Geology Reference
In-Depth Information
and consequently the velocity vector
x
(
t
can be
expressed as
The previous matrix
P
, according to Lyapu-
nov's method, satisfies
T
x
P A BG
−
+ −
A BG P
= −
C
d
(13)
=
Tz
( )
t
=
T
Φ
z
0
(8)
(
)
(
)
=
Consequently, the performance index
J
1
in
Equation 11 becomes
is a
(n×2n)
transformation
matrix. Substituting Equation 8 in Equation 6, it
is possible to express objective function
J
1
as
where
T
0
I
n
J
1
=
trace
[
P
(14)
1
=
∫
0
0
J
z
T T
z dt
Φ
C
Φ
(9)
The main advantage of this objective function
is that it is independent of external excitations.
d
0
Weighted Sum of Damping Ratios
for Dominant Modes (J2)
where
T
C
=
T C T
(10)
d
d
This objective function derives from Ashour and
Hanson's studies (1987), which have studied
the problem of optimal placement of dampers
by modeling the building as a continuous shear
beam and then using numerical procedures to
maximize the damping in the fundamental mode.
Their results show that the fundamental mode has
a maximum damping ratio when all the dampers
are placed in the first story unit. However, their
discovery is useful only for shear-type buildings
with identically constructed story units, so it is
possible to define a generalized objective func-
tion which is the weighted sum of damping ratios
for
q
dominant modes, valid for any kind of civil
engineering structure. The damping ratio of the
i
th
mode can be obtained as
As expressed in Equation 9, the Average Dis-
sipated Energy
J
1
depends on the initial conditions
z
0
, therefore the maximization of
J
1
assumes dif-
ferent values for different initial conditions. In
order to eliminate this dependence,
z
0
is assumed
like a random variable (uniformly distributed on
the surface of the
2n
-dimensional unit sphere), so
the expression of
J
1
becomes
∞
∫
T
J
=
E
[
J
]
=
trace
Φ
C
Φ
dt
(11)
1
d
0
It is possible to introduce the matrix
P
, de-
fined as
=
−
Re(
)
λ
∞
∫
Φ
ζ
i
(15)
T
P
=
C
Φ
dt
(12)
i
λ
d
i
0
where
λ
i
is obtained by the complex eigenvalue
analysis of the state Equation 5. In the case of sys-
tems with a small number of degrees of freedom,
the eigenvalues can be calculated easily, instead
for very large order systems, eigenvalue analysis
is very difficult.
that is definite because, since civil engineering
structures are stable and
u
(t)
always enhances
the stability of the structure, the system matrix
(
A
-
BG
)
will always be stable.
Search WWH ::
Custom Search