## 3 Instantaneous Contact between the Minor Flank and Transition Surface
Variation Characteristic of Instantaneous Friction Velocity Speed on the Minor Flank of the Cutter Tooth  
In order to reveal the change characteristics of the instantaneous friction speed vector on the minor flank of cutter tooth, according to the instantaneous friction upper and lower boundary criteria on the minor flank of the cutter tooth, the instantaneous contact friction area element between the minor flank of cutter tooth and the machining transition surface was identified. The friction speed size and direction angle of the surface element were solved. The trajectory parameter equation of any point on the cutter tooth was obtained by using Equation (6). The trajectory of any point on the cutter tooth is skewed to time, and the sub-velocities \(v_{nx}\), \(v_{ny}\) and \(v_{nz}\) along the \(x\), \(y\) and \(z\) axes of the workpiece coordinate were obtained, respectively. The relative speed \(v_{n}\) of any point is:  
\[v_{n}=\sqrt{v_{nx}^{2}+v_{ny}^{2}+v_{nz}^{2}} \tag{25}\]  
The unit vector of the relative speed \(v_{n}\) in the workpiece coordinate is \((v_{nx},v_{ny},v_{nz})\). The intersection point between the minor flank of the cutter tooth and the workpiece transition surface is \(\sigma_{r}\). The common tangent plane \(P_{i}\) passes through the or point and is tangent to the minor flank of the cutter tooth and the workpiece transition surface at the same time:  
\[P_{i}(x,y,z)=\frac{\partial A_{i}^{\prime}(x,y,z)}{\partial x}(x-x_{r})+ \frac{\partial A_{i}^{\prime}(x,y,z)}{\partial y}(y-y_{r})+\frac{\partial A_ {i}^{\prime}(x,y,z)}{\partial z}(z-z_{r}) \tag{26}\]  
Project \(v_{n}\) on the common tangent plane \(P_{i}\), and the unit vector \(\overset{\rightarrow}{l}_{p}\) in the projection direction through the point or is \((l_{px},l_{py},l_{pz})\). In the workpiece coordinate, the friction speed \(v_{m}\) at any point on the minor flank is as follows:  
\[v_{m}=v_{n}\cdot\cos\theta_{m} \tag{27}\]In the equation, \(\theta_{m}\) is the angle between the relative velocity \(v_{n}\) and the unit vector \(\overset{\rightarrow}{l}_{p}\). \(\theta_{m}\) and \(\theta_{c}\) are complementary angles to each other. The calculation method of \(\theta_{c}\) is as follows:  
\[\theta_{c}=\arccos(\frac{l_{px}\cdot v_{nx}+l_{py}\cdot v_{ny}+l_{yz}\cdot v_{nz }}{\sqrt{l_{px}^{2}+l_{py}^{2}+l_{pz}^{2}}\cdot\sqrt{v_{nx}^{2}+v_{ny}^{2}+v_{nz }^{2}}}) \tag{28}\]  
In Equation (28), \(\theta_{c}\) is the angle between the relative speed \(v_{n}\) and the friction speed \(v_{m}\). In the workpiece coordinate, the friction speed direction vector \(\overset{\rightarrow}{v}_{m}\) of the surface element on the minor flank is \((v_{mx},v_{my},v_{mz})\). The unit vectors of the \(y\)-axis and \(z\)-axis of the workpiece coordinate are \(\overset{\rightarrow}{l}_{ry}\) and \(\overset{\rightarrow}{l}_{kz}\), respectively. The angle \(\theta_{my}\) and \(\theta_{mz}\) between \(v_{m}\) and the \(y\)-axis and \(z\)-axis of the cutter tooth coordinate system are obtained by the two-vector angle solution method, respectively. According to the instantaneous transformation relationship matrix between the cutter tooth coordinate and the workpiece coordinate, the friction speed components \(v_{mXi}\), \(v_{mYi}\) and \(v_{mZi}\) of the instantaneous friction surface element of the minor flank of the cutter tooth were obtained, as shown in Equation (29). \[\left[ \begin{array}{ccccc}v_{mXi}&v_{mYi}&v_{mZi}&1\end{array} \right]^{T}= \theta^{-1}\cdot\left[ \begin{array}{ccccc}v_{mx}&v_{my}&v_{mz}&1\end{array}  \right]^{T} \tag{29}\]  
In the equation, \(v_{mXi}\) is the friction speed of the surface element on the minor flank of the cutter tooth along the \(X_{i}\) axis of the cutter tooth coordinate. \(v_{mYi}\) is the friction speed of the surface element on the minor flank of the cutter tooth along the \(Y_{i}\) axis of the cutter tooth coordinate. \(v_{mZi}\) is the friction speed of the surface element on the minor flank of the cutter tooth along the \(Z_{i}\) axis of the cutter tooth coordinate.