Research Article Detrended Fluctuation Analysis and Hough...

Research ArticleDetrended Fluctuation Analysis and HoughTransform Based Self-Adaptation Double-Scale FeatureExtraction of Gear Vibration Signals

JiaQing Wang Han Xiao Yong Lv Tao Wang and Zengbing Xu

Hubei Province Key Lab of Machine Transmission and Manufacturing Engineering Wuhan University of Science and TechnologyPO Box 222 Wuhan Hubei 430081 China

Correspondence should be addressed to Han Xiao coolxiaohan163com

Received 27 July 2015 Revised 2 December 2015 Accepted 7 December 2015

Academic Editor Evgeny Petrov

Copyright copy 2016 JiaQing Wang et al This 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

This paper presents the analysis of the vibration time series of a gear system acquired by piezoelectric acceleration transducerusing the detrended fluctuation analysis (DFA) The experimental results show that gear vibration signals behave as double-scalecharacteristics which means that the signals exhibit the self-similarity characteristics in two different time scales For furtherunderstanding the simulation analysis is performed to investigate the reasons for double-scale of gearrsquos fault vibration signalAccording to the analysis results a DFA double logarithmic plot based feature vector combined with scale exponent and interceptof the small time scale is utilized to achieve a better performance of fault identification Furthermore to detect the crossoverpoint of two time scales automatically a new approach based on the Hough transform is proposed and validated by a group ofexperimental testsThe results indicate that comparing with the traditional DFA the faulty gear conditions can be identified betterby analyzing the double-scale characteristics of DFA In addition the influence of trend order of DFA on recognition rate of faultgears is discussed

1 Introduction

Generally the gear transmission systems are characterizedwith periodic behaviors However the defects of gearsbearings or transmission shafts may cause the nonlinearvibration The gearbox vibration signals captured by thesensors are complicated nonlinear and nonstationary [1 2]Many researchers verified that the vibration time series ofthe gear transmission systems exhibit nonlinearity and self-similarity [1 3 4] Therefore a lot of the nonlinear timeseries analysis methods and several nonlinear characteristicquantities such as fractal dimension [5] entropy [6] andthe Lyapunov exponent [7] have been employed to detectthe faults Though these nonlinear based methods may besuitable to analyze the nonlinear characteristics of vibrationsignals they are difficult to obtain the more accurate resultswithout considering the real scale related features of thetime series which are characterized with multiexponents ornonlinear parameters

In the recent years the fractal or multifractal time serieshave been observed in many fields such as geophysics timeseries medical time series and technical time series [8] Thetraditional approaches for the fractal analysis such as Hurstrsquosrescaled-range analysis (119877119878) [9] and fluctuation analysis(FA) [8] always assume the time series as the stationarydata without considering the possible fluctuation caused bysome reasons The methods for the nonstationary time seriesinclude the wavelet analysis the discrete wavelet transform(WT) and the detrended fluctuation analysis (DFA) [8] TheDFAwhich was first introduced by Peng et al in 1992 is a newHurst exponent calculationmethod [10] based on the randomwalk theory Basically it represents a detrending version offluctuation analysis (FA) which is more reliable and suitablefor analyzing the nonstationary signal compared to the 119877119878or the FA analysis This is because the DFA can removethe external polynomial trends of the differential orders inorder to obtain the accurate intrinsic statistical characteristicsfrom the time series One advantage of the DFA is that

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 3409897 9 pageshttpdxdoiorg10115520163409897

2 Shock and Vibration

it can detect the long-range correlations embedded in theseemingly nonstationary time series and also avoid the spu-rious detection of the apparent long-range correlations whichare an artifact of nonstationarity The DFA has been widelyapplied to various fields such as meteorology [11] materialsscience [12] finance [13] biological signals [14] and hydro-graph [15] Several modified DFA methods have also beenproposed [8]

The DFA was also used in equipment fault diagnosisde Moura et al [16] employed the DFA and the principalcomponent analysis (PCA) to the cluster analysis of gearfaults Instead of using the long-range correlation or scaleexponents of time series the idea of Mourarsquos method is touse the fluctuation function as a mapping function fromdata space to characteristic space Afterwards De Mouraet al [17] used the DFA to analyze the bearing fault Sridharet al [18] combined the EEMD with DFA to denoise thenoise-corrupted signal The DFA are used to determine thenoise components in IMFs Through the DFA the crossoverphenomenons are found in finance [19] meteorology [20]medical science [21] and equipment fault diagnosis Linand Chen [22] found the interesting crossover propertiesin vibration signals captured from gearboxes and rollingbearingsThe scale exponents corresponding to different timescales in double logarithm plots were used as the featureparameters to describe the defective conditions of gears androlling bearing Liu [2] claimed that the DFA curves ofbearingrsquos vibration signals can be quantified by two scaleexponents and the exponents in a small time scale can beutilized to distinguish the faulty bearing conditions Theauthorrsquos previous research also showed that the gear vibrationsignals had crossover phenomenonThe scale exponents andintercepts ofDFA curveswere used for gear fault classification[23] Jiang et al [24] evaluated the optimal scaling intervalswith Quasi-Monte Carlo algorithm and the least squaresupport vector machine was used for multifault diagnosis ofgearbox Hough [25] combined the least squaresmethodwithsliding window to extract the scale exponent and the neuralnetwork algorithm was used for classification of gear fault

However the detail reasons for multiscales of faultvibration signals were not discussed in abovementioneddescription Furthermore as far as the authors know theinfluence of detrend order of DFA on fault recognitionwas not discussed in previous literatures and only limitedmethods for evaluating the crossover points of DFA weredeveloped In addition the scale exponents of different timescale intervals were used as the characteristic parameterin previous researches However the intercept of doublelogarithmic plot of the DFA was not utilized Acutally theintercept that is used in our research involves a lot ofinformation of vibration signal

In this paper the detrended fluctuation analysis (DFA)is employed to analyze the gear vibration signals Accordingto the double logarithmic plot of the DFA it is verified thatthe gear vibration signals exhibit self-similarity in two rangesof time scales The reason for the double-scale characteristicis discussed through the simulation analysis Furthermorethe scale exponents and intercepts corresponding to differentscale intervals are extracted as the feature vectors to describe

the fault condition of gears It is found that more pieces ofinformation about the gear faults are involved in the smalltime scale interval In order to detect the crossover point oftwo time scales and extract the parameters (scale exponentsand intercepts) automatically a new approach based onthe Hough transform is proposed The experiments wereperformed with the proposed parameters to classify the gearfaults Combining the Gaussian mixture model (GMM) andBayesian maximum likelihood classifiers the classification ofgear vibration signals achieved successfully

The remainder of the paper is organized as followsSections 2 and 3 overview the detrended fluctuation analysis(DFA) and the Hough transform (HT) By applying severalgear fault simulated signals the analysis and discussion aboutthe DFA are presented in Section 4 In Section 5 a self-adaptive feature extraction and classification method for thevibration signals based on DFA and HT is introduced andverified by the experiments Finally Section 6 contains theconclusions

2 Detrended Fluctuation Analysis

Considering 119909(119905) 119905 = 1 2 3 119873 is a time series of length119873

Step 1 Map 119909(119905) to time series 119910(119898) by integration

119910 (119898) =

119898

sum

119905=1

[119909 (119905) minus 119909] 119898 = 1 2 3 119873 (1)

where 119909 is the mean of the time series 119909(119905)

119909 =1

119873

119873

sum

119905=1

119909 (119905) (2)

Step 2 Divide 119910(119898) into 119873119904= int[119873119904] which are sub-time

series with equal length 119904 The length 119873 of a time series isusually not a multiple of the length 119904 and redundant data ofthe time series119910(119898)may be left Although the redundant datacan be deleted in the following analysis we suggest repeatingthe same process from the opposite end of the same timeseries For each sub-time series compute the correspondingleast squares 119901 order fits

119910119896(119898) =

119901

sum

119895=0

120573119895119905119895

(119896 = 1 2 3 2119873119904) (3)

where 119910119896(119898) is the trend of the 119896th sub-time series It is the

fitting polynomial in this sub-time series Linear quadraticcubic or higher-order polynomials can be used in the fittingprocedure (usually called DFA1 DFA2 DFA3 etc) 120573

119895is the

coefficient of 119895th order

Step 3 For each sub-time series compute the fluctuationfunction

119865 (119904) = (1

2119873119904

2119873119904

sum

119896=1

[1198652

(119904 119896)])

12

(4)

Shock and Vibration 3

where

1198652

(119904 119896) =1

119904

119904

sum

119894=1

119910 [(119896 minus 1) 119904 + 1] minus 119910119896(119898)2

if 119896 = 1 2 119873119904

1198652

(119904 119896) =1

119904

119904

sum

119894=1

119910 [119873 minus (119896 minus 119873119904) 119904 + 119894] minus 119910

119896(119898)2

if 119896 = 119873119904+ 1 2119873

119904

(5)

Step 4 Repeat Steps 1 through 3 for a broad range of sub-timeseries (ie box) with length 119904 If the time series are long-rangepower-law correlated the relationship between119865(119904) and 119904 canbe described as follows

119865 (119904) sim 119904120572

997904rArr

119865 (119904) = 119860119904120572

(6)

where 120572 is the scale exponent It can be calculated by takingthe logarithm of both sides of (6)

log (119865 (119904)) = log119860 + 120572 log 119904 (7)

and subsequently plotting log (119865(119904)) versus log 119904 to obtainscale exponent 120572 and intercept log119860 by linear regression

The scale exponent 120572 characterizes the long-range power-law correlation properties of the time series It has a closerelationship with the self-correlation function If 120572 = 05 1and 15 the characteristics of the time series correspond tothe independent random process (white noise) 1119891 processand Brownian motion respectively If 0 lt 120572 lt 05the correlations in the signal are antipersistent (negativecorrelations) If 05 lt 120572 le 1 correlations in the signal arepersistent (positive correlations)

3 Hough Transform

The Hough transform [25] is an automatic image analysistechnique which can be used to detect regular curves suchas straight lines circles and ellipses within an image Theplotting of log(119865(119904)) versus log 119904 of DFA can be seen as animage The linear relationship between log(119865(119904)) and log 119904means a series of straight lines in the plotting The Houghtransform for detecting straight lines is introduced as follows

Generally in a Cartesian coordinate plane (119883 119884) astraight line can be described as 119910 = 119886119909+119887 where parameters119886 and 119887 are the slope and intercept respectively Only whenthe values of 119886 and 119887 are known can we describe this lineaccurately The point (119909

0 1199100) on this line can be written as

1199100= 119886119909

0+ 119887 and it can be changed as 119887 = minus119909

0119886 + 119910

0

which indicate a straight line in the coordinate plane (119860 119861)That means a point in plane (119883 119884) corresponds to a line inplane (119860 119861) and vice versa If every point on a line 119910 =

119886119909 + 119887 in plane (119883 119884) is mapped to plane (119860 119861) the lineswill cross at one point (119886 119887) and the line in plane (119883 119884)will be identified If there are several crossed points in plane

X

Y

r

120579

O

Figure 1 The pair of parameters (119903 120579) used in polar coordinateswas used to replace the pair of parameters (119886 119887) used in Cartesiancoordinates as the Hough parameter space

Initialize accumulator119867(119903 120579) to all zerosFor each point (119909

119894 119910119894) in the (119883 119884) plane

For 120579 = 0 to 180119903 = 119909119894cos 120579 + 119910

119894sin 120579

119867(119903 120579) = 119867(119903 120579) + 1

EndEnd

Algorithm 1

(119860 119861) several straight lines will be identified in plane (119883 119884)However vertical lines in the (119883 119884) plane described as 119909 = 119886will give rise to unbounded values of the slope parameter 119886Thus Duda and Hart proposed the use of a different pairof parameters (119903 120579) used in polar coordinates which arereferred to as the Hough parameter space to replace the pairof parameters (119886 119887) used in Cartesian coordinates (Figure 1)Consider

119903 = 119909 cos 120579 + 119910 sin 120579 (8)

The outline of the Hough transform consists of thesteps shown in Algorithm 1

Find the values of (119903119894 120579119894) where 119867(119903 120579) is a local maxi-

mumThe detected lines in the (119909 119910) plane will be 119903

119894= 119909 cos 120579

119894+

119910 sin 120579119894

4 Detrended Fluctuation Analysis ofSimulated Signals and Discussion

Different vibration condition signals of gears contain differ-ent frequency components and amplitudes The main fre-quencies that should be paid more attention include rotationfrequency meshing frequencies their harmonic frequenciesand sidebands Usually the gearrsquos vibration signal can beregarded as a combination of a series of sinusoid signals withdifferent frequencies and random noise


Small timescale interval

Transitionalinterval

Large timescale interval

y(t)

x(t)

n(t)

m(t)

minus5

0

5

log F

(s)

3 4 5 62

log s

Figure 2 log119865(119904) versus log 119904 of 119910(119905) 119909(119905) 119899(119905) and119898(119905)

Consider the composite signal 119910(119905) = 119909(119905) + 119899(119905) and119898(119905) = 2119909(119905) where 119899(119905) is a Gaussian distributed randomsignal and 119909(119905) = 08 sin(10120587119905)+01 sin(20120587119905)+01 sin(40120587119905)

The signal-noise-ratio (SNR) of 119910(119905) is equal to 284 Thedefinition of SNR is as follows

SNR = 10 lg 119909119890

(9)

where 119909 and 119890 are the root-mean-squares of 119909(119905) and 119899(119905)respectively

The logarithm scale fluctuation function maps of 119910(119905)119909(119905)119898(119905) and 119899(119905) are shown in Figure 2

Figure 2 shows that the correlations of simulated data119910(119905) do not follow the same scaling law in all time scalesObviously there are two linear intervals and a transitionalinterval in a double logarithmic plot of the DFA for mixturedsignal 119910(119905) Comparing 119910(119905) with 119899(119905) and 119909(119905) in a smalltime scale interval 119910(119905) shows similar linear features withrandom signal 119899(119905) which corresponds to local fluctuationand high frequency components In a large time scaleinterval it shows similar features with periodic signal 119909(119905)which corresponds to the large fluctuation and low frequencyperiodic components Since the gearrsquos vibration signal can beregarded as a combination of a series of periodic signals withdifferent frequencies and random noise the gearrsquos vibrationwill present double-scale characteristic too

Moreover comparing the double logarithmic plot of 119909(119905)and 119898(119905) the scale exponents are the same However theintercept of 119898(119905) is larger than the intercept of 119909(119905) becausethe amplitude of119898(119905) is larger than that of 119909(119905)Therefore theintercept is a useful parameter which characterizes the signalintensity

For the gear vibration signal its characteristics analyzedby DFA are correlated with the fault conditions Differentfault patterns will cause different scale exponents Moreovera more severe defect will cause a larger vibration intensitywhich will cause a larger intercept of double logarithmic plot

Magneticpowder brake

AccelerometerGear box Gear box

Figure 3 The experimental setup of the gearbox fault detection

In our research the scale exponent 120572 and intercept 119887 of differ-ent time scale are utilized as the characteristic parameters todescribe the gear vibration conditions The aforementionedHough transform is used to locate the position of crossoverpoint and distinguish linear relation of small and large timescales and to get correct scale exponent and intercept

5 Application of DFA to GearboxFault Diagnostics

In this section the signals corresponding to four gear faultconditions obtained from gearbox experimental facility areanalyzed by the DFA The scale exponent and intercept areextracted as characteristic parameters to describe the gearconditions Combining Gaussian mixture model (GMM)with Bayesian maximum likelihood classifier these signalsare classified

51 Experimental Setup The experiment setup is shown inFigure 3 and its schematic diagram is shown in Figure 4 Theexperimental facility consists of an electric machine a singlestage gearbox with a pair of spur gears a magnetic powderbrake with necessary load and an IO Tech Wave Book516E16-bit 1MHz data acquisition system with Ethernet interfaceA 055 kW DC motor rotates the pinion which has 20 teethand the mating gear which has 37 teeth is loaded by amagnetic powder brake The vibration generated by thegearbox was picked up by a PCB piezoelectric vibrationaccelerometer The accelerometer is mounted on verticaldirection of the bearing block at input end For each kind ofgear failures including ldquoNormalrdquo ldquoToothlessrdquo ldquoScratchedrdquoand ldquoCircular pitch errorrdquo 150 groups of vibration signals areacquired The motorrsquos rotational speed and transferred loadwere random values which lie in the ranges of 300 rminndash1217 rmin and 0Nsdotmndash20Nsdotm respectively The samplingfrequency is 10 kHz


(4)

(3) (2)

Measuring point

YDF801-4

(1)

N = 055kW

(1) Motor(2) Acceleration transducer

(3) Single geared drive box(4) Magnetic powder brake

Figure 4 The schematic diagram of the experimental setup

52 Signals and Discussion The representative vibration sig-nal and log(119865(119904)) versus log 119904 map of ldquoNormalrdquo and ldquoTooth-lessrdquo gear conditions are shown in Figure 5 The detrendorder is one and the minimum and maximum windowsizes are 8 and 512 sample points respectively For capturedvibration signals Figures 5(c) and 5(d) show that there aretwo different scaling intervals in the double logarithmic plotof DFA fluctuation function As we discussed in Section 4the gear vibration signals can be seen as the combinationof random noise and a series of period signals which havedifferent amplitudes and frequencies The interval of smalltime scales corresponds to a random signal which includeslocal fluctuation and high frequency components and thelarge one corresponds to the periodic components in thesignal which are well correlated

Consider the double logarithmic plot as a binary imageon which the values of pixels at a given coordinate(log(119904) log[119865(119904)]) are one and the values of the rest of thepixels are equal to zero With Hough transform the strangeline corresponding to two ranges will be detected Accord-ingly the corner point the scale exponent120572 and the intercept119887 in different time scale will be extracted automaticallyA proposed feature vector mapping of vibration signals isshown in Figure 6

As Figure 6(a) shows the characteristic parameter mapsof four kinds of gear faults overlap weakly in the small timescale

Except for the ldquoCircular pitch errorrdquo there is a smalloverlap between the ldquoNormalrdquo ldquoScratchedrdquo and ldquoToothlessrdquogears In the large time scale the maps of ldquoScratchedrdquo andldquoToothlessrdquo gears are overlapped completely which indicatesthat the clustered results are better in the small time scalethan those in large time scale In theory even if the gearhas a tiny defect or the gear fault condition changes slightlythe vibration condition caused by the reduction of meshingstiffness will change However these changesmay be so subtle

that only local fluctuation in signals is affected The localfluctuation of signals corresponds to signalsrsquo morphologicalcharacters in small time scales interval or high frequencycomponents rather than the large time scales interval or lowfrequency components Only when the severity of defectsreaches a lever or the fault conditions change a lot will thevariation of large fluctuation in vibration signals and thedifference in large time scale intervals be observedThereforecomparing to large time scales there are more useful piecesof diagnostic information in small scales As a contrast thecharacteristic parameter maps of four kinds of gear faults bytraditional DFA without considering linear relationship indifferent time range are shown in Figure 6(c) Obviously themaps of Scratched and Toothless gears are overlappedmostlyThat means that the difference between ldquoScratchedrdquo andldquoToothlessrdquo conditions signals that cannot be distinguishedin traditional DFA can be identified by small time scaleparameters in double-scale logarithmic plot There is moreuseful diagnostic information in small scales In the followingsection the feature vector consisting of scale exponent 120572 andintercept 119887 in small time scale is used to characterize thegear fault vibration signal The Gaussian mixture model andmaximum Bayes classification will be employed to identifythe gear faults

53 Fault RecognitionAlgorithmsCombinedGaussianMixtureModel with DFA Considering we have a training dataset anda testing dataset which consist of a series of gearbox vibrationsignals the main steps of the proposed algorithm for gearfault classification are described as follows

Step 1 For a kind of fault condition employ DFA to plot thedouble logarithm graphs of all training signals and extract thefeature vector (120572

1 1198871) of small time scale as training space

by Hough transform where 1205721is scale exponents and 119887

1is

intercept of double logarithm graphs

Step 2 Build theGaussianmixturemodel (GMM) of trainingspace corresponding to this kind of fault by expectationmax-imum algorithm The GMM is defined by

119901 (119909) =

119872

sum

119896=1

119908119896119901119896(119909) =

119872

sum

119896=1

119908119896119873(119909 120583

119896 Σ119896) (10)

where119872 is the number of mixtures119908119896is the mixture weight

with the constraint that sum119908119896

= 1 and 119909 = (1205721 1198871)

119873(119909 120583119896 Σ119896) = (1(2120587)

12

|Σ119896|12

)119890minus(12)(119909minus120583

119896)119879

Σ119896

minus1

(119909minus120583119896) is

Gaussian probability function which describes the 119896th nor-mal distribution such as center width and direction [26]with mean 120583

119896and covariance matrix Σ

119896

Step 3 Repeat Steps 1 and 2 to build GMMs for all kinds ofgear fault condition signals

Step 4 Before a testing signal is classified the feature vector(1205722 1198872) should be extracted as mentioned in Step 1 Then

the signal is classified using a Bayesian maximum likelihoodclassifier This is accomplished by computing the conditional


minus6

minus4

minus2

0

2Ac

cele

ratio

n (m

ms2)

01 02 03 040Time (s)

(a)

2 4 6 800

1

2

3

4

log F

(s)

log s

(b)

minus4

minus2

0

2

4

Acce

lera

tion

(mm

s2)

01 02 03 040Time (s)

(c)

2 4 6 800

1

2

3

4

log F

(s)

log s

(d)

Figure 5The representative vibration signal power spectrum and the DFA curves obtained from the two types of gears with a rotation speedof 985 rpm (a) Signal from a ldquoNormalrdquo gear (b) DFA from a ldquoNormalrdquo gear (c) Signal from a ldquoToothlessrdquo gear (d) DFA from a ldquoToothlessrdquogear

likelihoods of the signal under each learned GMM and byselecting the model with the highest likelihood

= argmax119901 (119884 | 119888119894) (11)

where 119884 is the feature vector (120572 119887) of the testing signal and119901(119884 | 119888

119894) is the probability of 119884 with known 119894th gear fault

condition described by 119894th GMM

In our classification experiment for each gear condition100 signals were selected to constitute training dataset and 50signals were selected to constitute testing dataset to verify theproposed approach The minimum and maximum windowsizes are 8 and 512 sampling points respectively and themixture number of GMM is four In order to evaluate theinfluence of the trend order of DFA which may change theposition of crossover point the classification experimentswhen detrend order of DFA ranges from one to six are con-ducted and the classification results are listed in Tables 1ndash6The plotting of recognition rate versus trend order of DFA isshown in Figure 7

Tables 1ndash6 show that for the ldquoNormalrdquo ldquoScratchedrdquoand ldquoCircular pitch errorrdquo signals the recognition rates are

Table 1 Classification results with proposed method (DFA1)

Gear fault Diagnosis results Recognition rateNOR SCR TL CPE

NOR 49 0 1 0 98SCR 0 46 4 0 92TL 5 3 42 0 84CPE 0 0 0 50 100NOR ldquoNormalrdquo SCR ldquoScratchedrdquo TL ldquoToothlessrdquo CPE ldquoCircular pitcherrorrdquo



NOR 50 0 0 0 100SCR 0 45 5 0 90TL 6 2 42 0 84CPE 0 0 0 50 100

over 90 percent even the detrend order of DFA is changedThe recognition rate of ldquoToothlessrdquo signal is less than 90


NormalScratched

ToothlessCircular pitch error

03 04 05 06021205721

b 1

3

35

4

45

5

55

(a)

NormalScratched


08 09 1 11 12 13071205722

b 2

minus1

0

1

2

3

4

(b)

15

2

25

3

35

4

45

b

06 07 08 09 105120572

NormalScratched


(c)

Figure 6 (120572 119887)map of different ranges of scales (a) 1205721and 1198871in the first range of scales (b) 120572

2and 1198872in the second range of scales (c) 120572 and

119887 extracted by traditional DFA



NOR 49 0 0 1 98SCR 0 48 2 0 96TL 7 1 42 0 84CPE 0 0 0 50 100

percent in all the classification experiments Few of samplesof ldquoScratchedrdquo and ldquoToothlessrdquo signals are misclassified asthe ldquoNormalrdquo Theoretically if we study the signals from thefrequency domain the differences between ldquoNormalrdquo gearsldquoScratchedrdquo gears and ldquoToothlessrdquo gears are the vibrationamplitudes on some special frequencies such as rotationfrequency meshing frequency and their frequency multipli-cationWith the severity of gear fault increasing the vibration



NOR 46 0 3 0 92SCR 0 49 1 0 98TL 10 1 39 0 78CPE 0 0 0 50 100

amplitude will increase slowly and the noise will eventuallycause the fault characteristic overlap of three kinds of gearsrsquovibration signals That causes the identification of these threekinds of fault conditions difficult in our experiments Figure 7shows that when detrend order of DFA changes from oneto six the recognition rate of ldquoNormalrdquo ldquoScratchedrdquo andldquoCircular pitch errorrdquo signals keeps the range from 90 precentto 100 precent however the global trend of recognition




NOR 45 1 3 1 90SCR 0 47 3 0 94TL 10 1 39 0 78CPE 0 0 0 50 100



NOR 46 0 4 0 92SCR 1 46 3 0 92TL 14 4 32 0 64CPE 1 0 0 49 98

rate is decreasing with the increasing of detrend order ofDFAThe recognition rate of ldquoToothlessrdquo signals will decreasemonotonically with the detrend order of DFA increaseThat means that the detrend order of DFA influences theclassification results Higher detrend order of DFAmay causeless recognition rate As a suggestion the detrend order ofDFA should be low in order to get better classification result

6 Conclusion

In this study the detrended fluctuation analysis (DFA) whichcan deal with nonstationary signals was employed to analyzethe gear vibration signals acquired by piezoelectric acceler-ation transducer and the experimental results showed thatthe characteristics of all the gear condition signals turned outto be double-scale To further understand the phenomenonof the double-scale the experimental signals were analyzedby simulation The simulation results show that the double-scales correspond to high and low frequency componentsof signal respectively and the intercepts are determinedby signal intensity A feature vector which employed anexponent 120572 and an intercept 119887 in small time scale wasproposed to describe slight defect of gear Since the vibrationpattern caused by the reduction of meshing stiffness changeslocal fluctuation of vibration signal subtly due to a slightdefect it is more suitable to use double-scale parameters forgear conditions classification instead of the traditional DFAIn addition the Hough transform was used to estimate theposition of crossover point and extract the scale exponent andintercept of the small time scale from the double logarithmicplot of DFA automatically Moreover the Gaussian mixturemodel (GMM) and Bayesian maximum likelihood (BML)classifier were employed to describe the distribution of thefeature vector and identify the patterns of test data respec-tively The classification experiments with different detrendorder of DFA were conducted The experimental resultsdemonstrate that the proposed approach is the effectivenessand the detrend order ofDFA sholud be low formore accurateclassification results

NormalScratched


DFA2 DFA3 DFA4 DFA5 DFA6DFA1Detrend order of DFA

65

70

75

80

85

90

95

100

105

Reco

gniti

on ra

te (

)

Figure 7The plotting of recognition rate versus trend order ofDFA

Conflict of Interests

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

Acknowledgment

The reported research was partially based upon work sup-ported by National Natural Science Foundation of China(NSFC) awards (Grant nos 51105284 51475339 51375154 and51405353)

References

[1] S J Loutridis ldquoSelf-similarity in vibration time series applica-tion to gear fault diagnosticsrdquo Journal of Vibration andAcousticsvol 130 no 3 Article ID 031004 2008

[2] J Liu ldquoDetrended fluctuation analysis of vibration signals forbearing fault detectionrdquo in Proceedings of the IEEE InternationalConference on Prognostics and Health Management (PHM rsquo11)pp 1ndash5 IEEE Montreal Canada June 2011

[3] J Wang R Li and X Peng ldquoSurvey of nonlinear vibration ofgear transmission systemsrdquo Applied Mechanics Reviews vol 56no 3 pp 309ndash329 2003

[4] R Bajric D Sprecic and N Zuber ldquoReview of vibration signalprocessing techniques towards gear pairs damage identifica-tionrdquo International Journal of Engineering amp Technology IJET-IJENS vol 11 no 4 pp 124ndash128 2011

[5] X Wang C Liu F Bi X Bi and K Shao ldquoFault diagnosis ofdiesel engine based on adaptive wavelet packets and EEMD-fractal dimensionrdquo Mechanical Systems and Signal Processingvol 41 no 1-2 pp 581ndash597 2013

[6] 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 no 8 pp6077ndash6085 2010

[7] L-Y Wang W-G Zhao and D-H Wei ldquoApplication of Lya-punov exponent spectrum in pressure fluctuation of draft tuberdquoJournal of Hydrodynamics Series B vol 21 no 6 pp 856ndash8602009


[8] J W Kantelhardt S A Zschiegner E Koscielny-Bunde SHavlin A Bunde and H E Stanley ldquoMultifractal detrendedfluctuation analysis of nonstationary time seriesrdquo Physica AStatistical Mechanics and Its Applications vol 316 no 1ndash4 pp87ndash114 2002

[9] J B Bassingthwaighte and G M Raymond ldquoEvaluatingrescaled range analysis for time seriesrdquo Annals of BiomedicalEngineering vol 22 no 4 pp 432ndash444 1994

[10] C-K Peng S V Buldyrev A L Goldberger et al ldquoLong-rangecorrelations in nucleotide sequencesrdquoNature vol 356 no 6365pp 168ndash170 1992

[11] K Kocak ldquoExamination of persistence properties of wind speedrecords using detrended fluctuation analysisrdquo Energy vol 34no 11 pp 1980ndash1985 2009

[12] J M O Matos E P de Moura S E Kruger and J M ARebello ldquoRescaled range analysis and detrended fluctuationanalysis study of cast irons ultrasonic backscattered signalsrdquoChaos Solitons amp Fractals vol 19 no 1 pp 55ndash60 2004

[13] J C Reboredo M A Rivera-Castro J G V Miranda andR Garcıa-Rubio ldquoHow fast do stock prices adjust to marketefficiency Evidence from a detrended fluctuation analysisrdquoPhysica A Statistical Mechanics and Its Applications vol 392no 7 pp 1631ndash1637 2013

[14] J C Echeverrıa J Alvarez-Ramırez M A Pena E RodrıguezM J Gaitan and R Gonzalez-Camarena ldquoFractal and nonlin-ear changes in the long-term baseline fluctuations of fetal heartraterdquoMedical Engineering amp Physics vol 34 no 4 pp 466ndash4712012

[15] M A Little and J P Bloomfield ldquoRobust evidence for randomfractal scaling of groundwater levels in unconfined aquifersrdquoJournal of Hydrology vol 393 no 3-4 pp 362ndash369 2010

[16] E P de Moura A P Vieira M A S Irmao and A A SilvaldquoApplications of detrended-fluctuation analysis to gearbox faultdiagnosisrdquoMechanical Systems and Signal Processing vol 23 no3 pp 682ndash689 2009

[17] E P De Moura C R Souto A A Silva and M A SIrmao ldquoEvaluation of principal component analysis and neuralnetwork performance for bearing fault diagnosis from vibrationsignal processed by RS and DF analysesrdquo Mechanical Systemsand Signal Processing vol 25 no 5 pp 1765ndash1772 2011

[18] M Sridhar D Venkata Ratnam B Sirisha et al ldquoCharacteri-zation of low latitude ionospheric scintillations using EEMDmdashDFA methodrdquo in Proceedings of the IEEE International Confer-ence on Innovations in Information Embedded and Communi-cation Systems (ICIIECS rsquo15) pp 1ndash6 Coimbatore India March2015

[19] MA Shi-Hao ldquoCrossover phenomena in detrended fluctuationanalysis used in financial marketsrdquo Communications inTheoret-ical Physics vol 51 no 2 pp 358ndash362 2009

[20] R G Kavasseri and R Nagarajan ldquoEvidence of crossoverphenomena in wind-speed datardquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 51 no 11 pp 2255ndash22622004

[21] C-K Peng S Havlin H E Stanley and A L GoldbergerldquoQuantification of scaling exponents and crossover phenomenain nonstationary heartbeat time seriesrdquo Chaos vol 5 no 1 pp82ndash87 1995

[22] J Lin andQ Chen ldquoA novelmethod for feature extraction usingcrossover characteristics of nonlinear data and its applicationto fault diagnosis of rotary machineryrdquoMechanical Systems andSignal Processing vol 48 no 1-2 pp 174ndash187 2014

[23] H Xiao Y Lv and T Wang ldquoDetrended fluctuation analysis togearrsquos vibration signals and its application in fault classificationrdquoJournal of Vibration of Engineering vol 28 pp 331ndash336 2015

[24] X Jiang S Li and Y Wang ldquo1641 A novel method for self-adaptive feature extraction using scaling crossover character-istics of signals and combining with LS-SVM for multi-faultdiagnosis of gearboxrdquo Journal of Vibroengineering vol 17 no4 pp 1861ndash1878 2015

[25] P V C Hough ldquoMethod and means for recognizing complexpatternsrdquo US Patent no 3069654 US Patent and TrademarkOffice Washington DC USA 1962

[26] A C Lindgren M T Johnson and R J Povinelli ldquoSpeechrecognition using reconstructed phase space featuresrdquo in Pro-ceedings of the IEEE International Conference on Speech andSignal Processing (ICASSP rsquo03) vol 1 pp I-60ndashI-63 IEEE April2003

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



it can detect the long-range correlations embedded in theseemingly nonstationary time series and also avoid the spu-rious detection of the apparent long-range correlations whichare an artifact of nonstationarity The DFA has been widelyapplied to various fields such as meteorology [11] materialsscience [12] finance [13] biological signals [14] and hydro-graph [15] Several modified DFA methods have also beenproposed [8]

The DFA was also used in equipment fault diagnosisde Moura et al [16] employed the DFA and the principalcomponent analysis (PCA) to the cluster analysis of gearfaults Instead of using the long-range correlation or scaleexponents of time series the idea of Mourarsquos method is touse the fluctuation function as a mapping function fromdata space to characteristic space Afterwards De Mouraet al [17] used the DFA to analyze the bearing fault Sridharet al [18] combined the EEMD with DFA to denoise thenoise-corrupted signal The DFA are used to determine thenoise components in IMFs Through the DFA the crossoverphenomenons are found in finance [19] meteorology [20]medical science [21] and equipment fault diagnosis Linand Chen [22] found the interesting crossover propertiesin vibration signals captured from gearboxes and rollingbearingsThe scale exponents corresponding to different timescales in double logarithm plots were used as the featureparameters to describe the defective conditions of gears androlling bearing Liu [2] claimed that the DFA curves ofbearingrsquos vibration signals can be quantified by two scaleexponents and the exponents in a small time scale can beutilized to distinguish the faulty bearing conditions Theauthorrsquos previous research also showed that the gear vibrationsignals had crossover phenomenonThe scale exponents andintercepts ofDFA curveswere used for gear fault classification[23] Jiang et al [24] evaluated the optimal scaling intervalswith Quasi-Monte Carlo algorithm and the least squaresupport vector machine was used for multifault diagnosis ofgearbox Hough [25] combined the least squaresmethodwithsliding window to extract the scale exponent and the neuralnetwork algorithm was used for classification of gear fault

However the detail reasons for multiscales of faultvibration signals were not discussed in abovementioneddescription Furthermore as far as the authors know theinfluence of detrend order of DFA on fault recognitionwas not discussed in previous literatures and only limitedmethods for evaluating the crossover points of DFA weredeveloped In addition the scale exponents of different timescale intervals were used as the characteristic parameterin previous researches However the intercept of doublelogarithmic plot of the DFA was not utilized Acutally theintercept that is used in our research involves a lot ofinformation of vibration signal

In this paper the detrended fluctuation analysis (DFA)is employed to analyze the gear vibration signals Accordingto the double logarithmic plot of the DFA it is verified thatthe gear vibration signals exhibit self-similarity in two rangesof time scales The reason for the double-scale characteristicis discussed through the simulation analysis Furthermorethe scale exponents and intercepts corresponding to differentscale intervals are extracted as the feature vectors to describe

the fault condition of gears It is found that more pieces ofinformation about the gear faults are involved in the smalltime scale interval In order to detect the crossover point oftwo time scales and extract the parameters (scale exponentsand intercepts) automatically a new approach based onthe Hough transform is proposed The experiments wereperformed with the proposed parameters to classify the gearfaults Combining the Gaussian mixture model (GMM) andBayesian maximum likelihood classifiers the classification ofgear vibration signals achieved successfully

The remainder of the paper is organized as followsSections 2 and 3 overview the detrended fluctuation analysis(DFA) and the Hough transform (HT) By applying severalgear fault simulated signals the analysis and discussion aboutthe DFA are presented in Section 4 In Section 5 a self-adaptive feature extraction and classification method for thevibration signals based on DFA and HT is introduced andverified by the experiments Finally Section 6 contains theconclusions

2 Detrended Fluctuation Analysis

Considering 119909(119905) 119905 = 1 2 3 119873 is a time series of length119873

Step 1 Map 119909(119905) to time series 119910(119898) by integration

119910 (119898) =

119898

sum

119905=1

[119909 (119905) minus 119909] 119898 = 1 2 3 119873 (1)

where 119909 is the mean of the time series 119909(119905)

119909 =1

119873

119873

sum

119905=1

119909 (119905) (2)

Step 2 Divide 119910(119898) into 119873119904= int[119873119904] which are sub-time

series with equal length 119904 The length 119873 of a time series isusually not a multiple of the length 119904 and redundant data ofthe time series119910(119898)may be left Although the redundant datacan be deleted in the following analysis we suggest repeatingthe same process from the opposite end of the same timeseries For each sub-time series compute the correspondingleast squares 119901 order fits

119910119896(119898) =

119901

sum

119895=0

120573119895119905119895

(119896 = 1 2 3 2119873119904) (3)

where 119910119896(119898) is the trend of the 119896th sub-time series It is the

fitting polynomial in this sub-time series Linear quadraticcubic or higher-order polynomials can be used in the fittingprocedure (usually called DFA1 DFA2 DFA3 etc) 120573

119895is the

coefficient of 119895th order

Step 3 For each sub-time series compute the fluctuationfunction

119865 (119904) = (1

2119873119904

2119873119904

sum

119896=1

[1198652

(119904 119896)])

12

(4)


where

1198652

(119904 119896) =1

119904

119904

sum

119894=1

119910 [(119896 minus 1) 119904 + 1] minus 119910119896(119898)2

if 119896 = 1 2 119873119904

1198652

(119904 119896) =1

119904

119904

sum

119894=1


119896(119898)2

if 119896 = 119873119904+ 1 2119873

119904

(5)


119865 (119904) sim 119904120572

997904rArr

119865 (119904) = 119860119904120572

(6)


log (119865 (119904)) = log119860 + 120572 log 119904 (7)



3 Hough Transform




1199100= 119886119909


0119886 + 119910

0



X

Y

r

120579

O



119894 119910119894) in the (119883 119884) plane

For 120579 = 0 to 180119903 = 119909119894cos 120579 + 119910

119894sin 120579

119867(119903 120579) = 119867(119903 120579) + 1

EndEnd

Algorithm 1


119903 = 119909 cos 120579 + 119910 sin 120579 (8)




119894= 119909 cos 120579

119894+

119910 sin 120579119894







y(t)

x(t)

n(t)

m(t)

minus5

0

5

log F

(s)

3 4 5 62

log s




SNR = 10 lg 119909119890

(9)














(4)

(3) (2)

Measuring point

YDF801-4

(1)

N = 055kW













1is



119901 (119909) =

119872

sum

119896=1

119908119896119901119896(119909) =

119872

sum

119896=1

119908119896119873(119909 120583

119896 Σ119896) (10)



= 1 and 119909 = (1205721 1198871)

119873(119909 120583119896 Σ119896) = (1(2120587)

12

|Σ119896|12

)119890minus(12)(119909minus120583

119896)119879

Σ119896

minus1

(119909minus120583119896) is



119896





minus6

minus4

minus2

0

2Ac

cele

ratio

n (m

ms2)

01 02 03 040Time (s)

(a)

2 4 6 800

1

2

3

4

log F

(s)

log s

(b)

minus4

minus2

0

2

4

Acce

lera

tion

(mm

s2)

01 02 03 040Time (s)

(c)

2 4 6 800

1

2

3

4

log F

(s)

log s

(d)



= argmax119901 (119884 | 119888119894) (11)











NOR 50 0 0 0 100SCR 0 45 5 0 90TL 6 2 42 0 84CPE 0 0 0 50 100



NormalScratched


03 04 05 06021205721

b 1

3

35

4

45

5

55

(a)

NormalScratched


08 09 1 11 12 13071205722

b 2

minus1

0

1

2

3

4

(b)

15

2

25

3

35

4

45

b

06 07 08 09 105120572

NormalScratched


(c)






NOR 49 0 0 1 98SCR 0 48 2 0 96TL 7 1 42 0 84CPE 0 0 0 50 100




NOR 46 0 3 0 92SCR 0 49 1 0 98TL 10 1 39 0 78CPE 0 0 0 50 100





NOR 45 1 3 1 90SCR 0 47 3 0 94TL 10 1 39 0 78CPE 0 0 0 50 100



NOR 46 0 4 0 92SCR 1 46 3 0 92TL 14 4 32 0 64CPE 1 0 0 49 98


6 Conclusion


NormalScratched



65

70

75

80

85

90

95

100

105

Reco

gniti

on ra

te (

)




Acknowledgment


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










where

1198652

(119904 119896) =1

119904

119904

sum

119894=1

119910 [(119896 minus 1) 119904 + 1] minus 119910119896(119898)2

if 119896 = 1 2 119873119904

1198652

(119904 119896) =1

119904

119904

sum

119894=1


119896(119898)2

if 119896 = 119873119904+ 1 2119873

119904

(5)


119865 (119904) sim 119904120572

997904rArr

119865 (119904) = 119860119904120572

(6)


log (119865 (119904)) = log119860 + 120572 log 119904 (7)



3 Hough Transform




1199100= 119886119909


0119886 + 119910

0



X

Y

r

120579

O



119894 119910119894) in the (119883 119884) plane

For 120579 = 0 to 180119903 = 119909119894cos 120579 + 119910

119894sin 120579

119867(119903 120579) = 119867(119903 120579) + 1

EndEnd

Algorithm 1


119903 = 119909 cos 120579 + 119910 sin 120579 (8)




119894= 119909 cos 120579

119894+

119910 sin 120579119894







y(t)

x(t)

n(t)

m(t)

minus5

0

5

log F

(s)

3 4 5 62

log s




SNR = 10 lg 119909119890

(9)














(4)

(3) (2)

Measuring point

YDF801-4

(1)

N = 055kW













1is



119901 (119909) =

119872

sum

119896=1

119908119896119901119896(119909) =

119872

sum

119896=1

119908119896119873(119909 120583

119896 Σ119896) (10)



= 1 and 119909 = (1205721 1198871)

119873(119909 120583119896 Σ119896) = (1(2120587)

12

|Σ119896|12

)119890minus(12)(119909minus120583

119896)119879

Σ119896

minus1

(119909minus120583119896) is



119896





minus6

minus4

minus2

0

2Ac

cele

ratio

n (m

ms2)

01 02 03 040Time (s)

(a)

2 4 6 800

1

2

3

4

log F

(s)

log s

(b)

minus4

minus2

0

2

4

Acce

lera

tion

(mm

s2)

01 02 03 040Time (s)

(c)

2 4 6 800

1

2

3

4

log F

(s)

log s

(d)



= argmax119901 (119884 | 119888119894) (11)











NOR 50 0 0 0 100SCR 0 45 5 0 90TL 6 2 42 0 84CPE 0 0 0 50 100



NormalScratched


03 04 05 06021205721

b 1

3

35

4

45

5

55

(a)

NormalScratched


08 09 1 11 12 13071205722

b 2

minus1

0

1

2

3

4

(b)

15

2

25

3

35

4

45

b

06 07 08 09 105120572

NormalScratched


(c)






NOR 49 0 0 1 98SCR 0 48 2 0 96TL 7 1 42 0 84CPE 0 0 0 50 100




NOR 46 0 3 0 92SCR 0 49 1 0 98TL 10 1 39 0 78CPE 0 0 0 50 100





NOR 45 1 3 1 90SCR 0 47 3 0 94TL 10 1 39 0 78CPE 0 0 0 50 100



NOR 46 0 4 0 92SCR 1 46 3 0 92TL 14 4 32 0 64CPE 1 0 0 49 98


6 Conclusion


NormalScratched



65

70

75

80

85

90

95

100

105

Reco

gniti

on ra

te (

)




Acknowledgment


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













y(t)

x(t)

n(t)

m(t)

minus5

0

5

log F

(s)

3 4 5 62

log s




SNR = 10 lg 119909119890

(9)














(4)

(3) (2)

Measuring point

YDF801-4

(1)

N = 055kW













1is



119901 (119909) =

119872

sum

119896=1

119908119896119901119896(119909) =

119872

sum

119896=1

119908119896119873(119909 120583

119896 Σ119896) (10)



= 1 and 119909 = (1205721 1198871)

119873(119909 120583119896 Σ119896) = (1(2120587)

12

|Σ119896|12

)119890minus(12)(119909minus120583

119896)119879

Σ119896

minus1

(119909minus120583119896) is



119896





minus6

minus4

minus2

0

2Ac

cele

ratio

n (m

ms2)

01 02 03 040Time (s)

(a)

2 4 6 800

1

2

3

4

log F

(s)

log s

(b)

minus4

minus2

0

2

4

Acce

lera

tion

(mm

s2)

01 02 03 040Time (s)

(c)

2 4 6 800

1

2

3

4

log F

(s)

log s

(d)



= argmax119901 (119884 | 119888119894) (11)











NOR 50 0 0 0 100SCR 0 45 5 0 90TL 6 2 42 0 84CPE 0 0 0 50 100



NormalScratched


03 04 05 06021205721

b 1

3

35

4

45

5

55

(a)

NormalScratched


08 09 1 11 12 13071205722

b 2

minus1

0

1

2

3

4

(b)

15

2

25

3

35

4

45

b

06 07 08 09 105120572

NormalScratched


(c)






NOR 49 0 0 1 98SCR 0 48 2 0 96TL 7 1 42 0 84CPE 0 0 0 50 100




NOR 46 0 3 0 92SCR 0 49 1 0 98TL 10 1 39 0 78CPE 0 0 0 50 100





NOR 45 1 3 1 90SCR 0 47 3 0 94TL 10 1 39 0 78CPE 0 0 0 50 100



NOR 46 0 4 0 92SCR 1 46 3 0 92TL 14 4 32 0 64CPE 1 0 0 49 98


6 Conclusion


NormalScratched



65

70

75

80

85

90

95

100

105

Reco

gniti

on ra

te (

)




Acknowledgment


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(4)

(3) (2)

Measuring point

YDF801-4

(1)

N = 055kW













1is



119901 (119909) =

119872

sum

119896=1

119908119896119901119896(119909) =

119872

sum

119896=1

119908119896119873(119909 120583

119896 Σ119896) (10)



= 1 and 119909 = (1205721 1198871)

119873(119909 120583119896 Σ119896) = (1(2120587)

12

|Σ119896|12

)119890minus(12)(119909minus120583

119896)119879

Σ119896

minus1

(119909minus120583119896) is



119896





minus6

minus4

minus2

0

2Ac

cele

ratio

n (m

ms2)

01 02 03 040Time (s)

(a)

2 4 6 800

1

2

3

4

log F

(s)

log s

(b)

minus4

minus2

0

2

4

Acce

lera

tion

(mm

s2)

01 02 03 040Time (s)

(c)

2 4 6 800

1

2

3

4

log F

(s)

log s

(d)



= argmax119901 (119884 | 119888119894) (11)











NOR 50 0 0 0 100SCR 0 45 5 0 90TL 6 2 42 0 84CPE 0 0 0 50 100



NormalScratched


03 04 05 06021205721

b 1

3

35

4

45

5

55

(a)

NormalScratched


08 09 1 11 12 13071205722

b 2

minus1

0

1

2

3

4

(b)

15

2

25

3

35

4

45

b

06 07 08 09 105120572

NormalScratched


(c)






NOR 49 0 0 1 98SCR 0 48 2 0 96TL 7 1 42 0 84CPE 0 0 0 50 100




NOR 46 0 3 0 92SCR 0 49 1 0 98TL 10 1 39 0 78CPE 0 0 0 50 100





NOR 45 1 3 1 90SCR 0 47 3 0 94TL 10 1 39 0 78CPE 0 0 0 50 100



NOR 46 0 4 0 92SCR 1 46 3 0 92TL 14 4 32 0 64CPE 1 0 0 49 98


6 Conclusion


NormalScratched



65

70

75

80

85

90

95

100

105

Reco

gniti

on ra

te (

)




Acknowledgment


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus6

minus4

minus2

0

2Ac

cele

ratio

n (m

ms2)

01 02 03 040Time (s)

(a)

2 4 6 800

1

2

3

4

log F

(s)

log s

(b)

minus4

minus2

0

2

4

Acce

lera

tion

(mm

s2)

01 02 03 040Time (s)

(c)

2 4 6 800

1

2

3

4

log F

(s)

log s

(d)



= argmax119901 (119884 | 119888119894) (11)











NOR 50 0 0 0 100SCR 0 45 5 0 90TL 6 2 42 0 84CPE 0 0 0 50 100



NormalScratched


03 04 05 06021205721

b 1

3

35

4

45

5

55

(a)

NormalScratched


08 09 1 11 12 13071205722

b 2

minus1

0

1

2

3

4

(b)

15

2

25

3

35

4

45

b

06 07 08 09 105120572

NormalScratched


(c)






NOR 49 0 0 1 98SCR 0 48 2 0 96TL 7 1 42 0 84CPE 0 0 0 50 100




NOR 46 0 3 0 92SCR 0 49 1 0 98TL 10 1 39 0 78CPE 0 0 0 50 100





NOR 45 1 3 1 90SCR 0 47 3 0 94TL 10 1 39 0 78CPE 0 0 0 50 100



NOR 46 0 4 0 92SCR 1 46 3 0 92TL 14 4 32 0 64CPE 1 0 0 49 98


6 Conclusion


NormalScratched



65

70

75

80

85

90

95

100

105

Reco

gniti

on ra

te (

)




Acknowledgment


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










NormalScratched


03 04 05 06021205721

b 1

3

35

4

45

5

55

(a)

NormalScratched


08 09 1 11 12 13071205722

b 2

minus1

0

1

2

3

4

(b)

15

2

25

3

35

4

45

b

06 07 08 09 105120572

NormalScratched


(c)






NOR 49 0 0 1 98SCR 0 48 2 0 96TL 7 1 42 0 84CPE 0 0 0 50 100




NOR 46 0 3 0 92SCR 0 49 1 0 98TL 10 1 39 0 78CPE 0 0 0 50 100





NOR 45 1 3 1 90SCR 0 47 3 0 94TL 10 1 39 0 78CPE 0 0 0 50 100



NOR 46 0 4 0 92SCR 1 46 3 0 92TL 14 4 32 0 64CPE 1 0 0 49 98


6 Conclusion


NormalScratched



65

70

75

80

85

90

95

100

105

Reco

gniti

on ra

te (

)




Acknowledgment


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












NOR 45 1 3 1 90SCR 0 47 3 0 94TL 10 1 39 0 78CPE 0 0 0 50 100



NOR 46 0 4 0 92SCR 1 46 3 0 92TL 14 4 32 0 64CPE 1 0 0 49 98


6 Conclusion


NormalScratched



65

70

75

80

85

90

95

100

105

Reco

gniti

on ra

te (

)




Acknowledgment


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









Research Article Detrended Fluctuation Analysis and Hough...

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Transcript of Research Article Detrended Fluctuation Analysis and Hough...