# 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
### Relative Slip of the Ball-Billet with Inner and Outer Discs  
For the convenience of analysis, the ball-billet is hypothesized to make only relative slippage with the inner disc but has no slip with other grinding discs. In other words, \(V_{A}=V_{B}=0\) and \(V_{C}\
eq 0\), where \(V_{A}\), \(V_{B}\), and \(V_{C}\) are there lattice velocities of the grinding discs to the ball.  
Here, \(S_{C}=\frac{V_{C}}{R_{C}\Omega_{C}}\) is the sliding-friction ratio between the ball-billet and the inner disc. \(N_{t}=\frac{\Omega_{B}}{\Omega_{C}}\) is the rotating speed ratio between inner and outer discs. As shown in Equation (2) [21; 45],  
\[\Omega_{b}=\frac{R_{B}\Omega_{B}+R_{C}\Omega_{C}(1-S_{C})}{2R_{A}(1+\sin\alpha )}\Rightarrow\frac{\Omega_{b}}{\Omega_{C}}=\frac{R_{B}N_{t}+R_{C}(1-S_{C})}{2R _{A}(1+\sin\alpha)} \tag{6}\]  
\[\tan\theta=\frac{1}{R_{B}N_{t}+R_{C}(1-S_{C})}\left(\frac{1+\sin\alpha}{\cosThe grinding structural parameters in Table 2 were brought into Equations (6) and (7) for calculation. To disclose the influences of slip on the revolving speed of the ball-billet, a rotating speed relation map was plotted on the basis of the above calculation results by using \(\Omega_{b}/\Omega_{C}\) and the rotating angular velocity as the \(y\)-axis and the rotating speed ratio of inner and outer discs (\(N_{l}\)) as the \(x\)-axis. Calculation results are shown in Figures 6 and 7. On this basis, the relations between \(\Omega_{b}/\Omega_{C}\) and \(N_{t}\), as well as that between \(\theta\) and \(N_{t}\) under different sliding-friction ratios (\(S_{c}\)) between the ball-billet and inner disc, could be concluded. In Figure 6, \(N_{t}\) ranges within \([-10,10]\). When \(S_{c}\) = 0, it is the calculation result under no slip between the ball-billet and the grinding disc.  
As shown in Figure 7, the influencing degrees of \(S_{c}\) on the motion state of the ball were different when a slip occurred between the ball-billet and the inner disc. With the increase in \(S_{c}\), the influencing degrees of the rotating speed of the inner disc on the rotating angular velocity and the revolving speed of the ball-billet are decreased. When \(S_{c}\) = 1, the relative movement velocity between the ball-billet and the inner disc was equal to the linear velocity on the grinding disc. In other words, the rotation of the inner disc may not cause ball movement. The ball-billet and the grinding discs are considered in the state of complete slip. The relation curve in Figure 6 passes through the origin (0, 0), and \(\Omega_{b}/\Omega_{C}\)= 0. At this moment, the rotation of the outer disc directly determines the motion state of the ball-billet. In Figure 7, the rotating angular velocity may not change with \(N_{t}\) but is a constant. With the increase in \(S_{c}\), \(N_{t}\) can decrease relatively when \(\theta\) can change comprehensively within \([-90^{\circ},90^{\circ}]\). However, the variation degree of \(\theta\) is intensified.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Structural** & **Ball Diameter** & **Rollaway Radius \(R_{A}\)** & **V-Shaped Groove** \\
**Parameters** & **db (mm)** & **(mm)** & **Angle \(\alpha\) = \(\beta\)** \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters of the grinding structure.  
Figure 6: Relation between \(\Omega_{b}/\Omega_{C}\) and \(N_{t}\) (\(S_{c}\) changes between 0 and 1).  
As shown in Figure 7, when \(S_{c}=0\), it means that the ball blank and the grinding disc do not slide; when \(S_{c}=1\), it means that the ball blank and the grinding disc are in a state of complete skid, and the curve in the middle represents the intermediate transition state.