Using the discrete model for the processing of natural vibrations, tapered beams made of AFG materials carrying masses at different spots

Abstract

A discrete physical model based on modal analysis theory is used for the first time to investigate the linear free vibration of tapered Euler-Bernoulli beams constructed of axial functional gradient (AFG) materials and carrying masses concentrated at different locations, applied for CC, CF and SS end conditions. The considered beam will be modeled by the present model composed of N masses (N-DOF) connected by (N+1) small bars, the mass of the beam is totally distributed only on the set of N masses, this distribution is governed by the law of variation of the geometric and material properties of the beam. The components modeling the bending stiffness of the considered beam are (N+2) coil springs whose stiffness depends essentially on the law describing the variation of the squared moment and Young's modulus along the beam axis. After the construction of the stiffness matrix and the mass matrix, the Lagrange formalization is used to present the problem in mathematical form. Thus, the results of dimensional frequencies using several types of tapered beams carrying masses at different places have been discussed in this work. It is noted in the following that the discrete system is composed of (N+n)masses (taking into account the concentric masses carried by the beam).