## 5 Discussion  
### The Influence Mechanism of Cutting Parameters on Cutting Frequency  
Based on the construction of the new dynamic cutting geometric model, the periodicity of the cutting force is proposed and verified through numerical experiments. Subsequently, the influence of rake angle, cutting depth, and cutting speed on cutting frequency is studied. To reveal the underlying mechanisms of the above results, efforts are made to analyze the rock-breaking process from the perspective of energy conservation. The rock-cutting process is also a process of energy conversion. Before the shear surface is formed, the generation of the extrusion zone is accompanied by the accumulation of extrusion energy. Once the conditions for the formation of the shear surface are met, the energy accumulated in the extrusion zone dissipates. This energy is primarily used for the formation of the shear fracture surface and the dissipation of the cuttings [16]. Based on this, the energy conversion relationship in the single-cutting process is analyzed. Firstly, the energy dissipation term is analyzed. As shown in Figure 4, when the shear surface is formed, the newly generated fracture area is \(2 bh/\sin\psi\). According to fracture theory, the energy required to form the unit failure surface is the fracture energy density \(E_{s}\), and the total fracture energy \(W_{s}\) required to form the shear surface is as follows:  
\[W_{s}=\frac{2 bhE_{s}}{\sin\psi} \tag{7}\]  
Equation (7) shows that as the cutting depth increases, more energy is required to form the fracture surface [11]. In addition, when the shear surface is generated, the powdery debris in the extrusion zone and the wedge-shaped debris formed by shear failure will dissipate. Assuming that the dissipation velocity of all cuttings is equal to the cutting speed \(v\), the energy \(W_{m}\) converted into the kinetic energy of the cuttings can be expressed as follows:  
\[W_{m}=\frac{m_{s}v^{2}}{2} \tag{8}\]  
where \(m_{s}\) is the quality of the cuttings. Simultaneously, the friction between the tool and the rock will also cause energy dissipation in the form of thermal and acoustic energy, which is collectively referred to as the friction energy \(W_{f}\). Figure 16: The relationship between cutting frequency and cutting depth in the schemes of \(a=20^{\circ}\). According to previous literature [22; 26; 27], the uniform load \(\sigma_{0}\) imposed on the boundary by the cuttings in the extrusion zone is solely related to the rock mechanical parameters under the limiting cutting condition. In order for the stress on the boundary of the extrusion zone to reach \(\sigma_{0}\), the accumulation of cuttings is accompanied by energy accumulation. Considering that the energy accumulated in the extrusion zone will subsequently be converted into the three types of dissipative energy mentioned above, we infer that one of these types should be elastic energy. Therefore, under the limiting cutting conditions, we define the elastic energy absorbed by the unit volume of cuttings in the extrusion zone as the extrusion energy density \(E_{c}\), and assume that it is uniquely associated with the rock type. In this case, the accumulated extrusion energy \(W_{c}\) in the extrusion zone can be expressed as follows:  
\[W_{c}=\frac{bE_{c}h_{c}^{2}\tan a}{2} \tag{9}\]  
Now, considering that the extrusion energy is equal to the dissipation energy, we have the following:  
\[W_{c}=W_{s}+W_{m}+W_{f} \tag{10}\]  
Due to the small mass of cuttings generated in the single cutting process and the short duration of friction energy dissipation, the cuttings' kinetic energy \(W_{m}\) and friction energy \(W_{f}\) can be ignored.