Bearing Dynamics Modeling Based on the Virtual State-Space and Hammerstein–Wiener Model
Abstract
:1. Introduction
- This study uses multi-dimensional virtual state-spaces to generate sufficient data, improving the accuracy of model identification.
- This study constructs multi-dimensional models of the dynamic system by utilizing the virtual state-spaces, which allows for high-precision modeling, mainly through the dimension with a more vital characterization ability of the dynamic system.
- This study employs the HW model in the control field to effectively mitigate the challenges posed by the nonlinear part of the system in dynamic modeling.
2. Theoretical Foundation
2.1. Bearing 5-DoF Model for Dynamics Analysis
2.2. Virtual State-Space for Dynamics Modeling
2.3. Hammerstein–Wiener Method for Model Identification
3. Database and Data Preprocessing
4. Experimental Results
4.1. Model In- and Output
4.2. Nonlinear Module Estimator
4.3. Zeros and Poles of Transfer Function
4.4. Analysis
5. Verification
5.1. Validation of Optimized Configuration
5.2. Verification with Bearings under Other Conditions
5.3. Verification with Other Datasets
5.4. Discussion
6. Conclusions
- The concept of virtual states is introduced, highlighting its significance in exploring system dynamic characteristics from multiple dimensions. This study employs virtual states and multi-dimensional bearing state-space models derived from measurable states in the dynamic analysis of bearings.
- A bearing fault modeling method is developed using virtual state-space and HW models, utilizing acceleration signals and their multi-order derivatives to construct high-precision MISO-HW models based on virtual states.
- To optimize data generation and improve the quality of high-dimensional data, derivative estimators could be developed, enabling the construction of MIMO models. These MIMO models allow for the extraction of characteristic parameters that more effectively represent the dynamic degradation processes in multidimensional models.
- The state matrix in the models (refer to Figure 4) captures the relationships among states and the system’s dynamic characteristics across different dimensions. This matrix allows for the extraction of characteristic parameters highly correlated with bearing aging and faults, facilitating the development of high-precision diagnostic and prognostic methods.
- This method can be further applied to different data sets and objects. A test bench equipped with a bearing test module and high-quality data acquisition terminals has been constructed. Experiments are planned to create a comprehensive dataset to validate the proposed method further. Furthermore, gear and rotor data will be employed to test this method following the design of a new test bench and relevant experiments.
- Future improvements in the accuracy of this modeling method could be achieved by incorporating varying friction states. For example, integrating a friction coefficient that adapts to different friction conditions could enhance the model’s precision. Additionally, adopting an improved elastohydrodynamic lubrication modeling approach could serve as a superior alternative to the current Hertz contact theory-based modeling.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A
State Order n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|---|
Bearing 1_1 | 95.32 | 95.65 | 96.51 | 94.89 | 94.19 | 96.10 | 97.17 | 94.85 | 94.03 | 96.70 |
Bearing 1_2 | 94.26 | 95.81 | 96.07 | 96.92 | 97.79 | 95.78 | 96.06 | 95.74 | 96.42 | 97.37 |
Bearing 1_3 | 96.44 | 96.41 | 96.04 | 96.42 | 97.21 | 97.87 | 97.94 | 97.79 | 97.02 | 97.38 |
Bearing 1_4 | 95.86 | 95.22 | 96.29 | 96.29 | 94.50 | 92.41 | 93.44 | 94.54 | 95.42 | 95.91 |
Bearing 1_5 | 96.50 | 96.97 | 97.59 | 96.92 | 94.61 | 95.23 | 94.44 | 92.19 | 93.91 | 95.84 |
Bearing 1_6 | 97.35 | 97.10 | 98.52 | 97.22 | 97.58 | 98.07 | 97.49 | 94.32 | 96.65 | 98.12 |
Bearing 1_7 | 98.66 | 97.96 | 99.27 | 98.47 | 98.97 | 97.77 | 93.31 | 95.05 | 91.77 | 95.32 |
Bearing 2_1 | 99.75 | 98.17 | 94.96 | 92.30 | 94.32 | 91.84 | 90.25 | 93.99 | 90.82 | 93.87 |
Bearing 2_2 | 99.58 | 98.94 | 98.66 | 97.70 | 94.99 | 93.82 | 93.17 | 94.14 | 91.48 | 95.57 |
Bearing 2_3 | 98.33 | 98.14 | 98.81 | 97.41 | 95.91 | 97.25 | 97.82 | 95.21 | 95.78 | 96.17 |
Bearing 2_4 | 98.28 | 99.38 | 98.79 | 97.68 | 97.49 | 97.06 | 94.01 | 96.36 | 96.18 | 96.11 |
Bearing 2_5 | 97.00 | 96.83 | 95.84 | 97.71 | 98.06 | 98.14 | 97.57 | 95.49 | 96.24 | 97.66 |
Bearing 2_6 | 98.18 | 99.74 | 99.46 | 98.77 | 98.73 | 97.93 | 96.20 | 96.09 | 94.53 | 96.24 |
Bearing 2_7 | 97.11 | 98.91 | 98.41 | 98.79 | 98.74 | 99.05 | 97.81 | 95.05 | 93.52 | 95.85 |
Bearing 3_1 | 95.03 | 97.14 | 98.34 | 94.93 | 95.72 | 96.84 | 96.33 | 93.00 | 93.48 | 97.70 |
Bearing 3_2 | 97.88 | 98.48 | 99.35 | 97.88 | 96.95 | 96.30 | 98.20 | 96.29 | 94.90 | 92.74 |
Bearing 3_3 | 97.91 | 97.63 | 96.41 | 94.47 | 94.05 | 94.16 | 94.04 | 93.55 | 94.30 | 95.80 |
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Parameter | Value |
---|---|
Diameter of the outer race | 29.1 mm |
Diameter of inner race | 22.1 mm |
Bearing mean diameter | 25.6 mm |
Inside diameter | 20.0 mm |
Outside race diameter | 32.0 mm |
Diameter of rolling elements | 3.5 mm |
Thickness | 7.0 mm |
Number of rolling elements | 13 |
Operating Condition | Condition 1 | Condition 2 | Condition 3 |
---|---|---|---|
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 | ||
Rotational speed [rpm] | 1800 | 1650 | 1500 |
Force [N] | 4000 | 4200 | 5000 |
Dataset | CWRU | MFPT | PU | PHM Challenge |
---|---|---|---|---|
Number of rolling elements | 9 | 8 | 8 | 13 |
Rolling element diameter [mm] | 7.94 | 0.235 | 6.75 | 3.5 |
Pitch diameter [mm] | 39.04 | 1.245 | 28.55 | 25.6 |
Sample rate [kHz] | 12.0 | 48.8 | 64.0 | 25.6 |
Configuration | Type 1 | Type 2 | Type 3 | Type 4 |
---|---|---|---|---|
Input | ||||
Output |
Nonlinearity Estimator | Type 1 | Type 2 | Type 3 | Type 4 |
---|---|---|---|---|
Piecewise Linear function (PWL) | 28.48% | 7.97% | 53.49% | 76.74% |
Sigmoid Network (SN) | 29.46% | 3.39% | 68.19% | 64.86% |
Wavelet Network (WN) | 29.12% | 9.22% | 77.60% | 14.53% |
Saturation (Sat) | 34.24% | −11.88% | 71.93% | 8.67% |
Dead Zone (DZ) | 21.06% | 17.17% | −66.33% | −27.16% |
One-Dimensional Polynomial (1-DP) | 21.32% | 1.41% | 94.95% | 0.12% |
Unit Gain (UG) | 12.06% | −11.34% | 86.91% | 98.70% |
Input Order | Output | Inputs | Accuracy [%] | Duration [s] | Input Order | Output | Inputs | Accuracy [%] | Duration [s] |
---|---|---|---|---|---|---|---|---|---|
0 | 95.1475 | 43.42 | 5 | 96.3879 | 243.65 | ||||
1 | 95.4515 | 92.31 | 6 | 95.4982 | 270.04 | ||||
2 | 97.8659 | 182.01 | 6 | 95.4982 | 270.04 | ||||
3 | 95.5293 | 185.62 | 8 | 96.7953 | 323.06 | ||||
4 | 95.6098 | 222.27 | 9 | 95.8023 | 361.37 |
Input Module | Output Module | Accuracy [%] | Duration [s] | Input Module | Output Module | Accuracy [%] | Duration [s] |
---|---|---|---|---|---|---|---|
PWL | PWL | 57.0940 | 320.01 | PWL | UG | 96.2595 | 230.29 |
SN | SN | 71.8959 | 515.63 | SN | 96.7476 | 305.78 | |
WN | WN | 31.5333 | 530.03 | WN | 95.1923 | 298.02 | |
Sat | Sat | 74.4528 | 253.84 | Sat | 86.6872 | 209.66 | |
DZ | DZ | −165.9393 | 312.46 | DZ | 77.6185 | 263.64 | |
1-DP | 1-DP | 94.2116 | 238.85 | 1-DP | 97.8659 | 181.21 | |
UG | UG | 87.9686 | 157.12 | UG | 94.5437 | 250.42 | |
1-DP | PWL | 38.9662 | 243.43 | SN | 95.0212 | 398.24 | |
SN | 20.4830 | 349.05 | UG | WN | 57.3083 | 308.79 | |
WN | 29.2596 | 341.45 | Sat | 76.0919 | 229.48 | ||
Sat | 54.7752 | 184.98 | DZ | −308.0773 | 224.81 | ||
DZ | −20.0300 | 225.96 | 1-DP | 79.0184 | 134.87 | ||
PWL | 1-DP | 95.4379 | 253.76 | ||||
SN | 96.0861 | 445.57 | |||||
WN | 86.2918 | 365.10 | |||||
Sat | 78.9940 | 256.11 |
Test Group | Zeros Number | Poles Number | Accuracy [%] | Duration [s] | Test Group | Zeros Number | Poles Number | Accuracy [%] | Duration [s] |
---|---|---|---|---|---|---|---|---|---|
Group 1 | 0 | 1 | 95.7328 | 96.14 | Group 2 | 1 | 1 | 87.1260 | 164.44 |
0 | 2 | 96.2157 | 125.48 | 1 | 2 | 85.8251 | 207.27 | ||
0 | 3 | 97.8659 | 182.21 | 1 | 3 | 56.5755 | 230.76 | ||
0 | 4 | 96.5741 | 154.12 | 1 | 6 | 65.8701 | 234.77 | ||
0 | 5 | 95.8959 | 163.29 | 1 | 4, 5, 7, 8, 9 | \ | \ | ||
0 | 6 | 96.8135 | 186.20 | Group 3 | 2 | 2 | 85.8871 | 191.79 | |
0 | 7 | 95.4290 | 179.67 | 2 | 3, 4, …, 9 | \ | \ | ||
0 | 8 | 97.5603 | 177.53 | Group 4 | 3 | 3 | 0.0272 | 369.05 | |
0 | 9 | 96.5667 | 183.86 | 3 | 4, 5, …, 9 | \ | \ | ||
0 | 10 | 94.3603 | 192.23 | Group 5 | 4 | 4, 5, …, 9 | \ | \ |
State Order n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|---|
Min. accuracy [%] | 81.63 | 89.92 | 84.96 | 64.01 | 46.76 | 74.95 | 67.11 | 59.31 | 58.67 | 60.10 |
Max. accuracy [%] | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 |
Mean accuracy [%] | 98.58 | 98.98 | 98.29 | 96.21 | 95.34 | 96.44 | 94.89 | 93.73 | 93.72 | 94.92 |
Standard deviation | 3.64 | 2.14 | 3.03 | 6.38 | 8.56 | 5.24 | 6.59 | 7.88 | 8.27 | 8.03 |
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Jiang, G.; Zhou, K.; Li, Z.; Yan, J. Bearing Dynamics Modeling Based on the Virtual State-Space and Hammerstein–Wiener Model. Sensors 2024, 24, 5410. https://doi.org/10.3390/s24165410
Jiang G, Zhou K, Li Z, Yan J. Bearing Dynamics Modeling Based on the Virtual State-Space and Hammerstein–Wiener Model. Sensors. 2024; 24(16):5410. https://doi.org/10.3390/s24165410
Chicago/Turabian StyleJiang, Genghong, Kai Zhou, Zhaorong Li, and Jianping Yan. 2024. "Bearing Dynamics Modeling Based on the Virtual State-Space and Hammerstein–Wiener Model" Sensors 24, no. 16: 5410. https://doi.org/10.3390/s24165410