On numerically solving an eigenvalue problem arising in a resonator gyroscope

In 1890 G.H. Bryan observed that when a vibrating structure is rotated with respect to inertial space, the vibrating pattern rotates at a rate proportional to the inertial rate of rotation. This effect, called “Bryan’s effect”, as well as the proportionality constant, called “Bryan’s factor”, have numerous navigational applications. Using a computer algebra system, we present a numerically accurate method for determining fundamental eigenvalues (and some of the overtone eigenvalues) as well as the corresponding eigenfunctions for a linear ordinary differential equation (ODE) boundary value problem (BVP) associated with a slowly rotating vibrating disc. The method provides easy and accurate calculation of Bryan’s factor, which is used to calibrate the resonator gyroscopes used for navigation in deep space missions, stratojets and submarines. Bryan used “thin shell theory” to calculate Bryan’s factor for fundamental vibrations. Apart from the high accuracy achieved, the numerical routine used here is more robust than “thin shell theory” because it determines (at least for low frequencies) the fundamental as well as the first three overtone frequencies and each Bryan’s factor associated with these vibration modes. The theory involved and the calculation of results with this numerical method are quick, easy and accurate and might be applied in other disciplines that need to solve suitable eigenvalue problems. Indeed, results are obtained directly using commercial software to numerically solve a system of linear ODE BVPs without having to formulate the extremely technical solution that is traditionally used (viz: solve the governing system of partial differential equations via Helmholtz potential functions and the necessary numerical calculation of Bessel and Neumann functions).