# The Problem of Stability in Mechanical Systems Using the Example of Mine Hoist Installations
## 1 Introduction  
Hoist installations transport output from the bottom to the surface and are used in most mining industries. In every country, hoist installations depend on the mine location and how the bedrock works. The kind of mine and how much output is transported to the surface are also important. The hoist systems differ in every mine, depending on the location or the project task data. Different forces affect the hoist skip while it is moving. These forces come from the steel ropes connected to the top and bottom of the hoist skip and from the interactions between the hoist skip and the shaft steelwork. Other forces are generated due to the irregularities and misalignments on the guide shaft. In a conveyance-shaft steelwork system, it is assumed that vibrations are generated due to irregularities and misalignments of the guide strings [1]. A hoist installation driving at a constant velocity affects the shaft steelwork through horizontal forces. These forces come from the irregularities and misalignments of the guide strings, resulting from wear and tear, the rock mass movement, and the influence of conditions in the shaft. The vibrations of the vessel are generated through the irregularities and misalignments on the guide strings, as researched in [2; 3; 4; 5; 6; 7; 8]. The proper dynamic analysis for this system means that varying shaft steelwork stiffness is neglected because of its minor impact on the amplitude of vibrations and vessel stability. Despite that, research checking the impact of misalignments and irregularities on the vessel's vibration and vibration from the skip parameters is omitted. Stability is essential in different mechanical systems to provide safety for people and the whole industry. The problem of instability in hoist installations could occur in the form of parametric vibrations, which are very dangerous for the transporting system in mines. First of all, attempts were made to determine the impact on the stability of varying shaft steelwork stiffness in a conveyance-shaft steelwork system. Determining the impact allows the occurrence of parametric vibrations to be defined. Parametric vibrations couldbe dangerous for the system because they generate unstable work for different parameters, which can change during the process. Hoist installations are produced as specific units and have different parameters for the masses, building, and specifications for where they are. Parametric vibrations could be generated only by changing parameters in the system, which may be mass or mechanical parameters for the system. In hoist installations, while transporting output, the parameter that could change in a mechanical system is the stiffness coefficient for the steel shaft guide. Parametric vibrations affect the stability of a machine's work. The main goal of this work is to establish the appearance of parametric vibration for the hoist installation and determine if unstable work is possible. The publications about stability in hoist installations concern parametric vibrations in ropes [6; 9; 10; 11], which is not the primary goal of this article. The problem of parametric vibrations in hoist installations is hazardous for the whole system. A theoretical analysis showed that the problem is significant, but has not yet been checked by other authors. Parametric vibrations are checked in ropes connected to the skip's head or the bottom frame. Hoist vibration was analyzed in [12; 13; 14; 15], which is very important because many machines vibrate due to force from outside the machine. Parametric vibrations in hoist installations are more dangerous than forced vibrations because dynamic parameters for the whole system affect them. These parameters depend on physical and mathematical models and their values. The article's main goal was to answer the question of whether parametric vibrations are possible in the hoist installation using theoretical analysis.