## 2 Instantaneous Posture of the Milling Cutter and the Minor Cutting Edge
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Serial Numbers** & **Parameters** & **Parameters Explanation** \\ \hline
35 & \(A_{x}\)(t), \(A_{y}\)(t), \(A_{z}\)(t) & The vibration displacement of the milling cutter in three directions of \(x\), \(y\) and \(z\), respectively. \\ \hline
36 & \(l\) & Milling cutter overhang. \\ \hline
37 & \(\theta_{1}\)(t) & The projection angle of the milling cutter attitude angle \(\theta\)(t) on the \(v_{00}w\) plane. \\ \hline
38 & \(\theta_{2}\)(t) & The projection angle of the milling cutter attitude angle \(\theta\)(t) on the \(w_{00}w\) plane. \\ \hline
39 & \(\theta_{mX}\) & Angle between projection of \(X_{i}\) axis on _xoy_ plane and \(x\) axis. \\ \hline
40 & \(\theta_{mY}\) & Angle between \(Y_{i}\) axis and _xoy_ plane. \\ \hline
41 & \(\theta_{mZ}\) & The angle between projection of \(X_{i}\) axis and \(X_{i}\) axis on _xoy_ plane. \\ \hline
42 & \(\theta_{my}\) & Angle between friction velocity \(v_{m}\) and \(Y_{i}\) axis. \\ \hline
43 & \(\theta_{mz}\) & Angle between \(v_{m}\) and \(Z_{i}\) axis. \\ \hline
44 & \(\theta_{m}\) & The angle between \(v_{m}\) and relative motion velocity \(v_{n}\). \\ \hline
45 & \(\theta_{c}\) & Angle between \(sn\) and vector \(l_{p}\). \\ \hline
46 & \(\varphi^{\prime}_{i}(t_{0})\) & The position of the tip point of the \(i\)th tooth of the milling cutter at the initial cut-in moment during milling \\ \hline
47 & \(l_{p}\) & Projection vector of \(v_{n}\) on the common tangent plane of the cutter minor flank and workpiece transition surface. \\ \hline
48 & \(o_{r}\) & Center point of the surface element \\ \hline
49 & \(l_{u}\) & Friction upper boundary \\ \hline
50 & \(l_{d}\) & Friction lower boundary \\ \hline
51 & \(\mu\) & Friction coefficient \\ \hline
52 & \(f_{p}\) & Friction force \\ \hline
53 & \(v_{m}\) & Friction speed \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._  
Figure 2: Minor cutting edge and minor flank of the cutter teeth. \(A_{i}\) is the intersection equation of the minor flank and \(p_{0}\) plane, which could be expressed as:  
\[\frac{\partial A_{i}(X_{i},Y_{i},Z_{i})}{\partial Y_{i}}=\tan a_{0}^{\prime} \tag{1}\]  
\[A_{i}(X_{i},Y_{i},Z_{i})=\left\{ \begin{array}{l}X_{i}\\ Y_{i}=(X_{i}-l_{X_{1}})\cdot\tan\lambda_{s}^{\prime}+l_{Y_{1}}\\ Z_{i}=(X_{i}-l_{X_{1}})\cdot\tan k_{r}^{\prime}+l_{Y_{1}}\cdot\tan a_{0}^{ \prime}\end{array} \right. \tag{2}\]  
When \(\lambda_{s}^{\prime}=0\) and \(k_{r}^{\prime}=0\), the minor cutting edge \(l_{X_{1}}\) could be expressed as:  
\[l_{X_{1}}(X_{i},Y_{i},Z_{i})=\left\{ \begin{array}{l}X_{i}\\ Y_{i}=0\\ Z_{i}=0\end{array} \right. \tag{3}\]  
When \(\lambda_{s}^{\prime}\) and \(k_{r}^{\prime}\) are not 0, the minor cutting edge \(l_{X_{2}}\) could be expressed as:  
\[l_{X_{2}}(X_{i},Y_{i},Z_{i})=\left\{ \begin{array}{l}X_{i}\\ Y_{i}=(X_{i}-l_{X_{1}})\cdot\tan\lambda_{s}^{\prime}\\ Z_{i}=-(X_{i}-l_{X_{1}})\cdot\tan k_{r}^{\prime}\end{array} \right. \tag{4}\]  
The minor cutting edge equation \(l_{X_{3}}\) of the cutter tooth could be derived as:  
\[l_{X_{3}}(X_{i},Y_{i},Z_{i})=\left\{ \begin{array}{l}X_{i}\\ Y_{i}=l_{X_{2}}\cdot\tan\lambda_{s}^{\prime}\\ Z_{i}=-l_{X_{2}}\cdot\tan k_{r}^{\prime}\end{array} \right. \tag{5}\]  
The cutting trajectory of the milling cutter and cutter tooth were characterized by the motion of the milling cutter coordinate origin and the cutter tooth coordinate origin relative to the workpiece under vibration. The instantaneous cutting posture angle of the cutter and its tooth was characterized by the cone angle formed by the axis of the cutter and its tooth and the axis of the cutter at the predetermined position, as shown in Figure 3.  
In the workpiece coordinate system, the trajectory equation of any point on the minor cutting edge could be expressed as:  
\[\left[ \begin{array}{cccccc}x&y&z&1\end{array} \right]^{T}=\vartheta\cdot \left[ \begin{array}{cccccc}X_{i}&Y_{i}&Z_{i}&1\end{array}