# Analysis of the Influencing Factors of Grinding Uniformity under Three-Body Coupling Grinding Mode
## 5 Effects of Mechanism Error on the Rotating Angular Velocity of the Ball
### Effects of the Tilt of the Lower Grinding Disc  
Given that manufacturing and assembly errors and uneven stress during processing may cause some tilt of the grinding disc (nonperpendicularity to the axial direction of the principal axis), this may also change the motion state of the ball-billet and change the distribution of the grinding tracks on the spherical surface. In this section, the tilt of the inner disc was investigated, as shown in Figure 15. The situation of the outer disc tilt is similar to the situation of the inner disc tilt, and it is not discussed in this section. The geometric relation of the influences of the inner disc tilt (\(\gamma\)) on the contact points of the ball-billet is shown in Figure 16. In Figure 16, R is a point on the rotating axis of the grinding disc. D-E-R refers to the position of the inner disc under ideal conditions, and B-D-E refers to the V-shaped rollaway under ideal conditions. D'-H-R refers to the position of the inclined inner disc, and B'-D'-H is the tracks after the inclination of the inner disc. The ball center points before and after the inclination of the inner disc are O and O', respectively. When the inner disc has no tilt, the groove angles are \(\alpha=\beta=45^{\circ}\). EM is parallel to NH. Here, if \(\overset{-}{AE}=a\) and \(\overset{-}{ER}=b\), then \(a+b=R_{A}\), and \(\overset{-}{ED}=\sqrt{2}a\). According to Figure 16, it can obtain  
\[\overline{EH}=2\overline{ER}\sin\frac{\angle ERH}{2}=2b\sin\frac{\gamma}{2}\]  
\[\overline{EF}=\overline{EH}\sin\angle EHF=2b\sin\frac{\gamma}{2}\sin(\frac{ \pi}{4}-\frac{\gamma}{2})\]  
\[\overline{DM}=\overline{DE}\sin\angle DEM=\sqrt{2}a\sin\gamma\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Degree of Run-Out** & **STD** \\ \hline \(\Delta h_{z}=0\) mm (or \(\Delta h_{j}=0\) mm) & 1.44 \\ \(\Delta h_{z}=0.1\) mm (or \(\Delta h_{j}=0.1\) mm) & 1.5601 \\ \(\Delta h_{z}=1\) mm (or \(\Delta h_{j}=1\) mm) & 1.6883 \\ \(\Delta h_{z}=2\) mm (or \(\Delta h_{j}=2\) mm) & 1.8311 \\ \(\Delta h_{z}=-0.1\) mm (or \(\Delta h_{j}=0\) mm) & 1.7023 \\ \(\Delta h_{z}=-1\) mm (or \(\Delta h_{j}=0\) mm) & 1.9768 \\ \(\Delta h_{z}=-2\) mm (or \(\Delta h_{j}=0\) mm) & 1.9864 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Standard deviation (STD) of spherical grinding uniformity under different face run-out of the inner disc. Figure 15: Diagram of inner disc tilt. \[\overline{DD^{\prime}}=\frac{\overline{DN}}{\sin\angle DD^{\prime}N}=\frac{ \overline{DM}+\overline{EF}}{\sin\angle DD^{\prime}N}=\frac{\sqrt{2}a\sin\gamma+2b \sin\frac{\gamma}{2}\sin(\frac{\pi}{4}-\frac{\gamma}{2})}{\sin(\frac{\pi}{2}- \gamma)}\]  
\[\overline{B^{\prime}D^{\prime}}=\overline{B^{\prime}O^{\prime}}\cot\angle B^{ \prime}D^{\prime}O^{\prime}=r_{b}\cot(\frac{\pi}{4}-\frac{\gamma}{2})\]  
\[\overline{OO^{\prime}}=\overline{BB^{\prime}}=\overline{DD^{\prime}}-\overline {B^{\prime}D^{\prime}}+\overline{BD}=\frac{\sqrt{2}a\sin\gamma+2b\sin\frac{ \gamma}{2}\sin(\frac{\pi}{4}-\frac{\gamma}{2})}{\sin(\frac{\pi}{2}-\gamma)}- r_{b}\cot(\frac{\pi}{4}-\frac{\gamma}{2})+r_{b}\]  
It can deduce geometric parameters when the inner disc inclines:  
\[\left\{ \begin{array}{l}{R_{A}}^{\prime}={R_{A}}-\frac{\overline{OO^{\prime} }}{\sqrt{2}}={R_{A}}-\frac{\sqrt{2}a\sin\gamma+2b\sin\frac{\gamma}{2}\sin( \frac{\pi}{4}-\frac{\gamma}{2})}{\sqrt{2}\cos\gamma}+\frac{r_{b}\cot(\frac{ \pi}{4}-\frac{\gamma}{2})}{\sqrt{2}}-\frac{r_{b}}{\sqrt{2}}\\ {R_{B}}^{\prime}={R_{A}}^{\prime}+r_{b}\cos\alpha={R_{A}}-\frac{\sqrt{2}a\sin \gamma+2b\sin\frac{\gamma}{2}\sin(\frac{\pi}{4}-\frac{\gamma}{2})}{\sqrt{2} \cos\gamma}+\frac{r_{b}\cot(\frac{\pi}{4}-\frac{\gamma}{2})}{\sqrt{2}}\\ {R_{C}}^{\prime}={R_{A}}-\frac{\overline{DD^{\prime}}}{\overline{D^{\prime}} }-\overline{B^{\prime}D^{\prime}}\sin\angle D^{\prime}C^{\prime}P={R_{A}}- \frac{\sqrt{2}a\sin\gamma+2b\sin\frac{\gamma}{2}\sin(\frac{\pi}{4}-\frac{ \gamma}{2})}{\sqrt{2}\cos\gamma}-r_{b}\cot(\frac{\pi}{4}-\frac{\gamma}{2})\sin (\frac{\pi}{4}-\gamma)\end{array} \right.