## 4 Discussion: Analysis of Factors Affecting Drilling Forces
### Analysis of Drill Bit Size on Drilling Force
Figure 13 depicts the relationship curves between thrust force, torque, and rotational speed for the three drill bit sizes--59 #, 75 #, and 110 #--all at the same drilling speed of 2 mm/s. It can be observed that both thrust force and torque decrease in a power law manner with the increase in rotational speed. Under the same rotational speed conditions,  
Figure 11: Drilling maximum principal stress cloud image of different size bits. (**a**) 59# Drill bit (**b**) 75# Drill bit (**c**) 110# Drill bit. Figure 12: Thrust force, torque and drilling speed for different bit sizes at 200 rpm. larger drill bit sizes result in higher thrust force and torque. However, the correlation coefficient of the relationship curve between thrust force, torque, and rotational speed remains largely unaffected by drill bit size. The increase in drill bit size leads to an increase in thrust force and torque mainly due to the enlarged effective contact area between the drill bit and the rock during drilling. Therefore, it is possible to calculate the required thrust force and torque for a given drill bit size by considering the effective contact area between the drill bit and the rock. For various drill bit sizes, the following relationship expressions between thrust force, torque, and rotational speed, taking into account the drill bit size effect, can be established:  
\[\left\{ \begin{array}{l}F=\alpha(\frac{\eta_{2}}{\omega^{2}}+F_{0}^{\prime})= \alpha\frac{\eta_{1}}{\omega^{2}}+\alpha F_{0}^{\prime}\\ T=\alpha(\frac{\eta_{2}}{\omega^{2}}+M_{0}^{\prime})=\alpha\frac{\eta_{2}}{ \omega^{2}}+\alpha T_{0}^{\prime}\end{array} \right. \tag{17}\]  
where \(F\) is thrust force, \(T\) is torque, \(\omega\) is rotational speed, \(\eta_{1}\) is the slope of the relationship between the thrust force and the rotation speed, \(\eta_{2}\) is the slope of the relationship between the torque and the rotation speed, \(F_{0}^{\prime}\) is thrust force constant, \(T_{0}^{\prime}\) is the torque constant, and \(\alpha\) is the drill bit size influence factor. Figure 14 illustrates the relationship between torque and thrust force during the drilling process for different-sized drill bits. From the graph, it is evident that regardless of the drill bit size, the torque during the drilling process exhibits a linear increase with thrust force. Additionally, larger drill bit sizes correspond to steeper slopes in the torque-pressure curve. Similarly, both Equations (15) and (17) yield the relationship expression between torque and thrust force, taking into account the drill bit size effect as follows:  
\[\left\{ \begin{array}{l}T=\alpha\left[\frac{\xi_{2}}{\xi_{1}}(\frac{F}{\alpha }-F_{0})+T_{0}\right]\\ T=\alpha\left[\frac{\eta_{2}}{\eta_{1}}(\frac{F}{\alpha}-F_{0}^{\prime})+T_{0} ^{\prime}\right]\end{array} \right. \tag{18}\]  
From Equation  
The changes in the relationships between thrust force, torque, drilling speed, and rotational speed, as well as the alteration in the torque-thrust force relationship during the drilling process with different-sized drill bits, primarily stem from variations in the effective contact area between the drill bit and the rock. Therefore, by introducing the drill bit size influence factor through the conversion relationship between different-sized drill bits and the effective contact area with the rock, we obtain the relationship expressions that consider the drill bit size effect for thrust force and torque concerning drilling speed and rotational speed. Additionally, we acquire the expression for the torque-thrust force relationship while considering the drill bit size effect.