Key Engineering Materials Vol. 347 (2007) pp 271-276 online at http://www.scientific.net © (2007) Trans Tech Publications, Switzerland Online available since 2007/Sep/15
Bearing Fault Detection Using Higher-Order Statistics Based ARMA Model Fucai Li1, 2, a, Lin Ye2, b, Guicai Zhang1, c and Guang Meng1, d 1
State Key Laboratory of Mechanical System and Vibration Shanghai Jiao Tong University, 1954 Huashan Road, Shanghai 200030, P.R. China 2
School of Aerospace, Mechanical & Mechatronic Engineering The University of Sydney, NSW 2006, Australia
[email protected],
[email protected],
[email protected],
[email protected]
a
Keywords: ARMA model, higher-order statistics, machine fault detection, bearing
Abstract. Impulse response provides important information about flaws in mechanical system. Deconvolution is one system identification technique for fault detection when signals captured from bearings with and without flaw are both available. However effects of measurement systems and noise are obstacles to the technique. In the present study, a model, namely autoregressive-moving average (ARMA), is used to estimate vibration pattern of rolling element bearings for fault detection. The frequently used ARMA estimator cannot characterize non-Gaussian noise completely. Aimed at circumventing the inefficiency of the second-order statistics-based ARMA estimator, higher-order statistics (HOS) was introduced to ARMA estimator, which eliminates the effect of noise greatly and, therefore, offers more accurate estimation of the system. Furthermore, bispectrums of the estimated HOS-based ARMA models were subsequently applied to get clearer information. Impulse responses of signals captured from the test bearings without and with flaws and their bispectra were compared for the purpose of fault detection. The results demonstrated the excellent capability of this method in vibration signal processing and fault detection. Introduction Rolling element bearing is one of the most important parts of rotating machinery. Any flaw in bearing may result in malfunction and even lead to catastrophic failure. Hitherto, many techniques have been used in bearing fault diagnosis in the literature. Vibration signal processing is among the most frequently applied techniques. When a defect in one surface of a bearing encounters another surface, an extra vibration may be excited and extra vibration signal component is therefore introduced to measured signals. Hence, feature extraction for the extra vibration becomes the primary task of fault detection. However, measured vibration signals are always submerged in heavy background noise, such as sampling noise, unwanted vibration from other components and surrounding noise, which makes feature extraction more difficult. Effective vibration signal processing methods are therefore required for the purpose of fault detection. Being a branch of signal processing, pattern recognition is a significant fault detection technique. Deconvolution is one of the popular pattern recognition methods and it has been used in nondestructive evaluation (NDE) [1-3]. However, noise is obstacle to this technique and both input and output signals are required in application of deconvolution for system identification, which is infeasible in rotating machinery fault detection. Using measurements of the output signal only, autoregressive-moving average (ARMA) model estimator is therefore more applicable in bearing vibration system identification. Higher-order statistics (HOS) play an important role in system identification because they are capable of identifying unknown systems from output observation even when contaminated with additive Gaussian/non-Gaussian noise. Besides, they can track down the non-linearity in the system’s amplitude and/or phase characteristics. Extracted features using different processing methods may be different. Patterns of bearings with various health statuses are concerned in the present study. Aimed at fault detection, HOS were All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of the publisher: Trans Tech Publications Ltd, Switzerland, www.ttp.net. (ID: 130.203.133.33-17/04/08,14:16:02)
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introduced to estimate coefficients of ARMA model of a system, termed HOS-based ARMA model. The method was used to analyze vibration signals collected from bearings without and with flaws in races and ball. ARMA models of the test bearings with different health statuses were estimated, which were depicted using impulse response curves. To get more detail comparison, bispectrum, one of the higher-order spectra, was subsequently introduced to the estimated ARMA models. Comparison of the spectrograms proved the effectiveness of the method for bearing damage detection. HOS-Based ARMA Model ARMA Model. A typical single channel input-output system model is depicted in Fig. 1. In this model, u (k ) is input; x(k ) is theoretical output; n(k ) is additive noise; and y (k ) is real output of the system. The convolutional equation describing the noise-free output of this system is k
x(k ) = h(k ) ∗ u (k ) = ∑ h(i )u (k − i ) .
(1)
i =0
which can also be expressed in the complex z-domain or frequency domain, as X ( z ) = H ( z )U ( z ) or X (ω ) = H (ω )U (ω ) .
(2)
where z = exp( jωT ) , and sampling time T is assumed to equal unity. Hence, z = exp( jω ) .
u(k)
H(z)
x(k)
+
+
n(k)
+
y(k)
Fig. 1. Typical single channel system model. Three very popular parametric channel models are: (1) Moving Average (MA), in which H ( z ) = B( z ) and B(z ) is a rational polynomial in z ; (2) Autoregressive (AR), in which H ( z ) = 1 A( z ) and A( z ) is a rational polynomial in z ; and (3) Autoregressive-Moving Average (ARMA), in which H ( z ) = B( z ) A( z ) where A( z ) and B ( z ) are as described for the AR and MA models, respectively [4,5]. Higher-Order Statistics (HOS). HOS are defined in terms of cumulants of a signal, and higherorder spectra are defined in terms of HOS. Particular cases of higher-order spectra are the thirdorder spectrum also called the bispectrum which is, by definition, the Fourier transform (FT) of the third-order statistics, and the trispectrum (fourth-order spectrum) which is the FT of the fourthorder statistics of a stationary signal [4]. Besides possessing all the virtues of HOS, bispectrum is a more popular higher-order spectrum because it can be depicted in a three-dimensional reference frame in comparison with the four-dimensional chart requirement of trispectrum. Bispectrum of a signal x(t ) is defined by [6] B3 x ( f1 , f 2 ) = E[ X ( f1 ) X ( f 2 ) X * ( f1 + f 2 )] .
(3)
where X ( f ) is FT of x(t ) , f = ω 2π ; X * ( f ) is conjugation of X ( f ) ; and E [•] denotes statistical expectation. System Identification Using HOS-Based ARMA Model. The assumed system model of the present study is shown in Eq. 1 and Fig. 1. Real output of the system is y (k ) , y (k ) = x(k ) + n(k ) .
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Here additive noise n(k ) is assumed to be symmetric distributed. The system input process u (k ) is independent and identically-distributed (i.i.d.), non-Gaussian, independent of the noise, and satisfies 0 < C3u (0,0) < ∞ , in which, C3u is third-order cumulant of a zero-mean stationary process. It is defined by C3u (k , l ) = E[u ∗ (n)u (n + k )u (n + l )] .
(4)
Nikias and Pan [7] have developed an ARMA model estimation algorithm which was based on the logarithm of the bispectrum. Because of the aforementioned assumption, the output bispectrum B3 y ( f1 , f 2 ) exists and is given by B3 y ( f1 , f 2 ) = βH ( f1 ) H ( f 2 ) H ∗ ( f1 + f 2 ) .
(5)
where β is a constant depending on the i.i.d. input u (k ) . The bicepstrum is well defined [7] for purpose of estimating coefficients of ARMA model and is given by Bˆ (k , l ) = ∫∫ df1df 2 e j 2πf1k e j 2πf1l ln B3 x ( f1 , f1 ) = hˆ(k )δ (k ) + hˆ(l )δ (l ) + hˆ(− k )δ (k − l ) + ln C3u (0,0)δ (k )δ (l )
.
(6)
where hˆ(k ) is the cepstrum of H (z ) , defined by hˆ(k ) = ∫ df exp( j 2πfk ) ln H ( f ) . After some algebra, the following key cepstral equation is obtained and given by p
∑ mhˆ(m)[C
3y
(k − m, m) + C3 y (k + m, l + m)] = kC3 y (k , l ) .
(7)
m =− q
where p and q are AR and MA orders of the ARMA model, respectively. The Eq. 7 offers a set of linear equations for estimating a finite set of p + q parameters. The complex cepstrum, hˆ(k ) , is the inverse FT of the logarithm of the FT of h(n) . Hence, the estimated ARMA model, h(n) , is the IFT of the exponential of the FT of the complex cepstrum calculated from Eq. 7, detailed in [5].
timing belt
load
Processing Results Using HOS-Based ARMA Model gear box
motor
shaft
accelerometer
bearing
flange bearing for the tests
Fig. 2. Experiment apparatus. Non-stationary or singular vibrations always accompany the presence of defects in running bearings. Bearings with different health statuses have different vibration models. Therefore, changes in
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estimated model expose the working condition of a bearing. The experiment apparatus used in the present study is shown in Fig. 2. Vibration signals were collected using accelerometer at the sampling rate of fs = 6,400Hz when the motor ran at the speed of 1,200 revolutions per minute (RPM). Vibration signal captured from an intact bearing is regarded as the benchmark in bearing fault diagnosis, shown in Fig. 3 [8], which shows that noise is the dominant components. Hence, noise makes fault detection more difficult.
Amplitude (v)
0.10 0.05 0.00 -0.05 -0.10 0.0
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0.4
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Fig. 3. Vibration signal of an intact bearing. A ball element bearing consists of three parts, viz. balls, races (including inner race and outer race) and cage. Balls and races are of the damageable components. In this study, single point faults were introduced to ball and races of the test bearings to evaluate the effectiveness of the proposed method. Chen and Sam [1,2] have applied power spectrum based deconvolution method in system identification for NDE of materials. The power spectrum is, in fact, a member of the class of HOS, i.e., it is a second-order spectrum [4]. However, the major drawbacks of conventional second-order statistics based deconvolution techniques are their inability to identify non-minimum phase systems, and their sensitivity to additive Gaussian noise, detailed in [1,2]. Yamani, Bettayeb and Ghouti [3] extended the second-order based deconvolution method to higher-order statistics based deconvolution which overcomes the drawbacks of the former method. Nevertheless, system input should be known in deconvolution techniques, which is infeasible in rotating machinery. 8 4
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(d) Fig. 4. Impulse responses of bearings with different health statuses; (a) intact bearing; (b) ball damaged bearing; (c) inner race damaged bearing; (d) outer race damaged bearing.
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0.5
0.5
0.4
0.4
f2 (Normalized Frequency)
f2 (Normalized Frequency)
Impulse responses of estimated system identification ARMA models of bearings with different health status are shown in Fig. 4. Here, all the ARMA models are estimated using HOS-based ARMA estimator. Despite the dominant noise in original vibration signal, such as the vibration signal of intact bearing shown in Fig. 3, impulse response of estimated system of intact bearing is relatively smooth, shown in Fig. 4(a). Namely, noise is no longer barrier to system identification using the present method.
0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4
0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 f1 (Normalized Frequency)
-0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 f1 (Normalized Frequency)
(a)
(b)
0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 f1 (Normalized Frequency)
f2 (Normalized Frequency)
f2 (Normalized Frequency)
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0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 f1 (Normalized Frequency)
(c) (d) Fig. 5. Bispectra of estimated ARMA models of bearings with different health status; (a) intact bearing; (b) ball damaged bearing; (c) inner race damaged bearing; (d) outer race damaged bearing. Traditional impulse response curves are always difficult to understand in engineering practice. To make the impulse responses more understandable, bispectra of the estimated ARMA models in Fig. 4 are further calculated and shown in Fig. 5. For the convenience of comprehension, all the spectra are depicted using contour plots, and normalized frequency is used as unit of the two frequency axes. When the symmetric factor is ignored, only one crest locates between normalized frequency 0.3 and 0.4 in bispectrum of intact bearing, as shown in Fig. 5(a), which means that intact bearing system has the clearest and simplest pattern in comparison with the damaged cases shown in Fig. 5 (b), (c) and (d). Frequency coupling of estimated model appears only at the low-frequency region when the flaw locates on ball of the test bearing, as shown in Fig. 5(b). Moreover, two crests locate on f1 and f2 frequency axes in lower-frequency regions, respectively. On the other hand, when there are flaws in races of the test bearings, ARMA models become complicated. Coupling in higher-frequency region appear in the bispectra of race damaged bearings in comparison with ball damaged case, as shown in Fig. 5(c) and (d). Further conclusion can be made when the inner race damage case is compared with outer race damage case, which is that flaw
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in inner race produces more crests in the model bispectrum of the bearing system, as shown in Fig. 5(c). Complexities of vibration signals of damaged bearings are also studied in [9], in which, similar conclusions are also drawn. Conclusion Aimed at bearing fault detection, HOS-based ARMA model estimator was introduced in this study. Extracted features about damage of bearings may be different when using different methods. By processing the vibration signals collected from test bearings, coefficients of ARMA models of these bearings were estimated using HOS-based ARMA model. Impulse response was selected as the depicting way to show the extracted features. To make the estimated results more easily understood, bispectra of the ARMA models of bearings with different health statuses were further obtained. Results show that the bispectrum of system model of intact bearing is the clearest and simplest one in comparison with those of the damage cases; bispectra of race damaged bearings are more complicated than that of ball damaged bearing. Moreover, inner race damage makes the ARMA model of bearing system more complicated than outer race damage. Further work will be directed towards interpretation of bispectrum of the estimated ARMA model and the influence of different damage sizes on system impulse response using higher-order spectra. Acknowledgements Fucai Li, Guicai Zhang and Guang Meng are grateful for the supports received from the Natural Science Foundation of China (NSFC No. 50575146), NSFC Joint Research Fund for Overseas Chinese Young Scholars (No. 10528206) and Key International S&T Cooperation Project of China Ministry of Science and Technology (No. 2005DFA00110), respectively. Lin Ye is grateful for Discovery Project (DP) from the Australian Research Council. The authors would like to express their gratitude to Prof. Peng Chen of Mie University, Japan, for the experimental data he supplied. References [1] S.-K. Sin, C.-H. Chen: IEEE Transactions on Image Processing Vol. 1 (1992), p. 3. [2] C.H. Chen, W.-L. Hsu, S.-K. Sin, in: Acoustics, Speech, and Signal Processing, 1988. ICASSP-88., 1988 IEEE International Conference on, edited by A.E. Rosenburg, New York, USA (1988), p. 867. [3] A. Yamani, M. Bettayyeb, L. Ghouti, in: Proceedings of the IEEE Signal Processing Workshop on Higher-Order Statistics, Banff, Alberta, Canada (1997), p. 214. [4] C.L. Nikias, J.M. Mendel: IEEE Signal Processing Magazine Vol. 10 (1993), p. 10. [5] R. Pan, C.L. Nikias: IEEE Transactions on Acoustics Speech and Signal Processing Vol. 36 (1988), p. 186. [6] G.C. Zhang, J. Chen, F.C. Li, W.H. Li: Damage Assessment of Structures Vi (2005), p. 167. [7] C.L. Nikias, P. Renlong, in: Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP '87), Dallas, Tex, USA (1987), p. 980. [8] F.C. Li, J. Chen, G.C. Zhang, Z.J. He: Damage Assessment of Structures Vi (2005), p. 127. [9] F.C. Li, L. Ye, G.C.Zhang, G. Meng, in: 7th International Conference on Damage Assessment of Structures, edited by L. Garibaldi, Torino, Italy (2007).
