SPECTRAL METHODS ASSESSMENT IN JOURNAL BEARING FAULT DETECTION APPLICATIONS

Transcription

1 The 3 rd International Conference on DIAGNOSIS AND PREDICTION IN MECHANICAL ENGINEERING SYSTEMS DIPRE 12 SPECTRAL METHODS ASSESSMENT IN JOURNAL BEARING FAULT DETECTION APPLICATIONS 1) Ioannis TSIAFIS 1), K.-D. BOUZAKIS 2), Grigoris TSOLIS 2), Thomas XENOS 1) Laboratory for Machine Tools and Manufacturing Engineering, Mechanical Engineering Department, Aristoteles University of Thessaloniki, GREECE 2) Department of Electrical & Computer Engineering, Aristotle University of Thessaloniki Thessaloniki, GREECE tsiafis@eng.auth.gr, bouzakis@auth.gr, tdxenos@auth.gr ABSTRACT The scope of this work is to assess three spectral methods employed in journal bearing fault detection. To this end, two popular methods namely the Short Time Fourier Transform (STFT) and the Wavelet Transform and an innovative and very promising method, namely the Hilbert Huang transform, were employed. Five experimentations were carried out employing five pairs of journal bearings. Vibration time series, measured by an accelerometer assembled on the base of the bearing, were obtained. The time series were processed by means of Fourier transform, wavelet transform and the Hilbert Huang transform and the resulting spectra of each pair of bearings, sound and defective, were examined for possible differences. Both the Fourier Transform and the wavelet transform analysis did not reveal any differences between the spectra corresponding to sound and defective bearings. On the contrary, the analysis employing the Hilbert Huang Transform revealed significant differences between the respective first five intrinsic mode functions (IMF) which reduced in magnitude as the order of the IMF increased i.e. as the spectral frequency decreased, whereas the Hilbert spectra obtained from the time series corresponding to sound and defective bearings strongly differed. Keywords: Journal bearings, vibration, FFT, wavelet. 1. INTRODUCTION Journal bearings are important machine elements; they are characteristic example of friction systems the function of which are based on the laws concerning the flow field of the lubricant. Lubricants employed are mineral oils or pure graphite. Under normal lubricant conditions, the two parts of a journal bearing (stator and rotor) are separated by a thin lubricating layer through which the forces applied on the rotor are transferred to the stator, taking into account the main restriction that the loads applied are smaller than a limiting value. The purpose of this work is to assess spectral methods employed in non destructive fault detection. Two kinds of damages can be observed in journal bearings systems: transversal cracks due to fatigue of the axle and radial damage of the journal bearings. The first one can end to a complete damage of the setup when it extends to 60% of the axle diameter; the symptoms of the crack to the oscillatory performance of such systems have been extensively examined [1, 2, 3, 4, 5]. On the contrary, the diagnosis of the axial damages on the bearings, in spite of several attempts that can be found in the literature [6, 7, 8], have neither been extensively nor successfully investigated, especially in cases when the machine was operating. Moreover, the methods of spectral analysis employed in these studies were either Fourier or wavelet transform, which, as it will be shown in this paper can hardly give reliable and explicit results. 1

2 The scope of this paper is to assess the spectral methods usually employed in such problems i.e. the Fourier and wavelet transforms and to propose a new diagnostic method based on the Hilbert Huang transform, the superiority of which, with respect to reliability and accuracy, will be proved. 2. SPECTRAL ANALYSIS METHODS 2.1. The Fourier transform For a stationary signal, i.e. a signal the statistical parameters of which are not changing with time, its spectral content can be described by a Fourier Transform of its autocorrelation function. On the other hand, most of the signals obtained from the analysis of natural processes, can hardly be characterized stationary since their spectral content is a function of time. Consequently, the spectral density function of a non-stationary signal is both time and frequency dependant, and it is called evolutionary spectrum. The idea of a linear time-frequency transform for a non-stationary signal was proposed by Gabor [10], under the assumption that the spectral content of a non-stationary process can be described by applying a Fourier transform in successive temporal windows. As a result, a Short-Time Fourier Transform (STFT) was proposed j2 ft d SFx ( t, f ) x( )h( t )e (1) where, h is a window function, the selection of which depends on the application and its particularities regarding the signal analysis; usually it is not a trivial process. Moreover, the use of constant width windows together with the uncertainty principle, poses several restrictions regarding the accuracy of the analysis in the sense that an increase of the accuracy into the time domain ends to a reciprocal restriction of the accuracy into the frequency domain and vice versa The Wavelet Transform The wavelet transform (WT) attempts to cover the limitations and disadvantages of the STFT. In this case the temporal windows differ in duration and depend on the analysis requirements and the nature of the signal. Consequently, whenever higher precision is required in the lower frequencies, the window temporal length is larger and vise versa. The continuous wavelet transform for a real signal is determined as: 1 WT x (t, ) x( )w( t, ) d (2.1) A characteristic wavelet employed in this analysis is the Mexican hat the characteristic function of which is: 2 t 0,5 2 t w(t, ) 1 e (2.2) 1/ 2 The term is used for energy conservation reasons (e.g. it assures that all wavelets, in every scale have the same energy). The parameters and are the dilation reduction and transitional parameters respectively on the horizontal axis, whereas w(t, ) is the analyzing wavelet. The dilation is obtained by changing the value of which is a real and positive number. This value together with t gives a multi-resolution possibility. As in STFT the results of a WT analysis vary in resolution depending on the type of the wavelet chosen, On the other hand a multitude of different wavelets is available. Nevertheless, it has to be pointed out that WT is not a time-frequency transform; it is in fact a time scale transform given that it is materialized on different scales of the wavelet and of the temporal window The Hilbert Huang transform A very useful quantity in the analysis and processing of real life signals is the frequency. As it was mentioned before, in the case of non-stationary signals the frequency of the signal is a function of time and it can be only determined locally. Consequently, the parameter of interest and with physical sense is the instantaneous frequency; this parameter describes the spectral content of the signal as a function of time and theoretically it can be approximated by the frequency of a sinusoidal signal that each time approximates locally the signal under consideration [11]. On the other hand while the meaning of the frequency in stationary signals is strictly defined, in the case of non-stationary signals it is not. The definitions and the methods proposed in the literature vary with the application. The most acceptable definition was given by Carlson and Fry. According to this, instantaneous frequency is defined as the variation rate of the angular phase of the signal. To 2

3 this end Gabor [10] suggested a method for the angular phase calculation constructing an analytical signal through a Hilbert transform of the original signal. Thus, for an original signal x(t), the analytical signal is given by: z(t) = x(t) + j H(x(t)) (8) where H(x(t)) is Hilbert transform. Hilbert transform is an integral transform emphasizing the local properties of the signal, since it is a convolution of the signal with 1/t. For a real signal it is defined as: 1 x( ) H (t ) P d (9) t where P is the Cauchy principle value. Thus, the instantaneous frequency of the signal is given by: 1 d ( t ) argz( t ) (10) 2 dt On the other hand, the instantaneous frequency calculation, as a derivative of the angular phase of the analytical signal, makes sense only for a signal the frequency content of which is very narrow, i.e. for monocomponent signals. This means that such a calculation for multicomponent signals can be exclusively performed after analyzing the signal in monocomponents. It must be pointed out that even in monocomponent signals, any dc components may give erroneous results for the instantaneous frequency of the signal. Consequently, in the case of multicomponent signals the most common method to calculate instantaneous frequency is based on the use of the evolutionary spectrum obtained by the timefrequency distributions. The Empirical Mode Decomposition is the first stage of an algorithm known as Hilbert-Huang transform, where a real or complex signal is decomposed in a series of structural components, known as Intrinsic Mode Functions (IMF) [12, 13, 14, 15]. We define as IMF any function having the same number of zero-crossings and extrema, and also having symmetric envelopes defined by the local maxima and minima respectively. Since IMFs admit well-behaved Hilbert transforms, the second stage of the algorithm is to use the Hilbert transform to provide instantaneous frequencies as a function of time for each one of the IMF components. Depending on the application, only the first stage of the Hilbert-Huang Transform may be used. For a discrete time signal x(n) the EMD starts by defining the envelopes of its maxima and minima using cubic splines interpolation. Then, the mean of the two envelopes is calculated as: m 1 (n)=(e max (n)+e min (n))/2 (3) Accordingly, the mean m 1 (n) is then subtracted from the original signal: h 1 (n)=x(n)-m 1 (n) (4) and the residual h 1 (n) is examined for the IMF criteria of completeness. If it is an IMF then the procedure stops and the new signal under examination is expressed as: x 1 (n)=x(n)-h 1 (n) (5) However, if h 1 (n) if is not an IMF, the procedure, also known as sifting, is continued k times until the first IMF is realized. Thus: h 11 (n)=h 1 (n)-m 11 (n) (6) where the second subscript index corresponds to sifting number, and finally: IMF 1 =h 1k (n)=h k-1 (n)-m 1k (n) (7) In fact, the sifting process is continued until the last residual is either a monotonic function or a constant. It should be mentioned that as the sifting process evolves, the number of the extrema from one residual to the next drops, thus guaranteeing that complete decomposition is achieved in a finite number of steps. The final product is a wavelet-like decomposition going from higher to lower oscillation frequencies, with the frequency content of each mode decreasing as the order of the IMF increases [12, 14]. The big difference however, with the wavelet analysis is that while modes and residuals can intuitively be given a spectral interpretation in the general case, their high versus low frequency discrimination applies only locally and corresponds in no way to a predetermined sub-band filtering. Selection of modes instead, corresponds to an automatic and adaptive (signal-dependent) timevariant filtering [5]. After completion of EMD the signal can be written as follows, x n k IMF r( n ) (7) i1 i where k is the total number of the IMF components and r(n) is the residual. 3. EXPERIMENTAL SETUP The experimental setup (figures 1a and b) consisted of a shaft rotating through two journal bearings (type: PSM ), driven by a variable speed electric motor controlled by an inverter. An accelerometer was connected very close to the bearings and the electrical signals generated by it was digitized and fed to a laptop. Data collection and the subsequent signal processing were performed using Matlab. 3

4 This is a very satisfactory sample. The time series were analyzed by means of STFT, WT and the Hilbert Huang transform. Six sound and six defective journal bearings were used. Given that the results were almost identical, the results of bearing No 1 are presented here. Fig. 1a. Experimental setup Fig. 1b. Diagram of the experimental setup 4. RESULTS AND DISCUSSION Fig. 3a. FFT sound bearing The scope of this work is to assess three spectral methods employed in journal bearing fault detection. To this end, two popular methods namely the Short Time Fourier Transform (STFT) and the Wavelet Transform and one innovative and very promising method, namely the Hilbert Huang transform, were employed. In figs 2a and 2b the time series obtained after 180 s of sampling are presented. Fig 2a corresponds to a sound journal bearing whereas fig 2b corresponds to a defective one. Taking into account that the sampling frequency was Hz and the sampling time 180 s each experimentation gave a total of 11,796,480 samples. 4 Fig. 2a. Time series sound bearing Fig. 2b. Time series defective bearing Fig. 3b. FFT defective bearing From Figs 2a and 2b we can clearly see a level difference of the order of 12 db between sound and defective journals. Given that the calibration throughout the experimentations were kept identical, it can be deduced that the faults caused on the each journal bearing enhanced the signal level due to amplification of the vibrations. Figs 3a and 3b present the spectra obtained after the STFT method was applied. As it can be seen, the discrimination between sound (fig. 3a) and faulty (fig. 3b) bearing is very hazy. One may only distinguish a distortion (widening) of the spectrum corresponding to the faulty bearing at a frequency around Hz, which could be marginally employed as a diagnostic tool The Wavelet Transform used employed a Mexican Hat wavelet. A comparative study of the specta of sound (fig. 4a) and defective (fig. 4b) bearing cannot reveal any differences. Therefore, we can easily conclude that WT cannot be used as a diagnostic tool for application of this kind.

5 Fig. 4a. Mexican hat CWT sound bearing Fig. 5b. ΙΜF 1-4 defective bearing Fig. 4b. Mexican hat CWT defective bearing The time series analyses by means of the Hilbert Huang transform reveals strong differences between sound and defective journal bearings both in the IMFs (figs 5a, 6a, 7a and figs 5b, 6b, 7b respectively) and in the corresponding Hilbert spectra (figs 8a and 18b) [14]. In fact a comparative study between the relevant IMFs 1 to 5 of the sound and defective bearings (these IMFs correspond to the higher frequencies) reveals very strong differences; these differences tend to fade as the order of the IMF is getting higher ie as the frequency is decreasing. Fig. 6a. ΙΜF 5-8 sound bearing Fig. 6b. ΙΜF 5-8 defective bearing Fig. 5a. ΙΜF 1-4 sound bearing Fig. 7a. ΙΜF 9-11 sound bearing 5

6 Fig. 7b. ΙΜF 9-11 defective bearing Fig. 8a. Spectrum Hilbert sound bearing the Short Time Fourier Transform (STFT) and the Wavelet Transform and one innovative and very promising method, namely the Hilbert Huang transform, were employed. Five experimentations were carried on employing five pairs of journal bearings. Vibration time series, measured by an accelerometer assembled on the base of the bearing, were obtained. A comparison between the time series obtained from the sound bearings and the corresponding ones obtained from the defect bearing showed a 12 db difference in magnitude, apparently due to the increased vibrations induced exclusively by the defects. All time series were processed by means of Fourier transform, wavelet transform and the Hilbert Huang transform and the resulting spectra of each pair of bearings, sound and defective, were examined for possible differences. Both the Fourier Transform and the wavelet transform analysis did not reveal any differences between the spectra corresponding to sound and defective bearings. On the contrary, the analysis employing the Hilbert Huang Transform revealed significant differences between the respective first five intrinsic mode functions (IMF) which reduced in magnitude as the order of the IMF increased i.e. as the spectral frequency decreased, whereas the Hilbert spectra obtained from the time series corresponding to sound and defective bearings strongly differed. Consequently, it is concluded that only the Hilbert Huang transform method proved reliable both in detecting and accurately diagnosing the faults; therefore it can be a promising and powerful diagnostic tool in journal bearing fault detection applications. Fig. 8b. Spectrum Hilbert defective bearing Consequently, it is concluded that only the Hilbert Huang transform method proved reliable both in detecting and accurately diagnosing the faults; therefore the Hilbert-Huang method can be a promising and powerful diagnostic tool in journal bearing fault detection applications. 5. CONCLUSION The scope of this work is to assess three spectral methods employed in journal bearing fault detection. To this end, two popular methods namely 6 REFERENCES 1. Darpe A.K., Gupta K., Chawla A., 2003, Experimental investigations of the response of a cracked rotor to periodic axial excitation, Journal of Sound and Vibration, 2/13, 260(2), pp Darpe A.K., Gupta K., Chawla A., 2004, Coupled bending, longitudinal and torsional vibrations of a cracked rotor, Journal of Sound and Vibration, 269(1-2), pp Isermann R., 1995, Model based fault detection and diagnosis methods, Proceedings of the American Control Conference, vol. 3, pp Markert R., Platz R., Seidler M., 2001, Model Based Fault Identification in Rotor Systems by Least Squares Fitting, International Journal of Rotating Machinery, 7(5), pp Pennacchi P., Bachschmid N., Vania A., 2006, A model-based identification method of transverse cracks in rotating shafts suitable for industrial machines, Mechanical Systems and Signal Processing, 11;20(8), pp

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