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Figure 3. A schematic of the generalized Kelvin model
u t
( )
=
k q
[
( )
t
−
q t
( )],
and Moreschi (2002), Alkhatib et al. (2004) and
Wu et al. (1997). An interesting comparison be-
tween the properties of PSO and the genetic
method is presented by Plevris and Papadrakakis
(2011). Compared with other evolutionary algo-
rithms, like genetic algorithm and ant colony
optimization algorithm, PSO has some appealing
features including easy implementation, few
parameters tuning and fast convergence rate. Some
applications of ant colony optimization method
to solve structural optimization problems are
presented by Viana
et al.
(2008) and by Kaveh A.
and Talatahari S. (2009).
0
0
w
,
1
1
u t
( )
=
k q
[
( )
t
−
q
( )]
t
+
c q
[
( )
t
−
q
( )],
t
i
i
w i
,
+
1
w i
,
i
w i
,
+
1
w i
,
u t
( )
=
k q t
[
( )
−
q
( )]
t
+
c q t
[
( )
−
q
( )]
t
(9)
m
m
3
w m
,
m
3
w m
,
where
u t
i
( )
is the force in the
i-th
element of the
model (
i
=
0 , , ..,
). Symbols
k
i
and
c
i
are the
spring stiffness and the damping factor of the
dashpot of the
i-th
element of the model, respec-
tively, and
q t
m
1
( )
and
q t
3
( )
denote the external
nodes displacements given in the local coordinate
system. Moreover, the dot stands for differentia-
tion with respect to time
t
and the symbol
q
DESCRIPTION OF MODELS
OF VE DAMPERS
w
,
( )
denotes additional displacements, called “the
internal variable” (
i
t
The properties of VE dampers can be properly
captured using generalized rheological models like
the generalized Kelvin model and the generalized
Maxwell model, shown in Figure 3 and Figure
4, respectively. The generalized Kelvin model is
built from the spring and a set of the
m
Kelvin
elements connected in series while the general-
ized Maxwell model is built from the spring and
a set of the
m
Maxwell elements connected in
parallel. In this paper, a serially connected spring
and dashpot will be referred to as the Maxwell
element while the Kelvin element is the spring
and dashpot connected in parallel.
The behaviour of the generalized Kelvin
model of damper can be described by means of
the following equations:
=
1, ..,
).
After introducing the vector of external reac-
tions
m
T
and
utilizing the equilibrium conditions of the exter-
nal nodes:
R t
R
z
( )
t
=
[
R t R t R t R t
( ),
( ),
( ),
( )]
1
2
3
4
,
R t
2
( )
= −
u t
( )
( )
=
,
0
1
0
R t
and
R t
4
( )
=
the following matrix
equation can be written:
1
( )
=
u t
m
( )
0
R
t
=
K q
t
+
K q
t
+
C q
t
+
C q
t
( )
( )
( )
( )
( )
z
zz
z
zw w
zz
z
zw w
(10)
where
T
q
z
( )
t
=
[ ( ),
q t
q t
( ),
q t
( ),
q t
( )]
,
1
2
3
4
and the symbols
K
zz
,
K
zw
,
C
zz
and
C
zw
denote the stiffness and
( )
t
[
q
( ), ......,
t
q
( )]
t
T
q
w
=
w
,
1
w m
,
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