Quantifying squeeze film damping in four-leaf clover-coupled micro-resonators: A comprehensive study under variable vacuum degrees

Abstract

The squeeze film effect has a significant impact on the performance of micro-resonators when utilizing the electrostatic excitation and capacitive detection technique, due to the narrow air gaps present. It becomes the most critical damping effect that needs to be addressed. However, when micro-resonators are encapsulated in vacuum or rarefied air, squeeze film damping analysis can become considerably complex due to high-quality factors. In this study, we systematically investigate the squeeze film damping of a widely-used four-leaf clover-coupled micro-resonator under different vacuum degrees. The characteristic frequencies of micro-resonators are determined by applying the principle of mode superposition. Subsequently, the energy transfer theory is utilized to derive the theoretical formula for the quality factor caused by squeeze film damping, for varying levels of vacuum. Finite element analysis is then conducted to simulate the characteristic behavior and squeeze film damping of the resonator, confirming the accuracy of the proposed theory. In the experiment, a novel four-leaf silicon micro-resonator is fabricated using bulk anisotropic wet etching and other micromachining technologies. An experimental platform is established to study the squeeze film damping of the micro-resonator under various vacuum degrees, where the micro-resonator is excited electrostatically and detected capacitively. The frequency response and quality factor of the resonator under different vacuum degrees are measured using a house-made modal test circuit board. The experimental results demonstrate excellent agreement with the theoretical and simulated results, indicating that the proposed method can be a reliable and precise guide for understanding the squeeze film damping effect in advance, once the primary model of a micro-resonator has been designed.