entropy Article
Multi-Fault Diagnosis of Gearbox Based on Improved Multipoint Optimal Minimum Entropy Deconvolution Fuhe Yang, Xingquan Shen * and Zhijian Wang * School of Mechanical Engineering, The North University of China, Taiyuan 030051, China;
[email protected] * Correspondence:
[email protected] (X.S.);
[email protected] (Z.W.); Tel.: +86-135-136-26928 Received: 5 July 2018; Accepted: 9 August 2018; Published: 17 August 2018
Abstract: Under complicated conditions, the extraction of a multi-fault in gearboxes is difficult to achieve. Due to improper selection of methods, leakage diagnosis or misdiagnosis will usually occur. Ensemble Empirical Mode Decomposition (EEMD) often causes energy leakage due to improper selection of white noise during signal decomposition. Considering that only a single fault cycle can be extracted when MOMED (Multipoint Optimal Minimum Entropy Deconvolution) is used, it is necessary to perform the sub-band processing of the compound fault signal. This paper presents an adaptive gearbox multi-fault-feature extraction method based on Improved MOMED (IMOMED). Firstly, EEMD decomposes the signal adaptively and selects the intrinsic mode functions with strong correlation with the original signal to perform FFT (Fast Fourier transform); considering the mode-mixing phenomenon of EEMD, reconstruct the intrinsic mode functions with the same timescale, and obtain several intrinsic mode functions of the same scale to improve the entropy of fault features. There is a lot of white noise in the original signal, and EEMD can improve the signal-to-noise ratio of the original signal. Finally, through the setting of different noise-reduction intervals to extract fault features through MOMED. The proposed method is compared with EEMD and VMD (Variational Mode Decomposition) to verify its feasibility. Keywords: improved multipoint optimal minimum entropy deconvolution; multi-fault; fault diagnosis
1. Introduction The gearbox in mechanical equipment is one of the most important power transmission components; its health directly affects whether the mechanical equipment can work normally. If we can accurately predict the faults’ location, the huge human and financial losses that are caused by faults can be effectively avoided, so research of new composite fault-diagnosis methods play a decisive role in the normal operation of the gearbox. When the inner and outer ring or the rolling body of the gear and bearings fails, the failures are coupled to each other, which makes faults often be represented in the form of compound faults, and periodic pulses will appear in vibration signals [1–3]. The compound fault feature extraction of rotating machinery is still a big challenge [4,5], especially in strong-noise environments. So we are still in need of a lot of research to solve how to extract compound fault features in strong-noise environments. In the aspect of compound fault diagnosis, Ensemble Empirical Mode Decomposition (EEMD) can self-adaptively resolve different feature components into different modal functions [6]. But, there is often modal aliasing in EEMD; so-called modal aliasing refers to the same intrinsic mode function (IMF) containing different feature components, or the same timescale is broken down in different IMFs, which further leads to entropy loss. Although EEMD improves the precision of the decomposition by adding different white noises to the original signal and repeatedly
Entropy 2018, 20, 611; doi:10.3390/e20080611
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asking the mean of IMFs, the literature [7–9] confirmed EEMD is more effective and accurate to the fault feature extraction of rotating machinery; it can self-adaptively resolve a complex signal into several IMFs. If the signal frequency band is too wide or the signal–noise ratio (SNR) too low, it will affect the decomposition efficiency of the EEMD [10]. The reason is that decomposition accuracy is greatly affected by the level of added white noise. If the level of white noise selected is too large, it will lead to overdecomposition. Mode aliasing still exists while it is too small, but it will not be enough to change the distribution of extreme value points [11]. The literature [12] indicates that how to self-adaptively select the level of white noise has still not been resolved. Wang [13] optimize the choice of white noise in EEMD by the combined modal-function method and, at the same time, improve the efficiency of its decomposition, but through the analysis of the simulation signal and the measured signal, modal aliasing still can’t be avoided completely. To sum up the above analysis, EEMD has been successfully applied in fault diagnosis, but due to the inappropriate selection for white-noise levels, there is still modal aliasing, which leads to entropy losses; therefore, we put forward combined IMFs (CIMFs) so that the original signal can be self-adaptively resolved into a different frequency band when a white-noise level is given, by combining several adjacent IMFs with the same frequency band after being resolved by the EEMD. In 2016, McDonald [14] proposed a Multipoint Optimal Minimum Entropy Deconvolution (MOMED) method for rotating machine fault extraction. It is an improvement of minimum entropy deconvolution, which is to solve the optimal filter with the kurtosis maximum as the objective function, but it can only highlight a few pulses, whereas MOMED can highlight more pulses. MOMED uses a target to define the location and weightings of the impulse train obtained by deconvolution. These target goals are very suitable for feature extraction of a vibration source of a rotating machine that generates a pulse per rotation. However, when multiple faults coexist or under strong background noise, it is difficult to accurately extract the fault period components due to the influence of more than one fault period or background noise. Therefore, it is necessary to pretreat the vibration signal. When multiple faults coexist, coupled with white-noise pollution, the frequency band of complex vibration signals is relatively wide, so it is difficult for traditional FFT to identify each fault feature, and it is likely to cause incorrect diagnosis. The EEMD can adaptively modulate the original signal. In addition, EEMD can separate signals in order of high and low frequencies. In particular, when multiple faults coexist, different timescales are decomposed into different intrinsic functions, and the failure frequency is determined by solving the IMF of each layer. However, a large number of experiments have shown that EEMD cannot completely separate different timescales, and there are serious modal aliasing, which results in entropy leakage. We can determine the IMF with the same fault characteristics in advance by using FFT in each layer of the IMF; we can separate the different frequency bands in original signals by CIMF, and these operations can not only decompose the original signal into CIMF with different scales, but also make the optimization of the parameters for MOMED noise reduction. The CIMF can adaptively divide the original signal into different frequency bands and improve the entropy of the IMF. The period of the fault pulse in each CIMF can be determined by FFT. Finally, the appropriate period interval is input and the fault feature is extracted by MOMED. This paper explores a new method of fault feature extraction based on IMOMED, which can accurately identify the fault characteristics of the gearbox, and provides a new idea for fault feature extraction of rotating machinery. 2. Background and New Method 2.1. Introduction of EEMD To solve the problem of mode mixing, EEMD is introduced based on the statistical properties of white noise. The EEMD algorithm can be given as follows. (1) Given x (t) is an original signal, add a random white noise signal nj(t) to x (t) xj(t) = x (t) + nj(t)
(1)
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where xj(t) is the noise-added signal, j = 1, 2, 3, . . . , M and M is the number of trial. (2) Decompose x j (t) into I ( I MFs)Ci,j (i = 1, 2, 3, . . . , I) using EMD method. Where Ci,j denotes the ith IMF of the jth trial, and I is the number of IMFs. (3) If j < M, then go to step (1) with j = j + 1. Repeat steps (1) and (2) repeatedly, but with different white-noise series each time. (4) Calculate the ensemble means of corresponding IMFs of the decompositions as the final result: m
∑ Ci,j
Ci =
! /M
(2)
j =1
where i = 1, 2, 3 . . . , I. (5) Ci (i = 1, 2, 3, . . . , I ) is the ensemble mean of corresponding IMF of EEMD. 2.2. Introduction of Multipoint Optimal Minimum Entropy Deconvolution In 1984, Cabrelli [15] proposed a new norm for impulse deconvolution, called the norm, and proved the deconvolution problem geometrically. The D-norm deconvolution problem can solve filter coefficients by an exact noniterative process. In 2016, McDonald [14] proposed a deconvolution target of multiple impulses at known locations based on Multi D-Norm, allowing for periodic impulse train deconvolution target goals, and introduced this maximization problem as MOMED. Assuming that the collected response signal is as shown in Equation (3), y(n) = h(n) x (n) + e(n)
(3)
where e(n) is the white noise, x(n) is the impulse train, h(n) is the transfer function, and y(n) is the collected vibration signal. The essence of the MOMED algorithm is to find a FIR filter that resets the input signal y(n) through the output signal x(n). → →
Multi D-Norm = MDN ( y , t ) = MOMED:
1 →
→T →
t y
(4)
→
k t k kyk
→T →
→ →
max MDN ( y , t ) = max → f
t y
(5)
→
kyk
→
The expected output of impulse deconvolution is a continuous spike. Where the target vector, t , is a constant vector that defines the location and weightings of the goal impulses to be deconvolved, h iT → → for example, t = . The target vector t will aim to deconvolve 0 0 0 1 0 0 0 1 0 0 two impulses in the output signal: one impulse at n = 4 and the other at n = 8, which represent the maximum of entropy. This Multi D-Norm is normalized to between 0 and 1, where a value of 1 indicates that the optimal target solution was reached. With the optimal target solution, the fault period with different sampling rates can be extracted, and the period of different fault characteristics at →
the same sampling frequency can be identified. Therefore, the target vector t can be used to determine the separation and position of the impulse signal. We solve the extrema of Equation (5) by taking the →
derivative with respect the filter coefficients ( f = f 1 , f 2 , . . . , f L ): →T → d t y d t y d t y d t y = → 1→ 1 + → 2→ 2 + , . . . , + → N − L→ N − L → → kyk kyk df d f kyk d f kyk df N represents the number of sampling points, and L represents the size of the filter.
(6)
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As we know:
d tk yk →
→ −1
= kyk
→
d f kyk
→
→ −3
tk Mk − k y k
→ Mk =
→
t k y k X0 y
x k + L −1 x k + L −2 .. . xk
Therefore Equation (6) can be written: →T → → → → → −3 → T → → → −1 d t y M − k y k t y X0 y M + t M + , . . . , + t = k y k t 2 2 N−L N−L 1 1 → → kyk df
(7)
With the simplification: →
→
→
→
t 1 M 1 + t 2 M 2 + , . . . , + t N − L M N − L = X0 t →
and solving for extrema by equating to 0 , Equation (7) becomes: → −1
kyk that is:
→ −3 → T →
→
X0 t − k y k →T →
t y
→ 2
→
→
t y X0 y = 0
→
→
X0 y = X0 t
kyk →
→
Since y = X0T f and assuming ( X0 X0T ) →T →
t y
→ 2
kyk
→
−1
exists:
f = ( X0 X0T )
−1
→
→
X0 t
(8) →
−1
→
Since multiples of f are also solutions to Equation (8), multiples of f = ( X0 X0T ) X0 t are solutions to the MOMEDA problem. This method can completely avoid the iterative operation, regardless of whether the cycle is the integer and the length of the filter on the impact of noise reduction. The multistage transmission gearbox has wide frequency distribution and a large number of fault cycles. The noise reduction effect of MOMEDA is affected by the noise-reduction interval. When multiple faults coexist, only one periodic shock can be extracted for each interval of noise reduction. The smaller the interval, the more accurate the search, so you need to set the noise-reduction interval in advance. 3. Multi-Fault Feature Recognition under Strong Noise EEMD is an adaptive noise-reduction method, which can improve the SNR [3–8]. However, when the white-noise amplitude is chosen improperly, the same fault feature is decomposed into different intrinsic modal functions, resulting in the weakening of the fault characteristic entropy, especially since the weak component of the composite fault is more likely to disappear. In order to overcome mode aliasing and improve the entropy of the same modal function, the CIMF method was chosen to improve frequencies of the original signal [13]. The specific method is as follows: 1. Setting the two parameters of EEMD. If the added white-noise amplitude is small, the distribution of the extreme points cannot be changed, and the added noise is meaningless.
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The larger amplitude will have excessive influence on the original signal, and the correct decomposition result cannot be obtained. In this paper, white-noise amplitude of the EEMD was determined by comparing the SNR after reconstructing the normal components. The method was as follows: Noise amplitude gradually increased from zero, and the step size is 0.01. (1) Adding 100 times white noise with amplitudes to the simulated signal x; (2) Decomposing the signal after adding noise with EEMD; (3) Calculating the SNR after reconstructing the normal components; (4) When the SNR reached the maximum, the corresponding noise amplitude was the best amplitude. The number of integrations had little effect on the decomposition efficiency of EEMD. In this paper, it was set to 100. 2. Resolve the signals by EEMD, and the IMFs that have correlation with the original signals are determined by correlation coefficient; the correlation coefficient is defined as shown in Equation (9): ρ xy
E ( x − µ x )(y − µy = σx σy
(9)
The relevance threshold in the article is 0.3, which is greater than 0.3 for correlation, otherwise it is irrelevant. 3. Solve the modal function FFT of each layer, determine their central frequency, and recombine the frequency band of the same frequency or integer times. By combining the neighboring IMFs that contain the same frequency, we obtain the CMF as follows [13,16]: IMF1, IMF2 and IMF3 have the same fault feature, IMF4, and IMF5 have the same fault feature, IMF6 contains the same frequency. CI MF1 = I MF1 + I MF2 + I MF3
(10)
CI MF2 = I MF4 + I MF5
(11)
CI MF3 = I MF6
(12)
4. Determine the frequency of each CIMF above and calculate the corresponding period. Now that you’ve determined the component of frequency, why do you have to calculate the period? First, the result of EEMD decomposition is self-adaptive, but after EEMD decomposition, there exist noises in each layer of IMFs, and the noises consist of two parts: one is contained in the original signal, and the other is the adding noise of EEMD algorithm. These have not been completely neutralized, so the feature extraction of CIMF in entropy concentration is required. After the signal is decomposed by EEMD, the signal will be decomposed into high-order modal functions from high frequency to low frequency. Each layer has a fixed center frequency. The center-frequency reciprocal is the time period. Multiplying the period by the sampling frequency is the fault period (sampling point). Different frequencies represent different periods. 5. Set three different search intervals and use MOMED to extract the features of different fault cycles. It should be noted that MOMED can continuously extract a series of periodic shocks, but it has the following characteristics: (1) Given a random white-noise signal, a weak periodic pulse can be extracted by MOMED within a fixed noise-reduction interval, because the goal of MOMED is to solve the multipoint kurtosis in a series of signals and maximize them, then search for periodic pulses; there must be a result after each solution, and then it can extract the corresponding periodic pulses. As is shown in Figure 1, the original signal is a white noise.
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Figure 1. 1. Noise and Multipoint Optimal Entropy Minimum Entropy (MOMED) Deconvolution (MOMED) Figure Noise and Multipoint Optimal Minimum Deconvolution noise-reduction results. noise-reduction results. Figure 1. Noise and Multipoint Optimal Minimum Entropy Deconvolution (MOMED)
1. Noise and Multipoint Optimala Minimum Entropy pulse Deconvolution (MOMED) AfterFigure the noise reduction by MOMED, 13.3 periodical is extracted. Surprisingly, noise-reduction results. Afternoise-reduction the noise reduction by MOMED, a 13.3 periodical pulse is extracted. Surprisingly, after results. after running the program again, we would get another different periodic impact. In engineering running After the program again, weby would geta 13.3 another different periodic impact. In engineering the results noise reduction MOMED, periodical pulse is extracted. Surprisingly, after applications, such would lead to misdiagnosis. Therefore, a reasonable noise-reduction interval After the noise reduction by MOMED, a 13.3 periodical pulse is extracted. Surprisingly, after applications, such resultsagain, wouldwelead to misdiagnosis. Therefore, a reasonable noise-reduction running the program would get another different periodic impact. In engineering needsrunning to be determined to ensure that every noise-reduction interval has an impact component. thetoprogram again, we towould get that another different periodic impact. In engineering interval needs be results determined ensure every noise-reduction interval has an impact applications, such would lead to misdiagnosis. Therefore, a reasonable noise-reduction Regarding the selection method of the interval, the article gives the simulation signal and applications, such results would leadperiod to misdiagnosis. Therefore, a reasonable noise-reduction component. Regarding the selection method of the period interval, the article gives simulation interval needs to be determined to ensure that every noise-reduction interval has the an impact explains accordingly. First, MOMEDtocan onlythat extract thenoise-reduction continuous pulse of ahas single cycle, so it interval needs to be determined ensure every interval an impact Regarding the selection of the period gives the simulation signalcomponent. and explains accordingly. First,method MOMED can onlyinterval, extract the the article continuous pulse of a single component. Regarding thethe selection method of the period interval,fault the article givesinto the different simulationIMFs. is necessary to explains decompose signal and decompose different features signal and accordingly. First, MOMED candecompose only extractdifferent the continuous pulse ofinto a single cycle,signal so it is necessary to decompose the signal and fault features different andofexplains accordingly. First, MOMED can only extract the continuous pulse of a single The selection the cycle takes into account that, ifdecompose the selected range is too large, a different shock pulse cycle, so it is necessary to decompose the signal and different fault features into IMFs.cycle, The so selection of thetocycle takes the into account that, if thedifferent selected range is too a shock itfor is necessary decompose signal and decompose fault features intolarge, different that searches half, 0.5, or 0.75 times of the cycle will occur, resulting in misdiagnosis. Therefore, IMFs. The selection of the cycle takes into account that, if the selected range is too large, a shock pulseIMFs. that The searches forofhalf, 0.5, or 0.75into times of the occur, resulting in misdiagnosis. selection the cycle takes that,cycle if thewill selected range is too large, a shock the article a small asaccount possible, but the of the is greater pulse selects that searches forinterval half, 0.5,asormuch 0.75 times of the cycle willlower occur,limit resulting in interval misdiagnosis. Therefore, the article selects a small interval as much ascycle possible, but theresulting lower limit of the interval is pulse that searches for half, 0.5, or 0.75 times of the will occur, in misdiagnosis. Therefore, article selects a small as is much as possible, butcycle, the lower the intervalis is over, than 0.75 times.the The upper limit of theinterval interval greater than one but limit if theofovertaking Therefore, thetimes. article The selects a small interval asinterval much as possible, but theone lower limitbut of the interval is greater thanthan 0.75 upper limit greaterthan than cycle, if overtaking the overtaking greater 0.75 times. The upper limitofofthe the interval isisgreater one cycle, but if the the amount of calculation increases. greater than 0.75of times. The upper limit of the interval is greater than one cycle, but if the overtaking is over, the amount increases. is over, the amount calculation ofnoise-reduction calculation increases. (2) An appropriate interval needs to be set each time. For example, a period of is over, the amount of calculation increases. (2) An noise-reduction tobe beset seteach each time. example, a period (2) appropriate An appropriate noise-reductioninterval interval needs needs to time. ForFor example, a period of of impact oscillation is 100 (the number of sampling points); if set theeach noise-reduction interval is setof in (60, (2) An appropriate noise-reduction interval needs to be time. For example, a period impact oscillation is 100 (the number thenoise-reduction noise-reduction interval in (60, impact oscillation is 100 (the numberofofsampling sampling points); points); ififthe interval is setisinset (60, 100) or (20, 200), extracting the cycle of 100 and 25 by MOMED is shown in Figures 2 and 3, and impact oscillation is 100 (the number of sampling points); if the noise-reduction interval is set in (60,there 100) or 200), (20, 200), extracting cycleofof100 100and and 25 25 by by MOMED in in Figures 2 and 3, and 100) or (20, extracting thethe cycle MOMEDisisshown shown Figures 2 and 3, there and there 100) or (20, 200), extracting theincycle of 100 noise-reduction and 25 by MOMED is shown Figures 2 and 3, and there are multiples and factors of 100 the large interval, sointhe components with a period are multiples and factors of 100 in the large noise-reduction interval, so the components with a are multiples andand factors of of100 noise-reduction interval, the components with a multiples factors 100ininthe thelarge large noise-reduction interval, so so thesince components with a of 25 are are first extracted by MOMEDA. This result still leads to misdiagnosis, the period becomes period of 25 are first extracted by MOMEDA. This result still leads to misdiagnosis, since the period period of 25 are first extracted by MOMEDA. This result still leads to misdiagnosis, since the period period 25original are first extracted bythe MOMEDA. This result still leads to misdiagnosis, since a quarter of of the and corresponding frequency becomes four times, sothe it isperiod becomes a quarter of signal the original signal and the corresponding frequency becomes four times, sonecessary it is becomes a quarter of the original signal frequency becomes times, so it is becomes a quarter of the original signaland andthe the corresponding corresponding frequency becomes fourfour times, so it is to reduce the period interval as much as as possible topossible improve noisethe reduction accuracy. necessary to reduce the period interval much as to the improve noise reduction accuracy. necessary to reduce thethe period interval possibletotoimprove improve noise reduction accuracy . necessary to reduce period intervalas asmuch much as possible thethe noise reduction accuracy .
Figure 2. Periodic impact and MOMED noise reduction results.
Figure 2. Periodic noisereduction reduction results. Figure 2. Periodicimpact impactand and MOMED MOMED noise results.
Figure 2. Periodic impact and MOMED noise reduction results.
Figure 3. Different range of another impact noise component. Figure 3. Different range of another impact noise component.
Figure 3. Different range of another impact noise component. (3) For the complex vibration signal with multiple periodic pulses, such as the signal containing Figure 3. Different range anotherperiodic impact noise component. (3) For the complex vibration signal withofmultiple pulses, such as the signal containing three cycles which are 50, 80, and 120, respectively, when it is given a noise interval (20, 150), three which are 50, 80, and 120,with respectively, it ispulses, given asuch noise (20,containing 150), (3) Forcycles the complex vibration signal multiple when periodic as interval the signal perform the MOMED—perhaps we can only extract the impact signal whose cycle is 80. The reason perform the MOMED—perhaps we can only extract the impact signal whose cycle is 80. The reason (3) For the complex vibration signal with multiple periodic pulses, such as the signal containing threeiscycles which areenergy 50, 80,ofand respectively, it is given a noisethe interval 150),ofperform that the impact 80 is120, strong and, at thiswhen time, MOMED defaults periodic(20, impact 50 iscycles that thewhich impactare energy of 80only is strong and, at impact this time, MOMED defaults the periodic impact is ofthat 50 the three 50, 80, and 120, respectively, when it is given a noise interval the MOMED—perhaps we can extract the signal whose cycle is 80. The reason and 120 as noise. We will learn the specific instance analysis in the simulation signal for the (20, next 150), and 120 MOMED—perhaps asof noise. will and, learn the specific instance analysis in signal the signalisfor the next perform the we can only the impact whoseimpact cycle The reason impact energy 80 isWe strong at this time,extract MOMED defaults the simulation periodic of80. 50 and 120 as section . section . is that the impact energy of 80 is strong and, at this time, MOMED defaults the periodic impact of 50
noise. We will learn the specific instance analysis in the simulation signal for the next section. and 120 as noise. We will learn the specific instance analysis in the simulation signal for the next section.
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Entropy 2018, 20, x 7 of 15 (4) Based on the above analysis, when characteristic extraction of a complex multi-fault signal is carried(4) out, the first step is to reduce its noise adaptively by EEMD. Secondly, it is necessary Based on the above analysis, when characteristic extraction of a complex multi-fault signal is to determine is a step faultisperiod in each layeradaptively of the intrinsic function. Third,it recombine carriedthat out,there the first to reduce its noise by EEMD. Secondly, is necessarythe to IMF of thedetermine same IMF; determine the impact and, set different ranges and that there is a fault period inperiod; each layer offinally, the intrinsic function. period Third, recombine theMOMED IMF noiseofreduction to further extract fault information. Considering that noiseranges cannot completely the same IMF; determine the impact period; and, finally, set different period andbe MOMED noise reduction to further extractnoise-reduction fault information.intervals Considering that noise cannot be completely eliminated, it is necessary to reduce as much as possible. eliminated, is necessary reduce noise-reduction intervals as much as possible. In order to itimprove the to SNR, CIMF is used as the prefilter to reduce the noise, which not only In interference order to improve thebackground SNR, CIMF is used but as the prefilter to reduce the noise, which notfrequency only reduces the of the noise, also increases the energy of the same reduces the interference of the background noise, but also increases the energy of the same components. Then, the fault periods can be determined one by one through MOMED. frequency components. Then, the fault periods can be determined one by one through MOMED. The flowchart of the proposed fault feature extraction method based on IMOMED is shown The flowchart of the proposed fault feature extraction method based on IMOMED is shown in in Figure 4. 4. Figure
Figure 4. The flow chart of the proposed fault feature extraction method based on Improved
Figure 4. The flow chart of the proposed fault feature extraction method based on Improved Multipoint Multipoint Optimal Minimum Entropy Deconvolution (IMOMED). Optimal Minimum Entropy Deconvolution (IMOMED).
4. Performance Evaluation by Simulated Signals
4. Performance Evaluation by Simulated Signals
To evaluate the effectiveness of the proposed method to extract multiple faults, a typical
multiple impact is simulated, is shown in Figure 5. To evaluate the signal effectiveness of thewhich proposed method to extract multiple faults, a typical multiple Sampling points arewhich 2048 and the sampling frequency is 2000 Hz, the simulation signal contains impact signal is simulated, is shown in Figure 5. the noise signal is 0.5), the sinusoidal signal,isthe impact signal 1 (amplitude is contains 0.7, the the Sampling points(amplitude are 2048 and the sampling frequency 2000 Hz, the simulation signal period is 100, frequency is 20 Hz); impact signal 2 (amplitude is 0.7, the cycle is 33, frequency is 60 noise signal (amplitude is 0.5), the sinusoidal signal, the impact signal 1 (amplitude is 0.7, the period Hz); and impact signal 3 (amplitude is 1.0, the cycle is 15.3, frequency is 130 Hz). Our purpose is to is 100, frequency is 20 Hz); impact signal 2 (amplitude is 0.7, the cycle is 33, frequency is 60 Hz); extract each fault feature from the simulation signals with multiple impacts. We directly use and impact signal 3 (amplitude is 1.0, the cycle is 15.3, frequency is 130 Hz). Our purpose is to extract MOMED for the simulation signal to reduce noise and take different noise-reduction intervals, the each fault feature from the simulation with multiple impacts. Weobtained, directlythe useeffect MOMED for the result of which is shown in Figure 6,signals and each time an impact signal is of noise simulation signal to reduce noise and take different noise-reduction intervals, the result of which is reduction improves. The final extraction period of different noise-reduction intervals is also shown in Figure and eachimpact time an impactof signal is obtained, of noiseThe reduction different, and6,the three vibrations the original signal the are effect not extracted. reason isimproves. that the noise-reduction interval of the signal is not set properly. Therefore, the simulation needs The final extraction period of different noise-reduction intervals is also different, and signal the three impact to be processed by frequency division. The aim is to make sure it has a unique timescale in the IMF. vibrations of the original signal are not extracted. The reason is that the noise-reduction interval of the signal is not set properly. Therefore, the simulation signal needs to be processed by frequency division. The aim is to make sure it has a unique timescale in the IMF.
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Figure 5. Simulation signal. Figure 5. Simulation signal.
Figure 5. Simulation signal.
(a) (a)
(b) (b)
(c) (c) Figure 6. Different ranges after noise reduction by MOMED. (a) range = (40, 120), (b) range = (10, 200), Figure 6. Different ranges after noise reduction by MOMED. (a) range = (40, 120), (b) range = (10, 200), Figure 6. Different ranges after noise reduction by MOMED. (a) range = (40, 120), (b) range = (10, 200), (c) range = (31, 200). (c) range = (31, 200). (c) range = (31, 200).
EEMD decomposition the simulationsignal signal is shown ininFigure 7. 7. EEMD decomposition of of the simulation Figure EEMD decomposition of the simulation signalisis shown shown in Figure 7.
.
.
Figure 7. Simulation signal after Ensemble Empirical Mode Decomposition (EEMD) decomposition results.
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Figure 7. Simulation signal after Ensemble Empirical Mode Decomposition (EEMD) decomposition results. Figure 7. Simulation signal after Ensemble Empirical Mode Decomposition (EEMD) decomposition results.
The strong correlation with thethe original signal are are taken. From the The first firsteight eightlayers layersofofthe theIMF IMFwith with strong correlation with original signal taken. From The first eight layers of the IMF with strong correlation with the original signal are taken. From time-domain graph, it can be seen that each layer of the IMF does not contain the three impacts of the the time-domain graph, it can be seen that each layer of the IMF does not contain the three impacts of the graph, it can be seen thatthat each layer of the IMFindoes not contain the three impacts of original signal. The composition shows noise still exists layer ofofthe IMF. The noise the time-domain original signal. The composition shows that noise still exists ineach each layer the IMF. The noise in in the original signal. The composition shows that noise still exists in each layer of the IMF. The noise in this this area area consists consists of of two two parts. parts. The first is is the the noise noise of of the the original original signal, signal, and and the the second second is is that that the the this area consists of two parts. The neutralized. firstneutralized. is the noise of the perform original signal, and second isasthat the added white noise is is notnot completely Therefore, FFT on them separately, shown added white noise completely Therefore, perform FFT onthe them separately, as added white noise is not completely neutralized. Therefore, perform FFT on them separately, as in Figure shown in8.Figure 8. shown in Figure 8.
Figure 8. 8. Spectrum Spectrum of of intrinsic intrinsic mode mode functions functions (IMFs) (IMFs) decomposed decomposed after after EEMD. EEMD. Figure Figure 8. Spectrum of intrinsic mode functions (IMFs) decomposed after EEMD.
The first two layers contain the same timescale, the third and fourth contain the same scale, and The first two two layers layers containthe the sametimescale, timescale, third fourth contain same the same scale, The thethe third andand fourth contain and the fifth and sixth layerscontain contain thesame same scale. Recombine IMFs with the same the timescalescale, to obtain and the fifth and sixth layers contain the same scale. Recombine IMFsthe with the timescale same timescale to the fifth and sixth layers contain the same scale. Recombine IMFs with same to obtain CIMF1, CIMF2, CIMF3, as shown in Figure 9. obtain as shown in Figure 9. CIMF1,CIMF1, CIMF2,CIMF2, CIMF3,CIMF3, as shown in Figure 9.
Figure 9. Combined IMF (CIMF) results. Figure 9. Combined IMF (CIMF) results. Figure 9. Combined IMF (CIMF) results.
The new IMF contains only a single timescale. It is determined by calculation that their periods The new IMF a single timescale. It is determined calculation that 110), their (51, periods are 100, 60, and 15 contains (sample only points), and then the appropriate periodby intervals are (90, 70), The60, new IMF contains only a single timescale. It is determined byintervals calculation that their periods are 100, and 15 (sample points), and then the appropriate period are (90, 110), (51, 70), and (11, 19). The results of MOMED noise reduction are shown in Figures 10–12. MOMED (CIMF1) are 100, and 15results (sample points), and then the appropriate period intervals areMOMED (90, 110),(CIMF1) (51, 70), and (11,60, 19). of MOMED reduction are shown in 10–12. indicates thatThe MOMED denoising of noise CIMF1 and reconstructing X1Figures of the original signal apparently and (11, 19). results denoising of MOMED reduction are shown in 10–12. MOMED (CIMF1) indicates thatThe MOMED of noise CIMF1 and reconstructing X1Figures of the original signal apparently has extracted the X1 impact component. In addition to strong impact, the noise is still distributed indicates that MOMED denoising of CIMF1 and reconstructing X1 of the original signal apparently has X1domain, impact component. addition to strong impact, the noise still distributed overextracted the entirethe time but the copyIn is relatively small without affecting the is overall judgment. has extracted the X1domain, impact component. In addition to strong impact, the noise is still distributed over the entire time but the copy is relatively small without affecting the overall judgment. The noise-reduced signals and original signals of MOMED (CIMF2) and MOMED (CIMF3) are over the entire time domain, but original the copy signals is relatively small without affecting the overall judgment. The noise-reduced signals and of MOMED (CIMF2) and MOMED (CIMF3) are almost completely reconstructed. It is further illustrated that the proposed compound fault feature The noise-reduced signals and original of MOMEDthat (CIMF2) and MOMED (CIMF3) are almost almost completely reconstructed. It is signals further illustrated extraction method has strong engineering application value. the proposed compound fault feature completely reconstructed. It is further illustrated that the proposed compound fault feature extraction extraction method has strong engineering application value. method has strong engineering application value.
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Figure 10. MOMED MOMED (CIMF1). (CIMF1). Figure Figure 10. 10. Figure 10. MOMED (CIMF1). Figure 10. MOMED (CIMF1).
Figure 11. MOMED (CIMF2). Figure 11. MOMED (CIMF2). MOMED (CIMF2). Figure 11. MOMED Figure 11. MOMED (CIMF2).
Figure 12. MOMED (CIMF3). Figure 12. MOMED (CIMF3). Figure 12. MOMED (CIMF3). Figure Figure12. 12. MOMED MOMED (CIMF3). (CIMF3).
In order order to to compare compare the the effectiveness effectiveness of of the the proposed proposed method, method, the the simulation simulation signal signal is is analyzed analyzed In In orderThe to compare thethe effectiveness of the proposed method, the simulation signal is analyzed with VMD. K = 4 and penalty factor are 3000. Considering that the penalty factor has little little withIn VMD. K = 4 and the penalty factor areproposed 3000. Considering that the penalty factor has In orderThe to compare compare the effectiveness ofthe the proposed method,the the simulation signal is analyzed analyzed order to the effectiveness of method, simulation signal is with VMD. The K = 4 and the penalty factor are 3000. Considering that the penalty factor has little effect on on the the algorithm, algorithm, the the reason reason of of K K == 44 is is that that the the simulation simulation signal signal contains contains four four frequency frequency effect withVMD. VMD. TheKK 4 and penalty factor are 3000. Considering thatpenalty the penalty factor haseffect little with The = 4= and thethe penalty factor 3000. that the factorfour has frequency little effect on the algorithm, the reason of K =are 4 is thatConsidering the simulation signal contains components in order to decompose it into four eigenmode functions. The result is shown in Figure Figure components in order to decompose it into four eigenmode functions. The result is shown in effect on the algorithm, theofreason K =the 4 is that the simulation signal contains four frequency on the algorithm, the reason K = 4 it isofthat simulation signal contains frequency components components in order to decompose eigenmode functions. Thefour result is shown in Figure 13. The The number number of decomposed decomposed layers into is K K =four 4. The The corresponding spectrum is shown shown in Figure Figure 14, in in 13. of layers 4. corresponding spectrum is in 14, components in order to decompose it is into=four eigenmode functions. The in result is shown innumber Figure in order to decompose it into four eigenmode functions. The result is shown Figure 13. The 13. The number of decomposed layers is K = 4. The corresponding spectrum is shown in Figure 14, in which the the second second and and fourth fourth layers layers are are shown. shown. Corresponding Corresponding to to the the characteristic characteristic component component 130 130 which 13.decomposed Thethe number ofand decomposed layers K = 4. The corresponding spectrum is14, shown in Figure 14, in of layers isfourth K = 4. layers The corresponding spectrum is shown in Figure in which the second which second areisshown. Corresponding to the characteristic component 130 Hz, the first layer is 60 Hz, the characteristic component of the third layer can’t determine its Hz, the first layer is 60 Hz, the characteristic component of the third layer can’t determine its which the second andshown. fourth Corresponding layers are shown. Corresponding to the characteristic 130 and fourth layers are to the characteristic Hz,component the first layer Hz, the first layer is 60 the characteristic of thecomponent third layer130 can’t its composition, and 20 20 HzHz, vibration frequency component can’t be be extracted. extracted. Therefore, in aadetermine strong-noise composition, and Hz vibration frequency can’t Therefore, in strong-noise Hz, the the firstcharacteristic layer is 60 Hz, the characteristic component of the third layer can’t determine its is 60 Hz, component of the third layer can’t determine its composition, and 20 Hz composition, and 20 Hz vibration frequency can’t be extracted. Therefore, in a strong-noise environment, VMD VMD extraction extraction fault fault features features are are also also prone prone to to mode mode mixing. mixing. environment, composition, and 20 Hz vibration Therefore, frequency incan’t be extracted. Therefore, in extraction a strong-noise vibration frequency be extracted. a prone strong-noise environment, VMD fault environment, VMD can’t extraction fault features are also to mode mixing. environment, VMD extraction features are also prone to mode mixing. features are also prone to modefault mixing.
Figure 13. Simulation signal after VMD decomposition results. Figure 13. Simulation signal after VMD decomposition results. Figure Figure 13. Simulation Simulation signal signal after after VMD VMD decomposition results. Figure 13. Simulation signal after VMD decomposition results.
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Figure 14. Spectrum of IMFs decomposed after VMD.
Figure 14. Spectrum of IMFs decomposed after VMD.
5. Application Case
5. Application Case
Figure 14. Spectrum of IMFs decomposed after VMD.
The experimental device mainly includes speed display, three-way acceleration sensors, test The experimental mainly includes speed display, three-way sensors, 5. Application gears (18 teeth),Case test device bearings, a motor, shafts, and so on. The type of the testacceleration bearing is 32,212, the test ratio of the test agear is 1:1,shafts, the half-tooth engagement is adopted, the bearing speed is 1200 gearstransmission (18 The teeth), test bearings, motor, and so on. The type of the test is 32,212, experimental device Figure mainly display, three-way 14. includes Spectrum ofspeed IMFs decomposed after VMD. acceleration sensors, test r/min, and the ratio sampling frequency is 8000 Hz. The test load is carried out step is by adopted, step, the normal the transmission of the test gear is 1:1, the half-tooth engagement the speed gears (18 teeth), test bearings, a motor, shafts, and so on. The type of the test bearing is 32,212, the and faulty gears and bearings are loaded to the test load of 1000 N.M, and the type of the three-way 5. Application Case is 1200 r/min, and the frequency 8000 Hz.engagement The test load is carried out step by step, transmission ratio of sampling the test gear is 1:1, theishalf-tooth is adopted, the speed is 1200 2 acceleration sensor is YD77SA (themainly is speed 0.01 V/ms The parameters the motor are:of the the normal andthe faulty gears and bearings are loaded to the test load ofout 1000 N.M, and thenormal type r/min, and sampling frequency issensitivity 8000 Hz. The test load is).carried step byofstep, the The experimental device includes display, three-way acceleration sensors, test 2 test model 200L-4; 380bearings, V;YD77SA power 30(the KW, constant conversion is32,212, 50three-way –100 Hz;motor gears (18voltage teeth), test a motor, andload so power on. The frequency type of and the and faulty gears andsensor bearings are loaded toshafts, the test of 0.01 1000 N.M, thebearing type ofisthe three-way acceleration is sensitivity is V/ms ). The parameters ofthe the transmission ratiolocation of the test gear is 1:1, thepeeling, half-tooth engagement adopted, 1200The quality is 255 Kg. Fault including gear gear outer ringisdefect (0.2the cmspeed × 0.4iscm). 2 are: model 200L-4; voltage 380 V; power 30 KW, is constant power frequency conversion is 50–100 acceleration sensor is YD77SA (the sensitivity 0.01 V/ms ). The parameters of the motor are: Hz; r/min, and the sampling 8000 Hz. Thesensor test load carried out step by step, the outer normalring faulty bearing is located at the frequency three-wayisacceleration 1#.isThe fault frequency of the model 200L-4; voltage 380 V; power 30 KW, constant power frequency conversion is 50 –100 Hz; quality is 255 Kg. Fault location including gear peeling, gear outer ring defect (0.2 cm × 0.4 cm). andand faulty and bearings are loaded to is the2048. test load of 1000 N.M, calculation, and the type it of can the three-way is 160 Hz, thegears number of sampling points After a simple be obtained quality is 255 Kg. Fault location including gear peeling, gear outer ring defect (0.2 cm × 0.4 cm). The 2 The faulty bearing is located atYD77SA thering three-way acceleration sensor 1#. parameters The fault of frequency of the outer (the sensitivity is 0.01 V/ms ). The the motor are: that the acceleration fault periodsensor of theisouter is 50, the meshing frequency of the gear is 360 Hz, and the gear faulty bearing isthe located at the three-way acceleration sensor 1#. a The fault frequency of the outer ring ring is 160 Hz, and number of sampling points is 2048. After simple calculation, it can be obtained model 200L-4; voltage 380 V; power 30 KW, constant power frequency conversion is 50 – 100 Hz; meshing cycle is 22. The test rig is shown in Figure 15. The gear fault is pitting, and the outer ring is 160 Hz, and the number of sampling points is 2048. After a simple calculation, it can be obtained quality is 255 Kg. Fault location including gear peeling,frequency gear outer ring defect (0.2is cm360 × 0.4 cm).and Thethe gear that the fault period of the outer ring is 50, the meshing of the gear Hz, fault is generated by EDM (Electric discharge machining), as shown in Figure 16. that thefaulty fault period the outer ring is 50, the meshing sensor frequency of fault the gear is 360ofHz, bearing of is located at the three-way acceleration 1#. The frequency the and outerthe ringgear meshing cycle is 22. The test rig is shown in Figure 15. The gear fault is pitting, and the outer ring fault 160 Hz, the number points is 2048. Aftergear a simple can the be obtained meshingis cycle is and 22. The test rigofissampling shown in Figure 15. The faultcalculation, is pitting,itand outer ring is generated by the EDM (Electric discharge as shown in of Figure 16. that fault period the outerdischarge ringmachining), is 50, the meshing frequency theFigure gear is 16. 360 Hz, and the gear fault is generated by EDMof(Electric machining), as shown in meshing cycle is 22. The test rig is shown in Figure 15. The gear fault is pitting, and the outer ring fault is generated by EDM (Electric discharge machining), as shown in Figure 16.
Figure 15. Rig for gear-transmission testing (1: Speed-adjustable motor, 2: coupling, 3: accompanied gearbox, 4: speed-reversing instrument, 5: torsion bar, 6: test gearbox, 7: three-way acceleration sensor 1#, 8: three-way acceleration sensor 2#). Figure 15. Rig for gear-transmission testing (1: Speed-adjustable motor, 2: coupling, 3: accompanied Figure 15. Rig for 15. gear-transmission testing (1: Speed-adjustable motor, 2: coupling, 3: accompanied Rig for gear-transmission (1: Speed-adjustable motor, 2: coupling, 3: accompanied gearbox,Figure 4: speed-reversing instrument,testing 5: torsion bar, 6: test gearbox, 7: three-way acceleration gearbox, 4: gearbox, speed-reversing instrument, 5: torsion bar, 6: test gearbox, 7: three-way acceleration 4: speed-reversing instrument, 5: torsion bar, 6: test gearbox, 7: three-way acceleration sensor 1#, 8: three-way acceleration sensor 2#). 1#, 8: three-way acceleration sensor sensor 1#, 8: sensor three-way acceleration sensor 2#). 2#).
(a)
(b)
(a) (b) Figure 16. Bearing (a) and gear fault diagram. (a) Bearing outer ring(b) defect; (b) Gear peeling
Figure 16. Bearing gear fault (a)(a) Bearing ring defect; (b) Gear peeling. Figure 16.and Bearing and geardiagram. fault diagram. Bearing outer outer ring defect; (b) Gear peeling Figure 16. Bearing and gear fault diagram. (a) Bearing outer ring defect; (b) Gear peeling
The time-domain waveforms of vibration signals of the gearbox measured by the is shown in Figure 17. From the time-domain waveform a significant impact can be seen occurring periodically,
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Entropy 2018, 20, x 12 of 15 corresponding to the gear-meshing frequency; spectral analysis determines this judgment, a small The time-domain waveforms of vibration signals of the gearbox measured by the is shown in peak at 360 Hz, 720 Hz, and asymmetrical sideband, and there is no peak of the frequency of bearing The waveformswaveform of vibration signals of impact the gearbox by theperiodically, is shown in Figure 17.time-domain From the time-domain a significant can bemeasured seen occurring failure. EEMD is applied to obtain Figure 18, aand the corresponding spectrum is shown Figure 17. From waveform significant impact determines can be frequency seen this occurring periodically, corresponding tothe thetime-domain gear-meshing frequency; spectral analysis judgment, a small in Figure 19. The first four layers have the strongest correlation with the original signal. The first corresponding frequency; spectral determines this judgment, a small peak at 360 Hz, to 720the Hz,gear-meshing and asymmetrical sideband, and analysis there is no peak of the frequency of bearing three failure. layers have center frequencies of 720 Hz, 720 Hz, and 360 Hz, respectively. These are the peak at EEMD 360 Hz,is720 Hz, and asymmetrical sideband, there is no peak of thespectrum frequencyisof bearing applied to obtain Figure 18, and the and corresponding frequency shown insame failure. EEMD is applied to obtain 18, and the corresponding frequency spectrum shown infault timescale; the center frequency ishave 160Figure Hz, and EMMD has obvious mode mixing. Although the Figure 19. The first four layers the strongest correlation with the original signal. Theisfirst three Figure have 19. first four layers have the strongest correlation the original The first three lead information of The gears and bearings can beHz, determined by 360 the with spectrum, modesignal. mixing layers center frequencies of 720 720 Hz, and Hz, respectively. These arecan theeasily same layers have center frequencies 720 Hz, 720 Hz, the and 360 Hz,ofrespectively. These are the same timescale; theor center frequency isof160 Hz, and EMMD hasenergy obvious mode mixing. Although the faultto be to misdiagnosis leakage diagnosis, and therefore the fault information needs timescale; the center frequency is 160 Hz, and EMMD has obvious mode mixing. Although the fault information of gears bearings canof bethe determined by the mode mixing can easily to lead enhanced. According to and the innovation article, the firstspectrum, three layers were reorganized obtain information of gears and bearings can be determined by the spectrum, mode mixing can easily lead to misdiagnosis or leakage diagnosis, the energy fault 20. information CIMF1, and the fourth layer was CIMF2.and Thetherefore results are shownof inthe Figure The two needs layerstoofbemode to misdiagnosis or leakage andthe therefore the first energy of layers the fault information needs to be enhanced. According to the diagnosis, article, the three reorganized to obtain functions contain most of the innovation energy in of the original fault signal. Sincewere their periods are 22 and 55, enhanced. According to the innovation of the article, the first three layers were reorganized to obtain CIMF1, and the fourth layer was CIMF2. The results are shown in Figure 20. The two layers of mode respectively, the the noise-reduction interval isThe setresults to (15, 24)shown and (50, 70) in20. order to extract themode periodic CIMF1, and fourth CIMF2. in Since Figure two layers functions contain mostlayer of thewas energy in the original are fault signal. theirThe periods are 22 of and 55, information. Respectively obtained MOMED (CIMF1) and MOMED (CIMF2), the results of which functions contain most of the energy in the original signal. Since their periods arethe 22periodic and 55, are respectively, the noise-reduction interval is set to (15, fault 24) and (50, 70) in order to extract shown in Figures Respectively 21 and 22, and continuous has extracted. Each impact entropy respectively, the noise-reduction interval isperiodic set(CIMF1) to (15,impact 24) and (50,been 70) in order tothe extract the information. obtained MOMED and MOMED (CIMF2), results ofperiodic which information. Respectively obtained MOMED (CIMF1) and MOMED (CIMF2), the results of which is stronger than the EEMD decomposition results. Through envelope analysis, the results are shown are shown in Figures 21 and 22, and continuous periodic impact has been extracted. Each impact in are23 shown in Figures 21 the andEEMD 22, anddecomposition continuous periodic impact been impact Figures and 24. Their fundamental frequencies are 360 Hz andhas 160 Hz,extracted. respectively, entropy is stronger than results. Through envelope analysis,Each thewhich results fully entropy is stronger than the EEMD decomposition results. Through envelope analysis, the results are shown in Figures 23 and 24.and Their fundamental frequencies 360 Hz In and 160 Hz, respectively, attenuates the noise interference extracts compound fault are features. order to further compare are shown in Figures 23 and 24. Their fundamental frequencies are 360 Hz and 160 Hz, respectively, which fully attenuates noise interference and extracts compound fault features. In order to with the method proposedthe in the article, VMD is used to analyze the vibration signal. The penalty which fully attenuates themethod noise interference and extracts compound fault features.the In vibration order to further compare with the proposed in the article, VMD is used to analyze factor was 2000, and the number of layers for decomposition was 5. The results are shown in Figure 25. furtherThe compare with thewas method in the article, VMD is used to analyze the vibration signal. penalty factor 2000,proposed and the number of layers for decomposition 5. The results The corresponding frequency spectrum is shown in Figure 26. Only 720 Hz is thewas corresponding fault signal. The penalty factor was 2000, and the number of layers for decomposition was 5. The results are shown in Figure 25. The corresponding frequency spectrum is shown in Figure 26. Only 720 Hz information. Since each25. layer ratio must have a center frequency when VMD is decomposed, are shown in Figure The corresponding shown in the Figure 26. Only 720 the Hz is the corresponding fault information. Sincefrequency each layerspectrum ratio mustishave a center frequency when mixing occurs in other layers. In addition, the number of decomposition levels of the VMD needs to be is the corresponding fault information. Since each layer ratio must have a center frequency when the VMD is decomposed, mixing occurs in other layers. In addition, the number of decomposition levels determined artificially. Therefore, the results of the decomposition are indefinite and they are prone VMD is decomposed, mixing occurs in other layers. In addition, the number of decomposition levels of the VMD needs to be determined artificially. Therefore, the results of the decomposition are to of the VMD needs to be determined artificially. Therefore, the results of the decomposition are misdiagnosis or missed diagnosis. indefinite and they are prone to misdiagnosis or missed diagnosis.
indefinite and they are prone to misdiagnosis or missed diagnosis.
Figure 17. Time domain and spectrum of vibration signals.
Figure 17. Time domain and spectrum of vibration signals. Figure 17. Time domain and spectrum of vibration signals.
Figure 18. Time domain of vibration signals after EEMD.
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Figure 18. Time domain of vibration signals after EEMD. Figure18. 18.Time Timedomain domainof ofvibration vibrationsignals signalsafter afterEEMD. EEMD. Figure Figure 18. Time domain of vibration signals after EEMD.
Figure 19. Spectrum of vibration signals after EEMD. Figure 19. Spectrum of vibration signals after EEMD. Figure19. 19.Spectrum Spectrumof ofvibration vibrationsignals signalsafter afterEEMD. EEMD. Figure Figure 19. Spectrum of vibration signals after EEMD.
Figure 20. CIMF of vibration signals. Figure20. 20.CIMF CIMFof ofvibration vibrationsignals. signals. Figure 20. CIMF of vibration signals. Figure Figure 20. CIMF of vibration signals.
... .
Figure 21. MOMED Figure21. 21.MOMED MOMED (CIMF1). (CIMF1). Figure Figure MOMED (CIMF1). (CIMF1). Figure 21. 21. MOMED (CIMF1).
Figure 22. MOMED (CIMF2). Figure22. 22.MOMED MOMED(CIMF2). (CIMF2). Figure Figure 22. MOMED (CIMF2). Figure 22. MOMED (CIMF2).
Figure 23. The envelope spectrum of MOMED (CIMF1). Figure23. 23.The Theenvelope envelopespectrum spectrumof ofMOMED MOMED(CIMF1). (CIMF1). Figure Figure 23. The envelope spectrum of MOMED (CIMF1).
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Figure 22. MOMED (CIMF2).
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Figure 23. The envelope spectrum of MOMED (CIMF1). Figure 23. The envelope spectrum of MOMED (CIMF1).
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Figure 24. 24. The envelope envelope spectrum of of MOMED (CIMF2). (CIMF2). Figure Figure 24.The The envelopespectrum spectrum ofMOMED MOMED (CIMF2). Figure 24. The envelope spectrum of MOMED (CIMF2).
Figure 25. Vibration signals after VMD decomposition results. Figure 25. Vibration Vibration signalsafter after VMDdecomposition decomposition results. Figure Figure25. 25. Vibrationsignals signals afterVMD VMD decompositionresults. results.
Figure 26. Spectrum of IMFs decomposed after VMD of vibration signal. Figure 26. Spectrum of IMFs decomposed after VMD of vibration signal. Figure 26. Spectrum of IMFs decomposed after VMD of vibration signal. Figure 26. Spectrum of IMFs decomposed after VMD of vibration signal.
6. Conclusions 6. Conclusions 6. Conclusions 6. Conclusions (1) The decomposition results by the EEMD method are related to the added white noise with a (1) The decomposition results by the EEMD method are related to the added white noise with a (1) The which decomposition results by the EEMDofmethod are relatedaccuracy. to the added white noise with a noise(1) level, results in the degradation decomposition Under a strong-noise The decomposition the EEMDof method are relatedaccuracy. to the added white noise with noise level, which results results in the by degradation decomposition Under a strong-noise noise level, which results in the degradation of decomposition accuracy. Under strong-noise the artificially determined white noise cannot decompose fault Under signalsaawith different aenvironment, noise level, which results in the degradation of decomposition accuracy. strong-noise environment, the artificially determined white noise cannot decompose fault signals with different environment, the artificially determined white noise cannot decompose fault signals with different features into different IMFs. On the contrary, thenoise samecannot fault feature may be decomposed into several environment, the artificially white decompose fault signals with different features into different IMFs. determined On the contrary, the same fault feature may be decomposed into several features into different IMFs. On the contrary, the same fault feature may be decomposed into several layers of IMFs, resulting in entropy leakage. layers of IMFs, resulting in entropy leakage. layers IMFs, resulting in entropynoise leakage. (2)ofThe accuracy of MOMED reduction is determined by the noise-reduction interval. If (2) The accuracy of MOMED noise reduction is determined by the noise-reduction interval. If (2) The accuracy of MOMED noise determined by the noise-reduction interval. If you choose improperly, you will miss thereduction diagnosis.isMOMED can only extract a single fault message you choose improperly, you will miss the diagnosis. MOMED can only extract a single fault message you improperly, you will miss theseparates diagnosis.the MOMED cansignal only extract a single faultand message at a choose time. EEMD not only adaptively original in order of high low at a time. EEMD not only adaptively separates the original signal in order of high and low at a time. EEMD not adaptively the can original signal order of of the high andfault low frequencies, but also is only an adaptive filter.separates This article improve the in entropy same frequencies, but also is an adaptive filter. This article can improve the entropy of the same fault
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features into different IMFs. On the contrary, the same fault feature may be decomposed into several layers of IMFs, resulting in entropy leakage. (2) The accuracy of MOMED noise reduction is determined by the noise-reduction interval. If you choose improperly, you will miss the diagnosis. MOMED can only extract a single fault message at a time. EEMD not only adaptively separates the original signal in order of high and low frequencies, but also is an adaptive filter. This article can improve the entropy of the same fault feature by combining the IMFs of the same timescale. In different timescales, setting different noise-reduction intervals can extract complex fault features. (3) The validity of the IMOMED method is proved by the simulation signal and the measured signal, and the compound fault features can be extracted successfully by the proposed method, which has immunity even under strong background noise. (4) The advantage of this article is to improve the SNR by EEMD, overcome the modal aliasing phenomenon by combining the modal functions, and then extract the impact components with MOMED. The disadvantage is that the adaptation of the method proposed in the paper needs further analysis and discussion. Author Contributions: F.Y. conceived and designed the experiments; Z.W. performed the experiments; X.S. wrote the paper. All authors have read and approved the final manuscript. Funding: This work was supported by the National Natural Science Foundation of China (No. 51475318). Conflicts of Interest: The authors declare no conflict of interest.
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