Skip to main content
SpringerLink
Log in

Iterative HOEO fusion strategy: a promising tool for enhancing bearing fault feature

  • Research Article
  • Published:
Frontiers of Mechanical Engineering Aims and scope Submit manuscript

Abstract

As parameter independent yet simple techniques, the energy operator (EO) and its variants have received considerable attention in the field of bearing fault feature detection. However, the performances of these improved EO techniques are subjected to the limited number of EOs, and they cannot reflect the non-linearity of the machinery dynamic systems and affect the noise reduction. As a result, the fault-related transients strengthened by these improved EO techniques are still subject to contamination of strong noises. To address these issues, this paper presents a novel EO fusion strategy for enhancing the bearing fault feature nonlinearly and effectively. Specifically, the proposed strategy is conducted through the following three steps. First, a multi-dimensional information matrix (MDIM) is constructed by performing the higher order energy operator (HOEO) on the analysis signal iteratively. MDIM is regarded as the fusion source of the proposed strategy with the properties of improving the signal-to-interference ratio and suppressing the noise in the low-frequency region. Second, an enhanced manifold learning algorithm is performed on the normalized MDIM to extract the intrinsic manifolds correlated with the fault-related impulses. Third, the intrinsic manifolds are weighted to recover the fault-related transients. Simulation studies and experimental verifications confirm that the proposed strategy is more effective for enhancing the bearing fault feature than the existing methods, including HOEOs, the weighting HOEO fusion, the fast Kurtogram, and the empirical mode decomposition.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Abbreviations

EMD:

Empirical mode decomposition

EO:

Energy operator

FK:

Fast Kurtogram

HOEO:

Higher order energy operator

LTSA:

Local tangent space alignment

MDIM:

Multi-dimensional information matrix

NFER:

Normalized frequency energy ratio

PE:

Permutation entropy

REB:

Rolling element bearing

SIR:

Signal-to-interference ratio

SNR:

Signal-to-noise ratio

WHOEOF:

Weighting higher order energy operator fusion

A :

Amplitude of the fault bearing vibration

A :

Alignment matrix

C i :

Correlation matrix

Ej[•]:

Function of the jth order EO

Envsq[•]:

Function of squared envelope

e k :

Column vector of k ones

f b :

Rolling element fault frequency

f c :

Center frequency

f d :

Fault characteristic frequency

f i :

Inner race fault frequency

f o :

Outer race fault frequency

f RE :

Resonance frequency

g i :

Eigenvector of the alignment matrix

G :

Reorganizing result of the global representation G0

G 0 :

Global representation of manifold learning

H :

Number of harmonics used to calculate NFER

HOEO-F(•):

Function of WHOEOF

Id :

Number of the saved intrinsic dimensions

I :

Identity matrix

J :

Total number of orders

k :

Nearest neighboring size

k*:

Nearest neighborhood corresponding to the smallest P

K :

Number of spectral lines

Kurt(•):

Function of kurtosis

L j :

Amplitude of the jth interference component

m :

Embedded dimension

M :

MDIM

\(\overline {\boldsymbol{M}} \) :

Normalized MDIM

M(x(t)):

MDIM with the signal x(t)

M 1(x(t)):

HOEO matrix of signal x(t)

M 2(x(t)):

Once iterative of M1(x(t))

M 3(x(t)):

Twice iterative of M1(x(t))

n(t):

Noise component

N :

Number of data points of the analysis signal

P(f):

Amplitude of the envelope spectrum at frequency f

P i :

Relative frequency of the ith permutation

P k :

PE of the reorganizing result G at nearest neighbors k

r(t):

Fault bearing vibration

R i :

Row vector of the MDIM

\({\overline {\boldsymbol{R}} _i}\) :

Row vector of the normalized MDIM

S(t):

A transient with unit amplitude

R i :

0-1 selection matrix

SIR(•):

Function of SIR

t :

Time

T :

Total lasting time of analysis signal

T d :

Time interval between two adjacent transients

T p :

Time period of the fault characteristic frequency

Δt 1, Δt 2, Δt 3, Δt 4 :

Intervals of the repetitive transients in the simulated bearing, outer race, inner race, and rolling element fault signal, respectively

u j(t):

jth vibration interferences

V i :

Matrix composed by Id largest right singular vectors of centralized matrix

w b :

Bandwidth

x(t):

Continuous time signal

x(n):

Discrete form of x(t)

\(\dot x(t)\) :

First-order derivative of x(t) with respect to time t

\(\hat \dot x(t)\) :

Hilbert transform of \(\dot x(t)\)

\(\ddot x(t)\) :

Second-order derivative of x(t) with respect to time t

x (j)(t):

jth derivative of x(t)

y(t):

Preset transients

Z i :

Matrix combined by a set of k nearest neighbors of column mn

\({\overline {\boldsymbol{Z}} _i}\) :

Mean of Zi

α j :

Coefficient associated with the jth HOEO

\(\alpha _j^ \ast \) :

Optimal coefficient associated with the jth HOEO

α*:

Optimal coefficient vector

β :

Structural damping characteristic of the fault bearing vibration

ω j :

Frequency of the jth interference component

ω r :

Resonance frequency excited by the bearing defect

λ i :

Eigenvalue of the alignment matrix

ξ :

Decay rate of the transient

τ i :

A random variable to simulate the slip effect of transients

References

  1. An Z H, Jiang X X, Cao J, Yang R, Li X G. Self-learning transferable neural network for intelligent fault diagnosis of rotating machinery with unlabeled and imbalanced data. Knowledge-Based Systems, 2021, 230: 107374

    Article  Google Scholar 

  2. Chen X F, Wang S B, Qiao B J, Chen Q. Basic research on machinery fault diagnostics: past, present, and future trends. Frontiers of Mechanical Engineering, 2018, 13(2): 264–291

    Article  Google Scholar 

  3. Wang X, Gu H Y, Wang T Y, Zhang W, Li A H, Chu F L. Deep convolutional tree-inspired network: a decision-tree-structured neural network for hierarchical fault diagnosis of bearings. Frontiers of Mechanical Engineering, 2021, 16(4): 814–828

    Article  Google Scholar 

  4. Zhang X, Huang T, Wu B, Hu Y M, Huang S, Zhou Q, Zhang X. Multi-model ensemble deep learning method for intelligent fault diagnosis with high-dimensional samples. Frontiers of Mechanical Engineering, 2021, 16(2): 340–352

    Article  Google Scholar 

  5. Liu J, Zhou K B, Yang C Y, Lu G L. Imbalanced fault diagnosis of rotating machinery using autoencoder-based SuperGraph feature learning. Frontiers of Mechanical Engineering, 2021, 16(4): 829–839

    Article  Google Scholar 

  6. Ziaja A, Antoniadou I, Barszcz T, Staszewski W J, Worden K. Fault detection in rolling element bearings using wavelet-based variance analysis and novelty detection. Journal of Vibration and Control, 2016, 22(2): 396–411

    Article  Google Scholar 

  7. Syed S H, Muralidharan V. Feature extraction using discrete wavelet transform for fault classification of planetary gearbox—a comparative study. Applied Acoustics, 2022, 188: 108572

    Article  Google Scholar 

  8. An B T, Wang S B, Yan R Q, Li W H, Chen X F. Adaptive robust noise modeling of sparse representation for bearing fault diagnosis. IEEE Transactions on Instrumentation and Measurement, 2021, 70: 1–12

    Article  Google Scholar 

  9. Ahmed H O A, Wong M L D, Nandi A K. Intelligent condition monitoring method for bearing faults from highly compressed measurements using sparse over-complete features. Mechanical Systems and Signal Processing, 2018, 99: 459–477

    Article  Google Scholar 

  10. Xiao L, Bajric R, Zhao J S, Tang J X, Zhang X H. An adaptive vibrational resonance method based on cascaded varying stable-state nonlinear systems and its application in rotating machine fault detection. Nonlinear Dynamics, 2021, 103(1): 715–739

    Article  Google Scholar 

  11. López C, Naranjo Á, Lu S L, Moore K J. Hidden markov model based stochastic resonance and its application to bearing fault diagnosis. Journal of Sound and Vibration, 2022, 528: 116890

    Article  Google Scholar 

  12. Lu S L, Zheng P, Liu Y B, Cao Z, Yang H, Wang Q J. Sound-aided vibration weak signal enhancement for bearing fault detection by using adaptive stochastic resonance. Journal of Sound and Vibration, 2019, 449: 18–29

    Article  Google Scholar 

  13. Abdelkader R, Kaddour A, Bendiabdellah A, Derouiche Z. Rolling bearing fault diagnosis based on an improved denoising method using the complete ensemble empirical mode decomposition and the optimized thresholding operation. IEEE Sensors Journal, 2018, 18(17): 7166–7172

    Article  Google Scholar 

  14. Jiang X X, Huang Q, Shen C Q, Wang Q, Xu K, Liu J, Shi J J, Zhu Z K. Synchronous chirp mode extraction: a promising tool for fault diagnosis of rolling element bearings under varying speed conditions. Chinese Journal of Aeronautics, 2022, 35(1): 348–364

    Article  Google Scholar 

  15. Jiang X X, Song Q Y, Wang H E, Du G F, Guo J F, Shen C Q, Zhu Z K. Central frequency mode decomposition and its applications to the fault diagnosis of rotating machines. Mechanism and Machine Theory, 2022, 174: 104919

    Article  Google Scholar 

  16. Jiang X X, Wang J, Shen C Q, Shi J J, Huang W G, Zhu Z K, Wang Q. An adaptive and efficient variational mode decomposition and its application for bearing fault diagnosis. Structural Health Monitoring, 2021, 20(5): 2708–2725

    Article  Google Scholar 

  17. Maurya S, Singh V, Verma N K. Condition monitoring of machines using fused features from EMD-based local energy with DNN. IEEE Sensors Journal, 2020, 20(15): 8316–8327

    Article  Google Scholar 

  18. Liang M, Bozchalooi I S. An energy operator approach to joint application of amplitude and frequency-demodulations for bearing fault detection. Mechanical Systems and Signal Processing, 2010, 24(5): 1473–1494

    Article  Google Scholar 

  19. Liu Y, Jiang Z S, Huang H Z, Xiang J W. A TEO-based modified laplacian of gaussian filter to detect faults in rolling element bearing for variable rotational speed machine. IET Science, Measurement & Technology, 2021, 15(2): 193–203

    Article  Google Scholar 

  20. Wan S T, Zhang X. Bearing fault diagnosis based on Teager energy entropy and mean-shift fuzzy C-means. Structural Health Monitoring, 2020, 19(6): 1976–1988

    Article  Google Scholar 

  21. Han T, Liu Q N, Zhang L, Tan A C C. Fault feature extraction of low speed roller bearing based on Teager energy operator and CEEMD. Measurement, 2019, 138: 400–408

    Article  Google Scholar 

  22. Ma J, Wu J D, Wang X D. Incipient fault feature extraction of rolling bearings based on the MVMD and Teager energy operator. ISA Transactions, 2018, 80: 297–311

    Article  Google Scholar 

  23. Gu R, Chen J, Hong R J, Wang H, Wu W W. Incipient fault diagnosis of rolling bearings based on adaptive variational mode decomposition and Teager energy operator. Measurement, 2020, 149: 106941

    Article  Google Scholar 

  24. Feng Z P, Zuo M J, Hao R J, Chu F L, Lee J. Ensemble empirical mode decomposition-based teager energy spectrum for bearing fault diagnosis. Journal of Vibration and Acoustics, 2013, 135(3): 031013

    Article  Google Scholar 

  25. Singh J, Darpe A K, Singh S P. Rolling element bearing fault diagnosis based on over-complete rational dilation wavelet transform and auto-correlation of analytic energy operator. Mechanical Systems and Signal Processing, 2018, 100: 662–693

    Article  Google Scholar 

  26. Pei X L, Zheng X Y, Wu J L. Intelligent bearing fault diagnosis based on Teager energy operator demodulation and multiscale compressed sensing deep autoencoder. Measurement, 2021, 179: 109452

    Article  Google Scholar 

  27. Chen Z X, He C B, Liu Y B, Lu S L, Liu F, Li G L. Incipient fault feature extraction of rolling bearing based on optimized singular spectrum decomposition. IEEE Sensors Journal, 2021, 21(18): 20362–20374

    Article  Google Scholar 

  28. Zeng M, Yang Y, Zheng J D, Cheng J S. Normalized complex Teager energy operator demodulation method and its application to fault diagnosis in a rubbing rotor system. Mechanical Systems and Signal Processing, 2015, 50–51: 380–399

    Article  Google Scholar 

  29. Randall R B, Smith W A. Uses and mis-uses of energy operators for machine diagnostics. Mechanical Systems and Signal Processing, 2019, 133: 106199

    Article  Google Scholar 

  30. Liang M, Faghidi H. A non-parametric non-filtering approach to bearing fault detection in the presence of multiple interference. Measurement Science and Technology, 2013, 24(10): 105013

    Article  Google Scholar 

  31. Liang M, Faghidi H. Intelligent bearing fault detection by enhanced energy operator. Expert Systems with Applications, 2014, 41(16): 7223–7234

    Article  Google Scholar 

  32. Imaouchen Y, Kedadouche M, Alkama R, Thomas M. A frequency-weighted energy operator and complementary ensemble empirical mode decomposition for bearing fault detection. Mechanical Systems and Signal Processing, 2017, 82: 103–116

    Article  Google Scholar 

  33. O’Toole J M, Temko A, Stevenson N. Assessing instantaneous energy in the EEG: a non-negative, frequency-weighted energy operator. In: Proceedings of the 36th Annual International Conference of the IEEE Engineering in Medicine and Biology Society. Chicago: IEEE, 2014, 3288–3291

    Google Scholar 

  34. Jiang X X, Li S M, Cheng C. A novel method for adaptive multiresonance bands detection based on VMD and using MTEO to enhance rolling element bearing fault diagnosis. Shock and Vibration, 2016, 2016: 8361289

    Article  Google Scholar 

  35. Feng Z P, Qin S F, Liang M. Time-frequency analysis based on Vold-Kalman filter and higher order energy separation for fault diagnosis of wind turbine planetary gearbox under nonstationary conditions. Renewable Energy, 2016, 85: 45–56

    Article  Google Scholar 

  36. Xu Y B, Cai Z Y, Ding K. An enhanced bearing fault diagnosis method based on TVF-EMD and a high-order energy operator. Measurement Science and Technology, 2018, 29(9): 095108

    Article  Google Scholar 

  37. Faghidi H, Liang M. Bearing fault identification by higher order energy operator fusion: a non-resonance based approach. Journal of Sound and Vibration, 2016, 381: 83–100

    Article  Google Scholar 

  38. Faghidi H, Liang M. Detection of bearing fault detection from heavily contaminated signals: a higher-order analytic energy operator method. Journal of Vibration and Acoustics, 2015, 137(4): 041012

    Article  Google Scholar 

  39. Maragos P, Potamianos A. Higher order differential energy operators. IEEE Signal Processing Letters, 1995, 2(8): 152–154

    Article  Google Scholar 

  40. Wang J, Du G F, Zhu Z K, Shen C Q, He Q B. Fault diagnosis of rotating machines based on the EMD manifold. Mechanical Systems and Signal Processing, 2020, 135: 106443

    Article  Google Scholar 

  41. Li Q C, Ding X X, He Q B, Huang W B, Shao Y M. Manifold sensing-based convolution sparse self-learning for defective bearing morphological feature extraction. IEEE Transactions on Industrial Informatics, 2021, 17(5): 3069–3078

    Article  Google Scholar 

  42. Wang Y, Xu G H, Liang L, Jiang K S. Detection of weak transient signals based on wavelet packet transform and manifold learning for rolling element bearing fault diagnosis. Mechanical Systems and Signal Processing, 2015, 54–55: 259–276

    Article  Google Scholar 

  43. Wang J, He Q B. Wavelet packet envelope manifold for fault diagnosis of rolling element bearings. IEEE Transactions on Instrumentation and Measurement, 2016, 65(11): 2515–2526

    Article  Google Scholar 

  44. Wang J, He Q B, Kong F R. Multiscale envelope manifold for enhanced fault diagnosis of rotating machines. Mechanical Systems and Signal Processing, 2015, 52–53: 376–392

    Google Scholar 

  45. Hurley N, Rickard S. Comparing measures of sparsity. IEEE Transactions on Information Theory, 2009, 55(10): 4723–4741

    Article  MathSciNet  MATH  Google Scholar 

  46. Wang Y X, Xiang J W, Markert R, Liang M. Spectral kurtosis for fault detection, diagnosis and prognostics of rotating machines: a review with applications. Mechanical Systems and Signal Processing, 2016, 66–67: 679–698

    Article  Google Scholar 

  47. Miao Y H, Zhao M, Lin J. Improvement of kurtosis-guided-grams via Gini index for bearing fault feature identification. Measurement Science and Technology, 2017, 28(12): 125001

    Article  Google Scholar 

  48. Wang D, Zhong J J, Li C, Peng Z K. Box-Cox sparse measures: a new family of sparse measures constructed from kurtosis and negative entropy. Mechanical Systems and Signal Processing, 2021, 160: 107930

    Article  Google Scholar 

  49. Hou B C, Wang D, Xia T B, Xi L F, Peng Z K, Tsui K L. Generalized Gini indices: complementary sparsity measures to Box-Cox sparsity measures for machine condition monitoring. Mechanical Systems and Signal Processing, 2022, 169: 108751

    Article  Google Scholar 

  50. Hou B C, Wang D, Xia T B, Wang Y, Zhao Y, Tsui K L. Investigations on quasi-arithmetic means for machine condition monitoring. Mechanical Systems and Signal Processing, 2021, 151: 107451

    Article  Google Scholar 

  51. Wang D. Some further thoughts about spectral kurtosis, spectral L2/L1 norm, spectral smoothness index and spectral Gini index for characterizing repetitive transients. Mechanical Systems and Signal Processing, 2018, 108: 360–368

    Article  Google Scholar 

  52. Rostaghi M, Ashory M R, Azami H. Application of dispersion entropy to status characterization of rotary machines. Journal of Sound and Vibration, 2019, 438: 291–308

    Article  Google Scholar 

  53. Wang X Z, Si S B, Li Y B. Multiscale diversity entropy: a novel dynamical measure for fault diagnosis of rotating machinery. IEEE Transactions on Industrial Informatics, 2021, 17(8): 5419–5429

    Article  Google Scholar 

  54. Yan R Q, Liu Y B, Gao R X. Permutation entropy: a nonlinear statistical measure for status characterization of rotary machines. Mechanical Systems and Signal Processing, 2012, 29: 474–484

    Article  Google Scholar 

  55. Randall R B, Antoni J. Rolling element bearing diagnostics&# x2014;a tutorial. Mechanical Systems and Signal Processing, 2011, 25(2): 485–520

    Article  Google Scholar 

  56. Antoni J. Fast computation of the kurtogram for the detection of transient faults. Mechanical Systems and Signal Processing, 2007, 21(1): 108–124

    Article  Google Scholar 

  57. Lei Y G, Lin J, He Z J, Zuo M J. A review on empirical mode decomposition in fault diagnosis of rotating machinery. Mechanical Systems and Signal Processing, 2013, 35(1–2): 108–126

    Article  Google Scholar 

  58. Song Q Y, Jiang X X, Wang S, Guo J F, Huang W G, Zhu Z K. Self-adaptive multivariate variational mode decomposition and its application for bearing fault diagnosis. IEEE Transactions on Instrumentation and Measurement, 2022, 71: 1–13

    Google Scholar 

  59. Huang Y, Lin J H, Liu Z C, Wu W Y. A modified scale-space guiding variational mode decomposition for high-speed railway bearing fault diagnosis. Journal of Sound and Vibration, 2019, 444: 216–234

    Article  Google Scholar 

Download references

Acknowledgements

The authors declare that they have no competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. This research was supported by the National Natural Science Foundation of China (Grant Nos. 52172406 and 51875376), the China Postdoctoral Science Foundation (Grant Nos. 2022T150552 and 2021M702752), and the Suzhou Prospective Research Program, China (Grant No. SYG202111), which are highly appreciated by the authors.

Author information

Authors and Affiliations

  1. School of Rail Transportation, Soochow University, Suzhou, 215131, China

    Xingxing Jiang, Demin Peng, Changqing Shen & Zhongkui Zhu

  2. China Academy of Railway Sciences Co. Ltd., Beijing, 100094, China

    Jianfeng Guo

  3. School of Civil and Hydraulic Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China

    Jie Liu

Authors
  1. Xingxing Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Demin Peng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jianfeng Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jie Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Changqing Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Zhongkui Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhongkui Zhu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jiang, X., Peng, D., Guo, J. et al. Iterative HOEO fusion strategy: a promising tool for enhancing bearing fault feature. Front. Mech. Eng. 18, 9 (2023). https://doi.org/10.1007/s11465-022-0725-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11465-022-0725-z

Keywords

Search

Navigation