Non-stationary responses on hoisting cables with slowly varying length.
Cables in hoisting installations, due to their flexibility, are susceptible to vibrations. A common arrangement in industrial hoisting systems comprises a driving winder drum, a steel wire cable, a sheave mounted in headgear, a vertical shaft and a conveyance. This system can be treated as an assemblage of two connected interactive, continuous substructures, namely of the catenary and of the vertical rope, with the sheave acting as a coupling member, and with the winder drum regarded as an ideal energy source. The length of the vertical rope is varying during the wind so that the mean catenary tension is continuously varying. Therefore, the natural frequencies of both subsystems are time-dependent and the entire structure represents a non-stationary dynamic system. The main dynamic response, namely lateral vibrations of the catenary and longitudinal vibrations of the vertical rope, are caused by various sources of excitation present in the system. The most significant sources are loads due to the winding cycle acceleration/deceleration profile and a mechanism applied on the winder drum surface in order to achieve a uniform coiling pattern. The classical moving frame approach is used to derive a mathematical model describing the non-stationary response of the system. First the longitudinal response and passage through primary resonance is examined. The response is analyzed using a combined perturbation and numerical technique. The method of multiple scales is used to formulate a uniformly valid perturbation expansion for the response near the resonance, and a system of first order ordinary differential equations for the slowly varying amplitude and phase of the response results. This system is integrated numerically on a slow time scale. A model example is discussed, and the behaviour of the essential dynamic properties of the system during the transition through resonance is examined. Interactions between various types of vibration within the system exist. The sheave inertial coupling between the catenary and the vertical rope subsystems facilitates extensive interactions between the catenary and the vertical rope motions. The nature of these interactions is strongly non-linear. The lateral vibration of the catenary induces the longitudinal oscillations in the vertical system and vice-versa. In order to analyze dynamic phenomena arising due these interactions the nonlinear partial-differential equations of motion are discretised by writing the deflections in terms of the linear, free-vibration modes of the system, which result in a non-linear set of coupled, second order ordinary differential equations with slowly varying coefficients. Using this formulation, the dynamic response of an existing hoisting installation, where problematic dynamic behaviour was observed, is simulated numerically. The simulation predicts strong modal interactions during passage through external, parametric and internal resonances, confirming the autoparametric and non-stationary nature of the system recorded during its operation. The results of this research demonstrate the non-stationary and non-linear behaviour of hoisting cables with slowly varying length. It is shown that during passage through resonance a large response may lead to high oscillations in the cables' tensions, which in turn contribute directly to fatigue damage effects. The results obtained show also that the non-linear coupling in the system promotes significant modal interactions during the passage through the instability regions. The analysis techniques presented in the study form a useful tool that can be employed in determining the design parameters of hoisting systems, as well as in developing a careful winding strategy, to ensure that the regions of excessive dynamic response are avoided during the normal operating regimes.