Geology Reference
In-Depth Information
OPTIMIZATION AND FINE-TUNING
OF TLCGD IN STATE SPACE
(22), there is again a complete TLCGD-TMD-
analogy if
u u
*
=
κ
. The equivalent mass ratio
and the TLCGD frequency ratio can be identified
as
After a preliminary modal TLCGD design, fine-
tuning in state space is recommended as it allows
optimizing any dynamic system independent of
the number of degrees of freedom. Therefore it
is possible to split the TLCGD into smaller units
in case of a too large cross-sectional area to keep
the flow one-dimensional, to account for the influ-
ence of neighboring vibration modes during the
optimization and to select the performance crite-
ria as a linear combination of the system states.
Combining the structure's equation of motion
(18), however modally expanded (possibly using
some appropriate modal truncation) with the
dynamic absorber Eq. (13) via the modally ex-
panded linearized interaction forces (17) for all
installed TLCGD renders a coupled system in
state space formulation. With the state hyper vec-
(
)
2
*
κκ
V V
i j
,
i j
,
*
µ
=
µ
<
µ
,
j
j
j
2
−
(
)
1
1
V V
*
+
µ
κκ
j
i j
,
i j
,
*
ω
ω
δ
Aj opt
,
jopt
δ
=
jopt
−
(
)
2
*
Sj
1
+
µ
1
κκ
V V
j
i j
,
i j
,
(24)
*
=
the opti-
mal tuning values are again given by (9),
*
=
Together with
ζ
ζ
,
ω
ω
Aj
Aj
Aj
Aj
−
(
)
2
*
1
+
µ
1
κκ
V V
j
i j
,
i j
,
δ
=
,
T T T T
T
with generalized
modal coordinates collected in
q
and the liquid
strokes, collected in
u
, the entire system dynam-
ics can be given by the lightly coupled system of
first order differential equations,
=
jopt
1
+
µ
tor
z
q
u
q
u
(25)
j
(
)
2
*
3
µ κκ
V V
j
i j
,
i j
,
ζ
=
(
)
jopt
8 1
+
µ
j
The optimal tuning values are recommended
initial values for a subsequent fine-tuning in state
space which also takes the influence of the neigh-
bouring modes into account. From the definition
of the generalized equivalent mass ratio
µ
j
*
, the
damping coefficient
ζ
S
*
and the angular frequen-
cy
ω
S
*
of the equivalent TMD system it is appar-
ent that the TLCGD geometry (factor
κκ
), the
TLCGD position and direction (factor
V
z A z E x
=
+
g
(26)
r
g
where
A
r
denotes the system matrix containing
all system information, e.g. natural frequencies
and light damping of the host structure, mass and
stiffness distribution but also the TLCGD design
parameter (of all smaller units) to be optimized
during fine-tuning.
E
g
is the base excitation influ-
ence matrix and
x
g
the oblique uniaxial ground
excitation. Assuming
x
g
to be time harmonic with
amplitude
a
0
and forcing frequency
ω
,
2 2
)
as well as the earthquake angle of incidence
α
influence more or less the optimal tuning values.
V
γ
,
i j
i j
,
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