Induction Motor Bearing Fault Detection with Non-stationary Signal Analysis

Proceedings of International Conference on Mechatronics Kumamoto Japan, 8-1 May 7 ThA1-C-1 Induction Motor Bearing Fault Detection with Non-stationary Signal Analysis D.-M. Yang Department of Mechanical and Automation Engineering, Kao-Yuan University, Kaohsiung 81, Taiwan, R.O.C dmyang@cc.kyu.edu.tw Abstract The purpose of this research is to identify bearing fault features. This approach uses continuous wavelet transforms as a non-stationary signal preprocessor and the singular value decomposition (SVD) technique as salient feature extraction. Simulations of a model for bearing inner race defect as well as actual bearing vibration data from a normal bearing and the defective inner race bearing are used to demonstrate the proposed method for bearing fault detection and diagnosis. The results obtained have shown that this approach is effective for bearing fault detection and diagnosis. Index Terms Bearing fault, singular value decomposition, wavelet analysis I. INTRODUCTION The rolling element bearing is one of the most critical components in rotating electrical machinery due to the fact that the large majority of problems arise from faulty bearings. A previous report [1] on failed components of induction motors has pointed out that the most significant contributor to bearing failure is inadequate maintenance, and this can, in turn, result in winding failure within the machine. Therefore, proper monitoring of bearing condition is highly cost effective in reducing capital loss. Vibration-based monitoring techniques, both in the time and frequency domains, have been widely used for detection and diagnosis of bearing defects for several decades. A brieeview of vibration monitoring techniques can be found in []. These methods have traditionally been applied, separately, in the time and frequency domains. A time-domain analysis focuses principally on statistical characteristics of the vibration signal such as peak level, standard deviation, skewness, kurtosis and crest factor. A frequency domain approach uses Fourier methods to transform the time-domain signal to the frequencydomain where further analysis is carried out, conventionally using vibration amplitude and power spectra. It should be noted that use of either domain implicitly excludes the direct use of information present in the other. However, a relatively new signal processing technique, called wavelet analysis, is adopted here and used to develop vibration signal processing procedures for bearing fault detection. The wavelet approach has advantages over traditional Fourier methods for signal analysis, particularly for signals containing discontinuities and shape spike. When Fourier transformation is performed, the signal representation is moved from the time-domain to the frequency domain and this change of domain can lead to loss of information and to interpretation difficulties. This disadvantage is overcome in wavelet analysis which represents a signal by using shifted and scaled versions of a so-called mother wavelet and this enables examination of the frequency information of the signal as it evolves with time. This ability to represent simultaneously both time-domain and frequency-domain information is a significant advantage of the wavelet approach. The use of the continuous wavelet transform to study bearings with outer race and inner race defects is presented in [3]. Scale-wavelet power spectrum and time-wavelet energy spectrum autocorrelation analysis method were used to extract fault features. This approach provide a new way for rolling bearing fault diagnosis. Li and Ma [4] used wavelet transforms to study bearings with outer race and rolling element defects under a range of load and speed conditions. Different scales of wavelet transform can provide information about a bearing defect at various locations, and this property makes it more effective for the detection of bearing-localised defects than traditional time-domain or frequency-domain methods. The complex nature of information obtained from wavelet transforms make subjective interpretation and diagnostic use difficult. In order to effectively interpret the wavelet map, the time-frequency domain is used instead of time-scale domain. Hence, time-frequency distributions of two specifically simulated signals and the associated singular values performed by the SVD technique are presented in this paper. In the following sections, the non-stationary signal processing using the wavelet analysis will be introduced briefly in Section II. The SVD technique for extracting bearing defective features from the wavelet coefficients will be developed in Section III. The simulation results of the proposed approach for a model for a bearing defective inner race will be given in Section IV. Experimental studies of normal and defective bearing vibration signatures using the proposed approach will be presented in Section V and a conclusion will be presented in Section VI. 1-444-1184-X/7/$5. 7 IEEE 1

II. NON-STATIONARY SIGNAL ANALYSIS BY THE WAVELET ANALYSIS Wavelet analysis is an approach which decomposes a timedomain signal into components in different time windows and different frequency bands and presents the resulting information in the form of a surface in the time-frequency plane, sometimes referred to as a scalogram [5]. The scalogram is similar in concept to the spectrogram but differs from in that the frequency resolution of the scalogram is logarithmic rather than linear, as is the case for the spectrogram. Because of the nature of the frequency resolution, the wavelet approach is more effective in analyzing both the long-time, low-frequency and the short-time, high-frequency content of a time signal. This characteristic is very useful for analyzing pulse-like and non-stationary signals. The continuous wavelet transform of a square-integrable, continuous time signal s(t) is the inner product between and the analyzing wavelet ψ a,b (t), which gives the wavelet coefficients W x (a, b) = 1 a s(t)ψ ( t b a ) (1) where a is the dilation parameter governing the wavelet frequency, b is the parameter localizing the wavelet function in the time domain and ψ (t)is the complex conjugate of the analyzing wavelet ψ(t). The Fourier transform of s(t) is defines as S(f) = x(t)e jπft dt () Whereas Fourier decomposition is based on the harmonic wave e jπft, the wavelet analysis is based on an analyzing wavelet. There are a number of different complex or real valued functions used as analyzing wavelets. The analyzing wavelet used in this paper is the complex-valued Morlet wavelet given by ψ M (t) = e jπf t e t / where f is the central frequency of the Morlet wavelet and the value of f is taken 1 in this paper. An alternative formulation of the continuous wavelet transform can be obtained by transforming both the signal s(t) and the analyzing wavelet ψ a,b (t) in the frequency domain: (3) W x (a, b) = a S(f)Ψ (af)e jπfb df (4) where Ψ (af)e jπfb is the Fourier transform of ψ ( t b a ) respectively. For an easier implementation of wavelet transform, equation (4) can be expressed in a discrete form as: W x (a p, b q ) = a p S(f k )Ψ (a p f k )e jπf kb q (5) k III. SINGULAR VALUE DECOMPOSITION FOR FEATURE EXTRACTION The purpose of using a feature extraction procedure is to significantly reduce the quantity of data but, at the same time, retain the original salient information. It is well known that the SVD has optimal decorrelation properties and provides a means of detecting dominant characteristics on a set of data. The SVD theorem states that any m by n matrix W can be decomposed and written as the product of matrices [6], [7]. P W (α m, β n ) = U mm Σ mn Vnn t = λ k u k vk T (6) k=1 where the superscript, T, donates the transpose, U is an orthogonal m by m matrix made up of left singular vectors, u k, V is an orthogonal n by n matrix made up oight singular vectors, v k, and Σ is an m by n matrix of non-negative real singular values and decrease monotonically from the upper left to the lower right of Σ. The singular value of W are represented by λ 1 λ λ 3 λ P, where P = min(m, n). λ 1, λ, λ 3,..., λ P are the eigenvalues of W T W. For convenience the matrix, W, is termed the vibration characteristic matrix. In SVD, the singular values are usually arranged in descending order, so the first principal component, corresponding to the largest eigenvalue, characterizes most of the variation in the data set. From the point of view of PCA, it is the first principal component (PC1) which is the linear combination of the first eigenvalue and the associated eigenvector of the correlation matrix W T W that characterizes the maximum variance of W [8]. In the present application, only the first singular values will be considered since these account for the large majority of variations in the analyzed data set. IV. BEARING INNER RACE DEFECT MODEL A defect in one surface of a rolling element bearing striking another surface produces periodical impulses. A model for the impact train of a defective rolling element bearing under load can be the following form [9], [1], [11]. d(t) = k= d q(τ k + kt d )δ(t kt d τ k ) (7) where d is the amplitude of the impulse force characterizing the severity of the defect. δ(t) is the impulse function produced by a single impact of a point defect, T d is the period between the impulses, τ k is a random variable for the time lag between two impacts due to the presence of slip and q(t) is the Stribeck equation [1]. The distribution of the load q(t) around the load zone is defined approximately as follows: q(t) = { Q [1 1 ε (1 cosθ)]n, inner race defect 1, outer race defect where θ = π t, Q is the maximum load intensity (kn), ε is load distribution factor, n is constant ( 3 1 for ball, 9 for roller bearing), is the shaft rotating frequency. (8)

(a) (b) Amplitude of x(t) 3 1 1 Magnitude of load distribution, q(t) 5 4 3 1 (c)..4.6.8 1 (d)..4.6.8 1 15 1 Magnitude of y(t) 5 5 1 (e)..4.6.8 1 (f) Amplitude of u 1.11.1.9.8.7.6.5.4.3 Amplitude of v 1.8.7.6.5.4.3..1. 5 1 15 Frequency (Hz)..4.6.8 1 Fig. 1. Detection of a simulated bearing inner race fault with the wavelet analysis and the SVD technique. (a) time signal of x(t) computed by (9); (b) the distribution of the load q(t) calculated by (8); (c) the impulses generated by an inner race defect under a radial load using (1); (d) wavelet time-frequency distribution of y(t); (e) the first left singular values, u 1 of Fig. 1(d); (f) the first right singular values, v 1 of Fig. 1(d). When the inner race of the bearing passed through the load zone with each revolution of the shaft, the periodic function is shown in the time domain and a series of impulses in the frequency domain. When an impact from the localized bearing defect causes the bearing, the housing, and the shaft to vibrate. Approximating the structure as a second-order mass-spring-damper vibration system, the acceleration of the structure can be expressed as an exponentially time-decaying signal [1], [13], [14]. The excited structure is assumed to be a linear multidegree of freedom system with M excited modes, the structural response to each impulse with N repetition periods T can be expressed as: x(t) = M N 1 i=1 j= A i e ζπfn i (t jti)u(t jti) cos[πf di (t jt i ) φ i ]u(t jt i ) (9) 3

(a) (b) 6 Amplitude (G).6.4...4.6 Amplitude (G) 4 4 6 (c)..4.6.8.1.1.14.16 (d)..4.6.8.1.1.14.16 Fig.. Normal bearing vs. Defective bearing. (a) time signal for the normal bearing vibration; (b) time signal for the defective bearing vibration; (c) wavelet time-frequency distribution of the normal bearing vibration; (d) wavelet time-frequency distribution of the defective bearing vibration. where A is the gain factor, ζ is the damping ratio, f n is the undamped natural frequency of the structure, f d = f n 1 ζ, u(t) is a unit step function, and φ is the initial phase determined by the initial conditions. The defective bearing vibration signal shown in Fig. 1(a) was simulated by computing (9) with M = 1, N = 8, T =.18 seconds, A = 4, ζ =.1, f n = 88 Hz and φ = π/. The distribution of the inner race under radial load is plotted in Fig. 1(b). Now assume that the structural response to the inner race defect induced impact due to the modulation effect of the loading function. The impulses generated by an inner race defect under a radial load can be expressed by the product x(t)q(t), given by y(t): y(t) = { x(t)q [1 1 ε (1 cosθ)]n, for θ < θ max, elsewhere (1) The defective ball bearing inner race under load distribution, y(t) is depicted in Fig. 1(c) with Q =5 kn, ε=.5 and = Hz. The signatures produced by the product of x(t)q(t) are different from one shaft revolution to the next and can be clearly seen in Fig. 1(c). In this signal, the five largest positive and negative amplitude components calculated by equation (11) are marked with circles and diamond shapes, respectively. y (t) > on < t a, t c > and y (t) < on < t c, t b > = y(t c ) a local maximum y (t) < on < t a, t c > and y (t) > on < t c, t b > = y(t c ) a local minimum (11) The time-frequency distribution for the defective ball bearing inner race under load distribution is plotted in Fig. 1(d). A wide-band impulse frequency spectrum can be clearly seen from this wavelet time-frequency distribution analysis. In order to extract the simulated non-stationary bearing inner race defective vibration features, the SVD method is employed to 4

(a) (b) Amplitude of u 1.16.14.1.1.8.6.4 Amplitude of u 1..15.1.5. 5 1 15 Frequency (Hz) 5 1 15 Frequency (Hz) (c) (d).4.4.35.35 Amplitude of v 1.3.5 Amplitude of v 1.3.5...15.15.1.1..3.4.5.6.7.1.1..3.4.5.6.7 Fig. 3. The first singular values plots (normal bearing vs. defective bearing). (a) the first left singular values, u 1 of Fig. (c); (b) the first left singular values, u 1 of Fig. (d); (c) the first right singular values, v 1 of Fig. (c); (d) the first right singular values, v 1 of Fig. (d). achieve this goal. Fig. 1(e) shows the first left singular values, u 1 and the most significant frequency at 88 Hz is considered to be contributed by the undamped natural frequency of the structure, f n. By further investigations of Fig. 1(f) for the first right singular values, v 1, the times of the occurrence of the patterns with circular and diamond-shaped markers in this nonstationary signal are the same as those in Fig. 1(c). Because the analyzing wavelet is complex-valued, the resulting wavelet coefficients are complex numbers according to equation (1). Therefore, only absolute values of the wavelet modulus are considered and presented in the time-frequency domain. From the above simulated defective bearing vibration results, they have demonstrated that the SVD approach is capable of identifying salient patterns from non-stationary timefrequency wavelet vibration distribution. V. ACTUAL BEARING VIBRATION ANALYSIS In attempt to assess the suitability of the proposed wavelet analysis and the SVD approach for identifying features useful for describing defective bearing signatures, two bearings are used to investigate the bearing faults. Two bearings are of NTN-type NTN66. Their outer and inner diameters are 6 and 3 mm, respectively and each has 9 balls of diameter 9.5 mm. One bearing is used for normal test and the other with a hole in the inner race whose defect size is approximately 1. mm in diameter and.5 mm in depth is used for identifying the bearing faults. The output shaft bearing of a. kw/175 rpm three phase induction motor drives a generator via a flexible coupling. The generator is used to absorb the energy generated by the motor. The bearing vibration signatures of 48 points sampled at 1 khz were recorded from the normal and defective bearings. Fig. (a) shows the time signal for the normal bearing under full-load operation. An example time signal for the defective inner race under full-load operation is shown in Fig. (b). In comparison with normal bearing time signal, shown in Fig. (a), the overall magnitude for defective inner race time signal, shown in Fig. (b) is significantly increased. Figs. (c) and (d) represent the wavelet time-frequency distribution of normal and defective bearing vibration signatures, respectively. Fig. 3 shows the singular value patterns from the time-frequency 5

distribution data. As bearing defect progresses, the most significant frequency component, 67 Hz, shown in Fig. 3(a) has a swift change to the largest frequency component, 13 Hz, shown in Fig. 3(b). The 13 Hz is considered to be a harmonic of ball spin frequency. Figs. 3(c) and (d) represent the right singular value patterns from the time-frequency distribution vibration data of the normal bearing and the defective bearing inner race, respectively. In Fig. 3(d), the impulse patterns can be clearly identified in the SVD analysis for the defective inner race bearing vibration data. Those defective patterns depict in Fig. 3(d) as the bearing defect recurrence rate. The period of this defect is the time interval between balls encountering the same defect. When the ball encounters this defect, it is a metal crash, and a high-frequency shock pulse travels through the bearing housing as a pressure wave. While there is no such pattern for the normal bearing signal. With the aid of the SVD analysis, the distinctive features for the wavelet time-frequency vibration data from the normal and defective bearing can be easily identified automatically. [1] Y.-F. Wang and P. J. Kootsookos, Modeling of low shaft speed bearing faults for condition monitoring, Mechanical Systems and Signal Processing, vol. 1, no. 3, pp. 415 46, 1998. [11] D. Brie, Modelling of the spalled rolling element bearing vibration signal: An overview and some new results, Mechanical Systems and Signal Processing, vol. 14, no. 3, pp. 353 369,. [1] T. A. Harris, Rolling bearing analysis, Wiley, New York, 4th edition, 1. [13] S. Braun and B. Datern, Analysis ooller/ball bearing vibrations, Journal of Engineering for Industry,Transactions of the ASME, vol. 11, pp. 118 15, January 1979. [14] S. G. Braun, The signature analysis of sonic bearing vibrations, IEEE Transactions on Sonics and Ultrasonics, vol. SU-7, no. 6, pp. 317 38, November 198. VI. CONCLUSIONS This paper has presented an approach for identification of bearing fault patterns. This approach uses continuous wavelet transforms as a non-stationary signal preprocessor and a SVD technique as salient feature extraction. The simulated and experimental bearing results obtained have shown this signal approach can be used for bearing fault detection and diagnosis. Further studies are being undertaken to extend this work to establish an automatic intelligent diagnostic system. ACKNOWLEDGMENT The author wish to thank the National Science Council for their financial support under project number 94-1-E-44-. REFERENCES [1] Motor Reliability Working Group, Report of large motor reliability survey of industrial and commercial installations, part, IEEE Transactions on Industry Applications, vol. IA-1, no. 4, pp. 865 87, July/August 1985. [] N. Tandon and A. Choudhury, A review of vibration and acoustic measurement methods for the detection of defects in rolling element bearings, Tribology International, vol. 3, pp. 469 48, 1999. [3] J. Cheng, D. Yu, and Y. Yang, Application of an impulse response wavelet to fault diagnosis oolling bearings, Mechanical System and Signal Processing, vol. 1, pp. 9 99, 7. [4] C. J. Li and J. Ma, Wavelet decomposition of vibrations for detection of bearing-localized defects, NDT & E International, vol. 3, no. 3, pp. 143 149, 1997. [5] O. Rioul and M. Vetterli, Wavelets and signal processing, IEEE Signal Processing Magzine, vol. 8, pp. 14 38, October 1991. [6] G. H. Golub and C. F. Van Loan, Matrix computations, Johns Hopkins University Press, Baltimore, USA, 3rd edition, 1996. [7] V. Venkatachalam and J. L. Aravena, Nonstationary signal classification using pseudo power signatures: The matrix SVD approach, IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, vol. 46, no. 1, pp. 1497 155, December 1999. [8] I. T. Jollife, Principal Component Analysis, Springer-Verlag, New York, nd edition,. [9] P. D. McFadden and J. D. Smith, Model for the vibration produced by a single point defect in a rolling element bearing, Journal of Sound and Vibration, vol. 96, no. 1, pp. 69 8, 1984. 6