Bearing fault diagnosis based on amplitude and phase map of Hermitian wavelet transform

Transcription

1 Journal of Mechanical Science and Technology 5 (11) (011) 731~740 DOI /s Bearing fault diagnosis based on amplitude and phase map of Hermitian wavelet transform Hui Li 1,*, Lihui Fu 1 and Haiqi Zheng 1 Department of Electromechanical Engineering, Shijiazhuang Institute of Railway Technology, Shijiazhuang , China First Department, Shijiazhuang Mechanical Engineering College, Shijiazhuang , China (Manuscript Received January 6, 010; Revised June 8, 011; Accepted July 4, 011) Abstract The rolling element bearing characteristic frequencies contain very little energy and are usually overwhelmed by noise and higher level of structural vibrations. The continuous wavelet transform enables one to look at the evolution in the time scale joint representation plane. This makes it very suitable for the detection of singularity generated by localized defects in a mechanical system. However, most applications of the continuous wavelet transform have widely focused on the use of the Morlet wavelet transform. The complex Hermitian wavelet is constructed based on the first and the second derivatives of the Gaussian function to detect signal singularities. The Fourier spectrum of Hermitian wavelet is real, which the Fourier spectrum has no complex phase and the Hermitian wavelet does not affect the phase of a signal in complex domain. This gives the desirable ability to detect the singularity characteristic of a signal precisely. In this study, the Hermitian wavelet amplitude and phase map are used in conjunction to detect and diagnose the bearing fault. The Hermitian wavelet amplitude and phase map are found to show distinctive signatures in the presence of bearing inner race or outer race damage. The simulative and experimental results show that the Hermitian wavelet amplitude and phase map can extract the transients from strong noise signals and can effectively diagnose bearing faults. Keywords: Fault diagnosis; Hermitian wavelet transform; Bearing; Amplitude and phase map; Signal processing Introduction The rolling element bearing is an important component for power transmitting systems within the machine tool or gearbox drive train. Monitoring the condition of the bearing component provides advantages in the safety, operation and maintenance areas. Therefore, the predictive maintenance philosophy of using vibration information to lower operating costs and increase machinery availability has been the subject of intensive research throughout the industry. Since most of the machinery in a predictive maintenance program contains rolling element bearings, it is imperative to establish a suitable condition monitoring procedure to prevent malfunction and breakage during operation. The Hertzian contact stresses between the rolling elements and the races are one of the basic mechanisms that initiate a localized defect. When a rolling element strikes a localized defect, an impulse occurs which excites the resonance of the structure. Therefore, the vibration signature of the damaged This paper was recommended for publication in revised form by Associate Editor Eung-Soo Shin * Corresponding author. Tel.: , Fax.: address: huili68@163.com KSME & Springer 011 bearing consists of exponentially decaying sinusoid having the structure resonance frequency. The duration of the impulse is extremely short compared with the interval between impulses, and so its energy is distributed at a very low level over a wide range of frequencies and hence can be easily masked by noise and low-frequency effects. The periodicity and amplitude of the impulses are governed by the bearing operating speed, location of the defect, geometry of the bearing, and the type of the bearing load [1]. The rolling elements experience some slippage as the rolling elements enter and leave the bearing load zone. As a consequence, the occurrence of the impacts never reproduces exactly at the same position from one cycle to another; moreover, when the position of the defect is moving with respect to the load distribution of the bearing, the series of impulses are modulated in amplitude. However, the periodicity and the amplitude of the impulses experience a certain degree of randomness [, 3]. All these make the bearing defects very difficult to detect by conventional Fourier spectrum analysis that assumes the analyzed signal to be strictly periodic. A method of conditioning the signal before the spectrum estimation takes places is necessary. A commonly used method is envelope detection. To overcome the modulation problem, several signal envelope de-

2 73 H. Li et al. / Journal of Mechanical Science and Technology 5 (11) (011) 731~740 modulation techniques have been introduced. In highfrequency resonance technique (HFRT), an envelope detector demodulates the pass-band filtered signal and the frequency spectrum is determined by fast Fourier transform (FFT) technique [4]. Another well-established method is based on the Hilbert transform [5] or Teager Kaiser energy operator (TKEO) [6, 7]. The envelope analysis technique is widely accepted as a powerful tool in the detection and diagnosis of bearing faults. The resonance frequency oscillation of the impacts and the possibly suppressed impact periodicity are two modulations of the vibrations that both reduce the amplitude of the power spectrum peaks, which are more likely to be suppressed below the overall noise level. A popular countermove is to remove the resonance frequency modulation with the envelope method, which consists of a band-pass filter followed by a demodulation and a fast Fourier transform. In the traditional envelope analysis technique, the fault is identified through the peak value of envelope spectrum. However, the central frequency of the filter is determined with experience while forming an envelope signal, which will make great subjective influence on the diagnosis results [1]. The rolling element bearing faults by their nature are time localized transient events. To deal with non-stationary and non-linear signals, time-frequency analysis techniques such as the short time Fourier transform (STFT) [8], Wigner-Ville distribution (WVD) [9, 10] and wavelet transform (WT) [3, 11, 1, 13] are widely used. The STFT [8] uses sliding windows in time to capture the frequency characteristics as functions of time. Therefore, a spectrum is generated at discrete time instants. An inherent drawback with the STFT is the limitation between time and frequency resolutions. A finer frequency resolution can only be achieved at the expense of time resolution and vice-versa. Furthermore, this method requires large amounts of computation and storage for display. The WVD [14] is a basic time-frequency representation, which is part of the Cohen class of distribution. The difficulty with this method is the severe cross terms as indicated by the existence of negative power for some frequency ranges. In addition, the WVD of discrete time signals suffers from the aliasing problem, which may be overcome by employing various approaches. The wavelet transform provides powerful multi-resolution analysis in both time and frequency domain and thereby becomes a favored tool to extract the transitory features of nonstationary vibration signals produced by the faulty bearing [15, 16]. The wavelet analysis results in a series of wavelet coefficients, which indicate how close the signal is to the particular wavelet. To extract the fault feature of signals more effectively, an appropriate wavelet base function should be selected. Morlet wavelet is mostly applied to extract the rolling element bearing fault feature because of the large similarity with the impulse generated by the faulty bearing [3]. The impulse response wavelet was constructed and applied to extract the feature of fault vibration signal in Ref. [17]. A number of wavelet-based functions were proposed for mechanical fault detection with high sensitivity in Ref. [18], and the differences between single and double-sided Morlet wavelets were presented. An adaptive wavelet filter based on single-sided Morlet wavelet was introduced in Ref. [19]. An application of Morlet continuous wavelet transform for the detection of cracks in geared system was presented in Ref. [0]. The discrete and continuous wavelet transforms to detect abnormal transients generated by early gear damage were used in Ref. [1, ]. The Hermitian wavelet is a complex valued wavelet that is constructed based on the first and the second derivatives of the Gaussian function to detect signal singularities [3]. The Fourier spectrum of Hermitian wavelet is real, which the Fourier spectrum has no complex phase, and the Hermitian wavelet does not affect the phase of a signal in complex domain. This gives the desirable ability to detect the singularity characteristic of a signal precisely. However, it is only applied to diagnose the fault of the large air compressor [4]. In this study, an alternative approach for detecting localized faults in the outer or the inner races of a rolling element bearing using the Hermitian wavelet amplitude and phase map is investigated. The techniques are demonstrated by the experiments on a gearbox with a rolling element bearing under simulated crack on the inner race or the outer race. The characteristic periods related to the bearing defect can be effectively extracted. The experimental results show that this method based on the Hermitian wavelet amplitude and phase map can effectively diagnose the faults of rolling element bearings. This paper is organized as follows. Section 1 gives a brief introduction of bearing fault detection. Section introduces the Hermitian wavelet transform. Section 3 presents simulative signals that demonstrate the abilities of the Hermitian wavelet amplitude and phase map to detect the singularity characteristic. Section 4 describes the experimental set-up. Section 5 gives the applications of the Hermitian wavelet amplitude and phase map to fault diagnosis of rolling element bearing. Finally, the main conclusions of this paper are given in Section 6.. Hermitian wavelet transform The Hermitian wavelet was formulated by Harold Szu in 1997 [3] for the numerical estimation of function derivatives in the presence of noise. The Hermitian wavelet is a lowoscillation, complex valued wavelet. The real and imaginary parts of this wavelet are defined to be the second and first derivatives of a Gaussian function respectively. The Gaussian function is defined as follows: 1 t 1 gt () = e. (1) π The first and second derivatives of Gaussian function are given as follows:

3 H. Li et al. / Journal of Mechanical Science and Technology 5 (11) (011) 731~ ψ () t to its daughter wavelet in terms of the shift parameter b and the scale parameter a, 1 t b a 1 t b t b ψ ab() t = 1 + i e, a a a 1 ( ) aω ψˆ ab( ω) = [( aω) + aω] e. (7) (6) Eqs. (6) and (7) indicate that Hermitian wavelet is complex in time domain and real in frequency domain. The continuous wavelet transform (CWT) Wx(, t a ) of the signal x() t is defined as a convolution integral of x() t with scaled and dilated versions of Hermitian mother wavelet function ψ (), t Fig. 1. The Hermitian wavelet and its spectrum: (a) the real part and the imaginary part; (b) real part Fourier spectrum; (c) imaginary part Fourier spectrum; (d) its spectrum. 1 (1) dg() t 1 t ψ () t = = te, () dt π ψ 1 () d g() t 1 t t t e () = = (1 ). (3) dt π Then the Hermitian wavelet is given as follows: 1 () (1) 1 t ψ() t = ψ () t iψ () t = (1 + it t ) e, (4) π 1 ω ψω ˆ ( ) = ( ω + ω) e. (5) Fig. 1 shows the Hermitian wavelet, its real part, imaginary part, and its spectrum. In Fig. 1(a), the solid line stands for its real part and the dashed line stands for its imaginary part. The real part is known as the Mexican hat wavelet that it is an even function and has one-and-a-half oscillations in its support domain. The imaginary part is an odd function and has a single oscillation in its support domain. Therefore, the Hermitian wavelet is a single oscillation wavelet having much lower frequency bandwidth than that of Morlet wavelet. As a result of Nyquist theorem, a fewer number of grid points would be needed for the discrete convolution operation. Therefore, the singularity characteristic will not be overly smeared and the noise can be smoothed away. Obviously, the Fourier spectrum of Hermitian wavelet is real, which is another good property. Since the Fourier spectrum has no complex phase; the Hermitian wavelet does not affect the phase of a signal in complex domain. This gives the desirable ability to detect the singularity characteristic of a signal precisely. In addition, we can extend the Hermitian mother wavelet 1 t τ Wx(, t a) = x( τψ ) dτ= x() t ψ ab(). t (8) a a When the signal x() t is periodic, Eq. (8) becomes a circular convolution, and the continuous wavelet transform (CWT) is better calculated in the frequency domain according to 1 Wx (, t a) F [ X( ) aw( a )] a 1 = ω ω (9) where X ( ω ) and aw ( aω ) are the Fourier transform of the signal x() t and dilated wavelet ψ ab(), t respectively. The operator F 1 [] denotes inverse Fourier transform. In our computer implementation, X ( ω ) is calculated only once, and W( aω ) is calculated for all values of the scale parameter a. Since the Hermitian wavelet is complex-valued, the Hermitian wavelet transform Wx(, t a ) is also complex-valued. Thus, it gives the amplitude and phase modulation of the signal x() t over the entire time-scale plane. The absolute value of Wx(, t a ) indicates the transient energy of a signal x(), t which provides an amplitude plot, and its phase θ x(, t a ) shows in Eq. (10) indicates the transient phase of a signal x(), t which provides an phase plot. 1 Im[ WTx ( t, a)] θx(, ta) tan = Re[ WTx ( t, a)] (10) where the operator Re( ) and Im( ) are the real part and imaginary part of Wx(, t a ), respectively. The operator 1 tan ( ) denotes inverse tangent. Since the simplest possible complex Hermitian wavelet consists of single and double oscillations, then these filters ψ (1) () t in Eq. () and ψ () () t in Eq. (3) can track the singularity of noisy sequence through time by means of the bifurcation in the time scale joint representation plane.

4 734 H. Li et al. / Journal of Mechanical Science and Technology 5 (11) (011) 731~740 Fig.. Time domain simulative signal. Fig. 4. The Morlet wavelet and its spectrum. Fig. 3. The Hermitian wavelet amplitude and phase map of simulative signal. 3. Singularity detection simulation using amplitude and phase map of Hermitian wavelet Consider the Hermitian wavelet described in the previous section; two simulative examples have been demonstrated. The results of these signals simulation studying the performance of amplitude and phase map of Hermitian wavelet are presented to realize a better understanding of this numerical method. For example, a sine wave in which there is a small impulse is used to simulate singularity in each period. 3.1 Singularity detection of simulative signal without noise The time domain simulative signal is shown in Fig.. The sampling time is 0.5 s. We apply the Hermitian wavelet transform to the simulative signal shown in Fig.. The amplitude and phase wavelet map are illustrated in Fig. 3(a) and Fig. 3(b), respectively. In Fig. 3(a), the amplitude wavelet map demonstrates the period of a sine wave. The amplitude wavelet map shows great regularity. The white area corresponds to the positive amplitude of the simulative signal, and the black area corresponds to its negative amplitude. The high frequency components are concentrated on the scale parameter a is less than 5, while the low frequency components are concentrated on the scale parameter a is from 5 to 5. In the bottom of the amplitude wavelet map, there are five black bars which correspond to the small impulses embedded in the sine wave. In Fig. 3(b), the phase wavelet map also shows great regularity. The sharp transitions from white to black in the phase wavelet map correspond to jump of the wrapped phase. They exactly correspond to the inflexion points of the simulative signal. Particularly, there are distinct black bars located in where the scale a is from 1 to 5. These black bars correspond to the small impulses embedded in the sine wave. It is obvious that the wavelet phase map determines the exact positions of those small impulses and enlarges the weak information caused by the embedded small impulses. Therefore, the wavelet phase map in the time scale joint representation domain reveals the precise location of the singularity as well as the scale nature of the simulative signal. Therefore, we can draw a conclusion that the combination of the wavelet amplitude and phase map can illustrate the whole and local signal characteristics and reveal signal singularity effectively. As a comparison, we applied the Morlet wavelet transform to do the continuous wavelet transform on the identical simulative signal shown in Fig.. The Morlet wavelet is given as follows [4]: ω0 t 4 iω0t ϕ() t π e e = e, 1 1 ( ω ω0 ) 4 (11) ˆ( ϕω) = π e. (1) The Morlet wavelet, its real part, imaginary part, and its spectrum are displayed in Fig. 4, respectively. From Fig. 4, we can see that the real part or the imaginary part of Morlet wavelet has about six oscillations. We wish to comment on the comparison between Hermitian wavelet and Morlet wavelet from Nyquist sampling theorem viewpoint. The Fourier transform of Morlet wavelet in Eq. (1) shown in Fig. 4(b) which compared with Hermitian spectrum in Eq. (5) shown in Fig. 1(d) has a broader bandwidth than that of Hermitian wavelet.

5 H. Li et al. / Journal of Mechanical Science and Technology 5 (11) (011) 731~ Fig. 5. The Morlet wavelet amplitude and phase map of simulative signal. Fig. 6. Time domain simulative noisy signal. Since the bandwidth of Morlet wavelet is several times bigger than that of the Hermitian wavelet, then according to the Nyquist sampling theorem, the number of grids needs to sample the Morlet wavelet more than that of Hermitian wavelet. Therefore, Hermitian wavelet can be represented with fewer numbers of grid points which allow less smear of the singularity of the signals [5]. We apply the Morlet wavelet transform to the identical simulative signal displayed in Fig.. The amplitude and phase wavelet map are illustrated in Fig. 5(a) and Fig. 5(b), respectively. The high frequency components are concentrated on the scale parameter a is less than 0, while the low frequency components are concentrated on the scale parameter a is from 0 to 50. In the low frequency area, both the amplitude and phase wavelet map show great regularity. However, in the high frequency area, both the amplitude and phase wavelet map are very fuzzy. Particularly, the black bars located in where the scale parameter a is from 1 to 0 are very fuzzy. Comparing Fig. 5 with Fig. 3, one can draw the conclusion that for the ability of revealing the precise location of the singularity, the Hermitian wavelet is better than that of Morlet wavelet. Fig. 7. The Hermitian wavelet amplitude and phase map of simulative noisy signal. 3. Singularity detection of simulative signal with noise To demonstrate the effects of Hermitian and Morlet wavelet filters, we take a noisy sine wave in which there is a small impulse is used to simulate singularity in each period. The sine wave and the small impulse are the same as that of simulative signal shown in Fig.. The time domain signal with noise is given in Fig. 6. The sampling time is also 0.5 s. We also apply the Hermitian wavelet and Morlet wavelet transform to the simulative signal, respectively. The Hermitian wavelet amplitude and phase map, and the Morlet wavelet amplitude and phase map are shown in Fig. 7 and Fig. 8, respectively. In Fig. 7, the Hermitian wavelet amplitude and phase map show great regularity. The Hermitian wavelet amplitude and phase map clearly demonstrate the period of sine wave and the small impulses embedded in the sine wave. Fig. 8. The Morlet wavelet amplitude and phase map of simulative noisy signal. However, the Morlet wavelet amplitude and phase map cannot clearly demonstrate the period of sine wave and the small impulses embedded in the sine wave. Therefore, the Morlet wavelet is not suited to analyze the singularity of a noisy data. From the above simulative results, it can be observed that

6 736 H. Li et al. / Journal of Mechanical Science and Technology 5 (11) (011) 731~740 Motor Output shaft Note: 1 G1 G Gearbox Accelerometer 1,,3,4 bearing Fig. 9. Experimental set-up. the wavelet amplitude and phase map using the Hermitian wavelet transform of the sampled data of the simulative signal can clearly and precisely display the locations of each singularity and the relationship among them. These analyses demonstrate a better detection of the proposed Hermitian wavelet transform provides a single oscillation which can detect the local singularity more efficiently. 4. Experimental Set-up 3 4 Input shaft Generator Photoelectric sensor G1,G gear B&K3560 Spectrum Analyzer The test apparatus used in this paper is shown in Fig. 9. The experimental set-up consists of a single-stage gearbox, driven by a 4.5 kw AC governor motor. The driving gear has 8 teeth and the driven gear has 36 teeth. Therefore, the transmission ratio is 36/8, which means that a decrease in rotation speed is achieved. The module of the gear is.5 mm. The tested bearing was used to study only one kind of surface failure: the bearing was damaged on the inner race or the outer race. The rolling bearing tested has a groove on the inner race or the outer race. Localized defect was seed on the race by an electric-discharge machine to keep their size and depth under control. The size of the artificial defect was 1 mm in depth and the width of the groove was 1.5 mm. The rotating speed of the input shaft is 1500 r/min, that is, the rotating frequency f r of the input shaft is 5 Hz. The type of the ball bearing is 08. There are 10 balls in a bearing and the contact angle α = 0, ball diameter d=18.33 mm, bearing pitch diameter D=97.5 mm. The monitoring and diagnostic system is composed of three accelerometers, amplifiers, a speed and torque transducer, B&K 3560 spectrum analyzer and a computer. The sampling span is 1.8 khz, the sampling frequency is 3768 Hz and the sampling point is 819. After sampling, the measured vibration signals were loaded into MATLAB from data-files. 5. Bearing faults diagnosis using Hermitian wavelet amplitude and phase map Rolling element bearings are installed on many kinds of machinery. Many problems with those machines may be caused by defects of the rolling element bearing. Generally, localized defects may occur on the inner race, outer race, cage or rollers of bearing. A local fault may produce periodic Clutch impacts, the size and the repetition period which are determined by the shaft rotation speed, the type of fault and the geometry of the bearing. The successive impacts produce a series of impulse responses, which may be amplitude modulated because of the passage of fault through the load zone. The spectrum of such a signal would consists of a harmonics series of frequency components spaced at the component fault frequency with the highest amplitude around the resonance frequency. These frequency components are flanked by sidebands if there is an amplitude modulation due to the load zone. According to the period of the impulse, we can judge the location of the defect using characteristic defect frequency formulae. Because an inner race defect has more transfer segments when transmitting the impulse to the outer surface of the case, usually the impulse components are rather weak in the vibration signal. Each bearing element has its characteristic frequency of defect. Those frequencies can be calculated from the geometry of the bearing and its rotating speed. For a bearing with a stationary outer race, the defect characteristic frequency of its inner race, outer race, rollers and cage can be calculated according to the Eqs. (13)-(16), respectively. f f f f inner outer roller cage z d = 1 cos α fr, + D z d = 1 cos α fr, D D d = 1 cos α fr, d D 1 d = 1 cos α f D r (13) (14) (15) (16) where z is the number of rollers, d is the diameter of the rollers, D is the pitch diameter,α is the contact angle, and f r is the rotating speed of the input shaft. Therefore, according to Eqs. (13) and (14), the characteristic defect frequency of the inner race and outer race defect are calculated to be f inner =148.5 Hz and f outer =101.5 Hz, respectively. 5.1 Application of Hermitian wavelet amplitude and phase map to fault detection of inner race To demonstrate the performance of the proposed approach, this section presents the application examples for the detection and diagnosis of localized bearing defects. In all the examples, the Hermitian wavelet is as a wavelet transform mother function. The original vibration signal of inner race defect is displayed in Fig. 10. It is clear that there are periodic impacts in the vibration signal. There are significant fluctuations in the peak amplitude of the signal, and there are also considerable variations of frequency content. From Fig. 10, we can hardly

7 H. Li et al. / Journal of Mechanical Science and Technology 5 (11) (011) 731~ Fig. 10. Original vibration signal with bearing inner race fault. Fig. 13. The Hermitian wavelet amplitude and phase map of vibration signal with inner race fault. Fig. 11. FFT of original vibration signal with inner race fault. Fig. 1. Envelope spectrum of the vibration signal with inner race fault. find the characteristic defect frequency or cycle of the inner race defect. Fig. 11 shows the FFT of the vibration signal with inner race fault. There is no characteristic defect frequencies component around f inner and its harmonics. Therefore, classical Fourier analysis has some limitation, such as being unable to extract the characteristic defect frequency of inner effectively. Fig. 1 shows the envelope spectrum of the vibration signal. In Fig. 1, one can see the spectrum line at the characteristic defect frequency f inner (148.5 Hz) of the bearing with inner race fault. But harmonic frequencies of the characteristic defect frequency are not very clear. This is possibly because of the unsuitable pass-band frequency and the effect of underground noise. Therefore, the envelope spectrum has some limitation, such as being unable to process non-stationary signals. To the data of Fig. 10, the Hermitian wavelet transform is applied, resulting in wavelet amplitude and phase map. The Hermitian wavelet amplitude and phase map are shown in Fig. 13. In Fig. 13, we can clearly see the distinctive signature for the vibration signal with bearing inner race fault. The wavelet amplitude and phase map both show about 36 jumps which correspond to the bearing characteristic defect cycle of the inner race ( T inner = s). This can be explained as follows. The presences of inner race fault results in a sudden increase of vibration energy. For the defective bearing, transient vibrations caused by the rolling elements-defect interactions are clearly seen through the characteristic defect frequency and its high order harmonic frequencies. In addition, these transient vibrations have shown a repetitive pattern with T inner interval, which corresponds to a repetitive characteristic defect cycle of the inner race, resulting from the structural defect on the inner race. Such repetitive cycle reflects degradation of the inner race health condition as the defect propagated through the bearing inner race. Physically, impacts generated by the rolling ball-defect interactions excite intrinsic modes of the bearing system, giving rise to a train of transient vibrations at the mode-related resonant frequencies. The phase wavelet map showed in Fig. 13(b) is clearer when compared with amplitude wavelet map shown in Fig. 13(a). Therefore, the phase wavelet map is often much more sensitive than amplitude wavelet map. The phase wavelet map for scales below 0 does not show a clear pattern. This is expected for the low scales where many noise sources contaminate the signal. The simplicity of the Hermitian wavelet amplitude and phase representation can be put down to the ability of this signal processing method to eliminate undesirable modulation effects. Therefore, the Hermitian wavelet amplitude and phase

8 738 H. Li et al. / Journal of Mechanical Science and Technology 5 (11) (011) 731~740 Fig. 16. FFT of original vibration signal with outer race fault. Fig. 14. The Morlet wavelet amplitude and phase map of vibration signal with inner race fault. Fig. 17. Envelope spectrum of the vibration signal with outer race fault. Fig. 15. Original vibration signal of the bearing with outer race fault. map have been shown to provide an effective tool for bearing inner fault diagnosis. For a comparison study, the Morlet amplitude and phase map are displayed in Fig. 14. However, there are no strong and distinctive features since it is not suited to analyze the singularity of a noisy vibration signal. 5. Application of Hermitian wavelet amplitude and phase map to fault detection of outer race The original vibration signal of the outer race defect is displayed in Fig. 15. Fig. 16 shows the FFT of the vibration signal. The envelope spectrum of the high-pass filtered signal is displayed in Fig. 17. From Fig. 17, one can only see the spectrum line at the characteristic frequency f outer (101.5 Hz) of the bearing with outer race fault. But the high order harmonic frequencies of the characteristic defect frequency are not very clear. To the data of Fig. 15, the Hermitian wavelet transform is applied, resulting in wavelet amplitude and phase map. The Hermitian wavelet amplitude and phase map are shown in Fig. 18(a) and 18(b), respectively. In Fig. 18, we can clearly see Fig. 18. The Hermitian wavelet amplitude and phase map of vibration signal with outer race fault. the distinctive signature for the vibration signal with bearing outer race fault. The wavelet amplitude and phase map both show about 5 jumps which correspond to the bearing characteristic defect cycle of the outer race ( T outer = s). In addition, these transient vibrations have shown a repetitive pattern with interval, resulting from the structural defect on the outer race. Such a repetitive cycle reflects degradation of

9 H. Li et al. / Journal of Mechanical Science and Technology 5 (11) (011) 731~ Acknowledgment The authors are grateful to the National Natural Science Foundation of China (No and No ). The authors are also grateful to the editors and anonymous reviewers for their constructive comments. References Fig. 19. The Morlet wavelet amplitude and phase map of vibration signal with outer race fault. the outer race health condition as the defect propagated through the bearing outer race. For a comparison study, the Morlet amplitude and phase map are displayed in Fig. 19. However, there are also no strong and distinctive features since it is not suited to analyze the singularity of a noisy data. From the above experimental results, it can be observed that the Hermitian wavelet amplitude and phase map can clearly and precisely demonstrate the locations of each singularity. These analyses demonstrate that the Hermitian wavelet transform can detect the local singularity more effectively and can diagnose the rolling element bearing defect efficiently. 6. Conclusions Singularity recognition is always useful for mechanical fault detection and diagnosis. The Hermitian wavelet is a lowoscillation, complex valued wavelet. The real and imaginary parts of this wavelet are defined to be the second and first derivatives of a Gaussian function, respectively. Since the Fourier spectrum of Hermitian wavelet is real, which the Fourier spectrum has no complex phase, the Hermitian wavelet does not affect the phase of a signal in complex domain. This gives a desirable ability to detect the singularity characteristic of a signal precisely. This research shows that the Hermitian amplitude and phase map are very effective in detection and diagnosis bearing fault. The technique excels in extracting transients, which are often the indicators of incipient defects in a bearing system. Therefore, we can recognize the vibration modes that coexist in the system and have a better understanding of the nature of the fault information contained in the vibration signal. According to the Hermitian amplitude and phase map, the characteristic period of the bearing defect can be easily recognized. Practical vibration signal monitored from a gearbox with bearing fault is analyzed by the presented method. The experimental results show that the Hermitian amplitude and phase map can effectively diagnose the bearing fault. [1] N. Tandon and A. Choudhury, An analytical model for the prediction of the vibration response of rolling element bearings due to a localized defect, Journal of Sound and Vibration, 05 (3) (1997) [] 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, 96 (1) (1984) [3] J. Lin and L. Qu, Feature extraction based on Morlet wavelet and its application for mechanical fault diagnosis, Journal of Sound and Vibration, 34 (1) (000) [4] P. D. McFadden and J. D. Smith, Vibration monitoring of rolling element bearings by the high-frequency resonance technique a review, Tribology International, 17 (1) (1984) [5] D. Ho and R. B. Randall, Optimization of bearing diagnostic techniques using simulated and actual bearing fault signals, Mechanical Systems and Signal Processing, 14 (5) (000) [6] P. Maragos, J. F. Kaiser, and T. F. Quatieri, On amplitude and frequency demodulation using energy operators, IEEE Transactions on Signal Processing, 41 (5) (1993) [7] H. Li and H. Zheng, Bearing Fault Detection Using Envelope Spectrum Based on EMD and TKEO, Proceedings of the 5th International Conference on Fuzzy Systems and Knowledge Discovery, 3 (008) [8] L. Cohen, Time-frequency analysis, Prentice-Hall, Englewood Cliffs, NJ, [9] Q. Meng and L. Qu, Rotating machinery fault diagnosis using Wigner distribution, Mechanical Systems and Signal Processing, 5 (3) (1991) [10] W. J. Staszewski, K. Worden and G. R. Tomlinson, Thefrequency analysis in gearbox fault detection using the Wigner-Ville distribution and pattern recognition, Mechanical Systems and Signal Processing, 11 (5) (1997) [11] W. J. Staszewski, Wavelet based compression and feature selection for vibration analysis, Journal of Sound and Vibration, 11 (5) (000) [1] C. J. Li and J. Ma, Wavelet decomposition of vibration for detection of bearing-localized defects, NDT&E International, 30 (3) (1997) [13] S. Prabhakar, A. R. Mohanty and A. S. Sekhar, Application of discrete wavelet transform for detection of ball bearing race fault, Tribology International, 3 (1) (00) [14] H. Li, Y. Zhang and H. Zheng, Hilbert-Huang transform and marginal spectrum for detection and diagnosis of localized defects in roller bearings, Journal of Mechanical Science and Technology, 3 () (009)

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