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A morphogram with the optimal selection of parameters used in morphological analysis for
enhancing the ability in bearing fault diagnosis
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2012 Meas. Sci. Technol. 23 065001
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IOP PUBLISHING MEASUREMENT SCIENCE AND TECHNOLOGY
Meas. Sci. Technol. 23 (2012) 065001 (15pp) doi:10.1088/0957-0233/23/6/065001
A morphogram with the optimal selectionof parameters used in morphologicalanalysis for enhancing the ability inbearing fault diagnosis
Dong Wang, Peter W Tse1 and Yiu L Tse
Smart Engineering Asset Management Laboratory and Croucher Optical Non-destructive Testing andQuality Inspection Laboratory, Department of Systems Engineering and Engineering Management,City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, People’s Republic of China
E-mail: [email protected], [email protected] and [email protected]
Received 8 November 2011, in final form 6 March 2012Published 30 April 2012Online at stacks.iop.org/MST/23/065001
AbstractMorphological analysis is a signal processing method that extracts the local morphologicalfeatures of a signal by intersecting it with a structuring element (SE). When a bearing suffersfrom a localized fault, an impulse-type cyclic signal is generated. The amplitude and the cyclictime interval of impacts could reflect the health status of the inspected bearing and the cause ofdefects, respectively. In this paper, an enhanced morphological analysis called ‘morphogram’is presented for extracting the cyclic impacts caused by a certain bearing fault. Based on thetheory of morphology, the morphogram is realized by simple mathematical operators,including Minkowski addition and subtraction. The morphogram is able to detect all possiblefault intervals. The most likely fault-interval-based construction index (CI) is maximized toestablish the optimal range of the flat SE for the extraction of bearing fault cyclic features sothat the type and cause of bearing faults can be easily determined in a time domain. Themorphogram has been validated by simulated bearing fault signals, real bearing faulty signalscollected from a laboratorial rotary machine and an industrial bearing fault signal. The resultsshow that the morphogram is able to detect all possible bearing fault intervals. Based on themost likely bearing fault interval shown on the morphogram, the CI is effective in determiningthe optimal parameters of the flat SE for the extraction of bearing fault cyclic features forbearing fault diagnosis.
Keywords: morphological analysis, bearing fault diagnosis, morphogram, conditionmonitoring, parameter optimization, construction index
(Some figures may appear in colour only in the online journal)
1. Introduction
Rolling element bearings are one of the most importantcomponents in machinery. Their failure often results in
1 Author to whom any correspondence should be addressed.
economic loss and human injury. When localized faults occuron the surface of the inner race, outer race or rolling elements,the collected vibration signals manifest repetitive cyclic faultimpulses with exponential decay [1, 2]. The identificationof these cyclic fault intervals is beneficial to bearing faultdiagnosis. In fact, the cyclic fault intervals can be roughly
0957-0233/12/065001+15$33.00 1 © 2012 IOP Publishing Ltd Printed in the UK & the USA
Meas. Sci. Technol. 23 (2012) 065001 D Wang et al
regarded as the reciprocal of the bearing fault characteristicfrequencies. The identification of these frequencies is widelyused in bearing fault diagnosis [3, 4].
Morphological analysis is a new signal processing methodused in bearing fault diagnosis. The fundamental idea is toprocess a signal by using the intersection with a structuringelement (SE). The basic operators of morphological analysiscan be realized using Minkowski addition and subtraction.Nikolaou and Antoniadis [5] had investigated the feasibilityof morphological analysis for bearing fault diagnosis. Theyproposed a performance chart to assess the effect of the lengthof the flat SE for the envelope extraction of the repetitivetransient signal. However, this optimal method requires theinitial impulse repetition period of the original signal. As ageneral empirical rule, they suggested that the recommendedlength of the flat SE should range from 0.6 to 0.7 times thebearing fault characteristic period for approximate envelopeextraction. Zhang et al [6] used a triangular SE to performmulti-scale morphological analysis to extract bearing faultimpulse features. Although their results showed that theirproposed method could enhance the signal-to-noise ratio(SNR), a triangular SE was indeed not the impulsive featureitself. In order to extract the impulsive signal, a sinusoidal SEwith exponential decay was used by Wang et al [7]. This ideawas very good to extract the transient signal from noises. Butthe method has its own shortcoming. The search ranges for themaximum amplitude, decay coefficient and natural frequencyshould be well established before the optimal SE is derived.They did not clarify how to establish these search ranges.If improper ranges were selected, the computing time wouldbecome longer for the three-dimensional optimization problemand the SE would be inaccurate for the original transient signal.Following the work of Nikolaou and Antoniadis [5], Donget al [8] also used the flat SE for morphological analysis. Theirresults showed that the general rule provided by Nikolaouand Antoniadis was not suitable for all conditions. For thisreason, they introduced a new criterion, namely the maximumSNR, to optimize the length of the flat SE. However, theyused the average of closing and opening operators to extractbi-directional impulsive components instead of the impulsivesignal. In the research of Hao et al [9], they believed thatthe flat SE would result in a straightforward applicationfor the detection of peaks of the impulsive signal. Basedon the theory of pattern spectrum entropy, the quantity of thepattern spectrum was used to describe the fault characteristicfeatures. The barycenter of the pattern spectrum was usedas another index for the fault feature extraction. Then, thesupport vector machine (SVM) was employed to intelligentlyidentify different bearing faults. In their method, the openingoperation was used to calculate the pattern spectrum. Theirproposed method is good to distinguish different fault features.However, the SVM-based pattern spectrum method needs a setof historical data for fault pattern recognition. It means thattheir proposed method requires a large number of data for thetraining. Besides, it should be noted that the SVM was trainedby a particular bearing on a particular machine. It lacked robustapplication for different bearings and different machines. Fromthe above introduction, it can be seen that the current research
on morphological analysis for machinery fault diagnosis is stilllimited [5–9].
In this paper, following the idea of Nikolaou andAntoniadis, a new method called a morphogram is proposed todetect all possible bearing fault intervals in the time domain.Based on the most likely bearing fault interval reported bythe morphogram, a new construction index, CI, is proposedto automatically determine the optimal length range of theflat SE for the extraction of bearing fault cyclic features forbearing fault diagnosis. The flat SE is used in this paperbecause the results obtained in [5, 8, 9] illustrated that theflat SE is simple and useful to detect the peaks of the faulttransient signal. Being different from the work of Nikolaouand Antoniadis, which is based on the assumption that theinitial pulse repetition period of the desired signal is known toselect the optimal parameter prior to morphological analysis[5], the morphogram is able to detect all possible existingbearing fault intervals in advance. Then, the most likely fault-interval-based CI is maximized to establish the optimal rangeof the flat SE for the extraction of bearing fault cyclic features.It means that our proposed method does not require priorfault knowledge of a specific bearing fault signal to decidethe optimal parameter range prior to the use of morphologicalanalysis. On the other hand, the proposed method gives theaccurate optimal range of flat SE for a specific signal becausethe optimal length range of the flat SE is based on the specificinformation provided by the morphogram. Here, the specificinformation provided by the morphogram can be understood asthe evidence for a specific bearing fault signal. It is reasonableto regard the optimal length range of the flat SE establishedby the proposed method as the posterior parameter estimation.The empirical rule obtained from a population of results bythe closing operator with the flat SE in [5] can be regardedas the prior parameter estimation. According to Bayesianupdating theory, this is a reason why the optimal length rangeof the flat SE based on the evidence of a morphogram ismore accurate than the optimal length range recommendedby the prior empirical rule. The rest of this paper is organizedas follows. Section 2 introduces the fundamental theory ofmorphological analysis. Section 3 proposes the morphogramand the selection of the optimal parameters for bearing faultdiagnosis. In section 4, simulated bearing fault signals andreal bearing fault signals are used to validate the proposedmethod. Then, comparison with other methods is conductedon the same real fault signals. Comparison of the proposedCI with kurtosis is also done. Further discussion is providedin section 5 and an industrial bearing fault signal is furtheremployed to validate the proposed method. The final sectionconcludes the paper.
2. Fundamental theory of morphological analysis
Morphological analysis uses a SE to revise the shape of anoriginal signal and extract its local morphological features.Let f (n) be a one-dimensional signal over the domainF = (0, 1, 2, . . . , N−1). Furthermore, let g(n) be the SE overthe domain G = (0, 1, 2, . . . , M−1). The basic morphologyoperators include dilation and erosion, which can be expressed
2
Meas. Sci. Technol. 23 (2012) 065001 D Wang et al
by Minkowski addition and Minkowski subtraction in the timedomain. The equations for the erosion operator � and thedilation operator ⊕ are defined as [5]
( f � g)(n) = min[ f (n + m) − g(m)],
m ∈ 0, 1, 2, . . . , M − 1, (1)
( f ⊕ g)(n) = max[ f (n − m) + g(m)],
m ∈ 0, 1, 2, . . . , M − 1. (2)
The erosion operator is used to reduce the wave peaks andenlarge the signal minima. The dilation operator is able toincrease the wave valleys and enlarge the signal maxima [9].
Considering various combinations of equations (1) and(2), other kinds of morphological operators can be built forthe extraction of different morphological features [5–9]. Theopening operator f ◦ g and the closing operator f · g areintroduced as [5]
( f ◦ g)(n) = [( f � gs) ⊕ g](n), (3)
( f · g)(n) = [( f ⊕ gs) � g](n), (4)
where gs(n) is the reflection of g(n). The signal can besmoothed from the bottom by cutting the wave peaks usingthe opening operator and the signal can be smoothed from thetop by filling up the wave valleys using the closing operator [9].Therefore, the closing operator is usually used for extractingthe positive envelope and the opening operator is employed toextract the negative envelope [5].
3. The proposed method for bearing fault cyclicinterval identification and the optimal parameterselection of morphological analysis
Envelope analysis is important for bearing fault diagnosisbecause it transforms fault signatures from the high-frequencyband (the structural resonant frequency band amplifies the faultimpulses) to the low-frequency band (the desired diagnosticband contains repetition frequency) [4]. For this reason, theclosing operator is employed in this paper to extract the positiveenvelope. The flat SE is used for further analysis becausethe results showed that the flat SE is simple yet effective tobe employed in the morphological analysis for bearing faultdiagnosis [5, 8, 9]. The flowchart of the proposed method forbearing fault diagnosis is depicted in figure 1.
Step 1. Load the original vibration signal.Step 2. Initialize the length of the flat SE l(i) to be equalto 1; here, i = 1. In this step, due to the shortest length ofthe flat SE, it is difficult to smooth noise using the closingoperator.Step 3. A morphological closing algorithm is then appliedto the original vibration signal using equation (4). As aresult, the positive envelope is obtained.Step 4. Detect local maxima of the positive envelope andset the non-maxima values to 0. Consequently, a newsignal Si is constructed by only retaining the maxima. Itshould be noted that the new signal has the same temporallength as that of the original signal.
Load the original vibration signal
Initialize the length of the flat structuring element l(i)=1, here i=1
Process the original signal by using the closing operator and obtain the positive envelope
Let values at the non-local maxima equal to 0 and a new signal is nominated as Si
No
Increase the current length of flat
structuring element by 1, namely l(i+1)=
l(i)+1.
Exceeding an established threshold?
Generate morphogram and find all possible bearing fault intervals.
Yes
Find the most likely fault interval that has the longest unchanged distance along the normalized
length increased direction
Calculate the construction index
Find bearing fault cyclic features on the morphogram at the optimal length range of flat SE
for bearing fault diagnosis
Figure 1. The flowchart of the proposed method.
Step 5. If the current length of the flat SE does not exceedan established threshold, the current length of the flat SEl(i) will be increased by 1, namely l(i + 1) = l(i)+1.Repeat steps 3 and 4. Here, the established threshold ofthe flat SE can be determined by the sampling frequencyFs and the bearing fault characteristic frequencies, whichare related to the shaft rotating frequency and bearingsize. The outer race fault characteristic frequency fo andthe inner race fault characteristic frequency fI are given asfollows [4]:
fo = z × fs
2
(1 − d
Dcos α
), (5)
fI = z × fs
2
(1 + d
Dcos α
), (6)
where fs is the shaft rotating frequency in Hz; d is thediameter of the rolling element; D is the pitch diameterof the bearing; Z is the number of rolling elements; andα is the contact angle. Therefore, the calculated length insamplings for each kind of bearing fault is defined as
lcalculate(outer) = round
(Fs
fo
), (7)
lcalculate(inner) = round
(Fs
fI
), (8)
where the round ( ) function is to take the element to thenearest integer. In the case of bearing outer race and inner
3
Meas. Sci. Technol. 23 (2012) 065001 D Wang et al
The
normalized
length of
flat SE
1/ lmax S1(1) S1(2) S1(3) ........... S1(L-1) S1(L)
2/ lmax S2(1) S2(2) S2(3) ........... S2( L-1) S2(L)
............
............
............
............
............
............
............
(lmax-1)
/lmax
Slmax-1
(1)
Slmax-1 (2) Slmax-1 (3) ........... Slmax-1 (L) Slmax-1(L)
1 Slmax
(1)
Slmax (2) Slmax (3) ........... Slmax (L) Slmax (L)
Time The
norm
aliz
ed l
ength
incr
ease
d d
irec
tion
Figure 2. The proposed morphogram.
race faults, lcalculate(outer) is chosen as the establishedthreshold of the flat SE because lcalculate(outer) shownin equation (7) is larger than lcalculate(inner) shown inequation (8) according to equations (5) and (6). In otherwords, the length range of the flat SE varying from 1to lcalculate(outer) covers the length range of the flat SEvarying from 1 to lcalculate(inner). For the rest of this paper,we define the following rules: (a) for the computationof morphological analysis, the length range of the flatSE varies from 1 to lcalculate(outer); (b) for displaying theresults on the morphogram, the length range of the flatSE is normalized by dividing lcalculate(outer). It means thatthe normalized length range of the flat SE shown on themorphogram varies from 1/lcalculate(outer) to 1.
Step 6. Once the current length of the flat SE exceeds theestablished threshold lcalculate(outer), the morphogram canthen be constructed. The morphogram transforms a one-dimensional signal into a two-dimensional signal, whichis a function of the flat SE length and time. The functionvalues, namely the local maxima, are indicated by a two-dimensional color map, which is plotted in figure 2. Si( j)is the jth value of Si obtained using the closing operatorwith the length of the flat SE l(i) = i; L is the length ofthe original signal in samplings and Fs is the samplingfrequency. For instance, all of the possible bearing faultswere detected, and the intervals are denoted as T1, T2,. . .,Tp. These intervals have their own unchanged distanceslT1, lT2,. . ., lTp along the normalized length increaseddirection shown on the morphogram.
Step 7. Find the most likely bearing fault intervalTfault (only one interval) among all possible fault intervalsT1, T2,. . ., Tp. The most likely bearing fault interval,Tfault, that has the longest unchanged distance along thenormalized length increased direction is shown on themorphogram. (Examples can be found in the results shownin figures 5, 9 and 11.)
Step 8. Based on the most likely bearing fault intervalTfault, the optimal length range of the SE for the extractionof bearing fault cyclic features can be established by
maximizing the CI. The definition of the CI is given asfollows:
CI(i)
⎧⎪⎪⎨⎪⎪⎩
L
Tfault × Fs− Ci, Ci − L
Tfault × Fs> 0
Di, Ci − L
Tfault × Fs< 0
(9)
where Ci is the number of non-zero Si. Di is the numberof non-zero Si in the case of Ci − L
Tfault×Fs< 0. As
the length of the flat SE increases, noise is effectivelysmoothed. However, a longer length of the flat SE maysmooth the local maximum peaks and cause reduction ofthe number of bearing fault cyclic intervals. This willresult in poor bearing fault cyclic features (the cyclicfault features gradually disappear). Therefore, an optimallength range of the flat SE can be established to smoothnoise as much as possible and keep most of the bearingfault cyclic intervals by maximizing equation (9). Onthe other hand, given a bearing fault signal with lengthL, in some specific cases, the condition Ci − L
Tfault×Fsin
equation (9) has to be remedied by increasing ordecreasing 1 due to the random occurrence of bearingfault transients and the non-integer L
Tfault×Fs. It means that
the inspected maxima lines caused by bearing transientsignals will be one more (or less) than L
Tfault×Fs. The main
reason for condition amendment is that the bearing faultsignal is a second-order pseudo-cyclostationarity ratherthan a strict periodic signal. Therefore, if there is one moremaxima line shown on the morphogram which was causedby a transient, then Ci − L
Tfault×Fswill be decreased by 1. If
there is one less maxima line shown on the morphogramcaused by a transient, then Ci − L
Tfault×Fswill be increased
by 1.Step 9. After the optimal length range of the flat SEis established by maximizing equation (9), several localmaxima lines displaying bearing fault cyclic intervals willbe highlighted on the morphogram. The mean spacingof these local maxima lines is approximately equal tothe reciprocal of bearing fault characteristic frequencycalculated by equation (5) or (6).
To clarify the difference between Nikolaou andAntoniadis’s work [5] and the proposed method in this paper,
4
Meas. Sci. Technol. 23 (2012) 065001 D Wang et al
Assume the known bearing fault prior to morphological analysis
Scan all possible flat SE lengths for morphological analysis and calculate the intervals between
successive local maxima
Calculate the number of intervalsthat are close to the known fault
period for each flat SE
Find the optimal length of flat SE corresponding to the maximum
number of impulse intervals
Generate morphogram shown in Figure 2
Find all possible fault intervals shown on morphogram
Calculate the proposed index CI for each flat SE according to
Equation (9) and the most likely fault interval
Find the optimal length range of the flat SE by maximizing CI for
the extraction of fault cyclic features
Recommend an empirical rule for all cases of bearing fault signals
The prior parameter estimation
Evidence for a specific case
The posterior parameter estimationfor a specific case for the extractionof bearing fault cyclic features
(a) (b)
Figure 3. The optimal methods for the length selection of the flat SE(a) in the work of Nikolaou and Antoniadis [5] and (b) in this paper.
figure 3 shows the flowcharts of the optimal method for thelength selection of the flat SE proposed in [5] and this paper.As shown in figure 3(a), Nikolaou and Antoniadis assumedthat the initial pulse repetition period of the original bearingfault signal is known before the use of the morphologicalanalysis and the optimal selection of the flat SE length.Then, they suggested an empirical rule that a flat SE around0.6–0.7 times the cyclic fault interval could be used for theselection of the flat SE. In other words, it is reasonable toregard their recommending empirical rule obtained from apopulation of results by using morphological analysis as theprior parameter estimation for all cases of bearing fault signals.In figure 3(b), our proposed method reports a morphogram-based method to find all possible bearing fault intervals.The results shown on the morphogram can be regarded asthe evidence for a specific bearing fault signal. Then, basedon the most likely bearing fault interval displayed by themorphogram, the proposed CI is maximized to determine theoptimal length range of the flat SE. The optimal length range ofthe flat SE can be treated as the posterior parameter estimationfor a specific bearing fault signal.
Besides, the optimal length range of the flat SE is used toextract bearing fault cyclic features for bearing fault diagnosis.As a result, this is the reason why the parameter estimationproposed in this paper is more accurate than the empirical ruleproposed by Nikolaou and Antoniadis. On the other hand, in[8] it is reported that the empirical rule might be ineffectivein some cases. Based on the above discussion, we can explainwhy the empirical rule is sometimes ineffective. In general,there are only three possibilities. Recall that the empiricalrule is derived from a population of cases and the optimallength range obtained by the proposed method is a specificrange for a specific case. Firstly, if the optimal length rangerecommended by the empirical rule is less than the optimallength range obtained by the proposed method, then the result
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07-1
0
1
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07-1
0
1
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07-1
0
1
Am
plitu
de
Am
plitu
de
Am
plitu
de
Time (s)
(a)
(b)
(c)
Figure 4. Simulated signals: (a) the simulated signal consisting of aperiodic impulse signal with exponential decay; (b) the noise withnormal distribution with variance 0.1; (c) the mixed signal with theSNR equal to −11.16 dB.
obtained by the empirical rule will comprise much noisewhich is not completely smoothed by the closing operator.This means that the cyclic fault features are interrupted bynoise and it is not obvious to find the cyclic fault features.Secondly, if the optimal length range recommended by theempirical rule is covered by the optimal length range obtainedby the proposed method, then both methods would sharethe same effectiveness. Finally, if the optimal length rangerecommended by the empirical rule is larger than the optimallength obtained by the proposed method, then it means thatmany local maximum peaks related to cyclic fault intervalsare partly smoothed, and thus the fault cyclic features will bedestroyed and result in poor bearing fault cyclic features.
4. The validation of the proposed method by usingsimulated and experimental fault signals
4.1. A simulated fault signal with one resonant frequencyanalyzed by the morphogram
Firstly, a series of simulated impulses with exponential decaywithout considering the influence of slip is built to simulate abearing fault signal, which is given as
y(k) = e−α× mod (k/Fs,1/ fm ) × sin
(2π f1k
Fs
), (10)
where α is equal to 900 and fm is the fault characteristicfrequency (equal to 100 Hz). Fs is the sampling frequencywhich is set to 12 000 Hz and f 1 is the resonant frequency equalto 2500 Hz. Eight hundred and forty samplings are used. Anormally distributed random signal with mean 0 and variance0.1 is then added to equation (10). The SNR is defined as alogarithmic decibel scale of the ratio of the simulated signalpower to the added noise power.
The simulated signal, the noise signal and the mixedsignal with noise are plotted in figures 4(a)–(c). In this case,the SNR is equal to −11.16 dB for the simulated signal
5
Meas. Sci. Technol. 23 (2012) 065001 D Wang et al
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1-250 -200 -150 -100 -50 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-50 -40 -30 -20 -10 0 10
0.4
0.5
0.6
0.7
0.8
0.9
1
The
nor
mal
ized
leng
th o
f fl
at s
truc
turi
ng e
lem
ent
Time (s)
10ms The optimal
normalized length
range of flat SE
(a) (b)
(c)
CI
The local information of CI
The optimal normalized
length range of flat SE
Noise interruption
T2 T1
lT2
lT1
lT3
T3
lT4
T4
lT5
T5
lT6
T6
Cyclic features are partly smoothedT7
lT7
Figure 5. The morphogram for the extraction of cyclic fault intervals and selection of the optimal normalized length range of the flat SE inthe case of the simulated signal corrupted by normally distributed noise with variance 0.1. (a) The results shown on a morphogram; (b) CIvalues and the optimal length range; (c) local zoom of figure 5(b).
mixed with noise shown in figure 4(c). With the aid of themorphogram shown in figure 5(a), the most likely fault intervalT4 among all possible fault intervals can be observed when thenormalized length of the flat SE increases because the intervalT4 has the longest unchanged distance lT4 along the normalizedlength increased direction. Based on this evidence displayed infigure 5(a), the CI could be calculated for the determination ofthe optimal flat SE length range for bearing fault diagnosis. Theobtained CI values are plotted in figure 5(b). By maximizingthe proposed CI, the lower limit normalized length of the flatSE equals 0.49 in order to smooth as much noise as possible.The upper limit normalized length of the flat SE equals 0.82in order to retain the most frequent cyclic intervals. Therefore,an optimal length range of 0.49–0.82 (the corresponding reallength range for computation varied from 59 to 99 samplings)can be automatically determined. The local information of CIsshown in figure 5(b) is zoomed in figure 5(c). From the resultshown on the morphogram at the optimal length range of theflat SE, it is easy to find the bearing fault cyclic features.Besides, it is found that the proposed method does not requirethe known cyclic intervals of the simulated signal prior to themorphological analysis.
In order to investigate the influence of noise with differentvariances, normally distributed noise signals with variancesfrom 0.01 to 0.1 (increased by 0.01, SNR from 8.6 to –10.9)were added into equation (10), separately. Five realizations
Table 1. The normalized optimal length range of the flat SEobtained by the proposed method for processing the simulatedsignal with different variances.
Trials
SNR First Second Third Fourth Fifth(dB) realization realization realization realization realization
8.6 0.44–0.99 0.37 –0.99 0.36–0.99 0.33–0.95 0.43–0.953.0 0.40–0.99 0.36–0.95 0.40–0.99 0.38–0.99 0.44–0.95
−1.6 0.44–0.95 0.36–0.99 0.43–0.95 0.34–0.95 0.39–0.95−3.2 0.35–0.95 0.43–0.98 0.48–0.91 0.40–0.95 0.45–0.90−5.4 0.44–0.95 0.46–0.95 0.63–0.95 0.44–0.83 0.48–0.95−6.6 0.47–0.95 0.43–0.98 0.40–0.54 0.40–0.95 0.45–0.82−7.5 0.49–0.90 0.44–0.82 0.48–0.92 0.47–0.95 0.43–0.84−9.1 0.36–0.70 0.45–0.67 0.44–0.78 0.38–0.79 0.37–0.85
−10.1 0.45–0.91 0.46–0.69 0.46–0.87 0.35–0.62 0.35–0.80−10.9 0.5–0.75 0.42–0.8 0.37–0.76 0.54–0.71 0.47–0.83
for the normally distributed noise signal with each variancewere performed. For comparison with the empirical rule(a flat SE around 0.6–0.7 times the cyclic fault interval)given in [5], the optimal flat SE obtained by the proposedmethod is normalized by dividing the fault cyclic interval into120 samplings
(Fsfm
= 12000 Hz100 Hz
). The normalized length ranges
for the optimal selection of the flat SE are tabulated in table 1.It is obvious that the optimal length ranges of the flat SE
6
Meas. Sci. Technol. 23 (2012) 065001 D Wang et al
-200 -100 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07-1
-0.5
0
0.5
1
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1The
nor
mal
ized
leng
th o
f fl
at s
truc
turi
ng e
lem
ent
Time (s)
10ms
The optimal
normalized length
range of flat SE
(a)
(b) CI (c)
Noise interruption Time (s)
Cyclic features are partly smoothed
Figure 6. The case of the simulated signal with random slippage corrupted by normally distributed noise. (a) The simulated signal mixedwith normally distributed noise; (b) the morphogram of the signal shown in figure 6(a); (c) the optimal length range obtained by theproposed method.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.070
0.5
1
1.5
0.062 0.06250.4
0.45
0.5
0 0.01 0.02 0.03 0.04 0.05 0.06 0.070
0.5
1
1.5
0.062 0.06250.46
0.48
0.5
0 0.01 0.02 0.03 0.04 0.05 0.06 0.070
0.5
1
1.5
0.062 0.06250.46
0.48
0.5
Time (s)
(a)
(b)
(c)
Time (s)
Time (s)
Figure 7. Processing of the simulated signal shown in figure 6(a) by the method of Nikolaou and Antoniadis [5]. (a) The result obtained bythe length of the flat SE 0.5 times 120 samplings; (b) the result obtained by the length of the flat SE 0.6 times 120 samplings; (c) the resultobtained by the length of the flat SE 0.7 times 120 samplings.
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Meas. Sci. Technol. 23 (2012) 065001 D Wang et al
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
-2
0
2
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
-0.50
0.51
0 1000 2000 3000 4000 5000 6000
20406080
100
0 1000 2000 3000 4000 5000 6000
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Figure 8. Real bearing fault signals: (a) outer race fault signal; (b) inner race fault signal; (c) the frequency spectrum of the outer race faultsignal; (d) the frequency spectrum of the inner race fault signal.
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Figure 9. The morphogram for the extraction of cyclic fault intervals and selection of the optimal normalized length range of the flat SE inthe case of the outer race fault signal (the motor speed was 1797 rpm). (a) The results shown on a morphogram; (b) CI values and theoptimal length range; (c) local zoom of figure 9(b).
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Meas. Sci. Technol. 23 (2012) 065001 D Wang et al
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Figure 10. The morphogram for the extraction of cyclic fault intervals and selection of the optimal normalized length range of the flat SE inthe case of the outer race fault signal (the motor speed was 1725 rpm). (a) The results shown on the morphogram; (b) CI values and theoptimal length range; (c) local zoom of figure 10(b).
vary with different noise variances. In other words, the resultsindicated that the optimal length ranges of the flat SE varywith different blocks of bearing fault signals. But it should benoted that the optimal length range obtained by the proposedmethod for a specific fault signal is more accurate than theempirical rule that only takes part of the optimal length rangeobtained by the proposed method into account.
Another simulated fault signal is built by introducingrandom slippage into equation (10). The simulated signal withrandom slippage mixed with noise is plotted in figure 6(a). Themorphogram of the simulated signal shown in figure 6(a) isplotted in figure 6(b) and its corresponding optimal normalizedlength range of the flat SE is shown in figure 6(c). It is foundthat for this case the empirical rule (0.6–0.7 times the faultinterval (120 samplings)) is no longer included at the optimalnormalized length range (0.34–0.51) which was obtained bythe proposed method. For further investigating the influenceof different lengths of the flat SE, the results obtained by themethod of Nikolaou and Antoniadis with the length of 0.5, 0.6and 0.7 times the fault interval (120 samplings) are shown infigures 7(a)–(c). It is found that the cyclic features are onlypartly smoothed as the lengths of the flat SE are chosen as0.6 and 0.7 times the fault interval. Compared with the resultsshown in figures 7(a) and (b), the result obtained by the lengthof 0.5 times the fault interval (120 samplings) retains most ofthe cyclic features (a clearer result obtained by our proposedmethod is shown in figure 6(b)). Therefore, considering both
results shown in figures 6 and 7, the optimal length rangeobtained by the proposed method is more accurate than theempirical rule recommended by Nikolaou and Antoniadis forthe extraction of bearing fault cyclic features. In other words,the proposed method keeps most of the bearing fault cyclicfeatures.
4.2. Real bearing fault signals analyzed by the morphogram
A set of real motor bearing data, collected with a samplingfrequency of 12 kHz by an accelerometer at the drive end ofthe motor housing [10], was used to validate the proposedmethod. These signals are widely used in the validation ofproposed methods for bearing fault diagnosis [6, 11, 12].Single point faults were introduced to normal bearingsusing electro-discharge machining with a fault diameter of0.007 inches and a fault depth of 0.0011 inches. The outer raceand inner race fault characteristic frequencies were calculatedas 107 and 162 Hz, respectively. The motor speed was1797 rpm. Only 600 samplings were used for each faultsignal. The original bearing outer race and inner race faultsignals are plotted in figures 8(a) and (b), where theircorresponding frequency spectra are shown in figures 8(c)and (d).
The results obtained by the proposed method are shownin figure 9 for processing the outer race fault signal shown infigure 8(a). In figure 9(a), the most likely cyclic interval T2
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Figure 11. The morphogram for the extraction of cyclic fault intervals and selection of the optimal normalized length range of the flat SE inthe case of the inner race fault signal. (a) The results shown on the morphogram; (b) CI values and the optimal length range; (c) local zoomof figure 11(b).
among all possible fault intervals is observed. By using theproposed CI, the results of which are shown in figures 9(b)and (c), a value ranging from 0.35 to 0.83 (the correspondingreal length for computation varied from 39 to 93 samplings)is automatically obtained as the effective normalized length ofthe SE to suppress noise and keep most of the cyclic features.It is found that the potential periodic signal becomes moreobvious by smoothing the noise with longer lengths of the flatSE. However, the cyclic intervals of the fault are reduced whenthe normalized length of the flat SE is larger than 0.83 (the reallength is equal to 93 samplings). In addition, the bearing faultinformation can be easily found at the optimal length range ofthe flat SE on the morphogram shown in figure 9(a).
Furthermore, outer race fault signal with more than onedifferent speed was analyzed by the proposed method. Theresults are shown in figure 10. A value ranging from 0.34 to0.71 (the corresponding real length for computation variedfrom 40 to 83 samplings) is automatically obtained as theeffective normalized length of the SE. The results shown infigures 9 and 10 indicate that the optimal normalized lengthrange varies with different cases even though they belong tothe same bearing outer fault. The reason is that the principle ofmorphological analysis aims to modify the shape of a specificsignal. Different signals have different specific shapes. As aresult, the optimal length range varies with the shape of thespecific signal.
The same analyzing method is applied to the inner racefault signal. The morphogram for the inner race fault signal isshown in figure 11(a). The most likely fault interval T6 amongall possible fault intervals is observed. In order to suppressthe noise and retain most of the periodic features, a valueranging from 0.20 to 0.47 (the corresponding real length forcomputation varied from 23 to 53 samplings) is obtained as theeffective normalized length of the SE by using the most likelyfault-interval-based CI, which is shown in figures 11(b) and(c). The fault cyclic intervals around 6.2 ms are easily detectedat the optimal length range of the morphogram.
The results shown in figures 9–11 illustrate that theproposed method is effective in detecting bearing faults.Moreover, an alternative solution has been proposed todetermine the optimal length range of the flat SE withoutthe assumption of the known bearing fault pattern prior tomorphological analysis. In other words, the proposed methoddoes not require any knowledge of the fault cyclic intervalbefore the morphological analysis is performed. The mostlikely fault interval among all possible fault intervals isobserved by the morphogram. Based on the most likely faultinterval reported on the morphogram, the optimal lengthrange of the flat SE could be automatically determined bymaximizing the CI for the extraction of bearing fault cyclicfeatures for bearing fault diagnosis.
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Figure 12. The methods processing the real outer race fault signal. (a) The SNR values obtained by the method of Dong et al [8]; (b) theimpulsive signal obtained by the optimal length corresponding to the maximum SNR1 reported by figure 12(a); (c) the envelope extractedby the method of Nikolaou and Antoniadis [5]; (d) the signal extracted from the morphogram shown in figure 9(a) at the optimal flat SElength equal to 40.
4.3. Comparison with two other methods
The empirical rule (recommending length of the flat SE0.6 times the reciprocal of bearing fault characteristicfrequency) suggested by Nikolaou and Antoniadis [5] and themethod developed by Dong et al [8] were used as a comparisonwith the proposed method because both of them use flat SE. Inthe work of Nikolaou and Antoniadis [5], the closing operatorprovided by equation (4) was used to extract the envelope ofthe original signal. On the other hand, Dong et al [8] usedthe average of opening and closing operators provided byequations (3) and (4) to extract the bi-directional impulsivecomponents. The optimal length suggested by Dong et al [8]was simply reviewed as follows:
Step 1. Assume that the length of the flat SE is equal to0.1 × T. Here, T is the reciprocal of the bearing faultcharacteristic frequency.
Step 2. Perform closing and opening operators on theoriginal signal. Then two signals A and B were obtained.So, A is the positive envelope and B is the negativeenvelope. Take the average of the positive envelope Aand the negative envelope B as the impulsive signal.
Step 3. Calculate the power Poriginal of the original signaland the power PImpulse of the extracted impulsive signal.Then, the signal-to-noise ratio 1 (SNR1) is obtained by
SNR1 = 10 × log
(PImpluse
Poriginal − PImpluse
). (11)
Step 4. Repeat steps 1 to 3 by increasing the length of flatSE with the step length 0.1 × T until the length of theflat SE exceeds T. Ten SNR values are obtained after theoptimal process terminates.Step 5. Find the optimal length corresponding to themaximum of SNR values. Finally, use the optimal lengthto extract the impulsive signal as the step 2 does.
These two methods were used to analyze the same realsignals shown in figures 8(a) and (b), separately. The resultsfor processing the outer race fault signal are plotted infigure 12. First of all, figure 12(a) reports the SNR valuesobtained by the method of Dong et al for the selection of theoptimal flat SE. In this case, the optimal length suggested bythe maximum of SNR is 90. Then, the average of positive andnegative envelopes was regarded as the impulsive signal shownin figure 12(b). The fault cyclic intervals could be detected.The envelope signal obtained by the closing operator withthe length of the flat SE 0.6 times the reciprocal of bearing
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Figure 13. The methods processing the real inner race fault signal. (a) The SNR values obtained by the method of Dong et al [8]; (b) theimpulsive signal obtained by the optimal length corresponding to the maximum SNR1 reported by figure 13(a); (c) the envelope extractedby the method of Nikolaou and Antoniadis [5]; (d) the signal extracted from the morphogram shown in figure 11(a) at the optimal flat SElength equal to 40.
fault characteristic frequency is shown in figure 12(c). Thefault cyclic intervals are diagnosed by the method of Nikolaouand Antoniadis. However, their resolutions for identificationof bearing fault cyclic features are not as good as the resultshown in figure 9(a), where the fault cyclic intervals betweenthe lower limit and the upper limit of the optimal length of theflat SE are much clearer. The result obtained by the proposedmethod at the optimal length of the flat SE equal to 40 isgiven in figure 12(d) for further comparison with the resultsshown in figures 12(b) and (c). It is clear to see that the resultobtained by the proposed method has better visual inspectionability for identifying outer race fault cyclic intervals. Theresults for processing the inner race fault signal are givenin figure 13. In figure 13(a), the length 11 of the flat SEwas suggested to extract the impulsive signal. However, thefault cyclic intervals are not clear in figure 13(b). The closingoperator with the length of the flat SE 0.6 times the reciprocalof bearing fault characteristic frequency was employed toextract the envelope signal. Its result is shown in figure 13(c),where although the envelope was displayed, the fault cyclicintervals were not easily observed. Compared with the resultsshown in figures 13(b) and (c), the results shown in figure 11(a)at the optimal length range show better visual inspection effectfor the extraction of bearing fault cyclic features. For further
comparison, the result obtained by the proposed method atthe optimal length of the flat SE equal to 40 is shown infigure 13(d), where the signal clearly displays the inner racefault cyclic features. In conclusion, the proposed method hasbetter visual inspection ability than those methods proposedby Nikolaou and Antoniadis [5] and the method developed byDong et al [8].
4.4. Comparison with kurtosis used as an indicator for theselection of the optimal length of the flat SE
Kurtosis has been widely used to indicate the occurrenceof bearing faults and the selection of the proper resonantfrequency band for the enhancement of the bearing fault signal[4]. In this section, kurtosis was used to quantify the envelopesignals obtained by the closing operator with different flatSE lengths. The result for processing the inner race faultsignal is shown in figure 14, where it was found that kurtosisis ineffective to select the optimal length range of the flatSE because bearing inner race fault cyclic features are notobvious at the optimal length range of the flat SE establishedby maximizing the kurtosis. Compared with the result shownin figure 11(a), it is easy to find that the proposed CI is betterthan kurtosis for the selection of the optimal length range ofthe flat SE for the extraction of bearing fault cyclic features.
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Figure 14. The result obtained by replacing the proposed CI with kurtosis for processing inner race fault signal. (a) The results shown on themorphogram; (b) kurtosis values and the optimal length range.
5. Further discussion: application of the proposedmethod to a real industrial fault signal
Nevertheless, it must be pointed out that, in some cases,heavy noise and other strong vibration sources may mask themorphological features of the original bearing faulty signal.Hence, the band-pass filtering operation, such as the optimalwavelet filtering [12], fast kurtogram [13] and discrete wavelettransform [15], is required to retain the resonant frequencyband for removing the interruption caused by low-frequencyperiodic components prior to the morphological analysis. Abearing fault signal collected from an industrial machine[14] was used as an example for further discussion. Twofaulty components were confirmed. One was rotor eccentricproblem and the other one was bearing outer race defect.The sampling frequency is 32.77 kHz. In other words, theouter race fault characteristic frequency was about 100 Hzor outer race fault cyclic interval was about 10 ms [14].The temporal and frequency-dependent signals are shown infigures 15(a) and (b). It is difficult to find the bearing faultsignal which is overwhelmed by other strong vibration sourcesin figure 15(a). However, a resonant frequency band could befound at the higher frequency band in figure 15(b).
5.1. The analysis of an industrial fault signal by the proposedmethod without removing the low-frequency periodiccomponent
The proposed method was directly applied to the originalsignal shown in figure 15(a). The results are shown infigures 16(a) and (b), where it was found that even though thereis an interruption caused by rotor eccentric problem, bearingfault cyclic features (10 ms) are still detected. However, itshould be noted that bearing cyclic features are not obviousdue to interruption caused by the rotor eccentric fault. Here, itshould be pointed out that the empirical rule (0.6–0.7) is not asgood as the proposed method because the proposed method can
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find more bearing fault cyclic intervals at the optimal lengthrange.
5.2. The analysis of an industrial fault signal by the proposedmethod with removal of the low-frequency periodiccomponent
In order to extract much more bearing fault information,discrete wavelet transform [15] (the decomposition level 3 andDaubechies wavelet with the order 1) was used to remove the
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Figure 16. The morphogram for the extraction of cyclic fault intervals and selection of the optimal normalized length range of the flat SE inthe case of the industrial signal including rotor eccentric problem and bearing outer race fault. (a) The results shown on the morphogram;(b) CI values and the optimal length range.
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Figure 17. The morphogram for the extraction of cyclic fault intervals and selection of the optimal normalized length range of the flat SE inthe case of the industrial bearing fault signal. (a) The results shown on the morphogram; (b) CI values and the optimal length range.
interruption caused by the low-frequency periodic component.The bearing fault signal obtained by discrete wavelet transformis shown in figure 15(c). Then, the proposed method wasapplied to the bearing fault signal shown in figure 15(c).The results obtained by the proposed method are shown infigure 17. In figure 17(a), 10 ms cyclic features are readilydetected at the optimal normalized length range shown infigure 17(b). At this moment, a value of 0.44–0.56 (thecorresponding real length for computation varied from 144to 186 samplings) is obtained as the effective length of theflat SE. The reason why the real optimal length for calculationbecomes larger than those reported in the previous section isthat the sampling frequency in this case is 32.77 kHz and thusthe outer race fault cyclic interval is about 328 samplings.
In conclusion, the morphogram coupled with the proposedindex, CI, for deciding the optimal length range of the flat SEhas good potential to serve as a signal processing methodfor the extraction of bearing fault cyclic features. Besides,removing the interruption from the low-frequency componentsis beneficial for diagnosing bearing faults.
6. Conclusions
In this paper, an enhanced morphological analysis, whichhas been named as ‘morphogram’ is proposed for bearingfault diagnosis. The morphogram maps a one-dimensionalsignal onto a two-dimensional signal to serve as the functionof the length of the flat SE and time. The advantages of
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Meas. Sci. Technol. 23 (2012) 065001 D Wang et al
the morphogram are illustrated as follows: firstly, it onlyrequires a few samplings to determine bearing fault cyclicintervals. Compared with the methods depending on inspectionof bearing fault characteristic frequencies, the proposedmethod no longer considers frequency resolution, which isusually constrained by the length of samplings. Secondly, theproposed method is very simple and only requires addition andsubtraction operations. Thirdly, based on the most likely faultinterval observed by the morphogram, the optimal length rangeof the flat SE can be automatically determined by maximizingthe proposed index, CI, for the extraction of bearing fault cyclicfeatures. It means that the morphogram with the maximizedCI can smooth most of the noisy signals and extract thesignal that contains fault cyclic intervals. Besides, differentsignals have different optimal length ranges for the use ofthe flat SE. Fourthly, the use of the CI is better than kurtosisfor the selection of the optimal length range of the flat SE.Finally, the proposed method was compared with two othermethods. The results show that the proposed method hasbetter fault cyclic features for clear bearing fault identification.The morphogram provides a simple method to inexperiencedmaintenance workers in diagnosing the existence of outer raceand inner race bearing faults.
Acknowledgments
The work described in this paper was partly supported bya grant from the Research Grants Council of the HongKong Special Administrative Region, China (project no CityU122011) and partly supported from the Croucher Foundation(project no 9220027). The authors wish to thank Professor K ALoparo for his permission to use the bearing data. Finally, weappreciate the valuable comments from anonymous refereeswho helped us improve our work.
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