# Sideband Peak Count in a Vibro-Acoustic Modulation Method for Crack Detection

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Configuration of the Experimental Setup

## 3. Results

## 4. Bilinear Crack Model of Multiple Sideband Peak Interpretation

_{0}+ Dk(q), where the nonlinear part of the stiffness coefficient Dk(q) << K

_{0}.

_{1}and P

_{2}.

_{1}= P

_{2}= 1.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Buck, O. Nonlinear Acoustic Properties of Structural Materials—A Review. In Review of Progress in Quantitative Nondestructive Evaluation; Thompson, D.O., Chimenti, D.E., Eds.; Springer: Boston, MA, USA, 1990; pp. 1677–1684. ISBN 978-1-4684-5774-2. [Google Scholar]
- Zheng, Y.; Maev, R.G.; Solodov, I.Y. Review/Sythèse Nonlinear Acoustic Applications for Material Characterization: A Review. Can. J. Phys.
**2000**, 77, 927–967. [Google Scholar] [CrossRef] - Broda, D.; Staszewski, W.J.; Martowicz, A.; Uhl, T.; Silberschmidt, V.V. Modelling of Nonlinear Crack–Wave Interactions for Damage Detection Based on Ultrasound—A Review. J. Sound Vib.
**2014**, 333, 1097–1118. [Google Scholar] [CrossRef] - Sutin, A.M.; Salloum, H. Interaction of Acoustic and Electromagnetic Waves in Nondestructive Evaluation and Medical Applications. Radiophys. Quantum Electron.
**2020**, 63, 40–54. [Google Scholar] [CrossRef] - Klepka, A.; Pieczonka, L.; Dziedziech, K.; Staszewski, W.J.; Aymerich, F.; Uhl, T. Structural Damage Detection Based on Nonlinear Acoustics: Application Examples. In Nonlinear Ultrasonic and Vibro-Acoustical Techniques for Nondestructive Evaluation; Kundu, T., Ed.; Springer International Publishing: Cham, Switzerland, 2019; pp. 139–174. ISBN 978-3-319-94476-0. [Google Scholar]
- Krohn, N.; Pfleiderer, K.; Stoessel, R.; Solodov, I.; Busse, G. Nonlinear Acoustic Imaging: Fundamentals, Methodology, and NDE-Applications. In Proceedings of the Acoustical Imaging; Arnold, W., Hirsekorn, S., Eds.; Springer: Dordrecht, The Netherlands, 2004; pp. 91–98. [Google Scholar]
- Guyer, R.A.; Johnson, P.A. Nonlinear Mesoscopic Elasticity: The Complex Behaviour of Rocks, Soil, Concrete; John Wiley & Sons: Hoboken, NJ, USA, 2009; ISBN 978-3-527-40703-3. [Google Scholar]
- Kundu, T.; Eiras, J.N.; Li, W.; Liu, P.; Sohn, H.; Payá, J. Fundamentals of Nonlinear Acoustical Techniques and Sideband Peak Count. In Nonlinear Ultrasonic and Vibro-Acoustical Techniques for Nondestructive Evaluation; Kundu, T., Ed.; Springer International Publishing: Cham, Switzerland, 2019; pp. 1–88. ISBN 978-3-319-94476-0. [Google Scholar]
- Jin, J.; Rivière, J.; Ohara, Y.; Shokouhi, P. Dynamic Acousto-Elastic Response of Single Fatigue Cracks with Different Microstructural Features: An Experimental Investigation. J. Appl. Phys.
**2018**, 124, 075303. [Google Scholar] [CrossRef] - Bergman, R.H.; Shahbender, R.A. Effect of Statically Applied Stresses on the Velocity of Propagation of Ultrasonic Waves. J. Appl. Phys.
**1958**, 29, 1736–1738. [Google Scholar] [CrossRef] - Toupin, R.A.; Bernstein, B. Sound Waves in Deformed Perfectly Elastic Materials. Acoustoelastic Effect. J. Acoust. Soc. Am.
**1961**, 33, 216–225. [Google Scholar] [CrossRef] - Korotkov, A.S.; Slavinskij, M.M.; Sutin, A.M. Variations of acoustic nonlinear parameters with the concentration of defects in steel. Akust. Zurnal
**1994**, 40, 84–87. [Google Scholar] - Klepka, A.; Dziedziech, K.; Mrówka, J.; Górski, J. Experimental Investigation of Modulation Effects for Contact-Type Interfaces in Vibro-Acoustic Modulation Tests. Struct. Health Monit.
**2021**, 20, 917–930. [Google Scholar] [CrossRef] - Hu, H.F.; Staszewski, W.J.; Hu, N.Q.; Jenal, R.B.; Qin, G.J. Crack Detection Using Nonlinear Acoustics and Piezoceramic Transducers—Instantaneous Amplitude and Frequency Analysis. Smart Mater. Struct.
**2010**, 19, 065017. [Google Scholar] [CrossRef] - Li, N.; Wang, F.; Song, G. New Entropy-Based Vibro-Acoustic Modulation Method for Metal Fatigue Crack Detection: An Exploratory Study. Measurement
**2020**, 150, 107075. [Google Scholar] [CrossRef] - Ballad, E.M.; Vezirov, S.Y.; Pfleiderer, K.; Solodov, I.Y.; Busse, G. Nonlinear Modulation Technique for NDE with Air-Coupled Ultrasound. Ultrasonics
**2004**, 42, 1031–1036. [Google Scholar] [CrossRef] [PubMed] - Lee, S.E.; Lim, H.J.; Sohn, H.; Hong, J.W. Excitation Conditions for Nonlinear Ultrasonic Wave Modulation Technique. In KKHTCNN; Chulalongkorn University: Bangkok, Thailand, 2015. [Google Scholar]
- Dziedziech, K.; Klepka, A.; Roemer, J.; Pieczonka, L. Experimental Study of Thermo-Acoustic Wave Modulation in a Cracked Plate. J. Sound Vib.
**2021**, 498, 115970. [Google Scholar] [CrossRef] - Golchinfar, B.; Ramezani, M.G.; Donskoy, D.; Saboonchi, H. Vibro-Acoustic Modulation Technique Comparison with Conventional Nondestructive Evaluation Methods. In Health Monitoring of Structural and Biological Systems XIV; SPIE: Bellingham, WC, USA, 2020; Volume 11381, pp. 187–198. [Google Scholar]
- Korotkov, A.S.; Sutin, A.M. Modulation of Ultrasound by Vibrations in Metal Constructions with Cracks. Acoust. Lett.
**1994**, 18, 59–62. [Google Scholar] - Sutin, A.M.; Donskoy, D.M. Vibro-Acoustic Modulation Nondestructive Evaluation Technique. In Nondestructive Evaluation of Aging Aircraft, Airports, and Aerospace Hardware II; SPIE: Bellingham, WC, USA, 1998; Volume 3397, pp. 226–237. [Google Scholar]
- Duffour, P.; Morbidini, M.; Cawley, P. Comparison between a Type of Vibro-Acoustic Modulation and Damping Measurement as NDT Techniques. NDT E Int.
**2006**, 39, 123–131. [Google Scholar] [CrossRef] - Meyer, J.J.; Adams, D.E. Theoretical and Experimental Evidence for Using Impact Modulation to Assess Bolted Joints. Nonlinear Dyn.
**2015**, 81, 103–117. [Google Scholar] [CrossRef] - Donskoy, D.; Zagrai, A.; Chudnovsky, A.; Golovin, E.; Agarwala, V. Damage Assessment with Nonlinear Vibro-Acoustic Modulation Technique. In In American Society of Mechanical Engineers Digital Collection, Proceedings of the International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Las Vegas, NV, USA, 4–7 September 2007; ASME: New York, NY, USA, 2009; pp. 1949–1956. [Google Scholar]
- Liu, B.; Luo, Z.; Gang, T. Influence of Low-Frequency Parameter Changes on Nonlinear Vibro-Acoustic Wave Modulations Used for Crack Detection. Struct. Health Monit.
**2018**, 17, 218–226. [Google Scholar] [CrossRef] - Jia, J.; Hu, H.; Tao, L.; Yang, D. Analysis of Load Effect on Nonlinear Vibro-Acoustic Modulation Used in on-Line Monitoring of Fatigue Cracks. Smart Mater. Struct.
**2017**, 26, 095048. [Google Scholar] [CrossRef] - Boll, B.; Willmann, E.; Fiedler, B.; Meißner, R.H. Weak Adhesion Detection—Enhancing the Analysis of Vibroacoustic Modulation by Machine Learning. Compos. Struct.
**2021**, 273, 114233. [Google Scholar] [CrossRef] - Castellano, A.; Fraddosio, A.; Piccioni, M.D.; Kundu, T. Linear and Nonlinear Ultrasonic Techniques for Monitoring Stress-Induced Damages in Concrete. J. Nondestruct. Eval. Diagn. Progn. Eng. Syst.
**2021**, 4, 041001. [Google Scholar] [CrossRef] - Donskoy, D.; Golchinfar, B.; Ramezani, M.; Rutner, M.; Hassiotis, S. Vibro-acoustic amplitude and frequency modulations during fatigue damage evolution. In AIP Conference Proceedings; AIP Publishing LLC: Melville, NY, USA, 2019; Volume 2102, p. 040004. [Google Scholar]
- Ramezani, M.G.; Golchinfar, B.; Donskoy, D.; Hassiotis, S.; Venkiteela, G. Steel Material Degradation Assessment via Vibro-Acoustic Modulation Technique. Transp. Res. Rec.
**2019**, 2673, 579–585. [Google Scholar] [CrossRef] - Donskoy, D.M. Nonlinear Acoustic Methods. In Encyclopedia of Structural Health Monitoring; John Wiley & Sons, Ltd.: Hoboken, NJ, USA, 2009; ISBN 978-0-470-06162-6. [Google Scholar]
- Liu, P.; Sohn, H. Damage Detection Using Sideband Peak Count in Spectral Correlation Domain. J. Sound Vib.
**2017**, 411, 20–33. [Google Scholar] [CrossRef] - Alnuaimi, H.; Amjad, U.; Park, S.; Russo, P.; Lopresto, V.; Kundu, T. An Improved Nonlinear Ultrasonic Technique for Detecting and Monitoring Impact Induced Damage in Composite Plates. Ultrasonics
**2022**, 119, 106620. [Google Scholar] [CrossRef] [PubMed] - Frau, A.; Aymerich, F.; Porcu, M.; Pieczonka, L.; Klepka, A.; Staszewski, W. Probing WaveFrequency Selection for the Nonlinear Vibro-Acoustic Wave Modulation Tests. AIAS–Assoc. Ital. L’analisi Sollecitazioni
**2014**, 43, 9–12. [Google Scholar] - Zagrai, A.; Donskoy, D.; Lottiaux, J. N-Scan
^{®}: New Vibro-Modulation System for Crack Detection, Monitoring and Characterization. In AIP Conference Proceedings; AIP Publishing LLC: Melville, NY, USA, 2004; Volume 700, pp. 1414–1421. [Google Scholar] [CrossRef] - Zaitsev, V.Y.; Matveev, L.A.; Matveyev, A.L. Elastic-Wave Modulation Approach to Crack Detection: Comparison of Conventional Modulation and Higher-Order Interactions. NDT E Int.
**2011**, 44, 21–31. [Google Scholar] [CrossRef] - Dimarogonas, A.D. Vibration of Cracked Structures: A State of the Art Review. Eng. Fract. Mech.
**1996**, 55, 831–857. [Google Scholar] [CrossRef] - Bovsunovsky, A.; Surace, C. Non-Linearities in the Vibrations of Elastic Structures with a Closing Crack: A State of the Art Review. Mech. Syst. Signal Processing
**2015**, 62–63, 129–148. [Google Scholar] [CrossRef] - Wauer, J. On the Dynamics of Cracked Rotors: A Literature Survey. Appl. Mech. Rev.
**1990**, 43, 13–17. [Google Scholar] [CrossRef] - Chu, Y.C.; Shen, M.-H.H. Analysis of Forced Bilinear Oscillators and the Application to Cracked Beam Dynamics. AIAA J.
**1992**, 30, 2512–2519. [Google Scholar] [CrossRef] - Ostrovsky, L.A.; Starobinets, I.M. Transitions and Statistical Characteristics of Vibrations in a Bimodular Oscillator. Chaos
**1995**, 5, 496–500. [Google Scholar] [CrossRef]

**Figure 1.**Schema of the experimental VAM setup (

**left**) and a picture of the sample with glued transducers (

**right**).

**Figure 2.**Spectra in the frequency window of 60 Hz around the probe frequency f = 175 kHz for Sample #3: (

**a**) after 11,252 load cycles, no damage initiation; (

**b**) after 47,139 cycles, initial damage accumulation; (

**c**) after 51,219 cycles, visible crack. The number of cycles to failure was 52,106 (see Table 2).

**Figure 4.**Spectra in the frequency window of 2000 Hz around the probe frequency f = 175 KHz for Sample #3: (

**a**) after 11,252 load cycles, no damage initiation; (

**b**) after 47,139 cycles, initial damage accumulation; (

**c**) after 51,219 cycles, visible crack. The number of cycles to failure was 52,106 (see Table 2).

**Figure 6.**Spectra in the frequency window of 60 Hz around the probe frequency f = 195 kHz for Sample #5: (

**a**) after 0 normalized cycles, no visible crack; (

**b**) after 0.69 normalized cycles, initial damage accumulation; (

**c**) after 0.9 normalized cycles, visible crack.

**Figure 7.**Spectra in the frequency window of 2000 Hz around the probe frequency f = 195 kHz for Sample #5: (

**a**) after 0 normalized cycles, no visible crack; (

**b**) after 0.69 normalized cycles, initial damage accumulation; (

**c**) after 0.9 normalized cycles, visible crack.

**Figure 8.**Three specimens failed after being examined by the MI versus the number of cycles ratio for average frequencies.

**Figure 9.**Three specimens were successfully examined by the SPN versus the number of cycles ratio for average frequencies.

**Figure 10.**Variations of parameter MI (

**upper**) and SPN (

**lower**) for Sample #4 for different probe frequencies.

**Figure 12.**Examples of bilinear (red line) and quadratic (blue line) force–displacement spring relationship. The nonlinear parameter here is $\alpha =0.5,$ which is much larger than for the considered case.

**Figure 14.**Dependence of the first sideband peak (A+) on (

**a**) the probe wave amplitude and (

**b**) the pump wave amplitude calculated from the bilinear model. The dotted lines show the same dependences from the quadratic model.

**Figure 15.**Examples of the spectra calculated from the bilinear crack models: (

**a**) probe wave 0 dB and pump wave 10 dB; (

**b**) probe wave 10 dB and pump wave 0 dB.

Mechanical Properties | Value |
---|---|

Density | 0.284 Ib/in3 |

Ultimate Tensile Strength | 63,800 psi |

Yield Tensile Strength | 53,700 psi |

Shear Strength | 43,500 psi |

Melting Point | 2590–2670 °F |

Sample | Number of Cycles to Failure |
---|---|

Sample #1 | 46,102 |

Sample #2 | 120,074 |

Sample #3 | 52,106 |

Sample #4 | 67,637 |

Sample #5 | 67,712 |

Sample #6 | 70,603 |

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**

Alnutayfat, A.; Hassiotis, S.; Liu, D.; Sutin, A. Sideband Peak Count in a Vibro-Acoustic Modulation Method for Crack Detection. *Acoustics* **2022**, *4*, 74-86.
https://doi.org/10.3390/acoustics4010005

**AMA Style**

Alnutayfat A, Hassiotis S, Liu D, Sutin A. Sideband Peak Count in a Vibro-Acoustic Modulation Method for Crack Detection. *Acoustics*. 2022; 4(1):74-86.
https://doi.org/10.3390/acoustics4010005

**Chicago/Turabian Style**

Alnutayfat, Abdullah, Sophia Hassiotis, Dong Liu, and Alexander Sutin. 2022. "Sideband Peak Count in a Vibro-Acoustic Modulation Method for Crack Detection" *Acoustics* 4, no. 1: 74-86.
https://doi.org/10.3390/acoustics4010005