## 2 Construction of New Dynamic Rock-Cutting Model and Analysis of Cutting Periodicity
### Construction of a New Dynamic Cutting Model
Subsequently, as the wedge-shaped cuttings above the shear surface and the fine cuttings in the extrusion zone dissipate, the resistance to the forward movement of the tool decreases, causing the thrust \(F_{c}{}^{t}\) to gradually diminish. By \(t_{\mathbf{2}}\), the rock debris is completely expelled, and \(F_{c}{}^{t}\) is reduced to a minimum. Subsequently, the rock-breaking activities between \(t_{\mathbf{0}}\) to \(t_{\mathbf{2}}\) will be repeated between \(t_{\mathbf{2}}\) and \(t_{\mathbf{4}}\), where the cutting states at \(t_{\mathbf{2}}\), \(t_{\mathbf{3}}\) and \(t_{\mathbf{4}}\) correspond to those at \(t_{\mathbf{0}}\), \(t_{\mathbf{1}}\) and \(t_{\mathbf{3}}\), respectively. This repetition will continue until the cutting process stops. This phenomenon aligns with the findings of previous linear cutting tests [40; 41]. The rock-breaking activity from \(t_{\mathbf{0}}\) to \(t_{\mathbf{2}}\) can be regarded as the smallest unit of the entire cutting process. Therefore, it is defined as a single cutting process (SCP) in our study. Accordingly, when the cutting depth \(h\), cutting speed \(v\), and rake angle \(a\) are determined, we hypothesize that the tangential cutting force \(F_{c}{}^{t}\) will oscillate periodically with the cycle of the single cutting process. Similar observations have been made in previous studies [25]. Inspired by these phenomena, the periodicity of the tangential cutting force \(F_{c}{}^{t}\) is proposed in this study. The tangential cutting force \(F_{c}{}^{t}\) will oscillate periodically with the cycle of the single cutting process (SCP) [25]. The duration and frequency of the single  
Figure 4: The new dynamic rock-cutting model (NDCM) with \(h_{c}\
eq h\). cutting process are defined as the cutting cycle \(\mathbf{T_{c}}\) and the cutting frequency \(\mathbf{f_{c}}\), respectively, which satisfy the following relationship:  
\[f_{c}=\frac{1}{T_{c}} \tag{1}\]  
The cutter travels during the cutting cycle \(\mathbf{T_{c}}\), which is defined as a single cutting stroke \(\mathbf{L_{c}}\), and its expression is as follows:  
\[L_{c}=vT_{c} \tag{2}\]  
Then, the height of the extrusion zone \(\mathbf{h_{c}}\) can be obtained by the following:  
\[h_{c}=L_{c}\tan\psi \tag{3}\]  
where \(\mathbf{\psi}\) is the shear angle. Substituting Equations (1) and (2) into Equation (3) yields the following:  
\[h_{c}=\frac{v\tan\psi}{f_{c}} \tag{4}\]  
When the contact between the dissipated cuttings and the tooth is not considered, the length of the oblique edge of the extrusion zone represents the maximum contact range of tool-rock interactions (\(\mathbf{MCR}\)), and it can be expressed as follows:  
Figure 4.0  
Substituting Equation (4) into Equation (5), we have the following:  
Figure 4.1  
It can be seen from Equation (6) that the cutting frequency \(\mathbf{f_{c}}\) is directly related to the size of the extrusion zone. Based on the above analysis, the determination of the cutting cycle is crucial for the development of the cutting model and for understanding the tool-rock contact relationship. To obtain the cutting frequency \(\mathbf{f_{c}}\), the tangential cutting force \(\mathbf{F_{c}}^{t}\) is analyzed from the perspective of signal theory. According to the principle of signal analysis, the spectrum of \(\mathbf{F_{c}}^{t}\) can be obtained using fast Fourier transform (FFT) at the appropriate sampling frequency. If a peak appears in the spectrum, the corresponding frequency is considered to be the dominant frequency of the signal. This method has been employed to determine the frequency of the rock-breaking force of the disk cutter, and the results demonstrate its effectiveness [40]. The dominant frequency of the signal represents the primary component of the original signal, which dictates the overall signal's trend. Therefore, it is inferred that the dominant frequency of the tangential force \(\mathbf{F_{c}}^{t}\) is the cutting frequency \(\mathbf{f_{c}}\). Based on this, an FFT-based cutting frequency acquisition method (referred to as the FFT method) is proposed.