Construction of exact and asymptotic solutions for the problem of torsional vibrations of a radially inhomogeneous sphere

Abstract

In this paper, the torsional vibrations of a radially inhomogeneous isotropic open sphere containing no 0 and π poles are studied. It is considered that the elastic moduli and density of the material are linear functions of the radius of the sphere. Cases are considered when the lateral surface of the sphere is stress-free and fixed, and arbitrary boundary conditions are specified on conic sections. The formulated boundary value problems are reduced to spectral problems. After fulfilling boundary conditions specified on the lateral surfaces of the sphere, a dispersion equation is obtained. Exact solutions to the problem are constructed. Next, we analyzed the roots of the dispersion equation for a relatively small parameter characterizing the thickness of the sphere, depending on the oscillation frequency. For a sphere of small thickness, asymptotic zeros are constructed, and their characteristics are studied.