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Application Research of a New Adaptive Noise Reduction Method in Fault Diagnosis

Application Research of a New Adaptive Noise Reduction Method in Fault Diagnosis applied sciences Article Application Research of a New Adaptive Noise Reduction Method in Fault Diagnosis 1 1 , 1 2 Wenxiao Guo , Ruiqin Li *, Yanfei Kou and Jianwei Zhang School of Mechanical Engineering, North University of China, Taiyuan 030051, China; gwx041202@163.com (W.G.); kouyanfei@163.com (Y.K.) Department of Informatics, University of Hamburg, 22527 Hamburg, Germany; zhang@informatik.uni-hamburg.de * Correspondence: liruiqin@nuc.edu.cn Received: 18 June 2020; Accepted: 21 July 2020; Published: 23 July 2020 Abstract: The feature extraction of composite fault of gearbox in mining machinery has always been a diculty in the field of fault diagnosis. Especially in strong background noise, the frequency of each fault feature is di erent, so an adaptive time-frequency analysis method is urgently needed to extract di erent types of faults. Considering that the signal after complementary ensemble empirical mode decomposition (CEEMD) contains a lot of pseudo components, which further leads to misdiagnosis. The article proposes a new method for actively removing noise components. Firstly, the best scale factor of multi-scale sample entropy (MSE) is determined by signals with di erent signal to noise ratios (SNRs); secondly, the minimum value of a large number of random noise MSE is extracted and used as the threshold of CEEMD; then, the e ective Intrinsic mode functions(IMFs) component is reconstructed, and the reconstructed signal is CEEMD decomposed again; finally, after multiple iterations, the MSE values of the component signal that are less than the threshold are obtained, and the iteration is terminated. The proposed method is applied to the composite fault simulation signal and mining machinery vibration signal, and the composite fault feature is accurately extracted. Keywords: fault diagnosis; IMF; complementary ensemble empirical mode decomposition; multi-scale sample entropy 1. Introduction In industrial production, the normal operation of mining machinery is the guarantee for the economic growth of enterprises. The gear and bearing in the gearbox are the most important transmission parts. When one of them fails, complex mechanical behavior will appear. In the unbalanced mechanical environment, other parts will appear fatigued, and further develop into the weak fault and significant fault. At the same time, their vibration signals are coupled with each other. In the case of strong noise environments and complex transmission paths, weak fault noises are submerged, and further leakage or misdiagnosis occurs [1–5]. Mechanical intelligence fault diagnosis usually includes three links. (1) Signal acquisition: Obtain multi-physical monitoring signals radiated by mechanical equipment to reflect the health status of equipment. (2) Feature extraction: By analyzing the acquired monitoring data, extract features to reveal fault information. (3) Fault identification and prediction: Based on the extracted features, faults are identified and predicted through artificial intelligence models and methods. Over decades, with the development of computer science, a series of di erent time-frequency analysis methods have emerged. In 1998, empirical mode decomposition (EMD) was proposed, and then a large number of applications in signal processing [6–8], but it has endpoint e ects and mode aliasing phenomena. In 2005, ensemble empirical mode decomposition (EEMD) was proposed [9–11]; Appl. Sci. 2020, 10, 5078; doi:10.3390/app10155078 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 5078 2 of 18 it can not only overcome the endpoint e ects of EMD but also weaken the mode aliasing phenomenon, so it has been applied in industrial production by a large number of scholars. However, it needs to set the times of additions and the amplitude of white noise; so far, no formula can adaptively determine the white noise amplitude value. At the same time, the number of additions is also not adaptive. When the number is too large, the calculation amount is too large; otherwise, the noise cannot be e ectively averaged [12–14]. In order to improve the decomposition eciency, complementary ensemble empirical mode decomposition (CEEMD) was proposed, which weakens the e ect of the contained noise on the original signal by adding di erent positive and negative white noise multiple times [15], and is widely used in fault diagnosis [16–18]. CEEMD can improve the decomposition eciency of EEMD, but the noise part of the original signal still exists. In other words, the noise component of the original signal still exists in the multiple Intrinsic mode functions(IMFs) decomposed. If it is not removed in time, it will lead to leakage or misdiagnosis. So, it is necessary to propose an e ective method to eliminate noise components. In recent years, variable mode decomposition (VMD) [19–21] has been applied to fault diagnosis. It can decompose the original signal into several di erent time scale functions from low frequency to high frequency. The center frequency of each time scale is di erent, so its decomposition eciency is higher than the EEMD and CEEMD. However, the method is limited to the choice of the number of decomposition layers and the penalty factor. So far there is still no suitable method to adaptively determine these two parameters. Therefore, in order to adaptively reduce the noise of complex vibration signals, this article uses the CEEMD decomposition method. In strong background noise, each fault feature frequency is di erent [22–29], so an adaptive time-frequency analysis method is urgently needed to extract di erent types of faults. The article employs CEEMD to adaptively reduce the noise of complex vibration signals, but CEEMD has the problem of mode aliasing and the diculty of removing noise components. Considering that the multi-scale sample entropy (MSE) [30–32] can estimate the complexity of time series, accurately measuring the complexity of di erent modal functions with strong noise resistance ability and excellent consistency, and at the same time, a stable entropy value can be obtained by using fewer data segments. Therefore, the MSE is used as the basis for eliminating the noise component, but the accuracy of the MSE is related to the scale factor. So, a method is proposed to determine the best scale factor in the MSE by the simulation signal with di erent signal to noise ratio (SNR), and the best factor is placed in the multi-scale sample entropy to calculate the MSE value of random noise, simulation signal, and vibration signal. This is firstly done by calculating the multiscale sample entropy value of 10,000 sets of random noise, extracting the minimum value and using it as the threshold of the noise component in CEEMD; then, the e ective IMFs component is reconstructed, and the reconstructed signal is CEEMD decomposed again; this is followed by calculating the MSE value of each component signal, comparing the obtained MSE values with the threshold value, and if there are IMFs component signals greater than the threshold value, remove them, then reconstruct the remaining signal and perform CEEMD decomposition again; after multiple iterations, the MSE values of the component signal that are less than the threshold can be obtained, and the iteration is terminated. Finally, the resulting signal is analyzed by the spectrum to determine the fault characteristics. The proposed method is applied to the composite fault simulation signal and mining machinery vibration signal, and the composite fault feature is accurately extracted. At the same time, the feasibility of the method is verified by comparison with the traditional CEEMD. 2. Basic Theory 2.1. CEEMD To solve the mode aliasing problem caused by EMD, EEMD uses the characteristics that white noise can be evenly distributed in the spectrum. Adding white noise to EMD improves the mode aliasing phenomenon; however, the residual white noise also brings reconstruction error problems. ... ... ... ... ... ... ... ... ... Appl. Sci. 2020, 10, 5078 3 of 18 Inspired by EEMD, adding white noise to equalize the noise in the original signal, Yeh J R [15] proposed a new time-frequency analysis method CEEMD. The CEEMD method adds a pair of positive and negative white noise to the original signal for EMD decomposition to achieve the purpose of eliminating residual white noise in the reconstructed signal. In addition, the added white noise improves the mode aliasing problem, and the number of CEEMD noise addition is lower, which improves the calculation eciency. The CEEMD method is as follows: (1) Two new signals are formed by adding white noise with a certain standard deviation and equal length and opposite signs to the original signal. Appl. Sci. 2020, 10, x 3 of 19 S = S(t) + n +i +i (1) positive and negative white noise to the original signal for EMD decomposition to achieve the ( ) S = S t + n i i purpose of eliminating residual white noise in the reconstructed signal. In addition, the added white noise improves the mode aliasing problem, and the number of CEEMD noise addition is where S is the sum of the original signal and the positive white noise signal, S(t) is the original +i lower, which improves the calculation efficiency. signal, n is the positive white noise; S is the sum of the original signal and the negative white The CEEMD method is as follows: +i -i noise signal, and n is the negative white noise. (1) Two new sig -i nals are formed by adding white noise with a certain standard deviation and equal length and opposite signs to the original signal. (2) EMD decomposition of S and S to obtain IMF and IMF , then the first-order IMF is +i -i +(i,j) -(i,j) SS=+ ()t n ++ii (1) SS=+ ()t n 1−−ii IMF = IMF + IMF (2) 1 +(i,1) (i,1) 2N where S+i is the sum of the original signal and the positive white noise signal, S(t) is the i=1 original signal, n+i is the positive white noise; S-i is the sum of the original signal and the negative white noise signal, and n-i is the negative white noise. (3) Repeat the above steps to add Gaussian white noise 2N times. (2) EMD decomposition of S+i and S-i to obtain IMF+(i,j) and IMF-(i,j), then the first-order IMF is (4) Finally, the decomposition results of multiple component combinations are obtained: IMF=+ IMF IMF () (2) 1 +− ()ii ,1 () ,1 2 N i =1 (3) Repeat the above steps to add Gaussian white noise 2N times. IMF = IMF + IMF (3) +(i,j) (i,j) (4) Finally, the decomposition results of2 mN ultiple component combinations are obtained: i=1 IMF=+ IMF IMF () (3) j +− () ij,,() i j where N is the number of pairs added with white noise, IMF is the j-th component obtained by 2 N i =1 j CEEMD decomposition, and IMF is the i-th IMF component of the j-th signal. (i,j) where N is the number of pairs added with white noise, IMFj is the j-th component obtained by CEEMD decomposition, and IMF(i,j) is the i-th IMF component of the j-th signal. The decomposition process of CEEMD is shown in Figure 1. The decomposition process of CEEMD is shown in Figure 1. Original signal S(t) Add white Add white Add white Add white Add white Add white Add white Add white ... noise (+)n noise (-)n noise (+)n noise (-)n noise (+)n noise (-)n noise (+)n noise (-)n 1 1 2 2 N-1 N-1 N N S S S S ... S S S S +1 -1 +2 -2 +(N-1) -(N-1) +N -N EM EMD D Average ... value + + + + + + + IMF IMF IMF IMF IMF +11 IMF +21 IMF +(N-1)1 -(N-1)1 IMF IMF 1 -11 -21 +N1 -N1 Average + + + + + + + + ... value + + + + + + + IMF IMF IMF IMF IMF IMF IMF IMF IMF 2 +12 -12 +22 -22 +(N-1)2 -(N-1)2 +N2 -N2 Average + + + + + + + + value ... + + + + + + + IMF IMF IMF IMF IMF +13 IMF +23 IMF +(N-1)3 -(N-1)3 IMF IMF 3 -13 -23 +N3 -N3 + + + + + + + + ... + + + + + + + Average + + + + + + + + ... value + + + + + + + IMF IMF IMF IMF IMF IMF IMF IMF +1N -1N +2N -2N +(N-1)N -(N-1)N +NN -NN IMF Average + + + + + + + + ... value + + + + + + + r r r r +1N r +2N r +(N-1)N -(N-1)N r r -1N -2N +NN -NN r Figure 1. Complementary ensemble empirical mode decomposition (CEEMD) flowchart. Figure 1. Complementary ensemble empirical mode decomposition (CEEMD) flowchart. .... Appl. Sci. 2020, 10, 5078 4 of 18 2.2. Multi-Scale Sample Entropy (MSE) Fault signals detected in rotating machinery are usually embedded in multi-scale structures. However, the sample entropy (SE) [33,34] only analyzes the single scale signal and ignores much useful information. This limits its performance in extracting embedded fault features [35]. In order to solve the shortcomings of sample entropy, Costa [36] proposed multi-scale sample entropy as an e ective complexity measure of time series on di erent time scales. For a time series {x , x , ::: , 1 2 x , x }, MSE constructs multiple coarse-graining time series. The process of coarse-graining scale N-1 N factor Q is obtained by averaging the internal structure of the time series, but without overlapping the length of the scale factor. Figure 2 shows the coarse-graining examples of processes. Therefore, the coarse-graining time series can be obtained from Equation (4): jQ 1 N y = x , 1  j  (4) Q Q i=(j1)Q+1 where Q is the scale factor and is a positive integer. When Q = 1, the coarse-graining time series y is the original time series. j is the length of the coarse-graining time series. Appl. Sci. 2020, 10, x 5 of 19 Scale factor Q=1 x x x x x x x x x 1 2 3 4 5 6 i-1 i i+1 ▲ ▲ ▲ ▲ ▲ ▲ ¨¨¨ ▲ ▲ ▲ Scale factor Q=2 x x x x x x x x x 1 2 3 4 5 6 i-1 i i+1 ▲ ▲ ▲ ▲ ▲ ▲ ¨¨¨ ▲ ▲ ▲ \ ∕ \ ∕ \ ∕ \ ∕ ◆ ◆ ◆ ◆ (2) (2) (2) (2) y y y y =(x +x )/2 1 2 3 i/2 i-1 i Scale factor Q=3 x x x x x x x x x 1 2 3 4 5 6 i-1 i i+1 ▲ ▲ ▲ ▲ ▲ ▲ ¨¨¨ ▲ ▲ ▲ \ | ∕ \ | ∕ \ | ∕ ★ ★ ★ (3) (3) (3) y y y =(x +x +x /3 1 2 i/3 i-1 i i+1) Scale factor Q=n Figure 2. Coarse graining process. Figure 2. Coarse graining process. (5) Then calculate the probability of matching points The next step is to calculate the multi-scale sample entropy and scale factor Q for each newly reconstructed coarse-graining time-series signal y . NQ The −+ m mu 1 lti-scale sample entropy is calculated mm Br = B r () ()  (9) as follows: i NQ−+ m 1 i =1 (Q) (1) From N/Q-m+1 to vector y (i) NQ −m mm (10) ArA = r () ()  i h i NQ − m (Q) Q Q Q i =1 y (i) = y (i), y (i + 1), , y (i + m 1) , i = 1, , N/Q m + 1 (5) m m where B (r) and A (r) represent the probability of two sequences matching m and m + 1 points, where m is the length of the sequence to be compared. respectively. (6) The theoretical value of multiscale sample entropy is defined as  Ar()  SampEn() m,l r=− im ln  (11) NQ →∞ Br ()   When the data point N/Q is a finite number, the estimate of sample entropy is given by the following formula:  Ar () SampEn m,, r N =−ln . ()  (12) Br()   It can be seen from Equation (12) that the value of SampEn is related to the value of m,r. At present, there is no definite value, and the empirical value is m = 2, rÎ(0.1SD~0.5SD). SD is the standard deviation of the original data. In this paper, when studying the complexity of time series, m = 1, r = 0.25SD is taken. Appl. Sci. 2020, 10, 5078 5 of 18 (Q) (Q) (2) Calculate the maximum norm between two vectors d [y (i), y (j)]. m m h i Q Q Q Q d y (i), y (j) = max y (i + k) y (j + k) , 0  k  m 1 (6) m m m m (3) Define function B m m B (r) = u (i), i = 1, , N/Q m + 1 (7) N/Q m + 1 (Q) (Q) (Q) where r is the tolerance of the acceptance matrix, u (i) is equal to d [y (i), y (j)], and d [y (i), m m m m m (Q) y (j)] r, i , j. (4) Define function A m m+1 A (r) =  (i), i = 1, , N/Q m + 1 (8) N/Q m + 1 (Q) (Q) (Q) (Q) m+1 where  is equal to d [y (i), y (j)], and d [y (i), y (j)] r, i , j. m+1 m+1 m+1 m+1 m+1 m+1 (5) Then calculate the probability of matching points N/Qm+1 m m B (r) = B (r) (9) N/Q m + 1 i=1 N/Qm m m A (r) = A (r) (10) N/Q m i=1 m m where B (r) and A (r) represent the probability of two sequences matching m and m + 1 points, respectively. (6) The theoretical value of multiscale sample entropy is defined as ( ) A (r) ( ) SampEn m, r = lim ln . (11) N/Q!1 B (r) When the data point N/Q is a finite number, the estimate of sample entropy is given by the following formula: " # A (r) SampEn(m, r, N) = ln . (12) B (r) It can be seen from Equation (12) that the value of SampEn is related to the value of m,r. At present, there is no definite value, and the empirical value is m = 2, r2 (0.1SD~0.5SD). SD is the standard deviation of the original data. In this paper, when studying the complexity of time series, m = 1, r = 0.25SD is taken. 2.3. Optimized CEEMD Method CEEMD is an improved method based on the EMD method to add a pair of positive and negative white noise; there is also a mode aliasing phenomenon, and there is still noise in the decomposed IMF component. In order to eliminate the influence of noise on the decomposition results, this paper uses multi-scale sample entropy to remove the IMF with more noise components, reconstruct the signal, and perform the Fast Fourier transform(FFT) transform to obtain the fault feature signal to achieve the purpose of optimizing CEEMD. The flow chart is shown in Figure 3, the method is as follows: Appl. Sci. 2020, 10, x 6 of 19 2.3. Optimized CEEMD Method CEEMD is an improved method based on the EMD method to add a pair of positive and negative white noise; there is also a mode aliasing phenomenon, and there is still noise in the decomposed IMF component. In order to eliminate the influence of noise on the decomposition results, this paper uses multi-scale sample entropy to remove the IMF with more noise components, Appl. Sci. 2020, 10, 5078 6 of 18 reconstruct the signal, and perform the Fast Fourier transform(FFT) transform to obtain the fault feature signal to achieve the purpose of optimizing CEEMD. The flow chart is shown in Figure 3, the method is as follows: (1) Employing the CEEMD method to decompose the original signal X(t) containing noise to obtain a series of IMFs. (1) Employing the CEEMD method to decompose the original signal X(t) containing noise to (2) The selection of the scale factor Q of the MSE will a ect the multiscale sample entropy value of obtain a series of IMFs. each IMF. According to the decomposition characteristics of the IMF, the noise content of each (2) The selection of the scale factor Q of the MSE will affect the multiscale sample entropy value of layer is di erent. In order to better distinguish each layer of signals, this paper chooses a scale each IMF. According to the decomposition characteristics of the IMF, the noise content of each factor layer is differ of 1–20 and ent. In order to better distinguish ea signals with di erent SNRs forch simulation layer of signals, this pape to determine ther best chooses scale a scale factor Q. factor of 1--20 and signals with different SNRs for simulation to determine the best scale factor (3) Substitute m = 2, r = 0.25SD, and the Q that was determined in step (2) into the MSE, calculate the Q. multiscale sample entropy value of 10,000 sets of random noise signals, and obtain the minimum (3) Substitute m = 2, r = 0.25SD, and the Q that was determined in step (2) into the MSE, calculate value K of the noise multiscale sample entropy. the multiscale sample entropy value of 10,000 sets of random noise signals, and obtain the (4) Calculate the multi-scale sample entropy value of each IMF component, compare the obtained minimum value K of the noise multiscale sample entropy. value with K, remove components greater than K value, and reconstruct the remaining (4) Calculate the multi-scale sample entropy value of each IMF component, compare the obtained IMFs components. value with K, remove components greater than K value, and reconstruct the remaining IMFs (5) Perform CEEMD decomposition on the reconstructed signal again, and then calculate the components. multiscale sample entropy value of each IMF component. (5) Perform CEEMD decomposition on the reconstructed signal again, and then calculate the (6) Compare the MSE value of each IMF component with the minimum value K, if there is more than multiscale sample entropy value of each IMF component. K worth of IMF, repeat (2–6); otherwise, continue. (6) Compare the MSE value of each IMF component with the minimum value K, if there is more (7) The than signal K worth is FFT of IMF transformed , repeat (2–6); to obtain otherw the ise,fault continu signal. e. (7) The signal is FFT transformed to obtain the fault signal. Input the Feature Obtaining scale factor original signal extraction Q by signal simulation with noise X(t) with different SNR Remove IMFs greater than K, and CEEMD MSE reconstruct the decomposition Yes remaining IMFs No Calculate the MSEs value of 10000 sets of Obtain physically random noise signals, MSE of IMFs less meaningful IMFs and determine the than K component signals minimum value K Compare the MSEs of Calculate the Calculate the IMFs and K value, if it MSE of each MSE of each is less than K, it is the IMF component IMF component sensitive component Reconstruct CEEMD sensitive decomposition components IMFs Figure 3. Optimized CEEMD method flow chart. Figure 3. Optimized CEEMD method flow chart. 3. Simulation 3.1. Construct a Simulation Signal The collected gearbox signal appears in the form of a mixture of the modulation signal of the gear mesh and the impact signal of the bearing rotation, and the signal contains noise in the working environment. As a result, this paper establishes a mathematical model containing a mixture of impact Appl. Sci. 2020, 10, x 7 of 19 3. Simulation 3.1. Construct a Simulation Signal The collected gearbox signal appears in the form of a mixture of the modulation signal of the gear mesh and the impact signal of the bearing rotation, and the signal contains noise in the Appl. working Sci. 2020,environment 10, 5078 . As a result, this paper establishes a mathematical model containing a 7 of 18 mixture of impact signal, modulation signal, and noise signal for signal simulation. The mathematical model established is as follows: signal, modulation signal, and noise signal for signal simulation. The mathematical model established is as follows: x ()tA=× exp − sin() 2πft () 1 mc  T ( ) ( ) > x t = A  exp sin 2 f t 1 m c > T  m > xtf =+ 1cos 2ππt sin 2ft  ()() ( ) () 21nz (13) < x (t) = (1 + cos(2 f t)) sin(2 f t) 2 n1 (13) > noise() t = 0.5randn() t noise(t) = 0.5randn(t) >  Xt =+ x t x t+noise t () () () () X(t) = x (12 t) + x (t) + noise(t) 1 2 where A = 3.0, g = 0.01, T = 1/13, f = 130 Hz, f = 10 Hz, and f = 50 Hz. randn(t) is random noise. where mAm = 3.0, g = 0.01, Tm m = 1/13, fc = 130 Hz, c fn1 = 10 Hz, and fz = 50 Hz. z randn(t) is random noise. The n1 The number of s number ofasampling mpling point points s is is 101024 24 and and ththe e sampl sampling ing frefr quenc equency y is 2048 H is 2048zHz. . Figure Figur 4 is the time e 4 is the time domain diagram and frequency domain diagram of the simulation signal, where the frequency domain diagram and frequency domain diagram of the simulation signal, where the frequency domain domain diagram is obtained by performing FFT transformation on the time domain signal. diagram is obtained by performing FFT transformation on the time domain signal. (a) (b) Figure 4. Simulation Figure 4. Si signal. mulation si (a) Tgnal ime .domain (a) Time d of simulation omain of simul signal, ation (b si )gnal Frequency , (b) Frequenc domain y d of omai simulation n of signal. simulation signal. Among them, X1 is the impact signal of bearing rotation, X2 is the modulation signal of gear mesh, and X(t) is the mixed signal of impact signal, modulation signal, and random noise signal. It can be observed from the figure that the X(t) signal contains a lot of noise signals. 3.2. Optimized CEEMD Method The simulation signal X(t) is decomposed by CEEMD, and the decomposed time and frequency domain diagram is shown in Figure 5. Impact signals can be observed in the time-domain diagram. The signal in the frequency-domain diagram still has mode aliasing and contains some noise, which will a ect the extraction of fault feature signals. Therefore, this paper proposes to use MSE to optimize the decomposition of CEEMD. Appl. Sci. 2020, 10, x 8 of 19 Among them, X1 is the impact signal of bearing rotation, X2 is the modulation signal of gear mesh, and X(t) is the mixed signal of impact signal, modulation signal, and random noise signal. It can be observed from the figure that the X(t) signal contains a lot of noise signals. 3.2. Optimized CEEMD Method The simulation signal X(t) is decomposed by CEEMD, and the decomposed time and Appl. Sci. 2020, 10, 5078 8 of 18 frequency domain diagram is shown in Figure 5. Impact signals can be observed in the time-domain diagram. The signal in the frequency-domain diagram still has mode aliasing and contains some noise, which will affect the extraction of fault feature signals. Therefore, this paper The scale factor Q of the multi-scale sample entropy will a ect the MSE value, and di erent proposes to use MSE to optimize the decomposition of CEEMD. Q values have di erent MSE values. According to the decomposition characteristics of CEEMD, The scale factor Q of the multi-scale sample entropy will affect the MSE value, and different Q each layer of IMF has a di erent SNR. In order to accurately determine the MSE value of each IMF, values have different MSE values. According to the decomposition characteristics of CEEMD, each the di er lay ent er of IM IMFsF must has a diffe be accurately rent SNR. In ord distinguished. er to accura In tely determin this paper,e the the MSE value of each I values of the scaleM factor F, theQ range different IMFs must be accurately distinguished. In this paper, the values of the scale factor Q range from 1 to 20 are taken, and the SNR is respectively taken as30 dB,20 dB,10 dB, 1 dB, and 10 dB from 1 to 20 are taken, and the SNR is respectively taken as −30 dB, −20 dB, −10 dB, 1 dB, and 10 dB for simulation. The results are shown in Figure 6. The signals with the same SNR in the figure have for simulation. The results are shown in Figure 6. The signals with the same SNR in the figure have di erent MSE values due to di erent Q values. Taking the SNR30dB as an example, when the scale different MSE values due to different Q values. Taking the SNR −30dB as an example, when the factor takes di erent values, MSE only has large fluctuations, and the fluctuation range is 1.6–2.5. scale factor takes different values, MSE only has large fluctuations, and the fluctuation range is For MSE with the same scale factor, there are di erent values of MSE due to di erent SNRs. Multi-scale 1.6–2.5. For MSE with the same scale factor, there are different values of MSE due to different SNRs. Multi-scale sample entropy measures the complexity of the signal, the more noise in the signal, the sample entropy measures the complexity of the signal, the more noise in the signal, the greater the greater the MSE value. In Figure 6, when the scale factor is taken as 10, MSE values are arranged in MSE value. In Figure 6, when the scale factor is taken as 10, MSE values are arranged in the order the order that the noise content in the signal is more or less. There is no situation where the MSE that the noise content in the signal is more or less. There is no situation where the MSE value of the value of the low-noise signal is greater than the MSE of the high-noise signal, and the signal can be low-noise signal is greater than the MSE of the high-noise signal, and the signal can be distinguished. distinguished. Therefore, we choose the scale factor Q = 10 to calculate the MSE value of each layer Therefore, we choose the scale factor Q = 10 to calculate the MSE value of each layer of the IMF. of the IMF. Figure 5. Simulation signal decomposed by CEEMD. Appl. Sci. 2020, 10, x 9 of 19 Figure 5. Simulation signal decomposed by CEEMD. Figure 6. Scale factor Q. Figure 6. Scale factor Q. To eliminate the signal with larger MSE value in IMFs, it is necessary to determine the appropriate threshold. In this paper, the MSE values of 10,000 sets of noise signals (the signal length of each group of random noise is the same as the simulated signal) are calculated separately. The signal length of each group of random noise is the same as the vibration signal and the values are normalized, and the graph is drawn in Figure 7. The MSE value of the noise signal in Figure 7 varies from 0.55 to 1.0. In this paper, the minimum value of the MSE value in the 10,000 sets of noise signals is selected as the threshold for eliminating IMFs. 10,000 sets of random noise are randomly obtained, so the threshold in this paper is a dynamically changing value. After a lot of simulation, the minimum threshold fluctuates in a small range, so it is reasonable to choose the minimum value of 10,000 groups of noise as the threshold in this paper. Figure 7. Multi-scale sample entropy (MSE) values of 10,000 sets of random noise. In this paper, the scale factor Q = 10 is used to calculate the MSE values of IMFs and 10,000 sets of random noise, at the same time, the minimum value of 10,000 sets of random noise is used as the threshold, and Figure 8 is drawn. In Figure 8, the pink line is the minimum value of 10,000 groups of random noise, and the red bar is the MSE value of each IMF. In this paper, IMFs signals larger than the threshold are eliminated, so the IMF1–IMF4 signals are eliminated. Reconstruct the remaining IMF5–IMF10 signals and perform FFT transformation to obtain Figure 9. In Figure 9, the red line is the reconstructed signal, and the blue is the original signal. It can be seen that the high-frequency noise part of the signal is removed from 400 Hz to 1000 Hz, which achieves the purpose of noise reduction. Appl. Sci. 2020, 10, x 9 of 19 Appl. Sci. 2020, 10, 5078 9 of 18 Figure 6. Scale factor Q. To eliminate the signal with larger MSE value in IMFs, it is necessary to determine the appropriate To eliminate the signal with larger MSE value in IMFs, it is necessary to determine the threshold. In this paper, the MSE values of 10,000 sets of noise signals (the signal length of each group appropriate threshold. In this paper, the MSE values of 10,000 sets of noise signals (the signal length of random noise is the same as the simulated signal) are calculated separately. The signal length of each of each group of random noise is the same as the simulated signal) are calculated separately. The group of random noise is the same as the vibration signal and the values are normalized, and the graph signal length of each group of random noise is the same as the vibration signal and the values are is drawn in Figure 7. The MSE value of the noise signal in Figure 7 varies from 0.55 to 1.0. In this paper, normalized, and the graph is drawn in Figure 7. The MSE value of the noise signal in Figure 7 the minimum value of the MSE value in the 10,000 sets of noise signals is selected as the threshold for varies from 0.55 to 1.0. In this paper, the minimum value of the MSE value in the 10,000 sets of noise eliminating IMFs. 10,000 sets of random noise are randomly obtained, so the threshold in this paper is signals is selected as the threshold for eliminating IMFs. 10,000 sets of random noise are randomly a dynamically changing value. After a lot of simulation, the minimum threshold fluctuates in a small obtained, so the threshold in this paper is a dynamically changing value. After a lot of simulation, range, the mi soni itmum is reasonable threshold fl to uctua choose tes i the n a sma minimum ll range, value so itof is re 10,000 asona gr ble t oups o choose of noise the min as the imum threshold value in of 10,000 groups of noise as the threshold in this paper. this paper. Figure 7. Multi-scale sample entropy (MSE) values of 10,000 sets of random noise. Figure 7. Multi-scale sample entropy (MSE) values of 10,000 sets of random noise. In this paper, the scale factor Q = 10 is used to calculate the MSE values of IMFs and 10,000 sets In this paper, the scale factor Q = 10 is used to calculate the MSE values of IMFs and 10,000 sets of random noise, at the same time, the minimum value of 10,000 sets of random noise is used as the of random noise, at the same time, the minimum value of 10,000 sets of random noise is used as the threshold, and Figure 8 is drawn. In Figure 8, the pink line is the minimum value of 10,000 groups threshold, and Figure 8 is drawn. In Figure 8, the pink line is the minimum value of 10,000 groups of of random noise, and the red bar is the MSE value of each IMF. In this paper, IMFs signals larger random noise, and the red bar is the MSE value of each IMF. In this paper, IMFs signals larger than than the threshold are eliminated, so the IMF1–IMF4 signals are eliminated. Reconstruct the the threshold are eliminated, so the IMF1–IMF4 signals are eliminated. Reconstruct the remaining remaining IMF5–IMF10 signals and perform FFT transformation to obtain Figure 9. In Figure 9, the IMF5–IMF10 signals and perform FFT transformation to obtain Figure 9. In Figure 9, the red line is the red line is the reconstructed signal, and the blue is the original signal. It can be seen that the reconstructed signal, and the blue is the original signal. It can be seen that the high-frequency noise high-frequency noise part of the signal is removed from 400 Hz to 1000 Hz, which achieves the part of Appl. the Sci. signal 2020, 10is , x removed from 400 Hz to 1000 Hz, which achieves the purpose of noise 10 of reduction. 19 purpose of noise reduction. Figure 8. MSE of each IMF component. Figure 8. MSE of each IMF component. According to the method mentioned in this article, the reconstructed signal needs to be decomposed again by the CEEMD method, and the MSE value of each component signal is calculated. The MSE value is compared with the dynamic threshold value, and if there is an IMFs component signal greater than the threshold, it is removed, and the remaining signal is reconstructed and CEEMD decomposition is performed again. After multiple iterations, it is finally obtained that there is no case where the MSE value of the component signal is greater than the threshold, so the iteration is terminated. The time and frequency domain diagrams of signal decomposition at the end of the iteration are shown in Figure 10. The fault frequencies of 50 Hz and 130 Hz of the simulated signal in Figure 10 were extracted, which verified the effectiveness of the proposed method. Finally, the relationship between the MSE value of each signal component and the dynamic threshold is shown in Figure 11. In Figure 11, the MSE values of IMF1–IMF10 are all smaller than the dynamic threshold. Figure 9. Comparison of the original signal and reconstructed signal. Appl. Sci. 2020, 10, x 10 of 19 Figure 8. MSE of each IMF component. According to the method mentioned in this article, the reconstructed signal needs to be decomposed again by the CEEMD method, and the MSE value of each component signal is calculated. The MSE value is compared with the dynamic threshold value, and if there is an IMFs component signal greater than the threshold, it is removed, and the remaining signal is reconstructed and CEEMD decomposition is performed again. After multiple iterations, it is finally obtained that there is no case where the MSE value of the component signal is greater than the threshold, so the iteration is terminated. The time and frequency domain diagrams of signal decomposition at the end of the iteration are shown in Figure 10. The fault frequencies of 50 Hz and 130 Hz of the simulated signal in Figure 10 were extracted, which verified the effectiveness of the proposed method. Finally, the relationship between the MSE value of each signal component and Appl. Sci. 2020 the dyna , 10, 5078 mic threshold is shown in Figure 11. In Figure 11, the MSE values of IMF1–IMF10 are all 10 of 18 smaller than the dynamic threshold. Figure 9. Comparison of the original signal and reconstructed signal. Figure 9. Comparison of the original signal and reconstructed signal. According to the method mentioned in this article, the reconstructed signal needs to be decomposed again by the CEEMD method, and the MSE value of each component signal is calculated. The MSE value is compared with the dynamic threshold value, and if there is an IMFs component signal greater than the threshold, it is removed, and the remaining signal is reconstructed and CEEMD decomposition is performed again. After multiple iterations, it is finally obtained that there is no case where the MSE value of the component signal is greater than the threshold, so the iteration is terminated. The time and frequency domain diagrams of signal decomposition at the end of the iteration are shown in Figure 10. The fault frequencies of 50 Hz and 130 Hz of the simulated signal in Figure 10 were extracted, which verified the e ectiveness of the proposed method. Finally, the relationship between the MSE value of each signal component and the dynamic threshold is shown in Figure 11. In Figure 11, the MSE values of IMF1–IMF10 are all smaller than the dynamic threshold. Appl. Sci. 2020, 10, x 11 of 19 Figure 10. CEEMD decomposition reconstructed signals. Figure 10. CEEMD decomposition reconstructed signals. Figure 11. MSE value of the component signal (reconstructed signal is decomposed by CEEMD). 4. Experimental Verification To verify the feasibility of the proposed method in engineering application, it is planned to use the gearbox fault test bench for experiments. The experiment bench is shown in Figure 12, which includes a triple-phase asynchronous motor, connector, gear speed increaser, torque and rotational speed sensor, planetary reducer, triaxial acceleration sensor, and magnetic powder loader. The Appl. Sci. 2020, 10, x 11 of 19 Appl. Sci. 2020, 10, 5078 11 of 18 Figure 10. CEEMD decomposition reconstructed signals. Figure 11. MSE value of the component signal (reconstructed signal is decomposed by CEEMD). Figure 11. MSE value of the component signal (reconstructed signal is decomposed by CEEMD). 4. Experimental Verification 4. Experimental Verification To verify the feasibility of the proposed method in engineering application, it is planned to use To verify the feasibility of the proposed method in engineering application, it is planned to use the the gearbox fault test bench for experiments. The experiment bench is shown in Figure 12, which gearbox fault test bench for experiments. The experiment bench is shown in Figure 12, which includes includes a triple-phase asynchronous motor, connector, gear speed increaser, torque and rotational Appl. Sci. 2020, 10, x 12 of 19 a triple-phase asynchronous motor, connector, gear speed increaser, torque and rotational speed sensor, speed sensor, planetary reducer, triaxial acceleration sensor, and magnetic powder loader. The planetary reducer, triaxial acceleration sensor, and magnetic powder loader. The magnetic powder magnetic powder loader is used to adjust the load. The triaxial acceleration sensor model is loader is used to adjust the load. The triaxial acceleration sensor model is YD77SA, the sensitivity is -2 YD77SA, the sensitivity is 0.01 V/ms , and the sensor placement is shown in Figure 12. -2 0.01 V/ms , and the sensor placement is shown in Figure 12. During the test, the original bearing was replaced by a bearing with cracks in the inner ring During the test, the original bearing was replaced by a bearing with cracks in the inner ring and and flaking of the inner ring. The bearing types are NJ210 and NJ405, as shown in Figure 13. The flaking of the inner ring. The bearing types are NJ210 and NJ405, as shown in Figure 13. The bearing bearing speed is 1500 rmp. According to the calculation formula of the bearing inner ring fault speed is 1500 rmp. According to the calculation formula of the bearing inner ring fault frequency, frequency, the fault frequencies of NJ405 and NJ210 are 147.66 Hz and 231 Hz, respectively, as the fault frequencies of NJ405 and NJ210 are 147.66 Hz and 231 Hz, respectively, as shown in Table 1. shown in Table 1. Figure Figure 12 12. . Experiment Experiment bench, 1—Magnetic Powder Load bench, 1—Magnetic Powder Loader; er; 2—Gear 2—Gear speed Incr speed easer; 3—T Increaser; orque and 3—Torqu e Rotational Speed Sensor; 4—Planetary Reducer; 5—Triaxial Acceleration Sensor #1; 6—Triaxial and Rotational Speed Sensor; 4—Planetary Reducer; 5—Triaxial Acceleration Sensor #1; 6—Triaxial Acceleration Sensor #2; and 7—Triple-phase Asynchronous Motor. Acceleration Sensor #2; and 7—Triple-phase Asynchronous Motor. (a) (b) Figure 13. Faulty bearing. (a) NJ405 bearing. (b) NJ210 bearing. The time and frequency domain of vibration signals in the experiment are shown in Figure 14. In Figure 14a, the periodic impact of the collected vibration signal is not very obvious. Figure 14b is the frequency domain diagram obtained by the FFT transformation of the vibration signal. It can be seen that the fault period is submerged by noise and it failed to judge the existence of the fault. The vibration signals are processed by CEEMD and the method proposed in this article, and the effects of each method are compared. Appl. Sci. 2020, 10, x 12 of 19 magnetic powder loader is used to adjust the load. The triaxial acceleration sensor model is -2 YD77SA, the sensitivity is 0.01 V/ms , and the sensor placement is shown in Figure 12. During the test, the original bearing was replaced by a bearing with cracks in the inner ring and flaking of the inner ring. The bearing types are NJ210 and NJ405, as shown in Figure 13. The bearing speed is 1500 rmp. According to the calculation formula of the bearing inner ring fault frequency, the fault frequencies of NJ405 and NJ210 are 147.66 Hz and 231 Hz, respectively, as shown in Table 1. Figure 12. Experiment bench, 1—Magnetic Powder Loader; 2—Gear speed Increaser; 3—Torque and Appl. Sci.Rotational Spe 2020, 10, 5078 ed Sensor; 4—Planetary Reducer; 5—Triaxial Acceleration Sensor #1; 6—Triaxial 12 of 18 Acceleration Sensor #2; and 7—Triple-phase Asynchronous Motor. (a) (b) Figure 13. Figure 13.Faul Faulty ty beari bearing. ng. (a()a NJ ) NJ405 405 bbearing. earing. (b (b ) NJ ) NJ210 210 b bearing. earing. The time and frequency domain of vibration signals in the experiment are shown in Figure 14. The time and frequency domain of vibration signals in the experiment are shown in Figure 14. In In Fi Figur gur e 14 e 1 a, 4athe , the per periodic iodic impact impact o of f t the he collect collected ed vibr vibration ation signa signal l is not is not very very obvious obvious. . FiguFigur re 14b e i14 s b is the the f fr requency equency domai domain n dia diagram gram obtai obtained ned by the FFT by the FFT transformation transformation of the v of the ibrvibration ation signal. It c signal. an b It can e seen that the fault period is submerged by noise and it failed to judge the existence of the fault. The be seen that the fault period is submerged by noise and it failed to judge the existence of the fault. vibration signals are processed by CEEMD and the method proposed in this article, and the effects The vibration signals are processed by CEEMD and the method proposed in this article, and the e ects of each method are compared. of each method are compared. Appl. Sci. 2020, 10, x 13 of 19 (a) (b) Figure 14. Time-domain and frequency-domain graphs of experimentally acquired signals. (a) Time Figure 14. Time-domain and frequency-domain graphs of experimentally acquired signals. (a) Time domain. (b) Frequency domain. domain. (b) Frequency domain. Table 1. Failure frequency. Table 1. Failure frequency. Rotating Speed n NJ405 NJ210 Rotating Speed n NJ405 NJ210 1500 rpm 147.66 Hz 231.2 Hz 1500 rpm 147.66 Hz 231.2 Hz Figure 15a is the time domain diagram of the vibration signal decomposed by CEEMD, and Figure 15a is the time domain diagram of the vibration signal decomposed by CEEMD, and Figure 15b Figure 15b is the frequency domain diagram of the vibration signal obtained by FFT. The signal in is the frequency domain diagram of the vibration signal obtained by FFT. The signal in the figure is the figure is decomposed into 15 layers in total, and from the time domain diagram and frequency decomposed into 15 layers in total, and from the time domain diagram and frequency domain diagram in domain diagram in the corresponding diagram, we can observe that the component signals, the corresponding diagram, we can observe that the component signals, IMF1–IMF5, are high-frequency IMF1–IMF5, are high-frequency noise signals. Bearing failure signals are not obtained in the IMF6–IMF15 component signals, so the vibration signal fails to extract the bearing failure signal through traditional CEEMD decomposition. Employing the method proposed in this paper to decompose the vibration signal. The threshold needs to be determined, so this paper uses the scale factor Q = 10 to calculate the MSE value of 10,000 sets of random noise. The signal length of each set of random noise is consistent with the vibration signal. The result is shown in Figure 16. The MSE value of the noise signal in Figure 16 varies from 0.705 to 0.745. In this paper, the scale factor Q = 10 is used to calculate the MSE values of IMFs and 10,000 sets of random noise, and the minimum value of 10,000 sets of random noise is used as the threshold, and Figure 17 is drawn. The pink line in Figure 17 is the minimum value of 10,000 sets of random noise, and the red bar is the MSE value of each IMF. In this paper, IMFs signals larger than the threshold are eliminated, so the IMF1–IMF3 signals in the figure below are eliminated. Reconstruct the remaining IMF4–IMF15 signals and perform FFT transformation to obtain Figure 18. In Figure 18, red is the reconstructed signal and blue is the original signal. It can be seen that the high-frequency noise part of the signal from 3000 Hz to 5000 Hz has been removed, which achieves the purpose of noise reduction. In Figure 18, 148 Hz is closest to the fault signal 147.66 Hz; 230.6 Hz is closest to the fault signal 231.2 Hz. Considering that the bearing operating environment has errors, which are allowed, so the method proposed in this article is effective. Appl. Sci. 2020, 10, 5078 13 of 18 noise signals. Bearing failure signals are not obtained in the IMF6–IMF15 component signals, so the vibration signal fails to extract the bearing failure signal through traditional CEEMD decomposition. Employing the method proposed in this paper to decompose the vibration signal. The threshold needs to be determined, so this paper uses the scale factor Q = 10 to calculate the MSE value of 10,000 sets of random noise. The signal length of each set of random noise is consistent with the vibration signal. The result is shown in Figure 16. The MSE value of the noise signal in Figure 16 varies from 0.705 to 0.745. In this paper, the scale factor Q = 10 is used to calculate the MSE values of IMFs and 10,000 sets of random noise, and the minimum value of 10,000 sets of random noise is used as the threshold, and Figure 17 is drawn. The pink line in Figure 17 is the minimum value of 10,000 sets of random noise, and the red bar is the MSE value of each IMF. In this paper, IMFs signals larger than the threshold are eliminated, so the IMF1–IMF3 signals in the figure below are eliminated. Reconstruct the remaining IMF4–IMF15 signals and perform FFT transformation to obtain Figure 18. In Figure 18, red is the reconstructed signal and blue is the original signal. It can be seen that the high-frequency noise part of the signal from 3000 Hz to 5000 Hz has been removed, which achieves the purpose of noise reduction. In Figure 18, 148 Hz is closest to the fault signal 147.66 Hz; 230.6 Hz is closest to the fault signal 231.2 Hz. Considering that the bearing operating environment has errors, which are allowed, so the method proposed in this article is e ective. Appl. Sci. 2020, 10, x 14 of 19 (a) (b) Figure 15. CEEMD decomposes vibration signals. (a) Time domain. (b) Frequency domain. Figure 15. CEEMD decomposes vibration signals. (a) Time domain. (b) Frequency domain. Figure 16. MSE value of 10,000 sets of random noise signals (the signal length is consistent with the vibration signal). Appl. Sci. 2020, 10, x 14 of 19 (a) (b) Appl. Sci. 2020, 10, 5078 14 of 18 Figure 15. CEEMD decomposes vibration signals. (a) Time domain. (b) Frequency domain. Figure 16. MSE value of 10,000 sets of random noise signals (the signal length is consistent with the Figure 16. MSE value of 10,000 sets of random noise signals (the signal length is consistent with the vibration signal). Appl. Sci. 2020, 10, x 15 of 19 Appl. vibration Sci. 2020signal). , 10, x 15 of 19 Figure 17. MSE value of each component of vibration signal decomposed by CEEMD. Figure 17. MSE value of each component of vibration signal decomposed by CEEMD. Figure 17. MSE value of each component of vibration signal decomposed by CEEMD. Figure 18. Vibration signal reconstruction. Figure Figure 18. 18. V Vibration signal ibration signal r reconstruction. econstruction. In In Figur Figure 18 e 18, , the the failur failure e frfre equency quency o offthe the recon reconstr structed si ucted signal gnal iis s not ob not obvious. vious. Accordi According ng to the to the In Figure 18, the failure frequency of the reconstructed signal is not obvious. According to the method proposed in this paper, the reconstructed signal needs to be decomposed by the CEEMD method method proposed proposed inin this this paper paper, the reconstructe , the reconstructed d signal ne signal needs eds tto o be decomp be decomposed osed by the by the CEEMD CEEMD method, and the MSE value of each component signal be calculated. The MSE value is compared method, and the MSE value of each component signal be calculated. The MSE value is compared with the dynamic threshold value, and if there is an IMFs component signal greater than the with the dynamic threshold value, and if there is an IMFs component signal greater than the threshold, it is removed, the remaining signal is reconstructed, and CEEMD decomposition is threshold, it is removed, the remaining signal is reconstructed, and CEEMD decomposition is performed again. After multiple iterations, if it is finally obtained that there is no case where the performed again. After multiple iterations, if it is finally obtained that there is no case where the MSE value of the component signal is greater than the threshold, then the iteration is terminated. MSE value of the component signal is greater than the threshold, then the iteration is terminated. The time and frequency domain diagrams of signal decomposition at the end of the iteration are The time and frequency domain diagrams of signal decomposition at the end of the iteration are shown in Figure 19. In Figure 19, the reconstructed vibration signal is decomposed into 14 layers, of shown in Figure 19. In Figure 19, the reconstructed vibration signal is decomposed into 14 layers, of which IMF1–IMF5 are high-frequency noise signals; IMF6 is the failure frequency of the inner ring which IMF1–IMF5 are high-frequency noise signals; IMF6 is the failure frequency of the inner ring of NJ210 bearings; IMF7 is the failure frequency of the inner ring of NJ405 bearings; IMF8–IMF14 of NJ210 bearings; IMF7 is the failure frequency of the inner ring of NJ405 bearings; IMF8–IMF14 are meaningless interference components. Therefore, fault features are extracted, which verifies the are meaningless interference components. Therefore, fault features are extracted, which verifies the effectiveness of the proposed method. The relationship between the multi-scale sample entropy of effectiveness of the proposed method. The relationship between the multi-scale sample entropy of each signal component and the dynamic threshold is shown in Figure 20. In Figure 20, the MSE each signal component and the dynamic threshold is shown in Figure 20. In Figure 20, the MSE values of IMF1–IMF14 are all smaller than the dynamic threshold. values of IMF1–IMF14 are all smaller than the dynamic threshold. Appl. Sci. 2020, 10, 5078 15 of 18 method, and the MSE value of each component signal be calculated. The MSE value is compared with the dynamic threshold value, and if there is an IMFs component signal greater than the threshold, it is removed, the remaining signal is reconstructed, and CEEMD decomposition is performed again. After multiple iterations, if it is finally obtained that there is no case where the MSE value of the component signal is greater than the threshold, then the iteration is terminated. The time and frequency domain diagrams of signal decomposition at the end of the iteration are shown in Figure 19. In Figure 19, the reconstructed vibration signal is decomposed into 14 layers, of which IMF1–IMF5 are high-frequency noise signals; IMF6 is the failure frequency of the inner ring of NJ210 bearings; IMF7 is the failure frequency of the inner ring of NJ405 bearings; IMF8–IMF14 are meaningless interference components. Therefore, fault features are extracted, which verifies the e ectiveness of the proposed method. The relationship between the multi-scale sample entropy of each signal component and the dynamic threshold is shown in Figure 20. In Figure 20, the MSE values of IMF1–IMF14 are all smaller than the dynamic threshold. Appl. Sci. 2020, 10, x 16 of 19 (a) (b) Figure Figure 19. 19. TheThe time time and and frequency frequenc domain y domai of n of rec reconstr ons ucted tructed s signal ignal by by CEEMD CEEMD d decomposition. ecomposition. (a ()aT ) ime domain. (b) Frequency domain. Time domain. (b) Frequency domain. Appl. Sci. 2020, 10, x 16 of 19 (a) (b) Figure 19. The time and frequency domain of reconstructed signal by CEEMD decomposition. (a) Appl. Sci. 2020, 10, 5078 16 of 18 Time domain. (b) Frequency domain. Figure 20. MSE value of the component signal (reconstructed signal is decomposed by CEEMD). 5. Conclusions In this paper, the CEEMD method is used to adaptively reduce the noise of complex vibration signals, but the pseudo components after CEEMD decomposition seriously reduce the diagnostic eciency. The article proposes a new method of adaptively removing noise components, applying the proposed method to the composite fault simulation signal and the vibration signal of mining machinery, accurately extracting the composite fault characteristics, and verifying the feasibility of the method by comparing it with the traditional CEEMD. Finally, the following conclusions are obtained: (1) Determine the best scale factor Q = 10 in the MSE by simulation signals with di erent SNR, and then employ it to calculate the MSE values of noise signals, simulation signals, and vibration signals, which provide support for optimizing the CEEMD method. (2) Calculate the MSE value of 10,000 sets of random noise, and use the minimum value as the threshold, remove the IMFs component signal greater than the threshold, achieve the purpose of CEEMD noise reduction, and verify the feasibility of the minimum value as the threshold. (3) Through CEEMD decomposition of the reconstructed signal, the MSE value of each component signal is calculated. The MSE value is compared with the dynamic threshold value, and if there is an IMFs component signal greater than the threshold, it is removed, the remaining signal is reconstructed, and CEEMD decomposition is performed again. After multiple iterations, when it is finally shown that there is no case where the MSE value of the component signal is greater than the threshold, the iteration is terminated. Finally, the obtained signal is analyzed by the frequency spectrum to determine the fault characteristics. Author Contributions: W.G. conceived and designed the experiments; R.L. and J.Z. performed the experiments; and W.G. and Y.K. wrote the paper. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Key Research and Development Program of Shanxi Province (International Cooperation) (201903D421051, 201803D421028); Youth Fund Project of Shanxi Province (201901D211210). Conflicts of Interest: The authors declare no conflict of interest. Appl. Sci. 2020, 10, 5078 17 of 18 References 1. Xu, J.; Wang, Z.; Tan, C.; Si, L.; Zhang, L.; Liu, X. 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Costa, M.; Goldberger, A.L.; Peng, C.-K. Multiscale Entropy Analysis of Complex Physiologic Time Series. Phys. Rev. Lett. 2002, 89, 705–708. [CrossRef] © 2020 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/). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Sciences Multidisciplinary Digital Publishing Institute

Application Research of a New Adaptive Noise Reduction Method in Fault Diagnosis

Guo, Wenxiao; Li, Ruiqin; Kou, Yanfei; Zhang, Jianwei
Applied Sciences , Volume 10 (15) – Jul 23, 2020
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applied sciences Article Application Research of a New Adaptive Noise Reduction Method in Fault Diagnosis 1 1 , 1 2 Wenxiao Guo , Ruiqin Li *, Yanfei Kou and Jianwei Zhang School of Mechanical Engineering, North University of China, Taiyuan 030051, China; gwx041202@163.com (W.G.); kouyanfei@163.com (Y.K.) Department of Informatics, University of Hamburg, 22527 Hamburg, Germany; zhang@informatik.uni-hamburg.de * Correspondence: liruiqin@nuc.edu.cn Received: 18 June 2020; Accepted: 21 July 2020; Published: 23 July 2020 Abstract: The feature extraction of composite fault of gearbox in mining machinery has always been a diculty in the field of fault diagnosis. Especially in strong background noise, the frequency of each fault feature is di erent, so an adaptive time-frequency analysis method is urgently needed to extract di erent types of faults. Considering that the signal after complementary ensemble empirical mode decomposition (CEEMD) contains a lot of pseudo components, which further leads to misdiagnosis. The article proposes a new method for actively removing noise components. Firstly, the best scale factor of multi-scale sample entropy (MSE) is determined by signals with di erent signal to noise ratios (SNRs); secondly, the minimum value of a large number of random noise MSE is extracted and used as the threshold of CEEMD; then, the e ective Intrinsic mode functions(IMFs) component is reconstructed, and the reconstructed signal is CEEMD decomposed again; finally, after multiple iterations, the MSE values of the component signal that are less than the threshold are obtained, and the iteration is terminated. The proposed method is applied to the composite fault simulation signal and mining machinery vibration signal, and the composite fault feature is accurately extracted. Keywords: fault diagnosis; IMF; complementary ensemble empirical mode decomposition; multi-scale sample entropy 1. Introduction In industrial production, the normal operation of mining machinery is the guarantee for the economic growth of enterprises. The gear and bearing in the gearbox are the most important transmission parts. When one of them fails, complex mechanical behavior will appear. In the unbalanced mechanical environment, other parts will appear fatigued, and further develop into the weak fault and significant fault. At the same time, their vibration signals are coupled with each other. In the case of strong noise environments and complex transmission paths, weak fault noises are submerged, and further leakage or misdiagnosis occurs [1–5]. Mechanical intelligence fault diagnosis usually includes three links. (1) Signal acquisition: Obtain multi-physical monitoring signals radiated by mechanical equipment to reflect the health status of equipment. (2) Feature extraction: By analyzing the acquired monitoring data, extract features to reveal fault information. (3) Fault identification and prediction: Based on the extracted features, faults are identified and predicted through artificial intelligence models and methods. Over decades, with the development of computer science, a series of di erent time-frequency analysis methods have emerged. In 1998, empirical mode decomposition (EMD) was proposed, and then a large number of applications in signal processing [6–8], but it has endpoint e ects and mode aliasing phenomena. In 2005, ensemble empirical mode decomposition (EEMD) was proposed [9–11]; Appl. Sci. 2020, 10, 5078; doi:10.3390/app10155078 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 5078 2 of 18 it can not only overcome the endpoint e ects of EMD but also weaken the mode aliasing phenomenon, so it has been applied in industrial production by a large number of scholars. However, it needs to set the times of additions and the amplitude of white noise; so far, no formula can adaptively determine the white noise amplitude value. At the same time, the number of additions is also not adaptive. When the number is too large, the calculation amount is too large; otherwise, the noise cannot be e ectively averaged [12–14]. In order to improve the decomposition eciency, complementary ensemble empirical mode decomposition (CEEMD) was proposed, which weakens the e ect of the contained noise on the original signal by adding di erent positive and negative white noise multiple times [15], and is widely used in fault diagnosis [16–18]. CEEMD can improve the decomposition eciency of EEMD, but the noise part of the original signal still exists. In other words, the noise component of the original signal still exists in the multiple Intrinsic mode functions(IMFs) decomposed. If it is not removed in time, it will lead to leakage or misdiagnosis. So, it is necessary to propose an e ective method to eliminate noise components. In recent years, variable mode decomposition (VMD) [19–21] has been applied to fault diagnosis. It can decompose the original signal into several di erent time scale functions from low frequency to high frequency. The center frequency of each time scale is di erent, so its decomposition eciency is higher than the EEMD and CEEMD. However, the method is limited to the choice of the number of decomposition layers and the penalty factor. So far there is still no suitable method to adaptively determine these two parameters. Therefore, in order to adaptively reduce the noise of complex vibration signals, this article uses the CEEMD decomposition method. In strong background noise, each fault feature frequency is di erent [22–29], so an adaptive time-frequency analysis method is urgently needed to extract di erent types of faults. The article employs CEEMD to adaptively reduce the noise of complex vibration signals, but CEEMD has the problem of mode aliasing and the diculty of removing noise components. Considering that the multi-scale sample entropy (MSE) [30–32] can estimate the complexity of time series, accurately measuring the complexity of di erent modal functions with strong noise resistance ability and excellent consistency, and at the same time, a stable entropy value can be obtained by using fewer data segments. Therefore, the MSE is used as the basis for eliminating the noise component, but the accuracy of the MSE is related to the scale factor. So, a method is proposed to determine the best scale factor in the MSE by the simulation signal with di erent signal to noise ratio (SNR), and the best factor is placed in the multi-scale sample entropy to calculate the MSE value of random noise, simulation signal, and vibration signal. This is firstly done by calculating the multiscale sample entropy value of 10,000 sets of random noise, extracting the minimum value and using it as the threshold of the noise component in CEEMD; then, the e ective IMFs component is reconstructed, and the reconstructed signal is CEEMD decomposed again; this is followed by calculating the MSE value of each component signal, comparing the obtained MSE values with the threshold value, and if there are IMFs component signals greater than the threshold value, remove them, then reconstruct the remaining signal and perform CEEMD decomposition again; after multiple iterations, the MSE values of the component signal that are less than the threshold can be obtained, and the iteration is terminated. Finally, the resulting signal is analyzed by the spectrum to determine the fault characteristics. The proposed method is applied to the composite fault simulation signal and mining machinery vibration signal, and the composite fault feature is accurately extracted. At the same time, the feasibility of the method is verified by comparison with the traditional CEEMD. 2. Basic Theory 2.1. CEEMD To solve the mode aliasing problem caused by EMD, EEMD uses the characteristics that white noise can be evenly distributed in the spectrum. Adding white noise to EMD improves the mode aliasing phenomenon; however, the residual white noise also brings reconstruction error problems. ... ... ... ... ... ... ... ... ... Appl. Sci. 2020, 10, 5078 3 of 18 Inspired by EEMD, adding white noise to equalize the noise in the original signal, Yeh J R [15] proposed a new time-frequency analysis method CEEMD. The CEEMD method adds a pair of positive and negative white noise to the original signal for EMD decomposition to achieve the purpose of eliminating residual white noise in the reconstructed signal. In addition, the added white noise improves the mode aliasing problem, and the number of CEEMD noise addition is lower, which improves the calculation eciency. The CEEMD method is as follows: (1) Two new signals are formed by adding white noise with a certain standard deviation and equal length and opposite signs to the original signal. Appl. Sci. 2020, 10, x 3 of 19 S = S(t) + n +i +i (1) positive and negative white noise to the original signal for EMD decomposition to achieve the ( ) S = S t + n i i purpose of eliminating residual white noise in the reconstructed signal. In addition, the added white noise improves the mode aliasing problem, and the number of CEEMD noise addition is where S is the sum of the original signal and the positive white noise signal, S(t) is the original +i lower, which improves the calculation efficiency. signal, n is the positive white noise; S is the sum of the original signal and the negative white The CEEMD method is as follows: +i -i noise signal, and n is the negative white noise. (1) Two new sig -i nals are formed by adding white noise with a certain standard deviation and equal length and opposite signs to the original signal. (2) EMD decomposition of S and S to obtain IMF and IMF , then the first-order IMF is +i -i +(i,j) -(i,j) SS=+ ()t n ++ii (1) SS=+ ()t n 1−−ii IMF = IMF + IMF (2) 1 +(i,1) (i,1) 2N where S+i is the sum of the original signal and the positive white noise signal, S(t) is the i=1 original signal, n+i is the positive white noise; S-i is the sum of the original signal and the negative white noise signal, and n-i is the negative white noise. (3) Repeat the above steps to add Gaussian white noise 2N times. (2) EMD decomposition of S+i and S-i to obtain IMF+(i,j) and IMF-(i,j), then the first-order IMF is (4) Finally, the decomposition results of multiple component combinations are obtained: IMF=+ IMF IMF () (2) 1 +− ()ii ,1 () ,1 2 N i =1 (3) Repeat the above steps to add Gaussian white noise 2N times. IMF = IMF + IMF (3) +(i,j) (i,j) (4) Finally, the decomposition results of2 mN ultiple component combinations are obtained: i=1 IMF=+ IMF IMF () (3) j +− () ij,,() i j where N is the number of pairs added with white noise, IMF is the j-th component obtained by 2 N i =1 j CEEMD decomposition, and IMF is the i-th IMF component of the j-th signal. (i,j) where N is the number of pairs added with white noise, IMFj is the j-th component obtained by CEEMD decomposition, and IMF(i,j) is the i-th IMF component of the j-th signal. The decomposition process of CEEMD is shown in Figure 1. The decomposition process of CEEMD is shown in Figure 1. Original signal S(t) Add white Add white Add white Add white Add white Add white Add white Add white ... noise (+)n noise (-)n noise (+)n noise (-)n noise (+)n noise (-)n noise (+)n noise (-)n 1 1 2 2 N-1 N-1 N N S S S S ... S S S S +1 -1 +2 -2 +(N-1) -(N-1) +N -N EM EMD D Average ... value + + + + + + + IMF IMF IMF IMF IMF +11 IMF +21 IMF +(N-1)1 -(N-1)1 IMF IMF 1 -11 -21 +N1 -N1 Average + + + + + + + + ... value + + + + + + + IMF IMF IMF IMF IMF IMF IMF IMF IMF 2 +12 -12 +22 -22 +(N-1)2 -(N-1)2 +N2 -N2 Average + + + + + + + + value ... + + + + + + + IMF IMF IMF IMF IMF +13 IMF +23 IMF +(N-1)3 -(N-1)3 IMF IMF 3 -13 -23 +N3 -N3 + + + + + + + + ... + + + + + + + Average + + + + + + + + ... value + + + + + + + IMF IMF IMF IMF IMF IMF IMF IMF +1N -1N +2N -2N +(N-1)N -(N-1)N +NN -NN IMF Average + + + + + + + + ... value + + + + + + + r r r r +1N r +2N r +(N-1)N -(N-1)N r r -1N -2N +NN -NN r Figure 1. Complementary ensemble empirical mode decomposition (CEEMD) flowchart. Figure 1. Complementary ensemble empirical mode decomposition (CEEMD) flowchart. .... Appl. Sci. 2020, 10, 5078 4 of 18 2.2. Multi-Scale Sample Entropy (MSE) Fault signals detected in rotating machinery are usually embedded in multi-scale structures. However, the sample entropy (SE) [33,34] only analyzes the single scale signal and ignores much useful information. This limits its performance in extracting embedded fault features [35]. In order to solve the shortcomings of sample entropy, Costa [36] proposed multi-scale sample entropy as an e ective complexity measure of time series on di erent time scales. For a time series {x , x , ::: , 1 2 x , x }, MSE constructs multiple coarse-graining time series. The process of coarse-graining scale N-1 N factor Q is obtained by averaging the internal structure of the time series, but without overlapping the length of the scale factor. Figure 2 shows the coarse-graining examples of processes. Therefore, the coarse-graining time series can be obtained from Equation (4): jQ 1 N y = x , 1  j  (4) Q Q i=(j1)Q+1 where Q is the scale factor and is a positive integer. When Q = 1, the coarse-graining time series y is the original time series. j is the length of the coarse-graining time series. Appl. Sci. 2020, 10, x 5 of 19 Scale factor Q=1 x x x x x x x x x 1 2 3 4 5 6 i-1 i i+1 ▲ ▲ ▲ ▲ ▲ ▲ ¨¨¨ ▲ ▲ ▲ Scale factor Q=2 x x x x x x x x x 1 2 3 4 5 6 i-1 i i+1 ▲ ▲ ▲ ▲ ▲ ▲ ¨¨¨ ▲ ▲ ▲ \ ∕ \ ∕ \ ∕ \ ∕ ◆ ◆ ◆ ◆ (2) (2) (2) (2) y y y y =(x +x )/2 1 2 3 i/2 i-1 i Scale factor Q=3 x x x x x x x x x 1 2 3 4 5 6 i-1 i i+1 ▲ ▲ ▲ ▲ ▲ ▲ ¨¨¨ ▲ ▲ ▲ \ | ∕ \ | ∕ \ | ∕ ★ ★ ★ (3) (3) (3) y y y =(x +x +x /3 1 2 i/3 i-1 i i+1) Scale factor Q=n Figure 2. Coarse graining process. Figure 2. Coarse graining process. (5) Then calculate the probability of matching points The next step is to calculate the multi-scale sample entropy and scale factor Q for each newly reconstructed coarse-graining time-series signal y . NQ The −+ m mu 1 lti-scale sample entropy is calculated mm Br = B r () ()  (9) as follows: i NQ−+ m 1 i =1 (Q) (1) From N/Q-m+1 to vector y (i) NQ −m mm (10) ArA = r () ()  i h i NQ − m (Q) Q Q Q i =1 y (i) = y (i), y (i + 1), , y (i + m 1) , i = 1, , N/Q m + 1 (5) m m where B (r) and A (r) represent the probability of two sequences matching m and m + 1 points, where m is the length of the sequence to be compared. respectively. (6) The theoretical value of multiscale sample entropy is defined as  Ar()  SampEn() m,l r=− im ln  (11) NQ →∞ Br ()   When the data point N/Q is a finite number, the estimate of sample entropy is given by the following formula:  Ar () SampEn m,, r N =−ln . ()  (12) Br()   It can be seen from Equation (12) that the value of SampEn is related to the value of m,r. At present, there is no definite value, and the empirical value is m = 2, rÎ(0.1SD~0.5SD). SD is the standard deviation of the original data. In this paper, when studying the complexity of time series, m = 1, r = 0.25SD is taken. Appl. Sci. 2020, 10, 5078 5 of 18 (Q) (Q) (2) Calculate the maximum norm between two vectors d [y (i), y (j)]. m m h i Q Q Q Q d y (i), y (j) = max y (i + k) y (j + k) , 0  k  m 1 (6) m m m m (3) Define function B m m B (r) = u (i), i = 1, , N/Q m + 1 (7) N/Q m + 1 (Q) (Q) (Q) where r is the tolerance of the acceptance matrix, u (i) is equal to d [y (i), y (j)], and d [y (i), m m m m m (Q) y (j)] r, i , j. (4) Define function A m m+1 A (r) =  (i), i = 1, , N/Q m + 1 (8) N/Q m + 1 (Q) (Q) (Q) (Q) m+1 where  is equal to d [y (i), y (j)], and d [y (i), y (j)] r, i , j. m+1 m+1 m+1 m+1 m+1 m+1 (5) Then calculate the probability of matching points N/Qm+1 m m B (r) = B (r) (9) N/Q m + 1 i=1 N/Qm m m A (r) = A (r) (10) N/Q m i=1 m m where B (r) and A (r) represent the probability of two sequences matching m and m + 1 points, respectively. (6) The theoretical value of multiscale sample entropy is defined as ( ) A (r) ( ) SampEn m, r = lim ln . (11) N/Q!1 B (r) When the data point N/Q is a finite number, the estimate of sample entropy is given by the following formula: " # A (r) SampEn(m, r, N) = ln . (12) B (r) It can be seen from Equation (12) that the value of SampEn is related to the value of m,r. At present, there is no definite value, and the empirical value is m = 2, r2 (0.1SD~0.5SD). SD is the standard deviation of the original data. In this paper, when studying the complexity of time series, m = 1, r = 0.25SD is taken. 2.3. Optimized CEEMD Method CEEMD is an improved method based on the EMD method to add a pair of positive and negative white noise; there is also a mode aliasing phenomenon, and there is still noise in the decomposed IMF component. In order to eliminate the influence of noise on the decomposition results, this paper uses multi-scale sample entropy to remove the IMF with more noise components, reconstruct the signal, and perform the Fast Fourier transform(FFT) transform to obtain the fault feature signal to achieve the purpose of optimizing CEEMD. The flow chart is shown in Figure 3, the method is as follows: Appl. Sci. 2020, 10, x 6 of 19 2.3. Optimized CEEMD Method CEEMD is an improved method based on the EMD method to add a pair of positive and negative white noise; there is also a mode aliasing phenomenon, and there is still noise in the decomposed IMF component. In order to eliminate the influence of noise on the decomposition results, this paper uses multi-scale sample entropy to remove the IMF with more noise components, Appl. Sci. 2020, 10, 5078 6 of 18 reconstruct the signal, and perform the Fast Fourier transform(FFT) transform to obtain the fault feature signal to achieve the purpose of optimizing CEEMD. The flow chart is shown in Figure 3, the method is as follows: (1) Employing the CEEMD method to decompose the original signal X(t) containing noise to obtain a series of IMFs. (1) Employing the CEEMD method to decompose the original signal X(t) containing noise to (2) The selection of the scale factor Q of the MSE will a ect the multiscale sample entropy value of obtain a series of IMFs. each IMF. According to the decomposition characteristics of the IMF, the noise content of each (2) The selection of the scale factor Q of the MSE will affect the multiscale sample entropy value of layer is di erent. In order to better distinguish each layer of signals, this paper chooses a scale each IMF. According to the decomposition characteristics of the IMF, the noise content of each factor layer is differ of 1–20 and ent. In order to better distinguish ea signals with di erent SNRs forch simulation layer of signals, this pape to determine ther best chooses scale a scale factor Q. factor of 1--20 and signals with different SNRs for simulation to determine the best scale factor (3) Substitute m = 2, r = 0.25SD, and the Q that was determined in step (2) into the MSE, calculate the Q. multiscale sample entropy value of 10,000 sets of random noise signals, and obtain the minimum (3) Substitute m = 2, r = 0.25SD, and the Q that was determined in step (2) into the MSE, calculate value K of the noise multiscale sample entropy. the multiscale sample entropy value of 10,000 sets of random noise signals, and obtain the (4) Calculate the multi-scale sample entropy value of each IMF component, compare the obtained minimum value K of the noise multiscale sample entropy. value with K, remove components greater than K value, and reconstruct the remaining (4) Calculate the multi-scale sample entropy value of each IMF component, compare the obtained IMFs components. value with K, remove components greater than K value, and reconstruct the remaining IMFs (5) Perform CEEMD decomposition on the reconstructed signal again, and then calculate the components. multiscale sample entropy value of each IMF component. (5) Perform CEEMD decomposition on the reconstructed signal again, and then calculate the (6) Compare the MSE value of each IMF component with the minimum value K, if there is more than multiscale sample entropy value of each IMF component. K worth of IMF, repeat (2–6); otherwise, continue. (6) Compare the MSE value of each IMF component with the minimum value K, if there is more (7) The than signal K worth is FFT of IMF transformed , repeat (2–6); to obtain otherw the ise,fault continu signal. e. (7) The signal is FFT transformed to obtain the fault signal. Input the Feature Obtaining scale factor original signal extraction Q by signal simulation with noise X(t) with different SNR Remove IMFs greater than K, and CEEMD MSE reconstruct the decomposition Yes remaining IMFs No Calculate the MSEs value of 10000 sets of Obtain physically random noise signals, MSE of IMFs less meaningful IMFs and determine the than K component signals minimum value K Compare the MSEs of Calculate the Calculate the IMFs and K value, if it MSE of each MSE of each is less than K, it is the IMF component IMF component sensitive component Reconstruct CEEMD sensitive decomposition components IMFs Figure 3. Optimized CEEMD method flow chart. Figure 3. Optimized CEEMD method flow chart. 3. Simulation 3.1. Construct a Simulation Signal The collected gearbox signal appears in the form of a mixture of the modulation signal of the gear mesh and the impact signal of the bearing rotation, and the signal contains noise in the working environment. As a result, this paper establishes a mathematical model containing a mixture of impact Appl. Sci. 2020, 10, x 7 of 19 3. Simulation 3.1. Construct a Simulation Signal The collected gearbox signal appears in the form of a mixture of the modulation signal of the gear mesh and the impact signal of the bearing rotation, and the signal contains noise in the Appl. working Sci. 2020,environment 10, 5078 . As a result, this paper establishes a mathematical model containing a 7 of 18 mixture of impact signal, modulation signal, and noise signal for signal simulation. The mathematical model established is as follows: signal, modulation signal, and noise signal for signal simulation. The mathematical model established is as follows: x ()tA=× exp − sin() 2πft () 1 mc  T ( ) ( ) > x t = A  exp sin 2 f t 1 m c > T  m > xtf =+ 1cos 2ππt sin 2ft  ()() ( ) () 21nz (13) < x (t) = (1 + cos(2 f t)) sin(2 f t) 2 n1 (13) > noise() t = 0.5randn() t noise(t) = 0.5randn(t) >  Xt =+ x t x t+noise t () () () () X(t) = x (12 t) + x (t) + noise(t) 1 2 where A = 3.0, g = 0.01, T = 1/13, f = 130 Hz, f = 10 Hz, and f = 50 Hz. randn(t) is random noise. where mAm = 3.0, g = 0.01, Tm m = 1/13, fc = 130 Hz, c fn1 = 10 Hz, and fz = 50 Hz. z randn(t) is random noise. The n1 The number of s number ofasampling mpling point points s is is 101024 24 and and ththe e sampl sampling ing frefr quenc equency y is 2048 H is 2048zHz. . Figure Figur 4 is the time e 4 is the time domain diagram and frequency domain diagram of the simulation signal, where the frequency domain diagram and frequency domain diagram of the simulation signal, where the frequency domain domain diagram is obtained by performing FFT transformation on the time domain signal. diagram is obtained by performing FFT transformation on the time domain signal. (a) (b) Figure 4. Simulation Figure 4. Si signal. mulation si (a) Tgnal ime .domain (a) Time d of simulation omain of simul signal, ation (b si )gnal Frequency , (b) Frequenc domain y d of omai simulation n of signal. simulation signal. Among them, X1 is the impact signal of bearing rotation, X2 is the modulation signal of gear mesh, and X(t) is the mixed signal of impact signal, modulation signal, and random noise signal. It can be observed from the figure that the X(t) signal contains a lot of noise signals. 3.2. Optimized CEEMD Method The simulation signal X(t) is decomposed by CEEMD, and the decomposed time and frequency domain diagram is shown in Figure 5. Impact signals can be observed in the time-domain diagram. The signal in the frequency-domain diagram still has mode aliasing and contains some noise, which will a ect the extraction of fault feature signals. Therefore, this paper proposes to use MSE to optimize the decomposition of CEEMD. Appl. Sci. 2020, 10, x 8 of 19 Among them, X1 is the impact signal of bearing rotation, X2 is the modulation signal of gear mesh, and X(t) is the mixed signal of impact signal, modulation signal, and random noise signal. It can be observed from the figure that the X(t) signal contains a lot of noise signals. 3.2. Optimized CEEMD Method The simulation signal X(t) is decomposed by CEEMD, and the decomposed time and Appl. Sci. 2020, 10, 5078 8 of 18 frequency domain diagram is shown in Figure 5. Impact signals can be observed in the time-domain diagram. The signal in the frequency-domain diagram still has mode aliasing and contains some noise, which will affect the extraction of fault feature signals. Therefore, this paper The scale factor Q of the multi-scale sample entropy will a ect the MSE value, and di erent proposes to use MSE to optimize the decomposition of CEEMD. Q values have di erent MSE values. According to the decomposition characteristics of CEEMD, The scale factor Q of the multi-scale sample entropy will affect the MSE value, and different Q each layer of IMF has a di erent SNR. In order to accurately determine the MSE value of each IMF, values have different MSE values. According to the decomposition characteristics of CEEMD, each the di er lay ent er of IM IMFsF must has a diffe be accurately rent SNR. In ord distinguished. er to accura In tely determin this paper,e the the MSE value of each I values of the scaleM factor F, theQ range different IMFs must be accurately distinguished. In this paper, the values of the scale factor Q range from 1 to 20 are taken, and the SNR is respectively taken as30 dB,20 dB,10 dB, 1 dB, and 10 dB from 1 to 20 are taken, and the SNR is respectively taken as −30 dB, −20 dB, −10 dB, 1 dB, and 10 dB for simulation. The results are shown in Figure 6. The signals with the same SNR in the figure have for simulation. The results are shown in Figure 6. The signals with the same SNR in the figure have di erent MSE values due to di erent Q values. Taking the SNR30dB as an example, when the scale different MSE values due to different Q values. Taking the SNR −30dB as an example, when the factor takes di erent values, MSE only has large fluctuations, and the fluctuation range is 1.6–2.5. scale factor takes different values, MSE only has large fluctuations, and the fluctuation range is For MSE with the same scale factor, there are di erent values of MSE due to di erent SNRs. Multi-scale 1.6–2.5. For MSE with the same scale factor, there are different values of MSE due to different SNRs. Multi-scale sample entropy measures the complexity of the signal, the more noise in the signal, the sample entropy measures the complexity of the signal, the more noise in the signal, the greater the greater the MSE value. In Figure 6, when the scale factor is taken as 10, MSE values are arranged in MSE value. In Figure 6, when the scale factor is taken as 10, MSE values are arranged in the order the order that the noise content in the signal is more or less. There is no situation where the MSE that the noise content in the signal is more or less. There is no situation where the MSE value of the value of the low-noise signal is greater than the MSE of the high-noise signal, and the signal can be low-noise signal is greater than the MSE of the high-noise signal, and the signal can be distinguished. distinguished. Therefore, we choose the scale factor Q = 10 to calculate the MSE value of each layer Therefore, we choose the scale factor Q = 10 to calculate the MSE value of each layer of the IMF. of the IMF. Figure 5. Simulation signal decomposed by CEEMD. Appl. Sci. 2020, 10, x 9 of 19 Figure 5. Simulation signal decomposed by CEEMD. Figure 6. Scale factor Q. Figure 6. Scale factor Q. To eliminate the signal with larger MSE value in IMFs, it is necessary to determine the appropriate threshold. In this paper, the MSE values of 10,000 sets of noise signals (the signal length of each group of random noise is the same as the simulated signal) are calculated separately. The signal length of each group of random noise is the same as the vibration signal and the values are normalized, and the graph is drawn in Figure 7. The MSE value of the noise signal in Figure 7 varies from 0.55 to 1.0. In this paper, the minimum value of the MSE value in the 10,000 sets of noise signals is selected as the threshold for eliminating IMFs. 10,000 sets of random noise are randomly obtained, so the threshold in this paper is a dynamically changing value. After a lot of simulation, the minimum threshold fluctuates in a small range, so it is reasonable to choose the minimum value of 10,000 groups of noise as the threshold in this paper. Figure 7. Multi-scale sample entropy (MSE) values of 10,000 sets of random noise. In this paper, the scale factor Q = 10 is used to calculate the MSE values of IMFs and 10,000 sets of random noise, at the same time, the minimum value of 10,000 sets of random noise is used as the threshold, and Figure 8 is drawn. In Figure 8, the pink line is the minimum value of 10,000 groups of random noise, and the red bar is the MSE value of each IMF. In this paper, IMFs signals larger than the threshold are eliminated, so the IMF1–IMF4 signals are eliminated. Reconstruct the remaining IMF5–IMF10 signals and perform FFT transformation to obtain Figure 9. In Figure 9, the red line is the reconstructed signal, and the blue is the original signal. It can be seen that the high-frequency noise part of the signal is removed from 400 Hz to 1000 Hz, which achieves the purpose of noise reduction. Appl. Sci. 2020, 10, x 9 of 19 Appl. Sci. 2020, 10, 5078 9 of 18 Figure 6. Scale factor Q. To eliminate the signal with larger MSE value in IMFs, it is necessary to determine the appropriate To eliminate the signal with larger MSE value in IMFs, it is necessary to determine the threshold. In this paper, the MSE values of 10,000 sets of noise signals (the signal length of each group appropriate threshold. In this paper, the MSE values of 10,000 sets of noise signals (the signal length of random noise is the same as the simulated signal) are calculated separately. The signal length of each of each group of random noise is the same as the simulated signal) are calculated separately. The group of random noise is the same as the vibration signal and the values are normalized, and the graph signal length of each group of random noise is the same as the vibration signal and the values are is drawn in Figure 7. The MSE value of the noise signal in Figure 7 varies from 0.55 to 1.0. In this paper, normalized, and the graph is drawn in Figure 7. The MSE value of the noise signal in Figure 7 the minimum value of the MSE value in the 10,000 sets of noise signals is selected as the threshold for varies from 0.55 to 1.0. In this paper, the minimum value of the MSE value in the 10,000 sets of noise eliminating IMFs. 10,000 sets of random noise are randomly obtained, so the threshold in this paper is signals is selected as the threshold for eliminating IMFs. 10,000 sets of random noise are randomly a dynamically changing value. After a lot of simulation, the minimum threshold fluctuates in a small obtained, so the threshold in this paper is a dynamically changing value. After a lot of simulation, range, the mi soni itmum is reasonable threshold fl to uctua choose tes i the n a sma minimum ll range, value so itof is re 10,000 asona gr ble t oups o choose of noise the min as the imum threshold value in of 10,000 groups of noise as the threshold in this paper. this paper. Figure 7. Multi-scale sample entropy (MSE) values of 10,000 sets of random noise. Figure 7. Multi-scale sample entropy (MSE) values of 10,000 sets of random noise. In this paper, the scale factor Q = 10 is used to calculate the MSE values of IMFs and 10,000 sets In this paper, the scale factor Q = 10 is used to calculate the MSE values of IMFs and 10,000 sets of random noise, at the same time, the minimum value of 10,000 sets of random noise is used as the of random noise, at the same time, the minimum value of 10,000 sets of random noise is used as the threshold, and Figure 8 is drawn. In Figure 8, the pink line is the minimum value of 10,000 groups threshold, and Figure 8 is drawn. In Figure 8, the pink line is the minimum value of 10,000 groups of of random noise, and the red bar is the MSE value of each IMF. In this paper, IMFs signals larger random noise, and the red bar is the MSE value of each IMF. In this paper, IMFs signals larger than than the threshold are eliminated, so the IMF1–IMF4 signals are eliminated. Reconstruct the the threshold are eliminated, so the IMF1–IMF4 signals are eliminated. Reconstruct the remaining remaining IMF5–IMF10 signals and perform FFT transformation to obtain Figure 9. In Figure 9, the IMF5–IMF10 signals and perform FFT transformation to obtain Figure 9. In Figure 9, the red line is the red line is the reconstructed signal, and the blue is the original signal. It can be seen that the reconstructed signal, and the blue is the original signal. It can be seen that the high-frequency noise high-frequency noise part of the signal is removed from 400 Hz to 1000 Hz, which achieves the part of Appl. the Sci. signal 2020, 10is , x removed from 400 Hz to 1000 Hz, which achieves the purpose of noise 10 of reduction. 19 purpose of noise reduction. Figure 8. MSE of each IMF component. Figure 8. MSE of each IMF component. According to the method mentioned in this article, the reconstructed signal needs to be decomposed again by the CEEMD method, and the MSE value of each component signal is calculated. The MSE value is compared with the dynamic threshold value, and if there is an IMFs component signal greater than the threshold, it is removed, and the remaining signal is reconstructed and CEEMD decomposition is performed again. After multiple iterations, it is finally obtained that there is no case where the MSE value of the component signal is greater than the threshold, so the iteration is terminated. The time and frequency domain diagrams of signal decomposition at the end of the iteration are shown in Figure 10. The fault frequencies of 50 Hz and 130 Hz of the simulated signal in Figure 10 were extracted, which verified the effectiveness of the proposed method. Finally, the relationship between the MSE value of each signal component and the dynamic threshold is shown in Figure 11. In Figure 11, the MSE values of IMF1–IMF10 are all smaller than the dynamic threshold. Figure 9. Comparison of the original signal and reconstructed signal. Appl. Sci. 2020, 10, x 10 of 19 Figure 8. MSE of each IMF component. According to the method mentioned in this article, the reconstructed signal needs to be decomposed again by the CEEMD method, and the MSE value of each component signal is calculated. The MSE value is compared with the dynamic threshold value, and if there is an IMFs component signal greater than the threshold, it is removed, and the remaining signal is reconstructed and CEEMD decomposition is performed again. After multiple iterations, it is finally obtained that there is no case where the MSE value of the component signal is greater than the threshold, so the iteration is terminated. The time and frequency domain diagrams of signal decomposition at the end of the iteration are shown in Figure 10. The fault frequencies of 50 Hz and 130 Hz of the simulated signal in Figure 10 were extracted, which verified the effectiveness of the proposed method. Finally, the relationship between the MSE value of each signal component and Appl. Sci. 2020 the dyna , 10, 5078 mic threshold is shown in Figure 11. In Figure 11, the MSE values of IMF1–IMF10 are all 10 of 18 smaller than the dynamic threshold. Figure 9. Comparison of the original signal and reconstructed signal. Figure 9. Comparison of the original signal and reconstructed signal. According to the method mentioned in this article, the reconstructed signal needs to be decomposed again by the CEEMD method, and the MSE value of each component signal is calculated. The MSE value is compared with the dynamic threshold value, and if there is an IMFs component signal greater than the threshold, it is removed, and the remaining signal is reconstructed and CEEMD decomposition is performed again. After multiple iterations, it is finally obtained that there is no case where the MSE value of the component signal is greater than the threshold, so the iteration is terminated. The time and frequency domain diagrams of signal decomposition at the end of the iteration are shown in Figure 10. The fault frequencies of 50 Hz and 130 Hz of the simulated signal in Figure 10 were extracted, which verified the e ectiveness of the proposed method. Finally, the relationship between the MSE value of each signal component and the dynamic threshold is shown in Figure 11. In Figure 11, the MSE values of IMF1–IMF10 are all smaller than the dynamic threshold. Appl. Sci. 2020, 10, x 11 of 19 Figure 10. CEEMD decomposition reconstructed signals. Figure 10. CEEMD decomposition reconstructed signals. Figure 11. MSE value of the component signal (reconstructed signal is decomposed by CEEMD). 4. Experimental Verification To verify the feasibility of the proposed method in engineering application, it is planned to use the gearbox fault test bench for experiments. The experiment bench is shown in Figure 12, which includes a triple-phase asynchronous motor, connector, gear speed increaser, torque and rotational speed sensor, planetary reducer, triaxial acceleration sensor, and magnetic powder loader. The Appl. Sci. 2020, 10, x 11 of 19 Appl. Sci. 2020, 10, 5078 11 of 18 Figure 10. CEEMD decomposition reconstructed signals. Figure 11. MSE value of the component signal (reconstructed signal is decomposed by CEEMD). Figure 11. MSE value of the component signal (reconstructed signal is decomposed by CEEMD). 4. Experimental Verification 4. Experimental Verification To verify the feasibility of the proposed method in engineering application, it is planned to use To verify the feasibility of the proposed method in engineering application, it is planned to use the the gearbox fault test bench for experiments. The experiment bench is shown in Figure 12, which gearbox fault test bench for experiments. The experiment bench is shown in Figure 12, which includes includes a triple-phase asynchronous motor, connector, gear speed increaser, torque and rotational Appl. Sci. 2020, 10, x 12 of 19 a triple-phase asynchronous motor, connector, gear speed increaser, torque and rotational speed sensor, speed sensor, planetary reducer, triaxial acceleration sensor, and magnetic powder loader. The planetary reducer, triaxial acceleration sensor, and magnetic powder loader. The magnetic powder magnetic powder loader is used to adjust the load. The triaxial acceleration sensor model is loader is used to adjust the load. The triaxial acceleration sensor model is YD77SA, the sensitivity is -2 YD77SA, the sensitivity is 0.01 V/ms , and the sensor placement is shown in Figure 12. -2 0.01 V/ms , and the sensor placement is shown in Figure 12. During the test, the original bearing was replaced by a bearing with cracks in the inner ring During the test, the original bearing was replaced by a bearing with cracks in the inner ring and and flaking of the inner ring. The bearing types are NJ210 and NJ405, as shown in Figure 13. The flaking of the inner ring. The bearing types are NJ210 and NJ405, as shown in Figure 13. The bearing bearing speed is 1500 rmp. According to the calculation formula of the bearing inner ring fault speed is 1500 rmp. According to the calculation formula of the bearing inner ring fault frequency, frequency, the fault frequencies of NJ405 and NJ210 are 147.66 Hz and 231 Hz, respectively, as the fault frequencies of NJ405 and NJ210 are 147.66 Hz and 231 Hz, respectively, as shown in Table 1. shown in Table 1. Figure Figure 12 12. . Experiment Experiment bench, 1—Magnetic Powder Load bench, 1—Magnetic Powder Loader; er; 2—Gear 2—Gear speed Incr speed easer; 3—T Increaser; orque and 3—Torqu e Rotational Speed Sensor; 4—Planetary Reducer; 5—Triaxial Acceleration Sensor #1; 6—Triaxial and Rotational Speed Sensor; 4—Planetary Reducer; 5—Triaxial Acceleration Sensor #1; 6—Triaxial Acceleration Sensor #2; and 7—Triple-phase Asynchronous Motor. Acceleration Sensor #2; and 7—Triple-phase Asynchronous Motor. (a) (b) Figure 13. Faulty bearing. (a) NJ405 bearing. (b) NJ210 bearing. The time and frequency domain of vibration signals in the experiment are shown in Figure 14. In Figure 14a, the periodic impact of the collected vibration signal is not very obvious. Figure 14b is the frequency domain diagram obtained by the FFT transformation of the vibration signal. It can be seen that the fault period is submerged by noise and it failed to judge the existence of the fault. The vibration signals are processed by CEEMD and the method proposed in this article, and the effects of each method are compared. Appl. Sci. 2020, 10, x 12 of 19 magnetic powder loader is used to adjust the load. The triaxial acceleration sensor model is -2 YD77SA, the sensitivity is 0.01 V/ms , and the sensor placement is shown in Figure 12. During the test, the original bearing was replaced by a bearing with cracks in the inner ring and flaking of the inner ring. The bearing types are NJ210 and NJ405, as shown in Figure 13. The bearing speed is 1500 rmp. According to the calculation formula of the bearing inner ring fault frequency, the fault frequencies of NJ405 and NJ210 are 147.66 Hz and 231 Hz, respectively, as shown in Table 1. Figure 12. Experiment bench, 1—Magnetic Powder Loader; 2—Gear speed Increaser; 3—Torque and Appl. Sci.Rotational Spe 2020, 10, 5078 ed Sensor; 4—Planetary Reducer; 5—Triaxial Acceleration Sensor #1; 6—Triaxial 12 of 18 Acceleration Sensor #2; and 7—Triple-phase Asynchronous Motor. (a) (b) Figure 13. Figure 13.Faul Faulty ty beari bearing. ng. (a()a NJ ) NJ405 405 bbearing. earing. (b (b ) NJ ) NJ210 210 b bearing. earing. The time and frequency domain of vibration signals in the experiment are shown in Figure 14. The time and frequency domain of vibration signals in the experiment are shown in Figure 14. In In Fi Figur gur e 14 e 1 a, 4athe , the per periodic iodic impact impact o of f t the he collect collected ed vibr vibration ation signa signal l is not is not very very obvious obvious. . FiguFigur re 14b e i14 s b is the the f fr requency equency domai domain n dia diagram gram obtai obtained ned by the FFT by the FFT transformation transformation of the v of the ibrvibration ation signal. It c signal. an b It can e seen that the fault period is submerged by noise and it failed to judge the existence of the fault. The be seen that the fault period is submerged by noise and it failed to judge the existence of the fault. vibration signals are processed by CEEMD and the method proposed in this article, and the effects The vibration signals are processed by CEEMD and the method proposed in this article, and the e ects of each method are compared. of each method are compared. Appl. Sci. 2020, 10, x 13 of 19 (a) (b) Figure 14. Time-domain and frequency-domain graphs of experimentally acquired signals. (a) Time Figure 14. Time-domain and frequency-domain graphs of experimentally acquired signals. (a) Time domain. (b) Frequency domain. domain. (b) Frequency domain. Table 1. Failure frequency. Table 1. Failure frequency. Rotating Speed n NJ405 NJ210 Rotating Speed n NJ405 NJ210 1500 rpm 147.66 Hz 231.2 Hz 1500 rpm 147.66 Hz 231.2 Hz Figure 15a is the time domain diagram of the vibration signal decomposed by CEEMD, and Figure 15a is the time domain diagram of the vibration signal decomposed by CEEMD, and Figure 15b Figure 15b is the frequency domain diagram of the vibration signal obtained by FFT. The signal in is the frequency domain diagram of the vibration signal obtained by FFT. The signal in the figure is the figure is decomposed into 15 layers in total, and from the time domain diagram and frequency decomposed into 15 layers in total, and from the time domain diagram and frequency domain diagram in domain diagram in the corresponding diagram, we can observe that the component signals, the corresponding diagram, we can observe that the component signals, IMF1–IMF5, are high-frequency IMF1–IMF5, are high-frequency noise signals. Bearing failure signals are not obtained in the IMF6–IMF15 component signals, so the vibration signal fails to extract the bearing failure signal through traditional CEEMD decomposition. Employing the method proposed in this paper to decompose the vibration signal. The threshold needs to be determined, so this paper uses the scale factor Q = 10 to calculate the MSE value of 10,000 sets of random noise. The signal length of each set of random noise is consistent with the vibration signal. The result is shown in Figure 16. The MSE value of the noise signal in Figure 16 varies from 0.705 to 0.745. In this paper, the scale factor Q = 10 is used to calculate the MSE values of IMFs and 10,000 sets of random noise, and the minimum value of 10,000 sets of random noise is used as the threshold, and Figure 17 is drawn. The pink line in Figure 17 is the minimum value of 10,000 sets of random noise, and the red bar is the MSE value of each IMF. In this paper, IMFs signals larger than the threshold are eliminated, so the IMF1–IMF3 signals in the figure below are eliminated. Reconstruct the remaining IMF4–IMF15 signals and perform FFT transformation to obtain Figure 18. In Figure 18, red is the reconstructed signal and blue is the original signal. It can be seen that the high-frequency noise part of the signal from 3000 Hz to 5000 Hz has been removed, which achieves the purpose of noise reduction. In Figure 18, 148 Hz is closest to the fault signal 147.66 Hz; 230.6 Hz is closest to the fault signal 231.2 Hz. Considering that the bearing operating environment has errors, which are allowed, so the method proposed in this article is effective. Appl. Sci. 2020, 10, 5078 13 of 18 noise signals. Bearing failure signals are not obtained in the IMF6–IMF15 component signals, so the vibration signal fails to extract the bearing failure signal through traditional CEEMD decomposition. Employing the method proposed in this paper to decompose the vibration signal. The threshold needs to be determined, so this paper uses the scale factor Q = 10 to calculate the MSE value of 10,000 sets of random noise. The signal length of each set of random noise is consistent with the vibration signal. The result is shown in Figure 16. The MSE value of the noise signal in Figure 16 varies from 0.705 to 0.745. In this paper, the scale factor Q = 10 is used to calculate the MSE values of IMFs and 10,000 sets of random noise, and the minimum value of 10,000 sets of random noise is used as the threshold, and Figure 17 is drawn. The pink line in Figure 17 is the minimum value of 10,000 sets of random noise, and the red bar is the MSE value of each IMF. In this paper, IMFs signals larger than the threshold are eliminated, so the IMF1–IMF3 signals in the figure below are eliminated. Reconstruct the remaining IMF4–IMF15 signals and perform FFT transformation to obtain Figure 18. In Figure 18, red is the reconstructed signal and blue is the original signal. It can be seen that the high-frequency noise part of the signal from 3000 Hz to 5000 Hz has been removed, which achieves the purpose of noise reduction. In Figure 18, 148 Hz is closest to the fault signal 147.66 Hz; 230.6 Hz is closest to the fault signal 231.2 Hz. Considering that the bearing operating environment has errors, which are allowed, so the method proposed in this article is e ective. Appl. Sci. 2020, 10, x 14 of 19 (a) (b) Figure 15. CEEMD decomposes vibration signals. (a) Time domain. (b) Frequency domain. Figure 15. CEEMD decomposes vibration signals. (a) Time domain. (b) Frequency domain. Figure 16. MSE value of 10,000 sets of random noise signals (the signal length is consistent with the vibration signal). Appl. Sci. 2020, 10, x 14 of 19 (a) (b) Appl. Sci. 2020, 10, 5078 14 of 18 Figure 15. CEEMD decomposes vibration signals. (a) Time domain. (b) Frequency domain. Figure 16. MSE value of 10,000 sets of random noise signals (the signal length is consistent with the Figure 16. MSE value of 10,000 sets of random noise signals (the signal length is consistent with the vibration signal). Appl. Sci. 2020, 10, x 15 of 19 Appl. vibration Sci. 2020signal). , 10, x 15 of 19 Figure 17. MSE value of each component of vibration signal decomposed by CEEMD. Figure 17. MSE value of each component of vibration signal decomposed by CEEMD. Figure 17. MSE value of each component of vibration signal decomposed by CEEMD. Figure 18. Vibration signal reconstruction. Figure Figure 18. 18. V Vibration signal ibration signal r reconstruction. econstruction. In In Figur Figure 18 e 18, , the the failur failure e frfre equency quency o offthe the recon reconstr structed si ucted signal gnal iis s not ob not obvious. vious. Accordi According ng to the to the In Figure 18, the failure frequency of the reconstructed signal is not obvious. According to the method proposed in this paper, the reconstructed signal needs to be decomposed by the CEEMD method method proposed proposed inin this this paper paper, the reconstructe , the reconstructed d signal ne signal needs eds tto o be decomp be decomposed osed by the by the CEEMD CEEMD method, and the MSE value of each component signal be calculated. The MSE value is compared method, and the MSE value of each component signal be calculated. The MSE value is compared with the dynamic threshold value, and if there is an IMFs component signal greater than the with the dynamic threshold value, and if there is an IMFs component signal greater than the threshold, it is removed, the remaining signal is reconstructed, and CEEMD decomposition is threshold, it is removed, the remaining signal is reconstructed, and CEEMD decomposition is performed again. After multiple iterations, if it is finally obtained that there is no case where the performed again. After multiple iterations, if it is finally obtained that there is no case where the MSE value of the component signal is greater than the threshold, then the iteration is terminated. MSE value of the component signal is greater than the threshold, then the iteration is terminated. The time and frequency domain diagrams of signal decomposition at the end of the iteration are The time and frequency domain diagrams of signal decomposition at the end of the iteration are shown in Figure 19. In Figure 19, the reconstructed vibration signal is decomposed into 14 layers, of shown in Figure 19. In Figure 19, the reconstructed vibration signal is decomposed into 14 layers, of which IMF1–IMF5 are high-frequency noise signals; IMF6 is the failure frequency of the inner ring which IMF1–IMF5 are high-frequency noise signals; IMF6 is the failure frequency of the inner ring of NJ210 bearings; IMF7 is the failure frequency of the inner ring of NJ405 bearings; IMF8–IMF14 of NJ210 bearings; IMF7 is the failure frequency of the inner ring of NJ405 bearings; IMF8–IMF14 are meaningless interference components. Therefore, fault features are extracted, which verifies the are meaningless interference components. Therefore, fault features are extracted, which verifies the effectiveness of the proposed method. The relationship between the multi-scale sample entropy of effectiveness of the proposed method. The relationship between the multi-scale sample entropy of each signal component and the dynamic threshold is shown in Figure 20. In Figure 20, the MSE each signal component and the dynamic threshold is shown in Figure 20. In Figure 20, the MSE values of IMF1–IMF14 are all smaller than the dynamic threshold. values of IMF1–IMF14 are all smaller than the dynamic threshold. Appl. Sci. 2020, 10, 5078 15 of 18 method, and the MSE value of each component signal be calculated. The MSE value is compared with the dynamic threshold value, and if there is an IMFs component signal greater than the threshold, it is removed, the remaining signal is reconstructed, and CEEMD decomposition is performed again. After multiple iterations, if it is finally obtained that there is no case where the MSE value of the component signal is greater than the threshold, then the iteration is terminated. The time and frequency domain diagrams of signal decomposition at the end of the iteration are shown in Figure 19. In Figure 19, the reconstructed vibration signal is decomposed into 14 layers, of which IMF1–IMF5 are high-frequency noise signals; IMF6 is the failure frequency of the inner ring of NJ210 bearings; IMF7 is the failure frequency of the inner ring of NJ405 bearings; IMF8–IMF14 are meaningless interference components. Therefore, fault features are extracted, which verifies the e ectiveness of the proposed method. The relationship between the multi-scale sample entropy of each signal component and the dynamic threshold is shown in Figure 20. In Figure 20, the MSE values of IMF1–IMF14 are all smaller than the dynamic threshold. Appl. Sci. 2020, 10, x 16 of 19 (a) (b) Figure Figure 19. 19. TheThe time time and and frequency frequenc domain y domai of n of rec reconstr ons ucted tructed s signal ignal by by CEEMD CEEMD d decomposition. ecomposition. (a ()aT ) ime domain. (b) Frequency domain. Time domain. (b) Frequency domain. Appl. Sci. 2020, 10, x 16 of 19 (a) (b) Figure 19. The time and frequency domain of reconstructed signal by CEEMD decomposition. (a) Appl. Sci. 2020, 10, 5078 16 of 18 Time domain. (b) Frequency domain. Figure 20. MSE value of the component signal (reconstructed signal is decomposed by CEEMD). 5. Conclusions In this paper, the CEEMD method is used to adaptively reduce the noise of complex vibration signals, but the pseudo components after CEEMD decomposition seriously reduce the diagnostic eciency. The article proposes a new method of adaptively removing noise components, applying the proposed method to the composite fault simulation signal and the vibration signal of mining machinery, accurately extracting the composite fault characteristics, and verifying the feasibility of the method by comparing it with the traditional CEEMD. Finally, the following conclusions are obtained: (1) Determine the best scale factor Q = 10 in the MSE by simulation signals with di erent SNR, and then employ it to calculate the MSE values of noise signals, simulation signals, and vibration signals, which provide support for optimizing the CEEMD method. (2) Calculate the MSE value of 10,000 sets of random noise, and use the minimum value as the threshold, remove the IMFs component signal greater than the threshold, achieve the purpose of CEEMD noise reduction, and verify the feasibility of the minimum value as the threshold. (3) Through CEEMD decomposition of the reconstructed signal, the MSE value of each component signal is calculated. The MSE value is compared with the dynamic threshold value, and if there is an IMFs component signal greater than the threshold, it is removed, the remaining signal is reconstructed, and CEEMD decomposition is performed again. After multiple iterations, when it is finally shown that there is no case where the MSE value of the component signal is greater than the threshold, the iteration is terminated. Finally, the obtained signal is analyzed by the frequency spectrum to determine the fault characteristics. Author Contributions: W.G. conceived and designed the experiments; R.L. and J.Z. performed the experiments; and W.G. and Y.K. wrote the paper. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Key Research and Development Program of Shanxi Province (International Cooperation) (201903D421051, 201803D421028); Youth Fund Project of Shanxi Province (201901D211210). Conflicts of Interest: The authors declare no conflict of interest. Appl. Sci. 2020, 10, 5078 17 of 18 References 1. Xu, J.; Wang, Z.; Tan, C.; Si, L.; Zhang, L.; Liu, X. 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Journal

Applied Sciences – Multidisciplinary Digital Publishing Institute

Published: Jul 23, 2020

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