## 2 Design Theory of Multi-Rope Friction Hoisting System with Double Cages in Vertical Shaft
### Static Calculation Theory of Hoisting System  
#### 2.3.1 Static Tension Calculation of Hoisting Wire Rope  
The hoisting system of the shaft gravity energy storage system is a multi-rope friction hoisting system with double cages (floor-standing). The structure diagram of the hoisting system in the static state is shown in Figure 3. Since the system adopts a double-cage hoisting scheme, and the initial overhang heights of the hoisting wire ropes and tail ropes on both sides of the wheel are different, it is necessary to consider the force conditions of the hoisting system when the heavy load is loaded in different cages. Additionally, due to the use of an equal-weight balanced tail rope scheme (\(n_{1}p_{k}=n_{2}q_{k}\)), the static tension at the tangent point of the friction wheel remains approximately constant throughout the hoisting process (assuming the overhang height of hoisting wire rope at the heavy-load end is \(l_{c}\) at a certain moment, the overhang height of tail rope is \(h_{av}\), and the hoisting height is \(\Delta l\), then \(n_{1}p_{k}(h_{c}-\Delta l)+n_{2}q_{k}(h_{av}+\Delta l)=n_{1}p_{k}h_{c}+n_{2}q_ {k}h_{av}\), with the same principle applying to the light-load end). Therefore, when calculating the static tension of hoisting wire rope, it is only necessary to calculate the static tension at the tangent point of the friction wheel of the hoisting system in its initial position. When the left-side cage is the heavy-load end, the static tension at the tangent point of the friction wheel at the heavy-load end \(F_{jw}\) and the static tension at the tangent point of the friction wheel at the light-load end \(F_{jl}\) can be calculated through Equations (23) and (24). \[F_{jw}=[n_{1}q_{k}(h_{1}-L_{s}\sin\beta_{s})+n_{2}p_{k}h_{3}+1000\cdot(Q+Q_{2} )]g \tag{23}\]  
\[F_{jl}=[n_{1}q_{k}(h_{2}-L_{x}\sin\beta_{x})+n_{2}p_{k}h_{4}+1000\cdot Q_{2}]g \tag{24}\]When the right-side cage is the heavy-load end, the static tension at the tangent point of the friction wheel at the heavy-load end \(F_{jw}\) and the static tension at the tangent point of the friction wheel at the light-load end \(F_{jl}\) can be calculated through Equations (25) and (26):  
\[F_{jw}=[n_{1}q_{k}(h_{1}-L_{x}\sin\beta_{x})+n_{2}p_{k}h_{3}+1000\cdot(Q+Q_{z}) ]g \tag{25}\]  
\[F_{jl}=[n_{1}q_{k}(h_{2}-L_{s}\sin\beta_{s})+n_{2}p_{k}h_{4}+1000\cdot Q_{z}]g \tag{26}\]  
where \(n_{1}\) and \(n_{2}\) are the number of hoisting ropes and tail ropes, \(p_{k}\) is the unit weight of the hoisting rope, \(q_{k}\) is the unit weight of the tail rope, \(h_{1}\) is the initial overhang height of the hoisting rope at the heavy-load end, \(h_{2}\) is the initial overhang height of the tail rope at the heavy-load end, \(h_{4}\) is the initial overhang height of the tail rope at the light-load end, \(L_{s}\) and \(L_{x}\) are the chord lengths of the upper rope and lower rope, \(\beta_{s}\) and \(\beta_{x}\) are the elevation angles of the upper rope and lower rope, \(Q\) is the lifting load, \(Q_{z}\) is the weight of the hoisting container, and \(g\) is the acceleration due to gravity. The values \(h_{1},h_{2},h_{3},h_{4}\) are calculated by Equations (8)-(11) or (12)-(15) according to the position of the heavy-load. 3.2 Calculation of the Static Anti-Slip Safety Coefficient between Hoisting Rope and Friction Lining  
The multi-rope friction hoisting system utilizes the friction between the wire rope and the friction hinings on the friction wheel to transmit motion. According to [PERSON]'s formula for flexible body friction transmission, the ratio of the wire rope tensions on both sides of the friction wheel should satisfy Equation (27) to prevent slipping:  
\[\frac{F_{jl}}{F_{jw}}\leq e^{\mu\alpha} \tag{27}\]  
where \(F_{jw}\) is the static tension at the tangent point of the friction wheel at the heavy-load end, \(F_{jl}\) is the static tension at the tangent point of the friction wheel at the light-load end, \(\mu\)  
Figure 3: Schematic diagram of hoisting system of vertical shaft gravity energy storage system.