# Influence of Rake Angles of Multi-Position Tool on Cutting Characteristics
## 2 Experimental
### Mathematical Model Derivation
\begin{array}{c}\frac{\sin 2 \theta}{2}\\ \frac{1-\cos 2\theta}{2}\end{array} \right]dh+\left[ \begin{array}{ccc}1&k_{ rp}&0\\ -k_{rp}&1&0\\ 0&0&-k_{ap}\end{array} \right]k_{tp}\left[ \begin{array}{c}\cos\theta\\ \sin\theta\\ 1\end{array} \right]dh \tag{15}\]  
Figure 5: Diagram of shear force and ploughing force. \[=q_{s}p_{s}dh+q_{p}p_{p}dh \tag{16}\]  
The \(p_{s}(\theta),\ p_{p}(\theta)\) is limited by the initial cutting angle \(\theta_{1}\) and the final cutting angle \(\theta_{2}\) during machining. Therefore, the basic cutting function must be multiplied with a cutting window function W (\(\theta,h\)). During cutting, \(\theta\) will be changed by the height of milling \(h\) and the partial milling force mode when considering the cutting window is as follows:  
\[w(\theta)=\left\{ \begin{array}{ll}1&when...\theta\in[\theta_{1},\theta_{2}] \\ 0&otherwise\end{array} \right. \tag{17}\]  
The single blade cutting mode is obtained from the integration of the partial milling force function and the cutting height, which can be shown as follows:  
\[f_{j}(\varphi)=\int df=\int\limits_{0}^{da}\left\langle[q_{s}][p_{s}]+\left[q _{p}\right][p_{p}\right]\right\rangle w(\theta)dh \tag{18}\]  
\[ts(\varphi)=\sum\limits_{k=-\infty}^{\infty}\delta(\phi-(k-1)\beta_{p}) \tag{19}\]  
\[f(\phi)=ts(\phi)*f_{j}(\phi) \tag{20}\]  
The * stands for the operational symbol of convolution. During single blade cutting, it can be the convolution of the partial milling force function. General tools can do multi-blade milling, and the tools continuously rotate to cause the periodic change of the milling force. The periodicity is \(\beta_{p}=2\pi/Z\), therefore the force that is measured during processing is a periodic cutting force. The single blade mode is extended to be multi-blade periodic cutting force, by calculating the convolution of the single blade cutting force and the blade sequence function \(ts\) (\(\phi\)). The total cutting force mode is as follows:  
The average milling force is the important reference data during milling. The functions of the Fourier transform can be obtained when calculating the average value of the periodic function:  
\[F(\omega)=TS(\omega)F_{j}(\omega) \tag{21}\]  
\[=N\sum\limits_{k=-\infty}^{\infty}\delta(\omega-Nk)F_{j}(\omega) \tag{22}\]  
The Fourier transform equation for the \(TS\) (\(\omega\)) blade sequence function \(ts\) (\(\phi\)) is:  
\[TS(\omega)=N\sum\limits_{k=-\infty}^{\infty}\delta(\omega-NK) \tag{23}\]  
We make the Fourier transform based on the above equation. The total milling force function can be shown as the form of Fourier transform series:  
\[f(\phi)=\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}F(\omega)e^{j\omega\phi} d\omega=\sum\limits_{k=-\infty}^{\infty}A[Nk]e^{jNk\phi} \tag{24}\]  
\[A[Nk]=\frac{N}{2\pi}Q_{s}(Nk)P_{s}(Nk)+\frac{N}{2\pi}Q_{p}(Nk)P_{p}(Nk) \tag{25}\]The specific cutting force constant is the important parameter to calculate the cutting force. This can be used to calculate the six unknown specific cutting force constants \(k_{ts}\), \(k_{ts}\), \(k_{as}\), \(k_{tp}\), \(k_{rp}\), and \(k_{qp}\) by adopting slot milling and semi-slot milling. The equation is as follows [9; 10; 11]:  
\[\left[ \begin{array}{l}k_{ts}t_{x}\\ k_{rs}k_{ts}t_{x}\\ k_{as}k_{ts}t_{x}\\ k_{tp}\\ k_{rp}k_{tp}\\ k_{ap}k_{tp}\end{array} \right]=\frac{2\pi}{NR_{0}}T^{-1}\left[ \begin{array}{ l}A_{xls}[0]\\ A_{yls}[0]\\ A_{zls}[0]\\ A_{xs}[0]\\ A_{ys}[0]\\ A_{zs}[0]\end{array} \right] \tag{26}\]  
Therefore, the integrated convolution cutting force mode of milling cutter's slot milling can be set up. The convolution cutting force mode includes tool geometry (radius, helical angle), cutting conditions (axial and radial cutting depth), material parameters (specific cutting force constant), and other important parameters in the milling process. These can be applied to imitate cutting under various milling conditions [4; 12; 13].