On pull-in and stable vibration of a microplate due to symmetric electrostatic actuation

Abstract

The vibrations of electrically and symmetrically actuated micro-rectangular plates are considered. Coulomb and geometric nonlinearities are taken into account, as well as small linear damping. The Berger model and the Kantorovich procedure are used, which allows one to reduce the original PDE to a nonlinear ODE. Based on this ODE, the dynamic pull-in phenomenon is investigated, i.e., the transition from the oscillatory regime to an attracting one is studied. An effective numerical algorithm to determine the voltage for which the system collapses is proposed. Furthermore, the bounds for the regions of parametric instability can be computed. Applications with only an AC voltage and various variants with combined AC and DC voltages were considered. The study was carried out for a wide range of AC voltage frequency variations, that is, from zero to twice the natural frequency of the linear plate. In all cases, the instabilities that arise are parametric in nature. Along with the study of the original nonlinear equation, its simplified versions are also considered. Linearization of the Coulomb nonlinearity leads to a Mathieu–Duffing equation with double parametric forcing. Neglecting the geometric nonlinearity, a Mathieu equation with double parametric forcing is obtained. An interesting conclusion of the numerical experiments is that linearization of the original nonlinear equation and its replacement by the Mathieu equation allow one to determine quite accurately the dynamic pull-in of a microplate upon which a symmetric electrostatic actuation is applied. Since the theory for Mathieu equation is well developed, it is possible to analytically determine the frequency of the parametric resonances of the system. This is important, since in the vicinity of the corresponding AC voltage frequencies, sharp decreases in the dynamic pull-in values occur.