The Vibration of Shaft Ropes With Time-Variable Length, Treated by Means of Riemann’s Method

Abstract

Longitudinal vibrations in the ropes of a mining hoist, introduced by application of the emergency brake, are complicated by the change in rope length during the braking action. The usual method of the Laplace transform cannot be applied to the governing one-dimensional wave equation ytt − a2 yxx = 0, since the field of integration deviates from the semi-infinite strip 0 < x < l0 , 0 < t < ∞. In this paper it will be shown how the rigorous solution can be found by using Riemann’s method of the characteristic curves. Instead of the boundary values for the deviation y certain differential relations are given which describe the dynamic conditions at the discrete masses, e.g., the cages and the Koepe pulley.