# Automation of Experimental Modal Analysis Using Bayesian Optimization
## 3 Application
### Machine Tool Data  
A second data set was acquired on a DMG DMC duo Block 55H machine tool in order to demonstrate the robustness and effectiveness of the proposed modal parameter extraction method (i.e., the AutoEMA method). As illustrated in Figure 7, the machine tool consists of a machine tool bed and a workpiece table (WPT), enabling movement in the global \(z\)-axis. The machine tool's vibrational response was measured at 17 nodes in all three spatial directions. Four Kistler(r) triaxial accelerometers (two times type 8762A10 and two times type 8762A50) were used together with a National Instruments (NI)(r) CDAQ-9198 rack with three type NI(r)-9232 modules and one type NI(r)-9234 module. The measurements were repeated for four WPT positions z\({}_{1}\), z\({}_{2}\), z\({}_{3}\), and z\({}_{4}\) along the \(z\)-axis with impulse hammer excitations in \(x\)-, \(y\)-, and z-directions. The node points are shown in Figure 7 and described in Table 5. In particular, a corner of the machine tool bed (\(N_{28}\)) was chosen as the excitation node. For each WPT position, the data acquisition has resulted in 153 FRFs \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Mode** & \(\xi_{ref}\) & \(\xi_{ML}\) & \(\xi_{AE}\) & NDD\({}_{ref-ML}\) & NDD\({}_{ref-AE}\) \\ \hline
1 & 11.4\% & 11.3\% & 10\% & 0.7\% & 13.9\% \\
2 & 5.0\% & 5.0\% & 4.8\% & 0.3\% & 3.9\% \\
3 & 3.4\% & 3.4\% & 3.4\% & 0.2\% & 0.1\% \\
4 & 3.3\% & 3.6\% & 3.3\% & 8.6\% & 0.6\% \\
5 & 3.3\% & 3.4\% & 3.2\% & 4.4\% & 1.9\% \\
6 & 3.5\% & 3.7\% & 3.0\% & 6.1\% & 17.1\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Overview of the modal damping results for the synthetic data use case (see Section 3.1) showing the absolute values and the NDD for both the MATLAB® (ML) and AutoEMA (AE) approach. Figure 7: Rendering of the considered machine tool, the measurement positions z\({}_{1}\) to z\({}_{4}\), and the measured nodes; the WPT is depicted in position z\({}_{3}\) and a description of the node points can be found in Table 5.  
(17 nodes measured in three spatial directions for three excitation directions). The FFRFs were calculated as the average of three hammer hits for a measurement duration of 4 s with a sampling rate of 10.240 Hz. The approach presented in Section 2 was applied for each WPT position using the same parameter bounds as in the synthetic data use case (see Table 2). Because of the more complicated structure, the optimization was stopped after 100 iterations in this case. Table 6 shows the final values of the hyperparameters (see Table 1) for all considered WPT positions. It can be seen that most parameters have been estimated quite similarly over all WPT positions. However, some outliers exist as, for example, the maximum model order \(p_{max}\) for WPT position \(\mathrm{z_{4}}\). The data set contained redundant measurements since, for each WPT position, three measurements for each DOF resulting from three different excitation directions had been made. It is believed that redundancy has supported the estimation of the eigenfrequencies and the modal damping ratios. However, the mode shape information was only extracted from measurements with an excitation in the \(y\)-direction as the input force was the easiest to control in that direction, thus leading to the best mode shape estimations. The same data were also fed into the MATLAB(r) modalfit function, which requires the number of modes being determined as an input parameter. In order to obtain comparable results, it was set to the same number which the AutoEMA approach has found. For this use case, no meaningful results could be produced with the LSCE algorithm, leaving only the LSRF method for the evaluations. It is noteworthy that the modal parameter estimation using the AutoEMA method took less than one minute on a laptop with four Intel(r) i7-7700 HQ CPU cores, whereas a run of the LSRF algorithm required 22 h and 48 min on a simulation workstation with 24 Intel(r) Xeon(r) Gold 5220R CPU cores.