Dynamic stability of electrostatically actuated initially curved shallow micro beams

Micro and nano devices incorporating bistable structural elements have functional advantages including the existence of several stable configurations at the same actuation force, extended working range, and tunable resonant frequencies. In this work, after a short review of operational principles of bistable micro devices, results of a theoretical and numerical investigation of the transient dynamics of an initially curved, shallow, double-clamped micro beam, actuated by distributed electrostatic and inertial forces are presented. Due to the unique combination of mechanical and electrostatic nonlinearities, typically not encountered in large scale structures, the device exhibits sequential snap-through and electrostatic (pull-in) instabilities. A phase plane analysis, performed using a consistently derived lumped model along with the numerical results, indicate that critical voltages corresponding to the dynamic snap-through and pull-in instabilities are lower than their static counterparts, while the minimal curvature required for the appearance of the dynamic snap-through is higher than in the static case. The boundaries of the bistability region of a quasi-statically loaded beam are found in terms of the geometrical and loading parameters and are shown to be bounded from above by the dynamic pull-in instability. Some of the post-buckling states cannot be reached under suddenly applied or quasi-statically increasing voltages: specially tailored loading schemes are suggested for realization of these configurations often beneficial in applications.