# Automation of Experimental Modal Analysis Using Bayesian Optimization
## 3 Application
### Synthetic Test Data  
To test the proposed approach, an oscillator with six degrees of freedom (DOFs) was set up as shown in Figure 4 and used for simulating synthetic reference input data. Using a time-domain simulation, six FRFs were calculated from an input force at the first mass \(f_{1}\) to the displacement at all DOFs. To demonstrate the robustness of the AutoEMA method, normal (Gaussian) noise with zero mean was superimposed to the respective outputs \(x_{1},x_{2},\ldots,x_{6}\). Additionally, the (damped) eigenvalue problem was solved, resulting in estimates for the mode shapes, the eigenfrequencies, and the modal damping ratios. These modal parameters are, in the following, regarded as ground truth values and to be estimated using both the MATLAB(r) and the AutoEMA approach. Regarding the AutoEMA approach, Table 2 shows the value range constraining the hyperparameters of the modal parameter estimation in the Bayesian optimization process and their initial values. These boundaries were considered to be wide in order to keep the solution space as widely open as possible. Note that the minimum damping of a mode \(k_{Dmin}\) was set to the lowest physically meaningful value of 0%, so it was excluded from the optimization in this case. Per definition, the regularization parameter \(r\) cannot be optimized and was also fixed. However, the value \(r\) of 0.002 has been found to be suitable for many data sets, including the one used herein and the real-world data shown in Section 3.2. The initial values were chosen arbitrarily within the bounds, as the result of the Bayesian optimization was found to be widely insensitive. In total, 40 iterations, that is, evaluations of the cost function (see Section 2.2), were made. The same data were also fed into the MATLAB(r) function modalfit. It requires the number of modes to be determined as an input parameter. In this case, it was set to the true value of six modes, which was also identified automatically by the AutoEMA approach. The modalfit function has implemented both the LSCE and the LSRF algorithms. Here, the default LSCE method was used. Figure 4: Schematic illustration of an MDOF system used for generating synthetic measurement data in Section 3.1. Figure 5 shows the (simulated) reference FRF from a force input \(f_{1}\) at the first DOF to the displacement \(x_{1}\) at the same DOF alongside the reconstructed FRFs from both the MATLAB(r) and AutoEMA modal models. It can be seen that both approaches estimated the original FRF well despite the presence of noise. However, the AutoEMA approach slightly outperformed the MATLAB(r) approach in the low frequency region up to 150 Hz. Figure 6 shows the conformance of the estimated mode shapes with the (simulated) reference data quantified by the modal assurance criterion (MAC) [20], which is defined as  
\[\mathrm{MAC}(\boldsymbol{\varphi}_{1},\boldsymbol{\varphi}_{2})=\frac{| \boldsymbol{\varphi}_{1}^{T}\boldsymbol{\varphi}_{2}^{*}|^{2}}{\left( \boldsymbol{\varphi}_{1}^{T}\boldsymbol{\varphi}_{1}^{*}\right)\left( \boldsymbol{\varphi}_{2}^{T}\boldsymbol{\varphi}_{2}^{*}\right)}. \tag{5}\]  
In this case, \(\boldsymbol{\varphi}_{i}\) are mode shapes being compared, and, again, \(x^{*}\) denotes the complex conjugation and \(x^{T}\) the transpose of a vector \(x\). Similar to the FRAC, a value of 0% indicates no correlation at all, and a result of 100% states a perfect coincidence between the two mode \begin{table}
\begin{tabular}{c c c} \hline \hline
**Hyperparameter** & **Initial Value** & **Value Range** \\ \hline \(p_{max}\) & 80 & 60 to 120 \\ \(k_{Af}\) & 1\% & 0.1\% to 10\% \\ \(k_{AD}\) & 5\% & 5\% to 20\% \\ \(k_{Dmin}\) & 0\% & 20\% to 30\% \\ \(d_{max}\) & 2 Hz & 0.4 Hz to 4 Hz \\ \(q_{min}\) & 30\% & 20\% to 60\% \\ \(k_{eubl}\) & 0.5 & 0.1 to 0.8 \\ \(r\) & 0.002 & \\ \hline \hline \end{tabular}
\end{table}