Unified stability criteria of electrostatically actuated initially curved micro-beams in the presence of curved electrodes

Following increasing interest in electrostatic actuation of curved beams via curved electrodes, a rigorous limit point and bifurcation analyses are carried out to view how the beam reacts when it is driven by a bell-shaped electrode. The culmination of the study lies in a set of criteria, specifying the conditions for bistability and symmetry breaking as a function of beam and electrode geometries. The study is based on a single and a two degree-of-freedom (DOF) reduced order (RO) models, derived via Galerkin's decomposition, allowing an analytical extraction of a bistability condition, revealing the emergence of three limit points and a semi-analytical derivation of symmetry breaking criteria, for the emergence of bifurcation points. The former is based on the existence of a vanishing discriminant of a cubic equation, which forms a boundary in the geometrical space of both beam and electrode geometries. The extraction of the latter is carried out by demanding existence of two different bifurcation points on the one hand, and the crossing of either point to its respective stable region, on the other hand. The overall criteria show that while actuation voltages will indeed increase or decrease as a function of electrode curvature, along with its operational range, it plays a key role in determining both symmetric and asymmetric responses of the beam. Such results can serve researchers and engineers alike in designing curved beam-electrode systems for usage in future studies, promoting their usage in micro-electro-mechanical (MEMS) based applications.