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1、Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 22 (2008) 542556 Rotating machinery fault diagnosis using signal-adapted lifting scheme Zhen Lia,?, Zhengjia Heb, Yanyang Zia, Hongkai Jianga aSchool of Mechanical Engineering, Xian Jiaotong University, 710049 Xian, Ch

2、ina bState Key Lab for Manufacturing Systems Engineering, Xian Jiaotong University, 710049 Xian, China Received 24 March 2006; received in revised form 25 September 2007; accepted 26 September 2007 Available online 2 October 2007 Abstract Wavelet transform has been widely used for vibration-based ma

3、chine fault diagnosis. However, it is a diffi cult task to choose or design appropriate wavelet or wavelets for a given application. In this paper, a new signal-adapted lifting scheme for rotating machinery fault diagnosis is proposed, which allows us to construct a wavelet directly from the statist

4、ics of a given signal. The prediction operator based on genetic algorithms is designed to maximize the kurtosis of detail signal produced by the lifting scheme, and the update operator is designed to minimize a reconstruction error. The signal-adapted lifting scheme is applied to analyze bearing and

5、 gearbox vibration signals. The conventional diagnosis techniques and non-adaptive lifting scheme are also used to analyze the same signals for comparison. The results demonstrate that the signal-adapted lifting scheme is more effective in extracting inherent fault features from complex vibration si

6、gnals. r 2007 Elsevier Ltd. All rights reserved. Keywords: Adaptive lifting scheme; Fault diagnosis; Vibration signal analysis 1. Introduction Rotating machinery is very popular in industrial applications. An unexpected failure of rotating machinery may result in signifi cant economic losses. In ord

7、er to avoid the occurrence of abnormal events, vibration signal analysis is widely used in the rotating machinery condition monitoring and fault diagnosis. Usually, depending on machine operating conditions and severity of defects, the measured vibration signals are always complex and non-stationary

8、, and the useful fault information is buried in noises. Therefore it is diffi cult to detect the symptoms of a potential failure from such vibration signals. There are many vibration-based diagnosis techniques available for rotating machinery 1. The Hilbert envelope analysis 2 has been used successf

9、ully in rotating machinery fault diagnosis as one of the most common demodulation methods. Unfortunately, the central frequency of fi lter is determined by experience while forming an envelope signal, which will cause great infl uence on the results. Cepstrum analysis 3 ARTICLE IN PRESS 0888-3270/$-

10、see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2007.09.008 ?Corresponding author. Tel.: +862982667963; fax: +862982663689. E-mail address: (Z. Li). was developed to identify the sideband components and extract fault features of gearbox. However, if the signal-to-noise

11、 ratio (SNR) is very low and the fault signature is buried in strong background noises, the sideband components are not easy to be isolated using the cepstrum analysis. In the time domain, time domain averaging 4 may be one of the most popular traditional techniques for detecting gear- box faults; i

12、t is powerful in suppressing the noise and other non-synchronous components. However, this technique has a limitation that it usually requires a reference signal to obtain a synchronously averaged signal. Wavelet transform is well known for its ability to focus on localized structures in timefrequen

13、cy domain 5. In the past few years, many researchers have investigated the application of wavelet transform to vibration signals analysis for fault diagnosis of rotating machinery. For instance, Wang and McFadden 6 used orthogonal wavelets such as the Db4 and harmonic wavelet to detect the early gea

14、r failure. Lin and Qu 7 presented a denoising method based on Morlet wavelet to extract fault features of gearbox. Sun et al. 8 applied singularity analysis through the continuous wavelet transform (CWT) to capture the time of impacts in bearing vibration signals. Li and Ma 9 used continuous wavelet

15、 analysis to detect localized bearing defects based on vibration signals. Tse et al. 10 performed a comparison on the effectiveness of envelope detection and CWT in fault diagnosis of roller bearings. These wavelet techniques are effective for fault detection of rotating machinery. However, one comm

16、on approach taken by the wavelet techniques mentioned above is that the standard wavelets selected from a library of previously designed wavelet functions are used as the mother wavelets. Unfortunately, such standard wavelet functions are independent of a given signal. Since different types of wavel

17、ets have different timefrequency structures, it is always very diffi cult to choose the best wavelet function for extracting fault features from given signal. Moreover, an inappropriate wavelet will reduce the accuracy of the fault detection. To overcome the above limitations, it is necessary to dev

18、elop new methods to design signal-adapted wavelet functions for rotating machinery fault diagnosis. The lifting scheme is introduced by Sweldens as a powerful tool to construct biorthogonal wavelets in the spatial domain 11,12. It provides a great deal of fl exibility and freedom for the constructio

19、n of biorthogonal wavelets, and can be used to construct adaptive wavelets by the design of prediction operator and update operator. In this paper, we present a new signal-adapted lifting scheme for rotating machinery fault diagnosis. In the prediction step, kurtosis is used as the performance measu

20、rement of the prediction operator, and genetic algorithms are employed to design prediction operator in order to maximize the kurtosis of detail signals. In the update step, the update operator is designed to minimize the difference between the original and reconstructed signals when the high-freque

21、ncy signal is removed. We fi nd that the signal-adapted lifting scheme can closely match the characteristics of vibration signals, and it is very effective to extract the transient feature components from the complex vibration signals. The structure of the paper is organized as follows. In Section 2

22、, the theory of lifting scheme is reviewed briefl y. In Section 3, a new signal-adapted lifting scheme is set up. The constraints of the designed lifting scheme are described. The genetic algorithms are used to design the prediction operator based on kurtosis maximization principle. The update opera

23、tor is designed to minimize the reconstruction error. In Section 4, we discuss the redundant version of the lifting scheme. In Section 5, the signal-adapted lifting scheme is applied to detect localized defects of roller bearing and gearbox. Comparisons with conventional diagnosis techniques and non

24、-adaptive lifting scheme are also shown. Conclusions are given in Section 6. 2. The lifting scheme The lifting scheme is a spatial domain construction of biorthogonal wavelets. It does not rely on the Fourier transform. The lifting scheme consists in three main steps 11,12. In the split step, the or

25、iginal signal x xii2Zis split into even samples s0 s0 i i2Zand odd samples d0 d0 i i2Z, s0 i x2i;d0 i x2i1.(1) ARTICLE IN PRESS Z. Li et al. / Mechanical Systems and Signal Processing 22 (2008) 542556543 In the prediction step, we apply an operator P on s(0)to predict d(0). The prediction error d di

26、i2Zis regarded as the detail signal of x, di d0 i ? X N=2 r?N=21 prs0 ir, (2) where pr are coeffi cients of the prediction operator P and N is the number of coeffi cients of prediction operator. In the update step, an update of even samples s(0)is accomplished by using an update operator U on detail

27、 signal d and adding the result to s(0). The update sequence s sii2Zcan be regarded as the approximation signal of x, si s0 i X N=2 j?N=21 ujdij?1,(3) where uj are coeffi cients of the update operator U and N is the number of coeffi cients of update operator. Using approximation signal s again as th

28、e input to lifting scheme can generate detail signal and approximation signal at lower resolution level. 3. The design of signal-adapted lifting scheme 3.1. Constraints of the designed lifting scheme The theorems in 12, expressed by Sweldens in the lifting framework, ensure the biorthogonality of th

29、e fi lter bank relevant to the lifting scheme. However, different prediction operator P and update operator U can construct wavelet functions with different timefrequency structures. In order to optimize the prediction operator and update operator, Gouze et al. 13 introduced two meaningful condition

30、s for operators P and U. The symmetrical linear phase constraints are expressed as follows: pr p?r1;r 1;2;.;N=2,(4) uj u?j1j 1;2;.; N=2.(5) The fi ltering normalization constraints are expressed as below: X N=2 r1 pr 1 2 ,(6) X N=2 j1 uj 1 4 .(7) 3.2. Design of the prediction operator P The predicti

31、on step provides the detail signal d. To ensure that the derived lifting fi lters can effectively isolate feature components from the original signal, a criterion for the prediction operator is needed. Kurtosis is used in engineering to detect fault symptoms because it is sensitive to sharp changed

32、structures, such as impulses 3. In this paper, we use kurtosis as the performance measurement of prediction operator. The criterion of prediction operator P is defi ned as follows: KP Efd ?d4g s4 ,(8) ARTICLE IN PRESS Z. Li et al. / Mechanical Systems and Signal Processing 22 (2008) 542556544 where

33、d and s are the mean and standard deviation of detail signal d, and E? is the expectation. Our objective is to fi nd an optimal prediction operator Poptthat maximizes the kurtosis criterion KPwhile satisfying the constraints Eqs. (4) and (6). Many optimization methods have been presented and each ha

34、s its own advantages and limitations. The fl exible polyhedron method 14 involves substantive computation. Neural networks 15 can be used to fi nd global optimization, but the selection of the architecture for neural networks is a tedious task. From the optimization point of view, one of the main ad

35、vantages of genetic algorithms 16 is that it does not have mathematical requirements on the optimization problem. Moreover, genetic algorithms are effective in global optimization. Therefore, in this paper, genetic algorithms are used to optimize the prediction operator P. In the initialization popu

36、lation for prediction operator P, the coeffi cients (p?N/2+1,y,pN/2) of prediction operator P are coded using the real-coded mechanism. First, the coeffi cients (p2,y,pN/2) are generated randomly. Second, the coeffi cients (p?N/2+1,y,p?1 ) are given according to Eq. (4). Finally, the coeffi cients (

37、p0, p1) are obtained by the following formula: p0 p1 1 2 ? X N=2 r2 pr.(9) The arithmetic crossover and uniform mutation operators commonly used in genetic algorithms are employed for the optimization process 16. To increase the effi ciency of the process, the population scale is set to 50, the numb

38、er of iteration to 100, the probability of crossover to 0.7 and the probability of mutation to 0.025. 3.3. Design of the update operator U An effective update operator produces approximation signal, offering an accurate representation of the original signal at the lower resolution. To obtain the opt

39、imal update coeffi cients, the quadratic error of reconstruction is used as the criterion for the update operator 13. It is described as follows: JU Ef s0? s02g Efd 0 ? d02g,(10) where s0and d 0 represent even samples and odd samples of the reconstructed signal x without using the detail signal d. W

40、hen d 0, the inverse lifting scheme is shown in Fig. 1. From Fig. 1, s0and d 0 are given by the following equations: s 0 s,(11) d 0 Pns.(12) During the design of the update operator, our goal is to fi nd the optimal update operator Uoptthat could minimize the criterion JUwhile satisfying the constra

41、ints (5) and (7). Let l be the Lagrange operator, under the constraint (7), the new criterion can be expressed as follows: JUu;l Ef s0? s02g Efd 0 ? d02g l 1 4 ? X N=2 j1 uj 0 1 A. (13) ARTICLE IN PRESS (0) s -UP MERGE d = 0 s x d (0) (0) + + Fig. 1. The inverse lifting scheme with d 0. Z. Li et al.

42、 / Mechanical Systems and Signal Processing 22 (2008) 542556545 Considering Eqs. (3), (5), and (11), we obtain Ef s0? s02g E X N=2 j1 ujdij?1 di?j 2 4 3 5 2 8 : 9 = ;. (14) From Eqs. (2), (3), (5), and (12), we deduce the following result: Efd 0 ? d02g E X N=2 j1 ujs0ij s0i?j1 ? di 2 4 3 5 2 8 : 9 =

43、 ;, (15) where s0i X N=2 r1 prdi?r dir?1.(16) To minimize Eq. (13), partial derivatives of the criterion JU(u,l) with respect to the variables ujand l are expressed as follows: qJUu;l quj 0;j 1;2;.; N=2,(17) qJUu;l ql 0.(18) By setting Ak;l Efdik?1 di?kdil?1 di?l s0ik s0i?k1s0il s0i?l1g,(19) Bk;1 Ef

44、dis0ik s0i?k1g,(20) a linear system with (N=2 1) variables, which is combined by Eqs. (17) and (18), can be rewritten as AX B(21) with A A1;1?A1; N=2 ?1 . . . . . . . . . . . A N=2;1 ?A N=2;N=2 ?1 1?10 2 6 6 6 6 6 4 3 7 7 7 7 7 5 ,(22) X u1;.;u N=2;l? T, (23) B B1;1;.;B N=2;1;1=4? T. (24) According

45、to Eq. (5) and the vector X u1;u2;.;u N=2;l?, we can obtain the optimal update operator Uopt u? N=21;.;u1;.;uN=2?. Because longer time is consumed if the larger values are chosen for the number of prediction coeffi cients and update coeffi cients, the number of prediction operator should be chosen f

46、rom 6, 8, and 10, and the number of update operator should be selected from 4, 6 and 8. 4. Redundant lifting scheme For the classical wavelet transform, a solution for translation invariance is given by redundant wavelet transform, which eliminates the decimation step and retains the information of

47、low- and high-frequency signals. The redundant wavelet transform can also be translated into a redundant lifting scheme 17. ARTICLE IN PRESS Z. Li et al. / Mechanical Systems and Signal Processing 22 (2008) 542556546 Based on the design of signal-adapted lifting scheme mentioned in Section 3, we obt

48、ain the prediction operator P and update operator U which closely match the inspected signal. In the redundant lifting scheme, instead of partitioning a signal x into s(0)and d(0), we let both s(0)and d(0)be x. The redundant prediction operator P(k)and the redundant update operator U(k)are computed

49、by padding the prediction operator P and update operator U with zeros at the corresponding level k. The redundant decomposition results of an approximation signal s(k)at level k with lifting scheme are expressed by following equations: dk1 sk? Pksk,(25) sk1 sk Ukdk1,(26) where d(k+1)and s(k+1)are de

50、tail signal and approximation signal at level k+1. 5. Application of signal-adapted lifting scheme in rotating machinery fault diagnosis To demonstrate the performance of signal-adapted lifting scheme in rotating machinery fault diagnosis, this section presents two application examples for the detec

51、tion of localized defects in rolling bearing and gearbox. 5.1. Detection of the outer-race defect in rolling bearing We use the signal-adapted lifting scheme to identify localized defects on the outer raceway of rolling bearing. The geometric parameters of the tested bearing in the experiment are li

52、sted in Table 1. Fig. 2 shows the outer-race of bearing with localized defects. The vibration signals are picked up at a constant inner-race rotation speed of 385rpm. Based on the geometric parameters and rotating speed, the characteristic frequency can be calculated 3. The frequency of ball-passing

53、-outer-raceway (fbpo) is 46.30Hz. The signals are digitized at a sampling frequency of 25.6KHz. Fig. 3(a) shows the vibration signal of the inspected bearing. Its FFT spectrum on a log amplitude scale is given in Fig. 3(b). According to bearing kinematics and dynamics, impact occurs each time when a

54、 roller encounters the spalls. However, from ARTICLE IN PRESS Table 1 The geometric parameters of the tested bearing Ball diameter68mm Pitch diameter450mm Contact angle01 Number of rolling elements17 Fig. 2. The outer-race of bearing with the localized defects. Z. Li et al. / Mechanical Systems and

55、Signal Processing 22 (2008) 542556547 Fig. 3(a), these impacts are buried in the wideband noise and environmental noise. The meaningful information for detecting the failure is not given in Fig. 3(b). In order to extract the feature components caused by outer-raceway defects, we use the signal-adapt

56、ed lifting scheme to analyze bearing vibration signal shown in Fig. 3(a). The number of prediction coeffi cients is 8, and the number of update coeffi cients is 4. Using the signal-adapted lifting scheme introduced in Section 3, the prediction operator and update operator are calculated, which are a

57、dapted to the vibration signal shown in Fig. 3(a). The prediction operator is 0.1302, ?0.0934, ?0.0159, 0.4791, 0.4791, ?0.0159, ?0.0934, 0.1302, and the update operator is 0.1306, 0.1194, 0.1194, 0.1306. Fig. 4 shows the redundant decomposition results. Evenly spaced impulse clusters can be observe

58、d from the detail signal (d1). The periodic intervals of the impulse clusters are approximately equal to 21.7ms, which is equivalent to the inverse of the frequency of ball-passing-outer-raceway (fbpo). Hence it can be concluded that the impulses are caused by the outer-raceway defect of bearing. To

59、 demonstrate the effi ciency of the signal-adapted lifting scheme, the same vibration signal is analyzed by using Hilbert envelope analysis and non-adaptive lifting scheme. Fig. 5(a) shows the band-pass fi ltered signal (the pass band is 50007000Hz). Its envelope spectrum is shown in Fig. 5(b). From Fig. 5(b), there is not an obvious spectrum line at the characteristic frequency of outer-race defect (46.30Hz). The redundant decomposition results using non-adaptive lifting scheme are shown in Fig. 6, and the approxi