Figure 1.
Distribution map of sequence values: (a) random number generation, (b) tent mapping generation, (c) logistic mapping generation, (d) circle mapping generation.
Figure 1.
Distribution map of sequence values: (a) random number generation, (b) tent mapping generation, (c) logistic mapping generation, (d) circle mapping generation.
Figure 2.
Schematic of the reverse learning strategy for lens imaging.
Figure 2.
Schematic of the reverse learning strategy for lens imaging.
Figure 3.
Iterative curves of different optimization algorithms under the single modal benchmark function: (a) iteration curve for benchmark test function F1 and (b) iteration curve for benchmark test function F2.
Figure 3.
Iterative curves of different optimization algorithms under the single modal benchmark function: (a) iteration curve for benchmark test function F1 and (b) iteration curve for benchmark test function F2.
Figure 4.
Iterative curves of different optimization algorithms under multimodal benchmark functions: (a) iteration curve for benchmark test function F3 and (b) iteration curve for benchmark test function F4.
Figure 4.
Iterative curves of different optimization algorithms under multimodal benchmark functions: (a) iteration curve for benchmark test function F3 and (b) iteration curve for benchmark test function F4.
Figure 7.
Basic characteristic diagram of the periodic shock signal: (a) time-domain diagram and (b) Hilbert envelope spectrogram.
Figure 7.
Basic characteristic diagram of the periodic shock signal: (a) time-domain diagram and (b) Hilbert envelope spectrogram.
Figure 8.
Basic characteristic diagram of the simulated signal: (a) time-domain diagram and (b) Hilbert envelope spectrogram.
Figure 8.
Basic characteristic diagram of the simulated signal: (a) time-domain diagram and (b) Hilbert envelope spectrogram.
Figure 9.
IPOA-VMD optimization iteration curve (simulated signal).
Figure 9.
IPOA-VMD optimization iteration curve (simulated signal).
Figure 10.
Time-frequency domain diagram of the IMF component after IPOA-VMD decomposition (simulated signal): (a) time domain and (b) frequency domain.
Figure 10.
Time-frequency domain diagram of the IMF component after IPOA-VMD decomposition (simulated signal): (a) time domain and (b) frequency domain.
Figure 11.
Time-domain diagram of the processed IMF1 using MOMEDA.
Figure 11.
Time-domain diagram of the processed IMF1 using MOMEDA.
Figure 12.
TEO spectrum of IMF1 after MOMEDA processing.
Figure 12.
TEO spectrum of IMF1 after MOMEDA processing.
Figure 13.
Results of IMF1 processing by different methods: (a) TEO demodulation analysis and (b) MOMEDA processing.
Figure 13.
Results of IMF1 processing by different methods: (a) TEO demodulation analysis and (b) MOMEDA processing.
Figure 14.
Spectrum of each component of simulated signals with different SNRs after IPOA-VMD decomposition: (a) SNR equals −16 dB, (b) SNR equals −17 dB, and (c) SNR equals −18 dB.
Figure 14.
Spectrum of each component of simulated signals with different SNRs after IPOA-VMD decomposition: (a) SNR equals −16 dB, (b) SNR equals −17 dB, and (c) SNR equals −18 dB.
Figure 15.
TEO spectrum of optimal IMF of simulated signals with different SNRs after MOMEDA processing: (a) SNR equals −16 dB, (b) SNR equals −17 dB, and (c) SNR equals −18 dB.
Figure 15.
TEO spectrum of optimal IMF of simulated signals with different SNRs after MOMEDA processing: (a) SNR equals −16 dB, (b) SNR equals −17 dB, and (c) SNR equals −18 dB.
Figure 16.
Schematic of the rolling bearing fault experimental platform.
Figure 16.
Schematic of the rolling bearing fault experimental platform.
Figure 17.
Basic characteristic diagram of inner ring fault signals: (a) time domain and (b) Hilbert envelope spectrum.
Figure 17.
Basic characteristic diagram of inner ring fault signals: (a) time domain and (b) Hilbert envelope spectrum.
Figure 18.
IPOA-VMD optimization search iteration curve (actual signal).
Figure 18.
IPOA-VMD optimization search iteration curve (actual signal).
Figure 19.
Time- and frequency-domain waveforms of each IMF component after IPOA-VMD decomposition (actual signal): (a) time domain and (b) frequency domain.
Figure 19.
Time- and frequency-domain waveforms of each IMF component after IPOA-VMD decomposition (actual signal): (a) time domain and (b) frequency domain.
Figure 20.
Time-domain diagram of IMF2 processed by MOMEDA.
Figure 20.
Time-domain diagram of IMF2 processed by MOMEDA.
Figure 21.
TEO spectrum of IMF2 after MOMEDA processing.
Figure 21.
TEO spectrum of IMF2 after MOMEDA processing.
Figure 22.
Time-domain waveforms and TEO spectrum of each optimal IMF of the fault signal under different loads after processing via MOMEDA: (a) motor load is 50%, (b) motor load is 100%, and (c) motor load is 150%.
Figure 22.
Time-domain waveforms and TEO spectrum of each optimal IMF of the fault signal under different loads after processing via MOMEDA: (a) motor load is 50%, (b) motor load is 100%, and (c) motor load is 150%.
Figure 23.
Optimal IMF envelope spectra of rolling bearing inner race fault signals obtained using the EEMD algorithm under different loads: (a) motor load is 0%, (b) motor load is 50%, (c) motor load is 100%, and (d) motor load is 150%.
Figure 23.
Optimal IMF envelope spectra of rolling bearing inner race fault signals obtained using the EEMD algorithm under different loads: (a) motor load is 0%, (b) motor load is 50%, (c) motor load is 100%, and (d) motor load is 150%.
Figure 24.
Optimal IMF envelope spectra of rolling bearing inner race fault signals obtained using the FP-VMD algorithm under different loads: (a) motor load is 0%, (b) motor load is 50%, (c) motor load is 100%, and (d) motor load is 150%.
Figure 24.
Optimal IMF envelope spectra of rolling bearing inner race fault signals obtained using the FP-VMD algorithm under different loads: (a) motor load is 0%, (b) motor load is 50%, (c) motor load is 100%, and (d) motor load is 150%.
Figure 25.
Optimal IMF envelope spectra of rolling bearing inner race fault signals obtained using the EWT-ICA algorithm under different loads: (a) motor load is 0%, (b) motor load is 50%, (c) motor load is 100%, and (d) motor load is 150%.
Figure 25.
Optimal IMF envelope spectra of rolling bearing inner race fault signals obtained using the EWT-ICA algorithm under different loads: (a) motor load is 0%, (b) motor load is 50%, (c) motor load is 100%, and (d) motor load is 150%.
Table 1.
Benchmarking function parameters.
Table 1.
Benchmarking function parameters.
Benchmark Function Expression | Dimension | Value Range | Optimal Solution |
---|
| 30 | [−100,100] | 0 |
| 30 | [−100,100] | 0 |
| 30 | [−5.12,5.12] | 0 |
| 30 | [−32,32] | 0 |
Table 2.
Comparison of the test results of different optimization algorithms.
Table 2.
Comparison of the test results of different optimization algorithms.
Function | Statistic | DS | GOA | MPA | PSO | AOA | POA | IPOA |
---|
F1 | Avg | 5.9020 × 104 | 5.2200 × 103 | 1.5787 × 102 | 1.5218 × 103 | 1.0680 × 10−10 | 5.5570 × 10−17 | 0 |
MR | 0 | 0 | 0 | 0 | 53 | 51 | 97 |
F2 | Avg | 5.9990 × 101 | 1.9120 × 101 | 3.9619 × 10−1 | 5.4721 × 100 | 7.8481 × 10−7 | 1.0567 × 10−10 | 0 |
MR | 0 | 0 | 0 | 0 | 21 | 32 | 97 |
F3 | Avg | 1.9642 × 102 | 1.4376 × 102 | 1.7388 × 101 | 2.8763 × 102 | 3.4106 × 10−13 | 0 | 0 |
MR | 0 | 0 | 0 | 0 | 61 | 49 | 98 |
F4 | Avg | 9.7789 × 100 | 1.1971 × 101 | 8.5541 × 10−2 | 5.6286 × 100 | 1.9966 × 101 | 1.1888 × 10−10 | 8.8817 × 10−16 |
MR | 0 | 0 | 0 | 0 | 0 | 28 | 97 |
Table 3.
IPOA initialization parameters (simulated signal).
Table 3.
IPOA initialization parameters (simulated signal).
Parameter Name | Specific Value |
---|
population size | 80 |
spatial dimension | 2 |
number of iterations | 30 |
penalty factor range | [100, 20,000] |
range of values of decomposition layers | [2, 15] |
Table 4.
Evaluation parameters of each IMF component (simulated signal).
Table 4.
Evaluation parameters of each IMF component (simulated signal).
Index | IMF1 | IMF2 | IMF3 | IMF4 |
---|
Kurtosis | 3.9309 | 2.7432 | 2.6481 | 3.3938 |
SEGI | 0.5552 | 0.4590 | 0.4551 | 0.5636 |
K-SEGI | 3.7923 | 1.0787 | 1.0000 | 2.9924 |
Table 5.
Rolling bearing dataset.
Table 5.
Rolling bearing dataset.
Fault Condition | Fault Diameter(mm) | Motor Load (%) | Motor Speed (r/min) |
---|
Inner race fault | 0.3556 | 0 | 1797 |
50 | 1772 |
100 | 1750 |
150 | 1730 |
Table 6.
Fundamental characteristics of the rolling bearing.
Table 6.
Fundamental characteristics of the rolling bearing.
Bearing Model | Rolling Diameter | Pitch Diameter | Number of Balls | Contact Angle |
---|
SKF6205-2RS | 7.94 mm | 39.04 mm | 9 | 0° |
Table 7.
IPOA initialization parameters.
Table 7.
IPOA initialization parameters.
Parameter Name | Specific Value |
---|
population size | 80 |
spatial dimension | 2 |
number of iterations | 30 |
penalty factor range | [100, 20,000] |
range of values of decomposition layers | [2, 15] |
Table 8.
Evaluation parameters of each IMF component (actual signal).
Table 8.
Evaluation parameters of each IMF component (actual signal).
Index | IMF1 | IMF2 | IMF3 |
---|
Kurtosis | 2.8668 | 9.0051 | 5.3069 |
SEGI | 0.4953 | 0.6656 | 0.6294 |
K-SEGI | 1.0000 | 4.0000 | 2.2734 |