Tristable properties and limit point behaviour in electrostatically actuated initially curved coupled micro beams

Abstract

A limit point behaviour of a metastructure, composed of two double clamped, initially curved beams, coupled via a rigid truss at their respective centres, is studied when subjected to a distributed electrostatic load. The analysis is based on a reduced order (RO) model, resulting from Galerkin’s decomposition, with symmetric buckling modes used as the base functions. To better understand the behaviour of the structure, the electrostatic analysis is preceded by a study of the model under “mechanical”, displacement-independent, load, allowing validation against a finite elements (FE) model, which served as proving ground for the RO model. In addition, results obtained by the RO model are compared with those extracted via finite differences (FD) solutions, facilitating the usage of the latter as the reference in the electrostatic, displacement-dependent, analysis. To accommodate for winding equilibrium paths, all solutions employed the implicit arc-length “Riks” method. The analysis has produced complicated equilibrium paths, which necessitated a corresponding local stability analysis, conducted using the energy method, allowing for limit point characterisation. The analyses have shown that the double beam metastructure is able to possess bistable and tristable properties, usually found in highly complex structures, which can be cumbersome for design and miniaturisation, provided that it meets certain geometrical parameters. Several variations of tristability are disclosed in the study, hitherto unknown, providing insight to the complexity of the structure. The analysis indicates that a model with at least three degrees-of-freedom (DOF) is needed to predict the various critical thresholds with reasonable errors. In so doing, the model can be used for static characterisation and design of various applications, where a simple structure, possessing tristable properties, is required.