On the stability of coupled oscillations of the elastic bottom of a rigid rectangular channel and ideal liquid

Abstract

Normal oscillations of the elastic bottom of a rigid rectangular duct with an ideal non-compressive fluid, which completely fills it, were investigated. The elastic bottom is a membrane. 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 fluid. These conditions are independent of the liquid height and the membrane mass. Exact stability conditions that coincide with hydrostatic conditions are derived. It is shown that the approximate value of the critical tension for asymmetric frequencies is 4/5 times lower, and for symmetric frequencies, it is 0.818 times lower.