# 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  
The geometric contact relationship between the cutting tool and the workpiece surface when machining the workpiece is shown in Figure 4. The tool coordinate system is \(O\)-\(XY\)Z; in the local region of the current cutting contact point \(O\)', the design surface is approximated as the tangent plane \(X\)'-\(O\)'-\(Y\)' at the current cutting contact point \(O\)'. Meanwhile, the workpiece surface \(X_{w}\)-\(O_{w}\)-\(Y_{w}\) is approximated as the parallel plane of \(X\)'-\(O\)'-\(Y\)', and the inclination angle of the tangent plane \(X\)'-\(O\)'-\(Y\)' is denoted as \(\alpha\). According to the geometric structure of the DACT, when \(\alpha\in[0,\pi/6]\), the cutting contact point \(O\)' is located on the bottom arc \(L_{\sigma}\) of the DACT, and then \(O\)' can be expressed as \(\{3\pi\)sin\(\alpha\),\(0\),\(3\pi\)cos\(\alpha-\sqrt{3}r\}\); when \(\alpha\in[\pi/6,\pi/2]\), the cutting contact point \(O\)' is located on the ring arc \(L_{\tau}\) of the DACT, and then \(O\)' can be expressed as \(\{r\)sin\(\alpha\)+, \(0\), cos\(\alpha\}\). As shown in Figure 5, the inclination angle \(\alpha\), cutting depth \(a_{p}\), and tool geometry parameter \(r\) jointly determine which part of the cutting tool actually participates in the cutting process. Therefore, the cutting modes of the DACT can be divided into six cases. Figure 4: Geometric contact relationship. Figure 3: Comparison of effective cutting speeds. When \(\alpha\in[0,\pi/6\ -\arccos(1-a_{p}/3r)]\), only the bottom arc \(L_{e}\) is involved in cutting, as shown in area 1 in Figure 6.  
When \(\alpha\in[\pi/6\ -\arccos(1-a_{p}/3r),\pi/6+\arccos(1-a_{p}/r)]\) (\(a_{p}\in[0,r-\sqrt{3}r/2]\)) \(\cup\ \alpha\in[\pi/6\ -\arccos(1-a_{p}/3r),\pi/2-\arccos(1-a_{p}/r)]\) (\(a_{p}\in[r-\sqrt{3}r/2,r]\)), the bottom arc \(L_{e}\) and ring arc \(L_{r}\) are involved in cutting, as shown in area 2 in Figure 6.  
Figure 5: Schematic diagrams of cutting modes of DACT: (**a**) \(O^{\prime}\) is located on \(L_{e}\), and only \(L_{e}\) is involved in cutting; (**b**) \(O^{\prime}\) is located on \(L_{e}\), and \(L_{e}\) and \(L_{e}\) participate in cutting; (**c**) \(O^{\prime}\) is located on \(L_{e}\), and \(L_{e}\), \(L_{r}\), and the peripheral cutting edge participate in cutting simultaneously; (**d**) \(O^{\prime}\) is located on \(L_{r}\), and only \(L_{r}\) is involved in cutting; (**e**) \(O^{\prime}\) is located on \(L_{r}\), and \(L_{r}\) and the peripheral cutting edge participate in cutting; (**f**) \(O^{\prime}\) is located on \(L_{r}\), and \(L_{e}\), \(L_{r}\), and the peripheral cutting edge participate in cutting simultaneously. Figure 6: Relationship between cutting mode, inclination angle \(\alpha\), cutting depth \(a_{p}\), and tool parameter \(r\). When \(\alpha\in[\pi/6+\arccos(1-a_{p}/r),\pi/2-\arccos(1-a_{p}/r)]\) (\(a_{p}\in[0,r-\sqrt{3}r/2]\)), only the ring arc \(L_{r}\) is involved in cutting, as shown in area 3 in Figure 6.  
When \(\alpha\in[\pi/2-\arccos(1-a_{p}/r),\pi/2]\) (\(a_{p}\in[0,r-\sqrt{3}r/2]\)) \(\cup\alpha\in[\pi/6+\arccos(1-a_{p}/r),\pi/2]\) (\(a_{p}\in[r-\sqrt{3}r/2,r]\)), the ring arc \(L_{r}\) and peripheral cutting edge of the DACT are involved in cutting simultaneously, as shown in area 4 in Figure 6.  
When \(\alpha\in[\pi/2-\arccos(1-a_{p}/r),\pi/6+\arccos(1-a_{p}/r)]\) (\(a_{p}\in[r-\sqrt{3}r/2,r]\)), the bottom arc \(L_{e}\), ring arc \(L_{r}\), and peripheral cutting edge participate in cutting, as shown in area 5 in Figure 6.  
The section curve of the cutting tool in the plane \(X_{w}\)-\(O_{w}\)-\(Y_{w}\) is the intersection curve of the cutting tool and workpiece surface. The intersection curve is a closed curve, which determines the effective cutting width in the cutting process. Translate the origin \(O\) (0,0,0) of the tool coordinate system \(O\)-\(XYZ\) to the cutting contact point \(O^{\prime}\), and rotate the \(OX\) and \(OZ\) axes with angle \(\alpha\) counterclockwise around the \(OY\) axis simultaneously.