid mass
m m
f
=
φ
q
,
w
=
φ
q
,
(
)
(
)
i j
,
j
i j
,
j
j
,
3
i
−
2
j
,
3
i
−
1
=
φ
3
and inserted into the control forces,
Eq.(14), and into the absorber equation, Eq.(13),
the structure-TLCGD interaction is determined
with respect to the selected modal generalized
coordinate
q
j
. Considering base excitation only,
the dynamic structural behavior can be approxi-
mated by the isolated two degree of freedom
system in combination with Eq.(19), Fu et al.
(2010)
u
q
Ti j
,
j
,
i
j
(
)
1
κκ
reduces the fre-
quency of the main structure. The dead weight of
the rigid TLCGD pipe is neglected at this stage
of the tuning process. It is accounted for during
the fine tuning in state space. According to the
TLCGD-TMD analogy the equivalent TMD
system is obtained setting
κ
= =1
. If the
properties of the equivalent TMD system are again
denoted by a star
*
, Eqs.(20) and (21) render
=
−
m
m
fj
1
+
µ
κ
V
q
u
2
ζ ω
0
q
u
j
Sj
Sj
j
j
j
γ
i j
,
+
+
0
2
ζ ω
j
j
Aj Aj
j
κ
V
1
j
γ
i j
,
L
cos
α
+
L
sin
α
jy
jz
ω
2
0
q
u
Sj
j
a
g
,
= −
m
0
ω
2
j
(
)
j
Aj
κ
cos
α γ
−
j
j
(20)
*
m
m
*
*
*
Aj
q
u
2
ζ ω
0
q
u
1
+
µ
V
j
j
j
γ
i j
,
Sj
Sj
+
*
+
*
*
0
2
ζ ω
*
*
j
V
=
v
cos
γ
+
w
sin
γ
j
j
V
1
Aj Aj
γ
i j
,
Ai j
,
j
Ai j
,
j
γ
i j
,
L
*
cos
α
+
L
*
sin
α
*
2
jy
jz
ω
0
q
u
(
)
v
=
φ
−
φ
z
−
z
r
j
Sj
= −
*
a
g
m
Ai j
,
j
,(
3
i
−
2
)
j
,
3
i
Ai j
,
C i
Si
*
0
ω
*
2
j
M
(
α γ
)
j
Aj
cos
−
j
(
)
(22)
w
=
φ
+
φ
y
−
y
r
Ai j
,
j
,(
3
i
−
1
)
j
,
3
i
Ai j
,
C i
Si
M
µ
j
*
=
V m m
*
2
*
i j
,
Aj
j
2
µ
j
=
V m m
i j
,
fj
j
V
*
2
=
v
2
+
w
2
i j
,
Ai j
,
Ai j
,
2
H
φ
2
2
2
j
,
3
i
V
=
v
+
w
+
κ
i j
,
Ai j
,
Ai j
,
3
∑
N
r
*
*
*
L
=
m
φ
+
m v
Si
(
)
jy
Sn
j
,
3
n
−
2
Aj Ai j
,
n
=
1
∑
N
L
m
m v
=
φ
,
+
(
)
jy
Sn
fj Ai j
,
n
1
j
3
n
−
2
=
∑
N
*
*
*
L
=
m
φ
+
m w
(23)
∑
N
(
)
jz
Sn
j
,
3
n
−
1
Aj Ai j
,
L
=
m
φ
,
+
m w
(21)
n
=
1
(
)
jz
Sn
j
3
n
−
1
fj Ai j
,
n
=
1
Considering equal seismic excitation and
comparing the dynamic equations, Eqs.(20) and
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