complexity analysis of time-frequency features for...

Research ArticleComplexity Analysis of Time-Frequency Features for VibrationSignals of Rolling Bearings Based on Local Frequency

Youfu Tang 1 Feng Lin1 and Qian Zou2

1School of Mechanical Science and Engineering Northeast Petroleum University Daqing 163318 China2Research Institute of Petroleum Exploration and Development Beijing 100083 China

Correspondence should be addressed to Youfu Tang tang_youfu210163com

Received 30 March 2019 Revised 9 June 2019 Accepted 18 June 2019 Published 10 July 2019

Academic Editor Franck Poisson

Copyright copy 2019 Youfu Tang et al -is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

-e multisource impact signal of rolling bearings often represents nonlinear and nonstationary characteristics and quantitativedescription of the complexity of the signal with traditional spectrum analysis methods is difficult to be obtained In this studyfirstly a novel concept of local frequency is defined to develop the limitation of traditional frequency-en an adaptive waveformdecompositionmethod is proposed to extract the time-frequency features of nonstationary signals withmulticomponents Finallythe normalized LempelndashZiv complexity method is applied to quantitatively measure the time-frequency features of vibrationsignals of rolling bearings -e results indicate that the time-frequency features extracted by the proposed method have clearphysical meanings and can accurately distinguish the different fault states of rolling bearings Furthermore the normalizedLempelndashZiv complexity method can quantitatively measure the nonlinearity of the multisource impact signal So it supplies aneffective basis for fault diagnosis of rolling bearings

1 Introduction

Rolling bearings play an important role in rotatingmachineryHowever due to their loud sound poor damping and shockresistance features the failure rate is higher compared withother parts Scholars home and abroad have paid more at-tention on healthy monitoring and fault diagnosis of rollingbearings [1ndash4] Particularly under the long-term and high-speed operation the rolling bearingsmay easily produce somedifferent faults such as peeling abrasion and gluing As isknown to all in the field of mechanical fault diagnosis thefeature extraction method of vibration signal is continuouslyimproved with the development of signal processing tech-nology Various advanced fault feature extraction methodshave been extensively studied [5ndash8] For the vibration signalof rolling bearings it usually presents multisource impactcharacteristics which are typically nonlinear and non-stationary [9 10] Because of the special mapping relationshipbetween the excitation of equipment components and thefrequency of vibration signals the focus of mechanical faultdiagnosis research is mainly on spectral analysis and time-frequency analysis [11 12]

At present there are two frequency concepts that are wellknown One is the traditional frequency defined for har-monic signals [13] and the other is the instantaneous fre-quency defined for the narrowband signal [14] For a longtime the concept of frequency has only a clear physicalmeaning for the dynamic signal of a simple harmonic vi-bration Since Joseph Fourier proposed the mathematicalexpression of the Fourier transform the traditional fre-quency obtained a perfect physical interpretation in thefeature extraction of both the periodic and nonperiodicsignals But the Fourier transform can only obtain in-formation on which harmonic frequency components andrelative intensity information of each component are con-tained and cannot obtain information on how these fre-quency components evolve with time -erefore it is themost effective tool for analyzing and processing the sta-tionary signal When the signal is nonstationary the featureextracted by the Fourier transform only measures the degreeof similarity between the signal and the harmonic basisfunction rather than the actual frequency information of thevibration signal [15] -e concept of instantaneous fre-quency can effectively measure the time-varying features of

HindawiShock and VibrationVolume 2019 Article ID 7190568 13 pageshttpsdoiorg10115520197190568

nonstationary signals It has been successfully applied in thefields of radar sonar detection seismic monitoring elec-tronic communication and mechanical equipment faultdiagnosis [16 17] However the instantaneous frequencyhas only a clear physical meaning for the narrowband signalIt is considered that the narrowband signal has only onefrequency component at each instantaneous time point andmuch large-scale frequency information is lost Even formany nonstationary signals there is lack of clear physicalmeaning and even cannot be calculated

In the past few decades a series of theories and tech-niques of adaptive signal decomposition and transientfeature extraction have been developed and widely used invarious disciplines and engineering fields (eg Fouriertransform [18] short-time Fourier transform [19] wavelettransform [20] second-generation wavelet transform [21]multiwavelet transform [22] chirplet transform [23] andatomic decomposition [24]) All of them are based on thebasic function expansion of the inner product transformprinciple performing the similarity measure between thesignal and the basis function [25] However for the rollingbearings the vibration signal represents strong nonlinearitynonstationarity and multisource impact so it is unlikely toadapt all waveform features using one or several given basisfunctions In the absence of sufficient prior knowledge if afixed basis function is employed for signal decompositionerroneous information will be generated and the physicalmeaning will be unclear Feature information sufficient toidentify the fault is difficult to extract In recent years someadaptive signal decomposition methods including empiricalmode decomposition (EMD) [26] local mean de-composition (LMD) [27] and local characteristic-scale de-composition (LCD) [28] wavelet modulus maxima andsynchronous detection [29] improved orthogonal matchingpursuit and K-SVD algorithm with adaptive transient dic-tionary [30] have been gradually proposed -ose methodsare not required to predict the feature information containedin the analyzed signal and they completely follow thewaveform of the signal -us they exhibit good adaptabilityin signal processing

Although many related achievements have beenachieved in the field they also have the following problems(1) -e magnitude of the feature amplitudes under differentstates is greatly different the real physical meaning is am-biguous and the comparability is poor (2) -e featureamplitude under the same state may be fluctuated greatlywith different samples so the repeatability and stability ofthe results are poor (3) -e effects of noise interference aredifferent to the accuracy of the results -ose problems areclosely related to the nonlinear dynamic characteristics ofrolling bearings How to judge and quantitatively describethe nonlinearity and evolution of rolling bearings underdifferent states is an important prerequisite for fault di-agnosis In the field of nonlinearity some quantitative in-dicators including Lyapunov exponents [31] correlationdimension [32] entropy [33] and complexity [34] have beenwell applied Among of them the LempelndashZiv complexity(LZC) has many good advantages to measure the time seriesarose from the nonlinear dynamics system which have been

used widely in data compression [35] coding [36] gener-ation of test signals [37] and so on Additionally comparedwith the methods of spectral analysis and time-frequencyanalysis the LZC method can detect the long-range cor-relations embedded in the seemingly nonstationary timeseries and also avoid the spurious detection of apparentlong-range correlations that are an artifact of nonstationarity[38]

In order to improve the diagnosis accuracy of rollingbearings effective and quantitative features should beextracted firstly On the one hand the vibration signal ofrolling bearings represents the complex feature of non-linearity and nonstationarity the typical and useful featureinformation may be not significant On the other hand thevibration signal of rolling bearings may be strongly influ-enced by background noise For solving the above problemsan integration approach based on adaptive waveform de-composition (AWD) and LempelndashZiv complexity (LZC) wasproposed Using the AWD method the nonstationarymulticomponent signal of rolling bearings can be convertedinto a series of stationary single component signal which arerich in useful feature information -e LZC method is ap-plied to quantitatively describe the complexity of nonlinearsignal in the time-frequency domain -erefore both theAWD method and LZC algorithm have their own functionin improving the diagnosis accuracy of rolling bearings andthe integration methods of AWD and LZC will have the besteffect

2 Time-Frequency Feature Extraction Based onLocal Frequency and AdaptiveWaveform Decomposition

21 Definition of Local Frequency For an arbitrary signalx(t) assume that the corresponding discrete time series isx(i) ∣ i 1 2 3 n -e time domain waveform of thesignal x(t) is shown in Figure 1 and the local maximum ofthe x(t) satisfies the following equation

x(i)gex(iminus 1)

x(i)gex(i + 1)

x(iminus 1)nex(i + 1)

(1)

From the minimum waveform required to complete avibration the original signal can be approximated as con-sisting of a series of V waves containing two adjacentmaxima as shown in Figure 2

Assuming that the V wave start position tk is the time atwhich the kth local maximum value in the original signal x(t)is located and k 1 2 3 middot middot middot N according to the V wave thelocal period T(t) of the original signal x(t) can be defined asfollows

T(t) tk+1 minus tk tk le tlt tk+1 (2)

where T(t) represents the time required for the signal tocomplete a complete local vibration over a local time range

Similarly a local mean m(t) and a local amplitude a(t)are also given below

2 Shock and Vibration

m(t) 1nk

1113944

nk

i1x ti( 1113857 tk le tlt tk+1 (3)

a(t) x tk( 1113857 + x tk+1( 1113857

2 tk le tlt tk+1 (4)

where nk is the number of samples contained between thekth maximum and the k+ 1th maximum

According to formula (2) the local frequency is definedas the reciprocal of the local period ie

v(t) 1

T(t)

1tk+1 minus tk

tk le tlt tk+1 (5)

where the local frequency v(t) represents the number of timescompleting the vibration in a unit of local time It is used tomeasure the speed of local vibration and the unit is still Hz

-e characteristic curves of local amplitude a(t) localmean m(t) and local frequency v(t) calculated by equations(3) to (4) are all broken lines In order to improve the ac-curacy of analysis results and to get a smoother curve themoving average (MA) technology is applied for processingthe curves and the principle is as follows [39]

Let the original broken line sequence be y(i) ∣ i 1113864

1 2 3 n Every point after being smoothed shouldsatisfy the formula as follows

yprime(i) 1

2N + 1[y(iminusN) + y(iminusN + 1) + + y(i + Nminus 1)

+ y(i + N)]

(6)

where 2N+ 1 is a smooth interval span 2N+ 1lt nObviously the difference in span selection has a direct

influence on the smoothing effect which will cause differenterrors in the calculation results of the local feature quantitiesFor the impact signal in order to ensure that the impactextreme value is not lost during the smoothing process it isnecessary to adjust the span to 15 of the maximum intervalof the adjacent maximum value as the best effect -esmoothing result of the characteristic curve of the arbitrarysignal x(t) is shown in Figures 3 and 4

22 Construction of Time-Frequency Distribution Based onLocal Frequency In order to analyze the nonstationarysignal it is necessary to construct a method of time-fre-quency distribution based on local frequency According tothe local amplitude curve a(t) defined in equation (4) itrepresents the absolute amplitude of the V-wave in acomplete local period In order to eliminate the influence ofthe equilibrium position point fluctuation the local meancurve m(t) should be removed ie

aprime(t) |a(t)minusm(t)| (7)

where aprime(t) represents the fluctuation amplitude of the localfrequency component v(t) respected to the equilibriumposition with time

Plotting the time t the local frequency v(t) and the localamplitude aprime(t) on a three-dimensional map and the localamplitude aprime(t) is represented by a contour of a different color-en a typical time-frequency distribution diagram of thesignal x(t) based on the local frequency is formed Figure 5shows the time-frequency distribution of the signal x(t) isconsistent with the local frequency feature extracted in Figure 4

23 Adaptive Waveform Decomposition AlgorithmAccording to definition of local frequency and its time-frequency distribution we can only analyze the signal withsingle component For many test signals of machine itusually contains various components from different exci-tations So if we want to extract the time-frequency featureof each component the multicomponent signal should bedecomposed firstly In the calculation of the traditionalfrequency and instantaneous frequency of the multicom-ponent nonstationary signal the signal can be finally re-duced to the following two forms

x(t) 1113944infin

k1ck sin kωt + θk( 1113857

x(t) 1113944N

k1ak(t)cos ϕk(t) + R(t)

(8)

where R(t) is the residual term-e two forms of decomposition have great similarities

all of which express frequency information in a superpo-sition of multiple harmonic waveforms However theactual complex signals often do not have harmonicwaveform characteristics andmay be composed of a limited

t (s)

A (m

s)

Figure 2 -e V wave

0 02 04 06 08 1

ndash5

0

5

t (s)

A (m

s)

Figure 1 -e time domain waveform of the signal x(t)

Shock and Vibration 3

number of other shape waveforms Considering the effi-ciency accuracy and frequency physical meaning of signaldecomposition the harmonic waveform combination doesnot conform to the essential features of the signal-erefore based on the local frequency concept a newadaptive waveform decomposition (AWD) method isproposed to adapt to the feature extraction of non-stationary signal with multicomponents

x(t) 1113944N

m1sm(t) + W(t) (9)

where sm(t) represents themth waveform function containedin the original signal and W(t) is the residual term -eimplementation process is as follows

(1) Assume that the discrete time series correspondingto the signal x(t) is x(i) ∣ i 1 2 3 n -enaccording to formula (12) find the local maxima n1in the signal x(t) as the first layer extreme valuesequence p1(k) ∣ k 1 2 3 N11113864 1113865

(2) Calculate the local amplitude a1(t) the local meanm1(t) and the local frequency v1(t) corresponding tothe first-order extreme value sequence p1(t)according to formulas (3) to (5)

(3) Smooth a1(t) m1(t) and v1(t) using the movingaverage technique

(4) De-average the local amplitude to obtain the am-plitude envelope curve a1prime(t) which is

a1prime(t) a(t)minusm(t) (10)

(5) Similarly find the n2 local maxima as the secondlayer p2(k) ∣ k 1 2 3 N21113864 1113865 in the first layer ofextreme sequence p1(k) ∣ k 1 2 3 N21113864 1113865continue smoothing and de-averaging processing toobtain the amplitude envelope curve a2prime(t) accordingto steps (2) (3) and (4)

(6) Repeat the above steps until you find the m-th orderextremum sequence pm(k) ∣ k 1 2 3 Nm1113864 1113865When it satisfiesNmle 2 the decomposition is ended

According to the above method x(t) can obtain mminus 1amplitude envelope curves a1prime(t) a2prime(t) amminus1prime (t) so eachcomponent decomposed from x(t) is obtained as follows

0 02 04 06 08 1

ndash5

0

5

t (s)

A (m

s)

Local amplitude

Local mean

Figure 3 -e local amplitude and local mean curve of the signal x(t)

0 02 04 06 08 10

20

40

60

t (s)

v (H

z)

Local frequency

Figure 4 -e local frequency curve of the signal x(t)

60

50

40

30

20

10

0

t (s)

f (H

z)

0 05 10

1

2

3

4

Figure 5 -e time-frequency distribution of the signal x(t) basedon local frequency


sm(t) amminus1prime (t)

smminus1(t) amminus2prime (t)minus amminus1prime (t)

⋮

s1(t) xminus a1prime(t)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(11)

-e abovementioned decomposition process only de-pends on characteristics of the signal x(t) itself and the localfrequency of each components s1(t) s2(t) sm(t) doesnot depend on the harmonic form -erefore the proposedAWD method has good adaptability

24 Applicability Analysis of the Proposed Method In thissection some simulation examples are taken as the researchobject and the main characteristics of the proposed methodabove in time-frequency analysis of multicomponent signalsare described in detail by comparing with the EMD (em-pirical mode decomposition) method [40]

241 Combination of Multiple Harmonic Signals

x1(t) sin(2π middot 10 middot t) + sin(2π middot 30 middot t) + sin(2π middot 80 middot t)

(12)

Figure 6 shows the decomposition effects of the simu-lation signal x2(t) based on AWD and EMD Obviously bothof them can effectively extract the three harmonic wave-forms contained in the original signal which are decom-posed from high frequency to low frequency respectively Asthe number of decomposition layers increases the distancebetween the first extreme point of the layer and the first datapoint increases Whether it is the sliding average method orthe spline envelope method it is difficult to give a satis-factory fit Although many scholars have given various end-point extension methods it has always been ineffective fornonlinear and nonstationary signals

Figure 7 shows the time-frequency distribution result ofthe signal x1(t) based on the fusion method of local fre-quency and AWD and Figure 8 shows the result based onthe fusion method of instantaneous frequency and EMD Itcan be seen that both the local frequency and instantaneousfrequency can accurately represent the stationary distribu-tion of the three harmonic components in the original signalx1(t) But the time-frequency resolution of the fusionmethod of local frequency and AWD is higher than thefusion method of instantaneous frequency and EMD

242 Combination of Harmonic Signal and FM-AM Signal

x2(t) sin(2π middot 30t) +[1 + 05 sin(2π middot 10 middot t)]

middot cos[2π middot 150 middot t + 2 cos(2π middot 5 middot t)](13)

Figure 9 shows the decomposition effects of the sim-ulation signal x2(t) based on AWD and EMD Similarlythere is still a high degree of coincidence and the signalx2(t) is adaptively decomposed into three components Itcan be seen that the first layer of AWD and EMD result is

the FM-AM component the second layer is the harmoniccomponent and the third layer is the trend term

Figure 10 shows the time-frequency distribution result ofthe signal x2(t) based on the fusion method of local fre-quency and AWD and Figure 11 shows the result based onthe fusion method of instantaneous frequency and EMD

In the time-frequency distribution result of the signalx2(t) based on the fusion method of local frequency andAWD as shown in Figure 10 the 30Hz stationary harmonicand FM-AM with 150Hz carrier frequency are clearly

ndash4ndash2

024

ndash4ndash2

024

A (m

s)

0 005 01 015 02ndash4ndash2

024

t (s)

EMDAWD

Figure 6 -e decomposition result of the signal x1(t) based onAWD and EMD

200

150

100

50

0

f (H

z)

t (s)0 05 1

0

02

04

06

08

1

Figure 8 -e time-frequency distribution of the signal x1(t) basedon the instantaneous frequency and EMD

200

150

100

50

0

v (H

z)

t (s)0 05 1

0

02

04

06

08

1

12

Figure 7 -e time-frequency distribution of the signal x1(t) basedon the local frequency and AWD


extracted However in Figure 11 the fusion method ofinstantaneous frequency and EMD generate some in-terference information at the low frequency because of thetrend term Additionally comparing Figure 10 withFigure 11 it can be seen that the time-frequency featuresextracted by using the proposed method in this paper haveno false redundant frequency and the resolution is sig-nificantly higher

In summary by comparing the time-frequency analysisof the multicomponent signals it is shown that both the

AWD and the EMD can achieve a good adaptive de-composition effect and both of them have the same accu-racy However in time-frequency analysis the localfrequency is clearer in resolution than instantaneous fre-quency and it not only has clear physical meaning but alsocontains less interference components -erefore the pro-posed method can accurately obtain the essential features ofthe frequency components contained in the original signaland its validity and applicability is verified according tosimulation signals above

3 Quantitative Measure of the Complexity ofNonlinear Time Series

31 Algorithm of LZC -e definition of complex measure isproposed by Kolmogorov and is characterized by thenumber of bits of the shortest program required to produce asequence of symbols Later Lempel and Ziv proposed analgorithm to achieve this complexity called LempelndashZivcomplexity (LZC) which is widely used in nonlinear sci-entific research [41] LZC analysis is based on coarsegraining of the measurements Before the LZC measure iscalculated the time series must be transformed into a finitesymbol sequence Generally an arbitrary time seriesx(i) ∣ i 1 2 3 middot middot middot n is converted into a binary series Bycomparison with the threshold x(i) is converted into a 0-1series S(i) ∣ i 1 2 3 middot middot middot n as follows [42]

S(i) 0 x(i)lt xave

1 x(i)ge xave1113896 (14)

where xave is the mean of time series of x(i) -en the LZCmeasure can be estimated by using the following algorithmas shown in Figure 12

-e abovementioned algorithm is repeated until Q is thelast character -e measure of complexity is c(n) In order toobtain a complexity measure which is independent of theseries length c(n) must be normalized If the length of thesequence is n and the number of different symbols in thesymbol set is 2 it has been proved that the upper bound ofc(n) is given by

limnrarrinfin

c(n) b(n) n

log2(n) (15)

and c(n) can be normalized via b(n)

0leC c(n)

b(n)le 1 (16)

where the normalized complexity index C is called theLempelndashZiv complexity

Obviously equation (17) can be established only if thesample length n is large enough -e literature has given theexperience valve nge 3600 [43] -e complexity index Cpresents the random degree of time series For the com-pletely random time series C is close to 1 On the contrarythe periodic time series is close to 0 In practice the vibrationsignals of reciprocating compressor vary with each faultstates randomly so the LZC can be extracted as a usefulmeasure for machine health condition feature

ndash4ndash2

024

ndash4ndash2

024

A (m

s)

0 005 01 015 02ndash4ndash2

024

t (s)

EMDAWD


200

150

100

50

0

v (H

z)

t (s)0 05 1

02040608

121

14

Figure 10-e time-frequency distribution of the signal x2(t) basedon the local frequency and AWD

200

150

100

50

0

f (H

z)

t (s)0 05 1

0

02

04

06

08

1

Figure 11-e time-frequency distribution of the signal x2(t) basedon the instantaneous frequency and EMD


32 Performance Experiment of LZC To identify the effec-tiveness of LZC method the typical nonlinear dynamicalsystem is taken as an example to investigate the performanceof LZC Consider the following logistic map [44]

xn+1 λ middot xn middot 1minusxn( 1113857 (17)

where xn isin [0 1] λ isin [28 4] λ is the control parameters andthe step size is 001 -e sample length of nonlinear timeseries is N 5000

According to the LZC algorithm the complexity index Cof logistic map with different parameters λ can be calculated-e result is shown in Figure 13

Observe Figures 13(a) and 13(b) it indicates the dynamicalbehaviour of logistic map with different control parameters λ-e trends of Lyapunov exponent and the complexity index Care much synchronous When 28le λlt 357 the Lyapunovexponent is less than zero and it shows that the logistic systementers in the periodic or quasiperiodic region Moreover mostvalues of LZC are close to zero When 357lt λle 4 Lyapunovexponent is greater than zero it indicates that the systementers the chaos region Meanwhile the complexity index Ctends to be around 05 During this region the Lyapunovexponent less than zero including λ 363 λ 374 andλ 383 are called periodic windows In these regions thecomplexity index C is still close to zero

In summary LZC can be used as identifying the dy-namical behaviour of nonlinear system effectively -eanalysis results are stable and reliable when the number ofsamples increase -erefore the LZC value of differentsystem states can be compared together without any priorknowledge about fault mechanism

4 Application in Fault Feature Extraction ofRolling Bearings

41 Preprocessing Analysis of Vibration Signal To reduce theinfluence of noise on the nonlinear and nonstationary

measure analysis results of multisource impact signals of gasvalve the preprocessing based on AWD and mutual in-formation (MI) fusion noise reduction technology wasproposed -e definition of MI can be given as follows

I(X Y) H(X) + H(Y)minusH(X Y)

1113944n

iminus11113944

m

jminus1P xi yj1113872 1113873log

p xi yj1113872 1113873

p xi( 1113857p yj1113872 1113873

(18)

where p(x) p(y) and p(x y) denote the probabilitydensity function of two random variables X Y and the jointprobability distribution of (X Y) MI measures the corre-lation degree of two signal When they are absolute MI valuewill be 0 When they are fully the same the value will be 1

According to the proposed AWD method in this paperthe original vibration signal with background noise can beadaptively decomposed into a series of waveform compo-nent signals s1(t) s2(t) sm(t) -e MI value of eachcomponent signal and the original signal were calculatedfollowing the steps of equation (18) respectively By sortingone or more correlated component signals are selected to berecombined to form a noise reduction signal

42 LZC Feature Extraction of Rolling Bearing in TimeDomain In this section vibration signals of the rollingbearings under different faults are taken as an applicationobject -e standard data are obtained from the bearingdata center of the Electrical Engineering Laboratory of CaseWestern Reserve University and its experimental benchstructure is shown in Figure 14 In the experiment the deepgroove ball bearing was selected at the motor drive end-edetailed parameters are shown in Table 1 In this experi-ment the tiny pits of the inner ring the outer ring and therolling element of the bearing are processed by the elec-tric spark machine to simulate the single-point damagefault diameter and depth size are 01778mmtimes 02794mm

Symbol time series S = s1 s2

s3 sn

Initialization S1 = s1 Q1 = s2SQ1 = s1s2 SQP1 = s1 c(1) = 1

Q isin SQP subsets

Yes

Yes

No

i = i + 1

i gt n

End

Start

NoS is invariability update Q nextsymbol is added to Q SQP is updated

c(i) = c(i) + 1 Q is combined to Sset Q as next symbol

Figure 12 -e principle diagram of LZC


03556mm times 02794mm 05334mm times 02794mm and07112mm times 127mm respectively -e sampling fre-quency fs is 12000Hz the running speed is set separatelyinto 1797 rmin 1772 rmin 1750 rmin and 1730 rminwhich are corresponding to fundamental frequency f0 of therolling bearings 2995Hz 2953Hz 2917Hz and 2883HzWhere the fault position of the outer ring is selected at the 6orsquoclock direction which is the main load position

According to the parameters in Table 1 taking the speedof 1750 rmin as an example the frequency features of faultsin each part can be calculated separately as follows

fIR 12

z 1 +d

Dcos α1113888 1113889f0 15794Hz (19)

fOR 12

z 1minusd

Dcos α1113888 1113889f0 10456Hz (20)

fBE 12

D

d1minus

d

D1113888 1113889

2

cos2 α⎛⎝ ⎞⎠f0 13748Hz (21)

where fIR fOR fBE and f0 are corresponding to the faultfrequency of the inner ring the outer ring the rolling el-ement and the fundamental frequency of rotation speedrespectively

Figure 15 shows the time domain waveform of the vi-bration signal of rolling bearing in four states including thenormal state inner ring fault state outer ring fault state androlling element fault state -e rotation speed is 1750 rminand the damage size is 05334mmtimes 02794mm It can beseen that the vibration signals in the fault state of the innerring and the outer ring have obvious impact feature whilethe features of the rolling element fault state and the normalstate are not obvious It shows that the vibration amplitudeof the normal state is the smallest and the the amplitude ofthe outer ring state is the largest -is is mainly because theselected outer ring fault position is in the main bearingdirection and the load of the rotor increases the impactenergy of the rolling element falling into the outer ring

Figure 16 shows the AWD results of the above four states ofrolling bearings -e vibration signals in each state are adap-tively decomposed into six components As the number ofdecomposition layers increases the frequency graduallychanges from the high-frequency band to the low-frequencyband -e components of each layer in the normal state donot have obvious impact features the signal-to-noise ratio is

1

05

ndash05

0

ndash1

ndash15

ndash2

Lyapunov exponentLZC

λ

(a)

(b)

28 3 32 34 357 363 374 383 4

1

08

06

04

02

0Periodic

Periodic window

Quasiperiodic Chaos

Lyap

unov

Cx

Figure 13 -e logistic map with different parameters λ (a) Bifurcation diagram (b) comparison of Lyapunov exponent and LZC

Figure 14 -e fault simulation experimental bench of rollingbearing

Table 1 Parameters of rolling bear

No Parameter (unit) Value1 Number of ball roller z 92 Diameter of ball roller d (mm) 7943 Diameter of rolling bear D (mm) 394 Contact angle α (deg) 90


0 005 01 015 02ndash04

ndash02

0

02

04

t (s)

A (m

s2 )

(a)

0 005 01 015 02t (s)

A (m

s2 )

ndash4

ndash2

0

2

4

(b)

0 005 01 015 02t (s)

A (m

s2 )

ndash4

ndash2

0

2

4

(c)

ndash04

ndash02

0

02

04

0 005 01 015 02t (s)

A (m

s2 )

(d)

Figure 15 -e time domain waveform in four state of rolling bearings (a) normal state (b) inner ring fault state (c) outer ring fault stateand (d) rolling element fault state

S1S2

S3S4

S5S6

0

0

0

0

0

0

01ndash01

01

ndash01

ndash01ndash02

005

ndash00501

ndash03005

ndash01

t (s)0 005 01 015 02

(a)

S1S2

S3S4

S5S6

20

ndash2

05ndash1

10

ndash1

63010

ndash1

105

0

t (s)0 005 01 015 02

(b)

S1S2

S3S4

S5S6

40

ndash4

10

ndash1

20

ndash1

0ndash4

30

ndash1

20

ndash2

t (s)0 005 01 015 02

(c)

020

ndash0201

0ndash02

0201

0ndash01

01

ndash010

0

ndash03008

ndash002

S1S2

S3S4

S5S6

t (s)0 005 01 015 02

(d)

Figure 16 -e AWD results in four state of rolling bearings (a) normal state (b) inner ring fault state (c) outer ring fault state and(d) rolling element fault state


relatively low and the main feature of components is not easilyrecognized -e first three components of the inner ring faultstate present the impact features and the latter three compo-nents present the harmonic features However the outer ringfault state and the rolling element fault state have the significantperiodic impact features in the first four components and theremaining component are low-frequency harmonic periodicfeatures In addition the noise content of the first and secondlayer waveform components in each state is larger than that ofthe other layer components and the energy of the signal ismainly concentrated in the first two components

According to the analysis of AWD results we can easilyobtain the main components and then the LempelndashZivcomplexmeasure analysis is performed on each signal by usingthe LZC algorithm Table 2 shows the LZC features of the time-domain signal for the rolling bearing in four states at differentspeeds with the damage size 05334mmtimes 02794mm Table 3shows the LZC features of the time-domain signal for therolling bearings in fault states under different damage di-ameters with the speed 1750 rmin

It can be seen from Table 2 that the LZC values of rollingbear in the inner ring fault state are gradually decreased withthe speed decreased from 1797 rmin to 1730 rmin but theLZC values are increased in the other three states-e reasonmay be as follows With the decreasing of the rotational thebearing load increases relatively and the modulation effecton the signal is more significant -e contact between theouter ring and the rolling element is basically fixed and itsmain role is frequency modulation So the new frequencychange is increased -e complexity of the signal leads to ahigher LZC value However when the inner ring rotates withthe axis the contact area of pitting fault changes continu-ously and its main role is to generate a low-frequencyamplitude modulation It is different from frequencymodulation that the signal is affected globally in the low-frequency oscillation mode -erefore the periodicity isenhanced and the randomness is reduced which results in alower LZC value

In Table 3 with the increasing of the damage diameterfrom 01778mm to 07112mm the LZC values of rollingbear in the inner ring fault state are gradually decreasedwhile the LZC values are decreased in the other three states-at is because the rotation speed is constant the load isrelatively stable the damage area is enlarged the actualcontact area between the rolling element and the outer ringis reduced the contact pressure is rapidly increased anddecreased and the impact feature is more obvious -einfluence is dominant reflecting a wider frequency band andharmonics on the spectrum -e result indicates that therandomness of the original signal is more prominent and itresults in a higher LZC value However due to the influenceof the low frequency the vibration signal of inner ringpresents the amplitude modulation feature -e order oforiginal signal is enhanced and the randomness is reducedwhich results in a lower LZC value

43 LZCFeatureExtractionofRollingBearing inTime-FrequencyDomain According to the analysis above we can find that

the time domain features of vibration signal the rollingbearing as shown in Figure 15 cannot directly and accuratelyidentify the characteristic frequency especially for dis-tinguishing the normal state and the fault state of the rollingelement with low signal-to-noise ratio For the complexnonstationary signal of rolling bearing in this section thetime-frequency analysis based on the local frequency andAWD method is applied -e result is shown in Figure 17

In Figure 17 the time-frequency distribution diagram ofthe vibration signal in four states of the rolling bearing showsa typical AM-FM features and the local frequency fluctuatesaround a certain constant value with time -e carrierfrequency in the normal state is 88Hz which is near thetriple frequency of the fundamental frequency and thecarrier frequencies in the inner ring fault state the outer ringfault state and the rolling element fault state are 158Hz105Hz and 138Hz respectively -is is consistent with thetheoretical frequency of each fault state calculated byequations (18)ndash(20) which can further verify the effective-ness of the proposed method In addition according to thetime-frequency analysis it is obvious that the feature of therolling element fault is the most ambiguous the fluctuationis the most severe and the noise is the most seriousHowever the time-frequency distribution of the vibrationsignal under normal condition is relatively stable

In order to accurately quantify these time-frequencyfeature of rolling bearings the LempelndashZiv complexityanalysis method is applied Table 4 shows the LZC com-parison results of the rolling bearing in the time-frequencydomain at different speeds and Table 5 shows the LZCcomparison results of the rolling bearing in time-frequencydomain under different damage diameters It can be seenthat the structural complexity of time-frequency distributionin each state of the rolling bearing has a similar change lawwith the time domain With the decreasing of the rotationalspeed or the improvement of the damage degree the LZCvalues of time-frequency distribution in the inner ring faultgradually decrease but they are increased in the other threestates-e result indicates that the time-frequency transform

Table 2 -e LZC features of the time-domain signal for the rollingbearing at different speeds

Rotation speed(rmin) Normal Inner ring

faultOuter ring

faultRolling

element fault1797 01503 02089 00968 028791772 01962 02013 01655 029531750 02017 01936 02885 030321730 02115 01860 03614 03236

Table 3 -e LZC features of the time-domain signal for the rollingbearing under different damage diameters

Damage size(mm)

Inner ringfault

Outer ringfault

Rolling elementfault

01778 02726 02038 0277703556 02573 02404 0295505334 01936 02885 0303207112 01420 03013 03140


preserves the nature of randomness and nonlinearity of theoriginal signal itself which can easily and quickly quantifythe complexity of different fault states In addition com-pared with the LZC value in the time domain the LZC valuein the time-frequency domain is correspondingly reduced-erefore the time-frequency analysis can reduce thestructural complexity of the signal making the fault featureclearer and clearer and suppressing the randomness of thenoise to some extent -e role of sexual interference has laida good foundation for the fault diagnosis of rolling bearings

5 Conclusions

A novel local frequency concept and AWD algorithm isproposed for the purpose of time-frequency feature ex-traction for the vibration signal of rolling bearings pre-sented multisource impact nonlinear and nonstationarycharacteristics Compared with the commonly usedmethod of time-frequency analysis based on traditionalfrequency or instantaneous frequency the proposedmethod develops the limitation of frequency definition on

Table 4 -e LZC features of time-frequency distribution for the rolling bearing at different speeds

Rotation speed (rmin) Normal Inner ring fault Outer ring fault Rolling element fault1797 00459 00637 00764 008411772 00507 00588 00825 009121750 00611 00433 01188 011211730 00688 00312 01217 01172

Table 5 -e LZC features of time-frequency distribution for the rolling bearing under different damage diameters

Damage size (mm) Inner ring fault Outer ring fault Rolling element fault01778 00611 00866 0081503556 00543 00968 0092305334 00433 01188 0112107112 00341 01424 01510

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

006

005

004

003

002

001

0

(a)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

004

003

002

001

0

(b)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

001

0005

0

(c)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

01

005

0

(d)

Figure 17 -e time-frequency distribution in four state of rolling bearings (a) normal state (b) inner ring fault state (c) outer ring faultstate and (d) rolling element fault state


nonharmonic signal and the physical meaning is clearerAdditionally the time-frequency features extracted by theAWD and local frequency method have good adaptabilitythe decomposition result is completely independent andthe frequency band is high to low It can avoid the false orcross component caused by the forced decomposition ofthe traditional Fourier transform in the form of harmonicbasis function Based on the proposed method in thisstudy the LZC algorithm is appled to quantitativelymeasure the complexity of time-frequency features forvibration signals of the rolling bearings -erefore it canbe used as an effective method for fault diagnosis of rollingbearings

Data Availability

-e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

-is work was supported by the Natural Science Foundationof China (51505079) and University Nursing Program forYoung Scholars with Creative Talents in HeilongjiangProvince (UNPYSCT-2015078)

References

[1] Y Qin ldquoA new family of model-based impulsive wavelets andtheir sparse representation for rolling bearing fault diagnosisrdquoIEEE Transactions on Industrial Electronics vol 65 no 3pp 2716ndash2726 2018

[2] H Shao H Jiang Y Lin and X Li ldquoA novel method forintelligent fault diagnosis of rolling bearings using ensembledeep auto-encodersrdquo Mechanical Systems and Signal Pro-cessing vol 102 pp 278ndash297 2018

[3] A Glowacz W Glowacz Z Glowacz and J Kozik ldquoEarlyfault diagnosis of bearing and stator faults of the single-phaseinduction motor using acoustic signalsrdquo Measurementvol 113 pp 1ndash9 2018

[4] S Xiao S Liu F Jiang M Song and S Cheng ldquoNonlineardynamic response of reciprocating compressor system withrub-impact fault caused by subsidencerdquo Journal of Vibrationand Control vol 25 no 11 pp 1737ndash1751 2019

[5] M Cerrada R-V Sanchez C Li et al ldquoA review on data-driven fault severity assessment in rolling bearingsrdquo Me-chanical Systems and Signal Processing vol 99 pp 169ndash1962018

[6] H D Shao H K Jiang X Q Li and S P Wu ldquoIntelligentfault diagnosis of rolling bearing using deep wavelet auto-encoder with extreme learning machinerdquo Knowledge-BasedSystems vol 140 pp 1ndash14 2018

[7] L Wang Z Liu Q Miao and X Zhang ldquoComplete ensemblelocal mean decomposition with adaptive noise and its ap-plication to fault diagnosis for rolling bearingsrdquo MechanicalSystems and Signal Processing vol 106 pp 24ndash39 2018

[8] Q He E Wu and Y Pan ldquoMulti-scale stochastic resonancespectrogram for fault diagnosis of rolling element bear-ingsrdquo Journal of Sound and Vibration vol 420 pp 174ndash184 2018

[9] Y Cheng N Zhou W Zhang and Z Wang ldquoApplication ofan improved minimum entropy deconvolution method forrailway rolling element bearing fault diagnosisrdquo Journal ofSound and Vibration vol 425 pp 53ndash69 2018

[10] L Cui B Li J Ma and Z Jin ldquoQuantitative trend faultdiagnosis of a rolling bearing based on Sparsogram andLempel-Zivrdquo Measurement vol 128 pp 410ndash418 2018

[11] A Rai and S H Upadhyay ldquoA review on signal processingtechniques utilized in the fault diagnosis of rolling elementbearingsrdquo Tribology International vol 96 pp 289ndash306 2016

[12] Z Q Ma W Ruan M Chen and X Li ldquoAn improved time-frequency analysis method for instantaneous frequency es-timation of rolling bearingrdquo Shock and Vibration vol 2018Article ID 8710190 18 pages 2018

[13] M J Roberts Signals and Systems Analysis Using TransformMethods and MATLAB McGraw-Hill New York NY USA2nd edition 2011

[14] D Vakman ldquoOn the analytic signal the Teager-Kaiser energyalgorithm and other methods for defining amplitude andfrequencyrdquo IEEE Transactions on Signal Processing vol 44no 1 pp 791ndash797 1996

[15] S Xu L Feng Y Chai and Y He ldquoAnalysis of A-stationaryrandom signals in the linear canonical transform domainrdquoSignal Processing vol 146 pp 126ndash132 2018

[16] D Gabor ldquo-eory of communication Part 1 the analysis ofinformationrdquo Journal of the Institution of Electrical Engi-neersmdashPart III Radio and Communication Engineeringvol 93 no 26 pp 429ndash441 1946

[17] I Daubechies ldquoOrthonormal bases of compactly supportedwaveletsrdquo Communications on Pure and Applied Mathe-matics vol 41 no 7 pp 909ndash996 1988

[18] F J Harris ldquoOn the use of windows for harmonic analysiswith the discrete Fourier transformrdquo Proceedings of the IEEEvol 66 no 1 pp 51ndash83 1978

[19] D Griffin and J Lim ldquoSignal estimation frommodified short-time Fourier transformrdquo IEEE Transactions on AcousticsSpeech and Signal Processing vol 32 no 2 pp 236ndash243 1984

[20] I Daubechies ldquo-e wavelet transform time-frequency lo-calization and signal analysisrdquo IEEE Transactions on In-formation Feory vol 36 no 5 pp 961ndash1005 1990

[21] J Wang Q Wei L Zhao T Yu and R Han ldquoAn improvedempirical mode decomposition method using second gen-eration wavelets interpolationrdquo Digital Signal Processingvol 79 pp 164ndash174 2018

[22] S Mouatadid J F Adamowski M K Tiwari and J M QuiltyldquoCoupling the maximum overlap discrete wavelet transformand long short-term memory networks for irrigation flowforecastingrdquo Agricultural Water Management vol 219pp 72ndash85 2019

[23] J Wang Y Han L M Wang P Z Zhang and P ChenldquoInstantaneous frequency estimation for motion echo signalof projectile in bore based on polynomial chirplet transformrdquoRussian Journal of Nondestructive Testing vol 54 no 1pp 44ndash54 2018

[24] J Tan ldquoAtomic decomposition of variable Hardy spaces viaLittlewoodndashPaleyndashStein theoryrdquo Annals of Functional Anal-ysis vol 9 no 1 pp 87ndash100 2018

[25] Z J He Y Y Zi X-F Chen and X-D Wang ldquoTransformprinciple of inner product for fault diagnosisrdquo Journal ofVibration Engineering vol 20 no 5 pp 528ndash533 2007


[26] N E Huang Z Shen S R Long et al ldquo-e empirical modedecomposition and the Hilbert spectrum for nonlinear andnon-stationary time series analysisrdquo Proceedings of theRoyal Society of London Series A Mathematical Physicaland Engineering Sciences vol 454 no 1971 pp 903ndash9951998

[27] J S Smith ldquo-e local mean decomposition and its applicationto EEG perception datardquo Journal of the Royal Society Interfacevol 2 no 5 pp 443ndash454 2005

[28] J S Cheng J D Zheng and Y Yang ldquoA nonstationary signalanalysis approach-the local characteristic-scale de-composition methodrdquo Journal of Vibration Engineeringvol 25 no 2 pp 215ndash220 2012

[29] Y Qin Y Mao and B Tang ldquoMulticomponent de-composition by wavelet modulus maxima and synchronousdetectionrdquo Mechanical Systems and Signal Processing vol 91pp 57ndash80 2017

[30] Y Qin J Zou B Tang Y Wang and H Chen ldquoTransientfeature extraction by the improved orthogonal matchingpursuit and K-SVD algorithm with adaptive transientdictionaryrdquo IEEE Transactions on Industrial Informatics2019

[31] P C Muller ldquoCalculation of Lyapunov exponents for dy-namic systems with discontinuitiesrdquo Chaos Solitons ampFractals vol 5 no 9 pp 1671ndash1681 1995

[32] D Logan and J Mathew ldquoUsing the correlation dimension forvibration fault diagnosis of rolling element bearingsmdashI Basicconceptsrdquo Mechanical Systems and Signal Processing vol 10no 3 pp 241ndash250 1996

[33] D Wu S Zhang H Zhao and X Yang ldquoA novel fault di-agnosis method based on integrating empirical wavelettransform and fuzzy entropy for motor bearingrdquo IEEE Accessvol 6 pp 35042ndash35056 2018

[34] Y Li G Li Y Yang X Liang and M Xu ldquoA fault diagnosisscheme for planetary gearboxes using adaptive multi-scalemorphology filter and modified hierarchical permutationentropyrdquo Mechanical Systems and Signal Processing vol 105pp 319ndash337 2018

[35] L Liu D Li and F Bai ldquoA relative LempelndashZiv complexityapplication to comparing biological sequencesrdquo ChemicalPhysics Letters vol 530 pp 107ndash112 2012

[36] V Alexeenko J A Fraser A Dolgoborodov et al ldquo-eapplication of Lempel-Ziv and Titchener complexity analysisfor equine telemetric electrocardiographic recordingsrdquo Sci-entific Reports vol 9 no 1 p 2619 2019

[37] W Henkel G Muskhelishvili D Nigatu and P SobetzkoFeDNA from a Coding Perspective Information-and Commu-nication Feory in Molecular Biology Springer ChamSwitzerland 2018

[38] K Yu J Tan and T Lin ldquoFault diagnosis of rolling elementbearing using multi-scale Lempel-Ziv complexity andMahalanobis distance criterionrdquo Journal of Shanghai JiaotongUniversity (Science) vol 23 no 5 pp 696ndash701 2018

[39] X K Chai X H Weng Z M Zhang Y T Lu G T Liu andH J Niu ldquoQuantitative EEG in mild cognitive impairmentand Alzheimerrsquos disease by AR-spectral and multi-scale en-tropy analysisrdquo in Proceedings of the World Congress onMedical Physics and Biomedical Engineering 2018 SpringerPrague Czech Republic June 2018

[40] L Angrisani and M DrsquoArco ldquoA measurement method basedon a modified version of the chirplet transform for in-stantaneous frequency estimationrdquo IEEE Transactions onInstrumentation andMeasurement vol 51 no 4 pp 704ndash7112002

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

[42] J Ziv and A Lempel ldquoA universal algorithm for sequentialdata compressionrdquo IEEE Transactions on Information Feoryvol 23 no 3 pp 337ndash343 1977

[43] H Hong and M Liang ldquoFault severity assessment for rollingelement bearings using the LempelndashZiv complexity andcontinuous wavelet transformrdquo Journal of Sound and Vi-bration vol 320 no 1-2 pp 452ndash468 2009

[44] S C Phatak and S S Rao ldquoLogistic map a possible random-number generatorrdquo Physical Review E vol 51 no 4pp 3670ndash3678 1995


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Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

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SensorsJournal of



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wwwhindawicom Volume 2018

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Submit your manuscripts atwwwhindawicom

nonstationary signals It has been successfully applied in thefields of radar sonar detection seismic monitoring elec-tronic communication and mechanical equipment faultdiagnosis [16 17] However the instantaneous frequencyhas only a clear physical meaning for the narrowband signalIt is considered that the narrowband signal has only onefrequency component at each instantaneous time point andmuch large-scale frequency information is lost Even formany nonstationary signals there is lack of clear physicalmeaning and even cannot be calculated

In the past few decades a series of theories and tech-niques of adaptive signal decomposition and transientfeature extraction have been developed and widely used invarious disciplines and engineering fields (eg Fouriertransform [18] short-time Fourier transform [19] wavelettransform [20] second-generation wavelet transform [21]multiwavelet transform [22] chirplet transform [23] andatomic decomposition [24]) All of them are based on thebasic function expansion of the inner product transformprinciple performing the similarity measure between thesignal and the basis function [25] However for the rollingbearings the vibration signal represents strong nonlinearitynonstationarity and multisource impact so it is unlikely toadapt all waveform features using one or several given basisfunctions In the absence of sufficient prior knowledge if afixed basis function is employed for signal decompositionerroneous information will be generated and the physicalmeaning will be unclear Feature information sufficient toidentify the fault is difficult to extract In recent years someadaptive signal decomposition methods including empiricalmode decomposition (EMD) [26] local mean de-composition (LMD) [27] and local characteristic-scale de-composition (LCD) [28] wavelet modulus maxima andsynchronous detection [29] improved orthogonal matchingpursuit and K-SVD algorithm with adaptive transient dic-tionary [30] have been gradually proposed -ose methodsare not required to predict the feature information containedin the analyzed signal and they completely follow thewaveform of the signal -us they exhibit good adaptabilityin signal processing

Although many related achievements have beenachieved in the field they also have the following problems(1) -e magnitude of the feature amplitudes under differentstates is greatly different the real physical meaning is am-biguous and the comparability is poor (2) -e featureamplitude under the same state may be fluctuated greatlywith different samples so the repeatability and stability ofthe results are poor (3) -e effects of noise interference aredifferent to the accuracy of the results -ose problems areclosely related to the nonlinear dynamic characteristics ofrolling bearings How to judge and quantitatively describethe nonlinearity and evolution of rolling bearings underdifferent states is an important prerequisite for fault di-agnosis In the field of nonlinearity some quantitative in-dicators including Lyapunov exponents [31] correlationdimension [32] entropy [33] and complexity [34] have beenwell applied Among of them the LempelndashZiv complexity(LZC) has many good advantages to measure the time seriesarose from the nonlinear dynamics system which have been

used widely in data compression [35] coding [36] gener-ation of test signals [37] and so on Additionally comparedwith the methods of spectral analysis and time-frequencyanalysis the LZC method can detect the long-range cor-relations embedded in the seemingly nonstationary timeseries and also avoid the spurious detection of apparentlong-range correlations that are an artifact of nonstationarity[38]

In order to improve the diagnosis accuracy of rollingbearings effective and quantitative features should beextracted firstly On the one hand the vibration signal ofrolling bearings represents the complex feature of non-linearity and nonstationarity the typical and useful featureinformation may be not significant On the other hand thevibration signal of rolling bearings may be strongly influ-enced by background noise For solving the above problemsan integration approach based on adaptive waveform de-composition (AWD) and LempelndashZiv complexity (LZC) wasproposed Using the AWD method the nonstationarymulticomponent signal of rolling bearings can be convertedinto a series of stationary single component signal which arerich in useful feature information -e LZC method is ap-plied to quantitatively describe the complexity of nonlinearsignal in the time-frequency domain -erefore both theAWD method and LZC algorithm have their own functionin improving the diagnosis accuracy of rolling bearings andthe integration methods of AWD and LZC will have the besteffect

2 Time-Frequency Feature Extraction Based onLocal Frequency and AdaptiveWaveform Decomposition

21 Definition of Local Frequency For an arbitrary signalx(t) assume that the corresponding discrete time series isx(i) ∣ i 1 2 3 n -e time domain waveform of thesignal x(t) is shown in Figure 1 and the local maximum ofthe x(t) satisfies the following equation

x(i)gex(iminus 1)

x(i)gex(i + 1)

x(iminus 1)nex(i + 1)

(1)

From the minimum waveform required to complete avibration the original signal can be approximated as con-sisting of a series of V waves containing two adjacentmaxima as shown in Figure 2

Assuming that the V wave start position tk is the time atwhich the kth local maximum value in the original signal x(t)is located and k 1 2 3 middot middot middot N according to the V wave thelocal period T(t) of the original signal x(t) can be defined asfollows

T(t) tk+1 minus tk tk le tlt tk+1 (2)

where T(t) represents the time required for the signal tocomplete a complete local vibration over a local time range

Similarly a local mean m(t) and a local amplitude a(t)are also given below


m(t) 1nk

1113944

nk

i1x ti( 1113857 tk le tlt tk+1 (3)

a(t) x tk( 1113857 + x tk+1( 1113857




v(t) 1

T(t)

1tk+1 minus tk

tk le tlt tk+1 (5)





yprime(i) 1


+ y(i + N)]

(6)








x(t) 1113944infin


x(t) 1113944N


(8)



t (s)

A (m

s)

Figure 2 -e V wave

0 02 04 06 08 1

ndash5

0

5

t (s)

A (m

s)




x(t) 1113944N

m1sm(t) + W(t) (9)










0 02 04 06 08 1

ndash5

0

5

t (s)

A (m

s)

Local amplitude

Local mean


0 02 04 06 08 10

20

40

60

t (s)

v (H

z)

Local frequency


60

50

40

30

20

10

0

t (s)

f (H

z)

0 05 10

1

2

3

4





⋮


⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(11)





(12)










ndash4ndash2

024

ndash4ndash2

024

A (m

s)

0 005 01 015 02ndash4ndash2

024

t (s)

EMDAWD


200

150

100

50

0

f (H

z)

t (s)0 05 1

0

02

04

06

08

1


200

150

100

50

0

v (H

z)

t (s)0 05 1

0

02

04

06

08

1

12








S(i) 0 x(i)lt xave

1 x(i)ge xave1113896 (14)



limnrarrinfin

c(n) b(n) n

log2(n) (15)


0leC c(n)

b(n)le 1 (16)



ndash4ndash2

024

ndash4ndash2

024

A (m

s)

0 005 01 015 02ndash4ndash2

024

t (s)

EMDAWD


200

150

100

50

0

v (H

z)

t (s)0 05 1

02040608

121

14


200

150

100

50

0

f (H

z)

t (s)0 05 1

0

02

04

06

08

1













1113944n

iminus11113944

m


p xi yj1113872 1113873

p xi( 1113857p yj1113872 1113873

(18)





s3 sn


Q isin SQP subsets

Yes

Yes

No

i = i + 1

i gt n

End

Start







fIR 12

z 1 +d

Dcos α1113888 1113889f0 15794Hz (19)

fOR 12

z 1minusd

Dcos α1113888 1113889f0 10456Hz (20)

fBE 12

D

d1minus

d

D1113888 1113889

2

cos2 α⎛⎝ ⎞⎠f0 13748Hz (21)




1

05

ndash05

0

ndash1

ndash15

ndash2


λ

(a)

(b)

28 3 32 34 357 363 374 383 4

1

08

06

04

02

0Periodic

Periodic window

Quasiperiodic Chaos

Lyap

unov

Cx






0 005 01 015 02ndash04

ndash02

0

02

04

t (s)

A (m

s2 )

(a)

0 005 01 015 02t (s)

A (m

s2 )

ndash4

ndash2

0

2

4

(b)

0 005 01 015 02t (s)

A (m

s2 )

ndash4

ndash2

0

2

4

(c)

ndash04

ndash02

0

02

04

0 005 01 015 02t (s)

A (m

s2 )

(d)


S1S2

S3S4

S5S6

0

0

0

0

0

0

01ndash01

01

ndash01

ndash01ndash02

005

ndash00501

ndash03005

ndash01

t (s)0 005 01 015 02

(a)

S1S2

S3S4

S5S6

20

ndash2

05ndash1

10

ndash1

63010

ndash1

105

0

t (s)0 005 01 015 02

(b)

S1S2

S3S4

S5S6

40

ndash4

10

ndash1

20

ndash1

0ndash4

30

ndash1

20

ndash2

t (s)0 005 01 015 02

(c)

020

ndash0201

0ndash02

0201

0ndash01

01

ndash010

0

ndash03008

ndash002

S1S2

S3S4

S5S6

t (s)0 005 01 015 02

(d)













faultOuter ring

faultRolling

element fault1797 01503 02089 00968 028791772 01962 02013 01655 029531750 02017 01936 02885 030321730 02115 01860 03614 03236


Damage size(mm)

Inner ringfault

Outer ringfault


01778 02726 02038 0277703556 02573 02404 0295505334 01936 02885 0303207112 01420 03013 03140



5 Conclusions






300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

006

005

004

003

002

001

0

(a)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

004

003

002

001

0

(b)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

001

0005

0

(c)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

01

005

0

(d)




Data Availability




Acknowledgments


References

















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


m(t) 1nk

1113944

nk

i1x ti( 1113857 tk le tlt tk+1 (3)

a(t) x tk( 1113857 + x tk+1( 1113857




v(t) 1

T(t)

1tk+1 minus tk

tk le tlt tk+1 (5)





yprime(i) 1


+ y(i + N)]

(6)








x(t) 1113944infin


x(t) 1113944N


(8)



t (s)

A (m

s)

Figure 2 -e V wave

0 02 04 06 08 1

ndash5

0

5

t (s)

A (m

s)




x(t) 1113944N

m1sm(t) + W(t) (9)










0 02 04 06 08 1

ndash5

0

5

t (s)

A (m

s)

Local amplitude

Local mean


0 02 04 06 08 10

20

40

60

t (s)

v (H

z)

Local frequency


60

50

40

30

20

10

0

t (s)

f (H

z)

0 05 10

1

2

3

4





⋮


⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(11)





(12)










ndash4ndash2

024

ndash4ndash2

024

A (m

s)

0 005 01 015 02ndash4ndash2

024

t (s)

EMDAWD


200

150

100

50

0

f (H

z)

t (s)0 05 1

0

02

04

06

08

1


200

150

100

50

0

v (H

z)

t (s)0 05 1

0

02

04

06

08

1

12








S(i) 0 x(i)lt xave

1 x(i)ge xave1113896 (14)



limnrarrinfin

c(n) b(n) n

log2(n) (15)


0leC c(n)

b(n)le 1 (16)



ndash4ndash2

024

ndash4ndash2

024

A (m

s)

0 005 01 015 02ndash4ndash2

024

t (s)

EMDAWD


200

150

100

50

0

v (H

z)

t (s)0 05 1

02040608

121

14


200

150

100

50

0

f (H

z)

t (s)0 05 1

0

02

04

06

08

1













1113944n

iminus11113944

m


p xi yj1113872 1113873

p xi( 1113857p yj1113872 1113873

(18)





s3 sn


Q isin SQP subsets

Yes

Yes

No

i = i + 1

i gt n

End

Start







fIR 12

z 1 +d

Dcos α1113888 1113889f0 15794Hz (19)

fOR 12

z 1minusd

Dcos α1113888 1113889f0 10456Hz (20)

fBE 12

D

d1minus

d

D1113888 1113889

2

cos2 α⎛⎝ ⎞⎠f0 13748Hz (21)




1

05

ndash05

0

ndash1

ndash15

ndash2


λ

(a)

(b)

28 3 32 34 357 363 374 383 4

1

08

06

04

02

0Periodic

Periodic window

Quasiperiodic Chaos

Lyap

unov

Cx






0 005 01 015 02ndash04

ndash02

0

02

04

t (s)

A (m

s2 )

(a)

0 005 01 015 02t (s)

A (m

s2 )

ndash4

ndash2

0

2

4

(b)

0 005 01 015 02t (s)

A (m

s2 )

ndash4

ndash2

0

2

4

(c)

ndash04

ndash02

0

02

04

0 005 01 015 02t (s)

A (m

s2 )

(d)


S1S2

S3S4

S5S6

0

0

0

0

0

0

01ndash01

01

ndash01

ndash01ndash02

005

ndash00501

ndash03005

ndash01

t (s)0 005 01 015 02

(a)

S1S2

S3S4

S5S6

20

ndash2

05ndash1

10

ndash1

63010

ndash1

105

0

t (s)0 005 01 015 02

(b)

S1S2

S3S4

S5S6

40

ndash4

10

ndash1

20

ndash1

0ndash4

30

ndash1

20

ndash2

t (s)0 005 01 015 02

(c)

020

ndash0201

0ndash02

0201

0ndash01

01

ndash010

0

ndash03008

ndash002

S1S2

S3S4

S5S6

t (s)0 005 01 015 02

(d)













faultOuter ring

faultRolling

element fault1797 01503 02089 00968 028791772 01962 02013 01655 029531750 02017 01936 02885 030321730 02115 01860 03614 03236


Damage size(mm)

Inner ringfault

Outer ringfault


01778 02726 02038 0277703556 02573 02404 0295505334 01936 02885 0303207112 01420 03013 03140



5 Conclusions






300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

006

005

004

003

002

001

0

(a)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

004

003

002

001

0

(b)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

001

0005

0

(c)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

01

005

0

(d)




Data Availability




Acknowledgments


References

















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



x(t) 1113944N

m1sm(t) + W(t) (9)










0 02 04 06 08 1

ndash5

0

5

t (s)

A (m

s)

Local amplitude

Local mean


0 02 04 06 08 10

20

40

60

t (s)

v (H

z)

Local frequency


60

50

40

30

20

10

0

t (s)

f (H

z)

0 05 10

1

2

3

4





⋮


⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(11)





(12)










ndash4ndash2

024

ndash4ndash2

024

A (m

s)

0 005 01 015 02ndash4ndash2

024

t (s)

EMDAWD


200

150

100

50

0

f (H

z)

t (s)0 05 1

0

02

04

06

08

1


200

150

100

50

0

v (H

z)

t (s)0 05 1

0

02

04

06

08

1

12








S(i) 0 x(i)lt xave

1 x(i)ge xave1113896 (14)



limnrarrinfin

c(n) b(n) n

log2(n) (15)


0leC c(n)

b(n)le 1 (16)



ndash4ndash2

024

ndash4ndash2

024

A (m

s)

0 005 01 015 02ndash4ndash2

024

t (s)

EMDAWD


200

150

100

50

0

v (H

z)

t (s)0 05 1

02040608

121

14


200

150

100

50

0

f (H

z)

t (s)0 05 1

0

02

04

06

08

1













1113944n

iminus11113944

m


p xi yj1113872 1113873

p xi( 1113857p yj1113872 1113873

(18)





s3 sn


Q isin SQP subsets

Yes

Yes

No

i = i + 1

i gt n

End

Start







fIR 12

z 1 +d

Dcos α1113888 1113889f0 15794Hz (19)

fOR 12

z 1minusd

Dcos α1113888 1113889f0 10456Hz (20)

fBE 12

D

d1minus

d

D1113888 1113889

2

cos2 α⎛⎝ ⎞⎠f0 13748Hz (21)




1

05

ndash05

0

ndash1

ndash15

ndash2


λ

(a)

(b)

28 3 32 34 357 363 374 383 4

1

08

06

04

02

0Periodic

Periodic window

Quasiperiodic Chaos

Lyap

unov

Cx






0 005 01 015 02ndash04

ndash02

0

02

04

t (s)

A (m

s2 )

(a)

0 005 01 015 02t (s)

A (m

s2 )

ndash4

ndash2

0

2

4

(b)

0 005 01 015 02t (s)

A (m

s2 )

ndash4

ndash2

0

2

4

(c)

ndash04

ndash02

0

02

04

0 005 01 015 02t (s)

A (m

s2 )

(d)


S1S2

S3S4

S5S6

0

0

0

0

0

0

01ndash01

01

ndash01

ndash01ndash02

005

ndash00501

ndash03005

ndash01

t (s)0 005 01 015 02

(a)

S1S2

S3S4

S5S6

20

ndash2

05ndash1

10

ndash1

63010

ndash1

105

0

t (s)0 005 01 015 02

(b)

S1S2

S3S4

S5S6

40

ndash4

10

ndash1

20

ndash1

0ndash4

30

ndash1

20

ndash2

t (s)0 005 01 015 02

(c)

020

ndash0201

0ndash02

0201

0ndash01

01

ndash010

0

ndash03008

ndash002

S1S2

S3S4

S5S6

t (s)0 005 01 015 02

(d)













faultOuter ring

faultRolling

element fault1797 01503 02089 00968 028791772 01962 02013 01655 029531750 02017 01936 02885 030321730 02115 01860 03614 03236


Damage size(mm)

Inner ringfault

Outer ringfault


01778 02726 02038 0277703556 02573 02404 0295505334 01936 02885 0303207112 01420 03013 03140



5 Conclusions






300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

006

005

004

003

002

001

0

(a)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

004

003

002

001

0

(b)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

001

0005

0

(c)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

01

005

0

(d)




Data Availability




Acknowledgments


References

















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




⋮


⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(11)





(12)










ndash4ndash2

024

ndash4ndash2

024

A (m

s)

0 005 01 015 02ndash4ndash2

024

t (s)

EMDAWD


200

150

100

50

0

f (H

z)

t (s)0 05 1

0

02

04

06

08

1


200

150

100

50

0

v (H

z)

t (s)0 05 1

0

02

04

06

08

1

12








S(i) 0 x(i)lt xave

1 x(i)ge xave1113896 (14)



limnrarrinfin

c(n) b(n) n

log2(n) (15)


0leC c(n)

b(n)le 1 (16)



ndash4ndash2

024

ndash4ndash2

024

A (m

s)

0 005 01 015 02ndash4ndash2

024

t (s)

EMDAWD


200

150

100

50

0

v (H

z)

t (s)0 05 1

02040608

121

14


200

150

100

50

0

f (H

z)

t (s)0 05 1

0

02

04

06

08

1













1113944n

iminus11113944

m


p xi yj1113872 1113873

p xi( 1113857p yj1113872 1113873

(18)





s3 sn


Q isin SQP subsets

Yes

Yes

No

i = i + 1

i gt n

End

Start







fIR 12

z 1 +d

Dcos α1113888 1113889f0 15794Hz (19)

fOR 12

z 1minusd

Dcos α1113888 1113889f0 10456Hz (20)

fBE 12

D

d1minus

d

D1113888 1113889

2

cos2 α⎛⎝ ⎞⎠f0 13748Hz (21)




1

05

ndash05

0

ndash1

ndash15

ndash2


λ

(a)

(b)

28 3 32 34 357 363 374 383 4

1

08

06

04

02

0Periodic

Periodic window

Quasiperiodic Chaos

Lyap

unov

Cx






0 005 01 015 02ndash04

ndash02

0

02

04

t (s)

A (m

s2 )

(a)

0 005 01 015 02t (s)

A (m

s2 )

ndash4

ndash2

0

2

4

(b)

0 005 01 015 02t (s)

A (m

s2 )

ndash4

ndash2

0

2

4

(c)

ndash04

ndash02

0

02

04

0 005 01 015 02t (s)

A (m

s2 )

(d)


S1S2

S3S4

S5S6

0

0

0

0

0

0

01ndash01

01

ndash01

ndash01ndash02

005

ndash00501

ndash03005

ndash01

t (s)0 005 01 015 02

(a)

S1S2

S3S4

S5S6

20

ndash2

05ndash1

10

ndash1

63010

ndash1

105

0

t (s)0 005 01 015 02

(b)

S1S2

S3S4

S5S6

40

ndash4

10

ndash1

20

ndash1

0ndash4

30

ndash1

20

ndash2

t (s)0 005 01 015 02

(c)

020

ndash0201

0ndash02

0201

0ndash01

01

ndash010

0

ndash03008

ndash002

S1S2

S3S4

S5S6

t (s)0 005 01 015 02

(d)













faultOuter ring

faultRolling

element fault1797 01503 02089 00968 028791772 01962 02013 01655 029531750 02017 01936 02885 030321730 02115 01860 03614 03236


Damage size(mm)

Inner ringfault

Outer ringfault


01778 02726 02038 0277703556 02573 02404 0295505334 01936 02885 0303207112 01420 03013 03140



5 Conclusions






300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

006

005

004

003

002

001

0

(a)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

004

003

002

001

0

(b)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

001

0005

0

(c)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

01

005

0

(d)




Data Availability




Acknowledgments


References

















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







S(i) 0 x(i)lt xave

1 x(i)ge xave1113896 (14)



limnrarrinfin

c(n) b(n) n

log2(n) (15)


0leC c(n)

b(n)le 1 (16)



ndash4ndash2

024

ndash4ndash2

024

A (m

s)

0 005 01 015 02ndash4ndash2

024

t (s)

EMDAWD


200

150

100

50

0

v (H

z)

t (s)0 05 1

02040608

121

14


200

150

100

50

0

f (H

z)

t (s)0 05 1

0

02

04

06

08

1













1113944n

iminus11113944

m


p xi yj1113872 1113873

p xi( 1113857p yj1113872 1113873

(18)





s3 sn


Q isin SQP subsets

Yes

Yes

No

i = i + 1

i gt n

End

Start







fIR 12

z 1 +d

Dcos α1113888 1113889f0 15794Hz (19)

fOR 12

z 1minusd

Dcos α1113888 1113889f0 10456Hz (20)

fBE 12

D

d1minus

d

D1113888 1113889

2

cos2 α⎛⎝ ⎞⎠f0 13748Hz (21)




1

05

ndash05

0

ndash1

ndash15

ndash2


λ

(a)

(b)

28 3 32 34 357 363 374 383 4

1

08

06

04

02

0Periodic

Periodic window

Quasiperiodic Chaos

Lyap

unov

Cx






0 005 01 015 02ndash04

ndash02

0

02

04

t (s)

A (m

s2 )

(a)

0 005 01 015 02t (s)

A (m

s2 )

ndash4

ndash2

0

2

4

(b)

0 005 01 015 02t (s)

A (m

s2 )

ndash4

ndash2

0

2

4

(c)

ndash04

ndash02

0

02

04

0 005 01 015 02t (s)

A (m

s2 )

(d)


S1S2

S3S4

S5S6

0

0

0

0

0

0

01ndash01

01

ndash01

ndash01ndash02

005

ndash00501

ndash03005

ndash01

t (s)0 005 01 015 02

(a)

S1S2

S3S4

S5S6

20

ndash2

05ndash1

10

ndash1

63010

ndash1

105

0

t (s)0 005 01 015 02

(b)

S1S2

S3S4

S5S6

40

ndash4

10

ndash1

20

ndash1

0ndash4

30

ndash1

20

ndash2

t (s)0 005 01 015 02

(c)

020

ndash0201

0ndash02

0201

0ndash01

01

ndash010

0

ndash03008

ndash002

S1S2

S3S4

S5S6

t (s)0 005 01 015 02

(d)













faultOuter ring

faultRolling

element fault1797 01503 02089 00968 028791772 01962 02013 01655 029531750 02017 01936 02885 030321730 02115 01860 03614 03236


Damage size(mm)

Inner ringfault

Outer ringfault


01778 02726 02038 0277703556 02573 02404 0295505334 01936 02885 0303207112 01420 03013 03140



5 Conclusions






300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

006

005

004

003

002

001

0

(a)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

004

003

002

001

0

(b)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

001

0005

0

(c)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

01

005

0

(d)




Data Availability




Acknowledgments


References

















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia












1113944n

iminus11113944

m


p xi yj1113872 1113873

p xi( 1113857p yj1113872 1113873

(18)





s3 sn


Q isin SQP subsets

Yes

Yes

No

i = i + 1

i gt n

End

Start







fIR 12

z 1 +d

Dcos α1113888 1113889f0 15794Hz (19)

fOR 12

z 1minusd

Dcos α1113888 1113889f0 10456Hz (20)

fBE 12

D

d1minus

d

D1113888 1113889

2

cos2 α⎛⎝ ⎞⎠f0 13748Hz (21)




1

05

ndash05

0

ndash1

ndash15

ndash2


λ

(a)

(b)

28 3 32 34 357 363 374 383 4

1

08

06

04

02

0Periodic

Periodic window

Quasiperiodic Chaos

Lyap

unov

Cx






0 005 01 015 02ndash04

ndash02

0

02

04

t (s)

A (m

s2 )

(a)

0 005 01 015 02t (s)

A (m

s2 )

ndash4

ndash2

0

2

4

(b)

0 005 01 015 02t (s)

A (m

s2 )

ndash4

ndash2

0

2

4

(c)

ndash04

ndash02

0

02

04

0 005 01 015 02t (s)

A (m

s2 )

(d)


S1S2

S3S4

S5S6

0

0

0

0

0

0

01ndash01

01

ndash01

ndash01ndash02

005

ndash00501

ndash03005

ndash01

t (s)0 005 01 015 02

(a)

S1S2

S3S4

S5S6

20

ndash2

05ndash1

10

ndash1

63010

ndash1

105

0

t (s)0 005 01 015 02

(b)

S1S2

S3S4

S5S6

40

ndash4

10

ndash1

20

ndash1

0ndash4

30

ndash1

20

ndash2

t (s)0 005 01 015 02

(c)

020

ndash0201

0ndash02

0201

0ndash01

01

ndash010

0

ndash03008

ndash002

S1S2

S3S4

S5S6

t (s)0 005 01 015 02

(d)













faultOuter ring

faultRolling

element fault1797 01503 02089 00968 028791772 01962 02013 01655 029531750 02017 01936 02885 030321730 02115 01860 03614 03236


Damage size(mm)

Inner ringfault

Outer ringfault


01778 02726 02038 0277703556 02573 02404 0295505334 01936 02885 0303207112 01420 03013 03140



5 Conclusions






300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

006

005

004

003

002

001

0

(a)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

004

003

002

001

0

(b)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

001

0005

0

(c)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

01

005

0

(d)




Data Availability




Acknowledgments


References

















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




fIR 12

z 1 +d

Dcos α1113888 1113889f0 15794Hz (19)

fOR 12

z 1minusd

Dcos α1113888 1113889f0 10456Hz (20)

fBE 12

D

d1minus

d

D1113888 1113889

2

cos2 α⎛⎝ ⎞⎠f0 13748Hz (21)




1

05

ndash05

0

ndash1

ndash15

ndash2


λ

(a)

(b)

28 3 32 34 357 363 374 383 4

1

08

06

04

02

0Periodic

Periodic window

Quasiperiodic Chaos

Lyap

unov

Cx






0 005 01 015 02ndash04

ndash02

0

02

04

t (s)

A (m

s2 )

(a)

0 005 01 015 02t (s)

A (m

s2 )

ndash4

ndash2

0

2

4

(b)

0 005 01 015 02t (s)

A (m

s2 )

ndash4

ndash2

0

2

4

(c)

ndash04

ndash02

0

02

04

0 005 01 015 02t (s)

A (m

s2 )

(d)


S1S2

S3S4

S5S6

0

0

0

0

0

0

01ndash01

01

ndash01

ndash01ndash02

005

ndash00501

ndash03005

ndash01

t (s)0 005 01 015 02

(a)

S1S2

S3S4

S5S6

20

ndash2

05ndash1

10

ndash1

63010

ndash1

105

0

t (s)0 005 01 015 02

(b)

S1S2

S3S4

S5S6

40

ndash4

10

ndash1

20

ndash1

0ndash4

30

ndash1

20

ndash2

t (s)0 005 01 015 02

(c)

020

ndash0201

0ndash02

0201

0ndash01

01

ndash010

0

ndash03008

ndash002

S1S2

S3S4

S5S6

t (s)0 005 01 015 02

(d)













faultOuter ring

faultRolling

element fault1797 01503 02089 00968 028791772 01962 02013 01655 029531750 02017 01936 02885 030321730 02115 01860 03614 03236


Damage size(mm)

Inner ringfault

Outer ringfault


01778 02726 02038 0277703556 02573 02404 0295505334 01936 02885 0303207112 01420 03013 03140



5 Conclusions






300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

006

005

004

003

002

001

0

(a)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

004

003

002

001

0

(b)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

001

0005

0

(c)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

01

005

0

(d)




Data Availability




Acknowledgments


References

















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


0 005 01 015 02ndash04

ndash02

0

02

04

t (s)

A (m

s2 )

(a)

0 005 01 015 02t (s)

A (m

s2 )

ndash4

ndash2

0

2

4

(b)

0 005 01 015 02t (s)

A (m

s2 )

ndash4

ndash2

0

2

4

(c)

ndash04

ndash02

0

02

04

0 005 01 015 02t (s)

A (m

s2 )

(d)


S1S2

S3S4

S5S6

0

0

0

0

0

0

01ndash01

01

ndash01

ndash01ndash02

005

ndash00501

ndash03005

ndash01

t (s)0 005 01 015 02

(a)

S1S2

S3S4

S5S6

20

ndash2

05ndash1

10

ndash1

63010

ndash1

105

0

t (s)0 005 01 015 02

(b)

S1S2

S3S4

S5S6

40

ndash4

10

ndash1

20

ndash1

0ndash4

30

ndash1

20

ndash2

t (s)0 005 01 015 02

(c)

020

ndash0201

0ndash02

0201

0ndash01

01

ndash010

0

ndash03008

ndash002

S1S2

S3S4

S5S6

t (s)0 005 01 015 02

(d)













faultOuter ring

faultRolling

element fault1797 01503 02089 00968 028791772 01962 02013 01655 029531750 02017 01936 02885 030321730 02115 01860 03614 03236


Damage size(mm)

Inner ringfault

Outer ringfault


01778 02726 02038 0277703556 02573 02404 0295505334 01936 02885 0303207112 01420 03013 03140



5 Conclusions






300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

006

005

004

003

002

001

0

(a)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

004

003

002

001

0

(b)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

001

0005

0

(c)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

01

005

0

(d)




Data Availability




Acknowledgments


References

















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia












faultOuter ring

faultRolling

element fault1797 01503 02089 00968 028791772 01962 02013 01655 029531750 02017 01936 02885 030321730 02115 01860 03614 03236


Damage size(mm)

Inner ringfault

Outer ringfault


01778 02726 02038 0277703556 02573 02404 0295505334 01936 02885 0303207112 01420 03013 03140



5 Conclusions






300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

006

005

004

003

002

001

0

(a)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

004

003

002

001

0

(b)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

001

0005

0

(c)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

01

005

0

(d)




Data Availability




Acknowledgments


References

















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



5 Conclusions






300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

006

005

004

003

002

001

0

(a)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

004

003

002

001

0

(b)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

001

0005

0

(c)

300

250

200

150

100

50

0

t (s)

v (H

z)

0 01 02

01

005

0

(d)




Data Availability




Acknowledgments


References

















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Data Availability




Acknowledgments


References

















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia
























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


complexity analysis of time-frequency features for...

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