Comparison of the four rod theories of longitudinally vibrating rods

Abstract

Dynamics of a longitudinally vibrating rod are analyzed from a wave standpoint, in which the motion of the rod is described in terms of waves that propagate through the uniform rod and are reflected and transmitted at structural discontinuities and boundaries. This study is based on the four engineering theories, namely, the elementary, Love, Mindlin–Herrmann, and three-mode theories. The propagation relations that are governed by the equations of motion are derived. The reflection and transmission relations, which are dependent upon the continuity and equilibrium conditions at structural discontinuities, are obtained. Waves generated by externally applied forces are found. The wave propagation, reflection, and transmission relations are assembled to provide a concise and systematic approach to vibration analysis of rods. Numerical examples are presented. Comparisons and recommendations are made for meaningful engineering practice.