Geology Reference
In-Depth Information
The excitation of the TLGGD is due to projec-
tion of the floor absolute acceleration
a
A
in the
trace of the TLCGD-mid-plane, unit vector
e
'
,
as in Eq.(1).
Given the relative floor acceleration of the
floor's center of mass
C
M
with respect to ground,
a
C
point
C
f
to the center of floor mass
C
M
is under-
stood,
(
)
2
M
=
m H u r
κ
−
F z
−
z
C x
f T
T
S
Ay
A
C
M
M
(
)
+
F y
−
y
(15)
Az
A
C
M
=
, ,
, and a single point uniax-
ial obliquely incident seismic ground excitation
v
v
w
u
In case of strong asymmetry the application
of a more efficient torsional TLCGD (TTLCGD)
is recommended. A schematic view of the TTLC-
GD is given in Figure 2c. Its horizontal ring shaped
pipe system forms an almost closed loop enclos-
ing the modal center of velocity, ending with two,
sealed vertical pipe sections close to each other
at the optimally selected reference point
A
.
Based on the properly adapted generalized
Bernoulli equation, Eq.(1), the linearized equa-
tion of motion of the relative fluid flow in the
TTLCGD results,
M
M
T
M
=
cos
α
,
w
g
=
sin
α
, further assuming
that the TLCGD is installed with its trace in a
general direction
γ
with respect to the
y
-axis,
evaluation of Eq.(12) yields, for details see Fu et
al. (2010) or Fu (2008),
a
a
g
2
u
+
2
ζ ω
u
+
ω
u
= −
κ
{
a
cos(
α γ
−
)
A A
A
g
u
r
+
[
v
−
(
z
−
z
)
T
]cos
γ
M
A
C
M
S
u
r
[
+
w
+
(
y
−
y
)
T
]sin }
γ
M
A
C
M
2
u
+
2
ζ ω
u
+
ω
u
= −
κ
u
,
S
A A
A
T
0
TT
(13)
A
r L
2
2
(16)
p
κ
=
,
u
=
r
θ
,
=
m r
f
I
T
0
TT
f
fx
f
f
eff
For sake of substructure synthesis the general-
ized interaction forces are calculated applying the
conservation of momentum and angular momen-
tum of the fluid mass
m
f
, with respect to its
center of mass,
C
f
. Neglecting all undesired
nonlinear parts and assuming small floor rotations
θ
1
the linearized interaction forces become
where
A
p
denotes the area enclosed by the
TTLCGD-loop projected onto the floor plane.
The control and interaction moment with respect
to the floor's center of mass
C
M
becomes, see Fu
et al. (2010),
(
)
κ
y
κ
z
+
F
=
m a
cos
cos
α
+
v
−
z
−
z
u
r
M
=
m r u
+
T
3
A
a
−
T
3
A
a
m r
κ
u
,
Ay
f
g
M
A
C
T
S
C x
f
f
TT
Az
Ay
f
f T
0
r
r
M
M
f
f
+
κ
m u
γ
z
r
f
F
m a
v
u
A
=
cos
α
+
−
κ
3
,
(
)
C y
f
g
M
T
TT
F
=
m a
sin
sin
α
+
w
+
y
−
y
u
r
M
f
Az
f
g
M
A
C
T
S
M
y
r
+
κ
m u
γ
F
=
m
a
sin
α
+
w
+
κ
u
A
,
f
C z
f
g
M
T
3
TT
M
(14)
f
L
L
eff
κ
=
κ
,
κ
=
2
H L
T
0
T
0
T
3
1
1
(17)
κ
is defined in Eq.(4). The resulting moment about
the vertical axis becomes, transformation from
It can be concluded that the TTLCGD should
be installed on the floor with largest modal rota-
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