# Optimization of Sensor Placement for Modal Testing Using Machine Learning
## 5 Comparison of Proposed Methodology with Traditional Approaches and Discussion  
This section compares the proposed methodology with results from a traditional sensor placement methodology for modal analysis, specifically the Effective Independence Method (EIM) [23; 35]. In the case of the EIM, the sensors are chosen based on the numerical modes of the beam; therefore, they do not depend on the applied excitation. EIM is applied to a mesh with 100 elements, similar to case 1. The first ten nodes identified by the EIM as the best sensor locations for analysis of the beam are listed in Table 10, and the locations are plotted in Figure 18. \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Effective Independence Method**} \\ \hline
9 & \\ \hline
16 & \\ \hline
25 & \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Optimal sensor locations identified using a traditional approach (EIM). Figure 17: Sensor locations from chirp excitation for modes 5 through 7, case 3. (**a**): Raw ODS, (**b**): normalized ODS, (**c**): average FRF. A comparison between the natural frequencies identified using the optimal sensors' locations selected by the EIM and the random forest feature selection approach is provided in Tables 11 and 12. Both the proposed method and the traditional EIM method appear to perform similarly in the beam problem, yielding similar natural frequencies to each other in both loading conditions (Tables 11 and 12). The natural frequencies resulting from the choice of output a (ODS) are generally characterized by a lower error than those obtained for the use of \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**FEA** & \multicolumn{2}{c}{**EIM**} & \multicolumn{3}{c}{**(a) ODS**} & \multicolumn{3}{c}{**(b) Norm ODS**} & \multicolumn{3}{c}{**(c) Avg FRF**} \\ \hline \(f\) **(Hz)** & \(f\) **(Hz)** & \% Err** & \(f\) **(Hz)** & \% Err** & \(f\) **(Hz)** & \% Err** & \(f\) **(Hz)** & \% Err** \\ \hline
42.8 & – & 100\% & – & 100\% & 54.3 & 27\% & – & 100\% \\ \hline
268.3 & 278.9 & 4\% & 277.5 & 3\% & 281.6 & 5\% & 278.0 & 4\% \\ \hline
750.5 & 755.6 & 1\% & 755.7 & 1\% & 757.5 & 1\% & 751.7 & 0\% \\ \hline
1468.7 & 1482.5 & 1\% & 1471.0 & 0\% & 1462.8 & 0\% & 1462.6 & 0\% \\ \hline
2424.1 & 2396.0 & 1\% & 2397.1 & 1\% & 2395.1 & 1\% & 2398.2 & 1\% \\ \hline
3614.8 & 3523.6 & 3\% & 3526.0 & 2\% & 3531.6 & 2\% & 3499.3 & 3\% \\ \hline
5039.1 & 4837.4 & 4\% & 4834.3 & 4\% & 4747.1 & 6\% & – & – \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Comparison between natural frequencies identified using the optimal sensors’ locations selected by the EIM and the random forest feature selection approach for case 2. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**FEA** & \multicolumn{2}{c}{**EIM**} & \multicolumn{3}{c}{**(a) ODS**} & \multicolumn{3}{c}{**(b) Norm ODS**} & \multicolumn{3}{c}{**(c) Avg FRF**} \\ \hline \(f\) **(Hz)** & \(f\) **(Hz)** & \% Err** & \(f\) **(Hz)** & \% Err** & \(f\) **(Hz)** & \% Err** & \(f\) **(Hz)** & \% Err** \\ \hline
42.8 & 49.4 & 15\% & 49.5 & 16\% & 52.0 & 21\% & 51.1 & 19\% \\ \hline
268.3 & 279.7 & 4\% & 279.7 & 4\% & 291.8 & 9\% & 280.1 & 4\% \\ \outputs b (normalized ODS) and c (average FRF). Therefore, the proposed method is considered a feasible approach with traditional methodologies. The choice of global output a seems to be more reliable than global outputs b and c in its ability to identify a set of sensors that maximizes the larger number of natural frequencies that can be extracted through modal analysis. The approach is also robust with respect to the applied excitation; although the optimal sensor location changes slightly when different excitations are used to generate the database for input to the RF method, the identified natural frequencies do not differ markedly. Additionally, an excitation needs to be applied to perform modal testing; therefore, we could argue that the use of the expected excitation during testing to create the database will result in optimal sensor positioning for a given excitation.