Geometrical compensation for mode-matching of a (100) silicon ring resonator for a vibratory gyroscope

A geometrical compensation design method in a (100) single crystal silicon (SCS) vibratory ring gyroscope (VRG) has been proposed in order to decrease frequency splits (∆f) caused by anisotropic in-plane Young's modulus (E) of (100) SCS and realize mode-matching in VRG. The radial width as a function of the E was varied to equalize the effective bending stiffness along the different crystal directions. The finite element analysis (FEA) simulation verified that the ∆f decreased from 260 Hz to 145 Hz after compensation. By optimizing suspending beam dimensions, the simulated ∆f further decreased to 9 Hz. Both uncompensated and compensated resonators of 1-mm diameter were fabricated using a silicon-on insulator (SOI) wafer of a 22- m-thick device layer. The fabricated devices showed large ∆f because of fabrication error, which is adjusted by electrostatic tuning. After the electrostatic tuning, the measured minimum frequency splits for the compensated and uncompensated VRGs were about 54 Hz and 166 Hz, respectively. The small measured ∆f after the electrostatic tuning indicates that the proposed geometrical compensation is effective in eliminating frequency split caused by anisotropy in elastic constants.