Voltage Response for Parametrically Actuated MEMS Cantilever Beam Using Homotopy Analysis Method and Method of Multiple Scales

The purpose of this paper is to investigate the nonlinear dynamics governing the behavior of electrostatically actuated micro electro mechanical systems (MEMS) cantilever undergoing parametric resonance. The MEMS consists of a cantilever parallel to a ground plate. The beam is actuated via an A/C voltage with excitation frequency near first natural frequency of the cantilever. The model includes damping, electrostatic, and Casimir (or van der Waals) forces. The electrostatic force is modeled to include the fringe effect. The amplitude-voltage response of the parametric resonance and the effects of varying the magnitudes of the fringe, Casimir (or Van der Waals), and damping forces along with varying the detuning parameter are reported. The response is obtained using two different methods, namely the method of multiple scales (MMS), and the homotopy analysis method (HAM). In this study approximations up to a 2^nd order HAM are used. HAM is a deformation technique that begins with an initial guess and continuously deforms it to the exact answer. For the 1^st Order HAM, a softening effect is reported. The 1^st Order HAM matches the MMS results in low amplitude and begins to soften and deviate away from the MMS solution in higher amplitudes. For the 2^nd Order HAM deformation the softening effect is slightly more pronounced with a slightly lower prediction of the maximum deflection of the cantilever tip. For the 2^nd order deformation solution the stable branch of the amplitude-voltage response obtained by the HAM shifts leftward from the MMS solution with the unstable branches between the two methods continue to agree in low amplitudes and deviate in high amplitudes. As a remark, the higher order HAM solutions are obtained symbolically with the software Mathematica and numerically ran with the software Matlab.