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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]