# Strain Modal Analysis of Small and Light Pipes Using Distributed Fibre Bragg Grating Sensors

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. Strain Modal Analysis

^{ε}] is given by [20]:

_{i}represents the number of inputs and N

_{o}represents the number of output measurements; m is the total number of modes considered; ${\psi}_{ir}^{\epsilon}$ is the normalised value of the r-th strain mode at point i, and ϕ

_{lr}is the normalised value of the r-th displacement mode at point l; k

_{r}, m

_{r}, and c

_{r}are the r-th modal stiffness, modal mass and modal damping, respectively; ω is the frequency of excitation. The columns of the matrix correspond to the strain responses due to the excitation points along the rows of the matrix. The elements of the SFRF matrix can be expressed as:

_{r}, m

_{r}, and c

_{r}for all the modes. Therefore, it is only necessary to excite one selected point and acquire the strain responses at all the measurement points, which enable the modal parameters (frequency, damping, strain mode shape) to be obtained after data processing.

#### 2.2. Strain Modal Parameter Identification

_{ilr}and s

_{r}are the r-th modal residue and pole, respectively; ω

_{r}and ζ

_{r}are the r-th undamped natural frequency and damping ratio, respectively. The method of rational fraction orthogonal polynomial (RFOP) is used to estimate the poles of the system in this paper. The poles of the system can be obtained by power polynomial solutions and stabilisation diagrams. The stabilisation criterion about frequency, damping and mode shape can be utilised to acquire the number of the poles.

_{r}are applied to Equation (7), the strain frequency response function can be transferred to the linear equation related to residues, upper residuals and lower residuals, which could be solved by the least square method.

_{r}] of one column can be expressed as:

## 3. Experiment and Comparison with Finite Element Analysis (FEA) Simulation

#### 3.1. Experimental Setup

^{3}, 200 GPa, and 0.31, respectively.

#### 3.2. Finite Element Analysis

## 4. Results and Discussion

## 5. Application

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**Signals of impacting force and acceleration. (

**a**) impacting force; (

**b**) frequency spectrum of impacting force; (

**c**) acceleration; (

**d**) acceleration FRF.

**Figure 7.**Comparison of normalised strain mode shapes between the results of FEA simulation and experiment, the strain mode shapes along the Z direction of the pipe: (

**a**) mode 1; (

**b**) mode 2; (

**c**) mode 3; (

**d**) mode 4.

**Figure 8.**MAC correlation of strain mode shapes between the results of experimental testing and FEA calculation.

Sensor Number | Wavelength (nm) | Distance from Fixed End (mm) |
---|---|---|

FBG1 | 1295.388 | 35 |

FBG2 | 1290.417 | 125 |

FBG3 | 1288.557 | 215 |

FBG4 | 1309.855 | 305 |

FBG5 | 1304.282 | 395 |

FBG6 | 1298.596 | 485 |

FBG7 | 1309.965 | 575 |

FBG8 | 1288.398 | 665 |

FBG9 | 1290.245 | 755 |

FBG10 | 1304.615 | 845 |

FBG11 | 1298.676 | 935 |

Mode Number | Natural Frequency (Hz) | Damping Ratio (%) |
---|---|---|

1 | 14.8 | 0.471 |

2 | 92.8 | 0.204 |

3 | 259.0 | 0.195 |

4 | 507.3 | 0.221 |

Mode Number | Natural Frequency (Hz) | Damping Ratio (%) |
---|---|---|

1 | 74.6 | 0.665 |

2 | 209.8 | 0.305 |

3 | 410.4 | 0.145 |

4 | 670.6 | 0.195 |

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**MDPI and ACS Style**

Huang, J.; Zhou, Z.; Zhang, L.; Chen, J.; Ji, C.; Pham, D.T.
Strain Modal Analysis of Small and Light Pipes Using Distributed Fibre Bragg Grating Sensors. *Sensors* **2016**, *16*, 1583.
https://doi.org/10.3390/s16101583

**AMA Style**

Huang J, Zhou Z, Zhang L, Chen J, Ji C, Pham DT.
Strain Modal Analysis of Small and Light Pipes Using Distributed Fibre Bragg Grating Sensors. *Sensors*. 2016; 16(10):1583.
https://doi.org/10.3390/s16101583

**Chicago/Turabian Style**

Huang, Jun, Zude Zhou, Lin Zhang, Juntao Chen, Chunqian Ji, and Duc Truong Pham.
2016. "Strain Modal Analysis of Small and Light Pipes Using Distributed Fibre Bragg Grating Sensors" *Sensors* 16, no. 10: 1583.
https://doi.org/10.3390/s16101583