## 4 Results and Discussions
### Regression Models  
#### 4.3.1 Surface Roughness Model Based on Cutting Parameters  
The surface roughness (\(R_{a}\)) was determined by the individual cutting parameter, depth of cutting (\(a_{p}\), mm), feed rate (_f_, mm/min), and spindle speed (\(n\), rpm) in addition to interactive parameters, such as cutting depth and spindle speed. Based on the regression coefficients listed in Table 1, the mathematical model is given by  
\[\begin{array}{ccccc}R_{a}=&0.2235+&9.33\times 10^{-5}\times n&+0.2963\times a _{p}&-3.6\times 10^{-5}\times n\times&a_{p}&-2.6&\times 10^{-4}\times a_{p} \times f\\ &&-3.48\times 10^{-8}\times n&\times&f+4.74\times 10^{-8}\times a_{p}\times n& \times&f\end{array} \tag{11}\]  
In Table 2, the R-square and adjusted R-square are around 0.67, implying that the predicted values are close to the measurements, as shown in Figure 12. Further, the \(p\)-value of each input variable and their interactions against surface roughness are far less than 0.05, indicating all machining parameters have apparent influences on the roughness of the machined surface. The average percentage of prediction accuracy for all testing data is ~90.3% with an RMSE of 7.57%, indicating that the roughness prediction model is well fitted with the cutting parameters defined in wide ranges.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameters** & **Coefficients** & **Standard Deviations** & **\(p\)-Value** \\ \hline Intercept & \(2.235\times 10^{-1}\) & \(4.935\times 10^{-2}\) & \(9.459\times 10^{-6}\) \\ Spindle speed (\(n\)) & \(9.330\times 10^{-5}\) & \(1.530\times 10^{-5}\) & \(4.754\times 10^{-9}\) \\ Cutting depth (\(a_{p}\)) & \(2.964\times 10^{-1}\) & \(5.391\times 10^{-2}\) & \(1.010\times 10^{-7}\) \\ Depth \(\times\) speed (\(a_{p}\times n\)) & \(-3.600\times 10^{-5}\) & \(1.350\times 10^{-2}\) & \(7.991\times 10^{-3}\) \\ Depth \(\times\) feed (\(a_{p}\times f\)) & \(-2.600\times 10^{-4}\) & \(3.830\times 10^{-5}\) & \(8.078\times 10^{-11}\) \\ Feed \(\times\) speed (\(f\times n\)) & \(-3.400\times 10^{-8}\) & \(5.130\times 10^{-9}\) & \(3.956\times 10^{-10}\) \\ Depth \(\times\) speed \(\times\) feed (\(a_{p}\times n\times f\)) & \(4.740\times 10^{-8}\) & \(6.650\times 10^{-9}\) & \(1.312\times 10^{-11}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Regression parameters for surface roughness prediction model.  
Figure 11: Correlation between surface roughness and machining vibration level.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Prediction Model** & **R** & **R Square** & **Adjusted R Square** & **RMSE** & **MAPE** \\ \hline (a) Polynomial roughness model (\(R_{a}\)) & 0.82665 & 0.68336 & 0.67521 & 7.57\% & 9.97\% \\ (a) Power-law roughness model (\(R_{a}\)) & 0.747994 & 0.559496 & 0.55389 & 8.75\% & 11.8\% \\ (a) Polynomial Maching vibration (\(V_{b}\)) & 0.89478 & 0.80063 & 0.79549 & 0.47\% & 13.9\% \\ (b) Polynomial roughness model (\(R_{a}\)) & 0.82676 & 0.68353 & 0.67398 & 7.57\% & 9.95\% \\ (b) Power-law roughness model (\(R_{a}\)) & 0.749639 & 0.56204 & 0.554858 & 8.76\% & 11.8\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparisons of statistical parameters of prediction models.