A Timoshenko-like model for the study of three-dimensional vibrations of an elastic ring of general cross-section

Abstract

The present paper proposes analytical formulations of the eigenvalues and eigenfunctions (frequencies and modes) of vibrating rings of any cross-section shape, so as to be applied to engineering problems, like design optimization or life duration improvement. This is done by means of the Timoshenko beam theory accounting for the curved metric through new constitutive laws. An original non-dimensionalization reduces the number of independent parameters to only four. The motion is governed by a system of six differential equations applied to six independent variables. However, the reference curvature is found to induce fundamental changes to the mode structure. In that sense, a classical series decomposition is not efficient to provide analytical expressions of the mode shapes. The Chapman–Enskog method is proposed and explained to overcome this difficulty. Then, all modes (flexural, shear, torsional and longitudinal) are formulated explicitly, showing the coupling order of each of the six degrees of freedom of the cross-section. These results are compared to computations using a 3D elastodynamic model in order to validate the model and to point out its limitations. The latter is finally discussed with respect to other models proposed in the literature.