Modeling of nonlinear torsional vibrations of a truncated conical rod

In the present study, a nonlinear mathematical model of non-stationary torsional vibrations of a truncated conical rod made of elastic material taking into account the nonlinear relationship between stresses and strains has been developed. A nonlinear equation for torsional vibrations of the truncated conical rod has been derived with respect to the main part of the torsional displacement of the axis of symmetry of the rod. It has been demonstrated that the obtained equation for nonlinear torsional vibrations of the truncated conical elastic rod coincides with known equations obtained by other authors in particular cases. Using the derived equation, the stress-strain state of an arbitrary cross-section of the conical rod can be uniquely determined based on spatial coordinates and time. The problem of non-stationary torsional vibrations of the truncated conical rod under the action of axial and surface dynamic loads has been numerically due to the constructed model, when the wide end of the rod is rigidly fixed and the narrow end is free.