Analysing manufacturing imperfections in a spherical vibratory gyroscope

In a recent article in the Journal of Sound and Vibration (JSV) we discussed the influence of mass imperfections and isotropic (viscous) damping on the vibrating pattern of a slowly rotating spherical body. Using the mathematical tools developed in the JSV article, in addition to mass imperfections, we demonstrate how to introduce prestress using a non-linear theory of elasticity and how to introduce anisotropic (viscous) damping into the equations of motion using Rayleigh dissipation. The equations of motion thus obtained demonstrate that a “precession wave” is generated within the sphere and indicate what control measures (if any) might be taken in order to approximate an “ideal” situation where the vibration pattern within the spherical body rotates at a rate (called the precession rotation rate) that is proportional to the slow rotation rate of the sphere. The constant of proportionality referred to above is called Bryan´s constant and is used to calibrate the hemispherical resonator gyroscopes that are used in the space shuttle. The equations of motion demonstrate that four “slow” variables are present namely the principal and quadrature amplitudes of vibration, the precession angle and a phase angle. It appears that each slow variable is affected by mass-stiffness imperfections andor constant prestress and anisotropic damping. The phenomenon of “beats” is predicted and a numerical experiment indicates that two capture effects are possible with precession angle and that the same is true for the phase angle.