Bistability Condition for Electrostatically Actuated Initially Curved Micro-Beams in the Presence of Curved Electrodes

Abstract

Following increasing interest in electrostatic actuation of curved beams via curved electrodes. A rigorous limit point analysis is carried out to view how the beam reacts as a function of its geometry, as well as that of the electrode. The culmination of the study is in a bistability condition that describes what geometry both beam and electrode must have in order for bistability to be present. The study is based on a single-degree-of-freedom (DOF) reduced order (RO) model of a curved beam, derived from Galerkin’s decomposition. The extraction of a condition is based on the existence of a vanishing discriminant of a cubic equation, which formed a boundary in the parameters space of both beam and electrode geometries. The boundary describes a shift in behaviour, from mono- to bistability. Such a model and subsequent analysis have been used before for the study of curved beams, especially when it is on the verge of bistability, with high degree of fidelity. The condition shows that while actuation voltages will increase or decrease as a function of electrode curvature, as well as operational range, the curvature of an electrode plays a key role in determining the behaviour of the beam. Such results can serve researchers and engineers alike in designing curved beam-electrode configurations for usage in future studies, thus promoting their usage in micro-electro-mechanical (MEMS) based applications.