## 3 Design and Calculation Results of Vertical Shaft Multi-Rope Friction
### Static Analysis and Calculation of the Hoisting System  
The static calculation results of the hoisting system with different rope selection schemes are shown in Table 5. The maximum static tension at the tangent point of the friction wheel for the three rope selection schemes is consistently observed at the heavy-load end when the right side is the heavy-load end, with values of 2756 kN, 2576 kN, and 2797 kN and the ratio of static tensions is 1.56, 1.62, and 1.55 respectively. The static anti-slip safety coefficient and the pressure ratio between the hoist rope and the friction lining all meet the requirements of the Safety Regulations for Coal Mines. Also, there is little difference in the calculation results when lifting heavy loads on the left and right sides, primarily due to the small vertical distance between the sheave wheels on both sides, resulting in similar calculation results of initial overhang height on both sides of the hoist rope and tail rope.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Design Parameter** & \multicolumn{3}{c}{**Selection Scheme**} \\ \hline Calculated minimum breaking force, \(F_{0}/\)(kN) & 2422 & 2144 & 2787 \\ \hline Safety coefficient, \(m\) (Calculated value) & 6.9 & 5.8 & 7.6 \\ \cline{2-4} Load ratio = Effective lifting load/ & 1.54 & 1.88 & 1.48 \\ \cline{2-4}  & Rope weight & 4 & 3 \\ \hline Selection design of balance tail rope & \multicolumn{2}{c}{Unit weight, \(q_{k}/\)(kg/m)} & 24.9 & 20.4 & 34.2 \\ \cline{2-4}  & Elastic modulus, \(E_{w}/\)(GPa) & 120 & 120 & 120 \\ \hline \hline \end{tabular} Note: For the selection of the \(6V\times 37\) + FC type hoisting rope, since this model does not have a large diameter of 67.2 mm, the largest diameter available in the selection table, 58 mm, is chosen as the calculation case.  
\end{table}
Table 4: _Cont._  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Heavy-Load End** & **Calculation Parameter** & **Round Strand** & \
\begin{tabular}{c} **Triangular** \\ **Strand** \\ \end{tabular} &
\begin{tabular}{c} **Compacted** \\ **Strand** \\ \end{tabular} \\ \hline \multirow{6}{*}{Left cage as heavy-load end} & Static tension at friction wheel cut point at heavy-load end, \(F_{j1}/\)(kN) & 2752 & 2573 & 2793 \\ \cline{2-5}  & Static tension at friction wheel cut point at light-load end, \(F_{j2}/\)(kN) & 1776 & 1596 & 1805 \\ \cline{2-5}  & Ratio of static tension, \(F_{j1}/F_{j2}\) & 1.55 & 1.61 & 1.54 \\ \cline{2-5}  & Static anti-slip safety coefficient of the rope-lining, \(\delta\) & 2.19 & 1.97 & 2.20 \\ \cline{2-5}  & Pressure ratio of the rope-lining, \(P/\)(MPa) & 1.74 & 1.71 & 1.82 \\ \hline \multirow{6}{*}{Right cage as heavy-load end} & Static tension at friction wheel cut point at light-load end, \(F_{j4}/\)(kN) & 2756 & 2576 & 2797 \\ \cline{2-5}  & Static tension at friction wheel cut point at light-load end, \(F_{j4}/\)(kN) & 1771 & 1593 & 1802 \\ \cline{1-1} \cline{2-5}  & Ratio of static tension, \(F_{j1}/F_{j2}\) & 1.56 & 1.62 & 1.55 \\ \cline{1-1} \cline{2-5}  & Static anti-slip safety coefficient of the rope-lining, \(\delta\) & 2.17 & 1.96 & 2.18 \\ \cline{1-1} \cline{2-5}  & Pressure ratio of the rope-lining, \(P/\)(MPa) & 1.74 & 1.71 & 1.82 \\ \hline \hline \end{tabular} Note: According to the Safety Regulations for Coal Mines, the static anti-slip safety coefficient of wire ropes in multi-rope friction hoisting systems must not be less than 1.75. Additionally, the ratio pressure between the wire rope and the friction lining should not exceed 2 MPa.  
\end{table}
Table 5: Static calculations of different rope selection schemes in hoisting system.