Optimal Placement of Viscoelastic Dampers Represented by the Classical and Fractional Rheological Models - Design Optimization of Active and Passive Structural Control Systems - page 52

ing building with soft storeys. The method of
simultaneous optimal distribution of stiffness and
damping for the rehabilitation of existing build-
ings was also proposed in the paper by Cimellaro
(2007). The objective function of his method
combines the displacement, absolute acceleration,
and base shear transfer function. Lavan and Levy
(2005) proposed a method for the optimal design
of viscous dampers based on a global damage
index. In several papers, the active control theo-
ry is used to optimize the size and location of
dampers. For example, the H 2 method and the
H ∞ methods are used by Yang et al. (2002) while
the linear quadratic regulation (LQR) method is
used by Gluck et al. (1996), Agrawal and Yang
(2000), and by Loh et al. (2000) to determine
damper allocations. Interesting studies concerning
the allocation and sizing of viscous dampers were
presented by Main and Krenk (2005) and by
Engelen et al. (2007). Recently, the issue of the
optimal placement of VE dampers was considered
in two papers by Fujita et al. (2010a, 2010b) and
by Pawlak and Lewandowski (2010), who used
the fractional Kelvin model of VE dampers.
Moreover, the optimal connections of parallel
structures by VE dampers were considered by
Zhu et al. (2010). A study of the effect of hyster-
etic damper's stiffness on energy distribution
among building stories was presented in two
papers by Nakashima et al. (1996 a, b).
The aims of the chapter are to find the optimal
location of VE dampers and to determine their
optimal parameters. Several objective functions,
which are minimized, are taken into account.
One of the objective functions is the weighted
sum of amplitudes of the transfer functions of
interstorey drifts, evaluated at the fundamental
natural frequency of the frame with such damp-
ers. Another objective functions are taken as the
extreme displacement within the structure or as
the extreme bending moment in the supporting
columns. The optimization constraints are for-
mulated, based on the properties of dampers. The

considered optimization problem is solved using
the sequential optimization method (SOM) and
the particle swarm optimization (PSO) method.
Moreover, a mathematical formulation of the
problem of dynamics of structures with VE
dampers, modelled by classical and fractional
rheological models, is presented. The fractional
models of dampers have an ability to correctly
describe the behaviour of VE dampers using a
small number of model parameters. Advanced
classical rheological models of VE dampers are
also taken into account. The equations of motion
of the considered frame structures, expressed in
physical co-ordinates and in the state space, are
derived. The optimal distributions of dampers in
buildings are found for various objective func-
tions using the above mentioned damper models.

FORMULATION OF THE
OPTIMIZATION PROBLEM
AND DESCRIPTION OF THE
SOLUTION METHOD

In the considered optimization problem, the fol-
lowing objective functions to be minimized are
taken into account:

1. The weighted sum of amplitudes of the trans-
fer functions of interstorey drifts, evaluated
at the fundamental natural frequency of the
frame with the dampers
2. The weighted sum of amplitudes of the
transfer functions of displacements evalu-
ated at the fundamental natural frequency
of the frame with dampers
3. The extreme bending moment in columns
caused by a real earthquake

The above mentioned objective functions may
be described as follows:

T
= w h

F (1)