Detection of gear defects by resonance demodulation detected by wavelet transform and comparison with the kurtogram

Detection of gear defects by resonance demodulation detected by wavelet transform and comparison with the kurtogram K. BELAID a, A. MILOUDI b a. Département de génie mécanique, faculté du génie de la construction, université Mouloud Mammeri de Tizi-Ouzou, BP 17 RP, T-O. e-mail : bel_kamel78@yahoo.fr b. Laboratoire de mécanique avancé, Université des sciences et technologies Houari Boumediene, Alger, e- mail : amiloudi@yahoo.fr Abstract: The envelope analysis is a very effective tool for demodulating resonances in order to detect transients in a signal. However, the choice of a resonance and its optimal frequency band is not an easy task in practice. The kurtogram and wavelet transform are two high technics used to characterize non-stationarities hidden in a signal, the first one, namely the kurtogram, allows to respond to the given problem. It consists to determine the central frequency (resonance) and the appropriate bandwidth witch maximizes the kurtosis. However, it does not allow a combination of any center frequency with any frequency band. The aim of this paper is a comparison between this technic and the wavelet transform which has the advantage of multi-scales filter that can be adapted to particular situations like gear defects. Keywords : Resonance, Envelope analysis, Continuous wavelet transform, Kurtogram, Kurtosis, Spectral kurtosis. 1 Introduction The gearboxes are important elements in industry machines and they are subjected to various efforts and degradations. A failure not detected at the convenient time will affect the precision of the machine and cause serious accidents. Some gears faults (locating faults) induce a series of impulses in time signal, after transforming into frequency domain, here energy becomes distributed over a wide range of frequencies, therefore, it is judicious to choose the most adequate band in order to include only the sought components. The envelope analysis is a good tool in these situations [1], it is widely used for demodulating resonances detected in the basic spectrum of the time signal. But in practice, to choose a suitable resonance is not always easy, moreover, in some cases, secondary resonances that can give better results are not easily locatable [2]. The localization of useful resonances seems to be a difficult task and increase time consumption if we must analyze all resonances which can arise in the basic spectrum. Kurtosis can make this procedure less difficult; it is a scalar indicator which makes it possible to locate transients in the signal. Among the methods which use this principle is Spectral Kurtosis (SK), it is one of the new methods which is based on this indicator to locate the optimal band frequency [3]. This method (SK) consists of applying the kurtosis to the matrix coefficients computed by the short-time Fourier transform (STFT). The disadvantage of this technic is the window size remains constant during the analysis, to overcome this problem we analyzing the signal with several windows in order to identify the useful frequency band; this tool called Kurtogram [4]. This second method can be interpreted as a cascade of spectral kurtosis with different window lengths which giving rise a two-dimensional representation with in X-coordinate the frequencies and in Y-coordinate the decomposition levels N which represents the number of windows samples Nw. It has the advantage of deducing the optimal frequency band ( f) and center frequency (f c) necessary for a good interpretation of the envelope spectrum. The disadvantage with this tool is the impossibility of combining any center frequency with any frequency band. The continuous wavelet transform (CWT) is most used tool in research of the transients hidden in a signal. In certain work, an association of Kurtosis to this technic gave good results. In [5] it is used to seek the best form of the Morlet wavelet to be used in the decomposition. In [6,7], after decomposition to the wavelet 1

coefficients, the Kurtosis is calculated according to scales in order to locate the resonance frequencies. From the definition of the wavelet transform, the continuous version can be interpreted as an adapted multi-scale filtering in objective to research the moments where the signal are most similar to the wavelet and that for various dilated versions of this wavelet. In this paper, we will exploit this definition associated with the kurtosis compared to the kurtogram in order to diagnose gears defects simulated on a test bench. 2 The spectral kurtosis (SK) and the Kurtogram Spectral kurtosis provides distribution of transient components in the frequency domain. The principle of this method is to compute the STFT by moving window along the signal, and then we calculate the kurtosis (1) of the squared coefficients (, ) called spectrogram given by equation (2) [8]. ( ) = (, ) (, ) 2 (1) (, ) = ( ) ( ) (2) with x(n) the discrete version of the signal x(t) and W(s) the window analysis. FIG. 1 - Simulated time signal with high SNR, its Kurtosis spectral with Nw = 16 and a cascade of spectral kurtosis (Nw=16:10:156) of the same signal but with low SNR. Figure 1-a watch a simulated signal with transients compenents whose resonance frequency is 2048 Hz which was prepared in Matlab (composed from exponential decay 2048 Hz with amplitude of 2 and impacts repetition of 7.5 Hz with sampling frequency of 8192 Hz), its kurtosis spectral (figure 1-b) shows important values in the résonance range but the large values are around it; this is due to the high signal-to-noise ratio [8]. This drawback can be avoided by using the auto-regressive method AR [10]. The spectral kurtosis uses only one window, the choice of its length is not an easy task. The solution is to analyze with various window lengths and then to select that result with the maximum overall value, it is the kurtogram principle witch can be interpreted as a cascade of spectral kurtosis (figure 1-c) [3]. In figure 1c, we note the difficulty to choosing the step and the limits of the window lengths. The kurtogram overcomes this problem by proposing to decompose the signal into several frequency bands in several levels based on an arborescent multirate filter-bank structure. The solution is an even decomposition based on a 1/3-binary tree [4]. Each level n is divided by 2 n where n = 0, 1, 1.6, 2, 2.6, 3, 3.6, 4, 4.6, 5, etc (figure 2-a). It consists of a high-pass filter and low pass filter of each band followed by a down-sampling by factor 2. The width of the bands on a given level n is f = Fs [1/2 n+1 ], positioned on a center frequency f c,k = F e [(k + 2-1 ) 2 -n-1 ]; with Fs the sampling frequency (k is the band position). To get better results, it is necessary that the center frequency f c corresponds as well as possible to the resonant frequency, and to have an optimal band frequency f around f c will include only the transient components. But with this decomposition, it is not possible to obtain any frequency band around any center frequency. The values given in figure 2b are imposed by the decomposition and cannot represent exactly the resonance characteristics, at least for the resonance value. In [9], it was shown that the shifting center 2

frequency about 1 khz from the resonant frequency makes detection almost impossible and shifting by 500 Hz may degrades severely the clearness of the envelope spectrum. n f 0 1/2 1 1/4... 1.6 1/6 2 1/8...... 2.6 1/12 3 1/16 n 1/2 n+1 0 1/4 1/2 Frequency (Hz) FIG. 2 - The 1/3-binary tree decomposition and the kurtogram of the simulated signal (high SNR). 3 The continuous wavelet transform (CWT) The continuous wavelet transform (3) has the advantage of dilatation and compression of the window (4) in order to adapt it to different components of a signal:, = ( ), ( ) (3), ( ) =, ( ) is the complex conjugate of, ( ). 1 The continuous wavelet transform give redundant information; we will exploit this characteristic for detecting resonances. In [7] a step scales of 0.1 is used to locate resonances present in the signal; already with a step 1 [6], a redundancy of information exists, it is important in the case of the stationary signals but it is not really for the non-stationary signals. For this reason a step of 0.1 is proposed to increase the redundancy information in the objective to locate resonances. Once the wavelet coefficients are determined with this step, we calculate the kurtosis according to scales, which gives redundancy bumps around resonances, and after study of concerned wavelet coefficients, we can go back to the exciting defects. Equation 5 allows the passage between the frequency domain and the scales. By giving the center frequency of the mother wavelet f co and the sampling frequency Fs (1/ t), the scale a of any frequencies f will be determined. a f f co F e fco f. t The kurtosis of the wavelet coefficients (calculated by Morlet wavelet) of the time signal of figure 1a according to the scales is represented in figure 3-a, the range of the scales starts by 1.6 because all the values lower than this are not representative (the scale of 4096 Hz = 1.6). We note in this figure that the greatest values of kurtosis are around the scale 3.2 which corresponds to frequency 2048 Hz. With this method we have access directly to resonance frequencies contrary to the kurtogram which gives in certain cases approximate values. Second information which we can obtain with this approach concerns the frequency range in which we can seek the adequate band frequency. The kurtosis is important in the range scales around 3.2, it is roughly between 2.8 and 3.6 which corresponds to the frequency band extends from 1850 Hz to 2250 Hz. Figure 3-b show the spectrum of the wavelet coefficients at scale 3.2 where we can see that the frequency range around the resonance is the same as that given in figure 3-a. In addition this range corresponds very well to that given in the basic spectrum (figure 3-c). 3 (5) (4)

4 Experimental validation For the experimental validation of this approach, we simulated two gears defects on a test bench which its system transmission and technical features are respectively given in figure 4 and table 1. A stripe is carried out on both gear and pinion tooth, one on each gear (estimated defect area is 40 % in R1 and 10 % in R2). In this situation, we will have three shocks, the two first relate to the participation in contact of each defect (f 1 and f 2) and the third is made when the two teeth will enter the mesh area simultaneously and contact one another (f HT : hunting tooth frequency HFT). The studied test bench signal is a 0.8 s recorded with sampling frequency 5120 Hz. Figure 5a present the time signal at 2400 rpm and 5b illustrate the linear spectrum of this signal. The corresponding frequencies are f 1=40 Hz, f 2=32 Hz and f HT= 8 Hz. The basic spectrum reveals the presence of a comb lines at the frequency f 1 (40 Hz) at low frequencies and a great energy around frequencies 230 Hz and 480 Hz, probably, resonant frequencies. The fast kurtogram calculated to level 5 is given in figure 5c which gives a center frequency of 1920 Hz and a band frequency of 1280 Hz. FIG. 4 The test bench and its system transmission FIG. 3 The kurtosis of wavelet coefficients of the simulated signal with low SNR, the coefficients spectrum at scale 3.2 and the basic spectrum of the simulated signal D1 P4 P2 A2 A1 R2 R1 P3 P1 Ac D2 V M 4 N Designation Technical features 1 Electric motor M P = 1.5 kw 2 Frequency converter V 0 < f < 50 Hz 3 Input shaft A1 4 Output shaft A2 5 Inertia d iscs (02) Di 6 Input pinion R1 80 tooth 7 Ouput pinion R2 100 tooth 8 Bearing (04) Pi 9 Coupling Ac Tableau 1 - The features design of the test bench. The frequency 1920 Hz is the center of the band ( f = Fs(1/4), f c,1 = f + f /2), so this frequency cannot represent automatically the resonance frequency, all center frequencies represent the distance from the center of the concerned band to the origin, therefore if the resonance frequency is different from the one of these center frequencies, the results obtained will never be the best. The envelope spectrum of this band is given in figure 5d where we note a comb lines at frequency 40 Hz correspond to the input pinion defect. We will now analyze the signal with the continuous wavelet transform with scales discretization equal to 0.1 beginning through 1.7, which corresponds to the maximum frequency of signal (2560 Hz), until scale 25. We note the presence of redundancy bumps (figure 6-a) at the scales 1.8 and 8.5, determined by the energy concentration around the resonant frequencies detected by the kurtosis. Figures 6b and c respectively show the wavelet coefficients spectrums of the considered scales (1.8 and 8.5); the second spectrum clearly shows the second resonant present in the basic spectrum of the time signal. The resonance around 2000 Hz is not

apparent very well in the basic spectrum, an enlargement of the spectrum around this frequency shows clear comb lines of 40 Hz (figure 6-d). (d) FIG. 5 The time signal at 2400 rpm, the corresponding spectrum, its kurtogram and the envelope spectrum of the band (d). The envelope spectrum of the coefficients on the scale 1.8, given in figure 7-a, shows a comb lines at frequency 40 Hz. An enlarging at the low frequencies of this same spectrum reveals a comb lines of frequency 8 Hz corresponding to the hunting tooth frequency (figure 7-b) The envelope spectrum of the coefficients on the scale 8.5 (figure 7-c) show a comb lines at frequency 8 Hz corresponding to the coincidence of the gear defects. This last comb is not visible in the spectrum determined by the kurtogram (figure 7-d). (d) FIG. 6 - The coefficients kurtosis, the coefficients spectrum at scale 1.8, the coefficients spectrum at scale 8.5 and an enlargement in the basic spectrum around 2000 Hz (d). We have shown practically, with this example, that a secondary resonant frequency, that it is almost difficult to detect it in the basic spectrum, gives better information of two defects. In this practical example, the wavelet transform has indeed filtered the transient components exactly at the resonance scales and with an adequate band frequency in order to have a good envelope spectrum interpretation. 5

(d) FIg. 7 - The envelope spectrum of the coefficients at scale 1.8 and, The envelope spectrum of the coefficients at scale 8.5 and an enlargement in the kurtogram envelope spectrum (d). Conclusion The kurtogram makes it possible to have, in an approximate way, the resonant frequency and the spectral band useful for the demodulation. We have seen with this approach that the exact location of resonance is an important factor in the diagnostic procedure. With the wavelet transform associated with kurtosis, this task seems to be easier with the possibility of locating more than one resonance, contrary to the kurtogram which does not enjoy this ability because it gives only the band of the greatest value of the kurtosis without having the possibility of distinguishing that immediately below. The practical example has verified clearly the difficulty of choosing study resonances. This example showed that a secondary resonance conveys best information concerning two defects of gears, thing which is not also easy to realize with the traditional tools. References [1] Wang W., Gearly detection of gear tooth cracking using the resonance demodulation technique. Mechanical Systems and Signal Processing (2001) 15(5), 887}903 [2] Boulenger A., Pachaud C., Diagnostic vibratoire en maintenance préventive, Ed. Dunod,1998. [3] Antoni J., The spectral kurtosis: a useful tool for characterising non-statioary signals. Mechanical Systems and Signals Processing, 20 (2006). [4] Antoni J., Fast computation of the kurtogram for the detection of transients faults. Mechanical Systems and Signals Processing, 21 (2007). [5] Lin J., Zuo M. J., Gearbox fault diagnosis using adaptive wavelet filter, Mechanical Systems and Signals Processing, vol 17 issue 6, 2003 p1259-1269 [6] Belaid K., Miloudi A., Early detection of gear defects by wavelets transform. 4 th International Conference on Advances in Mechanical Engineering and Mechanics, 16-18 December, 2008, Sousse, Tunisia. [7] Belaid K., Miloudi A., Slimani M., Utilisation du Kurtosis dans le diagnostic des défauts combinés d engrenages par la transformée continue en ondelettes, Revue des sciences et des technologies, Synthèse N 22, 2010. [8] Randall R. B., Antoni J., Rolling element bearing diagnostics A tutorial, Mechanical Systems and Signals Processing 25 (2011). [9] Barszcz T., Jablonski A., A novel method for the optimal band selection for vibration signal demodulation and comparison with the kurtogram, Mechanical Systems and Signals Processing 25 (2011). [10] Swahi N., Randall R. B., Spectral kurtosis enhancement using autoregressive models, in : proceedings of the ACAM2005 conference, melbourne, february, 2005. 6