Relative stability of motion of a body with a conical cavity containing an ideal liquid

Abstract

The stability of the rotational motion of a body with a cylindrical cavity filled with an ideal incompressible liquid was considered in detail by Chetaev [1]. He assumed potential flow of the liquid, which is a good approximation to physical reality in the case of small oscillations of the body or in the case of additional baffles within the cavity. Analysis was made of the solutions of the inner Neumann problem obtained by Zhukovskii in [2], where he also determined the liquid velocity potential for certain other cavities. The velocity potential for a more general case of cylindrical cavities was found in [3]. The problem of the stability of a rotating top with ellipsoidal and cylindrical cavities has been considered in the studies of Sobolev, Rumyatsev, Ishlinskii, Temehenko, and others. The present paper comiders the stability of motion of a body with a conical cavity filled with an ideal liquid. The cavity may be separated by radial baffles passing through the axis of symmetry. As in [1], the liquid flowis assumed irrotational. 1. The motion of the body with a cavity will be defined by the rxanslational velocity v of some point O and the rotational velocity to about this point. The internal motion of the liquid changes only due to the rotational motion of the body about the point O, which may be considered fixed if we consider only relative motions. We introduce the body coordinate system xlxzx a with axes directed along the principal axes of the body's ellipsoid of inertia. We represent the absolute velocity potential function of the liquid particles, expressed in terms of the coordinates of a point in the moving coordinate system, in the form