Vibration analysis of nanorods by the Rayleigh-Ritz method and truncated Fourier series

Abstract

A solution procedure applicable for analyzing the free longitudinal vibration of nanorods is presented in this study. The longitudinal displacement of the nanorod is sought as a linear combination of a Fourier series and auxiliary trigonometric functions. The auxiliary functions are introduced to account for all the relevant discontinuities in the boundary displacement and its derivatives. The improved Fourier series represents a residual or conditioned displacement that is of at least C1 class. The basic theory of the nanorod is shown, which included the total potential energy and the kinematic energy of the nanorod system. The Rayleigh-Ritz method is applied to determine the coefficients of the improved Fourier series expansion for the displacements. The solution approach here is referred to as the improved Fourier-Ritz method. The generality, accuracy and efficiency of the presented approach are fully demonstrated and verified through benchmark examples involving classical and elastic boundary conditions.