Nonlinear torsional vibrations of a circular cylindrical elastic shell

Abstract

In this article, we develop a mathematical model of non-stationary torsional vibrations of a circular cylindrical elastic shell, taking into account nonlinear co-relation between stresses and deformations. We derive corrected, physically nonlinear equations of torsional vibrations of such a shell of homogeneous and isotropic material, from which, in a particular case, we can obtain some well-known equations of vibrations of classical type. An algorithm, which allows one-to-one determination of the stress-strain state of points of arbitrary cross-section of the considered system by spatial coordinate and time, is proposed. Some limiting and special cases following and obtained results are analyzed. In particular, as a limiting case of the equations of shell vibrations, the equations of torsional vibrations of a circular rod were derived from which, being limited by a few first terms follows the well-known equation of G. Kauderer.