# The Problem of Stability in Mechanical Systems Using the Example of Mine Hoist Installations
## 4 Results  
Figures 3-5 are the results of the solution for equation (24) as its roots. The system of Equation (19) has an unstable solution. When value points for \(n_{1}\) and \(n_{4}\) are found in the instability region (shaded area), the solution is unstable and parametric vibrations occur. The solution will be stable as long as the value points for \(n_{1}\) and \(n_{4}\) can be found in the stable region (unshaded area). The shape of these figures is similar to the theoretical analyses for parametric vibrations. When the damping in the system of the mining skip and steel shaft is considered (it was not included when stable and unstable regions in Figures 3 and 4 were established), the resonance of regions is narrowed down, and its edges should be shifted up toward the abscissa. Furthermore, the reduced value of \(n_{3}\) (Figures 3 and 4) could also cause the narrowing of resonance regions, which is safer for the whole steel construction. Numerical values of the \(n_{1}\), \(n_{3}\), and \(n_{4}\) parameters and their range are chosen as similar to those for typical constructions of hoist installations currently used in Poland [18; 19; 20; 28]. Numerical values of the individual parameters assume the following intervals [18; 19; 20; 29; 32]:  
\[ \begin{array}{l}n_{1}=\frac{n_{4}}{n_{6}}\cong 3\div 4,\\ n_{3}=\frac{k_{2}}{k_{5}}\cong 1\cdot 10^{-2}\div 1\cdot 10^{-3},\\ n_{4}=2\pi\frac{V_{1}}{l_{0}}\frac{1}{p_{10}}\cong 0.1\div 0.2.\end{array}  \tag{25}\]  
These values are similar to the values in other countries and for other researchers [12; 13; 14; 15]. It is evident that (Figures 3-5) the current shape of the resonance curve is not caused by the value of the \(\varepsilon\) parameter (which is defined by \(\varepsilon_{\min}\) to \(\varepsilon_{\max}\)). Analyzing the resonance curve's equation and its placement on the plane of \(n_{1}\) and \(n_{4}\) (Figures 3 and 4) in the context of the hoist installation's skip and shaft steel system, parametric vibrations are not probable. From engineering practice (considering vibration attenuation), resonance is usually impossible. The analysis shows that, in Polish mines, the conditions of work for the hoist installations are safe, and resonance is unnoticeable. After calculating the roots from the authors' derived Equation (24), we plotted charts to visually represent the influence of dimensionless parameters \(n_{4}\), \(n_{3}\), and \(\varepsilon\) on these roots. This method not only provided a clear and intuitive understanding of the relationship between the parameters and the equation's results but also reassured us of the accuracy of our calculations. The precision of our calculations is a testament to the thoroughness of our research. Based on the zero place for this equation obtained for parameter \(n_{4}\) being equal to 2, we can analyze it and check if this situation is probable. This point would mean that the \(n_{1}\) parameter equals 0, which is possible only when the vessel has no mass, which is impossible for a real object. Parameter \(n_{3}\) also influences the shape of the chart, shown in Figure 6. Figures 3-5 show different shapes of the equation's results, where we changed the values of \(n_{3}\) for the minimal, average, and maximal values. The shape is similar, and we can see that as the value of \(n_{3}\) increases, the line on the chart becomes sharper, indicating a more pronounced effect on the results. Figure 5: Stability range for operation of the hoist installation (\(n_{3}=0.005\), \(n_{2}=0\)). Next, Figure 6 clearly illustrates how the \(n_{3}\) parameter impacts the gradient of the chart in the range of \(n_{4}\) equals 2. As the value of the \(n_{3}\) parameter increases, the gradient of the chart becomes steeper, indicating a more rapid change in the results. These results have significant implications for understanding the practical influence of dimensionless parameters on the equation's results, underscoring the relevance of our research in the field and its potential to provide information for real-world applications.