Vibration and Electrical Interference Removal for Improved Bearing Fault Diagnosis

Transcription

1 Mechatronics 008, June 3 5, University o Limerick, Ireland ibration and Electrical Intererence Removal or Improved Bearing Fault Diagnosis Iman Soltani Bozchalooi and Ming Liang * Department o Mechanical Engineering University o Ottawa 770 King Edward Avenue, Ottawa, Ontario, Canada, K 65 ABSRAC he bearing vibration signal contains important inormation or bearing health assessment. However, the measured signal is oten heavily clouded by various noises due to the compounded eect o intererences o other machine elements and background noises. hereore, reliable condition monitoring would not be possible without proper de-noising. his paper presents a new de-noising scheme to enhance the vibration signals acquired rom aulty bearings. his de-noising scheme eatures a spectral subtraction to trim down the in-band noise prior to wavelet iltering. he Gabor wavelet is used in the wavelet transorm and its parameters, i.e., scale and shape actor are selected in separate steps. he proper scale is ound based on a novel resonance estimation algorithm. he shape actor value is then selected by minimizing a smoothness index. De-noising results are presented or experimental data acquired rom a aulty bearing with deective outer race.. IRODUCIO Monitoring and diagnosis o the health o machinery elements has attracted much attention over the past ew decades []. One o the important approaches in this area o research is based on the analysis o the vibration data measured through the accelerometers mounted on or near the critical mechanical components. However, the main problem aecting the perormance o the ault diagnosis methods is the corrupting noise. As a result, an eective de-noising method would be necessary to remove such corrupting noise and intererences. wo major de-noising approaches namely wavelet threshold based (also known as decomposition based) [] and wavelet ilter based [3-6] have been used to puriy the vibration signals measured rom aulty bearings or gears. he studies reported in [3, 4] are in avour o the wavelet ilter based de-noising. However, the perormance o this denoising scheme is greatly aected by the wavelet parameter selection strategy. We propose a new approach or the selection o the Gabor wavelet parameters namely scale and shape actor as they deine the center requency and the bandwidth o a Gaussian ilter respectively. he irst step o the selection process consists o a resonance requency estimation algorithm. he estimation result leads us to a desirable scale value. Shape actor value associated with the selected scale is then ound by minimizing a smoothness index (SI). hough the iltering process, i perormed properly, can increase the SR o the measured vibration, the in-band noise with requency content in the range covered by the bandpass ilter is not eliminated. For this reason, we propose to apply spectral subtraction [7, 8] prior to bandpass iltering such that certain in-band noise can be removed. his paper hereater is organized as ollows. Section provides a brie overview o the wavelet transorm and the wavelet ilter based de-noising method. he proposed wavelet parameter selection methods are presented in section 3. Section 4 details the spectral subtraction technique as a mean to improve the perormance o the wavelet ilter based de-noising method. he proposed algorithm is then experimentally evaluated using a aulty bearing with damaged outer race in Section 5. Section 6 concludes the paper.. SIGAL EHACEME USIG A ADAPIE WAELE FILER A major de-noising approach is bandpass iltering. In this method the high SR requency band o the signal is passed through the ilter and other requency components are discarded. Due to the resonance excited by the ault impacts, such a high SR band also exists or the vibration signal measured rom aulty bearings. By bandpass iltering the measured vibration signal around this requency band the impulsive eatures o these signals can be more clearly identiied. One iltering method would be wavelet transorming the signal at a ixed scale. he continuous wavelet transorm (CW) o (t) with respect to a mother wavelet ψ () t is obtained by [9, 0]: * Corresponding author: el.: (63) ext. 669; Fax: (63) ; liang@eng.uottawa.ca

2 th Mechatronics Forum Biennial International Conerence s, W (, s u) = () tψ u () t dt () + where ψ t u su, () t = ( ) s ψ, s and u are real and asterisk stands or complex conjugate. s A close look at Eq. () shows the resemblance o this equation to the convolution process. Hence, we can write: F W s u = F u F u () u[ (, )] u[ ( )]. u[ ψ s,0 ( )] where F x denotes Fourier transorm o a unction with respect to variable x. According to Eq. () the wavelet transorm W ( s, u) or a ixed scale can be interpreted as a iltering process using a ilter with an impulse response equal to ψ () s,0 t [0]. Due to its optimum time and requency resolutions, we choose Gabor wavelet as the mother wavelet. he Gabor wavelet is deined as: t i 0t = (3) ψ () t ce e π Eqs. () and (3) lead to: F [ W( s, u)] = sf [ ( u)] ˆ ( s) (4) u u ψ where ψˆ ( ) is the Fourier transorm o ψ () t and can be written as: π ( ).( 0 ) π ψˆ( ) = c e (5) he last two equations imply that or a constant scale, the Gabor wavelet transorm acts as a bandpass iltering process with a Gaussian ilter. he bandwidth and center requency o this ilter can be adjusted by changing shape actor and scale s, respectively. Furthermore, it can be shown that the iltered signal given by the wavelet ( π 0 ) transorm at a ixed scale will be analytic i >> [9]. hereore, the modulus o this analytic result would provide the envelope o the bandpass iltered signal [6, 9]. Denoting the Gabor wavelet transorm o a signal (t) at s, scale s and shape actor by W ( u), the envelope o the bandpass iltered (t) is obtained by ( ) ( ) s, s, s, ( ) = Re ( ) + Im ( ) W u W u W u (6) 3. WAELE PARAMEER SELECIO 3. SCALE SELECIO It is well known that the requency band corresponding to the resonance excited by the ault impacts orms the high SR band o the vibrations measured rom aulty bearings. Accordingly, a resonance requency estimation method should provide a proper scale value. Figure (a) shows a portion o the simulated aulty bearing vibrations. Such vibrations orm a periodic signal. Each period contains two ranges: a Resonance Active Range (RAR) and a Resonance Inactive Range (RIR) (Figure (a)). When the shat speed goes up, the RAR length remains unchanged or generally increases due to the increased impact intensity. his observation suggests that the RAR comprise an increased portion o a period when the shat speed increases (Figure (a) and (b)). On the other hand, or () t = a()cos t ϕ() t, the instantaneous requency o (t) is given by ϕ '( t) []. Considering the ault generated βt impulse modelled as St () = Ae. cos( ω0tut ) (), where u(t) is the units step unction, the instantaneous requency ound or the signal interval associated with the RAR length would be equal to the excited resonance requencyω 0. Hence, when the shat speed increases an instantaneous requency that corresponds to the largest proportion

3 Mechatronics 008, June 3 5, University o Limerick, Ireland 3 increase can be identiied as the resonance requency. o implement the above idea, we orm the empirical probability density o the instantaneous requency or two vibration datasets measured rom the same bearing at two dierent rotational requencies. With an increase in the rotational speed, we expect a larger probability increase or the resonance requency compared to the other requencies. RAR RIR RAR RIR Amplitude (arbitrary units) Amplitude (arbitrary units) (a) (b) Figure.Simulated aulty bearing vibrations at rotational requency o (a) 0 Hz and (b) 9 Hz m o elaborate, we deine ϕ ' ( n ) as the instantaneous requency o the vibration signal measured at sampling requency, rotational speed ωm and at sampling time n. We urther denote the empirical probability density unction o ϕ ' m ( n ) ( n=... ) by Prm ( ) given as Pr ( ) m =, where is the number o samples with instantaneous requency o m n ϕ ' ( ) =, and is the total number o samples. Following a rise in the shat rotational speed, the probability density change or each requency will be: i, j j i Δ Pr ( ) = Pr ( ) Pr ( ) or ω j > ωi hen the resonance requency can be identiied as: res = arg max ΔPr i, j ( ), where res is the estimated resonance requency. he scale associated with the above identiied resonance requency will then be used or the de-noising process. 3. SHAPE FACOR SELECIO Since bandpass iltering is usually treated as a pre-processing step or envelope spectrum analysis [], we can use the characterizing nature o this envelope to adjust the shape actor or the bandwidth o the corresponding bandpass ilter. A criterion or this adjustment is the smoothness o the envelope o the bandpass iltered result. A robust index quantiying such a characteristic is the ratio o the geometric mean to the arithmetic mean, i.e., smoothness index (SI). his index has been used as a measure o spectral latness in speech signal processing [] and is deined as: n= rg/ A = Sn ( ) n= Sn ( ) (7) or a positive time series S(n). r G / A r G / A is always in the range o [0,] or a positive time series. A useul property o is that it approaches unity or smooth time series and drops to zero or a highly impulsive series. When the shape actor value is properly adjusted and consequently, the ault generated impulses can be clearly detected, we

4 4 th Mechatronics Forum Biennial International Conerence expect the envelope o the bandpass iltered signal to orm a more impulsive time series. Replacing S(n) in Eq. (7) by the expression given in Eq. (6), ( ) η can be written as W n= η ( ) = W n= SP, ( n) SP, ( n) where S P is the selected scale, the sampling period and the number o samples. Accordingly, the proper shape actor value can be ound by minimizing η( ). (8) 4. SPECRAL SUBRACIO In the previous section an adaptive iltering approach was proposed. However, through this method the noise components contained in the requency band o the ilter would not be removed. hereore, the quality o the denoising result may still be aected by such noises. o oset such a drawback o the wavelet ilter-based de-noising method, spectral subtraction can be used prior to wavelet transorm. Spectral subtraction has been widely used in speech signal processing [7]. In this method the PSD (Power Spectral Density) o the pure signal is estimated. his estimate is then used to reconstruct the enhanced signal. As explained beore the requency band that would pass through the ilter corresponds to the resonance excited by the ault impacts. Such impacts usually excite a resonance in the system at much higher requency than the vibration generated by other machine elements []. hereore, it is reasonable to assume that the major part o the in-band noise consists o the background noise present in the measurement device and mainly caused by the wiring laws and electrical intererences. o elaborate, we consider: () t = v() t + () t + () t (9) where (t) is the measured vibration, v(t) is aulty bearing vibration, () t is the measured noise caused by other vibration (e.g., shat imbalance, gear meshing, etc.) and () t is the background noise due to wiring laws and electrical intererences, uncorrelated with v(t) and () t. From (9), we have: ˆ ˆ ˆ ( ω) = vˆ ( ω) + ( ω) + ( ω) According to this equation, with an estimate o ˆ ( ω ), ˆ ˆ G( ω) = vˆ ( ω) + ( ω) can be estimated by: % % G( ω) ( ω) ( ω) ˆ = ˆ ˆ where ˆ % ( ) ω and ˆ % G( ω) are estimates o ˆ ( ω ) and G ˆ( ω ) respectively. Given ˆ% G( ω), an estimate o vˆ ( ω) + ˆ ( ω), to estimate vt () + () t the phase inormation is also required. We assume that the phase inormation is relatively unimportant so that we can approximate vˆ( ω) ˆ ( ω) ˆ( ω). ( + ) by ( ) As the spectral subtraction is concerned with the in-band noise which would usually correspond to a narrow requency band, it is reasonable to assume that the energy o the background noise () t is uniormly distributed over such a narrow band. Consequently, we can urther assume () t as white. his assumption will simpliy the analysis since the PSD o such noise is lat over the requency domain and can be represented with a single value or all requencies. As a result, subtraction o this constant value rom the PSD o the measured signal would provide us with an estimate o G ˆ( ω ). Furthermore, we can apply the smoothness minimization criterion introduced earlier to ind the proper subtraction value. A discretized range o values min, max are considered or ˆ% ( ω). he value on this range which minimizes the SI calculated or the envelope o the spectral subtracted signal is chosen as the best estimate or ˆ ( ω ). As Eq. (0) also yields negative values, it is modiied as ollows to ensure a non-negative result: (0)

5 Mechatronics 008, June 3 5, University o Limerick, Ireland 5 % Gˆ ( ω) ˆ % ˆ ˆ ( ) ( ) ( ) ˆ% ω ω or ω > ( ω) = 0 Otherwise () % Finally vt () + () t is estimated using: ˆ % G( ω) = Gˆ( ω).exp j. ˆ ( ω) and Gt F % Gω % () = ˆ ( ) () Following the spectral subtraction step, the reconstructed time signal given by Eq. () is wavelet transormed using the scale and shape actor values ound through the wavelet parameter selection algorithms explained in previous sections to eliminate () t. 5. EXPERIMEAL EALUAIO he proposed method is evaluated by de-noising signals o a aulty bearing with outer race ault. he experiment is conducted using a SpectraQuest Machinery Fault Simulator (MFK-PK5M) (Figure ). wo well balanced mass rotors ( thick, 4 in diameter and. lbs each) are installed on a 5/8 steel shat and supported by two bearings o type ER0K. he simulator is powered by a 3-hp AC motor that is controlled by a Hitachi drive (SJ00-0FU). he shat speed was 630 RPM (0.5Hz). he let bearing has a pre-seeded ault on the outer race with a characteristic requency o 3 Hz (= 3.05r). An accelerometer (Montronix S00-00) with 00 m/g sensitivity and - khz sensitivity range is used to collect the signal. he signal is ed to an I A-MIO-6DE-0 DAQ card and then collected via LabIEW. he signal processing is done using MALAB on a Pentium 4/.5GHz PC. o create additional vibration intererence, a gearbox is also connected to the driving shat using a belt as shown in Figure. he vibration signal is sampled at 0000 samples/sec. Part o the raw data and the corresponding denoised result are plotted in Figure 3. o select the scale using the resonance requency estimation algorithm, an additional vibration signal is acquired at 780 RPM (3Hz) rotational speed. he proper (s- ) combination is ound as (50, 0.3) and the minimum SI is As shown in the igure the time interval between two consecutive impulses is about seconds which precisely relects the ault characteristic requency o 3Hz. AC Motor Coupling ormal Bearing Faulty Bearing Sensor Belt Gearbox Loads AC Drive Figure. Experimental setup 6. COCLUSIO In this paper, a de-noising scheme was suggested to enhance aulty bearing signals. he algorithm involves two steps: spectral subtraction and wavelet iltering. he main purpose o spectral subtraction is to reduce the in-band noise with requency content that is the same as the requency band covered by the daughter Gabor wavelet. he resulting time domain signal is then wavelet transormed to eliminate the interering vibration signal rom other sources, e.g., shat imbalance, gear meshing, etc. o ind the desirable scale value, a resonance requency estimation

6 6 th Mechatronics Forum Biennial International Conerence algorithm is developed. Corresponding to the selected scale, a proper shape actor is then determined in a separate step by minimizing a smoothness index (SI) calculated or the wavelet coeicient moduli. he proposed method has been successully applied to de-noise signal measured rom a bearing with outer race ault =0.033 Amplitude () Amplitude () (a) (b) Figure 3. (a) ibration signal measured at 630 rpm rotational speed, (b) De-noising result. ACKOWLEDGEME he major part o this study was supported by atural Science and Engineering Research Council o Canada. heir support is greatly appreciated. REFERECES [] P. D. McFadden and J. D. Smith, ibration monitoring o rolling element bearing by the high-requency resonance technique- a review, ribology International 7 (984) 3-0. [] J. Lin and L. Qu, Feature extraction based on Morlet wavelet and its application in mechanical ault diagnosis, Journal o Sound and ibration 34 (000) [3] H. Qiu, J. Lee, J. Lin, and G. Yu, Wavelet ilter-based weak signature detection method and its application on rolling element bearing prognosis, Journal o Sound and ibration 89 (006) [4] H. Qiu J. Lee, J. Lin, and G. Yu, Robust perormance degradation assessment methods or enhanced rolling element bearing prognostics, Advanced Engineering Inormatics 7 (003) [5] J. Lin and M. J. Zuo, Gearbox ault diagnosis using adaptive wavelet ilter, Mechanical Systems and Signal Processing 7 (003) [6]. G. ikolaou and I. A. Antoniadis, Demodulation o vibration signals generated by deects in rolling element bearings using complex shited Morlet wavelets, Mechanical Systems and Signal Processing 6 (00) [7] J. S. Lim and A.. Oppenheim, Enhancement and bandwidth compression o noisy speech, Proceedings o the IEEE 67 (979) [8] J. P. Dron, F. Bolaers, Rasoloondraibe, Improvement o the sensitivity o the scalar indicators (crest actor, kurtosis) using a de-noising method by spectral subtraction: application to the detection o deects in ball bearings, Journal o Sound and ibration 70 (004) [9] S. Mallat, A wavelet tour o signal processing, Academic Press, San Diego, 998. [0] R. M. Rao and A.S. Bopardikar, Wavelet transorms: Introduction to theory and application, Addison Wesley, Reading, MA, 998. [] L. Cohen, ime-frequency Analysis, Englewood Clis,.J.:Prentice Hall, 995. [] S. M. Kay, he eect o noise on the autoregressive spectral estimator, IEEE ransactions on Acoustics, Speech and Signal Processing 7 (979)