Using Fourier series to analyse mass imperfections in vibratory gyroscopes

Abstract

When a vibrating structure is subjected to a rotation, the vibrating pattern rotates at a rate (called the precession rate) proportional to the inertial angular rate. This is known as Bryanʼs effect and it is employed to calibrate the resonator gyroscopes that are used to navigate in outer space, the stratosphere and under the polar cap. We study Bryanʼs effect for a non-ideal resonator gyroscope, using the computer algebra system (CAS) Mathematica to do the analysis involved, rendering this work accessible to undergraduate students with a working knowledge of college calculus and basic physics or mechanics (such as senior Engineering Mathematics students). In this paper the density of a slowly rotating vibrating annular disc is assumed to have small variations circumferentially, enabling a Fourier series representation of the density function. Using a CAS, the Lagrangian of the system of vibrating particles in the disc is calculated and, employing the CAS on the Euler–Lagrange equations, the equations of motion of the vibrating, rotating system are calculated in terms of “fast” variables, enabling us to demonstrate that the mass anisotropy induces a frequency splitting (beats). Unfortunately the fast variables are difficult to analyse (even with the aid of a CAS) and consequently a transformation from fast to slow variables is achieved. These slow variables are the principal and quadrature vibration amplitudes, precession rate and a phase angle. The transformation yields a system of four nonlinear ordinary differential equations (ODEs). This system of ODE demonstrates that the Fourier coefficients of the density function influence the precession rate and consequently a gyroscope manufactured from such a disc cannot use Bryanʼs effect for calibration purposes. Indeed, the CAS visualises that a capture effect occurs with the precession angle that appears to vary periodically and not increase linearly (Bryanʼs effect) as it would for a perfect structure. Keeping in mind that manufacturing imperfections will always be present in the real-world, the analysis shows how such density variations may be minimised. Using a symbolic manipulator such as Mathematica to do the “book-keeping” eliminates the plethora of technical detail that arises during calculations of a highly technical nature. This allows the aforementioned students to focus on the salient parts of the analysis, producing results that might have been beyond their capabilities without the aid of a CAS.