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(
)
provided. The cut-off value of the equivalent
linearized damping turns out to be dependent on
both, the maximum stroke of the fluid motion and
the amplitude of the time harmonic vertical vibra-
tion of the most critical double frequency para-
metric resonance,
*
*
*
F
=
ma
+
m a
+
u
,
m
=
κκ
m
,
A
A
A
f
m m
m
*
=
−
f
(7)
Eq. (6) corresponds to that of an equivalent
TMD whose displacement is characterized by
u
*
and the reaction force
F
A
splits into two parts:
The first term corresponds to the reaction force
of a dead weight load of mass
m
attached to the
supporting host structure. The second term,
m a
4
max
u
max
λ
a
g
x
g
ζ
=
δ
>
ζ
=
(
)
A
L
A
,
0
3
π
4 1
+
h H
sin
β
0
a
(5)
+
, represents the reaction force of a
corresponding TMD of mass
m
*
and displacement
u
*
. Consequently, there is a simple TMD-TLCGD
analogy, see Figure 1b: The total liquid mass
m
f
is split into the equivalent TMD mass, the “active
mass”
m
*
, and the dead weight mass
m
.
From the analogy it follows that the geometry
factor
κκ
should be as large as possible to assure
that most liquid mass is used for vibration absorp-
tion. Although not obvious from the analogy, the
liquid part acting as dead weight load is mainly
moving vertically, thereby providing the gravita-
tional restoring force. Demanding
κκ
=1
would
lead to a degenerated TLCGD with no inclined
pipe sections. However, the ratio
B H
should be
maximized combined with an opening angle
β
≥ °
*
*
(
u
)
A
The gas spring effect of the TLCGD,
h H
0
,
lowers the required damping even further and any
effects of the vertical excitation become negli-
gible if the inequality (5) holds.
TLCGD-TMD ANALOGY
By inspection of the absorber control force and
the equations of motion it is possible to establish
a convenient analogy that proves the equivalence
of TLCGD and TMD with respect to vibration
reduction, Hochrainer (2001). Considering only
horizontal frame acceleration
a
A
, the TLCGD
equation of motion is given by Eq.(3), with the
corresponding interaction force
F
A
of Eq.(4). Set-
ting virtually
κ
= =1
, these expressions turn
out to be identical to those of a TMD of the spring-
mass-dashpot type. Since the TLCGD must have
inclined pipe sections, this condition is not pos-
sible from a physical point of view. However,
when considering the liquid displacement scaled
by
u u
30
to minimize the dead fluid mass. Having
selected a suitable TLCGD geometry and the
fluid mass, the analogy can be used to obtain
optimal frequency and damping parameter from
the large number of design and optimization
criteria that have been originally developed for
TMD.
If the TLCGD is attached to a generalized
SDOF main structure with mass
M
and natural
frequency
Ω
, the mass of the host structure is
increased by the dead fluid mass
m
. Given the
mass ratio
µ
=
m
f
, the properties of the
equivalent TMD-main system, become slightly
altered,
*
=
κ
the plane TLCGD motion can be
described by
u
*
u
*
2
u
*
a
+
2
ζ ω
+
ω
= −
(6)
A A
A
A
with the corresponding horizontal interaction
force changed to
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