Fractional lower order linear chirplet transform and its application to bearing fault analysis

Junbo Long; Haibin Wang; Hongshe Fan; Zefeng Lao

doi:10.1371/journal.pone.0276489

Abstract

The amplitude and frequency of the mechanical bearing fault vibration signals vary with time, and which are non-stationary and non-Gaussian process. The fault signals belong to α stable distribution, and the characteristic index 1 < α < 2, even the noises are α stable distribution in extreme cases. The existing linear chirplet transform (LCT) degenerates, even fails under α stable distribution environment. A fractional low order linear chirplet transform (FLOLCT) which takes advantage of fractional p order moment is presented for α stable distribution noise environment, and the corresponding FLOLCT time-frequency representation (FLOLCTTFR) is developed in this paper. By employing a series of polynomial chirp rate parameters instead of a single chirp rate of the FLOLCT method, a fractional low order polynomial linear chirplet transform (FLOPLCT) is developed to improve time frequency concentration of the signals. The improved FLOLCT and FLOPLCT methods are used to compare with the existing LCT and PLCT methods based on second order statistics, the results reveal performance advantages of the proposed methods. Finally, the FLOLCT and FLOPLCT methods are applied to analyze the fault signature of the bearing ball fault data in the position of DE (Drive end accelerometer) and extract their fault signature, the result illustrates their performances.

Citation: Long J, Wang H, Fan H, Lao Z (2022) Fractional lower order linear chirplet transform and its application to bearing fault analysis. PLoS ONE 17(10): e0276489. https://doi.org/10.1371/journal.pone.0276489

Editor: Yiming Tang, Hefei University of Technology, CHINA

Received: June 20, 2022; Accepted: October 7, 2022; Published: October 21, 2022

Copyright: © 2022 Long et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All files are available from https://engineering.case.edu/bearingdatacenter/.

Funding: This work was supported by the Natural Science Foundation of China (61962029), the Natural Science Foundation of Jiangxi Province China(20192BAB207002), and Science and Technology Project of Jiangxi Provincial Health Commission (202211976, 202131072).

Competing interests: The authors have declared that no competing interests exist.

Introduction

Real-time monitoring and fault diagnosis of rotating machinery is an unusual means for normal industrial production. The amplitude and frequency of the mechanical bearing fault vibration signals vary with time, and which are non-stationary. Time-frequency distribution is an effective tool to deal the non-stationary signals, which has been applied for analyzing and processing the bearing fault data [1–3]. A time extraction method was proposed based on S transform to effectively process the pulse signals, which could clearly display the occurrence time of the vibration pulse of rolling bearing fault on the premise of ensuring high time-frequency aggregation [4]. A deformable convolutional neural network time-frequency analysis method was proposed in [5], which could achieve high-precision fault classification for the bearing fault signals. Aimming at the variable speed signals containing multiple faults, Tang et al. proposed a multiple time-frequency curve classification method and classification standard for the fault detection of composite bearings with multiple faults under the condition of time-varying speed without speed resampling [6]. Gültekin et al. used time-frequency images obtained by short-time Fourier transform and multi-perception data fusion based on deep residual network to diagnose the mechanical fault signals, the method combined with different types of measurement signal models can monitor the bearing status more effectively [7].

In time-frequency analysis of the mechanical fault signals, the traditional short time Fourier transform (STFT) time frequency representation method has a low time-frequency resolution [8, 9]. Wavelet Transform (WT) time frequency representation method can take into account both time-domain and frequency-domain resolution, but the limited length of wavelet basis function will cause energy leakage, and it is difficult to obtain accurate time-frequency analysis. Limited by Heisenberg Gabor inequality, the time and frequency of the continuous wavelet transform (CWT) time frequency representation method can not achieve high resolution at the same time, high frequency resolution is necessarily at the expense of time resolution [10, 11]. However, Wigner Ville distribution (WVD) is a bilinear time-frequency analysis method with high time-frequency resolution, but it is easy to lead in cross terms [12]. Steve and Simon first proposed the concept of linear chirplet transform (LCT) based on wavelet transform [13], whereafter, they applied LCT time-frequency representation to a practical problem in Marine radar and achieved good analysis results [14]. The chirp rate of the LCT method is invariable, hence, many experts and scholars have presented many improved post-processing algorithms based on the traditional LCT in recent years [15–23]. Yu Gang et al. proposed a general linear chirplet transform (GLCT) method, but which has a fixed window length, and its resolution is settled [15]. Guan Yunpeng et al. proposed a robust adaptive velocity synchronous linear chirplet transform (VSLCT) time-frequency representation method, which utilized a time-varying window function to improve time-frequency resolution, at the same time, a set of linear frequency modulation wavelets was used to eliminate the smear effect [16]. A polynomial linear chirplet transform (PLCT) time frequency representation method and real-time frequency estimation method using polynomial chirp rate instead of the traditional fixed chirp rate were proposed in [17], which effectively improved the accuracy of frequency estimation. Afterwards, a time-frequency fusion method based on PLCT was proposed in [18], which can characterize the time frequency structure of the signals, nd has better energy concentratioan for the multicomponent signals. The time frequency resolution of PLCT can be improved by synchronous extrusion, and an improved synchrosqueezing polynomial chirplet transform method was proposed [19, 20]. A component matching chirplet transform method based on frequency-dependent chirp rate was proposed in [21], which generates a set of chirplet to match the fault frequency components, so as to realize high resolution time-frequency fault diagnosis. The polynomial estimation of the PLCT method is easily influenced by noise or interference in mechanical fault analysis, so the chirplet transform based on nuclear ridge regression was proposed in [22], which can accurately characterize the time-frequency characteristics of the non-stationary signals and generate the time frequency plane with energy concentration. A group filter-matched signal reconstruction method was proposed using the approximation of source signals with linear chirps at any local time [23].

Due to the repeated transient characteristics caused by local damage, it is easy to be drowned by various interference components and strong noise, so early identification and diagnose of the rolling bearing faults is still difficult. Recently, it is verified that probability density functions (PDFs) of the mechanical bearing fault vibration signals have obviously trailing process, they belong to non-stationary and non-Gaussian distribution α stable distribution (1<α<2), even the strong noises are also α stable distribution [24–26]. The performance of the above-mentioned methods based on second-order statistics degenerates under α stable distribution environment, even which fails.

Aiming at the fractional low-order time-frequency distribution methods in α stable distribution environment, linear fractional low order short time Fourier transform (FLOSTFT) time-frequency representation method has no cross term interference, but its resolution is not high [27]. The resolution of the linear fractional low order Stockwell transform (FLOST) time-frequency representation method is improved to some extent [28]. Bilinear class fractional low order Wigner Ville distribution (FLOWVD) and fractional low order Wigner Ville distribution (FLOPWVD) time-frequency representation methods have higher time frequency resolution, but there is some cross term interference, the bilinear class fractional low order adaptive kernel time-frequency representation can suppress the cross term interference to some extent, parameter class fractional low order autoregressive (FLOAR), fractional low order moving average (FLOMA) and fractional low order autoregressive moving average (FLOARMA) time-frequency methods have no cross term, but the calculation is large and their resolution is not high [29, 30]. Hence, we propose the improved franctional low order linear chirplet transform (FLOLCT) and franctional low order polynomial linear chirplet transform (FLOPLCT) methods inspired by the LCT and PLCT methods in this paper, the corresponding franctional low order linear chirplet transform time frequency representation (FLOLCTTFR) and franctional low order polynomial linear chirplet transform time frequency representation (FLOPLCTTFR) are introduced. The improved FLOLCT and FLOPLCT methods and the traditional LCT and PLCT methods employing second order statistics are compared under Gaussian and α stable distribution environments. The simulation results show that the proposed FLOLCTTFR and FLOPLCTTFR methods have performance advantages and higher time frequency resolution, and can better be suitable for the impulse noise environment than the LCT and PLCT methods. The estimated MSEs of the polynomial parameters employing the FLOPLCT time frequency representation method are smaller than the PLCT method under different α(α < 2) and MSNR. Finally, we apply the improved FLOLCTTFR and FLOPLCTTFR methods to analyze the bearing ball fault in the position of DE data contaminated by Gaussian noise and α stable distribution, the results indicate that the performance of the improved methods is better than the existing methods, and which are feasible, effective and robust for fault diagnosis.

In this paper, the improved FLOLCTTFR and FLOPLCTTFR technologies based on fractional lower order statistics and LCT are proposed for the bearing ball fault in the position of DE data analysis under Gaussian and α stable distribution environment. The paper is structured in the following manner. α stable distribution description, bearing fault data and their α stable distribution model parameter estimation are introduced in section 2. The improved fractional lower order linear chirplet transform and fractional lower order polynomial linear chirplet transform methods are demonstrated in Section 3, and simulation comparisons with the existing LCT and PLCT methods based on second order statistics are performed to demonstrate superiority of the improved methods. Applications of the improved methods for the bearing ball fault data in the position of DE are demonstrated in section 4. Finally, conclusions and future research are given in Section 5.

α stable distribution and bearing fault data

α stable distribution description

The characteristic function of α stable distribution can be written by (1) where α is the characteristic index, when α = 2, Eq (1) degenerate into Gaussian distribution, hence which is often called the generalized Gaussian distribution. μ, γ and β are the location parameter, the dispersion coefficient and the symmetry parameter, respectively. If μ = 0 and γ = 1, the α stable distribution is called the standard α stable distribution, and if μ = 0, γ = 1 and β = 0, it is called the standard symmetric α stable distribution (SαS). Probability density functions (PDFs) of SαS stable distribution under α = 0.5, 1.0, 1.5 and 2.0 are shown in Fig 1. It can be seen that the smaller the characteristic index α, the stronger the impulsivity, and their PDFs have a long tail.

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Fig 1. PDFs of SαS stable distribution under α = 0.5, 0.75, 1.25, 1.75 and 2.0.

(a. Full PDFs diagram; b. Local trailing PDFs diagram).

https://doi.org/10.1371/journal.pone.0276489.g001

Bearing fault data and their α stable distribution parameter estimation

The real bearing fault data are gotten from the Case Western Reserve University (CWRU) bearing data center [31], and the data includes the normal bearings data, drive end bearing fault data and fan end bearing fault data. The fault points include inner race fault, ball fault, and outer race position relative to load zone centered at 6:00, 3:00 and 12:00. The fault diameter is 0.021 inches, the sampling frequency of the data is 12000Hz, and the motor speed is set at 1746–1752 revolutions per minute (rpm). The acceleration data is measured at locations near to base accelerometer (BA), the drive end accelerometer (DE) and fan end accelerometer (FE).

The normal and drive end bearing fault data are selected as the test signals in this paper. The waveforms of the normal and drive end bearing fault data are shown in Fig 2. We apply α stable distribution statistical model in Eq (1) to estimate the α stable distribution parameters of the normal signals and the drive end bearing fault signals including the inner race fault, ball fault and outer race fault data, the results are given in Table 1. The result shows that the parameters α of the normal BE and FE data and the ball BA and FE fault data α = 2, and they belong to Gaussian distribution. However, the parameters α of the inner race fault and outer race position relative to load zone centered at 6:00, orthogonal at 3:00 and opposite at 12:00 α < 2, they are α stable distribution.

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Fig 2. The waveforms of the normal and drive end bearing fault data.

(a. The waveforms of the normal DE and FE data; b. The waveforms of the inner race BA, DE and FE fault data; c. The waveforms of the ball BA, DE and FE fault data; d. The waveforms of the outer race BA, DE and FE fault data position relative to load zone centered at 6:00; e. The waveforms of the outer race BA, DE and FE fault data position relative to load zone orthogonal at 3:00; f. The waveforms of the outer race BA, DE and FE fault data position relative to load zone opposite at 12:00.

https://doi.org/10.1371/journal.pone.0276489.g002

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Table 1. α stable distribution parameter estimation of the normal and drive end bearing fault data.

https://doi.org/10.1371/journal.pone.0276489.t001

PDFs of the normal and drive end bearing fault data in Table 1 are shown in Fig 3. It is seen that PDFs of the inner race fault and outer race data have long trails, and their pulse characteristics are quite remarkable. But PDFs of the normal data and ball fault data have no tails and pulse characteristics.

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Fig 3. PDFs of the normal data and drive end bearing fault data.

(a-b. PDFs of the normal data; c-d. PDFs of the inner race BA, DE and FE fault data; E-F. PDFs of the ball BA, DE and FE fault data; g-h. PDFs of the outer race BA, DE and FE fault data position relative to load zone centered at 6:00; i-j. PDFs of the outer race BA, DE and FE fault data position relative to load zone orthogonal at 3:00; k-l. PDFs of the outer race BA, DE and FE fault data position relative to load zone opposite at 12:00.).

https://doi.org/10.1371/journal.pone.0276489.g003

Fractional lower order linear chirplet transform methods

FLOLCT time-frequency representation method

Principle.

We have proposed fractional lower order short time Fourier transform (FLOSTFT) of a signal x(τ) in [29] (2) where < p > denotes p order moment of x(τ), p is a real coefficient and α < p ≤ 2, α is characteristic index of α stable distribution noise, and , . The corresponding fractional lower order short time Fourier transform time-frequency representation (FLOSTFTTFR) of the signal x(τ) can be written by (3)

According to the definition of the FLOSTFT in Eq (2) and liner chirplet transform (LCT) [13, 14], we define fractional lower order liner chirplet transform (FLOLCT) as (4) (5) (6) where, the parameters c and t denote the chirp rate and time, respectively. stands for a windowed fractional lower p order moment function, and h_σ(τ—t) is a real window function. Ψ_{(t, c)}(τ) is the discrete demodulated function. Fractional low order linear chirplet transform is the extension of fractional low order short time Fourier transform, and it has the same form of Fourier transform calculation as the fractional low order short time Fourier transform. FLOLCT is usually obtained by Gaussian frequency FM wavelet packet, so it is often called fractional low order Gaussian linear chirplet transform.

Eq (6) can be written by (7) where e^jctτ is frequency shift operating factor, and its amplitude is ct. is frequency rotation operation factor, and its rotation angle is θ = arctan(-c).

The corresponding fractional lower order linear chirplet transform time frequency representation (FLOLCTTFR) can be written by (8) (9)

The value of the chirp rate c directly affects energy concentration of the analytical signal x(τ) in time-frequency domain. The closer c is to the actual frequency of the analytical signal, the higher the energy concentration is in time-frequency domain. When the chirp rate c = 0, then ψ_{(t, c)}(τ) = 1 and , Eq (4) changes into Eq (3), and the FLOLCT time-frequency method degrades to FLOSTFT time-frequency method. When p = 2, FLOLCT in Eq (4) degenerates into LCT, and FLOCTTFR in Eq (8) changes into LCTTFR. Hence, the LCT method is a special case of the FLOLCT method, and FLOLCT is a generalized LCT.

Application review.

In this simulation, the mixed signal contaminated by v(t)(Gaussian or SαS stable distribution noise) is defined as (10)

If v(t) is Gaussian noise, we apply signal to Gaussian noise ratio (SNR), but if v(t) is α stable distribution noise, there is no finite second moment, the variance of α stable distribution noise becomes meaningless, and SNR is unusable. So mixed signal noise ratio (MSNR) or generalized signal noise ratio (GSNR) can be used to replace SNR, and MSNR and GSNR be written by and , respectively. Where σ_x is variance of the signal x(τ), N is sample size, and γ is dispersion coefficient of the α stable distribution noise. In this paper, we use MSNR to perform the following experiments.

We apply the existing LCT (c = 6) method and the improved FLOLCT (c = 6, p = 1.2) method to estimate the time-frequency representations of the signal x(t)(Eq 10) in Gaussian noise environment (SNR = 6dB) and α stable distribution noise environment (MSNR = 16dB, α = 0.8), the simulation results are shown in Figs 4 and 5.

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Fig 4. Time-frequency representations of the signal x(τ) in Gaussian noise environment (c = 6, SNR = 6dB).

(a. LCT time-frequency representation of the signal x(τ); b. FLOLCT time-frequency representation of the signal x(τ)).

https://doi.org/10.1371/journal.pone.0276489.g004

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Fig 5. Time-frequency representations of the signal x(τ) in α stable distribution noise environment (MSNR = 16dB, α = 0.8).

(a. LCT time frequency representation of the signal x(τ); b. FLOLCT time-frequency representation of the signal x(τ)).

https://doi.org/10.1371/journal.pone.0276489.g005

Remarks.

Fig 4A and 4B are the time frequency representations of the signal x(τ) employing LCT (c = 6) and FLOLCT (c = 6, p = 1.2) in Gaussian noise environment (SNR = 6dB), respectively. The LCT and FLOLCT time frequency distributions of the signal x(τ) are shown in Fig 5A and 5B in α stable distribution noise environment (α = 0.8, MSNR = 16dB and p = 1.2), respectively. It can be seen that the LCT method can only work in Gaussian noise environment, but the FLOLCT method can work in Gaussian noise environment and α stable distribution noise environment, and which has good toughness.

The FLOLCT can provide a well-concentrated time frequency representation for the liner modulated signals, the FLOLCT method has better resolution for the multi frequency modulation signals than the FLOSTFT method because of the use of chirp rate c. However, the time frequency resolution of the FLOLCT method is controlled by the chirp rate c, the closer the chirp rate c is to the true instantaneous frequency of the signal, the higher energy concentration of the signal in time-frequency domain. Aimming at the signals containing time varying frequency component, their energy concentration are not satisfactory because that the FLOLCT method has immobile chirp rate c, as shown in Figs 4B and 5B. In the part of time 0–8 seconds, the frequency resolution of the signal x(τ) is very high, but the signal x(τ) has very low time-frequency resolution in the part of time 8–14 seconds.