# Automation of Experimental Modal Analysis Using Bayesian Optimization
## 3 Application
### Machine Tool Data
As a result, the MATLAB(r) approach was only run for WPT position \(\mathrm{z_{2}}\). Figure 8 shows the correlation between the mode shapes found by MATLAB(r) and the proposed approach for the frequency range up to 300 Hz, which was determined by the MAC. It can be seen that most modes were found by both methods, indicated by MAC values higher than 80%. However, some modes were only found by the AutoEMA method but not by the MATLAB(r) function, and vice versa. Given that, as will be shown below, the match between the modal models and the input data was very high in both cases, this result indicates modes that were either not well observed or highly damped. \begin{table}
\begin{tabular}{c c c c c c c} \hline  & \(p_{max}\) & \(k_{\Delta f}\) & \(k_{\Delta D}\) & \(k_{Dmax}\) & \(d_{max}\) & \(q_{min}\) & \(k_{\textit{ex}kl}\) \\ \hline \(\mathrm{z_{1}}\) & 108 & 4.2\% & 15.1\% & 25.6\% & 0.20 Hz & 27.9\% & 0.50 \\ \(\mathrm{z_{2}}\) & 108 & 5.9\% & 6.9\% & 21.7\% & 0.52 Hz & 28.8\% & 0.61 \\ \(\mathrm{z_{3}}\) & 108 & 0.4\% & 6.2\% & 24.0\% & 0.36 Hz & 28.6\% & 0.55 \\ \(\mathrm{z_{4}}\) & 112 & 5.4\% & 11.0\% & 24.2\% & 0.58 Hz & 28.2\% & 1.64 \\ \hline \end{tabular}
\end{table} 
Table 6: Found values of the hyperparameters (see Table 1) in the Bayesian optimization step for different WPT positions. \begin{table}
\begin{tabular}{c l l} \hline \hline \multicolumn{1}{c}{**Node**} & **Description** \\ \hline \(\mathrm{N_{0}}\), \(\mathrm{N_{9}}\) & \multicolumn{1}{c}{Shoe and rail nodes of the first LGS shoe} \\ \(\mathrm{N_{1}}\), \(\mathrm{N_{11}}\) & \multicolumn{1}{c}{Shoe and rail nodes of the second LGS shoe} \\ \(\mathrm{N_{2}}\), \(\mathrm{N_{10}}\) & \multicolumn{1}{c}{Shoe and rail nodes of the third LGS shoe} \\ \(\mathrm{N_{3}}\), \(\mathrm{N_{8}}\) & \multicolumn{1}{c}{Shoe and rail nodes of the fourth LGS shoe} \\ \(\mathrm{N_{5}}\) & \multicolumn{1}{c}{WPT node} \\ \(\mathrm{N_{17}}\), \(\mathrm{N_{18}}\), \(\mathrm{N_{19}}\), \(\mathrm{N_{27}}\) & \multicolumn{1}{c}{Machine bed nodes} \\ \(\mathrm{N_{20}}\), \(\mathrm{N_{21}}\), \(\mathrm{N_{22}}\) & \multicolumn{1}{c}{ME nodes} \\ \(\mathrm{N_{28}}\) & \multicolumn{1}{c}{Excitation node} \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Model nodes considered; the node locations are illustrated in Figure 7.  
Figure 9 qualitatively illustrates the resulting match between the mean input FRFs and the reconstructed mean FRFs from the modal model for two WPT positions. It can be seen that there is a very high level of concordance between the input and fitted data for both positions for the AutoEMA approach, thus highlighting the validity of the model. However, there were higher deviations for the MATLAB(r) approach at position \(z_{2}\), especially in the high frequency region. This outcome was also confirmed by the mean FRAC values of 88.7% for the AutoEMA FRF at position z\({}_{2}\) and 64.8% for the MATLAB(r) FRF, respectively. In general, even FRAC values of 70% are considered to be a good match [22]. The same conclusion can be drawn by looking at the individual FRFs in Figure 10. It can be seen that, for both approaches, the direct driving point FRF from a force at node \(N_{28}\) in the \(y\)-direction to the measured displacement at the same node in the same coordinate direction and the FRF from a force at node \(N_{28}\) in the \(y\)-direction to the measured deflection at one of the edges of the machine tool (node \(N_{19}\)) in \(z\)-direction were approximated very well by the modal models. Regarding the AutoEMA approach, the same was also true for other FRFs, which is shown in Table 7: Only for 10% of the measured FRFs at all WPT positions, the modal model leads to a fitted FRF with, measured by the FRAC value, less than 82.6% concordance. However, the MATLAB(r) approach on average approximates the input FRFs only with a match of 64.8% and one-tenth of them even worse than with 18.9% concordance.