# Research on a Novel Improved Adaptive Variational Mode Decomposition Method in Rotor Fault Diagnosis

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. VMD

## 3. The Proposed IAVMD Method

#### 3.1. Waveform Matching Extension

#### 3.2. Parameter Optimization of VMD Using GWO

#### 3.3. Teager Energy Operator

#### 3.4. The Proposed IAVMD Method

## 4. Simulation Validation

#### 4.1. Case 1: Sinusoidal Superposition Signal

^{−6}, respectively. Seen from Figure 9, four frequency components of the numerical signal $y(t)$ are separated efficiently. Figure 10a,b plots the two-dimensional and three-dimensional TFR based on IAVMD, respectively. One can clearly see that the time-frequency trajectory obtained by IAVMD is very clear in Figure 10, which indicates that the proposed IAVMD can exactly decompose the sinusoidal superposition signal containing multiple components.

#### 4.2. Case 2: AM-FM Superposition Signal

## 5. Experimental Validation

#### 5.1. Experiment Platform and Data Description

#### 5.2. Case 1: Rotor Rub-Impact Fault Detection

#### 5.3. Case 2: Rotor Oil-Whirl Fault Detection

#### 5.4. Case 3: Rotor Oil-Whip Fault Detection

#### 5.5. Result and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Lei, Y.; Lin, J.; He, Z.; Zuo, M.J.; Zuo, M. A review on empirical mode decomposition in fault diagnosis of rotating machinery. Mech. Syst. Signal. Process.
**2013**, 35, 108–126. [Google Scholar] [CrossRef] - Zhang, F.; Huang, J.; Chu, F.; Cui, L. Mechanism and Method for Outer Raceway Defect Localization of Ball Bearings. IEEE Access
**2020**, 8, 4351–4360. [Google Scholar] [CrossRef] - Yan, X.; Liu, Y.; Jia, M. Research on an enhanced scale morphological-hat product filtering in incipient fault detection of rolling element bearings. Measurement
**2019**, 147, 106856. [Google Scholar] [CrossRef] - Yu, K.; Lin, T.R.; Ma, H.; Li, H.; Zeng, J. A combined polynomial chirplet transform and synchroextracting technique for analyzing nonstationary signals of rotating machinery. IEEE Trans. Instrum. Meas.
**2019**, 1. [Google Scholar] [CrossRef] - Feng, Z.; Liang, M.; Chu, F. Recent advances in time–frequency analysis methods for machinery fault diagnosis: A review with application examples. Mech. Syst. Signal. Process.
**2013**, 38, 165–205. [Google Scholar] [CrossRef] - Yan, S.; Nguang, S.K.; Zhang, L. Nonfragile Integral-Based Event-Triggered Control of Uncertain Cyber-Physical Systems under Cyber-Attacks. Complexity
**2019**, 2019, 1–14. [Google Scholar] [CrossRef] - Huang, Y.; Lu, R.; Chen, K. Detection of internal defect of apples by a multichannel Vis/NIR spectroscopic system. Postharvest Biol. Technol.
**2020**, 161, 111065. [Google Scholar] [CrossRef] - Yan, R.; Gao, R.X.; Chen, X. Wavelets for fault diagnosis of rotary machines: A review with applications. Signal. Process.
**2014**, 96, 1–15. [Google Scholar] [CrossRef] - Yan, X.; Liu, Y.; Jia, M. Multiscale cascading deep belief network for fault identification of rotating machinery under various working conditions. Knowl. Based Syst.
**2020**. [Google Scholar] [CrossRef] - Guo, W.; Tse, P.W.T.; Djordjevich, A. Faulty bearing signal recovery from large noise using a hybrid method based on spectral kurtosis and ensemble empirical mode decomposition. Measurement
**2012**, 45, 1308–1322. [Google Scholar] [CrossRef] - Xiang, L.; Yan, X. A self-adaptive time-frequency analysis method based on local mean decomposition and its application in defect diagnosis. J. Vib. Control
**2016**, 22, 1049–1061. [Google Scholar] [CrossRef] - Yan, X.; Liu, Y.; Jia, M. A Feature Selection Framework-Based Multiscale Morphological Analysis Algorithm for Fault Diagnosis of Rolling Element Bearing. IEEE Access
**2019**, 7, 123436–123452. [Google Scholar] [CrossRef] - Li, Y.; Xu, M.; Wei, Y.; Huang, W. Rotating machine fault diagnosis based on intrinsic characteristic-scale decomposition. Mech. Mach. Theory
**2015**, 94, 9–27. [Google Scholar] [CrossRef] - Dragomiretskiy, K.; Zosso, D. Variational Mode Decomposition. IEEE Trans. Signal. Process.
**2013**, 62, 531–544. [Google Scholar] [CrossRef] - Upadhyay, A.; Sharma, M.; Pachori, R.B. Determination of instantaneous fundamental frequency of speech signals using variational mode decomposition. Comput. Electr. Eng.
**2017**, 62, 630–647. [Google Scholar] [CrossRef] - Lahmiri, S.; Shmuel, A. Variational mode decomposition based approach for accurate classification of color fundus images with hemorrhages. Opt. Laser Technol.
**2017**, 96, 243–248. [Google Scholar] [CrossRef] - Xue, Y.-J.; Cao, J.-X.; Wang, D.-X.; Du, H.-K.; Yao, Y. Application of the Variational-Mode Decomposition for Seismic Time–frequency Analysis. IEEE J. Sel. Top. Appl. Earth Obs. Remote. Sens.
**2016**, 9, 3821–3831. [Google Scholar] [CrossRef] - Jiang, X.; Wang, J.; Shi, J.; Shen, C.; Huang, W.; Zhu, Z. A coarse-to-fine decomposing strategy of VMD for extraction of weak repetitive transients in fault diagnosis of rotating machines. Mech. Syst. Signal. Process.
**2019**, 116, 668–692. [Google Scholar] [CrossRef] - Wang, D.; Luo, H.; Grunder, O.; Lin, Y. Multi-step ahead wind speed forecasting using an improved wavelet neural network combining variational mode decomposition and phase space reconstruction. Renew. Energy
**2017**, 113, 1345–1358. [Google Scholar] [CrossRef] - Liu, H.; Mi, X.; Li, Y. Smart multi-step deep learning model for wind speed forecasting based on variational mode decomposition, singular spectrum analysis, LSTM network and ELM. Energy Convers. Manag.
**2018**, 159, 54–64. [Google Scholar] [CrossRef] - Upadhyay, A.; Pachori, R.B. Instantaneous voiced/non-voiced detection in speech signals based on variational mode decomposition. J. Frankl. Inst.
**2015**, 352, 2679–2707. [Google Scholar] [CrossRef] - Upadhyay, A.; Pachori, R. Speech enhancement based on mEMD-VMD method. Electron. Lett.
**2017**, 53, 502–504. [Google Scholar] [CrossRef] - An, X.; Pan, L.; Zhang, F. Analysis of hydropower unit vibration signals based on variational mode decomposition. J. Vib. Control
**2017**, 23, 1938–1953. [Google Scholar] [CrossRef] - Wang, Y.; Markert, R.; Xiang, J.; Zheng, W. Research on variational mode decomposition and its application in detecting rub-impact fault of the rotor system. Mech. Syst. Signal. Process.
**2015**, 60, 243–251. [Google Scholar] [CrossRef] - Yang, W.; Peng, Z.; Wei, K.; Shi, P.; Tian, W. Superiorities of variational mode decomposition over empirical mode decomposition particularly in time–frequency feature extraction and wind turbine condition monitoring. Iet Renew. Power Gener.
**2016**, 11, 443–452. [Google Scholar] [CrossRef] [Green Version] - Yao, J.; Xiang, Y.; Qian, S.; Wang, S.; Wu, S. Noise source identification of diesel engine based on variational mode decomposition and robust independent component analysis. Appl. Acoust.
**2017**, 116, 184–194. [Google Scholar] [CrossRef] - Zhang, M.; Jiang, Z.; Feng, K. Research on variational mode decomposition in rolling bearings fault diagnosis of the multistage centrifugal pump. Mech. Syst. Signal. Process.
**2017**, 93, 460–493. [Google Scholar] [CrossRef] [Green Version] - Liu, H.; Xiang, J. A Strategy Using Variational Mode Decomposition, L-Kurtosis and Minimum Entropy Deconvolution to Detect Mechanical Faults. IEEE Access
**2019**, 7, 70564–70573. [Google Scholar] [CrossRef] - Isham, M.F.; Leong, M.S.; Lim, M.H.; Ahmad, Z.A. Variational mode decomposition: Mode determination method for rotating machinery diagnosis. J. Vibroeng.
**2018**, 20, 2604–2621. [Google Scholar] [CrossRef] [Green Version] - Yang, H.; Liu, S.; Zhang, H. Adaptive estimation of VMD modes number based on cross correlation coefficient. J. Vibroeng.
**2017**, 19, 1185–1196. [Google Scholar] [CrossRef] - Liu, Y.; Yang, G.; Li, M.; Yin, H. Variational mode decomposition denoising combined the detrended fluctuation analysis. Signal. Process.
**2016**, 125, 349–364. [Google Scholar] [CrossRef] - Jiang, F.; Zhu, Z.; Li, W. An Improved VMD With Empirical Mode Decomposition and Its Application in Incipient Fault Detection of Rolling Bearing. IEEE Access
**2018**, 6, 44483–44493. [Google Scholar] [CrossRef] - Zhang, J.; He, J.; Long, J.; Yao, M.; Zhou, W. A New Denoising Method for UHF PD Signals Using Adaptive VMD and SSA-Based Shrinkage Method. Sensors
**2019**, 19, 1594. [Google Scholar] [CrossRef] [Green Version] - Li, Z.; Chen, J.; Zi, Y.; Pan, J. Independence-oriented VMD to identify fault feature for wheel set bearing fault diagnosis of high speed locomotive. Mech. Syst. Signal. Process.
**2017**, 85, 512–529. [Google Scholar] [CrossRef] - Zhao, C.; Feng, Z. Application of multi-domain sparse features for fault identification of planetary gearbox. Measurement
**2017**, 104, 169–179. [Google Scholar] [CrossRef] - Wang, Z.; Wang, J.; Du, W. Research on Fault Diagnosis of Gearbox with Improved Variational Mode Decomposition. Sensors
**2018**, 18, 3510. [Google Scholar] [CrossRef] [Green Version] - Lian, J.; Liu, Z.; Wang, H.; Dong, X. Adaptive variational mode decomposition method for signal processing based on mode characteristic. Mech. Syst. Signal Process.
**2018**, 107, 53–77. [Google Scholar] [CrossRef] - Shan, Y.; Zhou, J.; Jiang, W.; Liu, J.; Xu, Y.; Zhao, Y. A fault diagnosis method for rotating machinery based on improved variational mode decomposition and a hybrid artificial sheep algorithm. Meas. Sci. Technol.
**2019**, 30, 055002. [Google Scholar] [CrossRef] - Isham, M.F.; Leong, M.S.; Lim, M.H.; Bin Ahmad, Z.A. Intelligent wind turbine gearbox diagnosis using VMDEA and ELM. Wind Energy
**2019**, 22, 813–833. [Google Scholar] [CrossRef] - Zhu, J.; Wang, C.; Hu, Z.; Kong, F.; Liu, X. Adaptive variational mode decomposition based on artificial fish swarm algorithm for fault diagnosis of rolling bearings. Proc. Inst. Mech. Eng. Part. C J. Mech. Eng. Sci.
**2017**, 231, 635–654. [Google Scholar] [CrossRef] - Yi, C.; Lv, Y.; Dang, Z. A Fault Diagnosis Scheme for Rolling Bearing Based on Particle Swarm Optimization in Variational Mode Decomposition. Shock. Vib.
**2016**, 2016, 1–10. [Google Scholar] [CrossRef] [Green Version] - Yan, X.; Jia, M. Application of CSA-VMD and optimal scale morphological slice bispectrum in enhancing outer race fault detection of rolling element bearings. Mech. Syst. Signal Process.
**2019**, 122, 56–86. [Google Scholar] [CrossRef] - Wang, Z.; He, G.; Du, W.; Zhou, J.; Han, X.; Wang, J.; He, H.; Guo, X.; Wang, J.; Kou, Y. Application of Parameter Optimized Variational Mode Decomposition Method in Fault Diagnosis of Gearbox. IEEE Access
**2019**, 7, 44871–44882. [Google Scholar] [CrossRef] - Zhou, J.; Guo, X.; Du, W.; Han, X.; Wang, J.; He, H.; Xue, H.; Kou, Y.; Zhou, J.; Guo, X. Research on Fault Extraction Method of Variational Mode Decomposition Based on Immunized Fruit Fly Optimization Algorithm. Entropy
**2019**, 21, 400. [Google Scholar] [CrossRef] [Green Version] - Wei, D.; Jiang, H.; Shao, H.; Li, X.; Lin, Y. An optimal variational mode decomposition for rolling bearing fault feature extraction. Meas. Sci. Technol.
**2019**, 30, 055004. [Google Scholar] [CrossRef] - Miao, Y.; Zhao, M.; Lin, J. Identification of mechanical compound-fault based on the improved parameter-adaptive variational mode decomposition. Isa Trans.
**2019**, 84, 82–95. [Google Scholar] [CrossRef] [PubMed] - Mirjalili, S.; Mirjalili, S.M.; Lewis, A. Grey wolf optimizer. Adv. Eng. Softw.
**2014**, 69, 46–61. [Google Scholar] [CrossRef] [Green Version] - Yan, X.; Jia, M. A novel optimized SVM classification algorithm with multi-domain feature and its application to fault diagnosis of rolling bearing. Neurocomputing
**2018**, 313, 47–64. [Google Scholar] [CrossRef] - Yan, X.; Liu, Y.; Jia, M. Health condition identification for rolling bearing using a multi-domain indicator-based optimized stacked denoising autoencoder. Struct. Health Monit.
**2019**, 1475921719893594. [Google Scholar] [CrossRef] - He, W.; Jiang, Z.-N.; Feng, K. Bearing fault detection based on optimal wavelet filter and sparse code shrinkage. Measurement
**2009**, 42, 1092–1102. [Google Scholar] [CrossRef] - Li, C.; Zhou, J. Semi-supervised weighted kernel clustering based on gravitational search for fault diagnosis. Isa Trans.
**2014**, 53, 1534–1543. [Google Scholar] [CrossRef] [PubMed] - Zhang, X.; Miao, Q.; Liu, Z.; He, Z. An adaptive stochastic resonance method based on grey wolf optimizer algorithm and its application to machinery fault diagnosis. Isa Trans.
**2017**, 71, 206–214. [Google Scholar] [CrossRef] [PubMed] - Hu, A. New process method for end effects of HILBERT-HUANG transform. Chin. J. Mech. Eng.
**2008**, 44, 154. [Google Scholar] [CrossRef] - Rodríguez, P.H.; Alonso, J.B.; Ferrer, M.A.; Travieso, C.M.; Alonso-Hernández, J.B.; Travieso-González, C.M. Application of the Teager–Kaiser energy operator in bearing fault diagnosis. Isa Trans.
**2013**, 52, 278–284. [Google Scholar] [CrossRef] [PubMed] - Yan, X.; Liu, Y.; Jia, M.; Zhu, Y. A Multi-Stage Hybrid Fault Diagnosis Approach for Rolling Element Bearing Under Various Working Conditions. IEEE Access
**2019**, 7, 138426–138441. [Google Scholar] [CrossRef] - Yan, X.; Jia, M. Improved singular spectrum decomposition-based 1.5-dimensional energy spectrum for rotating machinery fault diagnosis. J. Braz. Soc. Mech. Sci. Eng.
**2019**, 41, 50. [Google Scholar] [CrossRef] - Zhang, X.; Miao, Q.; Zhang, H.; Wang, L. A parameter-adaptive VMD method based on grasshopper optimization algorithm to analyze vibration signals from rotating machinery. Mech. Syst. Signal Process.
**2018**, 108, 58–72. [Google Scholar] [CrossRef] - Hu, A.; Yan, X.; Xiang, L. A new wind turbine fault diagnosis method based on ensemble intrinsic time-scale decomposition and WPT-fractal dimension. Renew. Energy
**2015**, 83, 767–778. [Google Scholar] [CrossRef] - Peel, M.; Pegram, G.; McMahon, T. Empirical mode decomposition: Improvement and application. Proc. Int. Congr. Model. Simul.
**2007**, 1, 2996–3002. [Google Scholar]

**Figure 3.**Intuitive examples of the waveform matching extension: (

**a**) a cycle shock pulse signal and (

**b**) a modulated compound signal.

**Figure 4.**The time domain waveform of the simulation signal: (

**a**) No outlier, (

**b**) single outlier, and (

**c**) multiple outlier.

**Figure 9.**The decomposition results obtained by IAVMD for simulation 1 and their corresponding FFT spectrum.

**Figure 10.**The time-frequency representations obtained by IAVMD for simulation 1: (

**a**) Two-dimensional and (

**b**) three-dimensional.

**Figure 11.**The decomposition results obtained by different methods for simulation 1: (

**a**) VMD, (

**b**) LMD, (

**c**) EMD, and (

**d**) WT.

**Figure 12.**The time-frequency representations obtained by different methods for simulation 1: (

**a**) OVMD, (

**b**) LMD, (

**c**) EMD, and (

**d**) WT.

**Figure 14.**The decomposition results obtained by IAVMD for simulation 2 and their corresponding FFT spectrum.

**Figure 15.**The time-frequency representations obtained by IAVMD for simulation 2: (

**a**) Two-dimensional and (

**b**) three-dimensional.

**Figure 16.**The time-frequency representations obtained by different methods for simulation 2: (

**a**) OVMD, (

**b**) LMD, (

**c**) EMD, and (

**d**) WT.

**Figure 18.**The analyzed results of the rotor rub-impact signal: (

**a**) Temporal waveform, (

**b**) FFT spectrum, time-frequency representations obtained by (

**c**) IAVMD, (

**d**) OVMD, (

**e**) LMD, (

**f**) EMD, and (

**g**) WT.

**Figure 19.**The analyzed results of the rotor oil-whirl signal: (

**a**) Temporal waveform, (

**b**) FFT spectrum, time-frequency representations obtained by (

**c**) IAVMD, (

**d**) OVMD, (

**e**) LMD, (

**f**) EMD, and (

**g**) WT.

**Figure 20.**The analyzed results of the rotor oil-whip signal: (

**a**) Temporal waveform, (

**b**) FFT spectrum, time-frequency representations obtained by (

**c**) IAVMD, (

**d**) OVMD, (

**e**) LMD, (

**f**) EMD, and (

**g**) WT.

Signal | Kurtosis | Entropy | WK | WSK |
---|---|---|---|---|

No outlier | 3.19 | 6.89 | 3.19 | 9.18 |

Single outlier | 7.82 | 6.58 | 6.35 | 9.35 |

Multiple outlier | 16.41 | 6.32 | 2.87 | 9.29 |

Methods | $\mathbf{Energy}\text{}\mathbf{Error}\text{}\mathit{\theta}$ | $\mathbf{Orthogonal}\text{}\mathbf{Index}\text{}\mathit{O}\mathit{I}$ | Computing Time (s) | |||
---|---|---|---|---|---|---|

Case 1 | Case 2 | Case 1 | Case 2 | Case 1 | Case 2 | |

IAVMD | 0.0147 | 0.0151 | 0.0045 | 0.0048 | 56.7163 | 43.1932 |

OVMD | 0.0231 | 0.0209 | 0.0113 | 0.0126 | 32.4437 | 26.9957 |

LMD | 0.0409 | 0.0348 | 0.0304 | 0.0416 | 3.5421 | 3.8951 |

EMD | 0.0992 | 0.1053 | 0.0612 | 0.0585 | 3.2879 | 3.0517 |

WT | 0.0394 | 0.0572 | 0.0365 | 0.0403 | 2.3585 | 2.7138 |

Methods | $\mathbf{Energy}\text{}\mathbf{Error}\text{}\mathit{\theta}$ | $\mathbf{Orthogonal}\text{}\mathbf{Index}\text{}\mathit{O}\mathit{I}$ | Computing Time (s) | ||||||
---|---|---|---|---|---|---|---|---|---|

Case 1 | Case 2 | Case 3 | Case 1 | Case 2 | Case 3 | Case 1 | Case 2 | Case 3 | |

IAVMD | 0.0209 | 0.0174 | 0.0158 | 0.0034 | 0.0060 | 0.0054 | 32.0132 | 27.0215 | 24.9027 |

OVMD | 0.0238 | 0.0221 | 0.0253 | 0.0213 | 0.0133 | 0.0228 | 20.1082 | 16.8875 | 15.5641 |

LMD | 0.0273 | 0.0198 | 0.0169 | 0.0672 | 0.0195 | 0.0201 | 1.9617 | 2.8573 | 2.9322 |

EMD | 0.0319 | 0.0295 | 0.0703 | 0.0948 | 0.0602 | 0.0430 | 1.7810 | 2.6301 | 2.7835 |

WT | 0.0563 | 0.0702 | 0.0681 | 0.0513 | 0.0303 | 0.0448 | 1.3016 | 1.5158 | 1.6914 |

**Table 4.**The correlation coefficient among different approaches in the simulation and experiment signal.

Methods | Correlation Coefficient of Simulation | Correlation Coefficient of Experiment | |||
---|---|---|---|---|---|

Case 1 | Case 2 | Case 1 | Case 2 | Case 3 | |

IAVMD | 1.0000 | 0.9989 | 1.0000 | 0.9999 | 0.9999 |

OVMD | 0.9996 | 0.9925 | 0.9998 | 0.9992 | 0.9993 |

LMD | 0.9968 | 0.9897 | 0.9838 | 0.9985 | 0.9976 |

EMD | 0.9908 | 0.9888 | 0.9830 | 0.9878 | 0.9901 |

WT | 0.7989 | 0.8751 | 0.7291 | 0.7656 | 0.8586 |

**Table 5.**The percentage differences of the IAVMD with respect to other approaches in the simulation signal.

Method Comparison | $\mathbf{Energy}\text{}\mathbf{Error}\text{}\mathit{\theta}$ | $\mathbf{Orthogonal}\text{}\mathbf{Index}\text{}\mathit{O}\mathit{I}$ | ||
---|---|---|---|---|

Case 1 | Case 2 | Case 1 | Case 2 | |

IAVMD vs. OVMD | 0.84% ↓ | 0.58% ↓ | 0.68% ↓ | 0.78% ↓ |

IAVMD vs. LMD | 2.62% ↓ | 1.97% ↓ | 2.59% ↓ | 3.68% ↓ |

IAVMD vs. EMD | 8.45% ↓ | 9.02% ↓ | 5.67% ↓ | 5.37% ↓ |

IAVMD vs. WT | 2.47% ↓ | 4.21% ↓ | 3.20% ↓ | 3.55% ↓ |

**Table 6.**The percentage differences of the IAVMD with respect to other approaches in the experiment signal.

Methods | $\mathbf{Energy}\text{}\mathbf{Error}\text{}\mathit{\theta}$ | $\mathbf{Orthogonal}\text{}\mathbf{Index}\text{}\mathit{O}\mathit{I}$ | ||||
---|---|---|---|---|---|---|

Case 1 | Case 2 | Case 3 | Case 1 | Case 2 | Case 3 | |

IAVMD vs. OVMD | 0.29% ↓ | 0.47% ↓ | 0.95% ↓ | 1.79% ↓ | 0.73% ↓ | 1.74% ↓ |

IAVMD vs. LMD | 0.64% ↓ | 0.24% ↓ | 0.11% ↓ | 6.38% ↓ | 1.35% ↓ | 1.47% ↓ |

IAVMD vs. EMD | 1.10% ↓ | 1.21% ↓ | 5.45% ↓ | 9.14% ↓ | 5.42% ↓ | 3.76% ↓ |

IAVMD vs. WT | 3.54% ↓ | 5.28% ↓ | 5.23% ↓ | 4.79% ↓ | 2.43% ↓ | 3.94% ↓ |

Methods | Memory of Simulation (KByte) | Memory of Experiment (KByte) | Average Time (s) | |||
---|---|---|---|---|---|---|

Case 1 | Case 2 | Case 1 | Case 2 | Case 3 | ||

IAVMD | 8.85 | 8.87 | 8.91 | 8.92 | 8.91 | 36.76→High |

OVMD | 5.46 | 5.48 | 5.04 | 5.03 | 5.03 | 28.67→High |

LMD | 1.93 | 1.96 | 1.98 | 1.97 | 1.98 | 3.04→Low |

EMD | 1.71 | 1.73 | 1.75 | 1.76 | 1.76 | 2.27→Low |

WT | 1.35 | 1.38 | 1.41 | 1.41 | 1.42 | 1.92→Low |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yan, X.; Liu, Y.; Zhang, W.; Jia, M.; Wang, X.
Research on a Novel Improved Adaptive Variational Mode Decomposition Method in Rotor Fault Diagnosis. *Appl. Sci.* **2020**, *10*, 1696.
https://doi.org/10.3390/app10051696

**AMA Style**

Yan X, Liu Y, Zhang W, Jia M, Wang X.
Research on a Novel Improved Adaptive Variational Mode Decomposition Method in Rotor Fault Diagnosis. *Applied Sciences*. 2020; 10(5):1696.
https://doi.org/10.3390/app10051696

**Chicago/Turabian Style**

Yan, Xiaoan, Ying Liu, Wan Zhang, Minping Jia, and Xianbo Wang.
2020. "Research on a Novel Improved Adaptive Variational Mode Decomposition Method in Rotor Fault Diagnosis" *Applied Sciences* 10, no. 5: 1696.
https://doi.org/10.3390/app10051696