DYNAMICAL STABILITY OF A ROPE WITH SLOW VARIABILITY OF THE PARAMETERS

Abstract

The longitudinal-transversal vibrations of a rope with varying length are considered here. The analysis of parametric resonances is the primary purpose of this paper. The dynamic state of the investigated system is described by a non-linear set of partial differential equations with boundary conditions varying with time. The physical non-linearity and damping properties of the rope material as well as dry friction between flakes are taken into account. The determination of the unstable regions by balance harmonic method for the main, secondary and combination resonances has been performed. The spatial diagrams of the regions of instability and their cross-sections are presented. The influence of the physical non-linearity and the character of the kinematic excitation are considered. The starting and braking of the winding machine is taken into consideration.