Applying digital signal processing techniques to improve the signal to noise ratio in vibrational signals

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1 Applying digital signal processing techniques to improve the signal to noise ratio in vibrational signals ALWYN HOFFAN, THEO VAN DER ERWE School of Electrical and Electronic Engineering Potchefstroom University for CHE Private Bag X6, Potchefstroom, 5 SOUTH AFRICA Abstract: - This paper investigates the application of signal processing techniques to improve the signal to noise ratio in vibrational signals used for condition monitoring. Environmental conditions such as instantaneous speed variations as well as the presence of multipe fault conditions can however obscure the defect signals that are required for reliable diagnostics and can lead to faulty diagnostic decisions []. While these problems can be solved with the right combination of techniques, the difficulty of obtaining sufficiently large measured data sets on which to train these techniques remain. Artificially generated training data sets, by empirical modeling of defects, is hence investigated and a simple vibrational model, which includes the effect of period variation, is proposed for the bearing defect data set in []. Signal processing techniques that can reduce the noise caused by instantaneous angular speed variations of the shaft and provide better features for fault recognition are subsequently investigated. Key-Words: - neural networks, classifiers, bearing defect modeling, vibrational features, multiple fault conditions, signal processing Introduction This paper investigates the application of signal processing techniques to improve the signal to noise ratio in vibrational signals used for condition monitoring. The vibrational signals caused by bearing and gear defects both lead to instantaneous variations that are observable in the time and frequency domains. Various practical effects, e.g. the simultaneous presence of more than one fault mechanism, can however obscure the defect signals that are required for reliable diagnostics. It have for example been shown before that a deteriorating secondary fault condition can lead to faulty diagnostic decisions []. These problems can in principle be overcome by correct preprocessing, extraction of appropriate signal features and the application of automated classification. A major problem to achieve this goal is however the difficulty to obtain sufficiently large measured data sets on which to train these techniques. The empirical modeling of vibrational phenomena related to such defects is hence investigated in order to artificially generate training data sets. cfadden and Smith in [,] proposes a simple vibrational model for inner race defects, which is extended by Brie in [5] to account for the effect of speed variations (which occur even if the shaft speed is constan. A simple vibrational model, which includes the effect of period variation, is proposed for the bearing defect data set in []. Signal processing techniques that can reduce the noise caused by angular speed variations of the shaft and provide better features for fault recognition are subsequently investigated. Although a reference signal related to the angular position of the shafts can reduce this problem, such a signal is unfortunately not available in many industrial data sets. A speed compensation algorithm is proposed and applied successfully to a synthetic vibration signal with speed variation. Problem Formulation. odelling of multiple fault mechanisms.. odelling of bearing defects (including size of defec The model proposed for bearing defects by cfadden and Smith in [,] have been extended by other researchers to incorporate additional environmental conditions such as period variations caused by e.g. changing contact angle by Brie in [5].

2 It have been shown in [6] that the size of the demodulation peak in the vibrational spectrum is essentially linearly related to the defect size with vibration measurements. The normalised ratio of demodulation peak to the carpet level (noise in the demodulation spectra) hence provides a quantitative measure of the bearing defect condition. Bearing replacement is suggested when this ratio is above 5dB. Figure shows the influence of buffer length on the demodulation peak of the bearing date set of []. Increased averaging leads to higher signal to noise levels. It should be noted that determining the carpet level is no longer easy with no trending measurements available and when speed variation is present x Fig. : ratio of demodulation peak relative to carpet level (vertical axis in db) versus length of time domain buffer (horizontal axis in samples) cfadden and Smith proposed a model of an exponentially damped sinusoid as the response of the structure to an impulse d( caused by contact of a ball with a single point (inner race) defect []. The impulses d( are modulated by the load function q( and the transfer function to the sensor a(. In [5] alternative models are proposed, firstly using timefrequency analysis, allowing instantaneous changes in frequency (chirp signal). The second model, while based only on a first degree of freedom system, uses time-varying parameters (LTV system). It is also shown in [5] that variations in instantaneous speed can be modeled using a quasi-periodical impulse train and analytical expressions for the autocorrelation and PSD (power spectral density) are derived. It is clear that instantaneous variations in the contact angle causes timing jitter in the response of the system to a bearing defect, even if no speed variation in the rotating speed takes place. The timing jitter causes spectral smearing [5], complicating spectral and cepstral analysis. The goal of our research is not to develop the best vibrational model for defect detection, but rather to evaluate the effectiveness of various signal processing operations, usually implemented in a preprocessing stage, to support the extraction of characteristic features. A simple vibrational model for defect detection, with reasonable correspondence to the measured data set, is therefore required. We propose a model of the response of the system to a defect impulse as a Gaussian modulated sinusoidal signal with fractional bandwidth BW and resonance frequency f c. The bandwidth of the pulse is *BW% measured at a level of BWR db relative to the normalised peak of the signal. The signal can be expressed analytically as: t c () v Y b = Aexp t exp( jπf with amplitude A, Gaussian mean f c, Gaussian variances t v = f v /π, f v =-BW f c /(8log(r and reference level r= (BWR/) Fig. : compensated time domain signal (horizontal axis in samples) An example of a time domain signal, repeating at 5 Hz intervals, is given in figure with BW=.5. The effect of multiple defects can be accommodated by the model proposed in []... odelling of bearing defects (multiple defects) In [] the effect of multiple defects, is modeled as the superposition of single point defects. It is assumed that the defects occur at regularly spaced intervals corresponding to the defect time (inverse of the fundamental defect frequency). For the special case of defects of equal strength, = and the demodulated spectrum is given by: ( I (x() is the Fourier Transform of the signal x() ( + ) ) H = ( f ) = I t τ ) () = I( ) ) + I( ) + I( = H + I I = +

3 I = H = ) α) exp exp jπfα jπft dt dα exp j πfτ = f ) = H = τ + ( f )( + exp( j π f ( I assuming that is the response of the system to a bearing defect and H(f) denotes the demodulation spectrum of the fullwave rectified signal. If it is assumed that the response of the system to an impact caused by the bearing defect is zero before the time delay τ, say at time t, and also that τ<t/, the integral I is zero. Under these assumptions the expression simplifies to: H ( f )( + exp( j f = f ) = H = τ + ( π I () The above expression indicates that the minimum value (zero) of the demodulation peak will be reached when f=.5/τ, i.e. when the second defect is delayed by half the defect period. This is obvious, as the time between defects is now halved. The defect frequency is consequently doubled. When τ= the demodulation peak is doubled, as expected. A general expression for defects can now be postulated under the following assumptions: The amplitude and time delay of defect k is given by A k and τ k respectively The response t-τ k ) of defect k ends before the start of defect k+. This ensures that the cross products in () are zero. The response of the system to the defects are all determined by scaled and shifted replicas of The desired expression can now be determined by using a binomial expansion and with the assumptions above: H = I A ) I( A ) + A ) ( + KI A ) + ( f = H = K () ( ) ( ( ) ) jπfτ jπfτ ( f )( A exp KA exp ) = + K This expression () indicates that the amplitude of the demodulation peak due to defects will be given by the vector sum of the squared amplitude values A k θ k, with phase angle θ k determined by the ratio of time delay τ k to period T=/f (k varies from to defect ). From () it is evident that the combined effect of defects on the demodulation spectra is given by a single point defect with impulse response h ( = A ) + K A ) (5) The demodulation spectra of (5) would be identical to that of (). Hence it can be concluded that an approximation to the impulse response of (5) would approximate the demodulation spectra well. ore specifically, the proposed model given by () could be tuned in amplitude to best fit the demodulation spectra against time (refer to figure ). By adjusting the time position of the pulse, the effect of small speed variations could also be incorporated... Influence of environmental conditions The introduction of fault conditions by dismantling of machinery always change the vibration levels due to the changing transfer function of the small defect signal through the complex mechanical interface leading to the vibration sensor. It is therefore preferable to trend vibration measurements during fault evolution (with no dismantling). Unfortunately, it is very difficult to detect small incipient defects in a complex industrial setting. The component would also usually not be replaced at this early stage, causing production losses. Hence it is difficult to establish a machinery database of small defects. Smaller experimental setups, in which environmental factors (such as speed, load and temperature) can be controlled, can be used to discover certain general trends and relationships. The influence of period variation can be modeled as a smoothing of the spectrum by a function related to the probability density function of the signal [5]. Hence high frequency harmonics will tend to smear into each other, as shown in figure. The effect of time domain speed compensation, described in section., is shown in figure, in which a repeating series of harmonics are clearly visible... ultiple fault conditions A bearing frequently acts as a support mechanism for a complex mechanical assembly consisting of gearboxes etc. As it is often impossible to obtain access to the bearings, vibration measurements are often made on the gearbox casing. In [] the total response is described as a linear superposition of the bearing and gear excitations. The small bearing defect signal could hence be swamped by the large meshing frequencies of the gearbox, making detection very difficult. Furthermore, a deteriorating secondary fault mechanism, e.g. increasing

4 imbalance, can lead to faulty diagnostic decisions [].. Signal processing techniques.. Spectral analysis The direct use of spectral analysis and trending of spectral components have been used for a long time in machinery fault diagnosis [,9]. As it is difficult to analyse and interpret the phase information from spectral analysis (which often contain critical diagnostic information), much interest have recently been expressed in combined time-frequency analysis methods which preserves phase information (e.g. [])... Time domain analysis Indicators such as RS, crest factor, kurtosis and statistical moments are useful for bearing diagnosis []... HFRT (high frequency resonance technique) In the HFRT (high frequency resonance technique) the signal is bandpass filtered around a suitable demodulation frequency and then rectified and lowpass filtered (envelope detection) []. The normalised ratio of the demodulation peak in the demodulation spectra relative to the carpet level provides a measure of the defect growth. As this ratio is regarded as the best feature of bearing defect evolution [6,7,8], no time domain and spectral features have been used in this paper. Problem Solution. Influence of the buffer length on demodulation spectra Figure shows the influence of buffer length on the demodulation peak relative to the carpet level. It is evident that a small buffer length does not allow a sufficient threshold for detection of the demodulation peak above the carpet level. From samples to 8 very little contribution is made to this ratio in db. By accumulating the individual contributions over a substantial buffer length reliable detection of a defect becomes possible (refer to figure ).. Effect of speed variation on demodulation peak Table shows that small amounts of speed variation on a simulated data set have little effect on the magnitude of the demodulation peak, although speed variation does smear the high frequency harmonics making direct frequency analysis difficult. Table : effect of speed variation on demodulation peak BW Speed variation % Demodulation peak Time domain speed compensation Figures and shows the magnitude spectrum of the simulated time domain signal before and after compensation. The smearing of spectral peaks can be clearly seen in figure Fig. : agnitude spectrum uncompensated (fractional bandwidth of.5 and % random speed variation) Fig. : agnitude spectrum compensated (fractional bandwidth of.5 and % random speed variation) The compensation algorithm is described below: The parameter choice BW=. corresponds to an impulsive source, while BW=.5 seems to match the experimental signal more closely

5 Algorithm for time domain speed compensation (pseudo code) ) Peaks = vector with sorted peak values in time domain signal ) Determine SC (Samples to Compensate) SC=cumsum (diff (diff(peaks) SC = N k = x( k) x( k ) + x( k ) ) For lk= to lengtpeaks)- dt=peaks(lk+)-peaks(lk) %dt is samples between consecutive peaks %K is constant defining offset of inserted/ %deleted samples (e.g. K =.) If SC> then delete SC samples at position Peaks(lk)+ dt *K else insert SC samples (zeros) at position Peaks(lk)+dT*K end if end lk The compensation algorithm essentially computes the number of samples to compensate during each period and then insert or delete the required samples at a suitable position. As the algorithm uses a peak finding algorithm to locate significant repeating energy, it is often difficult in experimental data sets to locate the peaks separated by the defect frequency. Figure 5 shows the above problem in an experimental data set [7]. It can be seen that is difficult to determine the exact time between peaks, even though a clear demodulation peak is visible in the demodulation spectra. The compensation algorithm have been applied to the experimental data set in figure 7 (figure 6 shows the uncompensated magnitude spectrum for comparison). Unfortunately a clear line spectrum, as in figure, is not obtained. From figure 5 it is evident that the amplitude modulation, implying a large dynamic range variation, as well as multiple responses (peaks) per defect time contributes to the performance of the simple compensation algorithm on the experimental data set. The spectrum of the compensated signal is shown in figure 7. Figure 8 shows the ratio of demodulation peak to carpet level on an industrial motor bearing as described in [8], with an outer race bearing defect. The pickpeak function in atlab have been used. cumsum - cumulative summation of series x(k) diff difference between consecutive samples of series x(k) Figure 9 indicates that repetitive frequency components are already visible in the raw high frequency spectra (which is not the case in the data set of figure 6). This would seem to indicate that instantaneous speed variations, due to instantaneous changes in the contact angle, are smaller in the data set described in [8]. As the demodulation peak is only about db above adjacent frequency components in figure 8 at the end of the buffer, it is not so easy to form a clear diagnostic decision. The carpet level itself was estimated from small frequency components close to the demodulation peak. 8 x.86 ms Hz Fig. 5: plot of energy (square) of time domain signal, bandpass filtered around 8.5 khz (horizontal axis in samples) Fig. 6: Experimental data set uncompensated ms 5 Hz Fig. 7: Experimental data set compensated

6 Fig. 8: ratio of demodulation peak relative to carpet level (in db) versus length of time domain buffer (horizontal axis in samples) [8]. 6 5 x 6 8 Fig. 9: high frequency spectra (horizontal axis in Hz) [8]. Conclusion A simple vibrational model of a bearing defect have been proposed, building on previous work in this field. The model incorporates a Gaussian modulated sinusoidal source, which seems to mimic the observed vibrational signal of the measured data set more closely than an exponentially decaying sinusoid. The model also includes the effect of instantaneous speed variations as proposed in [], which explains the observed spectral smearing. This model is used to illustrate the effect of speed variations on the demodulation peak, which is the primary feature indicating bearing defect. The influence of multiple defects and buffer length on the demodulation peak is also investigated. An algorithm to compensate for speed variations is proposed, and the ability of this algorithm to resolve the periodic frequency spectra of simulated signals is demonstrated. It is however shown that the improvement obtained with experimental data sets is not as dramatic. 5 Acknowledgements The authors gratefully acknowledge the use of the data sets for this article ([,8]). The supplier of the data set in [8] is Prof. Antoniadis and were in the framework of a collaboration with Aluminum of Greece SA. References: []H. Oehlman, D. Brie,. Tomczak, A. Richard. A method for analysing gearbox faults using time frequency representations, echanical systems and signal processing, Vol., No., 997, pp [] N.T. van der erwe, A.J. Hoffman, C. Stander, S. P. Heyns. Identifying multiple faults in rotating machinery 5, Proceedings of SAUPEC, University of Cape Town, Vol., pp [] P.D. cfadden, J.D. Smith. odel for the vibration produced by a single point defect in a rolling element bearing, Journal of sound and vibration, Vol. 96, 98, pp [] P.D. cfadden, J.D. Smith. odel for the vibration produced by multiple point defects in a rolling element bearing, Journal of sound and vibration, Vol. 98, 985, pp [5] D.I. Brie. odelling of the spalled rolling element I, echanical systems and signal processing, Vol., No.,, pp [6] J. Shiroishi, Y. Li, S. Liang, T. Kurfess, S. Danyluk. Bearing condition diagnostics via vibration and signal processing, echanical systems and signal processing, Vol., No. 5, 997, pp [7] A.J. Hoffman, N.T. van der erwe. A comparative evaluation of neural classification techniques for identifying multiple fault conditions. Proceedings of the 5th WSES International Conference on Circuits, Systems, Communications and Computers (CSCC ), Rethymno, Greece, July 8-5, pp. 9-. [8] N.G. Nikolaou, I.A. Antoniadis. Application of Wavelet Packets in bearing fault diagnosis. Proceedings of the 5th WSES International Conference on Circuits, Systems, Communications and Computers (CSCC ), Rethymno, Greece, July 8-5, pp. -9. [9] J.I. Taylor. The vibration analysis handbook: A practical guide for solving rotating machinery problems, Vibration Consultants Inc., ISBN , Chosen as plenary session paper at SAUPEC.