Keywords: free vibration, rectangular container, liquid-coupled vibration, Rayleigh-Ritz method, orthogonal polynomials, mode shapes.

This paper presents a theoretical analysis for the free vibration of a rectangular container partially filled with an ideal liquid. It is assumed that the container open to the top is simply supported along the two top edges and its side edges are fixed. Wet dynamic displacements of the container are approximated by combining the orthogonal polynomials satisfying the boundary condition. As the facing vertical rectangular plates are assumed to be geometrically identical, the vibration modes of the container can be divided into two categories: symmetric and antisymmetric modes with respect to the vertical plane passing through the center of the container and perpendicular to the free liquid surface. The liquid displacement potentials satisfying the Laplace equation and liquid boundary conditions are derived, and the wet dynamic modal functions of a half of the container can be expanded by the finite Fourier transform for compatibility requirements along the contacting surfaces between the container and the liquid. An eigenvalue problem is derived using the Rayleigh-Ritz method. Consequently, the wet natural frequencies of the rectangular container can be extracted. The proposed analytical method is verified by observing an excellent agreement with three-dimensional finite element analysis results.

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