Geology Reference
In-Depth Information
TLCD are restricted to frequencies, say below 0.5
Hz in practical applications. Only by the invention
of the passive gas spring this serious limitation is
bypassed and TLCGD frequencies say up to 5Hz
are possible in practical implementations. For
proper application of the piston theory, the fre-
quency is actually limited by the (relative)
maximum fluid speed,
u
depends on the amplitude
U
max
of a time har-
monic vibration of the liquid column. This relation
is used for any forced vibration. Numerical
simulations of a main structure with a TLCGD
attached under earthquake load, see e.g. Ho-
chrainer (2001), have shown that the turbulent
damping performs even better when compared to
the linearized one. For small amplitude excitation
the TLCGD is lightly damped and thus promptly
starts to oscillate with comparatively large fluid
strokes, thereby absorbing energy and keeping
the structural vibrations small. When coming to
peak vibrations with maximum energy dissipation
the turbulent damping prevents the TLCGD from
excessive liquid strokes.
If the cross sectional dimensions are small
when compared to the liquid column length it is
straightforward to define a representative stream-
line to evaluate Eq.(1), thereby averaging the
liquid flow over the cross sectional area, absolute
acceleration
a
A
of the frame is in the horizontal
direction of the trace of the TLCGD-midplane,
=
ω
, which must stay
below the critical speed of
u
max
<
12m s
to keep
the fluid-gas interface intact, Ziegler (2008).
Hence, for a given fluid stroke, the practicable
frequency-range in Eq.(3) is limited.
For a subsequent substructure synthesis the
horizontal structure-TLCGD interaction force
F
A
is obtained applying the conservation of momen-
tum of the fluid mass
m
if
with respect to its cen-
ter of mass in horizontal direction of the trace,
A
A
A
(
)
F
=
m a
+
κ
u m
,
=
ρ
A L L
,
=
2
H
+
B
B
,
A
if
A
if
H
1
1
H
L
L
eff
κ
=
κ
1
(4)
2
g
π
ω
π
u
+
2
ζ ω
u
+
ω
2
u
= −
κ
a
,
if
=
A
=
,
A A
A
A
A
2
4
L
where
a
A
denotes the absolute horizontal accel-
eration of the TLCGD housing, and
κ
is a geom-
etry dependent force factor. Furthermore, there
exist undesired moments resulting from the dis-
placement of the fluid center of mass with respect
to the reference point
A
, and a contribution from
gravity forces acting at the displaced center of
mass. However, it is common practice to neglect
the influence of the undesired moments that
party also exist for TMD of the spring-mass-
dashpot type.
Any vertical floor acceleration, commonly
assumed proportional to the horizontal component,
λ
x
0
(
)
L
=
L
2
sin
β
+
h H
,
h
=
np
ρ
g
,
0
eff
0
a
0
0
B
+
2
H
L
eff
cos
β
κ
=
(3)
L
eff
is the equivalent length of the fluid column
with constant cross sectional area
A
H
rendering
the same natural circular frequency
ω
A
.
κ
is a
geometry dependent coupling factor determining
the TLCGD excitation. The
n
-multiplied static
pressure head,
h
0
, when related to the virtual
height
H
a
of the gas volume serves as the most
convenient frequency tuning parameter for the
TLCGD allowing for its application in an ex-
tended range of frequencies when compared to
the classical TLCD. Without the passive gas spring
λ
x
.
, depending on the site condi-
tion, adds parametric forcing to Eq.(3). Detailed
analytical and experimental investigations, Reit-
erer et al. (2004), have proven that parametric
resonance does not occur if sufficient damping is
a
,
0
< ≤
1 2
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