## 2 Design Theory of Multi-Rope Friction Hoisting System with Double Cages in Vertical Shaft
### Calculation Theory and Selection of the Structural Parameters of the Hoisting System
#### 2.1.2 Calculation of Parameters of Friction Wheel  
The calculation formula for the wrapping angle of the hoisting wire rope on the friction lining \(\alpha\) is shown in Equation (5):  
\[\alpha=180^{\circ}+\beta_{x}-\beta_{s} \tag{5}\]  
where \(\beta_{s}\) is the elevation angle of the upper rope and \(\beta_{x}\) is the elevation angle of the lower rope.  
According to Article 417 of China's Safety Regulations for Coal Mines [18], the ratio of the minimum diameter of the sheave wheel, drum, friction wheel, and guide wheel to the diameter of the steel wire rope should meet the requirements in Table 1.  
In addition, according to Safety Inspections--Testing Specifications of In-Service Friction Hoist System for Coal Mines [19], the ratio of the diameter of the sheave wheel, friction wheel, and guide wheel to the diameter of the thickest steel wire in the steel wire rope should not be less than 1200.  
A floor-standing multi-rope friction hoisting system was designed and the method of installation on the well was adopted. The ratio of the diameter of the friction wheel, sheave wheel, and hoisting rope in this system should all meet Equations (6) and (7):  
\[\frac{D_{1}}{d}\geq 90 \tag{6}\]  
\[\frac{D_{1}}{\delta}\geq 1200 \tag{7}\]  
where \(D_{1}\) is the diameter of the friction wheel and sheave wheel, \(d\) is the nominal diameter of the hoisting rope, and \(\delta\) is the diameter of the thickest steel wire in the hoisting rope.  
According to the requirements of Formulas (6) and (7) and combined with the basic parameters of the floor-standing multi-rope friction elevator in GB/T10599-2023 (Multi-rope friction hoist) [20], the appropriate diameter of friction wheel and the rope groove spacing of the friction wheel is selected.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Use Classification**} & \multicolumn{2}{c}{**Minimum Ratio**} \\ \hline Friction wheel and the sheave wheel of the floor-standing friction hoisting system, and the friction wheel of the tower friction hoisting system with & Aboveground & 90 \\ \cline{2-3} a wrapping angle greater than 180\({}^{\circ}\) & Underground & 80 \\ \hline Friction wheel of the tower friction hoisting system with & Aboveground & 80 \\ \cline{2-3} a wrapping angle of 180\({}^{\circ}\) & Underground & 70 \\ \hline Guide wheel of friction hoisting system & 80 \\ \hline Drum and the sheave wheel with a wrapping angle greater than 90\({}^{\circ}\) & 80 \\ \cline{2-3} of the ground winding hoisting system & 60 \\ \hline Drum of the underground winding hoist and drilling hoist, the leading wheel and tail guide wheel and & 60 \\ \cline{2-3} the sheave wheel with a wrapping angle greater than 90\({}^{\circ}\) of the underground overhead passenger device & 60 \\ \hline Sheave wheel with a wrapping angle smaller than 90\({}^{\circ}\) of the underground winding hoist, the drilling & 40 \\ \cline{2-3} hoist and the underground overhead passenger device & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Requirements for the ratio of the minimum diameter of the sheave wheel, drum, friction wheel, and guide wheel to the diameter of the steel wire rope.