Analytical Approximations to Nonlinear Vibration of a Clamped Nanobeam in Presence of the Casimir Force

Abstract

In this paper, combination of the Galerkin decomposition procedure and homotopy analysis method (HAM) is used to derive analytical approximate solutions for oscillatory behavior of a nanobeam under the effect of the Casimir force. The model accounts for the inherent nonlinearity of the Casimir force, and the geometric nonlinearity of midplane stretching. It is found that stable equilibrium does not exist if the Casimir force parameter defined in this paper exceeds its critical value and Casimir dynamic pull-in instability has been accrued. Using numerical integration, the critical value of the Casimir force parameter is also presented. Furthermore a parametric study investigated the effects of design parameters such as applied axial loads and the aspect ratio of the nanobeam. It is found that there is a linear relationship between critical value of the Casimir force parameter and applied axial loading in constant aspect ratio of nanobeam and the value of dynamic Casimir pull-in parameter is around 87% of the static counterpart. Additionally, the HAM is optimized to accelerate the convergence of approximate solutions. The presented results are found to agree well with those obtained by the fourth-order Runge-Kutta method as well as the available results in the literature for static pull-in case.