BEARING COMPOUND FAULT ACOUSTIC DIAGNOSIS BASED ON IMPROVED BLIND DECONVOLUTION ALGORITHM. Nan Pan, Xing Wu and Yu Guo

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1 BEARING COMPOUND FAULT ACOUSTIC DIAGNOSIS BASED ON IMPROVED BLIND DECONVOLUTION ALGORITHM Nan Pan, Xing Wu and Yu Guo Faculty of Mechanical & Electrical Engineering, Kunming University of Science & Technology, Kunming, P.R. China ICETI-2014 J5001_SCI No. 15-CSME-49, E.I.C. Accession 3824 ABSTRACT In the progress of bearing fault acoustic testing, signals picked up by acoustic sensors are usually mixed with fault source signals and other noise signals due to the complexity of mechanical signals and various interference sources. In order to solve the above problems, an improved blind deconvolution algorithm is put forward. The proposed algorithm applies adaptive generalized morphological filtering to the observed signals to retain their characteristic details, and then utilizes an OMP algorithm based on the minimum kurtosis to restore the periodical signals in the mixed signals in order to reduce the impact of the periodic components on blind separation. Finally, the improved Kullback Leibler (KL) distance algorithm is employed to calculate the distances between independent components, which is used as the clustering index, and then to perform fuzzy C-means clustering. The experiment results of bearing compound fault extraction in real working-environment demonstrate the accuracy and reliability of the proposed algorithm. Keywords: compound fault acoustic diagnosis; improved KL divergence; blind deconvolution; adaptive generalized morphological filter. DIAGNOSTIC DE DÉFAILLANCE DES ROULEMENTS PAR ÉMISSION DE SIGNAUX ACOUSTIQUES BASÉS SUR UN ALGORITHME AMÉLIORÉ DE DÉCONVOLUTION AVEUGLE RÉSUMÉ Dans les processus des tests de diagnostic de défaillance des roulements, les signaux repérés par des capteurs acoustiques sont habituellement confondus avec des signaux de source d erreurs et autres signaux de bruit. Ceci est dû à la complexité des signaux mécaniques et des sources variées d interférence. Pour résoudre ce problème, un algorithme amélioré de déconvolution aveugle est mis de l avant. L algorithme proposé est appliqué par filtrage morphologique général adaptif aux signaux observés pour retenir les détails caractéristiques, et ensuite on utilise un algorithme de type OMP basé sur un aplatissement minimum pour restaurer les signaux périodiques dans les signaux mixtes dans le but de réduire l impact des composants périodiques sur la séparation aveugle. Finalement, l amélioration de distance Kullback Leibler (KL) de l algorithme est employée pour calculer la distance entre les composants indépendants. Lesquels sont utilisés comme regroupement d indices, et permettent ensuite d effectuer la fusion des c-moyennes floues (fuzzy C-means clustering). Les résultats des expériences d extraction de défaillance des roulements dans un environnement de travail réel démontrent l exactitude et l efficacité de l algorithme proposé. Mots-clés : diagnostic de défaillance des roulements; divergence KL améliorée; déconvolution aveugle; filtrage morphologique général adaptif. Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 3,

2 1. INTRODUCTION Signals picked up by sensors are usually mixed with fault source signals and other noise signals due to the complexity of mechanical signals, strong background noise in real-time working conditions, and various interference sources, making it difficult to extract and identify the useful signals [1]. How to extract characteristic information that reflects the true status of the object in question from the vast amount of mechanical signals is therefore the key to a successful fault diagnosis [2]. As signals picked up onsite manifest typical non-stationary characteristics and have high-frequency modulations, morphological filtering, a nonlinear filtering technology, can effectively enhance the signal contour and morphological features as well as suppress background noises. Morphological operator demodulation was first used by Nikolaou in 2003 to analyze impulse signals [3]. Lu Shen et al. from Zhejiang University employed generalized morphological filtering to de-noise vibration signals of rotary machines, and achieved higher signal-to-noise ratios (SNR) through eliminating the vibration noise interference [4]. Comon proposed a criterion function based on high-order cumulants and derived an adaptive algorithm to obtain separation matrices, making blind source separation possible [5]. Koldovsky et al. suggested a time-domain blind deconvolution algorithm (TDBD) based on Independent Component Analysis (ICA) and cluster analysis, which delivers good results when processing short convolutive mixtures of non-stationary signals such as voice signals [6]. Wang Yu et al. proposed a genetic algorithm (GA) based on TDBD, which can search for optimal delay adaptively within a large range and can be applied to real acoustic environments [7]. This algorithm, however, falls short when performing complex mechanical fault signal separation due to order uncertainty during blind processing, large amount of computation and accumulative errors [8]. Based on the above research findings and the TDBD GA, an improved blind deconvolution algorithm based on AGMF, CS and improved Kullback Leibler (KL) divergence is put forward, which applies adaptive generalized morphological filtering to the observed signals to retain their characteristic details and to partially reduce noises, and then utilizes an OMP algorithm based on the minimum kurtosis to restore the periodical signals in the mixed signals in order to reduce the impact of the periodic components on blind separation. Finally, the improved KL distance algorithm which based on kernel function density ratio estimation is employed to calculate the distances between independent components, which is used as the clustering index, and then to perform fuzzy C-means clustering. As a result, the clustering accuracy and the reliability of blind deconvolution results are both improved for the bearing compound fault extraction. 2. MATHEMATICAL MODEL FOR THE TDBD ALGORITHM At present, the blind signal-processing algorithm is shifting from the blind separation of instantaneous linear mixture in the early days toward the blind separation of convolutive mixture that bears a closer resemblance to real working conditions. The linear convolution model is formularized as follows [10]: x i (t) = n K i j j=1 τ=0 h i j (τ)s j (t τ), i = 1,...m (1) where x(t) = [x 1 (t)...x m (t)] T stands for a mixture of m number of observed signals, s(t) = [s 1 (t)...s n (t)] T stands for n number of unknown source signals, and h i j stands for the compound filter of the system, which is the impulse response with a length of K i j. The flow of the TDBD GA based on the linear convolution model is shown as below [7]: 1. Set the optimal range of the variable L and determine the encoding length L en ; randomly generate a group of size of M. 658 Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 3, 2015

3 2. Expand the mixed signals x(n) to a subspace x (n) of m L-dimension and perform independent component analysis on x (n) using FastlCA algorithm to obtain the independent component c(n) of a dimension of m L. 3. Calculate the distance D between the components c(n) and perform clustering on D using the agglomerate layered clustering algorithm. 4. Restore various clustering components, obtain the estimated source signals t(n), and calculate the fitness function f. 5. Determine whether the end conditions are satisfied. If yes, proceed to Step (6); if not, perform cross mutation on the chromosome to generate a new-generation of group then proceed to Step (2). 6. Output the results, and the whole process is complete. However, the following problems arise when the TDBD GA is used to separate the real mechanical faults [7 10]: 1. Although the characteristic signals of a faulty bearing are non-stationary, they are easily buried in the strong environmental noises and other non-stationary signals, increasing the difficulty in blind signal separation. 2. When the algorithm is applied to real mechanical fault signals, the delay parameter L covers a wide range, and a huge amount of computation and increased cumulative errors are incurred. 3. All sorts of fault signals are mixed together in the compound fault signals, making it difficult to identify a given fault source. 4. The processed mixed signals still contain periodic components and Gaussian noise, severely compromising the separation result of the algorithm. 3. IMPROVEMENTS ON THE TDBD GA 3.1. Mixed Signal Preprocessing Generalized Morphological Filtering Adaptive generalized morphological filter Since noisy signals imply difficulties in blind deconvolution and undesirable separation results, whereas the morphological filtering algorithm is easy to use and fast to execute, morphological filtering algorithm is thus applied to mechanical signal processing in this study [4]. In a previous study, an on-off filter and an off-on filter were constructed to eliminate the positive/negative impulse interference on signals [11]. They are given as: F OC ( f (n)) = ( f g g)(n) (2) F CO ( f (n)) = ( f g g)(n) (3) To achieve a better filtering result, an average junction filter can be constructed by cascading the on operation with the off operation. It is given as Y (n) = [F OC ( f (n)) + F CO ( f (n))]/2 (4) The average junction filter can suppress both positive and negative impulse noises in signals (weighting), but employing identical structural elements would result in a severely statistically deviated filter-output. The Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 3,

4 Fig. 1. Time domain waveform of source signals. best weight coefficient could be determined LMS adaptive method. A generalized morphological filter has been proposed previously [4], which is given as y(n) = 1 2 [F OC( f (n)) + F CO ( f (n))] (5) F OC ( f (n)) = ( f g 1 g 2 )(n) (6) F CO ( f (n)) = ( f g 1 g 2 )(n) (7) where g 1 and g 2 stand for two different types of structural elements. The signals of a faulty bearing usually contain impulse noise interference and random noise interference. Researches indicate that triangular structural elements are suitable to filter out impulse noise interference whereas semicircular structural elements are suitable to filter out random noise interference. Triangular and semicircular structural elements are therefore used in this study to construct a generalized morphological filter for signal preprocessing Computer Simulation To verify the effectiveness of adaptive generalized morphological filtering, periodic and impulse source signals with N number of data points and a sampling frequency (f s ) of 4096 were constructed, as shown in Fig. 1, where n(t) is the Gaussian white noise interference. S 1 (t) = sin(2π10t) + cos(2π50t) + n(t) (8) S 2 (t) = 1 exp( 90t)sin(2π440t) + n(t) (9) 2 Due to the complexities in respect of real environment and machine structure, the acquired signals are extremely susceptible to noise interference and have a low SNR. To simulate a real environment, strong Gaussian white noise interference is applied to the source signals with a SNR of 0.14 and 28.7 db respectively. Figure 2 shows that the source signals are almost overwhelmed by the noises. Generalized morphological filtering is then applied to the noisy signals and the filtered waveforms are shown in Fig. 2(b). It can be seen that noises are mostly eliminated and the basic morphological signal characteristics are retained, with the SNR after filtering being and 6.23 db respectively, indicating a satisfactory filtering effect. 660 Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 3, 2015

5 Fig. 2. De-noising simulation OMP Algorithm Based on the Minimum Kurtosis OMP algorithm Since mixed signals contain a large number of periodic components, which will affect the taxonomic clustering of signals and subsequently affect the blind signal separation result [7]. It is thus crucial to eliminate the periodic components from the mixed signals. Kurtosis is a statistical tool that can detect transient signals sensitively and is usually used as a blind detection filter. It is calculated as follows: E { x(t) 4} K n = { 3 (10) E 2 x(t) 2} Whiten processing is performed on mixed signals to normalize the signal energy (E 2 { x(t) 2 } = 1), so that the amount of computation can be reduced. The OMP algorithm is a type of greedy reconstruction algorithm in CS theory that is fast to compute and able to deliver good results. Therefore, in this study, kurtosis is used in conjunction with the OMP algorithm to reduce the impact of periodic components upon signal component clustering and restoration. First, the minimum kurtosis of the mixed signals are calculated. Then, the periodic signals with the minimum kurtosis are restored through the OMP algorithm. Finally, the restored signals are subtracted from the mixed signals to eliminate the periodic components from the mixed signals Computer simulation The simulated signals restored following Eq. (11) are shown in Fig. 3(a); n(t) stands for Gaussian white noises. The OMP algorithm based on the minimum kurtosis is used to process the simulated signals and the results are shown in Fig. 3(b). It can be seen that the periodic signals with a frequency of 50 Hz are restored, proving that the OMP algorithm based on the minimum kurtosis can effectively restore periodic components contained in the signals. S(t) = sin(2π50t) + n(t) (11) Experimental verification of inner ring fault extraction To verify whether the proposed algorithm can effectively extract the periodic components in mechanical signals, fault signals are acquired for an antifriction bearing containing a faulty inner ring during its operation Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 3,

6 Fig. 3. Signal restoration simulation. Fig. 4. Signal restoration simulation. on a rotary machine fault simulation platform. A 1/4 TEDS array microphone (BSWA Tech Co., Beijing, China) is employed to acquire the acoustic signals from the bearing that contains the faulty inner ring and is operating at a speed of 800 r/min (f is Hz). These acoustic signals, as shown in Fig. 4(a), are processed with the algorithm put forward in this study and the restored signals are shown in Fig. 4(b). It can be seen that the frequency of the restored signal is 13 Hz, close to the base frequency f Improve KL distance A methodology based on the KL distance and mutual information parameter can overcome the order uncertainty arisen during blind deconvolution [8]. In this paper, distances between various components are measured through the KL distance and are used as clustering index. The KL distance serves as the higherorder statistical information for the signals in question, and can reflect the statistical characteristics of the signals more accurately. The KL distance represents the distance between the probability densities of the two corresponding sets of signals in two adjacent frequency bands. p j (w) indicates the signal probability 662 Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 3, 2015

7 Table 1. Operation time contrast between three clustering indexes. Index type Clustering running time t/s Overall program running time t/s Improved KL distance Kurtosis Cosine measure Fig. 5. Scatter plot of three different indexes. density function, and is formularized as follows: p j (w,m)= Yj (w,m) 2 Yj (w,r) 2 (12) M r=1 where w stands for the frequency band, m and r stand for sequence numbers of windows, and M stands for the number of windows. Since formula (12) can only reflect signal energy distribution, it needs to be revised as follows: Yj (w,m) 2 p j (w)= Y j (w,r) 2 p j, f re (w,m) (13) M r=1 where p j, f re (w,m) standards for the probability of which a given value Y j (w,m) appears in the range [Y j (w,1) Y j (w,m)]. Equation (13) can reflect not only the complex-value of signals but also the signal energy distribution. Therefore, it can deliver more effective and accurate results than Eq. (12). The KL distance is given as where KL(p j (w), p l (w + 1)) = M m=1 p j (w,m) log { p j (w)= [p j (w,1)... p l (w,m)] p j (w,m) p l (w + 1,m) (14) j = 1,...,J, l = 1,...,J In this study, the computation time for the following three cases are compared: using KL distance as the clustering index, using kurtosis as the index, and using cosine measure as the index. The results are shown in Table 1. Furthermore, distances between various components, which are computed with the improved KL distance, kurtosis, and cosine measure respectively, are clustered, and the clustering results are shown in Figs. 5(a), (b), and (c) respectively. Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 3,

8 Table 2. Bearing specifications and test parameters. Bearing pitch Roll diameter Roll Contact angle Speed Sampling rate Sampling diameter D (mm) d (mm) number Z α n (rev/min) f s (HZ) number N Fig. 6. Experiment site of collecting the acoustic signals. Based on the computation time and the cluster scatter plot, it can be concluded that the algorithm using the improved KL distance results in linearly scattered distance clusters and is fast to compute. Therefore, replacing kurtosis or cosine measure based algorithm with the KL distance based algorithm can achieve a superior outcome when calculating the distances between independent components. The overall flow of the improved algorithm herein is as follows: 1. Initialize the delay parameter L and the clustering number. 2. Apply adaptive generalized morphological filtering to the observed signals and obtain signal x(n). 3. Expand signal x(n) to x (n) of a dimension of m L, and perform component analysis on x (n) using the FastlCA algorithm to obtain the independent component c(n) of a dimension of m L. 4. Calculate the normalized kurtosis of each component, determine the minimum kurtosis K min, and restore the periodic signal P corresponding to K min using the OMP algorithm. 5. Subtract P from c(n) to obtain the impulse signal c(n), calculate the distances D between the components c(n) using the improved KL distance algorithm, and cluster the distances D. 6. Restore various clustering components and obtain the estimated source signal t(n). 7. Obtain the envelope demodulation spectrum of the source signal t(n), and make a fault diagnosis. 664 Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 3, 2015

9 Fig. 7. Acoustic signals generated by two groups of bearings with outer right faults. Fig. 8. Separation results of the TDBD GA. 4. EXPERIMENTAL VERIFICATION To verify that the improved algorithm is practical and effective, test was conducted on a real faulty antifriction bearing. The antifriction bearing specifications and the parameters relevant to this test are listed in Table 2. Based on these parameters the characteristic frequency of the faulty outer ring of the bearing is calculated and is Hz; that of the inner ring is Hz Fault Extraction Experiment of Bearing This experiment is conducted on a rolling bearing with a faulty outer ring on a rotary machine fault simulation platform. Two BSWA 1/4 TEDS array microphones are placed on the two sides of the faulty bearing respectively, and they each forms an angle of 45 with the rotation axis, as shown in Fig. 6. Figure 7 shows the two sets of acoustic signals generated by the rotating bearing with a faulty outer ring. Firstly, the TDBD GA is employed to process the fault signals and the spectral envelopes of separated signals are shown in Fig. 8, which indicates that the characteristic frequency of the faulty outer ring is not successfully extracted. Then the improved algorithm is used to process the fault signals and the spectral Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 3,

10 Fig. 9. Separation results of the improved algorithm. envelopes of the separated signals are shown in Fig. 9. In this plot the spectral line at 65 Hz and its double frequency coincide with the theoretical characteristic frequency of Hz of the outer ring, suggesting a successful extraction of the characteristic fault frequencies. 5. CONCLUSIONS On the basis of TDBD GA, a blind deconvolution algorithm is put forward based on adaptive generalized morphological filtering and clustering using an improved KL distance calculation method, which can reduce the complexity in blind noisy-signal separation. The proposed algorithm utilizes the KL distance, i.e., the distance between independent components, as the clustering index and benefits from enhanced reliability and accuracy in terms of component clustering. During the blind deconvolution experiment performed on signals of faulty bearings acquired in real working-environments, this algorithm has successfully separated/extracted the fault impulse signals and hence proves to be effective. ACKNOWLEDGEMENTS This article was supported by the National Natural Science Foundation of China (No ) and the personnel training fund of Kunming University of Science & Technology (No. KKZ ). REFERENCES 1. Wang, H.M., Chen, X., An, G. et al., Fault diagnosis of gearbox based on higher order cumulant, Journal of Mechanical Strength, Vol. 26, No. 3, pp , He, Z.J., Chen, J., Wang, T.Y. et al., Mechanical Fault Diagnosis Theory and Application (1st ed.), Higher Education Press, Beijing, Nikolaou, N.G. and Antoniadis, I.A., Application of morphological operators as envelope extractors for impulsive-type Periodic signals, Mechanical Systems and Signal Processing, Vol. 17, No. 6, pp , Shen, L., Zhou, X.J., Zhang, W.B. et al., De-noising for vibration signals of a rotating machinery based on generalized mathematical morphological filter, Journal of Vibration and Shock, Vol. 28, No. 9, pp , Comon, P., Independent component analysis, a new concept, Signal Processing, Vol. 36, No. 3, pp , Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 3, 2015

11 6. Koldovsky, Z. and Tichavsky, P., Time domain blind audio source separation using advanced component clustering and reconstruction, in Proceeding of 2008 Hands-free Speech Communication and Microphone Arrays, Trento, Italy, pp , May 6 8, Wang, Y., Chi, Y.L., Wu, X. et al., An improved blind in the diagnosis of bearing acoustic solution of convolution algorithm, Journal of Vibration and Shock, Vol. 29, No. 6, pp , Zhao, X., Yang, Y.M., Song, N. et al., Study on permutation in blind signal processing, Computer Engineering and Applications, Vol. 44, No. 20, pp , Pan, N., Wu, X., Chi, Y.L. et al., Mechanical equipment status detection and fault diagnosis based on frequency domain blind convolution, Journal of Vibration and Shock, Vol. 31, No. 12, pp , Pan, N., Wu, X., Chi, Y.L. et al., Underdetermined blind convolution is used for rolling bearings composite fault acoustic diagnosis, Journal of Vibration, Measurement & Diagnosis, Vol. 33, No. 2, pp , Li, Y.C., Wu, X., Chi, Y.L. et al., Blind separation for rolling bearing faults based on morphological filters and sparse component analysis, Journal of Vibration and Shock, Vol. 30, No. 12, pp , Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 3,