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.
