Flexural Vibrations of a Ring with Arbitrary Cross Section

Abstract

The paper deals with the flexural vibrations of a ring with arbitrary cross section and proposes an approximate theory to predict the natural frequencies by means of the Ritz's method. The frequency determinant is derived from the Hamiltonian principle by use of the stress-strain-displacement relations of three-dimensional theory of elasticity, and by expanding three components of the displacement in a finite double power series of the radial and axial coordinates with unknown coefficients. Compared with the experimental results for twenty three kinds of rings which are classified into six sets with respect to the cross section, the theory was ascertained to be available for the practical use. Furthermore, as an application to a general axi-symmetric elastic solid with free boundary surface, the frequencies of truncated conical shells and combined shells of hemispherical and circular cylindrical shells were calculated and compared with the already-known results, showing a good agreement between them.