# Analysis of the Influencing Factors of Grinding Uniformity under Three-Body Coupling Grinding Mode
## 3 Balling Principle of the Three-Body Coupling Grinding Mode  
On the basis of the motion equation of a ball-billet under the three-body coupling grinding mode [21; 22], the basic grinding principle of precision balls was analyzed in this section. This was used as the theoretical basis to analyze the influencing factors of the three-body coupling grinding mode. Moreover, the motion parameters of a ball-billet under grinding mode with positive control over the rotating angular velocity, traditional V-shaped groove grinding mode, and dual-disc grinding mode were compared. Geometric Motions of a Ball-Billet under the Three-Body Coupling Grinding Mode  
[PERSON] and [PERSON] analyzed the geometric motion of a ball-billet under the grinding mode with positive control over the rotating angular velocity [21; 22]. As shown in Figure 5, under the grinding mode with positive control over the rotating angular velocity, the processed ball-billet contacts with the grinding disc at points A, B, and C in the V-shaped groove. Specifically, Discs A, B, and C rotate independently, and their rotating speeds are \(\Omega_{A},\Omega_{B}\), and \(\Omega_{C}\), respectively. The ball-billet makes revolution and rotation under the effects of three points. Relevant parameters are shown in Table 1.  
Figure 4: Principle of grinding mode with positive control over rotating angular velocity. We suppose that the ball in this movement is a \"real ball\". In other words, it is a perfect circle at any cross-section and rolls simply. In accordance with this hypothesis, the equilibrium equation set of speed at Points \(A\), \(B\), and \(C\) was obtained:  
\[\left\{ \begin{array}{l}R_{A}\Omega_{A}=R_{A}\Omega_{b}-\omega_{b}r_{b}\cos \theta\\ R_{B}\Omega_{B}=R_{A}\Omega_{b}+\omega_{b}r_{b}\sin(\alpha+\theta)\\ R_{C}\Omega_{C}=R_{A}\Omega_{b}+\omega_{b}r_{b}\sin(\beta-\theta)\end{array} \right. \tag{1}\]  
In practical grinding discs, \(\alpha=\beta\); therefore,  
\[\left\{ \begin{array}{l}\tan\theta=\frac{1+\sin\alpha}{\cos\alpha}-\frac{R_{ B}\Omega_{B}-R_{C}\Omega_{C}}{R_{A}R_{B}+R_{C}\Omega_{C}-2R_{A}\Omega_{A}}\\ \Omega_{b}=\frac{R_{B}\Omega_{B}+R_{C}\Omega_{C}-2R_{A}\Omega_{A}\sin\alpha}{2 R_{A}(1+\sin\alpha)}\\ \omega_{b}=\frac{R_{B}\Omega_{B}+R_{C}\Omega_{C}-2R_{A}\Omega_{A}}{2r_{b}(1+ \sin\alpha)\cos\theta}\end{array} \right. \tag{2}\]  
When \(\Omega_{A}=0\) and \(\Omega_{B}=\Omega_{C}=\Omega\) in Equation (1), it is transformed into the velocity equilibrium equation, i.e., Equation (3), under the V-shaped groove grinding mode. Similarly, \(\theta\), \(\Omega_{b}\), and \(\omega_{b}\) can be calculated, as shown in Equation (4). \[\left\{ \begin{array}{l}0=R_{A}\Omega_{b}-\omega_{b}r_{b}\cos\theta\\ R_{B}\Omega=R_{A}\Omega_{b}+\omega_{b}r_{b}\sin(\alpha+\theta)\\ R_{C}\Omega=R_{A}\Omega_{b}+\omega_{b}r_{b}\sin(\alpha-\theta)\end{array} \right. \tag{3}\]  
\[\left\{ \begin{array}{l}\theta=\tan^{-1}(\frac{r_{b}}{R_{C}}\sin\alpha)\\ \Omega_{b}=\frac{R_{A}\Omega}{r_{b}\cos(1+\sin\alpha)}\\ \omega_{b}=\frac{\Omega}{1+\sin\alpha}\end{array} \right. \tag{4}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Name** & **Signs** \\ \hline Contact points between upper and lower grinding discs and the & \(A,B,C\) \\ precision ball & \\ Distance from three contact points to the rotating axis of the lower & \(R_{A},R_{B},R_{C}\) \\ grinding disc & \\ Rotating speeds of grinding disc & \(\Omega_{A},\Omega_{B},\Omega_{C}\) \\ \hline Ball-billet radius & \(r_{b}\) \\ Revolutionary angular velocity of the ball & \(\Omega_{b}\) \\ Rotating angular velocity of the ball & \(\omega_{b}\) \\ V-shaped groove angles & a and \(\beta\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter signs of the grinding mode with positive control over the rotating angular velocity. Figure 5: Geometric motion under the grinding mode with positive control over the rotating angular velocity.