Torsional vibrations and waves in a radially inhomogeneous cylinder

This paper studies the problem of torsional vibration of a radially inhomogeneous isotropic cylinder. It is assumed that the elastic moduli and the density of the cylinder material are power functions of the cylinder radius. Cases where the lateral surface of the cylinder is free from stresses and fixed are studied. After satisfying homogeneous boundary conditions specified on the lateral surfaces of the cylinder, dispersion equations are obtained. Exact solutions to the problem are constructed. For a cylinder of small thickness, an analysis of the roots of the dispersion equations with respect to a small parameter characterizing the cylinder thickness is performed. Asymptotic solutions are constructed that allow calculating the three-dimensional stress–strain state of a radially inhomogeneous isotropic cylinder of small thickness. The propagation of torsional elastic waves in a radially layered cylinder consisting of alternating hard and soft layers is investigated. A theorem on the stratification of the thickness resonance frequency is obtained. In the vicinity of the origin, in the vicinity of the thickness resonance frequency, for sufficiently large values of the wavenumber and frequency, when their ratio is finite, asymptotic curves for dispersion curves are determined. Using a combination of asymptotic and numerical analysis, dispersion curves for a three-layer cylinder are constructed.