# Structural Damage Identification of Composite Rotors Based on Fully Connected Neural Networks and Convolutional Neural Networks

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## Abstract

**:**

## 1. Introduction

#### 1.1. State-of-the-Art

#### 1.2. Aim and Outline of the Paper

## 2. Materials and Methods

#### 2.1. Investigated Composite Rotors

#### 2.2. Data Set Creation and Data Augmentation

#### 2.3. Labeling of the Damage States

#### 2.4. Dimensionality Reduction Methods

#### 2.5. Model Training, Validation and Testing

#### 2.6. Overview of the Generated Machine Learning Models

- Algorithm: FC, 1D-CNN,
- Data set: original (OD) or augmented (AD),
- Dimensionality reduction method: pure data, each 30th frequency, PCA, PCA+LDA, SVD+LDA and KPCA,
- Classifier: radial position (RAD), angular position (ANG), load magnitude (MAGN) and damage accumulation (DAM).

## 3. Results

#### 3.1. Classification with Fully Connected Neural Network and Convolutional Neural Network

#### 3.2. Influence of Reduced Dimensionality

#### 3.3. Influence of Synthetically Augmented Data Set

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ANG | Angular Position |

ANN | Artificial Neural Network |

CNN | Convolutional Neural Network |

DAM | Damage Accumulation |

FC | Fully Connected Neural Network |

FE | Finite Element |

FRF | Frequency Response Function |

HMM | Hidden Markov Model |

KPCA | Kernel Principal Component Analysis |

LDA | Linear Discriminanat Analysis |

MAGN | Load Magnitude |

ML | Machine Learning |

OD | Original Data |

PCA | Principal Component Analysis |

PSD | Power Spectral Density |

RAD | Radial Position |

ReLU | Rectified Linear Unit |

SHM | Structural Health Monitoring |

SVD | Singular Value Decomposition |

## Appendix A

**Table A1.**All trained models and combinations of network, data type and dimensionality. Test accuracy, sensitivity, specificity and validation accuracy with cross-validation error is displayed.

Network Type | Data Type | Dimensionality Reduction | Classifier | Name | Acc. [%] | Sens. [%] | Spec. [%] |
---|---|---|---|---|---|---|---|

FC | Original logarithmized data | Pure data | RAD | FC-OD-Pure-RAD | $99.2\pm 0.5$ | $99.4\pm 0.6$ | $98.9\pm 0.6$ |

FC | Original logarithmized data | Pure data | ANG | FC-OD-Pure-ANG | $32.7\pm 4.9$ | $32.9\pm 7.5$ | $34.6\pm 6.0$ |

FC | Original logarithmized data | Pure data | MAGN | FC-OD-Pure-MAGN | $94.0\pm 4.3$ | $99.0\pm 1.5$ | $91.1\pm 7.0$ |

FC | Original logarithmized data | Pure data | DAM | FC-OD-Pure-DAM | $87.1\pm 1.4$ | $96.2\pm 2.6$ | $83.4\pm 2.5$ |

FC | Original logarithmized data | Every 30th frequency | RAD | FC-OD-30thFreq-RAD | $97.6\pm 1.4$ | $99.2\pm 1.1$ | $96.1\pm 2.5$ |

FC | Original logarithmized data | Every 30th frequency | ANG | FC-OD-30thFreq-ANG | − | − | − |

FC | Original logarithmized data | Every 30th frequency | MAGN | FC-OD-30thFreq-MAGN | $97.1\pm 1.8$ | $98.4\pm 2.2$ | $96.0\pm 2.1$ |

FC | Original logarithmized data | Every 30th frequency | DAM | FC-OD-30thFreq-DAM | $87.6\pm 1.6$ | $92.3\pm 6.3$ | $85.9\pm 2.8$ |

FC | Original logarithmized data | PCA | RAD | FC-OD-PCA-RDA | $99.0\pm 0.7$ | $100.0\pm 0$ | $98.1\pm 1.3$ |

FC | Original logarithmized data | PCA | ANG | FC-OD-PCA-ANG | − | − | − |

FC | Original logarithmized data | PCA | MAGN | FC-OD-PCA-MAGN | $95.8\pm 1.6$ | $95.9\pm 2.7$ | $95.8\pm 2.7$ |

FC | Original logarithmized data | PCA | DAM | FC-OD-PCA-DAM | $87.6\pm 2.5$ | $94.2\pm 3.1$ | $84.5\pm 3.9$ |

FC | Original logarithmized data | PCA + LDA | RAD | FC-OD-PCA+LDA-RAD | $98.6\pm 0.6$ | $99.4\pm 0.7$ | $97.7\pm 1.3$ |

FC | Original logarithmized data | PCA + LDA | ANG | FC-OD-PCA+LDA-ANG | − | − | − |

FC | Original logarithmized data | PCA + LDA | MAGN | FC-OD-PCA+LDA-MAGN | $94.6\pm 2.0$ | $97.3\pm 2.0$ | $92.2\pm 2.1$ |

FC | Original logarithmized data | PCA + LDA | DAM | FC-OD-PCA+LDA-DAM | $84.2\pm 1.4$ | $93.5\pm 3.8$ | $80.3\pm 2.7$ |

FC | Original logarithmized data | SVD + LDA | RAD | FC-OD-SVD+LDA-RAD | $92.2\pm 2.5$ | $91.3\pm 2.7$ | $93.0\pm 3.8$ |

FC | Original logarithmized data | SVD + LDA | ANG | FC-OD-SVD+LDA-ANG | − | − | − |

FC | Original logarithmized data | SVD + LDA | MAGN | FC-OD-SVD+LDA-MAGN | $77.1\pm 4.5$ | $79.9\pm 3.4$ | $74.9\pm 6.9$ |

FC | Original logarithmized data | SVD + LDA | DAM | FC-OD-SVD+LDA-DAM | $73.6\pm 2.7$ | $89.3\pm 5.2$ | $67.6\pm 2.9$ |

FC | Original logarithmized data | KPCA | RAD | FC-OD-KPCA-RAD | $99.3\pm 6.2$ | $99.5\pm 1.0$ | $99.2\pm 1.0$ |

FC | Original logarithmized data | KPCA | ANG | FC-OD-KPCA-ANG | − | − | − |

FC | Original logarithmized data | KPCA | MAGN | FC-OD-KPCA-MAGN | $96.4\pm 1.5$ | $96.8\pm 1.5$ | $95.9\pm 2.5$ |

FC | Original logarithmized data | KPCA | DAM | FC-OD-KPCA-DAM | $89.2\pm 3.4$ | $94.3\pm 2.0$ | $86.9\pm 4.5$ |

FC | Augmented logarithmized data | Pure data | RAD | FC-AD-Pure-RAD | $99.5\pm 0.4$ | $99.7\pm 0.4$ | $99.3\pm 0.5$ |

FC | Augmented logarithmized data | Pure data | ANG | FC-AD-Pure-ANG | $73.9\pm 2.3$ | $81.4\pm 2.5$ | $71.1\pm 3.0$ |

FC | Augmented logarithmized data | Pure data | MAGN | FC-AD-Pure-MAGN | $97.3\pm 0.5$ | $96.7\pm 1.2$ | $98.0\pm 0.5$ |

FC | Augmented logarithmized data | Pure data | DAM | FC-AD-Pure-DAM | $96.8\pm 0.5$ | $98.5\pm 0.3$ | $96.1\pm 0.7$ |

1D-CNN | Original logarithmized data | Pure data | RAD | CNN-OD-Pure-RAD | $99.3\pm 0.6$ | $99.1\pm 0.9$ | $99.4\pm 0.6$ |

1D-CNN | Original logarithmized data | Pure data | ANG | CNN-OD-Pure-ANG | $32.6\pm 1.4$ | $32.4\pm 1.4$ | $32.9\pm 1.6$ |

1D-CNN | Original logarithmized data | Pure data | MAGN | CNN-OD-Pure-MAGN | $95.0\pm 2.0$ | $95.5\pm 4.2$ | $94.4\pm 3.9$ |

1D-CNN | Original logarithmized data | Pure data | DAM | CNN-OD-Pure-DAM | $87.2\pm 3.4$ | $97.0\pm 2.9$ | $83.3\pm 5.3$ |

1D-CNN | Augmented logarithmized data | Pure data | RAD | CNN-AD-Pure-RAD | $99.5\pm 0.2$ | $99.5\pm 0.3$ | $99.1\pm 0.2$ |

1D-CNN | Augmented logarithmized data | Pure data | ANG | CNN-AD-Pure-ANG | $70.7\pm 4.5$ | $68.8\pm 4.6$ | $71.7\pm 4.6$ |

1D-CNN | Augmented logarithmized data | Pure data | MAGN | CNN-AD-Pure-MAGN | $95.3\pm 0.6$ | $95.7\pm 1.2$ | $94.9\pm 1.6$ |

1D-CNN | Augmented logarithmized data | Pure data | DAM | CNN-AD-Pure-DAM | $94.9\pm 0.8$ | $97.3\pm 0.3$ | $93.8\pm 1.2$ |

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**Figure 1.**A schematic illustration of the rotor geometry is shown on the left. In the middle, the generated FE-mesh is presented with one element in the thickness direction and an approximate element thickness of 5 mm. On the right, the boundary conditions are illustrated.

**Figure 3.**The four physical-based classifiers and their labels. (

**a**) shows the radial position, (

**b**) the angular position and (

**c**) the load magnitude that describes the out-of-plane load. The out-of-plane load with damage accumulation is shown in (

**d**) [11].

**Figure 4.**Correlation matrices for 5000 frequencies from the spectrum. (

**a**) shows the Pearson correlation matrix for linear dependencies and (

**b**) the Spearman correlation matrix for nonlinear dependencies.

**Figure 5.**Architecture of NNs used for classification. (

**a**) shows the FC. Fully connected (dense) layers were used in combination with the rectified linear activation function (ReLU) and the softmax activation function in the output layer. Dropout layers were inserted to reduce overfitting. (

**b**) visualizes the structure of the CNN, with 1D-convolutional and pooling layers being fundamental to this network type.

**Figure 6.**Classification results of two different network architectures FC (

**a**) and CNN (

**b**). We compare the models FC-OD-Pure-RAD, FC-OD-Pure-ANG, FC-OD-Pure-MAGN and FC-OD-Pure-DAM (

**a**) with CNN-OD-Pure-RAD, CNN-OD-Pure-ANG, CNN-OD-Pure-MAGN and CNN-OD-Pure-DAM (

**b**). The abbreviations can be looked up in Appendix A. OD stands for original data.

**Figure 7.**Accuracy for each of the dimensionality reduction methods PCA, KPCA and using only every 30th frequency point depends on the number of components. Representative curves are shown for the classifier detecting the radial position. The encircled points indicate the chosen number of components.

**Figure 8.**Comparison of dimensionality reduction methods for the models FC-OD-30thFreq-RAD/-MAGN/-DAM, FC-OD-PCA-RAD/-MAGN/-DAM, FC-OD-PCA+LDA-RAD/-MAGN/-DAM, FC-OD-SVD+LDA-RAD/-MAGN/-DAM and FC-OD-KPCA-RAD/-MAGN/-DAM. Accuracy calculated with FC is shown. The dimensionality reduction methods are PCA, PCA+LDA, SVD+LDA, KPCA and using only every 30th data point. Angular position was omitted because there was no training effect at all.

**Figure 9.**Classification results with an augmented data set for the two network architectures: FC (

**a**) with models FC-AD-Pure-RAD/-ANG/-MAGN/-DAM and CNN (

**b**) with models CNN-AD-Pure-RAD/-ANG/-MAGN/-DAM. The data set contains in total 7920 spectra. Three bars corresponding to accuracy, sensitivity and specificity are shown.

Factor | Unit | Levels |
---|---|---|

impact load | kN | 8, 12, 16, 20 |

radius | mm | 75, 105, 135, 165, 195, 225 |

angle | ° | 0, 45, 90 |

centrifugal load | % | 51, 53, 56, 58, 59, 60, 63, 69, 76, 100 |

**Table 2.**Hyperparameters used for the networks FC and CNN depending on the four classifiers RAD, ANG, MAGN and DAM.

Parameter | FC-RAD | FC-MAGN | FC-DAM | CNN-RAD | CNN-DAM |
---|---|---|---|---|---|

FC-ANG | CNN-MAGN | ||||

CNN-ANG | |||||

optimizer | Adam | Adam | Adam | Adam | Adam |

learning rate | 0.0001 | 0.0001 | 0.0001 | 0.001 | 0.0001 |

epochs | 100 | 100 | 100 | 100 | 100 |

batch size | 64 | 64 | 64 | 64 | 16 |

filter | - | - | - | 40, 40, 20 | 90, 90, 60 |

kernel size | - | - | - | 9 | 3 |

pooling size | - | - | - | 6 | 3 |

dense layer | 1000, 100, 100 | 5010, 1000, 100 | 6000, 1000, 100 | 20 | 20 |

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## Share and Cite

**MDPI and ACS Style**

Scholz, V.; Winkler, P.; Hornig, A.; Gude, M.; Filippatos, A.
Structural Damage Identification of Composite Rotors Based on Fully Connected Neural Networks and Convolutional Neural Networks. *Sensors* **2021**, *21*, 2005.
https://doi.org/10.3390/s21062005

**AMA Style**

Scholz V, Winkler P, Hornig A, Gude M, Filippatos A.
Structural Damage Identification of Composite Rotors Based on Fully Connected Neural Networks and Convolutional Neural Networks. *Sensors*. 2021; 21(6):2005.
https://doi.org/10.3390/s21062005

**Chicago/Turabian Style**

Scholz, Veronika, Peter Winkler, Andreas Hornig, Maik Gude, and Angelos Filippatos.
2021. "Structural Damage Identification of Composite Rotors Based on Fully Connected Neural Networks and Convolutional Neural Networks" *Sensors* 21, no. 6: 2005.
https://doi.org/10.3390/s21062005