# 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
\tag{8}\]When a two-tooth revolving cycloid milling cutter is taken for an example, the contour surface of the revolving cycloid milling cutter and the orthogonal helix cutting edge curve model are simulated numerically on the basis of the Matlab software. The simulation result of the orthogonal helix cutting edge curve of the revolving cycloid milling cutter is shown in Figure 2. The radius of the revolving cycloid milling cutter is 5 mm, while curve 1, curve 2 and curve 3 are the orthogonal helix cutting edge curves when the helix angle \(\beta\) is \(50^{\circ}\), \(40^{\circ}\) and \(30^{\circ}\), respectively. Comparison of Cutting States of the Revolving Cycloid Milling Cutter and the Ball-end Milling Cutter  
(1) Comparison of equivalent helix angles at the tool-workpiece cutting zone  
Equivalent cutting helix angles of the revolving cycloid milling cutter and the ball-end milling cutter with the same radius are compared in Figure 3. In Figure 3, \(\psi\) presents the machining inclination angle, while point B\({}_{1}\) and point B\({}_{2}\) are the tool-workpiece contact points of the revolving cycloid milling cutter and the ball-end milling cutter, respectively. Based on the orthogonal helix edge curve, its helix angle can be generally expressed as:  
\[\beta=\arccos(\frac{ds\cdot\partial s}{|ds|\cdot|\partial s|}), \tag{9}\]  
Figure 3: Comparison of equivalent helix angle for ball-end milling cutter and revolving cycloid milling cutter: (**a**) Revolving cycloid milling cutter; (**b**) ball-end milling cutter. Figure 2: Cutting edge curve simulation of revolving cycloid milling cutter for different helical angles: (**a**) The cutting edge curve simulation for a cylindrical edge; (**b**) the cutting edge curve simulation for end edge. Then, the helix angle \(\beta_{1}\) of the cutting edge curve of the revolving cycloid milling cutter at point B\({}_{1}\) can be expressed as:  
\[\beta_{1}=\arccos\sqrt{\frac{2\pi^{2}(1-\cos\theta)}{2\pi^{2}(1-\cos\theta)+( \theta-\sin\theta+\pi)^{2}\tan^{2}\beta_{0}\sin^{2}\theta}}, \tag{10}\]  
of which, the relationship of \(\theta\) and the machining inclination angle \(\psi\) satisfies the following equation:  
\[\theta=\arccos\left(\frac{\tan^{2}\psi-1}{\tan^{2}\psi+1}\right), \tag{11}\]  
The helix angle \(\beta_{2}\) of the cutting edge curve of the ball-end milling cutter at point B\({}_{2}\) can be expressed as:  
\[\beta_{2}=\arccos\sqrt{\frac{R^{4}}{R^{4}+(R^{2}-R^{2}\cos^{2}\psi)^{2}\tan^{2 }\beta_{0}}}, \tag{12}\]  
With the machining inclination angle \(\psi\) as a parameter, \(\beta_{1}\) and \(\beta_{2}\) are simulated numerically. The result is shown in Figure 4.  
According to the simulation and cutting experiment results that have been completed, the larger helix angle can improve the cutting process. As can be seen from the figure, \(\beta_{1}\) is greater than \(\beta_{2}\) in the process of the machining inclination angle changing from \(0^{\circ}\) to \(90^{\circ}\). Specifically, when the machining inclination angle is greater than \(5^{\circ}\) and less than \(60^{\circ}\), an apparent advantage can be found in \(\beta_{1}\), indicating that the actual cutting helix angle of revolving cycloid milling cutter is larger than that of the ball-end milling cutter, the cutting edge is longer, and the cutting process will be more stable. (2) Comparison of scallop heights of the workpiece surface  
The scallop heights of the revolving cycloid milling cutter and the ball-end milling cutter with the same radius are compared in Figure 5. It should be noted that \(h_{1}\) and \(h_{2}\) are machining scallop heights of the revolving cycloid milling cutter and the ball-end milling cutter. \(r\) is the distance between the tool center point O\({}_{1}\) of the revolving cycloid milling cutter and the highest point of the scallop height \(h_{1}\). Figure 3.0 Comparison of the equivalent cutting helix angle under different machining inclination angle.