## 3 Polishing Pressure Calculation Model  
The contact between the workpiece and a single abrasive particle of the flap wheel can be regarded as the contact between the infinite plane and a single abrasive particle. The elastic contact deformation is shown in Figure 3. According to [PERSON] elastic contact theory [35], the elastic deformation can be calculated using Equation (1),  
\[\left\{ \begin{array}{l}\delta_{\text{mgt}}=\sqrt[3]{\frac{9F_{\text{mgt}}^{2} }{16R_{\text{m}}E_{\text{mgt}}^{2}}}\\ E_{\text{mgt}}=\frac{E_{\text{mgt}}[1-v_{\text{mt}}^{2}]+E_{\text{mgt}}\left(1 -v_{\text{gt}}^{2}\right)}{E_{\text{mgt}}\left(1-v_{\text{mt}}^{2}\right)+E_{ \text{mgt}}\left(1-v_{\text{gt}}^{2}\right)}\\ r_{\text{mgt}}=\sqrt[3]{\frac{3T_{\text{mgt}}^{2}R_{\text{m}}}{4E_{\text{mgt}}}} \\ F_{\text{mgt}}=\frac{4}{3}E_{\text{mgt}}\sqrt{R_{\text{m}}}\sqrt{\delta_{ \text{mgt}}^{3}}\end{array} \right. \tag{1}\]  
where \(F_{\text{mgt}}\) is the force between single abrasive particle and workpiece, \(R_{\text{m}}\) is the abrasive radius, \(E_{\text{mgt}}\) is the equivalent elastic modulus of abrasive particle and workpiece, \(\delta_{\text{mgt}}\) is the elastic deformation of abrasive particles and workpiece, \(E_{\text{mgt}}\) is the elastic modulus of abrasive particle, \(v_{\text{mt}}\) is the Poisson's ratio of abrasive particle, \(r_{\text{mgt}}\) is the radius of contact circle, and \(v_{\text{gt}}\) is the Poisson's ratio of workpiece. Through mechanical analysis of the abrasive particle, the elastic deformation of the abrasive particle and the substrate can be obtained, as shown in Equation (2),  
\[\left\{ \begin{array}{l}\delta_{\text{mbt}}=\sqrt[3]{\frac{9F_{\text{mgt}}^{2} }{16R_{\text{m}}E_{\text{mbt}}^{2}}}\\ E_{\text{mbt}}=\frac{E_{\text{mbt}}[1-v_{\text{mt}}^{2}]+E_{\text{mbt}}\left(1 -v_{\text{bt}}^{2}\right)}{E_{\text{mbt}}\left(1-v_{\text{mt}}^{2}\right)+E_{ \text{mbt}}\left(1-v_{\text{bt}}^{2}\right)}\end{array} \right. \tag{2}\] \begin{table}
\begin{tabular}{l l l} \hline \hline
**Axis Positioning Accuracy of X/Y/Z (\(\mu\))** & **Stroke of X/Y/Z-Axis (mm)** & **Dimension of Polished Workpiece (mm)** \\ \hline
7.046/22.704/9.468 & 1000/1600/1000 & \(\Phi\)20–\(\Phi\)100 \\ \hline Maximum speed of A/C-axis motor (r/min) & Machine tool structure & Dimensions of polishing machines (mm) \\ \hline
16.6/11.1 & Gantry and five-axis linkage & 3300 \(\times\) 3150 \(\times\) 1800 \\ \hline Maximum speed of spindle (r/min) & Rotating Angle of A/C-axis & Maximum speed of X/Y/Z-axis motor (r/min) \\ \hline
24,000 & \(-\)5”\(\sim\)95”\(/\pm\)360” & 3000/3000/3000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of CNC polishing machine tool. Figure 2: Schematic diagram of the machine tool control process. where \(E_{\text{mbt}}\) is the equivalent elastic modulus of flap wheel substrate and abrasive particle, \(E_{\text{bt}}\) is the elastic modulus of flap wheel substrate, \(v_{\text{bt}}\) is the Poisson's ratio of flap wheel substrate, and \(\delta_{\text{mbt}}\) is the elastic deformation of the abrasive particle and substrate. During the elastic-plastic deformation stage between the abrasive particle and the workpiece, the polishing pressure and deformation between the abrasive and the workpiece can be expressed as follows:  
\[\left\{ \begin{array}{l}\delta_{\text{o}}=\delta_{\text{mgmax}}=\frac{\pi^{2} H_{\text{bw}}^{2}R_{\text{m}}}{16E_{\text{mgt}}^{2}}\\ r_{\text{mgt}}=\left(R_{\text{m}}\delta_{\text{mgt}}\right)^{1/2}\\ A_{\text{mgt}}=\pi R_{\text{m}}\delta_{\text{mgt}}\\ F_{\text{pmgt}}=\frac{F_{\text{mgt}}}{A_{\text{mgt}}}=\sqrt{\delta_{\text{mgt}}/ R_{\text{m}}}\\ F_{\text{pmg}}=H_{\text{bw}}/3\\ \delta_{\text{mg}}=\delta_{\text{mgt}}\end{array} \right. \tag{3}\]  
where \(\delta_{\text{o}}\) is the critical deformation, \(F_{\text{pmgt}}\) is the average polishing pressure between the abrasive particle and workpiece, \(A_{\text{mgt}}\) is the contact area between the abrasive particle and workpiece, and \(H_{\text{bw}}\) is the Brinell hardness of the workpiece material.