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

doi:10.1007/s11465-022-0725-z

Front. Mech. Eng.

2023, Vol. 18

Issue (1) : 9 https://doi.org/10.1007/s11465-022-0725-z

RESEARCH ARTICLE

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

Xingxing JIANG¹, Demin PENG¹, Jianfeng GUO², Jie LIU³, Changqing SHEN¹, Zhongkui ZHU¹(

)

¹. School of Rail Transportation, Soochow University, Suzhou 215131, China
². China Academy of Railway Sciences Co. Ltd., Beijing 100094, China
³. School of Civil and Hydraulic Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

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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.

Keywords higher order energy operator fault diagnosis manifold learning rolling element bearing information fusion

Corresponding Author(s): Zhongkui ZHU

Just Accepted Date: 30 August 2022 Issue Date: 10 April 2023

Cite this article:

Xingxing JIANG,Demin PENG,Jianfeng GUO, et al. Iterative HOEO fusion strategy: a promising tool for enhancing bearing fault feature[J]. Front. Mech. Eng., 2023, 18(1): 9.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-022-0725-z
https://academic.hep.com.cn/fme/EN/Y2023/V18/I1/9

Fig.1 Simulated signal with SNR = ?8 dB: (a) time waveform and (b) envelope spectrum.

Fig.2 PE values of fused results of simulated signal under different nearest neighboring sizes. PE: permutation entropy.

Fig.3 Fusion result of simulated signal obtained by the proposed strategy: (a) time waveform and (b) envelope spectrum.

Fig.4 Results of higher order energy operators of simulated signal: (a) order 2, (b) order 3, (c) order 4, (d) order 5, and (e) order 6.

Fig.5 Analysis result of weighting higher order energy operator fusion of simulated signal: (a) time waveform and (b) envelope spectrum.

Fig.6 Fused results of the simulated signals with different SNRs. SNR: signal-to-noise ratio.

Tab.1 Geometrical parameters and fault characteristic frequencies of the experiment bearings

Fig.7 Test rig of rolling element bearing.

Fig.8 Bearing signal with an outer race fault: (a) time waveform and (b) envelope spectrum.

Fig.9 PE values of fused results of bearing signal with an outer race fault under different nearest neighboring sizes. PE: permutation entropy.

Fig.10 Fusion result of bearing signal with an outer race fault: (a) time waveform and (b) envelope spectrum.

Fig.11 Results of higher order energy operators of bearing signal with an outer race fault: (a) order 2, (b) order 3, (c) order 4, (d) order 5, and (e) order 6.

Fig.12 Analysis result of weighting higher order energy operator fusion of bearing signal with an outer race fault: (a) time waveform and (b) envelope spectrum.

Fig.13 Results of fast Kurtogram of bearing signal with an outer race fault: (a) Kurtogram, (b) envelope waveform of the optimal filtered signal, and (c) envelope spectrum of the optimal filtered signal.

Fig.14 Results of empirical mode decomposition of bearing signal with an outer race fault: (a) waveforms of the first four modes and (b) envelope spectra of the first four modes.

Fig.15 NFERs of the bearing signal with an outer race fault. NFER: normalized frequency energy ratio, WHOEOF: weighting higher order energy operator fusion, FK: fast Kurtogram. Mode1–Mode4 denote the first four modes of empirical mode decomposition.

Fig.16 Bearing signal with an inner race fault: (a) time waveform and (b) envelope spectrum.

Fig.17 PE values of fused results of bearing signal with an inner race fault under different nearest neighboring sizes. PE: permutation entropy.

Fig.18 Fusion result of bearing signal with an inner race fault: (a) time waveform and (b) envelope spectrum.

Fig.19 Results of higher order energy operators of bearing signal with an inner race fault: (a) order 2, (b) order 3, (c) order 4, (d) order 5, and (e) order 6.

Fig.20 Analysis result of weighting higher order energy operator fusion of bearing signal with an inner race fault: (a) time waveform and (b) envelope spectrum.

Fig.21 Results of fast Kurtogram of bearing signal with an inner race fault: (a) Kurtogram, (b) envelope waveform of the optimal filtered signal, and (c) envelope spectrum of the optimal filtered signal.

Fig.22 Results of empirical mode decomposition of bearing signal with an inner race fault: (a) waveforms of the first four modes and (b) envelope spectra of the first four modes.

Fig.23 NFERs of the bearing signal with an inner race fault. NFER: normalized frequency energy ratio, WHOEOF: weighting higher order energy operator fusion, FK: fast Kurtogram. Mode1–Mode4 denote the first four modes of empirical mode decomposition.

Fig.24 Bearing signal with a rolling element fault: (a) time waveform and (b) envelope spectrum.

Fig.25 PE values of fused results of bearing signal with a rolling element fault under nearest neighboring sizes. PE: permutation entropy.

Fig.26 Fusion result of bearing signal with a rolling element fault: (a) time waveform and (b) envelope spectrum.

Fig.27 Results of higher order energy operators of bearing signal with a rolling element fault: (a) order 2, (b) order 3, (c) order 4, (d) order 5, and (e) order 6.

Fig.28 Analysis result of weighting higher order energy operator fusion of bearing signal with a rolling element fault: (a) time waveform and (b) envelope spectrum.

Fig.29 Results of fast Kurtogram of bearing signal with a rolling element fault: (a) Kurtogram, (b) envelope waveform of the optimal filtered signal, and (c) envelope spectrum of the optimal filtered signal.

Fig.30 Results of empirical mode decomposition of bearing signal with a rolling element fault: (a) waveforms of the first four modes and (b) envelope spectra of the first four modes.

Fig.31 NFERs of the bearing signal with a rolling element fault. NFER: normalized frequency energy ratio, WHOEOF: weighting higher order energy operator fusion, FK: fast Kurtogram. Mode1–Mode4 denote the first four modes of empirical mode decomposition.

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
Variables
A	Amplitude of the fault bearing vibration
A	Alignment matrix
C_i	Correlation matrix
$E j [?]$	Function of the jth order EO
$E n v s q [?]$	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 $G 0$
$G 0$	Global representation of manifold learning
$H$	Number of harmonics used to calculate NFER
$H O E O - F (?)$	Function of WHOEOF
$I d$	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
$K u r t (?)$	Function of kurtosis
$L j$	Amplitude of the $j$ th interference component
m	Embedded dimension
M	MDIM
$M ˉ$	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 $M 1 (x (t))$
$M 3 (x (t))$	Twice iterative of $M 1 (x (t))$
$n (t)$	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
$R ˉ i$	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
?t₁, ?t₂, ?t₃, ?t₄	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 $I d$ largest right singular vectors of centralized matrix
$w b$	Bandwidth
$x (t)$	Continuous time signal
$x (n)$	Discrete form of $x (t)$
$x ˙ (t)$	First-order derivative of $x (t)$ with respect to time $t$
$x ˙^(t)$	Hilbert transform of $x ˙ (t)$
$x ¨ (t)$	Second-order derivative of $x (t)$ with respect to time $t$
$x (j) (t)$	$j$ th derivative of $x (t)$
$y (t)$	Preset transients
$Z i$	Matrix combined by a set of $k$ nearest neighbors of column $m n$
$Z ˉ i$	Mean of $Z i$
$α j$	Coefficient associated with the jth HOEO
$α j ∗$	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

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