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Research ArticleNonlinear Parameters for Monitoring Gear ComparisonBetween Lempel-Ziv Approximate Entropy and SampleEntropy Complexity

Mourad Kedadouche Marc Thomas Antoine Tahan and Raynald Guilbault

Department of Mechanical Engineering Ecole de Technologie superieure 1100 Notre-Dame Street WestMontreal QC Canada H3C 1K3

Correspondence should be addressed to Marc Thomas marcthomasetsmtlca

Received 15 September 2014 Revised 16 March 2015 Accepted 14 April 2015

Academic Editor Miguel Neves

Copyright copy 2015 Mourad Kedadouche et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Vibration analysis is themost used technique for defect monitoring failures of industrial gearboxes Detection and diagnosis of geardefects are thus crucial to avoid catastrophic failures It is therefore important to detect early fault symptomsThis paper introducessignal processing methods based on approximate entropy (ApEn) sample entropy (SampEn) and Lempel-Ziv Complexity (LZC)for detection of gears defects These methods are based on statistical measurements exploring the regularity of vibratory signalsApplied to gear signals the parameter selection of ApEn SampEn and LZC calculation is first numerically investigated andappropriate parameters are suggested Finally an experimental study is presented to investigate the effectiveness of these indicatorsand a comparative study with traditional time domain indicators is presented The results demonstrate that ApEn SampEn andLZC provide alternative features for signal processing A newmethodology is presented combining both Kurtosis and LZC for earlydetection of faults The results show that this proposed method may be used as an effective tool for early detection of gear faults

1 Introduction

Gearboxes play an important role in industrial applicationsand unexpected failures often result in significant economiclosses Numerous papers considering gear condition moni-toring through vibration measurements were published overthe years Compared to classical techniques such as statisticaltime indicators or Fast Fourier Transform advanced signalprocessing techniques like time-frequency analysis (STFTWigner-Ville) [1ndash4] or wavelet transform [5 6] have shown tobemore efficient for gear defect detection Baydar et al [7ndash10]proposed various methods such as the instantaneous powerspectrum Wigner-Ville distribution and the wavelet trans-form method for local tooth fault detection from vibrationand acoustic signals Yesilyurt [11] applied the spectrogramand scalogram approach for gearbox fault detection

Various researches are focused on the understanding ofdynamic gears behaviour Parey andTandon [12] give a review

of the dynamic modelling of spur gears including defectsThe effects of various nonlinearities such as variable meshstiffness mesh damping gear errors effect of friction andbacklash on gear dynamic behaviour have been discussedDue to instantaneous variations of these effects gear systemsare often characterised by nonlinear behaviours

Randall [13] classified the vibrations generated by gearmeshing as being caused by deviations from the ideal toothprofile producing a vibration signal at the tooth-meshingfrequency These variations in tooth loading generate ampli-tude modulations and fluctuations in the rotational speedNonuniform tooth spacing produces frequency modulationeffects and local tooth faults generate additional impulsesImperfections (errors) in the gear during manufacture willalso be a source of vibration The instantaneous variations ofthe effects cited above may be considered as the first stepsof degradation of the gear At this stage a modification ofthe characteristics of dynamics responses may be observed

Hindawi Publishing CorporationShock and VibrationVolume 2015 Article ID 959380 12 pageshttpdxdoiorg1011552015959380

2 Shock and Vibration

However in the early stages of fault development mainfrequencies have low amplitudes and can be masked by othervibration sources or buried in background noise

A large number of statistical metrics have been appliedfor monitoring gear defects such as the root mean squareCrest Factor Kurtosis FM0 FM4 FM4

lowast M6 M6A M6AlowastNA4 NA4lowast NB4 NB4lowast [14ndash16] Most of these measurescharacterize the statistical properties of vibration and givea global view to provide any recognition of localized phe-nomena Stevens et al [17] noted that these measures aresuitable for detection and diagnosis when mechanical faultstake the form of impulses which impose periodic pulses ofshort time duration However Feng et al [18] noted that theseparameters cannot give a perception into the gear dynamicsystemgenerating the vibration signalsWu et al [19] used thestandard deviation for monitoring tooth cracks of gear but ithas been proven to be ineffective for tooth crack detection

Therefore it is interesting to use others techniques asnonlinear estimations of parameters which may be a goodalternative to monitoring hidden faults in the measuredsignals Yan and Gao [20] give a review of a large numberof nonlinear parameter identification techniques Wang et al[21] researched the application of some nonlinearity anal-ysis methods including the pseudophase portrait singularspectrum and correlation dimension for monitoring anddiagnosis of rotating machinery Jiang et al [22] used thecorrelation dimension for monitoring gearbox Feng et al[18] investigated the irregularity of the gear signal using theregularisation dimensions

As said before the effects of various nonlinearities cause avariation in the dynamic behaviour of the gear As the defaultdevelops more contact pressure between two mating partschanges The contact pressure variation results in amplitudeand frequency modulations and hence more frequency willappear in the frequency domain with their harmonics andan increase of the noise can be observed Therefore thefault progress leads to complexity (randomness) changesbecause of a variation of the contact pressure between twomating partsThe complexity of a signal increases whenmorefrequency components exist On the other hand when thedefect increases we observe more frequency components inthe signal and we can use the complexity of the signal asa quantitative measurement to evaluate the severity of thedefect The complexity of the signal can be described by theapproximate entropy (ApEn) the sample entropy (SampEn)and the Lempel-Ziv complexity (LZC) These parameters arenonlinear parameters which can be used to characterise theregularity of the signal

The Lempel-Ziv complexity (LZC) sample entropy (Sam-pEn) and approximate entropy (ApEn) thus present alterna-tive tools for signal analysis involving nonlinear dynamicsThese methods are becoming more popular and have foundwide applications in various disciplines especially in the fieldof biomedical engineering ApEn has recently received moreattention Yan and Gao [20] investigated the application ofApEn for the health monitoring of rolling element bearingsHe and Zhang [23] applied the ApEn for monitoring signalof the acoustic emission from defects in rolling elementBearings Fu et al [24] used the ApEn method to fault

signal analysis in electric power system Using ApEn Xuet al [25] detected the looseness of the bearing bushingin turbo generator In all these work ApEn is used as anonlinear feature parameter for analysing the vibration signaland effectively identifying the conditions of the mechanicalsystem

On the other hand Yan and Gao [26] investigated theapplication of Lempel-Ziv complexity (LZC) for the healthmonitoring of rolling element bearingsWang et al [27] com-pare and analyse quantitative diagnosis methods based onLempel-Ziv complexity for bearing faults using continuouswavelet transform (CWT) Empirical Mode Decomposition(EMD) method and wavelet packet method for decomposi-tion of vibration signal Kedadouche et al [28] combined LZCand EMD for early detection of gears cracks

The use of SampEn was introduced by Dennis Wonget al [29] which was the improvement of a previousmethod namely approximate entropy (ApEn) Howeversample entropy was not often used in monitoring of rotatingmachine Dennis Wong et al [29] used sample entropy forclassification of ball bearing faults Zhang et al [30] usedmul-tiscale entropy and adaptive neurofuzzy inference for bearingfault classification The sample entropy was discussed andimplemented in many areas such as physiological time seriesanalysis [31] heart rate variability [32] and neural respiratorysignal [33] Sun et al [34] proposed a health auxiliarymethodbased on SampEn for lead-acid battery unit Widodo et al[35] proposed intelligent prognostics for battery health basedon sample entropy (SampEn) features of discharge voltageThe SampEn feature can depict degradation condition ofbattery and represents the health condition of battery

As illustrated above LZC SampEn andApEn are becom-ing more and more attractive in the field of detectionand fault diagnosis However no work has been found toapply ApEn LZC or SampEn measurement for diagnosisof gear faults Therefore in this paper ApEn SampEn andLZC are introduced to analyse vibration signals from gearand investigate their efficiency for the defect detection andseverity evaluation of gears faults

2 Theoretical Background

21 Approximate Entropy Consider a time series 119878(119894) 119894 =

0 119873 Its ldquoregularityrdquo may be measured by ApEn in amultiple dimensional space so that series of vectors areconstructed and expressed as follows

119883 (119873minus119898+ 1)

= 119909 (119873minus119898+ 1) 119909 (119873minus119898+ 2) 119909 (119873)

(1)

Each vector is composed of119898 consecutive and discrete pointdata of the time series 119878 The distance between two vectors119883(119894) and 119883(119895) can be defined as the maximum differencebetween their respective corresponding elements

119889 (119883 (119894) 119883 (119895))

= max119896=12119898

(1003816100381610038161003816119909 (119894 + 119896 minus 1) minus 119909 (119895 + 119896 minus 1)1003816100381610038161003816)

(2)

Shock and Vibration 3

where 119894 = 1 2 119873 minus 119898 + 1 119895 = 1 2 119873 minus 119898 + 1 and119873is the number of data points contained in the times series

For each vector 119883(119894) a measurement that describes thesimilarity between the vectors119883(119894) and all other vectors119883(119895)can be defined as

119862119898

119894(119903) =

1119873 minus (119898 minus 1)

sum

119895 =119894

Θ119903 minus 119889 (119883 (119894) 119883 (119895)) (3)

where

Θ 119909 =

1 119909 ge 0

0 119909 lt 0(4)

The symbol 119903 represents a predetermined tolerance value 119903 =119896 times std

119878 where 119896 is a positive constant (119896 gt 0) and std

119878is the

standard deviation of the time series 119878A finite time series consisting of119873 data points is used to

estimate the ApEn value of the time series which is definedas

ApEn (119898 119903119873) = 0119898

(119903) minus 0119898+1

(119903)

0119898

(119903) =1

119873 minus119898 + 1

119873minus119898+1sum

119894=1ln [119862119898119894(119903)]

(5)

22 Sample Entropy Sample entropy was developed becauseit has better representation of the entropy in the analysedsignals as comparing with the original approximate entropy(ApEn) [31] The motivation for this method is the classifica-tion of complex systems that includes both deterministic andstochastic features from time series with a limited numberof data points [32] Higher values of sample entropy describemore irregularities in the time series

For a given 119873 total number of data points we form the119873 minus119898 + 1 vectors119883

119898(119894) as

119883119898(119894) = 119909 (119894) 119909 (119894 + 1) 119909 (119894 +119898minus 1)

119894 = 1 to 119873 minus119898 + 1(6)

where119898 is the length of sequences to be comparedThe distance between such vectors is defined as

119889119898(119883119898(119894) 119883

119898(119895))

= max (1003816100381610038161003816119883119898 (119894 + 119896) minus119883119898(119895 + 119896)

1003816100381610038161003816)

0 le 119896 le 119898 minus 1

(7)

For 119894 = 1 to119873 minus119898 + 1 let

119861119898

119894(119903) =

1119873 minus119898 + 1

V119898 (119894) (8)

where 119903 is the tolerance for accepting matrices and V119898 is thenumber of 119889

119898(119883119898(119894) 119883119898(119895)) for 119894 = 119895

Similarly we define another function

119860119898

119894(119903) =

1119873 minus119898 + 1

119908119898+1

(119894) (9)

where119908119898+1 is the number of 119889119898+1(119883119898+1(119894) 119883119898+1(119895)) le 119903 for

119894 = 119895The symbol 119903 represents a predetermined tolerance value

where 119896 is a positive constant (119896 gt 0) and std119878is the standard

deviation of the time series

119903 = 119896 times std119878 (10)

We can determine the expression of probability matchingpoints

119861119898

(119903) =1

119873 minus119898 + 1

119873minus119898

sum

119894=1119861119898

119894(119903)

119860119898

(119903) =1

119873 minus119898 + 1

119873minus119898

sum

119894=1119860119898

119894(119903)

(11)

where 119861119898

(119903) represents the probability that two sequenceswill match for119898 points whereas119860119898(119903) is the probability thattwo sequenceswillmatch for119898+1 pointsThe sample entropyis then defined as

Sample Entropy = minus ln [119860119898

(119903)

119861119898(119903)

] (12)

23 Complexity Analysis The complexity analysis is based onthe Lempel-Ziv definition [36]This approach transforms theanalysed signal into a data sequence To illustrate the proce-dure consider a gear vibration signal with a known meanvalue A new sequence (119878) is reconstructed by comparingthe value of each sample of the previous sequence within themean value If the value of the sample is larger it is set to one(1) otherwise to zero (0)Therefore only two binary symbolsare present in the new data sequence This 119878 is subsequentlyscanned from its first sample to its end When a subsequencethat is not encountered in the previous scanning process isdiscovered the complexity value is increased by one (1)Thusthe Lempel-Ziv complexity reflects the number of all differentsubsequences contained in the original sequence Figure 1(reproduced from [26]) described the algorithm

For generality sake normalized complexity 119862(119899) is oftenused to obtain ameasure independent of the sequence length

119862 (119899) =119888 (119899)

119887 (119899)

119887 (119899) =119873

log2 (119873)

(13)

24 Parameters Selection of ApEn SampEn and LZC for GearSignals From the above described algorithm of ApEn andSampEn it can be seen that the calculated ApEn and SampEnvalues depend on two parameters which are the embeddeddimension119898 and the tolerance 119903

However no guidelines exist for optimizing their valuesIn order to simulate the vibratory signals of gearbox agear multiplicative model whose meshing is modulated in


Yes

No

No

End

Yes

Start

Data sequence S

construction N = length (S)

the character S(i minus 1) into SQ120587

SQ120587 contains Q

c(n) = c(n) + 1

i = i + 1

i gt N

c(n) = 0 i = 1)

(Q =

Read S(i) and score it into Q and

C(n) = c(n)b(n)

SQ120587 = null

null

Initialization

Figure 1 The flow chart of LZC algorithm

Table 1 Geared system data of the simulated signal

Frequency (Hz) AmplitudePinion number of teeth = 20 15 015Gear number of teeth = 21 1428 015Gear mesh the first harmonic 300 1Gear mesh the second harmonic 600 06Gear mesh the third harmonic 900 03

amplitude has been usedThe gear model as defined in [37] isused

119909 (119905) =

+infin

sum

119898=minusinfin

1198781199031 (119905 minus1198981205911199031) +

+infin

sum

119898=minusinfin

1198781199032 (119905 minus1198981205911199032)

+

+infin

sum

119899=minusinfin

119878119890(119905 minus 119899120591

119890)

sdot (1++infin

sum

119898=minusinfin

1198781199031 (119905 minus1198981205911199031))

(14)

where 120591119890 1205911199031 and 120591

1199032 represent the meshing period andthe rotational periods 119878

119890(119905) 1198781199031(119905) and 119878

1199032(119905) represent themeshing signal and its modulation

Table 1 gives the parameters simulation of the modeldescribed in (14)

Figures 2 and 3 represent the simulated signal and itsspectrum respectively

For a given dimension 119898 the key step of the ApEnalgorithm is to calculate the distance between two vectors andto count the number of distances less than 119903 (where 119903 = 119896

lowast

standard deviation of the time series) as discussed in [20]

0 1000 2000 3000 4000 5000 6000 7000 8000

0

1

2

3

Sample

minus3

minus2

minus1

Figure 2 The simulated signal

ZoomModulation Modulation

Meshing Frequency rotation

of the gear andpigon

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

08

1

12

14

Frequency (Hz)

10 12 14 16 18 200

002004006008

01012014016

Figure 3 Spectrum of the simulated signal


002 04 06 08 10

05

1

15

2

ApEn

m = 1

m = 2

m = 3

m = 4

m = 5

m = 6

m = 7

m = 8

k value

Figure 4 The calculated ApEn values by different parameters (119896value and119898)

The selections of 119896 and 119898 are based on a convergenceanalysis [20 23ndash26] The choice of 119898 and 119896 depends on thenature of the time series treated The best parameters to bechosen are those who present little fluctuation or on the otherhand a good reproducibility

From Figure 4 it can be seen that when119898 = 1 ApEn hasthe poorest convergence property (a big fluctuation) With119898increasing (more than 2) the convergence property of ApEnbecomes better for 119896 greater than 04

LikeApEn sample entropy (SampEn) depends on the twoparameters 119898 and 119896 Therefore the determination of thesetwo parameters is critical The same procedure was under-taken for the selection of 119896 and119898The same characteristic wasobtained as ApEn From Figure 5 we can see that for 119898 = 2the SampEn presents a good convergence when 119896 is greaterthan 05

However a larger 119898 will lead to much higher computa-tional cost Actually when 119898 = 2 the convergence propertyof ApEn and SampEn is already good enough Of course the119896 value should also be a compromise to avoid losing muchdetailed system information

By this investigation 119898 = 2 and 119896 value = 05 timesthe standard deviation are selected for ApEn and SampEncalculation of the vibration signal in this paper

The relationship between the ApEn SampEn and LZCvalues and the data length are illustrated in Figure 6 InFigures 7 and 8 seven simulated signals are comparativelydisplayed under sampling rates of 4 8 12 16 20 and 24 kHzrespectively It is seen in three cases (ApEn SampEn andLZC) that when the data length is greater than 5000 pointsthe variation of ApEn SampEn and LZCwith respect to eachfrequency sampling rate becomes insignificant

However the sampling frequency and the data lengthare linked So we must take consideration of the component

01 02 03 04 05 06 07 08 09 101

02

03

04

05

06

07

08

09

Sam

ple e

ntro

py

k value

m = 2

m = 3

m = 4

m = 5

m = 6

m = 7

m = 8

Figure 5 The calculated sample entropy values by different param-eters (119896 value and119898)

05 1 15 2 25 30Length data times10

4

4kHz8kHz12kHz

16kHz20kHz24kHz

015

02

025

03

035

04

ApEn

Figure 6 The calculated ApEn values by different parameters(length of the data and the sample frequency)

defining the signalThe sample frequency must be more than2 times the highest frequency presented in the signal forShannon respect In our case the sample frequencymust be 2times greater than the thirdmeshing frequency (900Hz)Thesimulation was conducted with 119865

119904greater than 4 kHz which

is sufficient enough On the other hand for a better detectionof the modulations we must choose a good resolutionfrequency which is defined by Δ119867 = 119865

119904length(signal) For

the simulated signal the modulations are about plusmn15Hz Soa frequency resolution of 1Hz is enoughThis means that the


01

015

02

025

03

035

Sam

ple e

ntro

py


4

4kHz8kHz12kHz

16kHz20kHz

Figure 7 The calculated SampEn values by different parameters(length of the data and the sample frequency)

0

01

02

03

04

05

LZC

16kHz20kHz24kHz


4

4kHz8kHz12kHz

Figure 8The calculated LZC values by different parameters (lengthof the data and the sample frequency)

minimum length of data to be analysed is equal to the samplefrequency (4000Hz)

Figure 9 shows a comparison of computing time betweenLZC ApEn and SampEn

It is clear that the LZC is the best method for timecomputingThe sample entropy algorithm is simpler than theApEn algorithm requiring approximately one-half time tocalculate

25 Influence of Noise It is well known that a white noisecontains most abundant frequency components comparedwith other kinds of signal If a signal is contaminated by awhite noise calculated ApEn SampEn and LZC values willalso be Using the simulated signal the ApEn SampEn and

0

20

40

60

40968192

1228816384

20480

Com

puta

tion

time (

s)

Length data

ApEnSample entropyLZC

Figure 9 Time computation of both LZC and ApEn for differentlength of data

Table 2 ApEn SampEn and LZC for different SNR of gear signals

Cases ApEn LZC SampEnThe simulated signal 03529 01294 02644SNR = 80 dB 03547 01364 02656SNR = 60 dB 03648 01692 02765SNR = 40 dB 04495 03056 03500SNR = 0 dB 13163 08909 12190

LZC values corresponding to different SNRs are calculatedas listed in Table 2

It may be noticed that the ApEn SampEn and LZCvalues increase as the SNR decreases which corresponds toa degradation of the data quality As discussed in [20] theworking condition of machine system deteriorates due to theinitiation andor propagation of the defects The number offrequency components contained in the signal will increaseand the SNRdecreases resulting in a decrease in its regularityThis allows a decrease of its ApEn SampEn and LZC valuesThe simulation results listed in Table 2 confirm that theApEnSampEn andLZCvalues provide a quantitativemeasurementfor characterising a dynamic signal which can be representedby the deterioration of a machinersquos health condition

From Table 2 it can be seen that ApEn and SampEn arevery sensitive to large noises however ApEn and SampEn arenearly unaffected by the noise which stay low Otherwise wecan say that the LZC ismore sensitive to noise thanApEn andSampEn

3 Experimental Study

Therecordings of vibration signal were carried out at CETIMFrance on a gear system with a train of gearing with aratio of 2021 functioning continuously until its destruction


default

day 11 day 11 Dent 16 day 12Dent 16Dent 2

Figure 10 Evolution of the chipping

Table 3 Geared system data

Parameter Pinion GearNumber of teeth 20 21Speed (rpm) 1000 95238Drive torque (Nm) 200Face width (m) 0015 003Module (m) 001 001Pressure angle 20∘ 20∘

Addendum coefficient 10 10Dedendum coefficient 14 14

Table 4 Daily mechanical appraisal

Day Observation1 No acquisition2 No anomaly3 No anomaly4 5 6 7 Chipping teeth 128 No evolution9 Tooth 12 no evolution tooth 1516 start chipping10 Evolution of the chipping of the teeth 151611 12 13 Chipping across the full width of the tooth

The sample frequency is equal to 20 kHz Table 3 gives thedetails of the gear test rig parametersThe test duration was 13days with a daily mechanical appraisal measurements werecollected every 24 h except on the first day Table 4 gives adescription of the state of the gear at each 24 h Figure 10presents some picture of the chipping observed in days 11 and12 The acceleration signals for days 2 5 7 9 10 and 12 areshown in Figure 17

Pareya et al [38] use the same signal for their ownresearches In time domain a multitude of indicators existsmore or less efficient for the observation of certain failures

The most commonly used indicators are scalars from timeanalysis described by the equations below

RMS = radic1119873

119873

sum

119894=1(119909119894minus 119909)

2

Peak = sup1le119894le119873

1003816100381610038161003816119909119894

1003816100381610038161003816

CF =PeakRMS

119870factor = PeaktimesRMS

Kurtosis =(1119873)sum

119873

119894=1 (119909119894 minus 119909)4

RMS4

Skewness =(1119873)sum

119873

119894=1 (119909119894 minus 119909)3

((1119873)sum119873

119894=1 (119909119894 minus 119909)2)32

(15)

The RMS Peak Kurtosis Crest Factor 119870 Factor andSkewness of the experimental signal were considered andcalculated from day 2 to day 13 (on day 1 no signal was taken)

The evolutions of these descriptors are displayed inFigures 11ndash16

It can be seen that the all indicators increase greatlyafter day 11 After day 11 the signal becomes impulsive TheCrest Factor (CF) observes a little growth after day 9 (17 ofevolution compared to day 9)

The same observations were observed for the Peak values(26) the 119870 Factor (35) and the Skewness (20) Thisis due to the evolution of the chipping of the teeth 1516observed in day 9 A visual chipping was observed in day 7The time indicators are only able to detect the deteriorationafter day 9

Consequently the classical time domains are ineffectivefor early detection of gear defects These measures areappropriate for detection and diagnosis when mechanicalfaults are in advanced state of the degradationAs discussed in[13] at an early stage of gear degradation thewear is precededby a variation of different phenomena like variations in toothloading nonuniform tooth spacing elastic deformation ofthe gear teeth and so forth The classical indicators cannottrack these variations For this reason it is interesting to


002

0025

003

0035

004

0045

005

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days

Figure 11 RMS (ms2) value during the test for all days

0

005

01

015

02

025

03

Days 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Figure 12 Peak (ms2) value during the test for all days

0

2

4

6

8

10

12

14

16

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days

Figure 13 Kurtosis value during the test for all days

investigate other methods which can give a perception intothe gear dynamic behaviour caused by the effects cited above

Therefore ApEn SampEn and LZC of the experimentalsignal were computed from day 2 to day 13 (on day 1 nosignal was taken) The values of ApEn and SampEn areplotted in Figure 18 The ApEn method does not present asignificant increase except at day 12 So the characteristicApEn seems to have equivalent performance as the classicKurtosis After day 11 the ApEn decreases significantly Thisis due to the fact that the signal becomes impulsive and theSNR becomes higher as explained in Section 2 The SampEn

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days

Figure 14 Crest Factor value during the test for all days

minus025

minus005

015

035

055

075

095

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Days

Figure 15 Skewness value during the test for all days

0

0005

001

0015

002

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days

Figure 16 119870 factor value during the test for all days

presents the same evolution as ApEn but it presents a highervalue after day 11 than ApEn On the other hand as explainedand exhibited in Figure 9 the SampEn algorithm requiresapproximately one-half time to calculate than ApEn So itis preferable to use SampEn than ApEn The LZC for allsignals is plotted in Figure 19 The LZC shows a significantincrease after day 4 revealing a clear anomaly due to gearwears Table 4 showed that the chipping was observed clearlyon day 7Thatmeans that the beginning of the processes of thedegradation was inevitably initiated before day 7 Figure 20compares the spectrum evolution between days 4 and 5


0 002 004 006 008 01 012Time (s)

Day 2

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

0 002 004 006 008 01 012Time (s)

Day 7

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

0 002 004 006 008 01 012Time (s)

Day 10

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

0 002 004 006 008 01 012Time (s)

Day 12

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

Time (s)0 002 004 006 008 01 012

Day 9

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

Time (s)0 002 004 006 008 01 012

Day 5

0

01

02

03

minus01

minus02Acce

lera

tion

(ms2)

Figure 17 Evolution of acceleration signal with wear

An increase of the amplitude at day 5 may be clearly noticedas compared with day 4 Effectively this day presents thebeginning of the degradation due to wear According toLZC measurements the beginning of the chipping was infact initiated at day 5 This initiation is characterised by thegrowth of the frequency components related to the meshingConsequently these results show that LZC measurement isbetter than approximate entropy and sample entropy for earlydetection of the gear defects in this history case

Figure 21 presents both Kurtosis and LZC of all daysKurtosis is reported in the abscised119883 and LZC on119884 It can beseen that this plot divides the twelve days into three regionsThe first region contains the three first days (day 2 day 3 andday 4) The second encloses all days from 6 to 11 The secondregion is separated from the first by day 5 Day 5 presents thechange in the characteristics of the gear signal

The last region is marked by an increase of Kurtosis andLZC stays in the same level as the second region At this stagethe signal becomes impulsive and the gear is damaged

Using this representation we can easily distinguish thedifferent steps of the process of the degradation This rep-resentation gives both information on the impulsiveness ofthe signal and on the effect of the number of frequencycomponents and noise into the signal Consequently thisrepresentation combining Kurtosis and LZC may be used asan efficient tool for early detection of faults

4 Conclusion

This paper introduces ApEn SampEn and LZC metrics toanalyse the vibration signal recorded from defected gearsWith respect to gear signals the parameter selection of ApEn


02

025

03

035

04

045

05

055

06

065

07

1 2 3 4 5 6 7 8 9 10 11 12 13 14

SampEnApEn

Days

Figure 18 ApEn and SampEn value during the test for all days

2 3 4 5 6 7 8 9 10 11 12 13 1405

052

054

056

058

Days

LZC

Figure 19 LZC value during the test for all days

and SampEn was investigated and the results show that119898 = 2 and 119896 value = 05 times the standard deviationwere suitable and a good compromise for the ApEn andSampEn calculation In addition the influence of whitenoise on ApEn SampEn and LZC calculation was alsoinvestigated The results show that ApEn and SampEn arenearly unaffected by the noise when staying at a small levelHowever ApEn and SampEn are very sensitive to the noiseat high levels LZC is more sensitive to noise as comparedwithApEn An experimental studywas conducted to evaluatethe effectiveness of these parameters The results show thatsample entropy is better than approximate entropy HoweverLZC can detect the defect of the gears earlier than SampEnand ApEnThe classical time indicators seem to be ineffectivefor early detection compared to LZCThe representation in aplan (Kurtosis LZC) is proposed as a new tool for effectivelymonitoring gear defects Although experimental results lookpromising the proposed vibration methodology has to be

0 1000 2000 4000 5000 60000

0005

001

0015

002

Frequency (Hz)

Day 4Day 5

3000

Acce

lera

tion

(ms2)

Figure 20 Spectrum of the experimental signal

2 4 6 8 10 12 14

052053054055056057058

LZC

Kurtosis

051

Day 2Day 3Day 4Day 5Day 6Day 7


Day 5The first three days

The last two days

From day 6 to 11

Figure 21 Kurtosis versus LZC value during the test for all days

tested on the other test rig also This research is thus beingcontinued to analyze vibration signals from different defecttypes and on different types of gears to systematically validatethe efficiency of this technique

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support of NSERC (Natural Sciences andEngineering Research Council of Canada) FQRNT (FondsQuebecois de la Recherche sur la Nature et les Tech-nologies) MITACS Canada Pratt amp Whitney Canada andCETIM which provided the experimental results are grate-fully acknowledged

References

[1] W J Wang and P D McFadden ldquoEarly detection of gearfailure by vibration analysis i calculation of the time-frequencydistributionrdquo Mechanical Systems and Signal Processing vol 7no 3 pp 193ndash203 1993


[2] W J Wang and P D McFadden ldquoEarly detection of gear failureby vibration analysismdashII Interpretation of the time-frequencydistribution using image processing techniquesrdquo MechanicalSystems and Signal Processing vol 7 no 3 pp 205ndash215 1993

[3] M S Safizadeh A A Lakis and M Thomas ldquoGear faultdiagnosis using time-frequency methodsrdquo in Proceedings of the20th Seminar on Machinery Vibration pp 719ndash729 CanadianMachinery Vibration Association Quebec Canada 2002

[4] M S Safizadeh A A Lakis and M Thomas ldquoUsing short-time fourier transform in machinery diagnosisrdquo in Proceedingsof the 4th WSEAS International Conference on Electronic SignalProcessing and Control (ESPOCO rsquo05) pp 494ndash200 Rio deJaneiro Brazil April 2005

[5] Z K Peng and F L Chu ldquoApplication of the wavelet trans-form in machine condition monitoring and fault diagnosticsa review with bibliographyrdquo Mechanical Systems and SignalProcessing vol 18 no 2 pp 199ndash221 2004

[6] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosis ofrotary machines a review with applicationsrdquo Signal Processingvol 96 pp 1ndash15 2014

[7] N Baydar and A Ball ldquoDetection of gear deterioration undervarying load conditions by using the instantaneous powerspectrumrdquoMechanical Systems and Signal Processing vol 14 no6 pp 907ndash921 2000

[8] N Baydar Q Chen A Ball and U Kruger ldquoDetection of incip-ient tooth defect in helical gears using multivariate statisticsrdquoMechanical Systems and Signal Processing vol 15 no 2 pp 303ndash321 2001

[9] N Baydar and A Ball ldquoA comparative study of acoustic andvibration signals in detection of gear failures usingWigner-VilledistributionrdquoMechanical Systems and Signal Processing vol 15no 6 pp 1091ndash1107 2001

[10] N Baydar and A Ball ldquoDetection of gear failures via vibrationand acoustic signals using wavelet transformrdquo MechanicalSystems and Signal Processing vol 17 no 4 pp 787ndash804 2003

[11] I Yesilyurt ldquoThe application of the conditional momentsanalysis to gearbox fault detectionmdasha comparative study usingthe spectrogram and scalogramrdquoNDTampE International vol 37no 4 pp 309ndash320 2004

[12] A Parey and N Tandon Spur Gear Dynamic Models IncludingDefects A Review Sage Thousand Oaks Calif USA 2003

[13] R B Randall ldquoA new method of modeling gear faultsrdquo ASMEJournal of Mechanical Design vol 104 no 2 pp 259ndash267 1982

[14] P D Samuel and D J Pines ldquoA review of vibration-basedtechniques for helicopter transmission diagnosticsrdquo Journal ofSound and Vibration vol 282 no 1-2 pp 475ndash508 2005

[15] M Lebold K McClintic R Campbell C Byington and KMaynard ldquoReview of vibration analysis methods for gearboxdiagnostics and prognosticsrdquo in Proceedings of the 54th Meetingof the Society for Machinery Failure Prevention Technology pp623ndash634 Virginia Beach Va USA 2000

[16] A S Sait and Y I Sharaf-Eldeen ldquoA review of gearboxcondition monitoring based on vibration analysis techniquesdiagnostics and prognosticsrdquo in Rotating Machinery StructuralHealth Monitoring Shock and Vibration Volume 5 ConferenceProceedings of the Society for Experimental Mechanics Seriespp 307ndash324 Springer New York NY USA 2011

[17] P Stevens D Hall and E Smith ldquoA multidisciplinary researchapproach to rotorcraft health and usage monitoringrdquo in Pro-ceedings of the 52nd Annual Forum of the American HelicopterSociety pp 1732ndash1751 Washington DC USA 1996

[18] Z Feng M J Zuo and F Chu ldquoApplication of regularizationdimension to gear damage assessmentrdquoMechanical Systems andSignal Processing vol 24 no 4 pp 1081ndash1098 2010

[19] S Wu M J Zuo and A Parey ldquoSimulation of spur geardynamics and estimation of fault growthrdquo Journal of Sound andVibration vol 317 no 3ndash5 pp 608ndash624 2008

[20] R Yan and R X Gao ldquoApproximate Entropy as a diagnostic toolfor machine health monitoringrdquoMechanical Systems and SignalProcessing vol 21 no 2 pp 824ndash839 2007

[21] W J Wang J Chen X K Wu and Z T Wu ldquoThe applicationof some non-linear methods in rotating machinery fault diag-nosisrdquo Mechanical Systems and Signal Processing vol 15 no 4pp 697ndash705 2001

[22] J D Jiang J Chen and L S Qu ldquoThe application of correlationdimension in gearbox condition monitoringrdquo Journal of Soundand Vibration vol 223 no 4 pp 529ndash541 1999

[23] Y He and X Zhang ldquoApproximate entropy analysis of theacoustic emission from defects in rolling element bearingsrdquoJournal of Vibration and Acoustics vol 134 no 6 Article ID061012 8 pages 2012

[24] L Fu Z-Y He R-K Mai and Q-Q Qian ldquoApplicationof approximate entropy to fault signal analysis in electricpower systemrdquo Proceedings of the Chinese Society of ElectricalEngineering vol 28 no 28 pp 68ndash73 2008

[25] Y G Xu L J Li and Z J He ldquoApproximate entropy and itsapplications in mechanical fault diagnosisrdquo Information andControl vol 31 no 6 pp 547ndash551 2002

[26] R Yan and R X Gao ldquoComplexity as a measure for machinehealth evaluationrdquo IEEE Transactions on Instrumentation andMeasurement vol 53 no 4 pp 1327ndash1334 2004

[27] J Wang L Cui H Wang and P Chen ldquoImproved complexitybased on time-frequency analysis in bearing quantitative diag-nosisrdquoAdvances inMechanical Engineering vol 2013 Article ID258506 11 pages 2013

[28] M Kedadouche T Kidar M Thomas and A Tahan ldquoCom-bining EMD and Lempel-Ziv complexity for early detectionof gear cracksrdquo in Proceedings of the International ConferenceSurveillance 7 pp 100ndash110 Chartres France October 2013

[29] M L Dennis Wong C Liu and K Asoke Nandi Classificationof Ball Bearing Faults Using Entropic Measures Surveillance 7Chartres France

[30] L Zhang G Xiong H Liu H Zou andW Guo ldquoBearing faultdiagnosis using multi-scale entropy and adaptive neuro-fuzzyinferencerdquo Expert Systems With Applications vol 37 pp 6077ndash6085 2010

[31] J S Richman and J R Moorman ldquoPhysiological time seriesanalysis using approximate entropy and sample entropyrdquo TheAmerican Journal of Physiology Heart and Circulatory Physiol-ogy vol 278 pp 2039ndash2049 2000

[32] V Tuzcu and S Nas ldquoSample entropy analysis of heart rhythmfollowing cardiac transplantationrdquo in Proceedings of the IEEEInternational Conference on Systems Man and Cybernetics pp198ndash202 IEEE October 2005

[33] X Chen S C Irene and C H Ki ldquoComparison of the use ofapproximate entropy and sample entropy applications to neuralrespiratory signalrdquo in Proceedings of the 27th Annual Conferenceon Engineering in Medicine and Biology pp 4212ndash4215 IEEEShanghai China September 2005

[34] Y-H Sun H-L Jou and J-CWu ldquoAuxiliary diagnosis methodfor lead-acid battery health based on sample entropyrdquo EnergyConversion and Management vol 50 no 9 pp 2250ndash22562009


[35] A Widodo M-C Shim W Caesarendra and B-S YangldquoIntelligent prognostics for battery health monitoring based onsample entropyrdquo Expert Systems with Applications vol 38 no 9pp 11763ndash11769 2011

[36] A Lempel and J Ziv ldquoOn the complexity of finite sequencesrdquoIEEE Transactions on Information Theory vol 22 no 1 pp 75ndash81 1976

[37] M El BadaouiContribution of vibratory diagnostic of gearbox byCepstral analysis [PhD thesis] Jean Monnet University Saint-Etienne France 1999 (French)

[38] A Pareya M El Badaoui F Guillet and N Tandon ldquoDynamicmodelling of spur gear pair and application of empirical modedecomposition-based statistical analysis for early detection oflocalized tooth defectrdquo Journal of Sound and Vibration vol 294no 3 pp 547ndash561 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



However in the early stages of fault development mainfrequencies have low amplitudes and can be masked by othervibration sources or buried in background noise

A large number of statistical metrics have been appliedfor monitoring gear defects such as the root mean squareCrest Factor Kurtosis FM0 FM4 FM4

lowast M6 M6A M6AlowastNA4 NA4lowast NB4 NB4lowast [14ndash16] Most of these measurescharacterize the statistical properties of vibration and givea global view to provide any recognition of localized phe-nomena Stevens et al [17] noted that these measures aresuitable for detection and diagnosis when mechanical faultstake the form of impulses which impose periodic pulses ofshort time duration However Feng et al [18] noted that theseparameters cannot give a perception into the gear dynamicsystemgenerating the vibration signalsWu et al [19] used thestandard deviation for monitoring tooth cracks of gear but ithas been proven to be ineffective for tooth crack detection

Therefore it is interesting to use others techniques asnonlinear estimations of parameters which may be a goodalternative to monitoring hidden faults in the measuredsignals Yan and Gao [20] give a review of a large numberof nonlinear parameter identification techniques Wang et al[21] researched the application of some nonlinearity anal-ysis methods including the pseudophase portrait singularspectrum and correlation dimension for monitoring anddiagnosis of rotating machinery Jiang et al [22] used thecorrelation dimension for monitoring gearbox Feng et al[18] investigated the irregularity of the gear signal using theregularisation dimensions

As said before the effects of various nonlinearities cause avariation in the dynamic behaviour of the gear As the defaultdevelops more contact pressure between two mating partschanges The contact pressure variation results in amplitudeand frequency modulations and hence more frequency willappear in the frequency domain with their harmonics andan increase of the noise can be observed Therefore thefault progress leads to complexity (randomness) changesbecause of a variation of the contact pressure between twomating partsThe complexity of a signal increases whenmorefrequency components exist On the other hand when thedefect increases we observe more frequency components inthe signal and we can use the complexity of the signal asa quantitative measurement to evaluate the severity of thedefect The complexity of the signal can be described by theapproximate entropy (ApEn) the sample entropy (SampEn)and the Lempel-Ziv complexity (LZC) These parameters arenonlinear parameters which can be used to characterise theregularity of the signal

The Lempel-Ziv complexity (LZC) sample entropy (Sam-pEn) and approximate entropy (ApEn) thus present alterna-tive tools for signal analysis involving nonlinear dynamicsThese methods are becoming more popular and have foundwide applications in various disciplines especially in the fieldof biomedical engineering ApEn has recently received moreattention Yan and Gao [20] investigated the application ofApEn for the health monitoring of rolling element bearingsHe and Zhang [23] applied the ApEn for monitoring signalof the acoustic emission from defects in rolling elementBearings Fu et al [24] used the ApEn method to fault

signal analysis in electric power system Using ApEn Xuet al [25] detected the looseness of the bearing bushingin turbo generator In all these work ApEn is used as anonlinear feature parameter for analysing the vibration signaland effectively identifying the conditions of the mechanicalsystem

On the other hand Yan and Gao [26] investigated theapplication of Lempel-Ziv complexity (LZC) for the healthmonitoring of rolling element bearingsWang et al [27] com-pare and analyse quantitative diagnosis methods based onLempel-Ziv complexity for bearing faults using continuouswavelet transform (CWT) Empirical Mode Decomposition(EMD) method and wavelet packet method for decomposi-tion of vibration signal Kedadouche et al [28] combined LZCand EMD for early detection of gears cracks

The use of SampEn was introduced by Dennis Wonget al [29] which was the improvement of a previousmethod namely approximate entropy (ApEn) Howeversample entropy was not often used in monitoring of rotatingmachine Dennis Wong et al [29] used sample entropy forclassification of ball bearing faults Zhang et al [30] usedmul-tiscale entropy and adaptive neurofuzzy inference for bearingfault classification The sample entropy was discussed andimplemented in many areas such as physiological time seriesanalysis [31] heart rate variability [32] and neural respiratorysignal [33] Sun et al [34] proposed a health auxiliarymethodbased on SampEn for lead-acid battery unit Widodo et al[35] proposed intelligent prognostics for battery health basedon sample entropy (SampEn) features of discharge voltageThe SampEn feature can depict degradation condition ofbattery and represents the health condition of battery

As illustrated above LZC SampEn andApEn are becom-ing more and more attractive in the field of detectionand fault diagnosis However no work has been found toapply ApEn LZC or SampEn measurement for diagnosisof gear faults Therefore in this paper ApEn SampEn andLZC are introduced to analyse vibration signals from gearand investigate their efficiency for the defect detection andseverity evaluation of gears faults

2 Theoretical Background

21 Approximate Entropy Consider a time series 119878(119894) 119894 =

0 119873 Its ldquoregularityrdquo may be measured by ApEn in amultiple dimensional space so that series of vectors areconstructed and expressed as follows

119883 (119873minus119898+ 1)

= 119909 (119873minus119898+ 1) 119909 (119873minus119898+ 2) 119909 (119873)

(1)

Each vector is composed of119898 consecutive and discrete pointdata of the time series 119878 The distance between two vectors119883(119894) and 119883(119895) can be defined as the maximum differencebetween their respective corresponding elements

119889 (119883 (119894) 119883 (119895))

= max119896=12119898

(1003816100381610038161003816119909 (119894 + 119896 minus 1) minus 119909 (119895 + 119896 minus 1)1003816100381610038161003816)

(2)




119862119898

119894(119903) =

1119873 minus (119898 minus 1)

sum

119895 =119894

Θ119903 minus 119889 (119883 (119894) 119883 (119895)) (3)

where

Θ 119909 =

1 119909 ge 0

0 119909 lt 0(4)



119878is the



ApEn (119898 119903119873) = 0119898

(119903) minus 0119898+1

(119903)

0119898

(119903) =1

119873 minus119898 + 1

119873minus119898+1sum

119894=1ln [119862119898119894(119903)]

(5)



119898(119894) as

119883119898(119894) = 119909 (119894) 119909 (119894 + 1) 119909 (119894 +119898minus 1)

119894 = 1 to 119873 minus119898 + 1(6)


119889119898(119883119898(119894) 119883

119898(119895))

= max (1003816100381610038161003816119883119898 (119894 + 119896) minus119883119898(119895 + 119896)

1003816100381610038161003816)

0 le 119896 le 119898 minus 1

(7)

For 119894 = 1 to119873 minus119898 + 1 let

119861119898

119894(119903) =

1119873 minus119898 + 1

V119898 (119894) (8)


119898(119883119898(119894) 119883119898(119895)) for 119894 = 119895


119860119898

119894(119903) =

1119873 minus119898 + 1

119908119898+1

(119894) (9)





119903 = 119896 times std119878 (10)


119861119898

(119903) =1

119873 minus119898 + 1

119873minus119898

sum

119894=1119861119898

119894(119903)

119860119898

(119903) =1

119873 minus119898 + 1

119873minus119898

sum

119894=1119860119898

119894(119903)

(11)

where 119861119898



(119903)

119861119898(119903)

] (12)



119862 (119899) =119888 (119899)

119887 (119899)

119887 (119899) =119873

log2 (119873)

(13)




Yes

No

No

End

Yes

Start

Data sequence S



SQ120587 contains Q

c(n) = c(n) + 1

i = i + 1

i gt N

c(n) = 0 i = 1)

(Q =


C(n) = c(n)b(n)

SQ120587 = null

null

Initialization





119909 (119905) =

+infin

sum

119898=minusinfin

1198781199031 (119905 minus1198981205911199031) +

+infin

sum

119898=minusinfin

1198781199032 (119905 minus1198981205911199032)

+

+infin

sum

119899=minusinfin

119878119890(119905 minus 119899120591

119890)

sdot (1++infin

sum

119898=minusinfin

1198781199031 (119905 minus1198981205911199031))

(14)

where 120591119890 1205911199031 and 120591


119890(119905) 1198781199031(119905) and 119878





lowast


0 1000 2000 3000 4000 5000 6000 7000 8000

0

1

2

3

Sample

minus3

minus2

minus1





0 100 200 300 400 500 600 700 800 900 10000

02

04

06

08

1

12

14

Frequency (Hz)

10 12 14 16 18 200

002004006008

01012014016



002 04 06 08 10

05

1

15

2

ApEn

m = 1

m = 2

m = 3

m = 4

m = 5

m = 6

m = 7

m = 8

k value









01 02 03 04 05 06 07 08 09 101

02

03

04

05

06

07

08

09

Sam

ple e

ntro

py

k value

m = 2

m = 3

m = 4

m = 5

m = 6

m = 7

m = 8



4

4kHz8kHz12kHz

16kHz20kHz24kHz

015

02

025

03

035

04

ApEn








01

015

02

025

03

035

Sam

ple e

ntro

py


4

4kHz8kHz12kHz

16kHz20kHz


0

01

02

03

04

05

LZC

16kHz20kHz24kHz


4

4kHz8kHz12kHz






0

20

40

60

40968192

1228816384

20480

Com

puta

tion

time (

s)

Length data











default











RMS = radic1119873

119873

sum

119894=1(119909119894minus 119909)

2


1003816100381610038161003816119909119894

1003816100381610038161003816

CF =PeakRMS



119873

119894=1 (119909119894 minus 119909)4

RMS4


119873

119894=1 (119909119894 minus 119909)3

((1119873)sum119873

119894=1 (119909119894 minus 119909)2)32

(15)







002

0025

003

0035

004

0045

005

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days


0

005

01

015

02

025

03

Days 1 2 3 4 5 6 7 8 9 10 11 12 13 14


0

2

4

6

8

10

12

14

16

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days




0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days


minus025

minus005

015

035

055

075

095

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Days


0

0005

001

0015

002

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days




0 002 004 006 008 01 012Time (s)

Day 2

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

0 002 004 006 008 01 012Time (s)

Day 7

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

0 002 004 006 008 01 012Time (s)

Day 10

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

0 002 004 006 008 01 012Time (s)

Day 12

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

Time (s)0 002 004 006 008 01 012

Day 9

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

Time (s)0 002 004 006 008 01 012

Day 5

0

01

02

03

minus01

minus02Acce

lera

tion

(ms2)






4 Conclusion



02

025

03

035

04

045

05

055

06

065

07

1 2 3 4 5 6 7 8 9 10 11 12 13 14

SampEnApEn

Days


2 3 4 5 6 7 8 9 10 11 12 13 1405

052

054

056

058

Days

LZC



0 1000 2000 4000 5000 60000

0005

001

0015

002

Frequency (Hz)

Day 4Day 5

3000

Acce

lera

tion

(ms2)


2 4 6 8 10 12 14

052053054055056057058

LZC

Kurtosis

051




The last two days

From day 6 to 11





Acknowledgments


References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119862119898

119894(119903) =

1119873 minus (119898 minus 1)

sum

119895 =119894

Θ119903 minus 119889 (119883 (119894) 119883 (119895)) (3)

where

Θ 119909 =

1 119909 ge 0

0 119909 lt 0(4)



119878is the



ApEn (119898 119903119873) = 0119898

(119903) minus 0119898+1

(119903)

0119898

(119903) =1

119873 minus119898 + 1

119873minus119898+1sum

119894=1ln [119862119898119894(119903)]

(5)



119898(119894) as

119883119898(119894) = 119909 (119894) 119909 (119894 + 1) 119909 (119894 +119898minus 1)

119894 = 1 to 119873 minus119898 + 1(6)


119889119898(119883119898(119894) 119883

119898(119895))

= max (1003816100381610038161003816119883119898 (119894 + 119896) minus119883119898(119895 + 119896)

1003816100381610038161003816)

0 le 119896 le 119898 minus 1

(7)

For 119894 = 1 to119873 minus119898 + 1 let

119861119898

119894(119903) =

1119873 minus119898 + 1

V119898 (119894) (8)


119898(119883119898(119894) 119883119898(119895)) for 119894 = 119895


119860119898

119894(119903) =

1119873 minus119898 + 1

119908119898+1

(119894) (9)





119903 = 119896 times std119878 (10)


119861119898

(119903) =1

119873 minus119898 + 1

119873minus119898

sum

119894=1119861119898

119894(119903)

119860119898

(119903) =1

119873 minus119898 + 1

119873minus119898

sum

119894=1119860119898

119894(119903)

(11)

where 119861119898



(119903)

119861119898(119903)

] (12)



119862 (119899) =119888 (119899)

119887 (119899)

119887 (119899) =119873

log2 (119873)

(13)




Yes

No

No

End

Yes

Start

Data sequence S



SQ120587 contains Q

c(n) = c(n) + 1

i = i + 1

i gt N

c(n) = 0 i = 1)

(Q =


C(n) = c(n)b(n)

SQ120587 = null

null

Initialization





119909 (119905) =

+infin

sum

119898=minusinfin

1198781199031 (119905 minus1198981205911199031) +

+infin

sum

119898=minusinfin

1198781199032 (119905 minus1198981205911199032)

+

+infin

sum

119899=minusinfin

119878119890(119905 minus 119899120591

119890)

sdot (1++infin

sum

119898=minusinfin

1198781199031 (119905 minus1198981205911199031))

(14)

where 120591119890 1205911199031 and 120591


119890(119905) 1198781199031(119905) and 119878





lowast


0 1000 2000 3000 4000 5000 6000 7000 8000

0

1

2

3

Sample

minus3

minus2

minus1





0 100 200 300 400 500 600 700 800 900 10000

02

04

06

08

1

12

14

Frequency (Hz)

10 12 14 16 18 200

002004006008

01012014016



002 04 06 08 10

05

1

15

2

ApEn

m = 1

m = 2

m = 3

m = 4

m = 5

m = 6

m = 7

m = 8

k value









01 02 03 04 05 06 07 08 09 101

02

03

04

05

06

07

08

09

Sam

ple e

ntro

py

k value

m = 2

m = 3

m = 4

m = 5

m = 6

m = 7

m = 8



4

4kHz8kHz12kHz

16kHz20kHz24kHz

015

02

025

03

035

04

ApEn








01

015

02

025

03

035

Sam

ple e

ntro

py


4

4kHz8kHz12kHz

16kHz20kHz


0

01

02

03

04

05

LZC

16kHz20kHz24kHz


4

4kHz8kHz12kHz






0

20

40

60

40968192

1228816384

20480

Com

puta

tion

time (

s)

Length data











default











RMS = radic1119873

119873

sum

119894=1(119909119894minus 119909)

2


1003816100381610038161003816119909119894

1003816100381610038161003816

CF =PeakRMS



119873

119894=1 (119909119894 minus 119909)4

RMS4


119873

119894=1 (119909119894 minus 119909)3

((1119873)sum119873

119894=1 (119909119894 minus 119909)2)32

(15)







002

0025

003

0035

004

0045

005

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days


0

005

01

015

02

025

03

Days 1 2 3 4 5 6 7 8 9 10 11 12 13 14


0

2

4

6

8

10

12

14

16

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days




0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days


minus025

minus005

015

035

055

075

095

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Days


0

0005

001

0015

002

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days




0 002 004 006 008 01 012Time (s)

Day 2

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

0 002 004 006 008 01 012Time (s)

Day 7

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

0 002 004 006 008 01 012Time (s)

Day 10

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

0 002 004 006 008 01 012Time (s)

Day 12

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

Time (s)0 002 004 006 008 01 012

Day 9

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

Time (s)0 002 004 006 008 01 012

Day 5

0

01

02

03

minus01

minus02Acce

lera

tion

(ms2)






4 Conclusion



02

025

03

035

04

045

05

055

06

065

07

1 2 3 4 5 6 7 8 9 10 11 12 13 14

SampEnApEn

Days


2 3 4 5 6 7 8 9 10 11 12 13 1405

052

054

056

058

Days

LZC



0 1000 2000 4000 5000 60000

0005

001

0015

002

Frequency (Hz)

Day 4Day 5

3000

Acce

lera

tion

(ms2)


2 4 6 8 10 12 14

052053054055056057058

LZC

Kurtosis

051




The last two days

From day 6 to 11





Acknowledgments


References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Yes

No

No

End

Yes

Start

Data sequence S



SQ120587 contains Q

c(n) = c(n) + 1

i = i + 1

i gt N

c(n) = 0 i = 1)

(Q =


C(n) = c(n)b(n)

SQ120587 = null

null

Initialization





119909 (119905) =

+infin

sum

119898=minusinfin

1198781199031 (119905 minus1198981205911199031) +

+infin

sum

119898=minusinfin

1198781199032 (119905 minus1198981205911199032)

+

+infin

sum

119899=minusinfin

119878119890(119905 minus 119899120591

119890)

sdot (1++infin

sum

119898=minusinfin

1198781199031 (119905 minus1198981205911199031))

(14)

where 120591119890 1205911199031 and 120591


119890(119905) 1198781199031(119905) and 119878





lowast


0 1000 2000 3000 4000 5000 6000 7000 8000

0

1

2

3

Sample

minus3

minus2

minus1





0 100 200 300 400 500 600 700 800 900 10000

02

04

06

08

1

12

14

Frequency (Hz)

10 12 14 16 18 200

002004006008

01012014016



002 04 06 08 10

05

1

15

2

ApEn

m = 1

m = 2

m = 3

m = 4

m = 5

m = 6

m = 7

m = 8

k value









01 02 03 04 05 06 07 08 09 101

02

03

04

05

06

07

08

09

Sam

ple e

ntro

py

k value

m = 2

m = 3

m = 4

m = 5

m = 6

m = 7

m = 8



4

4kHz8kHz12kHz

16kHz20kHz24kHz

015

02

025

03

035

04

ApEn








01

015

02

025

03

035

Sam

ple e

ntro

py


4

4kHz8kHz12kHz

16kHz20kHz


0

01

02

03

04

05

LZC

16kHz20kHz24kHz


4

4kHz8kHz12kHz






0

20

40

60

40968192

1228816384

20480

Com

puta

tion

time (

s)

Length data











default











RMS = radic1119873

119873

sum

119894=1(119909119894minus 119909)

2


1003816100381610038161003816119909119894

1003816100381610038161003816

CF =PeakRMS



119873

119894=1 (119909119894 minus 119909)4

RMS4


119873

119894=1 (119909119894 minus 119909)3

((1119873)sum119873

119894=1 (119909119894 minus 119909)2)32

(15)







002

0025

003

0035

004

0045

005

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days


0

005

01

015

02

025

03

Days 1 2 3 4 5 6 7 8 9 10 11 12 13 14


0

2

4

6

8

10

12

14

16

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days




0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days


minus025

minus005

015

035

055

075

095

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Days


0

0005

001

0015

002

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days




0 002 004 006 008 01 012Time (s)

Day 2

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

0 002 004 006 008 01 012Time (s)

Day 7

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

0 002 004 006 008 01 012Time (s)

Day 10

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

0 002 004 006 008 01 012Time (s)

Day 12

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

Time (s)0 002 004 006 008 01 012

Day 9

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

Time (s)0 002 004 006 008 01 012

Day 5

0

01

02

03

minus01

minus02Acce

lera

tion

(ms2)






4 Conclusion



02

025

03

035

04

045

05

055

06

065

07

1 2 3 4 5 6 7 8 9 10 11 12 13 14

SampEnApEn

Days


2 3 4 5 6 7 8 9 10 11 12 13 1405

052

054

056

058

Days

LZC



0 1000 2000 4000 5000 60000

0005

001

0015

002

Frequency (Hz)

Day 4Day 5

3000

Acce

lera

tion

(ms2)


2 4 6 8 10 12 14

052053054055056057058

LZC

Kurtosis

051




The last two days

From day 6 to 11





Acknowledgments


References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










002 04 06 08 10

05

1

15

2

ApEn

m = 1

m = 2

m = 3

m = 4

m = 5

m = 6

m = 7

m = 8

k value









01 02 03 04 05 06 07 08 09 101

02

03

04

05

06

07

08

09

Sam

ple e

ntro

py

k value

m = 2

m = 3

m = 4

m = 5

m = 6

m = 7

m = 8



4

4kHz8kHz12kHz

16kHz20kHz24kHz

015

02

025

03

035

04

ApEn








01

015

02

025

03

035

Sam

ple e

ntro

py


4

4kHz8kHz12kHz

16kHz20kHz


0

01

02

03

04

05

LZC

16kHz20kHz24kHz


4

4kHz8kHz12kHz






0

20

40

60

40968192

1228816384

20480

Com

puta

tion

time (

s)

Length data











default











RMS = radic1119873

119873

sum

119894=1(119909119894minus 119909)

2


1003816100381610038161003816119909119894

1003816100381610038161003816

CF =PeakRMS



119873

119894=1 (119909119894 minus 119909)4

RMS4


119873

119894=1 (119909119894 minus 119909)3

((1119873)sum119873

119894=1 (119909119894 minus 119909)2)32

(15)







002

0025

003

0035

004

0045

005

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days


0

005

01

015

02

025

03

Days 1 2 3 4 5 6 7 8 9 10 11 12 13 14


0

2

4

6

8

10

12

14

16

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days




0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days


minus025

minus005

015

035

055

075

095

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Days


0

0005

001

0015

002

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days




0 002 004 006 008 01 012Time (s)

Day 2

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

0 002 004 006 008 01 012Time (s)

Day 7

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

0 002 004 006 008 01 012Time (s)

Day 10

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

0 002 004 006 008 01 012Time (s)

Day 12

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

Time (s)0 002 004 006 008 01 012

Day 9

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

Time (s)0 002 004 006 008 01 012

Day 5

0

01

02

03

minus01

minus02Acce

lera

tion

(ms2)






4 Conclusion



02

025

03

035

04

045

05

055

06

065

07

1 2 3 4 5 6 7 8 9 10 11 12 13 14

SampEnApEn

Days


2 3 4 5 6 7 8 9 10 11 12 13 1405

052

054

056

058

Days

LZC



0 1000 2000 4000 5000 60000

0005

001

0015

002

Frequency (Hz)

Day 4Day 5

3000

Acce

lera

tion

(ms2)


2 4 6 8 10 12 14

052053054055056057058

LZC

Kurtosis

051




The last two days

From day 6 to 11





Acknowledgments


References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










01

015

02

025

03

035

Sam

ple e

ntro

py


4

4kHz8kHz12kHz

16kHz20kHz


0

01

02

03

04

05

LZC

16kHz20kHz24kHz


4

4kHz8kHz12kHz






0

20

40

60

40968192

1228816384

20480

Com

puta

tion

time (

s)

Length data











default











RMS = radic1119873

119873

sum

119894=1(119909119894minus 119909)

2


1003816100381610038161003816119909119894

1003816100381610038161003816

CF =PeakRMS



119873

119894=1 (119909119894 minus 119909)4

RMS4


119873

119894=1 (119909119894 minus 119909)3

((1119873)sum119873

119894=1 (119909119894 minus 119909)2)32

(15)







002

0025

003

0035

004

0045

005

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days


0

005

01

015

02

025

03

Days 1 2 3 4 5 6 7 8 9 10 11 12 13 14


0

2

4

6

8

10

12

14

16

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days




0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days


minus025

minus005

015

035

055

075

095

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Days


0

0005

001

0015

002

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days




0 002 004 006 008 01 012Time (s)

Day 2

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

0 002 004 006 008 01 012Time (s)

Day 7

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

0 002 004 006 008 01 012Time (s)

Day 10

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

0 002 004 006 008 01 012Time (s)

Day 12

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

Time (s)0 002 004 006 008 01 012

Day 9

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

Time (s)0 002 004 006 008 01 012

Day 5

0

01

02

03

minus01

minus02Acce

lera

tion

(ms2)






4 Conclusion



02

025

03

035

04

045

05

055

06

065

07

1 2 3 4 5 6 7 8 9 10 11 12 13 14

SampEnApEn

Days


2 3 4 5 6 7 8 9 10 11 12 13 1405

052

054

056

058

Days

LZC



0 1000 2000 4000 5000 60000

0005

001

0015

002

Frequency (Hz)

Day 4Day 5

3000

Acce

lera

tion

(ms2)


2 4 6 8 10 12 14

052053054055056057058

LZC

Kurtosis

051




The last two days

From day 6 to 11





Acknowledgments


References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










default











RMS = radic1119873

119873

sum

119894=1(119909119894minus 119909)

2


1003816100381610038161003816119909119894

1003816100381610038161003816

CF =PeakRMS



119873

119894=1 (119909119894 minus 119909)4

RMS4


119873

119894=1 (119909119894 minus 119909)3

((1119873)sum119873

119894=1 (119909119894 minus 119909)2)32

(15)







002

0025

003

0035

004

0045

005

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days


0

005

01

015

02

025

03

Days 1 2 3 4 5 6 7 8 9 10 11 12 13 14


0

2

4

6

8

10

12

14

16

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days




0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days


minus025

minus005

015

035

055

075

095

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Days


0

0005

001

0015

002

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days




0 002 004 006 008 01 012Time (s)

Day 2

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

0 002 004 006 008 01 012Time (s)

Day 7

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

0 002 004 006 008 01 012Time (s)

Day 10

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

0 002 004 006 008 01 012Time (s)

Day 12

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

Time (s)0 002 004 006 008 01 012

Day 9

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

Time (s)0 002 004 006 008 01 012

Day 5

0

01

02

03

minus01

minus02Acce

lera

tion

(ms2)






4 Conclusion



02

025

03

035

04

045

05

055

06

065

07

1 2 3 4 5 6 7 8 9 10 11 12 13 14

SampEnApEn

Days


2 3 4 5 6 7 8 9 10 11 12 13 1405

052

054

056

058

Days

LZC



0 1000 2000 4000 5000 60000

0005

001

0015

002

Frequency (Hz)

Day 4Day 5

3000

Acce

lera

tion

(ms2)


2 4 6 8 10 12 14

052053054055056057058

LZC

Kurtosis

051




The last two days

From day 6 to 11





Acknowledgments


References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










002

0025

003

0035

004

0045

005

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days


0

005

01

015

02

025

03

Days 1 2 3 4 5 6 7 8 9 10 11 12 13 14


0

2

4

6

8

10

12

14

16

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days




0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days


minus025

minus005

015

035

055

075

095

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Days


0

0005

001

0015

002

1 2 3 4 5 6 7 8 9 10 11 12 13 14Days




0 002 004 006 008 01 012Time (s)

Day 2

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

0 002 004 006 008 01 012Time (s)

Day 7

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

0 002 004 006 008 01 012Time (s)

Day 10

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

0 002 004 006 008 01 012Time (s)

Day 12

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

Time (s)0 002 004 006 008 01 012

Day 9

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

Time (s)0 002 004 006 008 01 012

Day 5

0

01

02

03

minus01

minus02Acce

lera

tion

(ms2)






4 Conclusion



02

025

03

035

04

045

05

055

06

065

07

1 2 3 4 5 6 7 8 9 10 11 12 13 14

SampEnApEn

Days


2 3 4 5 6 7 8 9 10 11 12 13 1405

052

054

056

058

Days

LZC



0 1000 2000 4000 5000 60000

0005

001

0015

002

Frequency (Hz)

Day 4Day 5

3000

Acce

lera

tion

(ms2)


2 4 6 8 10 12 14

052053054055056057058

LZC

Kurtosis

051




The last two days

From day 6 to 11





Acknowledgments


References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 002 004 006 008 01 012Time (s)

Day 2

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

0 002 004 006 008 01 012Time (s)

Day 7

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

0 002 004 006 008 01 012Time (s)

Day 10

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

0 002 004 006 008 01 012Time (s)

Day 12

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

Time (s)0 002 004 006 008 01 012

Day 9

0

01

02

03

minus01

minus02

Acce

lera

tion

(ms2)

Time (s)0 002 004 006 008 01 012

Day 5

0

01

02

03

minus01

minus02Acce

lera

tion

(ms2)






4 Conclusion



02

025

03

035

04

045

05

055

06

065

07

1 2 3 4 5 6 7 8 9 10 11 12 13 14

SampEnApEn

Days


2 3 4 5 6 7 8 9 10 11 12 13 1405

052

054

056

058

Days

LZC



0 1000 2000 4000 5000 60000

0005

001

0015

002

Frequency (Hz)

Day 4Day 5

3000

Acce

lera

tion

(ms2)


2 4 6 8 10 12 14

052053054055056057058

LZC

Kurtosis

051




The last two days

From day 6 to 11





Acknowledgments


References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










02

025

03

035

04

045

05

055

06

065

07

1 2 3 4 5 6 7 8 9 10 11 12 13 14

SampEnApEn

Days


2 3 4 5 6 7 8 9 10 11 12 13 1405

052

054

056

058

Days

LZC



0 1000 2000 4000 5000 60000

0005

001

0015

002

Frequency (Hz)

Day 4Day 5

3000

Acce

lera

tion

(ms2)


2 4 6 8 10 12 14

052053054055056057058

LZC

Kurtosis

051




The last two days

From day 6 to 11





Acknowledgments


References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Nonlinear Parameters for Monitoring Gear ...downloads.hindawi.com › journals › sv › 2015 › 959380.pdf · Lempel-Ziv complexity for bearing faults, using continuous