# Research on Double-Arc Cutting Tool Design and Cutting Performance
## 2 Design of DACT
### Cutting Edge Design and Numerical Simulation of DACT  
#### 2.3.1 Cutting Edge with Constant Helix Angle of DACT  
The general mathematical model of the cutting edge with a constant helix angle is shown as follows:  
\[\left\{\begin{array}{l}\{F(\theta)\cos\varphi,F(\theta)\sin\varphi,G(\theta )\}\\ \varphi=\tan\beta\int_{\theta_{0}}^{\theta}\frac{\sqrt{F^{\prime 2}(\theta)+G^{ \prime 2}(\theta)}}{F(\theta)}\mathrm{d}\theta\end{array}\right. \tag{19}\]  
where \(\beta\) is the helix angle of cutting edge.  
According to the mathematical models of the bottom spherical surface in Equation (3), the following formulas can be obtained as follows:  
\[F_{e}(\theta)=3r\cos\theta,G_{e}(\theta)=3r\sin\theta-\sqrt{3}r \tag{20}\]  
\[F^{\prime}_{e}(\theta)=-3r\sin\theta,G^{\prime}_{e}(\theta)=3r\cos\theta \tag{21}\]  
Based on the principles of differential geometry, by integrating Equations (19)-(21), parameter \(\varphi_{\beta e}\) can be calculated as follows:  
\[\varphi_{\beta e}=\tan\beta\bigg{[}\ln\tan\bigg{(}\frac{\pi}{4}+\frac{\theta }{2}\bigg{)}\bigg{]}+\varphi_{\beta e0} \tag{22}\]  
According to the mathematical models of the ring torus surface in Equation (2), the following formulas can be obtained as follows:  
\[F_{r}(\theta)=r\cos\theta+r,G_{r}(\theta)=r\sin\theta \tag{23}\]  
\[F^{\prime}_{r}(\theta)=-r\sin\theta,G^{\prime}_{r}(\theta)=r\cos\theta \tag{24}\]  
Based on the principles of differential geometry, by integrating Equations (19), (23), and (24), parameter \(\varphi_{\beta r}\) can be calculated as follows:  
\[\varphi_{\beta r}=\tan\beta\tan\frac{\theta}{2}+\varphi_{\beta r0} \tag{25}\]  
According to the mathematical models of the cylindrical surface in Equation (4), the following formulas can be obtained as follows:  
\[F_{c}(u)=2r,G_{c}(u)=u \tag{26}\]  
\[F^{\prime}_{c}(u)=0,G^{\prime}_{c}(u)=1 \tag{27}\]  
Based on the principles of differential geometry, by integrating Equations (19), (26), and (27), parameter \(\varphi_{\beta c}\) can be calculated as follows:  
\[\varphi_{\beta c}=\frac{u\tan\beta}{2r}+\varphi_{\beta c0} \tag{28}\]  
Considering the smooth transition of the cutting edge with a constant helix angle on the bottom spherical surface, the ring torus surface, and the cylindrical surface:  
Set when \(\theta=0\), \(\varphi_{\beta r}=\varphi_{\beta c}=0\), and then \(\varphi_{\beta r0}=\varphi_{\beta c0}=0\).  
Set when \(\theta=\frac{\pi}{3}\), \(\varphi_{\beta e}=\varphi_{\beta r}\) and then \(\varphi_{\beta e0}=\tan\beta\Big{[}\frac{\sqrt{3}}{3}-\ln(2+\sqrt{3})\Big{]}\).  
According to the principle of differential geometry, the solution of parameters \(\varphi_{\beta r}\), \(\varphi_{\beta e}\), and \(\varphi_{\beta c}\) can be introduced into Equations (2)-(4), respectively. Then, the cutting edge with a constant helix angle of the DACT can be obtained as follows:  
\[\left\{\begin{array}{l}R_{\beta r}=\left\{\left(r\cos\theta+r\right)\cos \varphi_{\beta r},\left(r\cos\theta+r\right)\sin\varphi_{\beta r},r\sin\theta \right\}\\ \varphi_{\beta r}=\tan\beta\left[\ln\tan\left(\frac{\pi}{4}+\frac{\theta}{2} \right)\right]+\tan\beta\left[\frac{\sqrt{3}}{3}-\ln(2+\sqrt{3})\right]\\ \left\{\begin{array}{l}R_{\beta e}=\left\{3r\cos\theta\cos\varphi_{\beta e},3 r\cos\theta\sin\varphi_{\beta e},3r\sin\theta-\sqrt{3}r\right\}\\ \varphi_{\beta e}=\tan\beta\tan\frac{\theta}{2}\\ \end{array}\right.\\ \left\{\begin{array}{l}R_{\beta c}=\left\{2r\cos\varphi_{\beta c},2r\sin \varphi_{\beta c},u\right\}\\ \varphi_{\beta c}=\frac{u\tan\beta}{2r}\end{array}\right.\left(u\in\left[-D_{ C},0\right]\right)\end{array}\right. \tag{29}\]