Transient vibration phenomena in deep mine hoisting cables. Part 1: Mathematical model

The classical moving co-ordinate frame approach and Hamilton's principle are employed to derive a distributed-parameter mathematical model to investigate the dynamic behaviour of deep mine hoisting cables. This model describes the coupled lateral-longitudinal dynamic response of the cables in terms of non-linear partial differential equations that accommodate the non-stationary nature of the system. Subsequently, the Rayleigh-Ritz procedure is applied to formulate a discrete mathematical model. Consequently, a system of non-linear non-stationary coupled second order ordinary differential equations arises to govern the temporal behaviour of the cable system. This discrete model with quadratic and cubic non-linear terms describes the modal interactions between lateral oscillations of the catenary cable and longitudinal oscillations of the vertical rope. It is shown that the response of the catenary-vertical rope system may feature a number of resonance phenomena, including external, parametric and autoparametric resonances. The parameters of a typical deep mine winder are used to identify the depth locations of the resonance regions during the ascending cycles with various winding velocities.