## 3 Instantaneous Contact between the Minor Flank and Transition Surface  
Based on the analysis results of the instantaneous posture solution model of the milling cutter and the minor cutting edge of the cutter teeth and the analysis results of the formation process of the workpiece surface morphology, the contact relationship between the cutter teeth and the workpiece was characterized, as shown in Figure 6.  
The equation of the workpiece machining transition surface in the workpiece coordinate system could be expressed as:  
\[B_{i}(x,y,z,t)=\left\{ \begin{array}{l}I^{\prime}_{f}(x,y,z,t)=[x(t),y(t),z(t)] \\ t^{s}_{i}\leq t\leq t^{e}_{i}\end{array} \right. \tag{12}\]  
\[I^{\prime}_{f}(x,y,z,t)=\vartheta\cdot l_{f}(X_{i},Y_{i},Z_{i}) \tag{13}\]  
\[A^{\prime}_{i}(x,y,z,t)=\vartheta\cdot A_{i}(X_{i},Y_{i},Z_{i}) \tag{14}\]  
where \(A^{\prime}_{i}\) is the minor flank of the cutter tooth in the workpiece coordinate system. \(I^{\prime}_{f}\) is the minor cutting edge in the workpiece coordinate system. \(I^{s}_{i}\) is the moment when the tool cuts into the workpiece instantaneously. \(t^{e}_{i}\) is the instant when the cutting tool cuts out the workpiece. Figure 5: Comparison of experimental and simulation results of the machined surface morphology. (**a**) Experimental results; (**b**) Solution result. Figure 6: Instantaneous contact relationship between the minor flank and transition surface. In the cutter tooth coordinate system, \(d_{s}\) are the surface elements of the minor flank of the cutter tooth, as shown in Equation (15). \[ds(X_{i},Y_{i},Z_{i})=\frac{1}{\cos(\gamma(X_{i},Y_{i},Z_{i}))}\cdot dX_{i}\cdot dY _{i} \tag{15}\]  
The angle between the normal vector and the \(Z\) axis in the coordinate system \(O_{i}\)-\(X_{i}Y_{i}Z_{i}\) was expressed as follows:  
\[\gamma(X_{i},Y_{i},Z_{i})=\arccos\frac{1}{\sqrt{1+\left(\frac{\partial A_{i}(X _{i}Y_{i}Z_{i})}{\partial X_{i}}\right)^{2}+\left(\frac{\partial A_{i}(X_{i}Y_ {i}Z_{i})}{\partial Y_{i}}\right)^{2}}} \tag{16}\]  
From Equations (12)-(16), because the vibration and cutter tooth errors change the instantaneous friction contact state of the cutter, this further leads to the changes in the instantaneous posture of the friction surface elements of the minor flank. Using the square shoulder milling cutter, cutting parameters, tool tooth error and vibration detection data in the experimental scheme for milling titanium alloys and the calculation results of the milling cutter's instantaneous cutting behavior, the finite element model and boundary conditions of the square shoulder milling cutter for milling titanium alloys were constructed. The Johnson-Cook constitutive parameters and boundary conditions of the TC4 titanium alloy material are shown in Tables 5 and 6. In Table 5, \(A\), \(B\), \(e\), \(c\) and m are the yield stress strength, strain hardening constant, strain hardening exponent, strain rate hardening parameter and temperature strain rate sensitivity, respectively. The analysis results for the instantaneous thermal-stress coupling field on the flank of the cutter tooth in the Deform-3D environment, as shown in Figure 7.  
According to the analysis results of material mechanics, if the equivalent stress on the minor cutting edge of the cutter tooth (\(\sigma\)) greater than the yield strength of the material (\(\sigma_{S}\)), the material in the cutting edge area would fall off. Therefore, the equivalent stress could be used as the upper boundary criterion for the instantaneous friction of the cutter teeth, as shown in Equation (17). \[\sigma\geq\sigma_{S} \tag{17}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Tool Material** & **Coating Material** & **Theral Conductivity \(\lambda\) (w/m-\({}^{\circ}\)C)** & **Rotating Speed \(n\) (r/min)** & **Feed Rate** & **Cutting Width \(a_{e}\) (mm)** & **Cutting Depth \(a_{p}\) (mm)** \\ \hline WC & Ti-CN & 45 & 687 & 275 & 16 & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Boundary conditions of analysis scheme 1.