## 4 Results and Discussions
### Regression Models
#### 4.3.3 Surface Roughness Model Based on Cutting Parameters with Machining Vibration  
In this case, the machining vibration was incorporated with cutting parameters in regression analysis to derive the mathematical model for estimating surface roughness. Based on the regression coefficients in Table 4, \(R_{a}\) is expressed as  
\[\begin{array}{ll}R_{a}=&0.225+\ 9.17\times 10^{-5}\times n\ +0.295\times a_{p}-3.7 10^{-5}\times a_{p}\times n-2.6\ \times 10^{-4}\times a_{p}\times f\\ &-3.3\times 10^{-8}\times n\ \times\ f+4.75\times 10^{-8}\times a_{p}\times n\ \times f+0.372\times\ V_{b}\end{array} \tag{13}\]  
Figure 12: Comparisons of the predicted and measured values: (**a**) surface roughness and (**b**) machining vibration.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Coefficients** & **Standard Deviations** & \(p\)**-Value** \\ \hline Intercept & \(-5.180\times 10^{-3}\) & \(3.105\times 10^{-3}\) & \(9.655\times 10^{-2}\) \\ Spindle speed (\(n\)) & \(4.290\times 10^{-6}\) & \(9.650\times 10^{-7}\) & \(1.350\times 10^{-5}\) \\ Cutting depth (\(a_{p}\)) & \(2.603\times 10^{-3}\) & \(3.392\times 10^{-3}\) & \(4.435\times 10^{-1}\) \\ Depth \(\times\) speed (\(a_{p}\times n\)) & \(3.210\times 10^{-6}\) & \(8.520\times 10^{-7}\) & \(2.140\times 10^{-4}\) \\ Depth \(\times\) feed (\(a_{p}\times f\)) & \(-6.300\times 10^{-6}\) & \(2.410\times 10^{-6}\) & \(9.133\times 10^{-3}\) \\ Feed \(\times\) speed (\(f\times n\)) & \(-4.200\times 10^{-10}\) & \(3.230\times 10^{-10}\) & \(1.931\times 10^{-1}\) \\ Depth \(\times\) speed \(\times\) feed (\(a_{p}\times n\times f\)) & \(-2.600\times 10^{-10}\) & \(4.180\times 10^{-10}\) & \(5.371\times 10^{-1}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Regression parameters of the vibration prediction model.  
In this case, the determination coefficients (i.e., R-square and adjusted R-square) between the predicted values and measured values are around 0.67. It is found that the input variables associated with the cutting parameters have significant influence on surface roughness, with a small \(p\)-value \(<\) 0.05; however, the effect of machining vibration is insignificant. The average percentage of prediction accuracy for all testing data is \(\sim\)90.05% with an RMSE of 7.57%, and this shows that the surface roughness can also be predicted with good accuracy based on the cutting parameters and the measured vibration level in machining.  
In addition, the surface roughness was also modeled by power law model, as follows.  
Surface roughness model based on cutting parameters  
\[R_{a}=\ 0.04552(a_{p}^{\ 0.152955}\times n^{\ 0.636362}\times f^{\ -0.40132}) \tag{14}\]  
Surface roughness model based on cutting parameters with machining vibration  
\[R_{a}=\ 0.09804(a_{p}^{\ 0.136115}\times n^{\ 0.543685}\times f^{\ -0.36599}\times V_{B}^{\ 0.0 0606}) \tag{15}\]  
In Table 2, the R-square and adjusted R-square for the two power law models are \(\sim\)0.55 and the average percentage of prediction accuracy is \(\sim\)88% with an RMSE of 8.87%. Comparisons of the statistical parameters clearly indicate that polynomial models show better prediction performance in modeling the surface roughness than power law models.