# Influence of Rake Angles of Multi-Position Tool on Cutting Characteristics
## 2 Experimental
### Mathematical Model Derivation  
In general, cylindrical coordinate system \(\rho\)-\(r\)-\(h\) changes with the rotation of the tool during milling, as shown in Figure 4. The angle \(\rho\) is Zero when the height (\(h\)) of one point at the edge of the milling cutter is zero (\(h=0\)), and when the relationship of \(\rho\) and h is as below [7; 8; 9]:  
\[h(\rho)=\frac{R}{\tan\alpha}[\rho-(k-1)\rho_{a}] \tag{3}\]  
\[(k-1)\rho_{p}\leq\rho\leq(k-1)\rho_{p}+\rho_{a},k=1,2,3,\ldots,N \tag{4}\]  
\(\rho_{p}\) is the included angle between the blades, while \(\rho_{a}\) shows the relationship between the tool and its cutting height \(da\). However, the main effect variables of milling are the axial cutting depth \(da\), the radius of milling cutter \(R\), the rake angle or helical angle \(\alpha\) of the tool, and other parameters. \[\rho_{p}=\frac{2\pi}{N} \tag{5}\]  
Figure 3: Impact of rake angle change on the rake angle and clearance angle. Figure 2: Mechanism to change the rake angle. When the rectangular coordinate system X-Y-Z is overlapped with the cylindrical milling cutter coordinate, the heading direction of the work-piece is the same as the X-axis during the test. In addition, the \(\theta\) angle is the cutting rotation angle between the tool and the work-piece during cutting, as shown in Figure 4.  
X-Y-Z coordinates are those for the work-piece during cutting when the milling cutter is fed along the Z axis, and when the rotation angle of the tool is \(\theta\). The relationship between the milling cutter \(p\) and the tool rotation angle \(\phi\) can be shown on the work-piece coordinates as below. During milling, the radial cutting-in \(dr\) and the offset \(C\) affect the relationship between the initial feed-rate angle \(\theta_{1}\) and the final cutting angle \(\theta_{2}\). The cutting thickness \(t_{c}\) is a function of \(\theta\), and the relation is as below:  
\[\theta_{1}=\cos^{-1}(\frac{R-C}{R}) \tag{7}\]  
\[\theta_{2}=\cos^{-1}(1-\frac{C+dr}{R}) \tag{8}\]  
\[t_{c}=t_{x}\sin(\theta) \tag{9}\]  
During cutting, there is a contact point on the fillet of the cutting edge, as shown in \[h(\rho)=\frac{R}{\tan\alpha}[\rho-(k-1)\rho_{a}] \tag{3}\]0 The material above this point will be pushed to produce chips, and the material under this point will be pushed into the bottom of the tool. Then, the work-piece surface will be formed after the flank leaves. The position of the contact point will change with the geometrical shape and material of the tool, the material of the work-piece, and the cutting parameters [3]. The cutting force \(F\) is divided into ploughing force \(F_{P}\) and shear force \(F_{S}\). \[h(\rho)=\frac{R}{\tan\alpha}[\rho-(k-1)\rho_{a}] \tag{3}\]1 Milling cutter coordinate system. The impact of shear force and ploughing force on milling force can be divided into tangential, radial, and axial partial milling force, and the expression is shown in the below Equations (17)-(19). The tangential, radial, and axial partial milling forces are shown in the following equation:  
\[h(\rho)=\frac{R}{\tan\alpha}[\rho-(k-1)\rho_{a}] \tag{3}\]2  
\[h(\rho)=\frac{R}{\tan\alpha}[\rho-(k-1)\rho_{a}] \tag{3}\]3  
\[h(\rho)=\frac{R}{\tan\alpha}[\rho-(k-1)\rho_{a}] \tag{3}\]4  
The partial cutting by sorting is shown as below:  
\[\left[ \begin{array}{c}df_{t}\\ df_{r}\\ df_{a}\end{array} \right]=\left[ \begin{array}{c}1\\ k_{rs}\\ k_{as}\end{array} \right]k_{ts}t_{x}\sin\theta dh+\left[ \begin{array}{c}1\\ k_{rp}\\ k_{ap}\end{array} \right]k_{tp}dh \tag{13}\]  
The process of converting the tangential force, radial force, and axial force into the X-Y-Z coordinated system is as follows:  
\[\left[ \begin{array}{c}df_{x}\\ df_{y}\\ df_{z}\end{array} \right]=\left[ \begin{array}{ccc}\cos\theta&\sin\theta&0\\ \sin\theta&-\cos\theta&0\\ 0&0&-1\end{array} \right]\left[ \begin{array}{c}df_{t}\\ df_{r}\\ df_{a}\end{array} \right] \tag{14}\]  
\[=\left[ \begin{array}{ccc}1&k_{rs}&0\\ -k_{rs}&1&0\\ 0&0&k_{as}\end{array} \right]k_{ts}t_{x}\left[