# Simulation of Surface Topography Considering Cut-in Impact and Tool Flank Wear
## 2 The Kinematic Trajectory Equation of Cutting Edges  
The establishment of the trajectory equation of any cutting point on the cutting edge in the milling process is the basis and key to the simulation modeling of the surface topography of the ball-end milling cutter. Therefore, the rectangular coordinate system as shown in Figure 1 is introduced. The workpiece coordinate system (\(O_{W}X_{W}Y_{W}Z_{W}\) (\(\{W\}\) for short) is a reference coordinate system fixed on the workpiece. The tool rotation coordinate system (\(O_{T}X_{T}Y_{T}Z_{T}\) (\(\{T\}\) for short) rotates with the tool. The tool coordinate system (\(O_{G}X_{G}Y_{G}Z_{G}\) (\(\{G\}\) for short) translates and vibrates with the tool. The spindle coordinate system (\(O_{S}X_{S}Y_{S}Z_{S}\) (\(\{S\}\) for short) moves along the feed direction with the spindle. \(F_{n}\) is the normal contact force. \(F_{s}\) is the tangential contact force. \(\phi_{st}\) and \(\phi_{ex}\) are the entry and exit angles of the cutter to and from the cut, respectively. \(\varphi\) is the axial position angle. Taking ball-end milling cutter as the research object, the homogeneous coordinates of any point P on the cutting edge in \(\{T\}\) can be expressed as follows:  
Figure 1: Reference coordinate system of the kinematic trajectory, (**a**) the cutter workpiece engagement; (**b**) the overhead view of the cutter workpiece engagement. \[r(\varphi)=\left[ \begin{array}{c}x_{TP}\\ y_{TP}\\ z_{TP}\end{array} \right]=\left[ \begin{array}{c}R\cdot\sin\varphi\cdot\cos(\tan \beta(1-\cos\varphi))\\ R\cdot\sin\varphi\cdot\sin(\tan\beta(1-\cos\varphi))\\ R\cdot(1-\cos\varphi)\end{array} \right], \tag{1}\]  
where \(\varphi\) is axial position angle, \(\beta\) is helix angle of tool, and \(R\) is tool radius. Meanwhile, tool wear will occur gradually on the flank face and rake face with the cutting progress. Since tool flank wear has a significant influence on the surface topography, the effect of tool flank wear on the surface topography is researched. The geometric relationship between the flank wear and the discrete points of the cutting edge is shown in Figure 2.  
The homogeneous coordinates of any point P on the cutting edge with wear in \(\{T\}\) can be expressed as follows:  
\[r_{vb}(\varphi)=\left[ \begin{array}{c}x^{\prime}_{TP}\\ y^{\prime}_{TP}\\ z^{\prime}_{TP}\end{array} \right]=\left[ \begin{array}{c}R^{\prime}\cdot\sin \varphi_{icb}\cdot\cos(\tan\beta\cdot(1-\cos\varphi_{icb}))\\ R^{\prime}\cdot\sin\varphi_{icb}\cdot\sin(\tan\beta\cdot(1-\cos\varphi_{icb}) )\\ R^{\prime}\cdot(1-\cos\varphi_{icb})\end{array} \right], \tag{2}\]  
\[\Delta R=\frac{VB\cdot\tan\gamma}{1-\tan\alpha\cdot\tan\gamma}, \tag{3}\]  
\[R^{\prime}=R^{2}+\Delta R^{2}-2\cdot\Delta R\cdot R\cdot\cos(\frac{\pi}{2}- \varphi_{i}), \tag{4}\]  
\[\varphi_{icb}=\frac{R\cdot\sin\varphi_{i}-\Delta R}{R}, \tag{5}\]  
where \(\gamma\) and \(\alpha\) are the clearance angle and the rake of the cutting tool. The coordinate transformation matrix from \(\{T\}\) to \(\{G\}\) at any point P on the cutting edge can be expressed as follows:  
\[T_{1}=\left[ \begin{array}{cccc}\cos\phi_{i}&-\sin\phi_{i}&0&0\\ \sin\phi_{i}&\cos\phi_{i}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array} \right], \tag{6}\]  
where \(\phi_{i}=\phi_{i,1}+2\pi(j-1)/N-\omega t\), \(\phi_{i,1}\) is the initial entrance angle of the tool, \(N\) is the teeth number, and \(\omega\) is the angular velocity of the tool. The stiffness of workpiece is much higher than that of the cutter-spindle cutting system for the milling process of automobile panel die. Therefore, the vibration of machining process mainly occurs  
Figure 2: Schematic diagram of flank wear (**a**) the position change of cutting edge with tool wear; (**b**) the relationship between the radial radius variation and tool flank wear. in the tool-spindle cutting system. Meanwhile, because the stiffness of the ball-nose cutter in the axial direction is far greater than that in the radial direction, the vibration of the cutter in the axial direction is neglected:  
\[T_{2}=\left[