## 2 Extended Pick Failure Probability Assessment Approach
### Pick Failure Probability during the Shearing-Down Stage
If the distance between the centre of the drum and the floor is not less than the drum's tip-to-tip diameter, the cutting sector of the drum in a revolution is 90 degrees; otherwise, the cutting sector will be greater than 90 degrees. From Figure 4, it can be found that the drum in the lastrevolution has the largest cutting sector. As the thickness and the length of any segment are very small compared with the drum's cutting radius, the influence of the segment on the cutting sector can be ignored. As such, triangle \(\Delta O_{\Pi 1}O_{\Pi 2}A_{l}\) is an equilateral triangle. This means that the cutting sector in the last revolution is 120 degrees. In other words, the end angle of the cutting sector in the last revolution in the shearing-down stage, \(\alpha_{\mathit{fcl}}\), is  
\[\alpha_{\mathit{fcl}}=120\ (\mathrm{degree}) \tag{11}\]  
The end angles of cutting sectors in other revolutions in the shearing-down stage will be between 90 degrees and 120 degrees. However, accurate calculation of the cutting sector in an arbitrary drum revolution is not further discussed as it is not the focus of this study. For this paper, the cutting sector in the \(i\)-th revolution, \(\alpha_{\mathit{fci}}\), is approximated to be  
\[\alpha_{\mathit{fci}}=\ \left\{ \begin{array}{c}90\,,\ 0\leq n_{\mathit{di}}\leq n _{\mathit{db}}\\ \vdots\\ 90+\frac{60v_{n}(n_{\mathit{di}}-n_{\mathit{db}})}{D_{\mathit{d}}},\ n_{ \mathit{db}}<\ n_{\mathit{di}}\ \leq\ \frac{h_{\mathit{t}}-D_{\mathit{d}}}{v_{n}}\end{array} \right. \tag{12}\]  
where  
\[n_{\mathit{db}}=\mathit{floor}\bigg{(}\frac{2h_{\mathit{t}}-3D_{\mathit{d}}} {2v_{n}}\bigg{)} \tag{13}\]  
Given these constraints, the pick tip failure probability over the shearing-down stage can be estimated using a similar approach for calculating the pick tip failure probability over the sump-in stage, with a consideration of the differences between these two stages. The first difference in the tip failure probability calculation is the segment rock condition. In the sump-in stage, if the \(n_{\mathit{i}}\)-th segment in the \(i\)-th revolution is not rock, then any \(k_{i}\)-th (\(k_{i}>n_{\mathit{i}}\)) segment in the same revolution will not be rock; while in the shearing-down stage, any segments within the drum's cutting sector in a drum revolution may or may not be rock. However, once a segment in a drum revolution is rock, all segments at the same angular position in the following drum revolutions will also be rock. The second difference is the rock cutting start angle. While the start angle of rock cutting sector in the sump-in stage is generally  
Figure 4: A segment in the shearing-down stage. equal to zero, the start angle of rock cutting sector in the shearing-down stage is often non-zero (see Figure 4). As the DOC is normally much smaller than the drum's tip-to-tip diameter, the start angle of rock cutting sector in the shearing-down stage, \(\alpha_{csi}\), can be estimated by  
\[\alpha_{csi}=\mathrm{H}(L_{i})\mathrm{arcsin}\left(\frac{2L_{i}}{D_{d}}\right) \text{, when }L_{i}\ \leq\ \frac{D_{d}}{2} \tag{14}\]  
where \(\mathrm{H}(\cdot)\) is the Heaviside step function, and  
\[L_{i}=h_{t}-\frac{D_{d}}{2}-v_{n}n_{di}-h_{fj} \tag{15}\]  
As \(h_{fj}\) is a random variable, both \(L_{i}\) and \(\alpha_{csi}\) are also random variables. If \(L_{i}>\frac{D_{d}}{2}\), the drum will not cut the floor rock. Due to the above-mentioned differences between the sum-in stage and the shearing-down stage, there are some variations in the approach to estimating the failure probability of pick tips in the shearing-down stage. In the sum-in stage, the pick tip failure probability for each individual drum revolution is estimated first, and then the pick tip failure probability over the whole stage is estimated. However, this approach is not suitable for handling the segment rock conditions in the shearing-down stage.