Geology Reference
In-Depth Information
The increase in effective structural damping is
demonstrated convincingly by numerical simula-
tions of a simply supported standard steel bridge
of span 50 meter, time-harmonically forced at
mid-span.
The modally tuned VTLCGD with
Consequently almost doubling maximum
liquid strokes the parametric excitation remains
insignificant, while the control becomes more
robust, Figure 8b. With a simulated maximum
relative fluid speed of 1.7 m/s < 12 m/s, the liquid
surface is expected to stay intact, and no problems
with respect to the piston theory are expected.
f
=
2000
is designed to suppress the funda-
mental bridge mode with a modal mass of
m
m
kg
/ 2 35720
within the critical fre-
quency window. With the pressure in equilib-
rium state 1 prescribed,
p
0
S
=
ml
=
kg
APPLICATION OF TLCGD
IN LARGE DAMS
= ×
. Pa
, the
surplus pressure head
H
0
=
0.70m chosen in the
equilibrium state 2,
κ
0
1 2 10
5
Another promising field of application of TLCGD
is the passive control of seismically activated arch
dams. The structural damping can be increased
significantly which is of particular importance for
the retrofit of existing structures which do not meet
the criteria of modern seismic civil engineering
codes. Assuming a low water level, neither the
hydrostatic pressure nor the hydrodynamic forces
are considered, and radiation damping into the
water is absent. Such a structure is conveniently
analyzed by commercially available finite ele-
ment programs, which will also perform a modal
analysis of the arch dam, thereby accounting for
the proper foundation in the surrounding bed-rock.
The structure of the modal model is identical to
the MDOF system, Eq.(18), and for well separated
natural frequencies in a critical resonance range,
the dam-dynamics can be represented by a single
mode of vibration, see Eq.(19). The low frequency
modes show maximum modal displacements at
the dam crest, but it is necessary to distinguish
between symmetric and anti-symmetric modes.
The first show maximum displacements close to
the center of the dam, the latter, depending on the
dam geometry, rather at the ends of the middle
third. For effective damping by the action of
TLCGD, the modal structural damping is as low
as about 1% (for the free-standing dam a result of
material damping and radiation damping into the
bed-rock). Due to the enormous modal masses of
a dam the necessary liquid column cross-sectional
areas are much too large for a single TLCGD.
H L
.
in-
serted in Eq. (50) to obtain the equivalent mass
ratio
µ
*
=
2
=
0 40
0
=
0 9%
, the optimal Den Hartog param-
eters of a single VTLCGD become
f
A opt
.
=
. Hz
2 64
1
and
ζ
A opt
=
. %
.
5 6
1
Assuming
n
=1.4 yields the height of the gas
volume
H
a
=
0.380m, Eq. (47), and with the cross-
sectional area of
A=
0.571m
2
the design of the
single VTLCGD is completed. The linearly esti-
mated maximum fluid stroke of
max
u
=
0 05m
is compatible within the design dimensions, and
the parametric forcing proven to be fully negli-
gible,
ζ
.
( )
w
. % .
,
. Converting
the single VTLCGD into four pairs of smaller
VTLCGD units with fine tuning in state space
renders slightly modified absorber frequencies
and heights of gas volumes as well as much
lower optimal damping ratios,
=
5 6
>
ζ
=
0 08
A opt
1
A
0
f
A
=
2 80
.
Hz
,
H
=
0 32
.
m
,
1 8
, ,
opt
A
1 8
,
ζ
A
=
1 85
.
%
,
f
A
=
2 51
.
Hz
,
1 8
, ,
opt
2 7
, ,
opt
H
=
0 40
.
m
,
ζ
A
=
1 73
.
%
,
A
2 7
,
2 7
, ,
opt
f
A
2 61
.
Hz
,
H
0 37
.
m
,
=
=
3 6
, ,
opt
A
3 6
,
ζ
A
=
1 70
. %
and
f
A
=
2 70
.
Hz
,
3 6
, ,
opt
4 5
, ,
opt
H
=
0 34
.
m
,
ζ
A
=
1 73
. %
.
A
4 5
,
4 5
, ,
opt
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