# 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 Orbital Geometric Accuracy  
During processing, given that the ball-billet is put in the orbit of the lower grinding disc, the geometric accuracy of the orbit directly influences the processing of the ball-billet. Given the existence of processing errors, the practical orbit is not an ideal circular ring. For the convenience of analysis, its hypothesized that several uplifts are found on the grinding disc, which influences the geometric accuracy of the grinding orbit. The orbits with two and three uplifts are shown in Figure 18.  
Given that the orbit has uplifts, the ball-billet is lifted up by these uplifts when it arrives at the positions; thus, ball-billets at these positions are higher than the others. Hence, ball-billets at these positions undertake more loads than the balls in other parts. To comprehend the effects of the out-of-roundness of the grinding disc on grinding track points, MATLAB simulation was conducted in accordance with the simulation conditions (Table 8). Simulation results are shown in Figure 19. The STD of spherical grinding uniformity is shown in Table 9. \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Simulation Parameters**} \\ \hline Radius of grinding disc \(R_{A}\) (mm) & 300 \\ \hline Radius of outer disc under the ideal state \(R_{outer}\) (mm) & 303 \\ \hline Ball diameter \(r_{b}\) (mm) & 15 \\ \hline V-shaped groove angles \(\alpha\), \(\beta\) (rad) & \(\pi/4\) \\ \hline Rotating speeds of grinding disc & \(\Omega_{C}\) (rpm) & 20 \\ \cline{2-3}  & \(\Omega_{B}\) (rpm) & Variations (Refer to Figures 2–4) \\ \hline Difference between the long and short axes of the ellipse & 0.1, 0.2, 0.3, 0.4, 0.5, 1 \\ outer disc d (mm) & & \\ \hline Simulation time (s) & 600 \\ \hline Time interval \(\Delta t\) (s) & 0.06 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Simulation conditions of influences of out-of-roundness of the outer disc on spherical grinding tracks. Figure 18: Non-perfect circle diagram of the lower grinding disc: (a) two uplifts on the outer disc; (b) three uplifts on the outer disc. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**Error Parameters**} & \(d=0\) **mm** & \(d=0.1\) **mm** & \(d=0.2\) **mm** & \(d=0.3\) **mm** & \(d=0.4\) **mm** \\ \hline
**STD** & 1.44 & 2.4742 & 1.6863 & 3.5959 & 1.9236 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: STD of spherical grinding uniformity under different geometric precisions of orbits. Table 9 shows that the effects of the geometric precision of orbits on the STD of spherical grinding uniformity show no laws. However, the worsening degree of sphericity influences spherical grinding uniformity more than disc tilt and face run-out, and it has adverse effects on grinding uniformity. In practical processing, the orbital preloading of the ball-billet was applied to improve the geometric precision of orbits and eliminateheadverse effects of roundness errors as much as possible. The orbit distribution of spherical surfaces when the outer disc is not round is shown in Figure 19.  
In the above-investigated mechanism error of dual-revolving grinding equipment, grinding uniformity was influenced greatly when the outer disc was not round, and it failed to grind into balls when the error parameter \(d\) was higher than 0.3 mm. The rest of the errors might also influence spherical grinding uniformity to some extent, and such influences were relatively small. When some mechanism errors occurred, the grinding uniformity of the ball-billet was even slightly better than that under the ideal three-body coupling grinding mode (e.g., when \(\Delta{}h_{2}=0.1\) mm or \(\Delta{}h_{j}=0.1\) mm; or when \(\gamma=0.5^{\circ}\) and \(\gamma=1^{\circ}\)). This result might be because, given a small mechanism error and no adverse effects on the ball-billet processing, the rotating sliding of the ball-billet in relation to the rolling of the grinding disc was intensified, which accelerated the variation of \(\theta\).