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Study on identification method of heat transfer deterioration of supercritical fluids in vertically heated tubes
Compound fault extraction method via self-adaptively determining the number of decomposition layers of the variational mode decomposition Ziying Zhang, Xi Zhang, Panpan Zhang, Fengbiao Wu, and Xuehui Li
Citation: Review of Scientific Instruments 89, 085110 (2018); doi: 10.1063/1.5037565 View online: https://doi.org/10.1063/1.5037565 View Table of Contents: http://aip.scitation.org/toc/rsi/89/8 Published by the American Institute of Physics
Articles you may be interested in A novel high thrust–weight ratio linear ultrasonic motor driven by single–phase signal Review of Scientific Instruments 89, 085001 (2018); 10.1063/1.5037407 A supersonic target jet mill based on the entrainment of annular supersonic flow Review of Scientific Instruments 89, 085104 (2018); 10.1063/1.5039589 Wide field of view, real time bioimaging apparatus for noninvasive analysis of nanocarrier pharmacokinetics in living model animals Review of Scientific Instruments 89, 085105 (2018); 10.1063/1.5026852 A 1-m non-resonant inelastic x-ray scattering spectrometer at BL15U, Shanghai Synchrotron Radiation Facility Review of Scientific Instruments 89, 085108 (2018); 10.1063/1.5030032 Design and experimental demonstration of a circularly polarized mode converter for high-power microwave applications Review of Scientific Instruments 89, 084701 (2018); 10.1063/1.5035179 TDLAS-WMS second harmonic detection based on spectral analysis Review of Scientific Instruments 89, 083106 (2018); 10.1063/1.5031683
REVIEW OF SCIENTIFIC INSTRUMENTS 89, 085110 (2018)
Compound fault extraction method via self-adaptively determining the number of decomposition layers of the variational mode decomposition Ziying Zhang,1,2,a) Xi Zhang,1 Panpan Zhang,2 Fengbiao Wu,2 and Xuehui Li1
1 School
of Mechanical, Electronic and Information Engineering, China University of Mining and Technology (CUMT), Xueyuan Road, Beijing 100083, China 2 Shanxi Institute of Energy, Daxue Road, Jinzhong 030600, China
(Received 25 April 2018; accepted 15 July 2018; published online 9 August 2018) Local mean decomposition (LMD) is a self-adaptive method, which has been widely applied to extract early fault signals from bearings. However, mode mixing occurs during the decomposition process. Moreover, in processing signals with strong noise, false frequency components can be generated by variational mode decomposition (VMD). To address these problems, a weak fault extraction method based on VMD is proposed for rolling bearings. This method regards LMD and the combination production function (CPF) as prefilters for VMD. First, LMD is used for denoising the original signal, and then the CPF components that contain the fault information are combined into a new signal. Second, this method determines the decomposition level K of the VMD from the spectral peaks of the recombined signal. Finally, this method decomposes the recombined signal using the VMD. The main contributions of the proposed method are (i) the CPF method is employed for adaptively de-noising, and the power of the fault feature can be improved; (ii) the decomposition level K of the VMD can be determined adaptively. After processing a simulated signal, fault information of the gears and rolling elements is successfully extracted, thereby demonstrating the feasibility of the presented method. Moreover, the feasibility of the proposed method is further demonstrated in a comparison of results with those obtained from the MOMEDA (Multipoint Optimal Minimum Entropy Deconvolution Adjusted) algorithm. Published by AIP Publishing. https://doi.org/10.1063/1.5037565
I. INSTRUCTION
The working status of bearings, the dynamic transmission parts of rotating machines, is directly related to the operation status of the whole machine. Hence, the health status of bearings is a concern.1 If the bearings fail, they may affect the quality of production and cause damage or even result in casualties. Because of influences from the working environment, early bearing-fault signals are weak and are easily drowned out by noise. Extracting fault signals from noise has always been a major problem in fault diagnosis. Local mean decomposition (LMD) is a recent time-frequency decomposition method proposed by Jonathan Smith in 2005 based on the empirical mode decomposition (EMD).2,3 Many researchers have found that LMD alleviates the issue of mode mixing in comparison with EMD, but mode mixing in LMD still affects the decomposition results and generates false components. Three methods—the LMD method based on the differential rational spline and K-L divergence proposed by Li,4 the fault diagnosis method based on an improved LMD,5 and a multifractal theory analysis of weak-fault diagnosis of rolling bearings6 —have achieved good results in mitigating mode mixing. Nevertheless, a mixing of noise in the actual signal is produced in the LMD process, which increases the difficulty of extracting fault information. LMD was compared with EMD a) Author
to whom correspondence should be addressed: tbp1600401010@ student.cumtb.edu.cn.
0034-6748/2018/89(8)/085110/7/$30.00
in regard to decomposition efficiency by Cheng;7 the results showed LMD to be superior. Zhang8 combined fuzzy C-mean clustering and LMD to produce a classification and recognition of the different types of bearing faults. Based on the wavelet packet energy entropy and the LMD proposed by Li,9 the signal-to-noise ratio of the processed vibration signal was improved. Variational mode decomposition (VMD) is a self-adaptive decomposition method proposed by Dragomiretskiy et al. in 2014. Compared with the existing fault diagnosis methods, VMD has the advantages of, for example, fast convergence speed and good noise robustness. However, VMD is affected by the decomposition level K, on which there is little research at present. Wang10 used correlation coefficients between the high-frequency modes and the original signal to determine the K value. Ling11 performed a VMD analysis using the principle of a similar frequency center to determine the K value; good results were obtained. VMD is unable to highlight weak fault signals when the signal-to-noise ratio is low. Wang12 completed a fault diagnosis by combining VMD and singular value decomposition (SVD). Yao13 used VMD and the robust independent component analysis to suppress noise and extract the fault information accurately. In strong noise environments, VMD decomposition is influenced by the K value, so a self-adaptive method for determining the K value is crucial. LMD is a self-adaptive decomposition method that can inhibit the noise, but mode mixing issues arise. In other words, the same feature components are
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decomposed into different production function (PF) components, causing energy leakages. To determine the number of fault features in the original vibration signal and increase the energy of fault features, the PF components that have the same frequencies are recombined using the combination production function (CPF) method. This paper proposes a fault diagnosis method based on LMD-CPF-VMD. The noise signal is first decomposed using LMD, and then the decomposed PFs are recombined. During this process, not only is the signal denoised but also mode mixing incurred using LMD is suppressed in the recombination process. At the same time, when performing an envelope analysis of the PF components that contain the fault information, the K value of VMD is determined from the number of peaks in the envelope results. After dealing with the fault signal, the fault information is successfully extracted. The main contributions of the proposed method are as follows: First, considering that every intrinsic mode function (IMF) decomposed by VMD has a center frequency and the center frequency is arranged in order from low frequency to high frequency, the frequency band of noise in the gearbox is relatively wide, so there are often some miscellaneous items in VMD decomposition. The frequency is the peak of the center frequency, resulting in misdiagnosis. Therefore, the LMD needs to pre-filter the original signal. Second, this method is better used for multi-fault coexistence because LMD is an adaptive decomposition method. Despite modal aliasing, it can decompose different time scales into different PF serious diseases. After CPF processing, the decomposition layer of VMD can be determined by the number of frequency, and the diagnostic efficiency of the composite fault can be improved. Therefore, the method used in this paper is suitable for feature extraction of strong noise and complex faults. The rest of the paper is organized as follows: Sec. II is the background and proposed method; Sec. III is the LMDCPF-VMD fault diagnosis method; simulation signal analysis is given in Sec. IV; analysis of measured signals is given in Sec. V; and finally, the conclusions are given in Sec. VI.
II. BACKGROUND AND PROPOSED METHOD A. LMD-CPF method
Multiple fault vibration signals from gearboxes are multicomponent signals. The LMD method adaptively decomposes such signals into a number of PF components. Each PF level contains the features of the original signal, noise, and single frequency components; components with multiple frequencies are also possible. The LMD steps of the original vibration signal x(t) are as follows: (1) Find all the local extremum points ni of the original signal x(t), and then calculate all the local extremum mean values mi and the envelope estimation values ai from the local extremum points; here, ni + ni+1 , (1) mi = 2
Rev. Sci. Instrum. 89, 085110 (2018)
n − n i+1 i ai = . (2) 2 Next, connect all the adjacent mean values to the envelope estimation values. The moving average method is applied to obtain the local mean function myz (t) and the envelope function ayz (t), where y is the number of local mean and envelope functions and z indicates the number of iterations. (2) Separate all the local mean functions from the original signal x(t). Once the function hyz (t) has been obtained, then the function syz (t) has been demodulated, h11 (t) = x(t) − m11 (t) h12 (t) = s11 (t) − m11 (t) , .. . h1n (t) = s1(n−1) (t) − m1n (t)
(3)
where syz (t) is defined by s11 (t) = h11 (t)/a11 (t) s12 (t) = h12 (t)/a12 (t) . .. .
(4)
s1n (t) = h1n (t) − a1n (t) Then the same method is employed to calculate the envelop estimation function a12 (t) of s11 (t), if a12 (t) , 1; then s11 (t) is not a pure frequency-modulated signal, which means that an iterative procedure needs to be repeated for s11 (t) until the calculated s1n (t) becomes a pure frequencymodulated signal, that is, −1 ≤ s1n (t) ≤ 1 and a1(n+1) (t) = 1 can be satisfied. (3) Multiply all the calculated envelope functions to obtain an envelope signal a1 (t) = a11 (t)a11 (t) · · · a1n (t) =
n X
a1p (t).
(5)
p=1
The first PF component can be obtained from the envelope signal a1 (t) and the pure frequency-modulated signal s1n (t), PF1 (t) = a1 (t)s1n (t).
(6)
(4) PF 1 (t) is separated from x(t), and hence u1 (t) is obtained. Then, the above calculation procedure is repeated using u1 (t) as a new original signal and iterated until uq (t) is a monotone function u1 (t) = x(t) − PF1 (t) u2 (t) = u1 (t) − PF2 (t) . .. .
(7)
uq (t) = uq−1 (t) − PFq (t) (5) Next the original signal is decomposed into a sum of q PF components and a residual component uq (t), x(t) =
q X
PFq (t) + uq (t).
(8)
l=1
The CPF can be understood as a combination of PF components. After LMD has generated all PF components, those
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that contain the fault information are recombined using the envelope analysis of the signal to form a new component, the role of which is to mitigate mode mixing during LMD and suppress noise in the signal. B. VMD method
The specific construction steps of the constrained variational model are as follows. Step 1: For the input signal x(t), through the Hilbert Transform (HT), we can get the analytic signal of each mode function uk (t). Step 2: The center frequency ωk of each mode function uk (t) is estimated, and its spectrum is moved to the baseband. Step 3: After step 2, the bandwidth is estimated through the H 1 Gauss smoothness. The final constraint variational model can be expressed by Eq. (10), #
2 X
"
∂t (σ(t) + j )uk (t) e−jωk t
min
(uk )(ωk ) πt 2 k
. (9) X uk = x(t) s.t. k In Eq. (9), ∂t indicates the partial derivative of t, σ(t) is the impulse function, and {uk } = {u1 , ..., uK } represents the K IMFs obtained by VMD of the original signal x(t), and {ωk } = {ω1 , ..., ωK } represents the central frequency of each IMF component. In order to find the optimal solution to the above variational problem, the following form of Lagrange function is introduced: ! # 2 X "
σ(t) + j u (t) ei−jωk t
L({uk }, {ωk }, λ) = a k
πt
Xm+1 . λ n+1 (ω) ← λ n (ω) + τ x(t) − k
Renew ukn+1 , ωkn+1 , and λ separately. (4) Repeat step 3; the criterion to stop the iteration is X
un+1 − un
2
un
2 < ε. (14) k 2 k 2
k k
The decomposition is then completed. III. LMD-CPF-VMD FAULT DIAGNOSIS METHOD
The LMD-CPF-VMD method involves the following procedural steps: First, in using LMD to decompose the original signal, LMD acts as a prefilter that suppresses noise contained in the signal. Second, summing the effective PF components using the CPF method mitigates mode mixing induced by the LMD. Suppose there are four levels of PF components produced by the LMD decomposition, and after the envelope analysis, assume that levels 1, 3, and 4 contain fault information but none in level 2. The CPF then outputs CPF1 = PF1 + PF3 + PF4 ,
(15)
CPF2 = PF2 ,
(16)
where CPF1 is retained for the reason that it is an effective integration signal, and CPF2 is discarded because it is not a
2
k
2
X
+
x(t) − uk (t)
k 2 + * X + λ(t), x(t) − uk (t) ,
(10)
k
where a is a penalty factor and λ is a Lagrange multiplier. The process of decomposing the original signal into K components using VMD is as follows: (1) {uk }, {ωk }, λ1 , and n are initialized to 0. (2) After n = n + 1, the cycle iteration begins. (3) k = k + 1, before k = K, according to x(ω) − ukn+1 (ω) ←
K X
uin+1 (ω) −
i=1,i
K X
uin (ω) +
i=1,i
1 + 2α(ω − ωkn )2 ∞ ωkn+1 ←
λ n (ω) 2 , (11)
2 ω ukn+1 (ω) dω
0
∞
,
(12)
2
un+1 (ω) dω k 0
(13)
FIG. 1. LMD-CPF-VMD method flow chart.
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valid signal. After obtaining the recombination signal CPF1 , the decomposition level K of VMD is determined depending on the numbers of peaks in the PF components obtained by the envelope analysis. Finally, VMD is used to decompose the new signal. Mode mixing occurs during the LMD process. Noise affects the VMD, the results of which become distorted from strong noisy signals. At the same time, if the number of the decomposition levels is unreasonable, energy leakages occur. The CPF-VMD method overcomes this mode-mixing issue, suppresses the noise in the signal, and determines a reasonable K value for the VMD process. First, we determine whether the PF components contain fault information using the correlation coefficient method and then perform an envelope analysis of all the PF components after the LMD process. Second, we recombine the PF components that contain the same frequency components and determine the K value in accordance with the number of frequency components. Finally, we conduct the VMD decomposition and perform an envelope analysis again to extract the fault features. The flow chart of the LMD-CPF-VMD method is given in Fig. 1.
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FIG. 3. VMD decomposition of the simulation signal.
IV. SIMULATION SIGNAL ANALYSIS
The simulation signal x(t) selected for the experiment is made up of two impulse signals and a random noise signal. In Fig. 2, from top to bottom, are plots of an 80-Hz impulse signal, a 150-Hz impulse signal, the noise signal, and the synthetic signal. A comparison of the synthetic signal with the noise signal shows that noise is strong with the impulse signals being completely submerged. Figure 3 shows the VMD results of the simulation signal, where K takes the value 5. The frequencies of the IMF components are 40 Hz, 150 Hz, 180 Hz, 295 Hz, and 370 Hz. Only the 150 Hz is a strong impact signal frequency; the other components are different from the frequencies of the impulse signals in the simulation signal. However, the strong impact signal can only be obtained from the decomposition results. The weak signal still has not be extracted, which proves that the noise has a great influence on the VMD result.
FIG. 2. Composition of the synthetic signals.
FIG. 4. LMD results from the simulation signal.
LMD has a certain denoising capability, but mode mixing has appeared during decomposition. Figure 4 shows that the LMD results are obtained for the simulation signal and proves that the LMD is capable of providing a certain level of noise reduction. Moreover, the periodic impact signals appear in the PF components. Figure 5 shows the results of the envelope analysis for the three PF components. The observations from these results are (i) The PF1 signal is complex, and although there are many peak values, the effective peak frequencies cannot be reached and (ii) From the PF2 and PF3 signals, peak values appear
FIG. 5. Results from the envelope analysis of the PF components.
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FIG. 6. Recombined signal of CPF1 . FIG. 8. Gearbox simulation test table. 1—speed adjusting motor, 2— coupling, 3—paternity test gearbox, 4—speed torsion instrument, 5—torsion bar, 6—test gearbox, 7—three-dimensional accelerometer 1, and 8—threedimensional accelerometer 2.
FIG. 7. VMD decomposition of the recombined signal CPF1 .
clearly at frequencies 80 Hz, 150 Hz, and 300 Hz. In the LMD process, mode mixing is obvious. LMD is first used in processing the original signal, and the decomposition results are shown in Fig. 4. The correlation coefficients between the PF components and the original signal are calculated next: PF1 yields 0.421, PF2 yields 0.683, and PF3 yields 0.644. From the definition of the correlation coefficient, if its value is greater than 0.5, the two components are strongly correlated.14 Hence PF2 and PF3 are dominant, and the envelope analysis results of the PF components show that the main fault information is decomposed into PF2 and PF3 , which are then recombined by CPF. The signal after recombination is shown in Fig. 6. With three frequency peaks in PF2 and PF3 , we assume that there are three effective frequency components in the signal. The K value is reset from 5 to 3. Labeling the recombined signal as CPF1 , it is decomposed by VMD. From the results of the envelope analysis for the IMFs obtained (Fig. 7), there are three components with frequencies at 80 Hz, 120 Hz, and 150 Hz. Recalling that the frequencies of the two impact signals in the simulation signal are 80 Hz and 150 Hz, it can be concluded that the effective fault components have been extracted.
FIG. 9. Fault signal.
Figures 9 and 10 present the time-domain plot and the LMD decomposition of the fault signal, respectively. In PF1 , there is a significant peak at the fault frequency of 90 Hz, and there is a weak peak at 180 Hz. In PF2 , there are significant peaks at 50 Hz and 90 Hz. The rolling elements’ fault frequency 72 Hz is not obtained from the decomposition results. After calculation, the correlation coefficients of the three PF components are 0.921, 0.501, and 0.215. The CPF signal obtained by recombining PF1 and PF2 (Fig. 11) shows that the signal has peaks at 50 Hz, 90 Hz, and 180 Hz, so the decomposition level K of VMD is determined to be 3. Figure 12 shows the VMD decomposition results of the recombined signal.
V. ANALYSIS OF MEASURED SIGNALS
A closed power flow gearbox test bed for loading devices with a torsion bar is shown in Fig. 8. The rotational speed of the torsion rod is regulated by the speed-adjusting motor. The rotational speed was set to 1200 r/min. Tapered roller bearings of type 32 212 were employed to mount the rod. To simulate an actual fault, a bearing in the outer-ring raceway was pitted under an electric discharge. Two three-dimensional accelerometers (YD77SA) were used to register the vibrational signal at a sampling rate of 8000 Hz. The calculated gear fault frequency is 90 Hz, and the rolling element fault frequency is 72 Hz.
FIG. 10. LMD decomposition of the fault signals.
FIG. 11. CPF results.
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Rev. Sci. Instrum. 89, 085110 (2018) TABLE I. Correlation coefficients of IMF and component signals obtained by two algorithms. Algorithm LMD-CPF-VMD CPF-MOMEDA
FIG. 12. VMD decomposition results of a recombined signal.
From the envelope analysis results of the recombined signals, we find that the three IMF components have peaks at 72 Hz, 90 Hz, and 120 Hz. Recalling that the gear fault frequency is 90 Hz and the rolling elements fault frequency is 72 Hz, the results of the analysis demonstrate that the fault information has been extracted successfully. To improve the diagnosis efficiency, a multiple-point kurtosis was used in a comparison of our results with those obtained using the MOMEDA algorithm in a simple calculation. The rotation period of the shaft is 400 (the unit is the number of points), the bearing fault period is 111.1, the gear meshing frequency is 180 Hz, and the gear meshing period is 22.4. The multiplepoint kurtosis of the recombined signal (Fig. 11) is presented in Fig. 13. Taking out the failure cycles of the gears and the bearings and then denoising via MOMEDA, we find the periodic intervals of [15–25] and [105–115] in accordance with the bearing fault cycle and gear meshing cycle. With a step length
FIG. 13. MKurt spectrum of CPF2 .
Re2
Re3
0.8949 0.7832
0.8742 0.6957
0.8344
of 0.1, the results (Fig. 14) show that we have extracted the fault cycle components. The feasibility of the proposed method has been further demonstrated by the measured signal. Correlation coefficients are obtained for the above two methods. The results are shown in Table I. It can be found that the correlation coefficients of the three IMFs obtained by LMD-CPF-VMD and the original signals are 0.8949, 0.8742, and 0.8344, respectively; the correlation coefficients of CPF-MOMEDA and the original signals are 0.7832 and 0.6957, respectively. Therefore, the decomposition result of the method proposed in this paper is closer to the characteristics of the original signal.
VI. CONCLUSIONS
LMD has adaptively decomposed the fault signal into several PF components based on frequency, but mode mixing occurs during the decomposition process. Before decomposing, VMD needs the decomposition level K, its value being directly related to the accuracy of the decomposition results. Determining the K value from experience lacks reliability. When confronted with strong noise, VMD distorts the signal and produces false IMF components. LMDCPF-VMD is based on the VMD method. After a LMD of the original signal, CPF recombination is used to mitigate mode mixing. The noise contained in the signal is suppressed, enabling the decomposition level K to be determined depending on the amount of frequency information in the PF components. The recombined signal is decomposed by VMD. This paper has demonstrated the reliability of the proposed method through separate analyses of a simulation signal and a measured fault signal. The fault information was extracted successfully. The proposed method lays the foundation for subsequent research employing VMD. 1 S.
FIG. 14. Denoising for CPF MOMEDA.
Re1
Renqing, L. Jing, H. Wentao, and D. Feifei, “Multi-fault clustering and diagnosis of gear system mined by spectrum entropy based on higher order cumulants,” Rev. Sci. Instrum. 84(2), 395–399 (2013). 2 N. Shaohui, H. Zhennan, W. Zhijian, and W. Xuefeng, “Application of sample entropy based LMD-TFPF de-noising algorithm for the gear transmission,” Entropy 18(11), 414 (2016). 3 S. Chong, C. Huiliang, L. Jie, T. Jun, Z. Xiaoming, S. Yunbo, Y. Wei, and L. Jun, “Hybrid de-noising approach for fiber optic gyroscopes combining improved empirical mode decomposition and forward linear prediction algorithms,” Rev. Sci. Instrum. 87(3), 033305 (2016). 4 Y. Li, X. Liang, Y. Yang, M. Xu, and W. Huang, “Early fault diagnosis of rotating machinery by combining differential rational spline-based LMD and K–L divergence,” IEEE Trans. Instrum. Meas. 66(11), 3077–3090 (2017). 5 Z. Wang, W. Junyuan, K. Yanfei, Z. Jiping, N. Shaohui, and Z. Zhifang, “Weak fault diagnosis of wind turbine gearboxes based on MED-LMD,” Entropy 19(6), 277 (2017).
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Shi, W. Song, and S. Taheri, “Improved LMD, permutation entropy and optimized K-means to fault diagnosis for roller bearings,” Entropy 18(3), 70 (2016). 7 J. Cheng, Y. Yang, and Y. Yang, “A rotating machinery fault diagnosis method based on local mean decomposition,” Digital Signal Process. 22(2), 356–366 (2012). 8 S. Zhang, S. Guoxiu, L. Xinxin, and J. Xiong, “Study on mechanical fault diagnosis method based on LMD approximate entropy and fuzzy C-means clustering,” Chin. J. Sci. Instrum. 3, 034 (2013). 9 X. Li, L. Yong, Z. Huimin, D. Wu, and S. Yannan, “A novel bearing fault diagnosis method based on LMD and wavelet packet energy entropy,” Int. J. Emerging Electr. Power Syst. 18(5), 1–16 (2017). 10 J. Wang, C. Shuai, and Z. Chao, “Fault diagnosis method of gear based on VMD and multi-feature fusion,” J. Mech. Transm. 3, 032 (2017).
Rev. Sci. Instrum. 89, 085110 (2018) 11 X.
Ling and Z. Lijia, “Rolling bearing fault feature extraction based on the VMD and 1.5-dimensional teager energy spectrum,” J. Vib. Shock 18, 015 (2017). 12 Z. Wang, J. Limin, and Q. Yong, “Adaptive diagnosis for rotating machineries using information geometrical kernel-ELM based on VMD-SVD,” Entropy 20(1), 73 (2018). 13 J. Yao, X. Yang, Q. Sichong, W. Shuai, and W. Shaowei, “Noise source identification of diesel engine based on variational mode decomposition and robust independent component analysis,” Appl. Acoust. 116, 184–194 (2017). 14 N. E. Huang, S. Zheng, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. A 454, 1971 (1998).
Citation: Review of Scientific Instruments 89, 085110 (2018); doi: 10.1063/1.5037565 View online: https://doi.org/10.1063/1.5037565 View Table of Contents: http://aip.scitation.org/toc/rsi/89/8 Published by the American Institute of Physics
Articles you may be interested in A novel high thrust–weight ratio linear ultrasonic motor driven by single–phase signal Review of Scientific Instruments 89, 085001 (2018); 10.1063/1.5037407 A supersonic target jet mill based on the entrainment of annular supersonic flow Review of Scientific Instruments 89, 085104 (2018); 10.1063/1.5039589 Wide field of view, real time bioimaging apparatus for noninvasive analysis of nanocarrier pharmacokinetics in living model animals Review of Scientific Instruments 89, 085105 (2018); 10.1063/1.5026852 A 1-m non-resonant inelastic x-ray scattering spectrometer at BL15U, Shanghai Synchrotron Radiation Facility Review of Scientific Instruments 89, 085108 (2018); 10.1063/1.5030032 Design and experimental demonstration of a circularly polarized mode converter for high-power microwave applications Review of Scientific Instruments 89, 084701 (2018); 10.1063/1.5035179 TDLAS-WMS second harmonic detection based on spectral analysis Review of Scientific Instruments 89, 083106 (2018); 10.1063/1.5031683
REVIEW OF SCIENTIFIC INSTRUMENTS 89, 085110 (2018)
Compound fault extraction method via self-adaptively determining the number of decomposition layers of the variational mode decomposition Ziying Zhang,1,2,a) Xi Zhang,1 Panpan Zhang,2 Fengbiao Wu,2 and Xuehui Li1
1 School
of Mechanical, Electronic and Information Engineering, China University of Mining and Technology (CUMT), Xueyuan Road, Beijing 100083, China 2 Shanxi Institute of Energy, Daxue Road, Jinzhong 030600, China
(Received 25 April 2018; accepted 15 July 2018; published online 9 August 2018) Local mean decomposition (LMD) is a self-adaptive method, which has been widely applied to extract early fault signals from bearings. However, mode mixing occurs during the decomposition process. Moreover, in processing signals with strong noise, false frequency components can be generated by variational mode decomposition (VMD). To address these problems, a weak fault extraction method based on VMD is proposed for rolling bearings. This method regards LMD and the combination production function (CPF) as prefilters for VMD. First, LMD is used for denoising the original signal, and then the CPF components that contain the fault information are combined into a new signal. Second, this method determines the decomposition level K of the VMD from the spectral peaks of the recombined signal. Finally, this method decomposes the recombined signal using the VMD. The main contributions of the proposed method are (i) the CPF method is employed for adaptively de-noising, and the power of the fault feature can be improved; (ii) the decomposition level K of the VMD can be determined adaptively. After processing a simulated signal, fault information of the gears and rolling elements is successfully extracted, thereby demonstrating the feasibility of the presented method. Moreover, the feasibility of the proposed method is further demonstrated in a comparison of results with those obtained from the MOMEDA (Multipoint Optimal Minimum Entropy Deconvolution Adjusted) algorithm. Published by AIP Publishing. https://doi.org/10.1063/1.5037565
I. INSTRUCTION
The working status of bearings, the dynamic transmission parts of rotating machines, is directly related to the operation status of the whole machine. Hence, the health status of bearings is a concern.1 If the bearings fail, they may affect the quality of production and cause damage or even result in casualties. Because of influences from the working environment, early bearing-fault signals are weak and are easily drowned out by noise. Extracting fault signals from noise has always been a major problem in fault diagnosis. Local mean decomposition (LMD) is a recent time-frequency decomposition method proposed by Jonathan Smith in 2005 based on the empirical mode decomposition (EMD).2,3 Many researchers have found that LMD alleviates the issue of mode mixing in comparison with EMD, but mode mixing in LMD still affects the decomposition results and generates false components. Three methods—the LMD method based on the differential rational spline and K-L divergence proposed by Li,4 the fault diagnosis method based on an improved LMD,5 and a multifractal theory analysis of weak-fault diagnosis of rolling bearings6 —have achieved good results in mitigating mode mixing. Nevertheless, a mixing of noise in the actual signal is produced in the LMD process, which increases the difficulty of extracting fault information. LMD was compared with EMD a) Author
to whom correspondence should be addressed: tbp1600401010@ student.cumtb.edu.cn.
0034-6748/2018/89(8)/085110/7/$30.00
in regard to decomposition efficiency by Cheng;7 the results showed LMD to be superior. Zhang8 combined fuzzy C-mean clustering and LMD to produce a classification and recognition of the different types of bearing faults. Based on the wavelet packet energy entropy and the LMD proposed by Li,9 the signal-to-noise ratio of the processed vibration signal was improved. Variational mode decomposition (VMD) is a self-adaptive decomposition method proposed by Dragomiretskiy et al. in 2014. Compared with the existing fault diagnosis methods, VMD has the advantages of, for example, fast convergence speed and good noise robustness. However, VMD is affected by the decomposition level K, on which there is little research at present. Wang10 used correlation coefficients between the high-frequency modes and the original signal to determine the K value. Ling11 performed a VMD analysis using the principle of a similar frequency center to determine the K value; good results were obtained. VMD is unable to highlight weak fault signals when the signal-to-noise ratio is low. Wang12 completed a fault diagnosis by combining VMD and singular value decomposition (SVD). Yao13 used VMD and the robust independent component analysis to suppress noise and extract the fault information accurately. In strong noise environments, VMD decomposition is influenced by the K value, so a self-adaptive method for determining the K value is crucial. LMD is a self-adaptive decomposition method that can inhibit the noise, but mode mixing issues arise. In other words, the same feature components are
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decomposed into different production function (PF) components, causing energy leakages. To determine the number of fault features in the original vibration signal and increase the energy of fault features, the PF components that have the same frequencies are recombined using the combination production function (CPF) method. This paper proposes a fault diagnosis method based on LMD-CPF-VMD. The noise signal is first decomposed using LMD, and then the decomposed PFs are recombined. During this process, not only is the signal denoised but also mode mixing incurred using LMD is suppressed in the recombination process. At the same time, when performing an envelope analysis of the PF components that contain the fault information, the K value of VMD is determined from the number of peaks in the envelope results. After dealing with the fault signal, the fault information is successfully extracted. The main contributions of the proposed method are as follows: First, considering that every intrinsic mode function (IMF) decomposed by VMD has a center frequency and the center frequency is arranged in order from low frequency to high frequency, the frequency band of noise in the gearbox is relatively wide, so there are often some miscellaneous items in VMD decomposition. The frequency is the peak of the center frequency, resulting in misdiagnosis. Therefore, the LMD needs to pre-filter the original signal. Second, this method is better used for multi-fault coexistence because LMD is an adaptive decomposition method. Despite modal aliasing, it can decompose different time scales into different PF serious diseases. After CPF processing, the decomposition layer of VMD can be determined by the number of frequency, and the diagnostic efficiency of the composite fault can be improved. Therefore, the method used in this paper is suitable for feature extraction of strong noise and complex faults. The rest of the paper is organized as follows: Sec. II is the background and proposed method; Sec. III is the LMDCPF-VMD fault diagnosis method; simulation signal analysis is given in Sec. IV; analysis of measured signals is given in Sec. V; and finally, the conclusions are given in Sec. VI.
II. BACKGROUND AND PROPOSED METHOD A. LMD-CPF method
Multiple fault vibration signals from gearboxes are multicomponent signals. The LMD method adaptively decomposes such signals into a number of PF components. Each PF level contains the features of the original signal, noise, and single frequency components; components with multiple frequencies are also possible. The LMD steps of the original vibration signal x(t) are as follows: (1) Find all the local extremum points ni of the original signal x(t), and then calculate all the local extremum mean values mi and the envelope estimation values ai from the local extremum points; here, ni + ni+1 , (1) mi = 2
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n − n i+1 i ai = . (2) 2 Next, connect all the adjacent mean values to the envelope estimation values. The moving average method is applied to obtain the local mean function myz (t) and the envelope function ayz (t), where y is the number of local mean and envelope functions and z indicates the number of iterations. (2) Separate all the local mean functions from the original signal x(t). Once the function hyz (t) has been obtained, then the function syz (t) has been demodulated, h11 (t) = x(t) − m11 (t) h12 (t) = s11 (t) − m11 (t) , .. . h1n (t) = s1(n−1) (t) − m1n (t)
(3)
where syz (t) is defined by s11 (t) = h11 (t)/a11 (t) s12 (t) = h12 (t)/a12 (t) . .. .
(4)
s1n (t) = h1n (t) − a1n (t) Then the same method is employed to calculate the envelop estimation function a12 (t) of s11 (t), if a12 (t) , 1; then s11 (t) is not a pure frequency-modulated signal, which means that an iterative procedure needs to be repeated for s11 (t) until the calculated s1n (t) becomes a pure frequencymodulated signal, that is, −1 ≤ s1n (t) ≤ 1 and a1(n+1) (t) = 1 can be satisfied. (3) Multiply all the calculated envelope functions to obtain an envelope signal a1 (t) = a11 (t)a11 (t) · · · a1n (t) =
n X
a1p (t).
(5)
p=1
The first PF component can be obtained from the envelope signal a1 (t) and the pure frequency-modulated signal s1n (t), PF1 (t) = a1 (t)s1n (t).
(6)
(4) PF 1 (t) is separated from x(t), and hence u1 (t) is obtained. Then, the above calculation procedure is repeated using u1 (t) as a new original signal and iterated until uq (t) is a monotone function u1 (t) = x(t) − PF1 (t) u2 (t) = u1 (t) − PF2 (t) . .. .
(7)
uq (t) = uq−1 (t) − PFq (t) (5) Next the original signal is decomposed into a sum of q PF components and a residual component uq (t), x(t) =
q X
PFq (t) + uq (t).
(8)
l=1
The CPF can be understood as a combination of PF components. After LMD has generated all PF components, those
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that contain the fault information are recombined using the envelope analysis of the signal to form a new component, the role of which is to mitigate mode mixing during LMD and suppress noise in the signal. B. VMD method
The specific construction steps of the constrained variational model are as follows. Step 1: For the input signal x(t), through the Hilbert Transform (HT), we can get the analytic signal of each mode function uk (t). Step 2: The center frequency ωk of each mode function uk (t) is estimated, and its spectrum is moved to the baseband. Step 3: After step 2, the bandwidth is estimated through the H 1 Gauss smoothness. The final constraint variational model can be expressed by Eq. (10), #
2 X
"
∂t (σ(t) + j )uk (t) e−jωk t
min
(uk )(ωk ) πt 2 k
. (9) X uk = x(t) s.t. k In Eq. (9), ∂t indicates the partial derivative of t, σ(t) is the impulse function, and {uk } = {u1 , ..., uK } represents the K IMFs obtained by VMD of the original signal x(t), and {ωk } = {ω1 , ..., ωK } represents the central frequency of each IMF component. In order to find the optimal solution to the above variational problem, the following form of Lagrange function is introduced: ! # 2 X "
σ(t) + j u (t) ei−jωk t
L({uk }, {ωk }, λ) = a k
πt
Xm+1 . λ n+1 (ω) ← λ n (ω) + τ x(t) − k
Renew ukn+1 , ωkn+1 , and λ separately. (4) Repeat step 3; the criterion to stop the iteration is X
un+1 − un
2
un
2 < ε. (14) k 2 k 2
k k
The decomposition is then completed. III. LMD-CPF-VMD FAULT DIAGNOSIS METHOD
The LMD-CPF-VMD method involves the following procedural steps: First, in using LMD to decompose the original signal, LMD acts as a prefilter that suppresses noise contained in the signal. Second, summing the effective PF components using the CPF method mitigates mode mixing induced by the LMD. Suppose there are four levels of PF components produced by the LMD decomposition, and after the envelope analysis, assume that levels 1, 3, and 4 contain fault information but none in level 2. The CPF then outputs CPF1 = PF1 + PF3 + PF4 ,
(15)
CPF2 = PF2 ,
(16)
where CPF1 is retained for the reason that it is an effective integration signal, and CPF2 is discarded because it is not a
2
k
2
X
+
x(t) − uk (t)
k 2 + * X + λ(t), x(t) − uk (t) ,
(10)
k
where a is a penalty factor and λ is a Lagrange multiplier. The process of decomposing the original signal into K components using VMD is as follows: (1) {uk }, {ωk }, λ1 , and n are initialized to 0. (2) After n = n + 1, the cycle iteration begins. (3) k = k + 1, before k = K, according to x(ω) − ukn+1 (ω) ←
K X
uin+1 (ω) −
i=1,i
K X
uin (ω) +
i=1,i
1 + 2α(ω − ωkn )2 ∞ ωkn+1 ←
λ n (ω) 2 , (11)
2 ω ukn+1 (ω) dω
0
∞
,
(12)
2
un+1 (ω) dω k 0
(13)
FIG. 1. LMD-CPF-VMD method flow chart.
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valid signal. After obtaining the recombination signal CPF1 , the decomposition level K of VMD is determined depending on the numbers of peaks in the PF components obtained by the envelope analysis. Finally, VMD is used to decompose the new signal. Mode mixing occurs during the LMD process. Noise affects the VMD, the results of which become distorted from strong noisy signals. At the same time, if the number of the decomposition levels is unreasonable, energy leakages occur. The CPF-VMD method overcomes this mode-mixing issue, suppresses the noise in the signal, and determines a reasonable K value for the VMD process. First, we determine whether the PF components contain fault information using the correlation coefficient method and then perform an envelope analysis of all the PF components after the LMD process. Second, we recombine the PF components that contain the same frequency components and determine the K value in accordance with the number of frequency components. Finally, we conduct the VMD decomposition and perform an envelope analysis again to extract the fault features. The flow chart of the LMD-CPF-VMD method is given in Fig. 1.
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FIG. 3. VMD decomposition of the simulation signal.
IV. SIMULATION SIGNAL ANALYSIS
The simulation signal x(t) selected for the experiment is made up of two impulse signals and a random noise signal. In Fig. 2, from top to bottom, are plots of an 80-Hz impulse signal, a 150-Hz impulse signal, the noise signal, and the synthetic signal. A comparison of the synthetic signal with the noise signal shows that noise is strong with the impulse signals being completely submerged. Figure 3 shows the VMD results of the simulation signal, where K takes the value 5. The frequencies of the IMF components are 40 Hz, 150 Hz, 180 Hz, 295 Hz, and 370 Hz. Only the 150 Hz is a strong impact signal frequency; the other components are different from the frequencies of the impulse signals in the simulation signal. However, the strong impact signal can only be obtained from the decomposition results. The weak signal still has not be extracted, which proves that the noise has a great influence on the VMD result.
FIG. 2. Composition of the synthetic signals.
FIG. 4. LMD results from the simulation signal.
LMD has a certain denoising capability, but mode mixing has appeared during decomposition. Figure 4 shows that the LMD results are obtained for the simulation signal and proves that the LMD is capable of providing a certain level of noise reduction. Moreover, the periodic impact signals appear in the PF components. Figure 5 shows the results of the envelope analysis for the three PF components. The observations from these results are (i) The PF1 signal is complex, and although there are many peak values, the effective peak frequencies cannot be reached and (ii) From the PF2 and PF3 signals, peak values appear
FIG. 5. Results from the envelope analysis of the PF components.
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FIG. 6. Recombined signal of CPF1 . FIG. 8. Gearbox simulation test table. 1—speed adjusting motor, 2— coupling, 3—paternity test gearbox, 4—speed torsion instrument, 5—torsion bar, 6—test gearbox, 7—three-dimensional accelerometer 1, and 8—threedimensional accelerometer 2.
FIG. 7. VMD decomposition of the recombined signal CPF1 .
clearly at frequencies 80 Hz, 150 Hz, and 300 Hz. In the LMD process, mode mixing is obvious. LMD is first used in processing the original signal, and the decomposition results are shown in Fig. 4. The correlation coefficients between the PF components and the original signal are calculated next: PF1 yields 0.421, PF2 yields 0.683, and PF3 yields 0.644. From the definition of the correlation coefficient, if its value is greater than 0.5, the two components are strongly correlated.14 Hence PF2 and PF3 are dominant, and the envelope analysis results of the PF components show that the main fault information is decomposed into PF2 and PF3 , which are then recombined by CPF. The signal after recombination is shown in Fig. 6. With three frequency peaks in PF2 and PF3 , we assume that there are three effective frequency components in the signal. The K value is reset from 5 to 3. Labeling the recombined signal as CPF1 , it is decomposed by VMD. From the results of the envelope analysis for the IMFs obtained (Fig. 7), there are three components with frequencies at 80 Hz, 120 Hz, and 150 Hz. Recalling that the frequencies of the two impact signals in the simulation signal are 80 Hz and 150 Hz, it can be concluded that the effective fault components have been extracted.
FIG. 9. Fault signal.
Figures 9 and 10 present the time-domain plot and the LMD decomposition of the fault signal, respectively. In PF1 , there is a significant peak at the fault frequency of 90 Hz, and there is a weak peak at 180 Hz. In PF2 , there are significant peaks at 50 Hz and 90 Hz. The rolling elements’ fault frequency 72 Hz is not obtained from the decomposition results. After calculation, the correlation coefficients of the three PF components are 0.921, 0.501, and 0.215. The CPF signal obtained by recombining PF1 and PF2 (Fig. 11) shows that the signal has peaks at 50 Hz, 90 Hz, and 180 Hz, so the decomposition level K of VMD is determined to be 3. Figure 12 shows the VMD decomposition results of the recombined signal.
V. ANALYSIS OF MEASURED SIGNALS
A closed power flow gearbox test bed for loading devices with a torsion bar is shown in Fig. 8. The rotational speed of the torsion rod is regulated by the speed-adjusting motor. The rotational speed was set to 1200 r/min. Tapered roller bearings of type 32 212 were employed to mount the rod. To simulate an actual fault, a bearing in the outer-ring raceway was pitted under an electric discharge. Two three-dimensional accelerometers (YD77SA) were used to register the vibrational signal at a sampling rate of 8000 Hz. The calculated gear fault frequency is 90 Hz, and the rolling element fault frequency is 72 Hz.
FIG. 10. LMD decomposition of the fault signals.
FIG. 11. CPF results.
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Rev. Sci. Instrum. 89, 085110 (2018) TABLE I. Correlation coefficients of IMF and component signals obtained by two algorithms. Algorithm LMD-CPF-VMD CPF-MOMEDA
FIG. 12. VMD decomposition results of a recombined signal.
From the envelope analysis results of the recombined signals, we find that the three IMF components have peaks at 72 Hz, 90 Hz, and 120 Hz. Recalling that the gear fault frequency is 90 Hz and the rolling elements fault frequency is 72 Hz, the results of the analysis demonstrate that the fault information has been extracted successfully. To improve the diagnosis efficiency, a multiple-point kurtosis was used in a comparison of our results with those obtained using the MOMEDA algorithm in a simple calculation. The rotation period of the shaft is 400 (the unit is the number of points), the bearing fault period is 111.1, the gear meshing frequency is 180 Hz, and the gear meshing period is 22.4. The multiplepoint kurtosis of the recombined signal (Fig. 11) is presented in Fig. 13. Taking out the failure cycles of the gears and the bearings and then denoising via MOMEDA, we find the periodic intervals of [15–25] and [105–115] in accordance with the bearing fault cycle and gear meshing cycle. With a step length
FIG. 13. MKurt spectrum of CPF2 .
Re2
Re3
0.8949 0.7832
0.8742 0.6957
0.8344
of 0.1, the results (Fig. 14) show that we have extracted the fault cycle components. The feasibility of the proposed method has been further demonstrated by the measured signal. Correlation coefficients are obtained for the above two methods. The results are shown in Table I. It can be found that the correlation coefficients of the three IMFs obtained by LMD-CPF-VMD and the original signals are 0.8949, 0.8742, and 0.8344, respectively; the correlation coefficients of CPF-MOMEDA and the original signals are 0.7832 and 0.6957, respectively. Therefore, the decomposition result of the method proposed in this paper is closer to the characteristics of the original signal.
VI. CONCLUSIONS
LMD has adaptively decomposed the fault signal into several PF components based on frequency, but mode mixing occurs during the decomposition process. Before decomposing, VMD needs the decomposition level K, its value being directly related to the accuracy of the decomposition results. Determining the K value from experience lacks reliability. When confronted with strong noise, VMD distorts the signal and produces false IMF components. LMDCPF-VMD is based on the VMD method. After a LMD of the original signal, CPF recombination is used to mitigate mode mixing. The noise contained in the signal is suppressed, enabling the decomposition level K to be determined depending on the amount of frequency information in the PF components. The recombined signal is decomposed by VMD. This paper has demonstrated the reliability of the proposed method through separate analyses of a simulation signal and a measured fault signal. The fault information was extracted successfully. The proposed method lays the foundation for subsequent research employing VMD. 1 S.
FIG. 14. Denoising for CPF MOMEDA.
Re1
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