Free Transverse Vibration Analysis of Functionally Graded Beams Coupled to a Double Mass-Spring Element

Abstract

In this article, we study the vibratory motion of structures in bending within the framework of the Euler-Bernoulli theory, presenting a work that concerns the dynamic response of geometrically linear beams, through a very relevant case rarely seen in the literature. This is a continuous system containing three identical FGM beams coupled by an elastic system consisting of double elastic masses vertically connecting each beam to the other. The main objective of this paper is to determine the natural vibration frequencies of such a structure. The first part of this work focuses on determining the natural linear vibration frequencies of two homogeneous and isotropic beams connected by a double mass-spring system, in order to validate the results with those found in the literature. The second part consists of finding the natural frequencies of the free vibrations of three FGM beams elastically connected by the double mass-spring linear mechanical system. Using the boundary conditions and continuity conditions, we obtain a system of equations that will be solved numerically by the Newton Raphson method, whose eigenvalues are the natural frequencies of the structure under study. The effects of the volume fration index and the positions of the coupling systems and boundary conditions will be presented and discussed.