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

The classical moving co-ordinate frame approach and Hamiltonʹs 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.