Damping Models for Microcantilevers, Bridges, and Torsional Resonators in the Free-Molecular-Flow Regime

The dominant damping mode for micromachined resonators operating at low pressures, typically 1000 Pa or less, is collisions with gas particles. This paper presents simple models for determining the damping and the quality factor of microcantilever, bridge, and paddle resonators in vertical, horizontal, and torsional motion, using a consistent model of gas-surface interaction, operating in the free-molecular-flow regime. These models incorporate effects such as wall temperature, accommodation coefficients, and high aspect ratios. Finally, the limiting case of vibrational velocities approaching the thermal velocity of the gas particles is considered, with analysis showing that these models are valid until the peak velocity of the vibration reaches 0.2c, where c is the thermal velocity of the gas.