Large amplitude free vibration of nanobeams with various boundary conditions based on the nonlocal elasticity theory

Abstract

In this paper, a non-classical beam model based on the Eringen's nonlocal elasticity theory is proposed for nonlinear vibration of nanobeams with axially immovable ends. This non-classical (nonlocal) beam model incorporates the length scale parameter (nonlocal parameter) which can capture the small scale effect. The Hamilton's principal is employed to derive the governing equations and the related boundary conditions together with Euler-Bernoulli beam theory and the von-Karman's nonlinear strain-displacement relationships. An approximate analytical solution is obtained for the nonlinear frequency of the nanobeam by utilizing the Galerkin method and He's variational method. In the numerical results, the ratio of nonlinear frequency to linear frequency is presented for three different boundary conditions. The effect of nonlocal parameter on the nonlinear frequency ratio is examined. Also, some illustrative examples are also presented to verify the present formulation and solutions. Good agreement is observed. These results can be used as benchmark for future studies. (C) 2013 Elsevier Ltd. All rights reserved.