Mechanical Systems and Signal Processing

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1 Mechanical Systems and Signal Processing 5 () Contents lists available at SciVerse ScienceDirect Mechanical Systems and Signal Processing journal homepage: An enhanced Kurtogram method for fault diagnosis of rolling element bearings Dong Wang a,n, Peter W. Tse a,b, Kwok Leung Tsui b a Smart Engineering Asset Management Laboratory (SEAM), and Croucher Optical Non-destructive Testing and Quality Inspection Laboratory (CNDT), Department of Systems Engineering & Engineering Management, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China b Center for System Informatics and Quality Engineering, Department of Systems Engineering & Engineering Management, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China article info Article history: Received May Received in revised form 9 September Accepted October Available online 5 October Keywords: Kurtogram Rolling element bearing Fault diagnosis Wavelet packet transform Low signal-to-noise ratio abstract The Kurtogram is based on the kurtosis of temporal signals that are filtered by the short-time Fourier transform (STFT), and has proved useful in the diagnosis of bearing faults. To extract transient impulsive signals more effectively, wavelet packet transform is regarded as an alternative method to STFT for signal decomposition. Although kurtosis based on temporal signals is effective under some conditions, its performance is low in the presence of a low signal-to-noise ratio and non-gaussian noise. This paper proposes an enhanced Kurtogram, the major innovation of which is kurtosis values calculated based on the power spectrum of the envelope of the signals extracted from wavelet packet nodes at different depths. The power spectrum of the envelope of the signals defines the sparse representation of the signals and kurtosis measures the protrusion of the sparse representation. This enhanced Kurtogram helps to determine the location of resonant frequency bands for further demodulation with envelope analysis. The frequency signatures of the envelope signal can then be used to determine the type of fault that has affected a bearing by identifying its characteristic frequency. In many cases, discrete frequency noise always exists and may mask the weak bearing faults. It is usually preferable to remove such discrete frequency noise by using autoregressive filtering before the enhanced Kurtogram is performed. At last, we used a number of simulated bearing fault signals and three real bearing fault signals obtained from an experimental motor to validate the efficiency of these proposed modifications. The results show that both the proposed method and the enhanced Kurtogram are effective in the detection of various bearing faults. & Elsevier Ltd. All rights reserved.. Introduction Rolling element bearings are widely used in rotating machinery to support rotating shafts, and the major cause of machinery breakdowns is bearing failure. Hence, it is necessary to detect bearing faults at an early stage. Rolling element bearings usually consist of an inner race, an outer race, several rollers and a cage. When the surface of one of these components develops a localised fault, the impacts generated excite the resonant frequencies of the bearing and adjacent components, and induce a modulating phenomenon []. Demodulation of the original signal with envelope analysis can n Corresponding author. Tel.: þ85 68; fax: þ addresses: wangdonguestc@yahoo.cn (D. Wang), meptse@cityu.edu.hk (P.W. Tse), kltsui@cityu.edu.hk (K.L. Tsui) /$ - see front matter & Elsevier Ltd. All rights reserved.

2 D. Wang et al. / Mechanical Systems and Signal Processing 5 () reveal additional fault-related signatures []. To enhance the signal-to-noise ratio of the original signal, a band-pass filter is usually set manually to maintain the desired resonance frequency band before demodulation is performed. Antoni [] recently analysed spectral kurtosis thoroughly. A short-time Fourier transform (STFT)-based spectral kurtosis was then investigated for the diagnosis of rotating machine faults []. To reduce computing time, a /-binary tree Kurtogram estimator was proposed to perform fast on-line fault detection [5]. Since that time, improvements of both spectral kurtosis and the Kurtogram have attracted a great deal of attention. To clarify the impulses, Sawalhi et al. [6] proposed a method that combined minimum entropy deconvolution with spectral kurtosis. Their results showed that this method can hone impulses and increase the values of kurtosis. Combet and Gelman [7] presented spectral kurtosis-based optimal filtering for the residual signal of gears. The results showed that their method could enhance small transients in gear vibration signals. Zhang and Randall [8] considered that the fast Kurtogram was only an approximate estimation and therefore combined it with a genetic algorithm to determine the optimal centre frequency and bandwidth for resonance demodulation with envelope analysis. Barszcz and Randall [9] reported that a spectral kurtosis-based method was more effective than others for the early detection of tooth crack. Wang and Liang [] proposed an adaptive spectral kurtosis method that could adaptively determine the bandwidth and centre frequency of a filter by merging right-expanded windows to make the kurtosis of the filtered signal maximum. Lei et al. [] thought that wavelet packet transform (WPT) filters could process non-stationary transient vibration signals more efficiently than STFT. Therefore, they replaced STFT with WPT to improve the original Kurtogram. Barszcz and Jab"oński [] found that temporal signal-based kurtosis can be considerably affected by noise, and proposed a novel method called the Protrugram, which calculated the kurtosis of envelope spectrum amplitudes. It is logical to measure kurtosis in the frequency domain. When a bearing is healthy, its envelope spectrum is randomly distributed over that of the whole frequency. However, when it has localised faults, bearing fault characteristic frequencies dominate the envelope spectrum [,]. Peak values of bearing fault characteristic frequency and its harmonics can be measured by kurtosis with higher values because kurtosis can be used to measure the protrusion of a signal. Moreover, other faults, such as misalignment, eccentric fault and so on, can also be measured by kurtosis in the frequency domain because the envelope spectrum actually is a sparse representation of the envelope. In contrast to the study of Lei et al. [], we replaced kurtosis of the temporal signals extracted from wavelet packet nodes with that of the power spectrum of the envelope of the signals extracted from wavelet packet nodes. Moreover, the power spectrum was used because it can enhance fault frequency more effectively than the Fourier spectrum used by Barszcz and Jab"oński []. The remainder of paper is organised as follows. Section describes the basic concept of spectral kurtosis and the binary WPT for the decomposition of the frequency support of the original signal. In Section, a new detection method and the enhanced Kurtogram are proposed for the detection of rolling element bearing faults. In Section, we used simulated data and real data obtained in a laboratory to validate the proposed method and the enhanced Kurtogram. In Section, the comparison of the proposed method with the fast Kurtogram proposed by Antoni [5] and the improved Kurtogram proposed by Lei et al. [] are conducted to analyse the same signals. Our conclusions are summarised in Section 5.. The review of spectral kurtosis and binary wavelet packet transform.. Spectral kurtosis According to Wold Cramér representation, a zero-mean non-stationary random process x(n) can be decomposed into [5]: xðnþ¼ Z þ = = Hn,f ð Þe jpfn dz x ðf Þ ðþ where dz x (f) is a spectral increment and H(n,f) is the complex envelope of x(n) (the time varying transfer function of the system) at frequency f. Therefore, the spectral kurtosis can be represented by the fourth-order normalized cumulant [5]: K x ðþ¼ f /9H ð n,f Þ9 S ðþ /9Hðn,f Þ9 S where /ds stands for the temporal averaging operator. isusedineq.() because H(n, f) iscomplex.consideringthe presence of stationary additive noise, spectral kurtosis of the signal y(n) is described by [5]: K y ðþ¼ f K xðf Þ ðþ ½þrðf ÞŠ where r(f) is the noise to signal ratio at frequency f. Therefore, the spectral kurtosis is able to detect and localize the presence of non-stationarities represented by a signal. Antoni [,5] proposed two methods to calculate spectral kurtosis. One is based on Short Time Fourier Transform (the called Kurtogram for finding the optimal filter) [] and the other is based on / binary filter banks (fast Kurtogram for on-line condition monitoring and fault diagnosis) [5]. More details can be found in the original papers [,5]. Spectral kurtosis has been successfully used in many cases [, 9]. It can be one of the solutions to blind component separation that is a useful concept proposed by Antoni [5], for decomposing a vibration signal into periodic components, random transient components and random stationary components. Recently, Lei et al. [] indicated that a short-time Fourier transform (STFT)-based or filters-based spectral kurtosis was not as precise as wavelet packet transform is.

3 78 D. Wang et al. / Mechanical Systems and Signal Processing 5 () Therefore, they used WPT to replace the STFT in extracting transient characteristics, and measured the kurtosis of the temporal signal filtered by WPT. In another work, Barszcz and Jab"oński [] proposed a novel Protugram to measure the kurtosis of the envelope spectrum in the case of lower signal to noise ratio. They also pointed out that Kurtogram could find the frequency band with high kurtosis as the optimal band, but sometimes being incorrect []... Binary wavelet packet transform (WPT) [6] Assume a space V j is decomposed into a lower resolution or coarser space V j þ and a detail space W j þ by multiresolution approximation. In the above process, the orthogonal basis f j (t j n) naz of V j should be divided into f j þ (t j þ n) naz of V j þ and W j þ (t j þ n) naz of W j þ. It is obvious that only the approximated spaces V j are recursively used to construct the detail spaces and wavelet bases. For further division of the detail spaces, it must consider orthogonal splitting (of detail spaces). Wavelet packets offset the disadvantage of wavelets with multiresolution approximations. Assume a space W p j and its orthonormal basis c p j ðt j nþ nz, where j is its depth and p is the number of nodes. Therefore, the orthogonal basis of the space at node (j, p) can be divided into two new orthogonal bases as follows: c p j þ ðtþ¼ Xþ n ¼ hðnþc p j ðt j nþ, c p þ Xþ j þ ðtþ¼ gðnþc p j ðt j nþ, n ¼ hðnþ¼/c p j þ ðuþ,cp j ðt j nþs, ðþ ð5þ ð6þ gðnþ¼/c p þ j þ ðuþ,cp j ðt j nþs ð7þ here, h(n) and g(n) is a pair of conjugate mirror filters, and /,S is the inner product. In terms of c p j þ ðt j þ nþ nz and c p þ j þ ðt j þ nþ nz, orthonormal bases of two orthogonal spaces W p þ and Wp, respectively, it is deduced that: j þ j þ W p j þ W p þ j þ ¼ W p j, ð8þ where is the direct sum. Consequently, the iterative splitting steps result in two orthogonal spaces for each node (j, p). For example, the maximum depth J is set to. According to Eq. (5), W can be recursively decomposed as: W ¼ p ¼ Wp ¼ p ¼ Wp ¼7 p ¼ Wp ð9þ To generalize Eq. (9), W can be recursively decomposed at the maximum depth J by Eq. (8) as: W ¼ p ¼ Wp ¼ p ¼ Wp ¼¼J p ¼ Wp J ðþ On the other hand, considering the Fourier transform of Eqs. () and (5), the orthogonal basis at note (j, p) can be shown as: c p j þ ðoþ¼h½j oš c p j ðoþ, c p þ j þ ðoþ¼g½j oš c p j ðoþ, ðþ Due to the energy concentration of h½ j oš and g ½ j oš in their own orthogonal frequency bands, Eqs. () and () can be interpreted as a division of frequency support of c p j ðoþ. Therefore, it is concluded that division of frequency support of ðoþ at maximum depth J can be given as follows: c ðþ c ðoþ¼ p ¼ c p ðoþ¼ p ¼ c p ðoþ¼ ¼ J p ¼ c p J ðoþ: ðþ Moreover, it should be pointed out that frequency support c p j ðoþ at the same depth j has the same bandwidth. For any node (j, p), wavelet packet coefficients of the original signal xðtþ W can be calculated by taking the inner product of the original signal with every wavelet packet basis: d p j ½nŠ¼/xðtÞ,cp j ðt j nþs ðþ For the fast wavelet transform algorithm, wavelet packet coefficients are calculated in the decomposition: d p j þ ½nŠ¼dp j nhð nþ, ð5þ d p þ j þ ½nŠ¼dp j ngð nþ, ð6þ where * is convolution operator. By applying the recursive splitting of Eqs. () (6), wavelet packet coefficients are calculated in the reconstruction: d p j ½nŠ¼e d p j þ nhðnþþe d p þ j þ ngðnþ, ð7þ

4 D. Wang et al. / Mechanical Systems and Signal Processing 5 () x(t) Node (,) Depth Node (,) -Fs/ Hz -Fs/ Hz Node (,) Fs/-Fs/ Hz Depth Node (,) -Fs/8 Hz Node (,) Fs/8-Fs/ Hz Node (,) Fs/-Fs/8 Hz Node (,) Fs/8-Fs/ Hz Depth Node (,) Node (,) -Fs/6 Hz Fs/6- Fs/8 Hz Node (,) Fs/8-Fs/6 Hz Node (,) Fs/6- Fs/ Hz Node (,) Fs/- 5Fs/6 Hz Node (,5) 5Fs/6-Fs/8 Hz Node (,6) Fs/8-7Fs/6 Hz Node (,7) 7Fs/6-Fs/ Hz Fig.. Three-depth binary wavelet packet decomposition tree. where e d means inserting a zero between each sample of d. The process of the frequency support decomposition using wavelet packet transform is illustrated in Fig., where maximum depth is equal to and Fs is the sampling frequency.. The proposed method for bearing fault diagnosis Kurtosis for the measurement of temporal signals in the diagnosis of machinery faults has attracted a great deal of attention [ ]. However, when the signal-to-noise ratio is low, the use of kurtosis to detect impulses hidden in the signal is difficult because potential periodic peak values are overwhelmed by unexpected heavy noises. In addition, kurtosis may have greater values for non-gaussian noises. Recently, Barszcz and Jab"oński [] proposed a novel method called the Protrugram to measure kurtosis in the frequency domain. Their results showed that this is an effective method of diagnosing fault signals with a low signal-to-noise ratio, and the use of kurtosis to measure the envelope spectrum is logical. When a bearing is healthy, the envelope spectrum is randomly distributed over that of the whole frequency, but once it develops a localised fault, the bearing fault characteristic frequency and its harmonics become the major components in the envelope spectrum [,]. Therefore, the envelope signal or its counterpart in the frequency domain usually contains much more fault information than the original signal [,]. Lei et al. [] believed that WPT had good local properties in the domains of both time and frequency, and could extract transient characteristics more efficiently than STFT. Thus, they proposed a method based on kurtosis of the temporal signals extracted from wavelet packet nodes. As mentioned above, it is difficult to identify fault-related signatures hidden in a signal with a low signal-to-noise ratio using kurtosis of the temporal signal. This paper proposes a new method, which is based on binary WPT and kurtosis measurements in the frequency domain. A flowchart of the proposed method is shown in Fig. and the details are described below. Step. The original vibration signal measured by an accelerometer is first loaded. In many cases, the discrete frequency noise caused by low frequency periodic components, such as shaft rotating frequency, misalignment and eccentric fault, always exists and may mask the weak bearing signals. Refs. [] and [5] suggest that it is preferable to separate the bearing fault signal from the discrete frequency noise before the bearing fault signal is analysed. Some other options [] for removing discrete frequency noise are linear prediction, adaptive noise cancellation, self-adaptive noise cancellation, discrete/random separation, time synchronous averaging and eigenvector algorithm [7]. In this paper, prewhitening processing could be performed prior to the analysis of bearing fault signal if there is a disturbance caused by discrete frequency noise. Autoregressive model (AR) is a popular method to establish the deterministic periodic components. Considering an additive noise term v(n), the mathematical equation of AR for an actual signal x(n) is expressed as follows [8]: xðnþ¼ X p ¼ q aðpþxðn pþþvðnþ, ð8þ p ¼ The coefficients a(p) are the parameters of AR model, which could be obtained by the solution of the Yule Walker equations through the Levinson Durbin recursion algorithm (LDR) or Burg s method (BM) [8]. The order of model q could be decided by minimizing Akaike information criterion (AIC) given as [8]: AICðqÞ¼ lnðvn ð ÞÞþ ðqþþ, ð9þ N

5 8 D. Wang et al. / Mechanical Systems and Signal Processing 5 () Start Load the original vibration signal and perform pre-whitening processing Perform binary wavelet packet transform on the original signal at different depths and reconstruct the obtained signals as the same temporal length with the original signal Generate the enhanced Kurtogram Select the valuable node signals from all of the node signals for further investigation Demodulate the selected node signal by Hilbert transform to obtain the envelope Perform power spectrum on the envelope signal and identify the bearing fault characteristics End Fig.. Flowchart of the proposed method. where N is the length of the signal x(n). Because the Fourier transform of the convolution of two signals is equal to the product of the Fourier transform of the two signals, the Fourier transform of Eq. (8) is given as follows: vðf Þ¼xðfÞaðfÞ: ðþ Eq. () shows that only random components are retained, which means that transient impulses and stationary noise are left. The temporal signal v(n) ofv(f) is said to be pre-whitened. The above process is usually called pre-whitening processing. Binary WPT was then conducted on random components v(n) at different depths for the enhancement of bearing fault signal. The minimum bandwidth decomposed by binary WPT at maximum depth needed to be three times longer than the inner race fault frequency so that sufficient bearing fault-related signatures could be retained in the desired frequency band, and thus provided an alternative method of determining the maximum depth for binary WPT. For wavelet packet coefficients at a specific node to have the same temporal length as the original signal, those obtained at a specific node were reconstructed by setting those at the other nodes to zero. For simplification, the same notation d p j ðnþ is used to express these reconstructed wavelet packet coefficients. Step. It should be noted that the proposed method differs from both of those in Refs. [] and []. The kurtosis of the envelope spectrum was used to improve that of the temporal signal in Ref. [] to determine fault-related signatures in a signal with a low signal-to-noise ratio. The power spectrum was used to map the envelope signal into the frequencydependent signal because the Fourier transform used in Ref. [] is affected by residual noise in the filtered signal. The core of the proposed method, the enhanced Kurtogram, was prepared as follows. First, it was necessary to obtain a desired envelope signal that contains many fault-related signatures. The envelope signal e p j ðnþ for each node at different depths was obtained by taking the modulus of an analytical signal z p j ðnþ¼zp j real ðnþþizp ðnþ. An alternative approach j imag [9] for calculating the analytic signal z p j ðnþ of a discrete signal dp j ðnþ is introduced in the following. Assume the length of

6 D. Wang et al. / Mechanical Systems and Signal Processing 5 () the discrete-time signal is N and even. The N-point discrete Fourier transform (DFT) of the discrete signal is given as: D p j ðmþ¼ XN d p j ðnþexp ipmn, m ¼,,...,N : ðþ N n ¼ In order to satisfy the analytic-like properties, two constraints are used: z p j real ðnþ¼dp j ðnþ, n ¼,,,...,N, ðþ XN n ¼ z p j real ðnþzp j imag ðnþ¼: ðþ Construct N-point one-side discrete analytic signal transform: 8 D p j ðþ m ¼ >< D p Z p j ðmþ m ¼,:...,N : ðmþ j D p N j, m ¼ N, >:, m ¼ N þ, N þ,...,n : ðþ Consider an N-point inverse DFT of Z p j ðmþ, then the analytic signal zp ðnþ is obtained by: j P N z p n ¼ j ðnþ¼ Zp j ðmþexp ipmn=n N ¼ z p j real ðnþþizp j imagðnþ, m ¼,,...,N : ð5þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi After the envelope signal e p j ðnþ¼ ðz p j real ðnþþ þðz p j imag ðnþþ is obtained by taking the modulus of Eq. (5), the power spectrum E p ðmþ can be calculated by either parametric or non-parametric approaches. In the non-parametric approach, j Fourier transform of the autocorrelation function and the segment averaging method are two popular methods to estimate the power spectrum. The bias and variance considerations of the power spectra obtained by different methods are discussed in Ref. []. When a bearing suffers from a localized fault (in this case it is the bearing outer race fault), a few coefficients (bearing fault characteristic frequency and its harmonics) of E p ðmþ dominate the envelope spectrum. Then, kurtosis is employed to quantify j the protrusion of E p ðmþ. Therefore, the value of kurtosis is high in the envelope spectrum of bearing fault signal. The same j approach can be applied to inner race fault signals and rolling element fault signals. On the other hand, if there is no bearing fault, the protrusion of E p ðmþ will not be so observed. The value of kurtosis in the power spectrum should be relatively low. j Fig. shows the power spectra of envelope of a normal bearing signal and fault bearing signals (The data are provided by Bearing Data Centre []). The protrusion of the power spectra of the envelope of these signals is quantified by kurtosis. Assume the envelope signal has zero mean. Their corresponding kurtosis values are shown in Fig..KurtosisofE p ðmþ for each j node was calculated as: P N m ¼ Kj,p ð Þ¼ ðep j ðmþþ =N ð P,rjrJ,rpr J, ð6þ N m ¼ ðep j ðmþþ =NÞ Since the power spectrum is a real and positive function, it is not necessary to subtract the mean value from the power spectrum. Once the kurtosis values of all nodes had been calculated, the enhanced Kurtogram is paved. For example, the enhanced Kurtogram at maximum depth is plotted in Fig.. Step. A colour map is used to represent kurtosis of all nodes in Fig.. The depth of the colour values was proportional to the level of the kurtosis values. In the case of one resonant frequency band, the node with the maximum kurtosis was deemed worthy of further analysis (a simulation with one resonant frequency band is analysed in Section.). When more than one resonant frequency band can be seen, the first few maximum values at different frequency bands can be considered for further analysis (a simulation with two resonant frequency bands is analysed in Section.). After the nodes that could be assessed for bearing fault diagnosis had been selected, demodulation with envelope analysis was performed on the selected nodes to change the high-frequency signal to a low-frequency signal (bearing fault characteristic frequency-related signal). Step. Autocorrelation was performed on the envelope of the desired signal filtered by WPT to extract potential periodic characteristics in the time domain. Its Fourier transform, namely the power spectrum, was used to map the time-dependent signal into the frequency-dependent signal to identify bearing fault characteristic frequencies, which are usually calculated by Eqs. (7) (). The outer-race fault characteristic frequency f O, the inner-race fault

7 8 D. Wang et al. / Mechanical Systems and Signal Processing 5 () Kurtosis= Sparse representation of outer race fault signal Kurtosis= Sparse representation of inner race fault signal Kurtosis= Fig.. Power spectra of the envelope of signals. (a) Normal bearing signal; (b) bearing outer race fault signal; (c) bearing inner race fault signal (Note: the resolution of all spectra is equal to.5 Hz). Depth K(j,p) K (,) K (,) K (,) K (,) K (,) K (,5) K (,6) K (,7) K (,) K (,) K (,) K (,) K (,) K (,) Fs/8 Fs/ Fs/8 Fs/ Fig.. The paving of the enhanced Kurtogram. characteristic frequency f I, the rolling element fault characteristic frequency f B, the fundamental cage frequency f C and the ball spinning frequency f BS were formulated as follows []: f O ¼ Z f s d D cos a, ð7þ f I ¼ Z f s þ d D cos a, ð8þ f B ¼ D f s d f C ¼ f s d D cos a f BS ¼ D f s d d! D cos a, ð9þ d! D cos a, ðþ ðþ

8 D. Wang et al. / Mechanical Systems and Signal Processing 5 () where f s is the shaft rotating frequency in Hz, d and D are diameters of the rolling element and pitch diameter, respectively. Z is the number of rolling elements and a is the contact angle.. The proposed method validated by both simulated and real case studies.. Case : Simulated bearing fault signals with single resonant frequency We used the similar simulated signal as that given in Refs. [,], because a bearing fault signal consists of periodic bursts of exponentially decaying sinusoidal vibration []. Signals with single resonant frequency enhanced by different signal-to-noise ratios were considered initially. The selection of mother wavelet function for wavelet analysis is a hot topic. Among the wavelet families, Daubechies family (db M) is the most attractive one for discrete wavelet analysis because their daughter wavelets are orthogonal, biorthogonal and compact supported. Associated scaling filters are minimum-phase filters. Its support width, filter length and the number of vanishing moments are M-, M and M. For bearing fault diagnosis, the low-order Daubechies wavelets, such as db [], db5 [5], db [,6] and db [7] are more preferable to be used in discrete wavelet analysis. A Daubechies wavelet used in Lei s work [] was employed in our method to implement binary WPT. The simulated signal with a resonant frequency is given as: yðkþ¼ X e b ð k rfs=f m tr Þ=Fs sin pf k r Fs=f m t r =FsÞ, ðþ r where b is equal to 9, f m is the fault characteristic frequency (equal to Hz), Fs is the sampling frequency set to, Hz, t r, which is subject to a discrete uniform distribution, is used to simulate the randomness caused by slippage and f is the resonant frequency, equal to 7 Hz. A total of, samplings were used for each simulated signal. To display the temporal signal clearly, only 5 samplings are shown in both Sections. and.. A normally distributed random signal with a mean of and a variance of.5 were added to Eq. (). It is noted that the simulated bearing fault signal is an ideal one that is not interrupted by low frequency periodic components. Therefore, in the simulated cases of Sections. and., pre-whitening processing is not performed on the simulated signal. The simulated signal, the noise signal and the mixed signal are shown in Fig. 5(a c). The proposed method was used to analyse the mixed signal shown in Fig. 5(c). The enhanced Kurtogram is paved in Fig. 6(a). Node (, ) has the highest kurtosis of all the nodes. Moreover, it should be noted that the most useful node at each depth can be clearly identified, such as node (, ) at depth one, node (, ) at depth two, node (, ) at depth three and node (, ) at depth four. The temporal signal corresponding to node Samplings Fig. 5. Signals in the time domain: (a) the simulated signal with one resonance frequency; (b) the noise signal with a normal distribution and a variance of.5 and (c) the mixed signal.

9 8 D. Wang et al. / Mechanical Systems and Signal Processing 5 () Depths fm fm fm fm Fig. 6. The results obtained by the proposed method for processing the mixed signal with one resonant frequency. (a) The enhanced Kurtogram used in this paper and (b) power spectrum of the envelope of the signal extracted from node (, ) by wavelet packet transform. (, ) was then extracted by WPT. The power spectrum of the envelope of the signal extracted from node (, ) by WPT is given in Fig. 6(b), in which the fault frequency, Hz, and its harmonics are evident. The original fast Kurtogram proposed by Antoni [5] and the improved Kurtogram proposed by Lei et al. [] are applied to analyse the same mixed signal with one resonant frequency in the case of heavy noise. The common point of these two methods is that the kurtosis is computed from the envelopes of the temporal signals obtained by a set of filters. One of their limitations is that a high kurtosis value may be caused by other components rather than by bearing faults []. Their diagrams are shown in Figs. 7 and 8(a), where it is found that both the fast Kurtogram and the improved Kurtogram fail to detect the single resonant frequency band. The fault characteristic frequency Hz and its harmonics are difficult to identify from their envelope spectra shown in Figs. 7 and 8(b). As a result, it is illustrated that the fast Kurtogram and the improved Kurtogram based on kurtosis measurement from the temporal signal is difficult to indicate the resonant frequency band in the case of a low signal-to-noise ratio. Finally, a group of normally distributed random signals with a mean of and different variances (from. to.9 with step length¼.5) were added to Eq. (). Fig. 9 shows 7 mixed signals with different noise variances. The potential periodic characteristics of these mixed signals cannot be detected even though the simulated signal was mixed with a noise variance of. (the top row of Fig. 9). The results obtained by the proposed method are shown in Figs. and. Fig. shows temporal signals (autocorrelation of the envelope of the desired signals filtered by WPT) obtained. It is clear that this method is effective in extracting the potential periodic characteristics of the fault even when large noise variances are added to the original simulated signal. Fig. shows the frequency-dependent results (power spectra of the envelope of the desired signals filtered by WPT) obtained by the proposed method, from which the fault frequency, Hz, and its harmonics are clearly visible. This illustrates that the proposed method can be applied to the detection of early bearing fault signals that are usually overwhelmed by heavy noise... Case : Simulated bearing fault signals with two resonant frequencies In the second case study, another resonant frequency f (9 Hz) was added to Eq. (). Its final formation was: yðkþ¼ X e b ð k rfs=f m tr Þ=Fs ½sinðpf ðk r Fs=f m t r Þ=FsÞþsinðpf k r Fs=f m t r =FsÞŠ, ðþ r

10 D. Wang et al. / Mechanical Systems and Signal Processing 5 () Level k x Fig. 7. The results obtained by the fast Kurtogram for processing the mixed signal with one resonant frequency. (a) The fast Kurtogram and (b) frequency spectrum of the envelope of the signal obtained by a filter (the center frequency of 5 Hz and the bandwidth of Hz). Owing to the increased amplitudes of the overlapping temporal signal shown in Fig. (a), normally distributed random signals with a mean of and a variance of.6 were introduced into Eq. (). The simulated signal, the noise signal and the mixed signal are shown in Fig. (a c). The enhanced Kurtogram was first applied to the mixed signal with two resonant frequencies. The paving of the enhanced Kurtogram shown in Fig. (a) illustrates that the node with the highest kurtosis is located at (, ). At the same depth of, the second highest kurtosis corresponds to node (, ), which indicates the existence of another resonant frequency band and provides an alternative node for further analysis. Fig. (a) also clearly shows the locations of the two resonant frequencies at each depth. The power spectrum of the envelope of the signal extracted from node (, ) by WPT is plotted in Fig. (b), where it can be seen that the proposed method successfully detects the fault frequency, Hz, and its harmonics. The fast Kurtogram and the improved Kurtogram are used to analyse the same mixed signal with two resonant frequencies, respectively. The paving of the fast Kurtogram is shown in Fig. (a), where it is indicated that the optimal filter has a centre frequency of 55 Hz and a bandwidth of Hz. The frequency spectrum of the envelope of the signal is plotted in Fig. (b), where the fault characteristic frequency Hz is difficult to be detected. The paving of the improved Kurtogram is given in Fig. 5(a). Different from the simulated resonant frequencies, the improved Kurtogram indicates a wrong location for selecting the most useful node. The frequency spectrum of the envelope of the signal extracted from node (, 8) by wavelet packet transform is shown in Fig. 5(b), in which it is found that the frequency spectrum cannot provide any fault information to indicate the existence of the fault characteristic frequency of Hz. Consequently, in the case of a low signal-to-noise ratio, the proposed method is better than the fast Kurtogram and the improved Kurtogram for indicating the existence of the resonant frequency bands. Besides, the enhanced Kurtogram can simultaneously indicate multiple resonant frequency bands in this case. Finally, a group of normally distributed random signals with a mean of and different variances (from. to.9 with a step length of.5) were added to Eq. (). Fig. 6 shows 7 mixed signals with 7 different noise variances in the case of two resonant frequencies. The results (autocorrelation of the envelope of the desired signals filtered by WPT) obtained by the proposed method are shown in Fig. 7. Although strong noise variances were added to Eq. (), the proposed method is still effectively able to detect fault frequency-related signals at different noise variances. The fault frequency of Hz and

11 86 D. Wang et al. / Mechanical Systems and Signal Processing 5 () Depths Fig. 8. The results obtained by the improved Kurtogram proposed by Lei et al. for processing the mixed signal with one resonant frequency. (a) The improved Kurtogram and (b) frequency spectrum of the envelope of the signal extracted from node (, 7) by wavelet packet transform. Noise Variances Samplings Fig. 9. The mixed signal with different noise variances in the case of one resonant frequency. Signal to Noise Raito its harmonics are also easy to identify in Fig. 8. These results demonstrate that the proposed method can effectively detect fault signatures with a low signal-to-noise ratio... Different bearing fault signals obtained from an experimental motor and analysed by the proposed method A small AC motor with a speed around rpm in the Smart Engineering Asset Management Laboratory was used as the basic drive. The experimental setup and bearings are shown in Fig. 9(a) and (b). Three types of localised bearing faults were seeded, including an outer race fault, an inner race fault and a rolling element fault. The locations of the bearing faults are given in Fig. 9(c e). The sampling frequency was set at 8 khz. Each fault signal with the length of 75, samplings was used and they are shown in Fig. (a c), respectively. Bearing outer race, inner race, fundamental cage frequency and ball spinning frequency were calculated as 6 Hz, 9 Hz, 9.7 Hz and 6 Hz by Eqs. (7) ().

12 D. Wang et al. / Mechanical Systems and Signal Processing 5 () Noise Variances Signal to Noise Raito Samplings Fig.. Results (autocorrelation signals of the envelope of the signal filtered by wavelet packet transform) obtained by the proposed method in the time domain in the case of one resonant frequency Noise Variances Signal to Noise Raito Frequency(Hz) Fig.. Results (power spectra of the envelope of the signal filtered by wavelet packet transform) obtained by the proposed method in the frequency domain in the case of one resonant frequency. In order to reduce the signal to noise ratio of the original bearing fault signals and remove the influence caused by the shaft rotating frequency, pre-whitening processing was conducted on all original fault signals. The values of the Akaike information criterion for the different orders of the AR for processing the bearing outer race, inner race and ball fault signals are plotted in Fig. (a c), separately. The parameter of the AR filtering, namely the order of AR model, was chosen as by minimizing the Akaike information criterion. In Fig. (a c), temporal signals obtained by pre-whitening processing contained the outer race, inner race and roller fault signatures, respectively. These pre-whitened signals were equally analysed in the following sections by using the enhanced Kurtogram, the improved Kurtogram and the fast Kurtogram, respectively.... Case : A real bearing outer race fault signal obtained from an experimental motor To identify the bearing fault characteristic frequency correctly, the enhanced Kurtogram was applied to the signal shown in Fig. (a) to determine the most useful node of all. The colour map obtained is shown in Fig. (a), where node (, ) is the best among all nodes. This node was selected for further analysis by the proposed method. Power spectrum of the envelope of the signal extracted from node (, ) by WPT is shown in Fig. (b). From the result shown in Fig. (b), bearing outer race fault-related signatures are easily identified by inspecting the outer race fault characteristic frequency and its harmonics. For comparison, the fast Kurtogram and the improved Kurtogram are applied to the same outer race fault signal shown in Fig. (a). The paving of the fast Kurtogram is plotted in Fig. (a), where an optimal filter with the optimal centre frequency of,5 Hz and the bandwidth of 5 Hz is found. In Fig. (b), envelope spectrum of the signal obtained by the optimal filter provides outer race fault signatures for bearing fault diagnosis. In Fig. 5(a), the improved Kurtogram indicates that the most valuable node is (, 7) which is obtained by maximizing the kurtosis of the temporal signal filtered by wavelet packet transform. In Fig. 5(b), the frequency spectrum of the envelope of the signal extracted from node (, 7) by wavelet packet transform reveals outer race fault signatures. In the case of outer race fault diagnosis, although the three methods are effective in detecting the outer race localized faults, frequency spectrum obtained by the proposed method is clearest to show outer race fault characteristic frequency and its harmonics. Frequency spectra obtained by the other methods contain heavy noise which corrupts the visual inspection ability.

13 88 D. Wang et al. / Mechanical Systems and Signal Processing 5 () Samplings Fig.. Signals in the time domain: (a) the simulated signal with two resonance frequencies; (b) the noise signal with a normal distribution and a variance of.6 and (c) the mixed signal. 5 Depths f m f m f m f m Fig.. The results obtained by the proposed method for processing the mixed signal with two resonant frequencies. (a) The enhanced Kurtogram used in this paper and (b) power spectrum of the envelope of the signal extracted from node (, ) by wavelet packet transform.

14 D. Wang et al. / Mechanical Systems and Signal Processing 5 () Level k x Fig.. The results obtained by the fast Kurtogram for processing the mixed signal with two resonant frequencies. (a) The fast Kurtogram and (b) frequency spectrum of the envelope of the signal obtained by a filter (the center frequency of 55 Hz and the bandwidth of Hz).... Case : A real bearing inner race fault signal obtained from an experimental motor The enhanced Kurtogram was then applied to the signal in Fig. (b). In Fig. 6(a), it can be seen that node (, 8) has the highest kurtosis and would thus be useful in extracting the bearing fault feature. The final result generated by the proposed method is plotted in Fig. 6(b). Fig. 6(b) displays an inner race fault characteristic frequency and its several harmonics, which indicate that the bearing inner race had localised faults. For comparison, the fast Kurtogram is applied to the same inner race fault signal. The paving of the fast Kurtogram is shown in Fig. 7(a), where an optimal filter with the centre frequency of 8, Hz and the bandwidth of Hz is automatically chosen. The envelope spectrum of the signal filtered by the optimal filter shows fault related signatures in Fig. 7(b). The improved Kurtogram is used to analyse the same inner race fault signal. The paving of the improved Kurtogram is plotted in Fig. 8(a), in which the node (, 8) is the most useful node for providing inner race fault signatures. The envelope spectrum of the signal extracted from node (, 8) by WPT provides information about inner race faults in Fig. 8(b). The most value node (, 8) obtained by the enhanced Kurtogram coincides with that obtained by the improved Kurtogram. The results shown in Figs. 6 and 8(b) are power spectrum and frequency spectrum of the envelope of the signal extracted from the same node (, 8), respectively. It is clearly found that the power spectrum is capable of keeping inner race fault dominating frequency and depressing heavy noise. In the case of inner race fault diagnosis, the three methods are effective in detecting inner race fault characteristic frequency. From the results shown in Figs. 6 8(b), envelope spectra obtained by the proposed method and the improved Kurtogram have better visual inspection ability than the envelope spectrum obtained by the fast Kurtogram.... Case 5: A real bearing ball fault signal obtained from an experimental motor Finally, the proposed method was applied to the signal in Fig. (c). The result provided by the enhanced Kurtogram in Fig. 9(a) shows that node (, ) has the highest kurtosis. The final power spectrum of the envelope of the signal extracted from node (, ) by WPT is shown in Fig. 9(b). A ball defect is successfully diagnosed through the identification of a rolling element fault characteristic frequency. For comparison, both the fast Kurtogram and the improved Kurtogram are applied to the same ball localized fault signal. In Fig. (a), the optimal filter indicated by the fast Kurtogram has the centre frequency of, Hz and the bandwidth of 6666 Hz. In Fig. (b), the frequency spectrum of the envelope of the signal filtered by the optimal filter indicates the existence of ball localized faults. However, the fault signatures are not obvious enough. Fault related fault frequencies are nearly overwhelmed by heavy noise. In Fig. (a), the paving of the improved Kurtogram indicates that the node (, ) is the most useful node among all nodes for bearing ball localized fault diagnosis.

15 9 D. Wang et al. / Mechanical Systems and Signal Processing 5 () Depths Fig. 5. The results obtained by the improved Kurtogram proposed by Lei et al. for processing the mixed signal with two resonant frequencies. (a) The improved Kurtogram and (b) frequency spectrum of the envelope of the signal extracted from node (, 8) by wavelet packet transform. Noise Variances Samplings Signal to Noise Raito Fig. 6. The mixed signal with different noise variances in the case of two resonant frequencies. Frequency spectrum of the envelope of the signal extracted from node (, ) by WPT is effective in detecting the ball fault characteristic frequency. The visual inspection ability of the result obtained by the improved Kurtogram is not as good as that of the result obtained by the proposed method. In the case of ball localized fault diagnosis, although the three methods are effective in detecting ball localized faults, the frequency spectrum obtained by the proposed method provides the best visual inspection ability for bearing ball localized fault diagnosis. In Section, the simulated bearing fault signals and the real bearing fault signals have been simultaneously analysed by the enhanced Kurtogram, the fast Kurtogram and the improved Kurtogram. The results of performance comparison of the enhanced Kurtogram, the fast Kurtogram and the improved Kurtogram are summarized in Table, where Case concerns the mixed signal with one resonant frequency; Case concerns the mixed signal with two resonant frequencies; Case concerns the real experimental outer race fault signal; Case concerns the real experimental inner race fault signal and Case 5 concerns the real experimental ball fault signal. From the results shown in Table, it is found that the enhanced

16 D. Wang et al. / Mechanical Systems and Signal Processing 5 () Noise Variances Signal to Noise Raito Samplings Fig. 7. Results (autocorrelation signals of the envelope of the signal filtered by wavelet packet transform) obtained by the proposed method in the time domain in the case of two resonant frequencies. Noise Variances Signal to Noise Raito Fig. 8. Results (power spectra of the envelope of the signal filtered by wavelet packet transform) obtained by the proposed method in the frequency domain in the case of two resonant frequencies. Fig. 9. Experiment equipment and faulty elements of the tested bearings. (a) An AC motor with speed rpm, (b) tested bearing (SKF 6 EKTN9), (c) an outer race fault, (d) an inner race fault and (e) a ball fault.

17 9 D. Wang et al. / Mechanical Systems and Signal Processing 5 () Samplings x Samplings x Samplings x Fig.. The original signals in time domain (a) bearing with localized outer race fault; (b) bearing with localized inner race fault and (c) bearing with localized rolling element fault. - AIC The order of AR - AIC The order of AR -6 AIC The order of AR Fig.. The values of AIC for the different orders of AR for processing (a) the bearing with localized outer race fault; (b) the bearing with localized inner race fault and (c) the bearing with localized rolling element fault.

18 D. Wang et al. / Mechanical Systems and Signal Processing 5 () Samplings x Samplings x Samplings x Fig.. Signals obtained by pre-whitening processing (a) bearing with localized outer race fault; (b) bearing with localized inner race fault and (c) bearing with localized rolling element fault..5 Depths x f O 5 5 f O f O Fig.. The results obtained by the enhanced Kurtogram in this paper for detecting an outer race fault. (a) The enhanced Kurtogram and (b) power spectrum of the envelope of the signal extracted from node (, ) by wavelet packet transform.

19 9 D. Wang et al. / Mechanical Systems and Signal Processing 5 () Level k x -6 f O f O f O f O 5fO Fig.. The results obtained by the fast Kurtogram for detecting an outer race fault. (a) The fast Kurtogram and (b) frequency spectrum of the envelope of the signal obtained by a filter (the center frequency of,5 Hz and the bandwidth of 5 Hz). Depths x f O f O f O Fig. 5. The results obtained by the improved Kurtogram proposed by Lei et al. for detecting an outer race fault. (a) The improved Kurtogram and (b) frequency spectrum of the envelope of the signal extracted from node (, 7) by wavelet packet transform.

20 D. Wang et al. / Mechanical Systems and Signal Processing 5 () Depths x 5 5 f I f s f I f I f I Fig. 6. The results obtained by the enhanced Kurtogram in this paper for detecting an inner race fault. (a) The enhanced Kurtogram and (b) power spectrum of the envelope of the signal extracted from node (, 8) by wavelet packet transform..6 Level k x -6 6 f I f I f I Fig. 7. The results obtained by the fast Kurtogram for detecting an inner race fault. (a) The fast Kurtogram and (b) frequency spectrum of the envelope of the signal obtained by a filter (the center frequency of 8, Hz and the bandwidth of Hz).

21 96 D. Wang et al. / Mechanical Systems and Signal Processing 5 () Depths x 8 6 f s f I f I f I f I Fig. 8. The results obtained by the improved Kurtogram proposed by Lei et al. for detecting an inner race fault. (a) The improved Kurtogram and (b) frequency spectrum of the envelope of the signal extracted from node (, 8) by wavelet packet transform Depths x 6 f BS f BS f C f BS f BS Fig. 9. The results obtained by the enhanced Kurtogram in this paper for detecting ball localized faults. (a) The enhanced Kurtogram and (b) power spectrum of the envelope of the signal extracted from node (, ) by wavelet packet transform.

22 D. Wang et al. / Mechanical Systems and Signal Processing 5 () Level k x -6 f BS f BS Fig.. The results obtained by the fast Kurtogram for detecting an ball localized fault. (a) The fast Kurtogram and (b) frequency spectrum of the envelope of the signal obtained by a filter (the center frequency of, Hz and the bandwidth of 6666 Hz). 8 Depths f C 5 5 f BS f BS f BS fbs 5f BS x Fig.. The results obtained by the improved Kurtogram proposed by Lei et al. for detecting a ball localized fault. (a) The improved Kurtogram and (b) frequency spectrum of the envelope of the signal extracted from node (, ) by wavelet packet transform.

23 98 D. Wang et al. / Mechanical Systems and Signal Processing 5 () Table Performance comparison of the enhanced Kurtogram, the fast Kurtogram and the improved Kurtogram. Cases Is it effective in detecting bearing faults? Visual inspection ability The enhanced Kurtogram The fast Kurtogram The improved Kurtogram The enhanced Kurtogram The fast Kurtogram The improved Kurtogram Yes No No High Null Null Yes No No High Null Null Yes Yes Yes High Medium Medium Yes Yes Yes High Medium High 5 Yes Yes Yes High low Medium Case concerns the mixed signal with one resonant frequency; Case concerns the mixed signal with two resonant frequencies; Case concerns the real laboratorial outer race fault signal; Case concerns the real laboratorial inner race fault signal and Case 5 concerns the real laboratorial ball fault signal. Kurtogram is capable of detecting bearing faults in all cases. In the cases of and, the enhanced Kurtogram not only established the most valuable wavelet packet node for further envelope spectrum analysis but also identified the most useful node at each depth. However, when the fast Kurtogram and the improved Kurtogram were applied to analyse the same simulated signals, they failed to provide any bearing fault related signatures. In the cases of 5, although the three methods had ability to successfully detect the three different kinds of bearing faults, their visual inspection ability varied with bearing fault signals. For bearing outer race fault diagnosis, the frequency spectrum obtained by the enhanced Kurtogram most clearly showed the outer race fault frequency and its harmonics. The frequency spectra obtained by the fast Kurtogram and the improved Kurtogram were corrupted by heavy noise. Therefore, in the case of real bearing outer race fault diagnosis, the enhanced Kurtogram has the best visual inspection ability because outer race fault frequency is the concerned topic. As for the harmonics of outer race fault frequency, they are usually used to describe the regularity of a signal. The more regular (i.e., less impulsive), the fast the decay of the amplitudes of the harmonics of the outer race fault frequency. For bearing inner race fault diagnosis, the same most valuable wavelet packet node was recommended by the enhanced Kurtogram and the improved Kurtogram. The comparison between power spectrum and frequency spectrum of the envelope extracted from the same node showed that the power spectrum suppresses heavy noise while the frequency spectrum retains the sidebands around the inner race fault frequency. Besides, the rotating frequency could be identified in both frequency spectra. The frequency spectrum obtained by the fast Kurtogram had difficulty in indicating the shaft rotating frequency. Moreover, the frequency spectrum was corrupted by heavy noise. In the case of real inner race fault diagnosis, it was shown that the visual inspection ability of the enhanced Kurtogram and the improved Kurtogram was better than that of the fast Kurtogram. For bearing ball fault diagnosis, it was easy to find that the frequency spectrum obtained by the proposed method had the better visual inspection ability than frequency spectra obtained by the fast Kurtogram and the improved Kurtogram. The reasons are given as follows. First, heavy noise is suppressed by the proposed method. Second, the fundamental cage frequency can be detected by the enhanced Kurtogram and the improved Kurtogram. Thirdly, sidebands extracted by the proposed method are very clear. As a result, after the comprehensive comparison has been done, the enhanced Kurtogram has the best performance among the three methods for bearing fault diagnosis in our case studies. 5. Conclusions This paper proposes a new method and an enhanced Kurtogram to detect bearing faults. The method uses the enhanced Kurtogram to select the assessable WPT nodes from among all nodes. To construct the Kurtogram, the kurtosis of the power spectrum of the envelope of the signals filtered by WPT at each node is calculated. The nodes corresponding to the highest kurtosis can then be considered for further analysis. In the case of simulated signals mixed with noises, the proposed method has a good detection rate for fault frequency even though the signal-to-noise ratio is low. Moreover, it is able to locate the resonant frequency bands at different depths. However, the fast Kurtogram and the improved Kurtogram fail to diagnose the simulated signal mixed with heavy noise. In the case of the various real bearing fault signals obtained in the laboratory, the proposed method is able to distinguish between different bearing faults, including outer race, inner race and rolling element faults after AR filtering is used to remove the disturbance caused by discrete frequency noise, such as the rotating frequency components. The pre-whitened bearing fault signals are equally analysed by the enhanced Kurtogram, the improved Kurtogram and the fast Kurtogram, separately. Compared with the fast Kurtogram and the improved Kurtogram, it is found that their visual inspection ability is not as good as that of the enhanced Kurtogram for showing fault characteristic frequencies. Acknowledgments The work described in this paper was partly supported by a grant from the National Natural Science Foundation of China and Research Grants Council of the HKSAR Joint Research Scheme (Project No: N_CityU6/8) and a grant from