Stability analysis of electrostatically actuated nano/micro-beams under the effect of van der Waals force, a semi-analytical approach

• A new procedure for pull-in analysis of beam-type N/MEMS under vdW force has been introduced.
• This procedure able us to represent analytical and semi-analytical solutions for pull-in problems.
• This method can extract all pull-in parameters simultaneously.
• It is found that this new method agrees better than previous ones with FE results.
• Some closed-form expressions are also presented for vdW and electrical pull-in parameters.

The objective of the present paper is to determine pull-in parameters (pull-in voltage and its corresponding displacement) of nano/micro-beams with clamped–clamped, clamped-free, clamped–hinged and hinged–hinged boundary conditions, when they are subjected to the electrostatics and van der Waals (vdW) attractions. The governing non-linear boundary value equation of equilibrium is derived, non-dimensionalized and reduced to an algebraic equation, which describes the position of the maximum deflection of the beam, utilizing the Galerkin decomposition method. The equation which governs on the stability condition of the system is also obtained by differentiating the reduced equilibrium equation with respect to the maximum deflection of the beam. These two equations are solved simultaneously to determine pull-in parameters. Closed-form solutions are provided for cases under electrical loading and vdW attraction alone. The combined effect of both electrostatic and vdW loadings are also investigated using the homotopy perturbation method (HPM). It is found that the present semi-analytical findings are in excellent agreement with those obtained numerically. In addition, it is observed that the present semi-analytical approach can provide results which agree better with available three-dimensional finite element simulations as well as those obtained by nonlinear finite element method than other available analytical or semi-analytical findings in the literature. Non-dimensional electrostatic and vdW parameters, which are defined in the text, are plotted versus each other at pull-in condition. It is found that there exists a linear relationship between these two parameters at pull-in condition. Using this fact, pull-in voltage, detachment length and minimum allowable gap of electrostatically actuated nano/micro-beams are determined explicitly through some closed-form expressions.