On steady rotations of a rigid body bearing a single-axis powered gyroscope whose precession axis is parallel to a principal plane of inertia

The set of steady motions of the system named in the title is represented parametrically via the gyro gimbal rotation angle for an arbitrary position of the gimbal axis.We study the set of steady motions for a system in which the gyro gimbal axis is parallel to a principal plane of inertia as well as for a system with a dynamic symmetry. We determine all motions satisfying sufficient stability conditions. In the presence of dissipation in the gimbal axis, we use the Barbashin-Krasovskii theorem to identify each steady motion as either conditionally asymptotically stable or unstable.