## 2 Instantaneous Wear Volume Model of the Tooth Flank

Table 1: Illustration of instantaneous milling process variables of high-feed milling cutter under vibration. From Figure 1 and Equation (1), the transient attitude angle of the micro-element of the cutter tooth in the workpiece coordinate system is:  
\[\delta_{1}(t)=\arccos(\partial G_{i}^{0}(x(t),y(t),z(t))/\partial z) \tag{2}\]  
\[\delta_{2}(t)=\arccos(\partial G_{i}^{0}(x(t),y(t),z(t))/\partial y) \tag{3}\]  
The instantaneous friction contact point, \(e\), between the cutter flank and the machined transition surface is:  
\[e(x(t),y(t),z(t))=\left\{ \begin{array}{l}G_{i}(x(t),y(t),z(t))=0\\ G_{i}^{0}(x(t-\Delta t),y(t-\Delta t),z(t-\Delta t))=0\end{array} \right. \tag{4}\]  
where \(t-\Delta t\) is the moment when the cutting edge forms the machined transition surface. The speed of the contact point in the three directions along the coordinate system of the workpiece is:  
\[v_{x}(t)=\partial x_{e}(t)/\partial t,v_{y}(t)=\partial y_{e}(t)/\partial t,v _{z}(t)=\partial z_{e}(t)/\partial t \tag{5}\]  
where \((x_{e},y_{e},z_{e})\) is the coordinate of the friction contact point, \(e\), along the feed, cutting width, and axial directions of the tool flank, respectively. Using Equations (4) and (5), the angle \(\beta(t)\) between the normal vector of the common tangent at point \(e\) and the velocity of motion \(v(t)\) can be obtained, and the instantaneous friction velocity \(v^{\prime}(t)\) at point \(e\) is:  
\[v(t)=-v(t)\cdot\sin(\pi-\beta(t)) \tag{6}\]  
From Figure 1, the instantaneous wear depth of the cutter's face after the micro-element of the cutter tooth is shown in Figure 2, where \(e^{\prime}\) is the location point of point \(e\) on the tool flank after wear, \(\eta_{1}\) is the angle between the common tangent normal vector of point \(e\) and the \(z_{i}\) axis space, and \(\eta_{2}\) is the angle between the projection of the common tangent normal vector of point \(e\) on the _xioiyi_ coordinate plane and the \(x_{i}\) axis plane. From Figure 2, the instantaneous growth rate of the wear depth, \(vh_{i}(t)\), and the instantaneous increment of the wear depth, \(\Delta h_{i}(t)\), at the friction contact point, \(e\), of the cutter flank after the micro-element of the cutter tooth are:  
\[v_{hi}(t)=\partial h_{i}(t)/\partial t,\Delta h_{i}(t)=v_{hi}(t)\cdot dt \tag{7}\]  
where \(h_{i}(t)\) is the cumulative wear depth of the tooth flank as a function of time in the thermal coupling field of the cutter tooth. From Figure 2 and Equation (7), the instantaneous coordinate increments \(\Delta x_{i}(t)\) and \(\Delta z_{i}(t)\) of point \(e^{\prime}\), with respect to point \(e\) along the \(x_{i}\) and \(z_{i}\) directions, can be derived, respectively:  
\[\left\{ \begin{array}{l}\Delta x_{i}(t)=\Delta h_{i}(t)\cdot\sin\eta_{1} \cdot\cos\eta_{2}\\ \Delta z_{i}(t)=\Delta h_{i}(t)\cdot\cos\eta_{1}\end{array} \right. \tag{8}\]  
From Equations (4), (7) and (8), the micro-element tool flank of the cutter tooth under the milling cutter's wear conditions can be illustrated as:\[G_{i}(x(t),y(t),z(t))=\left\{ \begin{array}{l}x_{G_{i}}(t)=x_{i}(t)+\Delta x_{i}(t )\\ y_{G_{i}}(t)=y_{i}(t)\\ z_{G_{i}}(t)=z_{i}(t)+\Delta z_{i}(t)\end{array} \right. \tag{9}\]  
From Equations (7)-(9), the curve formed by the point where the instantaneous wear depth on the tool flank is 0 is the lower boundary of transient friction wear on the tooth flank. Using Equations (1), (7) and (8), the Equations of the micro-element cutting edge under wear conditions could be obtained. Accordingly, the boundaries of the transient friction wear can be determined, as shown in Figure 3.  
From Equations (4)-(9), the instantaneous wear volume growth rate, \(V_{i}(t)\), and the accumulated wear volume, \(\Sigma V_{i}\), at the friction contact points within the friction wear boundaries are:  
\[V_{i}(t)=v_{hi}(t)\cdot\oint_{L}ds_{i},\sum V_{i}=\int_{t_{j}}^{t_{j+1}}V_{i}( t)\cdot dt \tag{10}\]  
where \(s_{i}\) is the instantaneous friction area at point \(e\), and \(L\) is the instantaneous friction area boundary.