# Analysis of the Influencing Factors of Grinding Uniformity under Three-Body Coupling Grinding Mode
## 4 Effects of Ball-Billet and Grinding Disc Slip on Rotating Angular Velocity
### Motion Characteristics of Balls under the Sliding State  
According to [PERSON] et al. (2013), the frictional force between the ball-billet and the grinding disc could be calculated in accordance with a relative sliding speed. Based on the above analysis, the relative sliding velocities at three contact points between the ball-billet and grinding disc are \(V_{A}\), \(V_{B}\), and \(V_{C}\). \[\left\{ \begin{array}{l}V_{A}=R_{A}\Omega_{b}-\omega_{b}r_{b}\\ V_{B}=R_{B}\Omega_{B}-R_{A}\Omega_{b}-r_{b}[\omega_{b}r\sin\alpha+\omega_{b 2}\cos\alpha]\\ V_{C}=R_{C}\Omega_{C}-R_{A}\Omega_{b}-r_{b}[\omega_{b}r\sin\beta-\omega_{b2} \cos\beta]\end{array} \right. \tag{13}\]  
Based on previous studies ([PERSON] et al., 2014; [PERSON] et al., 2015), the components of rotating speeds (perpendicular to the grinding disc surface) and rolling speeds (parallel to the grinding disc surface) at three contact points between the ball-billet and grinding disc are  
\[\left\{ \begin{array}{l}\omega_{s,A}=\omega_{b}\sin\theta\\ \omega_{s,B}=\omega_{b}\cos(\alpha+\theta)+\Omega_{B}\sin\alpha\\ \omega_{s,C}=\omega_{b}\cos(\alpha-\theta)+\Omega_{C}\sin\alpha\\ \omega_{r,A}=\omega_{b}\cos\theta\\ \omega_{r,B}=\omega_{b}\sin(\alpha+\theta)+\Omega_{B}\cos\alpha\\ \omega_{r,C}=\omega_{b}\sin(\alpha-\theta)+\Omega_{C}\cos\alpha\end{array} \right. \tag{14}\]  
The calculation formulas of frictional force (\(F_{i}\)) and torque (\(M_{i}\)) under sliding conditions are deduced in Reference ([PERSON] et al., 2015):  
\[\left\{ \begin{array}{l}F_{i}/(\mu_{i}W_{i})=\tanh[1.275(e_{i}/a_{i})(\tanh(0.47\chi_{i})^{n})^{1/n}]\\ [F_{i}/(\mu_{i}W_{i})]^{2}+\left\{M_{i}/[0.56(\tanh(0.3\chi_{i})^{n})^{1/n}] \mu_{i}u_{i}W_{i}\right\}^{1.5}=1.0\end{array} \right. \tag{15}\]  
where \(W_{i}\) denotes the processing loads, \(F_{i}\) is the frictional force, \(M_{i}\) is the friction torque, and \(n\) refers to a correction index. According to references ([PERSON] et al., 2015; [PERSON] et al., 2015), \(n=2\). Then,  
\[\left\{ \begin{array}{l}F_{i}=\mu_{i}W_{i}\tanh[1.275(e_{i}/a_{i})(\tanh(0.4 7\chi_{i})^{2})^{1/2}]\\ M_{i}=0.56\mu_{i}u_{i}W_{i}\left[1-\left(F_{i}\left/\mu_{i}W_{i}\right.\right)^ {2}\right]^{2/3}\left[\tanh(0.3\chi_{i})^{2}\right]^{1/2}\end{array} \right. \tag{16}\]  
Figure 9: Critical speeds of the inner and outer discs under different \(N_{t}\). where \(a\) refers to the Hertz contact radius on contact points between the ball-billet and grinding discs, and \(a_{i}=\left(0.75W_{i}r_{B}\right)\left/E_{i}^{\prime}\right.\)\(e\) refers to the eccentric moment of the spin axis and the contact point centers of the ball-billet in its contact regions with the grinding disc. Moreover, \(e_{i}=V_{i}/\omega_{s,i}\). Equation (14) was brought in, and Equation (16) could calculate the mechanical equilibrium equation when some relative slippage occurs between the ball-billet and the grinding disc. \[\left\{ \begin{array}{l}W_{B}\sin\alpha+W_{\rm C}\sin\alpha-G-W_{A}-F_{B}\cos \alpha+F_{\rm C}\cos a=0\\ R^{*}-F_{A}+F_{\rm C}\sin\alpha+F_{B}\sin\alpha+W_{B}\cos\alpha-W_{\rm C}\cos \alpha=0\\ (F_{A}+F_{B}+F_{\rm c})\cdot r_{b}-M^{*}=0\end{array} \right. \tag{17}\]  
In the above text, the frictional force and frictional torque on the contact points between the ball-billet and grinding disc were brought into Equation (17) to calculate the rotating angular velocity and revolving angular velocity of the ball-billet under this circumstance. The variation of the revolving speed of the ball-billet with the rotating speed of the outer disc under different loading conditions when the rotating speed of the inner disc is fixed at 10 rpm is shown in Figure 10. The ideal variations of the revolving speed with loads are the same curve, and the revolving speed increases with the increase in the rotating speed of the outer disc. Given frictional force, the critical rotating speeds of the grinding disc under different loads vary. The critical rotating speed is positively related to processing loads.