Keywords: viscoelastic dampers, fractional rheological models, particle swarm optimization, optimal dampers location.

In civil engineering viscoelastic (VE) dampers are successfully applied to reduce any excessive vibration of buildings, caused by wind or earthquakes [1]. In a classic approach, mechanical models consisting of springs and dashpots are used to describe the rheological properties of VE dampers [2]. In this approach, the rheological properties of VE dampers are described using the fractional calculus and the fractional mechanical models. A single equation is enough to describe the VE damper dynamics, which is an important advantage of the model discussed.

In this paper, planar frame structures with the VE dampers mounted on them are considered. A three-parameter fractional rheological Kelvin model is considered. The dynamic analysis of frame or building structures with dampers is presented in many papers [3] where the Kelvin model is used. The equations of motion of the whole system (the structure with dampers) are written in terms of both physical and state-space variables. The proposed approach in the state space formulation is new. This is the main advantage of the proposed formulation, which does not require matrices with huge dimensions. However, the resulting matrix equation of motion is a fractional differential equation.

It is the aim of this paper to find the optimal placement of the dampers and to determine their optimal parameters. The objective function, which we minimize, is the weighted sum of amplitudes of the transfer functions of interstorey drifts, evaluated at the fundamental, natural frequency of the frame with the dampers. The problem of optimal distribution of the VE dampers modelled using the fractional rheological Kelvin model is solved for the first time.

The solution to the considered optimization problem is arrived at using the sequential optimization method and the particle swarm optimization method (PSO), which is based on the study of social behaviour in a self-organized population system [4]. Numerical tests carried out for a multi-storey building structure modelled as a shear plane frame with VE dampers mounted on it show that the presented methods are simple and efficient.

1

T.T. Soong, B.F. Spencer, "Supplemental energy dissipation: state-of-the-art and state-of-the-practice", Engineering Structures, 24, 243-259, 2002. doi:10.1016/S0141-0296(01)00092-X

2

S.W. Park, "Analytical modelling of viscoelastic dampers for structural and vibration control", International Journal of Solids and Structures, 38, 8065-8092, 2001. doi:10.1016/S0020-7683(01)00026-9

3

M.P. Singh, L.M. Moreschi, "Optimal placement of dampers for passive response control", Earthquake Engineering and Structural Dynamics, 31, 955-976, 2002. doi:10.1002/eqe.132.abs

4

R.E. Perez, K. Behdinan, "Particle swarm approach for structural design optimization", Computers & Structures, 85, 1579-1588, 2007. doi:10.1016/j.compstruc.2006.10.013

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