Dynamical stability of a rope undergoing slow changes of the parameters

The longitudinal-transversal vibrations of a rope with varying length are considered. The analysis of parametrical resonances is a primary purpose of this paper. The dynamic state of investigated system is described by a nonlinear set of partial differential equations with boundary conditions varying in time. The physical nonlinearity and damping properties of the rope material as well as the dry friction between flakes are taken into account. Determination of the unstable regions by means of the balance harmonic method for the main, secondary and combination resonances respectively is done. Diagrams of the regions of instability are presented. Influence of the physical nonlinearity and the character of the kinematic excitation are considered. The starting and the braking of winding machine is taken into consideration.