An Approximate Theory Governing Symmetric Motions of Elastic Rods of Rectangular or Square Cross Section

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Abstract
Variational equations of motion are developed for symmetric motions of linear elastic bars of rectangular cross section. In the finite term approximation, sufficient terms are retained to allow a longitudinal mode, two thickness-stretch modes, and two thickness-shear modes of vibration in an infinite bar of square cross section. Modes for complex wave numbers are also investigated. Adjustment factors in the strain energy and kinetic energy potentials are used to match exact and experimental solutions. Experimental frequency versus wave number results for four modes are reduced by Fourier synthesis and compared both to the approximate theory and to the exact solution for circular cylinders. Theory is intended to predict behavior of thick rectangular bars for which the plane stress solution is not accurate.

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