Size-dependent nonlinear forced vibration and dynamic stability of electrically actuated micro-plates

Abstract

A size-dependent structural dynamic model that incorporates the effect of geometric nonlinearity is developed in this paper for the forced vibration and dynamic stability of thin rectangular micro-plates. The equations of motion for micro-plates are derived within the framework of classical plate theory, modified couple stress theory (MCST), and von Kármán geometric nonlinearity using Hamilton's principle. Galerkin method is used to convert the governing partial differential equations to a nonlinear second-order ordinary differential equation, which is solved by a Runge-Kutta method. The static instability analysis of the micro-plate is performed to determine the critical electrostatic voltages, and to avoid the pull-in instability. By tracking the static behavior of the microplate, and determining the electrostatic pull-in voltage, the frequency response curves are plotted. In dynamic response, primary, superharmonic, and subharmonic resonance are studied, and the frequency response equation is obtained for each case by the method of multiple scales. Further efforts are made to investigate the influence of size effect, electrical loading (DC and AC voltages), and excitation frequency on the static, and dynamic responses, critical AC voltages, and dynamic stability of micro-plates. It is found that the critical dynamic voltage is a function of the frequency of excitation force. It is shown that the stiffness of micro-plate decreases by increasing the constant DC voltage; however, the increase in the alternating AC voltage does not considerably affect the stiffness of the micro-plate.