# Shape Optimization of Discontinuous Armature Arrangement PMLSM for Reduction of Thrust Ripple
## 4 Optimal Design of the End Edge Auxiliary Teeth
### Auxiliary Teeth Shape Optimization Using RSM  
For design optimization using the approximate model, 15 simulation runs were conducted using the BBD method. The results are presented in Table 9. The impact of each shape parameter obtained from Table 9 is illustrated in Figure 16. As shown, \(A_{t}\) significantly impacts the detent force, and there is an interaction between this parameter and the detent force. \(A_{e}\) also affects the detent force, while \(A_{b}\) in the direction of movement has a negligible impact on the detent force. Therefore, \(A_{b}\) was fixed at the maximum value of 30 mm. Figure 16: Pareto chart for end effect detent force. Figure 15: End edge auxiliary teeth. Figure 17 shows the response surface for each design variable affecting the detent force. A\({}_{\text{t}}\) is in the form of a quadratic curve; in the range of A\({}_{\text{t}}\) = 10-15 mm, the detent force due to the end effect is minimized when A\({}_{\text{e}}\) is minimized. Table 10 shows the ANOVA results for each shape parameter affecting the objective functions. The R\({}^{2}\) value, i.e., the coefficient of determination of the objective function according to the shape parameters, excluding the non-significant terms, was 99.87%, confirming that the results were within 5% of the significance level. It can be deduced that the regression model adequately explains the relationship between the performance parameters and the design variables. The results obtained from calculating the values of the equation for the quadratic polynomial regression curve are as follows:  
\[Detentforce(end)=8.1054-0.30204A_{t}+0.01367A_{e}+0.012180A_{t}*A_{t} \tag{8}\]  
As indicated by the response optimization graph presented in Figure 18, when A\({}_{\text{t}}\) was 12.42 mm and A\({}_{\text{e}}\) was at the minimum value, a detent force of 6.23 N was generated. Figure 19a shows the detent force of the PMLSM with the installation of the optimized auxiliary teeth. In the absence of the auxiliary teeth, a detent force of \(\pm\)9.33 N was generated when the mover entered or exited the armature; however, with the installation of the auxiliary teeth, a detent force of \(\pm\)6.49 N, with a 30.44% reduction, was generated. Figure 19b shows a comparison between the harmonic components of the initial and optimized models. As shown, all the components except for the 1 st- and 5 th-order harmonic components decreased; additionally, the 6 th-order component (with the largest value) significantly decreased. \begin{table}
\begin{tabular}{c c c c c} \hline
**Run Order** & \(A_{t}\) & \(A_{b}\) & \(A_{e}\) & **Detent Force [N]** \\ \hline
1 & 15 & 15 & 15 & 6.55 \\
2 & 15 & 15 & 15 & 6.55 \\
3 & 30 & 0 & 15 & 10.26 \\
4 & 30 & 30 & 15 & 10.23 \\
5 & 0 & 15 & 0 & 8.06 \\  & & & & \\
11 & 15 & 30 & 0 & 6.34 \\
12 & 0 & 30 & 15 & 8.32 \\
13 & 30 & 15 & 30 & 10.48 \\
14 & 15 & 30 & 30 & 6.62 \\
15 & 0 & 0 & 15 & 9.33 \\ \hline \end{tabular}
\end{table} 
Table 9: Design variables and boundary of the auxiliary teeth. Figure 17: Response surface of A\({}_{\text{t}}\) (mm) vs. A\({}_{\text{e}}\) (mm), Detent force (N). Figure 19: FEM results of end effect detect force at no load (**a**) Detent force of optimized model; (**b**) Detent force harmonic components of optimized model. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Source** & **DF** & **Adj SS** & **Adj MS** & **F-Value** & \(p\)**-Value** \\ \hline Model & 3 & 35.6044 & 11.8681 & 2846.15 & 0 \\ Linear & 2 & 35.1074 & 17.5537 & 4209.63 & 0 \\ \(A_{t}\) & 1 & 34.7708 & 34.7708 & 8338.55 & 0 \\ \(A_{e}\) & 1 & 0.3366 & 0.3366 & 80.71 & 0 \\ Square & 1 & 28.0395 & 28.0395 & 6724.27 & 0 \\ \(A_{t}\)*\(A_{t}\) & 1 & 28.0395 & 28.0395 & 6724.27 & 0 \\ Error & 11 & 0.0459 & 0.0042 & & \\ Lack-of-Fit & 9 & 0.0459 & 0.0051 & & \\ Pure Error & 2 & 0 & 0 & & \\ Total & 14 & 35.6503 & & & \\ \hline \hline \end{tabular} Model summary  
\begin{tabular}{c c c c} \hline \hline S & R-sq & R-sq (adj) & R-sq (pred) \\
0.0645747 & 99.87\% & 99.84\% & 99.72\% \\ \hline \hline \end{tabular}
\end{table}