Dynamic stability of ring-based angular rate sensors

Dynamic stability of a rotating ring subjected to harmonic perturbations in input angular rate is examined using an asymptotic approach. The governing equations that represent the transverse and tangential in-plane motion of the ring are derived via Hamilton's principle. The equations of motion, after discretization and suitable linearization, represent a two-degree-of-freedom time-varying linear gyroscopic system. Such a system can exhibit instability behaviour characterized by exponential growth in response amplitudes. Employing the method of averaging, conditions for instability are obtained in closed-form. Instability boundaries for the ring in the excitation intensity-frequency space are then established for small excitation amplitudes. In addition, effects of damping, input angular rate variations, and imperfection due to the ring asymmetry are discussed.