Frequency-Tuning and Dynamic Simulation of Electrostatically Actuated Beams

Abstract

The resonance frequency of electrostatically actuated micromachined beams can be tuned substantially by applying a DC voltage bias, first by decreasing the frequency until the onset of pull-in and then by increasing it by the virtue of contact. With the objective of modeling and designing the micromechanical structures after pull-in, a semi-analytical method was developed to determine the length of the contact between the beam and the substrate. The semi-analytical method which is validated on the straight beams is extended for the folded beam structures. This method provides a tool to the microsystem designer to quickly evaluate the deformed configuration of the folded beams after pull-in without the time-intensive contact analysis. This tool is used to design the micro‐speaker elements suitable for emitting low frequency sounds. Multiple instabilities after the pull-in were numerically observed and it was shown that the resonant frequency of an L-shaped beam can be varied in different frequency bands. The speaker element can emit any frequency in a given range, as the resonant frequency of the beam structures can be tuned both before and after pull-in. Operating the speaker element at resonance maximizes the efficiency of the speaker design because the amplitude of vibration is maximum at the resonance frequency. Furthermore, the interplay between the torsional and bending loads is used to minimize the out-of-plane deflection under self weight. A selection criterion is employed to choose a beam structure with optimum stiffness and natural frequency. Beam-based micro-speaker element designs with single and multi-layered suspended structures are proposed. Practical considerations such as volume displacement, mode shapes and dynamic coupling are discussed, on the basis of which design guidelines for a speaker element are proposed. Squeeze film effects and nonlinearity due to the midplane stretching is integrated into the transient analysis model to analyze the effect on the stroke of beam operating at resonance. A comparison between various speaker elements is presented.