Three-dimensional exact solution of free vibrations of a simply supported rectangular plate in contact with a fluid

Understanding fluid-solid interactions between an elastic panel and the surrounding fluid is crucial to the design of marine structures. Here, we put forward an exact, three-dimensional, solution to the free vibrations of a simply supported plate separating a fluid column from vacuum. The framework extends the classical elasticity solution by Pagano to the study of the interactions between a plate and a fluid column. In this vein, an exact solution is derived for the plate vibrations through a Fourier series, and a general form for the fluid flow is determined through the solution of the Laplace equation for potential flows. The solid and the fluid phases are coupled through continuity in normal velocity and stress components at the fluid-solid interface. The proposed framework provides a first insight into three-dimensional deformation and stress experienced by a plate due to hydrodynamic loading from the interaction with a fluid column. We discover a number of counterintuitive phenomena due to the interplay between elasticity and added mass, including symmetry breaking of through-the-thickness stress profiles, emergence of a mode shape that is not present for in-vacuum vibrations, recirculating patterns in the fluid flow, and switching of modes in response to changes in the fluid column height. The proposed solution is expected to offer a basis for the design of thick panels to be used in marine structures.