Stability of the Coupled Liquid-Elastic Bottom Oscillations in a Rectangular Tank

Abstract

Eigenoscillations of the elastic bottom of a rigid (two-dimensional) rectangular tank with an ideal incompressible liquid with irrotational flows, which completely fills it, are were investigated. The elastic bottom is a clamped thin rectangular plate subject to tensile or compressive forces in its middle surface. It is shown that the frequency equation is divided into two equations describing symmetric (even) and antisymmetric (odd) frequencies, and can be written in a single form for these frequencies. For even and odd frequencies, an approximate formula is obtained, from which approximate conditions follow for stability of coupled vibrations of an elastic basis and a liquid. Exact stability conditions are obtained. The stability conditions of the static approach coincide with the exact stability conditions of the dynamic approach. It is shown that the approximate value of the critical bending stiffness for asymmetric frequencies is 0.952 times lower, and for symmetric frequencies – 0.930 times.