# Study on the Design and Cutting Performance of a Revolving Cycloid Milling Cutter
## 3 Fabrications and Detection of the Revolving Cycloid Milling Cutter  
### Establishment of the Grinding Model for the Rake Face of the Revolving Cycloid Milling Cutter  
In order to establish a mathematical model for the grinding trajectory of the rake face of the revolving cycloid milling cutter, the tangential vector and the normal vector along the end surface of the cutting edge are considered as basic components of the cutting edge. On this basis, the bottom boundary curve of the end tooth rake face can be obtained with the rake angle of the tool and the depth parameter of the chip hold groove of the end tooth, so as to derive the position and angle of grinding wheels when the rake face of the end tooth is grinding. Figure 5: Comparison of scallop height of two kinds of milling cutters. Figure 6: Scallop height under different path intervals. As shown in Figure 7, the tangent vector at any point \(S_{j}\) on the cutting edge is \(\overrightarrow{Q}_{j}\); the normal vector is \(\overrightarrow{F}_{j}\); the cross product of \(\overrightarrow{Q}_{j}\) and \(\overrightarrow{F}_{j}\) is the vector \(\overrightarrow{B}_{j}\). Vectors \(\overrightarrow{F}_{j}\), \(\overrightarrow{Q}_{j}\), and \(\overrightarrow{B}_{j}\) can be expressed with the cutting edge curve \(r\):  
\[\left\{ \begin{array}{l}\overrightarrow{Q}_{j}=\frac{i}{||r||}=\Big{[}Q_{xj},Q_{yj},Q_{zj}\Big{]},\\ \overrightarrow{F}_{j}=\frac{r}{||r||}=\Big{[}F_{xj},F_{yj},F_{zj}\Big{]},\\ \overrightarrow{B}_{j}=Q_{j}\times F_{j}=\Big{[}B_{xj},B_{yj},B_{zj}\Big{]}, \end{array} \right. \tag{15}\]  
To be specific, the normal forward angle \(\gamma\) is measured through the normal surface \(P_{n}\). The rake face can be deemed as an extending surface of \(S_{j}V_{j}\) on the normal surface. The grinding track curve can be determined by determining the functional relationship between parameter \(z\) and radial depth \(h(z)\). Parameter \(z\) is evaluated from 0 to 2a of the \(Z\)-axis in Figure 1a, and the relationship between parameter \(z\) and parameter \(\theta\) is shown in Equation (1). By taking the curve in Figure 8 for example, the functional relationship \(h(z)\) is established, which is a parabola. Its definition domain and value domain are as follows: \(z\in(-A,0);h(z)\in(0,h_{max})\). The vertex of the curve is \((-A,0)\). The curve passes point \((0,h_{max})\). Then \(h(z)\) can be expressed as:  
\[h(z)=h_{max}\Big{(}1+\frac{z}{A}\Big{)}^{2}, \tag{16}\]  
The bottom boundary curve equation is determined by the depth function \(h(z)\) of the rake face, while the position and the angle of the grinding wheel during grinding are determined by the bottom boundary curve equation. The position vector \(r_{ovj}\) between point \(V_{j}\) on the bottom boundary curve and point \(S_{j}\) on the cutting edge is:  
\[r_{ovj}=r_{osj}+S_{j}V_{j}=\Big{[}x_{vj},y_{vj},z_{vj}\Big{]}, \tag{17}\]  
where, \(S_{j}V_{j}=-h(u)cos\gamma\overrightarrow{F}_{j}-h(u)sin\gamma\overrightarrow {B}_{j}\). The bottom boundary curve is parameterized:  
\[r_{ovj}=r_{ov}(\varphi)=[x_{v}(\varphi),y_{v}(\varphi),z_{v}(\varphi)], \tag{18}\]  
Figure 7: Mathematical model of the rake face for revolving cycloid milling cutter. Then, the normal vector \(\overrightarrow{F}_{vj}\) of a point \(V_{j}\) on the bottom boundary curve of the rake face of the end tooth is:  
\[\overrightarrow{F}_{vj}=\frac{dr_{ovj}^{2}/d\varphi^{2}}{\|dr_{ovj}^{2}/d \varphi^{2}\|}, \tag{19}\]  
Thereby, the parametric equation of the rake face of the end tooth is:  
\[S_{R}(\varphi,m)=(1-m)r_{osj}+mr_{ovj}=r_{osj}-mh(u)\bigg{(}\cos\gamma \overrightarrow{F}_{j}+\sin\gamma\overrightarrow{B}_{j}\bigg{)}, \tag{20}\]  
where the coefficient \(m\) is between 0 and 1. In order to avoid interference between the grinding wheel and the bottom boundary curve, when the vector \(\overrightarrow{F}_{vj}\) enters the grinding wheel side plane at point \(V_{j}\), the projection on the principal normal vector of the bottom boundary curve must pass through the center of the grinding wheel side.