entropy Article
Shannon Entropy of Binary Wavelet Packet Subbands and Its Application in Bearing Fault Extraction Shuting Wan, Xiong Zhang *
ID
and Longjiang Dou
Department of Mechanical Engineering, North China Electric Power University, Baoding 071003, China;
[email protected] (S.W.);
[email protected] (L.D.) * Correspondence:
[email protected] Received: 28 February 2018; Accepted: 4 April 2018; Published: 9 April 2018
Abstract: The fast spectrum kurtosis (FSK) algorithm can adaptively identify and select the resonant frequency band and extract the fault feature via the envelope demodulation method. However, the FSK method has some limitations due to its susceptibility to noise and random knocks. To overcome this shortage, a new method is proposed in this paper. Firstly, we use the binary wavelet packet transform (BWPT) instead of the finite impulse response (FIR) filter bank as the frequency band segmentation method. Following this, the Shannon entropy of each frequency band is calculated. The appropriate center frequency and bandwidth are chosen for filtering by using the inverse of the Shannon entropy as the index. Finally, the envelope spectrum of the filtered signal is analyzed and the faulty feature information is obtained from the envelope spectrum. Through simulation and experimental verification, we found that Shannon entropy is—to some extent—better than kurtosis as a frequency-selective index, and that the Shannon entropy of the binary wavelet packet transform method is more accurate for fault feature extraction. Keywords: bearing diagnosis; FSK; BWPT; Shannon entropy
1. Introduction Rolling bearings are one of the most widely used parts in mechanical equipment. Their operation condition affects the working state of the whole system. The fault identification and diagnosis of rolling bearings are of great significance in ensuring the safe and reliable operation of mechanical equipment. The separation of faulty components from strong background noise is a difficult problem in the field of mechanical fault diagnosis. The concept of spectrum kurtosis (SK) was presented by Dwyer. The basic principle was to calculate the kurtosis value of each spectral line; the value of different kurtoses could respond to the size of the transient impact. SK was rarely used until Antony et al. proposed a formal definition. They also proposed an estimator based on a short-time Fourier transform. The estimator can help link theoretical concepts to practical applications [1]. After that, Anthony et al. continued to work on how the SK can be used in dealing with nonstationary signals. Additionally, the concept of the kurtogram was introduced in their paper. The optimal band-pass filters can be selected by the kurtogram as a prelude to envelope analysis [2]. Then, in order to simplify the operation process of the kurtogram and to improve the speed of the algorithm and make it meet the requirements of industrial monitoring, Antony et al. proposed a fast spectral kurtosis (FSK) algorithm, which simplifies the process of selecting the optimal band ( f , ∆ f ). The structure of the fast spectral kurtosis algorithm is as follows: the first layer of the algorithm decomposes the signal in a binary tree structure, and the second layer is decomposed into a 1/3 tree structure; the rest can be deduced by analogy [3].The paving of the fast spectrum kurtosis (FSK) is shown in Figure 1.
Entropy 2018, 20, 260; doi:10.3390/e20040260
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Figure 1. The paving of the FSK. Figure 1. The paving of the FSK.
Since then, scholars have begun to study how to improve the kurtogram. These improvements are Sinceaimed then, scholars have begun study how improve theimprovement kurtogram. These mainly at two aspects of the to algorithm. Onetoaspect is the of the improvements frequency bandare mainly aimed atmethod, two aspects the algorithm. One aspect is theforimprovement of the frequency band segmentation whichofseeks to find an excellent method making the selection of the center frequency and bandwidth more accurate adaptive. Thefor other aspectthe involves improving the segmentation method, which seeks to findand an more excellent method making selection of the center calculation method of the index tryingand to find a method to replace the traditional kurtosis. frequency and bandwidth more and accurate more adaptive. The other aspect involves improving the Wang and Liang improved thetrying kurtogram by aadapting size of the bandwidth calculation method of the index and to find method the to replace the adaptive traditional kurtosis.and center frequency. implemented the algorithm by expanding an adaptive initial window in the Wang and LiangThey improved the kurtogram by adapting the size of the bandwidth and frequency axis via successive attempts to merge it with its subsequent translated windows [4]. Lei center frequency. They implemented the algorithm by expanding an initial window in the frequency et via al. successive proposed an improved kurtogram, the wavelet method was [4]. usedLei toetreplace the axis attempts to merge it with in its which subsequent translated windows al. proposed time Fourier transform (STFT)method or finite impulse filters in the antraditional improved short kurtogram, in which the wavelet was used toresponse replace (FIR) the traditional short traditional kurtogram algorithm. The improvement can improve the accuracy of the selective time Fourier transform (STFT) or finite impulse response (FIR) filters in the traditional kurtogram resonance frequency band in the event of large noise [5]. Based on Lei’s improvement, Wang made algorithm. The improvement can improve the accuracy of the selective resonance frequency band in a further improvement to the kurtogram. The kurtosis value was calculated based on the power the event of large noise [5]. Based on Lei’s improvement, Wang made a further improvement to the spectrum of the envelope of the signals. This improvement had a good effect on the low kurtogram. The kurtosis value was calculated based on the power spectrum of the envelope of the signal-to-noise ratio and on non-Gaussian noise interference [6]. Then, Tse and Wang made a signals. This improvement had a good effect on the low signal-to-noise ratio and on non-Gaussian deeper study of the kurtogram. A new concept called a sparsogram was proposed, which was noise interference Then,measurements Tse and Wangofmade a deeper study of the theenvelopes kurtogram. A new concept constructed using[6]. sparsity the power spectra from of the signal [7]. called a sparsogram was proposed, which was constructed using sparsity measurements of the There are still two shortcomings in this algorithm: one is that the fixed central frequency power and spectra from cannot the envelopes of entire the signal [7]. frequency There areband. still two in this bandwidth cover the resonant The shortcomings other shortcoming is algorithm: that the one is that the fixed central frequency and bandwidth cannot cover the entire resonant frequency band. bearing resonant frequency band may be split into two adjacent imperfect orthogonal frequency The other shortcoming is that the bearing resonant frequency band may be split into two adjacent bands, which reduce the bearing fault features. Considering the above, Tse and Wang made an imperfect orthogonal frequency bands,They which reduce the fault features. Considering the above, improvement in their second paper. presented anbearing automatic selection process for finding the Tseoptimal and Wang madeMorlet an improvement in their They presented automatic selection complex wavelet filter with second the helppaper. of a genetic algorithman that maximizes the sparsity measurement value [8]. Chen presented an improved version of FSK named a “fast process for finding the optimal complex Morlet wavelet filter with the help of a genetic algorithm spatial-spectral kurtosis kurtogram”. quasi-analytic wavelet version tight frame that maximizes theensemble sparsity measurement value [8].Discrete Chen presented an improved of FSK (QAWTF) expansion methodsensemble were incorporated in the kurtogram algorithm as a detection filter named a “fast spatial-spectral kurtosis kurtogram”. Discrete quasi-analytic wavelet tight instead of as an STFT or FIR filter. Moreover, an enhanced signal impulsiveness evaluating frame (QAWTF) expansion methods were incorporated in the kurtogram algorithm as a detection indicator, a STFT “spatial-spectral ensemble kurtosis (SSEK)” signal was used as the quantitative filter insteadnamed of as an or FIR filter. Moreover, an enhanced impulsiveness evaluating measure to select optimal analysis parameters [9]. Borghesani et al. presented an optimal indicator, named a “spatial-spectral ensemble kurtosis (SSEK)” was used as the quantitative measure indicator—the ratio of cyclic content (RCC)—for the choice of the selectivity in the cyclic frequency to select optimal analysis parameters [9]. Borghesani et al. presented an optimal indicator—the ratio of domain. Additionally, they compared the RCC with the more traditional kurtosis-based indicators [10]. cyclic content (RCC)—for the choice of the selectivity in the cyclic frequency domain. Additionally, Barszcz et al. propose the protugram method, which is based on the kurtosis of the envelope
they compared the RCC with the more traditional kurtosis-based indicators [10]. Barszcz et al. propose
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the protugram method, which is based on the kurtosis of the envelope spectrum amplitudes of the demodulated signal, instead of the kurtosis of the time signal [11]. A new index called spectral L2/L1 was proposed by Wang. The calculation method of the index is changed by adjusting the values of p and q, which makes the index more general than spectral kurtosis [12]. Entropy is derived from the concept of thermodynamics. Entropy can reflect a direct relationship between the size of the information and its uncertainty. It is precisely because entropy can reflect the measure of probabilistic and repetitive events that it is used so widely in various fields. For instance, the concept of entropy can be used in the design process of complex mechanical systems. Villecco et al. gave the entropic measure of epistemic uncertainties, which can be used in multibody system models [13,14]. At the same time, entropy was also applied to the signal processing of rotating machinery [15–18]. Antoni then introduced entropy to the kurtogram algorithm. After comprehensively considering the periodic and impact characteristics of the signal, Antoni proposed the squared envelope (SE) infogram, squared envelope spectrum (SES) infogram, and SE1/2 /SES1/2 infogram. Specifically, the SE infogram was used to calculate the negative entropy of the square envelope as an index, and the same SES infogram was used to calculate the negative entropy of the square envelope spectrum as an index [19]. Following this, scholars have done a great deal of work on the improvement of the infogram. These scholars mainly include Li, Xu, and Wang. Li et al. proposed the multiscale clustering grey infogram (MCGI), which was based on combining both negentropies in a grey fashion using multiscale clustering. First, the signal was decomposed into a multiple scale according to the Fourier spectrum, and fine segments were grouped using hierarchical clustering in each scale. Meanwhile, both time-domain and frequency-domain spectral negentropies were taken into account to guide the clustering through a grey evaluation of both negentropies [20]. Xu et al. proposed the multiscale fractional order entropy (MSFE) infogram, which can help in the selection of the optimal frequency band for extracting the repetitive transients from the whole frequency bands [21]. Wang extended the infogram to novel Bayesian inference-based optimal wavelet filtering for bearing fault feature identification [22]. In the case of strong noise interference, the FSK is often inaccurate in determining the center frequency and bandwidth of the signal resonance. In order to further improve the accuracy and efficiency of the algorithm in determining the center frequency and bandwidth, a method based on computing the Shannon entropy of wavelet subbands is proposed in this article. The new method can achieve a good control of the noise, as well as a selection of the center frequency that is more accurate and faster, benefiting from the good frequency band segmentation characteristics of the wavelet packet and the reliability of the Shannon entropy index. 2. The Limitation of the Kurtosis Index The previous improvement of FSK mainly focused on the frequency band segmentation, where the ideal effect was achieved. However, there is still a problem with the kurtosis index. The limitation of the kurtosis index mainly manifests itself in two aspects. The first is that kurtosis can easily be disturbed by noise, and the second is that kurtosis is sensitive to random knocks. The deficiency of the kurtosis index is verified by a simulation signal, and the simulation signal is defined as Equation (1): n 2 y1 = ∑ Ae−ξ [t−qi (t)/ f outer ] · sin(2π f o t) i =1 y2 = A1 e[−b1 ×(t−t1 )] · sin[2π f 1 · (t − t1 )] + A2 e[−b2 ×(t−t2 )] · sin[2π f 2 · (t − t2 )] , y = n(t) 3 Y = y1 + y2 + y3
(1)
where c is the center of the resonance frequency band, f outer = 50 Hz is the characteristic frequency, ξ is the damping ratio, A is the amplitude, and qi (t) = bt · f i c(i = 1, 2, . . . , n). A1 = 3 and A2 = 6 are
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Entropy 2018, 20, x FOR PEER REVIEW of 16 the amplitudes, b1 = −260, b2 = −360, t1 = 0.18, t2 = 0.5, f 1 = 1000, f 2 = 2600. n(t) is the4 Gaussian 2018, 20, parameters x FOR PEER REVIEW 4 of 16 whiteEntropy noise. The simulation are fshown in Table 1. =2600 n (t) f =1000 b =-260 b =-3of 60thet = 0.18 t =0.5 signal
amplitudes, 1 , 2 , 1 , 2 , 1 , 2 . is the Gaussian white noise. The parameters of the simulation signal are shown in Table 1. ) is the Gaussian white noise. t2=0.5 , f1=1000 60 , t1. amplitudes, b1=-260 , b2=-3 , fsimulation 2 =2600 . n (tsignal. 1=0.18 Table The ,parameters of the The parameters of the simulation signal are shown in Table 1. Table 1. The parameters of the simulation signal.
Parameter Type Parameter Values Table 1. The parameters Parameter Type of the simulation Parametersignal. Values Amplitude A 2 Amplitude A 2 Parameter Parameter Values20 Rotating frequency f r (Hz) Type 20 Rotating frequencyA f r (Hz) Natural frequency (Hz) 1000 Amplitude 2 Natural frequency (Hz) 1000 Sampling frequency (Hz) 8192 f 20 Rotating frequency r (Hz) frequency (Hz) 8192 Sampling Sampling points (Hz) 8192 Natural frequency (Hz) 1000 Fault frequency (Hz) points (Hz) 50 Sampling 8192 Sampling frequency (Hz) 8192 Fault frequency (Hz) 50 Sampling points (Hz) 8192 Fault frequency (Hz)shown in Figure 50 2. The synthetic signal (Figure 2d) The waveforms of the simulation signal are The waveforms of the simulation signal are shown in Figure 2. The synthetic signal (Figure 2d) is composed of three parts. Figure signal,which which is used to simulate the fault is composed of three parts. Figure2a 2ashows shows aa periodic periodic signal, is used to simulate the fault The waveforms of the simulation signal are shown in Figure 2. The synthetic signal (Figure 2d) impact. Figure 2b represents the random knocks, and Figure 2c shows the Gaussian noise. impact. Figure 2b represents the random knocks, and Figure 2c shows the Gaussian noise. is composed of three parts. Figure 2a shows a periodic signal, which is used to simulate the fault impact. Figure 2b represents the random knocks, and Figure 2c shows the Gaussian noise.
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(b)
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(b)
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(c) of (a) fault impulses; (b) random knocks; (c) the (d) Figure 2. The waveforms noise signal; and (d) the Figure 2. The waveforms of (a) fault impulses; (b) random knocks; (c) the noise signal; and (d) the synthetic signal. Figuresignal. 2. The waveforms of (a) fault impulses; (b) random knocks; (c) the noise signal; and (d) the synthetic synthetic signal.
The synthetic signal was analyzed by the FSK. The results are shown in Figure 3. As can be seen from Figure 3a,signal the central frequency by of the resonant band determined by theAs FSK was The The synthetic was analyzed Thefrequency resultsare are shown in Figure can be seen synthetic signal was analyzed bythe the FSK. FSK. The results shown in Figure 3. As3.can be seen 2650 Hz, 3a, while the actual frequency resonant frequency band of the simulation signal was 1000 Hz. The band fromfrom Figure the central of the resonant frequency band determined by the FSK Figure 3a, the central frequency of the resonant frequency band determined by the FSK was was pass filter wasactual carriedresonant out to obtain the resonance frequency band shown in Figure 3a, andband the 2650 2650 Hz, while the frequency band ofof the simulation Hz, while the actual resonant frequency band the simulationsignal signalwas was1000 1000 Hz. Hz. The The band pass envelope spectrum of the filtered signal was analyzed. The results are shown in Figures 3b,c. It can wasout carried out tothe obtain the resonance frequency band shown in Figure 3a, the andenvelope the filterpass wasfilter carried to obtain resonance frequency band shown in Figure 3a, and be seen that the filtered signal is the random knocks component in the synthetic signal. Therefore, envelope spectrum of signal the filtered was analyzed. The results are shown in Figures It can spectrum of the filtered wassignal analyzed. The results are shown in Figure 3b,c.3b,c. It can be seen the FSK presented an error under the random knocks interference. be seen that the filtered signal is the random knocks component in the synthetic signal. Therefore, that the filtered signal is the random knocks component in the synthetic signal. Therefore, the FSK the FSK presented an error under the random knocks interference.
presented an error under the random knocks interference.
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(b)
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Figure 3. Cont.
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(c) (c) (b) the signal after band pass filtering; (c) the Figure 3. (a) The kurtogram of the synthetic signal; Figure 3. (a) The kurtogram of the synthetic signal; (b) the signal after band pass filtering; envelope spectrum of (b). Figure 3. (a)spectrum The kurtogram (c) the envelope of (b).of the synthetic signal; (b) the signal after band pass filtering; (c) the envelope spectrum of (b).
Following this, the random knocks (y2) in the synthetic signal were removed. As shown in
Following this, the random knocks (y2) in the synthetic signal were removed. As shown in Figure 4, in order tothe verify the sensitivity of thethe kurtosis index to noise, an FSK analysis of the this, random knocks (y2) synthetic signal were removed. shown in Figure 4, Following in order towas verify theout sensitivity of in the kurtosis index to noise, an FSKAs analysis of the simulated signal carried under different signal-to-noise ratio (SNR) conditions by adjusting Figure 4, in order to verify the sensitivity of the kurtosis index to noise, an FSK analysis of the simulated was carried out underof different signal-to-noise ratio (SNR) the conditions bysignal adjusting the sizesignal of the random noise. The the signal in Figure 4a was −12dB, SNR by of the simulated signal was carried out SNR under different signal-to-noise ratio (SNR)and conditions adjusting the size of the 4b random noise. The SNR of the signal inofFigure 4a was −which 12dB, aand the SNR of the signal in Figure was −16 dB. Figure 4c is the kurtogram Figure 4a from resonance frequency the size of the random noise. The SNR of the signal in Figure 4a was −12dB, and the SNR of the signal in Figure 4b was −16 dB. Figure isisthe ofFigure Figure 4a from a resonance frequency band of 1000 Hz can be obtained. 4d is the kurtogram of4aFigure 4bwhich from which the resonance in Figure 4b was −16 dB. Figure4c4cFigure thekurtogram kurtogram of from which a resonance frequency frequency band cannot be accurately obtained. This can be explained by the fact that the FSK method bandband of 1000 Hz can be be obtained. kurtogramofofFigure Figure from which the resonance of 1000 Hz can obtained.Figure Figure4d 4dis is the the kurtogram 4b4b from which the resonance cannot extract the fault feature information under strong noise interference due to the sensitivity frequency band cannot be be accurately obtained. This can beexplained explained the FSK method frequency band cannot accurately obtained. Thisthe can be byby thethe factfact thatthat the FSK method of the kurtosis index to the noise. cannot extract the fault feature information under the strong noise interference due to the sensitivity of cannot extract the fault feature information under the strong noise interference due to the sensitivity of the kurtosis the noise. the kurtosis index index to thetonoise.
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(b)
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(c) band detection using fast spectral kurtosis (FSK):(d) Figure 4. The resonance (a) the synthetic signal with fault impulses and noise (−12 dB); (b) the synthetic signal with fault impulses and noise (−16 dB); (c) Figure 4. The resonance band detection using fast spectral kurtosis (FSK): (a) the synthetic signal Figure The resonance band usingoffast the4.kurtogram of (a); and (d)detection the kurtogram (b).spectral kurtosis (FSK): (a) the synthetic signal with with fault impulses and noise (−12 dB); (b) the synthetic signal with fault impulses and noise (−16 dB); (c) fault impulses and noise (−12 dB); (b) the synthetic signal with fault impulses and noise (−16 dB); the kurtogram of (a); and (d) the kurtogram of (b).
3. Proposed Method (c) the kurtogram of (a); and (d) the kurtogram of (b). 3. Proposed Method
3.1. Binary Wavelet Packet Transform 3. Proposed Method 3.1. Binary Wavelet Packet Transform algorithm usually uses a fast Fourier transform (FFT) The traditional kurtogram
3.1. Binary Wavelet Packet Transform decomposition or an FIR filter as aalgorithm hierarchicalusually decomposition In ordertransform to improve the The traditional kurtogram uses a method. fast Fourier (FFT) accuracy and efficiency of the resonance frequency band selection, more precise filters need to be
decomposition an FIR filter as a hierarchical decomposition method. In order to decomposition improve the or The traditionalor kurtogram algorithm usually uses a fast Fourier transform (FFT) incorporated into the kurtogram. accuracy and efficiency of the resonance frequency band selection, more precise filters need to be an FIR filter as a hierarchical decomposition method. In order to improve the accuracy and efficiency of incorporated into the kurtogram. the resonance frequency band selection, more precise filters need to be incorporated into the kurtogram. Wavelet packet transform can also be called subband tree or optimal subtree structuring. Wavelet packet analysis can provide a more precise method for signal analysis, which overcomes the problem of
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the wavelet analysis being inaccurate with respect to the division of high frequency bands. Moreover, it introduces the concept of basis selection based on wavelet theory. After the multilevel division of the frequency band, the best base function is selected adaptively according to the characteristics of the Entropy 2018, 20, x FOR PEER REVIEW 6 of 16 analyzed signal. From the viewpoint of signal processing, the WPT allows the signal to pass through Wavelet packet also be called tree or optimal subtreeThe structuring. a series of filters which aretransform differentcan in frequency butsubband have the same bandwidth. process can be Wavelet packet analysis can provide a more precise method for signal analysis, which overcomes defined by the following Equation (2): the problem of the wavelet analysis being inaccurate with respect to the division of high frequency bands. Moreover, it introduces the concept√of basis selection based on wavelet theory. After the K Hn bfunction bi,j =the 2best multilevel division of the frequency ∑ base band, i −1,j+n is selected adaptively according to =0 viewpoint of signal the characteristics of the analyzed signal. From nthe processing, the WPT allows , √ K the signal to pass through a series of filters which are different in frequency but have the same bi,j+l = 2 ∑ Gn bi−1,j+n n =0 bandwidth. The process can be defined by the following Equation (2):
(2)
where bi,j denotes the jth decomposed frequency-band bi,j 2 H nbi 1, j nsignal at level i (j = 1, 2, . . . , J, where J is n 0 the number of the decomposed frequency-band signals and equals 2 I ; i = 1, 2, . . . , I, where (2) I is the K b 2 G b n i 1, low-pass j n number of the decomposition levels); Hn and the and high-pass filters based on the i , j l Gn are n 0 , wavelets, respectively. K
where bi,j denotes the jth decomposed frequency-band signal at level i (j = 1, 2, …, J, where J is the number of the decomposed frequency-band signals and equals 2 I ; i=1, 2,..., I , where I is the 3.2. Shannon Entropy number of the decomposition levels); H n and Gn are the low-pass and high-pass filters based on the Thewavelets, conceptrespectively. of entropy was put forward by Clausius, a German physicist in 1865. It was originally
used to describe the material state of “energy degradation”. It has been widely applied in 3.2. Shannon Entropy thermodynamics. In 1948, Shannon extended the concept of entropy in statistical physics to the process The of signal in order to solve the problem of quantitative measurements of conceptprocessing of entropy was put forward by Clausius, a German physicist in 1865. It was originally used to describe the material statetheory”. of “energy degradation”. It hascan been widely applied in information, thus creating the “information Shannon entropy measure the disorder or thermodynamics. In 1948, extended theapplied concept to of many entropy in statistical to the uncertainty of information, andShannon consequently it is fields. Given physics a random sequence process of signal processing in order to solve the problem of quantitative measurements of { x1 , x2 , . . . . . ., xn }, the Shannon entropy is calculated according to Equation (3): information, thus creating the “information theory”. Shannon entropy can measure the disorder or uncertainty of information, and consequently n it is applied to many fields. Given a random sequence x1, x2 ,......, xn , the Shannon entropy is calculated (3): Sh = −c paccording xi )], ( xi ) · log[top(Equation
∑
(3)
i =1n
Sh c p xi log p xi ,
(3)
i 1
where S is the Shannon entropy, p( xi ) is the probability mass associated with the value xi and c is where S is the Shannon entropy, p x is the probability mass associated with the value xi and c an arbitrary positive constant that dictates the units. If we allow that c equals 1/log(2), then the is an arbitrary positive constant that dictates the units. If we allow that c equals 1/log(2), then the logarithm base is 2 and the entropy is measured in bits. It should be noted that the Shannon entropy H logarithm base is 2 and the entropy is measured in bits. It should be noted that the Shannon entropy is non-negative and thatand thethat probabilities needneed to sum toto1.1. H is non-negative the probabilities to sum The different SNRsSNRs werewere simulated bybyadding varyingdegrees degrees of noise toperiodic the periodic The different simulated adding varying of noise to the impact impact signal (y1). The results are shown in Figure 5. It can be seen that with the addition of noise, the value signal (y1). The results are shown in Figure 5. It can be seen that with the addition of noise, the value of the Shannon entropy increased; that is, with the increase of the SNR, the Shannon entropy of the Shannon entropy increased; that is, with the increase of the SNR, the Shannon entropy decreased. decreased. This is because the uncertainty of the signal increases with the addition of the noise. It is This is because the uncertainty of the signal increases with the addition of the noise. It is fully proved fully proved that Shannon entropy can be used as an indicator to measure the periodic impact that Shannon entropy can be used as an indicator to measure the periodic impact strength of the signal. strength of the signal. i
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Figure 5. Cont.
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(e)
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(g) Figure 5. The value of the Shannon entropy under different signal-to-noise ratio (SNR): (a) −2 dB;
Figure 5. The value of the Shannon entropy under different signal-to-noise ratio (SNR): (a) −2 dB; (b) −4 dB; (c) −6 dB; (d) −8 dB; (e) −10 dB; (f) −12 dB; (g) the curve of the Shannon entropy with (b) −4 dB; (c) −SNR. 6 dB; (d) −8 dB; (e) −10 dB; (f) −12 dB; (g) the curve of the Shannon entropy with different different SNR. 3.3. Process of the Proposed Method
3.3. Process of thetheoretical Proposed characteristic Method I The frequency of the bearing fault is calculated according to the I
parameter of the bearing. The empirical formula is as follows:
The theoretical characteristic frequency of the bearing fault is calculated according to the Z d N Z d N f (1 cos ) f (1- cos ) parameter of the bearing. The empirical 2 D formula 60 2is as D follows: 60 , i
o
D
d
N
1
d
N
f (1- cos ) Zf 2d (1-d（ D ）cos ) N N 60 2 Z D 60 d (1 + cos β) f o = (1 − , cos β) , 2 D 60 2 D 60 f ball
fi =
2
2
cage
where f i , f o , f ball , and cage respectively represent the inner ring fault, outer ring fault, rolling body element fault, andD cage fault.d Z 2 is the number N of the ball1 element, d d is theNdiameter of the f = (1 −diameter, ( ) cosN2 βis)the spindle f cage = (1and − cos β) pressure , D rolling element,ball is the pitch speed, is the angle. 2d D 60 2 D 60 II The vibration signal is decomposed by binary wavelet packet transform (BWPT), and the whereShannon f i , f o , f ball , and( Shf cage inner ring fault, ringoffault, i，j ) ofrespectively entropy each subband represent after BWPT the is calculated. After this, outer the inverse each rolling subband Shannon entropy as number Equation (4): body element fault, and cage value fault.isZtaken is the of the ball element, d is the diameter of the
II
rolling element, D is the pitch diameter, N is the 1 spindle speed, and β is the pressure angle. Si，j = (4) Shi，j . The vibration signal is decomposed by binary wavelet packet transform (BWPT), and the Shannon entropy of each an subband BWPT is the calculated. After theShannon inverse entropy of eachofsubband j ) of the III We(Sh then entropyafter spectrum with inverse value ( Si，this, i,j ) generate eachentropy wavelet value packet.is The subband of the maximum Shannon taken as Equation (4): S value (the minimum Shannon entropy) is IV
III
IV
selected as the optimal resonance frequency band. 1 selected to filter the resonance frequency The appropriate central frequency and bandwidth are Si,j = . band determined in (III). The envelope spectrum Shofi,jthe filtered signal is analyzed. Then, the characteristic spectral lines in the spectrum are compared with the theoretical frequency from (I), and the type of the fault is determined.
(4)
We then generate an entropy spectrum with the inverse value (Si,j ) of the Shannon entropy of The paving of the The proposed method shown in Figure each wavelet packet. subband of is the maximum S 6. value (the minimum Shannon entropy) is selected as the optimal resonance frequency band. The appropriate central frequency and bandwidth are selected to filter the resonance frequency band determined in (III). The envelope spectrum of the filtered signal is analyzed. Then, the characteristic spectral lines in the spectrum are compared with the theoretical frequency from (I), and the type of the fault is determined. The paving of the proposed method is shown in Figure 6.
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Figure 6. The paving of the proposed method.
Figure 6. The paving of the proposed method. 4. Simulation Analysis
Figure 6. The paving of the proposed method.
4. Simulation Analysis
The simulation model of the inner ring fault signal is defined as Equation (5) [23]: 4. Simulation Analysis
The simulation model of the inner ring fault signal is defined as Equation (5) [23]:
(t ) s(ring t ) n(fault t) Ai h(t is iTdefined i ) n(tas ) Equation (5) [23]: The simulation model signal of the xinner
x (t)
=As(t)A+cos(2 n(t) f=t ∑ h(t − iT − τi ) + n(t) )AiC i
i Ai h(t iT i ) n(t ) x(t ) s-(Btt ) n(t ) , (t ) = e Acos(2 f nt i fr )t + φ ) + C hA 0 cos(2π A A i f r t A ) C A Ai A0 cos(2− Bt (5) , cos(2π f n t + φω ) h (t )- Bt= e h(t ) e cos(2 f nt ) where i is the minor fluctuation of the ith shock relative to the ,period T; Ai is the amplitude i
0
r
A
A
(5)
(5)
where τi is the minor fluctuation of the ith shock relative to the period T; Ai is the amplitude modulation modulation with 1/ f r as a cycle; h(t ) is an exponential decay pulse; B is the attenuation with where 1/ f r as ia iscycle; h(t) isfluctuation an exponential decay pulse; B is the attenuation of the system; Ai is the amplitude relative to The the period T;coefficient the coefficient ofthe theminor system; A and of areith theshock initial phases. parameters of the inner ring φ A and φω are the The parameters of the inner ring simulation signal are shown 1/ shown f r phases. modulation withinitial as ain cycle; simulation signal are Table 2.h(t ) is an exponential decay pulse; B is the attenuation in Table 2. coefficient of the system; A and are the initial phases. The parameters of the inner ring Table 2. The parameters of the inner fault simulation. simulation signal are shown in Table 2. Table 2. The parameters of the inner fault simulation. Parameter Type Parameter Values Table 2. The parameters of the inner fault simulation. Amplitude A 1 Parameter Type Parameter Values Parameter Type Parameter Values f 20 Rotating frequency r (Hz) Amplitude A 1 Amplitude A (Hz) 1 frequency 2000 Rotating frequencyNatural f r (Hz) 20 20 Rotating (Hz) Sampling frequencyf r(Hz) 8192 Natural frequency (Hz) frequency 2000 Natural (Hz) 2000 Sampling points (Hz) 8192 Sampling frequency (Hz) frequency 8192 Sampling pointsSampling (Hz) 8192 frequency (Hz) 8192 Fault frequency (Hz) 130 Fault frequency (Hz) 130 Sampling points (Hz) 8192 Fault frequency 130 The waveform of the simulation signal of(Hz) the inner ring is shown in Figure 7a; Figure 7b shows the Gaussian noise; Figure 7c shows the synthetic signal. Figure 7d shows the spectrum diagram of The waveform of the simulation signal of the inner ring is shown in Figure 7a; Figure 7b shows The thefrom simulation of the inner ring is shown in Figurewere 7a; Figure 7b shows Figure 7c.waveform It can be of seen Figuresignal 7d that impact characteristics submerged by the Gaussian noise; Figure 7c shows the synthetic signal. Figure 7d shows the spectrum diagram of the Gaussian Figurefrequency 7c showsof the synthetic signal. shows theHz. spectrum diagram of noise and thatnoise; the natural the synthetic signalFigure was at7d about 2000 Figure 7c. It7c. canIt be fromfrom Figure 7d that the the inner impact characteristics Figure canseen be seen Figure 7d that inner impact characteristicswere weresubmerged submergedby bynoise and that the natural frequency of the synthetic signal was at about 2000 Hz. noise and that the natural frequency of the synthetic signal was at about 2000 Hz.
(a)
(b)
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(b)
Figure 7. Cont.
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Figure 7. The waveforms of (a) the innerfault fault impulses; (b) the noise signal; (c)(c) the synthetic signal; Figure Figure 7. 7. The Thewaveforms waveformsof of(a) (a)the theinner inner faultimpulses; impulses;(b) (b)the thenoise noisesignal; signal; (c)the thesynthetic syntheticsignal; signal; and (d) the spectrum of (c). and and(d) (d)the thespectrum spectrum of of (c). (c).
The synthetic signal was analyzed by the FSK. The results are shown in Figure 8. As can be seen
The synthetic signalfrequency was analyzed by the FSK. The results are shownby in Figure 8. As3460 can be seen fromsynthetic (a), the central of the resonant frequency band determined FSK was The signal was analyzed by the FSK. The results are shown inthe Figure 8. As canHz, be seen from while (a), the central frequency of the resonant frequency band determined by the FSK was 3460 Hz, thecentral actual resonant frequency band of the simulationband signaldetermined was 2000 Hz. from (a), the frequency of the resonant frequency byThe theband FSKpass wasfilter 3460 Hz, whilewas the actual resonant frequency band offrequency the simulation signal in was 2000 8a, Hz. Thethe band pass filter out to obtain the resonance band shown Figure envelope while thecarried actual resonant frequency band of the simulation signal was 2000 Hz.and The band pass filter was carried to filtered obtain signal the resonance frequency band shown in Figure 8a, and envelope spectrumout of the was analyzed. The result is shown in Figure 8b. There was the no useful was carried out to obtain the resonance frequency band shown in Figure 8a, and the envelope spectrum information in the envelope spectrum. Therefore, the FSK method in didFigure not extract the fault spectrum of the filtered signal was analyzed. The result is shown 8b. There wasfeature no useful of the filtered signal was analyzed. The result is shown in Figure 8b. There was no useful information information. Then, the synthetic signalTherefore, was analyzed the method SE1/2/SES1/2 infogram, and the information in the envelope spectrum. thevia FSK did not extract theresults fault are feature in theshown envelope spectrum. Therefore, the Figure FSK method did notfrequency extract the fault feature information. in Figure 9. As can be seen from 9a, the central of the resonant frequency information. Then, the synthetic signal was analyzed via the SE1/2/SES1/2 infogram, and the results are Then,band the was synthetic signal analyzed via the SE1/2 and the shown 1/2 infogram, about 2100 Hz.was can be seen from Figure 9b/SES that the spectrum ofresults filtered are signal shown in Figure 9. As can beIt seen from Figure 9a, the central envelope frequency of the resonant frequency in Figure As can be seen from Figure 9a, the central frequency of thebut resonant frequency could9. extract the fault characteristic frequency and the frequency doubling, there was still largeband band was about 2100 Hz. It can be seen from Figure 9b that the envelope spectrum of filtered signal noise interference. this, the synthetic signal the kurtosis spectrum of thecould was about 2100 Hz. It Following can be seen from Figure 9b that was the analyzed envelopevia spectrum of filtered signal couldwavelet extractpacket, the fault characteristic frequency and the frequency doubling,Figure but there was still large and the results are shown in the Figure 10. As can be seen from 10a,still the central extract the fault characteristic frequency and frequency doubling, but there was large noise noise frequency interference. Following this, the synthetic was analyzed via the was kurtosis of of the resonant frequency band was signal 2500 analyzed Hz, while the the bandwidth Fs/8. spectrum There waswavelet a the interference. Following this, the synthetic signal was via kurtosis spectrum of the wavelet packet, and the results are shown in the Figure 10. Asand canthe be seen from Figure Hz, 10a,and thethe central certain between central band of actual value packet, and error the results arethe shown in Figure 10. Asresonance can be seen from Figure 10a, 2000 the central frequency frequency of thewas resonant was spectrum 2500 Hz,in while the10b bandwidth There was a bandwidth slightlyfrequency larger. Theband envelope Figure also showswas that,Fs/8. although the of the resonant frequency band was 2500 Hz, while the bandwidth was Fs/8. There was a certain fault feature frequency Hz was extracted, frequency doubling were notHz, obvious certain error between the 130 central band of theitsresonance and the components actual value 2000 and the error between the central band of the resonance and the actual value 2000 Hz, and the bandwidth due to was a lowslightly signal-to-noise Finally, the proposedin method in order to deal with the bandwidth larger. ratio. The envelope spectrum Figurewas 10bapplied also shows that, although was slightly larger.signal. The envelope spectrum in Figure 10b11.also shows that, frequency although band the fault feature the synthetic The results are shown in Figure The resonance of the fault feature frequency 130 Hz was extracted, its frequency doubling components were not obvious frequency 130 Hz was extracted, its frequency doubling components were not obvious due to maximum Shannon entropy value is represented by the red line area in Figure 11a, where thea low due to a low signal-to-noise ratio. Finally, the proposed method was applied in order to deal with signal-to-noise ratio. was Finally, proposed method applied inwas order to deal the synthetic frequency center closethe to 2000 Hz and where was the bandwidth Fs/32. Fromwith the envelope the synthetic signal. The results are shown in Figure 11. The resonance frequency band of the spectrum (Figure it can seen that the fault featurefrequency frequency band and frequency doubling (130 signal. The results are11b), shown in be Figure 11. The resonance of the maximum Shannon maximum Shannon entropy value is extracted. represented by the red line area in Figure 11a, where the Hz, 260 Hz, 390 Hz) were accurately entropy value is represented by the red line area in Figure 11a, where the frequency center was frequency center was close to 2000 Hz and where the bandwidth was Fs/32. From the envelope close to 2000 Hz and where the bandwidth was Fs/32. From the envelope spectrum (Figure 11b), spectrum (Figure 11b), it can be seen that the fault feature frequency and frequency doubling (130 it can be seen that the fault feature frequency and frequency doubling (130 Hz, 260 Hz, 390 Hz) Hz, 260 Hz, 390 Hz) were accurately extracted. were accurately extracted.
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(b)
Figure 8. The resonance band detection using FSK: (a) the kurtogram, and (b) the envelope spectrum of the signal after band pass filtering.
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(b)
Figure Figure8. 8.The Theresonance resonanceband banddetection detectionusing usingFSK: FSK:(a) (a)the thekurtogram, kurtogram,and and(b) (b)the theenvelope envelopespectrum spectrum of the signal after band pass filtering. of the signal after band pass filtering.
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Figure 9. The resonance band detection using the infogram: (a) the SE1/2/SES1/2 infogram; and (b) the
Figure 9. The resonance band detection usingthe theinfogram: infogram: (a) the the SE /SES 1/2 infogram; and (b) the Figure 9. The resonance band detection using SE1/2 /SES and (b) the 1/2 1/2 infogram; Figure 9. spectrum The resonance band detection using the infogram: (a) (a) the SE 1/2 /SES 1/2 infogram; and (b) the envelope of the signal after band pass filtering. envelope spectrum of the signal after band pass filtering. envelope spectrum of the signal after envelope spectrum of the signal afterband bandpass pass filtering. filtering.
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Figure Figure 10. 10. The The resonance resonance band band detection detection using using the the kurtosis kurtosis spectrum spectrum of of the the wavelet wavelet packet: packet: (a) (a) the the
Figure Theresonance resonance band using the kurtosis spectrum of the wavelet (a) the Figure 10. 10.The banddetection detection using the kurtosis spectrum of thepacket: wavelet packet: kurtogram; kurtogram; and and (b) (b) the the envelope envelope spectrum spectrum of of the the signal signal after after band band pass pass filtering. filtering. kurtogram; and (b) the envelope spectrum of the signal after band pass filtering. (a) the kurtogram; and (b) the envelope spectrum of the signal after band pass filtering.
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(b) (b) (b)
Figure Figure 11. 11. The The resonance resonance band band detection detection using using the the proposed proposed method: method: (a) (a) the the Shannon Shannon entropy entropy Figure 11.ofThe resonance bandand detection using thespectrum proposedof method: (a) theband Shannon entropy spectrum the wavelet packet; (b) the envelope the signal after filtering. Figure 11. Theofresonance band detection the proposed method: (a) theafter Shannon entropy spectrum spectrum the wavelet packet; and (b)using the envelope spectrum of the signal band pass pass filtering. spectrum of the wavelet packet; and (b) the envelope spectrum of the signal after band pass filtering.
of the wavelet packet; and (b) the envelope spectrum of the signal after band pass filtering. 5. Experimental 5. Experimental Analysis Analysis 5. Experimental Analysis
5. Experimental Analysis 5.1. Experiment 1
5.1. Experiment Experiment 11 5.1. In the effectiveness the 5.1. Experiment 1 to In order order to further further verify verify the effectiveness of of the proposed proposed method, method, the the simulation simulation experiment experiment In order to further verify the effectiveness ofcompleted the proposed method, the simulation experiment on the outer ring fault of the rolling bearing was on the simulation experiment platform. on the outer ring fault of the rolling bearing was completed on the simulation experiment on the outer ring fault of the rolling bearing was completed on the simulation experiment platform. platform.
In order to further verify the effectiveness of the proposed method, the simulation experiment on the outer ring fault of the rolling bearing was completed on the simulation experiment platform. The bearing model was 6205E. The grooves, which were 1.5 mm deep and 0.2 mm wide, were processed in the outer ring in order to simulate the bearing failure. The vibration signal was collected via the
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The bearing model was 6205E. The grooves, which were 1.5 mm deep and 0.2 mm wide, were processed in the outer in The ordergrooves, to simulate bearing failure. signal was The bearing model was ring 6205E. whichthe were 1.5 mm deepThe andvibration 0.2 mm wide, were Entropy 2018, 20, x FOR PEER REVIEW 11 ofand 16 the acceleration sensor mounted on the bearing seat. The sampling frequency was 12,800 Hz, collected sensor on the The The sampling frequency processedvia in the the acceleration outer ring in ordermounted to simulate the bearing bearingseat. failure. vibration signal was motor speed was 1466 r/min. Figure 12mounted shows the structural diagram ofsampling the test platform. 12,800 Hz, and theacceleration motor speed was r/min. Figure 12 shows thedeep structural diagram of theFigure test 13 collected via the sensor on thewere bearing The frequency was The bearing model was 6205E. The 1466 grooves, which 1.5 seat. mm and 0.2 mm wide, were shows the outer fault The bearing parameters are shown in Table platform. 13bearing. showsring the outer fault Thethe bearing parameters are3.shown in Table 3. was 12,800 Hz,Figure and the motor speed 1466 r/min. Figure 12 shows the structural diagram of the test processed in the outer in was order tobearing. simulate bearing failure. The vibration signal platform. shows the outer fault bearing.onThe parameters shown frequency in Table 3. was collected Figure via the13acceleration sensor mounted thebearing bearing seat. The are sampling 12,800 Hz, and the motor speed was 1466 r/min. Figure 12 shows the structural diagram of the test platform. Figure 13 shows the outer fault bearing. The bearing parameters are shown in Table 3.
Figure 12. The experiment platform. 1. The drive motor; 2. The coupling; 3. The bearing housing with
Figure 12. The experiment platform. 1. The drive motor; 2. The coupling; 3. The bearing housing with the normal bearing seat; 4. platform. The loading device; 5. motor; The bearing with the bearing fail bearing seat. with Figure 12. The experiment 1. The drive 2. Thehousing coupling; 3. The housing the normal bearing seat; 4. The loading device; 5. The bearing housing with the fail bearing seat. the normal bearing seat; 4. The loading device; 5. The bearing housing with the fail bearing seat.
Figure 12. The experiment platform. 1. The drive motor; 2. The coupling; 3. The bearing housing with the normal bearing seat; 4. The loading device; 5. The bearing housing with the fail bearing seat.
Figure 13. Bearing outer ring fault. Figure 13. Bearing outer ring fault. Figure Bearing of outer ring fault. Table 3. The13. parameters the rolling bearing. Table 3. The parameters of the rolling bearing.
Figure 13. Bearing outer ring fault. Type Parameter TableParameter 3. The parameters of the rolling Values bearing. Parameter Type Parameter Inside Diameter (mm) 25 Values Table 3.Diameter The parameters bearing. Outside (mm) of the rolling 52 Inside Diameter (mm) 25 Parameter Type Parameter Values Ball Diameter (mm) 7.9 Outside Diameter (mm) Parameter 52 Values Parameter Inside Diameter (mm) Type 25 Pitch Diameter (mm) 39 Ball Diameter (mm) 7.9 Diameter 25 OutsideInside Diameter (mm) (mm) 52 Number of balls 9 Pitch Diameter (mm) 39 Ball Diameter (mm) 7.9 Outside Diameter (mm) 52 Pitch Diameter (mm) 39 Contact angle (°) 09 Number of balls Ball Diameter (mm) 7.9 Number of balls 9 Contact angle (°) 0 Pitch Diameter (mm) 39 Contact angle (◦ ) 0 The characteristic frequencyNumber of the bearing of ballsouter ring fault9is calculated as: The characteristic frequencyContact of the bearing angle Z d(°)outerNring fault0is calculated as:
(1- cos ) 87.7Hz The characteristic frequency of thef obearing 2 D Z d outer60 Nring fault; is calculated as:
(6) (6) The characteristic frequency of the bearing 2 D outer60ring fault;is calculated as: 8192 data points of the collected d signals N were selected for analysis, and the time Z vibration d cosin (1isZ−shown α)N 87.7Hz = 14. 87.7selected Hz; for analysis, and the time fring o =fault domain waveform of the 8192 data points of outer the collected (1-D cossignals ) Figure 2f o vibration 60 were (6) 2 D 60 ; domain waveform of the outer ring fault is shown in Figure 14. fo
(1-
cos )
87.7Hz
(6)
8192 8192 data points of theofcollected vibration signals werewere selected for analysis, andand the time domain data points the collected vibration signals selected for analysis, the time waveform of waveform the outer of ring shown Figurein14. domain thefault outerisring fault in is shown Figure 14.
Figure 14. The waveforms of the outer ring fault. Figure 14. The waveforms of the outer ring fault. Figure 14. The waveforms of the outer ring fault.
Figure 14. The waveforms of the outer ring fault.
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The vibration signal was analyzed via the FSK. The results are shown in Figure 15. As can be Thevibration vibrationsignal signalwas wasanalyzed analyzedvia viathe theFSK. FSK.The Theresults resultsare areshown shownin inFigure Figure15. 15.As Ascan canbe be The seen from Figure 15a, the central frequency of the resonant frequency band determined by the FSK seen from Figure 15a, the central frequency of the resonant frequency band determined by the FSK seen from Figure 15a, the central frequency of the resonant frequency band determined by the FSK was 890 Hz. The band pass filter was carried out to obtain the resonance frequency band shown in was890 890Hz. Hz.The Theband bandpass passfilter filterwas wascarried carriedout outto toobtain obtainthe theresonance resonancefrequency frequencyband bandshown shownin in was Figure 15a, and the envelope spectrum of the filtered signal was analyzed. The result isis shown in Figure 15a, and the envelope spectrum of the filtered signal was analyzed. The result shown Figure 15a, and the envelope spectrum of the filtered signal was analyzed. The result is shown inin Figure 15b. There was no useful information in the envelope spectrum. Therefore, the FSK method Figure15b. 15b.There Therewas wasno nouseful usefulinformation informationin inthe theenvelope envelopespectrum. spectrum.Therefore, Therefore,the theFSK FSKmethod method Figure did not extract the fault fault featureinformation. information. Followingthis, this, the vibration signal was analyzed via didnot notextract was analyzed viavia the did the fault feature feature information. Following Following this,the thevibration vibrationsignal signal was analyzed the kurtosis spectrum of the wavelet packet, and the results are shown in Figure 16.can Asbe can be seen kurtosis spectrum of the wavelet packet, and the results are shown in Figure 16. As seen from the kurtosis spectrum of the wavelet packet, and the results are shown in Figure 16. As can be seen from Figure 16a, the frequency central frequency of the resonant band frequency band was 2300 Hz, and was the Figure 16a, the central of the resonant was 2300 Hz,was and2300 the bandwidth from Figure 16a, the central frequency of the frequency resonant frequency band Hz, and the bandwidth was Fs/8. It can also beenvelope seen from the envelope spectrum in Figure 16b that the frequency Fs/8. It canwas alsoFs/8. be seen from spectrum in Figure 16b that frequency and frequency bandwidth It can alsothe be seen from the envelope spectrum in the Figure 16b that theitsfrequency and its frequency doubling components of the fault were by seriously affectedofby the frequency of the doubling components of the fault were seriously affected the frequency the noise, and that and its frequency doubling components of the fault were seriously affected by the frequency of they the noise, and that they were not easily discernible. Finally, the proposed method was used to deal with were not easily discernible. Finally, the proposed method was used to deal with the vibration signal. noise, and that they were not easily discernible. Finally, the proposed method was used to deal with the vibration signal. The results are shown in Figure 17. The resonance frequency band of the Thevibration results are shownThe in Figure The resonance frequency bandresonance of the maximum Shannon the signal. results17.are shown in Figure 17. The frequency band entropy of the maximum Shannon entropy value is showncenter in Figure 17a; the frequency center was close towas 2750 Hz, value is shown in Figure 17a; the frequency was close to 2750 Hz, and the bandwidth maximum Shannon entropy value is shown in Figure 17a; the frequency center was close to 2750Fs/16. Hz, and thethe bandwidth was Fs/16.(Figure From the envelope spectrum (Figure 17b), it can be seenand thatfrequency the fault From envelope spectrum 17b), it can be seen that the fault feature frequency and the bandwidth was Fs/16. From the envelope spectrum (Figure 17b), it can be seen that the fault feature frequency and frequency doubling (87.5 Hz, 176.6 Hz, 265.6 Hz) were accurately extracted. doubling (87.5 Hz, 176.6 Hz, 265.6 Hz) were accurately extracted. feature frequency and frequency doubling (87.5 Hz, 176.6 Hz, 265.6 Hz) were accurately extracted.
(a) (a)
(b) (b)
Figure 15. Theresonance resonance band detection using FSK: (a) the kurtogram; and (b) the envelope Figure15. 15. using FSK:FSK: (a) the and (b)and the envelope spectrum Figure The resonanceband banddetection detection using (a)kurtogram; the kurtogram; (b) the envelope spectrum of the signal after band pass filtering. of the signal after band pass filtering. spectrum of the signal after band pass filtering.
(a) (a)
(b) (b)
Figure 16. The resonance band detection using the kurtosis spectrum of the wavelet packet: (a) the Figure 16. The resonance band detection using the kurtosis spectrum of the wavelet packet: (a) the Figure 16. and The band detection using theafter kurtosis spectrum of the wavelet packet: kurtogram; (b)resonance the envelope spectrum of the signal band pass filtering. kurtogram; and (b) the envelope spectrum of the signal after band pass filtering. (a) the kurtogram; and (b) the envelope spectrum of the signal after band pass filtering.
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Figure 17. The resonance band detection using the proposed method: (a) the Shannon entropy
Figure 17. The resonance band Shannon entropy spectrum (a) detection using the proposed method: (a) the(b) spectrum of the wavelet packet; and (b) the envelope spectrum of the signal after band pass filtering. of the wavelet packet; and (b) the envelope spectrum of the signal after band pass filtering. Figure 17. The resonance band detection using the proposed method: (a) the Shannon entropy
5.2.spectrum Experiment 2 wavelet packet; and (b) the envelope spectrum of the signal after band pass filtering. of the
5.2. Experiment 2
The effectiveness of the proposed algorithm was further verified via the bearing experiment 5.2. Experiment 2 data the rolling of bearings at Case algorithm Western Reserve University in the United Statesexperiment [24]. The data The of effectiveness the proposed was further verified via the bearing The effectiveness of the proposed algorithm was further verified via the bearing experiment experimental equipment is shown in Figure 18. The bearing model was JEMSKF6023-2RS. The most of the rolling bearings at Case Western Reserve University in the United States [24]. The experimental data of the rolling bearings at Case Western Reserve University in the United States [24]. The minor dimensions of the 0.007-inch fault data were analyzed. The sampling frequency was 12 kHz,minor equipment is shown in Figure 18. The bearing model was JEMSKF6023-2RS. The most experimental equipment is shown inwas Figure 18.r/min. The bearing modelparameter was JEMSKF6023-2RS. The mostin and the rotation speed of the shaft 1772 The bearing dimensions are shown dimensions of the 0.007-inch fault data were analyzed. The sampling frequency was 12 kHz, minor of the 0.007-inch were analyzed. The sampling frequency Tabledimensions 4, and the frequency of each fault fault data characteristic of the bearing is shown in Table was 5. 12 kHz, and the rotation speed of the shaft was 1772 r/min. The bearing parameter dimensions are shown in and the rotation speed of the shaft was 1772 r/min. The bearing parameter dimensions are shown in Table 4, 4, and eachfault faultcharacteristic characteristic of the bearing is shown in Table 5. Table andthe thefrequency frequency of of each of the bearing is shown in Table 5.
Figure 18. The bearing test stand at Case University. Parameters of theat rolling bearing. Figure Table 18. The4.bearing test stand Case University.
Figure 18. The bearing test stand at Case University. Parameter Type Parameter Values Table 4. Parameters of the rolling bearing. Inside Diameter (mm) 17bearing. Table 4. Parameters of the rolling Parameter Type (mm) Parameter40 Values Outside Diameter Inside Diameter (mm) 17 Ball Diameter (mm) 6.7 Parameter Type Parameter Values Outside 40 Pitch Diameter Diameter (mm) (mm) 28.5 Inside Diameter (mm) 17 BallNumber Diameterof(mm) 6.78 balls Outside Diameter (mm) 40 PitchContact Diameter (mm) 28.50 angle (°) Ball Diameter (mm) 6.7 Number(mm) of balls 8 Pitch Diameter 28.5 Table 5. Fault feature frequency of the rolling bearing. Contact angle (°) 0 Number of balls 8
angle (◦ )Outer Ring Inner Ring Rolling 0 Element Types Contact of Failures Table 5. Fault feature frequency of the rolling bearing. Defect frequencies (Hz) 90 145 118 5. Fault feature frequency of the rolling bearing. Types ofTable Failures Outer Ring Inner Ring Rolling Element Defect frequencies (Hz) 90 145 118 Types of Failures Outer Ring Inner Ring Rolling Element Defect frequencies (Hz)
90
145
118
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The time domain waveform of the 0.007-inch rolling element fault data isisshown shown in Figure 19. The time time domain domainwaveform waveformof ofthe the0.007-inch 0.007-inchrolling rolling element elementfault faultdata datais shownin inFigure Figure19. 19. The The time domain waveform of the 0.007-inch rolling element fault data is shown in Figure 19. The analysis of the the vibration vibration signal signalby byFSK FSKisis isshown showninin inFigure Figure20. 20.The The center frequency determined The analysis vibration signal by FSK shown Figure 20. The center frequency determined The analysis of center frequency determined by The analysis of the vibration signal by FSK is shown in Figure 20. The center frequency determined by the kurtogram method was about 1600 Hz. The selected frequency band was filtered, and the thethe kurtogram method waswas about 16001600 Hz. The frequency band band was and the failure by kurtogram method about Hz. The frequency was and the by the kurtogram method was about 1600 Hz. selected The selected selected frequency bandfiltered, was filtered, filtered, and the failure frequency information was not obtained obtained from the spectrum. spectrum. Following this, the vibration vibration failure frequency information was not from the Following this, the frequency information was not obtained from the spectrum. Following this, the vibration signal failure frequency information was not obtained from the spectrum. Following this, the vibration signal was analyzed analyzed via the bothinfogram the infogram infogram (SE 1/2/SES /SES 1/2and ) andthe theproposed proposedalgorithm. algorithm. As As can can be seen seen was analyzed via both (SE1/2 /SES seen signal was via (SE 1/2 1/2 )1/2 signal was analyzed via both both the the infogram (SE 1/2/SES 1/2)) and and the the proposed proposed algorithm. algorithm. As As can be be seen from Figure 21a and Figure 22a, the two methods could both find the main resonance frequency from 22a, the two both find the main frequency from Figure Figures21a 21aand andFigure 22a, the two methods could could both main frequency bands. from Figure 21a and Figure 22a, the two methods methods couldfind boththe find theresonance main resonance resonance frequency bands. Additionally, it can can befrom seen the fromenvelope the envelope envelope spectrum (Figures 21b and and Figure 22b) that the the bands. Additionally, it be seen from the spectrum (Figures 21b Figure 22b) that Additionally, it can be seen spectrum (Figures 21b and 22b) that the frequency bands. Additionally, it can be seen from the envelope spectrum (Figures 21b and Figure 22b) that the frequency information of the fault feature could be found via the two methods. Compared with the frequency information of the feature could via the Compared the information of the fault could be found viafound the two Compared with the with analysis frequency information offeature the fault fault feature could be be found viamethods. the two two methods. methods. Compared with the analysis results obtained obtained through the two two methods, methods, it can can bethe seen that the thedoubling frequencycomponent doubling analysis results through the it be seen that frequency doubling results obtained through the two methods, it can be seen that frequency analysis results obtained through the two methods, it can be seen that the frequency doubling component (2f o,, 3f 3fo) in in Figure Figure 21b obvious, was more morebut obvious, but thatinterference the noise noise interference was larger.22b, In component o 21b obvious, that larger. (2f o , 3f o ) in(2f Figure was more that thebut noise was larger. was In Figure component (2f o, 3foo))21b in Figure 21b was was more obvious, but that the the noise interference interference was larger. In In Figure 22b, the noise control was better and the frequency of the fault was clearer. However, the Figure 22b, the was better and frequency the fault clearer. the the noise wascontrol better and frequency the faultof was However, the frequency Figure 22b,control the noise noise control was the better and the the of frequency of theclearer. fault was was clearer. However, However, the frequency doubling component component was because not clear clear because of the the frequency division of the the wavelet wavelet packet. packet. frequency doubling was of division of doubling component was not clear the frequency division of the wavelet frequency doubling component was not not clearofbecause because of the frequency frequency division of the packet. wavelet packet.
Figure 19. 19. The The time time domain domain waveform waveform of of the the rolling rolling element element fault. fault. Figure Figure Figure 19. 19. The The time time domain domain waveform waveform of of the the rolling rolling element element fault. fault.
(a) (a) (a)
(b) (b) (b)
Figure 20. 20. The resonance resonance band detection detection using FSK: FSK: (a) the the kurtogram; (b) (b) the envelope envelope spectrum of of Figure Figure 20. The The resonance band band detection using using FSK: (a) (a) the kurtogram; kurtogram; (b) the the envelope spectrum spectrum of the signal after band pass filtering. the the signal signal after after band band pass pass filtering. filtering.
(a) (a) (a)
(b) (b) (b)
Figure 21. The Theresonance resonance band detection using the infogram: infogram: (a) the/SES SE1/2 1/2/SES /SES 1/2 infogram; infogram; and Figure 21. band detection using the the 1/2 Figure 21. band detection using the infogram: (a) the(a) SE1/2 infogram; and (b)and the 21. The The resonance resonance band detection using the infogram: (a) the SE SE1/2 /SES 1/2 infogram; and 1/2 (b) the envelope spectrum of the signal after band pass filtering. (b) the envelope spectrum of the signal after band pass filtering. envelope spectrum of the signal band filtering. (b) the envelope spectrum of theafter signal afterpass band pass filtering.
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(b)
Figure 22. The Theresonance resonance band detection using the proposed method: (a) the Shannon entropy Figure 22. band detection using the proposed method: (a) the Shannon entropy spectrum spectrum of the wavelet packet; and (b) the envelope spectrum of the signal after band pass filtering. of the wavelet packet; and (b) the envelope spectrum of the signal after band pass filtering.
By comparing the proposed method with the infogram, we found that the proposed method By comparing the proposed method with the infogram, we found that the proposed method has has two advantages. One of them is that the wavelet packet decomposition has better spectral two advantages. One of them is that the wavelet packet decomposition has better spectral segmentation segmentation properties than Fourier decomposition (or FIR filter), so the envelope spectrum of this properties than Fourier decomposition (or FIR filter), so the envelope spectrum of this method has method has higher SNR. On the other hand, by calculating the Shannon entropy of wavelet packet higher SNR. On the other hand, by calculating the Shannon entropy of wavelet packet subband instead subband instead of calculating the negentropy of the squared envelope (SE) and the squared of calculating the negentropy of the squared envelope (SE) and the squared envelope spectrum (SES), envelope spectrum (SES), the calculation method in this paper is more direct and efficient, which the calculation method in this paper is more direct and efficient, which can avoid envelope error can avoid envelope error generated by envelope process, and save operation time. generated by envelope process, and save operation time. 6. Conclusions 6. Conclusions Fast spectral kurtosis is one of the most classical methods in bearing fault diagnosis. However, Fast spectral kurtosis is one of the most classical methods in bearing fault diagnosis. However, because of the limitations of its own algorithm, it is not possible to accurately extract fault because of the limitations of its own algorithm, it is not possible to accurately extract fault information information in the presence of strong background noise and random knocks. In view of the in the presence of strong background noise and random knocks. In view of the limitation of FSK, limitation of FSK, we proposed a new method of spectral analysis. The signal is segmented with a we proposed a new method of spectral analysis. The signal is segmented with a certain central certain central frequency and bandwidth by using the binary wavelet packet transform, which has frequency and bandwidth by using the binary wavelet packet transform, which has excellent frequency excellent frequency segmentation characteristics. After this, the Shannon entropy (reciprocal) of each segmentation characteristics. After this, the Shannon entropy (reciprocal) of each subband is calculated. subband is calculated. The appropriate center frequency and bandwidth have been chosen for The appropriate center frequency and bandwidth have been chosen for filtering, the envelope filtering, the envelope spectrum of the filtered signal is analyzed, and the faulty feature information spectrum of the filtered signal is analyzed, and the faulty feature information is obtained via the is obtained via the envelope spectrum. Through simulation and experimental verification, the envelope spectrum. Through simulation and experimental verification, the following conclusions were following conclusions were obtained. Shannon entropy is—to some extent—better than kurtosis as a obtained. Shannon entropy is—to some extent—better than kurtosis as a frequency selective index, frequency selective index, and the Shannon entropy of the binary wavelet packet transform method and the Shannon entropy of the binary wavelet packet transform method is more accurate for fault is more accurate for fault feature extraction. feature extraction. Acknowledgments: Acknowledgments: This This work work was was supported supported by by the the Fundamental Fundamental Research Research Funds Funds for for the the Central Central Universities Universities (No.2017XS133, No.2018QN093) and the National Natural Science Foundation of China (No. (No. 2017XS133, No. 2018QN093) and the National Natural Science Foundation of China (No.51777075). 51777075). Contributions: Shuting ShutingWan Wanconceived conceivedand anddesigned designed experiments; Longjiang performed Author Contributions: thethe experiments; Longjiang DouDou performed the the experiments; Zhang Longjiang Douanalyzed analyzedthe the data; data; Xiong Xiong Zhang Zhang contributed experiments; XiongXiong Zhang and and Longjiang Dou contributed reagents/materials/analysis tools and wrote the paper. reagents/materials/analysis tools and wrote the paper. Conflicts of Interest: The authors declare no conflict of interest. Conflicts of Interest: The authors declare no conflict of interest.
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