Dynamic snap-through motion and chaotic attractor of electrostatic shallow arch micro-beams

Abstract

In this paper, we investigate numerically the dynamic snap-through motion of a shallow arch micro-beam that is electrostatically excited through three separated electrodes. It is excited by a dc voltage and a time-varying signal with a frequency tuned near its first in-plane frequency in both initial curvature and initial-counter curvature configurations. A reduced-order model based on the Galerkin method is utilized to discretize the governing equation of motion and to analyze the static and dynamic responses of the micro-beam. The phase portraits obtained from simulations reveal the presence of a stable response that oscillates back and forth around single and double equilibria until it eventually settles in a steady state zone near the first equilibrium point or encloses both equilibria. This can be tracked using a Long-Time Integration (LTI) and recording the time–history signal for a long period. Remarkably, this behavior can be controlled by adjusting the initial operating conditions mainly the signal frequency. We also observe the emergence of a chaotic attractor in the vicinity of the resonance frequency when the shallow arch beam is excited while it is in its initial-counter curvature. The simulated FFTs indicate the presence of the chaotic attractor in the vicinity of primary resonance due to dynamic snap-through motion or large excitation between two equilibria. We observed the existence of periodic orbits with single loop (P-1) and an full chaotic behavior around the two equilibria. The existence of this phenomenon can be attributed to the fact that the excitation voltage is positioned close to the snap-through threshold.