Experimental and Theoretical Investigation of the Nonlinear Dynamics of an Electrostatically-Actuated Device

Abstract

We present modeling, analysis, and experimental investigation for dynamic instabilities and bifurcations in electrostatically actuated resonators. These instabilities are induced by exciting a microstructure with a nonlinear forcing composed of a DC parallel-plate electrostatic load superimposed to an AC harmonic load. Because of the dominant effect of the electrostatic nonlinearity, several resonances and nonlinear phenomena are induced. Examples of these are the excitation of secondary-resonances, superharmonic and subharmonic, at half and twice the natural frequency of the microstructure. Also, local bifurcations, such as saddle-node and pitchfork, and global bifurcations, such as the escape phenomenon and the homoclinic tangling may occur. These lead to undesirable jumps, hysteresis, and dynamic pull-in instabilities in MEMS devices and structures. The present work represents an attempt to explore these topics in more depth. The first part of this paper is focused on analyzing and studying the nonlinear dynamics of a capacitive device both theoretically and experimentally with a focus on the case of primary-resonance excitation (near the fundamental natural frequency of the structure). The device is made up of two cantilever beams with a proof mass attached to their tips. A nonlinear spring-mass-damper model is utilized, which accounts for squeeze-film damping. Long-time integration for the equation of motion is used to compare with the obtained experimental data. Then, global dynamic analysis is conducted using a finite difference method (primary resonance) and shooting method (subharmonic resonance) combined with the Floquet theory to capture periodic orbits and analyze their stability. The domains of attraction (basins of attraction) for selected data are calculated numerically. Experimental data revealing primary and sub-harmonic resonances, dynamic pull-in, and the escape-from-a-potential-well phenomenon are shown and compared with the theoretical results.