Trivariate Empirical Mode Decomposition via Convex ... - MDPI

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Trivariate Empirical Mode Decomposition via Convex Optimization for Rolling Bearing Condition Identification Yong Lv 1,2 , Houzhuang Zhang 1,2 and Cancan Yi 1,2, * 1

2

*

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] (H.Z.) Hubei Key Laboratory of Mechanical Transmission and Manufacturing Engineering, Wuhan University of Science and Technology, Wuhan 430081, China Correspondence: [email protected]; Tel.: +86-027-68-862-857

Received: 30 May 2018; Accepted: 13 July 2018; Published: 18 July 2018







Abstract: As a multichannel signal processing method based on data-driven, multivariate empirical mode decomposition (MEMD) has attracted much attention due to its potential ability in self-adaption and multi-scale decomposition for multivariate data. Commonly, the uniform projection scheme on a hypersphere is used to estimate the local mean. However, the unbalanced data distribution in high-dimensional space often conflicts with the uniform samples and its performance is sensitive to the noise components. Considering the common fact that the vibration signal is generated by three sensors located in different measuring positions in the domain of the structural health monitoring for the key equipment, thus a novel trivariate empirical mode decomposition via convex optimization was proposed for rolling bearing condition identification in this paper. For the trivariate data matrix, the low-rank matrix approximation via convex optimization was firstly conducted to achieve the denoising. It is worthy to note that the non-convex penalty function as a regularization term is introduced to enhance the performance. Moreover, the non-uniform sample scheme was determined by applying singular value decomposition (SVD) to the obtained low-rank trivariate data and then the approach used in conventional MEMD algorithm was employed to estimate the local mean. Numerical examples of synthetic defined by the fault model and real data generated by the fault rolling bearing on the experimental bench are provided to demonstrate the fruitful applications of the proposed method. Keywords: trivariate empirical mode decomposition; convex optimization; low-rank matrix approximation; rolling bearing condition identification

1. Introduction With the development of industrial production, the key equipment and structure always have the characteristics of complex structure, variable operation status and continuous online service. Especially in the metallurgical industry, most parts such as rolling bearing are working under an environment of high speed, overloading and high temperatures. As an important approach of structural health monitoring, the damage feature extraction algorithm for rolling bearings based on signal processing technology has important significance [1–3]. Naturally, the vibration signal is expressed as nonlinear, non-stationary and time-varying [4–8]. Moreover, the noise components directly affect the performance of the signal analysis [9–11]. Thus, improved signal processing methods are required. In order to improve the signals to noise ratio (SNR) of actual measured vibration signal, many scholars have done a lot of research work [12–16]. Short-time Fourier transform (STFT) Sensors 2018, 18, 2325; doi:10.3390/s18072325

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is presented as a modified edition of conventional Fourier transform, which is used to analysis non-stationary signals [17]. Since the size and shape of analysis window are fixed, it cannot achieve adaptability with the change of the target signal. Based on it, Wavelet Transform (WT) is proposed by the inner product operator between the analysis signal and the mother wavelet function [18,19]. Moreover, the multi-scale analysis is conducted by choosing different factors. The performance is still related to the basis function and decomposition level selection. So far, the research has focused on two radically different techniques. The first set of techniques is known as reassignment methods, while the second set of algorithms is referred to as data driven algorithms. Synchronous wavelet transform (SWT) is regarded as the most common application reassignment method. It can produce highly localized time-frequency representations in the scale domain [20]. However, it may be unsuitable for fast varying instantaneous frequency. Empirical Mode Decomposition (EMD) is a typical signal adaptive decomposition algorithm. If the signal contains intermittent modes, its performance is affected by the mode-mixing problem (MMP) [21,22]. Thus, by adding random white noises, Ensemble EMD (EEMD) and its improved edition have been proposed [23–25]. In essence, the problem of MMP, namely the stability of decomposition, determines the ultimate performance of signal decomposition. The above-mentioned signal processing method of time-frequency analysis is mainly oriented to the single sensor. The single sensor can only provide limited useful information, and the characteristics of the early equipment running status information are hard to be inspected. On the contrary, the multi-channel signals processing technology can be employed to identify the multichannel correlation, improve the accuracy and reliability of diagnosis. In short, it can provide more scientific evaluation to equipment online service quality [26]. Especially the sensor and measurement technology have made tremendous progress, multivariate signal processing method is the focus of this paper. Since EMD has presented a better potential, it can be set up as a powerful way in multi-channel signal processing. The concept of bivariate empirical mode decomposition (BEMD) was firstly proposed by Rilling [27]. By using the rotation property of quaternions, Empirical mode decomposition for trivariate signals (TEMD) is presented, which is employed to obtain multiple projections on the sphere [28]. Then, a multivariate extension of EMD has been proposed by Rehman, namely multivariate empirical mode decomposition (MEMD) [29]. As a critical step in signal decomposition, the uniformly sampled scheme is used to estimate the local mean envelope of BEMD and MEMD input signals in multiple directions in multi-dimensional space. In solving the problem of the decomposition stability, MEMD has more superiority than BEMD and conventional TEMD, it has been widely used in signal processing and other fields [30,31]. When the MEMD is used to identify the early damage status identification of the rolling bearing, the multi-channel data in high-dimensional space is usually inhomogeneous, and the feature information will be inferred by noise components. The noise components are generated by thermal shock, acoustic noise, base strain, the thermal noise inside the amplifier circuit and so on. It will be added to the original true data and produce measurement error. This phenomenon usually indicates that it is difficult to identify the real fault characteristic frequency information under strong noise environment. Therefore, the MEMD method has some shortcomings. For instance, it is sensitive to the noise components and has poor signal decomposition ability for complex signal. Generally, the vibration signal is measured by three sensors located in different positions and the trivariate signal is expressed as the form of matrix. Distinguished from MEMD, trivariate empirical mode decomposition based on low-rank matrix approximation via convex optimization framework and non-convex regularization term is proposed firstly to achieve better denoising performance. It has been proved that the non-convex penalty functions can obtain more accurate low-rank structure [32]. Then, considering the inhomogeneity of trivariate data distribution in high-dimensional, a non-uniformly sampled scheme based on singular value decomposition (SVD) is presented to obtain low-rank trivariate data matrix. The obtained principal component is corresponding to the directions of highest curvature, and the main features of the data can be sampled. Subsequently, a novel trivariate empirical mode decomposition is put forward in this paper for rolling bearing condition identification. In order

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to verify the validity of the method, the simulation signal model and measured experimental data of fault rolling bearing are conducted to analysis. This paper was organized as follows: the basic principles and characteristics of the trivariate empirical mode decomposition via convex optimization were introduced in the second chapter. In the third section, the numerical simulation analysis was conducted to confirm the effectiveness of the algorithm. In the fourth part, the validity of the proposed method was verified by the bearing data of the experimental bench. The conclusions of the study were given in section fifth. 2. Theoretical Descriptions Given a trivariate signal Y(t) = {y1 (t), y2 (t), y3 (t)} generated by the mechanical equipment using triaxial acceleration sensor or three sensors installed in different positions, it is a common fact that the actual measured trivariate signal is composed by useful signal and noise components. The problem of estimating a low-rank matrix X from its noisy observation Y is considered as follows: Y = X + W Y, X, W ∈ R3×n

(1)

where W represents zero-mean additive white Gaussian noise. It was reported that the non-convex regularization methods outperformed the convex regularization methods on matrix approximation and could enhance sparsity [32]. Thus, in this paper, a low-rank matrix approximation [33] for the trivariate signal Y can be expressed as the following model: k m n 1 arg min{ψ(X) = ||Y − X||2F + λ0 ∑ φ(σi (X); a0 ) + λ1 ∑ ∑ φ(Xij ; a1 )} (2) 2 X i =1 i =1 j =1 where k = min(m, n), φ is the parameterized non-convex penalty function, σi (X) represents the singular values of the matrix X. a0 and a1 represent the parameters of non-convex penalty functions, which are associated with the convexity of the objective function. The weights of two parameterized non-convex penalty functions are adjusted by the regularization parameters λ0 and λ1 . Obviously, if λ1 = 0 and φ( x, a) = | x |, Equation (2) is transformed into a typical nuclear norm minimization problem. Typical non-convex penalty functions should be 2nd order differentiable, continuous and symmetric. An example of a non-convex penalty function is the rational penalty function, which is defined as: |x| φ( x, a) = (3) 1 + a| x |/2 Equation (3) is parameterized by the parameter a ≥ 10. The proximal operator of φ( x, a) is expressed as:   1 (4) proxφ (y, λ, a) = argmin (y − x )2 + λφ( x, a) x∈R 2 It is noted that the proximity operator is associated with the non-convex function φ( x, a). In order to ensure that the objective function ψ(X) is strictly convex, the parameter a0 and a1 should meet the following requirement: 0 ≤ a0 λ0 + a1 λ1 < 1 (5) Then, the alternating direction method of multipliers (ADMM) algorithm in conjunction with variable splitting is employed to solve the trivariate data matrix of the low-rank approximation problem. The ADMM algorithm is applied to Equation (2), and yields the following iterative procedure. Firstly, initialize the model with Z = 0 and D = 0. Then, the iterative optimization is conducted as following: X ← proxφ (

1 λ (Y + µ(Z + D)), 1 , a1 ) 1+µ 1+µ

(6)

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[U, Σ, V] ← SVD(X − D)

(7)

Z ← U · proxφ (Σ, λ0 /µ, a0 ) · VT

(8)

D ← D − (X − Z)

(9)

where µ > 1 is the scalar augmented Lagrangian parameter to ensure the strictly convex of the algorithm; SVD indicates the operator of singular value decomposition. The trivariate data preprocessing method based on low-rank matrix approximation via convex optimization requires the specification of two regularization parameters λ0 and λ1 , two penalty parameters a0 and a1 , and the scalar augmented Lagrangian parameter µ. Generally, the regularization parameters λ0 and λ1 are determined by the follow formulation: λi = β i σ (10) where β i is chosen so as to maximize the signal-to-noise ratio. σ is corresponding to the spectral noise variance, which can be estimated by the wavelet coefficients by using the local variance analysis [34]. The values of two penalty parameters a0 and a1 can be simplified chosen by Equation (5). It’s also noted that the value of µ does not affect the solution due to the iterative algorithm converges, and µ = 1.5 is always suitable for algorithm convergence. Through the processing of the proposed method, a low-rank approximation matrix X is obtained from the observed spectral data matrix X. Then, the singular value decomposition (SVD) of the covariance matrix to X is performed,  E XXT = VΛV T , where V is the eigenvector matrix and Λ = diag{λ1 , λ2 , λ3 } is the eigenvalues corresponding to the eigenvalue matrix. The corresponding azimuth angle φH and inclination angle θ H of the Hammerseley projections are obtained by uniformly sampling a sphere using the Hammerseley sequence [29]. It is aimed to identify the Cartesian coordinates of the uniformly sampled sphere. Thus, the non-uniform sample e p on a sphere can be determined by constructing an ellipsoid with the following parameters:   1 λ13 cos θ H sin φH  1   3 el =  (11)  λ2 sin θ H cos φH  1

λ33 cos φH The direction of the highest curvature is sampled by rotating the ellipsoid el . Therefore, the novel non-uniform samples scheme can be defined as e p = Vel = {e1 , e2 , e3 }. As a critical step of signal decomposition, the local mean estimation according to the conventional MEMD algorithm using the non-uniform samples e p is implemented and the computational procedure of trivariate empirical mode decomposition via convex optimization can be described in Algorithm 1. Algorithm 1. Trivariate empirical mode decomposition via convex optimization 1. 2.

3. 4.

Perform the low-rank matrix approximation via convex optimization framework to a trivariate signal Y(t) and the new observed signal X(t) can be obtained. The projection pk (t)(k = 1, 2, . . . , K) can be calculated of the input low-rank trivariate signal X(t) along n o the direction vector e1 k , e2 k , e3 k . It should also be noted that K is the number of the directional vector sets. The time instants tkm corresponding to the maxima of the set of projected signals pk (t) is determined. Interpolate [tkm , X(tkm )] to obtain multivariate envelope curves Ek (t). Then, the envelop mean can be calculated M(t) =

5.

1 K

K

∑ E k ( t ).

k =1

Calculate the residual by R(t) = X(t) − M(t). If the stopping criterion condition of iteration can be satisfied, then R(t) is set as one IMF and repeat the above steps to X(t) − R(t) until the next IMF is isolated. If it does not satisfy the stopping criterion, then repeat the above steps to R(t) until it meets the criterion.

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Cauchy-type convergence condition is chosen as the stopping criterion of the sifting iterative process, which indicates the iterative will stop under the circumstance that the discrepancy between adjacent sifting results is less than a threshold with a range 0.2–0.3. For clarify, K = 32 is always suitable for the application of trivariate signal empirical mode decomposition. 3. Simulation Signal Analysis Rolling bearing is an important transmission component supporting the rotating part of the mechanical equipment. If a bearing with an outer race that is fixed to the structure, a typical fault rolling bearing model can be simplified as follows [35]: xi (t) = α sin(2π f i t)[1 + β sin(2π f r t)]

(12)

where f i is the inner ring failure frequency and f r is corresponding to rotational frequency, respectively. Then, the following three source signals X = { x1 , x2 , x3 } are employed to simulate the collected vibration signals in this section: x1 (t) = 0.1 cos(2π f 1 t + 10) (13) x2 (t) = 0.2 sin(2π f 2 t − 15)

(14)

x3 (t) = 0.3 sin(2π f i t)[1 + sin(2π f r t)]

(15)

where the characteristics frequency about three simulation source signals are chosen as f 1 = 10 Hz, f 2 = 45 Hz, f i = 110 Hz, f r = 10 Hz. The sampling number and sampling frequency are set as N = 1024 and f s = 1024 Hz, respectively. In the analog process of sensor collecting signal, the vibration signal will be collected by any sensor simultaneously, and the trivariate signal is the instantaneous mixed signal composed by three above mentioned simulation signal. Considering the influence of noise, the zero-mean additive white Gaussian noise with variance 0.5 is added to the observation signal, and it is denoted as S N . A random matrix of 3 × 3 is chosen to mix the simulation source signal together and it can be expressed as follows:   0.6241 0.3674 0.8852   A =  0.6791 0.9880 0.9133  (16) 0.3955 0.0377 0.7962 Then, the instantaneous mixed signal model can be described as Y = AX + S N and Y = {y1 , y2 , y3 }, which indicates trivariate signal with noisy generated by three source signal using three sensors. The analysis results of trivariate signal in time-domain and frequency spectrum provided by FFT are shown in Figures 1 and 2.

Then, Then, the the instantaneous instantaneous mixed mixed signal signal model model can can be be described described as as Y and Y == AX AX ++ SSNN and Y Y == {{yy1 ,, yy2 ,, yy3 }} ,, which which indicates indicates trivariate trivariate signal signal with with noisy noisy generated generated by by three three source source signal signal using using 1

2

3

three three sensors. sensors. The The analysis analysis results results of of trivariate trivariate signal signal in in time-domain time-domain and and frequency frequency spectrum spectrum Sensors 2018, 18, 2325 6 of 14 provided by FFT are shown in Figures 1 and 2. provided by FFT are shown in Figures 1 and 2.

Figure The time response mixed trivariate signal with noise. Figure Figure1.1. 1.The Thetime timeresponse responseofof ofmixed mixedtrivariate trivariatesignal signalwith withnoise. noise.

Figure 2. The frequency spectrum of mixed trivariate signal with noise. Figure Figure 2. 2. The The frequency frequency spectrum spectrum of of mixed mixed trivariate trivariate signal signal with with noise. noise.

As shown in Figures and hard identify the frequency characteristics and modulation As Asshown shownin inFigures Figures111and and2,2, 2,itit itisis ishard hardtoto toidentify identifythe thefrequency frequencycharacteristics characteristicsand andmodulation modulation phenomenon of the observed trivariate in phenomenon Only the the inner innerrace racefault faultfrequency frequency ffi in phenomenonof ofthe theobserved observed trivariate trivariate signals signals with with noisy. noisy. Only Only the inner race fault frequency i in channel#3 can be inspected, and other frequency components are not obvious for all the channels. It is channel#3 channel#3 can can be be inspected, inspected, and and other other frequency frequency components components are are not not obvious obvious for for all all the the channels. channels. It It illustrated that the influence of noise should not be neglected and more advanced methods are required. is illustrated that the influence of noise should not be neglected and more advanced methods are is illustrated that the influence of noise should not be neglected and more advanced methods are Then, the conventional MEMD methods are methods employedare to employed deal with the signal and the result required. Then, MEMD to deal simulation signal required. Then, the the conventional conventional MEMD methods are employed to simulation deal with with the the simulation signal isand plotted in Figure 3. From the analysis result of the three channels shown in Figure 3, it is proved and the the result result is is plotted plotted in in Figure Figure 3. 3. From From the the analysis analysis result result of of the the three three channels channels shown shown in in Figure Figure 3, 3, that MEMD has advantages in guaranteeing the stability of the decomposition results. However, it is proved that MEMD has advantages in guaranteeing the stability of the decomposition results. it is proved that MEMD has advantages in guaranteeing the stability of the decomposition results. the characteristic frequencies f 1 , f 2 and the frequency modulation f i ± f r still cannot be identified.

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However, However,the thecharacteristic characteristicfrequencies frequenciesf1f,1 , f 2f 2 and andthe thefrequency frequencymodulation modulation f i f±i ±f r f r still stillcannot cannotbebe Sensors 2018, 18, 2325 7 of 14 identified. identified.

Figure3.3.The The resultprovided provided by theMEMD MEMD method. Figure Figure 3. Theresult result providedbybythe the MEMDmethod. method.

Eventually, Eventually, the proposed method applied simulate trivariate signal and the result drawn Eventually,the theproposed proposedmethod methodisis isapplied appliedtoto tosimulate simulatetrivariate trivariatesignal signaland andthe theresult resultisis isdrawn ininFigure 4. For clarify the denosing performance, the 3th, 4th, and 7th IMFs are chosen according Figure 4. For clarify the denosing performance, the 3th, 4th, and 7th IMFs are chosen according Figure 4. accordingtoto to the maximum similarity with the original signal shown in Figure 5. It is obvious that the the maximum frequency maximum similarity with the original signal shown in Figure 5. It is obvious obvious that thefrequency frequency modulation found ininthe the 3th IMF. addition, the characteristic frequencies modulation ffiif±±±f rffr can can bebefound 3th IMF. InInIn addition, the characteristic frequencies f 1 , f1ff2, also canbe foundin the 3th IMF. addition, the characteristic frequencies , f 2f i

r

1

2

also bebeviewed ininthe IMF, respectively. The result provided the proposed method can be in the 4th and 7th7th IMF, respectively. The result provided byby the proposed method is alsocan canviewed viewed the4th 4thand and 7th IMF, respectively. The result provided by the proposed method iscoincides with practical situation and the effective can demonstrated. with practical situation and the effective can bebebe demonstrated. iscoincides coincides with practical situation and the effective can demonstrated.

Figure4.4.The The resultprovided provided by theproposed proposed method. Figure Figure 4. Theresult result providedbybythe the proposedmethod. method.

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3th, 4th, and Figure Figure5.5.3th, 3th,4th, 4th,and and7th 7thIMFs IMFsofofdifferent differentsignal signalininthe thefrequency frequencydomain. domain.

4.4.Analysis Analysisofofthe theRoll RollBearing Bearingon onthe theExperimental ExperimentalBench Bench Commonly, complicated than the simulation signal ininthe the measured vibration signalisismore more complicated than the simulation signal in Commonly,the themeasured measuredvibration vibrationsignal more complicated than the simulation signal the process ofofactual signal analysis. InInorder totoverify the effectiveness ofofthe proposed algorithm ofof the process of actual signal analysis. In order to verify the effectiveness ofthe the proposed algorithm process actual signal analysis. order verify the effectiveness proposed algorithm trivariate empirical mode decomposition via oftrivariate trivariate empirical mode decomposition viaconvex convexoptimization, optimization,the theouter outer ring failure signal empirical mode decomposition via convex outerring ringfailure failuresignal signalofof rolling bearing in the experimental device is analyzed. The experimental apparatus and its structure rolling bearing in the experimental device is analyzed. The experimental apparatus and its structure diagram bench comprises a adrive shaft which isisdriven byby550 are drawn Figure where bench comprises adrive drive shaft which is driven by 550 W diagramare aredrawn drawninin inFigure Figure6,6,where where bench comprises shaft which driven 550W W(220 (220 V/50 Hz) AC motor. (220 V/50 Hz) AC motor. V/50 Hz) AC motor.

(a)(a)

1-1-electric electricmotor motor 2-2-coupler coupler 3-3-Replaceable Replaceablefault faultrolling rollingbearing bearingwith with6207 6207 4-4-belt beltpulley pulley5-5-gearbox gearbox 6-6-transmission transmissionshaft shaft 7-7-Fault gear sets Fault gear sets (b) (b)

Figure 6.6.Fault experimental roll bearing. test The structure Figure6. Faultexperimental experimentaltable tableof rollbearing. bearing.(a) (a)The Thephysical physicalmap mapof testrig; rig;(b) (b)The Thestructure structure Figure Fault table ofofroll (a) The physical map ofoftest rig; (b) diagram of test rig. diagramofoftest testrig. rig. diagram

The Theelectric electricspark sparkmachining machiningmethod methodisisused usedtotocarry carryout outpitting pittingtreatment treatmenton onthe theouter outerring ringofof the thereplaceable replaceablebearing bearingtotosimulate simulatethe theouter outerfaults. faults.AAPCB-352C33 PCB-352C33acceleration accelerationsensor sensorisisused usedtoto

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The electric spark machining method is used to carry out pitting treatment on the outer ring of the collect the bearing acceleration signals. installation position ofacceleration sensor is located thetohorizontal replaceable to simulate theThe outer faults. A PCB-352C33 sensor isin used collect the direction, vertical direction and axial direction of the bearing on the right side of the experimental acceleration signals. The installation position of sensor is located in the horizontal direction, vertical platform. Theaxial bearing type is grooveonball with model number of 6207. The direction and direction of deep the bearing the bearing right side of the experimental platform. Theexternal bearing diameter of removable bearing was D = 72 mm and the inner diameter was d = 35 mm, respectively. type is deep groove ball bearing with model number of 6207. The external diameter of removable The testing flow for the experimental bench in Figure and the bearing was D = 72chart mm and inner diameter wasisd demonstrated = 35 mm, respectively. The7testing flowsampling chart for frequency is determined as 16,000 Hz. Vibration detection in horizontal (x), vertical (y) axial Hz. (z) experimental bench is demonstrated in Figure 7 and the sampling frequency is determinedand as 16,000 directions should be considered as far as possible, which is aimed to collect the trivariate signal. Vibration detection in horizontal (x), vertical (y) and axial (z) directions should be considered as far as Suppose there isisno relative sliding raceway surface and therelative rollingsliding elements, and the possible, which aimed to collect thebetween trivariatethe signal. Suppose there is no between the outer ring is fixed. Then, the outer fault frequency 87.01 Hz can be obtained. The detailed raceway surface and the rolling elements, and the outer ring is fixed. Then, the outer fault frequency experimental parameters in Table 1. 87.01 Hz can be obtained.are Theshown detailed experimental parameters are shown in Table 1.

electric motor

Replaceable fault rolling bearing with 6207

vertic al dir

y

ectio n

horiz ontal d

x

irectio n

acceleration sensor PCB-352C33

z axial direction

Fault gear sets

gearbox

Signal amplifier DHF8 Computer

Data acquistion USB-4716

plot of measured trivariate signal

Figure 7. 7. Testing Testing flow flow chart for experimental bench. Figure

Table Table 1. 1. The The experimental experimental parameters parameters and and fault fault frequency. frequency. Rotating Speed Rotating Speed r/min r/min 1450 1450

Rotating Rotating Frequency/Hz Frequency/Hz 24.17 24.17

Sampling Sampling Frequency/Hz Frequency/Hz 16,000 16,000

Sampling Sampling Time/s Time/s 0.3661 0.3661

Outer Fault Outer Fault Frequency/Hz Frequency/Hz 87.01 87.01

It can be seen from Table 1 that the rotation frequency is f r = 24.17 HZ. According to the It can be seen from 1 that the the rotation frequency f r = fault 24.17 signal HZ. According the theory theory of bearing fault Table diagnosis, frequency of outeris ring of rollingto bearing is of bearing fault diagnosis, the frequency of outer ring fault signal of rolling bearing is calculated as calculated as f o = 87.01 HZ. Then, the time domain and frequency domain plots of the original f o = 87.01 vibration HZ. Then,signal the time and frequency plots of the original measured vibration measured are domain shown in Figures 8 anddomain 9, respectively. signal are shown in Figures 8 and 9, respectively.

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Figure 8. The time domain plot of measured trivariate signal. Figure8.8.The Thetime timedomain domainplot plotofofmeasured measuredtrivariate trivariatesignal. signal. Figure

Figure 9. The frequency domain plot of measured trivariate signal. Figure9.9.The Thefrequency frequencydomain domainplot plotofofmeasured measuredtrivariate trivariatesignal. signal. Figure

The spectral spectral analysis analysis by by FFT FFT isis applied applied to to the the measured measured trivariate trivariate signal signal in in Figure 99 and and the the The The spectral analysis by FFT is applied are to the measured trivariate the signal in characteristic FigureFigure 9 and the result result shows that the signal components complex. In addition, fault frequency result that shows the components signal components are complex. In addition, the characteristic fault characteristic frequency shows thethat signal are complex. In addition, the fault frequency and and frequency frequency modulation phenomenon cannot be identified. identified. Then, the the conventional conventional MEMD and modulation phenomenon cannot be Then, MEMD frequency phenomenon cannotand be the identified. the conventional MEMD method methodisismodulation usedto toanalysis analysis thefault faultsignal signal resultin inThen, frequency-domain plotted inFigure Figure 10. used the and theinresult frequency-domain isisplotted in ismethod used to analysis the fault signal and the result frequency-domain is plotted in Figure 10.highest It 10. is It is also noted that all the analyses employed 32 projection vectors to capture the direction of It is noted also noted that all analyses the analyses employed 32 projection vectors to capture thedirection directionofofhighest highest also that all the employed 32 projection vectors tofeatures capture the curvature of trivariate signals. By observing the frequency of each IMFs, the fault curvatureof of trivariate signals. By observing the frequency features of the each IMFs, the fault curvature trivariate signals. By observing the frequency features of each IMFs, fault characteristic characteristicfrequency frequencystill stillcannot cannotbe beinspected. inspected. characteristic frequency still cannot be inspected.

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Figure 10. The result result provided by MEMD in frequency domain. Figure 10. 10. The result provided provided by by MEMD MEMD in in frequency frequency domain. domain. Figure

Finally, the proposed method is employed to the measured trivariate vibration signal and the Finally, the the proposed proposed method method is is employed employed to to the the measured measured trivariate trivariate vibration vibration signal signal and and the the Finally, result in time-domain is shown in Figure 11. For the sake of clarity, the FFT operators to the IMF3 for result in time-domain is shown in Figure 11. For the sake of clarity, the FFT operators to the IMF3 for result in time-domain is shown in Figure 11. For the sake of clarity, the FFT operators to the IMF3 for all the channels are conducted, and the corresponding results are plotted in Figure 12. Fortunately, all the channels are conducted, and the corresponding results are plotted in Figure 12. Fortunately, all the channels are conducted, and the corresponding results are plotted in Figure 12. Fortunately, the outer ring fault frequency can found. the special ff o and the outer outerring ringfault fault frequency and its its doubling doubling 22 ffoo found. can be beAdditionally, found. Additionally, Additionally, special o its doubling 2 f can be the frequency f o and the specialthe frequency o f ± f 2 f ± f and can also frequency modulation phenomenon of fault rolling bearing such as o r o r f o ± fbe also be be frequency modulation phenomenon of fault rollingsuch bearing modulation phenomenon of fault rolling bearing as f osuch ± f r as andf o2±f o f± f r can2 also identified. r and r can identified. Thus, we can firmly believe that the fault is located on the outer ring, which is consistent identified. Thus, we can firmly thatisthe fault on is located on ring, the outer ring, which is consistent Thus, we can firmly believe thatbelieve the fault located the outer which is consistent with the with actual with the the actual situation situation shown in Figure 13. 13. actual situation shown inshown Figure in 13.Figure

Figure 11. The result provided by the proposed method in time domain. Figure 11. 11. The result provided provided by by the the proposed proposed method method in in time time domain. domain.

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Figure Figure 12. 12. The Thefrequency frequency analysis analysis result result about about the the IMF3 IMF3 by by the the proposed proposed method. method. Figure 12. The frequency analysis result about the IMF3 by the proposed method.

Figure 13. The picture of faulty bearing located in outer ring. Figure 13. The picture of faulty bearing located in outer ring. Figure 13. The picture of faulty bearing located in outer ring.

By comparing the results respectively provided by MEMD and the method proposed in this By comparing the results respectively provided by MEMD and the method proposed in this paper, can make conclusion that the proposed methodand has better performance in this trivariate Bywe comparing theaaresults respectively provided by MEMD method proposed in paper, paper, we can make conclusion that the proposed method hasthe better performance in trivariate signal mode decomposition and fault featuremethod extraction. we can make a conclusion that the proposed has better performance in trivariate signal mode signal mode decomposition and fault feature extraction. decomposition and fault feature extraction. 5. Conclusions 5. Conclusions 5. Conclusions Based on the traditional MEMD method, a novel method of trivariate empirical mode Based on the traditional MEMD method, a novel method of trivariate empirical mode decomposition optimization rolling abearing identification is proposedmode and Based on via theconvex traditional MEMD for method, novel condition method of trivariate empirical decomposition via convex optimization for rolling bearing condition identification is proposed and tested in this paper. Theoptimization main research works bearing are listed as follows: (1) a low-rank matrix decomposition via convex for rolling condition identification is proposed and tested in this paper. The main research works are listed as follows: (1) a low-rank matrix approximation via convex optimization framework forathe trivariate signal denoising. tested in this paper. The main research works are listedisasproposed follows: (1) low-rank matrix approximation approximation via convex optimization framework is proposed for the trivariate signal denoising. By parameterized non-convex function, its performance is improved; principal via introducing convex optimization framework is proposed for the trivariate signal denoising.(2) Bythe introducing By introducing parameterized non-convex function, its performance is improved; (2) the principal component obtained by SVD operator is employed to indicate sample directions, the parameterized non-convex function, its performance is improved; (2) the principal component thus obtained component obtained by SVD operator is employed to indicate the sample directions, thus the non-uniform sample scheme to isindicate presented meetdirections, the requirement inhomogeneity of scheme the data by SVD operator is employed the to sample thus theof non-uniform sample is non-uniform sample scheme is presented to meet the requirement of inhomogeneity of the data distribution in high-dimensional space; (3) through the analysis of the simulation signal and the presented to meet the requirement of inhomogeneity of the data distribution in high-dimensional space; distribution in high-dimensional space; (3) through the analysis of the simulation signal and the measured signal, it simulation is demonstrated the measured proposed vibration method is superior to MEMD in (3) throughvibration the analysis of the signal that and the signal, it is demonstrated measured vibration signal, it is demonstrated that the proposed method is superior to MEMD in multi-scale feature extraction. It can be regarded as powerful tool in trivariate signal processing. multi-scale feature extraction. It can be regarded as powerful tool in trivariate signal processing.

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that the proposed method is superior to MEMD in multi-scale feature extraction. It can be regarded as powerful tool in trivariate signal processing. Author Contributions: Y.L. and C.Y. conceived and designed the experiments; Y.L. and H.Z. performed the experiments; H.Z. and C.Y. analyzed the data; C.Y. contributed to materials/analysis tools development; C.Y. and Y.L. wrote the paper. Funding: This work was supported by the National Natural Science Foundation of China under Grants No. 51475339, the Natural Science Foundation of Hubei province under Grant No. 2016CFA042, the Applied Basic Research Programs of Wuhan science and technology bureau under Grants No. 2017010201010115, the Engineering Research Center for Metallurgical Automation and Measurement Technology of Ministry of Education (Wuhan University of Science and Technology) under Grants No. MARC201701, and the Guangxi Key Laboratory of Optoelectronic Information Processing Open Foundation of China under Grant No. KFJJ2017-01. Acknowledgments: Thank you for reviewers’ valuable comments and suggestions. Conflicts of Interest: The authors declare no conflict of interest.

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