# Dynamic Deformation Reconstruction of Variable Section WING with Fiber Bragg Grating Sensors

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## Abstract

**:**

## 1. Introduction

## 2. A Deformation–Reconstruction Model for Wing with Variable Cross-Section

#### 2.1. Inverse Finite Element Model for Variable Cross-Section Beam

^{T}of the beam element can be expressed with the above section strain vector $\mathit{e}\left(\mathit{u}\right)$ in theory:

#### 2.2. Calculation of Section Strain of Variable Section Beam Element

## 3. Strain Error Correction

#### 3.1. SSILSVRFN Structure Learning

**P**as

**P**. The iteration number is updated as $\text{}{N}_{ite}={N}_{ite}+1$.

#### 3.2. SSILSVRFN Parameter Learning

#### 3.2.1. Consequent Parameter Learning

#### 3.2.2. Antecedent Parameter Learning

## 4. Verifications through Simulations and Experimentation

#### 4.1. Simulation Test

#### 4.2. Physical Model Test

- By comparing between the FE analysis and the reconstructing with using IFEM, numerical studies show that the percent error of the deformation reconstruction along the main direction remains below 6.0%.
- Because of the strain measurement system error and the model error, experimental studies show that the percent error of the deformation reconstruction along the main direction computed from the unmodified strain measurements with iFEM remains below 13%.
- Experimental application of the proposed method shows that: the percent error of the deformation reconstruction along the main direction computed from the modified strain measurements remains below 6.7%.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## References

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**Figure 5.**Structure of the self-structuring linear support vector regression algorithm fuzzy network (SSILSVRFN).

**Figure 8.**Structural block diagram of dynamic deformation reconstruction method for variable section wing.

**Figure 9.**The finite element (FE) model and the CAD model of the variable section wing. (

**A**) The FE model; (

**B**) The CAD model.

**Figure 16.**Comparison between deformation displacements of point C for the different time. (

**A**) X-axis; (

**B**) Z-axis; and (

**C**) Y-axis.

**Table 1.**The comparison between the FE analysis and the reconstructing using inverse finite element method (iFEM). The displacements are expressed in millimeter and rotations are expressed in radian.

Loading | ${\mathit{u}}_{\mathit{x}}$ | ${\mathit{v}}_{\mathit{y}}$ | ${\mathit{w}}_{\mathit{z}}$ | ${\mathit{\theta}}_{\mathit{x}}$ | ${\mathit{\theta}}_{\mathit{y}}$ | ${\mathit{\theta}}_{\mathit{z}}$ | |
---|---|---|---|---|---|---|---|

135 N | FE analysis | 0.81 | 2.43 | 124.06 | −0.1292 | 0.0039 | 0.0011 |

IFEM | 0.63 | 2.08 | 116.93 | −0.0846 | 0.0035 | 0.0032 | |

Absolute error | 0.18 | 0.35 | 7.13 | 0.0446 | 0.0004 | 0.0021 | |

Percent error | 22.2% | 14.4% | 5.7% | 34.5% | 10.3% | 190.9% | |

200 N | FE analysis | 1.52 | 4.27 | 168.18 | −0.2154 | 0.0066 | 0.0018 |

IFEM | 1.17 | 3.72 | 158.09 | −0.1492 | 0.0058 | 0.0047 | |

Absolute error | 0.35 | 0.55 | 10.09 | 0.0662 | 0.0008 | 0.0029 | |

Percent error | 23.0% | 12.8% | 6.0% | 30.7% | 12.1% | 161.1% |

**Table 2.**Comparison of deformation along the y-axis for different time by removing the load of 20 kg loaded at the wing tip, the measured strain is obtained from the fiber Bragg grating (FBG) sensor measurement, and the actual strain is calculated from the NDI measurement with iFEM.$\text{}{\mathit{C}}_{\mathit{Y}}^{\mathit{U}}$ and ${\mathit{C}}_{\mathit{Y}}^{\mathit{M}}$ are, respectively, the deformation computed from the unmodified and modified strain measurements with iFEM at point C.

Time/s | Measured Strain | Actual Strain | Modified Strain | ${\mathit{C}}_{\mathit{Y}}^{\mathit{U}}/\mathbf{mm}$ | ${\mathit{C}}_{\mathit{Y}}^{\mathit{N}\mathit{D}\mathit{I}}/\mathbf{mm}$ | ${\mathit{C}}_{\mathit{Y}}^{\mathit{M}}/\mathbf{mm}$ | Percentage Error |
---|---|---|---|---|---|---|---|

0.04 | 0.001671 | 0.002137 | 0.002183 | 132.01 | 149.16 | 138.89 | 6.7% |

0.14 | 0.001428 | 0.001945 | 0.00187 | 113.07 | 127.52 | 124.97 | 5.2% |

0.24 | 0.001334 | 0.001659 | 0.001603 | 95.32 | 105.29 | 99.72 | 4.5% |

0.34 | 0.001079 | 0.001271 | 0.001261 | 78.48 | 89.13 | 86.29 | 4.2% |

0.44 | 0.001235 | 0.001357 | 0.001391 | 67.86 | 76.11 | 69.70 | 5.3% |

0.54 | 0.000934 | 0.001099 | 0.001087 | 57.46 | 64.37 | 62.79 | 5.6% |

0.64 | 0.00083 | 0.001075 | 0.001035 | 48.95 | 56.28 | 52.86 | 4.5% |

0.75 | 0.000644 | 0.000723 | 0.000714 | 46.53 | 53.30 | 49.95 | 5.1% |

0.85 | 0.000654 | 0.000779 | 0.000742 | 38.67 | 44.32 | 40.75 | 3.9% |

0.95 | 0.000627 | 0.000696 | 0.0007 | 33.83 | 37.78 | 36.58 | 5.2% |

**Table 3.**Comparison of deformation along the x-axis and z-axis for different time by removing the load of 20 kg loaded at the wing tip.

Time/s | ${\mathit{C}}_{\mathit{X}}^{\mathit{U}}/\mathbf{mm}$ | ${\mathit{C}}_{\mathit{X}}^{\mathit{N}\mathit{D}\mathit{I}}/\mathbf{mm}$ | ${\mathit{C}}_{\mathit{X}}^{\mathit{M}}/\mathbf{mm}$ | Percentage Error | ${\mathit{C}}_{\mathit{Z}}^{\mathit{U}}/\mathbf{mm}$ | ${\mathit{C}}_{\mathit{Z}}^{\mathit{N}\mathit{D}\mathit{I}}/\mathbf{mm}$ | ${\mathit{C}}_{\mathit{Z}}^{\mathit{M}}/\mathbf{mm}$ | Percentage Error |
---|---|---|---|---|---|---|---|---|

0.04 | 1.82 | 1.26 | 1.42 | 12.70% | 4.75 | 3.87 | 4.11 | 6.20% |

0.14 | 1.54 | 0.89 | 1.03 | 15.73% | 2.37 | 1.92 | 2.06 | 7.29% |

0.24 | 1.46 | 0.92 | 0.99 | 7.61% | 3.29 | 2.87 | 3.15 | 9.76% |

0.34 | 1.27 | 0.75 | 0.82 | 9.33% | 2.46 | 2.14 | 2.33 | 8.88% |

0.44 | 1.14 | 0.67 | 0.73 | 8.96% | 2.34 | 2.03 | 2.18 | 7.39% |

0.54 | 1.08 | 0.56 | 0.65 | 16.07% | 2.26 | 1.98 | 2.09 | 5.56% |

0.64 | 1.03 | 0.51 | 0.6 | 17.65% | 2.12 | 1.85 | 1.97 | 6.49% |

0.75 | 0.89 | 0.47 | 0.52 | 10.64% | 2.06 | 1.74 | 1.88 | 8.05% |

0.85 | 0.82 | 0.43 | 0.51 | 18.60% | 1.84 | 1.58 | 1.69 | 6.96% |

0.95 | 0.78 | 0.37 | 0.44 | 18.92% | 1.79 | 1.62 | 1.72 | 6.17% |

**Table 4.**The maximum percentage error among the individual nodal displacements,$\text{}dis{p}_{Y}^{U}$ and $\text{}dis{p}_{Y}^{M}$ are, respectively, the displacement computed form the unmodified and modified strain measurements with iFEM.

Node | $\mathit{d}\mathit{i}\mathit{s}{\mathit{p}}_{\mathit{Y}}^{\mathit{U}}/\mathbf{mm}$ | $\mathit{d}\mathit{i}\mathit{s}{\mathit{p}}_{\mathit{Y}}^{\mathit{N}\mathit{D}\mathit{I}}/\mathbf{mm}$ | $\mathit{d}\mathit{i}\mathit{s}{\mathit{p}}_{\mathit{Y}}^{\mathit{M}}/\mathbf{mm}$ | Maximum Percentage Error |
---|---|---|---|---|

1 | 132.01 | 149.06 | 138.97 | 6.7% |

2 | 79.57 | 94.38 | 88.63 | 6.0% |

3 | 41.6 | 55.42 | 52.83 | 4.6% |

4 | 19.41 | 29.31 | 27.79 | 5.1% |

5 | 9.78 | 11.5 | 10.96 | 4.7% |

6 | 3.66 | 4.77 | 4.57 | 4.1% |

**Table 5.**Comparison of deformation along the y-axis for different time, $RM{S}_{U}$ and $RM{S}_{M}$ are, respectively, the accuracy of the deformation computed form the unmodified and modified strain measurements with iFEM.

Time/s | $\mathit{R}\mathit{M}{\mathit{S}}_{\mathit{U}}/\mathbf{mm}$ | $\mathit{R}\mathit{M}{\mathit{S}}_{\mathit{M}}/\mathbf{mm}$ | Percentage Reduced |
---|---|---|---|

0.04 | 12.01 | 3.97 | 66.8% |

0.14 | 11.13 | 3.18 | 71.4% |

0.24 | 10.22 | 3.38 | 66.9% |

0.34 | 9.42 | 3.71 | 60.5% |

0.44 | 9.51 | 3.20 | 66.3% |

0.54 | 9.31 | 3.29 | 64.6% |

0.64 | 7.05 | 2.16 | 69.4% |

0.75 | 6.21 | 2.03 | 67.2% |

0.85 | 6.15 | 2.50 | 59.2% |

0.95 | 5.83 | 2.18 | 62.5% |

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**MDPI and ACS Style**

Fu, Z.; Zhao, Y.; Bao, H.; Zhao, F. Dynamic Deformation Reconstruction of Variable Section WING with Fiber Bragg Grating Sensors. *Sensors* **2019**, *19*, 3350.
https://doi.org/10.3390/s19153350

**AMA Style**

Fu Z, Zhao Y, Bao H, Zhao F. Dynamic Deformation Reconstruction of Variable Section WING with Fiber Bragg Grating Sensors. *Sensors*. 2019; 19(15):3350.
https://doi.org/10.3390/s19153350

**Chicago/Turabian Style**

Fu, Zhen, Yong Zhao, Hong Bao, and Feifei Zhao. 2019. "Dynamic Deformation Reconstruction of Variable Section WING with Fiber Bragg Grating Sensors" *Sensors* 19, no. 15: 3350.
https://doi.org/10.3390/s19153350