International Journal of Aerospace and Lightweight Structures

doi: 10.3850/S2010428612000372

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

A. R. Askari¹ and M. Tahani^{2 1}Department of Mechanical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran. amaskari@gmail.com ²Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA. mtahani@um.ac.ir

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 mid- plane 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.

Keywords: Galerkin decomposition method, Homotopy analysis method, Casimir force, Static and dynamic pull-in instability.

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