# The Study of Localized Crack-Induced Effects of Nonlinear Vibro-Acoustic Modulation

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Models of Vibro-Acoustic Interactions in Cracked Beams

#### 2.1. Crack Models

- (1)
- Bi-linear model of elasticity

- (2)
- Open crack model

- (3)
- Breathing crack model

#### 2.2. Modal and Ultrasonic Excitation

#### 2.3. Vibro-Acoustic Responses and Indicators of Nonlinearity

_{HF}is the amplitude of the ultrasonic HF carrier component assessed from the response spectrum; the above parameters were used to study the crack localization effect.

## 3. Numerical Simulation Results

#### 3.1. Bi-Linear Crack Model

#### 3.2. Open Crack Model

#### 3.3. Breathing Crack Model

#### 3.4. Summary of Numerical Investigations

- The open crack model is not able to describe the nonlinear vibro-acoustic modulations, as expected, and cannot be used for the analysis of the crack localization effect.
- The proportionality between the input excitation amplitude and modulation intensity level was not observed in the numerically simulated results that involved the bi-linear and breathing crack models.
- The amplitudes of the first left and right sidebands were found to be similar in the breathing crack model and different in the bi-linear model.
- The proportionality between the crack depth and the modulation intensity level was observed for the bi-linear crack model. This relation is more complex in the breathing crack model and strongly depends on the ultrasonic HF excitation modeling approach.
- The crack localization effect was found in the bi-linear and breathing crack models, although the distribution of nonlinearity along the beam is significantly different for both crack models. The distribution of nonlinearity is also affected by the ultrasonic HF excitation modeling approach.

## 4. Experimental Validation

#### 4.1. Experimental Setup and Test Procedure

#### 4.2. Level of Nonlinearity

#### 4.3. Excitation Amplitude

#### 4.4. Crack Depth and Location

#### 4.5. Summary of Experimental Results

- The amplitudes of the first two pairs of sidebands are similar. The free–free beam produces similar results compared to the fixed–free beam results in the repeatability tests.
- The values of the R coefficient are proportional to the LF vibration excitation amplitudes for the cracked and intact beams. However, the dependency of the R coefficient on the HF ultrasonic excitation amplitudes has not been found.
- The amplitude of modulation sidebands cannot be used reliably to detect cracks in beams with fixed–free boundary conditions, regardless of the crack size and position. In contrast, cracks can be detected for all investigated crack sizes and locations in beams with free–free boundary conditions.
- The crack localization effect was found in the beam with the fixed–free boundary conditions but not observed in the beams with free–free boundary conditions.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram illustrating the principle of the nonlinear vibro-acoustic modulation technique used for crack detection: (

**a**) uncracked structure; (

**b**) cracked structure.

**Figure 2.**Schematic view of the beam used in FE simulations. Please note that the size and location of the crack, as well as the locations of sensors along the beam, depend on various simulation arrangements used.

**Figure 3.**Stress–strain relations applied for the bi-linear model of elasticity for three different crack depths simulated.

**Figure 4.**Breathing crack model in MSC Marc. The vicinity of the crack meshed for (

**a**) closed crack; (

**b**) open crack.

**Figure 6.**Parametric study of excitation amplitude for the bi-linear crack model and different amplitude levels of (

**a**) vibration LF excitation; (

**b**) ultrasonic HF excitation.

**Figure 7.**Parametric study of nonlinear vibro-acoustic modulations for the bi-linear crack model: (

**a**) modulation intensity; (

**b**) crack depth. Small, medium, and large nonlinearity levels correspond to three different crack depths modeled using the stress–strain curves given in Figure 3.

**Figure 8.**Crack localization effect for the bi-linear crack model. Three different single-crack locations are investigated, i.e., the cracks positioned 100, 150, and 200 mm away from the free end of the beam.

**Figure 9.**Crack localization effect for two different double-crack scenarios. Numerical simulations performed for the bi-linear crack model.

**Figure 10.**Crack localization effect for three different crack positions. The bi-linear crack model was used. The simplified piezo model was applied for the ultrasonic HF excitation.

**Figure 11.**Distribution of modulation intensity coefficient R along the beam for the open crack model used in numerical simulation. The results are compared with the bi-linear crack model.

**Figure 12.**Parametric study of excitation amplitude for the breathing crack model and different amplitude levels of (

**a**) ultrasonic HF excitation; (

**b**) vibration LF excitation.

**Figure 13.**Parametric study of nonlinear vibro-acoustic modulations for the breathing crack model: (

**a**) modulation intensity; (

**b**) crack depth.

**Figure 14.**Crack localization effect for the breathing crack model. Three different single-crack locations are investigated, i.e., the cracks positioned 100, 150, and 200 mm away from the free end of the beam.

**Figure 15.**Parametric study of nonlinear vibro-acoustic modulations for the breathing crack model and the piezo model of ultrasonic HF excitation: (

**a**) crack depth; (

**b**) crack localization effect.

**Figure 18.**Experimental Test 1—a parametric study of nonlinear vibro-acoustic modulations analyzed using the first (top figures) and second (bottom figures) pairs of sidebands: (

**a**) intact beam; (

**b**) cracked beam. The crack position is indicated by the vertical black solid line. The horizontal axis corresponds to the measurement locations.

**Figure 19.**Experimental Test 1—repeatability test for the R coefficient: (

**a**) intact beam; (

**b**) cracked beam. The crack position is indicated by the vertical solid black line.

**Figure 20.**Values of modulation intensity coefficient R measured along the intact and cracked beams (Experimental Test 1) for free–free boundary conditions: (

**a**) intact beam; (

**b**) cracked beam.

**Figure 21.**Study of LF vibration (top figures) and HF ultrasonic (bottom figures) excitation amplitude study for (

**a**) intact beam; (

**b**) cracked beam. Voltage excitation levels are indicated in the legends.

**Figure 22.**Values of R coefficient for the fixed–free boundary conditions: (

**a**) Experimental Test 2; (

**b**) Experimental Test 3. Both results are compared with the intact beam results. The crack location is indicated by the vertical solid black line.

**Figure 23.**Level of nonlinearity estimated for the fixed–free beam, for two different crack locations indicated by the vertical solid black (Experimental Test 1) and dashed red (Experimental Test 3) lines.

**Figure 24.**Values of R coefficient for the free–free boundary conditions: (

**a**) Experimental Test 2; (

**b**) Experimental Test 3. Both results are compared with the intact beam results. The crack location is indicated by the vertical solid black line.

**Figure 25.**Level of nonlinearity estimated for the free–free beam for two different crack locations indicated by the vertical solid black (Experimental Test 1) and dashed red (Experimental Test 3) lines.

Experimental Test | Crack Depth [mm] | Crack Position [mm] |
---|---|---|

Test 1 | 10 | 100 |

Test 2 | 4 | 100 |

Test 3 | 10 | 150 |

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**MDPI and ACS Style**

Broda, D.; Mendrok, K.; Silberschmidt, V.V.; Pieczonka, L.; Staszewski, W.J.
The Study of Localized Crack-Induced Effects of Nonlinear Vibro-Acoustic Modulation. *Materials* **2023**, *16*, 1653.
https://doi.org/10.3390/ma16041653

**AMA Style**

Broda D, Mendrok K, Silberschmidt VV, Pieczonka L, Staszewski WJ.
The Study of Localized Crack-Induced Effects of Nonlinear Vibro-Acoustic Modulation. *Materials*. 2023; 16(4):1653.
https://doi.org/10.3390/ma16041653

**Chicago/Turabian Style**

Broda, Dariusz, Krzysztof Mendrok, Vadim V. Silberschmidt, Lukasz Pieczonka, and Wieslaw J. Staszewski.
2023. "The Study of Localized Crack-Induced Effects of Nonlinear Vibro-Acoustic Modulation" *Materials* 16, no. 4: 1653.
https://doi.org/10.3390/ma16041653