# Transfer-Learning-Based Estimation of the Remaining Useful Life of Heterogeneous Bearing Types Using Low-Frequency Accelerometers

^{*}

## Abstract

**:**

## 1. Introduction

- It is the first deep transfer learning approach that combines the possibility of using data from low-cost sensors with low sampling frequencies and the usage of transfer learning between different bearing types. This contribution is an important requirement for real-world applications.
- It is a novel RUL approach that combines a hybrid feature extraction approach (intermediate domain) with a data-driven feature extraction approach (convolutional layers).

## 2. Preliminaries

#### 2.1. Scoring of an RUL Approach

_{i}) of the predictions have to be calculated according to Equation (1), where RUL_Act is the actual RUL and RUL_Est is the estimated RUL. The index i is for the selected test dataset.

_{i}(see Equation (2)). There are two different weights. If Er

_{i}> 0, which means that the estimated RUL is less than the actual RUL, the deviations are less serious than in cases where Er

_{i}< 0. In the first case, a component is replaced too early, resulting only in increased material costs and a short, planned downtime, while the second case leads to an unforeseen and thus unplanned breakdown. Therefore, the two cases are weighted differently.

_{i}s of all N test datasets.

#### 2.2. Convolutional Neural Network

#### 2.3. Long Short-Term Memory Network

## 3. Intermediate-Domain-Based RUL Estimation

#### 3.1. Feature Extraction: Intermediate Domain

#### 3.1.1. Overview

#### 3.1.2. Windowed Envelope

#### 3.1.3. De-Noising

#### 3.1.4. Normalization

_{Normalized}).

#### 3.1.5. Intermediate Domain Image

#### 3.2. Feature Extraction: Convolutional Layers

#### 3.3. Proposed LSTM Architecture

_{RUL}is the estimated RUL, T

_{CUR}is the current lifetime of the bearing, and HI is the health indicator, which is the output of the RUL network and is in a range between 1.0 for new and 0.0 for a defect [37]. Using a health indicator means that the network is trained with labels between 1.0 and 0.0 instead of a time. Therefore, before the beginning of the training, all training samples must be relabeled.

- Layout 1: This layout reflects an LSTM without any intermediate fully connected layer. It is based only on the feature extraction part of the CNN, followed by the LSTM layout proposed by Sahoo [44], which has 32 outputs after the last LSTM layer. These outputs are directly fed into a fully connected output layer made of one neuron, giving the final health indicator as the output.
- Layout 2: This layout reflects the common usage of several (deep) fully connected layers [45]. Therefore, in addition to Layout 1, another fully connected layer with 32 outputs and a dropout layer with a dropout rate of 0.5 are inserted directly after the last LSTM layer. The use of 32 outputs for the fully connected layer is based on the success of the double convolutional layers in the classification model, where two identical layers are used in a row. Since the previous LSTM layer has 32 outputs, 32 outputs are also chosen here. A dropout factor of 0.5 is chosen based on recommendations in the literature, such as Géron [18].
- Layout 3: This layout only differs from Layout 2 in that the last LSTM layer with 32 outputs is removed. This layout was chosen to analyze the impact of the LSTM layers. A less complex model with fewer layers could be used if this approach was superior.

#### 3.4. Transfer Learning Approach

#### 3.5. Constraint

#### 3.6. Generalization

## 4. Benchmark

#### 4.1. Benchmark Description

#### 4.2. Supplements to the Test Positions

#### 4.2.1. Bearing Dataset 1_4

#### 4.2.2. Bearing Dataset 2_5

#### 4.2.3. Bearing Dataset 1_6

#### 4.3. Benchmark Execution

- Used environment: The relevant components are an NVIDIA P100 GPU in combination with Python (version 3.6.8) and the TensorFlow machine learning library (version 2.2).
- Intermediate domain: The used bearing, in combination with the used process parameters, results in the following characteristic fault frequencies: outer ring fault: 168 Hz, inner ring fault: 222 Hz, ball fault: 108 Hz, and cage fault: 13 Hz. This results in a maximum frequency of 888 Hz, which is required for the fourth harmonics of the inner ring fault. The intermediate domain thus fulfills the requirement of having a solution that can be used in use cases with current industrial triaxial accelerometers with low sampling rates.
- LSTM layout: The intermediate domain images were used in an LSTM, according to Section 3.3. Among other parameters, a window size based on 85 measurements (n = 85) and a step size of two (s = 2) were used.
- Training settings: A batch size of 120 was used for the training. A larger batch size could not be used because of hardware limitations. In addition, a learning rate of 0.0005 was used. An Adam optimizer with the mean squared error (MSE) as a loss function was used during the training. This was based on the recommendation of Liu et al. [12] that, of all common loss functions, MSE is the most sensitive to measurement errors. For the training itself, 300 epochs were used since no improvements in the result of the loss function could be achieved afterward.

#### 4.4. Conclusion

## 5. Discussion and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The weighted error Ai (Equation (2)) as a function of the relative error Er

_{i}. A negative Er

_{i}represents a longer estimated RUL than the actual RUL.

**Figure 2.**The different stages of the presented transfer learning-based RUL approach. This process is used for the transfer learning step but also for the pretraining of the network.

**Figure 3.**The different steps to obtain the proposed intermediate domain image from the raw sensor data.

**Figure 4.**The steps of the intermediate domain creation: (

**a**) raw sensor data; (

**b**) windowed envelope; (

**c**) the frequency selective filter; (

**d**) final intermedia domain image (after normalization and scaling). The layers in (

**c**,

**d**) correspond to the following fault areas of a bearing: 1: outer; 2: inner; 3: ball; 4: cage.

**Figure 5.**An exemplary intermediate domain image for bearing data with four harmonics for each of the four fault frequencies.

**Figure 6.**Different tasks of the parts of the presented CNN. They start with low-level feature extraction and end with classification with the help of fully connected layers.

**Figure 8.**Example of the window size with 10 measured values. The step size s = 2 and the number of used steps n = 3.

**Figure 9.**Evaluation of different numbers of used steps (n) and step sizes (s) for the training of the proposed RUL framework. The best result is achieved with an input size of 85 and a step size of 2.

**Figure 10.**The scores for the RUL estimation with and without pretraining of the convolutional layers. The score achieved with pretrained layers is much higher than that without pretraining.

**Figure 11.**The four bearing degradation stages: (

**a**) stage 1, contains the rotation frequencies and ultrasonic frequencies; (

**b**) stage 2, the natural frequencies of the bearing become visible; (

**c**) stage 3, the characteristic fault frequencies appear; (

**d**) stage 4, the frequencies in area B and area C are replaced with random noise.

**Figure 12.**The sensor values of bearing dataset 1_4 in the time domain (

**a**) and in the time–frequency domain (

**b**). Both figures show increased amplitudes at the test position.

**Figure 13.**The sensor values of bearing dataset 2_5 in the time domain (

**a**) and the time–frequency domain (

**b**). Both plots do not indicate a degradation at the test position.

**Figure 14.**The sensor values of bearing dataset 1_6 in the time domain (

**a**) and in the time–frequency domain (

**b**). Both plots show no increased measured values at the test position after 23,020 s.

**Table 1.**The three evaluated layouts for the RUL approach. The number of outputs of each layer is in parentheses. The evaluation was conducted with the IEEE PHM 2012 Data Challenge dataset. Except for the differences in the table, all three evaluations were conducted with the same settings.

Layout 1 | Layout 2 | Layout 3 | |
---|---|---|---|

Used Layers | CNN (8192) | CNN (8192) | CNN (8192) |

LSTM (128) | LSTM (128) | LSTM (128) | |

LSTM (64) | LSTM (64) | LSTM (64) | |

LSTM (32) | LSTM (32) | Fully Connected (32) | |

Fully Connected (1) | Fully Connected (32) | Dropout (rate = 0.5) | |

Dropout (rate = 0.5) | Fully Connected (1) | ||

Fully Connected (1) | |||

PHM Score | 0.1094 | 0.35 | 0.05647 |

**Table 2.**Detailed results of the proposed RUL framework with and without transfer learning. For each of the 11 bearings, the relative error Er according to Equation (1) is shown. In addition, the mean of all values is given.

Bearing | Without Pretraining [Er] | With Pretraining [Er] |
---|---|---|

1_3 | −70.56 | 29.27 |

1_4 | −172.46 | −78.35 |

1_5 | −968.26 | −159.24 |

1_6 | −10,570.0 | −6413.71 |

1_7 | −268.18 | 35.37 |

2_3 | 16.34 | −0.7 |

2_4 | −323.12 | −124.18 |

2_5 | −384.43 | −919.58 |

2_6 | −251.93 | 8.16 |

2_7 | −86.63 | 12.13 |

3_3 | −234.16 | −0.96 |

Mean | 1213.28 | 707.42 |

Datasets | Operating Conditions | ||
---|---|---|---|

1800 rpm; 4000 N Load | 1650 rpm; 4200 N Load | 1500 rpm; 5000 N Load | |

Learning set | Bearing1_1 | Bearing2_1 | Bearing3_1 |

Bearing1_2 | Bearing2_2 | Bearing3_2 | |

Test set | Bearing1_3 | Bearing2_3 | Bearing3_3 |

Bearing1_4 | Bearing2_4 | ||

Bearing1_5 | Bearing2_5 | ||

Bearing1_6 | Bearing2_6 | ||

Bearing1_7 | Bearing2_7 |

**Table 4.**This table shows the relative error (Er), its mean, and the PHM score of the different RUL approaches and the proposed approach (with and without bearing datasets 1_6 and 2_5).

Bearing | Sutrisno et al. [51] (%) | Porotsky and Bluvband [52] (%) | Zheng [53] (%) | Zhang et al. [54](%) | Proposed RUL Framework (%) | Proposed RUL Framework without 1_6 and 2_5 (%) |
---|---|---|---|---|---|---|

Bearing 1_3 | 97 | N/A | 92.44 | 2.27 | 29.27 | 29.27 |

Bearing 1_4 | 80 | N/A | 100 | 5.6 | −78.35 | −78.35 |

Bearing 1_5 | 9 | N/A | 20.43 | 12.42 | −159.24 | −159.24 |

Bearing 1_6 | −5 | N/A | 7.76 | 10.96 | −6413.71 | N/A |

Bearing 1_7 | −2 | N/A | 82.29 | −22.46 | 35.37 | 35.37 |

Bearing 2_3 | 64 | N/A | 82.93 | 0.99 | −0.7 | −0.7 |

Bearing 2_4 | 10 | N/A | 3.22 | 5.76 | −124.18 | −124.18 |

Bearing 2_5 | −440 | N/A | 58.77 | 25.89 | −919.58 | N/A |

Bearing 2_6 | 49 | N/A | 5.63 | −10.85 | 8.16 | 8.16 |

Bearing 2_7 | −317 | N/A | −121.94 | 1.72 | 12.13 | 12.13 |

Bearing 3_3 | 90 | N/A | −54.38 | −3.66 | −0.96 | −0.96 |

Mean | 105.73 | N/A | 57.25 | 9.32 | 707.42 | 40.76 |

Score | 0.3066 | 0.28 | 0.2992 | 0.64 | 0.35 | 0.43 |

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**MDPI and ACS Style**

Schwendemann, S.; Sikora, A.
Transfer-Learning-Based Estimation of the Remaining Useful Life of Heterogeneous Bearing Types Using Low-Frequency Accelerometers. *J. Imaging* **2023**, *9*, 34.
https://doi.org/10.3390/jimaging9020034

**AMA Style**

Schwendemann S, Sikora A.
Transfer-Learning-Based Estimation of the Remaining Useful Life of Heterogeneous Bearing Types Using Low-Frequency Accelerometers. *Journal of Imaging*. 2023; 9(2):34.
https://doi.org/10.3390/jimaging9020034

**Chicago/Turabian Style**

Schwendemann, Sebastian, and Axel Sikora.
2023. "Transfer-Learning-Based Estimation of the Remaining Useful Life of Heterogeneous Bearing Types Using Low-Frequency Accelerometers" *Journal of Imaging* 9, no. 2: 34.
https://doi.org/10.3390/jimaging9020034