Theory and error analysis of vibrating-member gyroscope

There is a general class of gyroscopic instruments that use a vibrating member as the sensitive element. Instruments in this class are governed by the differential equations
\ddot{x} + \Omega^{2}x - c\omega\dot{y} = 0
\ddot{y} + \Omega^{2}y - c\omega\dot{x} = 0
which describe the position of a reference point on the vibrating member relative to a coordinate system fixed in the instrument case. The point
(x,y)
moves in an elliptical orbit with poriod
2\pi/\Omega
. The orbit precesses at a rate
-c\omega/2
(with
c \leq 2
) relative to the coordinate system and, hence, tends to remain fixed in inertial space. The differential equations for the orbital elements (
a
=semimajor axis
b
=semiminor axis, φ=inclination, and
\theta
=orbital angle) are derived for a nonideal gyroscope with damping and anisoelasticity present. The difforential equations for the long-term effects are obtained by averaging the coefficients over the approximate period of oscillation. These equations can he transformed into a fourth-order system of linear differential equations with constant coefficients in the average energy and angular momentum, and their derivatives. These equations are solved explicitly for several cases of practical interest and the results are interpreted physically.