## 2 Extended Pick Failure Probability Assessment Approach
### Pick Failure Probability during the Shearing-Down Stage  
When the operation of the drum shifts from the sump-in stage to the shearing-down stage, its advance direction changes from horizontal (along the roof) to vertical (towards the floor). In the shearing-down stage, the drum primarily cuts the coal seam, as well as some floor rock (Figure 2). This stage may also involve cutting roof rock if the thickness of the roof rock is greater than the cutting radius of the drum (Figure 2). However, the roof rock thickness is generally less than two thirds of the tip-to-tip diameter of the drum, and the coal is much softer than the rock. Therefore, in the following analysis, the impact of cutting roof rock and the coal seam on the pick tip failure probability in the shearing-down stage is ignored, and only the impact of cutting the floor rock on the pick tip failure probability is considered. The scenario where the roof rock thickness can take any value will be studied in due course. Three issues need to be addressed in the analysis of the pick tip failure probability during the shearing-down stage. The first issue is that although the location of the pick at the drum transition point from the sump-in stage to the shearing-down stage can be calculated, it is not convenient to use this location to count the drum revolutions. To address this, the following simplification is possible if the roof rock thickness is less than the 2/3 of the tip-to-tip diameter of the drum. As the rock cutting in the last cycle at the sump-in stage has been fully considered, it is acceptable to assume that the first revolution in the shearing-down stage always starts from the point with the maximum sump-in distance (rightmost point) at the last revolution in the sump-in stage. The second issue is that similarly to the roof rock, the thickness of the floor rock also generally varies from place to place. As a result, both the number of the drum revolutions involving floor hard rock cutting and the rock length cut in each drum revolution can be uncertain (i.e., their values can change randomly). However, this process is different from the sump-in stage. To solve this problem, the calculation of the pick tip failure probability always starts from the first drum revolution in the shearing-down stage, because  
\[F_{fi}=0,\text{ when }n_{di}v_{n}\ \leq h_{t}-D_{d}-h_{fmax} \tag{9}\]  
In Equation (9), \(h_{fmax}\) is the largest mean thickness of the floor rock (mm), \(F_{fi}\) is the pick tip failure probability over the \(i\)-th drum revolution in the shearing-down stage, \(h_{t}\) is the height of the roadway (mm), and \(n_{di}\) is the number of revolutions from the 1 st drum revolution to the \(n_{di}\)-th drum revolution in the shearing-down stage. Equation (9) holds because the drum will not start to cut the floor rock until the \(n_{di}\)-th drum revolution satisfies the following condition:  
\[n_{di}>\frac{\left(h_{t}-D_{d}-h_{fmax}\right)}{v_{n}} \tag{10}\]  
The third issue is that the thickness of the floor rock can vary over a drum revolution in the shearing-down stage, and thus it is required to identify whether a segment being cut is rock or not. Figure 4 shows an arbitrary segment (the \(s\)-th segment) in the shearing-down stage. It can be seen from this figure that a segment being cut is rock only when its distance from the floor is shorter than \(h_{fi}\) and its angular position (central angle) is less than the cutting sector. Note that, unlike in the sump-in stage where the angular position of a segment is measured from vertical, the angular position of a segment in the shearing-down stage is the angle between the horizontal direction and the line to the middle point of the segment. The cutting sector in the shearing-stage is also measured from the horizontal direction. As previously mentioned, the cutting sectors of the revolutions in the shearing-down stage are not all the same.