Sparse dictionary design based on edited cepstrum and its application in rolling bearing fault diagnosis

https://doi.org/10.1016/j.jsv.2020.115704Get rights and content

Highlights

  • A novel dictionary design method is proposed for impact feature extraction.

  • Modal parameters identification errors are corrected by quantitative compensation.

  • A segmental matching pursuit algorithm with fast calculation speed is applied.

  • Simulations and experimental tests verify the effectiveness of proposed method.

Abstract

Rolling bearing with a localized defect usually generates periodically impact vibration responses, which carry important information for bearing fault diagnosis. Due to the inevitable noise disturbances, extracting accurate impact features of faulty bearing is still a hard task. In view of the superiority of sparse decomposition on feature extraction, a novel sparse dictionary design method is proposed based on edited cepstrum to improve the precision of feature extraction. The impulse response function is selected as sparse atom, which better reflects the structure and inherent modal characteristics of the faulty bearing. The modal parameters are directly identified from the deconvolved fault signal by edited cepstrum. Identification errors caused by the cepstrum windowing are corrected by quantitative compensation, which further improves the accuracy of dictionary design. A segmental matching pursuit algorithm is applied to speed sparse coefficients solving and fault features reconstruction. A series of simulation analyses verify the proposed method's effectiveness, anti-noise performance and robustness. Experimental tests on pure rolling bearing and gearbox bearing further verify the method's effectiveness under different working conditions. Additionally, comparisons with an improved spectral kurtosis method and an edited cepstrum methodshow the proposed method be more reliable in diagnostic performance.

Introduction

Rolling bearings are used in almost all types of rotating machinery. Its failure will damage mechanical performance and even lead to fatal breakdowns [1]. More and more scholars have been processing the collected vibration signal to evaluatemechanicalperformance. However, the vibration signals contain not only fault information but also strong background noise [2]. Additionally,factors including speed fluctuation, rolling element sliding and nonuniformly distributed loads cause vibration signals more complex, such as characteristics of amplitude modulation and non-stationary phenomenon. It further increases the difficulty of bearing fault diagnosis.

In term of fault feature extraction of rolling bearings, spectral kurtosis (SK) is the most commonly used method due to its superiority of locating transients in spectral domain, which attracts scholars' attention. Antoni et al. [3] firstly proposed fast kurtogram to select the optimal frequency band for demodulation analysis. Jia et al. [4] proposed an improved SK based on maximum correlated kurtosis deconvolution (MCKD) to detect early fault of bearing. The MCKD firstly lifts the kurtosis value of signal and further suppresses noise interferences. Then the SK is applied on the filtered signal to find the optimal frequency band for demodulation. Some studies are focused on combining SK with different signal processing techniques to better select optimal frequency band [5,6]. Basically, the SK methods is to select an optimal frequency band for analysis, but ignores other resonance bands containing fault information. And its diagnosis accuracy and reliability are easily influenced by background noise.

Recently, signal sparse decomposition has become a more and more popular approach in rotating machinery diagnosis [7], [8], [9], [10], [11], [12], [13], due to its excellent performance of feature extraction. Studies on signal sparse decomposition range from sparse coefficient solving [7], [8], [9] to sparse dictionary design [10], [11], [12], [13]. On the aspect ofsparse coefficients solving, Yang et al. [7] proposed a sliding window denoising K-Singular Value Decomposition (K-SVD) method to sparely extract impact features of rolling bearings; Ren et al. [8] utilized a modified Majorzation-Minimization (MM) algorithm to address the convex optimization problem. On the aspect of sparse dictionary design, Li et al. [12] applied adaptive Q-factor wavelet transform to construct complete dictionary, which shows better results than both dictionaries based on discretecosine transform and discrete Hart transform. Cui et al. [13] proposed a concatenation dictionary composed of an impact dictionary as the higher level and step dictionary as the lower level to analysis rolling bearing fault signals. In sum, sparse coefficients solving mainly contributes to feature reconstruction speed, but sparse dictionary design fundamentally affects feature reconstruction accuracy. Therefore, constructing a more accurate dictionary is of great importance in improving the precision of signal sparse decomposition, which is worth of further research.

The cepstrum technique was originally applied to detect earthquake echoes and speech signal tones. Given the deconvolution ability of cepstrum, Oppenheim et al. [14] presented the homomorphic system, namely transforming the nonlinear convolution relationship into a linear superposition relationship. Smith et al. [15] proposed a simple polynomial curve-fitting approach to fit the pole-zero model of transfer function in the cepstrum, thereby obtained the modal parameters. Badaoui et al. [16] defined a diagnosis indicator by calculating the sum of first rahmonic peaks for gear diagnostic. Randall et al. [17] pointed that edited cepstrum can be used to generate fault-related time signals by retaining the modal information and combining with its original phase information. Furthermore, envelope analysis is applied on the obtained time signals to detect bearing fault. Peeters et al. [18] compared the cepstrum pre-whitening (CPW) with automated cepstrum editing procedure (ACEP), which are both signalpre-processing techniques for bearingfault detection. Even though cepstrum has the property of deconvolution, it is mostly used as pre-processing techniques to remove modal information for detecting fault characteristic quefrency, which may be severely affected by noise. However, modal information also contains important fault features, which should be fully utilized.

As above mentioned, the sparse dictionary needs to reflect signal characteristics to the greatest extent, especially under low signal-to-noise ratio (SNR) conditions. The more similar the dictionary atom is to the original signal, the better the sparse decomposition effect gets. For the vibration signal of faulty rolling bearing, its transient impact responses are determined by modal parameters including natural frequencies and damping ratios. The cepstrum technique provides an effective approach to identify modal parameters directly from the collected vibration signal. However, the identified parameters are still affected by noise, which needs improvement. To solvethese problems, a novel method combining cepstrum technique and sparse decomposition is proposed to extract fault features of rolling bearings. The proposed method can exact more accurate modal parameters under low SNR and establish high-precision sparse dictionary for bearing fault diagnosis. A brief flow chart of the proposed method is shown in Fig. 1. In the process of sparse dictionary design, the extraction precision of transfer function is improved through removing the interference of excitation force based on the deconvolution property of editing cepstrum; Multi-order modal parameters are identified by the rational fractional polynomial fitting (RFPF) method to take full use of the fault information. The sparse coefficients are solved by a segmental matching pursuit (MP) algorithm to speed feature reconstruction efficiency.

In rest sections, Section 2 introduces the vibration signal model of rolling bearings; Section 3 gives the dictionary design method based on edited cepstrum, as well as the novel bearing fault diagnosismethod using sparse representation; Section 4 is the simulation verifications; Section 5 is the experiment tests and comparisons with an improved SK method and an edited cepstrum method. Section 6 is discussions and conclusions.

Section snippets

Vibration signal model of rolling bearing faults

Since rollingbearing fault generates transient impact force whose spectrum is broadband, multiple natural frequencies of the bearing system will be excited out, thus each single impulse response signal should be characterized by multi-order modal parameters. Considering repetitive occurrences of impact force along with the rotation speed, the faulty vibration signal collected by the sensor mounted on the bearing pedestal could be represented by Eq. (1) [10].x(t)=iIjJAijexp[2πξj1ξj2fdj(tτii

Impact feature extraction of rolling bearing based on MP

MP is a simple and effective approximate solution method to sparsely decompose a signal. It selects the best matched atoms from the redundant dictionary to linearly construct feature components in the original signal [20]. In other words, a signal can be decomposed into a linear superposition of atoms dr0(dr0=1)in the dictionary D ∈ Rn × q. After each greedy search, an atom dr0 that best matches the signal can be obtained, and the original signal x can be decomposed by Eq. (5).x=|x,dr0|dr0+R

Simulation analysis

To verify the effectiveness of the proposed method, a noiseless simulation signal of rolling bearingis first analyzed, and then Gaussian white noises with different SNRs are added into prior simulation signal to research its anti-noise performance and robustness. Comparison analyses with an improvedspectral kurtosis method and an edited cepstrum methodare conducted to illustrate the advantage of the proposed method.

Outer race fault on a pure rolling bearing

The experimental device and the faulty outer race are shown in Fig. 14. The bearing designation is N205M (pitch diameter D = 38 mm, ball diameter d = 6.5 mm, ball number z = 13, contact angle α = 0°). An artificial fault (width 0.2 mm and depth 0.5 mm) was produced on the outer race of the bearing. The rotation speed of the shaft was set as 800 rpm, so fn = 13.33 Hz and the fault characteristic frequencies of outer race and inner race fo = 71.84 Hz and fi = 101.52 Hz, respectively. The

Discussion and Conclusion

A novel edited cepstrum combined with MP method is proposed to extract impact features of rolling bearing faults. Taking advantage of cepstrum's deconvolution property, the FRF of bearing is accurately recognized. The RFPF method is utilized to fit the recognized FRF for calculating the specific numerical value of modal parameters. Based on the identified multiple-order modal parameters, the impact sparse dictionary is constructed with higher precision and much closer to the practical signal. A

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was partly supported by the National Natural Science Foundation of China (No. 51705156 and 51875207); the Natural Science Foundation of Guangdong Province-China (No. 2017A030310557 and 2018A030313947); and the key project of Guangzhou Science and Technology Program-China (No. 201904010387). Additionally, thanks for the support of CSC scholarship.

References (25)

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    Citation Excerpt :

    The frequency-domain methods and time-frequency reconstruction methods usually decompose the signal by constructing a filter bank. However, the parameters of the filter bank in these methods are needed to be pre-set (such as the decomposition level and basis function of wavelet transform decomposition [6], the decomposition level and penalty coefficient of [7] the sparse dictionary of the sparse decomposition [8] and the decomposition level and filter order of the 1/3-binary tree filter bank used in kurtogram [9]) and this nonadaptive approach limits their decomposition performance in nonlinear and nonstationary vibration signals. The time-domain decomposition methods directly decompose the signal by designing an adaptive iterative sifting algorithm and perform better in decomposing strong nonlinear and nonstationary vibration signals [10].

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