# Precision Manufacturing of Patterned Beryllium Bronze Leaf Springs via Chemical Etching
## 3 Process Parameter Optimization
### Mean Value Analysis  
To find the optimal combination of the minimum roundness, straightness and offset, the mean values of the S/N ratio at the same level for each factor were calculated as shown in Tables 3 and 4. For example, the mean S/N ratio for roundness at level 2 of factor A is calculated using  
\[\overline{\eta_{RA2}}=\frac{1}{3}(\eta_{R}(2)+\eta_{R}(4)+\eta_{R}(9)) \tag{3}\]  
where \(R\) and \(A\) in the subscript denote the roundness and factor A, respectively. Factor A is set to level 2 only in experiments 2, 4 and 9. The mean S/N ratios for all levels of all factors can be obtained in a similar way, as shown in Table 5.  
According to the Taguchi principle, the maximum S/N ratio value should be considered when determining the main levels and main factors. From Table 5, we can see that the combination of factor levels (A\({}_{3}\), B\({}_{1}\), C\({}_{2}\)), (A\({}_{1}\), B\({}_{1}\), C\({}_{1}\)) and (A\({}_{3}\), B\({}_{3}\), C\({}_{2}\)) contribute to a minimization of the roundness, straightness and offset, respectively. Using the optimal factor-level combination, three groups experiments were conducted separately for the minimum roundness, straightness and offset, with each group repeated five times. Table 6 shows the results. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**No.** & **1** & **2** & **3** & **4** & **5** & **Combination of Factor Levels** \\ \hline roundness (\(\mu\)m) & 52 & 46 & 33 & 38 & 49 & A\({}_{3}\), B\({}_{1}\), C\({}_{2}\) \\ straightness (\(\mu\)m) & 22 & 15 & 30 & 16 & 22 & A\({}_{1}\), B\({}_{1}\), C\({}_{1}\) \\ offset (\(\mu\)m) & 9 & 10 & 8 & 10 & 9 & A\({}_{3}\), B\({}_{3}\), C\({}_{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Measured results of three variables. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**No.** & **Average Roundness (\(\mu\)m)** & \(\eta_{R}\) & **Average Straightness (\(\mu\)m)** & \(\eta_{S}\) & **Average Offset (\(\mu\)m)** & \(\eta_{o}\) \\ \hline
1 & 70 & –37.80 & 21 & –26.93 & 15 & –23.77 \\
2 & 73 & –37.38 & 57 & –35.40 & 15 & –22.16 \\
3 & 54 & –33.67 & 21 & –27.39 & 6 & –16.19 \\
4 & 32 & –31.08 & 29 & –27.58 & 7 & –15.16 \\
5 & 54 & –34.99 & 50 & –34.74 & 12 & –22.32 \\
6 & 59 & –35.66 & 40 & –32.76 & 10 & –20.87 \\
7 & 41 & –33.16 & 37 & –31.65 & 9 & –19.48 \\
8 & 34 & –30.49 & 29 & –30.39 & 6 & –16.26 \\
9 & 44 & –32.76 & 34 & –30.81 & 10 & –19.57 \\ mean value & 51.2 & –34.11 & 35.3 & –30.58 & 10.0 & –19.53 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Mean values of the response variable and its S/N ratio. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Response Variables**} & \multirow{2}{*}{**Level**} & \multicolumn{4}{c}{**Mean S/N Ratio**} \\ \cline{3-5}  & & **Factor A** & **Factor B** & **Factor C** \\ \hline \multirow{3}{*}{roundness} & 1 & –36.28 & –34.01 & –34.65 \\  & 2 & –33.91 & –34.29 & –33.74 \\  & 3 & –32.14 & –34.03 & –33.94 \\ \hline \multirow{3}{*}{straightness} & 1 & –29.91 & –28.72 & –30.03 \\  & 2 & –31.69 & –33.51 & –31.26 \\  & 3 & –30.95 & –30.32 & –31.26 \\ \hline \multirow{3}{*}{offset} & 1 & –20.71 & –19.47 & –20.30 \\  & 2 & –19.45 & –20.25 & –18.96 \\ \cline{1-1}  & 3 & –18.72 & –19.33 & –19.33 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Mean values at the same level of each factor and variable.