Damped Free Oscillations of a Gyroscopic System

Abstract

THE equations of motion of a gyroscopic system consisting of a symmetric rotor of axial moment of inertia C mounted in heavy gimbals are: where n is the constant angular velocity of the rotor about its axis; θ1 and θ2 are small angles denoting rotations about the outer and inner gimbal axes respectively; c 1 and c 2 are the corresponding coefficients of viscous friction; A is the moment of inertia of the entire system about the outer gimbal axis and B that of the inner gimbal plus rotor about the inner gimbal axis. These equations are based on three principal assumptions: (1) the centres of mass of the rotor and of the gimbals lie at the intersection of the gimbal axes, (2) the axis of the rotor is nearly perpendicular to the axis of the outer gimbal, and (3) the axes of rotation are principal axes of inertia. If there is no friction at the gimbal axes, equations (1) define a free, undamped oscillation in which the rotor axis performs a conical whirl about its central position and the gimbals vibrate at frequency p = Cn/√AB.