## 4 Result  
### Verification of Tangential Force Periodicity and Cutting Frequency Extraction Method  
In the numerical simulation, the equivalent plastic zone cloud map indicates the deformation degree of each part of the rock. During the single cutting process, the penetration of the shear surface is represented by the complete entry of the rock on the potential failure surface into the plastic zone. Based on this, the penetration of the plastic zone serves as an indicator. By performing a frame-by-frame analysis of the evolution of the plastic zone cloud map, the time at which each shear surface penetration occurs can be determined. As shown in Figure 4, the time interval between two consecutive plastic zone penetrations is theoretically equal to the cutting cycle \(\mathbf{T_{c}}\), and the cutting frequency can be obtained using Equation (1). This method for determining the cutting frequency is referred to as the plastic zone method, and the result is recorded as \(\mathbf{f_{c}}\)-\(\mathbf{\mathrm{P}}\). Here, a numerical experiment with \(\mathbf{a}=20^{\circ}\), \(\mathbf{v}=35\) mm/s and \(\mathbf{h}=1.0\) mm is used as an example for the following analysis.  
Figure 9 shows the variation in tangential cutting force with time. It can be observed that after 0.043 s, the cutting process enters a stable phase, and the tangential cutting force fluctuates smoothly. Similar phenomena have also been reported in other studies [11,34].  
Figure 10 presents the cloud diagram of the equivalent plastic zone when the shear failure surface extends to the free surface of the rock for six consecutive times. The moments shown in Figure 10a-f are 0.2000 s, 0.2263 s, 0.2525 s, 0.2788 s, 0.3063 s, and 0.3350 s, respectively, with time intervals of 0.0263 s, 0.0262 s, 0.0263 s, 0.0275 s, and 0.0287 s. The mean and standard deviation of the time intervals are 0.0270 s and 0.001 s, respectively, which indicates that the generation of shear surfaces follows a regular pattern and confirms the repeatability of the single cutting process in rock cutting. Accordingly, the average interval time is taken as the cutting cycle \(T_{c}\), and the corresponding cutting frequency \(\mathbf{f_{c}}\)-\(\mathbf{\mathrm{P}}\) is 37.04 Hz.  
Moreover, the tangential cutting force in Figure 9 was analyzed using the fast Fourier transform (FFT), and its spectrum is shown in Figure 11. A significant peak appears at \(\mathbf{f}=35.85\) Hz, indicating that the tangential cutting force \(\mathbf{F_{c}}^{t}\) has a dominant frequency of 35.85 Hz. This frequency, obtained using the FFT method, differs by 3% from the \(\mathbf{f_{c}}\)-\(\mathbf{\mathrm{P}}\) value obtained through the plastic zone method. Additionally, it is noteworthy that there is some amplitude at frequencies other than the peak, suggesting the presence of signal components in other frequency bands. This can be attributed to the fact that, while the  
Figure 9: The variation in tangential cutting force with time in the scheme of \(\mathbf{a}=20^{\circ}\), \(\mathbf{v}=35\) mm/s and \(\mathbf{h}=1.0\) mm.  
To systematically explore the difference in cutting frequency between the two methods, the distribution of \(f_{\text{c}}\)-P and \(f_{\text{c}}\)-F across all schemes is shown in Figure 12. It can be observed that the cutting frequency data points are clustered near the contour line, and the root mean square error (RMSE) between the two methods is 1.48 Hz. The small difference in cutting frequency between the two methods demonstrates the reliability of the FFT method. Given the difficulty of monitoring the plastic deformation of rock in practice, the FFT method is simpler and more reliable. Therefore, in the subsequent study on the influence of cutting parameters on cutting frequency, the analysis will be based on \(f_{\text{c}}\)-F.