Generalized parametric resonance in electrostatically actuated microelectromechanical oscillators

Abstract

This paper investigates the dynamic response of a class of electrostatically driven microelectromechanical (MEM) oscillators. The particular systems of interest are those which feature parametric excitation that arises from forces produced by fluctuating voltages applied across comb drives. These systems are known to exhibit a wide range of behaviors, some of which have escaped explanation or prediction. In this paper we examine a general governing equation of motion for these systems and use it to provide a complete description of the dynamic response and its dependence on the system parameters. The defining feature of this equation is that both the linear and cubic terms feature parametric excitation which, in comparison to the case of purely linear parametric excitation (e.g. the Mathieu equation), significantly complicates the system's dynamics. One consequence is that an effective nonlinearity for the overall system cannot be defined. Instead, the system features separate effective nonlinearities for each branch of its nontrivial response. As such, it can exhibit not only hardening and softening nonlinearities, but also mixed nonlinearities, wherein the response branches in the system's frequency response bend toward or away from one another near resonance. This paper includes some brief background information on the equation of motion under consideration, an outline of the analytical techniques used to reach the aforementioned results, stability results for the responses in question, a numerical example, explored using simulation, of a MEM oscillator which features this nonlinear behavior, and preliminary experimental results, taken from an actual MEM device, which show evidence of the analytically predicted behavior. Practical issues pertaining to the design of parametrically excited MEM devices are also considered.