## 5 Results and Discussion
### Effect of Rotational Angle  
As illustrated in Figure 15a, there may be two working states of 3-RDE cutter. Position 1(solid line in Figure 15a) is the ideal working position. In this state, the convex ridge is perpendicular to the upper surface of the rock. However, due to the welding or design errors, the 3-RDE cutter might be placed at Position 2 (dash line in Figure 15a), where the convex ridge is not perpendicular to the upper surface of the rock any more. In this paper, the angle between the convex ridge and the rock is defined as the rotational angle \(\delta\). To investigate the effect of rotational angles, six levels of rotational angles are respectively set as \(0^{\circ}\),\(5^{\circ}\), \(10^{\circ}\), \(15^{\circ}\), \(20^{\circ}\), \(25^{\circ}\) to carry out the numerical simulation when the cutting depth is 1.5 mm and the back-rake angle is \(15^{\circ}\).  
The contact stress nephogram in rock breaking process with the 3-RDE cutter is shown in Figure 16 under different rotational angles. Here, the polar angle \(\theta\), which represents the cutter's region involved in the cutting, is defined (Figure 15b). The polar angle coincides with the convex ridge at \(0^{\circ}\), the left (right) side of which is negative (positive). It can be seen from Figure 16 that under the ideal condition (\(\delta=0^{\circ}\)) the contact stress is symmetrically distributed with the peak located at the symmetry plane of the cutter. As the rotational angle increases, the contact region gradually transitions from the area near the polar angle of \(0^{\circ}\) to the non-zero region of the polar angle. The distribution is no longer symmetrically with respect to the \(0^{\circ}\) polar angle. The unsymmetrical distribution of contact stress would result in non-uniform wear of 3-RDE cutter and reduce the cutter's life.  
Figure 16: Contact stress distribution of 3-RDE cutter under different rotational angles.  
Figure 14: Effect of cutting depth on: (**a**) F\({}_{\text{n}}\); (**b**) F\({}_{\text{h}}\); (**c**) MSE.  
Figure 15: Illustration of (**a**) Rotational angle (**b**) Polar angle.  
The effect of rotational angles on the cutting forces and MSE is depicted in Figure 17. It can be seen that both the cutting forces and the fluctuations, as well as the MSE, increases gradually with the increase of the rotational angle, which indicates that the increasing of the rotational angle leads to the reduction of the cutter's penetration capacity and the breaking efficiency, as well as the deterioration of cutting stability. Particularly, when the rotational angle exceeds 15\({}^{\circ}\), the axial and the tangential force, along with MSE, of 3-RDE cutter are larger than those of the conventional cutter (Figure 7). This suggests that the actual breaking efficiency is not as good as conventional cutter, and the advantages of 3-RDE cutter are not shown. In addition, the radial force also increases as the rotational angle increases. The larger radial force can cause the imbalance forces on the 3-RDE PDC bit and exacerbate the lateral vibration, which may results in the premature failure of the bit. The reason for the above phenomenon is that with the increasing of the rotational angle, the balance of the radial force on the working faces are broken, and the actual back-rake angle of the working face becomes larger, and the regions of the convex edge interacts with the rock descales. As a consequence, the capacity of balancing the radial force on working forces gradually weakens.  
In summary, the rotational angle has a significant impact on the rock breaking process. More attention should be focus on the brazing of 3-RDE cutter to minimize the welding error.