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In this paper, the coupled longitudinal and lateral vibrations of a rectangular parallelepiped composed of a uniform, isotropic, elastic body are examined. The one-dimensional model for the body was developed by Green, Naghdi and several of their co-workers. It is based on a Cosserat or directed rod theory. The frequency spectrum is found to divide naturally into three regions which are separated by degenerate cases. When the cross-section of the body is square, a natural splitting of the modes occurs. One-half of this split involves motions coupling the axial extension and symmetric lateral deformation of the body. The other half involves asymmetric lateral deformations. The cases of a free–free rod and a cantilevered rod are discussed and the results compared to existing works in the literature which use the three-dimensional theory of elasticity to model the vibrations.

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