# 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
### Critical Conditions of Relative Slippage Betweenthe Ball-Billet and the Grinding Discs  
The only effects of relative slippage between the ball-billet and grinding discs on the motion of the ball-billet have been discussed in the above section. In practical processing, complicated factors, including the friction coefficient between ball-billet and grinding discs, loads of ball-billet, processing precision of rollaway of the grinding disc, distribution of grinding liquid, viscosity of grinding liquid, and so on, may cause such relative motion. In this study, stress relation during the motion of ball-billet was analyzed to investigate the causes and critical conditions for relative motion between the ball-billet and grinding discs. The frictional forces between the ball-billet and three contact positions of the grinding discs were \(F_{A}\), \(F_{B}\), and \(F_{C}\):  
\[\left\{ \begin{array}{l}F_{A}=W_{A}\cdot\mu_{A}\\ F_{B}=W_{B}\cdot\mu_{B}\\ F_{C}=W_{C}\cdot\mu_{C}\end{array} \right. \tag{10}\]  
To analyze the critical conditions for the sliding of the ball-billet, it is hypothesized that frictional force can ensure a slip-free motion of the ball-billet, and the frictional force between the ball-billet and the grinding disc reaches the maximum. According to the relation equation of gyroscopic moment during the motion of the ball-billet,  
\[\left[M^{*}\right]\ =\ I\Omega_{b}\omega_{b}\text{cos}\ \theta\ =\ \left(F_{A}\ +\ F_{B}\ +\ F_{C}\right)\ r_{b}\ =\ \left(W_{A}\ \mu_{A}\ +\ W_{B}\ \mu_{B}\ +\ W_{C}\ \mu_{C}\right)\ r_{b}, \tag{11}\]  
Where [\(M^{*}\)] is the critical gyroscopic moment when the frictional force can guarantee the slip-free motion of the ball-billet. Given the fixed processing loads, the frictional force produced by the grinding disc and ball-billet has some limits. This decides that the gyroscopic moment on the ball-billet has some limit. In other words, the rotating angular velocity and revolving speed of the ball-billet cannot be increased infinitely during processing. However, the rotating speed (it is not increased infinitely, but it can achieve a high rotating speed for the ball-billet) cannot be increased continuously during grinding under the effect of an external driving device. In summary, some critical value of rotating speed exists between the ball-billet and grinding discs under some processing conditions. Under ideal conditions, no relative slippage occurs between the ball-billet and grinding disc when the rotating speed of the grinding disc is lower than the critical value. Under this circumstance, the rotating angular velocity and revolving speed of the ball-billet may increase with the increase in the rotating speed of the grinding disc. When the rotating speed of the grinding disc exceeds such a critical value, the rotating angular velocity and revolving speed of the ball-billet may not increase with the increase in rotating speed, thus having relative slippage. This critical speed is related to the frictional coefficient and processing loads. The higher frictional coefficient and processing loads lead to higher critical speed. For the convenience of analysis, the processing structure is hypothesized to be an ideal mechanism without errors. Hence,  
\[ \begin{array}{l}I\Omega_{b}w_{b}\cos\theta=(W_{A}\mu_{A}+W_{B}\mu_{B}+W_{C} \mu_{C})r_{b}\\ \Rightarrow\frac{m_{b}\left(R_{C}\Omega_{C}\right)^{2}\left(\frac{R_{B}}{R_{C} }N_{l}+1\right)^{2}}{10R_{A}\left(1+\sin\alpha\right)^{2}}=(W_{A}\mu_{A}+W_{ B}\mu_{B}+W_{C}\mu_{C})r_{b}\end{array}  \tag{12}\]  
Here, the \(\Omega\)10 mm ball was chosen as the analysis object (\(m_{b}\) = 10 g). Parameters in Table 2 were brought in to analyze under loading conditions of 1 N/ball and 2 N/ball. In this section, the frictional coefficients between the ball and three grinding discs are the same on the three contacting points. With references to the frictional coefficient among various materials [45] and previous research results [46], the frictional coefficient between the ball-billet and the grinding disc was set to 0.1.