DYNAMIC STABILITY OF A ROPE WITH SLOW VARIABILITY OF THE PARAMETERS

The longitudinal-transverse vibration of a rope of varying length is considered. Special emphasis is placed on the analysis of the parametric resonances of the rope. The dynamic state of the investigated system is described by a set of non-linear coupled partial differential equations with boundary conditions varying in time. The damping properties of the non-linear rope material as well as dry friction between particular strands are taken into account. The unstable regions for the main, secondary and combination resonances have been found by applying the harmonic balance method. General results are illustrated by a numerical example in which the effect of starting and braking of the winding machine is included. The influence of the material non-linearity and the character of kinematic excitation are also considered.