# Automation of Experimental Modal Analysis Using Bayesian Optimization
## 4 Conclusions and Outlook  
This paper presented an approach for automatically estimating the modal parameters of a system based on input measurements. In contrast to the state-of-the-art, no user input is needed for evaluating the stabilization diagram, nor is the solution space restricted by fixed rules derived from the input data set. Instead, a Bayesian optimization process ensures that modal parameters are found, thus leading to the model best representing the original input data. The proposed AutoEMA approach was first theoretically derived in Section 2, and was validated using simulated input data from an MDOF oscillator. The approach was benchmarked against ground truth reference values and the MATLAB(r) modalfit function in this context. Finally, the proposed approach was also applied to EMA data from a machine tool test bench. It was found that the AutoEMA approach led to modal parameters which effectively represent the original input data for different machine \begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & \(f_{1}^{e}\) & \(f_{2}^{e}\) & \(f_{3}^{e}\) & \(f_{4}^{e}\) & \(f_{5}^{e}\) & \(f_{6}^{e}\) & \(f_{7}^{e}\) & \(f_{8}^{e}\) \\ \hline \(z_{1}\) & 28.7 Hz & 31.0 Hz & - & 52.0 Hz & 76.1 Hz & 91.9 Hz & 109.1 Hz & 117.3 Hz \\ \(z_{2}\) & 28.8 Hz & 31.0 Hz & 41.8 Hz & 52.1 Hz & 75.7 Hz & 95.0 Hz & 108.1 Hz & 120.0 Hz \\ \(z_{3}\) & 28.7 Hz & 31.0 Hz & 41.4 Hz & 52.1 Hz & 75.2 Hz & 97.9 Hz & 107.0 Hz & 122.4 Hz \\ \(z_{4}\) & - & 31.0 Hz & - & 52.1 Hz & 75.0 Hz & - & 106.1 Hz & 127.3 Hz \\ \hline \end{tabular}
\end{table} 
Table 10: Identified first eight eigenfrequencies \(f_{i}^{e}\) for the WPT position \(z_{2}\) with the lowest number of found modes and the equivalent modes’ eigenfrequencies at the other positions; “-” indicates that no matching mode with a MAC value of at least 80% was found. \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & \(f_{1}^{e}\) & \(f_{2}^{e}\) & \(f_{3}^{e}\) & \(f_{4}^{e}\) & \(f_{5}^{e}\) & \(f_{6}^{e}\) & \(f_{7}^{e}\) & \(f_{8}^{e}\) \\ \hline AutoEMA & 28.8 Hz & 31.0 Hz & 41.8 Hz & 52.1 Hz & 75.7 Hz & 95.0 Hz & 108.1 Hz & 120.0 Hz \\ MATLAB® & - & 31.3 Hz & 42.3 Hz & 52.3 Hz & 76.1 Hz & 95.5 Hz & 108.2 Hz & 120.1 Hz \\ \(\mathrm{NFD}_{AE-ML}\) & - & 1.1\% & 1.2\% & 0.3\% & 0.4\% & 0.6\% & 0.1\% & 0.1\% \\ \hline \end{tabular}
\end{table} 
Table 8: Comparison of eigenfrequencies \(f_{i}^{e}\) found by MATLAB® and AutoEMA for WPT position z\({}_{2}\); “-” indicates that no matching mode with a MAC value of at least 80% was found. \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & \(f_{1}^{e}\) & \(f_{2}^{e}\) & \(f_{3}^{e}\) & \(f_{4}^{e}\) & \(f_{5}^{e}\) & \(f_{6}^{e}\) & \(f_{7}^{e}\) & \(f_{8}^{e}\) \\ \hline AutoEMA & 1.05\% & 0.88\% & 1.67\% & 1.04\% & 1.98\% & 3.82\% & 0.29\% & 1.50\% \\ MATLAB® & - & 2.78\% & 2.75\% & 1.68\% & 2.22\% & 3.20\% & 0.28\% & 3.02\% \\ \(\mathrm{NDD}_{AE-ML}\) & - & 213.7\% & 64.8\% & 62.2\% & 12.2\% & 19.2\% & 5.8\% & 101.0\% \\ \hline \end{tabular}
\end{table} 
Table 10: Identified first eight eigenfrequencies \(f_{i}^{e}\) for the WPT position \(z_{2}\) with the lowest number of found modes and the equivalent modes’ eigenfrequencies at the other positions; “-” indicates that no matching mode with a MAC value of at least 80% was found. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & \(f_{1}^{e}\) & \(f_{2}^{e}\) & \(f_{3}^{e}\) & \(f_{4}^{e}\) & \(f_{5}^{e}\) & \(f_{6}^{e}\) & \(f_{7}^{e}\) & \(f_{8}^{e}\) \\ \hline AutoEMA & 28.8 Hz & 31.0 Hz & 41.8 Hz & 52.1 Hz & 75.7 Hz & 95.0 Hz & 108.1 Hz & 120.0 Hz \\ MATLAB® & - & 31.3 Hz & 42.3 Hz & 52.3 Hz & 76.1 Hz & 95.5 Hz & 108.2 Hz & 120.1 Hz \\ \(\mathrm{NFD}_{AE-ML}\) & - & 1.1\% & 1.2\% & 0.3\% & 0.4\% & 0.6\% & 0.1\% & 0.1\% \\ \hline \end{tabular}
\end{table} 
Table 9: Comparison of modal damping ratios \(\xi_{i}\) found by MATLAB® and AutoEMA for WPT position z\({}_{2}\); “-” indicates that no matching mode with a MAC value of at least 80% was found.