## 3 Results
### Influence of Drilling Parameters on Bit Force
#### 3.2.2 Effect of Rotation Speed on Drill Bit Force  
During the drilling process, considering the impact of rotational speed on the force on the drill bit, a specific drilling speed is first fixed, and then the rotational speed is changed. The thrust force and torque values at different rotational speeds are obtained, as shown in Figure 7a-d. Test parameter values are determined based on experimental test results. The drilling speeds are set at 0.5 mm/s, 1.0 mm/s, 1.5 mm/s, and 2.0 mm/s, and under each fixed drilling speed, and the rotational speeds are varied at 50 rpm, 100 rpm, 200 rpm, and 300 rpm, resulting in corresponding thrust force and torque curves. From the graph, it is evident that at any given fixed drilling speed, both the thrust force and torque curves exhibit noticeable disparities, gradually decreasing with the increase in rotational speed. Due to the continuous interplay between drilling speed and rotational speed during the drilling process, significant impact pressure and torque are generated shortly after the drill bit initially contacts the rock. At this point, drilling is in an unstable state. Subsequently, the thrust force and torque maintained steady fluctuations, marking the drilling entering a stable stage.  
In the same manner, taking the average of the thrust force and torque curves during the stable phase for each rotational speed, we obtain the relationships between thrust force, torque, and rotational speed at different drilling speeds, as shown in Figure 8. For each fixed drilling speed, thrust force and torque both decrease with increasing rotational speed in a power law fashion. Moreover, as drilling speed increases, the influence of rotational speed on thrust force and torque becomes more pronounced. This behavior is a consequence of the drill bit's vertical motion along the drill rod axis, its rotational movement around the drill rod axis, and the frictional resistance experienced by the drill rod's wall and the drill bit's end, both axially and laterally [44]. The increase in drilling speed leads to an accumulation of energy, resulting in higher thrust force and torque. Conversely, higher rotational speeds reduce the thickness of rock cut by each rotation, which in turn reduces thrust force and torque. It is evident that increasing rotational speed can greatly enhance drilling efficiency. Therefore, the relationship expressions for thrust force, torque, and rotational speed can be established as follows:  
\[\left\{\begin{array}{l}F=\frac{\eta_{1}}{\omega^{2}}+F_{0}^{\prime}\\ T=\frac{\eta_{2}}{\omega^{2}}+T_{0}^{\prime}\end{array}\right. \tag{12}\]  
where \(F\) is thrust force, \(T\) is the torque, \(v\) is the drilling speed, \(\xi_{1}\) is the slope of the relationship between the thrust force and the drilling speed, \(\xi_{2}\) is the slope of the relationship between the torque and the drilling speed, \(F_{0}\) is thrust force constant, and \(T_{0}\) is the torque constant.  
Figure 7: _Cont._Figure 8: Thrust force, torque and rotational speed at different drilling speeds.  
Figure 7: Curves of thrust force and torque at different rotational speeds during drilling. **(a)** Drilling speed 0.5 mm/s **(b)** Drilling speed 1.0 mm/s **(c)** Drilling speed 1.5 mm/s **(d)** Drilling speed 2.0 mm/s.