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Health Degradation Monitoring and Early Fault Diagnosis of a Rolling Bearing Based on CEEMDAN and Improved MMSE Yong Lv 1,2 , Rui Yuan 1,2, * 1

2 3 4

*

ID

, Tao Wang 1,2 , Hewenxuan Li 3 and Gangbing Song 4

Key Laboratory of Metallurgical Equipment and Control Technology, Wuhan University of Science and Technology, Ministry of Education, Wuhan 430081, China; [email protected] (Y.L.); [email protected] (T.W.) Hubei Key Laboratory of Mechanical Transmission and Manufacturing Engineering, Wuhan University of Science and Technology, Wuhan 430081, China Department of Mechanical, Industrial and Systems Engineering, University of Rhode Island, Kingston, RI 02881, USA; [email protected] Smart Material and Structure Laboratory, Department of Mechanical Engineering, University of Houston, Houston, TX 77204, USA; [email protected] Correspondence: [email protected]; Tel.: +86-027-68862857; Fax: +86-027-68862212

Received: 16 May 2018; Accepted: 11 June 2018; Published: 14 June 2018







Abstract: Rolling bearings play a crucial role in rotary machinery systems, and their operating state affects the entire mechanical system. In most cases, the fault of a rolling bearing can only be identified when it has developed to a certain degree. At that moment, there is already not much time for maintenance, and could cause serious damage to the entire mechanical system. This paper proposes a novel approach to health degradation monitoring and early fault diagnosis of rolling bearings based on a complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) and improved multivariate multiscale sample entropy (MMSE). The smoothed coarse graining process was proposed to improve the conventional MMSE. Numerical simulation results indicate that CEEMDAN can alleviate the mode mixing problem and enable accurate intrinsic mode functions (IMFs), and improved MMSE can reflect intrinsic dynamic characteristics of the rolling bearing more accurately. During application studies, rolling bearing signals are decomposed by CEEMDAN to obtain IMFs. Then improved MMSE values of effective IMFs are computed to accomplish health degradation monitoring of rolling bearings, aiming at identifying the early weak fault phase. Afterwards, CEEMDAN is performed to extract the fault characteristic frequency during the early weak fault phase. The experimental results indicate the proposed method can obtain a better performance than other techniques in objective analysis, which demonstrates the effectiveness of the proposed method in practical application. The theoretical derivations, numerical simulations, and application studies all confirmed that the proposed health degradation monitoring and early fault diagnosis approach is promising in the field of prognostic and fault diagnosis of rolling bearings. Keywords: health degradation monitoring; early fault diagnosis; CEEMDAN; improved MMSE; smoothed coarse graining process; rolling bearing

1. Introduction Rolling bearings are a crucial part of the mechanical system, and its operational state directly affects the normal operation of the entire system. When the failure of a rolling bearing occurs, the change of the dynamic characteristics can be reflected from collected vibration signals [1–4], however, in most cases, the early fault of the rolling bearing is difficult to identify during the initial Materials 2018, 11, 1009; doi:10.3390/ma11061009

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phase of the fault. The dynamic characteristics generated by early fault are usually submerged in strong background noise, thus, the weak fault is difficult to extract, and the exact time when the fault starts to happen is difficult to identify [5–8]. In most cases, the fault of the rolling bearing can only be identified when it has developed to a certain degree. At that moment, there is already not much time for maintenance, and could cause serious damage to the whole mechanical system in a short time thereafter. Therefore, to find an effective way to identify early faults of rolling bearings and conduct fault diagnosis in advance of serious faults is critical [9]. The early weak fault phase, namely when the fault begins to occur initially, needs to be identified, then some technical means could be implemented to avoid a serious fault from happening, for instance, by replacing machine parts or conducting maintenance after shutting down the mechanical system. The health degradation monitoring of rolling bearings can spare a great deal of time for maintenance, avoid unnecessary losses, and reduce the risk of catastrophic consequences to great extents [10,11]. The early weak fault phase can be identified during health degradation monitoring processes. Afterwards, early fault diagnosis can detect the weak fault of the rolling bearing, and determine the fault type of the rolling bearing, thereby, the parts can be maintained or replaced in time [12]. Hence, the health degradation monitoring and early fault diagnosis is significant in the field of prognostic and fault diagnosis of rolling bearings. Many nonlinear signal processing methods have been proposed and developed in recent years, such as wavelet packet decomposition (WPD) [13], short-time Fourier transform (STFT) [14], independent component analysis (ICA) [15], empirical mode decomposition (EMD) [16], and empirical wavelet transform (EWT) [17]. Among all of the above methods, EMD [18] was proposed to adaptively decompose a time series into several approximating stationary time series, which is called the intrinsic mode function (IMF). Compared to other signal processing methods in the field of mechanical fault diagnosis, EMD is an adaptive nonlinear and nonstationary signal processing method, and has no requirement of basic functions. However, EMD still has modal aliasing and end effect problems, and the ensemble empirical mode decomposition (EEMD) [19–21] was proposed to alleviate the mode aliasing problem by utilizing the property of frequency uniform distribution of Gaussian white noise. Then complementary ensemble empirical mode decomposition (CEEMD) [22] was proposed to improve EEMD by adding positive and negative Gaussian white noise. In this way, it can guarantee the same decomposition effect as EEMD, and also reduce the sequence reconstruction errors caused by added white noise. However, CEEMD still cannot solve the problem of different orders of obtained IMFs caused by adding different noise signals, which promotes the emergence of the complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) [23,24]. CEEMDAN is an important development of the EEMD method, and CEEMDAN can realize the approximate perfect reconstruction of the decomposed signal, while also avoiding the problem that different noise signals generate different orders of IMFs. Entropy is a method of measuring the complexity of a time series. The approximate entropy [25] was proposed at the first beginning, and sample entropy (SE) [26] was proposed afterwards as an improvement method. SE analysis can reflect the single scale time series information. In the related traditional entropy-based analysis methods, such as permutation entropy [27,28], information entropy [29,30], and wavelet packet entropy [31], they can both measure the regularity and orderliness of the time series. The SE value decreases along with the decrease of disorder of the time series. The SE algorithm is a typical method proposed to measure the complexity and quantization of time series [26] and has been successfully applied to physiological time series analysis. The SE algorithm compares the data with itself, namely self-matching. Hence, the approximate entropy is the measure of new information of the generated time series, which makes the results occasionally contain false information. Compared with other nonlinear dynamic methods, such as the Lyapunov exponent [32] and fractal dimension [33], the SE can obtain a stable estimation value with less data, and tolerate a larger range of parameter values which need to be selected. To analyze the complexity of time series on different time scales, multiscale sample entropy (MSE) [34] was proposed based on SE. In the MSE algorithm, the coarse graining process is adopted to

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obtain the multiscale time series, instead of the original single-scale time series. Then the obtained signal is analyzed at each scale. The obtained MSE values can better reflect the dynamic characteristic changes of the time series. MSE has been widely applied to nonlinear and nonstationary data analysis [35,36], and it only requires short data to gain stable entropy values and has better anti-noise ability. The MSE greatly enriches the meaning of SE. The greater the probability of generating new patterns, and the more complex the time series is, the larger the entropy value is. Hence, the value can be used as the judgmental index and characteristic parameter to characterize the complexity of fault signals. MSE has a good performance in analyzing scalar time series, but when it comes to multivariate time series, it can only calculate the data of each channel separately. It cannot reflect the correlations and relationships between multivariate data, which can reflect the dynamic characteristics of multivariate time series more clearly and accurately. Later multivariate multiscale sample entropy (MMSE) [37] was proposed as a follow up study of MSE, and it can compute the MSE of multivariate time series, and deal with different embedding dimensions, delay time, and range of data channels in a strict and unified manner. On account of the internal essence of SE that it can measure the regularity and complexity of dynamic systems, MMSE can be adopted to continuously detect the characteristic change of time series [38,39]. Hence, it can be used to monitor the health condition and performance degradation of rolling bearings. This paper proposed a novel approach to health degradation monitoring and early fault diagnosis of rolling bearings based on CEEMDAN and improved MMSE. The conventional MMSE is improved by the smoothed coarse graining process. During the health degradation monitoring of rolling bearings, all signals continuously collected from the mechanical system can be decomposed by CEEMDAN to obtain IMFs, and MMSE values of effective IMFs are computed. By analyzing the MMSE values of all signals in the long-term, the operating condition of the mechanical systems can be monitored. The improved MMSE of IMFs obtained by CEEMDAN can amplify the dynamic change characteristics to solve the problem that an early weak fault is difficult to identify and extract. After identifying the early weak fault stage, the early fault diagnosis of the rolling bearing can be accomplished by CEEMDAN. CEEMDAN can remove the strong background noise of the early weak fault signal, while extracting the fault characteristic frequency accurately. The proposed approach mainly aims at conducting health degradation monitoring of rolling bearings in complex mechanical systems, to determine the early weak fault phase in the whole wear-out process of a rolling bearing, and, thus, conduct early fault diagnosis of rolling bearings. The rest of this paper is organized as follows: Section 2 introduces the methodology of CEEMDAN and MMSE improved by the smoothed coarse graining process and the proposed novel health degradation monitoring and early fault diagnosis approach. Section 3 presents the numerical simulations of CEEMDAN and MMSE, including the comparison between CEEMDAN and EEMD, MMSE with smoothed coarse graining process, and conventional MMSE. Section 4 presents the application studies of the proposed method to run-to-failure fault rolling bearing signals, to verify its effectiveness and validity. Section 5 concludes the paper. 2. Methodology 2.1. Complete Ensemble Empirical Mode Decomposition with Adaptive Noise EMD is an adaptive signal decomposition method for analyzing nonlinear and non-stationary signals [18]. EMD is analogous to wavelet analysis, while EMD can overcome the difficulty that wavelet decomposition requires a reasonable choice of wavelet basis functions. Its essence is to decompose the original signal in order of different fluctuations. A series of IMFs with different amplitudes are obtained. The IMF in the EMD method must satisfy: (1) The number of extreme points and the number of zero crossings must be equal or, at most, one difference; and (2) the upper envelope consists of the local maximum points, the lower envelope consists of the local minimum points, and the average values of the upper and lower envelops are all 0. The detailed procedures of EMD are below:

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(1) (2) (3) (4) (5)

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Given a signal x, and calculate all the maximum and minimum values of rk (k = 0), here rk = x. Use the cubic spline to interpolate all maximum and minimum points of rk to obtain the upper and lower envelopes emax and emin , respectively. Materials 2018, 11, x FOR PEER REVIEW 4 of 22 Calculate the average of the upper and lower envelopes m = (emax + emin )/2. (1) Given a signal the maximum minimum of rk (k = 0), here rk = x. if not, repeat Calculate the IMF byx,rkand − calculate m = hk+1all , and decide ifand hk+1 satisfiesvalues the conditions of IMF, (2) Use the cubic spline to interpolate all maximum and minimum points of rk to obtain the upper (2)–(3) to obtain the envelope average that satisfies the conditions. and lower envelopes emax and emin, respectively. Separate hk+1 from x to getof rthe , thenand letlower k = kenvelopes + 1, repeat (2)–(4) with regarding rk as the (3) Calculate the average m = (esteps max + emin )/2. k+1upper (4) Calculate the IMF by r k − m = h k+1 , and decide if h k+1 satisfies the conditions of IMF, if not, repeat original time series. (2)–(3) to obtain the envelope average that satisfies thekconditions. (5) Separate hk+1 from x to get rk+1, then letrkk+=1 k=+ x 1, − repeathsteps (2)–(4) with regarding rk as the i original time series. i =1

∑

(6)

(1)

k

Repeat the above steps until the residualrk +that meets the stop condition is obtained.(1) 1= x − ∑ hi i =1

To solve the mode mixing problem exists EMD, which would result in IMF distortion, (6) Repeat the above steps until the residual that in meets the stop condition is obtained. EEMD was proposed [19–21]. The EEMD algorithm is a noise-assisted signal processing method, To solve the mode mixing problem exists in EMD, which would result in IMF distortion, EEMD and EEMD EMD The multiple times onis the signal superimposed Gaussian white noise. wasperforms proposed [19–21]. EEMD algorithm a noise-assisted signal processingwith method, and EEMD performs EMD multiple times on the signal superimposed with Gaussian white that noise. The The superimposed signal has continuity on various frequency scales, on account Gaussian white superimposed has continuity various distribution. frequency scales,Hence, on account that can Gaussian noise has the statisticalsignal characteristics ofon uniform EEMD help white to alleviate the noise has the statistical characteristics of uniform distribution. Hence, EEMD can help to alleviate the mode mixing problem in the IMF component. The computational framework of the EEMD is basically mode mixing problem in the IMF component. The computational framework of the EEMD is basically the same as EMD, different white noise w(i) 1,…, . . .I),, where I), where I is the ensemble size, can provide the same asand EMD, and different white noise w(i)(i(i = = 1, I is the ensemble size, can provide a consistent reference structure timedomain domain distribution of the components after a consistent reference structure for for thethe time distribution ofremaining the remaining components after each decomposition. The illustration of EEMD is shown in Figure 1, and the ensemble size determines each decomposition. The illustration of EEMD is shown in Figure 1, and the ensemble size determines the times of EMD conducted on the superimposed signal. The detailed procedures of EEMD are the times of EMD conducted on the superimposed signal. The detailed procedures of EEMD are below: below: Original signal Gaussian white noise Superimposed signal

EMD

Multi-sets of IMFs Zero-mean principle IMFs of EEMD Figure 1. The illustration of EEMD.

Figure 1. The illustration of EEMD.

(1) (2) (3)

(1) Add Gaussian white noise to the signal to form a superimposed signal x(i) = x + εw(i). (i). (2) Add Gaussian white noise to the signal to form a superimposed signal x(i) = x + εw Add Gaussian white noise to the signal to form a superimposed signal x(i) = (3) Perform EMD of x(i) to obtain IMFs dk(i) (k = 1, …, K), K is the number of all IMFs:

x + εw(i) . K Add Gaussian white noise to the signal to form a superimposed signal x(i) = x + εw(i) . = x (i ) (i)∑ d k(i ) + r (i ) (2) (i) Perform EMD of x to obtain IMFs dk k =(k 1 = 1, . . . , K), K is the number of all IMFs:

(4) Adopt the zero-mean principle of Gaussian white noise to eliminate the influence by taking K Gaussian white noise as a time domain distribution reference structure. Then the IMFs can be (i ) x (i ) = d k + r (i ) expressed as: 1 I k =1 d k = ∑ d k(i ) (3)

∑

I

(4)

(2)

i =1

Adopt the zero-mean principle of Gaussian white noise to eliminate the influence by taking Gaussian white noise as a time domain distribution reference structure. Then the IMFs can be expressed as: dk =

1 I (i ) dk I i∑ =1

(3)

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Among the above procedures, the input signal of each EMD is rk (i) = rk+1 (i) − dk (i) , and there is no correlation between the rk (i) of the EMD of different noise signals. It would make EEMD suffer from reconstruction error and generate different numbers of IMFs due to different EMD of various noise signals. To solve the reconstruction error in EEMD, CEEMD was proposed [22], in which Gaussian white noise is added into x in pairs (one positive and one negative) for twice EEMD, as shown in: "

(i )

y1

(i )

y2

#

"

=

1 1

1 −1

#"

x w (i )

# (4)

Such a method can reduce the reconstruction error to great extents, however, it still cannot guarantee the same number of IMFs generated by EMD of y1 (i) and y2 (i) , which makes it difficult to compute the average. Thus, CEEMDAN [23,24] was put forward to ensure the same number of IMFs can be obtained, and reconstruction error can be eliminated at the same time. The first-order IMF obtained by CEEMDAN equals the first-order IMF obtained by EEMD, and the first residual r1 can be obtained by adding specific noise to make rk of each decomposition invariable. The detailed procedures of CEEMDAN can be expressed as follows, among them Ek (·) is defined as the operator of kth IMF, and w(i) is defined as zero-mean Gaussian white noise: (1)

Perform EMD towards x(i) = x + β0 w(i) (i = 1, . . . , I), and the first order IMF is: 1 I (i ) dˆ1 = ∑ d1 = d1 I i =1

(2) (3)

Compute the first residual: r1 = x − dˆ1 Perform EMD to obtain the first IMF of r1 + β1 E1 (w(i) ) (i = 1, . . . , I), and the second IMF is: 1 I dˆ2 = ∑ E1 (r1 + β 1 E1 (w(i) )) I i =1

(4)

(7)

Perform EMD to obtain the first IMF of rk + βk Ek (w(i) ) (i = 1, . . . , I), and the (k + 1)th IMF is: 1 I dˆ(k+1) = ∑ E1 (rk + β k Ek (w(i) )) I i =1

(6)

(6)

Compute kth residual for k = 2, 3, . . . , K: rk = r(k+1) − dˆk

(5)

(5)

(8)

Return to step (4) to compute the next order IMF, and repeat steps (4)–(6) until the residual cannot be decomposed by EMD. The coefficient βk = εk std(rk ) allows the SNR to be selected during each iteration, and std(·) is the standard deviation operator.

2.2. Improved Multivariate Multiscale Sample Entropy The SE [26] is an improved algorithm of approximate entropy, which can reflect the dissimilar probability of two similar sequences. The detailed procedures of SE are below: (1) (2) (3)

For the original time series X = {x1 , x2 , . . . , xN }, X(i) = [xi , xi+1 , . . . , xi+m−1 ], (1 ≤ i ≤ N − m) can be defined, m is the embedding dimension. Compute dij (1≤ j ≤ N − m, j 6= i) of X(i) and X(j), and calculate num(dij < r) when dij < r. dij is the maximum absolute value of difference of X(i) and X(j). Define Bim (r) = num(dij < r)/(N − m − 1). Compute the mean value of Bim (r), denoted by Bm (r).

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As for the dimension of m + 1, repeat above procedures to obtain Bim+1 (r), then Bm+1 (r) can be obtained. The SE can be expressed as: Materials 2018, 11, x FOR PEER REVIEW Materials 2018, 11, x FOR PEER REVIEW

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SE(m, r, N ) = InBm (r ) − InBm+1 (r )

Define the time scale on account of the coarse graining aiming at time series {x k,i}, in Define the time scaleon onaccount accountof ofthe thecoarse coarse graining graining process process aiming atatthe the time {x{x k,i}, in Define the time scale process aiming the timeseries series k,i }, in which i = 1, 2, …, N, and k = 1, 2, …, p. p is the number of variables, and N is the number of the points which i = 1, 2, …, N, and k = 1, 2, …, p. p is the number of variables, and N is the number of the points which i = 1, 2, . . . , N, and k = 1, 2, . . . , p. p is the number of variables, and N is the number of the of of each each variable. variable. As As for for any any scale scale φ, φ, the the obtained obtained multiple multiple variable variable time time series series by by the the coarse coarse graining graining points of each variable. As for any scale ϕ, the obtained multiple variable time series by the coarse process process is is as as follows: follows: graining process is as follows: jϕ

1 jϕjϕ ϕ x i 11 ϕy kϕ, j = = y yk,jk ,= j ,i ϕ ( j −1)ϕ +1 xxkk ,k,i ϕϕi=(ii= =( j −1)ϕ +1 j −1) ϕ +1

∑ ∑ ∑

(10) (10)(10)

where 11 ≤≤ jj ≤≤ N/φ, k = 1, 2, p. Take scale == 33 as an the coarse process where 2, …, …, scale scale as = an3example, example, the conventional conventional coarse graining graining where 1≤ j ≤ N/φ, N/ϕ,k k= 1, = 1, 2, .p. . .Take , p. Take as an example, the conventional coarseprocess graining is illustrated in Figure 2. is illustrated in Figure 2. process is illustrated in Figure 2.

xx1 1

xx2 2

xx3 3

xx4 4

xx5 5

xx6 6

xx7 7

xx8 8

xx9 9

xxN − 2 xxNN −−11 xxN −5 x xxN N −2 N −5 xN − 4 x N N − 4 xN −3 N −3

 

yy3 3

yy2 2

yy1 1

yy N −1 N −1 ϕ

yy N N ϕ ϕ

ϕ

Figure 2. The illustration of the conventional coarse graining process. Figure2.2.The Theillustration illustration of of the the conventional conventional coarse Figure coarsegraining grainingprocess. process.

It It can can be be seen seen from from Figure Figure 22 that that the the conventional conventional coarse coarse graining graining process process compresses compresses the the time time It can be seen from Figure 2 that the conventional coarse graining process compresses the time series series by by scale scale factors. factors. Along Along with with the the increase increase of of scale, scale, the the size size of of the the coarse coarse grained grained time time series series series by scale factors. Along with the increase of scale,is the size of themultiple coarse grained time series decreases, decreases, and and when when the the length length of of the the original original time time series series is not not an an integral integral multiple of of the the scale scale factor, factor, decreases, anddata when thebelength of the original time seriesprocess. is not an integral multiple of the scale factor, some some of of the the data will will be lost lost during during the the coarse coarse graining graining process. All All of of the the above above phenomena phenomena would would some of the data will be lost during the coarse graining process. All of the above phenomena would inevitably affect the calculation accuracy of the MMSE algorithm [37]. Aiming at improving such inevitably affect the calculation accuracy of the MMSE algorithm [37]. Aiming at improving such disadvantages, the coarse graining process, which adopts sliding average during inevitably affect the calculation accuracy of the MMSE algorithm Aiming at method improving such disadvantages, the smoothed smoothed coarse graining process, which adopts aa[37]. sliding average method during the coarse graining process, is proposed. This method avoids data loss, and ensures the coarsedisadvantages, the smoothed coarse graining process, which adopts a sliding average method during the coarse graining process, is proposed. This method avoids data loss, and ensures the coarsetime are length as original time series at each scale, significantly thegrained coarse graining process, proposed. method avoids loss, the coarse-grained grained time series series are the theissame same lengthThis as the the original timedata series at and eachensures scale, both both significantly improving the accuracy of subsequent algorithms. The smoothed coarse graining is time series arethe the accuracy same length as the original time series at each scale, coarse both significantly improving improving of subsequent algorithms. The smoothed graining process process is illustrated in Figure 3. theillustrated accuracy of algorithms. The smoothed coarse graining process is illustrated in Figure 3. in subsequent Figure 3.

xx1 1

xx2 2

xx3 3

xx4 4

xx5 5

xx6 6

xx7 7

xx8 8

xx9 9

xxN − 2 xxNN −−11 xxN −5 x xxN N −2 N −5 xN − 4 x N N − 4 xN −3 N −3

 

yy1 1

yy2 2

yy3 3

yy4 4

yy5 5

yy6 6

yy7 7

y yyN −5 y y −3 yNN −−22 − 4 yN N − 5 yN N −3 N −4

Figure3.3.The The illustration of of proposed smoothed Figure smoothedcoarse coarsegraining grainingprocess. process. Figure 3. Theillustration illustration of proposed proposed smoothed coarse graining process. φ Then Then the the multivariate multivariate SE SE of of each each multiple multiple variable variable yykjkjφ is is computed, computed, the the multivariate multivariate embedded embedded vector needs to be constructed in advance. Embedding theorem [40,41] is used to obtain vector needs to be constructed in advance. Embedding theorem [40,41] is used to obtain the the embedded embedded vectors vectors of of the the multivariate multivariate time time series. series. For For time time series series of of p-variate p-variate time time series, series, the the multivariate multivariate embedding embedding reconstruction reconstruction is is as as follows: follows:

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Then the multivariate SE of each multiple variable ykj ϕ is computed, the multivariate embedded vector needs to be constructed in advance. Embedding theorem [40,41] is used to obtain the embedded vectors of the multivariate time series. For time series of p-variate time series, the multivariate embedding reconstruction is as follows: Xm (i ) = [ x1,i , · · · x1,i+(m1 −1)λ1 , x2,i , · · · x2,i+(m2 −1)λ2 , x p,i , · · · x p,i+(m p −1)λ p ]

(11)

where M = [m1 , m2 , . . . , mp ] ∈ Rp is the embedding vector, and λ = [λ1 , λ2 , . . . , λp ] is the time delay vector. X m (i) ∈ Rm (m = m1 + m2 + . . . mp ). As for the above multivariate time series, the MMSE can be computed as follows: (1)

Constitute multivariate embedding vector X m (i), and define the distance of any two vectors X m (i) and X m (j) as the maximum norm as follows: D [Xm (i ), Xm ( j)] = maxl =1,2,··· ,m {| x (i + l − 1) − x ( j + l − 1)|}

(2)

(3)

As for the composite delay vector X m (i) and a threshold r, determine the number of instances Pi , where D[X m (i), X m (j)] ≤ r, j 6= i. Then compute the occurrence frequency Bi m (r) = Pi /(N − n − 1), where n = max{M} × max{λ}. Compute the average of B, denoted by Bm (r). B m (r ) =

(4)

(5)

1 N −n m Bi (r ) N − n i∑ =1

(13)

Extend the dimension of multivariate delay factor in (2) from m to (m + 1). Then, as for one embedding vector M = [m1, m2, . . . , mk . . . , mp], it can be converted into random space with the embedding vector of M = [m1, m2, . . . , mk+1 . . . , mp] in p different ways. Thus, p × (N − n) vectors Xm+1(i) can be obtained in Rm+1, where Xm+1(i) represents any embedding vector which increases embedding dimension mk to (mk + 1) for specific k. Due to the constant of the embedding dimension of other data channels in this process, the overall embedding dimension of multivariate time series increases from m to (m +1). m +1 Repeat procedures of (1)–(4) to compute all Bi k (r ), and calculate the mean value Bi m+1 (r) upon k. Then compute the mean value Bm+1 (r) upon i in the (m + 1) dimensional space as: B m +1 ( r ) =

(6)

(12)

1 p( N − n)

p( N −n)

∑

i =1

Bim+1 (r )

(14)

Here, Bm (r) represents the similar possibility in m dimensional space of any two composite delay vectors, whereas Bm+1 (r) represents the similar possibility upon (m + 1) dimensional space of two composite delay vectors. Then the MMSE can be expressed as: MMSE(M, λ, r, N ) = InBm (r ) − InBm+1 (r )

(15)

2.3. The Proposed Novel Health Degradation Monitoring Approach of Rolling Bearings Studies on the IMFs of EMD and the derived methods demonstrate IMFs of EMD-derived methods can be adopted to accurately depict signal dynamics, and the unique properties of decoupling frequency information [42–46]. Hence, the intrinsic analysis method can be applied to detect the change regularity of complex dynamic systems. In this paper, a novel health degradation monitoring and early fault diagnosis approach is proposed based on CEEMDAN and improved MMSE, where the smoothed coarse graining process was proposed to improve the conventional MMSE. All continuously-collected

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signals of rolling bearings can be decomposed by CEEMDAN to obtain IMFs, and MMSE values of effective IMFs can be computed to accomplish health degradation monitoring of mechanical systems. After identifying the early weak fault phase, the fault frequency of rolling bearings can be extracted by CEEMDAN. The schematic diagram of the proposed method in this paper is illustrated in Figure 4. Materials 2018, 11, x FOR PEER REVIEW

Collected Signals X (t )  ( x1 (t ), x2 (t ),

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CEEMDAN , xn (t ) )

(Test To Failure signals) CEEMDAN

Improved MMSE

Health Degradation Monitoring

Theoretic Highlights:  EMD — EEMD — CEEMD — CEEMDAN  SE — MSE — MMSE (Smoothed Coarse graining process)

(Effective IMFs) IMFi (t )

Health Degradation Monitoring Improved MMSE

IMFk (t )

MMSE value of x1 (t ) (scale=20)

…… Time Index /h X (t )  ( x1 (t ), x2 (t ), , xn (t ) )

Scale

Faulty Signal of Phase 2 (Health Degradation process)

Early Fault Diagnosis

CEEMDAN

Early Fault Diagnosis

Original Signal CEEMDAN

IMFs Reconstruction

fr fo

Frequency Index /Hz

Frequency Index /Hz

Figure 4. The proposed health degradation monitoring and early fault diagnosis scheme of a rolling

Figure bearing. 4. The proposed health degradation monitoring and early fault diagnosis scheme of a rolling bearing. 3. Numerical Simulations

3. Numerical Simulations

3.1. Simulation Research of Complete Ensemble Empirical Mode Decomposition with Adaptive Noise

3.1. Simulation Research of Complete Ensemble Empirical Mode Decomposition with Adaptive Noise Mechanical operation signals usually have characteristics of frequency modulated signals, thus,

Mechanical operation usually have characteristics of frequency modulated the frequency modulatedsignals signals are used here to verify the effectiveness of the proposed method in signals, thus, the frequency modulated signals used here toThe verify the effectiveness of bearing the proposed tracking the characteristic change of theare simulated signal. vibrational signal of rolling methodcan in be tracking thetocharacteristic of the simulated signal. The vibrational signal of rolling simplified a signal model change [47] as follows: bearing can be simplified to a signalxmodel [47] as follows: (t )   sin(2 f t )[1   cos(2 f t )] (16) b

r

where fb denotes the characteristic the rolling x (t) = frequency α sin(2π fofb t)[ 1+ β cosbearing, (2π f r t)]and fr denotes the rotational (16) frequency. α and β denote the power size. The collected rolling bearing signal usually has additive noise, and Gaussian white noise is adopted here to simulate the practical situation [48]. The filter where fb denotes the characteristic frequency of the rolling bearing, and fr denotes the rotational bank property of EEMD and CEEMDAN is important in alleviating the current mode mixing frequency. α and β denote the power size. The collected rolling bearing signal usually has additive problem, which is realized by taking advantage of the frequency uniformly-distributed property of noise, and Gaussian whiteHence, noisethe is simulated adopted faulty here rolling to simulate practical situation The filter Gaussian white noise. bearingthe signal can be simulated as [48]. follows:

bank property of EEMD and CEEMDAN is important in alleviating the current mode mixing problem, x1 (t )  cos(2 f1t )  sin(2 f 2t )[1  cos(2 f3t )]  s (17) which is realized by taking advantage of the frequency uniformly-distributed property of Gaussian whereHence, f1 = 35 Hz, = 80 Hz, f3 =faulty 10 Hz. rolling s denotesbearing the Gaussian white added to the with white noise. thef2 simulated signal can noise be simulated as signal, follows: the variance of 2, and mean of 0. The sampling point is 2048, and the sampling frequency is 2048 Hz during the numerical simulation. The EEMD and CEEMDAN are applied to the simulated signal.

x1 (t) = cos(2π f 1 t) + sin(2π f 2 t)[1 + cos(2π f 3 t)] + s

(17)

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where f 1 = 35 Hz, f 2 = 80 Hz, f 3 = 10 Hz. s denotes the Gaussian white noise added to the signal, with the variance of 2, and mean of 0. The sampling point is 2048, and the sampling frequency is Materials 2018, 11,the x FOR PEER REVIEW 9 of 22 2048 Hz during numerical simulation. The EEMD and CEEMDAN are applied to the simulated signal. The standard deviation of the added noise is 0.2, and the ensemble size is 50 in EEMD. pairs (positive and negative) of added of noise are noise adopted CEEMDAN. The time domain of Twenty-five pairs (positive and negative) added are in adopted in CEEMDAN. The timeplots domain IMFs of EEMD and CEEMDAN are shown in Figure 5. plots of IMFs of EEMD and CEEMDAN are shown in Figure 5.

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It can be seen from Figure 5 that 10 orders of IMFs are obtained by EEMD, while eight orders of It can be seen from Figure 5 that 10 orders of IMFs are obtained by EEMD, while eight orders IMFs are obtained by CEEMDAN. Obviously, CEEMDAN generates fewer orders of IMFs than of IMFs are obtained by CEEMDAN. Obviously, CEEMDAN generates fewer orders of IMFs than EEMD does. To illustrate the effectiveness of CEEMDAN in alleviating the mode mixing problem, EEMD does. To illustrate the effectiveness of CEEMDAN in alleviating the mode mixing problem, the correlation analysis is conducted to find the effective IMFs, and such a method was elaborated in the correlation analysis is conducted to find the effective IMFs, and such a method was elaborated in authors’ previous studies [49,50]. During the correlation analysis, the 3rd, 4th, and 5th IMFs of EEMD, authors’ previous studies [49,50]. During the correlation analysis, the 3rd, 4th, and 5th IMFs of EEMD, and the 3rd and 4th of CEEMDAN, are effective. The frequency domain plots of effective IMFs of EEMD and CEEMDAN are shown in Figure 6.

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and the 3rd and 4th of CEEMDAN, are effective. The frequency domain plots of effective IMFs of Materials 2018, 11, x FOR PEER REVIEW in Figure 6. 10 of 22 EEMD and CEEMDAN are shown

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It can be seen from Figure 6 that f1 = 35 Hz can both be well extracted by EEMD and CEEMDAN

Itwith can abeslight seen difference. from Figure 6 that f 1 = 35 Hz can both be well EEMD and CEEMDAN Furthermore, it can be observed fromextracted Figure 6aby that the characteristic with frequencies a slight difference. Furthermore, it can beextracted observed from 6a thatwhile the fcharacteristic f2 (±f3), namely 80 Hz (±10 Hz), can be in the 3rdFigure IMF of EEMD, 2 can also be found the 4th IMF of 80 EEMD. disturbed by some additive noise frequencies f 2in(± f 3 ), namely Hz (All ±10characteristic Hz), can befrequencies extracted are in the 3rd IMF of EEMD, while f 2 can frequencies the4th 3rdIMF andof 4thEEMD. IMF of All EEMD. This is exactly the mode are mixing phenomenon existing also be found inin the characteristic frequencies disturbed by some additive the EMD derived forIMF the performance of CEEMDAN, canmode be observed from Figure noiseinfrequencies in themethod. 3rd andAs4th of EEMD. This is exactlyitthe mixing phenomenon 6b that characteristic frequencies f2 (±f3) can be extracted accurately in 3rd IMF of CEEMDAN, existing in the EMD derived method. As for the performance of CEEMDAN, it can be observed from without additive noise disturbance. It indicates that there is no mode mixing problem appears in the Figure 6b that characteristic frequencies f 2 (±f 3 ) can be extracted accurately in 3rd IMF of CEEMDAN, obtained IMFs of CEEMDAN. It can be concluded based on the above analysis that CEEMDAN is without additive noise disturbance. It indicates that there is no mode mixing problem appears in superior to EEMD in alleviating the mode mixing problem, and can enable more accurate IMFs. the obtained IMFs of CEEMDAN. It can beproposed concluded based to onhealth the above analysis that CEEMDAN Hence, CEEMDAN can be adopted in the approach degradation monitoring and is superior to EEMD in of alleviating the mode mixing problem, and can enable more accurate IMFs. early fault diagnosis rolling bearings. Hence, CEEMDAN can be adopted in the proposed approach to health degradation monitoring and Simulation of Improved early 3.2. fault diagnosisResearch of rolling bearings. Multivariate Multiscale Sample Entropy To verify the effectiveness and superiority of improved MMSE with smoothed coarse graining

3.2. Simulation Research of Improved Multivariate Multiscale process, here trivariate signal with additive Gaussian whiteSample noise isEntropy adopted for numerical simulation. Theverify sampling point is 153,600and andsuperiority the samplingoffrequency is 1024 Hz;with namely, the sampling time is To the effectiveness improved MMSE smoothed coarse graining 150 s. The simulated trivariate rolling bearing signals are shown as follows: process, here trivariate signal with additive Gaussian white noise is adopted for numerical simulation. x1 (tsampling )  cos(2 f1tfrequency )[1  sin(2 f 2is t )]1024 Hz; namely, the sampling time is The sampling point is 153,600 and the (t )  sin(2 f3t )[1  0.5cos(2  f 4t )]as follows: 150 s. The simulated trivariate rollingx2bearing signals are shown (18) x3 (t )  cos(2 f5t )

x1 (t) = cos(2π f 1 t)[1 + sin(2π f 2 t)] where f1 = 40 Hz, f2 = 15 Hz,xf3 (=t)120 Hz,(2π f4 =f35t)[Hz, = 50 the fault characteristic (18) = sin 1 +f50.5 cosHz. (2πTo f 4 tsimulate )] 2 3 change during the health degradation monitoring process, different amplitudes of characteristic x3 (t) = cos(2π f 5 t)

signals containing noise are set at different periods. The time series can be divided into 150 trivariate faulty rolling bearing signal of multiple wheresubsequences f 1 = 40 Hz, 1024 f 2 = points 15 Hz,inf 3length. = 120 The Hz,simulated f 4 = 35 Hz, f 5 = 50 Hz. To simulate the fault characteristic periods are given here as follows:

change during the health degradation monitoring process, different amplitudes of characteristic signals 5 xi (t )  s, The t  1 time 50s series can be divided into 150 subsequences  periods. containing noise are set at different  2, 3, n = 1,signal 2, 3. of multiple periods 51 100rolling s , i = 1, bearing 1024 points in length. The X simulated (19) are n (t )   10 xtrivariate i (t )  s, t  faulty  given here as follows: 15 xi (t )  s, t  101 150s  where s denotes the Gaussian noise with a variance of 1 and mean of 0. Then, the MMSE of the  5xwhite  i ( t ) + s, t = 1 ∼ 50s trivariate signal of different periods are computed to simulate the health degradation monitoring Xn ( t ) = (19) 10xi (t) + s, t = 51 ∼ 100s , i = 1, 2, 3, n = 1, 2, 3. process. Here m = 5, τ sequence length l = 1024, scale φ = 20 were adopted in the simulation = 1,15x i ( t ) + s, t = 101 ∼ 150s

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where s denotes the Gaussian white noise with a variance of 1 and mean of 0. Then, the MMSE of the trivariate signal of different periods are computed to simulate the health degradation monitoring Materials 2018, 11, x FOR PEER REVIEW 11 of 22 process. Here m = 5, τ = 1, sequence length l = 1024, scale ϕ = 20 were adopted in the simulation studies. The calculated the 1st 1st second secondsimulated simulatedtrivariate trivariate signal studies. The calculatedMMSE MMSEresults resultsat atdifferent different scales scales of the signal areare shown in Figure 7, respectively adopting the conventional coarse graining process and smoothed shown in Figure 7, respectively adopting the conventional coarse graining process and smoothed coarse graining process. coarse graining process.

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It can be seen from Figure 7a that the MMSE values of the time series monotonically decrease as It can be from 7a that MMSE values of the timeare series monotonically decrease a whole, butseen when the Figure scales are 5, 8, the 10, and 15, the MMSE values greater than the former ones. as a whole, but when the scales are 5, 8, 10, and 15, the MMSE values are greater than the former ones. This phenomenon indicates that MMSE values with the conventional coarse graining process have This phenomenon indicates that MMSE values with the conventional coarse graining process have obvious fluctuations at certain scales, which could be caused by random noise, and would potentially obvious at certain scales, which could bebe caused by random noise, would values potentially lead tofluctuations inaccurate and unstable results. While it can seen from Figure 7b thatand the MMSE of lead inaccurate and unstable results. While it can be seen from Figure 7b that thecoarse MMSEgraining values of alltoscales monotonically decrease, it indicates MMSE values with the smoothed can reflect intrinsic dynamic characteristics of with the faulty rolling bearing signals process more all process scales monotonically decrease, it indicates MMSE values the smoothed coarse graining and steadily. Apart from that, it can also implyrolling that the smoothed coarse graining process canaccurately reflect intrinsic dynamic characteristics of the faulty bearing signals more accurately and can alleviate negative of random which makes such a method morecan robust. Hence, steadily. Apart the from that, it influence can also imply that noise, the smoothed coarse graining process alleviate the the proposed improved MMSE can be adopted in the application studies for health degradation negative influence of random noise, which makes such a method more robust. Hence, the proposed monitoring of rolling bearing. improved MMSE can be adopted in the application studies for health degradation monitoring of Based on the methodology and current studies of MMSE, if the MMSE values of a time series rolling bearing. monotonically it means this studies time series has lowif self-similarity, and of only contains Based on thedecrease, methodology andthat current of MMSE, the MMSE values a time series information at the smallest scale. Namely, the MMSE value of the smallest scale can be adopted as monotonically decrease, it means that this time series has low self-similarity, and only contains the indicator during health degradation monitoring of rolling bearings. Therefore, all the 1st (smallest information at the smallest scale. Namely, the MMSE value of the smallest scale can be adopted as the scale) MMSE values of multiple periods of the simulated trivariate rolling bearing signal are indicator during health degradation monitoring of rolling bearings. Therefore, all the 1st (smallest computed, as shown in Figure 8. scale) MMSE values of multiple periods of the simulated trivariate rolling bearing signal are computed, It can be seen from the Figure 8 that the 1st (smallest scale) MMSE values can recognize the as shown in Figure 8. dynamic characteristic of the time series, when the characteristic signal grows larger, namely the It can seengets from the Figure that the scale) 1st (smallest can degradation recognize the simulatedbefault worse, the 1st8(smallest MMSE scale) values MMSE indicatevalues the heath dynamic characteristic of the time series, when the characteristic signal grows larger, namely process. When the fault of rolling bearing happens or grows, the less complex the time series is, thethe simulated fault gets worse, the 1st (smallest scale) MMSE values indicate the intrinsic heath degradation less the MMSE value is. This means that MMSE values can be used for reflecting dynamic process. When the fault of rolling bearing happens or grows, the less complex the time series characteristics of the faulty rolling bearing. Hence, the proposed approach by adopting MMSE can is, thebe less theinMMSE is. Thismonitoring means thatof MMSE can be used for reflecting intrinsic dynamic used healthvalue degradation rollingvalues bearings. characteristics of the faulty rolling bearing. Hence, the proposed approach by adopting MMSE can be used in health degradation monitoring of rolling bearings.

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Figure 8. 1st MMSE valuesofofall allsubsequences subsequencesof of simulated simulated trivariate Figure 8. 1st MMSE values trivariatefaulty faultyrolling rollingbearing bearingsignal. signal. Figure 8. 1st MMSE values of all subsequences of simulated trivariate faulty rolling bearing signal.

4. Application Studies HealthDegradation DegradationMonitoring Monitoring and and Early 4. Application Studies ononHealth Early Fault FaultDiagnosis Diagnosisof Rolling Bearings 4. Application Studies on Health Degradation Monitoring and Early Fault Diagnosis of Rolling of Rolling Bearings Bearings

Application Studies ProposedHealth HealthDegradation Degradation Monitoring Monitoring Method 4.1.4.1. Application Studies of of thetheProposed MethodofofRolling RollingBearings Bearings 4.1. Application Studies of the Proposed Health Degradation Monitoring Method of Rolling Bearings

To verify the effectiveness the proposed method in the application tobearing rolling signal bearing signal To verify the effectiveness of theofproposed method in the application to rolling processing, To verify the effectiveness of the proposed method in the application to Intelligent rolling bearing signal processing, the run-to-failure fault rolling bearing signals provided by the Maintenance the run-to-failure fault rolling bearing signals provided by the Intelligent Maintenance Systems (IMS) Center processing, theCenter run-to-failure rolling bearing signals provided by the Intelligent Maintenance Systems (IMS) of (Cincinnati, the fault University of Cincinnati (Cincinnati, OR, USA) are adopted for of the University of Cincinnati OR, USA) are adopted for application studies. The recommended Systems (IMS) Center of the University of Cincinnati (Cincinnati, OR, USA) are adopted for of application studies. The recommended reference paper is given here [51]. The schematic diagram reference paper studies. is givenThe here [51]. The schematic diagram of thehere experimental apparatus and aofphoto recommended paper isisgiven The9.schematic theapplication experimental apparatus and a photo reference of the apparatus shown in [51]. Figure There arediagram four Rexnord of thethe apparatus is shown in Figure 9. There are four Rexnord ZA-2115 double-row (Rexnord, experimental apparatus and a photo of the apparatus is shown in Figure 9. There arebearings four Rexnord ZA-2115 double-row bearings (Rexnord, Milwaukee, WI, USA) installed on the shaft. A PCB 353B33 Milwaukee, USA) installed on (Rexnord, the shaft. Milwaukee, A PCB 353B33 quartz ICP accelerometer ZA-2115WI, double-row bearings WI,high USA)sensitivity installed on the shaft. A PCB 353B33(PCB, high sensitivity quartz ICP accelerometer (PCB, Buffalo, NY, USA) was vertically installed on the highNY, sensitivity quartz ICP accelerometer Buffalo, NY, USA) was vertically on the Buffalo, USA) was vertically installed on (PCB, the bearing house. The position of theinstalled sensor placement is bearing house. The position of the sensor placement is shown in Figure 9b. Then a test-to-failure bearing house. The position of the sensor placement is shown in Figure 9b. Then a test-to-failure shown in Figure 9b. Then a test-to-failure experiment was conducted, and all data were collected by a NI experiment was conducted, and all data were collected by a NI 6062E DAQ card (National experiment was conducted, and all data TX, were collected by2adataset NI 6062E DAQ card (National 6062E DAQ card (National Instrument, Austin, The No. is used inexperiment this paper, during the Instrument, datasetUSA). used thispaper, paper,during during Instrument,Austin, Austin,TX, TX,USA). USA).The The No. No. 22 dataset isisused ininthis thethe experiment 984984 setssets experiment 984 sets of data files were obtained with a collection every 10 min (164 h in total), and each file ofof data files every10 10min min(164 (164hhinintotal), total), and each contains 20,480 data fileswere wereobtained obtainedwith with aa collection collection every and each filefile contains 20,480 contains 20,480 points at the sampling rateThe of 20 kHz. The rotational speed wasr/min set atfacilitated 2000 r/min facilitated points at the sampling rate of 20 kHz. rotational speed was set at 2000 by an points at the sampling rate of 20 kHz. The rotational speed was set at 2000 r/min facilitated by an ACAC by an AC which motor which was coupled to theThus, shaft.the Thus, the rotational frequency fr Hz. is Hz. 33.33 Hz. Atofthe motor the rotational frequency 33.33 ofend the of motor whichwas wascoupled coupledto tothe the shaft. shaft. Thus, rotational frequency fr fisr is 33.33 AtAt thethe endend the theexperiment, experiment, bearing failure, which indicated the experiment recorded wear-outfailure, failure,which whichindicated indicated the experiment recordedthe experiment,bearing bearing111had hadan an outer outer ring ring wear-out wear-out the experiment recorded complete life testlife data. thethe complete complete lifetest testdata. data. Accelerationsensor sensor Acceleration

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(b) (a) Figure 9. (a) The schematic diagram of the apparatus; and (b) a picture of the apparatus. Figure 9. (a) The schematic diagram of the apparatus; and (b) a picture of the apparatus. Figure 9. (a) The schematic diagram of the apparatus; and (b) a picture of the apparatus.

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As mentioned above, a run-to-failure experiment of a rolling bearing was conducted. The entire lifetime of the rolling bearing wear-out process can be described in four steps, denoted by Phases 1–4, as shown in Table 1. The self-balancing and self-healing process during Phase 3 is similar to the retardation effect during the crack growth of the beam, owing to the reason they have similar mechanical characteristics. Table 1. Four phases of the entire time of the rolling bearing wear-out process. Phase 1

The bearing is operating normally, prior to the occurrence of the early weak fault.

Phase 2

The early weak fault occurs on the rolling bearing and interferes with its running condition slightly; this is the initial phase of early weak fault.

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The fault develops into the middle stage, generating the self-balancing and self-healing phenomenon, also called the retardation effect. This is the phase that the rolling bearing fault tends to be serious.

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The rolling bearing deteriorates promptly and has a serious fault. It usually results in the final breakdown of the rolling bearing.

To validate the effectiveness of the proposed health degradation monitoring and early fault diagnosis approach, EEMD and CEEMDAN of all vibration signals from 984 sets of data files are performed for comparison in subsequent analysis. Each data file has 20,480 points, and the sampling frequency is 20 KHz, as mentioned above. The sampling point is chosen as 8192 here, namely the first 8192 points are chosen for subsequent analysis out of 20,480 points. The EEMD and CEEMDAN of one set of data are shown in Figure 10. The standard deviation of the added noise is 0.2, and the ensemble size is 50 in EEMD, while 25 pairs (positive and negative) of added noise are adopted in CEEMDAN. It can be seen from Figure 10 that 14 orders of IMFs are obtained by EEMD, while 11 orders of IMFs are obtained by CEEMDAN. Obviously CEEMDAN can generate fewer orders of IMFs than EMD, alleviating the mode mixing problem to obtain accurate IMFs denoting characteristic frequencies. In the process of extracting accurate frequencies, correlation analysis is adopted here. The correlation analysis has been studied in authors’ previous studies [49,50]. The noise component and the effective IMFs denoting fault features can be determined in the correlation analysis, to analyze the main component of the vibration signal to extract fault frequencies. During the correlation analysis, the top three effective IMFs of EEMD and CEEMDAN are adopted to obtain the MMSE values, and the same strategy is employed hereinafter. The MMSE values of the 100th set of data is given here as an example, as shown in Figure 11. Owing to the reason that MMSE values of a time series monotonically decrease, which has been illustrated in Section 3.2, the MMSE value of the smallest scale can be adopted as the indicator during the health degradation monitoring. To validate the superiority of CEEMDAN over EEMD, and the advantage of MMSE with the smoothed coarse graining process than the conventional one, six contrastive methods are shown in Figure 12 regarding the degradation data of the rolling bearing. Parameter values of m = 5, τ = 1, scale ϕ = 20, were selected for the algorithm. They are, respectively, the health degradation monitoring of the rolling bearing adopting MSE and EEMD, MSE and CEEMDAN, MMSE (2nd MMSE values) and EEMD, MMSE (1st MMSE values) and EEMD, MMSE with conventional coarse graining process and CEEMDAN, and MMSE with the smoothed coarse graining process and CEEMDAN. All six contrastive methods are shown as follows. To illustrate the effectiveness and superiority of the proposed method, the specific analysis of the proposed method, which is also presented in Figure 12f, is shown in Figure 13.

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Figure 10. (a) IMFs obtained by EEMD; and (b) IMFs obtained by CEEMDAN. Figure 10. (a) IMFs obtained by EEMD; and (b) IMFs obtained by CEEMDAN.

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120 140 0 20 40 60 80 100 160 180 60 100 160 180 Figure 12. Comparative study of120the 140 six different methods. (a)0 The20MSE40values of80the optimal IMF obtained Time (/h) Time(b) (/h) The MSE values of the optimal IMF obtained by CEEMDAN; obtained by EEMD; (c) The 2nd by EEMD; (b) The MSE (e) values of the optimal IMF obtained by CEEMDAN; (c) The 2nd MMSE values MMSE values of effective IMFs obtained by EEMD; (d) The 1st MMSE values(f)of effective IMFs of effective IMFs obtained by EEMD; (d) The 1st MMSE values of effective IMFs obtained by EEMD; obtained by EEMD; (e) The 1st MMSE values of effective IMFs obtained by CEEMDAN (conventional (e) The 1st values of effective IMFs by CEEMDAN (conventional coarse graining process Figure 12. MMSE Comparative study of MMSE); the six obtained different methods. (a)values The MSE values of the optimal IMF coarse graining process within and (f) The 1st MMSE of effective IMFs obtained by within MMSE); and (f) The 1st MMSE values of effective IMFs obtained by CEEMDAN (smoothed coarse obtained by EEMD; (b) The MSE values of the optimal IMF obtained by CEEMDAN; (c) The 2nd CEEMDAN (smoothed coarse graining process within MMSE). grainingvalues processofwithin MMSE). MMSE effective IMFs obtained by EEMD; (d) The 1st MMSE values of effective IMFs obtained by EEMD; (e) The 1st MMSE values of effective IMFs obtained by CEEMDAN (conventional coarse graining process within MMSE); and (f) The 1st MMSE values of effective IMFs obtained by CEEMDAN (smoothed coarse graining process within MMSE).

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Figure 13. The 1st MMSE values effectiveIMFs IMFs obtained (smoothed coarse graining Figure 13. The 1st MMSE values ofofeffective obtainedby byCEEMDAN CEEMDAN (smoothed coarse graining process with MMSE). process with MMSE).

During the health degradation monitoring process, Phase 3 and Phase 4 are relatively easy to

During health monitoring process, Phase 3develops and Phase 4 are relatively easy to recognizethe since thesedegradation are two phases when the fault, respectively, to the middle stage and recognize since promptly. these areThe twoproposed phases approach when the fault, respectively, develops to theafter middle deteriorates in this paper aims at early fault diagnosis healthstage degradation monitoring, Phaseapproach 3 and Phase are paper not research only and deteriorates promptly. which The means proposed in 4this aims priorities. at early Hence, fault diagnosis Phase 1 and Phase 2 are marked in the six contrastive methods in Figure 12 to compare the after health degradation monitoring, which means Phase 3 and Phase 4 are not research priorities. effectiveness of all methods, where Phase 2 denotes the early weak fault phase. It can be observed Hence, only Phase 1 and Phase 2 are marked in the six contrastive methods in Figure 12 to compare from Figure 12a–c that the health degradation monitoring method respectively adopting MSE values the effectiveness of all methods, where Phase 2 denotes the early weak fault phase. It can be observed of the optimal IMF obtained by EEMD, MSE values of the optimal IMF obtained by CEEMDAN, and from Figure 12a–c that the health degradation monitoring method respectively adopting MSE values 2nd MMSE values of effective IMFs obtained by EEMD, cannot identify both Phase 1 and Phase 2 of theseparately. optimal It IMF obtained by EEMD, MSE values of the optimal IMF obtained by CEEMDAN, means these three methods fail to reveal the initial phase when an early fault happened; and 2nd MMSE values of effective IMFsThe obtained by EEMD, cannot identify both Phase 1 and Phase 2 namely Phase 2 cannot be identified. characteristic changes are more likely to be submerged by separately. It means these three methods fail be to seen reveal the phase when an values early fault happened; noise disturbances. From Figure 12d, it can that theinitial drop of the 1st MMSE of effective namely Phase 2 cannot be identified. characteristic changes are more likely submerged IMFs obtained by EEMD can indicateThe Phase 2, the value change is more distinct thanto thebe former three by methods but, still, the characteristic change is not obvious. From Figure 12e,f, it can be seen that health noise disturbances. From Figure 12d, it can be seen that the drop of the 1st MMSE values of effective monitoring based on CEEMDAN improved MMSE) can than both clearly IMFsdegradation obtained by EEMD can indicate Phase 2,and theMMSE value (also change is more distinct the former show Phase 1 and Phase 2, respectively. three methods but, still, the characteristic change is not obvious. From Figure 12e,f, it can be seen that To illustrate the effectiveness of the proposed approach that integrates CEEMDAN and MMSE health degradation monitoring based on CEEMDAN and MMSE (also improved MMSE) can both with smoothed coarse graining process, the enlarged results of Figure 12f are presented in Figure 13. clearly show Phase 1 and Phase 2, respectively. Phases 1–4 are all marked in Figure 13 to show the whole health degradation monitoring process To illustrate effectiveness the proposed approach that integrates CEEMDAN MMSE clearly. It can the be clearly seen thatof the proposed approach in this paper can monitor the wholeand health with smoothed process, the verify enlarged results of Figure are presented in From Figure 13. degradationcoarse processgraining distinctly, which can the effectiveness of the12f proposed approach. Phases 1–4 are all marked in Figure 13 to show the whole health degradation monitoring process Figure 13, it can be obtained that Phase 1 consist of 1–520 sets of data, and Phase 2 consists of 521– clearly. can be clearly seen that the proposed approach in thisMMSE paperwith can smoothed monitor coarse the whole 700 It sets of data. Furthermore, to show the superiority of improved graining processprocess over the distinctly, conventional MMSE, the comparison between the of results shown in Figure health degradation which can verify the effectiveness the proposed approach. based numerical analysisthat are Phase given 1inconsist Table 2.ofThe slopes are Phase obtained by From12e,f Figure 13, on it can be obtained 1–520 setsofofPhase data,2and 2 consists computing the slopes of their approximate linear fitting curves. of 521–700 sets of data. Furthermore, to show the superiority of improved MMSE with smoothed coarse graining process over the conventional MMSE, the comparison between the results shown in Table 2. The comparison between health degradation monitoring methods adopting MMSE with a Figure 12e,f based oncoarse numerical are given incoarse Tablegraining 2. Theprocess. slopes of Phase 2 are obtained by conventional graininganalysis process and a smoothed computing the slopes of their approximate linear fitting curves. Phase 1 Phase 2 Variance Variance Slope Table 2. The comparison between health degradation monitoring methods adopting MMSE with MMSE (Conventional coarse graining) 0.78 × 10−3 2.21 × 10−3 −6.80 × 10−4 a conventional coarse graining process a smoothed −3 MMSE (Smoothed coarseand graining) 0.34coarse × 10−3 graining 1.16 × 10process. −7.92 × 10−4 Methods

Methods MMSE (Conventional coarse graining) MMSE (Smoothed coarse graining)

Phase 1

Phase 2

Variance

Variance

Slope

0.78 × 10−3 0.34 × 10−3

2.21 × 10−3 1.16 × 10−3

−6.80 × 10−4 −7.92 × 10−4

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It can be seen from the above numerical analysis of the proposed health degradation monitoring approaches, respectively adopting conventional and smoothed coarse graining processes, that the variances Materials 2018, 11, x FOR PEER REVIEW 17 of 22 than of Phase 1 and Phase 2 by using MMSE with adopting smoothed coarse graining are both smaller adopting conventional coarse graining. This indicates that MMSE with a smoothed coarse graining process It can be seen from the above numerical analysis of the proposed health degradation monitoring contributes to more stable and robust results during the health degradation monitoring process. The absolute approaches, respectively adopting conventional and smoothed coarse graining processes, that the value of the slope of Phase 2 by using MMSE by adopting smoothed coarse graining is greater than adopting variances of Phase 1 and Phase 2 by using MMSE with adopting smoothed coarse graining are both conventional graining.conventional This indicates that the smooth coarse graining amplify the change smaller coarse than adopting coarse graining. This indicates thatprocess MMSEcan with a smoothed characteristic during process the health degradation monitoring process, contributes obtaining more coarse graining contributes to more stable and robust which results also during the healthtodegradation accurate results. process. Based on abovevalue analysis, can of bePhase seen2that the proposed approachsmoothed to the health monitoring Thethe absolute of theitslope by using MMSE by adopting coarse graining is greater than adopting conventional coarse graining. This indicates that more the smooth degradation monitoring process adopting MMSE with smoothed coarse graining can obtain stable and coarse graining process can amplify the change characteristic during the health degradation legible results than conventional MMSE, and can also obtain much better results than four other methods process, which also contributesapproach to obtaining more accurate Based on theinabove shownmonitoring in Figure 12. In summary, the proposed in this paper showsresults. good performance revealing analysis, it can be seen that the proposed approach to the health degradation monitoring process the fault change characteristics of rolling bearings in health degradation monitoring. adopting MMSE with smoothed coarse graining can obtain more stable and legible results than conventional MMSE,ofand can also obtain muchDiagnosis better results thanoffour other methods shown in 4.2. Application Research the Proposed Early Fault Method Rolling Bearings Figure 12. In summary, the proposed approach in this paper shows good performance in revealing Itthe can bechange concluded from Section 4.1 bearings that Phase 2 denotes the early weak fault stage, which is fault characteristics of rolling in health degradation monitoring.

when the early weak fault happened for the very first time. Then, in this section, CEEMDAN is 4.2.to Application of the Proposed Early Fault of Diagnosis Method of Rolling Bearingsthe early weak fault applied the signalResearch collected from the beginning Phase 2, aiming at extracting characteristic Thefrom 520th set of4.1 data the signal around 86.7th h)which is theisset of It canfrequency. be concluded Section that(denoting Phase 2 denotes the early weakthe fault stage, whenearly the early happened for theatvery first time. Then, in this section, CEEMDAN is of data that weakweak faultfault started to happen the very beginning during the wear-out process applied to the signal collected from the beginning of Phase 2, aiming at obvious, extractingor the early weak the rolling bearing, and the fault characteristic frequency would not be may have a very characteristic The 520th set of data (denoting the signal around the 86.7th h) iswith the the slightfault amplitude. The frequency. fault characteristic frequency would have greater amplitude along set of data that early weak fault started to happen at the very beginning during the wear-out process time. Hence, the 530th set of data (denoting the signal around the 88.3th h) is selected to extract the of the rolling bearing, and the fault characteristic frequency would not be obvious, or may have a early weak fault. The sampling points for frequency spectrum analysis hereinafter is chosen as 8192. very slight amplitude. The fault characteristic frequency would have greater amplitude along with The detailed parameters of the faulty rolling bearing are shown in Table 3. The calculating method of the time. Hence, the 530th set of data (denoting the signal around the 88.3th h) is selected to extract the characteristic frequency computed are shown in Table the early weak fault. Theand sampling pointscharacteristic for frequency frequency spectrum analysis hereinafter is 4. chosen as 8192. The detailed parameters of the faulty rolling bearing are shown in Table 3. The calculating Table 3. Detailed parameters of faulty rolling bearing. method of the characteristic frequency and computed characteristic frequency are shown in Table 4. Detailed Parameters of RexnordofZA-2115 Rolling Bearing Table 3. Detailed parameters faulty rolling bearing. Pitch diameter Dw Ball number n Contact angle α Ball diameter dr Detailed Parameters of Rexnord ZA-2115 Rolling Bearing 15.17 0.331 2.815 Dw Ball 16 number n Contact angle α Ball diameter dr Pitch diameter 16 15.17 0.331 2.815

Table 4. The calculating method and characteristic frequency of a Rexnord ZA-2115 rolling bearing. Table 4. The calculating method and characteristic frequency of a Rexnord ZA-2115 rolling bearing. Fault Type Fault Frequency Computation Fault Frequency Fault Type Fault Frequency Computation Fault Frequency Outer ring fault fofo==0.5n(1 cosα/Dww)f)f 236.4 Outer ring fault 0.5n(1−− ddrrcosα/D r r fof=o = 236.4

The time frequency domain plots selecteddata datamentioned mentioned above above are The time and and frequency domain plots of of thethe selected areshown shownininFigure Figure 14, 14, as follows. as follows.

Figure 14. Time and frequency domain plots of the 530th set of data (denoting the signal around the

Figure 14. Time and frequency domain plots of the 530th set of data (denoting the signal around the 88.3th h), during the beginning of Phase 2 of the rolling bearing wear-out process. 88.3th h), during the beginning of Phase 2 of the rolling bearing wear-out process.

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EEMD and CEEMDAN are applied to the chosen data mentioned above. The standard deviation EEMD and CEEMDAN are applied to the chosen data mentioned above. The standard deviation of the added noise is 0.2, and the ensemble size is 50 in EEMD, while 25 pairs (positive and negative) of the added noise is 0.2, and the ensemble size is 50 in EEMD, while 25 pairs (positive and negative) of added noise are adopted in CEEMDAN. The frequency domain plots of the optimal IMFs of EEMD of added noise are adopted in CEEMDAN. The frequency domain plots of the optimal IMFs of EEMD and CEEMDAN are are shown in in Figure and CEEMDAN shown Figure15. 15.

Figure 15. (a) IMFs obtained by EEMD; and (b) IMFs obtained by CEEMDAN.

Figure 15. (a) IMFs obtained by EEMD; and (b) IMFs obtained by CEEMDAN.

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It be seen seen from fromFigure Figure1515that that orders of IMFs obtained by EEMD, 11 orders of It can can be 1414 orders of IMFs areare obtained by EEMD, and and 11 orders of IMFs IMFs are obtained by CEEMDAN. The advantages of CEEMDAN have also been elaborated in are obtained by CEEMDAN. The advantages of CEEMDAN have also been elaborated in Section 3.1 and Section 3.1 andofthe beginning ofcorrelation Section 4.1.analysis The correlation analysis in thestudies authors’ previous studies the beginning Section 4.1. The in the authors’ previous [49,50] is adopted to [49,50] is adopted to determine the effective IMFs, then IMFs denoting the noise components will be determine the effective IMFs, then IMFs denoting the noise components will be discarded during the IMF’s discarded during the IMF’s reconstruction. To illustrate the effectiveness of processed CEEMDAN, the reconstruction. To illustrate the effectiveness of CEEMDAN, the frequency domain plot by WPD frequency domain plot processed by WPD is also presented. The wavelet function db15 with 11 layers is also presented. The wavelet function db15 with 11 layers is adopted during WPD. During EEMD and is adopted during WPD. During EEMD andtoCEEMDAN, the signal. effective IMFs are selected to reconstruct CEEMDAN, the effective IMFs are selected reconstruct the The frequency domain plots of the the signal. The frequency domain plots of the original signal and the reconstructed signals original signal and the reconstructed signals respectively processed by WPD, EEMD, and CEEMDAN are respectively processed by WPD, EEMD, and CEEMDAN are shown in Figure 16. shown in Figure 16. 0.005

0.015

0.01

fr

fr fo

0.005

fo 0

0 0

200

400

600

0

800

200

400

600

800

600

800

Frequency Index (/Hz)

Frequency Index (/Hz)

(b)

(a) 0.01

0.01

0.005

0.005

fr

fr fo

fo 0

0 0

200

400

Frequency Index (/Hz) (c)

600

800

0

200

400

Frequency Index (/Hz) (d)

Figure Figure 16. 16. Frequency Frequencydomain domain plots plots of of the theoriginal originalsignal signal and andthe thereconstructed reconstructed signals. signals. (a) (a)Frequency Frequency domain domain plot plot of of the the original original early early weak weak fault fault signal; signal; (b) (b) frequency frequency domain domain plot plot of of the the reconstructed reconstructed signal signal processed processed by by WPD; WPD; (c) (c)frequency frequencydomain domainplot plotof ofthe thereconstructed reconstructedsignal signalprocessed processedby byEEMD; EEMD; and (d) (d) frequency frequency domain domain plot plot of of the the reconstructed reconstructed signal signal processed processed by and by CEEMDAN. CEEMDAN.

It the original original early early weak weak fault fault frequency frequency ffo == 33.33 It can can be be seen seen from from Figure Figure 16a 16a that that the 33.33 Hz Hz is is o interfered by background noises to a great extent. There are great deal of redundant or disturbing interfered by background noises to a great extent. There are great deal of redundant or disturbing frequencies, frequencies, and and fault fault characteristic characteristic frequency frequency ffoo ==236.4 236.4Hz Hzisis submerged submerged in in those those frequencies. frequencies. It It can can be seen from Figure 16b that, after WPD denoising, the fault characteristic frequency f o and rotational be seen from Figure 16b that, after WPD denoising, the fault characteristic frequency fo and rotational frequency frequency ffrr can canbe befound, found,but butthe theresults resultsare arefar farfrom fromgood. good.Figure Figure16c 16cshows showsafter afterEEMD EEMDprocessing, processing, the frequency domain plot has less disturbing frequencies, but the fault characteristic the frequency domain plot has less disturbing frequencies, but the fault characteristic frequency frequency ffrr cannot cannot be be well well extracted, extracted, while while in in Figure Figure 16d, 16d, after after the the proposed proposed method method in in this this paper, paper, which which is is CEEMDAN processing, the fault characteristic frequency f o and rotational frequency fr can be well CEEMDAN processing, the fault characteristic frequency fo and rotational frequency fr can be well extracted. which indicates that CEEMDAN is extracted. There Thereare arefew fewredundant redundantorordisturbing disturbingfrequencies, frequencies, which indicates that CEEMDAN superior to EEMD in alleviating the mode mixing problem, and can extract the accurate fault is superior to EEMD in alleviating the mode mixing problem, and can extract the accurate fault characteristic early weak outer ringring fault can can be extracted by CEEMDAN, and characteristicfrequency. frequency.Hence, Hence,the the early weak outer fault be extracted by CEEMDAN, the results can also ascertain the health degradation monitoring results hereinbefore. Based on and the results can also ascertain the health degradation monitoring results hereinbefore. Based on the the above analysis, it can be concluded that the proposed early fault diagnosis by adopting CEEMDAN above analysis, it can be concluded that the proposed early fault diagnosis by adopting CEEMDAN can can achieve achieve aa good good result result in in application application studies, studies, and and the the proposed proposed approach approach is is promising promising in in the the field field of fault diagnosis of rolling bearings. of fault diagnosis of rolling bearings.

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5. Conclusions In this paper, the research work elaborates the effectiveness of the proposed health degradation monitoring and early fault diagnosis of rolling bearings based on CEEMDAN and improved MMSE, which utilizes a smoothed coarse graining process. The theoretical derivation can demonstrate the significance of the novel approach in this paper, and the effectiveness is verified by numerical simulations and practical applications. The numerical simulation results indicate that CEEMDAN can alleviate the mode mixing problem and enable accurate IMFs, and improved MMSE can reflect intrinsic dynamic characteristics of rolling bearings more accurately and steadily. In the application research of the proposed health degradation monitoring method of rolling bearings, the results indicate that Phases 1–4 can be well distinguished by the proposed approach. The change characteristic can be reflected and the deterioration of the early weak fault phase can be amplified. It contributes a great deal to clearly identifying Phase 2 during the whole wear-out process, which is significant during the health degradation monitoring of rolling bearings. The superiority of improved MMSE with the adopted smoothed coarse graining process to conventional MMSE is also illustrated by numerical analysis. Afterwards, during the application research of the proposed early fault diagnosis method of rolling bearings, CEEMDAN can accurately extract the early weak fault characteristic frequency of the rolling bearing. The results of frequency spectrum analysis indicate that CEEMDAN is superior to EEMD in alleviating the mode mixing problem, and can enable more accurate IMFs to extract an early outer ring fault characteristic frequency fo and also the rotational frequency fr . The results of the early weak fault diagnosis can also ascertain the health degradation monitoring. Based on the analysis results of the simulated signal and practical experimental data, it can be concluded that the proposed health degradation monitoring and early fault diagnosis approach is promising in the field of prognostic and fault diagnosis of rolling bearings. Author Contributions: Y.L. and R.Y. conceived and designed the experiments; Y.L. and R.Y. performed the numerical simulations and application researches; Y.L., R.Y., G.S., and H.L. analyzed the data and contributed to the analysis; Y.L. and R.Y. wrote the paper; and G.S. and T.W. revised the manuscript. All authors read and approved the manuscript. Funding: This research project is funded by National Natural Science Foundation of China under grant no. 51475339 and no. 51105284, the Natural Science Foundation of Hubei province under grant no. 2016CFA042, Wuhan Science and Technology Project under grant no. 2017010201010115. Acknowledgments: The authors also appreciate the free download of the original bearing failure data and one photo picture provided by the Intelligent Maintenance Systems Center of University of Cincinnati. Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publication of this article.

References 1. 2.

3. 4. 5. 6.

Lei, Y.; Lin, J.; He, Z.; Zuo, M.J. A review on empirical mode decomposition in fault diagnosis of rotating machinery. Mechan. Syst. Signal Process. 2013, 35, 108–126. [CrossRef] Cerrada, M.; Sánchez, R.V.; Li, C.; Pacheco, F.; Cabrera, D.; de Oliveira, J.V.; Vásquez, R.E. A review on data-driven fault severity assessment in rolling bearings. Mechan. Syst. Signal Process. 2018, 99, 169–196. [CrossRef] Li, B.; Chow, M.Y.; Tipsuwan, Y.; Hung, J.C. Neural-network-based motor rolling bearing fault diagnosis. IEEE Trans. Ind. Electron. 2000, 47, 1060–1069. [CrossRef] Zhao, M.; Lin, J.; Xu, X.; Lei, Y. Tacholess envelope order analysis and its application to fault detection of rolling element bearings with varying speeds. Sensors 2013, 13, 10856–10875. [CrossRef] [PubMed] Yuan, J.; Wang, Y.; Peng, Y.; Wei, C. Weak fault detection and health degradation monitoring using customized standard multiwavelets. Mechan. Syst. Signal Process. 2017, 94, 384–399. [CrossRef] Yan, R.; Liu, Y.; Gao, R.X. Permutation entropy: A nonlinear statistical measure for status characterization of rotary machines. Mechan. Syst. Signal Process. 2012, 29, 474–484. [CrossRef]

Materials 2018, 11, 1009

7.

8. 9.

10. 11. 12. 13. 14. 15. 16. 17. 18.

19. 20.

21. 22. 23.

24. 25. 26. 27. 28. 29.

21 of 22

Shen, Z.; He, Z.; Chen, X.; Sun, C.; Liu, Z. A monotonic degradation assessment index of rolling bearings using fuzzy support vector data description and running time. Sensors 2012, 12, 10109–10135. [CrossRef] [PubMed] Xiao, H.; Zheng, J.; Song, G. Severity evaluation of the transverse crack in a cylindrical part using a PZT wafer based on an interval energy approach. Smart Mater. Struct. 2016, 25, 035021. [CrossRef] Lu, S.; Zhou, P.; Wang, X.; Liu, Y.; Liu, F.; Zhao, J. Condition monitoring and fault diagnosis of motor bearings using undersampled vibration signals from a wireless sensor network. J. Sound Vib. 2018, 414, 81–96. [CrossRef] Yu, J. Adaptive hidden Markov model-based online learning framework for bearing faulty detection and performance degradation monitoring. Mechan. Syst. Signal Process. 2017, 83, 149–162. [CrossRef] Xue, L.; Li, N.; Lei, Y.; Li, N. Incipient Fault Detection for Rolling Element Bearings under Varying Speed Conditions. Materials 2017, 10, 675. [CrossRef] [PubMed] Zhang, S.; Zhang, Y.; Li, L.; Wang, S.; Xiao, Y. An effective health indicator for rolling elements bearing based on data space occupancy. Struct. Health Monit. 2018, 17, 3–14. [CrossRef] Zhao, L.Y.; Wang, L.; Yan, R.Q. Rolling bearing fault diagnosis based on wavelet packet decomposition and multi-scale permutation entropy. Entropy 2015, 17, 6447–6461. [CrossRef] Zhong, J.; Huang, Y. Time-frequency representation based on an adaptive short-time Fourier transform. IEEE Trans. Signal Process. 2010, 58, 5118–5128. [CrossRef] Guo, Y.; Na, J.; Li, B.; Fung, R.F. Envelope extraction based dimension reduction for independent component analysis in fault diagnosis of rolling element bearing. J. Sound Vib. 2014, 333, 2983–2994. [CrossRef] Flandrin, P.; Rilling, G.; Goncalves, P. Empirical mode decomposition as a filter bank. IEEE Signal Process. Lett. 2004, 11, 112–114. [CrossRef] Gao, Z.; Lin, J.; Wang, X.; Xu, X. Bearing Fault Detection Based on Empirical Wavelet Transform and Correlated Kurtosis by Acoustic Emission. Materials 2017, 10, 571. [CrossRef] [PubMed] Huang, N.E.; Shen, Z.; Long, S.R.; Wu, M.C.; Shih, H.H.; Zheng, Q.; Yen, N.C.; Tung, C.C.; Liu, H.H. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. Math. Phys. Eng. Sci. 1998, 454, 903–995. [CrossRef] Wu, Z.; Huang, N.E. Ensemble empirical mode decomposition: A noise-assisted data analysis method. Adv. Adapt. Data Anal. 2009, 1. [CrossRef] Sweeney, K.T.; McLoone, S.F.; Ward, T.E. The use of ensemble empirical mode decomposition with canonical correlation analysis as a novel artifact removal technique. IEEE Trans. Biomed. Eng. 2013, 60, 97–105. [CrossRef] [PubMed] Han, L.; Li, C.; Liu, H. Feature extraction method of rolling bearing fault signal based on EEMD and cloud model characteristic entropy. Entropy 2015, 17, 6683–6697. [CrossRef] Yeh, J.R.; Shieh, J.S.; Huang, N.E. Complementary ensemble empirical mode decomposition: A novel noise enhanced data analysis method. Adv. Adapt. Data Anal. 2010, 2, 135–156. [CrossRef] Torres, M.E.; Colominas, M.A.; Schlotthauer, G.; Flandrin, P. A complete ensemble empirical mode decomposition with adaptive noise. In Proceedings of the Acoustics, Speech and Signal Processing (ICASSP), 2011 IEEE International Conference, Prague, Czech Republic, 22–27 May 2011; pp. 4144–4147. Xu, Y.; Luo, M.; Li, T.; Song, G. ECG Signal De-noising and Baseline Wander Correction Based on CEEMDAN and Wavelet Threshold. Sensors 2017, 17, 2754. [CrossRef] [PubMed] Pincus, S.M. Approximate entropy as a measure of system complexity. Proc. Natl. Acad. Sci. USA 1991, 88, 2297–2301. [CrossRef] [PubMed] Alcaraz, R.; Rieta, J.J. A review on sample entropy applications for the non-invasive analysis of atrial fibrillation electrocardiograms. Biomed. Signal Process. Control 2010, 5, 1–14. [CrossRef] Bandt, C.; Pompe, B. Permutation entropy: A natural complexity measure for time series. Phys. Rev. Lett. 2002, 88, 174102. [CrossRef] [PubMed] Shi, Z.; Song, W.; Taheri, S. Improved LMD, Permutation Entropy and Optimized K-Means to Fault Diagnosis for Roller Bearings. Entropy 2016, 18, 70. [CrossRef] Ai, Y.T.; Guan, J.Y.; Fei, C.W.; Tian, J.; Zhang, F.L. Fusion information entropy method of rolling bearing fault diagnosis based on n-dimensional characteristic parameter distance. Mechan. Syst. Signal Process. 2017, 88, 123–136. [CrossRef]

Materials 2018, 11, 1009

30. 31. 32.

33. 34. 35. 36. 37. 38. 39.

40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50.

51.

22 of 22

Chen, B.; Yan, Z.; Chen, W. Defect Detection for Wheel-Bearings with Time-Spectral Kurtosis and Entropy. Entropy 2014, 16, 607–626. [CrossRef] Chen, X.; Liu, D.; Xu, G.; Jiang, K.; Liang, L. Application of wavelet packet entropy flow manifold learning in bearing factory inspection using the ultrasonic technique. Sensors 2014, 15, 341–351. [CrossRef] [PubMed] Caesarendra, W.; Kosasih, B.; Tieu, A.K.; Moodie, C.A. Application of the largest Lyapunov exponent algorithm for feature extraction in low speed slew bearing condition monitoring. Mechan. Syst. Signal Process. 2015, 50, 116–138. [CrossRef] Wang, B.; Hu, X.; Li, H. Rolling bearing performance degradation condition recognition based on mathematical morphological fractal dimension and fuzzy C-means. Measurement 2017, 109, 1–8. [CrossRef] Humeau-Heurtier, A. The Multiscale Entropy Algorithm and Its Variants: A Review. Entropy 2015, 17, 3110–3123. [CrossRef] Wu, Y.; Shang, P.; Li, Y. Modified generalized multiscale sample entropy and surrogate data analysis for financial time series. Nonlinear Dyn. 2018, 92, 1335–1350. [CrossRef] Wu, S.D.; Wu, C.W.; Lin, S.G.; Wang, C.C.; Lee, K.Y. Time series analysis using composite multiscale entropy. Entropy 2013, 15, 1069–1084. [CrossRef] Ahmed, M.U.; Mandic, D.P. Multivariate Multiscale Entropy Analysis. IEEE Signal Proces. Lett. 2012, 19, 91–94. [CrossRef] Yin, Y.; Shang, P. Multivariate multiscale sample entropy of traffic time series. Nonlinear Dyn. 2016, 86, 479–488. [CrossRef] Wei, Q.; Liu, D.H.; Wang, K.H.; Liu, Q.; Abbod, M.F.; Jiang, B.C.; Chen, K.P.; Wu, C.; Shieh, J.S. Multivariate Multiscale Entropy Applied to Center of Pressure Signals Analysis: An Effect of Vibration Stimulation of Shoes. Entropy 2012, 14, 2157–2172. [CrossRef] Wang, G.F.; Li, Y.B.; Luo, Z.G. Fault classification of rolling bearing based on reconstructed phase space and Gaussian mixture model. J. Sound Vib. 2009, 323, 1077–1089. [CrossRef] Barnard, J.P.; Aldrich, C.; Gerber, M. Embedding of multidimensional time-dependent observations. Phys. Rev. E 2001, 64, 046201. [CrossRef] [PubMed] Looney, D.; Hemakom, A.; Mandic, D.P. Intrinsic multi-scale analysis: A multi-variate empirical mode decomposition framework. Proc. R. Soc. A 2015, 471, 20140709. [CrossRef] [PubMed] Sharma, R.; Pachori, R.; Acharya, U. Application of Entropy Measures on Intrinsic Mode Functions for the Automated Identification of Focal Electroencephalogram Signals. Entropy 2015, 17, 669–691. [CrossRef] Sharma, R.; Pachori, R.B. Classification of epileptic seizures in EEG signals based on phase space representation of intrinsic mode functions. Expert Syst. Appl. 2015, 42, 1106–1117. [CrossRef] Hu, M.; Liang, H. Intrinsic mode entropy based on multivariate empirical mode decomposition and its application to neural data analysis. Cognit. Neurodyn 2011, 5, 277–284. [CrossRef] [PubMed] Rai, V.K.; Mohanty, A.R. Bearing fault diagnosis using FFT of intrinsic mode functions in Hilbert–Huang transform. Mechan. Syst. Signal Process. 2007, 21, 2607–2615. [CrossRef] Wu, W.; Chen, X.; Su, X. Blind Source Separation of Single-channel Mechanical Signal Based on Empirical Mode Decomposition. J. Mechan. Eng. 2011, 47, 12. [CrossRef] Lee, S.K.; White, P.R. The enhancement of impulsive noise and vibration signals for fault detection in rotating and reciprocating machinery. J. Sound Vib. 1998, 217, 485–505. [CrossRef] Lv, Y.; Yuan, R.; Song, G. Multivariate empirical mode decomposition and its application to fault diagnosis of rolling bearing. Mechan. Syst. Signal Process. 2016, 81, 219–234. [CrossRef] Yuan, R.; Lv, Y.; Song, G. Multi-Fault Diagnosis of Rolling Bearings via Adaptive Projection Intrinsically Transformed Multivariate Empirical Mode Decomposition and High Order Singular Value Decomposition. Sensors 2018, 18, 1210. [CrossRef] [PubMed] Qiu, H.; Lee, J.; Lin, J.; Yu, G. Wavelet filter-based weak signature detection method and its application on rolling element bearing prognostics. J. Sound Vib. 2006, 289, 1066–1090. [CrossRef] © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).