# The Problem of Stability in Mechanical Systems Using the Example of Mine Hoist Installations
## 2 Dynamic Analysis of the Hoist Installation  
Based on the dynamic analysis in [1], we attempted to define the influence of variable shaft steelwork stiffness on horizontal vibration displacements of a hoist installation in the shaft and this device's work stability. The dynamic equations were created based on the scheme in Figure 1, where a 2D model of the mining skip is presented. The hoist installation model, meticulously constructed as a three-mass model, is a testament to the depth of our research. The superiority of a three-mass model over a one-mass model is evident, as it allows for the investigation of myriad phenomena. The entire installation, a marvel of engineering, is a highly complex machine that necessitates a comprehensive analysis. The bottom frame and the head of the skip, where ropes are attached, are crucial elements. The vessel, the most critical element, is a complex structure represented by one mass with inertia moment and mass. The elements, like connection rods, are represented in the model as the spring and damper system, which are represented by parameters such as elasticity and damping coefficients. These three masses are connected by rods attached to the bottom frame, the head, and the vessel. The vessel by the bottom frame and the head are connected with the guides by the roller guide. The roller guides, which roll on the guide shaft, are not connected with the guide shaft at every point, allowing us to neglect the friction. In this model, the authors assumed that at least one roller guide rolls on the guide shaft at the bottom or top of the hoist vessel. The spring and damper system represents the connection between roller guides and guide shafts with parameters like \(k\) and \(h\). This is determined by the motion equation based on the assumption that random irregularities and misalignments on the shaft guide cause horizontal displacements of the system mass. This hoist installation model is also used in [21; 22]. The railroad vehicle also used a similar model for the mechanical scheme as in [23]. Horizontal vibration displacement affects the entire device, as well as elements like the drive, cables, and shaft steelwork. They are dangerous because of parametric vibrations, which are checked in this work. Figure 1: Mechanical model of the hoist installation [24]. The mechanical model in Figure 1 is a three-mass model, where the masses are the head of the hoist, the skip hopper, and the bottom frame. The main masses are connected to the shaft guide by the roller guides at the top and bottom of the hoist installation. The main assumptions for this model are: the elasticity of the roller guides and the guide shaft is linear, the irregularities and misalignments for the front planes of the guide shafts are parallel to each other, and the masses are connected by the weightless, laterally elastic, and longitudinally non-deformable cables. For this model, it is also assumed that the roller guides do not lose connection with the guide shafts. These assumptions form the basis of our model and help us understand the dynamics of the hoist installation. The irregularities are termed \(x_{1}\) and \(x_{2}\).