Optimal Placement of Viscoelastic Dampers Represented by the Classical and Fractional Rheological Models - Design Optimization of Active and Passive Structural Control Systems - page 53

w r T

where w = [

1 2 is the vector of weight
coefficients, and r stands for the number of quan-
tities taken into account in the objective function.
The vector h = [

w w , , ..., ] where C d is the assumed total amount of main

damping factors and i = 1 , , .., . Moreover,

c min = 0 or c min is the low-value positive number
if the particle swarm optimization method is used
to optimize the structure with dampers modelled
using the fractional Maxwell model.
The equations of motion of a structure with
dampers are treated as additional implicit con-
straints. Moreover, it is assumed that the damper's
damping factors are continuous design variables.
However, in practical applications, the damper's
capacity and size can be found only from a set
of actually manufactured dampers. Dampers are
fixed to a structure with the help of braces, which
are treated as elastic elements or as rigid elements
when the shear frame is used as the model of a
real structure.
The considered optimization problem is for-
mulated as follows:
For a given set of m possible damper locations,
find the positions of dampers and the value of
their main damping factors c d , which minimize
the objective function (1) and fulfill the explicit
constraints of Equation (2) and other implicit
constraints mentioned above.
The solution is obtained using the sequential
optimization method and the particle swarm op-
timization method (see, Kennedy and Eberhart
(2001), Clerc (2006), Gazi and Passino (2011)).
In the first method, for each possible location of
one damper the values of the objective function are
calculated. The optimal, most appropriate location
of the damper is the position for which the mini-
mum value of the objective function is obtained.
When the first damper location is determined, the
procedure is repeated until all locations for the
dampers are found. This procedure is very simple.
However, there is no proof for the solution's
convergence although many examples show that
the method is efficient in a number of engineer-
ing applications (see, for example, the papers by
Zhang and Soong (1992) and by Lewandowski
(2008)). Moreover, the order of convergence of

m

h r T

1 2 consists of the
values of the above mentioned amplitudes of the
transfer functions of interstorey drifts (case 1),
displacements (case 2), or values of the bending
moments in columns (case 3). The weight factor
w i could be chosen more or less arbitrarily or,
which is more reasonable, it could reflect the
designer's preferences.
The considered optimization problem is sub-
jected to some constraints. Due to limitations
resulting from the building's functionality and
manufacturing constraints, the dampers' positions
cannot be freely chosen. Therefore, it is reasonable
to assume that during the building design process
some, say m , places in the building are chosen as
acceptable damper locations. Moreover, it is
reasonable to assume that the properties of VE
dampers cannot be freely changed and only the
size of dampers is changeable. This means that
all parameters in the i-th VE damper's model are
proportional to one parameter, say, the damping
factor c d , which will be called the main damping
factor. In the case of the fractional model, it is
assumed that the parameter α , which describes
the order of the fractional derivative in the
model, is constant and cannot be changed during
the optimization procedure. If the damper model
contains more than one damping parameter, one
of them may be chosen as the main damping fac-
tor. For the above-mentioned reasons, it is assumed
that the sum of damping coefficients is known
and constant. Moreover, the values of the main
damping factor c d , of every damper must be non-
negative. The above constraints are written as:

h h , , ......., ]

m

∑ =

c C , c ≥ c

(2)

d i , d d i , min

i = 1