Implementation of numerical approximations in studying vibration of functionally graded beams

Abstract

In this investigation, a brief review on three efficient computational techniques viz. Finite Element Method, Differential Quadrature Method and Rayleigh–Ritz Method along with their mathematical formulation to study free vibration of thin Functionally Graded (FG) beams subject to various classical boundary supports have been presented. The deformation of FG beam is based on the framework of classical beam theory. Three different FG beam constituents assumed in this study are Al/Al $$_{2}$$ O $$_{3}$$ , Al/ZrO $$_2$$ and SUS304/Si $$_3$$ N $$_4$$ , in which the first component is meant for the metal constituent and the second for ceramic constituent respectively. The material properties of FG beam are assumed to vary continuously along thickness direction in a power-law form. The objective is to outline exemplary works carried out by various researchers on the concerned problem and also to find the effect of volume fraction of FG constituents on natural frequencies. The natural frequencies of different FG beams under four sets of classical edge supports have been evaluated along with two-dimensional mode shapes after finding the convergence with reference to concerned numerical methods and validation with available literature.