## 3 Instantaneous Contact between the Minor Flank and Transition Surface
The friction speed at any point in the cutter tooth coordinate is as follows:  
\[v_{m}(X_{i},Y_{i},Z_{i})=\sqrt{v^{2}_{mXi}+v^{2}_{mYi}+v^{2}_{mZi}} \tag{30}\]  
Using the unit vectors of the cutter tooth coordinate for the \(Y_{i}\) axis and \(Z_{i}\) axis of \(\overset{\rightarrow}{l}_{r}\) and \(\overset{\rightarrow}{l}_{k}\), the \(\theta_{mYi}\) and \(\theta_{mZi}\) of the angle between \(v_{m}\) and the tool tooth coordinate \(Y_{i}\) axis and \(Z_{i}\) axis were obtained, as shown in Equations (31) and (32). \[\theta_{mYi}=\arccos(\frac{l_{ryi}\cdot v_{mYi}}{|l_{ryi}|\cdot\sqrt{v_{mXi}^{2 }+v_{mYi}^{2}+v_{mZi}^{2}}}) \tag{31}\]  
\[\theta_{mZi}=\arccos(\frac{l_{kZi}\cdot v_{mZi}}{|l_{kZi}|\cdot\sqrt{v_{mXi}^{2 }+v_{mXi}^{2}+v_{mZi}^{2}}}) \tag{32}\]  
Using Equations (25)-(32), the friction velocity vector declination angles and their magnitudes for the three tooth minor flank area units \(ds\)(0.3, 0.05, 0) in experimental scheme 1 were calculated, and the results are shown in Figure 9.  
It could be seen from Figure 9 that under the milling vibration and cutter tooth error, the overall change trend of the friction speed of each cutter tooth was the same, and it gradually increases to the maximum value and then gradually decreases. However, there were obvious differences in the variation characteristics of the friction speed vector declination angle of each cutter tooth. This directly affects the variation characteristics of the friction speed of each tooth. According to the analysis results of the thermal-stress coupling field of the square shoulder milling cutter milling titanium alloy, Equations (25)-(30) were used to calculate the friction speed of the minor flank of the cutter tooth point by point during the cutting process, and the minor flank friction of the cutter tooth was obtained. The velocity distribution results are shown in Figure 10.  
From Figures 4, 9 and 10, in different milling periods determined by the rotating speed of the milling cutter, although the instantaneous position angle of the same cutter tooth was the same, the friction velocity distribution of the minor flank of the cutter tooth was obviously different. The main reason was tha, the milling vibrations at different milling moments corresponding to the same instantaneous position angle of the cutter teeth were different, which makes the instantaneous posture of the cutter tooth minor flank significantly different. It can be seen from the comparison between Figure 10 and Table 4, in the same milling period and at the same instantaneous position angle, the friction velocity distribution of the flank surface of different teeth was significantly different. The main reason was that, due to the influence of the cutter tooth error and the angle between the cutter teeth, the instantaneous contact between the minor flank of each cutter tooth and the machining transition surface has axial and radial offsets. Eacg cutter tooth also corresponds to the same instantaneous position angle with the milling vibration of different milling moments. Figure 9: Variation in the friction velocity vector of the tooth minor flank area unit ds(0.3, 0.05, 0). (**a**) Cutter tooth 1; (**b**) Cutter tooth 2; (**c**) Cutter tooth 3. The above analysis results showed that the instantaneous friction velocity vector of the minor flank would be obviously affected by cutter tooth error and the milling vibration, and the variations of the instantaneous friction velocity vector were different. The result will lead to the variability of the friction process of the minor flank of each tooth of the milling cutter.