## 3 Design and Calculation Results of Vertical Shaft Multi-Rope Friction  
### 3.1. Operational and Structural Parameters of Typical Hoisting System  
Three different types of hoisting wire ropes (round strand, triangular strand, and compacted strand) have been selected for design calculations. Their specific performance parameters are detailed in Table 3. The hoisting system employs a trapezoidal five-stage acceleration curve, as shown in Figures 7-9. Taking Figure 7a,b as examples, the hoisting system first undergoes a trapezoidal acceleration process, and the calculation formulas for its acceleration slope can be found in Equations (47)-(51). Subsequently, the system maintains the maximum speed for uniform operation until it reaches the predetermined deceleration position. Next, the system undergoes a trapezoidal deceleration process, and the speed gradually slows down to a crawling speed (\(v=0.5\) m/s). After maintainingfor 3 s in the crawling stage, the system enters the parking stage and finally smoothly reduces the speed to zero. The design of this five-segment acceleration curve aims to enhance the stability and efficiency of the overall operation. The maximum hoisting acceleration/deceleration of three different types of hoisting wire ropes are 0.74 m/s\({}^{2}\), 0.61 m/s\({}^{2}\), and 0.75 m/s\({}^{2}\) respectively. The maximum hoisting speed is 18 m/s and the hoisting distance is 1000 m. The running time of a single hoisting/lowering of three different types of hoisting wire ropes is 85.9 s, 90.7 s, and 85.7 s, respectively. The diameter of the sheave wheel and the spacing of the friction wheel groove are 7000 mm and 425 mm, respectively. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**Hoisting Rope**} \\ \cline{2-5}  & **Round Strand** & **Triangular Strand** & **Compacted Strand** \\ \hline \multirow{4}{*}{Operational Parameters} & Maximum operating speed, \(v_{max}\)/(m/s) & & 18 & \\ \cline{2-5}  & Hoisting acceleration/deceleration, & & & \\  & \(a\)/(m/s\({}^{2}\)) & 0.74 & 0.61 & 0.75 \\ \cline{2-5}  & Operating time, \(T\)/(s) & 85.9 & 90.7 & 85.7 \\ \hline \multirow{4}{*}{Structural parameters} & Upper rope length, \(L_{s}\)/(m) & & 62.2 & \\ \cline{2-5}  & Lower rope length, \(L_{x}\)/(m) & & 49.49 & \\ \cline{2-5}  & Upper rope angle, \(\beta_{s}\)/(\({}^{\circ}\)) & & 53.04 & \\ \cline{2-5}  & Lower rope angle, \(\beta_{x}\)/(\({}^{\circ}\)) & & 54.39 & \\ \cline{2-5}  & Wrapping angle of hoisting rope on & & 181.35 & \\ \cline{2-5}  & friction lining, \(\alpha\)/(rad) & & & \\ \cline{2-5}  & The calculation & & & \\  & result of & \(\geq\)5580 & \(\geq\)5220 & \(\geq\)5400 \\ \cline{2-5}  & Sheave wheel & Equation (6)/(mm) & & & \\ \cline{2-5}  & diameter ratio & The calculation & & \\ \cline{2-5}  & result of & \(\geq\)3936 & \(\geq\)3804 & \(\geq\)4400 \\ \cline{2-5}  & Equation (7)/(mm) & & & \\ \cline{2-5}  & Friction wheel-groove spacing/(mm) & & 425 & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Operational and structural parameters of typical hoisting system. Figure 7: Acceleration and velocity control curve of round strand wire rope: (**a**) acceleration control curve during the heavy-load lifting phase, (**b**) velocity control curve during the heavy-load lifting phase. The propagation velocity \(j\) of the elastic wave can be calculated by Equation (47):  
\[j=\sqrt{\frac{EF}{P_{k}}} \tag{47}\]  
where \(E\) is the elastic modulus of the hoisting rope, \(F\) is the cross-sectional area of the hoisting rope, and \(P_{k}\) is the unit weight of the hoisting rope. Calculate the ratio of the weight of the first rope to the load at the end of the rope, and construct the transcendental equation, as shown in Equation (48):  
\[ \begin{array}{l}\alpha=\frac{N_{1}P_{k}H_{c}}{Q_{d}}\\ \alpha=\lambda_{1}\tan\lambda_{1}\end{array}