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Modelling the dynamic behaviour of elevators with high lifting velocities (contemporary elevators in building construction and mine elevators) is a complex task and an important step in the design process and creating conditions for safe and reliable exploitation of these machines. Due to high heights and lifting velocities, the standard procedures for dynamic exploitation are not adequate. The study presents the method of forming a dynamic model to analyse nonstationary vibrations of a rope with time-varying length with nonholonomic boundary conditions in the position where the rope is connected with the cabin (cage) and in the upcoming point of its winding onto the pulley (drum). A unique method was applied to identify the basic parameters of the dynamic model (stiffness and damping) based on experimental measures for a concrete elevator. Due to the verification of this procedure, the experiment was conducted on a mine elevator in RTB Bor, Serbia. Using the obtained computer-experimental results, the simulations of the dynamic behaviour of an empty and loaded cage were shown. In addition, the study shows the specific method as the basis for forming a control program that would enable the decrease in vertical vibrations during an elevator starting and braking mode. © 2021 Journal of Mechanical Engineering. All rights reserved.