## 4 Discussion
### New Theory of Friction Transmission  
Figure 19 shows the contact force and friction force between the 369 th rope element (the initial position is 400 m above the cage) and the friction pulley under different lifting side loads. Among them, \(1.2\times 10^{5}\) kg of lifting side load corresponds to heavy lifting, \(0.95\times 10^{5}\) kg is equal-weight lifting, and \(0.7\times 10^{5}\) kg is light load lifting. It can be found that the initial value of the contact force just entering the pulley depends on the load ratio on both sides. Because there is a constant load at the lowering side, the contact force at different lifting sides changes to the same value after bypassing the wrap angle. However, under any load, the wire rope will produce positive effect friction as soon as it enters the wrap angle, and will provide reverse friction when it is about to be wound out. The greater the lifting side load is, the smaller the reverse friction.  
Integrating the friction force of the three rope elements in Figure 16 with time, as shown in Figure 20, the friction impulse of each rope element appears to have a downward inflection point when wrapping the friction pulley. That means that the friction drive of the pulley to the rope has changed from positive to the reverse. Until this point, friction has been beneficial to friction transmission; after this point, it has adverse effects. Combined with Figure 16, the friction force in the wrap angle can be divided into two regions by the judgement of the demarcation point (JDP). This is essentially different from the traditional static and dynamic angle theory--the static angle does not provide friction, and the dynamic angle provides a positive friction effect [19; 21; 24; 25]. The theory holds that the tension does not change when the rope of the lifting side just enters the wrap angle (static angle), i.e., the rope does not provide any friction due to deformation and slippage. When the rope reaches the dynamic angle, the friction is caused by the creep between the rope and pulley [26; 27].  
Figure 19: Contact and friction force between the 369 th rope element (400 m) and the friction pulley. (**a**) Contact; (**b**) friction.  
Figure 18: A sketch of load variation along rope length.  
Based on the above dynamic behavior, a new transmission theory of \"global dynamic wrap angle\" for wire rope is proposed (Figure 21). There is no static angle, and the entire wrap angle is dynamic. The rope entering the wrap angle is immediately subjected to the positive friction provided by the pulley, which is beneficial for the transmission. After that, when the rope reaches a certain position, it starts to provide reverse friction until it is wound out of the wrap angle. The division of positive and reverse friction area depends on the load of wire ropes on both sides. When m1<m2, the reverse effect area increases, and when m1>m2, the positive friction effect area occupies absolute predominance. The traditional static and dynamic angle theories may be more suitable for design calculation in shallow mine hoists. The new transmission theory of \"global dynamic wrap angle\" is more suitable for the dynamic analysis of deep mine friction hoisting with a high speed and a heavy load.