## 2 Prediction of Cutter Edge Temperature in Milling
### Engagement Patterns in Trochoidal Milling  
In trochoidal milling, the cutter is heated up and cooled down periodically. When the cutting edge cuts the material, the cutting edge is heated up because the heat generated flows into the cutter. While on the moments that the cutting edge leaves the material, the temperature starts to decrease. The engagement between cutting edges and material forms a complex pattern, and the pattern has a significant influence on cutter temperature. For circular model, trochoidal toolpath is composed of lines and circles alternately. For illustration, a cycle of the line-segment \(SE\) and the circle centered at \(O_{1}\) is plotted in Figure 6, in which a workpiece with a slot is being machined along the trochoidal toolpath. The cutter radius is \(R\), the segment length  
Figure 5: An illustration of the heat transfer problems in cutting process. is \(c\), and the circle radius is \(r\). To figure out the engagement, a coordinate system \(xO_{c}y\) is established here. The origin \(O_{o}\) is located straight above the point \(S\) with a distance \(r\), and the \(x\)-axis is parallel to line \(SE\). While the cutter feeds along \(SE\) (see Figure 6), cutter center \(O_{c}\) is formulated as  
\[\left[ \begin{array}{c}x_{c}\\ y_{c}\end{array} \right]=\left[ \begin{array}{c}\frac{f}{60}\cdot t\\ -r\end{array} \right],\;0\leq t\leq\frac{60c}{f} \tag{17}\]  
where \(f\) is feed speed (mm/min), and \(t\) is time parameter. Assume the cutter center is located on point \(S\) at the moment \(t=0\), and this moment is used as the reference to calculate the engagement. After \(O_{c}\) reaches point \(E\), the cutter starts to feed along the circle, and \(O_{c}\) is formulated as  
\[\left[ \begin{array}{c}x_{c}\\ y_{c}\end{array} \right]=\left[ \begin{array}{c}c+r\cdot\sin\left(\frac{f}{60 }\cdot t-\frac{c}{r}\right)\\ -r\cdot\cos\left(\frac{f}{60r}\cdot t-\frac{c}{r}\right)\end{array} \right], \;\text{and}\;\frac{60\cdot c}{f}\leq t\leq\frac{60\cdot(c+2\pi r)}{f} \tag{18}\]  
It is evident that when the cutter reaches point \(H\) (see Figure 6), the whole cutter starts to leave the material, and no cutting edge engages with the material. The moment \(t_{c}\) when the cutter reaches point \(H\) is calculated as  
\[t_{c}=\left[\arcsin(c/(2(r+R)))+\pi+\frac{c}{r}\right].\frac{60r}{f} \tag{19}\]  
While at a moment \(t\) before \(t_{c}\), the cutter is formulated as  
\[\left\{ \begin{array}{c}x=x_{c}+R\cdot\cos\beta\\ y=y_{c}+R\cdot\sin\beta\end{array} \right.,\;\text{and}\;\beta\in[0,2\pi] \tag{20}\]  
and the cutter intersects the workpiece with points \(P_{2}\) and \(P_{1}\) (see Figure 6). \(P_{2}\) is the location where the cutting edges start to cut into the material, and \(P_{1}\) is the location where the cutting edges leave the material. For the technical details of computing the location of \(P_{2}\) and \(P_{1}\), the readers can refer to our previous work [PERSON] et al. [3]. The angle between \(P_{2}O_{c}\) and the \(y\)-axis is called the entrance angle \(\theta_{st}\), and the angle between \(P_{1}O_{c}\) and the \(y\)-axis is called the exit angle \(\theta_{cx}\). The immersion angle is calculated as  
\[\theta_{imm}=\theta_{ex}-\theta_{st} \tag{21}\]  
Figure 6: Geometrical relationship in trochoidal milling. To determine whether a specified cutting edge \(P_{e}\) is engaged into cut, its equation is formulated as  
\[\left\{ \begin{array}{l}x_{\rm e}=x_{c}+R\cdot\cos(n\cdot t\cdot\pi/30)\\ y_{\rm e}=y_{c}+R\cdot\sin(n\cdot t\cdot\pi/30)\end{array} \right.,0\leq t\leq \frac{60(c+2\pi r)}{f} \tag{22}\]  
where \(n\) is spindle speed (rev/min). The angle \(\theta_{en}\) that lies between \(P_{e}O_{c}\) and \(y\) axis (see Figure 6) is the instantaneous engagement angle. The following two conditions are applied to determine the engagement between the cutting edge and the material. **Condition 1. **\(0\leq t\leq\mathrm{t_{c}}\).