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

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 max-imum 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 in-vestigated 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 non-linear 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 relation-ship between these two parameters at pull-in condition. Using this fact, pull-in voltage, de-tachment length and minimum allowable gap of electrostatically actuated nano/micro-beams are determined explicitly through some closed-form expressions.