Nonlinear free vibration analysis of nanobeams under magnetic field based on nonlocal elasticity theory

Tai-Ping Chang

doi:10.21595/jve.2015.16751

Abstract

In the present study, we perform the nonlinear free vibration analysis of nanobeams under longitudinal magnetic field based on Eringen’s nonlocal elasticity and Euler-Bernoulli beam theory. To capture the small scale effect of the nanobeams, we adopt the nonlocal beam model with the nonlocal parameter. We use the Hamilton’s principle and von-Kármán’s nonlinear strain-displacement relationships to derive the governing equations of nanobeams subject to magnetic field. In order to solve the equations, we utilize the Galerkin method and He’s variational method to come up with an approximate analytical solution for the nonlinear frequency of the nanobeams under magnetic field. In the numerical results, the nonlinear frequency ratio is presented for various values of the dimensionless nonlocal parameter and the dimensionless amplitude. Finally, we investigate the effect of nonlocal parameter on the nonlinear frequency ratio; moreover, we study and discuss the effect of magnetic field on the nonlinear free vibration behavior of nanobeams.

Highlights

Carbon nanotubes (CNTs) has attracted worldwide attention due to their potential use in the fields of chemistry, physics, nano-engineering, electrical engineering, materials science, reinforced composite structures and construction engineering
He’s variational method is adopted to the problem of nonlinear free vibration of nanobeams
An approximate analytical solution is obtained for the nonlinear frequency of the nanobeam by using He’s variational method

Summary

Introduction

Carbon nanotubes (CNTs) has attracted worldwide attention due to their potential use in the fields of chemistry, physics, nano-engineering, electrical engineering, materials science, reinforced composite structures and construction engineering. In most cases, the exact solution of the nonlinear equations is not possible, and some approximate solution methods are needed In this context, significant advances have been made recently in developing various analytical and numerical techniques to solve different type of nonlinear problems in structural mechanics, i.e., the multiple scales method, the elliptic Lindstedt-Poincare method, the harmonic balance method [7], the variational iteration method [8], the homotopy perturbation method [9, 10]. Significant advances have been made recently in developing various analytical and numerical techniques to solve different type of nonlinear problems in structural mechanics, i.e., the multiple scales method, the elliptic Lindstedt-Poincare method, the harmonic balance method [7], the variational iteration method [8], the homotopy perturbation method [9, 10] He [11] has proposed a novel variational method to obtain a simple and efficient approximate closed form solution for nonlinear differential equations. Some numerical examples are presented for the nonlinear frequency ratio for nanobeams with longitudinal magnetic field

Governing equation

Analytical solution based on He’s variational method

Numerical results and discussion

Conclusions