## 3 A Case Study
### Discussion  
Figure 5 shows that in a cutting cycle, the pick tip failure probability changes over time in a style of constant-increasing-constant-increasing-constant. The first constant period (zero period) occurs in the sump-in stage and lasts about 1.2 s, mainly due to the small DOC in the first revolution as mentioned in Section 2.2. In the first revolution, the maximum DOC is only 5.875 mm. The bending force generated from cutting the given sandstone with this small DOC will not damage the tips in the consideration. The first increase period happens in the sump-in stage due to hard rock cutting. The second constant period and the second increase period both occur in the shearing-down stage. In this stage, the tip cuts the coal seam first and then cuts hard floor rock. When cutting the coal seam only, the tip has a failure probability of 0 (zero) because as assumed, tips will not be subjected to bending-caused failure when they are cutting coal. As a result, the tip failure probability over the coal cutting period remains constant. Once the pick involves floor rock cutting, its tip failure probability increases over time again. The third constant period corresponds to the cutting-back stage. As discussed in Section 2.4.1, the failure probability of the tip at this stage can be ignored (i.e., it is regarded to be zero). Figure 8: TSDC tip failure probability in the sump-in stage–Scenario 3. Figure 9: TSDC tip failure probability in the shearing-down stage–Scenario 3. A comparison between Figures 6 and 8 indicates that the variation in the rock thickness can significantly affect the pick tip failure probability in the sump-in stage. Although the rock thickness in Scenario 3 is the expected value of the rock thickness in Scenario 2, the tip failure probability over the sump-in stage in Scenario 2 is much higher than that in Scenario 3. The reason is that there is 65% chance for the roof rock mean thickness to be 460 mm, which is much higher than the expected value of 442.5 mm. As all the rock thickness values are less than the drum's cutting radius (500 mm), a larger value of rock thickness means that the pick needs to cut rock segments with larger DOC. According to the formula given by [PERSON] [14] and Equation (20) in [11], the bending force acting on the tip is proportional to the square of DOC, and therefore a larger DOC will result in a higher failure probability of tips. However, a comparison between Figures 7 and 9 shows that the variation in the rock thickness has little influence on the pick tip failure probability in the shearing-down stage. The reason for this phenomenon is that the drum advance direction in the sharing-down stage is perpendicular to that in the sump-in stage. When the drum advances towards the floor rock, an increase in the floor rock thickness will change the number of drum revolutions involving hard rock cutting, rather than the DOC of rock segments being cut. The changed number of drum revolutions involving hard rock cutting will also change the tip failure probability, but the magnitude of the change in the tip failure probability heavily depends on the ratio of the rock thickness variation to the average rock thickness. In the above examples, the variation of the maximum rock thickness in Scenario 2 (460 mm) to the expected mean rock thickness in Scenario 3 (442.5 mm), and that of the minimum rock thickness in Scenario 2 (410 mm) to the expected mean rock thickness in Scenario 3 are respectively 17.5 mm and -32.5 mm. As the absolute values of the variations are both less than the drum's advance distance per revolution (47 mm), the difference between drum revolutions involving hard rock cutting in Scenario 2 and Scenario 3 is less than 1 (one) revolution. This small change in the number of the drum revolutions involving hard rock cutting has a trivial influence on the tip failure probability, because the ratios of the variations of the maximum and the minimum rock thickness to the expected value are small (3.95% and -7.34% respectively).