On Extensional Vibration Modes of Elastic Rods of Finite Length Which Include the Effect of Lateral Deformation

Extensional vibrations are analyzed in the linear theory of an elastic rod of finite length which accounts for lateral deformation. A simplified version of a theory of elastic rods is used; it is based on the concept of a directed curve as developed by Green et al. [1–3]. This simplified version is similar to Mindlin and Herrmann’s theory [4], and has been used recently by Krishnaswamy and Batra [5] to study wave propagation and pure thickness oscillations in an infinitely long rod. It may be recalled that for an infinitely long circular rod whose lateral surface is free of traction, the frequencies of extensional vibration are obtained by solving the well-known Pochhammer–Chree equation (see e.g., reference [6]). However, for a finite rod, this equation is invalid as it cannot satisfy the stress-free end boundary conditions. Various approximate solutions for extensional oscillations of a free finite rod have been obtained by Rumerman and Raynor [7], Hutchinson [8, 9], and Rasband [10], all of whom based their work on the classical three-dimensional theory of elasticity. McNiven and Perry [11] presented a solution based on an approximate set of equations which take into account the coupling between longitudinal, axial shear, and radial modes in a rod of infinite length. An experimental study on vibrations of solid cylinders can be found in reference [12]. In the present paper, free extensional vibrations of a finite rod subjected to various boundary conditions are analyzed. It is shown that the analysis can be separated into three cases depending on the range in which the natural frequency lies. Such a situation also arises in the Timoshenko theory of flexural vibrations of a beam (see, e.g., reference [13]). The differential equations of the extensional theory are very similar to those of the Timoshenko theory but there are some differences which we will allude to when appropriate. Our analysis of Case 3 (see section 3) may be regarded as the extensional counterpart of a recent paper by O’Reilly and Turcotte [14] on Timoshenko beams.