Coupled free vibrations of liquid in a three-dimensional rectangular container with an elastic cover

Abstract

The coupled free vibration of liquid and its elastic cover, such as a plate or a membrane, in a three-dimensional rectangular tank is investigated through an analytical scheme based on the velocity potential theory for the flow and the linear elastic theory for the cover. For the fluid domain, the velocity potential is expanded into double cosine series along the longitudinal and transverse directions, respectively, with the corresponding eigenvalues determined from the impermeable conditions on the side walls. The vertical modes of the potential are obtained from the Laplace equation. The deflection of the rectangular cover is expanded into the same double cosine series to match the potential, together with additional terms for satisfying the edge conditions. The polynomials are used for these additional terms, which are then expanded into cosine series. For the expansions of the higher-order derivatives of the deflection, the derivatives of these polynomial terms are expanded into cosine series directly, rather than being obtained through differentiating the cosine series of the deflection, to avoid the non-convergent series. Through imposing the boundary conditions on the fluid–plate interface and edge conditions, an infinite matrix equation for the unknown coefficients can be established. The natural frequencies can be obtained when the determinant of the matrix is zero. In practical computation, the infinite matrix equation is truncated into finite size. Results are first provided for natural frequencies. This is followed by the corresponding natural mode shapes and principal strains distribution on the cover. The underlying physics of these results is then provided.