Simulation of oscillations and study of resonant properties of a rope of variable length lying on an elastic foundation, taking into account dissipation

Currently, matters of dependability in the design of technical systems with moving boundaries require an increasingly complete consideration of underlying dynamic phenomena. Aim. The aim of the study is to develop a mathematical model and an approximate analytical method for studying the transverse vibrations and resonant properties of a viscoelastic rope of variable length lying on an elastic foundation, taking into account energy dissipation. The relevance of the work is due to the widespread use of technical systems with moving boundaries (lifting mechanisms, flexible transmissions, railway contact networks, rail tracks, belt conveyors, drill strings, etc.), for which dynamic loads and resonance are dangerous. The existing methods do not allow for a complete consideration of a system of factors, i.e., changes in the object’s length, resistance of the medium, elastic properties of the foundation and internal friction. Methods. To solve the problem, the Kantorovich–Galerkin method, effective for systems with moving boundaries, was applied. The original boundary value problem for a partial differential equation was reduced to a system of ordinary differential equations. The solution procedure included the transition to dimensionless variables, the selection of coordinate functions in the form of eigenmodes and the application of the Galerkin procedure. The small parameter method was used to analyze non–stationary processes. In the considered model, the drag force of the rope movement is assumed to be proportional to the velocity, and the bending rigidity of the structure is also taken into account. Results. Calculation expressions are presented for the amplitude of oscillations corresponding to the n–th dynamic mode. Particular attention is paid to the study of the phenomena of steady–state resonance and passage through resonance. The solution covers the most common case in practice of the action of external disturbances on the moving boundary of the system. It is established that the amplitude significantly depends on the velocity of the boundary, dissipation parameters and the rigidity of the foundation. The conditions for steady–state resonance are determined for a certain ratio of the frequency of the external influence and the natural frequency of the system. The phenomenon of passage through resonance is studied. The resulting analytical expressions were verified by comparison with known special cases, confirming the method’s validity with an error of up to 5% for the fundamental modes. Conclusions. The resulting analytical expressions for the oscillation amplitude, steady–state resonance conditions, and resonance passage parameters enable the formulation of a number of practical recommendations for design engineers aimed at increasing the dependability and durability of technical systems with moving boundaries and preventing resonant failures in variable–length systems. Key applied problems solved using this model include fatigue life assessment, residual life prediction, and emergency prevention. Consideration of dissipation and an elastic foundation is critical for assessing resonant properties. To prevent resonance, it is recommended to optimize the boundary velocity, use materials with increased friction or dampers, and increase the foundation rigidity. The results have practical significance for improving the dependability of systems with moving boundaries. Research prospects are related to taking into account nonlinear effects and non–harmonic influences.