# Non-Contact Damage Detection under Operational Conditions with Multipoint Laservibrometry

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Damage Index

- Step1: Measure the velocities at the measurement points on the intact and damaged structure by the MPV. These are the time response vectors ${}_{}{}^{}h\left(t\right)$ and ${}_{}{}^{\mathrm{d}}h\left(t\right)$.
- Step 2: Calculate the spectral density matrices ${}_{}{}^{}G{}_{h}\left(\omega \right)$ and ${}_{}{}^{\mathrm{d}}G{}_{h}\left(\omega \right)$ of the time response vectors ${}_{}{}^{}h\left(t\right)$ and ${}_{}{}^{\mathrm{d}}h\left(t\right)$. Use Equation (4) to calculate the singular value spectrum of the intact structure ${s}_{1}^{2}\left(\omega \right)$, the damaged structure ${}_{}{}^{\mathrm{d}}s{}_{1}^{2}\left(\omega \right)$, and their corresponding singular vectors ${u}_{1}\left(\omega \right)$ and ${}_{}{}^{\mathrm{d}}u{}_{1}\left(\omega \right)$.
- Step 3: Find the peaks in the singular value spectrum ${s}_{1}^{2}\left(\omega \right)$ and their corresponding frequency points ${\omega}_{i}$. Then, calculate the RVAC values at each frequency point ${\omega}_{i}$ by using ${u}_{1}\left(\omega \right)$ and ${}_{}{}^{\mathrm{d}}u{}_{1}\left(\omega \right)$ instead of $h\left(\omega \right)$ and ${}_{}{}^{d}h\left(\omega \right)$ in Equation (1):$$\mathrm{RVAC}\left({\omega}_{i}\right)=\frac{{\Vert {u}_{1}^{\mathrm{H}}\left({\omega}_{i}\right){}_{}{}^{\mathrm{d}}u{}_{1}^{}\left({\omega}_{i}\right)\Vert}^{2}}{\left({u}_{1}^{\mathrm{H}}\left({\omega}_{i}\right){u}_{1}^{}\left({\omega}_{i}\right)\right)\left({}_{}{}^{\mathrm{d}}u{}_{1}^{\mathrm{H}}\left({\omega}_{i}\right){}_{}{}^{\mathrm{d}}u{}_{1}^{}\left({\omega}_{i}\right)\right)}.$$
- Step 4: Calculate the DRQ value by using Equation (3).

#### 2.2. Measurement Setup

#### 2.3. Experiment

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 10.**ODSs comparison from the cantilever beam in damage levels 2 and 3 at two frequencies with measurement points (circled in blue) as an example.

Damage Level 1 | Damage Level 2 | Damage Level 3 | |
---|---|---|---|

Diameter of the blind hole in mm | 0 | 4.30 | 4.30 |

Depth in mm | 0 | 2.66 | 4.09 |

**Table 2.**DRQ values from the four measurement pairs in Figure 5.

Measurements 1 and 2 | Measurements 1 and 3 | Measurements 1 and 4 | Measurements 1 and 5 |
---|---|---|---|

0.9904 | 0.9896 | 0.9905 | 0.9894 |

Frequency Range in kHz | 0–35 | 35–70 | 70–100 |

Number of Resonance Frequency Point | 1–70 | 71–120 | 121–143 |

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Cao, X.; Rembe, C.
Non-Contact Damage Detection under Operational Conditions with Multipoint Laservibrometry. *Sensors* **2020**, *20*, 732.
https://doi.org/10.3390/s20030732

**AMA Style**

Cao X, Rembe C.
Non-Contact Damage Detection under Operational Conditions with Multipoint Laservibrometry. *Sensors*. 2020; 20(3):732.
https://doi.org/10.3390/s20030732

**Chicago/Turabian Style**

Cao, Xiaodong, and Christian Rembe.
2020. "Non-Contact Damage Detection under Operational Conditions with Multipoint Laservibrometry" *Sensors* 20, no. 3: 732.
https://doi.org/10.3390/s20030732