Dynamics and control of vibratory gyroscopes with special spherical symmetry

Abstract

It was shown in 1985 by Acad. V. Zhuravlev that the angular rate of a pure vibrating mode excited in a vibratory gyroscope with spherical symmetry is proportional to an inertial angular rate of the gyroscope. The effect is three dimensional and hence, it could be potentially used as a conception of a spatial inertial rotational sensor. Furthermore these effects are important in acoustics, geophysics and astrophysics. The effect was investigated qualitatively without specifying of a coordinate system and determination of the scale factors. In the present paper the effect of vibrating pattern precession is considered in a spherical coordinates. On the basis of exact solution of 3-D equations of motion of thick isotropic sphere, which are obtained in the spherical Bessel and the associated Legendre functions, the effects of rotation are investigated and scales factors are determined for different vibrating modes of the spherical body, spheroidal and torsional. Corresponding scales factors are calculated depending on nature of vibrating modes and their number. For realization of a three axes sensor it is necessary to realize three orthogonal spherical coordinate systems. Elements of control of vibrating spherically symmetric body are considered and possible imperfections are discussed