A new blind fault component separation algorithm ... - Semantic Scholar

Journal of Sound and Vibration 331 (2012) 4956–4970

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A new blind fault component separation algorithm for a single-channel mechanical signal mixture Dong Wang n, Peter W. Tse 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

a r t i c l e in f o

abstract

Article history: Received 22 October 2011 Received in revised form 1 May 2012 Accepted 25 May 2012 Handling Editor: K. Worden Available online 28 June 2012

A vibration signal collected from a complex machine consists of multiple vibration components, which are system responses excited by several sources. This paper reports a new blind component separation (BCS) method for extracting different mechanical fault features. By applying the proposed method, a single-channel mixed signal can be decomposed into two parts: the periodic and transient subsets. The periodic subset is related to the imbalance, misalignment and eccentricity of a machine. The transient subset refers to abnormal impulsive phenomena, such as those caused by localized bearing faults. The proposed method includes two individual strategies to deal with these different characteristics. The first extracts the sub-Gaussian periodic signal by minimizing the kurtosis of the equalized signals. The second detects the super-Gaussian transient signal by minimizing the smoothness index of the equalized signals. Here, the equalized signals are derived by an eigenvector algorithm that is a successful solution to the blind equalization problem. To reduce the computing time needed to select the equalizer length, a simple optimization method is introduced to minimize the kurtosis and smoothness index, respectively. Finally, simulated multiple-fault signals and a real multiple-fault signal collected from an industrial machine are used to validate the proposed method. The results show that the proposed method is able to effectively decompose the multiple-fault vibration mixture into periodic components and random non-stationary transient components. In addition, the equalizer length can be intelligently determined using the proposed method. & 2012 Elsevier Ltd. All rights reserved.

1. Introduction The diagnosis of machine faults that might lead to failure is a hot topic in mechanical engineering [1–5]. The identification of vibration-based faults using signature analysis is especially promising due to the simplicity of the collection method and explicit physical explanation. A vibration signal collected from a complex machine usually contains multiple system responses excited by several vibration sources. The development of blind source separation (BSS) has resulted in a number of applications for diagnosing machine faults [6–8]. To identify each potential fault, BSS requires a set of vibration signals collected from different locations. However, in a situation such as an enclosed space where only one transducer can be installed, the blind source separation problem is reduced to a single-channel blind source separation problem. In other words, a vibration signal collected by only one sensor is analyzed to assess the condition of multiple

n

Corresponding author. Tel.: þ 852 34424602; fax: þ 852 34420415. E-mail addresses: [email protected] (D. Wang), [email protected] (P.W. Tse).

0022-460X/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2012.05.035

D. Wang, P.W. Tse / Journal of Sound and Vibration 331 (2012) 4956–4970

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components. Zuo et al. [9] proposed a good idea that wavelet transform (WT) was used to decompose a single-channel signal into a series of sub-signals with different frequency bands. These signals can then be considered as multiple inputs to conduct independent component analysis or principal component analysis. The results showed that wavelet transform based on independent component analysis was better than wavelet transform based on principal component analysis [9,10]. Fan et al. [11] extended this idea by replacing traditional wavelet transform with a wavelet lifting scheme and then applied their proposed method to the diagnosis of bearing faults. Compared with wavelet transform, empirical mode decomposition (EMD) is a more adaptive method for deriving a series of sub-signals with different frequency bands [1,4]. Considering this advantage, Miao et al. [12] performed EMD prior to independent component analysis to extract the features of bearing faults. He et al. [13] used a similar method to analyze mechanical watch movements using a joint algorithm of continuous wavelet transform and independent component analysis. Antoni recently introduced a new concept called blind component separation (BCS) [14]. The major difference between blind component separation and blind source separation is that the former identifies the different system responses (components) excited by the different sources rather than seeking the sources themselves, whereas the latter blindly deconvolutes the collected signals and identifies the sources. This new concept alleviates some of the difficulties of blind source separation, such as convolutive mixtures, an insufficient number of collected mixtures, and an unknown number of sources. Based on BCS, the vibration signal is decomposed into periodic components and random components. The derived random components can be further explained as the summation of random non-stationary (transient) and random stationary signals. Prediction filters based on short-time Fourier transform [14] and spectral kurtosis [15] have been suggested to extract the periodic components and random transient components separately. Randall and Antoni [16] summarized a number of methods, such as an autoregressive model, adaptive noise cancellation, self-adaptive noise cancellation, discrete separation and time synchronous averaging, that could remove the periodic components prior to spectral kurtosis analysis [15] and cyclic spectral analysis [17] for the non-stationary transient components. Hong and Liang [18] proposed a joint wavelet transform and Fourier transform to identify multiple sources from a single-channel mixture. Wavelet transform was employed to obtain the number of sub-signals with different frequency bands. Fourier transform was then used to determine the number of sources and their corresponding frequency locations. In the case of a simulated signal, Tan et al. [19] illustrated the effectiveness of using an eigenvector algorithm (EVA) to extract bearingfault signals from periodic sinusoidal noises. To further explore the performance of the EVA and its application to a real industrial case, Tse et al. [20,21] used the EVA and a generalized EVA to recover the bearing-fault signal from a signal mixture containing a rotor eccentric fault and a bearing fault. Although the results were promising, they did not illustrate how to select the equalizer length properly. The results showed that the length of equalizer was very important in their proposed algorithm because different lengths resulted in different equalized signals that made the absolute of kurtosis locally maximum in the case of a multiple-fault signal. Li et al. [22] suggested that it was possible to scan all potential equalizer lengths to select the correct length. They proposed an index based on the change in kurtosis to indicate the best equalizer length to separate the bearing-fault signal from the periodic components. However, this optimization process was time-consuming because it involved selecting the correct length from all available lengths, and also depended largely on an experienced person inspecting a visual diagram. This paper describes a new and intelligent method that integrates two strategies to extract the periodic components and random non-stationary transient components separately. The first strategy minimizes the kurtosis of the equalized signal to extract the periodic components that usually exhibit sub-Gaussian characteristics. The second strategy minimizes the smoothness index of the equalized signal to extract the random transient components after removing the periodic components using a prediction filter, such as an autoregressive model. Finally, cyclic spectral analysis can be used to analyze such a second-order pseudo-cyclostationary signal. It should be pointed out that the equalizer length is intelligently established by the proposed method. The rest of this paper is organized as follows. In Section 2, the basic theories underlying the proposed method are reviewed. The proposed method integrated with two strategies is presented in Section 3 for the purpose of blind component separation. Section 4 validates the effectiveness of the proposed method by the analysis of a simulated multiple-fault signal and a real multiple-fault signal collected from an industrial machine. Conclusions are summarized in Section 5. 2. The basic theories underlying the proposed method 2.1. Autoregressive model for linear prediction An autoregressive model (AR) is a classic way to establish the deterministic periodic components [23]. The mathematic formula of the AR is described as vðkÞ ¼

pX ¼q

aðpÞvðkpÞ þ tðkÞ,

(1)

p¼1

where t(k) is a noise part. a(p) is the parameters of the AR model, which could be obtained by finding the solution of the Yule–Walker equations [23] through the Levinson–Durbin recursion algorithm (LDR) or Burg’s method (BM) [24]. q is the

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order of the model, which is determined by minimizing Akaike information criterion (AIC) [24]. Perform Fourier transform on Eq. (1): tðf Þ ¼ vðf Þaðf Þ:

(2)

It is found that only random components (transient impulses and stationary noise) are retained after the AR filter convolutes with the original signal. The temporal counterpart of t(f) is a pre-whitened signal. 2.2. Eigenvector algorithm for blind equalization In digital communication, blind equalization infers to an equalized signal x(k) obtained from a received signal v(k) by removing the influence coming from a channel h(k) with only considering some statistical information of a transmitted signal d(k). Assume the equalizer coefficients is e(k). It is necessary to satisfy the following requirement of blind equalization [25]: sðkÞ ¼ vðkÞneðkÞ ¼ dðkk0 Þ:

(3)

The explanation of Eq. (3) is that the equalized signal is a certain time-delay k0 of the transmitted signal d(k). A specific criterion by making the absolute of kurtosis maximum for blind equalization is given as [25] 2 maxcx4 ð0,0,0Þ ¼ maxkurtosisðxÞ, provided that r xx ð0Þ ¼ r dd ð0Þ ¼ Eðd Þ, (4) where cx4 ðl1 , l2 , l3 Þ is the 4th-order (auto) cumulant. In order to find an alternative solution to Eq. (4), a ‘‘cross-kurtosis’’ criterion (4th-order (cross-) cumulants) for blind equalization was proposed by Jelonnek and Kammeyer [26]: (5) maxcxy 4 ð0,0,0Þ , provided that r xx ð0Þ ¼ r dd ð0Þ, where cxy 4 ðl1 , l2 , l3 Þ is the 4th-order (cross-) cumulant and a reference signal y(k) is equal to v(k) * f(k). Here, f(k) is a reference system. According to x(k)¼v(k) * e(k), an eigenvector equation is derived [26]: (6) Cyv 4 e ¼ lRvv e choosing l ¼ max l1 ,. . ., lq , where Cyv 4 is the Hermitian cross-cumulant matrix. An iterative algorithm for finding the solution of the eigenvector equation is given in Ref. [27]. ð0Þ Step 1: Set an initial reference system f ðkÞ ¼ dðkroundð‘=2ÞÞ and i ¼0. The round function is to take the nearest integer and ‘ is the equalizer length. Then, estimate the ð‘ þ1Þ ð‘ þ 1Þ autocorrelation matrix Rvv of the received signal. yv Step 2: Calculate y(k)¼v(k) * f(i)(k) and estimate the ð‘ þ 1Þ ð‘ þ 1Þ Hermitian cross-cumulant matrix C4 . yv Step 3: Replace Cyv and R used in Eq.(6) with the estimated values C and R . By solving Eq. (6), the most significant vv vv 4 4 eigenvector eðiÞ ðkÞ is calculated. EVA ðiÞ Step 4: Let f ðkÞ ¼ eðiÞ EVA ðkÞ, increase i¼iþ1 and go to Step 2 until the termination condition i¼I is reached. 2.3. Cyclic spectral analysis for bearing fault signal The symmetric instantaneous auto-correlation function R2v(k,t) of a signal v(k) is [28] t t : R2v ðk, tÞ ¼ E v k þ v k 2 2

(7)

If Eq. (7) satisfies R2v(k,t)¼ R2v(kþN,t), v(k) is a second-order cyclostationary signal [28]. Its Fourier series is given as follows: X R2v ðk, tÞ ¼ R ða , tÞexpðj2pai kDÞ, (8) a 2v i i

where R2v(ai,t) is the non-zero Fourier coefficients and ai is the cyclic frequency. In Eq. (8), R2v(ai,t) is called the cyclic auto-correlation function of the signal v(k). The definition of spectral correlation density R2v(a,f) is described as [28] X X R ða,f Þsðaai Þ ¼ R ða , tÞ, (9) R2v ða,f Þ ¼ a 2v a 2v i i

i

where R2v(ai, f) is R2v ðai ,f Þ ¼

þ1 X

R2v ðai , tÞexpðj2pf tDÞ:

(10)

t ¼ 1

It is found that the spectral correlation density is a continuous function of spectral frequency f and a discrete function of cyclic frequency ai. The cyclic coherence function Y2v(a,f) of the spectral correlation density R2v(a,f) is defined as [17] R2v ða,f Þ Y 2v ða,f Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : R2v ð0,f þða=2ÞÞR2v ð0,f ða=2ÞÞ

(11)


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It is found that the cyclic coherence function is able to show pure second-order cyclo-stationarity and its (squared) magnitude is limited to between 0 and 1. The closer Eq.(11) is to 1, the stronger the linear dependence. Consequently, the cyclic coherence function effectively measures the degree of cyclo-stationarity. 3. The proposed method for blind component separation Antoni [14] proposed a new concept called blind component separation. In contrast to blind source separation, blind component separation is concerned with the vibration responses excited by a number of vibration sources rather than the sources themselves. Based on this concept, a vibration signal can be decomposed into three parts: the periodic components, the random non-stationary (transient) components and the random stationary components. The periodic components are usually caused by imbalances, misalignments and eccentricities. The random non-stationary components are those that have the highest degree of impulsiveness, such as localized bearing faults. This paper proposes a new method that integrates these two strategies to extract the periodic components and non-stationary transient components separately. The flowcharts of the two strategies for blind component separation are shown in Figs. 1 and 2. 3.1. The first strategy for the extraction of periodic components From Eqs. (4) and (5), it can be seen that the objective cost function of the EVA is the absolute of auto-kurtosis or cross-kurtosis, which means that the maximum of this objective cost function could reflect both sub-Gaussian and

Load the original vibration signal

Initialization for the eigenvector algorithm

Calculate the kurtosis of the equalized signals at ith iteration for each equalizer length

Generate some potential solutions and perform the EVA with each equalizer length

Find the equalized signal corresponding to the minimum kurtosis

The extraction of the periodic components

Fig. 1. The proposed method integrated with the first strategy for extracting the periodic components.

Load the original vibration signal

Initialization for eigenvector algorithm

Minimize AIC to select the proper order for the AR model

Perform the AR filtering on the original signal

Generate some potential solutions and perform EVA with each equalizer length

Calculate the smoothness indexf o the equalized signals at ith iteration for each length equalizer

Perform cyclic coherence function on the optimal equalized signal for the demodulation analysis

Find the equalized signal corresponding to the minimum smoothness index

Bearing fault identification by inspecting bearing fault frequency

Fig. 2. The proposed method integrated with the second strategy for bearing-fault diagnosis.

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super-Gaussian signals. The original blind equalization in a digital communication is actually blind deconvolution for a single-input single-output system to remove the influence of the transmission channel. A classic report on the use of the blind equalization method for a single-input single-output vibration system was given in [29]. In a complex machine, multiple vibration sources excite multiple system responses. If the correct equalizer length is chosen, it is possible to maximize the absolute kurtosis of one of the vibration components. In other words, the sub-Gaussian or super-Gaussian of one of the vibration components can be reflected in the measurement of the absolute kurtosis. Assuming the periodic components follow a sub-Gaussian distribution, the minimization of kurtosis may result in the recovery of the periodic components from the vibration signal mixture. In this paper, a new method is proposed to recover the periodic components, as shown in Fig. 1. The process is described in detail in the following section. Step 1: Load the original vibration signal and initialize the parameters of the eigenvector algorithm. In order to intelligently select the correct equalizer length, some potential solutions (different equalizer length) are randomly generated. These potential solutions are subject to a uniform distribution because any potential solution can be generated between the shortest equalizer length and the longest equalizer length. Assume the longest equalizer length is 120, the shortest equalizer length is 8, the number of potential solution (the length of equalizer) is 20 and the maximum iteration number of EVA for each equalizer length is set to 10. Step 2: Perform the eigenvector algorithm for each potential solution. Here, the autocorrelation matrix Rvv of the yv received signal v(k) and the cross-cumulant matrix C4 , are estimated by the following equations: PL1 T

k ¼ ‘kðjÞ vv Rvv ¼ with vT ¼ vðkÞ,vðk1Þ,. . .,vðk‘kðjÞ Þ , (12) L‘kðjÞ yv

PL1

C4 ¼

k ¼ ‘kðjÞ

yikðjÞ ðkÞ2 vvT

ð

PL1

k ¼ ‘kðjÞ

P yikðjÞ ðkÞ2 Þð L1 k¼‘

kðjÞ

vvT Þ

2

LlkðjÞ ðLlkðjÞ Þ 0 10 1 L1 L1 X X yikðjÞ ðkÞvA@ yikðjÞ ðkÞvT A þ 2@ k ¼ ‘kðjÞ

,

(13)

k ¼ ‘kðjÞ

where k(j) is the jth potential solution and i is the ith iteration. ‘ is the equalizer length. L is the length of the received signal. Eq. (13) is different from that used in Ref. [28] because we hope to slow down the convergence of EVA and try more different equalizer cofficients for the same equalizer length. Step 3: The optimal ‘kðjÞ and i are found to minimize the kurtosis: ð‘opt1 ,iopt1 Þ ¼ argð‘

kðjÞ

,iÞ minðkurtosisðf ð‘kðjÞ ,iÞ ðkÞnvðkÞÞÞ,

(14)

where f is the equalizer coefficients and * means convolution operator. Step 4: The signal obtained by f ð‘opt 1 ,iopt 1 Þ ðkÞnvðkÞ is the periodic components. 3.2. The second strategy for the extraction of random non-stationary transient components It is better to remove the periodic components before extracting the random non-stationary transient components, and then analyze what causes the impulsive transients. In this paper, we assume that the random non-stationary transient components are caused by localized bearing faults. Therefore, in this section, the aim of the second strategy is to diagnose the bearing fault. The proposed method integrated with the second strategy is shown in Fig. 2. The process is illustrated in detail as follows. Step 1: Load the vibration signal. In order to remove the influence of the periodic components, the autoregressive (AR) filtering introduced in Section 2.1 is used. The order of AR model is determined by minimizing Akaike information criterion ~ (AIC). Here, denote the signal obtained by the AR filtering as vðkÞ. Step 2: Initialize the parameters of the eigenvector algorithm. Here, some potential solutions subject to a uniform distribution (different equalizer length) are randomly generated. The longest equalizer length, the shortest length equalizer, the number of potential solutions (the equalizer length) and the maximum iteration number of EVA for each equalizer length are the same with those used in the previous section. Step 3: Perform the eigenvector algorithm for each potential solution. The autocorrelation matrix Rv~ v~ of the received yv~ ~ signal vðkÞ and cross-cumulant matrix C4 , are estimated by both Eqs. (12) and (13). Step 4: The optimal ‘kðjÞ and i are found to minimize the smoothness index (SI): 0 1 P ~ exp ð1=Lc Þ k ln f ð‘ ðjÞ ,iÞ ðkÞnvðkÞ B C k ð‘opt 2 ,iopt 2 Þ ¼ argð‘ ðjÞ ,iÞ min@ (15) A: P k ~ ð1=Lc Þ k f ð‘ ,iÞ ðkÞnvðkÞ kðjÞ

~ where Lc is the length of f ð‘ ðjÞ ,iÞ ðkÞnvðkÞ. k Here, the maximum of kurtosis is not used for the extraction of the impulsive transient signal because kurtosis is more sensitive to the outliers unrelated to periodic transients [30]. The smoothness index has been applied to the measurement of spectral flatness in speech signal processing [31]. Bozchalooi and Liang [30] demonstrated that the smoothness index can indicate the peaky degree of the signal filtered by the complex Morlet wavelet transform (a band-pass filter). Then,


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they varied the parameters of complex Morlet wavelet to make the smoothness index of the modulus of complex Morlet wavelet minimum for the optimal parameter selection. The smoothness index is limited to between 0 and 1. The closer the value is to 1, the flatter the signal. ~ Step 5: The signal recovered by f ð‘opt2 ,iopt2 Þ ðkÞnvðkÞ is the cyclic transient components. Step 6: Bearing fault identification. Even though bearing fault signal is isolated, it is necessary to recognize the cause of localized bearing faults. Bearing fault characteristic frequencies are effective to indicate the reason of the localized bearing faults. In other words, bearing fault characteristic frequencies are calculated as references for bearing fault pattern recognition by human visual inspection. Once bearing fault signal has one of bearing fault characteristic frequencies, a specific fault pattern can be identified. Outer race fault characteristic frequency fO, the inner race fault characteristic frequency fI and the rolling element fault characteristic frequency fB are given as [16]

Zf d f O ¼ S 1 cosy , (16) D 2 fI ¼

fB ¼

Zf S d 1þ cosy , D 2

Df S d

!

2

1

d

D2

(17)

cos2 y ,

(18)

where fS is the shaft rotation frequency in Hz; d is the diameter of the rolling element; D is the pitch diameter of the bearing; Z is the number of rolling elements; and y is the contact angle. Once a bearing suffers from a localized fault, the rolling elements strike the local fault either on the outer race or the inner race, resulting in a series of repetitive impact forces that excite the resonant frequencies of the structure between the bearing and the transducer [16]. Therefore, the bearing-fault signatures are more likely to be found in higher frequency bands. To detect the bearing fault’s characteristic frequencies at a lower frequency band more effectively, demodulation is necessary before a visual inspection of the bearing-fault frequencies is carried out. Cyclic spectral analysis [17,28] is a powerful tool for detecting cyclic frequencies hidden in a second-order cyclo-stationary signal. Furthermore, cyclic spectral analysis is a frequency–frequency (spectral frequency–cyclic frequency) plane tool that efficiently shows the relationship between the resonant frequency bands and modulating frequencies (cyclic frequencies). This is the major reason for employing cyclic spectral analysis in this paper. Rather than using the spectral correlation density, the cyclic coherence function of the spectral correlation density is a dimensionless measure of the degree of cyclo-stationarity. In addition, its (squared) magnitude is limited to between 0 and 1; the closer the cyclic coherence function is to 1, the stronger the linear dependence. As a result, the cyclic coherence function of the spectral correlation density is applied to the recovered bearing-fault signal obtained in Step 5 to identify the bearing faults. 4. Validation of the proposed method 4.1. A simulated multiple-fault signal analyzed by the proposed method First, a classic simulated multiple-fault signal consisting of low-frequency periodic components and high-frequency transient components [32] was built. The mathematical formula for the simulated multiple-fault signal is given as follows: X nðkÞ ¼ elðkrFs=f m tr Þ=Fs sinð2pf 0 ðkr Fs=f m tr Þ=FsÞ þ sinð2pf 1 k=FsÞ þ 0:8 sinð2pf 2 k=FsÞ: (19) r

where l is equal to 900;fm is the fault characteristic frequency (equal to 100 Hz); Fs is the sampling frequency set to 12,000 Hz; f0 is the resonant frequency, equal to 3700 Hz. f1 and f2 are low frequency components, equal to 60 Hz and 90 Hz, respectively. 2400 samples from the simulated signal are used. An amount for a normally distributed random signal is added to Eq. (19). The simulated multiple-fault temporal signal is plotted in Fig. 3(a) and its corresponding frequency spectrum is plotted in Fig. 3(b). To recover the low-frequency periodic components, the proposed method is integrated with the first strategy. The kurtosis value at each iteration is given in Fig. 4(a). There are 200 iterations in all: 20 potential solutions multiplied by 10 iterations for each length. The resulting temporal periodic components extracted by the proposed method integrated with the first strategy are shown in Fig. 4(b). The low-frequency periodic components are highlighted in the figure and the transient components shown in Fig. 3(a) are depressed. The validity of the conclusion can be confirmed by inspecting the frequency spectrum shown in Fig. 4(c), in which the frequencies of the transient components are largely depressed. Second, to recover the transient components using the proposed method integrated with the second strategy, AR filtering is employed to remove the interruption from the low frequency periodic signal. Before the filtering, the proper order should be established for the of AR filtering, which can be determined by the minimum AIC value. The values are plotted in Fig. 5(a), where the AIC values tend to be stable when the length of the AR filter is larger than 20. The order of the AR model was chosen as 41 in the case of the simulated signal. The residual signal filtered by the AR model is plotted in Fig. 5(b). Its corresponding frequency spectrum is given in Fig. 5(c), which shows that the low frequency components are

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Amplitude

2 1 0 -1 -2 0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Time (s) 1200

Amplitude

1000 800 600 400 200 0

1000

2000

3000 Frequency (Hz)

4000

5000

6000

Fig. 3. The simulated signal for two faulty components: (a) the temporal signal and (b) the frequency counterpart.

kurtosis

2.29 2.28 2.27 2.26 20

40

60

80

100

120

140

160

180

200

0.1 0.12 Time (s)

0.14

0.16

0.18

0.2

Iterations

Amplitude

2

0

-2

Amplitude

0.02

0.04

0.06

0.08

1200 1000 800 600 400 200 0

1000

2000

3000

4000

5000

6000

Frequency (Hz) Fig. 4. The results obtained using the proposed method integrated with the first strategy in the case of the simulated signal: (a) the minimum kurtosis at each iteration; (b) the low frequency periodic signal recovered from the original signal; and (c) the frequency part of (b).


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AIC

-4000 -6000 -8000 -10000

0

10

20

30

0.02

0.04

0.06

40 50 60 The order of AR model

70

80

90

0.14

0.16

0.18

100

Amplitude

1 0.5 0 -0.5 0.08

0.1

0.12

Time (s)

Amplitude

30 20 10 0

1000

2000

3000 Frequency (Hz)

4000

5000

6000

Fig. 5. The results obtained by AR filtering in the case of the simulated signal: (a) the AIC values for each order; (b) the signal filtered by the AR model with the order 41; and (c) the frequency spectrum of (b).

largely depressed. To further retain the relevant components (transient components) of the resulting signal shown in Fig. 5(b), the equalized signal with the minimum SI needs to be obtained. The minimum SI at each iteration is plotted in Fig. 6(a). The final equalized signal with the minimum SI is plotted in Fig. 6(b), which clearly indicates the existence of the repetitive impulsive transients. The frequency spectrum of the repetitive impulses is given in Fig. 6(c), where the resonant frequency band around 3700 Hz is extracted. To further identify the simulated bearing-fault pattern, cyclic spectral analysis is employed to analyze the resulting signal shown in Fig. 6(b). The squared modulus of the cyclic coherence function and its local zoom are plotted in Fig. 7(a) and (b), where the cyclic frequency 100 Hz and its harmonics are easily detected. In addition, there is a clear relationship between the resonant frequency band from 3400 Hz to 4000 Hz and the modulating frequency 100 Hz. Therefore, the proposed method integrated with the second strategy is able to recover the transient components from the simulated multiple-fault signal. It should be pointed out that the equalizer length is intelligently determined by the proposed method. There is no need for an experienced person to select the correct equalizer length. Two more simulated multiple-fault signals are analyzed for further discussion. One of them is built by adding two different resonant frequencies Eq. (19). It is noted that the modulating frequency 100 Hz is fixed. The simulated multiplefault signal with three resonant frequencies is described as X nðkÞ ¼ elðkrFs=f m tr Þ=Fs ðsinð2pf 0 ðkr Fs=f m tr Þ=FsÞ þ sinð2pf 3 ðkr Fs=f m tr Þ=FsÞ r

þsinð2pf 4 ðkr Fs=f m tr Þ=FsÞÞ þ sinð2pf 1 k=FsÞ þ 0:8 sinð2pf 2 k=FsÞ:

(20)

where f3 and f4 are equal to 2100 Hz and 5100 Hz, respectively. The minimum SI at each iteration is plotted in Fig. 8(a), after AR filtering is performed. The equalized signal with the minimum SI is shown in Fig. 8(b), where it is seen that the correct equalizer length is helpful to extract the repetitive impulsive transients. The frequency spectrum of the repetitive impulses is given in Fig. 8(c), where three resonant frequency bands around 2100 Hz, 3700 Hz and 5100 Hz are extracted. Another new simulated multiple-fault signal is built by adding a new modulating frequency into Eq. (20). It means that there are two different modulating frequencies. The mathematical formula for the simulated multiple-fault signal is given as X nðkÞ ¼ elðkrFs=f m tr Þ=Fs ðsinð2pf 3 ðkr Fs=f m tr Þ=FsÞ þ sinð2pf 4 ðkr Fs=f m tr Þ r

X þ elðkrFs=f m1 tr Þ=Fs sinð2pf 0 ðkr Fs=f m1 tr Þ=FsÞ þ sinð2pf 1 k=FsÞ þ 0:8 sinð2pf 2 k=FsÞ:

(21)

r

where fm1 is another fault characteristic frequency (equal to 160 Hz). In Fig. 9(a), the minimum SI at each iteration is given. The equalized signal with the minimum SI is plotted in Fig. 9(b). Its corresponding frequency spectrum is shown in Fig. 9(c), where

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0.52 SI

0.5 0.48 0.46 0.44 20

40

60

80

100

120

140

160

180

200

Amplitude

Iterations

2 0 -2 -4

Amplitude

0.02

0.04

0.06

0.08

0.1 Time (s)

0.12

0.14

0.16

0.18

500 400 300 200 100 0

1000

2000

3000

4000

5000

6000

Frequency (Hz) Fig. 6. The results obtained by EVA with the minimum SI in the case of the simulated signal: (a) the minimum SI at each iteration; (b) the recovered bearing-fault signal; and (c) the frequency part of (b).

two resonant frequency bands around 2100 Hz and 5100 Hz are retained. From the results shown in Figs. 8 and 9, it is concluded that the resonant frequency bands having the same modulating frequency can be kept.

4.2. A multiple-fault industrial signal analyzed by the proposed method The multiple fault signal [20–22][33] collected from a typical traction motor was used for the validation of the proposed method in this section. The schematic diagram is shown in Fig. 10(a). 250 kg rotor was supported by two rolling element bearings. One was a single row deep groove ball bearing (SKF 6215) at the drive end (here, denote its position as B). Another was a single row cylindrical roller bearing (NU 210) at the non-drive end (here, denote its position as A). Four accelerometers from four different directions (C–F) were used to sample the vibration signals. Fig. 10(b) shows the actual view for the relationship of four directions. Vibration signals from the sensors were amplified by a coupler (Kistler 5134), and a digital cassette recorder (Sony PC 204AX) was utilized to record the signals at the sampling rate of 48 kHz for each of the channels. Then, the signals were transmitted to PC with a data acquisition card at sampling rate of 32.77 kHz. Vibration data captured by an accelerometer closing to bearing location (D) is used. Shaft rotation speed fr was around 1498 rev/min. The fault intervals are approximately equal to 10 ms. The original multiple-fault signals are shown in Fig. 11(a) and (b); the signal shown in Fig. 11(b) is the frequency spectrum of the signal shown in Fig. 11(a). The rotor eccentric fault clearly dominates the whole signal and the bearingfault transient signal is hard to recognize. To separate the two fault components, first, the proposed method integrated with the first strategy was used to extract the periodic components (the rotor eccentric fault responses). The minimum kurtosis for each iteration is plotted in Fig. 12(a), which displays a total of 200 iterations (20 potential solutions and 10 iterations for each length). The final periodic components extracted by the proposed method are shown in Fig. 12(b) and (c). The proposed method integrated with the first strategy retains only the periodic components (the rotor eccentric fault responses). Moreover, its period is approximately 40 ms. Therefore, the proposed method with the first strategy is effective in recovering the rotor eccentric periodic signal from the signal mixture shown in Fig. 11(a). Second, the proposed method integrated with the second strategy was employed to recover the bearing-fault signal from the original signal. The AIC optimization algorithm from the AR filter was used to select the correct order for AR filtering. The optimal length for the AR filter was determined by the minimum AIC value. The AIC values, shown in Fig. 13(a), tend to be stable when the length of the AR filter is larger than 30. In the case of the real industrial fault signal,

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Roller bearing NU 210

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the order of the AR model was chosen as 54 as a compromise between long and short length. The residual signal filtered by the AR model is plotted in Fig. 13(b) and its corresponding frequency is in Fig. 13(c). After AR filtering, the frequency spectrum shown in Fig. 13(c) is almost uniform. It is still difficult to identify the bearing-fault signatures at this stage because of the low signal-to-noise ratio. Next, the signal obtained by the AR filtering was further processed by EVA by minimizing the SI. The minimum SI at each iteration is plotted in Fig. 14(a). In Fig. 14(b), the final equalized signal with the lowest SI shows that the repetitive impulsive transients were clearly detected. The frequency spectrum of the repetitive impulses is given in Fig. 14(c), where the resonant frequency bands are indicated. However, the bearing-fault pattern cannot be directly diagnosed by visual inspection of the bearing-fault characteristic frequency. To further identify the bearing fault, cyclic spectral analysis was conducted on the signal shown in Fig. 14(b). The squared modulus of the cyclic coherence function and its local zoom are plotted in Fig. 15(a) and (b). The cyclic frequency


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is about 100 Hz, which is approximately equal to the reciprocal of the outer-race fault interval mentioned earlier. Moreover, several harmonics exist. The results shown in Fig. 15 illustrate that the resonant frequencies are modulated by the characteristic frequency of the outer race fault because only the approximate cyclic frequency 100 Hz is detected as the modulating frequency. As a result, the proposed method integrated with the second strategy effectively recovered the bearing-fault signal from other strong vibration components. The proposed method can also intelligently decide the correct equalizer length for the first and second strategies when EVA is used.

5. Conclusions This paper reports a new blind component separation method. The proposed method includes two separate strategies. In the first strategy, the periodic components are recovered by minimizing the kurtosis of the equalized signal filtered by EVA. In the second strategy, AR filtering is first used to remove the influence of the periodic components before implementing EVA. Once the random non-stationary transient signal (the bearing-fault signal in this paper) is obtained, EVA with minimizing SI is employed to enhance the signal-to-noise ratio so that the bearing fault’s repetitive impulses stand out. Finally, cyclic spectral analysis is performed to identify the cyclic frequency (the modulating frequency or bearing-fault characteristic frequency) to diagnose the bearing fault. The results obtained by the proposed method integrated with the first and second strategies show that the method is effective for recovering the periodic components and the bearing-fault components, respectively. The proposed method also provides an intelligent method of determining the correct equalizer length for EVA.

Acknowledgments The work described in this paper was partly supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project no. CityU 122011) and partly supported by a grant of Croucher Foundation (Project no. 9220027). Finally, we want to express our deepest appreciation for the valuable comments from anonymous referees who helped me to improve my work.

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