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.

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