On the solution of a complicated biharmonic equation in a hydroelasticity problem

A hydroelastic problem of free vibrations of a thin plate that horizontally separates ideal in-compressible liquids of different densities in a rigid cylindrical tank with an arbitrary cross-section has been considered in the linear formulation. To solve the corresponding complicated inhomogeneous biharmonic equation, the fundamental system of the solutions of biharmonic equation (FSS) and the eigenmodes of ideal liquid oscillations in a cylindrical cavity were used. The frequency equation was obtained for arbitrary fixation of the plate contour. On the example of a clamped plate, the frequency equation was simplified by decomposing the corresponding homogeneous biharmonic equation into two harmonic equations and using Green’s formula for the Laplace operator. It was shown that in this case the frequency equation does not depend on the FSS and becomes greatly simplified because the FSS depends on the unknown frequency. The resulting equation has a single form for the cases of a right circular cylinder and a rectangular channel; in particular cases, it coincides with the previously obtained equations. Research of asymmetric vibration frequencies of a plate and a membrane, as well as axisymmetric vibration frequencies of a membrane in a circular cylinder, has been carried out. An approximation formula for high frequencies and approximate conditions for the stability of the plate and membrane vibrations were obtained. © 2023, Springer Nature Switzerland AG.