On one solution to the problem of resonant oscillations of a lifting rope taking into account its slippage when winding on the surface of a drum

Abstract

In this paper, we develop a mathematical model and methods for analyzing a non-classical problem of longitudinal oscillations of a rope, the upper end of which is wound on a drum, and a load is fixed to its second end. Taking into account the slippage of rope elements along the drum rim allows one to adequately describe the real dynamic processes in the system. Representing the problem of rope oscillations with moving boundaries as a differential equation with non-integrable boundary conditions provides a non-classical generalization of the hyperbolic problem. The linearization of boundary conditions by the method of averaged estimates simplifies the initial statement of the problem. The construction of equivalent integro-differential equations with symmetric and time-dependent kernels, as well as with time-varying integration limits, creates the basis for applying the apparatus of integral equations. Consideration of cases both with and without slippage provides a comprehensive analysis of the dynamic characteristics of the system. Reducing the integro-differential equation to a dimensionless form by introducing new variables unifies the mathematical model. The solution of the obtained integro-differential equation without taking into account slippage using a combination of the approximate method for constructing solutions of integro-differential equations and the Kantorovich-Galerkin method demonstrates the acceptable level of accuracy for calculations. The expression for the amplitude of oscillations corresponding to the n-th dynamic mode, obtained by the asymptotic method, is an analytical basis for analyzing the resonance properties. The study of the phenomenon of steady-state resonance and passage through resonance of the system by numerical methods and the developed software package has practical value for the dynamic calculation of lifting devices.