Analysis and design the bell-shaped vibratory gyroscope under parabolic coordinates system

The bell-shaped vibratory angular rate gyro (abbreviated as BVG) which is inspired by the Chinese traditional bell is a novel shell vibratory gyroscope. It sensitizes angular rate by the standing wave precession effects. This work presents the gyroscopic effect of paraboloidal bell-shaped resonator under rotating condition and analyzes the feasibility of it to make axisymmetric shell resonator gyroscopes. Firstly, based on the vibratory characteristics of bell-shaped resonator, it derives the transformation matrix that transforms from parabolic coordinates to rectangular coordinates with tensor analysis theory, by which it solves the elastic field equations under the parabolic coordinates, establishes the dynamic equations of bell-shaped resonator. It analyzes the gyroscopic effect of bell-shaped resonator due to coriolis force, proves the equivalency between bell-shaped vibratory gyro and traditional coriolis vibratory gyro, gets the formulas of natural frequency and Bryan's factor. Compared to traditional analysis methods, it simplify the analysis process and is more intuitive to describe the vibratory process of paraboloidal bell-shaped resonator. Then, It analyzes how the bell-shaped resonator vibrates under parabolic coordinates using finite element analysis method and investigates the relationship between the coordinates parameters and nature frequency. It demonstrates the correctness about the analysis method.