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The paper numerically investigates the problem of nonlinear torsional vibrations under the action of kinematic and dynamic loads acting on the end and surface of a round elastic rod based on the equation of its torsional vibrations, taking into account the physical nonlinearity developed earlier by the authors. The numerical method of finite differences is applied to solve the problem. Approximation of the vibration equation and boundary conditions leads to a system of algebraic equations, the solution of which is not mathematically difficult. On the basis of the obtained numerical data of the problem, graphs of the dependences of torsional displacement and stresses on time and the longitudinal coordinate are constructed. [ABSTRACT FROM AUTHOR]

Conical elements that make unsteady oscillations are found in the frames of fastening engines, in the columns of construction structures, supporting floors on which unbalanced units are installed, when installing suspensions, etc. Such elements are important structural elements of aerospace vehicles and are widely used in various applications, ranging from beam domes to space launch vehicles with large fuel tanks. Conical sections are found on the details of propeller shafts and shafts of powerful drives. The task of calculating a shaft with a variable cross-section for torsional vibrations is carried out approximately, replacing the shaft with a system of elements with a finite number of degrees of freedom, which leads to complex and cumbersome calculations. In the proposed article, a non-linear mathematical model of unsteady longitudinal vibrations of a truncated conical rod made of elastic material is developed, taking into account the nonlinear relationship between stresses and deformations. Refined, physically nonlinear equations of longitudinal vibrations of a truncated conical rod from a homogeneous and isotropic material are derived, from which, in a particular case, some well-known classical-type rod oscillation equations can be obtained. [ABSTRACT FROM AUTHOR]