# Time Domain Strain/Stress Reconstruction Based on Empirical Mode Decomposition: Numerical Study and Experimental Validation

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## Abstract

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## 1. Introduction

## 2. Strain and Stress Reconstruction Methodology

#### 2.1. Extraction of Modal Responses from Measurement Data Using EMD Method

_{i}(t) is the ith IMF and r(t) is the residue. The term $r(t)$ also represents the mean trend or constant for this signal [28].

_{int}must be imposed in the sifting process. The idea is to remove all frequency components lower and larger than ω

_{int}, and this can be done prior to or within the sifting process using a band-pass filter. The process to obtain the modal response corresponding to the ith natural frequency ω

_{i}is discussed in [28]. These IMFs have several characteristics: (1) Each IMF contains the intrinsic characteristics of the signal; (2) Once an IMF is obtained, the next IMF will not have the same frequency at the same time instant [29,30]; and (3) The first IMF for each IMFs series is considered to be the approximation of modal response. Using the sifting process with intermittency criteria, the original signal expression can be written as Equation (2):

_{i}(t) is the modal response (that is also an IMF) for the ith mode. Terms f

_{i}(t) (i = 1,2,…,n–m) are other IMFs but not modal responses.

#### 2.2. Transformation Equations for Strain and Stress Responses

**M**,

**K**and

**C**are the mass, stiffness, and damping matrices, respectively.

**X**is the displacement vector and

**F**is the load vector. For practical structures subject to stochastic excitations,

**F**is unknown and directly solving Equation (3) to obtain the dynamical responses of a sensor inaccessible location is not possible. However, the finite element method allows for correlating displacement responses of two different DOFs in the modal coordinates through the mode shape matrix. The mode shape matrix can readily be obtained by solving the eigenvalue problem of:

**f**is the vector of natural frequencies:

_{ij}represents the displacement contribution from DOF j under mode i. Since the model shape matrix is a constant once the number of the DOF and the discretization topology of the structure are determined, the ratio of displacement contribution of one DOF to that of another DOF is also a constant. This characteristic indicates that the responses of one DOF under modal coordinates allows for the calculation of responses of another DOF under modal coordinates. Denote the responses under modal coordinates as δ

_{ij}, where i and j represent the mode index and the DOF index, respectively, the physical meaning of the modal response relationship between two DOFs can be expressed as:

_{ij}(t) corresponds to the modal responses components for the overall physical displacement responses of

**X**

_{j}(t) for DOF j at a time index t. If the physical displacement responses of the DOF e having sensor measurements can be decomposed into its modal responses, i.e., ${X}_{e}\left(t\right)\approx \sum _{i=1\dots m}{\delta}_{ie}\left(t\right)$, using Equation (6), the physical displacement responses of the sensor inaccessible DOF u can be obtained as:

**ε**

^{(k)}and

**σ**

^{(k)}. From the finite element formulation, the strain and displacement has the following relationship:

**B**

^{(k)}is the strain-displacement matrix for element k and

**X**

^{(k)}is the displacement response vector consisting of all DOFs of element k. The expression of

**B**

^{(k)}usually has the form:

**L**is the differential operator and

**N**

^{(k)}is the matrix of shape functions for element k. Using Equation (8), the following equation is obtained under modal coordinates:

**c**is the material matrix. The constitutive equation for isotropic materials can be written explicitly as:

## 3. Numerical Examples

#### 3.1. Example 1: A Numerical Beam Structure Example

^{3}. The beam structure is divided into 10 equal segments in the FE model, as shown in Figure 2. Random forces are applied at all vertical direction DOFs of the FE model. The random forces are modeled as Gaussian white noise processes passed through a sixth order low-pass Butterworth filter with a 100 Hz cutoff. One percent of modal damping is considered. Displacement responses are calculated by solving the equation of motion of the beam based on its finite element model using mode superposition method. The sampling frequency is 1000 Hz. Strain responses are calculated using the strain-displacement matrix and Equation (9). The beam is modeled using Euler–Bernoulli beam theory and the 1 × 4 strain-displacement matrix is given by:

#### 3.2. Example 2: A Simplified Airfoil Structure Model Example

#### 3.2.1. Strain and Stress Response Reconstruction

#### 3.2.2. Effect of Measurement Noise to Reconstructed Strain and Stress Responses

## 4. Experimental Validation

#### 4.1. Experimental Setup

#### 4.2. Results and Discussion

#### Case 1. Effect of Sensor Number

#### Case 2. Effect of Sensor Location

#### Case 3. Effect of Mode Number

## 5. Conclusions

- (1)
- In this study, a time domain strain/stress reconstruction method based on EMD is proposed. According to numerical analysis results, the proposed method can produce results which are very close to theoretical solutions considering a practical noisy measurement system. The reconstructed results have an overall correlation coefficient larger than 0.975 under 10% RMS noise settings. The discrepancy between actual measurements and reconstruction results at the boundary region are possibly caused by the end boundary effect of EMD method.
- (2)
- Four sets of experiments, associated with basic example, sensor number, sensor location and mode number, verified the effectiveness of the time domain strain/stress reconstruction method in successfully reconstructing the strain response in location of interested. The results indicate that increasing the number of the measurement points has trivial effects on the reconstruction accuracy under ideal experimental circumstance. However, increasing the number of the measurement points may decrease the uncertainty (imposed by measurement noise or mishandling) for real engineering applications. Thus, more sensor measurements will commonly lead to higher reconstruction accuracy.
- (3)
- For the sensor location, two specific sensor locations should be avoided for reliable strain/stress response reconstruction: (a) locations where have low signal-to-noise ratio. In such case, the measured strain data are corrupted by noise, which will lead to inaccurate reconstruction results; (b) locations at or near the nodal points. In such locations, it may not capture all excited modes of the strain responses.
- (4)
- For the mode number, the higher modes will have little influence on the accuracy of the reconstruction, because of their low participation factors in Fourier spectra. Only dominant modes are efficient for accurate reconstructions.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Flowchart of the overall strain reconstruction procedure using remote strain measurements.

**Figure 2.**FE diagram of a beam structure with both ends fixed and applied forces. Synthesized strain measurement locations and the three reconstructed locations are the geometry centers of the surfaces of elements 1, 5, 8, and 10, respectively.

**Figure 3.**The synthesized strain measurement data and the Fourier spectra of the data. (

**a**) Entire 10 s measurement data; (

**b**) Measurement data (0–1 s); and (

**c**) Fourier spectra of the measurement data.

**Figure 4.**Four modal responses of the strain measurement data obtained using EMD method with intermittency criteria.

**Figure 5.**Reconstructed and theoretical strain responses for three locations (Loc. 1–3, Loc. in the paper are short form of location number) in Figure 2. Results are concentrated on 5–6 s for clear presentation. (

**a**) Results for Loc. 1; (

**b**) results for Loc. 2; and (

**c**) results for Loc. 3.

**Figure 6.**Reconstructed stress responses for three locations (Loc. 1–3) in Figure 2. Results are concentrated on 5–6 s for clear presentation. (

**a**) Results for Loc. 1; (

**b**) Results for Loc. 2; and (

**c**) Results for Loc. 3.

**Figure 8.**Strain measurement data and Fourier spectra of the data. (

**a**) Strain measurement data (0–3 s); and (

**b**) Fourier spectra of the measurement data (0–3 s).

**Figure 9.**Modal responses of the strain measurement data obtained by EMD method with intermittency criteria.

**Figure 10.**Reconstructed and theoretical strain responses for the location of interest shown in Figure 7.

**Figure 11.**Reconstructed and theoretical bending stress responses for the Loc. I labeled in Figure 7.

**Figure 12.**Reconstruction performance measured in correlation coefficient under different noise levels.

**Figure 13.**Schematic diagram of the experimental setup for the validation of the presented strain/stress reconstruction method.

**Figure 15.**Strain sensor measurement data and Fourier spectra of the data. (

**a**) strain sensor measurement data (0~8 s), and (

**b**) Fourier spectra of the measurement data (0~8 s).

**Figure 16.**Modal responses of the 15-th optical fiber measurement data obtained by EMD method with intermittency criteria.

**Figure 17.**(

**a**) Reconstructed and theoretical strain responses for the location of interest; (

**b**) Results are concentrated on 4–6 s for clear presentation.

**Figure 18.**(

**a**) Measured strain responses and reconstructed strain responses (using 1,2,3,14 measurement points) of the 5-th optical fiber measurement point on first 4 s; (

**b**) Results are concentrated on 1.2–1.6 s for clear presentation.

**Figure 19.**Modal strain responses of the 15-th optical fiber measurement data obtained by EMD method with intermittency criteria.

**Figure 20.**(

**a**) Measured strain responses and reconstructed strain responses of the 1st optical fiber measurement data; (0~4 s) (

**b**) Results are concentrated on 1.5~2 s for clear presentation.

**Figure 21.**(

**a**) Measured strain responses and reconstructed responses using six modes and four modes; (0~4 s) (

**b**) Results are concentrated on 1.5~2 s for clear presentation.

Mode | 1 | 2 | 3 | 4 |
---|---|---|---|---|

Identified frequency | 10.38 | 28.69 | 56.22 | 93.15 |

Passband corner frequency (Hz) | [8–9] | [22–26] | [46–52] | [80–87] |

Stopband corner frequency (Hz) | [11.5–13] | [31–36] | [61–67] | [98–105] |

Property | Value |
---|---|

Material | Aluminum 7075 |

Element type | Solid185 |

Young’s modulus E (GPa) | 72 |

Poisson’s ratio ν | 0.33 |

Mass per unit volume ρ (kg/m^{3}) | 2.81 × 103 |

Number of elements | 14,951 |

Property | Value |
---|---|

Material | Aluminum 7050 |

Length | 1.36 m |

Width | 0.12 m |

Thick | 0.01 m |

Young’s modulus E (GPa) | 7.17 |

Poisson’s ratio ν | 0.33 |

Mass per unit volume ρ (kg/m^{3}) | 2.81 × 103 |

Mode | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

Identified frequency | 4.39 | 27.23 | 75.69 | 148.22 | 244.15 | 365.26 |

Passband corner frequency (Hz) | [3–4] | [25–26] | [65–70] | [135–140] | [226–231] | [350–355] |

Stopband corner frequency (Hz) | [5–6] | [27.5–28.5] | [80–85] | [160–165] | [257–262] | [375–380] |

Case 1 | Case 2 | Case 3 | Case 4 | |
---|---|---|---|---|

Measurement points for reconstruction | 7-th | 3-th, 12-th | 3-th, 7-th, 12-th | the rest points except 5-th |

Correlation coefficient | 0.9497 | 0.9516 | 0.9519 | 0.9519 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

He, J.; Zhou, Y.; Guan, X.; Zhang, W.; Zhang, W.; Liu, Y.
Time Domain Strain/Stress Reconstruction Based on Empirical Mode Decomposition: Numerical Study and Experimental Validation. *Sensors* **2016**, *16*, 1290.
https://doi.org/10.3390/s16081290

**AMA Style**

He J, Zhou Y, Guan X, Zhang W, Zhang W, Liu Y.
Time Domain Strain/Stress Reconstruction Based on Empirical Mode Decomposition: Numerical Study and Experimental Validation. *Sensors*. 2016; 16(8):1290.
https://doi.org/10.3390/s16081290

**Chicago/Turabian Style**

He, Jingjing, Yibin Zhou, Xuefei Guan, Wei Zhang, Weifang Zhang, and Yongming Liu.
2016. "Time Domain Strain/Stress Reconstruction Based on Empirical Mode Decomposition: Numerical Study and Experimental Validation" *Sensors* 16, no. 8: 1290.
https://doi.org/10.3390/s16081290