# The Problem of Stability in Mechanical Systems Using the Example of Mine Hoist Installations
## 5 Conclusions  
The influence of the variable steel shaft elasticity on the stability of horizontal mass displacements in the skip and steel shaft system has been thoroughly analyzed and evaluated in this research. Based on the typical parameters for hoist installations currently working and designed in Poland, this comprehensive study builds on previous studies and the newest findings like the Equation (24). The authors of this work have presented an in-depth analysis of parametric vibrations for a hoist installation, with a more complicated model than in other research. After rigorous testing of parametric vibrations for the hoisting ropes, the stability analysis results demonstrate that instability regions for the investigated displacements are highly unlikely to occur, instilling confidence in the thoroughness of this analysis.  
Analogous points for the values of \(n_{1}\) and \(n_{4}\) are found in stable regions in the bottom right section in Figure 3. Conditions are improving because the \(n_{2}\) and \(n_{3}\) parameters cause unstable regions to narrow. The narrow unstable regions are better for the whole construction. The changing value of the \(n_{3}\) parameter, a key factor in hoist installation, directly impacts the instability regions. This means that as the \(n_{3}\) parameter varies, the regions where the hoist installation is prone to instability also change. Therefore, understanding and measuring the elasticity factor of the shaft guide and the flexible connector between the head and the hopper is crucial to predict and potentially prevent these changes in instability regions.  
Figure 6: The impact of the \(n_{3}\) parameter on the shape of stable operation ranges (\(\varepsilon=0.1\), \(n_{2}=0\)).  
Considering the simplifications of the assumptions, the influence of the variable shaft steelwork elasticity on the stability of vibrations in the skip and shaft steelwork system, which can potentially bring about the effect of parametric resonance, can be excluded from the analysis of the dynamics of hoist installations. This shows that the assumption is fully justified and valuable for practical use, enhancing the relevance of this research. The theoretical analysis presented in this work is also beneficial for researchers working on this topic. The primary aim of this study was to explore the potential for parametric vibrations in hoist installations. The authors conducted a comprehensive analysis, leading to the development of an equation. This equation enables a quantitative assessment of how the parameters of a hoist installation affect its performance. It provides a systematic approach to understanding and mitigating parametric vibrations, with significant implications for the safety and efficiency of hoist operations. The vibration of the hoist installation, presented in the introduction, is the topic that is currently being researched. A practical use will also be designing and modeling a new hoist installation that will be safer than before.  
Conceptualizations, S.W.; methodology, S.W. and A.D.; validation and analysis, A.D.; writing-original draft preparation, S.W.; writing and editing, A.D.; visualization, A.D.; supervision, S.W. All authors have read and agreed to the published version of the manuscript.  
This research was funded by subvention 16.16.130.942/B 403 KPiEM.  
Data Availability Statement: Data is contained within the article.  
The authors declare no conflict of interest.