Rolling bearing plays an important role in the overall operation of the mechanical system; therefore, it is necessary to monitor and diagnose the bearings. Kurtosis is an important index to measure impulses. Fast Kurtogram method can be applied to the fault diagnosis of rolling bearings by extracting maximum kurtosis component. However, the final result may disperse the effective fault information to different frequency bands or find wrong frequency band, resulting in inaccurate frequency band selection or misdiagnosis. In order to find the maximum component of kurtosis accurately, an algorithm of frequency band multidivisional and overlapped based on EWT (MDO-EWT) is proposed in this paper. This algorithm changes the traditional Fast Kurtogram frequency bands division method and filtering method. It builds the EWT boundaries based on the maximum kurtosis component in each iteration and finally obtains the maximum kurtosis component. Through the simulation signal and the rolling bearing inner and outer ring fault signals verification, it is proved that the proposed method has a good performance on accuracy and effectiveness.
Modern mechanical equipment tends to be universal, and rolling bearing is one of the important parts of rotating machinery. Bearing damage will further lead to equipment failure or even greater danger. Therefore, it is necessary to conduct real-time monitoring and fault diagnosis of bearing operation state [
Periodic shocks can be reflected by the kurtosis value. In 1983, Dwyer [
Empirical wavelet transform (EWT) is a fast developing fault extraction method in recent years. EWT is used to construct the filter and reconstruct its components in this article. In recent years, many scholars have studied and optimized the empirical wavelet transform. Gilles [
To obtain the maximum kurtosis component, a new algorithm of frequency band multidivisional and overlapped based on EWT (MDO-EWT) is proposed in this paper. The method is to establish the boundaries of EWT filter according to the maximum kurtosis component in each iteration, and the final maximum kurtosis component can be obtained to extract the fault information. It improves the problems of fast kurtogram in frequency domain boundary division, such as rigid division, and it is easy to be affected by noise. Compared to the FK method, the results showed the proposed method is more effective and accurate. The contents followed are organized as follows: Section
For nonstationary processes, the calculation of spectral kurtosis depends on the selection of frequency resolution. Spectral kurtosis can be regarded as a function Obtain the corresponding component of the maximum kurtosis and conduct envelope demodulation for fault diagnosis.
Frequency-band division using the 1/3-binary tree filter bank.
In order to elaborate the deficiency of the FK method, the following simulated signal with equal intervals and noise are constructed. It is defined as the following equation, and the parameters of the simulated signal are shown in Table
Parameters of the simulation signal.
Natural frequency ( |
Damping factor | Fault characteristic frequency | Noise |
---|---|---|---|
|
|
100 Hz | SNR = −8 db |
Suppose the sampling frequency is 12000 Hz, and take The maximum kurtosis component is just noise, and no effective information can be obtained (Figures The maximum kurtosis component can be found inaccurately. The corresponding frequency band is too narrow, leading to the dispersion of the real frequency band into different frequency bands (Figures
(a) Results by the FIR filter; (b) spectrum and the division method of FIR mode; (c) results by the STFT filter; (d) spectrum and the division method of the STFT mode.
In view of the above shortcomings of FK, the article proposes an algorithm of frequency band multidivisional and overlapped by using EWT for filtering, which is more accurate and effective to obtain the maximum kurtosis component. The flowchart of the proposed method is shown in Figure Define the original signal as Divide the frequency band into 6 segments uniformly and merge each segment with the next segment. At this time, the frequency band is divided into 5 overlapping segments. Select Meyer wavelet as the basis function, and then use the corresponding scaling function and empirical wavelet to establish two sets of adaptive filter. We can get 5 segments and 5 reconstructed time-domain components after this. Calculate kurtosis of all the reconstructed time-domain components, respectively. Then, the corresponding frequency band of the maximum kurtosis component and the frequency band overlapped with it are found. They are regarded as the new frequency band to be divided. At this time, According to the cyclic cutoff condition, the envelope spectrum is obtained for the finally found maximum kurtosis component, and corresponding fault information is extracted.
The flowchart of the proposed method.
When
(a)The division of the spectrum when
When
(a)Kurtosis polyline graph and new frequency band to be divided; (b) the division of the spectrum when
In each iteration, the frequency band corresponding to the maximum kurtosis component can be summarized into two categories. Category 1: frequency band
The empirical wavelet transform is actually the adaptive division of the Fourier spectrum and the establishment of a set of wavelet filters suitable for the signal to be processed. The empirical wavelet transform method can be simply described in the following three steps:
And, the transition function
The approximate coefficients
After obtaining the empirical mode of the signal, the Hilbert transform can be used to diagnose faults.
The FK method cannot correctly find the signal with center frequency as
The equal-interval impulse signal
Parameters of simulation signal in case study 1.
Natural frequency ( |
Damping factor | Fault characteristic frequency | Noise |
---|---|---|---|
|
|
100 Hz | SNR = −10 db |
(a) Red: the impulsive signal; blue: the simulated signal mixed with impulsive signal and white Gaussian noise with SNR −10 dB; (b) the spectrum of the simulated signal.
The mode FIR is used to deal with the signal (
(a) Fast kurtogram based on FIR; (b) signal component and its envelope sequence.
(a) Fast kurtogram based on STFT; (b) signal component and its envelope sequence.
Figure
(a) Time-domain waveform diagram of the signal component with the maximum kurtosis found by the new method and its frequency band; (b)its envelope sequence.
The collected engineering signals may contain similar nonimpulse but edge-frequency modulated signals in other frequency bands, which may cause interference. In the case study 2, a modulating signal
Parameters of the simulation signal in case study 2.
Natural frequency ( |
Signal components | Fault characteristic frequency | Noise |
---|---|---|---|
|
|
100 Hz | SNR = −12 db |
Figures
(a) The new impulsive signal
Figure
Fast kurtogram based on (a) FIR and (c) STFT; signal component and its envelope sequence by (b) FIR and (d) STFT.
Figure
(a) Time-domain waveform diagram of the signal component with the maximum kurtosis found by the new method and its frequency band and (b) its envelope sequence.
Simulated signals were used to verify the accuracy of the proposed method quantitatively from the point of signal energy. The original simulated signal can be described as the effective signal with noise, the effective signal energy accounts for the vast majority of the total energy, and the noise energy accounts for a small part. After the extracting process, the component obtained under ideal conditions should be all effective signals plus a small amount of noise; due to the presence of a small amount of noise, the energy of the extracted signal must be higher than the pure effective signal energy. If the extracting component is less effective, the following two situations can occur: Partly effective signal is extracted Failing to extract the effective signal, and the extracted part is all noise
When the extracted signal energy in situation (1) is compared to the pure effective signal energy, due to the presence of noise, we cannot tell whose energy is higher. However, when compared with the extracted signal energy at ideal extraction conditions, the former signal energy must be lower than the latter, since the effective part in situation (1) is less and the effective signal accounts for the vast majority of the total energy. In situation (2), the extracted signal (noise) energy must be lower than the pure effective signal energy.
The discrete signal energy can be expressed by the sum of the squares of the discrete points. According to the SNR formula, the energy ratio (ER) is defined as follows:
From the energy point of the simulated signal, we can conclude if the extraction effect is better, the energy ratio is higher; and ER must be < 1 in situation (2), ER > 1 in ideal situation, and ER (ideal situation) > ER (situation (2)).
The energy ratio histograms (Figure
(a) Energy ratio histogram of case 1 by three methods; (b) energy ratio histogram of case 2 by three methods.
In Figure
In this paper, two groups of experimental data were used to preliminarily verify the effectiveness of the proposed method. The first group was the bearing outer ring fault data, and the second group was the bearing inner ring fault data. The analysis and comparison with FK methods are given here in detail.
In this part, the test data of faulty bearing from Xi’an Jiaotong university laboratory were used. Figure
Bearing fault test rig.
Parameters of the outer ring fault bearing.
Fault characteristic frequency | Ball diameter | Pitch diameter | Number of ball |
---|---|---|---|
101.6 Hz | 0.3125 in | 1.318 in | 8 |
Sampling conditions.
Sensor position | Sampling frequency | Sampling points | Motor Speed (r/min) |
---|---|---|---|
Motor end | 12000 Hz | 24000 | 2000.9 |
Figure
(a) Original signal; (b) frequency spectrum.
Figures
(a) Fast kurtogram based on FIR; (b) signal component and its envelope sequence.
(a) Fast kurtogram based on STFT; (b) signal component and its envelope sequence.
The proposed method is used to process the signal, and the final result is shown in Figure
(a) Time-domain waveform diagram of the signal component with the maximum kurtosis found by the new method and its frequency band and (b) its envelope sequence.
The failure data of bearing inner ring were collected in the laboratory of Beijing university of technology by the failure test bench as shown in Figure
Bearing fault test rig.
Parameters of the inner ring fault bearing.
Fault characteristic frequency | Ball diameter | Pitch diameter | Number of ball |
---|---|---|---|
122 Hz | 0.5313 in | 2.286 in | 8 |
Sampling conditions.
Sensor position | Sampling frequency | Sampling points | Motor Speed (r/min) |
---|---|---|---|
Shaft end | 15360 Hz | 8192 | 2000.9 |
Figure
(a) Original signal; (b) frequency spectrum.
Similarly, the two methods of FK are used to process the signal, and the results are shown in Figure
(a) Fast kurtogram based on FIR; (b) signal component and its envelope sequence; (c) fast kurtogram based on STFT; (d) signal component and its envelope sequence.
To further illustrate the advantages of the proposed method, Figure
(a) Time-domain waveform diagram of the signal component with the maximum kurtosis found by the new method and its frequency band and (b) its envelope sequence.
The accuracy of the proposed method is further verified by using 48 groups of the bearing inner and outer ring failure data published by Case Western Reserve University. The experimental data are obtained by artificially manufacturing damage to different parts.
Outer ring data were selected from “Drive End Bearing Fault Data (fault diameter: 0.007in),” and the data sampling frequency of this group is
Parameters of the bearing.
Type | Inside diameter | Outside diameter | Ball diameter | Pitch diameter |
---|---|---|---|---|
6205-2RS | 0.9843 in | 2.0472 in | 0.3126 in | 1.537 in |
Based on any difference of the following conditions: motor speed, fault position, and sensor position, we selected and obtained 24 groups of outer ring fault signals (number of signal points per group,
Different conditions statement of 24 groups data.
Motor Speed (rpm/min) | Fault position | Sensor position |
---|---|---|
1797 | Centered to loading area | Drive end |
1772 | Orthogonal to loading area | Fan end |
1750 | Base | |
1730 |
We used MOD-EWT, FK-FIR, and FK-STFT to deal with the above data, respectively. Figure
Number of groups with (a) diagnostic capability by using the results of three methods and (b) better diagnostic capability by using the results of three methods.
As can be seen from Figure
The inner ring data were selected from “Fan End Bearing Fault Data (fault diameter: 0.007 in and 0.021 in).” The data sampling frequency of this group is
Parameters of the bearing.
Type | Inside diameter | Outside diameter | Ball diameter | Pitch diameter |
---|---|---|---|---|
6205-2RS | 0.6693 in | 1.5748 in | 0.2656 in | 1.122 in |
Based on any difference of the following conditions, fault diameter, motor speed, and sensor position, we selected and obtained 24 groups of outer ring fault signals (number of signal points per group,
Different conditions statement of 24 groups data.
Fault diameter | Motor speed (rpm/min) | Sensor position |
---|---|---|
0.007 | 1797 | Drive end |
0.021 in | 1772 | Fan end |
1750 | Base | |
1730 |
In the same way as dealing with the inner ring data, the histograms of the three methods are obtained as given in Figure
Number of groups with (a) diagnostic capability by using the results of three methods and(b) better diagnostic capability by using the results of three methods.
Due to the complexity of the inner ring fault signal and the influence of noise, the inner ring diagnostic effect is generally lower than that of the outer ring. Figure
In this paper, a novel bearing fault diagnosis method is proposed, which uses EWT for filtering. The proposed algorithm of frequency band multidivisional and overlapped (MDO-EWT) is to replace the FK bisection method. By verification of simulated signals, it is shown that the proposed method can correctly find the signal with center frequency as
In the future, we will be committed to the optimization of the algorithm, so that its use can be extended.
The experimental data [.txt] and [.mat] used to support the findings of this study were supplied by the research group of Key Laboratory of Advanced Manufacturing Technology under license and so they cannot be made publicly available, and the requests for access to these data should be made to the corresponding author.
The authors declare no conflicts of interest related to this manuscript.
The authors would like to gratefully acknowledge the National Natural Science Foundation of China (grant nos. 51775005 and 51675009) and the Key Laboratory of Advanced Manufacturing Technology for their support. Finally, the authors would like to appreciate anonymous reviewers and editors for their valuable comments and constructive suggestions. Funding was provided by the National Natural Science Foundation of China (grant nos. 51775005 and 51675009).