# A Data-Driven Damage Identification Framework Based on Transmissibility Function Datasets and One-Dimensional Convolutional Neural Networks: Verification on a Structural Health Monitoring Benchmark Structure

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## Abstract

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## 1. Introduction

## 2. Fundamental Theories

#### 2.1. One-Dimensional Convolutional Neural Networks (1D CNNs): Convolution and Pooling

#### 2.2. Transmissibility Function (TF)

**K, M**, and

**C**are the stiffness, mass, and damping matrices of the system.

**,**corresponding to the DOFs where the excitations are located; (*) represents the complex conjugate transpose. From Equation (7), it can be deduced that a TF can be represented as a function of the FRF matrix, which contains rich information about structural dynamic characteristics, but without any involvement of the influence of excitation.

## 3. Construction of the TF-1D CNN Damage Identification Framework

#### 3.1. Construction of Massive TF Datasets

#### 3.2. Construction of the 1D CNN Model

## 4. Damage Identification in the ASCE Benchmark Structure

#### 4.1. Numerical Model

#### 4.2. Dynamic Response Analysis

#### 4.3. Damage Identification Using the TF-1D CNN Framework

#### 4.4. Noise Effect Analysis

## 5. Comparison Study

#### 5.1. Comparison of TS- and FFT-Based 1D CNN Methods

#### 5.2. Comparison with the TF-ANN Method

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The designed structure of the one-dimensional convolutional neural network (1D CNN) model.

**Figure 3.**American Society of Civil Engineers (ASCE) structural health monitoring benchmark structure.

**Figure 4.**Finite element model of the ASCE structural health monitoring benchmark structure with marked positions of (

**a**) the damaged braces and response measurement and (

**b**) the excitations.

**Figure 5.**Dynamic responses in terms of (

**a**) time series (TS), (

**b**) fast Fourier transform (FFT)-based, and (

**c**) TF signals, subject to different damage scenarios and the same excitation condition.

**Figure 6.**Dynamic responses in terms of (

**a**) TS, (

**b**) FFT-based, and (

**c**) TF signals, subject to the same damage scenario and different excitation conditions.

**Figure 7.**TFs between accelerations b and a under (

**a**) stiffness loss of brace A, ranging from 5% to 50% at 5% intervals; (

**b**) stiffness loss of brace B, ranging from 5% to 50% at 5% intervals.

**Figure 9.**The locations at which (

**a**) reference and (

**b**) non-reference acceleration dynamic responses were captured from the structure along the x and y directions.

**Figure 10.**Visualization results of the damage features in the TF signals extracted by the 1D CNN model: (

**a**) two- and (

**b**) three- dimensional maps (different damage scenarios are labeled in the figure from 1 to 16).

**Figure 11.**Visualization results of the damage features in the TFs extracted by the 1D CNN model, under the noise levels of the signal-to-noise ratio (SNR) = (

**a**) 50, (

**b**) 40, (

**c**) 35, (

**d**) 30, (

**e**) 25, (

**f**) 20, (

**g**) 15, and (

**h**) 10 dB.

**Figure 12.**Visualization results of signal features in the (

**a**) TS and (

**b**) FFT-based signals extracted by the 1D CNN model.

**Figure 13.**Visualization results of signal features in the TSs extracted by the 1D CNN model under the noise levels of SNR = (

**a**) 50, (

**b**) 40, (

**c**) 35, (

**d**) 30, (

**e**) 25, (

**f**) 20, (

**g**) 15, and (

**h**) 10 dB.

**Figure 14.**Visualization results of the features in the FFT-based signals extracted by the 1D CNN model under the noise levels of SNR = (

**a**) 50, (

**b**) 40, (

**c**) 35, (

**d**) 30, (

**e**) 25, (

**f**) 20, (

**g**) 15, and (

**h**) 10 dB.

**Table 1.**Accuracy of damage identification using the transmissibility function (TF)-one-dimensional convolutional neural network (1D CNN) framework under noise influence. Signal-to-noise ratio (SNR).

Noise | SNR (dB) | |||||||
---|---|---|---|---|---|---|---|---|

50 | 40 | 35 | 30 | 25 | 20 | 15 | 10 | |

Accuracy (%) | 100.00 | 100.00 | 100.00 | 97.70 | 83.03 | 70.42 | 60.91 | 41.27 |

**Table 2.**Accuracy of damage identification based on time series (TS)- and fast Fourier transform (FFT)-1D CNN frameworks under different noise levels.

Noise | SNR (dB) | ||||||||
---|---|---|---|---|---|---|---|---|---|

50 | 40 | 35 | 30 | 25 | 20 | 15 | 10 | ||

Accuracy (%) | TS-1D CNN | 11.33 | 11.45 | 11.27 | 11.58 | 11.88 | 10.73 | 10.79 | 11.33 |

FFT-1D CNN | 45.33 | 45.27 | 45.27 | 45.45 | 45.21 | 44.97 | 44.85 | 45.45 |

**Table 3.**Comparison between the TF-1D CNN and TF-artificial neural network (ANN) framework in noisy environments under different noise levels.

Noise | SNR (dB) | ||||||||
---|---|---|---|---|---|---|---|---|---|

50 | 40 | 35 | 30 | 25 | 20 | 15 | 10 | ||

Accuracy (%) | TF-1D CNN | 100.00 | 100.00 | 100.00 | 97.70 | 83.03 | 70.42 | 60.91 | 41.27 |

TF-ANN | 96.36 | 95.15 | 90.18 | 71.88 | 43.21 | 30.91 | 23.52 | 18.91 |

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## Share and Cite

**MDPI and ACS Style**

Liu, T.; Xu, H.; Ragulskis, M.; Cao, M.; Ostachowicz, W.
A Data-Driven Damage Identification Framework Based on Transmissibility Function Datasets and One-Dimensional Convolutional Neural Networks: Verification on a Structural Health Monitoring Benchmark Structure. *Sensors* **2020**, *20*, 1059.
https://doi.org/10.3390/s20041059

**AMA Style**

Liu T, Xu H, Ragulskis M, Cao M, Ostachowicz W.
A Data-Driven Damage Identification Framework Based on Transmissibility Function Datasets and One-Dimensional Convolutional Neural Networks: Verification on a Structural Health Monitoring Benchmark Structure. *Sensors*. 2020; 20(4):1059.
https://doi.org/10.3390/s20041059

**Chicago/Turabian Style**

Liu, Tongwei, Hao Xu, Minvydas Ragulskis, Maosen Cao, and Wiesław Ostachowicz.
2020. "A Data-Driven Damage Identification Framework Based on Transmissibility Function Datasets and One-Dimensional Convolutional Neural Networks: Verification on a Structural Health Monitoring Benchmark Structure" *Sensors* 20, no. 4: 1059.
https://doi.org/10.3390/s20041059