Análise de vibrações em estruturas otimizadas com materiais piezelétricos

Abstract

The objective of this work is to solve a topology optimization problem to find the most rigid structure possible for a given volume constraint and compute the vibrations of this structure. Structures are obtained with only one material (aluminum) and with two materials (aluminum and piezoelectric material). The topic of piezoelectricity is addressed, citing some important historical contributions, in addition to the mechanical behavior of piezoelectric ceramics. The equations of piezoelectricity are presented, which will be used in the optimization problems with piezoelectric materials, to define the best location for the piezoelectric material. The theory of topology optimization is presented. Important historical contributions to structural optimization are presented, with main focus on topology optimization. The problem of minimal compliance is presented, which aims to obtain the topology with maximal stiffness satisfying the equilibrium condition and the volume constraint. To solve the problem, the concepts of the density method and optimality criterion are presented, in addition to the iterative scheme used. To solve the problem, the domain is discretized using the finite element method. Possible problems of numerical instability that can occur are presented together with the sensitivity filter, a resource to avoid these instabilities. Some concepts of the dynamics of structures are presented, that are necessary to compute the frequencies and modes of vibration. The results are presented by means of figures of the optimized structures and the figures corresponding to the first three vibration modes.