entropy Article
Rolling Bearing Fault Diagnosis Based on Optimal Notch Filter and Enhanced Singular Value Decomposition Bin Pang
ID
, Yuling He *, Guiji Tang, Chong Zhou and Tian Tian
School of Energy, Power and Mechanical Engineering, North China Electric Power University, Baoding 071000, China;
[email protected] (B.P.);
[email protected] (G.T.);
[email protected] (C.Z.);
[email protected] (T.T.) * Correspondence:
[email protected] Received: 18 May 2018; Accepted: 19 June 2018; Published: 21 June 2018
Abstract: The impulsive fault feature signal of rolling bearings at the early failure stage is easily contaminated by the fundamental frequency (i.e., the rotation frequency of the shaft) signal and background noise. To address this problem, this paper puts forward a rolling bearing weak fault diagnosis method with the combination of optimal notch filter and enhanced singular value decomposition. Firstly, in order to eliminate the interference of the fundamental frequency signal, the original signal was processed by the notch filter with the fundamental frequency as the center frequency and with a varying bandwidth to get a series of corresponding notch filter signals. Secondly, the Teager energy entropy index was adopted to adaptively determine the optimal bandwidth to complete the optimal notch filter analysis on the raw vibration signal and obtain the corresponding optimal notch filter signal. Thirdly, an enhanced singular value decomposition de-nosing method was employed to de-noise the optimal notch filter signal. Finally, the envelope spectrum analysis was conducted on the de-noised signal to extract the fault characteristic frequencies. The effectiveness of the presented method was demonstrated via simulation and experiment verifications. In addition, the minimum entropy deconvolution, Kurtogram and Infogram methods were employed for comparisons to show the advantages of the presented method. Keywords: Teager energy entropy; notch filter; enhanced singular value decomposition; rolling bearings; fault diagnosis
1. Introduction Rolling bearings are one of the most critical, but one of the most easily damaged parts of rotating machinery [1]. If the defects of bearings can not be detected in time, it will cause severe safety accidents [2]. Developing effective techniques to diagnose the faults of rolling bearings as early as possible is significant in preventing the adverse effects caused by the deterioration of bearing faults [3]. The actual vibration signals of bearings with defects consist of multiple components, and the impulsive fault feature signals are easily affected by the background noise, harmonic interference signals and transmission path [4]. Therefore, plenty of signal processing methods, such as de-noising analysis, deconvolution, signal decomposition and resonance demodulation, have been introduced to address the issue of early fault diagnosis of bearings under different situations. To reduce the effect of noise on the fault feature signals, different kinds of signal de-noising methods such as singular value decomposition (SVD) [5], mathematical morphology filter [6], nonlocal means de-noising [7], and stochastic resonance [8] have been widely used to de-noise the vibration signal of bearings. Although these developed de-noising methods can pure the vibration signal, they can’t remove the
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other interference signals except background noise. It is still very difficult to identify the fault features when other interference signals except background noise exist. Therefore, the signal de-noising approaches are always combined with some other signal analysis approaches to get a good fault diagnosis result [9]. Deconvolution approaches such as minimum entropy deconvolution (MED) [10], maximum correlated kurtosis deconvolution (MCKD) [11], sparse maximum harmonics-to-noise-ratio deconvolution (SMHD) [12], Lucy-Richardson deconvolution [13] and multipoint optimal minimum entropy deconvolution [14] have been researched and widely used to recover the impulsive fault feature signal from the measured vibration signal. Some indicators, such as kurtosis, maximum correlated kurtosis and harmonics-to-noise-ratio are taken as the targets and a designed FIR filter is performed on the vibration signal to obtain the filtered output reaches a maximum value of the target [15,16]. The deconvolution methods have shown their effectiveness in detecting bearing faults, but they have some limitations. MED takes the kurtosis indicator as the target. Kurtosis is sensitive to random impulses, which can cause erroneous deconvolution. For MCKD, complex parameter problems must be solved in order to achieve superior results. In the principle of SMHD, the harmonics-to-noise-ratio index should be accurately estimated by using prior knowledge. Lucy-Richardson deconvolution seems to be ineffective under the condition of strong background noise. To separate the fault feature signal from the interferences, various self adaptive decomposition techniques, such as empirical mode decomposition [17], variational mode decomposition [18], and empirical wavelet decomposition [19] have been adopted for bearing failure detection. The fault features may be carried by one of the decomposed component signals, but the abnormal events, including noise and other types of interferences, can cause the problem of mode mixing. When mode mixing occurs, the fault characteristics are difficult to identify [20]. Resonance demodulation methods such as Kurtogram [21], Autogram [22], and Infogram [23] have been widely employed to detect transient vibration signatures. The analysis results of the resonance demodulation methods are mainly affected by two aspects: (a) the segmentation of the frequency bands; (b) the selection indicator of the optimal frequency band. If the fault feature signal can’t be divided into one of the frequency bands successfully or the optimal frequency band is errousneously selected, the resonance demodulation methods will lose efficiency [24,25]. As a supporting component, the rolling bearing contacts with the rotation shaft directly so that the fundamental frequency signal, i.e., the rotating frequency signal, performs very prominent in the acquired vibration signals. The fundamental frequency signal is one of the most common harmonic interference signals. In addition, the background noise seems to be always an influence factor. With the compound influence of the fundamental frequency and the noise, the impulsive fault feature information tends to be easily submerged and difficult to be recognized and traditional bearing fault diagnosis methods lack pertinence for this fault situation. It is of great significance to develop an effective technique to eliminate the compound interference of the fundamental frequency signal and background noise. Notch filter with a designed center frequency and bandwidth can directly suppress the harmonic components of the signal [26]. Considering the excellent characteristics of notch filter in restraining the harmonics, the notch filter analysis is adopted to inhibit the interference of the fundamental frequency signal by setting the fundamental frequency as the center frequency in this paper. Compared with the de-noising, deconvolution, signal decomposition and resonance demodulation methods, the notch filter is more targeted for suppressing the fundamental frequency signal. Previous studies demonstrate that the bandwidth of notch filter is a key parameter that can directly impact the analysis results [27]. To set the optimal bandwidth of the notch filter adaptively, it is necessary to find an effective index to measure the richness of the fault feature information contained in the notch filter signals corresponding to different bandwidths. Teager energy operator (TEO) has the function of tracking and strengthening the impulsive shocks [28]. Entropy is able to reflect the regularity of the impulsive shocks [29]. Combing the advantages of TEO and entropy, Teager energy entropy (TEE), as an impulsive characteristic evaluation index, is presented in this paper to determine the optimal bandwidth of the notch filter adaptively. The filtered signal corresponding to the optimal
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bandwidth is called optimal notch filter signal in this paper. Singular value decomposition (SVD) is an effective approach for signal de-noising, which has been widely used in bearing fault diagnosis due to its superiority over other de-noising methods [30]. Based on the principle of SVD, we proposed the enhanced singular value decomposition (ESVD) method to suppress the noise carried by the optimal notch filter signal more effectively. A novel bearing fault diagnosis method with the combination of optimal notch filter based on TEE index and ESVD de-noising was presented to eliminate the compound interference of the fundamental frequency signal and background noise. The validity of the proposed method was verified by simulated and experimental signals. The remained contents are organized as follows: Section 2 gives a specific introduction of the theory background, the principle of the notch filter is presented, a TEE index is introduced and evaluated, detailed steps of the presented method are exhibited. The presented method is investigated by simulated and experimental signals in Sections 3 and 4 individually. Finally, Section 5 presents a few conclusions. 2. Theory Background 2.1. Optimal Notch Filter Based on Teager Energy Entropy Index 2.1.1. Notch Filter Notch filter is an effective tool to eliminate the harmonic interferences. The basic principle of notch filter to eliminate the harmonic interferences is conducting a narrow bandwidth filter on the original. By setting the center frequency of the notch filter as the rotation frequency of the shaft, the fundamental frequency signal can be inhibited. So, the notch filter is adopted in this paper for processing the vibration signal of bearings. The second order digital notch filter is discussed in this part. To filter the harmonic interference signal with the frequency of ω 0 , the frequency- amplitude curve of the filter should be zero at ω 0 and almost be constant at the other frequencies [26]. When the zero point satisfies, z = e±jω0 , the frequency-amplitude curve of the filter is zero at the frequencies of ω = ±ω 0 . In order to increase the amplitude to a constant instantly once the frequencies are not ±ω 0 , two extreme points are set near the zero points. The extreme point is described as: z = re±jω0 , and the transfer function of the notch filter is shown in Equation (1): H (z) =
b0 (z − ejω0 )(z − ejω0 ) b0 [1 − (2 cos ω0 )z−1 + z−2 ] = 1 − 2r (cos ω0 )z−1 + r2 z−2 (z − rejω0 )(z − rejω0 )
(1)
where 0 ≤ r ≤ 1, the value of which depends on the bandwidth of the notch filter, b0 denotes the gain factor. The frequency band of notch filter at −3 dB is defined as the bandwidth (Bw). The bandwidth Bw and the gain coefficient b0 can be expressed as: Bw = (1 − r ) f s , b0 =
1 − 2r cos ω0 + r2 2|1 − cos ω0 |
.
(2)
where fs represents the sampling frequency. When normalized frequency is used, the sampling frequency fs is 1 and the bandwidth Bw equals to 1 − r. The frequency-amplitude curve of notch filter varies with Bw. Figure 1 displays the frequency-amplitude curve of the notch filter with the bandwidth of 0.01, 0.05 and 0.1, respectively. As depicted in Figure 1, the amplitude of the frequency components around ω = ±ω 0 varies severely when Bw changes. Once Bw decreases, the depth of notch will decrease as well. The frequency-amplitude curve of the notch filter demonstrates that the selection of Bw has a remarkable influence on the analysis results of notch filter. A proper value of Bw should be selected to achieve the best result when applying the notch filter to the vibration signals of bearings.
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0
Amplitude (dB)
-50 -100
Bw=0.1 Bw=0.05
-150
Bw=0.01
-200 -250 -300 0.3
0.35 0.4 0.45 Frequency (Normalization frequency)
0.5
Figure 1. Frequency‐amplitude curve of the notch with different bandwidths. Figure 1. Frequency-amplitude curve of the notch with different bandwidths.
2.1.2. Teager Energy Entropy Index 2.1.2. Teager Energy Entropy Index When local defects defects appear in bearings, periodic impulses be generated. Some fault When local appear in bearings, periodic impulses will bewill generated. Some fault diagnosis diagnosis methods of bearings are parametric‐based analysis methods. In order to get the optimal methods of bearings are parametric-based analysis methods. In order to get the optimal results, results, a number of evaluation indexes have been employed by scholars to adaptively choose the a number of evaluation indexes have been employed by scholars to adaptively choose the parameters. parameters. as a cumulative fourth‐order cumulative is one of known the most known used and Kurtosis, as aKurtosis, fourth-order statistic, is onestatistic, of the most well andwell frequently frequently higher value of represents the kurtosis represents the features periodic indexes. A used higherindexes. value ofA the kurtosis index theindex periodic impulsive of impulsive the signal features of the signal are richer. But kurtosis is very sensitive to the accidental impulses [31]. When are richer. But kurtosis is very sensitive to the accidental impulses [31]. When an accidental impulse an accidental impulse appears, the kurtosis value will increase sharply, which may lead to erroneous appears, the kurtosis value will increase sharply, which may lead to erroneous evaluations. In order evaluations. In order to overcome the shortcoming of the kurtosis index, a novel evaluation index, to overcome the shortcoming of the kurtosis index, a novel evaluation index, called Teager energy called Teager energy entropy (TEE), which combines the advantages of Teager energy operator entropy (TEE), which combines the advantages of Teager energy operator (TEO) and Shannon entropy, (TEO) and Shannon entropy, is introduced to select the optimal Bw of the notch filter to implement is introduced to select the optimal Bw of the notch filter to implement the optimal notch filter analysis. the optimal notch filter analysis. The TEO of one dimensional signal s(t) has the expression as [28]: The TEO of one dimensional signal s(t) has the expression as [28]: .
2
..
Ψ[s(t)] = [s(t)] 2 − s(t)s(t). [ s (t )] [ s (t )] s (t ) s (t ) .
(3) (3)
And Equation (4) shows the corresponding discrete form: And Equation (4) shows the corresponding discrete form: 2 (n)] 1) Ψ[ s([ns)] =[s[(sn(n)])]2 −ss((nn + 1)ss((nn−1)1. ).
(4) (4)
As reported in [28], TEO has a good quality of tracking the instantaneous energy of the signal, As reported in [28], TEO has a good quality of tracking the instantaneous energy of the signal, and this quality has been widely used for enhancing the impulsive features of the vibration signal. and this quality has been widely used for enhancing the impulsive features of the vibration signal. The entropy is able to reflect the sparsity of the signal and has shown its potentiality for fault The entropy is able to reflect the sparsity of the signal and has shown its potentiality for fault diagnosis [32,33]. By calculating the Shannon entropy [34] of the instantaneous energy signal diagnosis [32,33]. By calculating the Shannon entropy [34] of the instantaneous energy signal obtained obtained by performing TEO on the signal, a new index, that is the TEE index, can be described by by performing TEO on the signal, a new index, that is the TEE index, can be described by Equation (5): Equation (5): N TEE = −N∑ p j ln p j TEE j=1p j ln p j . (5) j 1 N . (5) p j = Q( j)/ ∑ N Q ( j ), Q ( j ) = abs ( Ψ [ s ( j )]) p Q( j ) / j=1 Q( j ), Q( j ) abs ([ s ( j )]) j j 1 When the original signal carries more impulse shocks, the sparsity of the signal is higher and When the original signal carries more impulse shocks, the sparsity of the signal is higher and the TEE value is smaller. In this paper, the TEE index is used to evaluate the impulsive features of the TEE value is smaller. In this paper, the TEE index is used to evaluate the impulsive features of the notch filtered signals to determine the optimal bandwidth and the corresponding optimal notch the notch filtered signals to determine the optimal bandwidth and the corresponding optimal notch filter signal. filter signal. To validate the effectiveness of the TEE index in reflecting the richness of the periodic impulsive To the and effectiveness of the TEE index reflecting the richness the periodic shocks ofvalidate the signal the advantage of the TEE indexin over the kurtosis index in of overcoming the impulsive shocks of the signal and the advantage of the TEE index over the kurtosis index in
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influence of the accidental impulses, five simulated signals were designed with the sampling frequency of overcoming the influence of the accidental impulses, five simulated signals were designed with the 8192 Hz for evaluation. The first simulated signal x1 (t) is used to simulate the periodic impulse signal of the outer race sampling frequency of 8192 Hz for evaluation. fault, which is shown in Equation1(t) is used to simulate the periodic impulse signal of the outer race (6) [35]: The first simulated signal x fault, which is shown in Equation (6) [35]: M −1
1 ( t − iT )] ∗ [ Ae− ξ2π f n t cos(2π f t − θ )], x1 (t) = [ ∑M Dh o x1 (t ) i=[ 0 Dh(t iTo )] [ Ae- 2 fn t cos(2 f d t d )] ,
(6)
i 0
(6)
where M denotes the number of the impulsive pulses, D is the pulse amplitude, h(t) indicates the where M denotes the number of the impulsive pulses, D is the pulse amplitude, h(t) indicates the unit pulse function, Too is the interval of the impulsive pulses, ξ is the interval of the impulsive pulses, ξ is the structural damping factor, unit pulse function, T is the structural damping factor, f n p 2 fn represents the natural frequency of the system, f d = f n 12 − ξ is the damping natural frequency, represents the natural frequency of the system, f d f n 1 is the damping natural frequency, A A and θ represent the amplitude and initial phase respectively. and θ represent the amplitude and initial phase respectively. The parameters were set as: M = 25, D = 1, To = 0.01 s, A = 2, ξ = 0.06, fn = 2000 Hz and θ = 0. The parameters were set as: M = 25, D = 1, To = 0.01 s, A = 2, ξ = 0.06, fn = 2000 Hz and θ = 0. Equation (7) is the expression of the second simulated signal x2 (t), which is used to simulate the Equation (7) is the expression of the second simulated signal x2(t), which is used to simulate the fundamental frequency signal: fundamental frequency signal: x2 (t) = 6 sin(2π f 1 t), (7) x2 (t ) 6sin(2 f1t ) , (7) where f 1 = 15 Hz is the fundamental frequency. where f 1 = 15 Hz is the fundamental frequency. The third simulated signal x3 (t) is used to simulate the noise, and its value can be generated in The third simulated signal x 3(t) is used to simulate the noise, and its value can be generated in MATLAB using A*randn(1,N), where A = 1.5 and N = 2048. MATLAB using A*randn(1,N), where A = 1.5 and N = 2048. The fourth simulated signal x4 (t) can be described as: x4 (t) = x1 (t) + x3 (t). The fifth simulated The fourth simulated signal x4(t) can be described as: x4(t) = x1(t) + x3(t). The fifth simulated signal x5 (t) was obtained by adding a single impulse to x3 (t). signal x5(t) was obtained by adding a single impulse to x3(t). Figure 2 depicts the waveforms of the five simulated signals and their calculated values of kurtosis Figure 2 depicts the waveforms of the five simulated signals and their calculated values of and TEE. As shown in Figure 2, the kurtosis value of x1 (t) is 14.4072, which is the highest and the TEE kurtosis and TEE. As shown in Figure 2, the kurtosis value of x1(t) is 14.4072, which is the highest value of x1 (t) is 5.9126, which is the smallest. x2 (t) and x3 (t) have small kurtosis values and high TEE and the TEE value of x1(t) is 5.9126, which is the smallest. x2(t) and x3(t) have small kurtosis values values. Both the kurtosis and TEE indexes measure the impulsive characteristics of x1 (t), x2 (t) andof x3 (t) and high TEE values. Both the kurtosis and TEE indexes measure the impulsive characteristics effectively. Whereas, the kurtosis value of x (t) is higher than that of x (t), which demonstrates that the 5 4 x1(t), x2(t) and x3(t) effectively. Whereas, the kurtosis value of x 5(t) is higher than that of x 4(t), which accidental pulse results in an erroneous evaluation when using the kurtosis index. The TEE value demonstrates that the accidental pulse results in an erroneous evaluation when using the kurtosis of x5 (t) is higher than that of x54(t) is higher than that of x (t), which demonstrates4that the TEE index can be also effective when an index. The TEE value of x (t), which demonstrates that the TEE index can accidental pulse appears. And it should be noticed that a smaller TEE value means that the periodic be also effective when an accidental pulse appears. And it should be noticed that a smaller TEE value pulse characteristics of the signal are more prominent. means that the periodic pulse characteristics of the signal are more prominent. Kurtosis=14.4072, T EE=5.9126
Kurtosis=1.5138, T EE=7.6217 10 x2(t)
x1(t)
5 0 -5
0
0.05
0.1 0.15 T ime (s)
0.2
0 -10
0.25
0
(a) Kurtosis=1.8094, T EE=7.2913
0.2
0.25
Kurtosis=5.3471, T EE=7.0848 5 x4(t)
x3(t)
0.1 0.15 T ime (s)
(b)
2 1 0
0.05
0
0.05
0.1 0.15 T ime (s)
0.2
0 -5
0.25
(c)
0
0.05
0.1 0.15 T ime (s)
(d) Figure 2. Cont.
0.2
0.25
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Kurtosis=13.3093, T EE=7.1173
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x5(t)
Kurtosis=13.3093, T EE=7.1173
x5(t)
06 -6 0 0 -6
0.05 0
0.1 0.15 T ime (s)
0.2
0.25
0.1 (e) 0.15 T ime (s)
0.2
0.25
0.05
Figure 2. The Five simulation signals: (a) x 4(t); (e) x5(t). Figure 2. The Five simulation signals: (a)(e) x11(t); (b) x (t); (b) x22(t); (c) x (t); (c) 3x(t); (d) x 3 (t); (d) x4 (t); (e) x5 (t). Figure 2. The Five simulation signals: (a) x1(t); (b) x2(t); (c) x3(t); (d) x4(t); (e) x5(t).
Then varying degrees of noises were added to the periodic shock feature signal of Figure 2a to Then varying degrees of noises were added to the periodic shock feature signal of Figure 2a to obtain the signals with different signal‐to‐noise ratios (SNRs). The periodic fault characteristics are Then varying degrees of noises were added to the periodic shock feature signal of Figure 2a to obtain the signals with different signal-to-noise ratios (SNRs). The periodic fault characteristics are more obvious when the SNR of the signal is higher. By calculating the TEE values of these signals, obtain the signals with different signal‐to‐noise ratios (SNRs). The periodic fault characteristics are more obvious when the SNR of the signal is higher. By calculating the TEE values of these signals, the the effectiveness of the TEE index for evaluating the degree of the periodic shock feature signal can more obvious when the SNR of the signal is higher. By calculating the TEE values of these signals, effectiveness of the TEE index for evaluating the degree of the periodic shock feature signal can be be further verified as Figure 3 shows. the effectiveness of the TEE index for evaluating the degree of the periodic shock feature signal can further verified as Figure 3 shows. be further verified as Figure 3 shows. T EE=6.9422
T EE=6.6077
3 Amplitude Amplitude
T EE=6.9422 3
Amplitude
Amplitude
3 0
-3
0
0 -3
0.05 0
0.05
0.1 0.15 T0.1 ime (s) 0.15
(a)
0.2 0.2
0.25
T ime (s)
(a) T EE=6.2811
0
0 -3
0
0.05
0.05
0.1 0.15 0.1 0.15 T ime (s) T ime (s)
0.1
0.1T ime 0.15 (s) T ime (s)
0.2
0.2
0.15
0.2 0.2
0.25 0.25
(b) T EE=6.0690 T EE=6.0690
0.25
0.25
3
0
0
-3
-30 0
0.05
0.1
0.05
0.15
0.1 0.15 T ime (s) T ime (s)
(d) (d)
77
0.2
0.2
0.25
0.25
TEE index
TEE index
(c) (c)
0.05 0.05
(b)
Amplitude Amplitude
Amplitude
Amplitude
-3
-3
3
3
0
0
T EE=6.2811
3
3
0
-30 0
0.25
T EE=6.6077
6.56.5
66 55
10 10
15 15 SNR(dB) (dB) SNR
(e) (e)
20 20
Figure 3. The evaluation results of of the the simulated simulated signals signals with signal‐to‐noise ratios (SNRs) via Figure The evaluation results signal‐to‐noise ratios (SNRs) via Figure 3. 3. The evaluation results of the simulated signals withwith signal-to-noise ratios (SNRs) via Teager Teager energy entropy (TEE) index: (a) 5 dB; (b) 10 dB; (c) 15 dB; (d) 20 dB; (e) the cure of the TEE Teager energy entropy (TEE) energy entropy (TEE) index: (a)index: (a) 5 dB; (b) 10 dB; (c) 15 dB; (d) 20 dB; (e) the cure of the TEE 5 dB; (b) 10 dB; (c) 15 dB; (d) 20 dB; (e) the cure of the TEE index with index with different SNRs. index with different SNRs. different SNRs.
From Figure 3, it is noticeable that the value of the TEE index is smaller when the SNR is higher. From Figure 3, it is noticeable that the value of the TEE index is smaller when the SNR is higher. This is because the impact regularity of the signal decreases due to the effect of noise. The evaluation From Figure 3, it is noticeable that the value of the TEE index is smaller when the SNR is higher. This is because the impact regularity of the signal decreases due to the effect of noise. The evaluation This isresults in this section demonstrate the TEE index is effective for evaluating the shock fault feature because the impact regularity of the signal decreases due to the effect of noise. The evaluation results in this section demonstrate the TEE index is effective for evaluating the shock fault feature resultssignals. in this section demonstrate the TEE index is effective for evaluating the shock fault feature signals.
signals.
2.1.3. Optimal Bandwidth Selection Based on Teager Energy Entropy Index 2.1.3. Optimal Bandwidth Selection Based on Teager Energy Entropy Index 2.1.3. Optimal Bandwidth Selection Based on Teager Energy Entropy Index
Motivated by the above analysis, an optimal notch filter analysis method is proposed by
Motivated by thethe above analysis, an optimal notch filterfilter analysis method is proposed by selecting Motivated analysis, an optimal notch method is 4 proposed by selecting the by optimal above bandwidth of notch filter based on the analysis TEE index. Figure shows the theselecting optimalthe bandwidth of notch filter based on the TEE index. Figure 4 shows the flowchart optimal bandwidth of notch filter based on the TEE index. Figure 4 shows of the the optimal notch filter approach and the specific implementation processes are as follows:
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flowchart of the optimal notch filter approach and the specific implementation processes are as follows: (1) Measure the vibration signal of the defective bearing. (1) Measure the vibration signal of the defective bearing. (2) Set the fundamental frequency as the center frequency of the notch filter and perform the notch (2) Set the fundamental frequency as the center frequency of the notch filter and perform the notch filter analysis with varying Bws (Bw = [0.01fs , 0.99fs ], the step length is 0.01fs ) to achieve a series filter analysis with varying Bws (Bw = [0.01f s, 0.99fs], the step length is 0.01fs) to achieve a series of notch filter signals. of notch filter signals. (3) Calculate the TEE value of each notch filter signal, determine the optimal bandwidth with (3) Calculate the TEE value of each notch filter signal, determine the optimal bandwidth with the the smallest TEE value and select the corresponding notch filter signal as the optimal notch smallest TEE value and select the corresponding notch filter signal as the optimal notch filter filter signal. signal.
Figure 4. Flowchart of the optimal notch filter method. Figure 4. Flowchart of the optimal notch filter method.
If the fundamental frequency signal plays a role as an interference signal in the raw vibration If the fundamental frequency signal plays a role as an interference signal in the raw vibration signal, the amplitudes of the fundamental frequency and its harmonics can be generally identified signal, the amplitudes of the fundamental frequency and its harmonics can be generally identified from the envelope spectrum of the raw vibration signal. So we can judge that whether the from the envelope spectrum of the raw vibration signal. So we can judge that whether the fundamental fundamental frequency signal is an interference signal by observing the envelope spectrum of the frequency signal is an interference signal by observing the envelope spectrum of the raw vibration raw vibration signal. As reported in [18,36], the amplitude energy ration is an effective indicator to signal. As reported in [18,36], the amplitude energy ration is an effective indicator to evaluate evaluate the richness of one signal component in the raw vibration signal. Inspired by the ideas in the richness of one signal component in the raw vibration signal. Inspired by the ideas in [18,36], [18,36], the fundamental frequency amplitude energy ration (FFAER) indicator was introduced to the fundamental frequency amplitude energy ration (FFAER) indicator was introduced to automatically automatically judge whether the fundamental frequency signal is an interference: judge whether the fundamental frequency signal is an interference:
A2 ( f ) A2 (2 f ) A2 (3 f ) FFAER A2 ( f r) + A2 (2 f r ) + A2 (3 fr ) , r r Wr FFAER = ,
(8) (8)
W where fr is the fundamental frequency, A(fr), A(2fr), A(3fr) represent the amplitude of fr, 2fr, 3fr where fr is the fundamental frequency,of A(f A(2fr ), A(3f the amplitude of fenergy r ),original r ) represent r , 2fr , 3f r respectively in the envelope spectrum the signal, and W represents the total of respectively in the envelope spectrum of the original signal, and W represents the total energy of the the local envelope spectrum in the frequency band [0, 3fr]. local Equation (8) is an empirical formula, and a threshold (10% was recommended in this paper) is envelope spectrum in the frequency band [0, 3fr ]. Equation (8) is an empirical formula, and a threshold (10% was recommended in this paper) is used to evaluate whether the fundamental signal is an interference based on the results of many used to evaluate whether the fundamental signal is an interference based on the results of many tests. tests. If the value of FFAER of the signal is greater than the threshold, the original signal would be If the valueby of FFAER of the notch signal filter is greater than the threshold, the original signal would be frequency processed processed the optimal to eliminate the interference of the fundamental by the optimal notch filter to eliminate the interference of the fundamental frequency signal. signal. 2.2. Enhanced Singular Value Decomposition 2.2. Enhanced Singular Value Decomposition For a discrete signal, s(i), i = 1, 2, . . . , N, its Hankel matrix can be constructed as [37]: For a discrete signal, s(i), i = 1, 2, …, N, its Hankel matrix can be constructed as [37]:
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H=
s (1) s (2) .. . s(m)
s (2) s (3) .. . s ( m + 1)
··· s(n) · · · s ( n + 1) .. .. . . ···
s( N )
,
(9)
m×n
where 1 < n < N, m = N − n + 1. The values of n and m are respectively set as n = N/2 and m = N/2 + 1 according to the selection method presented in [37]. The essence of SVD is the orthogonal decomposition of the Hankel matrix, which can be expressed as: H = UDVT , (10) where U Rm ×m and V Rn ×n are orthogonal matrices, D Rm ×n is the obtained diagonal matrix. The expressions of the three matrices are as follows: U = [u1 , u2 , · · · , um ],
(11)
V = [v1 , v2 , · · · , vn ], D = diag σ1 , σ2 , · · · , σq , 0 ,
(12) (13)
where σi (i = 1, 2, . . . , q) denotes the singular values of the matrix H, q is the number of the nonzero singular values. With a comprehensive consideration of Equations (10)–(13), H can be described as Equation (14): H = σ1 u1 v1 T + σ2 u2 v2 T + . . . + σq uq vq T . (14) Let H i = σi ui vi T (i = 1, 2, . . . , q), denotes the corresponding matrix component of the singular value σi . Therefore, the matrix H can be expressed as: H = H1 + H2 + · · · + Hq .
(15)
In order to decrease the redundancy of matrix H, it is necessary to select the effective singular values and corresponding matrix components to reconstruct the matrix. A series of criterions have been proposed by scholars to select the proper singular components [38], and the difference spectrum of singular values (DSSV) [37] is the most widely used one. The number of the effective singular values can be determined by observing the difference spectrum sequence. The DSSV is achieved by backward reduction of singular values: bi = σi − σi+1 , i = 1, 2, · · · , q − 1. (16) All bi sets {b1 , b2 , . . . , bq −1 }, is called the DSSV. As pointed out in [38], the maximum peak point bk is the cut-off point of effective and useless singular values. Then the matrix H is reconstructed using the former k singular value components: H0 = H1 + H2 + · · · + Hk ,
(17)
where H0 is the reconstructed matrix, and the de-noised signal y(i), i = 1, 2, . . . , N, could be generated by the diagonal mean of H0 . Though the SNR of the vibration signal can be increased by using the selection criterion of DSSV, the impact features may be lost by reconstructing the de-noised signal only using the former k singular values based on DSSV. To avoid missing the useful information, this work adopts the method proposed in [39], which determines the number of the effective singular values according to the peak value group of DSSV and selects the last relative maximum peak point as the demarcation point of the useful
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identify the last maximum peak of DSSV adaptively, the definition of relative amplitude ration (RAR) of DSSV was given: components and noise to conduct the SVD de-noising method. In order to identify the last maximum bi amplitude ration (RAR) of DSSV was given: peak of DSSV adaptively, the definition ) relative ,i 1, 2,, q 1 , RAR(iof (18) max(bi ) bi RAR(i )i. If RAR(j) is smaller than a threshold, for j > m, j = m + 1, m + 2, = , i = 1, 2, · · · , q − 1, (18) where RAR(i) represents the RAR of b max(bi ) …, q − 1, the last maximum peak of DSSV can be identified as bm. When the threshold is set to a where RAR(i) the RAR of bi .are If RAR(j) is smaller than a threshold, for j > m, j=m + 1, m + 2, smaller value, represents more singular values selected for reconstruction, that means more information .and more noise of the raw signal will be retained. On the contrary, when the threshold is set to a . . , q − 1, the last maximum peak of DSSV can be identified as bm . When the threshold is set to agreater value, the reconstructed signal will contains less noise and some useful information may be smaller value, more singular values are selected for reconstruction, that means more information and more noise of the raw signal will be retained. On the contrary, when the threshold is set to a greater also lost. The threshold was set to 10% in this paper based on many tests. value,To further eliminate the remaining noise of the SVD de‐noised signal y(i), i = 1, 2, …, N, the SVD the reconstructed signal will contains less noise and some useful information may be also lost. The threshold was set to 10% in this paper based on many tests. de‐noised signal is multiplied a weight factor of amplitude to obtain the ESVD de‐noised signal as To further eliminate the remaining noise of the SVD de-noised signal y(i), i = 1, 2, . . . , N, the SVD follows: de-noised signal is multiplied a weight factor of amplitude to obtain the ESVD de-noised signal y (i ) 2 as follows: h(i ) y (i) [ ] , i 1, 2,, N , (19) y(pi ) 2 h (i ) = y (i ) × [ ] , i = 1, 2, . . . , N, (19) p where p is the maximum of y(i), i = 1, 2, …, N, and h(i), i = 1, 2, …, N, represents the ESVD de‐noised where p is the maximum of y(i), y (i )i = 1, 2, . . . , N, and h(i), i = 1, 2, . . . , N, represents the ESVD de-noised signal. The weight factor [ y(i) ]22 (i = 1, 2, …, N) is a positive number less than 1. Because the signal. The weight factor [ pp ] (i = 1, 2, . . . , N) is a positive number less than 1. Because the amplitudes of the noise interferences are relatively smaller than the fault feature signal carried by the amplitudes of the noise interferences are relatively smaller than the fault feature signal carried by SVD de-noised signal, the noise interferences can becan suppressed furtherfurther by multiplying the weight the SVD de‐noised signal, the noise interferences be suppressed by multiplying the factor, and the main components can become more prominent. weight factor, and the main components can become more prominent. 2.3. The Presented Method 2.3. The Presented Method To the compound interference of the fundamental frequency signal and background To suppress suppress the compound interference of the fundamental frequency signal and background noise, the optimal notch filter was combined with the ESVD de-noising for bearing failure detection. noise, the optimal notch filter was combined with the ESVD de‐noising for bearing failure detection. First, the raw vibration signal was analyzed by optimal notch filter analysis to get the optimal notch First, the raw vibration signal was analyzed by optimal notch filter analysis to get the optimal notch filter signal. Then, the ESVD de-noising analysis was conducted on the optimal notch filter signal filter signal. Then, the ESVD de‐noising analysis was conducted on the optimal notch filter signal to to get the ESVD de-noised signal.Finally Finallythe thede‐noised de-noisedsignal signalwas was analyzed analyzed by by envelope get the ESVD de‐noised signal. envelope spectrum spectrum analysis to obtain the fault features. Figure 5 describes the principle of the proposed method. analysis to obtain the fault features. Figure 5 describes the principle of the proposed method.
Figure 5. Framework of the presented method. Figure 5. Framework of the presented method.
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To investigate the proposed method, a multi-component signal constructed based on Equation (20) 3. Simulated Analysis 3. Simulated Analysis is used for analysis in this section.
To investigate the proposed method, a multi‐component signal constructed based on Equation To investigate the proposed method, a multi‐component signal constructed based on Equation S ( t ) = x1 ( t ) + x2 ( t ) + x3 ( t ). (20) (20) is used for analysis in this section. (20) is used for analysis in this section. As mentioned in Section 2.1.2, x1 (t)S represents simulated outer race (OR) fault signal(20) and the (t ) x1 (t ) xthe 2 (t ) x3 (t ) . S (t ) Hz. x1 (xt ) (t) xand (t )are . used for simulating the fundamental (20) 2 (t ) xx3(t) OR characteristic frequency fo = 1/To = 100 2 3 As mentioned in Section 2.1.2, x1(t) represents the simulated outer race (OR) fault signal and the frequency signal and noise, respectively. The simulated OR fault signal, fundamental frequency signal As mentioned in Section 2.1.2, x 1(t) represents the simulated outer race (OR) fault signal and the OR characteristic frequency fo = 1/To = 100 Hz. x2(t) and x3(t) are used for simulating the fundamental andOR characteristic frequency f noise component were respectively shown in(t) and x Figure 32a–c in above analysis. A sampling frequency = 1/To = 100 Hz. x (t) are used for simulating the fundamental frequency signal and noise, orespectively. The 2simulated OR fault signal, fundamental frequency of 8192 Hz was usedand andnoise, the signal length of S(t)simulated is 2048. OR fault signal, fundamental frequency frequency signal respectively. The signal and noise component were respectively shown in Figure 2a–c in above analysis. A sampling signal and noise component were respectively shown in Figure 2a–c in above analysis. A sampling Figure 6 depicts the waveform in time domain and envelope spectrum of S(t). It is hard to frequency of 8192 Hz was used and the signal length of S(t) is 2048. frequency of 8192 Hz was used and the signal length of S(t) is 2048. identify the impulsive features of OR fault from the waveform. Thespectrum value of the FFAER Figure 6 depicts the waveform in time domain and envelope of S(t). It is indicator hard to of Figure 6 depicts the waveform in time domain and envelope spectrum of S(t). It is hard the simulated signal is 73.18%, which indicates that the fundamental frequency signal performsto very identify the impulsive features of OR fault from the waveform. The value of the FFAER indicator of identify the impulsive features of OR fault from the waveform. The value of the FFAER indicator of the simulated signal is 73.18%, indicates that the fundamental performs prominent in the simulated signal.which Meanwhile, only the frequency of x2frequency (t), that issignal f 1 , could be found the simulated signal is simulated 73.18%, which indicates that the fundamental frequency signal performs in the Meanwhile, only the frequency of x 2(t), that is f fromvery the prominent envelope spectrum. Then, signal. the simulated signal was processed by notch filters1, could be with f 1 as the very prominent in the simulated signal. Meanwhile, only the frequency of x2(t), that is f1, could be found from the envelope spectrum. Then, the simulated signal was processed by notch filters with f center frequency, and with varying Bws (from 0.01fs to 0.99fs and the step length was 0.01fs ). The1 TEE found from the envelope spectrum. Then, the simulated signal was processed by notch filters with f 1 as the center frequency, and with varying Bws s to 0.99fs and the step length was 0.01fs). values of the notch filter signals under different(from 0.01f Bws were exhibited in Figure 7a. From Figure 7a, as the center frequency, and with varying Bws (from 0.01f s to 0.99fs and the step length was 0.01fs). The TEE values of the notch filter signals under different Bws were exhibited in Figure 7a. From it can beTEE found thatof the TEE value is minimum when Bw = 0.43f optimal Bw is 0.43f s . So the s and the The values the notch filter signals under different Bws were exhibited in Figure 7a. From Figure 7a, it can be found that the TEE value is minimum when Bw = 0.43f s. So the optimal Bw is 0.43fs corresponding optimal notch filter signal was depicted in Figure 7b. The optimal notch filter signal Figure 7a, it can be found that the TEE value is minimum when Bw = 0.43f s. So the optimal Bw is 0.43fs and the corresponding optimal notch filter signal was depicted in Figure 7b. The optimal notch filter reflects the same impulsive feature with x (t). However, the interferences are very obvious. Figure 7c and the corresponding optimal notch filter signal was depicted in Figure 7b. The optimal notch filter 1 signal reflects the same impulsive feature with x 1(t). However, the interferences are very obvious. shows the envelope spectrum of the optimal notch1(t). However, the interferences are very obvious. filter signal, some apparent peaks can be visible at signal reflects the same impulsive feature with x Figure 7c shows the envelope spectrum of the optimal notch filter signal, some apparent peaks can Figure 7c shows the envelope spectrum of the optimal notch filter signal, some apparent peaks can the frequencies of f , 2f , 3f , 4f , 5f and 6f , but the noise interferences are also visible. o o o o o o be visible at the frequencies of fo, 2fo, 3fo, 4fo, 5fo and 6f o, but the noise interferences are also visible. be visible at the frequencies of fo, 2fo, 3fo, 4fo, 5fo and 6fo, but the noise interferences are also visible. 1 1 Amplitude Amplitude
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Figure 7. The analysis results of S(t) using the optimal notch filter analysis: (a) the curve of the TEE index Figure 7. The analysis results of S(t) using the optimal notch filter analysis: (a) the curve of the TEE index under different bandwidths; (b) the optimal notch filter signal; (c) the envelope spectrum of (b). Figure 7. The analysis results of S(t) using the optimal notch filter analysis: (a) the curve of the TEE under different bandwidths; (b) the optimal notch filter signal; (c) the envelope spectrum of (b).
index under different bandwidths; (b) the optimal notch filter signal; (c) the envelope spectrum of (b).
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To decrease the influence of the noise interference on the optimal notch filter signal, the ESVD To decrease the influence of the noise interference on the optimal notch filter signal, the ESVD de-noising analysis was was performed performed on on the the optimal optimal notch notch filter Figure 8a 8a shows shows the the DSSV DSSV de‐noising analysis filter signal. signal. Figure obtained by conducting SVD on the optimal notch filter signal, from which we can see that the last obtained by conducting SVD on the optimal notch filter signal, from which we can see that the last relatively apparent maximum peak point appears when the serial number of the singular value is 19. relatively apparent maximum peak point appears when the serial number of the singular value is 19. Therefore, the first 19 singular components were adopted to construct the SVD de-noised signal as Therefore, the first 19 singular components were adopted to construct the SVD de‐noised signal as shown in Figure 8c 8c displays the the ESVD de-noised signal. The ESVD de-noised signal reflects shown in Figure Figure 8b. 8b. Figure displays ESVD de‐noised signal. The ESVD de‐noised signal more prominent impact features compared with the SVD de-noised signal and the optimal notch reflects more prominent impact features compared with the SVD de‐noised signal and the optimal filter signal. Consequently, the envelope spectrum of the ESVD de-noised signal shown in shown Figure 8d notch filter signal. Consequently, the envelope spectrum of the ESVD de‐noised signal in reflects8d more harmonics than the envelope the optimal notch filter signal, meanwhile, Figure reflects more harmonics than the spectrum envelope ofspectrum of the optimal notch filter signal, no amplitudes of noise could benoise could be found from Figure 8d. The simulation signal analysis found from Figure 8d. The simulation signal analysis results reflect the meanwhile, no amplitudes of validityreflect of the presented method overcoming the compound interference of noise and fundamental results the validity of the inpresented method in overcoming the compound interference of frequency signal, and enhancing the impulsive feature signal. noise and fundamental frequency signal, and enhancing the impulsive feature signal.
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The proposed method was compared with three methods, i.e., minimum entropy The proposed method was compared with three methods, i.e., minimum entropy deconvolution deconvolution (MED), Kurtogram and Infogram, to further validate its advantages. For an equitable (MED), Kurtogram and Infogram, to further validate its advantages. For an equitable comparison, comparison, the ESVD de‐noising was also conducted on the filtered signals obtained using the the ESVD de-noising was also conducted on the filtered signals obtained using the three methods. three methods. Figure 9a,b respectively display the waveform and envelope spectrum of the filtered Figure 9a,b respectively display the waveform and envelope spectrum of the filtered signal obtained by signal obtained by applying MED to S(t). From Figure 9a, a few impulse shocks can be detected, but applying MED to S(t). From Figure 9a, a few impulse shocks can be detected, but the noise interferences the noise interferences are very obvious. The first four harmonics of fo and some amplitudes of noise are very obvious. The first four harmonics of fo and some amplitudes of noise interferences can be interferences can be found from Figure 9b. Figure 9c shows the ESVD de‐noised signal of Figure found from Figure 9b. Figure 9c shows the ESVD de-noised signal of Figure 9a,d displays its envelope 9a,d displays its envelope spectrum. Some peaks can be found at the frequencies of fo and its spectrum. Some peaks can be found at the frequencies of fo and its harmonics, but less harmonics harmonics, but less harmonics can be detected from Figure 9d compared with Figure 8d. Figure 10a can be detected from Figure 9d compared with Figure 8d. Figure 10a displays the Kurtogram of S(t), displays the Kurtogram of S(t), the optimal narrowband can be identified at level 4 with center the optimal narrowband can be identified at level 4 with center frequency of 2048 Hz. A designed frequency of 2048 Hz. A designed band‐pass filter based on the information of the optimal band-pass filter based on the information of the optimal narrowband was conducted on the raw narrowband was conducted on the raw vibration signal and the filtered signal is plotted in Figure vibration signal and the filtered signal is plotted in Figure 10b. The filtered signal is also contaminated 10b. The filtered signal is also contaminated by noise. Figure 10c shows envelope spectrum of Figure by noise. Figure 10c shows envelope spectrum of Figure 10b. From Figure 10c, the first five harmonics 10b. From Figure 10c, the first five harmonics of fo and some amplitudes of noise interference can be of fo and some amplitudes of noise interference can be visible. Figure 10d depicts the ESVD de-noised visible. Figure 10d depicts the ESVD de‐noised signal of Figure 10b. Figure 10e shows the envelope signal of Figure 10b. Figure 10e shows the envelope spectrum of Figure 10d, which displays less spectrum of Figure 10d, which displays less harmonics of fo compared with Figure 8d. Figure 11 harmonics of fo compared with Figure 8d. Figure 11 reflects the analysis results obtained using the reflects the analysis results obtained using the Infogram method. We can find that the proposed Infogram method. We can find that the proposed method shows a better performance than the method shows a better performance than the Infogram method in enhancing the impulsive features Infogram method in enhancing the impulsive features and extracting more harmonics of fo . and extracting more harmonics of fo.
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Figure 9. The analysis results of S(t) using MED: (a) the filtered signal; (b) its envelope spectrum; (c) Figure 9. The analysis results of S(t) using MED: (a) the filtered signal; (b) its envelope spectrum; (c) the Figure 9. The analysis results of S(t) using MED: (a) the filtered signal; (b) its envelope spectrum; (c) the ESVD de‐noised signal of (a); (d) the envelope spectrum of (c). ESVD de-noised signal of (a); (d) the envelope spectrum of (c). the ESVD de‐noised signal of (a); (d) the envelope spectrum of (c).
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Figure 10. The analysis results of S(t) using Kurtogram: (a) the Kurtogram; (b) the filtered signal of Figure 10. The analysis results of S(t) using Kurtogram: (a) the Kurtogram; (b) the filtered signal of Figure 10. The analysis results of S(t) using Kurtogram: (a) the Kurtogram; (b) the filtered signal of the the optimal narrowband; (c) the envelope spectrum of (b); (d) the ESVD de‐noised signal of (b); (e) the optimal narrowband; (c) the envelope spectrum of (b); (d) the ESVD de‐noised signal of (b); (e) optimal narrowband; (c) the envelope spectrum of (b); (d) the ESVD de-noised signal of (b); (e) the the envelope spectrum of (d). the envelope spectrum of (d). envelope spectrum of (d).
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Figure 11. The analysis results of S(t) using Infogram: (a) the average infogram ∆I1/2(f; ∆f); (b) the Figure 11. The analysis results of S(t) using Infogram: (e) (a) the average infogram ∆I1/2 (f ; ∆f ); (b) the filtered signal of the optimal narrowband; (c) the envelope spectrum of (b); (d) the ESVD de‐noised filtered signal of the optimal narrowband; (c) the envelope spectrum of (b); (d) the ESVD de-noised signal of (b); (e) the envelope spectrum of (d). Figure 11. The analysis results of S(t) using signal of (b); (e) the envelope spectrum of (d). Infogram: (a) the average infogram ∆I1/2(f; ∆f); (b) the filtered signal of the optimal narrowband; (c) the envelope spectrum of (b); (d) the ESVD de‐noised
From the simulated analysis, we can find that the optimal notch filter analysis can eliminate signal of (b); (e) the envelope spectrum of (d). From the simulated analysis, we can find that the optimal notch filter analysis can eliminate the the interference of the fundamental frequency signal directly and effectively. The ESVD de‐noising shows a better performance than the SVD de‐noising in inhibiting the noise buried in the optimal interference ofthe thesimulated fundamental frequency signal directly and effectively. Theanalysis ESVD de-noising shows From analysis, we can find that the optimal notch filter can eliminate notch filter signal. the interference of the fundamental frequency signal directly and effectively. The ESVD de‐noising a better performance than the SVD de-noising in inhibiting the noise buried in the optimal notch filtershows a better performance than the SVD de‐noising in inhibiting the noise buried in the optimal signal. 4. Experimental Analysis notch filter signal.
4. Experimental Analysis
4.1. Experiment 1 4. Experimental Analysis
4.1. Experiment 1 Experiment 1 is conducted in the vibration test laboratory of North China Electric Power
4.1. Experiment 1 University (NCEPU). The test stand of experiment 1 is shown in Figure 12. Figure 13 displays the Experiment 1 is conducted in the vibration test laboratory of North China Electric Power artificial inner race (IR) fault on the defective bearing. Table 1 gives the specific parameters of the Experiment 1 is conducted in of the experiment vibration test laboratory of Figure North China Electric University (NCEPU). The test stand 1 is shown in 12. Figure 13Power displays faulty bearing. During the test, the rotating frequency (f r ) was kept at 24 Hz. The vibration signals were University (NCEPU). The test stand of experiment 1 is shown in Figure 12. Figure 13 displays the the artificial inner race (IR) fault on the defective bearing. Table 1 gives the specific parameters of the collected by eddy current sensors as Figure 12 reflects. The sampling frequency is 12,800 Hz and the artificial inner race (IR) fault on the defective bearing. Table 1 gives the specific parameters of the faulty bearing. During the test, the rotating frequency (fr ) was kept at 24 Hz. The vibration signals number of sampling points is 6400. The theoretical characteristic frequency of IR fault (f i) is 172 Hz. faulty bearing. During the test, the rotating frequency (fr) was kept at 24 Hz. The vibration signals were were collected by eddy current sensors as Figure 12 reflects. The sampling frequency is 12,800 Hz and collected by eddy current sensors as Figure 12 reflects. The sampling frequency is 12,800 Hz and the the number of sampling points is 6400. The theoretical characteristic frequency of IR fault (fi ) is 172 Hz. number of sampling points is 6400. The theoretical characteristic frequency of IR fault (f i) is 172 Hz.
Eddy current sensors
Defective bearing
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Figure 12. Test stand of experiment 1.
Figure 12. Test stand of experiment 1.
Figure 12. Test stand of experiment 1.
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Figure 13. Bearing inner race defect. Figure 13. Bearing inner race defect. Table 1. The details of the defective bearing of experiment 1. Table 1. The Figure 13. Bearing inner race defect. details of the defective bearing of experiment 1. Roller Diameter (mm) Pith Diameter (mm) Number of Rollers Contact Angle (°) Table 1. The details of the defective bearing of experiment 1. ◦ Roller Diameter Pith Diameter Number Contact 7.5 (mm) 38.5 (mm) 12 of Rollers 0° Angle ( ) Roller Diameter (mm) Pith Diameter (mm) Number of Rollers Contact Angle (°) 7.5 38.5 12 0◦ 7.5 38.5 12 0° The experimental IR fault signal and its envelope spectrum are shown in Figure 14. From Figure
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14, we can find that the signal collected by the eddy current sensor is similar to a sinusoidal signal The experimental IR fault signal and its envelope spectrum are shown in Figure 14. From Figure 14, The experimental IR fault signal and its envelope spectrum are shown in Figure 14. From Figure with burr. value of the is sensor 90.38% the tomain components the we can findThe that FFAER the signal collected byIR thefault eddysignal current isand similar a sinusoidal signalof with 14, we can find that the signal collected by the eddy current sensor is similar to a sinusoidal signal envelope spectrum are the fundamental frequency (24 and Hz) the and main its harmonics. Whereas, the fault burr. The FFAER value ofvalue the IR signal 90.38% components of the envelope with burr. The FFAER of fault the IR fault issignal is 90.38% and the main components of the features of the rolling bearing could not be found from the waveform and envelope spectrum of the spectrum arespectrum the fundamental frequency (24 Hz) and (24 its harmonics. theWhereas, fault features of the envelope are the fundamental frequency Hz) and its Whereas, harmonics. the fault experimental fault signal. rolling bearing could not be found from the waveform and envelope spectrum of the experimental features of the rolling bearing could not be found from the waveform and envelope spectrum of the fault signal. experimental fault signal. 150 1000
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Figure 14. The inner race fault signal: (a) its time waveform; (b) its envelope spectrum. Figure 14. The inner race fault signal: (a) its time waveform; (b) its envelope spectrum. Figure 14. The inner race fault signal: (a) its time waveform; (b) its envelope spectrum.
The IR fault signal was processed by the proposed method and Figure 15 displays the results. The IR fault signal was processed by the proposed method and Figure 15 displays the results. By setting the center frequency as 24 Hz, and setting the Bw as varying values, we performed the The IR fault signal was processed by the proposed method and Figure 15 displays the results. By setting the center frequency as 24 Hz, and setting the Bw as varying values, we performed the notch filter analysis on the IR fault signal. The TEE values of the output notch filter signals under Bynotch filter analysis on the IR fault signal. The TEE values of the output setting the center frequency as 24 Hz, and setting the Bw as varyingnotch filter values, wesignals under performed the different bandwidths are shown in Figure 15a. The optimal bandwidth is 0.16f s based on the results notch filter analysis on the IR fault signal. The TEE values of the output notch filter signals under different bandwidths are shown in Figure 15a. The optimal bandwidth is 0.16fs based on the results of Figure 15a. Figure 15b shows the corresponding optimal notch signal. Prominent impact features different bandwidths are shown in Figure 15a. The optimal bandwidth is 0.16f based on the results of of Figure 15a. Figure 15b shows the corresponding optimal notch signal. Prominent impact features can be detected in Figure 15b. However, the noise is also visible. From the s envelope spectrum of Figure 15a. Figure in 15b shows theHowever, corresponding optimal notch signal. Prominent impact featuresof can can be detected Figure 15b. the noise is also visible. From the envelope spectrum optimal notch filter signal shown in Figure 15c, there are two peak points at the frequencies of 172 beoptimal notch filter signal shown in Figure 15c, there are two peak points at the frequencies of 172 detected in Figure 15b. However, the noise is also visible. From the envelope spectrum of optimal Hz and 344 Hz, corresponding to fi and 2fi, respectively. Figure 15d reflects the signal obtained by Hz and 344 Hz, corresponding to f and 2f i, respectively. Figure 15d the signal of obtained notch filter signal shown in Figure i15c, there are two peak points atreflects the frequencies 172 Hzby and performing ESVD de‐noising on the optimal notch filter signal. Compared with the optimal notch ESVD de‐noising optimal notch filter signal. Compared with the optimal notch 344performing Hz, corresponding to fi and on 2fi ,the respectively. Figure 15d reflects the signal obtained by performing filter signal, Figure 15d presents more obvious and cleaner impact features. Figure 15e illustrates the filter signal, Figure 15d presents more obvious and cleaner impact features. Figure 15e illustrates the ESVD de-noising on the optimal notch filter signal. Compared with the optimal notch filter signal, envelope spectrum of Figure 15d. Obvious spectral lines can be detected at the frequencies of f i, 2fi, envelope spectrum of Figure 15d. Obvious spectral lines can be detected at the frequencies of f i, 2fi, Figure 15d presents more obvious and cleaner impact features. Figure 15e illustrates the envelope 3f i, 4fi and 5fi. Moreover, the sidebands with the interval of fr are also very clear. After the ESVD 3fi, 4fi and 5fi. Moreover, the sidebands with can the be interval of fat r are also very clear. After the ESVD spectrum ofanalysis, Figure 15d. Obvious spectral lines detected thesignal frequencies of fi , 2fi , 3fi , and 4fi and 5fi . de‐nosing optimal notch was suppressed more de‐nosing analysis, the the noise noise interferences interferences of of the the optimal notch signal was suppressed and more Moreover, the sidebands with the interval of fr are also very clear. After the ESVD de-nosing analysis, harmonic frequencies of can be detected. harmonic frequencies of can be detected. the noise interferences of the optimal notch signal was suppressed and more harmonic frequencies of can be detected.
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Figure 15. The analysis results of inner race fault signal using the proposed method: (a) the curve of
Figure 15. The analysis results of inner race fault signal using the proposed method: (a) the curve of Figure 15. The analysis results of inner race fault signal using the proposed method: (a) the curve of the TEE index under different bandwidths; (b) the optimal notch filter signal; (c) the envelope the TEE index under different bandwidths; (b) the optimal notch filter signal; (c) the envelope spectrum the spectrum of (b); (d) the ESVD de‐noised signal of (b); (e) the envelope spectrum of (d). TEE index under different bandwidths; (b) the optimal notch filter signal; (c) the envelope of (b); (d) the ESVD de-noised signal of (b); (e) the envelope spectrum of (d). spectrum of (b); (d) the ESVD de‐noised signal of (b); (e) the envelope spectrum of (d). The IR fault signal was further analyzed by MED, Kurtogram and Infogram for comparisons. The MED filtered signal plotted analyzed in Figure by 16a,b shows its envelope Obvious noise The IR fault signal was further analyzed by MED, Kurtogram and Infogram for comparisons. The IR fault signal was is further MED, Kurtogram andspectrum. Infogram for comparisons. interferences can be visible from Figure 16a. Although, the frequencies of f i and 2f i can be detected The MED filtered signal signal isis plotted plotted in in Figure Figure 16a,b 16a,b shows noise The MED filtered shows its its envelope envelopespectrum. spectrum.Obvious Obvious noise from Figure 16b, some peaks that reflect the noise can also be visible. Figure 16c displays the signal interferences can be visible visible from from Figure Figure 16a. Although, the frequencies of f can be detected interferences can be 16a. Although, the frequencies ofi and 2f fi and i2f can be detected i obtained by performing ESVD de‐noising on the MED filtered signal and Figure 16d shows its from Figure 16b, some peaks that reflect the noise can also be visible. Figure 16c displays the signal from Figure 16b, some peaks that reflect the noise can also be visible. Figure 16c displays the signal envelope spectrum. Only the first three harmonics of fi can be detected from Figure 16d 16d. shows Figure its obtained by performing performing ESVD de‐noising on the MED filtered signal and Figure obtained ESVD de-noising the filtered and Figure 16d shows 17a,b by respectively illustrate the Kurtogram on and the MED filtered signal signal obtained based on Figure 17a. its envelope spectrum. Only the first three harmonics of fi can be detected from Figure 16d. Figure envelope spectrum. Only the first three harmonics of f can be detected from Figure 16d. Figure Figure 17c shows the envelope spectrum of Figure i17b, which exhibits the first four harmonic 17a,b 17a,b respectively illustrate the Kurtogram the filtered signal obtained on 17a. Figure 17a. respectively illustrate the Kurtogram and the theand filtered signal obtained based onbased Figure Figure frequencies of fi. Figure 17d displays ESVD de‐noised signal of Figure 17b,e displays its 17c Figure 17c shows the envelope spectrum of Figure 17b, which exhibits the first four harmonic shows the envelope spectrum of Figure 17b, which exhibits the first four harmonic frequencies of envelope spectrum. The first five harmonics of f i can be identified in Figure 17e. But the fourth and frequencies of fi. Figure 17d displays the ESVD de‐noised signal of Figure 17b,e displays its fifth harmonics of f i and their sidebands shown in Figure 17e are less prominent than that shown in fi . Figure 17d displays the ESVD de-noised signal of Figure 17b,e displays its envelope spectrum. envelope spectrum. The first five harmonics of f i can be identified in Figure 17e. But the fourth and Figure 15d obtained using the proposed method. Figure 18 shows the process results obtained via The first five harmonics of fi can be identified in Figure 17e. But the fourth and fifth harmonics of fi fifth harmonics of f i and their sidebands shown in Figure 17e are less prominent than that shown in the Infogram method. By comparing Figure 18 and Figure 15, we can find that the Infogram method and their sidebands shown in Figure 17e are less prominent than that shown in Figure 15d obtained Figure 15d obtained using the proposed method. Figure 18 shows the process results obtained via is not as effective as the proposed method. The comparison results demonstrate the advantages of using the proposed method. Figure 18 shows the process results obtained via the Infogram method. the Infogram method. By comparing Figure 18 and Figure 15, we can find that the Infogram method the proposed method.
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Figure 16. The analysis results of inner race fault signal using MED: (a) the filtered signal; (b) the Figure 16. The analysis results of inner race fault signal using MED: (a) the filtered signal; (b) the Figure 16. The analysis results of inner race fault signal using MED: (a) the filtered signal; (b) the envelope spectrum of (a); (c) the ESVD de‐noised signal of (a); (d) the envelope spectrum of (c). Figure 16. The analysis results of inner race fault signal using MED: (a) the filtered signal; (b) the envelope spectrum of (a); (c) the ESVD de‐noised signal of (a); (d) the envelope spectrum of (c). envelope spectrum of (a); (c) the ESVD de-noised signal of (a); (d) the envelope spectrum of (c). envelope spectrum of (a); (c) the ESVD de‐noised signal of (a); (d) the envelope spectrum of (c).
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Figure 17. The analysis results of inner race fault signal using Kurtogram: (a) the Kurtogram; (b) the Figure 17. The analysis results of inner race fault signal using Kurtogram: (a) the Kurtogram; (b) the Figure 17. The analysis results of inner race fault signal using Kurtogram: (a) the Kurtogram; (b) the Figure 17. The analysis results of inner race fault signal using Kurtogram: (a) the Kurtogram; (b) the filtered signal of the optimal narrowband; (c) the envelope spectrum of (b); (d) the ESVD de‐noised filtered signal of the optimal narrowband; (c) the envelope spectrum of (b); (d) the ESVD de‐noised filtered signal of the optimal narrowband; (c) the envelope spectrum of (b); (d) the ESVD de-noised signal of (b); (e) the envelope spectrum of (d). filtered signal of the optimal narrowband; (c) the envelope spectrum of (b); (d) the ESVD de‐noised signal of (b); (e) the envelope spectrum of (d). signal of (b); (e) the envelope spectrum of (d). signal of (b); (e) the envelope spectrum of (d).
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Figure 18. The analysis results inner race fault signal using Infogram: (a) the average infogram Figure 18. The analysis results of inner race fault signal using Infogram: (a) average infogram Figure 18. The analysis results ofof inner race fault signal using Infogram: (a) the the average infogram 1/2(f; ∆f); (b) the filtered signal of the optimal narrowband; (c) the envelope spectrum of (b); (d) the 1/2(f; ∆f); (b) the filtered signal of the optimal narrowband; (c) the envelope spectrum of (b); (d) the ∆I ∆I∆I 1/2 (f ; ∆f ); (b) the filtered signal of the optimal narrowband; (c) the envelope spectrum of (b); (d) the ESVD de‐noised signal of (b); (e) the envelope spectrum of (d). ESVD de‐noised signal of (b); (e) the envelope spectrum of (d). ESVD de-noised signal of (b); (e) the envelope spectrum of (d).
4.2. Experiment 2 4.2. Experiment 2 4.2. Experiment 2 To further investigate the presented method, the data from Case Western Reserve University To further investigate the presented method, the data from Case Western Reserve University To further investigate the presented method, the data from Case Western Reserve University (CWRU) bearing data center [40] areare studied. The The test stand in experiment 2 and its structure diagram (CWRU) bearing data center [40] are studied. The test stand experiment and its structure (CWRU) bearing data center [40] studied. test stand in in experiment 2 2 and its structure are shown in Figure 19. An electric motor, a torque transducer, and a dynamometer constitute the test diagram are shown in Figure 19. An electric motor, a torque transducer, and a dynamometer diagram are shown in Figure 19. An electric motor, a torque transducer, and a dynamometer stand. The samples under consideration cover rolling element fault and outer race fault of the fan end constitute the test stand. The samples under consideration cover rolling element fault and outer constitute the test stand. The samples under consideration cover rolling element fault and outer bearing, the bearing type of which is SKF6203 deep groove ball. The detailed parameters of the faulty race fault of the fan end bearing, the bearing type of which is SKF6203 deep groove ball. The race fault of the fan end bearing, the bearing type of which is SKF6203 deep groove ball. The bearing are shown in Table 2. The vibration data was collected with a sampling rate of 12,000 Hz. detailed parameters of the faulty bearing are shown in Table 2. The vibration data was collected detailed parameters of the faulty bearing are shown in Table 2. The vibration data was collected The size defect of the studied bearing is 0.007 inches. with a sampling rate of 12,000 Hz. The size defect of the studied bearing is 0.007 inches. with a sampling rate of 12,000 Hz. The size defect of the studied bearing is 0.007 inches.
(a) (a) Figure 19. Cont.
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(b) (b) Figure 19. (a) Test stand of experiment 2; (b) its structure diagram. Figure 19. (a) Test stand of experiment 2; (b) its structure diagram. Figure 19. (a) Test stand of experiment 2; (b) its structure diagram. Table 2. The details of the defective bearing of experiment 2. Table 2. The details of the defective bearing of experiment 2. Table 2. The details of the defective bearing of experiment 2.
Ball Diameter (mm) Pith Diameter (mm) Number of Balls Contact Angle (°) Ball Diameter (mm) Pith Diameter (mm) Number of Balls Contact Angle (°) ◦ Ball Diameter (mm) Pith Diameter Number of Balls Contact Angle ( ) 6.75 28.5 (mm) 8 0° 6.75 28.5 8 0° 6.75 28.5 8 0◦
4.2.1. Case 1: Detection of Rolling Element Defect 4.2.1. Case 1: Detection of Rolling Element Defect 4.2.1.The Caseconsidered 1: Detection of Rolling Element Defect rolling element fault sample is is B007_3 X285_FE_time. To To improve the The considered rolling element fault sample B007_3 X285_FE_time. improve the efficiency of calculation, we excerpted 8192 points from the 20,000th point to the 28,191th point in The considered rolling element fault sample is B007_3 X285_FE_time. To improve the efficiency of efficiency of calculation, we excerpted 8192 points from the 20,000th point to the 28,191th point in the raw vibration signal. The vibration data was collected with the motor speed of 1730 rpm. The calculation, we excerpted 8192 points from the 20,000th point to the 28,191th point in the raw vibration the raw vibration signal. The vibration data was collected with the motor speed of 1730 rpm. The fault characteristic frequency of rolling element (f signal. The vibration data was collected with theb) is about 114.97 Hz. motor speed of 1730 rpm. The fault characteristic fault characteristic frequency of rolling element (f b) is about 114.97 Hz. Figure 20 displays the rolling element fault signal and its envelope spectrum. The FFAER value frequency of rolling element (f ) is about 114.97 Hz. b Figure 20 displays the rolling element fault signal and its envelope spectrum. The FFAER value of the rolling element fault signal was calculated as 16.6%, and only the rotating frequency (f Figure 20 displays the rolling element fault signal and its envelope spectrum. The FFAERr) and value of the rolling element fault signal was calculated as 16.6%, and only the rotating frequency (f r) and its harmonics can be detected in the envelope spectrum. It is suggested that the fundament signal ofits harmonics can be detected in the envelope spectrum. It is suggested that the fundament signal the rolling element fault signal was calculated as 16.6%, and only the rotating frequency (fr ) and plays a role as an an interference signal. Then the raw vibration data was analyzed by by the proposed itsplays a harmonics can beinterference detected in the envelope spectrum. It is suggested that the fundament signal role as signal. Then the raw vibration data was analyzed the proposed method and the analysis results are shown in Figure 21. As Figure 21a reflects, the TEE index has plays a role as an interference signal. Then the raw vibration data was analyzed by the proposed method and the analysis results are shown in Figure 21. As Figure 21a reflects, the TEE index has the minimum value when Bw = 0.85f s, so we set the optimal bandwidth as 0.85f s. Figure 21b shows method and the analysis results are shown in Figure 21. As Figure 21a reflects, the TEE index has the the minimum value when Bw = 0.85f s, so we set the optimal bandwidth as 0.85f s. Figure 21b shows the waveform of the optimal notch filter signal corresponding to the optimal bandwidth, and its its minimum value when Bw = 0.85f , so we set the optimal bandwidth as 0.85f . Figure 21b shows the s s the waveform of the optimal notch filter signal corresponding to the optimal bandwidth, and envelope spectrum is plotted in Figure 21c. From Figure 21c, the fault characteristic frequency of of waveform the optimal notch filter signal corresponding to the optimal bandwidth, and its envelope envelope ofspectrum is plotted in Figure 21c. From Figure 21c, the fault characteristic frequency rolling element f b can be visible, but we can find some interference peaks which make it hard to to spectrum is plottedfbin Figure 21c. From 21c, thesome fault characteristic frequency of rolling element rolling element can be visible, but Figure we can find interference peaks which make it hard identify b. Figure displays the signal obtained by by performing ESVD de‐noising on on the fb identify can befvisible, but21d we can find some interference peaks which make itthe hard to identify fb . Figure 21d fb. Figure 21d displays the signal obtained performing the ESVD de‐noising the optimal notch filter signal. From Figure 21d, obvious shock features can be visible. And from the displays signal obtained performing the obvious ESVD de-noising on thecan optimal notch And filterfrom signal. optimal the notch filter signal. by From Figure 21d, shock features be visible. the envelope spectrum of of the ESVD de‐noised signal, the spectral lines corresponding the From Figure 21d, obvious shock features can be visible. And from the envelope spectrumto ofto the envelope spectrum the ESVD de‐noised signal, the spectral lines corresponding the frequencies of f b , 2f b , 3f b , 4f b and 5f b are apparent. Moreover, no interferences can be visible from ESVD de-noised corresponding to the frequencies of fb , 2fb ,can 3fb , be 4fb visible and 5fb are frequencies of signal, fb, 2fb, the 3fb, spectral 4fb and lines 5fb are apparent. Moreover, no interferences from Figure 21e. apparent. Moreover, no interferences can be visible from Figure 21e. Figure 21e.
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Figure 20. The rolling element fault signal: (a) its time waveform; (b) its envelope spectrum. Figure 20. The rolling element fault signal: (a) its time waveform; (b) its envelope spectrum. Figure 20. The rolling element fault signal: (a) its time waveform; (b) its envelope spectrum.
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(e) Figure 21. The analysis results of rolling element fault signal using the proposed method: (a) the
Figure 21. The analysis results of rolling element fault signal using the proposed method: (a) the curve of 21. the TEE index results under of different bandwidths; (b) the optimal notch filter signal; (c) Figure The analysis rolling element (b) fault using the proposed (a) the the curve of the TEE index under different bandwidths; thesignal optimal notch filter signal;method: (c) the envelope envelope spectrum of (b); (d) the ESVD de‐noised signal of (b); (e) the envelope spectrum of (d). curve of index under different bandwidths; optimal notch filter signal; (c) the spectrum of the (b); TEE (d) the ESVD de-noised signal of (b); (e) (b) the the envelope spectrum of (d). envelope spectrum of (b); (d) the ESVD de‐noised signal of (b); (e) the envelope spectrum of (d).
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The processing results of the raw vibration obtained using MED, Kurtogram and Infogram are The processing results of the raw vibration obtained using MED, Kurtogram and Infogram respectively displayed in Figures 22–24. None of the three methods detected the characteristic The processing results of the raw vibration obtained using MED, Kurtogram and Infogram are arefrequency of rolling element fault. respectively displayed in Figures 22–24. None of the three methods detected the characteristic respectively displayed in Figures 22–24. None of the three methods detected the characteristic frequency of rolling element fault. frequency of rolling element fault. 0.5 0 0 -0.5
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(c) (d) Figure 22. The analysis results of rolling element fault signal using MED: (a) the filtered signal; (b) the envelope spectrum of (a); (c) the ESVD de‐noised signal of (a); (d) the envelope spectrum of (c). Figure 22. The analysis results of rolling element fault signal using MED: (a) the filtered signal; (b) Figure 22. The analysis results of rolling element fault signal using MED: (a) the filtered signal; (b) the the envelope spectrum of (a); (c) the ESVD de‐noised signal of (a); (d) the envelope spectrum of (c). envelope spectrum of (a); (c) the ESVD de-noised signal of (a); (d) the envelope spectrum of (c).
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Figure 23. The analysis results of rolling element fault signal using Kurtogram: (a) the Kurtogram; (b) Figure 23. The analysis results of rolling element fault signal using Kurtogram: (a) the Kurtogram; Figure 23. The analysis results of rolling element fault signal using Kurtogram: (a) the Kurtogram; (b) the filtered signal signal of the the optimal narrowband; narrowband; (c) the envelope envelope spectrum (b) the (c) the spectrum of of (b); (b); (d) (d) the the ESVD ESVD the filtered filtered signal of of the optimal optimal narrowband; (c) the envelope spectrum of (b); (d) the ESVD de‐noised signal of (b); (e) the envelope spectrum of (d). de-noised signal of (b); (e) the envelope spectrum of (d). de‐noised signal of (b); (e) the envelope spectrum of (d).
2 2 Amplitude (m/s Amplitude (m/s ) )
=1 @ Bw= 93.75, fc=46.875Hz
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Figure 24. Cont.
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200 400 600 Frequency (Hz) 0 0 200(e) 400 600 0 0 200Frequency (Hz) 400 600 Figure 24. The analysis results of rolling element fault signal using Infogram: (a) the average Frequency (Hz) (e) Figure 24. The analysis results of rolling element fault signal using Infogram: (a) the average infogram infogram ∆I1/2(f; ∆f); (b) the filtered signal of the optimal narrowband; (c) the envelope spectrum of (e) narrowband; (c) the envelope spectrum of (b); (d) the ∆I1/2 (f ; ∆f ); (b) the filtered signal of the optimal Figure 24. The analysis results of rolling element fault signal using Infogram: (a) the average (b); (d) the ESVD de‐noised signal of (b); (e) the envelope spectrum of (d). ESVD de-noised signal of (b); (e) the spectrum (d). using Infogram: (a) the average Figure 24. The analysis results of envelope rolling element fault ofsignal 1/2(f; ∆f); (b) the filtered signal of the optimal narrowband; (c) the envelope spectrum of infogram ∆I 1/2 (f; ∆f); (b) the filtered signal of the optimal narrowband; (c) the envelope spectrum of infogram ∆I (b); (d) the ESVD de‐noised signal of (b); (e) the envelope spectrum of (d). 4.2.2. Case 2: Detection of Outer Race Defect (b); (d) the ESVD de‐noised signal of (b); (e) the envelope spectrum of (d). 4.2.2. Case 2: Detection of Outer Race Defect r
Amplitude(m/s2)
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2 2 Amplitude/(m·s )2 Amplitude/(m·s ) Amplitude/(m·s )
The selected outer race (OR) fault sample is OR007@12_1 X305_FE_time. We excerpted the 4.2.2. Case 2: Detection of Outer Race Defect The selected outer race (OR) fault sample is OR007@12_1 X305_FE_time. We excerpted the 4.2.2. Case 2: Detection of Outer Race Defect 40,000th point to the 48,191th point of the original data to improve the efficiency of calculation. The The selected outer race point (OR) fault is OR007@12_1 X305_FE_time. We excerpted the 40,000th point tois the 48,191th of thesample original data to improve the efficiency of calculation. motor speed 1772 rpm. can calculate fault characteristic frequency outer race The selected outer race We (OR) fault sample the is OR007@12_1 X305_FE_time. We of excerpted the (fo) as 40,000th point to the 48,191th point of the original data to improve the efficiency of calculation. The The motor speed is 1772 rpm. We can calculate the fault characteristic frequency of outer race (fo ) as 90.17 Hz. 40,000th point to the 48,191th point of the original data to improve the efficiency of calculation. The motor speed is 1772 rpm. We can calculate the fault characteristic frequency of outer race (fo) as 90.17motor Hz.Figure 25 shows the time waveform and envelope spectrum of the raw vibration signal. From speed is 1772 rpm. We can calculate the fault characteristic frequency of outer race (fo) as 90.17 Hz. Figure 25 shows the time waveform and envelope spectrum of the raw vibration signal. From the the envelope spectrum, the rotating frequency and its third and fifth harmonics can be identified, 90.17 Hz. Figure 25 shows the time waveform and envelope spectrum of the raw vibration signal. From envelope spectrum, the rotatingfrequency frequencyis and its third and fifth harmonics can be vibration identified,signal but the but the fault characteristic invisible. The FFAER value of the raw is Figure 25 shows the time waveform and envelope spectrum of the raw vibration signal. From the envelope spectrum, the rotating frequency and its third and fifth harmonics can be identified, 43.84%, which also reflects that the fundamental frequency signal performs very prominent. faultthe envelope spectrum, the rotating frequency and its third and fifth harmonics can be identified, characteristic frequency is invisible. The FFAER value of the raw vibration signal is 43.84%, but the fault characteristic frequency is invisible. The FFAER value of the raw vibration signal is but the reflects fault characteristic frequency is frequency invisible. The FFAER value of the prominent. raw vibration signal is which also that the fundamental signal performs very 43.84%, which also reflects that the fundamental frequency signal performs very prominent. 1 0.06 43.84%, which also reflects that the fundamental frequency signal performs very prominent. f 0
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r
(b)
Figure 25. The outer race fault signal: (a) its time waveform; (b) its envelope spectrum. (a) (b) Figure 25. The outer race fault signal: (a) its time waveform; (b) its envelope spectrum. Figure 25. The outer race fault signal: (a) its time waveform; (b) its envelope spectrum. Figure 25. The outer race fault signal: (a) its time waveform; (b) its envelope spectrum.
Amplitude/(m·s2)
The raw vibration signal was analyzed by the proposed method and Figure 26 shows the The raw raw vibration vibration signal signal was was analyzed the proposed method and Figure shows results. We can determine that the optimal bandwidth of notch filter for the OR fault signal is 0.27f The analyzed by by the proposed method and Figure 26 26 shows the the s The raw vibration signal was analyzed by the proposed method and Figure 26 shows the results. We can determine that the optimal bandwidth of notch filter for the OR fault signal is 0.27f s based on Figure 26a. Figure 26b displays the optimal notch filter signal corresponding results. to the results. We can determine that the optimal bandwidth of notch filter for the OR fault signal is 0.27f s We can determine that the optimal bandwidth of notch filter for the OR fault signal is 0.27f based on based on on Figure Figure 26a. 26a. Figure Figure 26b 26b displays displays the optimal notch filter signal corresponding s to optimal bandwidth and Figure 26c shows its envelope spectrum. Two characteristic frequencies, f based the optimal notch filter signal corresponding to the the o Figure 26a.o, can be detected from Figure 26c, whereas the noise interferences are very obvious. Figure Figure 26b displays the optimal notch filter signal corresponding to the optimal bandwidth optimal bandwidth and Figure 26c shows its envelope spectrum. Two characteristic frequencies, f o and 2f optimal bandwidth and Figure 26c shows its envelope spectrum. Two characteristic frequencies, f o and 2fo, can be detected from Figure 26c, whereas the noise interferences are very obvious. Figure o26c , can be detected from Figure 26c, whereas the noise interferences are very obvious. Figure and and 2f Figure shows its envelope spectrum. Two characteristic frequencies, fo and 2fo , can be detected 26d reflects the de‐nosed signal by performing ESVD on the optimal notch filter signal and Figure 26d reflects the de‐nosed signal by performing ESVD on the optimal notch filter signal and Figure from26d reflects the de‐nosed signal by performing ESVD on the optimal notch filter signal and Figure Figure 26c, whereas the noise interferences are very obvious. Figure 26d reflects the de-nosed 26e shows its envelope spectrum. Some prominent peaks corresponding to the frequencies of f o, 2fo, 26e shows its envelope spectrum. Some prominent peaks corresponding to the frequencies of f o , 2f o, 26e shows its envelope spectrum. Some prominent peaks corresponding to the frequencies of f o, fault 3fo, by 4foperforming , 5fo and 6fo ESVD can be from Figure proposed method OR signal onvisible the optimal notch 26e. filterThe signal and Figure 26edetects showsthe itso, 2f envelope 3f o , 4f o , 5f o and 6f o can be visible from Figure 26e. The proposed method detects the OR fault 3f o, 4fo, Some 5fo and 6fo can be visible from Figure 26e. The proposed of method effectively. spectrum. prominent peaks corresponding to the frequencies fo , 2fo ,detects 3fo , 4fo ,the 5fo OR andfault 6fo can be effectively. effectively. visible from Figure 26e. The proposed method detects the OR fault effectively. Amplitude/(m·s2) Amplitude/(m·s2)
TEE TEE TEE
8.6 8.6 8.6 (0.27,8.4646)
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Figure 26. Cont.
(b)
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Figure 26. The analysis results of outer race fault signal using the proposed method: (a) the curve of (e) Figure 26. The analysis results of outer race fault signal using the proposed method: (a) the curve of (e) optimal the TEE index under different bandwidths; (b) the notch filter signal; (c) the envelope Figure 26. The analysis results of outer race fault signal using the proposed method: (a) the curve of the TEE index under different bandwidths; (b) the optimal notch filter signal; (c) the envelope spectrum spectrum of (b); (d) the ESVD de‐noised signal of (b); (e) the envelope spectrum of (d). Figure 26. The analysis results of outer race fault signal using the proposed method: (a) the curve of index under different bandwidths; (b) envelope the optimal notch filter signal; (c) the envelope of (b);the (d)TEE the ESVD de-noised signal of (b); (e) the spectrum of (d). the TEE index under different bandwidths; (b) the optimal notch filter signal; (c) the envelope spectrum of (b); (d) the ESVD de‐noised signal of (b); (e) the envelope spectrum of (d).
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The OR fault signal was also analyzed by the MED, Kurtogram and Infogram methods for spectrum of (b); (d) the ESVD de‐noised signal of (b); (e) the envelope spectrum of (d). comparisons. show It can and be found that the three for The OR faultFigures signal27–29 was respectively also analyzed bythe theanalysis MED, results. Kurtogram Infogram methods The OR fault signal was also analyzed by the MED, Kurtogram and Infogram methods for methods failed to diagnose the OR fault. The OR fault signal was also analyzed by the MED, Kurtogram and Infogram methods comparisons. Figures 27–29 respectively show the analysis results. It can be found that thefor three comparisons. Figures 27–29 respectively show the analysis results. It can be found that the three comparisons. Figures 27–29 respectively show the analysis results. It can be found that the three methods failed to diagnose the OR fault. methods failed to diagnose the OR fault. 0.03 0.3 methods failed to diagnose the OR fault.
(b) (b)
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2 Amplitude/(m·s ) 2 Amplitude/(m·s ) Amplitude/(m·s2)
signal using MED: (a) the filtered signal; (b) the Figure 27. The analysis results of outer race fault (c) (d) (c) (d) envelope spectrum of (a); (c) the ESVD de‐noised signal of (a); (d) the envelope spectrum of (c). Figure 27. The analysis results of outer race fault signal using MED: (a) the filtered signal; (b) the Figure Figure 27. The analysis results of outer race fault signal using MED: (a) the filtered signal; (b) the 27. The analysis results of outer race fault signal using MED: (a) the filtered signal; (b) the envelope spectrum of (a); (c) the ESVD de‐noised signal of (a); (d) the envelope spectrum of (c). 7.5 stft-kurt.2 - Kmax=3.1 @ Nw=2 , fc=2250Hz envelope spectrum of (a); (c) the ESVD de-noised signal of (a); (d) the envelope spectrum of (c). envelope spectrum of (a); (c) the ESVD de‐noised signal of (a); (d) the envelope spectrum of (c).
(a) (a)
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Figure 28. Cont.
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Figure 28. The analysis results of outer race fault signal using Kurtogarm: (a) the Kurtogram; (b) the
Figure 28. The analysis results of outer race fault signal using Kurtogarm: (a) the Kurtogram; (b) the Figure 28. The analysis results of outer race fault signal using Kurtogarm: (a) the Kurtogram; (b) the filtered signal of the optimal narrowband; (c) the envelope spectrum of (b); (d) the ESVD de‐noised filtered signal of the optimal narrowband; (c) the envelope spectrum of (b); (d) the ESVD de-noised filtered signal of the optimal narrowband; (c) the envelope spectrum of (b); (d) the ESVD de‐noised signal of (b); (e) the envelope spectrum of (d). signal of (b); (e) the envelope spectrum of (d). signal of (b); (e) the envelope spectrum of (d). stft-info - I max
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2 Amplitude(m/s ) 2) Amplitude(m/s
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=1.3 @ Bw= 1000, fc=515.625Hz
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=1.3 @ Bw= 1000, fc=515.625Hz
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stft-info 1 -I
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0 0 200 400 600(a) the average infogram Figure 29. The analysis results of outer race fault signal using Infogram: Frequency (Hz) ∆I1/2(f; ∆f); (b) the filtered signal of the optimal narrowband; (c) the envelope spectrum of (b); (d) the ESVD de‐noised signal of (b); (e) the envelope spectrum of (d). (e)
Figure 29. The analysis results of outer race fault signal using Infogram: (a) (a) the average infogram 5. Conclusions Figure 29. The analysis results of outer race fault signal using Infogram: the average infogram ∆I 1/2(f; ∆f); (b) the filtered signal of the optimal narrowband; (c) the envelope spectrum of (b); (d) the ∆I1/2 (fThe ; ∆f );fundamental frequency (b) the filtered signal of signal the optimal narrowband; the envelope spectrum of (b); (d) the is one of the most (c) common harmonic interference signals, ESVD de‐noised signal of (b); (e) the envelope spectrum of (d). ESVD de-noised signal of (b); (e) the envelope spectrum of (d). and the background noise can increase the difficulty of bearing fault diagnosis. To suppress the
5. Conclusions The fundamental frequency signal is one of the most common harmonic interference signals, and the background noise can increase the difficulty of bearing fault diagnosis. To suppress the
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5. Conclusions The fundamental frequency signal is one of the most common harmonic interference signals, and the background noise can increase the difficulty of bearing fault diagnosis. To suppress the compound interference of the fundamental frequency signal and background noise, this paper proposes a novel bearing failure detection method by combing the optimal notch filter based on Teager energy entropy index and enhanced singular value decomposition de-noising. The major contributions of this work can be summarized as follows: (1)
(2)
(3)
To adaptively determine the optimal bandwidth of the notch filter and implement the optimal notch filter analysis, a new indicator for evaluating the periodic impulsive features named Teager energy entropy index was presented. The Teager energy entropy index performs better in overcoming the accidental shocks than the kurtosis index. The optimal notch filter analysis based on Teager energy entropy index (with the fundamental frequency as the center frequency) shows its ability in inhibiting the interference of the fundamental frequency signal and enhancing the shock feature signal. An enhanced singular value decomposition de-noising method was proposed to improve the noise reduction of singular value decomposition.
Both the simulated and experimental signals validate the efficiency of the proposed method for overcoming the compound interference of the fundamental frequency and the noise components. It should be noticed that the center frequency is another key parameter of notch filter, which is set as the fundamental frequency in this paper to eliminate the interference of the fundament frequency signal. The fault feature signal may be affected by other harmonic interference signals besides the fundamental frequency signal. In our future work, the use of the notch filter analysis in suppressing other harmonics interference caused by other parts of the machine, especially the gears, will be studied. Some intelligent optimization methods such as genetic algorithm, particle swarm optimization and ant colony algorithm can be tested to adaptively set the optimal parameters of notch filter to implement the optimal notch filter analysis. The selection of the number of the effective singular values has an effect on the analysis results of the enhanced singular value decomposition. In this paper, the number of the effective singular values was determined by identifying the last relative maximum peak of the difference spectrum of singular values. In the future, we’ll consider more reasonable methods to adaptively select the number of the effective singular values. Author Contributions: Y.H. provided the experiment equipments and conceived the experiment; C.Z. and T.T. performed the experiment; B.P. put forward the method and wrote the paper; G.T. gave some valuable suggestions. Funding: This research was funded by [Fundamental Research Funds for the Central Universities] grant number [No. 2017XS134] and [the National Natural Science Foundation of China] grant number [No. 51777074, 51475164]. Conflicts of Interest: The authors declare no conflicts of interest.
References 1. 2.
3. 4. 5.
Glowacz, A.; Glowacz, W.; Glowacz, Z.; Kozik, J. Early fault diagnosis of bearing and stator faults of the single-phase induction motor using acoustic signals. Measurement 2018, 113, 1–9. [CrossRef] Yuan, R.; Lv, Y.; Song, G. Multi-Fault diagnosis of rolling bearings via adaptive projection intrinsically transformed multivariate empirical mode decomposition and high order singular value decomposition. Sensors 2018, 18, 1210. [CrossRef] [PubMed] Adamczak, S.; Stepien, K.; Wrzochal, M. Comparative study of measurement systems used to evaluate vibrations of rolling bearings. Procedia Eng. 2017, 192, 971–975. [CrossRef] Pang, B.; Tang, G.; Tian, T.; Zhou, C. Rolling bearing fault diagnosis based on an improved HTT transform. Sensors 2018, 18, 1203. [CrossRef] [PubMed] Zhao, M.; Jia, X. A novel strategy for signal denoising using reweighted SVD and its applications to weak fault feature enhancement of rotating machinery. Mech. Syst. Signal Process. 2017, 94, 129–147. [CrossRef]
Entropy 2018, 20, 482
6. 7. 8. 9.
10. 11.
12. 13. 14. 15. 16. 17. 18. 19.
20. 21. 22. 23. 24. 25. 26. 27. 28. 29.
25 of 26
Li, Y.; Zuo, M.J.; Lin, J.; Liu, J. Fault detection method for railway wheel flat using an adaptive multiscale morphological filter. Mech. Syst. Signal Process. 2017, 84, 642–658. [CrossRef] Lv, Y.; Zhu, Q.; Yuan, R. Fault diagnosis of rolling bearing based on fast nonlocal means and envelop spectrum. Sensors 2015, 15, 1182–1198. [CrossRef] [PubMed] He, Q.; Wu, E.; Pan, Y. Multi-scale stochastic resonance spectrogram for fault diagnosis of rolling element bearings. J. Sound Vib. 2018, 420, 174–184. [CrossRef] Liu, Z.; He, Z.; Guo, W.; Tang, Z. A hybrid fault diagnosis method based on second generation wavelet de-noising and local mean decomposition for rotating machinery. ISA Trans. 2016, 61, 211–220. [CrossRef] [PubMed] Li, J.; Li, M.; Zhang, J. Rolling bearing fault diagnosis based on time-delayed feedback monostable stochastic resonance and adaptive minimum entropy deconvolution. J. Sound Vib. 2017, 401, 139–151. [CrossRef] Miao, Y.; Zhao, M.; Lin, J.; Lei, Y. Application of an improved maximum correlated kurtosis deconvolution method for fault diagnosis of rolling element bearings. Mech. Syst. Signal Process. 2017, 92, 173–195. [CrossRef] Miao, Y.; Zhao, M.; Lin, J.; Xu, X. Sparse maximum harmonics-to-noise-ratio deconvolution for weak fault signature detection in bearings. Meas. Sci. Technol. 2016, 27, 10. [CrossRef] Raj, A.S. A novel application of Lucy-Richardson deconvolution: Bearing fault diagnosis. J. Vib. Control 2015, 21, 1055–1067. [CrossRef] Wang, Z.; Wang, J.; Zhao, Z.; Wang, R. A novel method for multi-fault feature extraction of a gearbox under strong background noise. Entropy 2018, 20, 10. [CrossRef] Li, G.; Zhao, Q. Minimum entropy deconvolution optimized sinusoidal synthesis and its application to vibration based fault detection. J. Sound Vib. 2017, 390, 218–231. [CrossRef] Yi, Z.; Pan, N.; Guo, Y. Mechanical compound faults extraction based on improved frequency domain blind deconvolution algorithm. Mech. Syst. Signal Process. 2017. [CrossRef] Zhang, X.; Liang, Y.; Zhou, J.; Zang, Y. A novel bearing fault diagnosis model integrated permutation entropy, ensemble empirical mode decomposition and optimized svm. Measurement 2015, 69, 164–179. [CrossRef] Yan, X.; Jia, M.; Xiang, L. Compound fault diagnosis of rotating machinery based on OVMD and a 1.5-dimension envelope spectrum. Meas. Sci. Technol. 2016, 27, 075002. [CrossRef] Song, Y.; Zeng, S.; Ma, J.; Guo, J. A fault diagnosis method for roller bearing based on empirical wavelet transform decomposition with adaptive empirical mode segmentation. Measurement 2018, 117, 266–276. [CrossRef] Zhang, L.; Wang, Z.; Quan, L. Research on weak fault extraction method for alleviating the mode mixing of LMD. Entropy 2018, 20, 387. [CrossRef] Antoni, J. Fast computation of the kurtogram for the detection of transient faults. Mech. Syst. Signal Process. 2007, 21, 108–124. [CrossRef] Moshrefzadeh, A.; Fasana, A. The autogram: An effective approach for selecting the optimal demodulation band in rolling element bearings diagnosis. Mech. Syst. Signal Process. 2018, 105, 294–318. [CrossRef] Antoni, J. The infogram: Entropic evidence of the signature of repetitive transients. Mech. Syst. Signal Process. 2016, 74, 73–94. [CrossRef] Wang, Y.; Xiang, J.; Markert, R.; Liang, M. Spectral kurtosis for fault detection, diagnosis and prognostics of rotating machines: A review with applications. Mech. Syst. Signal Process. 2016, 66–67, 679–698. [CrossRef] Xu, X.; Qiao, Z.; Lei, Y. Repetitive transient extraction for machinery fault diagnosis using multiscale fractional order entropy infogram. Mech. Syst. Signal Process. 2018, 103, 312–326. [CrossRef] Mojiri, M.; Karimi-Ghartemani, M.; Bakhshai, A. Time-domain signal analysis using adaptive notch filter. IEEE. Trans. Signal Process. 2007, 55, 85–93. [CrossRef] Koshita, S.; Noguchi, Y.; Abe, M.; Kawamata, M. Analysis of frequency estimation mse for all-pass-based adaptive iir notch filters with normalized lattice structure. Signal Process. 2016, 132, 85–95. [CrossRef] Zhao, H.; Li, L. Fault diagnosis of wind turbine bearing based on variational mode decomposition and Teager energy operator. IET Renew. Power Gen. 2016, 11, 453–460. [CrossRef] Wan, S.; Zhang, X.; Dou, L. Shannon Entropy of Binary Wavelet Packet Subbands and Its Application in Bearing Fault Extraction. Entropy 2018, 20, 260. [CrossRef]
Entropy 2018, 20, 482
30.
31. 32. 33. 34. 35. 36. 37. 38. 39. 40.
26 of 26
Zheng, K.; Li, T.; Zhang, B.; Zhang, Y.; Luo, J.; Zhou, X. Incipient Fault Feature Extraction of Rolling Bearings Using Autocorrelation Function Impulse Harmonic to Noise Ratio Index Based SVD and Teager Energy Operator. Appl. Sci. 2017, 7, 1117. [CrossRef] Sun, P.; Liao, Y.; Lin, J. The Shock Pulse Index and Its Application in the Fault Diagnosis of Rolling Element Bearings. Sensors 2017, 17, 535. [CrossRef] [PubMed] Wu, T.-Y.; Yu, C.-L.; Liu, D.-C. On Multi-Scale Entropy Analysis of Order-Tracking Measurement for Bearing Fault Diagnosis under Variable Speed. Entropy 2016, 18, 292. [CrossRef] Feng, Z.; Ma, H.; Zuo, M.J. Spectral negentropy based sidebands and demodulation analysis for planet bearing fault diagnosis. J. Sound Vib. 2017, 410, 124–150. [CrossRef] Eguiraun, H.; Casquero, O.; Martinez, I. The Shannon Entropy Trend of a Fish System Estimated by a Machine Vision Approach Seems to Reflect the Molar Se:Hg Ratio of Its Feed. Entropy 2018, 20, 90. [CrossRef] Cong, F.; Zhong, W.; Tong, S.; Tang, N.; Chen, J. Research of singular value decomposition based on slip matrix for rolling bearing fault diagnosis. J. Sound Vib. 2015, 344, 447–463. [CrossRef] Yan, X.; Jia, M.; Zhang, W.; Zhu, L. Fault diagnosis of rolling element bearing using a new optimal scale morphology analysis method. ISA Trans. 2018, 73, 165–180. [CrossRef] [PubMed] Zhao, X.; Ye, B. Selection of effective singular values using difference spectrum and its application to fault diagnosis of headstock. Mech. Syst. Signal Process. 2011, 25, 1617–1631. [CrossRef] Alharbi, N.; Hassani, H. A new approach for selecting the number of the eigenvalues in singular spectrum analysis. J. Frankl. Inst. 2016, 353, 1–16. [CrossRef] Guo, Y.; Wei, Y.; Zhou, X.; Fu, L. Impact feature extracting method based on S transform time-frequency spectrum denoised by SVD. J. Vib. Eng. 2015, 27, 621–628. Case Western Reserve University Bearing Data Center Website. Available online: https://csegroups.case. edu/bearingdatacenter/home (accessed on 8 June 2016). © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).