Analysis of the air-damping effect on a micromachined beam resonator

An analysis of the air-damping effect on the frequency response of a micromachined beam resonator is presented. The motion of the beam is analyzed based on the linear elastic beam theory. The air drag to the beam vibration is characterized based on the Oseen solution of the drag force acting on an infinite long cylinder that moves in incompressible viscous fluids at low Reynolds numbers. The conditions for the validation of this characterization are discussed. The analytical results show that air-damping generally shifts the resonant frequency downward and degrades the quality factor, and that this effect increases as the dimension of the beam decreases. The dependence of this effect on the dimension of the beam is illustrated. The presented analysis provides a simplified analytical approach to estimate the length-scale-dependent air-damping effect for designing high-performance micromachined beam resonators.