In this paper, the homotopy analysis method (HAM) is used to derive analytical approximate solutions for investigating the nonlinear vibrations of nano-electro-mechanical beam under the Casimir force without any electrostatic excitation. This model is nonlinear due to inherent nonlinearity of distributed Casimir force and geometrical nonlinearity which is taking into account in the present study such as mid-plane stretching. The critical value for the Casimir force parameter is also presented as an important parameter for designing nano-electro-mechanical systems (NEMS) based on nano-electro-mechanical beam by numerical integration method. Furthermore the effective parameters on presented pull-in parameter are investigated. The HAM is also optimized to accelerate the convergence of approximate solutions. The present results are found to agree well with those obtained by the fourth-order Runge-Kutta method as well as presented results in the literature for static pull-in.

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