On the Mathematical Model of Nonlinear Vibrations of a Biologically Active Rod with Consideration of the Rheological Factor

Qualitative and numerical methods of researching nonlinear vibration systems are used to study the mathematical model of nonlinear vibrations of a biologically active rod. This model is widely used in biomechanics and medical research for designing new materials with biofactor elements that possess certain preset features. Conditions are established for the existence of a unique solution of the boundary value problem for the beam vibration nonlinear differential equation, in which there is an integral summand with the fourth derivative by the spatial variables. This summand models the rheological factor in the system. The existence of classes of nonlinear rheological vibration systems with dissipation that have blow-up regimes is stated theoretically. The relation between nonlinearity indices in such regimes is obtained. The theoretical possibility of using the Runge-Kutta method for numerical solution of the corresponding boundary value problem is shown. The results are illustrated by a model example. The importance of the obtained theoretical assumptions for the practical modeling, analysis, and synthesis of parameters of technological vibration systems is shown.