Oscillations of a system of rigid bodies partially filled with viscous fluids under the action of an elastic damping device

In this paper, we investigate the linearized two-dimensional problem on small motions of a system of rigid bodies partially filled with viscous incompressible fluids and connected in series by springs. The first and last bodies are attached by springs to two supports with a given law of motion. The trajectory of the system is perpendicular to the direction of gravity, and the damping forces acting on the hydrodynamical system are generated by the friction of bodies against a stationary horizontal support. For the described system, the law of total energy balance is formulated. Using the orthogonal projection method and a number of auxiliary boundary problems, the original initial-boundary value problem is reduced to the Cauchy problem for a first-order differential-operator equation in the orthogonal sum of some Hilbert spaces. The properties of operator matrices, which are coefficients of the obtained differential equation, are investigated. A theorem on the unique solvability of the resulting Cauchy problem on the positive semi-axis is proved. On the basis of the proved theorem, sufficient conditions for the existence of a strong with respect to time solution of an initial-boundary value problem describing the evolution of a hydrodynamical system, are found. From a mathematical point of view, the system under consideration is a finite-dimensional perturbation of the well-known S.G. Krein's problem on small motions of a viscous fluid in an open vessel.