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which usually lead to a different damper size at
each story. The method can be referred to as a
Sequential Search algorithm (SS) and it can be
included in a broader category of methods called
heuristic search methods. These methods are very
flexible, because they provide designer and engi-
neers a number of possible choices: if the damper
size is constrained, then the number of damper
can be adjusted and if the number of dampers
is subjected to limitations, then the damper size
can be conveniently selected. Therefore in this
chapter, three simple heuristic search methods
that can be easily used by practical engineers are
compared using four different objective functions
and applied to three different building typologies.
I
M K M C
0
0
0
,
,
( )
A
=
−
B
=
E
t
=
−
1
−
1
−
1
−
1
−
M H
M
(2)
in which
H
is a (
n×r
) location matrix for passive
dampers. The damper force
u
(t) can be written as
u
t
= −
Gz
t
(3)
( )
( )
where
G
is a (
r×2n
) damper parameter matrix
given as
k
0
.
.
.
0
c
0
.
.
.
0
d
1
d
1
0
k
0
.
0 0
0
c
.
0
0 0
d
2
d
2
G
=
.
.
.
.
.
.
.
.
.
.
.
.
OBJECTIVE FUNCTIONS
0
0
.
k
.
0
0
0
.
c
.
0
dr
dr
(4)
Average Dissipated Energy (J1)
By substituting Equation 3 in Equation 1 and
neglecting
w(t)
, Equation 1 can be written as
This objective function represents the average
energy dissipated by the dampers; in this case, to
determine the optimal placement of passive energy
dissipation systems, it is necessary to maximize
J1
.
Consider an
n
story linear shear-type building
structure, equipped with r passive energy dis-
sipation systems in various story units. Masses,
stiffnesses and damping coefficients for differ-
ent floors of the building are contained in
(n×n)
matrices, respectively called
M, K, C
.
The state equation of motion is
z
( )
t
=
(
A BG z
−
=
) ( );
t
(5)
z
(
t
= 0
z
0
)
where z
0
expresses any initial condition. The
average dissipated energy is given by
∞
∫
T
J
=
x C x
d
t
(6)
1
d
0
in which C
d
is a (n×n) diagonal matrix that con-
tains the damping coefficients of the dampers
installed in each story units.
J
1
is the objective
function that represents the total energy dissipated
by the dampers when the structure is subjected
to a free vibration. Integrating Equation 5, it can
be written as
z
t
=
Az
t
+
Bu
t
+
Ew
t
(1)
( )
( )
( )
( )
where
z
=
T
is a
2n
state vector,
u
(t)
is a
r
vector whose each element is a function of
nonlinear stiffness and damping forces from the
damper installed in the ith story unit,
w
(
t
) is an
n
vector of external excitations,
A
is a (
2n×2n
)
system matrix,
B
is (
2n×r
) damper location matrix
and
E
is a (
2n×n
) matrix.
A, B
and
E
can be writ-
ten as
( )
t
[ ( ), ( )]
x
t
x
t
z
( )
t
=
=
Φ
( )
t
z
;
0
(7)
Φ
( )
t
e
(
A BG
−
)
t
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