# Study on the Design and Cutting Performance of a Revolving Cycloid Milling Cutter
## 2 Design of the Revolving Cycloid Milling Cutter
### Establishment and Simulation of Geometric Model for the Revolving Cycloid Milling Cutter  
Figure 1a is the forming principle of a cycloid. When a cycle with the diameter of \(2a\) rolls along the \(X\)-axis in the XOZ coordinate system, the trajectory formed by any selected point on the circumference is called a cycloid. Cycloids are widely used in industry, and their width and height correspond to each other. Figure 1: Geometric model of a revolving cycloid milling cutter: (**a**) The forming principle of a cycloid; (**b**) contour surface and cutting edge curve of a revolving cycloid milling cutter. The coordinate of the circle center is set as point \((-\pi a,a)\), point A coinciding with the \(X\)-axis at \((-\pi a,0)\) is defined as the initial position of the circle. After the circle rolls clockwise along the \(X\)-axis in one round, the position where point A coincides again with the \(X\)-axis and is defined as the end position of the circle. On this basis, the trajectory of point A can be expressed as:  
\[\left\{ \begin{array}{l}x=a(\theta-\sin\theta-\pi),\\ z=a(1-\cos\theta),\end{array} \right.\quad(0\leq\theta\leq 2\pi), \tag{1}\]  
To be specific, \(\theta\) is the included angle between the radius of point A and the \(Z\)-axis; \(\theta\) is equal to 0 at the initial position, while \(\theta\) is equal to \(2\pi\) at the end position. In the XOZ coordinate system, the \(Z\)-axis is regarded as the tool axis. The tool radius is equal to \(\pi a\); the height of the end tooth is the diameter \(2a\) of the selected circle. As shown in Figure 1b, a space rectangular coordinate system \(\delta_{t}(X_{t},Y_{t},Z_{t})\) is established with the axis direction of the tool as the \(Z\)-axis. The revolving surface formed by the cycloidal profile revolving around \(Z\)-axis is the contour surface of the revolving cycloid milling cutter, and can be expressed as:  
\[S_{A}=\{a(\theta-\sin\theta-\pi)\cos\varphi,a(\theta-\sin\theta-\pi)\sin\varphi,a(1-\cos\theta)\}, \tag{2}\]  
Among them, \(\varphi\) and \(\theta\) are surface parameters. Then, in the coordinate system \(\delta_{t}\), the contour surface of the revolving cycloid milling cutter can be expressed as:  
\[\left\{ \begin{array}{l}S_{0}=\{a(\theta-\sin\theta-\pi)\cos\varphi_{1},a( \theta-\sin\theta-\pi)\sin\varphi_{1},a(1-\cos\theta)\},\\ S_{1}=\{\pi a\cos\varphi_{2},\pi a\sin\varphi_{2},u\}\;(u>0),\end{array} \right. \tag{3}\]  
where, \(S_{0}\) and \(S_{1}\) are the revolving cycloid contour surface and the cylinder contour surface, respectively. The orthogonal helix cutting edge is applied in the revolving cycloid milling cutter, while the intersection line between the orthogonal helix surface and the tool rotating surface is used as a cutting edge curve. The orthogonal helicoid can be expressed as:  
\[S_{B}=\left\{R_{S}\cos\varphi,R_{S}\sin\varphi,\frac{P_{Z}\varphi}{2\pi}\right\} (0\leq R_{S}\leq R), \tag{4}\]  
Regarding the orthogonal helix cutting edge, the helical pitch \(P_{Z}\) should satisfy the following equation:  
\[\frac{P_{z}}{2\pi R}=\frac{1}{\tan\beta}\, \tag{5}\]  
Combining Equations (3)-(5), upon calculation, \(\varphi\) can be expressed as:  
\[\varphi=\frac{\tan\beta_{0}(1-\cos\theta)}{\pi}, \tag{6}\]  
where \(\beta_{0}\) is the helix angle of the cutting edge of the cylindrical edge of the revolving cycloid milling cutter. Substituting Equation (6) into Equation (3), one can obtain a mathematical model for the orthogonal helix cutting edge curve of the revolving cycloid milling cutter, as shown below:  
\[r_{1}=\left\{ \begin{array}{l}\{a(\theta-\sin\theta-\pi)\cos\varphi_{1},a( \theta-\sin\theta-\pi)\sin\varphi_{1},a(1-\cos\theta)\},\\ \varphi_{1}=\frac{\tan\beta_{0}(1-\cos\theta)}{\pi},\end{array} \right. \tag{7}\]  
The mathematical model of the cutting edge curve of the cylindrical edge of the tool is:  
\[r_{2}=\left\{ \begin{array}{l}\{R\sin\varphi_{2},R\cos\varphi_{2},u\},\\ \varphi_{2}=\frac{\tan\beta_{0}}{R}u,\end{array} \right.\quad(u>0).