Refined theories of longitudinal vibration of rods through unified framework, and their generalization

This study presents a detailed formulation of a unified framework for the Żak–Krawczuk refined theories addressing the longitudinal vibration of rods. Building upon this framework, we systematically derive governing differential equations for the higher-order two-mode, three-mode, and four-mode theories. These refined models incorporate higher-order effects, enabling more accurate characterization of longitudinal wave propagation in elastic rods. To assess the accuracy and validity of the proposed theories, the results are compared with the Pochhammer–Chree's exact solution based on three-dimensional theory of elasticity. The predictive capability of the Żak–Krawczuk models goes on improving with the increase in number of retained modes, particularly for higher frequency regimes. Finally, we use this framework to obtain higher order five-mode theory. As predicted, the solutions of this five-mode theory are more accurate than above lower-mode formulations.