In the proposed article, a mathematical model of unsteady torsional vibrations of a conical elastic shell is developed taking into account the nonlinear relationship between stresses and deformations. Refined, physically nonlinear equations of torsional vibrations of a conical shell from a homogeneous and isotropic material are derived, from which, in a particular case, some well-known classical-type vibration equations can be obtained. An algorithm is proposed that makes it possible to

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... determine the stress-strain state of points of an arbitrary cross-section of the shell in question by spatial coordinate and time from the field of the desired functions. Some limiting and special cases, the following and the obtained results are analyzed. In particular, as the limiting case of the shell vibration equations, the equations of torsional vibrations of a circular cylindrical shell are derived, and from it, as a special case, the physically nonlinear torsional vibration equations of a round rod, from which, when limited in it by the first few terms, follows the well-known equation of G.Kauderer, Prof. I.G.Filippov and others.