On the stability of the vertical rotation of a solid suspended on a rod (Q1820628)

The problem of the motion of a dynamically symmetric solid suspended from a fixed point by a weightless rod and two ball and socket joints one of which is fixed at the fixed point O', and the other is on the body axis of symmetry at the point 0 is considered. The question of the stability of the uniform body rotation when points O' and O, and the body centre of inertia C lie on the same vertical, and at the same time point O may be either above or below point O', and point C either above or below point O, is discussed. An analysis of the necessary and sufficient conditions for stability is given. The set of all the system's parameters is reduced to three- independent dimensionless parameters L,\(\Omega\) and \(\beta\), and in the plane (L,\(\Omega)\), for fixed values of \(\beta\), the regions for which the unperturbed rotation is steady, or steady to a first approximation, or non-steady are indicated. The regions for which the body rotation is steady to a first approximation when the point O is situated higher than the point O', and the point C lies higher or lower than the point O are determined.