# Automation of Experimental Modal Analysis Using Bayesian Optimization

*iwb*), TUM School of Engineering and Design, Technical University of Munich, Boltzmannstr. 15, 85748 Garching, Germany

^{*}

## Abstract

**:**

## 1. Introduction

^{®}function

`modalfit`. Last, Section 4 summarizes the main content of the paper and offers an outlook to future research.

## 2. Proposed Method

#### 2.1. Modal Parameter Estimation

#### 2.2. Hyperparameter Optimization

## 3. Application

`modalfit`from the signal processing toolbox of the well-established numerical computation program MATLAB

^{®}.

#### 3.1. Synthetic Test Data

^{®}and the AutoEMA approach.

^{®}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.

^{®}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

^{®}approach in the low frequency region up to 150 $\mathrm{Hz}$.

#### 3.2. Machine Tool Data

^{®}triaxial accelerometers (two times type 8762A10 and two times type 8762A50) were used together with a National Instruments (NI)

^{®}cDAQ-9198 rack with three type NI

^{®}-9232 modules and one type NI

^{®}-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 (17 nodes measured in three spatial directions for three excitation directions). The FRFs were calculated as the average of three hammer hits for a measurement duration of 4 $\mathrm{s}$ with a sampling rate of 10,240 $\mathrm{Hz}$.

^{®}

`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

^{®}i7-7700HQ CPU cores, whereas a run of the LSRF algorithm required 22 $\mathrm{h}$ and 48 $\mathrm{min}$ on a simulation workstation with 24 Intel

^{®}Xeon

^{®}Gold 5220R CPU cores. As a result, the MATLAB

^{®}approach was only run for WPT position z${}_{2}$.

^{®}and the proposed approach for the frequency range up to 300 $\mathrm{Hz}$, 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

^{®}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.

^{®}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

^{®}FRF, respectively. In general, even FRAC values of 70% are considered to be a good match [22].

^{®}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.

^{®}and the AutoEMA approach for the machine tool data use case. It can be seen that both approaches estimate very similar eigenfrequencies with a maximum NFD (see Equation (6)) of only $1.2$%. This holds especially true considering that even a mean relative eigenfrequency difference of $3.3$% is sufficient for many applications [23]. However, the MATLAB

^{®}approach seemed to generally estimate higher modal damping ratios with NDDs up to $213.7$%, as can be seen in Table 9.

## 4. Conclusions and Outlook

^{®}

`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 tool axis positions, thereby outperforming the MATLAB

^{®}method in both accuracy and calculation time.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

DOF | degree of freedom |

EMA | experimental modal analysis |

FRAC | frequency response assurance criterion |

FRF | frequency response function |

GA | genetic algorithm |

LGS | linear guiding system |

LSCE | least squares complex exponential |

LSCF | least squares complex frequency-domain |

LSRF | least squares rational function |

MAC | modal assurance criterion |

MDOF | multi-degree-of-freedom |

ME | mounting element |

NDD | natural damping difference |

NFD | natural frequency difference |

NI | National Instruments |

OMA | operational modal analysis |

SSI | stochastic subspace identification |

SVD | singular value decomposition |

WPT | workpiece table |

## References

- Lau, J.; Lanslots, J.; Peeters, B.; van der Auweraer, H. Automatic Modal Analysis—Myth or Reality? In Proceedings of the 25th International Modal Analysis Conference, Orlando, FL, USA, 19–22 February 2007.
- Verboven, P. Frequency-Domain System Identification for Modal Analysis. Ph.D. Thesis, Vrije Universiteit Brussel, Brussels, Belgium, 2019. [Google Scholar]
- Ellinger, J.; Zaeh, M.F. Automated Identification of Linear Machine Tool Model Parameters Using Global Sensitivity Analysis. Machines
**2022**, 10, 535. [Google Scholar] [CrossRef] - Reynders, E. System Identification Methods for (Operational) Modal Analysis: Review and Comparison. Arch. Comput. Methods Eng.
**2012**, 19, 51–124. [Google Scholar] [CrossRef] - Ewins, D. Modal Testing: Theory, Practice and Application, 2nd ed.; Mechanical Engineering Research Studies; Research Studies Press: Baldock, UK, 2000; Volume 10. [Google Scholar]
- Altintas, Y. Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design; Cambridge University Press: Cambridge, UK, 2012. [Google Scholar]
- Ozdemir, A.; Gumussoy, S. Transfer Function Estimation in System Identification Toolbox via Vector Fitting. IFAC-PapersOnLine
**2017**, 50, 6232–6237. [Google Scholar] [CrossRef] - Vold, H.; Kundrat, J.; Rocklin, T.; Russell, R. A Multi-Input Modal Estimation Algorithm for Mini-Computers; SAE Technical Paper Series; SAE International: Warrendale, PA, USA, 1982. [Google Scholar] [CrossRef]
- Peeters, B.; van der Auweraer, H.; Guillaume, P.; Leuridan, J. The PolyMAX Frequency-Domain Method: A New Standard for Modal Parameter Estimation? Shock Vib.
**2004**, 11, 395–409. [Google Scholar] [CrossRef] - van Overschee, P.; de Moor, B. Subspace Identification for Linear Systems; Springer: Boston, MA, USA, 1996. [Google Scholar] [CrossRef]
- Scionti, M.; Lanslots, J. Stabilisation Diagrams: Pole Identification Using Fuzzy Clustering Techniques. Adv. Eng. Softw.
**2005**, 36, 768–779. [Google Scholar] [CrossRef] - Reynders, E.; Houbrechts, J.; de Roeck, G. Fully Automated (Operational) Modal Analysis. Mech. Syst. Signal Process.
**2012**, 29, 228–250. [Google Scholar] [CrossRef] - Scionti, M.; Lanslots, J.; Goethals, I.; Vecchio, A.; van der Auweraer, H.; Peeters, B.; de Moor, B. Tools to Improve Detection of Structural Changes from In-Flight Flutter Data. In Proceedings of the 8th International Conference on Recent Advances in Structural Dynamics, Southampton, UK, 14–16 July 2003. [Google Scholar]
- van der Auweraer, H.; Peeters, B. Discriminating Physical Poles from Mathematical Poles in High Order Systems: Use and Automation of the Stabilization Diagram. In Proceedings of the 21st IEEE Instrumentation and Measurement Technology Conference (IEEE Cat. No.04CH37510), Como, Italy, 18–20 May 2004; pp. 2193–2198. [Google Scholar] [CrossRef]
- Neu, E.; Janser, F.; Khatibi, A.; Orifici, C. Fully Automated Operational Modal Analysis Using Multi-Stage Clustering. Mech. Syst. Signal Process.
**2017**, 84, 308–323. [Google Scholar] [CrossRef] - Mugnaini, V.; Zanotti, L.; Civera, M. A Machine Learning Approach for Automatic Operational Modal Analysis. Mech. Syst. Signal Process.
**2022**, 170, 108813. [Google Scholar] [CrossRef] - Zaletelj, K.; Bregar, T.; Gorjup, D.; Slavič, J. pyEMA. 2020. Available online: https://zenodo.org/record/4016671#.Y7vYsBVBzIU (accessed on 26 December 2022).
- Garnett, R. Bayesian Optimization; Cambridge University Press: Cambridge, UK, 2022. [Google Scholar]
- Heylen, W.; Lammens, S. FRAC: A Consistent Way of Comparing Frequency Response Functions. In Identification in Engineering Systems; Friswell, M.I., Mottershead, J., Eds.; University of Wales: Swansea, UK, 1996; pp. 48–57. [Google Scholar]
- Allemang, R. The Modal Assurance Criterion – Twenty Years of Use and Abuse. J. Sound Vib.
**2003**, 37, 14–23. [Google Scholar] - Imamovic, N. Validation of Large Structural Dynamics Models Using Modal Test Data. Ph.D. Thesis, Imperial College of Science, Technology & Medicine, London, UK, 1998. [Google Scholar]
- Semm, T.; Sellemond, M.; Rebelein, C.; Zaeh, M.F. Efficient Dynamic Parameter Identification Framework for Machine Tools. J. Manuf. Sci. Eng.
**2020**, 142, 081003. [Google Scholar] [CrossRef] - Hernandez-Vazquez, J.; Garitaonandia, I.; Fernandes, M.H.; Munoa, J.; Lacalle, L.N. A Consistent Procedure Using Response Surface Methodology to Identify Stiffness Properties of Connections in Machine Tools. Materials
**2018**, 11, 1220. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Vacher, P.; Jacquier, B.; Bucharles, A. Extensions of the MAC Criterion to Complex Modes. In Proceedings of the 24th International Conference on Noise and Vibration Engineering, 2010, Proceedings of ISMA2010 Including USD2010, Leuven, Belgium, 20–22 September 2010; pp. 2713–2726. [Google Scholar]

**Figure 1.**Flowchart showing the AutoEMA method; the modal parameter estimation is based on a set of hyperparameters explained in Table 1. Their value is determined by a Bayesian optimization strategy such that the resulting model optimally replicates the original input data.

**Figure 2.**Exemplarily FRF and potentially physical poles identified after applying the PolyMAX algorithm and criteria (a) to (d) (see Section 2.1) for the given hyperparameters ${k}_{\mathrm{\Delta}f}$, ${k}_{\mathrm{\Delta}D}$, ${k}_{Dmin}$, and ${k}_{Dmax}$.

**Figure 3.**Identified modes by agglomerative clustering with a maximum frequency distance ${d}_{max}$ of poles within a cluster; different clusters, that is, distinct modes, are indicated by different colors. The finally selected poles are marked with a cross.

**Figure 4.**Schematic illustration of an MDOF system used for generating synthetic measurement data in Section 3.1.

**Figure 5.**Comparison of the receptance amplitude and the phase between the input (solid) and the fitted FRF (AutoEMA: dashed, MATLAB

^{®}: dash-dotted) from the modal model from a force ${f}_{1}$ to the displacement ${x}_{1}$ of the first mass; normal (Gaussian) noise was added to the (synthetic) displacement data to simulate noisy real-world measurements.

**Figure 6.**Comparison of the found mode shapes using the MAC; here, the mode shapes from solving the damped eigenvalue problem resulting from the MDOF oscillator in Section 3.1 were used as ground truth, (

**a**) MATLAB

^{®}

`modalfit`, (

**b**) AutoEMA.

**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.

**Figure 8.**Correlation between the modes up to 300 Hz found by the AutoEMA approach and MATLAB

^{®}

`modalfit`determined by the MAC for the WPT position z${}_{2}$.

**Figure 9.**Comparison of the mean receptance amplitude of the original input FRFs (solid) with the fitted FRFs from the modal model (MATLAB

^{®}: dash-dotted, AutoEMA: dashed); the comparison is shown for the WPT positions having the lowest number of found modes z${}_{2}$ and the one at the greatest distance, that is, position z${}_{4}$. Due to the high calculation times, MATLAB

^{®}results are only available for position z${}_{2}$.

**Figure 10.**Comparison of the receptance amplitude and the phase between measured (solid) and fitted FRFs (MATLAB

^{®}: dash-dotted, AutoEMA: dashed) from the modal model for position z${}_{2}$; N${}_{{\alpha}_{1}}^{\left({\beta}_{1}\right)}$→ N${}_{{\alpha}_{2}}^{\left({\beta}_{2}\right)}$ denotes the FRF between a force excitation at node N${}_{{\alpha}_{1}}$ in coordinate direction ${\beta}_{1}$ and the measured displacement at node N${}_{{\alpha}_{2}}$ in coordinate direction ${\beta}_{2}$ (see Table 5).

Step | Symbol | Description |
---|---|---|

Pole calculation | ${p}_{max}$ | Maximum model order |

Classification | ${k}_{\mathrm{\Delta}f}$ | Maximum relative frequency difference of two poles |

${k}_{\mathrm{\Delta}D}$ | Maximum relative damping difference of two poles | |

${k}_{Dmin}$ | Minimum damping of a mode | |

${k}_{Dmax}$ | Maximum damping of a mode | |

Clustering | ${d}_{max}$ | Maximum frequency distance within a cluster |

${q}_{min}$ | Minimum percentage of the model order of poles per cluster | |

${k}_{eukl}$ | Maximum euclidean frequency distance factor between poles | |

Pole selection | - | - |

**Table 2.**Value range and initial values of the hyperparameters used in the optimization stage of the modal parameter optimization for both use cases; a description of the parameters can be found in Table 1 and Section 2.1. The regularization parameter r and the minimum modal damping ${k}_{Dmin}$ were set constant.

Hyperparameter | Initial Value | Value Range |
---|---|---|

${p}_{max}$ | 80 | 60 to 120 |

${k}_{\mathrm{\Delta}f}$ | 1% | 0.1% to 10% |

${k}_{\mathrm{\Delta}D}$ | 5% | 5% to 20% |

${k}_{Dmin}$ | 0% | |

${k}_{Dmax}$ | 20% | 20% to 30% |

${d}_{max}$ | 2 $\mathrm{Hz}$ | 0.4 $\mathrm{Hz}$ to 4 $\mathrm{Hz}$ |

${q}_{min}$ | 30% | 20% to 60% |

${k}_{eukl}$ | 0.5 | 0.1 to 0.8 |

r | 0.002 |

**Table 3.**Overview of the eigenfrequency results for the synthetic data use case (see Section 3.1) showing the absolute values and the NFD for both the MATLAB

^{®}(ML) and AutoEMA (AE) approach.

Mode | ${\mathit{\omega}}_{\mathit{ref}}$ | ${\mathit{\omega}}_{\mathit{ML}}$ | ${\mathit{\omega}}_{\mathit{AE}}$ | ${\mathbf{NFD}}_{\mathit{ref}-\mathit{ML}}$ | ${\mathbf{NFD}}_{\mathit{ref}-\mathit{AE}}$ |
---|---|---|---|---|---|

1 | 33.8 Hz | 33.8 Hz | 33.7 Hz | 0.0% | 0.2% |

2 | 86.6 Hz | 86.6 Hz | 86.6 Hz | 0.0% | 0.0% |

3 | 186.7 Hz | 186.6 Hz | 186.6 Hz | 0.0% | 0.0% |

4 | 237.3 Hz | 236.6 Hz | 237.3 Hz | 0.3% | 0.0% |

5 | 250.5 Hz | 250.3 Hz | 250.5 Hz | 0.1% | 0.0% |

6 | 318.9 Hz | 318.4 Hz | 319.5 Hz | 0.1% | 0.2% |

**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.

Mode | ${\mathit{\xi}}_{\mathit{ref}}$ | ${\mathit{\xi}}_{\mathit{ML}}$ | ${\mathit{\xi}}_{\mathit{AE}}$ | ${\mathbf{NDD}}_{\mathit{ref}-\mathit{ML}}$ | ${\mathbf{NDD}}_{\mathit{ref}-\mathit{AE}}$ |
---|---|---|---|---|---|

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 | $3.5$% | $3.7$% | $3.0$% | $6.1$% | $17.1$% |

**Table 5.**Model nodes considered; the node locations are illustrated in Figure 7.

Node | Description |
---|---|

N${}_{0}$, N${}_{9}$ | Shoe and rail nodes of the first LGS shoe |

N${}_{1}$, N${}_{11}$ | Shoe and rail nodes of the second LGS shoe |

N${}_{2}$, N${}_{10}$ | Shoe and rail nodes of the third LGS shoe |

N${}_{3}$, N${}_{8}$ | Shoe and rail nodes of the fourth LGS shoe |

N${}_{5}$ | WPT node |

N${}_{17}$, N${}_{18}$, N${}_{19}$, N${}_{27}$ | Machine bed nodes |

N${}_{20}$, N${}_{21}$, N${}_{22}$ | ME nodes |

N${}_{28}$ | Excitation node |

**Table 6.**Found values of the hyperparameters (see Table 1) in the Bayesian optimization step for different WPT positions.

${\mathit{p}}_{\mathit{max}}$ | ${\mathit{k}}_{\mathbf{\Delta}\mathit{f}}$ | ${\mathit{k}}_{\mathbf{\Delta}\mathit{D}}$ | ${\mathit{k}}_{\mathit{D}\mathit{max}}$ | ${\mathit{d}}_{\mathit{max}}$ | ${\mathit{q}}_{\mathit{min}}$ | ${\mathit{k}}_{\mathit{eukl}}$ | |
---|---|---|---|---|---|---|---|

z${}_{1}$ | 108 | 4.2% | 15.1% | 25.6% | 0.20 $\mathrm{Hz}$ | 27.9% | 0.50 |

z${}_{2}$ | 108 | 5.9% | 6.9% | 21.7% | 0.52 $\mathrm{Hz}$ | 28.8% | 0.61 |

z${}_{3}$ | 108 | 0.4% | 6.2% | 24.0% | 0.36 $\mathrm{Hz}$ | 28.6% | 0.55 |

z${}_{4}$ | 112 | 5.4% | 11.0% | 24.2% | 0.58 $\mathrm{Hz}$ | 28.2% | 1.64 |

**Table 7.**FRAC value statistics for FRFs measured for WPT position z${}_{2}$ for the machine tool use case.

Minimum | 10% Percentile | Mean | Median | Maximum | |
---|---|---|---|---|---|

AutoEMA | 61.0% | 82.0% | 88.6% | 90.1% | 94.7% |

MATLAB^{®} | 0.1% | 18.9% | 64.8% | 73.3% | 99.7% |

**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.

${\mathit{f}}_{1}^{\mathit{e}}$ | ${\mathit{f}}_{2}^{\mathit{e}}$ | ${\mathit{f}}_{3}^{\mathit{e}}$ | ${\mathit{f}}_{4}^{\mathit{e}}$ | ${\mathit{f}}_{5}^{\mathit{e}}$ | ${\mathit{f}}_{6}^{\mathit{e}}$ | ${\mathit{f}}_{7}^{\mathit{e}}$ | ${\mathit{f}}_{8}^{\mathit{e}}$ | |
---|---|---|---|---|---|---|---|---|

AutoEMA | 28.8 $\mathrm{Hz}$ | 31.0 $\mathrm{Hz}$ | 41.8 $\mathrm{Hz}$ | 52.1 $\mathrm{Hz}$ | 75.7 $\mathrm{Hz}$ | 95.0 $\mathrm{Hz}$ | 108.1 $\mathrm{Hz}$ | 120.0 $\mathrm{Hz}$ |

MATLAB^{®} | - | 31.3 $\mathrm{Hz}$ | 42.3 $\mathrm{Hz}$ | 52.3 $\mathrm{Hz}$ | 76.1 $\mathrm{Hz}$ | 95.5 $\mathrm{Hz}$ | 108.2 $\mathrm{Hz}$ | 120.1 $\mathrm{Hz}$ |

$\mathrm{N}\mathrm{F}{\mathrm{D}}_{AE-ML}$ | - | 1.1% | 1.2% | 0.3% | 0.4% | 0.6% | 0.1% | 0.1% |

**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.

${\mathit{\xi}}_{1}$ | ${\mathit{\xi}}_{2}$ | ${\mathit{\xi}}_{3}$ | ${\mathit{\xi}}_{4}$ | ${\mathit{\xi}}_{5}$ | ${\mathit{\xi}}_{6}$ | ${\mathit{\xi}}_{7}$ | ${\mathit{\xi}}_{8}$ | |
---|---|---|---|---|---|---|---|---|

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% |

$\mathrm{N}\mathrm{D}{\mathrm{D}}_{AE-ML}$ | - | 213.7% | 64.8% | 62.2% | 12.2% | 19.2% | 5.8% | 101.0% |

**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.

${\mathit{f}}_{1}^{\mathit{e}}$ | ${\mathit{f}}_{2}^{\mathit{e}}$ | ${\mathit{f}}_{3}^{\mathit{e}}$ | ${\mathit{f}}_{4}^{\mathit{e}}$ | ${\mathit{f}}_{5}^{\mathit{e}}$ | ${\mathit{f}}_{6}^{\mathit{e}}$ | ${\mathit{f}}_{7}^{\mathit{e}}$ | ${\mathit{f}}_{8}^{\mathit{e}}$ | |
---|---|---|---|---|---|---|---|---|

${z}_{1}$ | 28.7 $\mathrm{Hz}$ | 31.0 $\mathrm{Hz}$ | - | 52.0 $\mathrm{Hz}$ | 76.1 $\mathrm{Hz}$ | 91.9 $\mathrm{Hz}$ | 109.1 $\mathrm{Hz}$ | 117.3 $\mathrm{Hz}$ |

${z}_{2}$ | 28.8 $\mathrm{Hz}$ | 31.0 $\mathrm{Hz}$ | 41.8 $\mathrm{Hz}$ | 52.1 $\mathrm{Hz}$ | 75.7 $\mathrm{Hz}$ | 95.0 $\mathrm{Hz}$ | 108.1 $\mathrm{Hz}$ | 120.0 $\mathrm{Hz}$ |

${z}_{3}$ | 28.7 $\mathrm{Hz}$ | 31.0 $\mathrm{Hz}$ | 41.4 $\mathrm{Hz}$ | 52.1 $\mathrm{Hz}$ | 75.2 $\mathrm{Hz}$ | 97.9 $\mathrm{Hz}$ | 107.0 $\mathrm{Hz}$ | 122.4 $\mathrm{Hz}$ |

${z}_{4}$ | - | 31.0 $\mathrm{Hz}$ | - | 52.1 $\mathrm{Hz}$ | 75.0 $\mathrm{Hz}$ | - | 106.1 $\mathrm{Hz}$ | 127.3 $\mathrm{Hz}$ |

**Table 11.**Identified first eight modal damping ratios ${\xi}_{i}$ for the WPT position ${z}_{2}$ with the lowest number of found modes and the equivalent modes’ damping ratios at the other positions; “-” indicates that no matching mode with a MAC value of at least 80% was found.

${\mathit{\xi}}_{1}$ | ${\mathit{\xi}}_{2}$ | ${\mathit{\xi}}_{3}$ | ${\mathit{\xi}}_{4}$ | ${\mathit{\xi}}_{5}$ | ${\mathit{\xi}}_{6}$ | ${\mathit{\xi}}_{7}$ | ${\mathit{\xi}}_{8}$ | |
---|---|---|---|---|---|---|---|---|

${z}_{1}$ | 1.04% | 0.85% | - | 1.18% | 1.93% | 3.59% | 0.33% | 1.54% |

${z}_{2}$ | 1.05% | 0.88% | 1.67% | 1.04% | 1.98% | 3.82% | 0.29% | 1.50% |

${z}_{3}$ | 1.12% | 0.86% | 2.34% | 1.03% | 1.98% | 2.27% | 0.29% | 2.66% |

${z}_{4}$ | - | 0.96% | - | 1.01% | 2.16% | - | 0.32% | 2.93% |

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**MDPI and ACS Style**

Ellinger, J.; Beck, L.; Benker, M.; Hartl, R.; Zaeh, M.F.
Automation of Experimental Modal Analysis Using Bayesian Optimization. *Appl. Sci.* **2023**, *13*, 949.
https://doi.org/10.3390/app13020949

**AMA Style**

Ellinger J, Beck L, Benker M, Hartl R, Zaeh MF.
Automation of Experimental Modal Analysis Using Bayesian Optimization. *Applied Sciences*. 2023; 13(2):949.
https://doi.org/10.3390/app13020949

**Chicago/Turabian Style**

Ellinger, Johannes, Leopold Beck, Maximilian Benker, Roman Hartl, and Michael F. Zaeh.
2023. "Automation of Experimental Modal Analysis Using Bayesian Optimization" *Applied Sciences* 13, no. 2: 949.
https://doi.org/10.3390/app13020949