Rolling Bearing Fault Diagnosis Based on Optimal

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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6

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

Amplitude Amplitude

10 10 0 0 -10 -10 0 0

0.05 0.05

0.1 0.15 0.1 0.15 T ime (s) T ime (s)

0.2 0.2

0.25 0.25

 

0.5 0.5 0 00 0

f1 f1

100 100

200 300 200 300 Frequency (Hz) Frequency (Hz)

400 400

7.1 7.1 7 7 6.9 6.9 6.8 6.8 0 0

   

3 3 Amplitude Amplitude

TEE TEE

  (a)  (b)  (a)  (b)  Figure 6. The simulated signal S(t): (a) its time waveform; (b) its envelope spectrum.  Figure 6. The simulated signal S(t): (a) its time waveform; (b) its envelope spectrum. Figure 6. The simulated signal S(t): (a) its time waveform; (b) its envelope spectrum. 

500 500

(0.43,6.8256) (0.43,6.8256)

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 Bw (Normalization frequency) Bw (Normalization frequency)

(a)  (a) 

Amplitude Amplitude

0.4 0.4 0.2 0.2 0 00 0

f o f o

1 1

   

0 0 -3 -3 0 0

0.05 0.05

Noise interferences Noise interferences 0.1 0.15 0.2 0.25 0.1 0.15 0.2 0.25 T ime (s)   T ime (s)

(b)  (b) 

 

Amplitudes of noise Amplitudes of noise inteferences 2f o inteferences 2 f 3 fo 4 f o 3f o 5f o 6f o 4f o o 5f o 6f o

200 200

400 600 400 600 Frequency (Hz) Frequency (Hz)

(c)  (c) 

800 800

1000 1000

   

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. 

DSSV

Amplitude

Amplitude

2 40 20

i=19 0

0

20

Noise interferences 40 60 80 Singular value number i

-2

100

0.05

0.1 0.15 T ime (s)

0.2

0.25

 

(b) 

2

0.3 Amplitude

Amplitude

0

 

(a)  1 0 -1

0

f

0.05

0.1 0.15 T ime (s)

(c) 

0.2

0.25

 

2f

o

3f

o

0.1 0

0

o

0.2

0

200

4f

o

5f

o

6f

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Figure  8. The The ESVD ESVD  de‐noising  results  of optimal the  optimal  (a)  the (b) DSSV;  (b) de-noised the  SVD  Figure 8. de-noising results of the notch notch  signal:signal:  (a) the DSSV; the SVD de‐noised signal; (c) the ESVD de‐noised signal; (d) the envelope spectrum of (c).  signal; (c) the ESVD de-noised signal; (d) the envelope spectrum of (c).

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

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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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Accelerometers Accelerometers Fan end Fan end bearing bearing

Torque Torque transducer transducer &encoder &encoder

Drive end Drive end bearing bearing

Electric motor Electric motor

Dynamometer Dynamometer

   

(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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(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). 

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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

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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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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)

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Figure 26. Cont.

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Entropy 2018, 20, x FOR PEER REVIEW    22 of 26  -3 0.05 x 10 Entropy 2018, 20, x FOR PEER REVIEW    22 of 26  3 -3 0.05 x 10 -3 0.05 23 x 10f 2f 0 o o 3 12 f 2f 0 o o 2 f 2f 01 -0.05 0 o o 01 100 200 300 400 500 600 0 0.1 0.2 0.3 0.4 0.5 0.6 Frequency(Hz) T ime (s) 0 -0.05 0 100 200 300 400 500 600  0 0.1 0.2 0.3 0.4 0.5 0.6   -0.05 (c)  300 400 (d) T 0.3 0 100 200 500 600 0 0.1 0.2 0.5 0.6 Frequency(Hz) ime (s)0.4 -3     Frequency(Hz) T ime (s) x 10     (c)  (d)  2 -3 (c)  (d)  5 f x 10 -3 o 2 x 10 3f o 1 2 5f f 2f 4f 6f o o o o o 3f 5f o o 1 3f f 2f 4 f 0 1 o 6f o o 0 100 200 300 4o400 500 6of600 f 2f f o o o o Frequency (Hz) 0 100 200 300 400 500 600  0 (e)  300(Hz) 400 500 600 0 100 200 Frequency   Frequency (Hz)

  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. 

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  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). 

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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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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.

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