Vibration displacement on substrate due to time-harmonic stress sources from a micromechanical resonator

This paper investigates the vibration displacement on substrate due to time-harmonic stress sources from a micromechanical resonator fabricated using Microelectromechanical Systems (MEMS) technology. The vibrations of a micromechanical resonator exert a time-harmonic stress across the clamped region on its substrate and further excite elastic waves propagating into the substrate, therefore leading to the dissipation of vibration energy—commonly referred to as support loss. Support loss can be quantitatively evaluated in terms of this time-harmonic stress and its corresponding vibration displacement on substrate. To calculate this vibration displacement, this work treats the substrate as either a semi-infinite or infinite elastic medium. Then, the classic elastic wave theories are utilized to describe its behavior, and the Fourier transform or Hankel transform is employed for dealing with the geometrical infinity of the substrate. Consequently, the mathematical expressions for this vibration displacement, under several typical time-harmonic stress sources in micromechanical resonators, are derived. As a result, this work provides the basis for quantitatively evaluating support loss in a micromechanical resonator and predicts the relation of support loss versus some key resonator design parameters and substrate materials.