## 5 Conclusions  
This study presents the prediction models for estimating the surface roughness of machined parts under various cutting conditions from low-speed to high-speed stable machining. The prediction modes were developed using multiple regression analysis and an ANN modeling approach. According to the analyzed results, the following conclusions were drawn. 1. The response surface graphs indicate that the surface roughness and machining vibration level are affected to different extents by the combined effect of the spindle speed and cutting depth. Overall, roughness scatters at lower values under a lower cutting depth and spindle speed. Further, vibration amplitude increases with an increase in spindle speed and cutting depth; in particular, vibration apparently increases with an increase in the spindle speed from 3000 to 8000 rpm, with a correlation coefficient (R = 0.92-0.96). Overall within the whole testing conditions, the influence of feed rate on machining vibration and roughness cannot be clearly demonstrated. However, current results reveal that for machining with given cutting depth, the lower level of cutting speed and lower feed rate generated lower roughness than the higher speed and fast feed rate. Such interaction effect of spindle speed and feed rate on machining vibration is more significant than on surface roughness. Figure 16: Monitoring system for machining vibration and prediction of surface roughness. Figure 15: Contour surface milling sample for verification of predictive model. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Points**} & **Spiable** & **Cutting** & \multirow{2}{*}{**Feed Rate**} & **Machining** & **Measured** & **Predicted** & \multirow{2}{*}{**Error**} \\  & **Speed** & **Depth** & **(mm/min)** & **(g)** & **(mm)** & **(mm)** & **(mm)** & \\ \hline
1 & 6000 & 1.0 & 1800 & 0.0239 & 0.524 & 0.540 & 3.00\% \\
2 & 6000 & 1.0 & 1800 & 0.0256 & 0.563 & 0.580 & 2.25\% \\
3 & 6000 & 1.0 & 1800 & 0.0234 & 0.547 & 0.535 & 2.27\% \\
4 & 6000 & 1.0 & 1800 & 0.0280 & 0.586 & 0.545 & 7.02\% \\
5 & 6000 & 1.0 & 1800 & 0.0261 & 0.571 & 0.551 & 3.47\% \\
6 & 6000 & 1.0 & 1800 & 0.0198 & 0.526 & 0.485 & 7.88\% \\ \hline \multicolumn{5}{c}{Average} & 0.02446 & 0.553 & 0.534 & 4.31\% \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Predictions of surface roughness in end milling process by ANN. Both the surface roughness and machining vibration can be modeled with nonlinear quadratic forms based on cutting parameters and their interactions via multiple regression analysis. * In the ANN modeling of surface roughness, predictive models (MLP 3-12-1 and MLP 3-13-1) based on three cutting parameters--spindle speed, cutting depth, and feed rate--have high correlation coefficients (R \(\approx\) 0.9285 to 0.9693) between the predicted and measured values within training, validation, and testing data. The average percentage prediction accuracy of all datasets is about 94.5%. For neural models (MLP 4-15-1 and MLP 4-22-1) with additional variables (machining vibration), a higher correlation coefficient (R) is found in the range of 0.9611 to 0.9785. The average percentage prediction accuracy of all datasets is about 96.2%. * In the final validation of the predictive models through another machining test, the ANN models were found to demonstrate more accurate predictions of surface roughness. Based on the RMSE and MAPE values, the neural model MLP 4-15-1 with four input variables (three cutting parameters and machining vibration) shows a higher average prediction accuracy of 93.14% than that of the model MLP 3-13-1 with three cutting parameters. This result clearly verifies that the prediction accuracy can be increased if the spindle vibration in machining was also fed in the prediction algorithm. The predictive model can be the base to develop an on-line surface roughness recognition system, in which cutting parameters and spindle tool vibration assessed in machining can be fed into system to estimate the surface roughness in process.