Geology Reference
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tion to get the most effective energy absorption:
from inspection of the excitation term on the right
hand side of Eq.(16) large angular floor accel-
erations maximise the forced liquid motion and
consequently the energy absorption and thus the
effective damping. Similarly, for plane TLCGD,
a large component of the floor acceleration
a
A
parallel to the horizontal pipe section will induce
strong TLCGD liquid movements and conse-
quently the desired high energy absorption. It has
been shown that the TLCGD should be installed
at the maximum normal distance from the floor's
center of velocity allowed by the floor's plan.
asymmetric structure. Modal expansion of Eq.
(18) using the mass normalized (ortho-normalized)
modal transformation matrix
φφ φφ φφ
1
renders the modal equations of motion with light
modal structural damping
2
ζ ω
Sj
,
, ...,
N
2
Sj
added, thus
decoupled on their left hand side,
F
(19)
q
q
2
q
T
ME x
T
F
T
+
2
ζ ω
+
ω
= −
φφ
+
φφ
+
φφ
j
Sj
Sj
j
Sj
j
j
N g
j
j
A
However, the right hand side decouples only
approximately with respect to in- wind or lateral
wind forces and with respect to the control
forces of the absorbers as well. The floor displace-
ments (and rotations) in Cartesian coordinates
can subsequently be determined by modal super-
position,
x
=
MODAL TRANSFORMATION
OF BUILDING VIBRATIONS
N
∑
φφ
j
q
.
j
j
1
Considering a discretized asymmetric building
properly condensed to account for rigid floor
motions, the undamped dynamic equations, in
matrix form become
OPTIMIZATION BY MODAL
TUNING BY MEANS OF THE
TLCGD-TMD ANALOGY
Mx Kx
+
= −
ME x
+ +
F
F
,
N g
A
During the design stage, optimization in a first step
is proposed applying the TLCGD-TMD analogy.
Assuming that the geometrical constraints are
known from the building construction and dimen-
sion the maximum physical size of the absorber
can be determined in terms of pipe section lengths
and cross sectional areas. For given design earth-
quakes the TLCGD is optimized in an iterative
process using the TLCGD-TMD analogy. This
renders the maximum absolute liquid displacement
U
max
needed to determine the virtual height
H
T
1 0 0 1 0 0 1
0 1 0 0 1 0 0
0 0 0 0 0 0 0
(18)
E
=
N
T
where
x
=
, , , , , , ,
de-
notes the vector of floor displacements, with
u
y
z u
y
z u
u
1
1
T
1
2
2
T
2
TN
=
θ
.
r
Si
is the radius of inertia with respect
to the floors center of mass, storey number
i
,
M
and
K
are the mass and stiffness matrix, respec-
tively,
F
=
r
Ti
i Si
a
≥
3
max
. However, even if this inequality
cannot be met for excessive liquid strokes, the
TLCGD will not be damaged since the nonlinear
air spring effect renders increased restoring
forces and the smooth liquid flow breaks down
temporarily when the water column enters the
slightly inclined horizontal pipe sections of the
air spring.
U
T
F F M r F
,
,
,
, ...,
M r
y
1
z
1
1
x
S
1
y
2
Nx
SN
and
F
A
are the wind (inwind or lateral) and
TLCGD-structure interaction (control) force,
respectively, and
T
=
x
g
v w
,
, 0
is the vector of
g
g
horizontal ground acceleration.
E
N
is the influence
matrix of the ground excitation for the N-storey
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