# Experimental Shape Sensing and Load Identification on a Stiffened Panel: A Comparative Study

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. The Modal Method

#### 2.2. The Inverse Finite Element Method

**1**when the corresponding strain is measured or to a small value (${\mathbf{10}}^{-\mathbf{4}},{\mathbf{10}}^{-\mathbf{5}},{\mathbf{10}}^{-\mathbf{6}}$) when it is not measured. In the last case, the corresponding ${\mathit{\epsilon}}_{\mathit{k}}^{\mathit{m}}$ is set to 0 (as a consequence, the terms ${\mathit{\lambda}}_{\mathbf{7},\mathbf{8}}^{\mathit{e}}$ are always set to a small value and the terms ${\mathit{\epsilon}}_{\mathbf{7},\mathbf{8}}^{\mathit{\epsilon}}$ are always set to 0). The integral over the area of the element, ${\mathit{A}}^{\mathit{e}}$, in Equation (9), is numerically computed using Gaussian quadrature. Therefore, it is transformed into a summation over the $\mathit{n}\times \mathit{n}$ quadrature points:

#### 2.3. The 2-Step Method

## 3. Experimental Setup and Preliminary Computations

#### 3.1. Experimental Setup

#### 3.2. Models

^{®}.

#### 3.3. Configuration of Sensors

^{®}high-definition distributed fibre optic strain sensing system. The sensor is based on Rayleigh scattering and Optical Frequency Domain Reflectometry (OFDR) [9,10] and allows the measurement of the strain component along the fibre optic direction with an impressive density. Considering a 10 m long fibre, it is possible to measure the strain for every 1.3 mm. The use of this kind of fibre allows to follow complex paths on the structure and, in the case of the stiffened panel, to measure the strain along the $\mathit{x}$ direction on five sensing lines along the panel’s length in a back-to-back configuration (i.e., every measurement point on the top surface of the panel has a corresponding one on the bottom surface of the panel. The five optimal sensing lines have been searched between the 25 lines identified by the centroids of the elements of the inverse mesh (Figure 7). On each of the lines lying on the panel, 38 centroidal locations have been considered as measurement points (whereas 34 centroidal locations have been considered for the lines lying on the stiffeners). The centroids of the inverse elements too close to the supported boundaries, where the presence of the iron bars does not allow the application of the fibre, have not been considered.

**53,130**configurations of 5 lines out of the 25 possible ones, for each method, have been computed and the relative percent Root Mean Squared Errors ($\%{\mathit{ERMS}}_{\mathit{v}}$), with respect to the reference vertical displacements of all the 978 nodes of the iFEM mesh, have been collected:

## 4. Experimental Results

**HF-FEM**). The results show a good reproducibility of the experiment over the three tests and a consequent low level of variability for the experimentally measured quantities and the reconstructed ones. Therefore, the three tests have been considered representative of the experimental behaviour of the structure and no additional tests have been performed. The absolute value of the percent errors, averaged over the three tests ($\%{\overline{\mathit{Err}}}_{{\mathit{F}}_{\mathit{y}}}$$\%{\overline{\mathit{Err}}}_{{\mathit{v}}_{\mathbf{1}-\mathbf{4}}}$), have been reported in Table 3. The average error over the four reconstructed displacements ($\%{\overline{\mathit{Err}}}_{\mathit{v}}$) has also been computed for each method. In Figure 10, Figure 11, Figure 12 and Figure 13, the contours plots for the full transverse displacement field, reconstructed by the four shape sensing methods, are shown for only one representative test, the Test 3.

**iFEM**. This method shows an average error ($\%{\overline{\mathit{Err}}}_{\mathit{v}}$) that is $\mathbf{1}.\mathbf{5}\%$ and a maximum error of $\mathbf{2}.\mathbf{2}\%$, thus proving to be highly accurate in the reconstruction of the whole transverse displacement field. Moreover, it is important to remind that this method is able to reach this level of accuracy without the need of any knowledge of the material properties of the structure.

**MM**(1–22)), shows a good reconstruction of the first three displacements ($\%{\overline{\mathit{Err}}}_{{\mathit{v}}_{\mathbf{1}-\mathbf{3}}}\le \mathbf{7}.\mathbf{5}\%$) but a really poor reconstruction of the fourth displacement ($\%{\overline{\mathit{Err}}}_{{\mathit{v}}_{\mathbf{4}}}=\mathbf{31}.\mathbf{2}\%$). The second configuration (

**MM**$(\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{8},\mathbf{12})$), with only five modes selected, shows a better and more consistent overall accuracy, with an average error of $\mathbf{4}.\mathbf{3}\%$. Also in this case, the fourth displacement is reconstructed with lower accuracy ($\%{\overline{\mathit{Err}}}_{{\mathit{v}}_{\mathbf{4}}}=\mathbf{7}.\mathbf{9}\%$). These local phenomena, especially for the

**MM**(1–22) configuration, can be explained by analysing the working principles of the method. The method tries to reconstruct the deformed shape of the structure as a combination of the modal shapes, by using the strain information as the weights of the combination. Therefore, some mode shapes that exhibit buckles in some areas, if not sufficiently smoothed by the strain information given by the sensors, can bias the overall results obtained by the method, thus generating local inaccuracies. By comparing Figure 11 and Figure 12 with the most accurate reconstruction in Figure 13, it is possible to observe how the transverse displacement field is biased in the areas on the sides of the centre point, where the ${\mathit{v}}_{\mathbf{4}}$ sensor is located.

**2-step**approach is able to accurately identify the applied load, with an average error of $\mathbf{2}\%$ in the first step of the procedure. The application of the identified load to the refined FE model adds this error to the ones already present in the computation of the displacements using this model. In fact, for each displacement, the percent error is the sum of the percent errors coming from the refined model (

**HF-FEM**) and the percent error in the identification of the load. The analysis of these errors shows an overall accuracy that is slightly lower than the one obtained by the

**MM**$(\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{8},\mathbf{12})$. Nevertheless, it is important to notice that the accuracy of the 2-step method can be increased by adopting a refined model that is more representative of the experimental set-up, thus reducing the error coming from the model’s inaccuracy. Moreover, the 2-step method is the only one, within this study, that can simultaneously reconstruct the displacements and the applied load.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Experimental Strains

**Figure A1.**Experimental strains from the stiffened panel—The numbering of the fibres is the one reported in Figure 8.

**Figure A2.**Experimental strains from the stiffened panel—The numbering of the fibres is the one reported in Figure 8.

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**Figure 9.**LVDTs’ configuration—The location of the four LVDTs (${\mathit{v}}_{\mathbf{1}-\mathbf{4}}$) on the surface of the panel are shown. All dimensions are expressed in [mm].

**Figure 11.**Test 3—Transverse displacement contour for the Model Method (1-22) with all dimensions in mm.

**Figure 12.**Test 3—Transverse displacement contour for the Modal Method $(\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{8},\mathbf{12})$ with all dimensions in mm.

Al-Li Alloy | |
---|---|

$\mathit{E}\phantom{\rule{4pt}{0ex}}\left[\mathbf{MPa}\right]$ | 75,958 |

$\mathbf{\nu}$ | 0.300 |

$\mathbf{\rho}\phantom{\rule{4pt}{0ex}}[\mathbf{g}/{\mathbf{cm}}^{\mathbf{3}}]$ | 2.78 |

**Table 2.**Shape sensing and load identification results for the stiffened panel. In parenthesis, the percentage errors with respect to the experimental values are reported. The errors are computed considering the absolute value of the displacements.

Experimental | HF-FEM | 2-Step | MM (1–22) | MM $(1,2,3,8,12)$ | iFEM | |
---|---|---|---|---|---|---|

Test 1 | ||||||

${\mathit{F}}_{\mathit{y}}\phantom{\rule{4pt}{0ex}}\left[\mathbf{N}\right]$ | −865.0 | −883.1 | ||||

($\%{\mathit{Err}}_{{\mathit{F}}_{\mathit{y}}}$) | (+2.1%) | |||||

${\mathit{v}}_{\mathbf{1}}\phantom{\rule{4pt}{0ex}}\left[\mathbf{mm}\right]$ | −3.000 | −3.078 | −3.142 | −3.081 | −3.047 | −2.916 |

($\%{\mathit{Err}}_{{\mathit{v}}_{\mathbf{1}}}$) | (+2.6%) | (+4.7%) | (+2.7%) | (+1.6%) | (−2.8%) | |

${\mathit{v}}_{\mathbf{2}}\phantom{\rule{4pt}{0ex}}\left[\mathbf{mm}\right]$ | −2.644 | −2.752 | −2.809 | −2.823 | −2.705 | −2.624 |

($\%{\mathit{Err}}_{{\mathit{v}}_{\mathbf{2}}}$) | (+4.1%) | (+6.2%) | (+6.8%) | (+2.3%) | (−0.8%) | |

${\mathit{v}}_{\mathbf{3}}\phantom{\rule{4pt}{0ex}}\left[\mathbf{mm}\right]$ | −1.614 | −1.660 | −1.695 | −1.653 | −1.627 | −1.641 |

($\%{\mathit{Err}}_{{\mathit{v}}_{\mathbf{3}}}$) | (+2.9%) | (+5.0%) | (+2.4%) | (+0.8%) | (+1.7%) | |

${\mathit{v}}_{\mathbf{4}}\phantom{\rule{4pt}{0ex}}\left[\mathbf{mm}\right]$ | −1.610 | −1.600 | −1.633 | −1.058 | −1.475 | −1.562 |

($\%{\mathit{Err}}_{{\mathit{v}}_{\mathbf{4}}}$) | (−0.6%) | (+1.4%) | (−34.3%) | (−8.4%) | (−3.0%) | |

Test 2 | ||||||

${\mathit{F}}_{\mathit{y}}\phantom{\rule{4pt}{0ex}}\left[\mathbf{N}\right]$ | −882.0 | −899.9 | ||||

($\%{\mathit{Err}}_{{\mathit{F}}_{\mathit{y}}}$) | (+2.0%) | |||||

${\mathit{v}}_{\mathbf{1}}\phantom{\rule{4pt}{0ex}}\left[\mathbf{mm}\right]$ | −3.002 | −3.138 | −3.202 | −3.139 | −3.104 | −2.973 |

($\%{\mathit{Err}}_{{\mathit{v}}_{\mathbf{1}}}$) | (+4.5%) | (+6.7%) | (+4.6%) | (+3.4%) | (−1.0%) | |

${\mathit{v}}_{\mathbf{2}}\phantom{\rule{4pt}{0ex}}\left[\mathbf{mm}\right]$ | −2.634 | −2.806 | −2.863 | −2.825 | −2.761 | −2.657 |

($\%{\mathit{Err}}_{{\mathit{v}}_{\mathbf{2}}}$) | (+6.5%) | (+8.7%) | (+7.3%) | (+4.8%) | (+0.9%) | |

${\mathit{v}}_{\mathbf{3}}\phantom{\rule{4pt}{0ex}}\left[\mathbf{mm}\right]$ | −1.603 | −1.693 | −1.727 | −1.605 | −1.662 | −1.631 |

($\%{\mathit{Err}}_{{\mathit{v}}_{\mathbf{3}}}$) | (+5.6%) | (+7.7%) | (+0.1%) | (+3.7%) | (+1.7%) | |

${\mathit{v}}_{\mathbf{4}}\phantom{\rule{4pt}{0ex}}\left[\mathbf{mm}\right]$ | −1.613 | −1.631 | −1.644 | −1.130 | −1.481 | −1.638 |

($\%{\mathit{Err}}_{{\mathit{v}}_{\mathbf{4}}}$) | (+1.1%) | (+1.9%) | (−29.9%) | (−8.2%) | (+1.5%) | |

Test 3 | ||||||

${\mathit{F}}_{\mathit{y}}\phantom{\rule{4pt}{0ex}}\left[\mathbf{N}\right]$ | −882.0 | −899.7 | ||||

($\%{\mathit{Err}}_{{\mathit{F}}_{\mathit{y}}}$) | (+2.0%) | |||||

${\mathit{v}}_{\mathbf{1}}\phantom{\rule{4pt}{0ex}}\left[\mathbf{mm}\right]$ | −3.004 | −3.138 | −3.201 | −3.138 | −3.104 | −2.975 |

($\%{\mathit{Err}}_{{\mathit{v}}_{\mathbf{1}}}$) | (+4.5%) | (+6.6%) | (+4.5%) | (+3.3%) | (−1.0%) | |

${\mathit{v}}_{\mathbf{2}}\phantom{\rule{4pt}{0ex}}\left[\mathbf{mm}\right]$ | −2.649 | −2.806 | −2.862 | −2.834 | −2.760 | −2.644 |

($\%{\mathit{Err}}_{{\mathit{v}}_{\mathbf{2}}}$) | (+5.9%) | (+8.0%) | (+7.0%) | (+4.2%) | (−0.2%) | |

${\mathit{v}}_{\mathbf{3}}\phantom{\rule{4pt}{0ex}}\left[\mathbf{mm}\right]$ | −1.622 | −1.693 | −1.727 | −1.626 | −1.662 | −1.608 |

($\%{\mathit{Err}}_{{\mathit{v}}_{\mathbf{3}}}$) | (+4.4%) | (+6.5%) | (+0.2%) | (+2.5%) | (−0.9%) | |

${\mathit{v}}_{\mathbf{4}}\phantom{\rule{4pt}{0ex}}\left[\mathbf{mm}\right]$ | −1.609 | −1.631 | −1.664 | −1.137 | −1.493 | −1.644 |

($\%{\mathit{Err}}_{{\mathit{v}}_{\mathbf{4}}}$) | (+1.4%) | (+3.4%) | (−29.3%) | (−7.2%) | (+2.2%) |

**Table 3.**Absolute value of the percent errors of the reconstructed quantities averaged over the three tests.

2-Step | MM (1–22) | MM $(1,2,3,8,12)$ | iFEM | |
---|---|---|---|---|

$\%{\overline{\mathit{Err}}}_{{\mathit{F}}_{\mathit{y}}}$ | 2.0% | |||

$\%{\overline{\mathit{Err}}}_{{\mathit{v}}_{\mathbf{1}}}$ | 6.0% | 3.9% | 2.8% | 1.6% |

$\%{\overline{\mathit{Err}}}_{{\mathit{v}}_{\mathbf{2}}}$ | 7.7% | 7.0% | 3.8% | 0.6% |

$\%{\overline{\mathit{Err}}}_{{\mathit{v}}_{\mathbf{3}}}$ | 6.4% | 0.9% | 2.3% | 1.4% |

$\%{\overline{\mathit{Err}}}_{{\mathit{v}}_{\mathbf{4}}}$ | 2.3% | 31.2% | 7.9% | 2.2% |

$\%{\overline{\mathit{Err}}}_{\mathit{v}}$ | 5.6% | 10.8% | 4.2% | 1.5% |

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**MDPI and ACS Style**

Esposito, M.; Mattone, M.; Gherlone, M.
Experimental Shape Sensing and Load Identification on a Stiffened Panel: A Comparative Study. *Sensors* **2022**, *22*, 1064.
https://doi.org/10.3390/s22031064

**AMA Style**

Esposito M, Mattone M, Gherlone M.
Experimental Shape Sensing and Load Identification on a Stiffened Panel: A Comparative Study. *Sensors*. 2022; 22(3):1064.
https://doi.org/10.3390/s22031064

**Chicago/Turabian Style**

Esposito, Marco, Massimiliano Mattone, and Marco Gherlone.
2022. "Experimental Shape Sensing and Load Identification on a Stiffened Panel: A Comparative Study" *Sensors* 22, no. 3: 1064.
https://doi.org/10.3390/s22031064