On gyroscopic effects in vibrating and axially rotating solid and annular discs

Abstract

“Bryan’s effect" - that is, the effect of a vibrating pattern’s precession in the direction of inertial rotation of a vibrating ring - was discovered by G. Bryan in 1890. This effect has several applications in navigational instruments, such as cylindrical, hemispherical and planar circular disc rotational sensors. The model of a thin circular disc vibrating in its plane and subjected to inertial rotation is considered. The dynamics of the disc gyroscope are considered in terms of linear elasticity. Two models are considered: solid discs and a composite disc consisting of concentric annular discs with various boundary conditions on the inner and outer circumferences. It is assumed that the angular rate of inertial rotation of the composite disc is constant and has axial orientation. It is also assumed that this angular rate is much smaller than the lowest eigenvalue of the composite disk. Hence any centrifugal effects and quantities that are proportional to the square of the angular rate are negligible. Our model is formulated in general terms and then compared to a formulation in terms of Novozhilov-Arnold-Warbur-ton’s theory of thin shells. The system of equations of motion of the disc is separated and transformed into a pair of wave equations in polar coordinates. A solution is obtained in terms of Bessel and Neumann functions. Various non-axisymmetric modes of the composite disc are considered and the dependence of Bryan’s effect on eigenvalues, mass densities of the composite disc, its modulii of elasticity, Poisson ratios, outer and inner radii of the disc, and for various types of boundary conditions, are investigated