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A combined approach for weak fault signature extraction of rolling element bearing using Hilbert envelop and zero frequency resonator
Journal of Sound and Vibration 419 (2018) 436e451
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Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
A combined approach for weak fault signature extraction of rolling element bearing using Hilbert envelop and zero frequency resonator Keshav Kumar a, Sumitra Shukla b, Sachin Kumar Singh a, * a b
Department of Mechanical Engineering, Indian Institute of Technology (ISM) Dhanbad, Dhanbad, Jharkhand, 826004, India Department of Electronics Engineering, Indian Institute of Technology (ISM) Dhanbad, Dhanbad, Jharkhand, 826004, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 14 April 2017 Received in revised form 9 December 2017 Accepted 9 January 2018
Periodic impulses arise due to localised defects in rolling element bearing. At the early stage of defects, the weak impulses are immersed in strong machinery vibration. This paper proposes a combined approach based upon Hilbert envelop and zero frequency resonator for the detection of the weak periodic impulses. In the first step, the strength of impulses is increased by taking normalised Hilbert envelop of the signal. It also helps in better localization of these impulses on time axis. In the second step, Hilbert envelope of the signal is passed through the zero frequency resonator for the exact localization of the periodic impulses. Spectrum of the resonator output gives peak at the fault frequency. Simulated noisy signal with periodic impulses is used to explain the working of the algorithm. The proposed technique is verified with experimental data also. A comparison of the proposed method with Hilbert-Haung transform (HHT) based method is presented to establish the effectiveness of the proposed method. © 2018 Elsevier Ltd. All rights reserved.
Keywords: Bearing fault Weak signal detection Zero frequency resonator Hilbert envelop
1. Introduction Rolling element bearings are used widely in rotating machinery. Detection of bearing defects at early stage is important for scheduling the preventive maintenance and hence to ensure the safe running of machinery. The vibration signal measured from the bearing contains the fault signature. Periodic impulses are generated due to localised faults in bearing races. These impulses occur due to passing of rolling element over the localised faults. Detection of these periodic impulses is relatively easier for a grown up defect. However, at the early stage of the defect, the amplitude of the impulses is weak and the impulses are buried in the strong machinery vibration making it difficult to be detected. Hence, an appropriate signal processing method is required to extract the fault signature at early stage. FFT based methods are used extensively to establish the presence of fault signatures in the spectrum [1,2]. However, when the fault signal is weak, the FFT based methods are unable to capture the transient features of the non-stationary weak fault signal. Several time-frequency-energy based approaches have been proposed to detect the bearing faults at early stage. Among these, wavelet transforms (WT) and HHT based approaches are predominant [3e7].
* Corresponding author. E-mail addresses: [email protected] (K. Kumar), [email protected] (S. Shukla), [email protected] (S.K. Singh). https://doi.org/10.1016/j.jsv.2018.01.022 0022-460X/© 2018 Elsevier Ltd. All rights reserved.
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In wavelet transform, the signal is decomposed at different levels to capture the transient features of the weak fault signal. Successful feature extraction using wavelet transform depends upon the location and shape matches between the wavelet bases and the fault signal. Hence, selection of proper wavelet parameters depends upon fault signal characteristics. In this sense the wavelet transform in not adoptive. However, recently some adoptive wavelet based methods have been proposed. Zhang et al. [8] proposed an adoptive methodology using the flexible analytical wavelet transform (FAWT) for the extraction of weak fault signature. The optimal parameters for the FAWT were obtained by optimising the characteristic of kurtosis spectral entropy using genetic algorithms. Li et al. [9] proposed an adaptive morphological gradient lifting wavelet for the detection of bearing faults. The method was based upon selecting the average filter and the morphological gradient filter appropriately depending upon the local features of the analysed signal. Wavelet transform based adoptive algorithms are good at adjusting its parameters as per the fault signal requirement but at the same time it increases the complexity of already computationally demanding wavelet transform. Hilbert envelop is an alternative approach used for analysis of non-linear and non-stationary signal and it has been used widely for fault diagnostic in mechanical system [10e12]. Guo et al. [13] used Hilbert envelop spectrum for the feature extraction along with support vector machine (SVM) for classification of the bearing faults. However, the traditional Hilbert envelop fails to diagnose the early stage fault, whose fault feature is very weak and contaminated by noise [14]. HHT is developed for nonlinear and nonstationary data analysis by employing empirical mode decomposition (EMD) to the signal [15]. In EMD, the investigated signal is decomposed into a finite and a series of nearly mono-component modes with local characteristic time scales, and then the instantaneous frequency of each IMF is represented through Hilbert transform. HHT based algorithms have been widely used for the detection of bearing faults. Yu et al. [16] introduced local Hilbert spectrum and marginal local Hilbert spectrum for the fault diagnosis of rolling element bearings. For the marginal local Hilbert spectrum, the vibration signal is first translated in time scale by using wavelet bases, then the Hilbert transform to the envelope of the wavelet coefficients give the marginal local spectrum. HHT based methods are the most desirable methods among the time-frequencyeenergy based methods because of their computational efficiency and adaptive property. However, they have some limitations while detecting the bearing fault at incipient stage. The classical HHT suffers from end effects and redundant IMFs associated with EMD processes [17]. The energy leakage due to end effect is a serious concern because the fault signal energy is weak at the incipient stage. Recently, many schemes have been proposed to limit the end effect [18e20]. These schemes employ prediction approaches to extend each side of the signal for reducing the end effect. However, for bearing fault at incipient stage, the empirical extension of the signal cannot reflect the real feature of the weak fault signal. The fault in the bearing race generates quasi-periodic impulses. The information of impulse is present at all frequency range including at zero frequency. The impulse information can be obtained by passing a signal through a filter which retain only zero frequency component of the signal. A zero frequency resonator is a filter whose output contains impulse information in addition to DC components which grows/decays as a polynomial function of time. Hence, a zero frequency resonator can be used for impulse detection. Impulses in the bearing excite higher resonances of the structure. This gives rise to amplitude modulated signal where high frequency carrier signal is modulated by envelop of the periodic impulses. It is difficult to extract impulse like characteristic due to the bipolar nature (i.e. both positive and negative amplitudes) of amplitude modulated signal. An approximate localization of impulse can be obtained by exploiting unipolar nature of Hilbert envelop of the signal [21]. A more precise location of impulse can be obtained by passing Hilbert envelop of the signal to the zero frequency resonator. In this paper, a combined approach is proposed to detect the weak impulses using Hilbert envelope and zero frequency resonator. Working of the present algorithm is first explained using simulated signal. Thereafter, effectiveness of the algorithm is verified with real bearing vibration datasets for early stage detection of the bearing faults. A comparison of the proposed algorithm with HHT based method is also evaluated.
2. Zero frequency resonator and Hilbert envelop Brief introductions to zero frequency resonator and Hilbert envelop are presented in this section. The working of the zero frequency resonator and Hilbert envelop is explained using simulated signal. Periodic impulses are generated as the rolling element passes over localised faults on the inner or outer races. The impulses excite the high frequency resonances of the structures giving rise to amplitude modulated signal at the location of measurement. The bearing fault signal can be represented as [22]
xðnÞ ¼ bðnÞsinðw0 n þ ∅0 Þ;
ðn ¼ 0; …; N 1Þ
(1)
where, N is the signal length, bðnÞ is the periodic amplitude modulation signal, w0 is the carrier frequency and ∅0 is the phase of the carrier signal. In the next section, the basic principle of zero frequency resonator and Hilbert envelop to estimate periodic impulse is explained briefly with the help of a simulated signal.
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2.1. Zero frequency resonator A zero frequency resonator is a second order infinite impulse response (IIR) filter with a complex conjugate pair of poles located in unit circle [23e25]. The centre frequency of the resonator is chosen at zero Hz. The difference equation corresponding to zero frequency resonator is given by
y1 ½n ¼
2 X
ak y1 ½n k þ x½n
(2)
k¼1
and the corresponding transfer function is given by
H½z ¼
Y1 ½z 1 ¼ X½z 1 þ a1 z1 þ a2 z2
(3)
where, a1 ¼ 2 and a2 ¼ 1. Y1 ½z) and X½z are the z-transforms of filter output y1 ½n and filter input x½n; respectively. The pole of this transfer function lies on z ¼ 1. The output of zero frequency resonator is basically equivalent to two times successive integration of the input signal, as given in Eq. (4). Since the output of resonator is the cumulative summation of past two output samples and present input sample, the resonator output grows/decays approximately as a polynomial function of time.
y1 ½n ¼
n X l X
bðkÞsinðw0 k þ ∅0 Þ
(4)
l¼0 k¼0
Considering a zero frequency resonator, when impulses occur in the input signal, the output of the resonator gets fluctuation at the locations of impulses. However, these fluctuations in the output signal due to the impulse in the input signal are small in amplitude and these are overridden by large amplitude value of the filtered output. Therefore, it is difficult to extract the location of an impulse. Hence, a residual signal is obtained from the output of the resonator to extract the discontinuities due to impulses. The residual signal is computed by taking difference of the filtered output from its local mean. A window length is required to compute the local mean of the signal. Ideal length of the window will depend on the average duration between fluctuations in the output of the resonator. The window length should not be too short or too long. As, it may cause an increase in spurious zero crossing in the filtered output. The average duration between fluctuations can be obtained by any traditional pitch estimation technique such as autocorrelation method [26]. This technique performs well when signal strength is prominent. In case of weak signal, the autocorrelation technique may give a rough estimate of average duration between fluctuations. In the present work, the average duration between fluctuations is computed using autocorrelation method. Since the bearing fault signal is a non-stationary signal; the concept of long term autocorrelation function on this signal is not meaningful. Therefore, short-term autocorrelation function is defined on short segments of the signal. For the fault signal x(n), the short-term autocorrelation function can be written as,
rl ðmÞ ¼
0 1 1 NX ½xðn þ lÞwðnÞ½xðn þ l þ mÞwðn þ mÞ; N0 n¼0 0
0 m M0 1
(5) 0
where, w(n) is the window for analysis, N is the length of the window, i.e. N is the number of samples used in the computation of short-term autocorrelation function, l is the index of the starting sample of the frame and M0 is the number of autocorrelation points to be computed. Since the fault impulses are periodic in nature, the autocorrelation function will also be periodic. The first major peak after centre peak in the autocorrelation function indicates the fundamental period (T0 ) of the signal. For the present work, the average T0 (in samples) is computed from each segment of the fault signal and it is considered as the window length (2Lþ1) for computing the residual signal. The resulting residual signal is given by
y½n ¼ y1 ½n
m¼L X 1 ðy ½n þ mÞ 2L þ 1 m¼L 1
(6)
2.2. Impulse detection in simulated sequence Here, working of the zero frequency resonator is explained with the help of a simulated sequence. For this, a sequence consisting of periodic impulses is first passed through the zero frequency resonator and then the local mean of the sequence is subtracted from the output sequence of the zero frequency resonator. The subtracted output is termed as residual signal. Two cases are considered, (a) a sinusoid sequence modulated by periodic exponential impulses with a period of 50 samples, as
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discussed in Eq. (1), and (b) a sinusoid sequence modulated by periodic exponential impulses with a period of 50 samples and added with white noise. Case 1.
Sinusoidal sequence modulated with periodic exponential impulses
Fig. 1(a) shows a sequence consisting of a sinusoid modulated by periodic exponential impulses. The output of zero frequency resonator, given in Eq. (4), is shown in Fig. 1(b). From the figure, it can be seen that the output of the filter grows polynomial with time. In this case, the fluctuation in the resonator output occurs at every period of 50 samples. Therefore, 50 samples are considered as a window length for the local mean subtraction. The residual signal, given in Eq. (6), is shown in Fig. 1(c). It can be observed that the residual sequence has negative peaks corresponding to the impulse locations. Case 2.
White noise added to the modulated periodic exponential impulsive sequence
In this case, a sequence of white noise is added to the modulated periodic exponential impulse sequence given by,
b x ðnÞ ¼ bðnÞsinðw0 n þ ∅0 Þ þ dðnÞ
(7)
where, dðnÞ is N sample zero mean white noise sequence. It is assumed that the noise dðnÞ is uncorrelated with the amplitude modulated periodic impulsive signal. The noisy modulated exponential impulsive sequence, as shown in Fig. 2(a), is passed through the zero frequency resonator. The residual signal is shown in Fig. 2(c). Again the negative peaks are occurring at some instant of impulses. However, due to noise present in the sequence, some spurious negative peaks, shown in Fig. 2(c) by dotted arrows, are also observed in between successive impulses. Also, the output of the zero frequency resonator does not have negative peaks at some genuine impulse locations, shown in Fig. 2(c) by dotted circles. In this case, the genuine impulses are not observable easily due to more negative peaks in residual sequence. The number of spurious negative peaks need to be reduced. This can be achieved by utilising the unipolar nature of Hilbert envelop. Hilbert envelop of the signal will reduce the number of zero crossings due to noise and high frequency carrier signal which in turn will reduce the spurious negative peaks in the residual signal. Therefore Hilbert envelop of the input sequence is obtained to reduce the number of spurious zero crossings. Hilbert envelop of a signal is computed as follows: The complex analytic signal xa ½n corresponding to a real signal x[n] is defined as,
xa ½n ¼ x½n þ jxh ½n
(8)
where, xh ½n is the Hilbert transform of the x[n] and is given as
xh ½n ¼ IDFT½Xh ðwÞ
(9)
where, Xh ðwÞ is given as
Xh ðwÞ ¼
þjXðwÞ w < 0 jXðwÞ w > 0
(10)
and,
XðwÞ ¼ DFT½x½n
(11)
where, DFT and IDFT refer to discrete Fourier transform and inverse discrete Fourier transform, respectively.
Fig. 1. Output of zero frequency resonator (y-axis depicts amplitude of the signal and x-axis depicts duration of the signal in samples) (a) sinusoidal sequence modulated with periodic exponential impulses, (b) output of zero frequency resonator, and (c) residual sequence.
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Fig. 2. Output of zero frequency resonator for noisy periodic impulse sequence (y-axis depicts amplitude of the signal and x-axis depicts duration of the signal in samples). (a) noisy modulated exponential impulsive sequence, (b) output of zero frequency resonator and (c) residual signal.
The complex analytic signal contains only positive frequencies. The analytic signal can also be written as
xa ½n ¼ xa ½nejf½n
(12)
where jxa ½nj is called the amplitude envelop, and is given as,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xa ½n ¼ x2 ½n þ x2 ½n h
(13)
In the bearing fault signal, sinusoidal signal is modulated by the impulse signal. Therefore, the Hilbert transform of fault signal x½n given in Eq. (1) is given by,
xh ðnÞ ¼ bðnÞcosðw0 n þ ∅0 Þ
(14)
Hilbert envelop of the amplitude modulated signal eliminates the oscillations arising due to high frequency carrier signal. It helps in extracting the impulse like characteristics of the signal to be used as input to zero frequency resonator. Further, it eliminates the effect of carrier signal phase in accurate localization of impulses. In order to demonstrate this, consider an unfavourable phase, say ∅0 ¼ 0, of carrier signal at the instant of impulse occurrence. Assuming the impulse to occur at n ¼ 0, the situation can be simulated by putting n; ∅0 ¼ 0 in Eq. (1). At the instant of impulse occurrence, the fault signal x½n has sinð0Þ component which is equal to zero. Hence the signal magnitude at the instant of impulse will be zero. On the other hand, xh ðnÞ is the quadrature part of the input sequence contains cosð0Þ term which is unity. Therefore, envelop of the fault signal becomes jbðnÞj at the instant of impulse. For any other phase angle, ∅0 ¼ ∅1 , the Hilbert envelop of the signal is
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xa ðnÞ ¼ b2 ðnÞsin2 ðw0 n þ ∅1 Þ þ b2 ðnÞcos2 ðw0 n þ ∅1 Þ jxa ðnÞj ¼ jbðnÞj
(15)
From Eq. (15), it can be inferred that the envelop of the signal eliminates the effect of carrier signal phase in determining the location of impulse on time axis. It helps in two ways, first it increases the magnitude of the signal at the occurrence of impulse and secondly, localization of impulses on the time axis is better. Better localization on time axis will result in accurate estimation of impulse period. In case the input sequence xðnÞ is contaminated by uncorrelated noise dðnÞ, the analytic signal of noisy modulated periodic exponential impulse sequence can be given by Ref. [21]:
b x a ðnÞ ¼ xa ðnÞ þ da ðnÞ ¼ xa ðnÞ þ dðnÞ þ jdh ðnÞ
(16)
where, dh ðnÞ is the Hilbert transform of the uncorrelated noise sequence. Envelop of noisy modulated exponential signal obeys the triangle inequality [27], i.e.
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x a ðnÞj ¼ jxa ðnÞ þ da ðnÞj jxa ðnÞj þ jda ðnÞj jb
441
(17)
Therefore, envelop of the noisy modulated exponential periodic impulse can be written as,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x a ðnÞj ¼ bðnÞ þ d2 ðnÞ þ d2h ðnÞ jb
(18)
Since dh ðnÞ is the Hilbert transform of noise dðnÞ, which is basically the quadrature component of the noise dðnÞ, the Hilbert transform of a noise will also be a noise. It can also be observed in Fig. 3(b) with dotted circle. From Eqn. (18), envelop contains two components. First one is the slow varying and decaying impulse envelop which can be observed from Fig. 3(c) in solid circle. The second one is an envelope of noise which can be observed from Fig. 3(c) in dotted circle. An envelope normalization scheme is applied to the resulting Hilbert envelop of the input sequence in order to further reduce the effect of noise around the region of impulse and to increase the strength of impulse in the sequence [28]. For envelop normalization, a suitable window is chosen. The window length should be less than the average duration between the impulses around the sample. For bearing fault case, the fault frequency is approximately 236 Hz i. e. 4.5 msec approximately. Therefore, in the paper, 4.0 msec window length is kept to compute Normalised Hilbert envelop. The envelope normalization scheme is given by,
b x a ðnÞj x a ðnÞ ¼ jb c
1 2Mþ1
jb x a ðnÞj b m¼M j x a ðn þ mÞj
PM
(19)
Here 2Mþ1 samples, correspond to 4.0 ms, are required to compute the local mean of the enveloped signal i. e. M is taken to be 2*Fs=1000. In the normalised enveloped sequence, the strength of noise reduces around the impulse locations, as shown in Fig. 3(d) with dotted circles. The peakness of the impulses in the normalised envelope sequence is more compared to that in envelope sequence. Now for the processing of the sequence to extract the impulse period, first the normalised Hilbert envelope of the sequence is obtained from Eq. (19). The approximate locations of impulses are observable in the normalised Hilbert envelop, as shown in Fig. 4(b) by arrows. Thereafter, the Hilbert envelope of the sequence is passed through the zero frequency resonator. The residual sequence after local mean subtraction is shown in Fig. 4(d). By comparing Figs. 2(c) and 4(d), it can be seen that the spurious negative peaks are almost vanished and the resulting filter output contains negative peaks only at the location of impulses. The normalization of Hilbert envelope sequence increases the energy around the impulse at zero frequency. As a result, the identification of impulses is better when the zero frequency resonator is applied to the normalised Hilbert envelop.
Fig. 3. (a) Sequence of noisy modulated periodic impulse, (b) Hilbert transform of input sequence, (c) Hilbert envelop of the input sequence, and (d) normalised Hilbert enveloped sequence.
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Fig. 4. Output of zero frequency resonator after using Hilbert envelop (a) Sequence of noisy modulated exponential periodic impulse, (b) Hilbert envelop of the input sequence (c) output of zero frequency resonator and (d) residual sequence.
3. Algorithm for detection of bearing fault In this section, an algorithm based on Hilbert envelop and zero frequency resonator is proposed for the detection of bearing faults. The proposed method is applied for the detection of faults in rolling element bearings. 3.1. Experimental datasets for the bearing faults Working of the algorithm is explained with the help of real experimental dataset, obtained from Prognostics Center of Excellence (PCoE) contributed by Intelligent Maintenance System (IMS), University of Cincinnati [29]. Here, the bearing faults are not seeded. A healthy experimental setup is run up to bearing failure. The setup is shown in Fig. 5. Four double row bearings were installed on the shaft. The bearings had 16 rollers in each row, a pitch diameter of 2.815 in., roller diameter of 0.331 in., and a tapered contact angle of 15.17. Accelerometers were installed on the bearing housing for the measurement of vibration data. An AC motor was used to run the shaft at 2000 rpm. A radial load of 6000 lbs was applied onto the shaft. Sampling frequency of the measurements was 20 KHz. The vibration signals were collected every 20 min. The characteristics frequencies corresponding to the inner race defect, the rolling element defect, and the outer race defect are 296.93 Hz, 279.83 Hz, and 236.40 Hz, respectively [8]. The proposed algorithm is also verified with the Case Western Reserve University (CWRU) bearing dataset for the detection of seeded faults in rolling element [30e33]. Here the faults are introduced artificially using the electric discharge machining. Fault sizes ranging from 0.007 in. in diameter to 0.040 in. in diameter were introduced at the inner raceway, the rolling element (ball) and the outer raceway of the bearing. Only one fault is considered at a particular time. Accelerometer data, collected at 12 KHz sampling frequency, is used for the validation. The experimental setup is shown in Fig. 6. 3.2. The algorithm When localised fault occurs in a bearing, quasi-periodic impulses appear in time domain vibration signal. The vibration signal when passed through the zero frequency resonator will contain negative peaks corresponds to impulses due to fault. At the early stage, the strength of impulses due to the fault is weak and the fault signal is hidden in strong machinery vibration and hence the resonator output will also contain spurious negative peaks. In order to enhance the impulse like characteristic and to reduce spurious negative peaks, Hilbert envelop of the signal is obtained before passing the signal through zero frequency resonator. A block diagram of the proposed fault detection technique is shown in Fig. 7. First, difference of the fault signal is computed to remove any time varying low frequency bias in the signal. This signal is referred to as differenced signal subsequently. The proposed method consists of two steps. The first step determines normalised Hilbert Envelop of the signal to emphasise the impulse like characteristics and to improve the localization of impulses by eliminating the carrier frequency phase. The second step computes accurate fault location in time domain using zero frequency resonator. In the second step, the normalised Hilbert envelop is passed through the zero frequency resonator. Output of the zero frequency resonator is referred to
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Fig. 5. Bearing test rig for the IMS dataset [14].
Fig. 6. Bearing test rig for the CWRU dataset [30].
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Fig. 7. Block diagram of fault detection using Hilbert envelop and zero frequency resonator based method.
as filtered output. Now to extract the impulse locations, the residual signal is computed by subtracting the filtered output from its local mean, as given in Eq. (6). The differenced fault signal is segmented into short segments. Window length for the calculation of local mean is taken to be the average period T0 obtained from the short segments. The mean subtracted output signal is termed as residual signal. The spectrum of the residual signal is computed to estimate the faults characteristic frequencies. Fourier transform of a segment of 0.025 s of the residual signal is considered to compute the spectrum of the residual signal. Experiments with seeded faults do not mimic the natural defects at early stage [14]. Therefore the proposed algorithm is presented with IMS dataset where the bearing fault has grown naturally [8]. 3.3. Detection of serious defect To see the effect of Hilbert envelop, initially, the algorithm is applied without using the Hilbert envelop block in Fig. 7. The vibration signals at serious stage were collected at the end of the IMS bearing dataset [8]. The zero frequency resonator output and the residual signal are shown in Fig. 8(b) and (c), respectively. Figures show that the residual output signal contains negative peaks at regular intervals at the locations of impulses. However, these negative peaks are not observed easily due to the spurious negative peaks in between successive strong negative peaks. The multiple spurious negative peaks in residual output signal occurred due to the noise present in the signals. The spectrum of the residual signal is shown in Fig. 9. The strongest peak in the spectrum occurred at 228 Hz which is close to the characteristic frequency of outer race defect. The spectrum contains some more peaks at different frequencies. In order to reduce the effect of noise in the residual signal, now the Hilbert envelop of the signal is included in the algorithm. Hilbert envelop of the signal, shown in Fig. 10(b), contains prominent peaks at the impulse locations and spurious peaks due to noise are suppressed. Normalised Hilbert envelop of the signal is passed through zero frequency resonator and the output of the resonator is shown in Fig. 10(d). A comparison of residual signal from Figs. 8(c) and 10(d) shows that the use of Hilbert envelop in the algorithm reduces the number of spurious zero crossings. After including the Hilbert envelop in the algorithm, the negative peaks are observed only corresponding to the location of impulses occurring due to the fault. The spectrum corresponding to the residual output is shown in Fig. 11. The major peak in the spectrum is observed at 236 Hz, which is the characteristic frequency corresponding to outer race defect. Hence, the serious defect in the outer race is easily detected using the proposed algorithm. 3.4. Detection of early stage defect In the early stage, the vibration signal was collected nearly three days before the starting of serious defect [8]. The proposed algorithm, shown in Fig. 7, is applied to the vibration signal. The Hilbert envelop and the residual output of the signal are shown in Fig. 12(b) and (d), respectively. The impulses are weak during early stage defect. Therefore the strengths of impulses in Hilbert envelop of the signal (Fig. 12(b)) are not as prominent as in the case of serious defect stage (Fig. 10(b)).
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0.5
(a)
0 −0.5 0
200
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800
1000
1200
1400
1600
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2000
200
(b)
0 −200 0
200
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800
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1400
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2 1
(c)
0 −1 0 1
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(d)
0 −1 400
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Fig. 8. Output of zero frequency resonator at serious defect without including Hilbert envelop of the signal. Here, y-axis depicts amplitude of the signal and x-axis depicts duration of the signal (in samples). (a) Original fault signal, (b) filtered signal, (c) residual output and (d) zoomed portion of residual output.
60
228Hz
50
Magnitude
40
30
20
10
0 0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Frequency (in Hz)
Fig. 9. Spectrum of residual signal at serious defect without including the Hilbert envelop of the signal.
Although the strength of impulses are not prominent in Hilbert envelop of the signal, the residual signal detects negative peaks at the instants of impulses. The spectrum corresponding to the residual signal is shown in Fig. 13. The major peak in the spectrum is observed at 239 Hz which is close to the characteristic frequency of outer race defect. Hence, the proposed algorithm is detecting the serious stage as well as the early stage bearing faults.
4. Comparison with Hilbert-Huang transform In this section, the effectiveness of proposed technique to detect the fault signature is compared with the state of the art HHT technique. HHT is developed to decompose a signal into nearly mono component modes called as intrinsic mode function (IMF). Hilbert transform of each IMF gives instantaneous frequency and instantaneous amplitude. Instantaneous amplitude of the corresponding IMFs gives signal energy distribution in time domain with diversified scale.
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Fig. 10. Output of zero frequency resonator at serious defect after including Hilbert envelop of the signal. (a) Original fault signal, (b) Hilbert envelop of the signal, (c) filtered signal and (d) residual output.
Fig. 11. Spectrum of residual signal at serious defect after including the Hilbert envelop of the signal.
4.1. Comparison using IMS dataset Instantaneous amplitude of each IMF of the fault signal at serious stage (as described in Section 3.3) is computed and these are shown in Fig. 14. The instantaneous amplitude of first IMF shows the periodic impulses. Fourier transform of a segment of 0.025 s of the instantaneous amplitude is considered to compute the spectrum of first four IMFs and these are shown in Fig. 15. The spectrums of the first two IMFs show peaks at 236 Hz and 228 Hz, respectively. By comparing Figs. 11 and 15(a), it can be observed that both the methods, the HHT as well as the proposed method, have the ability to detect the fault impulse signature when the fault is at serious stage. Now, the two methods are compared for early stage detection of bearing defects (as described in Section 3.4). The Instantaneous amplitude of IMF and the spectrum of instantaneous amplitude, for early stage of fault, are shown in Figs. 16 and 17, respectively. Fig. 16(a) shows periodic impulses in the instantaneous amplitude of first IMF. However, the impulses at early stage is comparatively less observable in comparison to the impulses at serious stage defect, as shown in Fig. 14(a).
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Fig. 12. Output of zero frequency resonator at early defect. (a) Original fault signal, (b) Hilbert envelop of the signal, (c) filtered signal, and (d) residual signal.
Fig. 13. Spectrum of residual signal at early stage defect.
By comparing Figs. 13 and 17(a), it can be observed that both the techniques are detecting the bearing defects at early stage. However, the strength of peak at the fault characteristic frequency in case of HHT based method is less prominent compared to that in case of proposed method, based on zero frequency resonator and Hilbert envelop. In order to find out the consistency of the proposed method in detecting early stage fault, the fault identification algorithm is evaluated on several vibration signals taken at early stage of the fault. Approximately 200 vibration signals, starting from three days before the serious fault i.e., 4th day of bearing test [8], are considered for the analysis. Performance of the two methods in detecting the fault frequency from these vibration signals is shown in Fig. 18. The continuous line is the reference line corresponding to the fault frequency. The dotted line in Fig. 18(a) shows the performance of the present method and the dotted line in Fig. 18(b) shows the performance of HHT method. When the fault has grown a little, both the methods are consistent in detecting the fault frequency. However, the proposed method is more consistent in detecting the fault frequency at the incipient stage of the fault, as the dotted line is more close to the reference fault frequency line in Fig. 18(a) compared to that in Fig. 18(b). 4.2. Verification of the proposed algorithm for seeded fault using the CWRU bearing dataset Now the algorithm is verified with the Case Western Reserve University bearing dataset. Here the faults are seeded. The algorithm is verified for the inner race fault, the outer race fault and the ball fault. For the inner race and outer race faults,
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Fig. 14. Instantaneous amplitude of IMFs of the fault signal at serious stage of fault. (a) to (i) is for IMF1 to IMF9, respectively.
Fig. 15. Spectrum of instantaneous amplitude of IMFs of the fault signal at serious stage of fault. (a) to (d) is for IMF1 to IMF4, respectively.
diameter and depth of the faults are 0.007 in. and 0.011 in., respectively (record 106DE for the inner race fault and record 131 DE for the outer race fault). Diameter and depth for the ball fault are 0.028 in. and 0.150 in., respectively (record 3008 DE). The shaft speed varies from 1797 rpm to 1730 rpm depending upon the loads applied (0e3 hp). The characteristic fault frequencies corresponding to the rolling element defect, the inner race defect and the outer race defect for the selected conditions are 68 Hz, 148 Hz, and 90.3 Hz, respectively. Fig. 19(a) and (b) shows the spectrum for the rolling element fault in the bearing using the HHT and the proposed algorithm, respectively. For the rolling element fault, the even harmonics of ball spin frequency (BSF) are often dominant in the spectra [32]. BSF is the frequency with which the fault strikes the same race (inner or outer), so that in general there are two shocks for one rotation of the ball. One corresponds to its impact with the inner race and the other corresponds to its impact with the outer race. Therefore, the peak at fault frequency is missing in both the spectrums for the case of ball fault (Fig. 19(a) and (b)). However, the second harmonic can be observed in the two spectrums. The present algorithm shows a clear dominant peak at the second harmonic. For the inner and outer race fault detection also (Fig. 19(c)e(f)), considering a single dominant peak at characteristics fault frequency, the detection is better in case of the present algorithm.
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Fig. 16. Instantaneous amplitude of IMFs of the fault signal at early stage of fault. (a) to (i) is for IMF1 to IMF9, respectively.
Fig. 17. Spectrum of Instantaneous amplitude of IMFs of the fault signal at early stage of fault.
The proposed algorithm is capable of detecting periodic impulses in noisy conditions. Such conditions exist in case of bearing and gear faults of a rotating machinery. The proposed method can also be applied in biomedical signal such as in electrocardiogram (ECG) and phonocardiogram (PCG) signal, where signal contains impulses at the locations of heart sound. In case of civil structures, such as bridges and aircraft wings, impulse like characteristics due to slope discontinuity can be observed in the spatial second derivatives at the locations of localised damages. 5. Conclusions A new technique is proposed for the detection of localised bearing faults at early stage. The technique is based upon passing the Hilbert envelop of the signal through the zero frequency resonator to detect the weak impulses due to localised bearing defects. The unipolar nature of Hilbert envelop facilitates the extraction of impulse like characteristics of the signal. Additionally, Hilbert envelop of the signal reduces the effect of carrier frequency phase during localization of impulses, thus giving more accurate estimation of the impulse period. The zero frequency resonator locates the local discontinuity in the signal arising due to the impulses. The proposed method is tested with bearing vibration data. The method identifies very well
K. Kumar et al. / Journal of Sound and Vibration 419 (2018) 436e451
Frequency (Hz)
Frequency (Hz)
450
1000 Fault frequency using Proposed method Reference fault frequency at 236 Hz 500
0
(a)
4/3
4/6
4/9
1000
4/13 4/16 4/19 days/hours------>
4/23
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Fault frequency using HHT method Reference fault frequency at 236 Hz 500
0
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4/9
4/13 4/16 4/19 days/hours------>
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Fig. 18. Detection of fault signature in vibration signals at early stage, (a) using proposed method, and (b) using HHT method.
Fig. 19. Bearing fault identification using Case Western Reserve University Bearing Dataset, (a), (c), (e) spectra of HHT for ball fault, inner race fault and outer race fault, respectively, and (b), (d), (f) spectra of proposed algorithm for ball fault, inner race fault and outer race fault, respectively.
the bearing defects at early stage. The proposed algorithm has the merit that it does not involve any optimisation of parameters and gives a single dominant peak at the fault frequency. This will enable for an easy adaptation of the algorithm by machine operators.
Acknowledgment This work is supported by research grant from Science & Engineering Research Board India, through project File No. YSS/ 2015/001555.
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Appendix A. Supplementary data Supplementary data related to this article can be found at https://doi.org/10.1016/j.jsv.2018.01.022. References [1] Y.T. Su, S.J. Lin, On initial fault detection of a tapered roller bearing: frequency domain analysis, J. Sound Vib. 155 (1) (1992) 75e84. [2] C. Kar, A.R. Mohanty, Vibration and current transients monitoring for gearbox fault detection using multiresolution Fourier transform, J. Sound Vib. 311 (1e2) (2008) 109e132. [3] P. Shakya, M.S. Kulkarni, A.K. Darpe, Bearing diagnosis based on MahalanobiseTaguchieGrameSchmidt method, J. Sound Vib. 337 (2015) 342e362. [4] X. Chiementin, F. Bolaers, J.P. Dron, Early detection of fatigue damage on rolling element bearings using adapted wavelet, J. Vib. Acoust. 129 (2007) 495e506. [5] N.H. Chandra, A.S. Sekhar, Fault detection in rotor bearing systems using time frequency techniques, Mech. Syst. Signal Process. 72e73 (2016) 105e133. [6] V.K. Rai, A.R. Mohanty, Bearing fault diagnosis using FFT of intrinsic mode functions in HilberteHuang transform, Mech. Syst. Signal Process. 21 (2007) 2607e2615. [7] C. Mishra, A.K. Samantaray, G. Chakraborty, Rolling element bearing fault diagnosis under slow speed operation using wavelet de-noising, measurement 103 (2017) 77e86. [8] C. Zhang, B. Li, B. Chen, H. Cao, Y. Zi, Z. He, Weak fault signature extraction of rotating machinery using flexible analytic wavelet transform, Mech. Syst. Signal Process. 64e65 (2015) 162e187. [9] B. Li, P. Zhang, S. Mi, R. Huc, D. Liu, An adaptive morphological gradient lifting wavelet for detecting bearing defects, Mech. Syst. Signal Process. 29 (2012) 415e427. [10] R.B. Randall, J. Antoni, Rolling element bearing diagnostics- A tutorial, Mech. Syst. Signal Process. 25 (2011) 485e520. [11] M. Feldman, Hilbert transform in vibration analysis, Mech. Syst. Signal Process. 25 (2011) 735e802. [12] S.K. Singh, S. Shukla, Weak Signature Extraction of Bearing Fault Using Hilbert Envelop, VIRM 11, Manchester, UK, 2016, pp. 585e592. [13] L. Guo, J. Chen, X. Li, Rolling bearing fault classification based on envelope spectrum and support vector machine, J. Vib. Control 15 (2009) 1349e1363. [14] H. Qiu, J. Lee, J. Lin, G. Yu, Wavelet filter based weak signature detection method and its application on rolling element bearing prognostics, J. Sound Vib. 289 (2006) 1066e1090. [15] N.E. Huang, Z. Shen, S.R. Long, M.C. Wu, H.H. Shih, Q. Zheng, N.C. Yen, C.C. Tung, H.H. Liu, The Empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, Proceed. R. Soc. Lond. Ser. A 454 (1998) 903e995. [16] D. Yu, J. Cheng, Y. Yang, Application of EMD method and Hilbert spectrum to the fault diagnosis of roller bearings, Mech. Syst. Signal Process. 19 (2005) 259e270. [17] Z.K. Peng, Peter W. Tse, F.L. Chu, A comparison study of improved HilberteHuang transform and wavelet transform: application to fault diagnosis for rolling bearing, Mech. Syst. Signal Process. 19 (2005) 974e988. s, On empirical mode decomposition and its algorithms, in: Proceedings of IEEE EURASIP Workshop on Nonlinear [18] G. Rilling, P. Flandrin, P. Gonçalve Signal and Image Processing NSIP, Grado, Italy, June, 2003. [19] J. Xun, S. Yan, A revised HilberteHuang transformation based on the neural networks and its application in vibration signal analysis of a deployable structure, Mech. Syst. Signal Process. 22 (2008) 1705e1723. [20] J. Cheng, D. Yu, Y. Yang, Application of support vector regression machines to the processing of end effects of HilberteHuang transform, Mech. Syst. Signal Process. 21 (2007) 1197e1211. [21] A.V. Oppenheim, R.W. Schafer, Digital Signal Processing, Prentice-Hall, Englewood Cliffs, NJ, 1975. Ch-7. [22] J. Urbanek, J. Antoni, T. Barszcz, Detection of signal component modulations using modulation intensity distribution, Mech. Syst. Signal Process. 28 (2012) 399e413. [23] B. Yegnanarayana, K.S.R. Murty, Event-based instantaneous fundamental frequency estimation from speech signals, IEEE Trans. Audio Speech Lang. Process. 17 (2009) 614e624. [24] K.S.R. Murty, B. Yegnanarayana, Epoch extraction from speech signals, IEEE Trans. Audio Speech Lang. Process. 16 (2008) 1602e1613. [25] A. Oppenheim, R. Schafer, Discrete-time Signal Processing, second ed., Prentice Hall, Englewood Cliffs, New Jersey, 1999, pp. 382e386. [26] L.R. Rabiner, On the use of autocorrelation analysis for pitch detection, IEEE Trans. Acoust. Speech Signal Process. 25 (1) (1977) 24e33. [27] T.M. Apostol, One-variable Calculus with an Introduction to Linear Algebra, second ed., vol. 1, Blaisdell, Waltham, MA, 1967, p. 42. [28] B. Yegnanarayana, R.K. Swamy, K.S.R. Murty, Determining mixing parameters from multispeaker data using speech-specific information, IEEE Trans. Audio Speech Lang. Process. 17 (6) (2009) 1196e1207. [29] J. Lee, H. Qiu, G. Yu, J. Lin, and Rexnord Technical Services (2007). IMS, University of Cincinnati. “Bearing Data Set”, NASA Ames Prognostics Data Repository (http://data-acoustics.com/measurements/bearing-faults/bearing-4/), NASA Ames Research Center, Moffett Field, CA. [30] Case Western Reserve University Bearing Data Center Website http://csegroups.case.edu/bearingdatacenter/home. [31] W.A. Smith, R.B. Randall, Rolling element bearing diagnostics using the Case Western Reserve University data: a benchmark study, Mech. Syst. Signal Process. 64e65 (2015) 100e131. [32] R.B. Randall, Vibration-based Condition Monitoring, Industrial, Aerospace, and Automotive Applications, first ed., John Wiley & Sons, Ltd, Chichester, United Kingdom, 2011, pp. 47e49. [33] H. Al-Bugharbee, I. Trendafilova, A fault diagnosis methodology for rolling element bearings based on advanced signal pretreatment and autoregressive modelling, J. Sound Vib. 369 (2016) 246e265.
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
A combined approach for weak fault signature extraction of rolling element bearing using Hilbert envelop and zero frequency resonator Keshav Kumar a, Sumitra Shukla b, Sachin Kumar Singh a, * a b
Department of Mechanical Engineering, Indian Institute of Technology (ISM) Dhanbad, Dhanbad, Jharkhand, 826004, India Department of Electronics Engineering, Indian Institute of Technology (ISM) Dhanbad, Dhanbad, Jharkhand, 826004, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 14 April 2017 Received in revised form 9 December 2017 Accepted 9 January 2018
Periodic impulses arise due to localised defects in rolling element bearing. At the early stage of defects, the weak impulses are immersed in strong machinery vibration. This paper proposes a combined approach based upon Hilbert envelop and zero frequency resonator for the detection of the weak periodic impulses. In the first step, the strength of impulses is increased by taking normalised Hilbert envelop of the signal. It also helps in better localization of these impulses on time axis. In the second step, Hilbert envelope of the signal is passed through the zero frequency resonator for the exact localization of the periodic impulses. Spectrum of the resonator output gives peak at the fault frequency. Simulated noisy signal with periodic impulses is used to explain the working of the algorithm. The proposed technique is verified with experimental data also. A comparison of the proposed method with Hilbert-Haung transform (HHT) based method is presented to establish the effectiveness of the proposed method. © 2018 Elsevier Ltd. All rights reserved.
Keywords: Bearing fault Weak signal detection Zero frequency resonator Hilbert envelop
1. Introduction Rolling element bearings are used widely in rotating machinery. Detection of bearing defects at early stage is important for scheduling the preventive maintenance and hence to ensure the safe running of machinery. The vibration signal measured from the bearing contains the fault signature. Periodic impulses are generated due to localised faults in bearing races. These impulses occur due to passing of rolling element over the localised faults. Detection of these periodic impulses is relatively easier for a grown up defect. However, at the early stage of the defect, the amplitude of the impulses is weak and the impulses are buried in the strong machinery vibration making it difficult to be detected. Hence, an appropriate signal processing method is required to extract the fault signature at early stage. FFT based methods are used extensively to establish the presence of fault signatures in the spectrum [1,2]. However, when the fault signal is weak, the FFT based methods are unable to capture the transient features of the non-stationary weak fault signal. Several time-frequency-energy based approaches have been proposed to detect the bearing faults at early stage. Among these, wavelet transforms (WT) and HHT based approaches are predominant [3e7].
* Corresponding author. E-mail addresses: [email protected] (K. Kumar), [email protected] (S. Shukla), [email protected] (S.K. Singh). https://doi.org/10.1016/j.jsv.2018.01.022 0022-460X/© 2018 Elsevier Ltd. All rights reserved.
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In wavelet transform, the signal is decomposed at different levels to capture the transient features of the weak fault signal. Successful feature extraction using wavelet transform depends upon the location and shape matches between the wavelet bases and the fault signal. Hence, selection of proper wavelet parameters depends upon fault signal characteristics. In this sense the wavelet transform in not adoptive. However, recently some adoptive wavelet based methods have been proposed. Zhang et al. [8] proposed an adoptive methodology using the flexible analytical wavelet transform (FAWT) for the extraction of weak fault signature. The optimal parameters for the FAWT were obtained by optimising the characteristic of kurtosis spectral entropy using genetic algorithms. Li et al. [9] proposed an adaptive morphological gradient lifting wavelet for the detection of bearing faults. The method was based upon selecting the average filter and the morphological gradient filter appropriately depending upon the local features of the analysed signal. Wavelet transform based adoptive algorithms are good at adjusting its parameters as per the fault signal requirement but at the same time it increases the complexity of already computationally demanding wavelet transform. Hilbert envelop is an alternative approach used for analysis of non-linear and non-stationary signal and it has been used widely for fault diagnostic in mechanical system [10e12]. Guo et al. [13] used Hilbert envelop spectrum for the feature extraction along with support vector machine (SVM) for classification of the bearing faults. However, the traditional Hilbert envelop fails to diagnose the early stage fault, whose fault feature is very weak and contaminated by noise [14]. HHT is developed for nonlinear and nonstationary data analysis by employing empirical mode decomposition (EMD) to the signal [15]. In EMD, the investigated signal is decomposed into a finite and a series of nearly mono-component modes with local characteristic time scales, and then the instantaneous frequency of each IMF is represented through Hilbert transform. HHT based algorithms have been widely used for the detection of bearing faults. Yu et al. [16] introduced local Hilbert spectrum and marginal local Hilbert spectrum for the fault diagnosis of rolling element bearings. For the marginal local Hilbert spectrum, the vibration signal is first translated in time scale by using wavelet bases, then the Hilbert transform to the envelope of the wavelet coefficients give the marginal local spectrum. HHT based methods are the most desirable methods among the time-frequencyeenergy based methods because of their computational efficiency and adaptive property. However, they have some limitations while detecting the bearing fault at incipient stage. The classical HHT suffers from end effects and redundant IMFs associated with EMD processes [17]. The energy leakage due to end effect is a serious concern because the fault signal energy is weak at the incipient stage. Recently, many schemes have been proposed to limit the end effect [18e20]. These schemes employ prediction approaches to extend each side of the signal for reducing the end effect. However, for bearing fault at incipient stage, the empirical extension of the signal cannot reflect the real feature of the weak fault signal. The fault in the bearing race generates quasi-periodic impulses. The information of impulse is present at all frequency range including at zero frequency. The impulse information can be obtained by passing a signal through a filter which retain only zero frequency component of the signal. A zero frequency resonator is a filter whose output contains impulse information in addition to DC components which grows/decays as a polynomial function of time. Hence, a zero frequency resonator can be used for impulse detection. Impulses in the bearing excite higher resonances of the structure. This gives rise to amplitude modulated signal where high frequency carrier signal is modulated by envelop of the periodic impulses. It is difficult to extract impulse like characteristic due to the bipolar nature (i.e. both positive and negative amplitudes) of amplitude modulated signal. An approximate localization of impulse can be obtained by exploiting unipolar nature of Hilbert envelop of the signal [21]. A more precise location of impulse can be obtained by passing Hilbert envelop of the signal to the zero frequency resonator. In this paper, a combined approach is proposed to detect the weak impulses using Hilbert envelope and zero frequency resonator. Working of the present algorithm is first explained using simulated signal. Thereafter, effectiveness of the algorithm is verified with real bearing vibration datasets for early stage detection of the bearing faults. A comparison of the proposed algorithm with HHT based method is also evaluated.
2. Zero frequency resonator and Hilbert envelop Brief introductions to zero frequency resonator and Hilbert envelop are presented in this section. The working of the zero frequency resonator and Hilbert envelop is explained using simulated signal. Periodic impulses are generated as the rolling element passes over localised faults on the inner or outer races. The impulses excite the high frequency resonances of the structures giving rise to amplitude modulated signal at the location of measurement. The bearing fault signal can be represented as [22]
xðnÞ ¼ bðnÞsinðw0 n þ ∅0 Þ;
ðn ¼ 0; …; N 1Þ
(1)
where, N is the signal length, bðnÞ is the periodic amplitude modulation signal, w0 is the carrier frequency and ∅0 is the phase of the carrier signal. In the next section, the basic principle of zero frequency resonator and Hilbert envelop to estimate periodic impulse is explained briefly with the help of a simulated signal.
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2.1. Zero frequency resonator A zero frequency resonator is a second order infinite impulse response (IIR) filter with a complex conjugate pair of poles located in unit circle [23e25]. The centre frequency of the resonator is chosen at zero Hz. The difference equation corresponding to zero frequency resonator is given by
y1 ½n ¼
2 X
ak y1 ½n k þ x½n
(2)
k¼1
and the corresponding transfer function is given by
H½z ¼
Y1 ½z 1 ¼ X½z 1 þ a1 z1 þ a2 z2
(3)
where, a1 ¼ 2 and a2 ¼ 1. Y1 ½z) and X½z are the z-transforms of filter output y1 ½n and filter input x½n; respectively. The pole of this transfer function lies on z ¼ 1. The output of zero frequency resonator is basically equivalent to two times successive integration of the input signal, as given in Eq. (4). Since the output of resonator is the cumulative summation of past two output samples and present input sample, the resonator output grows/decays approximately as a polynomial function of time.
y1 ½n ¼
n X l X
bðkÞsinðw0 k þ ∅0 Þ
(4)
l¼0 k¼0
Considering a zero frequency resonator, when impulses occur in the input signal, the output of the resonator gets fluctuation at the locations of impulses. However, these fluctuations in the output signal due to the impulse in the input signal are small in amplitude and these are overridden by large amplitude value of the filtered output. Therefore, it is difficult to extract the location of an impulse. Hence, a residual signal is obtained from the output of the resonator to extract the discontinuities due to impulses. The residual signal is computed by taking difference of the filtered output from its local mean. A window length is required to compute the local mean of the signal. Ideal length of the window will depend on the average duration between fluctuations in the output of the resonator. The window length should not be too short or too long. As, it may cause an increase in spurious zero crossing in the filtered output. The average duration between fluctuations can be obtained by any traditional pitch estimation technique such as autocorrelation method [26]. This technique performs well when signal strength is prominent. In case of weak signal, the autocorrelation technique may give a rough estimate of average duration between fluctuations. In the present work, the average duration between fluctuations is computed using autocorrelation method. Since the bearing fault signal is a non-stationary signal; the concept of long term autocorrelation function on this signal is not meaningful. Therefore, short-term autocorrelation function is defined on short segments of the signal. For the fault signal x(n), the short-term autocorrelation function can be written as,
rl ðmÞ ¼
0 1 1 NX ½xðn þ lÞwðnÞ½xðn þ l þ mÞwðn þ mÞ; N0 n¼0 0
0 m M0 1
(5) 0
where, w(n) is the window for analysis, N is the length of the window, i.e. N is the number of samples used in the computation of short-term autocorrelation function, l is the index of the starting sample of the frame and M0 is the number of autocorrelation points to be computed. Since the fault impulses are periodic in nature, the autocorrelation function will also be periodic. The first major peak after centre peak in the autocorrelation function indicates the fundamental period (T0 ) of the signal. For the present work, the average T0 (in samples) is computed from each segment of the fault signal and it is considered as the window length (2Lþ1) for computing the residual signal. The resulting residual signal is given by
y½n ¼ y1 ½n
m¼L X 1 ðy ½n þ mÞ 2L þ 1 m¼L 1
(6)
2.2. Impulse detection in simulated sequence Here, working of the zero frequency resonator is explained with the help of a simulated sequence. For this, a sequence consisting of periodic impulses is first passed through the zero frequency resonator and then the local mean of the sequence is subtracted from the output sequence of the zero frequency resonator. The subtracted output is termed as residual signal. Two cases are considered, (a) a sinusoid sequence modulated by periodic exponential impulses with a period of 50 samples, as
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discussed in Eq. (1), and (b) a sinusoid sequence modulated by periodic exponential impulses with a period of 50 samples and added with white noise. Case 1.
Sinusoidal sequence modulated with periodic exponential impulses
Fig. 1(a) shows a sequence consisting of a sinusoid modulated by periodic exponential impulses. The output of zero frequency resonator, given in Eq. (4), is shown in Fig. 1(b). From the figure, it can be seen that the output of the filter grows polynomial with time. In this case, the fluctuation in the resonator output occurs at every period of 50 samples. Therefore, 50 samples are considered as a window length for the local mean subtraction. The residual signal, given in Eq. (6), is shown in Fig. 1(c). It can be observed that the residual sequence has negative peaks corresponding to the impulse locations. Case 2.
White noise added to the modulated periodic exponential impulsive sequence
In this case, a sequence of white noise is added to the modulated periodic exponential impulse sequence given by,
b x ðnÞ ¼ bðnÞsinðw0 n þ ∅0 Þ þ dðnÞ
(7)
where, dðnÞ is N sample zero mean white noise sequence. It is assumed that the noise dðnÞ is uncorrelated with the amplitude modulated periodic impulsive signal. The noisy modulated exponential impulsive sequence, as shown in Fig. 2(a), is passed through the zero frequency resonator. The residual signal is shown in Fig. 2(c). Again the negative peaks are occurring at some instant of impulses. However, due to noise present in the sequence, some spurious negative peaks, shown in Fig. 2(c) by dotted arrows, are also observed in between successive impulses. Also, the output of the zero frequency resonator does not have negative peaks at some genuine impulse locations, shown in Fig. 2(c) by dotted circles. In this case, the genuine impulses are not observable easily due to more negative peaks in residual sequence. The number of spurious negative peaks need to be reduced. This can be achieved by utilising the unipolar nature of Hilbert envelop. Hilbert envelop of the signal will reduce the number of zero crossings due to noise and high frequency carrier signal which in turn will reduce the spurious negative peaks in the residual signal. Therefore Hilbert envelop of the input sequence is obtained to reduce the number of spurious zero crossings. Hilbert envelop of a signal is computed as follows: The complex analytic signal xa ½n corresponding to a real signal x[n] is defined as,
xa ½n ¼ x½n þ jxh ½n
(8)
where, xh ½n is the Hilbert transform of the x[n] and is given as
xh ½n ¼ IDFT½Xh ðwÞ
(9)
where, Xh ðwÞ is given as
Xh ðwÞ ¼
þjXðwÞ w < 0 jXðwÞ w > 0
(10)
and,
XðwÞ ¼ DFT½x½n
(11)
where, DFT and IDFT refer to discrete Fourier transform and inverse discrete Fourier transform, respectively.
Fig. 1. Output of zero frequency resonator (y-axis depicts amplitude of the signal and x-axis depicts duration of the signal in samples) (a) sinusoidal sequence modulated with periodic exponential impulses, (b) output of zero frequency resonator, and (c) residual sequence.
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Fig. 2. Output of zero frequency resonator for noisy periodic impulse sequence (y-axis depicts amplitude of the signal and x-axis depicts duration of the signal in samples). (a) noisy modulated exponential impulsive sequence, (b) output of zero frequency resonator and (c) residual signal.
The complex analytic signal contains only positive frequencies. The analytic signal can also be written as
xa ½n ¼ xa ½nejf½n
(12)
where jxa ½nj is called the amplitude envelop, and is given as,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xa ½n ¼ x2 ½n þ x2 ½n h
(13)
In the bearing fault signal, sinusoidal signal is modulated by the impulse signal. Therefore, the Hilbert transform of fault signal x½n given in Eq. (1) is given by,
xh ðnÞ ¼ bðnÞcosðw0 n þ ∅0 Þ
(14)
Hilbert envelop of the amplitude modulated signal eliminates the oscillations arising due to high frequency carrier signal. It helps in extracting the impulse like characteristics of the signal to be used as input to zero frequency resonator. Further, it eliminates the effect of carrier signal phase in accurate localization of impulses. In order to demonstrate this, consider an unfavourable phase, say ∅0 ¼ 0, of carrier signal at the instant of impulse occurrence. Assuming the impulse to occur at n ¼ 0, the situation can be simulated by putting n; ∅0 ¼ 0 in Eq. (1). At the instant of impulse occurrence, the fault signal x½n has sinð0Þ component which is equal to zero. Hence the signal magnitude at the instant of impulse will be zero. On the other hand, xh ðnÞ is the quadrature part of the input sequence contains cosð0Þ term which is unity. Therefore, envelop of the fault signal becomes jbðnÞj at the instant of impulse. For any other phase angle, ∅0 ¼ ∅1 , the Hilbert envelop of the signal is
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xa ðnÞ ¼ b2 ðnÞsin2 ðw0 n þ ∅1 Þ þ b2 ðnÞcos2 ðw0 n þ ∅1 Þ jxa ðnÞj ¼ jbðnÞj
(15)
From Eq. (15), it can be inferred that the envelop of the signal eliminates the effect of carrier signal phase in determining the location of impulse on time axis. It helps in two ways, first it increases the magnitude of the signal at the occurrence of impulse and secondly, localization of impulses on the time axis is better. Better localization on time axis will result in accurate estimation of impulse period. In case the input sequence xðnÞ is contaminated by uncorrelated noise dðnÞ, the analytic signal of noisy modulated periodic exponential impulse sequence can be given by Ref. [21]:
b x a ðnÞ ¼ xa ðnÞ þ da ðnÞ ¼ xa ðnÞ þ dðnÞ þ jdh ðnÞ
(16)
where, dh ðnÞ is the Hilbert transform of the uncorrelated noise sequence. Envelop of noisy modulated exponential signal obeys the triangle inequality [27], i.e.
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x a ðnÞj ¼ jxa ðnÞ þ da ðnÞj jxa ðnÞj þ jda ðnÞj jb
441
(17)
Therefore, envelop of the noisy modulated exponential periodic impulse can be written as,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x a ðnÞj ¼ bðnÞ þ d2 ðnÞ þ d2h ðnÞ jb
(18)
Since dh ðnÞ is the Hilbert transform of noise dðnÞ, which is basically the quadrature component of the noise dðnÞ, the Hilbert transform of a noise will also be a noise. It can also be observed in Fig. 3(b) with dotted circle. From Eqn. (18), envelop contains two components. First one is the slow varying and decaying impulse envelop which can be observed from Fig. 3(c) in solid circle. The second one is an envelope of noise which can be observed from Fig. 3(c) in dotted circle. An envelope normalization scheme is applied to the resulting Hilbert envelop of the input sequence in order to further reduce the effect of noise around the region of impulse and to increase the strength of impulse in the sequence [28]. For envelop normalization, a suitable window is chosen. The window length should be less than the average duration between the impulses around the sample. For bearing fault case, the fault frequency is approximately 236 Hz i. e. 4.5 msec approximately. Therefore, in the paper, 4.0 msec window length is kept to compute Normalised Hilbert envelop. The envelope normalization scheme is given by,
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Here 2Mþ1 samples, correspond to 4.0 ms, are required to compute the local mean of the enveloped signal i. e. M is taken to be 2*Fs=1000. In the normalised enveloped sequence, the strength of noise reduces around the impulse locations, as shown in Fig. 3(d) with dotted circles. The peakness of the impulses in the normalised envelope sequence is more compared to that in envelope sequence. Now for the processing of the sequence to extract the impulse period, first the normalised Hilbert envelope of the sequence is obtained from Eq. (19). The approximate locations of impulses are observable in the normalised Hilbert envelop, as shown in Fig. 4(b) by arrows. Thereafter, the Hilbert envelope of the sequence is passed through the zero frequency resonator. The residual sequence after local mean subtraction is shown in Fig. 4(d). By comparing Figs. 2(c) and 4(d), it can be seen that the spurious negative peaks are almost vanished and the resulting filter output contains negative peaks only at the location of impulses. The normalization of Hilbert envelope sequence increases the energy around the impulse at zero frequency. As a result, the identification of impulses is better when the zero frequency resonator is applied to the normalised Hilbert envelop.
Fig. 3. (a) Sequence of noisy modulated periodic impulse, (b) Hilbert transform of input sequence, (c) Hilbert envelop of the input sequence, and (d) normalised Hilbert enveloped sequence.
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Fig. 4. Output of zero frequency resonator after using Hilbert envelop (a) Sequence of noisy modulated exponential periodic impulse, (b) Hilbert envelop of the input sequence (c) output of zero frequency resonator and (d) residual sequence.
3. Algorithm for detection of bearing fault In this section, an algorithm based on Hilbert envelop and zero frequency resonator is proposed for the detection of bearing faults. The proposed method is applied for the detection of faults in rolling element bearings. 3.1. Experimental datasets for the bearing faults Working of the algorithm is explained with the help of real experimental dataset, obtained from Prognostics Center of Excellence (PCoE) contributed by Intelligent Maintenance System (IMS), University of Cincinnati [29]. Here, the bearing faults are not seeded. A healthy experimental setup is run up to bearing failure. The setup is shown in Fig. 5. Four double row bearings were installed on the shaft. The bearings had 16 rollers in each row, a pitch diameter of 2.815 in., roller diameter of 0.331 in., and a tapered contact angle of 15.17. Accelerometers were installed on the bearing housing for the measurement of vibration data. An AC motor was used to run the shaft at 2000 rpm. A radial load of 6000 lbs was applied onto the shaft. Sampling frequency of the measurements was 20 KHz. The vibration signals were collected every 20 min. The characteristics frequencies corresponding to the inner race defect, the rolling element defect, and the outer race defect are 296.93 Hz, 279.83 Hz, and 236.40 Hz, respectively [8]. The proposed algorithm is also verified with the Case Western Reserve University (CWRU) bearing dataset for the detection of seeded faults in rolling element [30e33]. Here the faults are introduced artificially using the electric discharge machining. Fault sizes ranging from 0.007 in. in diameter to 0.040 in. in diameter were introduced at the inner raceway, the rolling element (ball) and the outer raceway of the bearing. Only one fault is considered at a particular time. Accelerometer data, collected at 12 KHz sampling frequency, is used for the validation. The experimental setup is shown in Fig. 6. 3.2. The algorithm When localised fault occurs in a bearing, quasi-periodic impulses appear in time domain vibration signal. The vibration signal when passed through the zero frequency resonator will contain negative peaks corresponds to impulses due to fault. At the early stage, the strength of impulses due to the fault is weak and the fault signal is hidden in strong machinery vibration and hence the resonator output will also contain spurious negative peaks. In order to enhance the impulse like characteristic and to reduce spurious negative peaks, Hilbert envelop of the signal is obtained before passing the signal through zero frequency resonator. A block diagram of the proposed fault detection technique is shown in Fig. 7. First, difference of the fault signal is computed to remove any time varying low frequency bias in the signal. This signal is referred to as differenced signal subsequently. The proposed method consists of two steps. The first step determines normalised Hilbert Envelop of the signal to emphasise the impulse like characteristics and to improve the localization of impulses by eliminating the carrier frequency phase. The second step computes accurate fault location in time domain using zero frequency resonator. In the second step, the normalised Hilbert envelop is passed through the zero frequency resonator. Output of the zero frequency resonator is referred to
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Fig. 5. Bearing test rig for the IMS dataset [14].
Fig. 6. Bearing test rig for the CWRU dataset [30].
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Fig. 7. Block diagram of fault detection using Hilbert envelop and zero frequency resonator based method.
as filtered output. Now to extract the impulse locations, the residual signal is computed by subtracting the filtered output from its local mean, as given in Eq. (6). The differenced fault signal is segmented into short segments. Window length for the calculation of local mean is taken to be the average period T0 obtained from the short segments. The mean subtracted output signal is termed as residual signal. The spectrum of the residual signal is computed to estimate the faults characteristic frequencies. Fourier transform of a segment of 0.025 s of the residual signal is considered to compute the spectrum of the residual signal. Experiments with seeded faults do not mimic the natural defects at early stage [14]. Therefore the proposed algorithm is presented with IMS dataset where the bearing fault has grown naturally [8]. 3.3. Detection of serious defect To see the effect of Hilbert envelop, initially, the algorithm is applied without using the Hilbert envelop block in Fig. 7. The vibration signals at serious stage were collected at the end of the IMS bearing dataset [8]. The zero frequency resonator output and the residual signal are shown in Fig. 8(b) and (c), respectively. Figures show that the residual output signal contains negative peaks at regular intervals at the locations of impulses. However, these negative peaks are not observed easily due to the spurious negative peaks in between successive strong negative peaks. The multiple spurious negative peaks in residual output signal occurred due to the noise present in the signals. The spectrum of the residual signal is shown in Fig. 9. The strongest peak in the spectrum occurred at 228 Hz which is close to the characteristic frequency of outer race defect. The spectrum contains some more peaks at different frequencies. In order to reduce the effect of noise in the residual signal, now the Hilbert envelop of the signal is included in the algorithm. Hilbert envelop of the signal, shown in Fig. 10(b), contains prominent peaks at the impulse locations and spurious peaks due to noise are suppressed. Normalised Hilbert envelop of the signal is passed through zero frequency resonator and the output of the resonator is shown in Fig. 10(d). A comparison of residual signal from Figs. 8(c) and 10(d) shows that the use of Hilbert envelop in the algorithm reduces the number of spurious zero crossings. After including the Hilbert envelop in the algorithm, the negative peaks are observed only corresponding to the location of impulses occurring due to the fault. The spectrum corresponding to the residual output is shown in Fig. 11. The major peak in the spectrum is observed at 236 Hz, which is the characteristic frequency corresponding to outer race defect. Hence, the serious defect in the outer race is easily detected using the proposed algorithm. 3.4. Detection of early stage defect In the early stage, the vibration signal was collected nearly three days before the starting of serious defect [8]. The proposed algorithm, shown in Fig. 7, is applied to the vibration signal. The Hilbert envelop and the residual output of the signal are shown in Fig. 12(b) and (d), respectively. The impulses are weak during early stage defect. Therefore the strengths of impulses in Hilbert envelop of the signal (Fig. 12(b)) are not as prominent as in the case of serious defect stage (Fig. 10(b)).
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Although the strength of impulses are not prominent in Hilbert envelop of the signal, the residual signal detects negative peaks at the instants of impulses. The spectrum corresponding to the residual signal is shown in Fig. 13. The major peak in the spectrum is observed at 239 Hz which is close to the characteristic frequency of outer race defect. Hence, the proposed algorithm is detecting the serious stage as well as the early stage bearing faults.
4. Comparison with Hilbert-Huang transform In this section, the effectiveness of proposed technique to detect the fault signature is compared with the state of the art HHT technique. HHT is developed to decompose a signal into nearly mono component modes called as intrinsic mode function (IMF). Hilbert transform of each IMF gives instantaneous frequency and instantaneous amplitude. Instantaneous amplitude of the corresponding IMFs gives signal energy distribution in time domain with diversified scale.
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Fig. 10. Output of zero frequency resonator at serious defect after including Hilbert envelop of the signal. (a) Original fault signal, (b) Hilbert envelop of the signal, (c) filtered signal and (d) residual output.
Fig. 11. Spectrum of residual signal at serious defect after including the Hilbert envelop of the signal.
4.1. Comparison using IMS dataset Instantaneous amplitude of each IMF of the fault signal at serious stage (as described in Section 3.3) is computed and these are shown in Fig. 14. The instantaneous amplitude of first IMF shows the periodic impulses. Fourier transform of a segment of 0.025 s of the instantaneous amplitude is considered to compute the spectrum of first four IMFs and these are shown in Fig. 15. The spectrums of the first two IMFs show peaks at 236 Hz and 228 Hz, respectively. By comparing Figs. 11 and 15(a), it can be observed that both the methods, the HHT as well as the proposed method, have the ability to detect the fault impulse signature when the fault is at serious stage. Now, the two methods are compared for early stage detection of bearing defects (as described in Section 3.4). The Instantaneous amplitude of IMF and the spectrum of instantaneous amplitude, for early stage of fault, are shown in Figs. 16 and 17, respectively. Fig. 16(a) shows periodic impulses in the instantaneous amplitude of first IMF. However, the impulses at early stage is comparatively less observable in comparison to the impulses at serious stage defect, as shown in Fig. 14(a).
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Fig. 12. Output of zero frequency resonator at early defect. (a) Original fault signal, (b) Hilbert envelop of the signal, (c) filtered signal, and (d) residual signal.
Fig. 13. Spectrum of residual signal at early stage defect.
By comparing Figs. 13 and 17(a), it can be observed that both the techniques are detecting the bearing defects at early stage. However, the strength of peak at the fault characteristic frequency in case of HHT based method is less prominent compared to that in case of proposed method, based on zero frequency resonator and Hilbert envelop. In order to find out the consistency of the proposed method in detecting early stage fault, the fault identification algorithm is evaluated on several vibration signals taken at early stage of the fault. Approximately 200 vibration signals, starting from three days before the serious fault i.e., 4th day of bearing test [8], are considered for the analysis. Performance of the two methods in detecting the fault frequency from these vibration signals is shown in Fig. 18. The continuous line is the reference line corresponding to the fault frequency. The dotted line in Fig. 18(a) shows the performance of the present method and the dotted line in Fig. 18(b) shows the performance of HHT method. When the fault has grown a little, both the methods are consistent in detecting the fault frequency. However, the proposed method is more consistent in detecting the fault frequency at the incipient stage of the fault, as the dotted line is more close to the reference fault frequency line in Fig. 18(a) compared to that in Fig. 18(b). 4.2. Verification of the proposed algorithm for seeded fault using the CWRU bearing dataset Now the algorithm is verified with the Case Western Reserve University bearing dataset. Here the faults are seeded. The algorithm is verified for the inner race fault, the outer race fault and the ball fault. For the inner race and outer race faults,
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Fig. 14. Instantaneous amplitude of IMFs of the fault signal at serious stage of fault. (a) to (i) is for IMF1 to IMF9, respectively.
Fig. 15. Spectrum of instantaneous amplitude of IMFs of the fault signal at serious stage of fault. (a) to (d) is for IMF1 to IMF4, respectively.
diameter and depth of the faults are 0.007 in. and 0.011 in., respectively (record 106DE for the inner race fault and record 131 DE for the outer race fault). Diameter and depth for the ball fault are 0.028 in. and 0.150 in., respectively (record 3008 DE). The shaft speed varies from 1797 rpm to 1730 rpm depending upon the loads applied (0e3 hp). The characteristic fault frequencies corresponding to the rolling element defect, the inner race defect and the outer race defect for the selected conditions are 68 Hz, 148 Hz, and 90.3 Hz, respectively. Fig. 19(a) and (b) shows the spectrum for the rolling element fault in the bearing using the HHT and the proposed algorithm, respectively. For the rolling element fault, the even harmonics of ball spin frequency (BSF) are often dominant in the spectra [32]. BSF is the frequency with which the fault strikes the same race (inner or outer), so that in general there are two shocks for one rotation of the ball. One corresponds to its impact with the inner race and the other corresponds to its impact with the outer race. Therefore, the peak at fault frequency is missing in both the spectrums for the case of ball fault (Fig. 19(a) and (b)). However, the second harmonic can be observed in the two spectrums. The present algorithm shows a clear dominant peak at the second harmonic. For the inner and outer race fault detection also (Fig. 19(c)e(f)), considering a single dominant peak at characteristics fault frequency, the detection is better in case of the present algorithm.
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Fig. 16. Instantaneous amplitude of IMFs of the fault signal at early stage of fault. (a) to (i) is for IMF1 to IMF9, respectively.
Fig. 17. Spectrum of Instantaneous amplitude of IMFs of the fault signal at early stage of fault.
The proposed algorithm is capable of detecting periodic impulses in noisy conditions. Such conditions exist in case of bearing and gear faults of a rotating machinery. The proposed method can also be applied in biomedical signal such as in electrocardiogram (ECG) and phonocardiogram (PCG) signal, where signal contains impulses at the locations of heart sound. In case of civil structures, such as bridges and aircraft wings, impulse like characteristics due to slope discontinuity can be observed in the spatial second derivatives at the locations of localised damages. 5. Conclusions A new technique is proposed for the detection of localised bearing faults at early stage. The technique is based upon passing the Hilbert envelop of the signal through the zero frequency resonator to detect the weak impulses due to localised bearing defects. The unipolar nature of Hilbert envelop facilitates the extraction of impulse like characteristics of the signal. Additionally, Hilbert envelop of the signal reduces the effect of carrier frequency phase during localization of impulses, thus giving more accurate estimation of the impulse period. The zero frequency resonator locates the local discontinuity in the signal arising due to the impulses. The proposed method is tested with bearing vibration data. The method identifies very well
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Fig. 19. Bearing fault identification using Case Western Reserve University Bearing Dataset, (a), (c), (e) spectra of HHT for ball fault, inner race fault and outer race fault, respectively, and (b), (d), (f) spectra of proposed algorithm for ball fault, inner race fault and outer race fault, respectively.
the bearing defects at early stage. The proposed algorithm has the merit that it does not involve any optimisation of parameters and gives a single dominant peak at the fault frequency. This will enable for an easy adaptation of the algorithm by machine operators.
Acknowledgment This work is supported by research grant from Science & Engineering Research Board India, through project File No. YSS/ 2015/001555.
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