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Multi-feature optimized VMD and fusion index for bearing fault diagnosis method

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Abstract

In order to effectively solve the parameter selection problem of variational mode decomposition (VMD) and accurately extract the bearing fault features, a bearing fault diagnosis method based on multi-feature optimized VMD and fusion index is proposed. Considering the multiple features of fault pulse when the bearing fails, the objective functions and fusion index of information entropy, correlation coefficient, and kurtosis are established, and the parameter optimization problem of VMD is transformed into a multi-objective optimization problem. Firstly, the multi-objective particle swarm optimization (MOPSO) algorithm is used to optimize the three objective functions, and the optimal Pareto frontier solution set of VMD parameter combination is obtained. Secondly, the fusion index is used to evaluate the Pareto frontier solution set, from which the optimal parameter combination of VMD is determined. The bearing fault signal is decomposed by VMD based on the optimal parameter combination, and several intrinsic mode functions (IMFs) are obtained. Then, the fusion index is used to select the optimal IMF, and fault features are extracted. Finally, the analysis results of the simulation signal and actual bearing vibration signals show the effectiveness of the proposed method.

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Abbreviations

f(t):

Original signal desired for decomposition

u k(t):

kth decomposed IMF

A k(t):

Instantaneous amplitude of uk(t)

ϕ k(t):

Derivative of the instantaneous frequency of uk(t)

K :

Number of IMF

ω k :

Central frequency of the kth IMF

α :

Penalty factor

λ :

Lagrange multiplication operator

\(v_i^{z + 1}\) :

Velocity of ith particle

\(loc_i^{z + 1}\) :

Position of ith particle

c 1 and c 2 :

Learning factors

it :

Number of iterations

r 1 and r 2 :

Random numbers in [0, 1]

\(p_i^{it}\) :

Individual optimal solution

\(p_g^{it}\) :

Global optimal solution

x :

Decision variable

fun d(x):

dth objective function

RS :

Random sequence

P :

Probability distribution of RS

N :

Sampling length

σ :

Standard deviation of uk(t)

\(ku{r_{{u_k}}}\) :

Kurtosis of uk(t)

\({r_{f{u_k}}}\) :

Correlation coefficient between uk(t) and f(t)

fun :

Fusion index

\({\bar f}\) :

Mean value of f(t)

np :

Population number

w :

Inertia weight

it m :

Maximum number of iterations

s(t):

Periodic shock signal

n(t):

Gaussian white noise

h(t):

Oscillating signal with an attenuated amplitude

C :

Attenuation coefficient

f i :

Fault feature frequency of the bearing inner race

f n :

Resonant frequency

f s :

Sampling frequency

S(df):

Sum of envelope spectrum amplitudes of bearing feature frequency and its harmonics

S:

Sum of signal envelope spectrum amplitudes

R f :

Fault feature ratio

P signal :

Target signal power

P noise :

Noise signal power

F(df):

Envelope spectrum amplitude of df

N(df):

Average amplitude of noise bands on both sides of df

f r :

Bearing rotation frequency

f o :

Fault feature frequency of the bearing outer race

f b :

Fault feature frequency of the bearing rolling element

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Acknowledgments

This work is supported by the National Natural Science Foundation of China (No: 61463028); and the Natural Science Foundation of Gansu Province (20JR10RA209).

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Authors and Affiliations

  1. School of Mechanical Engineering, Lanzhou Jiaotong University, Lanzhou, 730070, China

    Zhen Liu, Zhenrui Peng & Pei Liu

Authors
  1. Zhen Liu
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Corresponding author

Correspondence to Zhenrui Peng.

Additional information

Zhen Liu received the Bachelor’s in Mechanical Engineering and Automation from Lanzhou Institute of Technology, Lanzhou, China, in 2017. He received the Master’s in Vehicle Engineering from Lanzhou Jiaotong University, Lanzhou, China, in 2020, where he is currently pursuing the Ph.D. in Mechatronic Engineering. His main research areas include signal processing and fault diagnosis of mechanical equipment.

Zhenrui Peng received the Bachelor’s in Mechanical Engineering from Lanzhou Jiaotong University, Lanzhou, China, in 1995, and the Ph.D. in Control Science and Engineering from Zhejiang University, Hangzhou, China, in 2007. He presided over three projects of the National Natural Science Foundation of China and published over 90 papers up to now, and over 30 papers have been indexed by SCI, EI, and ISTP. His major research fields include fault diagnosis of mechanical equipment, finite element model updating, and structural response reconstruction.

Pei Liu received the Bachelor’s in Industrial Engineering from Lanzhou Jiaotong University, Lanzhou, China, in 2019, where he is currently pursuing the Master’s in Mechanical Engineering. His main research interests include signal processing and fault diagnosis of mechanical equipment.

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Liu, Z., Peng, Z. & Liu, P. Multi-feature optimized VMD and fusion index for bearing fault diagnosis method. J Mech Sci Technol 37, 2807–2820 (2023). https://doi.org/10.1007/s12206-023-0508-4

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  • DOI: https://doi.org/10.1007/s12206-023-0508-4

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