Envelope extraction based dimension reduction for independent component analysis in fault diagnosis of rolling element bearing

Journal of Sound and Vibration 333 (2014) 2983–2994

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Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi

Envelope extraction based dimension reduction for independent component analysis in fault diagnosis of rolling element bearing Yu Guo a,n, Jing Na a, Bin Li a, Rong-Fong Fung b a

Faculty of Mechanical and Electrical Engineering, Kunming University of Science and Technology, Kunming 650500, PR China Department of Mechanical and Automation Engineering, National Kaohsiung First University of Science and Technology, Kaohsiung 824, Taiwan b

a r t i c l e i n f o

abstract

Article history: Received 11 July 2013 Received in revised form 9 February 2014 Accepted 28 February 2014 Handling Editor: L.G. Tham Available online 21 March 2014

A robust feature extraction scheme for the rolling element bearing (REB) fault diagnosis is proposed by combining the envelope extraction and the independent component analysis (ICA). In the present approach, the envelope extraction is not only utilized to obtain the impulsive component corresponding to the faults from the REB, but also to reduce the dimension of vibration sources included in the sensor-picked signals. Consequently, the difficulty for applying the ICA algorithm under the conditions that the sensor number is limited and the source number is unknown can be successfully eliminated. Then, the ICA algorithm is employed to separate the envelopes according to the independence of vibration sources. Finally, the vibration features related to the REB faults can be separated from disturbances and clearly exposed by the envelope spectrum. Simulations and experimental tests are conducted to validate the proposed method. & 2014 Elsevier Ltd. All rights reserved.

1. Introduction The incipient faults detection plays an important role in the fault diagnosis of the REBs. To address this issue, the envelope analysis or high-frequency resonance demodulation technique has been developed and widely used for decades [1]. However, in some harsh conditions the envelope analysis may be invalid, because it can be spoiled by other impulsive or broadband vibration sources, such as a gear with cracked or broken teeth and the turbine and pump flow signals [2], which may lead to underestimating the defect severity or misrepresenting the defect types. To eliminate the errors caused by the disturbances from other vibration sources, an intuitive way is to separate the interested component from the noise components in the raw vibration before performing the envelope analysis. Independent components analysis (ICA) [3] is a newly developing blind sources separation (BSS) scheme. It has the salient advantage of decomposing the observed mixtures into different independent components (ICs) according to the discriminative statistical properties of sources. Recently, the ICA has been successfully applied in many applications, such as linguistic data analysis [4], image recognition [5], and mechanical fault detection [6,7]. However, there are still some

n

Corresponding author. Tel.: þ86 871 65930723; fax: þ 86 871 65920005. E-mail address: [email protected] (Y. Guo).

http://dx.doi.org/10.1016/j.jsv.2014.02.038 0022-460X & 2014 Elsevier Ltd. All rights reserved.

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limitations involved in the mixed vibration separation of rotating machinery by general BSS algorithms, which include the prior-known total number of individual sources in the mixture and noisy measurements [8]. These difficulties limit the applications of the ICA algorithms in engineering practices. To the best of authors' knowledge, the application of the ICA for the REB faults detection has not been fully addressed. Note that the envelope extraction in the envelope analysis has the ability of extracting the impulsive vibration features generated by the incipient REB faults and isolating the strong noises beyond the demodulation frequency band, which can actually be endowed a dimension1 reduction strategy for the ICA algorithms. By combining the advantages of the envelope extraction and the ICA, a robust envelope analysis scheme for the feature extraction of the faulty REBs is proposed in this paper. It has two prominent advantages: Firstly, different from most conventional ICA based schemes [10,11], which are designed to separate the vibrations generated by the faulty REBs, the proposed approach is employed to separate the envelopes of the vibrations, and is more reliable to obtain the incipient fault related weak impulsive feature in applications. Secondly, the resonance demodulation in the envelope extraction is employed to perform the dimension reduction for the ICA. It plays an important role in guaranteeing a successful ICs separation by the ICA algorithm, and then only a few vibration transducers are required for the REB fault diagnosis with unknown source number. The experimental results support the present approach positively. It is worth mentioning that an envelope order tracking has been introduced in our previous work [12], which is focused on addressing the REB fault detection under the varying speed condition of a rotating machinery by the order envelope analysis. In this paper, we will introduce a dimension reduction scheme for the ICA by the envelope extraction. The main contributions included in the two papers are totally different. However, the presented envelope order tracking obviously can be used to eliminate the potential speed fluctuation in this paper. The paper is arranged as follows. Firstly, the briefs on the envelope extraction and the ICA principles will be introduced in Sections 2 and 3, respectively. Subsequently, the proposed approach will be presented in Section 4. Simulation and experiment will be shown in Section 5 and conclusions will be drawn in Section 6.

2. Brief on envelope extraction Envelope analysis [1,2] is a popular incipient fault detection technique for REBs. Its main idea is to detect the feature frequencies corresponding to localized faults of REBs, which are included in the quasi-periodic impulsive train. Due to the impulsive feature, even the weak vibration from the incipient fault of a REB has a wide frequency band. Then, some resonances of local structures in the rotating machinery such as bearings, bearing houses, and sensors can be excited. As a result, the high-frequency carriers are generated, which are modulated by the weak vibrations. For this reason, the vibration containing the feature frequencies of the faults can be demodulated in a suitable frequency band by the envelope extraction and most disturbances are also isolated simultaneously.

2.1. Main steps of envelope analysis Generally, there are analog and digital forms for extracting the envelope from the REB vibration, of which the digital form is more popular due to the prominent improvement on digital signal processing techniques. Then, the discussion of envelope analysis in the following will be limited to the digital form. The main steps involved in the envelope analysis can be described briefly as follows: (1) Data acquisition: The raw vibration is picked up by a vibration sensor from a bearing house. Then the analog signal is conditioned, acquired and transformed into data series by a data acquisition device. (2) Band-pass filtering: A digital band-pass filtering is performed on the data series in a suitable resonant frequency band, by which most disturbances beyond the frequency band are removed or greatly suppressed, and the weak impulsive components become prominent in the rest components. (3) Envelope extraction: The envelope of the filtered data series is extracted by an envelope extracting algorithm, which could be the Hilbert transform [13], the wavelet transform [14] and the kurtogram [12,15]. (4) Envelope spectrum: The extracted envelope is calculated by spectrum techniques for the envelope spectrum, and the feature frequencies related to the incipient faults can be exposed. Among different envelop extraction approaches, the newly developed kurtogram method takes the advantage of determining the optimal demodulation frequency band adaptively for the band-pass filtering. In particular, the fast kurtogram [15] algorithm can be calculated efficiently, and thus it will be utilized in our study. A brief introduction on the kurtogram is presented in the next subsection. 1 In the statistics, the dimension of the data is the number of variables that are measured on each observation [9]. In the ICA, the dimension can be explained as the number of independent components included in the observed mixtures.

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Fig. 1. Schematic of the binary frequency/frequency resolution plane of the fast kurtogram.

2.2. Envelope extraction based on kurtogram As well known, the spectral kurtosis (SK) is a useful indicator to indicate non-Gaussian or randomly occurring signals with the frequency locations [16]. The higher the SK value, the stronger the impulsive components in the corresponding frequency band. Then, it can be utilized to indicate the frequency band which includes the most impulsive components. In various formulations of the SK, the new version developed by Antoni [15] is more suitable for the real word applications [17]. The SK is defined as SKðf Þ ¼

〈jHðt; f Þj4 〉  2; 〈jHðt; f Þj2 〉2

(1)

where Hðt; f Þ denotes the time–frequency complex envelope of the band-pass filtered signal x(t) at a center frequency f, and can be calculated by the fast kurtogram algorithm. In addition, from the viewpoint of a band-pass filter, the SK can be explained as the function of the center frequency f and the bandwidth B. The determination of the optimal combination ðf o ; Bo Þ leads to the development on the so-called kurtogram. The fast kurtogram realizes the optimal demodulation frequency band, and determines the progress by designing a treelike multi-rate filter-bank structure with quasi-analytic band-pass filters, where the filter banks at different levels are composed by binary or 1/3-binary filter structures. The schematic of the binary filter structure is shown in Fig. 1. The detailed instruction of the fast kurtogram algorithm can be found in [12,15]. The filtered sequence with the maximum SK value is determined as the envelope extraction result, and the corresponding combination (f o , Bo ) is the corresponding optimal filtering parameters. The step can be expressed by k

½ f o ; Bo ; co ðnÞ ¼ argmaxfSKð f i ; Bki Þg;

(2)

where f o , Bo , co ðnÞ represent the optimal center frequency, the optimal bandwidth and the optimal complex envelope, respectively; the notion argmax is denoted to obtain the parameters pertaining to the SK's maximum value calculated by the digital version of Eq. (1), see [15] for details. Then, the optimal envelope jco ðnÞj is determined. 3. Brief on independent component analysis As aforementioned, the ICA can be utilized to separate the observed mixtures into different independent components (ICs) according to the different statistical properties of sources. The initial model of the simplest linear noiseless ICA [3] can be expressed as x ¼ As;

(3) T

where x denotes an observed m-dimensional vector, x ¼ ðx1 ; x2 ; …; xm Þ , the element xi with i ¼ 1; …; m denotes the ith observed mixture; s represents an n-dimensional random vector, s ¼ ðs1 ; s2 ; …; sn ÞT , the element sj with j ¼ 1; …; n represents the jth (latent) independent component or original source, i.e., the ICj; A stands for an m  n mixing matrix, and its element aij with i ¼ 1; …; m; j ¼ 1; …; n is determined by the transfer path between the ith sensor and the jth source. According to the principles of matrix operations, if x and A can be obtained, the original sources or ICs can be obtained by s ¼ A  1 x ¼ Wx;

(4) 1

where W, the separating matrix, is the inverse of matrix A i.e., W ¼ A . In Eq. (4), the observed vector x can be obtained directly from a multi-channel synchronous acquisition by sensors, and the data from each channel represents an element (mixture) of (x). Generally, the matrix W is not obvious and the solution for s seems not to exist, but the ICA is possible to estimate the separating transfer matrix W based on the statistical properties of the element sj in the vector s. Once W is obtained, the vector s can be calculated by Eq. (4). It is worth pointing out that there are two key constraints on utilizing the

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separability of the popular ICA algorithms. Firstly, the dimension of independent components is required not more than the number of the observed vectors viz. m Zn in the standard ICA algorithms. Although there were some papers introducing approaches to realize the ICA under the special conditions m on [18,19], but these are rarely used in real applications. Secondly, the independent component must have non-Gaussian distributions (at most one of the sources sj is Gaussian). See [3] for details about the ICA. 4. Dimension reduction for the ICA based on envelope extraction The novel separation ability of the ICA is a potential tool to solve the difficulty in the vibration based REB faults analysis. The original idea is to extract the faulty REB vibration directly by the ICA separation. Some related researches on the topic can be found in [11]. However, there is a restrictive assumption in the most popular ICA algorithms that the number of observed vectors cannot be less than the dimension of independent components, viz. m Z n, which limits the applicability of direct extraction schemes. As aforementioned, the total number of the sources in the mixture may not be known in most applications. To address this issue, the dimension reduction will play a key role for the faulty REB feature extraction in the ICA. This will be studied in the following. 4.1. Traditional dimension reduction approaches for the ICA In some applications, there are sufficient channels to ensure m Zn. For example, the ICA has been utilized for artifacts removal in the EEG [20], where the number of sensors is dozens. However, there are also many conditions that we cannot ensure m Z n in the original measurement. In this case, what we have to do is to utilize the so-called dimension reduction strategy. It can simplify and reduce the complexity of the problem by reducing the number of sources under consideration. The time filtering and the principle component analysis (PCA) are two popular dimension reduction methods [21] in the ICA. The time filtering may be the most intuitive dimension reduction strategy in the ICA due to its noisy blocking property. The difficulty in the time filtering is how to determine a suitable filtering band, which depends on the data, and general answers are impossible to be obtained [21]. The PCA is another popular ICA dimension reduction approach in the sense of mean square error. It seeks to reduce the dimension of the data by finding a few orthogonal linear combinations of the original variables with the largest variance. However, as pointed out in [21], the weak ICs may be lost in the dimension reduction process, and will also be verified by our following experiments. 4.2. Dimension reduction by envelope extraction As well known, the feature extraction of the REB's incipient fault plays an important role in the faulty REB diagnosis. However, the vibration generated by an incipient fault is usually too weak to detect directly, because it is buried in strong background noises. This makes the PCA based dimension reduction strategy unsuitable in the ICA for the faulty REB feature extraction. On the other hand, the strong disturbances (most of which are the low-frequency harmonics contributed by other parts in the machinery, such as shafts and gears) beyond the selected resonant or bandpass filtering frequency band can be isolated. However, the envelope analysis results may be confused even incomprehensible for the identification of the faulty of the REB [2], where a wide-frequency-band disturbance exists. To address these difficulties in the feature extraction of the faulty REBs, a combination of the envelope extraction and the ICA for the faulty REBs feature extraction is proposed in the paper. It is noticed that the dimensions of the vibration can be reduced dramatically by the envelope extraction, which removes most disturbances beyond the resonant zone. Our proposed approach makes the ICA qualified to be employed for extracting the feature frequency components of the faulty REBs from the envelope mixtures. The main steps in the proposed approach are presented as follows (see Fig. 2): (1) Data acquisition: The vibrations are acquired synchronously from the bearing house by the piezoelectric acceleration sensors, which are the original observed mixtures. (2) Envelope extraction: The kurtogram based adaptive envelope extraction scheme is employed to extract the envelopes of the vibration. Through the envelope extraction, the strong noise beyond the demodulation frequency band is isolated and the impulsive components are enhanced. This step actually plays an important role of source dimension reduction for the ICA algorithms, by which only a few vibration channels (e.g. three vibration channels are used in our study) are sufficient to most applications. It is worth pointing out that only one observed vibration is used to calculate the kurtogram for the optimum filtering band. All the observed signals use the optimum filtering band for envelope's extraction, and keep the same lengths of the obtained envelopes. After this step, the envelope mixtures for the ICA are generated. (3) Separation of envelope mixtures by the ICA: Even though most disturbance components can be isolated by the envelope extraction scheme, some disturbances from other units of the rotating machinery still have the opportunity to spoil the analysis results. To solve the problem, the ICA is employed to separate the envelope mixtures according to the natural properties of the sources, such as the random impulsive train generated by the REB, and the periodic components

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Fig. 2. Schematic of the envelope dimension reduction based ICA.

caused by gears and shafts. Then, the disturbance components are removed and the interesting component can be extracted. (4) Envelope spectrum analysis: The spectrum analysis is applied to the extracted envelope signals one by one, and the frequency features can be exposed clearly for the diagnosis of the REBs. It is worth mentioning that there are two novel advantages included in the proposed scheme. Firstly, different from most ICA based schemes designed to extract the weak vibrations generated by incipient REB faults directly, the proposed approach is employed to extract the envelopes of these weak vibrations, and is more reliable to obtain the incipient faults related characteristic information in applications. Secondly, only a few vibration transducers are required for the robust monitoring of the REBs when the source number is unknown. Simulation and experiment results will show that the proposed approach is more robust to perform the envelope analysis for the REB fault diagnosis of the rotating machinery. 5. Simulation and experiment To show the advantage on the robustness of the proposed scheme, the comparisons between the conventional methods and the proposed scheme are introduced in the following simulation studies and practical experiments on a test rig, respectively. In the simulation studies, the comparison between the envelope analysis after direct vibration ICA separation and the proposed scheme is presented at first. Then, the comparison among the traditional envelope analysis, the envelope analysis after direct vibration ICA separation and the proposed approach for the data from the real world is introduced by the test experiments on a gearbox test rig. 5.1. Simulation Firstly, we simulate an impulsive response derived from a rotor system's single REB incipient fault, which can be expressed by rðtÞ ¼ 0:02 e  500t cos ð2πf r tÞ;

(5)

where r(t) denotes the impulsive response, f r ¼ 6500 Hz represents the oscillated frequency of the vibration, or the resonant frequency of the system, the constant 0.02 is the vibration amplitude, and constant 500 is the damping coefficient. By repeating the r(t) about 105 times per s (the fundamental frequency approximates 105 Hz), the impulse train R(t) generated by the REB fault can be simulated, where 1.5 percent random frequency fluctuation is added for considering the random slip of each rolling element. Secondly, we generate some disturbances in the simulation, which include an impulsive disturbance, two harmonic families and a Gaussian noise e(t) (power spectral density:  20 dB). The impulsive response is employed to simulate the meshing vibration of a broken tooth gear and given by  π d1 ðt Þ ¼ 0:5 e  500t cos 2πf r t þ : (6) 5 In the same way, the impulsive disturbance train D1 ðtÞ can be realized by repeating the d1 ðtÞ expressed by Eq. (6), 20 times per s (fundamental frequency is 20 Hz) for the simulation of a gear tooth broken fault. In this paper, two harmonic families are utilized to simulate the low-frequency disturbances for shafts, and are, respectively, given by d2 ðtÞ ¼ cos ð2πf 1 tÞ þ 0:5 cos ð4πf 1 tÞ þ0:25 cos ð6πf 1 tÞ;

(7)

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π π þ0:1 cos 10πf 2 t þ ; d3 ðt Þ ¼ cos 2πf 2 t þ 0:2 cos 6πf 2 t þ 19 17

(8)

where f 1 ¼ 20 Hz and f 2 ¼ 34:2 Hz denote the fundamental frequencies of the two harmonic families. Finally, the observed mixtures are given by the following equations: x1 ðtÞ ¼ RðtÞ þ D1 ðtÞ þd2 ðtÞ þ d3 ðtÞ þ eðtÞ;

(9)

x2 ðtÞ ¼ 0:7RðtÞ þ 0:9D1 ðtÞ þ1:5d2 ðtÞ þ 1:1d3 ðtÞ þ 0:6eðtÞ;

(10)

x3 ðtÞ ¼ 0:9RðtÞ þ 0:7D1 ðtÞ þ1:9d2 ðtÞ þ 1:2d3 ðtÞ þ 0:5eðtÞ;

(11)

where x1 ðtÞ, x2 ðtÞ, x3 ðtÞ represent the observed mixtures in three channels, respectively. Note that there are five independent sources in the mixtures, and they are R(t), D1 ðtÞ, d2 ðtÞ, d2 ðtÞ and e(t). If the number of the observed mixtures are not less than the number of independent sources, which can be obtained by the ICA separation. This will be verified in our study. However, if the prerequisite cannot be satisfied, for example, only three observed mixtures used viz. x1 ðtÞ, x2 ðtÞ and x3 ðtÞ, the ICA separation results are shown in Fig. 3, where two or more independent sources can be found in a separation result. Furthermore, to obtain the envelope spectra of both the three mixtures and the ICA separation results, the fast kurtogram algorithm is employed to calculate the optimum demodulation frequency band and the result is shown in Fig. 4, where the optimal filtering parameters are f o ¼ 6506:67 Hz and Bo ¼ 213:33 Hz.

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The squared envelope spectra [13] of x1 ðtÞ, x2 ðtÞ, x3 ðtÞ, IC1 , IC2 and IC3 are shown in Fig. 5, where the 20.02 Hz component related to the feature frequency family of D1 ðtÞ or the simulated broken tooth fault is exposed clearly. However, the interesting frequency lines associated with the simulated REB fault cannot be found. Note that there are some spectral lines in the high frequency part, which are caused by the aliasing phenomenon2 in Fig. 5. For example, the frequency lines enclosed by a red circle (after the down-sampling process in the fast kurtogram algorithm, where the corresponding sampling rate for the envelopes is 426.67 Hz and the Nyquist frequency or folding frequency is 213.33 Hz). Subsequently, the proposed approach is employed for a comparison. We still assume that the observed mixtures are x1 ðtÞ, x2 ðtÞ and x3 ðtÞ. The obtained kurtogram with the same optimal filtering band is almost similar to that shown in Fig. 4 and the corresponding envelopes are shown in Fig. 6. Then, the ICA is performed by taking the envelopes as the observed mixtures. The obtained envelope ICs and the corresponding squared envelope spectrum are shown in Figs. 7 and 8, respectively. It is clear that the proposed approach is valid, and the feature frequency (105.1 Hz) corresponding to the faulty REB's weak vibration is clearly exposed in Fig. 8(c). Note that the minor difference between the setting frequency 105 Hz and the prominent frequency line 105.1 Hz shown in Fig. 8(c) is due to the frequency resolution in the FFT algorithm. From the above comparative simulations, some conclusions can be drawn according to the results in the envelope analysis. Firstly, the weak pulse train and its feature frequencies are possible to be contaminated by strong background noises especially when there are some impulsive disturbances. Secondly, the ICA separation for the observed mixtures may fail if the number of observations is less than the number of sources. Thirdly, the proposed method is robust for the feature extraction of the faulty REB when the number of sources is unknown. 5.2. Test To verify the proposed approach for the data from the real world, an experiment on a test rig has been performed and introduced as follows. The picture of the test rig is shown in Fig. 9, where the REB with an artificial outer race fault, a broken tooth gear and a close view about the installation positions in the gearbox are shown in Fig. 10. The test conditions include that the faulty REB type is N1007 (the number of rolling elements Z¼15, the contact angle α ¼ 0, the diameter of the rolling element d¼6.75 mm, and the pitch circle diameter D¼48.5 mm); the rotating speed of the shaft is about 900 rev/min (the rotating frequency of the shaft f r ¼ 15 Hz), on which the faulty REB and the broken tooth gear are mounted; the data acquisition card is NI USB9234 (4 channels synchronous sampling) with the sampling rate of 20 kHz; three piezoelectric accelerometers (4530, sensitivity 69.7 pC/g, 3212, 87.8 pC/g and 3237, 95.2 pC/g) are mounted on the bearing house and a charge amplifier is employed. In theory, the feature frequency (ballpass frequency, outer race: BPFO) of the faulty REB can be calculated as in [22] by   Zf d BPFO ¼ r 1  cos α D 2   15  15 6:75 1 cos 0  96:84ðHzÞ: (12) ¼ 2 48:5

2 To reduce the aliasing occurred in the simulation, we can generate the signal by using a very high sampling frequency and then design a digital low-pass filter (acting as a anti-aliasing filter) to filter the signal before re-sampling it at a lower rate.

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The waveforms picked by the accelerometers are shown in Fig. 11. The common envelope analysis without the ICA is tested at first. The kurtogram of the vibration from channel 1 is shown in Fig. 12, where the maximum kurtosis is located at level 3 with the value 1.4, and the corresponding optimal filtering parameters are the center frequency f o ¼ 5769:23 Hz and the frequency band Bo ¼ 1282:05 Hz. By applying Eq. (2) for the envelope extraction and following the squared envelope spectrum analysis (remove the mean values of the three envelopes before performing the squared envelope spectrum analysis), the squared envelope spectrum is shown in Fig. 13, respectively. In Fig. 13, the frequency lines with 15 Hz and 30.01 Hz corresponding to 1  and 2  rotating speed of the gear, respectively, can be found in the envelope spectra. However, the interesting outer race fault corresponding frequency component (about 96.84 Hz in theory) is hardly to be found. Subsequently, the envelope analysis after the traditional ICA vibration separation for the interesting frequency component extraction of the REB fault is also performed for a comparison with the present approach. The FastICA [3] algorithm is utilized to separate the original vibrations shown in Fig. 11 to obtain the ICs. Then, the envelopes of the separated ICs are also obtained and the corresponding envelope spectrum is shown in Fig. 14, where the 2  frequency components of the rotating speed are still prominent and the interesting frequencies related to the outer race fault are also hardly found in all ICs' squared envelope spectrum. It is clear that the ICA separation fails to extract the faulty REB corresponding vibration.

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Finally, the envelope analysis by the proposed approach is introduced. At first, the envelopes are extracted by the fast kurtogram algorithm. Then, the FastICA algorithm is applied to the three envelopes to obtain the ICs. After that the squared envelope spectrum analysis is performed to the envelope ICs and the results are shown in Fig. 15. In Fig. 15(c), the outer race fault corresponding frequency component 95.34 Hz is clearly exposed. It is seen that the proposed envelope ICA separation is more robust than the traditional direct vibration ICA separation for the feature extraction of incipient REB faults.

6. Conclusions This paper proposed a robust feature extraction scheme for faulty REBs by combining the envelope extraction and the independent component analysis (ICA). In this approach, the envelope is not only utilized to obtain the impulsive vibration from the REBs, but also employed to perform the dimension reduction for the ICA. Consequently, the ICA algorithms can be

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successfully performed under the complex condition that the sensor number is limited and the source number is unknown. Then, the proposed approach can be used for the REB fault feature extraction in the condition that the traditional ways are invalid. The simulation results and practical test clearly show that the proposed approach provides a robust way to perform the REB fault detection in a rotating machinery. Acknowledgment Project supported by the National Natural Science Foundation of China (Grant no. 51365023) and (Grant no. 61203066). References [1] P.D. McFadden, J.D. Smith, Vibration monitoring of rolling element bearings by the high-frequency resonance technique—a review, Tribology International 17 (1) (1984) 3–10. [2] A.Y. Azovtsev, A.V. Barkov, D.L. Carter, Improving the accuracy of rolling element bearing condition assessment, Proceedings of the 20th Annual Meeting of the Vibration Institute, Saint Louis, June 1996, pp. 27–30. [3] A. Hyvärinen, E. Oja, Independent component analysis: algorithms and applications, Neural Networks 13 (2000) 411–430. [4] T. Honkela, A. Hyvärinen, Linguistic feature extraction using independent component analysis, Proceedings of IEEE International Joint Conference on Neural Networks, vol. 1, Budapest, July 2004, pp. 279–284. [5] J. Kim, J. Choi, J. Yi, M. Turk, Effective representation using ICA for face recognition robust to local distortion and partial occlusion, IEEE Transactions on Pattern Analysis and Machine Intelligence 27 (12) (2005) 1977–1981. [6] M.J. Roan, J.G. Erling, L.H. Sibul, A new, non-linear, adaptive, blind source separation approach to gear tooth failure detection and analysis, Mechanical Systems and Signal Processing 16 (5) (2002) 719–740. [7] Y. Guo, K.K. Tan, Order-crossing removal in Gabor order tracking by independent component analysis, Journal of Sound and Vibration 325 (1–2) (2009) 471–488. [8] J. Antoni, Blind separation of vibration components: principles and demonstrations, Mechanical Systems and Signal Processing 19 (6) (2005) 1166–1180. [9] I.K. Fodor, A Survey of Dimension Reduction Techniques, Lawrence Livermore National Laboratory Technical Report, UCRL-ID-148494, University of California, 2002. [10] N.C. Komgom, N. Mureithi, L. Aouni, T. Marc, On the use of time synchronous averaging, independent component analysis and support vector machines for bearing fault diagnosis, Proceedings of the 1st International Conference on Industrial Risk Engineering, Montreal, December 2007, pp. 610–624. [11] Z. Wang, J. Chen, G. Dong, Y. Zhou, Constrained independent component analysis and its application to machine fault diagnosis, Mechanical Systems and Signal Processing 25 (7) (2011) 2501–2512. [12] Y. Guo, T.W. Liu, J. Na, R.F. Fung, Envelope order tracking for fault detection in rolling element bearings, Journal of Sound and Vibration 331 (25) (2012) 5644–5654. [13] D. Ho, R.B. Randall, Optimisation of bearing diagnostic techniques using simulated and actual bearing fault signals, Mechanical Systems and Signal Processing 14 (5) (2000) 763–788. [14] N.G. Nikolaou, I.A. Antoniadis, Demodulation of vibration signals generated by defects in rolling element bearings using complex shifted Morlet wavelets, Mechanical Systems and Signal Processing 16 (4) (2002) 677–694. [15] J. Antoni, Fast computation of the kurtogram for the detection of transient faults, Mechanical Systems and Signal Processing 21 (1) (2007) 108–124. [16] R.F. Dwyer, Detection of non-Gaussian signals by frequency domain kurtosis estimation, Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing, Boston, April 1983, pp. 607–610. [17] J. Antoni, R.B. Randall, The spectral kurtosis: application to the vibratory surveillance and diagnostics of rotating machines, Mechanical Systems and Signal Processing 20 (2) (2006) 308–331. [18] T.W. Lee, M.S. Lewicki, M. Girolami, T.J. Sejnowski, Blind source separation of more sources than mixtures using overcomplete representations, IEEE Signal Processing Letters 6 (4) (1999) 87–90. [19] P. Bofill, M. Zibulevsky, Underdetermined blind source separation using sparse representations, Signal Processing 81 (11) (2001) 2353–2362. [20] J. Iriarte, E. Urrestarazu, M. Valencia, M. Alegre, A. Malanda, et al., Independent component analysis as a tool to eliminate artifacts in EEG: a quantitative study, Journal of Clinical Neurophysiology 20 (4) (2003) 249–257. [21] A. Hyvärinen, J. Karhunen, E. Oja, Independent Component Analysis, A Wiley-Interscience Publication, New York, 2001. [22] B.R. Randall, J. Antoni, Rolling element bearing diagnostics—a tutorial, Mechanical Systems and Signal Processing 25 (2) (2011) 485–520.