On the motion and stability of a gyroscope on gimbals in the Newtonian central force field

Abstract

In [1–5] the authors have investigated the motion and also the stability of a certain stationary motion of symmetric gyroscopes, with the axis of the outer gimbal ring vertiacl, in the uniform problems arising in the motion of a rigid body about a fixed point in the case of Lagrange. The author of [6] has investigated a rigid body moving about a fixed point, assuming that the dimensions of the body are small compared with the distance from the fixed of the center of attraction. His case, similar to the case of Lagrange, has beenn reduced to quadratures. The necessary conditions for stability of permanent rotations for the above case have been presented in [7]. The problem investigated here is described in the title. It is assumed that the direction of the axis of the outer gimbal ring coincides with the direction of the line from the attraction center to the point of intersection of the gimbal axes. This assumption permits, as might be expected, an analogy wtih the case of Lagrange. The dimensions of the body are assumed, as in [6], to be relatively small. The integration of the equations of motion in reduced to quadratures. When investigating the stability of stationary solutions (regular precession and “vertical rotation”) the method of Chetaev has been applied.