Vibration of a hemispherical shell gyro excited by an electrostatic field

Abstract

In this paper the stability and precession phenomenon of the modal pattern of a rotating hemispherical shell gyro excited parametrically by electrostatic forces is investigated. Nonlinear modal equations of the rotating shell under small strain and moderate rotation are derived by following Niordson's thin shell theory. The method of multiple time scales is used to determine quantitatively the vibration displacement of the shell from which the precession rate of the modal vibration pattern can be extracted. Floquet theory is employed to study the stability of the analytical solution. Numerical simulation verifies the analytical results. Recent work by Matthews and Rybak (1), and Chou and Chang (2,3) showed that the hemispherical shell gyroscope becomes attractive and competitive in comparison with other types of gyros. The work- ing principle is that the modal vibration pattern of the shell moves when the shell is rotated about its axis and this movement provides a measure of the applied rate of turn. An important design feature of the hemispherical resonator gyroscope is that one is able to excite the shell without having a direct mechan- ical coupling between the shell and its supporting frame. In order to maintain the precessing vibration modal pattern, an electrostatic excitation is employed. Linear analysis (3) of a rotating hemispherical shell gyro excited parametrically by electrostatic forces showed that the vibration amplitude is decayed to zero in the stable region and becomes divergent in the unstable region under excitation. Therefore the geometrically nonlinear (4) effects of the shell under small strain and moderate rotation (5) are studied. Reissner (6) derived the geometrical nonlinear shell equations with Kirchhoff-Love assumption. Since that, many people continuously improved the nonlinear shell theories (7). In this paper, the nonlinear modal equations of the rotating shell under small strain and moderate rotation are derived by following Niordson's (8) thin shell theory. The method of multiple time scales (9) is used to determine quantitatively the vibration amplitude of the shell from which the precessional rate of the modal vibration pattern can be extracted. Floquet (10) theory is employed to study the stability of the analytical solution. Numerical simulation verifies the analytical results.