Longitudinal vibrations of a Rayleigh-Bishop rod

Abstract

In this work, for analyzing the longitudinal vibrations of a conic rod, the authors used the Rayleigh–Bishop model, which generalizes the Rayleigh model and takes into account both lateral displacements and the shear stress in the transverse cross section. The rod vibrations are described by the linear partial differential equation with the variable coefficients containing the mixed fourth-order derivative. The free vibrations of cylindrical and conic rods are considered. It is shown that the classical model of longitudinal vibrations of the rod described by the second-order wave equation can substantially overestimate the frequencies of the rod free of vibrations in comparison with the Rayleigh–Bishop model. It should be noted that transverse vibrations of a rod described by a linear partial differential equation of the fourth order were considered in many works.