<table><caption>Table 12. *Cont.*</caption><thead><tr><th>Technique</th><th>What Is Needed?</th><th>How to?</th><th>Drawbacks</th><th>Damage</th><th>Detected Size</th></tr></thead><tbody><tr><td rowspan="3">Damping</td><td rowspan="3">The vibration of the structure is needed, and the damping can be obtained with several techniques, but the most classical one is the -3 dB method.</td><td rowspan="3">Send a white noise or a sweep signal with a large frequency band. Compare the damping parameter of intact and damaged structures.</td><td rowspan="3">Mainly influenced by operational factors and uncertainty in the damping characterization.</td><td>Delamination</td><td>>48%</td></tr><tr><td>Fatigue</td><td>3rd cycle</td></tr><tr><td>Impact</td><td>BVID</td></tr><tr><td rowspan="2">FRF</td><td rowspan="2">Input and output are needed to build the FRF.</td><td rowspan="2">Some metrics can be used to observe differences in the FRF of intact and damaged structures for a large frequency range.</td><td rowspan="2"></td><td>Impact</td><td></td></tr><tr><td>Hole</td><td></td></tr><tr><td rowspan="2">TF</td><td rowspan="2">The displacements or velocities at several locations of the structure for the same input are needed.</td><td rowspan="2">A ratio of two displacements or velocities has to be made, and a damage indicator is calculated from this ratio.</td><td rowspan="2">A large number of sensors or heavy processes are needed.</td><td>Bonding steel plate</td><td>>20%</td></tr><tr><td>Impact</td><td>>5 J</td></tr><tr><td>WT</td><td>Vibration output in the time domain.</td><td>The general vibration of the structure is needed, and the time domain signal should be divided into several sub-signals to compare each sub-signal energy.</td><td>Extra processing is needed, and the correct choice of wavelets should be made.</td><td>Delamination</td><td>>0.12%</td></tr></tbody></table>

<figure><img src="image_45.png" /><figcaption>Figure 43. Decision tree of vibration methods for specific damage in PMC structure.</figcaption></figure>

<table><caption>Table 13. Nonlinear vibration methods for SHM (Detected Size represents the minimum percentage of the detected damaged area divided by the total area).</caption><thead><tr><th>Technique</th><th>What Is Needed?</th><th>How to?</th><th>Drawbacks</th><th>Damage</th><th>Detected Size</th></tr></thead><tbody><tr><td rowspan="2">SFE</td><td rowspan="2">Global knowledge of the vibration of the structure to choose the best frequency for the harmonic signal excitation.</td><td rowspan="2">Compare damaged and intact structures through distortions in time-domain and phase portraits, sub- and super-harmonics, and damage indicators.</td><td rowspan="2">Small peaks can be hidden in the noise in operational detection.</td><td>Impact</td><td>BVID (&gt;7 J)</td></tr><tr><td>Delamination</td><td>>3%</td></tr><tr><td rowspan="3">Nonlinear resonance (NLR) and damping</td><td rowspan="3">Global knowledge of the vibration of the structure to choose the natural frequencies used for this technique with a harmonic or a sweep signal around these frequencies.</td><td rowspan="3">Compare the resonance curves obtained from different amplitudes of excitation for damaged and intact structures. Use nonlinear elastic and dissipative parameters as damage indicators.</td><td rowspan="3">A lot of tests have to be performed to obtain the resonance curves (10 in general) for 1 natural frequency. The frequency shift is also influenced by the boundary condition.</td><td>Debonding</td><td>>8%</td></tr><tr><td>Fatigue crack</td><td>1st cycle</td></tr><tr><td>Impact</td><td>BVID (10 J and 2400 times 3 J)</td></tr></tbody></table>
<table><caption>Table 12. *Cont.*</caption><thead><tr><th>Technique</th><th>What Is Needed?</th><th>How to?</th><th>Drawbacks</th><th>Damage</th><th>Detected Size</th></tr></thead><tbody><tr><td rowspan="3">Damping</td><td rowspan="3">The vibration of the structure is needed, and the damping can be obtained with several techniques, but the most classical one is the -3 dB method.</td><td rowspan="3">Send a white noise or a sweep signal with a large frequency band. Compare the damping parameter of intact and damaged structures.</td><td rowspan="3">Mainly influenced by operational factors and uncertainty in the damping characterization.</td><td>Delamination</td><td>>48%</td></tr><tr><td>Fatigue</td><td>3rd cycle</td></tr><tr><td>Impact</td><td>BVID</td></tr><tr><td rowspan="2">FRF</td><td rowspan="2">Input and output are needed to build the FRF.</td><td rowspan="2">Some metrics can be used to observe differences in the FRF of intact and damaged structures for a large frequency range.</td><td rowspan="2"></td><td>Impact</td><td></td></tr><tr><td>Hole</td><td></td></tr><tr><td rowspan="2">TF</td><td rowspan="2">The displacements or velocities at several locations of the structure for the same input are needed.</td><td rowspan="2">A ratio of two displacements or velocities has to be made, and a damage indicator is calculated from this ratio.</td><td rowspan="2">A large number of sensors or heavy processes are needed.</td><td>Bonding steel plate</td><td>>20%</td></tr><tr><td>Impact</td><td>>5 J</td></tr><tr><td>WT</td><td>Vibration output in the time domain.</td><td>The general vibration of the structure is needed, and the time domain signal should be divided into several sub-signals to compare each sub-signal energy.</td><td>Extra processing is needed, and the correct choice of wavelets should be made.</td><td>Delamination</td><td>>0.12%</td></tr></tbody></table>

<figure><img src="image_46.png" /><figcaption>Figure 43. Decision tree of vibration methods for specific damage in PMC structure.</figcaption></figure>

<table><caption>Table 13. Nonlinear vibration methods for SHM (Detected Size represents the minimum percentage of the detected damaged area divided by the total area).</caption><thead><tr><th>Technique</th><th>What Is Needed?</th><th>How to?</th><th>Drawbacks</th><th>Damage</th><th>Detected Size</th></tr></thead><tbody><tr><td rowspan="2">SFE</td><td rowspan="2">Global knowledge of the vibration of the structure to choose the best frequency for the harmonic signal excitation.</td><td rowspan="2">Compare damaged and intact structures through distortions in time-domain and phase portraits, sub- and super-harmonics, and damage indicators.</td><td rowspan="2">Small peaks can be hidden in the noise in operational detection.</td><td>Impact</td><td>BVID (&gt;7 J)</td></tr><tr><td>Delamination</td><td>>3%</td></tr><tr><td rowspan="3">Nonlinear resonance (NLR) and damping</td><td rowspan="3">Global knowledge of the vibration of the structure to choose the natural frequencies used for this technique with a harmonic or a sweep signal around these frequencies.</td><td rowspan="3">Compare the resonance curves obtained from different amplitudes of excitation for damaged and intact structures. Use nonlinear elastic and dissipative parameters as damage indicators.</td><td rowspan="3">A lot of tests have to be performed to obtain the resonance curves (10 in general) for 1 natural frequency. The frequency shift is also influenced by the boundary condition.</td><td>Debonding</td><td>>8%</td></tr><tr><td>Fatigue crack</td><td>1st cycle</td></tr><tr><td>Impact</td><td>BVID (10 J and 2400 times 3 J)</td></tr></tbody></table>
<table><caption>Table 13. *Cont.*</caption><thead><tr><th>Technique</th><th>What Is Needed?</th><th>How to?</th><th>Drawbacks</th><th>Damage</th><th>Detected Size</th></tr></thead><tbody><tr><td rowspan="2">VAM</td><td rowspan="2">Global knowledge of the vibration of the structure to choose the best frequency of excitation for the pumping signal and, in some cases, also for probing one.</td><td rowspan="2">Compare the occurrence of super-harmonics and sidebands between damaged and intact structures. Damage indicators based on their amplitude are also used.</td><td rowspan="2">This method is influenced a lot by the boundary conditions, so special care should be made with VAM.</td><td>Impact</td><td>BVID (&gt;1.8 J)</td></tr><tr><td>Crack</td><td></td></tr><tr><td rowspan="2">Nonlinear Damping (NLD)</td><td rowspan="2">Time-frequency domain signal is needed in order to perform the CWT.</td><td rowspan="2">Extract the instantaneous amplitude and phase of the response to calculate the instantaneous damping. Compare the variation of it between intact and damaged structures.</td><td rowspan="2">Post-processing techniques (CWT) have to be used to obtain the instantaneous parameters, which are necessary for the damping calculation.</td><td>Delamination</td><td>&gt;32%</td></tr><tr><td>Fatigue crack</td><td>2nd cycle</td></tr></tbody></table>

## 5. Conclusions

An important amount of techniques and the main methods of vibration analysis for SHM have been presented in this article, with a focus on PMC materials. The goal of this review was to study the techniques, used equipment, and configurations, as well as their effectiveness and, for some techniques, their drawbacks and limitations. We also reviewed the different damage mechanisms detected or not with these techniques, which allows for selecting the best method according to the studied damage.

Firstly, the classical linear methods have been introduced, even though they are considered insufficient to detect small and complex damage in the composite structures, and secondly, the nonlinear methods have shown interesting capacities in the last few years in detecting and locating damage with nonlinear behavior of the structure. The nonlinear methods are still in development, and nowadays there already exist three main techniques that can easily be used to perform SHM of PMC: the SFE, the VAM, and the nonlinear resonance.

This review highlighted the possibility of using vibration analysis for SHM of PMC. However, some limitations have been observed, such as relatively low sensitivity to early-stage damage, environmental effects, optimal sensor position, signal processing, etc. In the introduction, the concept of SHM has been presented, and from the cited articles, it is clear that the step of qualifying the damage mechanism is not yet well-developed in vibration analysis, whereas detection and localization are possible through several techniques. This indicates that further advancements are needed in this field, whether through the development of more optimized and/or internal sensors within the structure (smart materials), improved signal processing, or the development of new vibration analysis methods. A growing interest in Artificial Intelligence (AI) [151] has showed promising potential in the context of data processing in SHM and vibration analysis. This could become a significant asset for advancing the qualification aspect of SHM by enabling real-time data interpretation and adaptive damage detection strategies and thus providing a more robust and automated assessment of service life estimation. In addition, most of them investigated the presence of damage on laboratory-scale composite structures. This is important to take into consideration as the transposition to the operational scale still needs improvement and development.

To counteract these limitations for SHM diagnostics, current studies focus on combining the results of several vibration methods and also with other NDT results (such as AE, US, etc.). Indeed, this combination permits us to obtain a multi-sources and multi-physics response of the structure, thus enhancing the diagnostics. Indeed, a large range of NDT exists, which allows for combining them and obtaining different information about the damaged structure. Here, the AI algorithms have also demonstrated that they can become a major element in this combination in order to visualize similar patterns induced by the different types of damage through all of these techniques. This will help to avoid fake detection.
<table><caption>Table 13. *Cont.*</caption><thead><tr><th>Technique</th><th>What Is Needed?</th><th>How to?</th><th>Drawbacks</th><th>Damage</th><th>Detected Size</th></tr></thead><tbody><tr><td>VAM</td><td>Global knowledge of the vibration of the structure to choose the best frequency of excitation for the pumping signal and, in some cases, also for probing one.</td><td>Compare the occurrence of super-harmonics and sidebands between damaged and intact structures. Damage indicators based on their amplitude are also used.</td><td>This method is influenced a lot by the boundary conditions, so special care should be made with VAM.</td><td>Impact</td><td>BVID (&gt;1.8 J)</td></tr><tr><td>Nonlinear Damping (NLD)</td><td>Time-frequency domain signal is needed in order to perform the CWT.</td><td>Extract the instantaneous amplitude and phase of the response to calculate the instantaneous damping. Compare the variation of it between intact and damaged structures.</td><td>Post-processing techniques (CWT) have to be used to obtain the instantaneous parameters, which are necessary for the damping calculation.</td><td>Crack</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td>Delamination</td><td>&gt;32%</td></tr><tr><td></td><td></td><td></td><td></td><td>Fatigue crack</td><td>2nd cycle</td></tr></tbody></table>

## 5. Conclusions

An important amount of techniques and the main methods of vibration analysis for SHM have been presented in this article, with a focus on PMC materials. The goal of this review was to study the techniques, used equipment, and configurations, as well as their effectiveness and, for some techniques, their drawbacks and limitations. We also reviewed the different damage mechanisms detected or not with these techniques, which allows for selecting the best method according to the studied damage.

Firstly, the classical linear methods have been introduced, even though they are considered insufficient to detect small and complex damage in the composite structures, and secondly, the nonlinear methods have shown interesting capacities in the last few years in detecting and locating damage with nonlinear behavior of the structure. The nonlinear methods are still in development, and nowadays there already exist three main techniques that can easily be used to perform SHM of PMC: the SFE, the VAM, and the nonlinear resonance.

This review highlighted the possibility of using vibration analysis for SHM of PMC. However, some limitations have been observed, such as relatively low sensitivity to early-stage damage, environmental effects, optimal sensor position, signal processing, etc. In the introduction, the concept of SHM has been presented, and from the cited articles, it is clear that the step of qualifying the damage mechanism is not yet well-developed in vibration analysis, whereas detection and localization are possible through several techniques. This indicates that further advancements are needed in this field, whether through the development of more optimized and/or internal sensors within the structure (smart materials), improved signal processing, or the development of new vibration analysis methods. A growing interest in Artificial Intelligence (AI) [151] has showed promising potential in the context of data processing in SHM and vibration analysis. This could become a significant asset for advancing the qualification aspect of SHM by enabling real-time data interpretation and adaptive damage detection strategies and thus providing a more robust and automated assessment of service life estimation. In addition, most of them investigated the presence of damage on laboratory-scale composite structures. This is important to take into consideration as the transposition to the operational scale still needs improvement and development.

To counteract these limitations for SHM diagnostics, current studies focus on combining the results of several vibration methods and also with other NDT results (such as AE, US, etc.). Indeed, this combination permits us to obtain a multi-sources and multi-physics response of the structure, thus enhancing the diagnostics. Indeed, a large range of NDT exists, which allows for combining them and obtaining different information about the damaged structure. Here, the AI algorithms have also demonstrated that they can become a major element in this combination in order to visualize similar patterns induced by the different types of damage through all of these techniques. This will help to avoid fake detec-
tion/localization and to increase accuracy in the SHM process, with the goal of qualifying the damage mechanisms efficiently.

**Author Contributions:** Conceptualization, W.H., Z.A. and L.D.; methodology, W.H., Z.A. and L.D.; formal analysis, W.H. and L.D.; investigation, W.H. and L.D.; data curation, L.D.; writing—original draft preparation, W.H. and L.D.; writing—review and editing, W.H., Z.A. and L.D.; supervision, W.H., Z.A. and L.D.; project administration, W.H. and Z.A.; funding acquisition, W.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** The authors would like to thank the Ministère de l'Enseignement Supérieur et de la Recherche of France for funding this work as part of the doctoral thesis of Loan Dolbachian. This research was funded by Ministère de l'Enseignement Supérieur et de la Recherche of France.

**Data Availability Statement:** The raw/processed data required to reproduce these findings cannot be shared at this time as the data also form part of an ongoing study.

**Conflicts of Interest:** The authors declare no conflicts of interest.

## Glossary

<table><tr><td>AE</td><td>Acoustic Emission</td></tr><tr><td>AI</td><td>Artificial Intelligence</td></tr><tr><td>BVID</td><td>Barely Visible Impact Damage</td></tr><tr><td>CFRP</td><td>Carbon-Fiber Reinforced Polymer</td></tr><tr><td>CMC</td><td>Ceramic-Matrix Composite</td></tr><tr><td>DaDI</td><td>Damping Damage Indicator</td></tr><tr><td>DI</td><td>Damage Indicator</td></tr><tr><td>DIC</td><td>Digital Image Correlation</td></tr><tr><td>FEP</td><td>Fluorinated Ethylene Propylene</td></tr><tr><td>FFT</td><td>Fast Fourier Transform</td></tr><tr><td>FRF</td><td>Frequency Response Function</td></tr><tr><td>GFRP</td><td>Glass-Fiber-Reinforced Polymer</td></tr><tr><td>LDR</td><td>Local Defect Resonance</td></tr><tr><td>LV</td><td>Laser Vibrometer</td></tr><tr><td>MAC</td><td>Modal Assurance Criterion</td></tr><tr><td>MF</td><td>Modal Flexibility</td></tr><tr><td>MFC</td><td>Macro-Fiber Composite</td></tr><tr><td>MMC</td><td>Metal-Matrix Composite</td></tr><tr><td>MSC</td><td>Modal Shape Curvature</td></tr><tr><td>MSD</td><td>Modal Shape Displacement</td></tr><tr><td>MSE</td><td>Modal Strain Energy</td></tr><tr><td>NDT</td><td>Non-Destructive Techniques</td></tr><tr><td>NWMS</td><td>Nonlinear Wave Modulation Technique</td></tr><tr><td>PFC</td><td>Piezo-Fiber Composite</td></tr><tr><td>PM</td><td>Process Monitoring</td></tr><tr><td>PMC</td><td>Polymer-Matrix Composite</td></tr><tr><td>PSF</td><td>Plane Shape Function</td></tr><tr><td>PVC</td><td>PolyVinyl Chloride</td></tr><tr><td>PVDF</td><td>Polyvinylidene Fluoride</td></tr><tr><td>PZT</td><td>Lead Zirconate Titanate</td></tr><tr><td>SFE</td><td>Single Frequency Excitation</td></tr><tr><td>SHM</td><td>Structural Health Monitoring</td></tr><tr><td>SWP</td><td>Sweep Signal Excitation</td></tr><tr><td>TF</td><td>Transmittance or Transmissibility Function</td></tr><tr><td>UT</td><td>Ultrasonic Testing</td></tr><tr><td>VAM</td><td>Vibro-Acoustic Modulation</td></tr></table>
tion/localization and to increase accuracy in the SHM process, with the goal of qualifying the damage mechanisms efficiently.

**Author Contributions:** Conceptualization, W.H., Z.A. and L.D.; methodology, W.H., Z.A. and L.D.; formal analysis, W.H. and L.D.; investigation, W.H. and L.D.; data curation, L.D.; writing—original draft preparation, W.H. and L.D.; writing—review and editing, W.H., Z.A. and L.D.; supervision, W.H., Z.A. and L.D.; project administration, W.H. and Z.A.; funding acquisition, W.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** The authors would like to thank the Ministère de l'Enseignement Supérieur et de la Recherche of France for funding this work as part of the doctoral thesis of Loan Dolbachian. This research was funded by Ministère de l'Enseignement Supérieur et de la Recherche of France.

**Data Availability Statement:** The raw/processed data required to reproduce these findings cannot be shared at this time as the data also form part of an ongoing study.

**Conflicts of Interest:** The authors declare no conflicts of interest.

## Glossary

<table><tr><td>AE</td><td>Acoustic Emission</td></tr><tr><td>AI</td><td>Artificial Intelligence</td></tr><tr><td>BVID</td><td>Barely Visible Impact Damage</td></tr><tr><td>CFRP</td><td>Carbon-Fiber Reinforced Polymer</td></tr><tr><td>CMC</td><td>Ceramic-Matrix Composite</td></tr><tr><td>DaDI</td><td>Damping Damage Indicator</td></tr><tr><td>DI</td><td>Damage Indicator</td></tr><tr><td>DIC</td><td>Digital Image Correlation</td></tr><tr><td>FEP</td><td>Fluorinated Ethylene Propylene</td></tr><tr><td>FFT</td><td>Fast Fourier Transform</td></tr><tr><td>FRF</td><td>Frequency Response Function</td></tr><tr><td>GFRP</td><td>Glass-Fiber-Reinforced Polymer</td></tr><tr><td>LDR</td><td>Local Defect Resonance</td></tr><tr><td>LV</td><td>Laser Vibrometer</td></tr><tr><td>MAC</td><td>Modal Assurance Criterion</td></tr><tr><td>MF</td><td>Modal Flexibility</td></tr><tr><td>MFC</td><td>Macro-Fiber Composite</td></tr><tr><td>MMC</td><td>Metal-Matrix Composite</td></tr><tr><td>MSC</td><td>Modal Shape Curvature</td></tr><tr><td>MSD</td><td>Modal Shape Displacement</td></tr><tr><td>MSE</td><td>Modal Strain Energy</td></tr><tr><td>NDT</td><td>Non-Destructive Techniques</td></tr><tr><td>NWMS</td><td>Nonlinear Wave Modulation Technique</td></tr><tr><td>PFC</td><td>Piezo-Fiber Composite</td></tr><tr><td>PM</td><td>Process Monitoring</td></tr><tr><td>PMC</td><td>Polymer-Matrix Composite</td></tr><tr><td>PSF</td><td>Plane Shape Function</td></tr><tr><td>PVC</td><td>PolyVinyl Chloride</td></tr><tr><td>PVDF</td><td>Polyvinylidene Fluoride</td></tr><tr><td>PZT</td><td>Lead Zirconate Titanate</td></tr><tr><td>SFE</td><td>Single Frequency Excitation</td></tr><tr><td>SHM</td><td>Structural Health Monitoring</td></tr><tr><td>SWP</td><td>Sweep Signal Excitation</td></tr><tr><td>TF</td><td>Transmittance or Transmissibility Function</td></tr><tr><td>UT</td><td>Ultrasonic Testing</td></tr><tr><td>VAM</td><td>Vibro-Acoustic Modulation</td></tr></table>
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67. Idriss, M.; El Mahi, A.; El Guerjouma, R. Characterization of Sandwich Beams with Debonding by Linear and Nonlinear Vibration Method. *Compos. Struct.* **2015**, *120*, 200–207. [CrossRef]

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85. Qiao, P.; Lestari, W.; Shah, M.G.; Wang, J. Dynamics-Based Damage Detection of Composite Laminated Beams Using Contact and Noncontact Measurement Systems. *J. Compos. Mater.* **2007**, *41*, 1217–1252. [CrossRef]

86. Sampaio, R.P.C.; Maia, N.M.M.; Silva, J.M.M. Damage Detection Using the Frequency-Response-Function Curvature Method. *J. Sound. Vib.* **1999**, *226*, 1029–1042. [CrossRef]

87. Pandey, A.K.; Biswas, M.; Samman, M.M. Damage Detection from Changes in Curvature Mode Shapes. *J. Sound. Vib.* **1991**, *145*, 321–332. [CrossRef]

88. Radzieński, M.; Krawczuk, M.; Palacz, M. Improvement of Damage Detection Methods Based on Experimental Modal Parameters. *Mech. Syst. Signal Process.* **2011**, *25*, 2169–2190. [CrossRef]

89. Govindasamy, M.; Kamalakannan, G.; Kesavan, C.; Meenashisundaram, G.K. Damage Detection in Glass/Epoxy Laminated Composite Plates Using Modal Curvature for Structural Health Monitoring Applications. *J. Compos. Sci.* **2020**, *4*, 185. [CrossRef]

90. Lestari, W.; Qiao, P.; Hanagud, S. Curvature Mode Shape-Based Damage Assessment of Carbon/Epoxy Composite Beams. *J. Intell. Mater. Syst. Struct.* **2007**, *18*, 189–208. [CrossRef]

91. Hamey, C.S.; Lestari, W.; Qiao, P.; Song, G. Experimental Damage Identification of Carbon/Epoxy Composite Beams Using Curvature Mode Shapes. *Struct. Health Monit.* **2004**, *3*, 333–353. [CrossRef]

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93. Mehdizadeh, M. Curvature Mode Shape Analyses of Damage in Structures. Ph.D. Thesis, School of Aerospace, Mechanical & Manufacturing Engineering College of Science, Engineering and Technology, Bundoora, Australia, 2009.

94. Sazonov, E.; Klinkhachorn, P. Optimal Spatial Sampling Interval for Damage Detection by Curvature or Strain Energy Mode Shapes. *J. Sound. Vib.* **2005**, *285*, 783–801. [CrossRef]

95. Cha, Y.J.; Buyukozturk, O. Structural Damage Detection Using Modal Strain Energy and Hybrid Multiobjective Optimization. *Comput. Aided Civ. Infrastruct. Eng.* **2015**, *30*, 347–358. [CrossRef]

96. Eraky, A.; Anwar, A.M.; Saad, A.; Abdo, A. Damage Detection of Flexural Structural Systems Using Damage Index Method—Experimental Approach. *Alex. Eng. J.* **2015**, *54*, 497–507. [CrossRef]

97. Wang, S.; Xu, M. Modal Strain Energy-Based Structural Damage Identification: A Review and Comparative Study. *Struct. Eng. Int.* **2019**, *29*, 234–248. [CrossRef]

98. Hu, H.; Wang, J. Damage Detection of a Woven Fabric Composite Laminate Using a Modal Strain Energy Method. *Eng. Struct.* **2009**, *31*, 1042–1055. [CrossRef]

99. Hu, H.; Lee, C.H.; Wu, C.B.; Lu, W.J. Detection of Matrix Cracks in Composite Laminates by Using the Modal Strain Energy Method. *Mech. Compos. Mater.* **2010**, *46*, 117–132. [CrossRef]

100. dos Santos, F.L.M.; Peeters, B.; van der Auweraer, H.; Góes, L.C.S. Modal Strain Energy Based Damage Detection Applied to a Full Scale Composite Helicopter Blade. *Key Eng. Mater.* **2013**, *569–570*, 457–464.

101. Kumar, M.; Shenoi, R.A.; Cox, S.J. Experimental Validation of Modal Strain Energies Based Damage Identification Method for a Composite Sandwich Beam. *Compos. Sci. Technol.* **2009**, *69*, 1635–1643. [CrossRef]

102. Meng, F.; Yu, J.; Alaluf, D.; Mokrani, B.; Preumont, A. Modal Analysis and Damage Detection for Suspension Bridges: A Numerical and Experimental Investigation. *Smart Struct. Syst.* **2019**, *23*, 15–29.

103. Yang, J.S.; da Liu, Z.; Schmidt, R.; Schröder, K.U.; Ma, L.; Wu, L.Z. Vibration-Based Damage Diagnosis of Composite Sandwich Panels with Bi-Directional Corrugated Lattice Cores. *Compos. Part. A Appl. Sci. Manuf.* **2020**, *131*, 105781. [CrossRef]

104. Cao, M.S.; Sha, G.G.; Gao, Y.F.; Ostachowicz, W. Structural Damage Identification Using Damping: A Compendium of Uses and Features. *Smart Mater. Struct.* **2017**, *26*, 043001. [CrossRef]

105. Inman, D.J. *Engineering Vibration*; Pearson Education Limited: London, UK, 1994; ISBN 978-0-13-287169-3.

106. Slavič, J.; Simonovski, I.; Boltežar, M. Damping Identification Using a Continuous Wavelet Transform: Application to Real Data. *J. Sound. Vib.* **2003**, *262*, 291–307. [CrossRef]

107. Montalvão, D.; Ribeiro, A.M.R.; Duarte-Silva, J. A Method for the Localization of Damage in a CFRP Plate Using Damping. *Mech. Syst. Signal Process.* **2009**, *23*, 1846–1854. [CrossRef]

108. Montalvão, D.; Ribeiro, A.M.R.; Duarte-Silva, J.A.B. Experimental Assessment of a Modal-Based Multi-Parameter Method for Locating Damage in Composite Laminates. *Exp. Mech.* **2011**, *51*, 1473–1488. [CrossRef]

109. Montalvão, D.; Karanatsis, D.; Ribeiro, A.M.; Arina, J.; Baxter, R. An Experimental Study on the Evolution of Modal Damping with Damage in Carbon Fiber Laminates. *J. Compos. Mater.* **2015**, *49*, 2403–2413. [CrossRef]

110. Kessler, S.S.; Spearing, S.M.; Atalla, M.J.; Cesnik, C.E.S.; Soutis, C. Damage Detection in Composite Materials Using Frequency Response Methods. *Compos. Part B* **2002**, *33*, 87–95. [CrossRef]

111. Sampaio, R.P.C.; Maia, N.M.M.; Silva, J.M.M. The Frequency Domain Assurance Criterion as a Tool for Damage Detection. *Key Eng. Mater.* **2003**, *245–246*, 69–76. [CrossRef]

112. Chen, A. Frequency Response Assurance Criterion and Applications to Model Correlation of Body Structures. *SAE Tech. Pap.* **2003**. [CrossRef]
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114. Chesné, S.; Deraemaeker, A. Damage Localization Using Transmissibility Functions: A Critical Review. *Mech. Syst. Signal Process.* **2013**, *38*, 569–584. [CrossRef]

115. Zhang, H.; Schulz, M.J.; Ferguson, F.; Pai, P.F. Structural Health Monitoring Using Transmittance Functions. *Mech. Syst. Signal Process.* **1999**, *13*, 765–787. [CrossRef]

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117. Poulimenos, A.G.; Sakellariou, J.S. A Transmittance-Based Methodology for Damage Detection under Uncertainty: An Application to a Set of Composite Beams with Manufacturing Variability Subject to Impact Damage and Varying Operating Conditions. *Struct. Health Monit.* **2019**, *18*, 318–333. [CrossRef]

118. Liu, C.-L. A Tutorial of the Wavelet Transform. *NTUEE* **2010**, *21*, 22.

119. Yan, Y.J.; Yam, L.H. Detection of Delamination Damage in Composite Plates Using Energy Spectrum of Structural Dynamic Responses Decomposed by Wavelet Analysis. *Comput. Struct.* **2004**, *82*, 347–358. [CrossRef]

120. Singh, A.K. Detection of Defects in Composite Laminates with Vibro-Ultrasonic Methods. Ph.D. Thesis, National University of Singapore, Singapore, 2019.

121. Andreaus, U.; Baragatti, P. Experimental Damage Detection of Cracked Beams by Using Nonlinear Characteristics of Forced Response. *Mech. Syst. Signal Process.* **2012**, *31*, 382–404. [CrossRef]

122. Dolbachian, L.; Harizi, W.; Aboura, Z. Structural Health Monitoring (SHM) Study of Polymer Matrix Composite (PMC) Materials Using Nonlinear Vibration Methods Based on Embedded Piezoelectric Transducers. *Sensors* **2023**, *23*, 3677. [CrossRef]

123. Andreades, C.; Ciampa, F. CFRP Composites with Embedded PZT Transducers for Nonlinear Ultrasonic Inspection of Space Structures. In Proceedings of the EWSHM2018, Manchester, UK, 10–13 July 2018.

124. Andreades, C.; Malfense Fierro, G.P.; Meo, M. A Nonlinear Ultrasonic SHM Method for Impact Damage Localisation in Composite Panels Using a Sparse Array of Piezoelectric PZT Transducers. *Ultrasonics* **2020**, *108*, 106181. [CrossRef]

125. Polimeno, U.; Meo, M.; Almond, D.P. Smart Nonlinear Acoustic Based Structural Health Monitoring System. *Adv. Sci. Technol.* **2008**, *56*, 426–434. [CrossRef]

126. Loi, G.; Porcu, M.C.; Aymerich, F. Impact Damage Detection in Composite Beams by Analysis of Non-Linearity under Pulse Excitation. *J. Compos. Sci.* **2021**, *5*, 39. [CrossRef]

127. Ooijevaar, T.H.; Rogge, M.D.; Loendersloot, R.; Warnet, L.L.; Akkerman, R.; Tinga, T. Nonlinear Dynamic Behavior of an Impact Damaged Composite Skin-Stiffener Structure. *J. Sound. Vib.* **2015**, *353*, 243–258. [CrossRef]

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129. Meo, M.; Polimeno, U.; Zumpano, G. Detecting Damage in Composite Material Using Nonlinear Elastic Wave Spectroscopy Methods. *Appl. Compos. Mater.* **2008**, *15*, 115–126. [CrossRef]

130. Polimeno, U.; Meo, M. Understanding the Effect of Boundary Conditions on Damage Identification Process When Using Nonlinear Elastic Wave Spectroscopy Methods. *Int. J. Non Linear Mech.* **2008**, *43*, 187. [CrossRef]

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132. Wei, Q.; Zhu, L.; Zhu, J.; Zhuo, L.; Hao, W.; Xie, W. Characterization of Impact Fatigue Damage in CFRP Composites Using Nonlinear Acoustic Resonance Method. *Compos. Struct.* **2020**, *253*, 112804. [CrossRef]

133. van den Abeele, K.E.A.; Sutin, A.; Carmeliet, J.; Johnson, P.A. Micro-Damage Diagnostics Using Nonlinear Elastic Wave Spectroscopy (NEWS). *NDT E Int.* **2001**, *34*, 239–248. [CrossRef]

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114. Chesné, S.; Deraemaeker, A. Damage Localization Using Transmissibility Functions: A Critical Review. *Mech. Syst. Signal Process.* **2013**, *38*, 569–584. [CrossRef]

115. Zhang, H.; Schulz, M.J.; Ferguson, F.; Pai, P.F. Structural Health Monitoring Using Transmittance Functions. *Mech. Syst. Signal Process.* **1999**, *13*, 765–787. [CrossRef]

116. Sundaresan, M.; Ferguson, F.; Pai, P.; Sundaresan, M.J.; Schulz, M.J.; Hill, J.; Wheater, E.A.; Ferguson, F.; Pai, P.F. Damage Detection on a Wind Turbine Blade Section. In Proceedings of the International Society for Optical Engineering, San Jose, CA, USA, 26–28 January 1999.

117. Poulimenos, A.G.; Sakellariou, J.S. A Transmittance-Based Methodology for Damage Detection under Uncertainty: An Application to a Set of Composite Beams with Manufacturing Variability Subject to Impact Damage and Varying Operating Conditions. *Struct. Health Monit.* **2019**, *18*, 318–333. [CrossRef]

118. Liu, C.-L. A Tutorial of the Wavelet Transform. *NTUEE* **2010**, *21*, 22.

119. Yan, Y.J.; Yam, L.H. Detection of Delamination Damage in Composite Plates Using Energy Spectrum of Structural Dynamic Responses Decomposed by Wavelet Analysis. *Comput. Struct.* **2004**, *82*, 347–358. [CrossRef]

120. Singh, A.K. Detection of Defects in Composite Laminates with Vibro-Ultrasonic Methods. Ph.D. Thesis, National University of Singapore, Singapore, 2019.

121. Andreaus, U.; Baragatti, P. Experimental Damage Detection of Cracked Beams by Using Nonlinear Characteristics of Forced Response. *Mech. Syst. Signal Process.* **2012**, *31*, 382–404. [CrossRef]

122. Dolbachian, L.; Harizi, W.; Aboura, Z. Structural Health Monitoring (SHM) Study of Polymer Matrix Composite (PMC) Materials Using Nonlinear Vibration Methods Based on Embedded Piezoelectric Transducers. *Sensors* **2023**, *23*, 3677. [CrossRef]

123. Andreades, C.; Ciampa, F. CFRP Composites with Embedded PZT Transducers for Nonlinear Ultrasonic Inspection of Space Structures. In Proceedings of the EWSHM2018, Manchester, UK, 10–13 July 2018.

124. Andreades, C.; Malfense Fierro, G.P.; Meo, M. A Nonlinear Ultrasonic SHM Method for Impact Damage Localisation in Composite Panels Using a Sparse Array of Piezoelectric PZT Transducers. *Ultrasonics* **2020**, *108*, 106181. [CrossRef]

125. Polimeno, U.; Meo, M.; Almond, D.P. Smart Nonlinear Acoustic Based Structural Health Monitoring System. *Adv. Sci. Technol.* **2008**, *56*, 426–434. [CrossRef]

126. Loi, G.; Porcu, M.C.; Aymerich, F. Impact Damage Detection in Composite Beams by Analysis of Non-Linearity under Pulse Excitation. *J. Compos. Sci.* **2021**, *5*, 39. [CrossRef]

127. Ooijevaar, T.H.; Rogge, M.D.; Loendersloot, R.; Warnet, L.L.; Akkerman, R.; Tinga, T. Nonlinear Dynamic Behavior of an Impact Damaged Composite Skin-Stiffener Structure. *J. Sound. Vib.* **2015**, *353*, 243–258. [CrossRef]

128. Wei, L.; Chen, J. Far-End Delamination Characterization of CFRP Laminates Using Nonlinear Ultrasonic Spectra of Scaling Subtracted Vibration Response. *Compos. Commun.* **2024**, *45*, 101811. [CrossRef]

129. Meo, M.; Polimeno, U.; Zumpano, G. Detecting Damage in Composite Material Using Nonlinear Elastic Wave Spectroscopy Methods. *Appl. Compos. Mater.* **2008**, *15*, 115–126. [CrossRef]

130. Polimeno, U.; Meo, M. Understanding the Effect of Boundary Conditions on Damage Identification Process When Using Nonlinear Elastic Wave Spectroscopy Methods. *Int. J. Non Linear Mech.* **2008**, *43*, 187. [CrossRef]

131. Bentahar, M. Nonlinear Acoustics: Application to Ultrasonic Characterisation of Damage in Heterogeneous Materials and Remaining Life Determination. Ph.D. Thesis, INSA Lyon, Lyon, France, 2005.

132. Wei, Q.; Zhu, L.; Zhu, J.; Zhuo, L.; Hao, W.; Xie, W. Characterization of Impact Fatigue Damage in CFRP Composites Using Nonlinear Acoustic Resonance Method. *Compos. Struct.* **2020**, *253*, 112804. [CrossRef]

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147. Prevorovsky, Z.; Krofta, J.; Kober, J.; Chlada, M.; Kirchner, A. Non-Linear Ultrasonic Spectroscopy of 3D Printed Metallic Samples. *Insight Non-Destr. Test. Cond. Monit.* **2019**, *61*, 157–161. [CrossRef]

148. Loi, G.; Uras, N.; Porcu, M.C.; Aymerich, F. Damage Detection in Composite Materials by Flexural Dynamic Excitation and Accelerometer-Based Acquisition. *IOP Conf. Ser. Mater. Sci. Eng.* **2022**, *1214*, 012007. [CrossRef]

149. Kim, S.; Adams, D.E.; Sohn, H.; Rodriguez-Rivera, G.; Myrent, N.; Bond, R.; Vitek, J.; Carr, S.; Grama, A.; Meyer, J.J. Crack Detection Technique for Operating Wind Turbine Blades Using Vibro-Acoustic Modulation. *Struct. Health Monit.* **2014**, *13*, 660–670. [CrossRef]

150. Willmann, E.; Boll, B.; Mikaelyan, G.; Wittich, H.; Meißner, R.H.; Fiedler, B. Vibro-Acoustic Modulation Based Measurements in CFRP Laminates for Damage Detection in Open-Hole Structures. *Compos. Commun.* **2023**, *42*, 101659. [CrossRef]

151. Ruiz, D.V.; de Bragança, C.S.C.; Poncetti, B.L.; Bittencourt, T.N.; Futai, M.M. Vibration-Based Structural Damage Detection Strategy Using FRFs and Machine Learning Classifiers. *Structures* **2024**, *59*, 105753. [CrossRef]

**Disclaimer/Publisher's Note:** The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.
140. Solodov, I.; Bai, J.; Bekgulyan, S.; Busse, G. A Local Defect Resonance to Enhance Acoustic Wave-Defect Interaction in Ultrasonic Nondestructive Evaluation. *Appl. Phys. Lett.* **2011**, *99*, 211911. [CrossRef]

141. Solodov, I.; Bai, J.; Bekgulyan, S.; Busse, G. A Local Defect Resonance to Enhance Wave-Defect Interaction in Nonlinear Spectroscopy and Ultrasonic Thermography. In Proceedings of the World Conference on Nondestructive Testing, Durban, South Africa, 16–20 April 2012.

142. Aymerich, F.; Staszewski, W.J. Impact Damage Detection in Composite Laminates Using Nonlinear Acoustics. *Compos. Part A Appl. Sci. Manuf.* **2010**, *41*, 1084–1092. [CrossRef]

143. Aymerich, F.; Staszewski, W.J. Experimental Study of Impact-Damage Detection in Composite Laminates Using a Cross-Modulation Vibro-Acoustic Technique. *Struct. Health Monit.* **2010**, *9*, 541–553. [CrossRef]

144. Ooijevaar, T.H.; Loendersloot, R.; Rogge, M.D.; Akkerman, R.; Tinga, T. Vibro-Acoustic Modulation Based Damage Identification in a Composite Skin-Stiffener Structure. *Struct. Health Monit.* **2016**, *15*, 458–472. [CrossRef]

145. Ooijevaar, T.H.; Loendersloot, R.; Warnet, L.L.; de Boer, A.; Akkerman, R. Vibration Based Structural Health Monitoring of a Composite T-Beam. *Compos. Struct.* **2010**, *92*, 2007–2015. [CrossRef]

146. Chen, B.Y.; Soh, S.K.; Lee, H.P.; Tay, T.E.; Tan, V.B.C. A Vibro-Acoustic Modulation Method for the Detection of Delamination and Kissing Bond in Composites. *J. Compos. Mater.* **2016**, *50*, 3089–3104. [CrossRef]

147. Prevorovsky, Z.; Krofta, J.; Kober, J.; Chlada, M.; Kirchner, A. Non-Linear Ultrasonic Spectroscopy of 3D Printed Metallic Samples. *Insight Non-Destr. Test. Cond. Monit.* **2019**, *61*, 157–161. [CrossRef]

148. Loi, G.; Uras, N.; Porcu, M.C.; Aymerich, F. Damage Detection in Composite Materials by Flexural Dynamic Excitation and Accelerometer-Based Acquisition. *IOP Conf. Ser. Mater. Sci. Eng.* **2022**, *1214*, 012007. [CrossRef]

149. Kim, S.; Adams, D.E.; Sohn, H.; Rodriguez-Rivera, G.; Myrent, N.; Bond, R.; Vitek, J.; Carr, S.; Grama, A.; Meyer, J.J. Crack Detection Technique for Operating Wind Turbine Blades Using Vibro-Acoustic Modulation. *Struct. Health Monit.* **2014**, *13*, 660–670. [CrossRef]

150. Willmann, E.; Boll, B.; Mikaelyan, G.; Wittich, H.; Meißner, R.H.; Fiedler, B. Vibro-Acoustic Modulation Based Measurements in CFRP Laminates for Damage Detection in Open-Hole Structures. *Compos. Commun.* **2023**, *42*, 101659. [CrossRef]

151. Ruiz, D.V.; de Bragança, C.S.C.; Poncetti, B.L.; Bittencourt, T.N.; Futai, M.M. Vibration-Based Structural Damage Detection Strategy Using FRFs and Machine Learning Classifiers. *Structures* **2024**, *59*, 105753. [CrossRef]

**Disclaimer/Publisher's Note:** The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.