# Identification of Material Properties of Elastic Plate Using Guided Waves Based on the Matrix Pencil Method and Laser Doppler Vibrometry

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Determination of Guided Waves Characteristics

## 3. Experimental Data Extraction Using the Matrix Pencil Method

## 4. Objective Functions for Material Properties Characterization

#### 4.1. Objective Function Using Residual of Slownesses

#### 4.2. Objective Function Based on the Fourier Transform of Green’s Matrix

## 5. Generation of Test Data Sets

## 6. Numerical Analysis

#### 6.1. Analysis of the Properties of Objective Functions

#### 6.2. Inverse Problem Solution Using Synthesized Data

#### 6.3. Inverse Problem Solution Using LDV Experimental Data

## 7. Discussions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

NDT | non-destructive testing |

SHM | structural health monitoring |

SCs | slowness curves |

GWs | guided waves |

LDV | laser Doppler vibrometer |

## References

- Pagnotta, L. Determining elastic constants of materials with interferometric techniques. Inverse Probl. Sci. Eng.
**2006**, 14, 801–818. [Google Scholar] [CrossRef] - Kam, T.; Chen, C.; Yang, S. Material characterization of laminated composite materials using a three-point-bending technique. Compos. Struct.
**2009**, 88, 624–628. [Google Scholar] [CrossRef] - Tam, J.; Ong, Z.; Ismail, Z.; Ang, B.; Khoo, S. Identification of material properties of composite materials using nondestructive vibrational evaluation approaches: A review. Mech. Adv. Mater. Struct.
**2017**, 24, 971–986. [Google Scholar] [CrossRef] - Tam, J.H. Identification of elastic properties utilizing non-destructive vibrational evaluation methods with emphasis on definition of objective functions: A review. Struct. Multidiscip. Optim.
**2020**, 61, 1677–1710. [Google Scholar] [CrossRef] - Chen, Q.; Xu, K.; Ta, D. High-resolution Lamb waves dispersion curves estimation and elastic property inversion. Ultrasonics
**2021**, 115, 106427. [Google Scholar] [CrossRef] - Okumura, S.; Nguyen, V.H.; Taki, H.; Haïat, G.; Naili, S.; Sato, T. Phase velocity estimation technique based on adaptive beamforming for ultrasonic guided waves propagating along cortical long bones. Jpn. J. Appl. Phys.
**2017**, 56, 07JF06. [Google Scholar] [CrossRef] [Green Version] - Okumura, S.; Nguyen, V.H.; Taki, H.; Haïat, G.; Naili, S.; Sato, T. Rapid High-Resolution Wavenumber Extraction from Ultrasonic Guided Waves Using Adaptive Array Signal Processing. Appl. Sci.
**2018**, 8, 652. [Google Scholar] [CrossRef] [Green Version] - Bochud, N.; Laurent, J.; Bruno, F.; Royer, D.; Prada, C. Towards real-time assessment of anisotropic plate properties using elastic guided waves. J. Acoust. Soc. Am.
**2018**, 143, 1138–1147. [Google Scholar] [CrossRef] [Green Version] - Webersen, M.; Johannesmann, S.; Düchting, J.; Claes, L.; Henning, B. Guided ultrasonic waves for determining effective orthotropic material parameters of continuous-fiber reinforced thermoplastic plates. Ultrasonics
**2018**, 84, 53–62. [Google Scholar] [CrossRef] - Tian, Z.; Yu, L. Lamb wave frequency-wavenumber analysis and decomposition. J. Intell. Mater. Syst. Struct.
**2014**, 25, 1107–1123. [Google Scholar] [CrossRef] - Eremin, A.; Glushkov, E.; Glushkova, N.; Lammering, R. Evaluation of effective elastic properties of layered composite fiber-reinforced plastic plates by piezoelectrically induced guided waves and laser Doppler vibrometry. Compos. Struct.
**2015**, 125, 449–458. [Google Scholar] [CrossRef] - Eremin, A.A.; Golub, M.V.; Wilde, M.V.; Pleshkov, V.N. Influence of retroreflective films on the behaviour of elastic guided waves measured with laser Doppler vibrometry. Measurement
**2022**, 190, 110572. [Google Scholar] [CrossRef] - Takahashi, V.; Lematre, M.; Fortineau, J.; Lethiecq, M. Elastic parameters characterization of multilayered structures by air-coupled ultrasonic transmission and genetic algorithm. Ultrasonics
**2022**, 119, 106619. [Google Scholar] [CrossRef] - Lu, L.; Charron, E.; Glushkov, E.; Glushkova, N.; Bonello, B.; Julien, F.H.; Gogneau, N.; Tchernycheva, M.; Boyko, O. Probing elastic properties of nanowire-based structures. Appl. Phys. Lett.
**2018**, 113, 161903. [Google Scholar] [CrossRef] [Green Version] - Pagnotta, L.; Stigliano, G. Elastic characterization of isotropic plates of any shape via dynamic tests: Practical aspects and experimental applications. Mech. Res. Commun.
**2009**, 36, 154–161. [Google Scholar] [CrossRef] - Wilde, M.V.; Golub, M.V.; Eremin, A.A. Experimental observation of theoretically predicted spectrum of edge waves in a thick elastic plate with facets. Ultrasonics
**2019**, 98, 88–93. [Google Scholar] [CrossRef] - Tu, J.H.; Rowley, C.W.; Luchtenburg, D.M.; Brunton, S.L.; Kutz, J.N. On dynamic mode decomposition: Theory and applications. J. Comput. Dyn.
**2014**, 1, 391–421. [Google Scholar] [CrossRef] [Green Version] - Schöpfer, F.; Binder, F.; Wöstehoff, A.; Schuster, T.; von Ende, S.; Föll, S.; Lammering, R. Accurate determination of dispersion curves of guided waves in plates by applying the matrix pencil method to laser vibrometer measurement data. CEAS Aeronaut. J.
**2013**, 4, 61–68. [Google Scholar] [CrossRef] - Chamaani, S.; Akbarpour, A.; Helbig, M.; Sachs, J. Matrix Pencil Method for Vital Sign Detection from Signals Acquired by Microwave Sensors. Sensors
**2021**, 21, 5735. [Google Scholar] [CrossRef] - Wilde, M.V.; Golub, M.V.; Eremin, A.A. Elastodynamic behaviour of laminate structures with soft thin interlayers: Theory and experiment. Materials
**2022**, 15, 1307. [Google Scholar] [CrossRef] - Thelen, M.; Bochud, N.; Brinker, M.; Prada, C.; Huber, P. Laser-excited elastic guided waves reveal the complex mechanics of nanoporous silicon. Nat. Commun.
**2021**, 12, 3597. [Google Scholar] [CrossRef] [PubMed] - Liu, Z.; Xu, K.; Li, D.; Ta, D.; Wang, W. Automatic mode extraction of ultrasonic guided waves using synchrosqueezed wavelet transform. Ultrasonics
**2019**, 99, 105948. [Google Scholar] [CrossRef] [PubMed] - Fairuschin, V.; Brand, F.; Backer, A.; Drese, K.S. Elastic Properties Measurement Using Guided Acoustic Waves. Sensors
**2021**, 21, 6675. [Google Scholar] [CrossRef] [PubMed] - Lan, B. Non-iterative, stable analysis of surface acoustic waves in anisotropic piezoelectric multilayers using spectral collocation method. J. Sound Vib.
**2018**, 433, 16–28. [Google Scholar] [CrossRef] - Grünsteidl, C.; Murray, T.W.; Berer, T.; Veres, I.A. Inverse characterization of plates using zero group velocity Lamb modes. Ultrasonics
**2016**, 65, 1–4. [Google Scholar] [CrossRef] [Green Version] - Neumann, M.N.; Hennings, B.; Lammering, R. Identification and Avoidance of Systematic Measurement Errors in Lamb Wave Observation With One-Dimensional Scanning Laser Vibrometry. Strain
**2013**, 49, 95–101. [Google Scholar] [CrossRef] - Fletcher, R. Practical Methods of Optimization, 2nd ed.; John Wiley & Sons: New York, NY, USA, 2000; p. 436. [Google Scholar]
- Golub, M.V.; Doroshenko, O.V.; Wilde, M.V.; Eremin, A.A. Experimental validation of the applicability of effective spring boundary conditions for modelling damaged interfaces in laminate structures. Compos. Struct.
**2021**, 273, 114141. [Google Scholar] [CrossRef] - Kammer, C. Aluminium Taschenbuch: Band 1: Grundlagen und Werkstoffe; Beuth Verlag: Berlin, Germany, 2021; p. 790. [Google Scholar]
- Rogers, W.P. Elastic property measurement using Rayleigh-Lamb waves. Res. Nondestruct. Eval.
**1995**, 6, 185–208. [Google Scholar] [CrossRef] - Fomenko, S.I.; Golub, M.V.; Doroshenko, O.V.; Wang, Y.; Zhang, C. An advanced boundary integral equation method for wave propagation analysis in a layered piezoelectric phononic crystal with a crack or an electrode. J. Comput. Phys.
**2021**, 447, 110669. [Google Scholar] [CrossRef] - Glushkov, E.; Glushkova, N.; Eremin, A.; Lammering, R. Group velocity of cylindrical guided waves in anisotropic laminate composites. J. Acoust. Soc. Am.
**2014**, 135, 148–154. [Google Scholar] [CrossRef]

**Figure 3.**Surfaces of the dispersion maps for $|{K}_{22}^{-1}(f,s,0,\mathit{\theta})|$ (

**a**) and ${log}_{10}\left|{K}_{22}^{-1}(f,s,0,\mathit{\theta})\right|$ (

**b**) at $\mathit{\theta}=\{70\phantom{\rule{3.33333pt}{0ex}}\mathrm{GPa},0.33,1.9\phantom{\rule{3.33333pt}{0ex}}\mathrm{mm}\}$.

**Figure 4.**Examples of artificially generated corrupted slowness-frequency pairs (${\stackrel{\u02d8}{s}}_{k}({\mathit{\theta}}^{*},{f}_{n}),{f}_{n}$) for ${\mathit{\theta}}^{*}=\{70\phantom{\rule{3.33333pt}{0ex}}\mathrm{GPa},0.33,1.9\phantom{\rule{3.33333pt}{0ex}}\mathrm{mm}\}$.

**Figure 5.**Surfaces of objective functions $F\left(\tilde{\mathit{\theta}}\right)$ (

**a**–

**d**) and ${G}_{1}\left(\tilde{\mathit{\theta}}\right)$ (

**e**–

**h**) at $\tilde{\mathit{\theta}}=(E,\nu =0.33,H)$ and for different degrees of corruptness $\sigma $ ($\delta =0.4$).

**Figure 6.**Surfaces of objective functions $F\left(\tilde{\mathit{\theta}}\right)$ (

**a**–

**d**) and ${G}_{1}\left(\tilde{\mathit{\theta}}\right)$ (

**e**–

**h**) at $\tilde{\mathit{\theta}}=(70\phantom{\rule{3.33333pt}{0ex}}\mathrm{GPa},\nu ,H)$ for different degrees of corruptness ($\delta =0.4$).

**Figure 7.**Surfaces of objective functions $F\left(\tilde{\mathit{\theta}}\right)$ (

**a**–

**d**) and ${G}_{1}\left(\tilde{\mathit{\theta}}\right)$ (

**e**–

**h**) at $\tilde{\mathit{\theta}}=(E,\nu ,1.9\phantom{\rule{3.33333pt}{0ex}}\mathrm{mm})$ for different degrees of corruptness $\sigma $ ($\delta =0.4$).

**Figure 8.**The length of confidence intervals (

**a**,

**c**,

**e**,

**g**) and the relative error ${\widehat{\u03f5}}_{1}$ (

**b**,

**d**,

**f**,

**h**) for Young’s modulus E identification obtained using five objective functions $F,{G}_{1},{G}_{100},{H}_{1},{J}_{150}$ for different levels of noise $\sigma $ at $\delta =40$%.

**Figure 9.**The length of confidence intervals (

**a**,

**c**,

**e**,

**g**) and the relative error ${\widehat{\u03f5}}_{2}$ (

**b**,

**d**,

**f**,

**h**) for Poisson’s ratio $\nu $ identification obtained using five objective functions $F,{G}_{1},{G}_{100},{H}_{1},{J}_{150}$ for different levels of noise $\sigma $ at $\delta =40$%.

**Figure 10.**The length of confidence intervals (

**a**,

**c**,

**e**,

**g**) and the relative error ${\widehat{\u03f5}}_{3}$ (

**b**,

**d**,

**f**,

**h**) for plate thickness H identification obtained using five objective functions $F,{G}_{1},{G}_{100},{H}_{1},{J}_{150}$ for different levels of noise $\sigma $ at $\delta =40$%.

**Figure 11.**Experimentally observed (circles) and theoretically predicted slownesses (lines) calculated at $\widehat{\mathit{\theta}}$ estimated using objective function F based on the slowness residuals (solid lines) and ${G}_{1}$ based on Green’s matrix (dash-dotted and dashed lines) for GW 1–GW 2 (dashed and thick solid lines) and GW 1–GW 5 (dash-dotted and thin solid lines).

**Table 1.**The cumulative percentages and the mean for the results calculated using objective function F based on the slowness residuals.

GW Modes | Statistics | |||
---|---|---|---|---|

2.5% | 97.5% | 50% | mean | |

Young’s modulus | ||||

69.44067 | 70.02778 | 69.53528 | 69.57336 | |

70.85072 | 70.93444 | 70.89058 | 70.86935 | |

70.57222 | 70.99317 | 70.94851 | 70.86905 | |

Poisson’s ratio | ||||

0.2814911 | 0.3088387 | 0.2851565 | 0.2868929 | |

0.3426350 | 0.345837 | 0.3443798 | 0.345837 | |

0.3498317 | 0.352299 | 0.3513940 | 0.3512925 | |

Thickness | ||||

2.038286 | 2.045689 | 2.044301 | 2.043785 | |

2.019506 | 2.022150 | 2.020586 | 2.020512 | |

2.001522 | 2.014291 | 2.013035 | 2.010527 |

**Table 2.**The cumulative percentages and the mean for the results calculated using objective function ${G}_{1}$ based on Green’s matrix.

GW Modes | Statistics | |||
---|---|---|---|---|

2.5% | 97.5% | 50% | mean | |

Young’s modulus | ||||

70.51223 | 70.55632 | 70.53855 | 70.54622 | |

68.29054 | 70.90552 | 70.88853 | 70.71145 | |

68.94975 | 71.68033 | 71.39863 | 71.13951 | |

Poisson’s ratio | ||||

0.3130311 | 0.3199611 | 0.3156035 | 0.3164631 | |

0.2584863 | 0.3394943 | 0.3394111 | 0.3352184 | |

0.2730250 | 0.3862568 | 0.3508503 | 0.3454789 | |

Thickness | ||||

2.050283 | 2.058020 | 2.056627 | 2.055417 | |

1.985751 | 2.081555 | 2.029845 | 2.029940 | |

2.008208 | 2.090551 | 2.030139 | 2.034077 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Golub, M.V.; Doroshenko, O.V.; Arsenov, M.A.; Bareiko, I.A.; Eremin, A.A.
Identification of Material Properties of Elastic Plate Using Guided Waves Based on the Matrix Pencil Method and Laser Doppler Vibrometry. *Symmetry* **2022**, *14*, 1077.
https://doi.org/10.3390/sym14061077

**AMA Style**

Golub MV, Doroshenko OV, Arsenov MA, Bareiko IA, Eremin AA.
Identification of Material Properties of Elastic Plate Using Guided Waves Based on the Matrix Pencil Method and Laser Doppler Vibrometry. *Symmetry*. 2022; 14(6):1077.
https://doi.org/10.3390/sym14061077

**Chicago/Turabian Style**

Golub, Mikhail V., Olga V. Doroshenko, Mikhail A. Arsenov, Ilya A. Bareiko, and Artem A. Eremin.
2022. "Identification of Material Properties of Elastic Plate Using Guided Waves Based on the Matrix Pencil Method and Laser Doppler Vibrometry" *Symmetry* 14, no. 6: 1077.
https://doi.org/10.3390/sym14061077