Analysis of the flexural vibrations of variable density spheroids immersed in an ideal fluid, with application to ship structural dynamics

Abstract
An analysis is given of the characteristic flexural modes and frequencies of a linearly elastic free-free spheroid in an ideal fluid. The finite element method is used to represent the structural properties of a slender spheroid, employing a special element formulated for this purpose on the basis of Euler-Bernoulli beam theory. A consistent added mass matrix is derived from the exact solution of the infinite fluid potential problem, truncated at a suitable number of terms. A consistent added stiffness matrix is obtained for the buoyancy forces on a spheroid floating with its axis in a free surface, but other free surface effects (associated with wave generation) are assumed negligible. Solutions are computed for different aspect-ratio variable density spheroids vacuo , deeply submerged, and floating. The results indicate the possibility of considerable distortions in the lowest (‘rigid’) modes of slender floating bodies vibrating in a vertical plane, and illustrate the difficulty of defining three dimensional reduction factors for use with a simplified two dimensional theory. Derivation of the classical reduction factors for uniform density spheroids is given by way of comparison. The paper provides an illustration of use of a finite element formulation, in conjunction with consistent added mass and stiffness matrices, for a rational analysis of the structural dynamics of ships and other marine vehicles.

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