# A Study on Improving the Machining Performance of Scrolls
## 5 Experimental Results and Discussion
### Experiments on the Optimization of Milling Parameters
#### 5.3.2 ANOVA and Verification of the Profile Errors  
Table 12 shows the results of the ANOVA for the profile errors of scrolls. In this table, the degree of freedom (\(DF\)) for each factor is equal to its number of levels minus 1. \begin{table}
\begin{tabular}{c c c c} \hline
**Level** & **Factor** & **\(A\)** & **\(B\)** & **\(C\)** \\
**(Spindle Speed)** & **(Feed Rate)** & **(Tool Path)** \\ \hline
1 & 0.007388 & 0.005913 & 0.007333 \\
2 & 0.009088 & 0.007667 & 0.006837 \\
3 & 0.008763 & 0.011658 & 0.011067 \\ \hline \(\Delta_{\text{max}-\text{min}}\) & 0.001700 & 0.005746 & 0.004229 \\ Rank & 3 & 1 & 2 \\ \hline \end{tabular}
\end{table} 
Table 11: Response table for mean values for profile errors. Figure 19: Graph of mean values for profile errors. Figure 18: Graph of S/N ratios for profile errors. The sum of squares of the S/N ratios for the control factors (\(SS_{X}\)) and the total sum of squares of the S/N ratios (\(SS_{T}\)) can be obtained with the following equations, respectively [30]:  
\[SS_{X}=\frac{1}{m}{\sum_{i=1}^{p}}\left({\sum_{j=1}^{m}}{\eta_{ij}}\right)^{2}- \frac{1}{n}\left({\sum_{i=1}^{n}}{\eta_{i}}\right)^{2} \tag{8}\]  
\[SS_{T}={\sum_{i=1}^{n}}{\eta_{i}^{2}}-\frac{1}{n}\left({\sum_{i=1}^{n}}{\eta_{ i}}\right)^{2} \tag{9}\]  
where the subscript of the symbol \(SS_{X}\) represents the control factor, \(m\) is the number of S/N ratios at each level for each control factor, \(p\) is the number of levels, and \(n\) is the total number of S/N ratios; i.e., \(n=mp\). The error sum of squares is obtained as the difference between \(SS_{T}\) and the total sum of \(SS_{X}\), which can be expressed as:  
\[SS_{Error}=SS_{T}-{\sum_{X}}SS_{X} \tag{10}\]  
The mean squares for each factor (\(MS_{X}\)) and error (\(MS_{Error}\)) are obtained from the ratios of the sum of squares and their degrees of freedom. The ANOVA results shown in Table 12 reveal that the control factor (i.e., the spindle speed) is much less significant than the other two factors; therefore, we pooled this factor into the error term. The results also show that the degree of freedom of the pooled error was 4, the sum of squares was 38.86, and the mean square (\(MS_{Pooled\ error}\)) was 9.72. The \(F\)-ratio (\(F_{X}\)) and percentage of contribution (\(P_{X}\)) can be obtained with the following equations [30]:  
\[F_{X}=\frac{MS_{X}}{MS_{Pooled\ error}} \tag{11}\]  
\[P_{X}=\frac{SS_{X}-DF_{x}.MS_{Pooled\ error}}{SS_{T}}.100\% \tag{12}\]  
In accordance with the table of percentage points of the \(F\) distribution [30], where the value of \(F_{0,1,\ 2,\ 4}\) is 4.32 at a 90% confidence level, we concluded that the feed rate and tool path have much greater influences on the profile errors since their \(F\) values are larger than 4.32. Based on the previously revealed results, we determined the optimum combination of milling parameters for fine machining processes: a spindle speed of 10,000 rpm, feed rate of 500 mm/min, and B-spline tool path. For the verification of the recurrence of the optimum combination, we utilized the additive forecast model (i.e., Equation (13)) to predict the S/N ratio of the optimum combination for profile errors. In Equation (13), the symbol \(\eta_{opt}\) indicates the predicted value of the S/N ratio for the optimum combination, \(\overline{\eta}\) is the mean value of all the S/N ratios (here equal to 38.93 dB), and \(\eta_{factor}\) represents the response values of all factors and levels with the optimum combination shown in Table 10.  
\[\eta_{opt}=\overline{\eta}+\sum(\eta_{factor}-\overline{\eta}) \tag{13}\]  
According to Equation (13), the predicted S/N ratio of the optimum combination of milling parameters was 47.1 dB. We then conducted verification experiments, the results of which are shown in Figure 19:0 Table 13 shows that the verified average S/N ratio was