The stability of the steady motions of a symmetrical gyrostat on an absolutely rough horizontal plane

Abstract

The problem of the motion, without slipping, of a dynamically symmetrical gyrostat on a fixed horizontal plane is investigated. It is shown that Chaplygin's equations express, in projections on the axes of a semimobile system of coordinates, the theorem on the angular momentum of the gyrostat about the point of contact with the plane. All possible steady motions of a heavy symmetrical gyrostat on an absolutely rough plane including the case when the nutation angle is equal to zero are investigated using the generalized Routh theorem. The effect of the rotor on the stability of steady motions is also examined. The case when the body of the gyroscope is a solid with a circular base, in particular, a disc with a rotor, is considered. It is shown that the rotating rotor has a stabilizing influence on equilibrium of the gyrostat and a destabilizing influence on the rolling of the gyrostat about a straight line, provided the axial angular momentum of the gyrostat is zero.