Analysis of Geometrically Non-Linear Free Vibrations of Functional Graded Beams in a Thermal Environment

Abstract

The thermal non-linear vibrational behavior of functional graded beams is analyzed using Euler-Bernoulli beam theory and Hamilton's theorem, combined with spectral analysis. A set of non-linear equations is derived based on the von Kármán deformation-displacement relationship and the properties of the functional graded material are assumed to be temperature dependent and to follow a simple power distribution in the thickness direction. Numerical solutions of the non-linear dynamic equations of functional graded beams are obtained by an approximate method called the second formulation. In addition, numerical examples were performed to highlight the accuracy of the method used in this study, and the results are very consistent with those found in the literature. In the numerical examples, the influences played by thermal environmental conditions and volume fraction index are discussed in detail.