## 2 Research Method
### Analysis of the Roller Trajectory  
The novel non-circular spinning method is based on hypocycloid theory and relies on the relative motion between the workpiece and the rollers to process the workpiece. Analysis of the roller trajectory is key to determining whether the simulation achieves accurate results.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Mechanical Properties** & **Value** \\ \hline Yield Strength & 62.05 MPa \\ Ultimate Strength & 151.68 MPa \\ Young’s Modulus & 69,000 MPa \\ Poisson’s Ratio & 0.3 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Mechanical properties of the 6061 aluminum alloy.  
Figure 1: Finite element model of the novel non-circular spinning method.  
In mathematics, the hypocycloid can be defined as follows: when a moving circle is inscribed on a fixed circle and rolls along the fixed circle without sliding, the trajectory of any point on the moving circle is a hypocycloid [14; 15]. If the roller is regarded as a mass point and set as point P, then the trajectory of the roller can be derived by setting the radius of the fixed circle as R, the radius of the moving circle as r, P as a point fixed on the moving circle and the distance from P to the center of the moving circle as e. As shown in Figure 2, the center of the fixed circle is O, the center of the moving circle is C, the unit vector of the X axis is \(\overset{\rightarrow}{i}\) and the unit vector of the Y axis is \(\overset{\rightarrow}{j}\). Initially, point P coincides with the X axis. Upon rotation, the contact point of the fixed circle and the moving circle is A, the center of the circle C becomes C\({}^{\prime}\) and point P moves to \(P^{\prime}\). Then, C\({}^{\prime}\) must be at the radius OA of the fixed circle. The following expressions can be defined:  
\[\overrightarrow{OP^{\prime}}=\overrightarrow{OC^{\prime}}+\overrightarrow{ C^{\prime}P^{\prime}} \tag{1}\]  
If  
\[\theta\ =\ \measuredangle\left(\overset{\rightarrow}{i},\overrightarrow{OC^{ \prime}}\right),\ \beta\ =\ \measuredangle\left(\overrightarrow{C^{\prime}P^{\prime}}, \overrightarrow{C^{\prime}A}\right) \tag{2}\]  
Then  
\[\overrightarrow{OC^{\prime}}\ =\ \overset{\rightarrow}{i}\left(R-r\right) \cos\theta\ +\ \overset{\rightarrow}{j}\left(R-r\right)\sin\theta, \tag{3}\]  
\[R\theta\ =\ \widehat{AB}\ =\ \widehat{AB}\ :=\ r\beta \tag{4}\]  
Thus,  
\[\beta=\ \frac{R}{r}\theta \tag{5}\]  
The directionality angle between \(\overrightarrow{C^{\prime}P^{\prime}}\) and the X axis is  
\[\measuredangle\left(\overset{\rightarrow}{i},\overrightarrow{C^{\prime}P^{ \prime}}\right)=\ \ \theta-\beta=\ \frac{r-R}{r}\theta \tag{6}\]  
\[\left|\overrightarrow{C^{\prime}P^{\prime}}\right|=\ \mathrm{e}, \tag{7}\]  
Therefore,  
\[\overrightarrow{C^{\prime}P^{\prime}}\ =\ \overset{\rightarrow}{i}e\cos \frac{r-R}{r}\theta+\overset{\rightarrow}{j}e\sin\frac{r-R}{r}\theta=\ \overset{ \rightarrow}{i}e\cos\frac{R-r}{r}\theta-\overset{\rightarrow}{j}e\sin \frac{R-r}{r}\theta \tag{8}\]  
\[\overrightarrow{OP^{\prime}}\ =\ \overrightarrow{OC^{\prime}}+\overrightarrow{C^{ \prime}P^{\prime}}=\ \left[\left(R-r\right)\cos\theta\ +\ e\cos\frac{R-r}{r}\theta\right] \overset{\rightarrow}{i}\ +\ \left[\left(R-r\right)\sin\theta-\ e\sin\frac{R-r}{r}\theta \right]\overset{\rightarrow}{j} \tag{9}\]  
Equation (9) is the hypocycloid equation of the rollers. By setting the coordinates of reference point P as (x, y), Equation (9) can be used to obtain the parameter equations of reference point P (which is also the parameter equation of the hypocycloid equation), expressed as  
\[\left\{\begin{array}{l}x=\left(R-r\right)\cos\theta+\ e\cos\frac{R-r}{r} \theta\\ y=\left(R-r\right)\sin\theta-\ e\sin\frac{R-r}{r}\theta\end{array}\left(- \infty<\theta<+\infty\right)\right.\]  
When \(R=2r\), the equation becomes  
\[\left\{\begin{array}{l}x=\left(r+e\right)\cos\theta\\ y=\left(r-e\right)\sin\theta\end{array}\left(-\infty<\theta<+\infty\right)\right.\]  
Additionally, the parameter equation of reference point P becomes the elliptic equation.