Large oscillation of electrostatically actuated curved beams

This paper investigates the responses of initially curved micro-beams subjected to an electrostatic excitation. A Euler-Bernoulli beam theory is utilized to derive the governing equation of motion. A reduced-order model was developed by discretizing the equation of motion using straight beam mode shapes as basis functions in a Galerkin expansion. The results show evidence of the superharmonic resonances of order-three and two in addition to primary resonance. The co-existence of multiple stable orbits observed around only one stable equilibrium and under excitation waveforms with RMS voltage less than the snap-back voltage. These branches are a branch of small orbits within a narrow potential well and two branches of medium-sized and large orbits within a wider potential well. The transition between them results in the characteristic of the double peaks appearing in the frequency-response curve. We also report a bubble structure along the medium-sized branch consists of a cascade of period-doubling bifurcations and a cascade of reverse period-doubling bifurcations. A chaotic attractor develops within that structure at larger excitation levels. It demonstrates evidence of chaos with a wide-based spectrum and an elevated noise-floor. Odd-periodic windows appear also within the attractor including period-three (P-3), period-five (P-5) and period-six (P-6) windows. The chaotic attractor terminates in a cascade of reverse period-doubling bifurcations of period-four (P-4) orbits and period-two (P-2) orbits.