Long moving vertical ropes and cables are used in various engineering systems. In particular, in high-rise structures traction drive elevators they are employed as a means of car and counterweight suspension and for compensation of tensile forces over the traction sheave. Also, ropes are applied in the elevator governor systems to activate the safety gear in order to bring the car and counterweight safely to rest in the event of emergency and failure of the normal stopping system. An adverse situation arises when the host structure is excited near its fundamental natural frequency and vibrates harmonically. This often results in a passage through resonance conditions in the rope system when the slowly varying natural frequencies of the ropes approach the frequency of the inertial load resulting from the building sway. The nature of such a loading is usually nondeterministic and it is necessary to apply stochastic models to analyze the dynamic responses of the ropes. In this paper a model to describe the lateral dynamic behaviour of a vertical moving rope is developed. The model takes into account the fact that the longitudinal elastic stretching of the ropes is coupled with their transverse motions which results in cubic nonlinear terms. The governing nonstationary nonlinear equations are then solved numerically to investigate the passage through resonance conditions arising during the elevator travel. Then, the methodology to account for the stochastic nature of the building sway is discussed and the differential equations governing the second-order statistical moments of the state vector are developed. The stochastic differential equations are treated numerically to predict the variance of the rope response
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