# Research on Double-Arc Cutting Tool Design and Cutting Performance
## 2 Design of DACT
### Cutting Geometry Model of DACT
#### 2.2.2 Cutting Residual Height of DACT
Through the above two transformations, the local coordinate system \(O^{\prime}\)-\(X^{\prime}Y^{\prime}Z^{\prime}\) can be obtained. When the cutting contact point \(O^{\prime}\) is {\(C_{1}\),0,\(C_{2}\)}, thus, when \(\alpha\in[0,\pi/6]\), \(C_{1}=3r\text{sin}\alpha\), \(C_{2}\) = \(3r\text{cos}\alpha-\sqrt{3}r\); when \(\alpha\in[\pi/6,\pi/2]\), \(C_{1}=r\text{sin}\alpha+\pi\), \(C_{2}=r\text{cos}\alpha\). The cutting tool rotation surface in coordinate system \(O^{\prime}\)-\(XY^{\prime}Z^{\prime}\) can be expressed as follows:  
\[\left[ \begin{array}{c}x^{\prime}\\ y^{\prime}\\ z^{\prime}\end{array} \right]=\left[ \begin{array}{ccc}\cos\alpha&0&-\sin \alpha\\ 0&1&0\\ \sin\alpha&0&\cos\alpha\end{array} \right]\left[ \begin{array}{c}x-C_{1}\\ y\\ z-C_{2}\end{array} \right] \tag{8}\]  
Make \(z^{\prime}=a_{p}\), and then the general expression of the intersection curves of the cutting tool and plane \(X_{w}\)-\(O_{w}\)-\(Y_{w}\) can be obtained as follows:  
\[\left[\pm\sqrt{F^{2}(\theta)-\left(\left(C_{2}-G(\theta)\right)\cot\alpha+a_{ p}\csc\alpha+C_{1}\right)^{2}}\right] \tag{9}\]  
where \(\left\{ \begin{array}{c}F_{e}(\theta)=3r\cos\theta_{e}\\ G_{e}(\theta)=3r\sin\theta_{e}-\sqrt{3}r\end{array} \right.\) \(\left(\theta_{e}\in\left[\frac{\pi}{3},\frac{\pi}{2}\right]\right)\); \(\left\{ \begin{array}{c}F_{r}(\theta)=r\cos\theta_{r}+r\\ G_{r}(\theta)=r\sin\theta_{r}\end{array} \right.\)\(\left(\theta_{r}\in\left[0,\frac{\pi}{3}\right]\right)\). As shown in Figure 7, the green curve is the intersection curve of the bottom spherical surface of the DACT and the workpiece surface; the red curve is the intersection curve of the ring torus of the DACT and the workpiece surface; and the blue curve is the intersection curve of the cylindrical part of the DACT and the workpiece surface. The included angle \(\gamma\) between the feed direction of the cutting tool and the \(X_{w}\) axis, the geometric parameter \(r\), the inclination angle \(\alpha\), and the cutting depth \(a_{p}\) are the main factors that affect the effective cutting width of the DACT, as shown in Figure 7.  
When only the bottom spherical part is involved in the cutting, the effective cutting width of the DACT is:  
\[W_{DACT}=2\sqrt{6\pi a_{p}-{a_{p}}^{2}} \tag{10}\]  
When cutting with a BEM, the intersection curve between the cutting tool and the workpiece is approximately circular. Thus, the effective cutting width of the BEM can be obtained as follows:  
\[W_{BEM}=2\sqrt{2 Ra_{p}-{a_{p}}^{2}} \tag{11}\]  
Above all, the effective cutting width of the DACT is larger than that of the BEM by selecting an appropriate feed direction angle \(\gamma\) so that more materials can be removed. When the feed direction angle \(\gamma\) is 0, the cutting width of the DACT is the largest. When \(\gamma=0\), the normal section at the current cutting contact point \(O^{\prime}\) is \(Y^{\prime}\)-\(O^{\prime}\)-\(Z^{\prime}\). Then, the general expression of the intersection curves of the cutting tool and plane \(Y^{\prime}\)-\(O^{\prime}\)-\(Z^{\prime}\) can be obtained as follows:  
\[\left[\pm\sqrt{F^{2}(\theta)-\left(\left(G(\theta)-C_{2}\right)\tan\alpha+C_ {1}\right)^{2}}\right] \tag{12}\]  
where \(\left\{ \begin{array}{l}F_{\epsilon}(\theta)=3r\cos\theta_{\epsilon}\\ G_{\epsilon}(\theta)=3r\sin\theta_{\epsilon}-\sqrt{3}r\end{array} \right.\)\(\left(\theta_{\epsilon}\in\left[\frac{\pi}{3},\frac{\pi}{2}\right]\right)\); \(\left\{ \begin{array}{l}F_{r}(\theta)=r\cos\theta_{r}+r\\ G_{r}(\theta)=r\sin\theta_{r}\end{array} \right.\)\(\left(\theta_{r}\in\left[0,\frac{\pi}{3}\right]\right)\). The intersection curves of the DACT in the normal section \(Y^{\prime}\)-\(O^{\prime}\)-\(Z^{\prime}\) are shown in Figure 8.  
Therefore, when the tool path interval is \(L\), the intersection curves of the cutting tools in \(Y^{\prime}\)-\(O^{\prime}\)-\(Z^{\prime}\) determines the cutting scallop height, as shown in Figure 9.  
Figure 8: Intersection curves of DACT and normal section \(Y^{\prime}\)-\(O^{\prime}\)-\(Z^{\prime}\).