Analysis of a nutation damper for a two-degree-of-freedom gyroscope

Abstract

The dynamics of a two-degree-of-freedom gyroscope with a ball-in-tube nutation damper mounted in the gimbal is analyzed. The nonlinear coupled equations of motion are solved by using the method of multiple scales. The results show that, if the natural frequency of the ball-in-tube damper is tuned to equal the nutation frequency of the gyroscope with the damper removed, the wobble motion of the gyroscope will be eliminated most rapidly and the decay time constant of the nutation angle is a concave function of the damping coefficient. If the nutation damper is oriented perpendicular to the spin axis of the rotor, the residual precession due to the gravity effect can be avoided. HE gyroscope is one of the two basic components of all inertial navigation systems (the other is an accelerometer). It serves as directional reference in an inertial frame. Gyro- scopes are also used as motion sensors for stabilizing the mo- tion of ships, aircraft, and other mechanical systems. Its de- sign and control has been an important technological problem for half a century. The two-degree-of-freedom gyroscope has a support that permits the spin axis to have two degrees of rotational freedom with respect to the case of the gyroscope. Such a device is shown schematically in Fig. 1. The Cartesian coordinate system X,Y,Z is fixed on the case of the gyroscope with origin O coinciding with the center of mass of the rotor. The system x,y,z is fixed on the inner gimbal. The inner gimbal can rotate freely about the x axis. The outer gimbal can rotate freely about the Z axis. In the navigation system, the spin axis of the rotor is used to indicate one of the directions of the axes of the navigational frame, i.e., north, east, and down. These physical directions rotate with respect to inertial space due to Earth rotation and vehicle motion. Therefore, the spin axis of the rotor must be rotatable on command. This can be done by applying a calibrated torque to the inner gimbal and the outer gimbal. The rotor will inevitably result in a coning motion (precession and nutation) of the spin axis about the angular momentum vector of the gyroscope (which is fixed in space in the absence of subsequent external torque) after a torque with finite duration has been applied. Therefore, the gyroscope will be of practical value only if the nutation and precession are damped.1 The phenomenon of coning motion also appears for the gyroscope, which serves as a motion sensor. The elimina- tion of the coning motion will not only enhance the gyro- scope's accuracy, but will also reduce the loading of the con- trol unit.2 The coning motion can be suppressed by attaching a vibra- tion damper to the gyroscope at some appropriate place. Phys- ically, the vibration damper might be any device that can dissi- pate the energy of coning motion of the rotor and convert the transverse angular momentum of the gyroscope into spin an- gular momentum. The spin-stabilized satellite will also result in such coning motion after a reorientation maneuver. The commonly used damping devices for removing this coning motion of the satellite are viscous ring nutation dampers,35