Geology Reference
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cos
sin
α
α
sum of its diagonal elements. The minimization
of the scalar function
J
is generally performed
numerically as unconstrained quadratic optimiza-
tion with initial tuning parameters obtained by
Den Hartog's formula. The parameter optimization
of Eq.(28) may also include structural uncer-
tainty by generalizing the performance criterion.
If e.g. the extreme variations in the mass and
stiffness distribution (due to structural deteriora-
tion) are considered in addition to the ideal struc-
ture by adding the associated performance indices
J
A
r
min
and
J
A
r
max
x
i t
ω
=
a
e
e
,
e
=
(27)
g
0
S
S
0
the complex steady state solution depends on the
angle of incidence
α
of the earthquake and on the
excitation frequency,
z
)
−
1
(
)
=
(
0
. The optimal
tuning parameter are found by minimizing the
response for given
α
over the entire forcing fre-
quency range. Introducing the positive definite
weighing matrix
S
a suitable quadratic perfor-
mance index, expressed by the system state vec-
ω α
,
i
ω
I A
−
E e
a
r
g S
J
=
J
+
J
+
J
(30)
A
A
A
r
r
min
r
max
T
T
=
T
can be given by
tor
z
q
q
S
where
J
A
r
refers to the performance index of the
ideal main system. The minimum of Eq.(30)
renders a robust optimization accounting for all
uncertainties considered in the extended perfor-
mance criterion, Hochrainer et al. (2006). All
structural modifications and uncertainties are
hidden in the performance indices
J
A
r
min
and
J
A
r
max
as they include the influence of the corresponding
system matrices
A
r
min
and
A
r
max
. If required the
summation in Eq.(30) can be extended to addi-
tional performance indices and even weighing
factors to account for further structural uncertain-
ties in the optimization process.
∞
∫
T
J
=
z
( )
ω
Sz
( )
ω ω
d
S
S
−∞
T
=
2
π
E e
a
PE e
a
→
min
(28)
g S
0
g S
0
The positive definite weighing matrix is chosen
to optimize the structural response with respect
to certain system states, e.g. floor displacements.
The covariance matrix
P
is defined as the solution
of the Ljapunov matrix equation, see e.g. Müller
et al. (1976),
T
A P PA
+
= −
S
(29)
r
r
Although derived for harmonic excitation the
state space optimization can also be interpreted
in terms of stochastic quantities. Assuming the
ground excitation to be a stationary random white
noise process, the structural vibration response
can be characterized by a random process with
zero mean, and a covariance matrix given by
P
,
see again Müller et al. (1976). The covariance
matrix is an important response measure, since
the standard deviation of the states is given by
diagonal elements. This property of the covariance
matrix can be applied directly for a stochastic
optimization by minimizing a properly weighed
APPLICATION TO ASYMMETRIC
THIRTY STOREY BUILDING
To demonstrate the effectiveness of the proposed
damping system, a thirty-storey moderately asym-
metric building with rectangular cross section is
analysed under seismic excitation. The building
data were obtained by Huo et al. (2001), the dy-
namic analysis and optimization is described in
detail in Fu et al. (2010). The building mass is
homogenously distributed over the storeys with
a floor mass of
m
=
384 10
3
kg and a moment
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