On the theory of small vibrations of a rigid body with a cavity partially filled with a viscous incompressible liquid

Abstract

boundary value problems. We introduce the o*x*y* z* coordinate system; its o*x*-axis is directed opposite to the gradient j of the field of mass forces of the nonperturbed motion relative to the inertial o x y z coordinate system, while the o* y* - and o* z* -axes are oriented in a certain manner relative to the body performing this motion. The vector-j'is assumed to maintain an unchanged direction (at the same time its modulus can be an arbitrary function of time bounded from below). We further introduce the oxyz coordinate system with the axes rigidly connected to the body. In the figure we have shown the position of oxyz relative to o*x*y*z*, given by the vectors of small displacement ~(t) and small rotation 0(t); these vectors give the perturbed motion Of the body. The wave motions of the liquid are described by the generalized coordinates Sn(t) (n = 1, 2 .... ), each of which represents the displacement of the surface of the liquid at a certain point of its contour, relative to the nonperturbed surface, during the n-th mode of vibration. The use of the usual assumptions, characteristic for the linear formulation of problems of hydrodynamics of a viscous liquid [2], leads to the following boundary value problem: