Harmonic and sub-harmonic resonance of MEMS subjected to a weakly non-linear parametric and external excitations

Abstract

This paper presents a simplified mathematical model for the purpose of studying the resonant responses of a nonlinear dynamical system, (micro - electro – mechanical systems (MEMS)), which represented by a Van-der Pol equation subjected to a weakly non-linear parametric and forcing excitations. Using Multiple scales method, the Van-der Pol equation is transformed to a system of second order differential equation up to first order of small parameter " . Three types of resonances are studied (harmonic resonance and subharmonic resonances of even order (one - half and one - fourth )). The modulation equations for each resonances, steady state solutions, frequency-response equations, stability analysis are determined. Numerical analysis for frequency-response equations and stability conditions are carried out. Results are presented graphically by group of  figures. Finally discussion for these figures are given.