Abstract Summary
Homogenization of periodic media, which are characterized by a modular repetition along one direction of a unitary cell, can be conveniently used to model beam-like structures as equivalent continua. This is the case of lattice structures, as well as multi-story tower-buildings. As regards the latter, in several recent papers, buildings and towers with a uniform geometry along the height were macroscopically modelled as shear-shear-torsional beams. Especially, in a literature paper of Luongo and Zulli (Meccanica, 55(4), 907-925), a homogeneous shear beam model was proposed to address the free and forced dynamic behaviour of a multi-story building. The equations of motion for a damped system were derived, where only the special case of proportional damping was considered, according to the Rayleigh model. First, the free dynamics was investigated, highlighting the interesting organization of the natural frequencies and modes in triplets, as functions of the wave-number. Then, on the base of the latter, a solution strategy was proposed in the framework of the perturbation theory. Here, the case of a multi-story building is analysed, where visco-elastic devices are added to passively control its response. To this aim, the equations of motion of the shear-shear-torsional beam model (described in the paper of Luongo and Zulli) are consistently modified, involving a non-proportional damping model. It derives by the addition of spring-dashpot systems placed in-parallel to the columns; it is found that, although dashpots lead to a proportional damping matrix, the resultant damping is of non-proportional type, as they are added to a building. As a consequence, the equations of motion result in coupled partial differential equations. The solution is assumed as trial series in the normal coordinates, from which a set of infinite triplets of ordinary differential equations is derived. Each triplet is formed by coupled scalar equations and governs the evolution of the three cross-section modes associated to the same wave-number; however, triplets are uncoupled among them. It is observed that damping couples the cross-section dependence of modes, i.e. inside the single triplet, differently than what happens in the proportional damping case. A numerical example on a case study is proposed, where the analytical results of the homogeneous model are compared with those of a refined model implemented in a commercial F.E.M. software.
