Small oscillations of an ideal liquid contained in a vessel closed by an elastic circular plate, in uniform rotation

Abstract

The problem of the small oscillations of an ideal liquid contained in a vessel in uniform rotation has been studied by Kopachevskii and Krein in the case of an entirely rigid vessel [3]. We propose here, a generalization of this model by considering the case of a vessel closed by an elastic circular plate. In this context, the linearized equations of motion of the system plateliquid are derived. Functional analysis is used to obtain a variational equation of the small amplitude vibrations of the coupled system around its equilibrium position, and then two operatorial equations in a suitable Hilbert space are presented and analyzed. We show that the spectrum of the system is real and consists of a countable set of eigenvalues and an essential continuous spectrum filling an interval. Finally the existence and uniqueness theorem for the solution of the associated evolution problem is proved by means the semigroups theory.

Keywords

• small oscillations
• plates theory
• variational and operatorial methods
• spectral theory
• semi-groups