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Application of an improved kurtogram method for fault diagnosis of rolling element bearings
Mechanical Systems and Signal Processing 25 (2011) 1738–1749
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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp
Application of an improved kurtogram method for fault diagnosis of rolling element bearings Yaguo Lei n, Jing Lin, Zhengjia He, Yanyang Zi State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, China
a r t i c l e in f o
abstract
Article history: Received 26 July 2010 Received in revised form 13 December 2010 Accepted 21 December 2010 Available online 8 January 2011
Kurtogram, due to the superiority of detecting and characterizing transients in a signal, has been proved to be a very powerful and practical tool in machinery fault diagnosis. Kurtogram, based on the short time Fourier transform (STFT) or FIR filters, however, limits the accuracy improvement of kurtogram in extracting transient characteristics from a noisy signal and identifying machinery fault. Therefore, more precise filters need to be developed and incorporated into the kurtogram method to overcome its shortcomings and to further enhance its accuracy in discovering characteristics and detecting faults. The filter based on wavelet packet transform (WPT) can filter out noise and precisely match the fault characteristics of noisy signals. By introducing WPT into kurtogram, this paper proposes an improved kurtogram method adopting WPT as the filter of kurtogram to overcome the shortcomings of the original kurtogram. The vibration signals collected from rolling element bearings are used to demonstrate the improved performance of the proposed method compared with the original kurtogram. The results verify the effectiveness of the method in extracting fault characteristics and diagnosing faults of rolling element bearings. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Kurtogram Wavelet packet transform Rolling element bearings Fault diagnosis
1. Introduction Rolling element bearings are widely used in modern machinery, and faults occurring in the bearings may lead to fatal breakdown of machines. Therefore, it is significant to be able to accurately detect and diagnose the existence of the faults occurring in the bearings. Vibration signals collected from bearings carry rich information on machine health conditions. Therefore, the vibration-based methods have received intensive study during the past decades. It is possible to obtain vital characteristic information from the vibration signals through the use of signal processing techniques [1]. In order to effectively diagnose faults occurring in rolling element bearings, researchers have extensively investigated different signal processing techniques to accurately extract fault characteristics from vibration signals. Since the envelope analysis focuses on the low-amplitude high-frequency broadband signals characterizing bearing conditions and may minimize the effects of interfering signals within the selected frequency band, it has been widely applied in detecting faults of rolling element bearings [2]. Spectral kurtosis (SK), as an envelope analysis technique, was originally presented by Dwyer [3] to complement the classical power spectral density in detecting and characterizing transients in a signal. The basic idea of this technique is to exploit the possibility of using the kurtosis at each frequency line as a measure to discover the presence of non-Gaussian components and to indicate in which frequency bands these occur. Such non-Gaussian
n
Corresponding author. E-mail address: [email protected] (Y. Lei).
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components are the structural responses to an excitation caused by, e.g. a local flaking on the outer race in the case of bearing faults. Additionally, SK can determine the frequency of the excited component. Therefore, SK can indicate not only the non-Gaussian components in signals, but also their locations in the frequency domain. Subsequently, Antoni [4,6] and Antoni and Randall [5] made a very thorough and interesting study on SK; first they generalized it and proposed the definition of the ‘‘kurtogram’’ to a wider class of non-stationary signals. Because of the superiority in extracting transient components masked in a noisy signal, kurtogram has been proved to be a very powerful and practical tool in machinery fault diagnosis [6–9]. Kurtogram based on the short time Fourier transform (STFT) or FIR filters, however, limits the performance improvement of the kurtogram in extracting transient characteristics from a noisy signal and identifying machinery faults. To overcome the above shortcomings of kurtogram and to further enhance its accuracy in discovering characteristics and detecting faults, more precise filters need to be developed and incorporated into the kurtogram algorithm. As a special filter, WT possesses good local property in both time and frequency spaces [10], and therefore it has been studied extensively and applied successfully to vibration signal analysis for fault diagnosis of rolling element bearing fault diagnosis [11]. WT provides evident advantages in filtering out noise and keeping the transient characteristics if the wavelet used is similar to the feature components hidden in a signal. Therefore, some researchers have adopted WT as filters to implement and improve the kurtogram. For example, Shi et al. [12] used the complex Morlet wavelet transform to implement kurtogram and proposed fault diagnosis method of rolling element bearings. This method can find the best frequency band for demodulation automatically by the criterion of SK based on the complex Morlet wavelet transform [12]. Sawalhi and Randall [13] presented a two-step technique to optimize SK and got the kurtogram for diagnosing rolling element bearing faults using the complex Morlet wavelet transform to decompose bearing signals [13]. However, WT cannot effectively split the high frequency band (detail signal) where the fault modulation information always exists. The way of overcoming this difficulty is to extend WT to wavelet packet transform (WPT) [14]. Therefore, an improved method using WPT based on the Daubechies wavelet to implement kurtogram is proposed in this paper. The method adopts WPT as the filter to overcome the shortcomings of the original kurtogram based on STFT or FIR filters. In addition, compared with the methods based on the complex Morlet wavelet transform in Refs. [12,13], this improved kurtogram method has the following three advantages: (1) It is faster because the kurtogram based on the complex Morlet wavelet transform is of computational complexity. (2) It used WPT based on Daubechies wavelet being orthogonal, compact supported and approximately symmetric, and WPT guarantees feature extraction of transient characteristics without redundant or omitted information. (3) WPT may effectively split the high frequency band where the bearing fault modulation information exists. Thus, it is expected that a more accurate result of characteristic discovery can be obtained by the improved kurtogram method. Experiments on an experimental rolling element bearing and a real locomotive rolling element bearing are conducted to test the improved performance of the improved method compared with the original kurtogram. The results demonstrate the effectiveness of the method in extracting fault characteristics from noisy vibration signals.
2. Review of kurtogram and wavelet packet transform 2.1. Kurtogram Kurtogram was first introduced by Antoni and Randall [5], which comes from spectral kurtosis (SK) [3]. The SK method selects kurtosis as a measure of distance between an arbitrary random process and a Gaussian one with the aim of detecting the existence of transients in a signal. The SK value of the signal is obtained by calculating kurtosis of each frequency component contained in the signal. SK finally represents the transient characteristics of the signal as a function of frequency. SK can indicate not only transient components in the signal but also their locations in the frequency domain, and therefore overcome the disadvantages of the power spectral density in detecting and characterizing signal transients. Considering the Wold–Grame´r decomposition of a non-stationary signal, we can define signal y(t) as the response of a system with time varying impulse response h(t,s) excited by signal x(t). Then y(t) can be expressed as [4] Z þ1 yðtÞ ¼ ej2pft Hðt,f Þ dXðf Þ ð1Þ 1
where H(t,f) is the time varying transfer function of the considered system and can be interpreted as the complex envelope of signal y(t) at frequency f and dX(f) is the spectral process associated with x(t). In real cases of machinery vibration, H(t,f) is stochastic and should be presented as H(t,f,o), where o is the random variable presenting time variations of filters. When H is time stationary and independent of process x, we obtain a conditionally non-stationary process (CNS). SK is based on the fourth-order spectral cumulant of a CNS process. C4y ðf Þ ¼ S4y ðf Þ2S22y ðf Þ
ð2Þ
where S2y ðf Þ is the time-averaged result of S2y ðt,f Þ, S2y ðt,f Þ is an instantaneous moment and measures the energy of the complex envelope. Then, the SK is generated by normalizing the fourth-order cumulant given in Eq. (2). It is a measure of
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the peakiness of the probability density function Ky ðf Þ ¼
C4y ðf Þ S4y ðf Þ ¼ 2 2 S22y ðf Þ S2y ðf Þ
ð3Þ
The estimation of the SK values for particular frequency f is implemented by calculating the STFT of signal y(t) and is given by Eq. (4). Fy ðt,f Þ ¼
Z
þ1
yðtÞw ðttÞej2pft dt ¼ /yðtÞ,wðttÞej2pft S
ð4Þ
1
where w*(t t) represents the window function and superscript n denotes the conjugation. The STFT is actually an inner product operation /y(t),w(t t)e j2pftS of signal y(t) and filter w(t t)ej2pft, which is the triangle basis function ej2pft enveloped by window function w(t t). It is noted that the selected window length and the percent of overlap used for the STFT calculation significantly influence the value of SK. Therefore, for the purpose of selecting the optimal window length the spectral kurtosis is calculated for several different window lengths. The implementation result of these calculations is the so-called kurtogram. The window parameters for which the spectral kurtosis has its maximum value determine the band-pass filter parameters. Therefore, kurtogram is a 2D map and presents values of SK calculated for various parameters of frequency f and bandwidth Df. Let Kk,l be the SK value of the filtered signal issued from the lth filter at the kth decomposition level, and then the kurtogram paving of the (f/Df) plane is shown in Fig. 1. A more detailed explanation of kurtogram can be found in Refs. [5,6].
2.2. Wavelet packet transform Wavelets have perfect local property in both time and frequency spaces, and can be used as an effective filter to filter out noise and preserve signal characteristics [10]. WT, however, does not process the high frequency bands where the modulation information of machine fault exists. Wavelet packet transform (WPT) furthers decomposition of the high frequency bands and may generate a more precise frequency-band partition over the whole analyzed frequency bands. As a result, the frequency resolution may be enhanced. Thus, WPT can be implemented based on wavelet filters, which is
Levels 0
1/2
1
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k
1/2
K0, 1 K1, 1
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K3, 5
K2.6, 5 K3, 6
K3, 7
Kk, 1
Kk, l
f 0
1/8
1/4
(Δf)k
3/8
Fig. 1. Paving of the original kurtogram.
1/2
k+1
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defined by the following equation: 8 K > pffiffiffi X > > > Hn bi1,j þ n b ¼ 2 > < i,j n¼0
ð5Þ
K > pffiffiffi X > > > Gn bi1,j þ n > : bi,j þ l ¼ 2 n¼0
where bi,j denotes the jth decomposed frequency-band signal at level i (j ¼ 1,2,. . .,J, where J is the number of the decomposed frequency-band signals and equals 2I; i ¼ 1,2,. . .,I, where I is the number of decomposition levels); Hn and Gn are the low- and high-pass filters based on wavelets, respectively. 3. Proposed method After the WPT decomposition, a set of frequency-band signals is obtained. Then, kurtosis is calculated for each of the frequency-band signals according to the following equation: PN ðbi,j ðnÞmb Þ4 b Ki,j ¼ n¼1 ð6Þ ðN1Þs4b where mb, sb and N are the mean value, the standard deviation and the point number of the frequency-band signal bi,j, respectively. b Replacing Kk,l of the original kurtogram by Ki,j of WPT, we propose an improved kurtogram method. Correspondingly, the kurtogram paving of the (f/Df) plane of the proposed method is shown in Fig. 2. Employing WPT as the filter of kurtogram to overcome the disadvantages of the original kurtogram using the STFT or FIR filters, an improved kurtogram method is proposed in this paper. The flow chart of the proposed method is presented in Fig. 3. It contains the following procedural steps. First, the wavelet packet transform is implemented and performed on a vibration signal, and as a result, a series of frequency-band signals is produced. Second, the value of kurtosis is calculated for each of the decomposed frequency-band signals. Third, kurtogram is constructed based on these kurtosis values. In terms of the kurtogram, the frequency-band signal having the maximum kurtosis is selected and the envelope analysis is performed on this signal. Then the envelope spectrum of the signal with the maximum kurtosis is obtained. Using the envelope spectrum, the fault characteristic frequencies are discovered and the faults can be finally diagnosed. As mentioned above, the frequency-band signal having the maximum kurtosis is selected in the proposed method and is further processed to detect faults. This band selection method is based on sensitivity of kurtosis to impulses caused by local defects of rolling element bearings. Therefore the proposed method can simply point out a band signal, but it does not mean that the selected band must be optimal especially for signals with quite a small signal-to-noise ratio. In this study, we are not focusing on the selection method of the optimal band. However, a novel method for selection of the optimal
Levels b K 0, 1 b
b
K 1, 1
K1, 2 b
b
K 2, 1 b
K 3, 1
b
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b
K 3, 2
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b
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b
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b
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i
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Fig. 2. Paving of the improved kurtogram method.
(Δf)i
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Start
A vibration signal to be analyzed
Perform WPT on the signal
Calculate kurtosis of the decomposed frequency-band signals
Generate kurtogram using the values of kurtosis
Select the frequency-band signal with the maximum kurtosis
Demodulate the frequency-band signal
Calculate the envelope spectrum of the frequency-band signal
Diagnose faults using the envelope spectrum
End Fig. 3. Flow chart of the proposed method.
frequency band is recently presented to overcome the aforementioned drawbacks. The application demonstrates that it is effective in detecting transients with small signal-to-noise ratio [15]. 4. Experiments, diagnosis results and discussions In this section, two vibration signals collected from an experimental rolling element bearing and a real locomotive rolling element bearing will be used to demonstrate the effectiveness of the proposed method. 4.1. Case 1: fault diagnosis of an experimental rolling element bearing The experimental system of a rolling element bearing is shown in Fig. 4(a), and (b) is the structure sketch of the test rig. A tested bearing is installed on a motor driven shaft. The EDM method is used to introduce faults on the outer race of the tested bearing. The defect depth is 0.3 mm and the width is 1.2 mm. The picture of the faulty bearing is shown in Fig. 5. An accelerometer is mounted on the shaft box to acquire vibration signals from the bearings. The data acquisition and analysis system by Sony EX is used to collect data. Experimental parameters are listed in Table 1. The characteristic frequency of the fault on bearing outer race (fO) can be calculated by the following equation and is equal to 174.46 Hz. fr B ð7Þ fO ¼ NR 1 cos a C 2 where fr is the bearing rotational frequency, NR is the roller number, a is the contact angle and B and C are the roller and pitch diameters, respectively. A vibration signal measured from the experimental bearing is plotted in Fig. 6(a), and (b) is the envelope spectrum of the signal. Several impulses can be observed from the vibration signal; however, they are masked by noise and not evident enough to detect the existence of faults. From the envelope spectrum shown in Fig. 6(b), the rotating frequency of the bearing and its harmonics are quite dominant but the fault frequency of the outer race (fO) cannot been seen. This means that the envelope analysis fails to discover the fault characteristics from a noisy signal with the outer race fault. Then the proposed method is applied to processing the vibration signal. Because the Daubechies wavelet is orthogonal, compact supported and approximately symmetric [16], the WPT via db10 (Daubechies wavelet with the vanishing moment 10) is selected and utilized to decompose the signal shown in Fig. 6(a) at levels 1, 2, 3 and 4 in the proposed
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Gearbox Speed controller
Timing belt Accelerometer
Motor
Load
Bearings
Accelerometer
Sony EX
Shaft Laptop
Sensor cable
Flange Tested bearing
Test rig Fig. 4. (a) Experimental system and (b) structure sketch.
Defect on the outer race
Fig. 5. Fault on the outer race of the experimental bearing.
Table 1 Parameters in the experiments. Parameter
Bearing specs Outer race diameter (mm) Inner race diameter (mm) Roller diameter (mm) Roller number Contact angle (deg) Rotating speed (rpm) Sampling frequency (kHz) Sampling points
Value Experimental rolling element bearing
Locomotive rolling element bearing
N205 52 25 7.5 13 0 2000 20 16,384
552732QT 160 290 34 17 0 653 12.8 16,384
method. The decomposed frequency-band signals are obtained at each of the four decomposition levels. Then the value of b kurtosis (Ki,j ) is calculated for each frequency-band signal at each level. As a result, we get the kurtogram and show it in b Fig. 7. The maximum kurtosis is K3,4 in the figure and issued from the forth frequency-band signal at the third decomposition level marked by the black rectangle. The frequency-band signal is plotted in Fig. 8(a) and reveals clearer impulses compared with the original signal shown in Fig. 6(a). The envelope analysis is performed on this frequency-band signal and the corresponding envelope spectrum is presented in Fig. 8(b). It is seen from the envelope spectrum that the fault frequency of the bearing outer race (fO) and its harmonics are rather evident. It suggests that the defect on the outer race of the experimental bearing is effectively detected by the proposed method.
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1
Amplitude (v)
0
-1 0.1
0.03
0.2
0.3
0.4 Time (s)
0.5
0.6
0.7
0.8
2fr
fr
0.02 0.01 0 0
50
100
150
200 250 300 Frequency (Hz)
350
400
450
500
Fig. 6. Experimental rolling element bearing: (a) vibration signal and (b) envelope spectrum.
0
16
1
14 12
Level
2 10 3 8 4
5
6 4 0
1250 2500 3750 5000 6250 7500 8750 10000 Frequency (Hz)
Fig. 7. Kurtogram of the proposed method for the experimental bearing.
For comparison, the original kurtogram is also applied to analyzing the vibration signal in Fig. 6(a) and the decomposition levels are also from 1 to 4. The corresponding kurtogram is shown in Fig. 9(a) and the maximum kurtosis is K3:6,6 , indicated by the black rectangle in the figure. The filtered signal with the maximum kurtosis is further processed using the envelope analysis, and Fig. 9(b) shows its envelope spectrum. Although the envelope spectrum reflects the fault frequency and its harmonics of the bearing outer race, the signal-to-noise ratio in Fig. 9(b) is much lower than that of Fig. 8(b). It means that the proposed method is able to more clearly extract fault characteristics than the original kurtogram. 4.2. Case 2: fault diagnosis of a real locomotive rolling element bearing In this case, a real locomotive rolling element bearing having flaking fault on the outer race shown in Fig. 10 is tested on a test bench to collect vibration signals and to validate the performance of the proposed method. The test bench displayed in Fig. 11 consists of a hydraulic motor, two supporting pillow blocks, a tested bearing, a hydraulic radial load application system and a tachometer for shaft speed measurement. The bearing is installed in a hydraulic motor driven mechanical system. An accelerometer is mounted on the load module adjacent to the outer race of the bearing for measuring its vibration. Parameters in the test are given in Table 1. Fig. 12 displays a collected vibration signal and the corresponding envelope spectrum. Because of heavy noise, we cannot see the impulse components characterizing bearing faults from the vibration signal. Moreover, only the bearing rotating frequency is obvious and the fault characteristic frequency is completely buried by heavy noise in the envelope spectrum.
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0.1
Amplitude (v)
0
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x 10-3 4
fO
3
0.4 Time (s)
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0.8
2fO
2
3fO
1 0
0.5
100
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300
4fO
400 500 600 Frequency (Hz)
700
800
900
1000
Fig. 8. Diagnosis results using the proposed method for the experimental bearing: (a) frequency-band signal having the maximum kurtosis and (b) envelope spectrum.
0 5 1 4.5 1.6
4 3.5
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2
3 2.6 2.5 3 2 3.6
1.5 1
4
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4000 6000 Frequency (Hz)
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10000
x 10-5 fO
1
3fO
2fO
4fO
0.5 0 0
100
200
300
400 500 Frequency (Hz)
600
700
800
Fig. 9. Diagnosis results using the original kurtogram for the experimental bearing: (a) kurtogram and (b) envelope spectrum of the frequency-band signal having the maximum kurtosis.
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Flaking on the outer race
Fig. 10. Fault on the outer race of the locomotive bearing.
Hydraulic motor Coupling
Supporting pillow blocks
Shaft
Tested bearing Load module Accelerometer Hydraulic cylinder
Tachometer
Fig. 11. Test bench of a locomotive rolling element bearing.
5
Amplitude (v)
0
-5
0.2
0.15
0.4
0.6 Time (s)
0.8
1
1.2
fr
0.1 0.05 0 0
50
100
150
200 250 300 Frequency (Hz)
350
400
450
500
Fig. 12. Locomotive rolling element bearing: (a) vibration signal and (b) envelope spectrum.
To demonstrate the effectiveness of the proposed method in extracting weak features from noisy signals, the method is applied to analyze the signal in Fig. 12(a). The result produced by the proposed method on the signal of the locomotive rolling element bearing is presented in Fig. 13. It can be seen from the kurtogram in Fig. 13 that the first frequency-band
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0 5.4 5.2
1
5 4.8
Level
2
4.6 4.4
3
4.2 4
4
3.8 5
3.6 0
800 1600 2400 3200 4000 4800 5600 6400 Frequency (Hz)
Fig. 13. Kurtogram of the proposed method for the locomotive bearing.
0.2 0
Amplitude (v)
-0.2 -0.4 0.2
0.4
0.6 Time (s)
0.8
1
1.2
x 10-3 4
fO
3
2fO
2
3fO
1
4fO 0
50
100
150
200 250 300 Frequency (Hz)
350
400
450
500
Fig. 14. Diagnosis results using the proposed method for the locomotive bearing: (a) frequency-band signal having the maximum kurtosis and (b) envelope spectrum.
b signal at the second decomposition level obtains the maximum kurtosis (K2,1 ) and it is marked by the black rectangle. Fig. 14 shows this frequency-band signal and its envelope spectrum. A series of impulses reflecting bearing faults is highlighted in the frequency-band signal displayed in Fig. 14(a), which is totally masked by heavy noise in the original signal shown in Fig. 12(a). Furthermore, the envelope spectrum of the frequency-band signal in Fig. 14(b) provides a satisfied result in which noise is almost completely filtered out and the fault characteristic frequency and its harmonics are quite effectively extracted. Thus, it may be concluded that the proposed method can successfully discover the weak characteristics from signals with heavy noise, and therefore can effectively detect the fault in the real locomotive rolling element bearing. The original kurtogram is utilized to discover fault characteristics by analyzing the same noisy signal in Fig. 12(a) and detect the fault of the real locomotive rolling element bearing. The result of the locomotive rolling element bearing using the original kurtogram is presented in Fig. 15(a). According to the kurtogram, it is found that the maximum kurtosis (K1:6,3 ) is calculated from the third filtered signal at the 1.6th decomposition level. The envelope spectrum of the filtered signal having maximum kurtosis is computed and shown in Fig. 15(b). Similar to the envelope spectrum of the original vibration signal in Fig. 12(b), Fig. 15(b) only extracts the rotating frequency of the bearing, but the fault characteristic frequency of the outer race is vague due to heavy noise. Therefore, we believe that based on the original kurtogram, it is difficult to discover weak characteristics from noisy signals and to detect the fault of the real locomotive rolling element bearing.
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0 1.2 1
1.1 1
1.6 0.9
Level
2
0.8 0.7
2.6
0.6 3 0.5 3.6
0.4 0.3
4 0.2 4.6 0
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2000
3000 4000 Frequency (Hz)
5000
6000
Amplitude (v)
0.04
0.02
0 0
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200 250 300 Frequency (Hz)
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Fig. 15. Diagnosis results using the original kurtogram for the locomotive bearing: (a) kurtogram and (b) envelope spectrum of the frequency-band signal having the maximum kurtosis.
4.3. Discussions Through the comparisons between the proposed method and the original kurtogram in the above two cases, it is appropriate to draw the following conclusions that in the case of the experimental rolling element bearing, both the original kurtogram and the proposed method can discover fault characteristics and diagnose faults due to its relative ease, while the proposed method obtains clearer fault characteristics with higher signal-to-noise ratio than the original kurtogram. In the case of the real locomotive rolling element bearing, the original kurtogram cannot discover useful information and totally fails to detect the fault. However, the proposed method still filters out heavy noise and precisely matches the fault characteristics. Thus, the proposed method is suitable and effective to weak characteristics extraction from noisy signals and fault diagnosis of rolling element bearings. The success obtained by the proposed method can be mainly attributed to the powerful filtering property of the WPT filter, which can match the signal characteristics of noisy signals precisely and filter out noise effectively. The proposed method, however, still has its drawbacks. (1) In the proposed method, WPT is based on a binary tree of filter banks, and therefore frequency leakage happens for the components having half of the analytical frequency. For example, if the analytical frequency is 1000 Hz for a given signal, then based on a binary tree decomposition, the frequency component of 500 Hz is always split into two adjacent frequency bands whatever the number of the decomposition level is. Consequently, frequency leakage occurs at the frequency of 500 Hz. Thus, if the fault characteristic frequency is 500 Hz, then we will fail to accurately extract the whole information of fault characteristics. It is interesting that Antoni [6] extended a binary tree to 1/3-binary tree of filter banks in the kurtogram and therefore the above issue of the frequency leakage does not appear in his implementation of kurtogram. (2) The proposed method is not suitable for automatic fault diagnosis. There is a demand in industrial applications for developing condition monitoring indicators that can track the
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degradation trends of bearings automatically. This idea can be introduced into the method presented in this paper. A new research topic is to develop equipment health indicators based on the proposed method for automatic fault detection. The authors will investigate the above two issues in their future work.
5. Conclusions Aiming at the shortcomings of the original kurtogram, this paper presents an improved kurtogram method to overcome the shortcomings by replacing the filter of the original kurtogram with wavelet packet transform (WPT) filter. The WPT filter has good local property in both time and frequency spaces and can filter out noise and precisely match the fault characteristics for various noisy signals. The vibration signals collected from two kinds of rolling element bearings are used to test the improved performance of the proposed method compared with the original kurtogram. The results validate the effectiveness of the proposed method in extracting weak characteristics and diagnosing faults of rolling element bearings.
Acknowledgements This research is supported by the National Natural Science Foundation of China (51005172), the Fundamental Research Funds for the Central Universities and the Preferred Support Funds of the Scientific Research Project of the Ministry of Human Resources and Social Security of China for Returned Scholars. References [1] Y.G. Lei, Z.J. He, Y.Y. Zi, Fault diagnosis of rotating machinery based on multiple ANFIS combination with GAs, Mechanical Systems and Signal Processing 21 (2007) 2280–2294. [2] B. Samanta, K.R. Al-Balushi, Artificial neural network based fault diagnostics of rolling element bearings using time-domain features, Mechanical Systems and Signal Processing 17 (2003) 317–328. [3] D.F. Dwyer, Detection of non-Gaussian signals by frequency domain kurtosis estimation, in: Proceedings of the International Conference on Acoustic, Speech, and Signal Processing, Boston 1984, pp. 607–610. [4] J. Antoni, The spectral kurtosis: a useful tool for characterizing non-stationary signals, Mechanical Systems and Signal Processing 20 (2006) 282–307. [5] J. Antoni, R.B. Randall, The spectral kurtosis: application to the vibratory surveillance and diagnostics of rotating machines, Mechanical Systems and Signal Processing 20 (2006) 308–331. [6] J. Antoni, Fast computation of the kurtogram for the detection of transient faults, Mechanical Systems and Signal Processing 21 (2007) 108–124. [7] N. Sawalhi, R.B. Randall, H. Endo, The enhancement of fault detection and diagnosis in rolling element bearings using minimum entropy deconvolution combined with spectral kurtosis, Mechanical Systems and Signal Processing 21 (2007) 2616–2633. [8] T. Barszcz, R.B. Randall, Application of spectral kurtosis for detection of a tooth crack in the planetary gear of a wind turbine, Mechanical Systems and Signal Processing 23 (2009) 1352–1365. [9] P. Boˇskoski, J. Petrovcˇicˇ, B. Musizza, D. Juricˇicˇ, Detection of lubrication starved bearings in electrical motors by means of vibration analysis, Tribology International 43 (2010) 1683–1692. [10] J. Lin, Ridges reconstruction based on inverse wavelet transform, Journal of Sound and Vibration 294 (2006) 916–926. [11] H. Qiu, J. Lee, J. Lin, G. Yu, Wavelet filter-based weak signature detection method and its application on rolling element bearing prognostics, Journal of Sound and Vibration 289 (2006) 1066–1090. [12] L. Shi, R.B. Randall, J. Antoni, Rolling element bearing fault detection using improved envelope analysis, in: Proceedings of the Eighth International Conference on Vibrations in Rotating Machinery, Swansea, Wales, September 7–9, 2004, pp. 301–311. [13] N. Sawalhi, R.B. Randall, Spectral kurtosis optimization for rolling element bearings, in: Proceedings of the Eighth International Symposium on Signal Processing and its Applications, Sydney, August 28–31, 2005, pp. 839–842. [14] Z.J. He, Y.Y. Zi, Q.F. Meng, et al., Fault Diagnosis Principle of Non-stationary Signal and Applications to Mechanical Equipment, Higher Education Press, Beijing, 2001, (in Chinese). [15] T. Barszcz, A. Jab"on´ski, A novel method for the optimal band selection for vibration signal demodulation and comparison with the Kurtogram, Mechanical Systems and Signal Processing 25 (2011) 431–451. [16] I. Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics Philadelphia, PA, USA, 1992, (Chapter 1), pp. 3–5.
Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp
Application of an improved kurtogram method for fault diagnosis of rolling element bearings Yaguo Lei n, Jing Lin, Zhengjia He, Yanyang Zi State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, China
a r t i c l e in f o
abstract
Article history: Received 26 July 2010 Received in revised form 13 December 2010 Accepted 21 December 2010 Available online 8 January 2011
Kurtogram, due to the superiority of detecting and characterizing transients in a signal, has been proved to be a very powerful and practical tool in machinery fault diagnosis. Kurtogram, based on the short time Fourier transform (STFT) or FIR filters, however, limits the accuracy improvement of kurtogram in extracting transient characteristics from a noisy signal and identifying machinery fault. Therefore, more precise filters need to be developed and incorporated into the kurtogram method to overcome its shortcomings and to further enhance its accuracy in discovering characteristics and detecting faults. The filter based on wavelet packet transform (WPT) can filter out noise and precisely match the fault characteristics of noisy signals. By introducing WPT into kurtogram, this paper proposes an improved kurtogram method adopting WPT as the filter of kurtogram to overcome the shortcomings of the original kurtogram. The vibration signals collected from rolling element bearings are used to demonstrate the improved performance of the proposed method compared with the original kurtogram. The results verify the effectiveness of the method in extracting fault characteristics and diagnosing faults of rolling element bearings. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Kurtogram Wavelet packet transform Rolling element bearings Fault diagnosis
1. Introduction Rolling element bearings are widely used in modern machinery, and faults occurring in the bearings may lead to fatal breakdown of machines. Therefore, it is significant to be able to accurately detect and diagnose the existence of the faults occurring in the bearings. Vibration signals collected from bearings carry rich information on machine health conditions. Therefore, the vibration-based methods have received intensive study during the past decades. It is possible to obtain vital characteristic information from the vibration signals through the use of signal processing techniques [1]. In order to effectively diagnose faults occurring in rolling element bearings, researchers have extensively investigated different signal processing techniques to accurately extract fault characteristics from vibration signals. Since the envelope analysis focuses on the low-amplitude high-frequency broadband signals characterizing bearing conditions and may minimize the effects of interfering signals within the selected frequency band, it has been widely applied in detecting faults of rolling element bearings [2]. Spectral kurtosis (SK), as an envelope analysis technique, was originally presented by Dwyer [3] to complement the classical power spectral density in detecting and characterizing transients in a signal. The basic idea of this technique is to exploit the possibility of using the kurtosis at each frequency line as a measure to discover the presence of non-Gaussian components and to indicate in which frequency bands these occur. Such non-Gaussian
n
Corresponding author. E-mail address: [email protected] (Y. Lei).
0888-3270/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2010.12.011
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components are the structural responses to an excitation caused by, e.g. a local flaking on the outer race in the case of bearing faults. Additionally, SK can determine the frequency of the excited component. Therefore, SK can indicate not only the non-Gaussian components in signals, but also their locations in the frequency domain. Subsequently, Antoni [4,6] and Antoni and Randall [5] made a very thorough and interesting study on SK; first they generalized it and proposed the definition of the ‘‘kurtogram’’ to a wider class of non-stationary signals. Because of the superiority in extracting transient components masked in a noisy signal, kurtogram has been proved to be a very powerful and practical tool in machinery fault diagnosis [6–9]. Kurtogram based on the short time Fourier transform (STFT) or FIR filters, however, limits the performance improvement of the kurtogram in extracting transient characteristics from a noisy signal and identifying machinery faults. To overcome the above shortcomings of kurtogram and to further enhance its accuracy in discovering characteristics and detecting faults, more precise filters need to be developed and incorporated into the kurtogram algorithm. As a special filter, WT possesses good local property in both time and frequency spaces [10], and therefore it has been studied extensively and applied successfully to vibration signal analysis for fault diagnosis of rolling element bearing fault diagnosis [11]. WT provides evident advantages in filtering out noise and keeping the transient characteristics if the wavelet used is similar to the feature components hidden in a signal. Therefore, some researchers have adopted WT as filters to implement and improve the kurtogram. For example, Shi et al. [12] used the complex Morlet wavelet transform to implement kurtogram and proposed fault diagnosis method of rolling element bearings. This method can find the best frequency band for demodulation automatically by the criterion of SK based on the complex Morlet wavelet transform [12]. Sawalhi and Randall [13] presented a two-step technique to optimize SK and got the kurtogram for diagnosing rolling element bearing faults using the complex Morlet wavelet transform to decompose bearing signals [13]. However, WT cannot effectively split the high frequency band (detail signal) where the fault modulation information always exists. The way of overcoming this difficulty is to extend WT to wavelet packet transform (WPT) [14]. Therefore, an improved method using WPT based on the Daubechies wavelet to implement kurtogram is proposed in this paper. The method adopts WPT as the filter to overcome the shortcomings of the original kurtogram based on STFT or FIR filters. In addition, compared with the methods based on the complex Morlet wavelet transform in Refs. [12,13], this improved kurtogram method has the following three advantages: (1) It is faster because the kurtogram based on the complex Morlet wavelet transform is of computational complexity. (2) It used WPT based on Daubechies wavelet being orthogonal, compact supported and approximately symmetric, and WPT guarantees feature extraction of transient characteristics without redundant or omitted information. (3) WPT may effectively split the high frequency band where the bearing fault modulation information exists. Thus, it is expected that a more accurate result of characteristic discovery can be obtained by the improved kurtogram method. Experiments on an experimental rolling element bearing and a real locomotive rolling element bearing are conducted to test the improved performance of the improved method compared with the original kurtogram. The results demonstrate the effectiveness of the method in extracting fault characteristics from noisy vibration signals.
2. Review of kurtogram and wavelet packet transform 2.1. Kurtogram Kurtogram was first introduced by Antoni and Randall [5], which comes from spectral kurtosis (SK) [3]. The SK method selects kurtosis as a measure of distance between an arbitrary random process and a Gaussian one with the aim of detecting the existence of transients in a signal. The SK value of the signal is obtained by calculating kurtosis of each frequency component contained in the signal. SK finally represents the transient characteristics of the signal as a function of frequency. SK can indicate not only transient components in the signal but also their locations in the frequency domain, and therefore overcome the disadvantages of the power spectral density in detecting and characterizing signal transients. Considering the Wold–Grame´r decomposition of a non-stationary signal, we can define signal y(t) as the response of a system with time varying impulse response h(t,s) excited by signal x(t). Then y(t) can be expressed as [4] Z þ1 yðtÞ ¼ ej2pft Hðt,f Þ dXðf Þ ð1Þ 1
where H(t,f) is the time varying transfer function of the considered system and can be interpreted as the complex envelope of signal y(t) at frequency f and dX(f) is the spectral process associated with x(t). In real cases of machinery vibration, H(t,f) is stochastic and should be presented as H(t,f,o), where o is the random variable presenting time variations of filters. When H is time stationary and independent of process x, we obtain a conditionally non-stationary process (CNS). SK is based on the fourth-order spectral cumulant of a CNS process. C4y ðf Þ ¼ S4y ðf Þ2S22y ðf Þ
ð2Þ
where S2y ðf Þ is the time-averaged result of S2y ðt,f Þ, S2y ðt,f Þ is an instantaneous moment and measures the energy of the complex envelope. Then, the SK is generated by normalizing the fourth-order cumulant given in Eq. (2). It is a measure of
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the peakiness of the probability density function Ky ðf Þ ¼
C4y ðf Þ S4y ðf Þ ¼ 2 2 S22y ðf Þ S2y ðf Þ
ð3Þ
The estimation of the SK values for particular frequency f is implemented by calculating the STFT of signal y(t) and is given by Eq. (4). Fy ðt,f Þ ¼
Z
þ1
yðtÞw ðttÞej2pft dt ¼ /yðtÞ,wðttÞej2pft S
ð4Þ
1
where w*(t t) represents the window function and superscript n denotes the conjugation. The STFT is actually an inner product operation /y(t),w(t t)e j2pftS of signal y(t) and filter w(t t)ej2pft, which is the triangle basis function ej2pft enveloped by window function w(t t). It is noted that the selected window length and the percent of overlap used for the STFT calculation significantly influence the value of SK. Therefore, for the purpose of selecting the optimal window length the spectral kurtosis is calculated for several different window lengths. The implementation result of these calculations is the so-called kurtogram. The window parameters for which the spectral kurtosis has its maximum value determine the band-pass filter parameters. Therefore, kurtogram is a 2D map and presents values of SK calculated for various parameters of frequency f and bandwidth Df. Let Kk,l be the SK value of the filtered signal issued from the lth filter at the kth decomposition level, and then the kurtogram paving of the (f/Df) plane is shown in Fig. 1. A more detailed explanation of kurtogram can be found in Refs. [5,6].
2.2. Wavelet packet transform Wavelets have perfect local property in both time and frequency spaces, and can be used as an effective filter to filter out noise and preserve signal characteristics [10]. WT, however, does not process the high frequency bands where the modulation information of machine fault exists. Wavelet packet transform (WPT) furthers decomposition of the high frequency bands and may generate a more precise frequency-band partition over the whole analyzed frequency bands. As a result, the frequency resolution may be enhanced. Thus, WPT can be implemented based on wavelet filters, which is
Levels 0
1/2
1
1/4
1.6
1/6
2
1/8
K2.6, 6
2.6
1/12
K3, 8
3
1/16
k
1/2
K0, 1 K1, 1
K1, 2
K1.6, 1
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K2, 2 K2.6, 2
K3, 2
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K2, 4
K2.6, 4
K3, 4
K3, 5
K2.6, 5 K3, 6
K3, 7
Kk, 1
Kk, l
f 0
1/8
1/4
(Δf)k
3/8
Fig. 1. Paving of the original kurtogram.
1/2
k+1
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defined by the following equation: 8 K > pffiffiffi X > > > Hn bi1,j þ n b ¼ 2 > < i,j n¼0
ð5Þ
K > pffiffiffi X > > > Gn bi1,j þ n > : bi,j þ l ¼ 2 n¼0
where bi,j denotes the jth decomposed frequency-band signal at level i (j ¼ 1,2,. . .,J, where J is the number of the decomposed frequency-band signals and equals 2I; i ¼ 1,2,. . .,I, where I is the number of decomposition levels); Hn and Gn are the low- and high-pass filters based on wavelets, respectively. 3. Proposed method After the WPT decomposition, a set of frequency-band signals is obtained. Then, kurtosis is calculated for each of the frequency-band signals according to the following equation: PN ðbi,j ðnÞmb Þ4 b Ki,j ¼ n¼1 ð6Þ ðN1Þs4b where mb, sb and N are the mean value, the standard deviation and the point number of the frequency-band signal bi,j, respectively. b Replacing Kk,l of the original kurtogram by Ki,j of WPT, we propose an improved kurtogram method. Correspondingly, the kurtogram paving of the (f/Df) plane of the proposed method is shown in Fig. 2. Employing WPT as the filter of kurtogram to overcome the disadvantages of the original kurtogram using the STFT or FIR filters, an improved kurtogram method is proposed in this paper. The flow chart of the proposed method is presented in Fig. 3. It contains the following procedural steps. First, the wavelet packet transform is implemented and performed on a vibration signal, and as a result, a series of frequency-band signals is produced. Second, the value of kurtosis is calculated for each of the decomposed frequency-band signals. Third, kurtogram is constructed based on these kurtosis values. In terms of the kurtogram, the frequency-band signal having the maximum kurtosis is selected and the envelope analysis is performed on this signal. Then the envelope spectrum of the signal with the maximum kurtosis is obtained. Using the envelope spectrum, the fault characteristic frequencies are discovered and the faults can be finally diagnosed. As mentioned above, the frequency-band signal having the maximum kurtosis is selected in the proposed method and is further processed to detect faults. This band selection method is based on sensitivity of kurtosis to impulses caused by local defects of rolling element bearings. Therefore the proposed method can simply point out a band signal, but it does not mean that the selected band must be optimal especially for signals with quite a small signal-to-noise ratio. In this study, we are not focusing on the selection method of the optimal band. However, a novel method for selection of the optimal
Levels b K 0, 1 b
b
K 1, 1
K1, 2 b
b
K 2, 1 b
K 3, 1
b
K 2, 2 b
b
K 3, 2
K 3, 3
K 2, 3 b
b
K 3, 4
K 3, 5
b
K 2, 4 b
b
K 3, 6
K 3, 7
b
b
K 3, 8
b
K i, 1
K i, J
0
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1
1/4
2
1/8
3
1/16
i
1/2i+1
f 0
1/8
1/4
3/8
1/2
Fig. 2. Paving of the improved kurtogram method.
(Δf)i
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Start
A vibration signal to be analyzed
Perform WPT on the signal
Calculate kurtosis of the decomposed frequency-band signals
Generate kurtogram using the values of kurtosis
Select the frequency-band signal with the maximum kurtosis
Demodulate the frequency-band signal
Calculate the envelope spectrum of the frequency-band signal
Diagnose faults using the envelope spectrum
End Fig. 3. Flow chart of the proposed method.
frequency band is recently presented to overcome the aforementioned drawbacks. The application demonstrates that it is effective in detecting transients with small signal-to-noise ratio [15]. 4. Experiments, diagnosis results and discussions In this section, two vibration signals collected from an experimental rolling element bearing and a real locomotive rolling element bearing will be used to demonstrate the effectiveness of the proposed method. 4.1. Case 1: fault diagnosis of an experimental rolling element bearing The experimental system of a rolling element bearing is shown in Fig. 4(a), and (b) is the structure sketch of the test rig. A tested bearing is installed on a motor driven shaft. The EDM method is used to introduce faults on the outer race of the tested bearing. The defect depth is 0.3 mm and the width is 1.2 mm. The picture of the faulty bearing is shown in Fig. 5. An accelerometer is mounted on the shaft box to acquire vibration signals from the bearings. The data acquisition and analysis system by Sony EX is used to collect data. Experimental parameters are listed in Table 1. The characteristic frequency of the fault on bearing outer race (fO) can be calculated by the following equation and is equal to 174.46 Hz. fr B ð7Þ fO ¼ NR 1 cos a C 2 where fr is the bearing rotational frequency, NR is the roller number, a is the contact angle and B and C are the roller and pitch diameters, respectively. A vibration signal measured from the experimental bearing is plotted in Fig. 6(a), and (b) is the envelope spectrum of the signal. Several impulses can be observed from the vibration signal; however, they are masked by noise and not evident enough to detect the existence of faults. From the envelope spectrum shown in Fig. 6(b), the rotating frequency of the bearing and its harmonics are quite dominant but the fault frequency of the outer race (fO) cannot been seen. This means that the envelope analysis fails to discover the fault characteristics from a noisy signal with the outer race fault. Then the proposed method is applied to processing the vibration signal. Because the Daubechies wavelet is orthogonal, compact supported and approximately symmetric [16], the WPT via db10 (Daubechies wavelet with the vanishing moment 10) is selected and utilized to decompose the signal shown in Fig. 6(a) at levels 1, 2, 3 and 4 in the proposed
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Gearbox Speed controller
Timing belt Accelerometer
Motor
Load
Bearings
Accelerometer
Sony EX
Shaft Laptop
Sensor cable
Flange Tested bearing
Test rig Fig. 4. (a) Experimental system and (b) structure sketch.
Defect on the outer race
Fig. 5. Fault on the outer race of the experimental bearing.
Table 1 Parameters in the experiments. Parameter
Bearing specs Outer race diameter (mm) Inner race diameter (mm) Roller diameter (mm) Roller number Contact angle (deg) Rotating speed (rpm) Sampling frequency (kHz) Sampling points
Value Experimental rolling element bearing
Locomotive rolling element bearing
N205 52 25 7.5 13 0 2000 20 16,384
552732QT 160 290 34 17 0 653 12.8 16,384
method. The decomposed frequency-band signals are obtained at each of the four decomposition levels. Then the value of b kurtosis (Ki,j ) is calculated for each frequency-band signal at each level. As a result, we get the kurtogram and show it in b Fig. 7. The maximum kurtosis is K3,4 in the figure and issued from the forth frequency-band signal at the third decomposition level marked by the black rectangle. The frequency-band signal is plotted in Fig. 8(a) and reveals clearer impulses compared with the original signal shown in Fig. 6(a). The envelope analysis is performed on this frequency-band signal and the corresponding envelope spectrum is presented in Fig. 8(b). It is seen from the envelope spectrum that the fault frequency of the bearing outer race (fO) and its harmonics are rather evident. It suggests that the defect on the outer race of the experimental bearing is effectively detected by the proposed method.
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1
Amplitude (v)
0
-1 0.1
0.03
0.2
0.3
0.4 Time (s)
0.5
0.6
0.7
0.8
2fr
fr
0.02 0.01 0 0
50
100
150
200 250 300 Frequency (Hz)
350
400
450
500
Fig. 6. Experimental rolling element bearing: (a) vibration signal and (b) envelope spectrum.
0
16
1
14 12
Level
2 10 3 8 4
5
6 4 0
1250 2500 3750 5000 6250 7500 8750 10000 Frequency (Hz)
Fig. 7. Kurtogram of the proposed method for the experimental bearing.
For comparison, the original kurtogram is also applied to analyzing the vibration signal in Fig. 6(a) and the decomposition levels are also from 1 to 4. The corresponding kurtogram is shown in Fig. 9(a) and the maximum kurtosis is K3:6,6 , indicated by the black rectangle in the figure. The filtered signal with the maximum kurtosis is further processed using the envelope analysis, and Fig. 9(b) shows its envelope spectrum. Although the envelope spectrum reflects the fault frequency and its harmonics of the bearing outer race, the signal-to-noise ratio in Fig. 9(b) is much lower than that of Fig. 8(b). It means that the proposed method is able to more clearly extract fault characteristics than the original kurtogram. 4.2. Case 2: fault diagnosis of a real locomotive rolling element bearing In this case, a real locomotive rolling element bearing having flaking fault on the outer race shown in Fig. 10 is tested on a test bench to collect vibration signals and to validate the performance of the proposed method. The test bench displayed in Fig. 11 consists of a hydraulic motor, two supporting pillow blocks, a tested bearing, a hydraulic radial load application system and a tachometer for shaft speed measurement. The bearing is installed in a hydraulic motor driven mechanical system. An accelerometer is mounted on the load module adjacent to the outer race of the bearing for measuring its vibration. Parameters in the test are given in Table 1. Fig. 12 displays a collected vibration signal and the corresponding envelope spectrum. Because of heavy noise, we cannot see the impulse components characterizing bearing faults from the vibration signal. Moreover, only the bearing rotating frequency is obvious and the fault characteristic frequency is completely buried by heavy noise in the envelope spectrum.
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0.1
Amplitude (v)
0
-0.1 0.1
0.2
0.3
x 10-3 4
fO
3
0.4 Time (s)
0.6
0.7
0.8
2fO
2
3fO
1 0
0.5
100
200
300
4fO
400 500 600 Frequency (Hz)
700
800
900
1000
Fig. 8. Diagnosis results using the proposed method for the experimental bearing: (a) frequency-band signal having the maximum kurtosis and (b) envelope spectrum.
0 5 1 4.5 1.6
4 3.5
Level
2
3 2.6 2.5 3 2 3.6
1.5 1
4
0.5 4.6
Amplitude (v)
1.5
0
2000
4000 6000 Frequency (Hz)
8000
10000
x 10-5 fO
1
3fO
2fO
4fO
0.5 0 0
100
200
300
400 500 Frequency (Hz)
600
700
800
Fig. 9. Diagnosis results using the original kurtogram for the experimental bearing: (a) kurtogram and (b) envelope spectrum of the frequency-band signal having the maximum kurtosis.
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Flaking on the outer race
Fig. 10. Fault on the outer race of the locomotive bearing.
Hydraulic motor Coupling
Supporting pillow blocks
Shaft
Tested bearing Load module Accelerometer Hydraulic cylinder
Tachometer
Fig. 11. Test bench of a locomotive rolling element bearing.
5
Amplitude (v)
0
-5
0.2
0.15
0.4
0.6 Time (s)
0.8
1
1.2
fr
0.1 0.05 0 0
50
100
150
200 250 300 Frequency (Hz)
350
400
450
500
Fig. 12. Locomotive rolling element bearing: (a) vibration signal and (b) envelope spectrum.
To demonstrate the effectiveness of the proposed method in extracting weak features from noisy signals, the method is applied to analyze the signal in Fig. 12(a). The result produced by the proposed method on the signal of the locomotive rolling element bearing is presented in Fig. 13. It can be seen from the kurtogram in Fig. 13 that the first frequency-band
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0 5.4 5.2
1
5 4.8
Level
2
4.6 4.4
3
4.2 4
4
3.8 5
3.6 0
800 1600 2400 3200 4000 4800 5600 6400 Frequency (Hz)
Fig. 13. Kurtogram of the proposed method for the locomotive bearing.
0.2 0
Amplitude (v)
-0.2 -0.4 0.2
0.4
0.6 Time (s)
0.8
1
1.2
x 10-3 4
fO
3
2fO
2
3fO
1
4fO 0
50
100
150
200 250 300 Frequency (Hz)
350
400
450
500
Fig. 14. Diagnosis results using the proposed method for the locomotive bearing: (a) frequency-band signal having the maximum kurtosis and (b) envelope spectrum.
b signal at the second decomposition level obtains the maximum kurtosis (K2,1 ) and it is marked by the black rectangle. Fig. 14 shows this frequency-band signal and its envelope spectrum. A series of impulses reflecting bearing faults is highlighted in the frequency-band signal displayed in Fig. 14(a), which is totally masked by heavy noise in the original signal shown in Fig. 12(a). Furthermore, the envelope spectrum of the frequency-band signal in Fig. 14(b) provides a satisfied result in which noise is almost completely filtered out and the fault characteristic frequency and its harmonics are quite effectively extracted. Thus, it may be concluded that the proposed method can successfully discover the weak characteristics from signals with heavy noise, and therefore can effectively detect the fault in the real locomotive rolling element bearing. The original kurtogram is utilized to discover fault characteristics by analyzing the same noisy signal in Fig. 12(a) and detect the fault of the real locomotive rolling element bearing. The result of the locomotive rolling element bearing using the original kurtogram is presented in Fig. 15(a). According to the kurtogram, it is found that the maximum kurtosis (K1:6,3 ) is calculated from the third filtered signal at the 1.6th decomposition level. The envelope spectrum of the filtered signal having maximum kurtosis is computed and shown in Fig. 15(b). Similar to the envelope spectrum of the original vibration signal in Fig. 12(b), Fig. 15(b) only extracts the rotating frequency of the bearing, but the fault characteristic frequency of the outer race is vague due to heavy noise. Therefore, we believe that based on the original kurtogram, it is difficult to discover weak characteristics from noisy signals and to detect the fault of the real locomotive rolling element bearing.
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0 1.2 1
1.1 1
1.6 0.9
Level
2
0.8 0.7
2.6
0.6 3 0.5 3.6
0.4 0.3
4 0.2 4.6 0
1000
2000
3000 4000 Frequency (Hz)
5000
6000
Amplitude (v)
0.04
0.02
0 0
50
100
150
200 250 300 Frequency (Hz)
350
400
450
500
Fig. 15. Diagnosis results using the original kurtogram for the locomotive bearing: (a) kurtogram and (b) envelope spectrum of the frequency-band signal having the maximum kurtosis.
4.3. Discussions Through the comparisons between the proposed method and the original kurtogram in the above two cases, it is appropriate to draw the following conclusions that in the case of the experimental rolling element bearing, both the original kurtogram and the proposed method can discover fault characteristics and diagnose faults due to its relative ease, while the proposed method obtains clearer fault characteristics with higher signal-to-noise ratio than the original kurtogram. In the case of the real locomotive rolling element bearing, the original kurtogram cannot discover useful information and totally fails to detect the fault. However, the proposed method still filters out heavy noise and precisely matches the fault characteristics. Thus, the proposed method is suitable and effective to weak characteristics extraction from noisy signals and fault diagnosis of rolling element bearings. The success obtained by the proposed method can be mainly attributed to the powerful filtering property of the WPT filter, which can match the signal characteristics of noisy signals precisely and filter out noise effectively. The proposed method, however, still has its drawbacks. (1) In the proposed method, WPT is based on a binary tree of filter banks, and therefore frequency leakage happens for the components having half of the analytical frequency. For example, if the analytical frequency is 1000 Hz for a given signal, then based on a binary tree decomposition, the frequency component of 500 Hz is always split into two adjacent frequency bands whatever the number of the decomposition level is. Consequently, frequency leakage occurs at the frequency of 500 Hz. Thus, if the fault characteristic frequency is 500 Hz, then we will fail to accurately extract the whole information of fault characteristics. It is interesting that Antoni [6] extended a binary tree to 1/3-binary tree of filter banks in the kurtogram and therefore the above issue of the frequency leakage does not appear in his implementation of kurtogram. (2) The proposed method is not suitable for automatic fault diagnosis. There is a demand in industrial applications for developing condition monitoring indicators that can track the
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degradation trends of bearings automatically. This idea can be introduced into the method presented in this paper. A new research topic is to develop equipment health indicators based on the proposed method for automatic fault detection. The authors will investigate the above two issues in their future work.
5. Conclusions Aiming at the shortcomings of the original kurtogram, this paper presents an improved kurtogram method to overcome the shortcomings by replacing the filter of the original kurtogram with wavelet packet transform (WPT) filter. The WPT filter has good local property in both time and frequency spaces and can filter out noise and precisely match the fault characteristics for various noisy signals. The vibration signals collected from two kinds of rolling element bearings are used to test the improved performance of the proposed method compared with the original kurtogram. The results validate the effectiveness of the proposed method in extracting weak characteristics and diagnosing faults of rolling element bearings.
Acknowledgements This research is supported by the National Natural Science Foundation of China (51005172), the Fundamental Research Funds for the Central Universities and the Preferred Support Funds of the Scientific Research Project of the Ministry of Human Resources and Social Security of China for Returned Scholars. References [1] Y.G. Lei, Z.J. He, Y.Y. Zi, Fault diagnosis of rotating machinery based on multiple ANFIS combination with GAs, Mechanical Systems and Signal Processing 21 (2007) 2280–2294. [2] B. Samanta, K.R. Al-Balushi, Artificial neural network based fault diagnostics of rolling element bearings using time-domain features, Mechanical Systems and Signal Processing 17 (2003) 317–328. [3] D.F. Dwyer, Detection of non-Gaussian signals by frequency domain kurtosis estimation, in: Proceedings of the International Conference on Acoustic, Speech, and Signal Processing, Boston 1984, pp. 607–610. [4] J. Antoni, The spectral kurtosis: a useful tool for characterizing non-stationary signals, Mechanical Systems and Signal Processing 20 (2006) 282–307. [5] J. Antoni, R.B. Randall, The spectral kurtosis: application to the vibratory surveillance and diagnostics of rotating machines, Mechanical Systems and Signal Processing 20 (2006) 308–331. [6] J. Antoni, Fast computation of the kurtogram for the detection of transient faults, Mechanical Systems and Signal Processing 21 (2007) 108–124. [7] N. Sawalhi, R.B. Randall, H. Endo, The enhancement of fault detection and diagnosis in rolling element bearings using minimum entropy deconvolution combined with spectral kurtosis, Mechanical Systems and Signal Processing 21 (2007) 2616–2633. [8] T. Barszcz, R.B. Randall, Application of spectral kurtosis for detection of a tooth crack in the planetary gear of a wind turbine, Mechanical Systems and Signal Processing 23 (2009) 1352–1365. [9] P. Boˇskoski, J. Petrovcˇicˇ, B. Musizza, D. Juricˇicˇ, Detection of lubrication starved bearings in electrical motors by means of vibration analysis, Tribology International 43 (2010) 1683–1692. [10] J. Lin, Ridges reconstruction based on inverse wavelet transform, Journal of Sound and Vibration 294 (2006) 916–926. [11] H. Qiu, J. Lee, J. Lin, G. Yu, Wavelet filter-based weak signature detection method and its application on rolling element bearing prognostics, Journal of Sound and Vibration 289 (2006) 1066–1090. [12] L. Shi, R.B. Randall, J. Antoni, Rolling element bearing fault detection using improved envelope analysis, in: Proceedings of the Eighth International Conference on Vibrations in Rotating Machinery, Swansea, Wales, September 7–9, 2004, pp. 301–311. [13] N. Sawalhi, R.B. Randall, Spectral kurtosis optimization for rolling element bearings, in: Proceedings of the Eighth International Symposium on Signal Processing and its Applications, Sydney, August 28–31, 2005, pp. 839–842. [14] Z.J. He, Y.Y. Zi, Q.F. Meng, et al., Fault Diagnosis Principle of Non-stationary Signal and Applications to Mechanical Equipment, Higher Education Press, Beijing, 2001, (in Chinese). [15] T. Barszcz, A. Jab"on´ski, A novel method for the optimal band selection for vibration signal demodulation and comparison with the Kurtogram, Mechanical Systems and Signal Processing 25 (2011) 431–451. [16] I. Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics Philadelphia, PA, USA, 1992, (Chapter 1), pp. 3–5.