Geology Reference
In-Depth Information
VTLCGD insensitive to overloads and to the
parametric forcing caused by the vertical motion.
Considering a force and base excited SDOF
host structure with mass, damping and natural
frequency denoted
M
S
,
ζ
S
and
Ω
S
, respectively,
with a VTLCGD attached renders, with proper
linearization, a linear coupled matrix equation of
motion of the two DOF isolated system, see Ziegler
(2008),
critical double frequency parametric resonance
becomes negligible with the cut-off value in this
case given by (the vertical ground acceleration is
assumed proportional to the horizontal component,
λ
z
a
,
0
< ≤
λ
z
1 2
.
(
)
<<
( )
,
w
ζ
>
ζ
=
max
λ
a
+
w L
2
1
A
A
0
z
g
(49)
Comparing Eq.(46) with that of a TMD at-
tached to a properly altered main SDOF-system
Den Hartog tuning is possible with the equivalent
mass ratio defined by
1
+
µ
−
µκ
w
u
2
ζ
Ω
0
w
u
0
S
S
+
+
−
κ
1
0
2
ζ ω
0
A A
( )
F t
M
0
Ω
2
0
w
u
1
+
−
µ
S
= −
λ
z
a
+
2
κ
g
0
ω
S
m
M
*
κ
µ
2
0
A
µ
*
=
A
=
µ
(50)
0
+
(
)
*
2
κ
1
1
-
(46)
S
0
2
g
π
m
M
2
H
L
ω
π
µ
=
F
,
κ
=
0
,
f
=
A
=
,
where all the relevant parameter are denoted by
a star
*
. The formula remains applicable for
modal tuning if the normal mode is normalized
to one at the position of the VTLCGD. The opti-
mal tuning values are simply given by the trans-
formation, Eq. (9), together with
ζ
0
A
2
4
L
S
0
L
2
np
L
=
,
h
=
0
(
)
0
0
ρ
g
1
+
h
+
nH H
0
0
a
(47)
*
=
ζ
,
Aj
Aj
Assuming constant cross sectional areas of the
piping system, the linearized VTLCGD-structure
interaction force
F
z
, the liquid column length
L
as well as the active fluid mass
m
*
can be given
by
ω
*
=
ω
they become
Aj
Aj
(
)
1
1
2
+
µ
−
κ
ω
δ
*
0
A opt
,
opt
δ
=
=
)
=
,
opt
Ω
(
1
+
µ
1
+
µ
1
−
κ
2
S
0
2
3
8 1
µ κ
ζ
=
0
(
)
(
)
jopt
+
µ
F
=
m w
+
w
-
κ
0
u
z
f
g
(51)
L
eff
= =
L
2
H B
+
Equation (46) takes on a hyper matrix form
for a multiple-degree-of-freedom main system
(preferably described in modal coordinates) with
several VTLCGDs attached at properly selected
positions, and possibly converted into smaller
units in parallel action at one and the same loca-
tion. In such a case, fine-tuning in state space is
recommended.
m
*
=
κ
2
m
(48)
A
f
Like all TLCGD, the vertical absorber is sus-
ceptible to parametric excitation, but with suffi-
cient absorber damping understood, the most
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