Research ArticleRail Corrugation Detection of High-Speed Railway Using WheelDynamic Responses

Jianbo Li 12 and Hongmei Shi 12

1School of Mechanical Electronic and Control Engineering Beijing Jiaotong University Beijing 100044 China2Key Laboratory of Vehicle Advanced Manufacturing Measuring and Control Technology (Beijing Jiaotong University)Ministry of Education Beijing 100044 China

Correspondence should be addressed to Hongmei Shi hmshibjtueducn

Received 4 January 2019 Accepted 11 February 2019 Published 25 February 2019

Guest Editor Krzysztof Holak

Copyright copy 2019 Jianbo Li and Hongmei Shi-is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Rail corrugation often occurs on the high-speed railway which will affect ride comfort and even the train operation safety in severecondition Detection of rail corrugation wavelength and depth is absolutely essential for maintenance and safety A novel methodusing wheel vibration acceleration is proposed in this paper in which ensemble empirical mode decomposition (EEMD) isemployed to estimate the wavelength and bispectrum features are extracted to recognize the depth with support vector machine(SVM) Firstly a vehicle-track coupling model considering the rail corrugation of high-speed railway is established to calculate thewheel vibration acceleration Secondly the estimation algorithm of wavelength is studied by analyzing the main frequency withEEMD -e optimal parameters of EEMD are selected according to the orthogonal coefficient of decomposition results and thedistribution of the extreme points of signal -e depth detection is transformed to a classification problem with SVM Bispectrumfeatures which are extracted from the reconstructed signal using the high-frequency components of wheel vibration accelerationcombining with train speed and corrugation wavelength are input into SVM to recognize the rail corrugation depth Finallynumerical simulation is carried out to verify the accuracy of the proposed estimationmethod-e simulation results show that theproposed detection algorithm can accurately identify rail corrugation the estimation error of rail corrugation wavelength is lessthan 025 and the classification accuracy of rail corrugation depth is more than 99

1 Introduction

Rail corrugation is a type of wavy wear formed longitudi-nally along the top of the rail which appears in all types ofrail systems [1] With the rapid development of high-speedrailway rail corrugation has been found everywhere es-pecially with a fixed rail corrugation wavelength due to thesame train operation speed and vehicle type -e results offield tests on a high-speed railway in China [2] show that railcorrugation is distributed in many discontinuous places-elength of rail corrugation is usually about 10m to 15m alongthe longitudinal direction of rail and the wavelength is from120mm to 150mm -e depth of rail corrugation is muchsmaller generally in the range of 004mm to 01mm-e railcorrugation causes aggravated interactions between vehicle

and rail and terrible noises which severely influences thesafety and ride comfort of a running high-speed train It isnecessary for maintenance and safety to detect the wave-length and depth of rail corrugation effectively

In recent years research of track structure healthmonitoring utilizing vehicle vibration responses hasattracted more and more attention -ere are many studieson rail corrugation monitoring based on operating vehiclesRail corrugation detection method based on vehicle vibra-tion responses is a noncontact detection method Moreovervibration acceleration signal is easy to obtain and real-timemonitoring of track status can be realized without affectingrailway operation Hopkins and Taheri [3] proposed a defectdetection algorithm for rail health monitoring with wavelettransform -e vibration acceleration signal of the bogie is

HindawiShock and VibrationVolume 2019 Article ID 2695647 12 pageshttpsdoiorg10115520192695647

decomposed by wavelet transform and the Lipschitz ex-ponent of each scale is calculated to identify wheel flat railcrack and rail corrugation Molodova et al [4] used axle boxacceleration to measure short wave track defects includingrail corrugation -e amplitude and power spectral densityof the vibration acceleration signal of axle box are analyzedto detect defects Huang et al [5] proposed a rail corrugationdetection technology based on fiber laser accelerometer -efiber laser accelerometer is installed on the bogie to detectthe vertical acceleration of train axle box -e vibrationacceleration signal is denoised by wavelet transform andthen the waveform of rail corrugation is estimated by doubleintegral Kaewunruen [6] used dynamic wheel-rail in-teraction to monitor rail corrugation on curved track forguiding maintenance Corrugation roughness data are alsoobtained by double integration of axle box accelerationsignals Salvador et al [7] made time-frequency analysis ofthe axle box vibration acceleration signal with short-timeFourier transform which can detect rail corrugation iso-lated rail defects and loss of track vertical alignment Most ofthe above studies only identify if there is rail corrugationdefect without giving a detailed diagnosis of the wavelengthand depth of rail corrugation

EEMD is a signal decomposition method which is animprovement algorithm of empirical mode decomposition(EMD) and widely used in signal analysis and mechanicalfault diagnosis Shen et al [8] proposed fast EEMD opti-mization algorithm which applies nonlinear correlationcoefficient and accuracy requirements to obtain the bestparameters of EEMD -e proposed EEMD optimizationalgorithm is used to extract the accent of emotional speechGuo and Tse [9] used EEMD to analyze bearing vibrationsignals and proposed a method to automatically select ap-propriate EEMD parameters using the relative root meansquare error between the decomposition result and theoriginal signal Xue et al [10] applied the same method todiagnose the fault of rolling bearing Kedadouche et al [11]used the Pearson coefficient of correlation between IMFs toselect the parameters of EEMD

In this paper a rail corrugation detection method ispresented based on the vertical vibration response of wheel-e basic principle of detection algorithm is analyzing thehigh frequency of wheel vibration acceleration with EEMDmethod to estimate the wavelength and depth of rail cor-rugation -e paper is organized as follows A vehicle-trackcoupling model considering rail corrugation is described inSection 2 to calculate the wheel vibration acceleration usedin the detection algorithm -e detection algorithm of railcorrugation is presented in Section 3 in which the esti-mation method of wavelength using EEMD to acquire themain frequency and the bispectrum features extraction toclassify the depth are respectively described in detail -enumerical simulation and detection results of rail corru-gation are demonstrated and analyzed in Section 4

2 Simulation Model

In order to obtain the wheel dynamic responses used todetect rail corrugation a vertical vehicle-track coupling

model is established as shown in Figure 1 consisting ofvehicle model track model track irregularity model and railcorrugation model

21 VehicleModel of High-Speed Railway -e vehicle modelis a multibody system including a car body two bogies andfour wheelsets which are connected by stiffness-dampingelements [12]-ere are 10 degrees of freedom which are thevertical motion yc and nodding motion βc of the car bodythe vertical motion yt and nodding motion βt of two bogiesand the vertical motion yw of four wheelsets -e vehicleparameters of Chinese high-speed train CRH3 used in themodel are represented in Table 1

22 Slab Track Model -e slab ballastless track structure isapplied into the track model for its wide use in China high-speed railway which is constituted of rail fasteners trackslabs CA mortar layer and foundation [14] -e rail isregarded as an infinite beam with continuous support thetrack slab is regarded as a free beam and the fasteners andmortar layer are represented as equivalent stiffness-dampingelements -e parameters of the slab track model for sim-ulation are shown in Table 2

23 Vehicle-Track Coupling Model -e vehicle model andthe track model are coupled by wheel-rail interactionAccording to Hertzian nonlinear elastic theory [12] thevertical force between the wheel and rail is defined as

p(t)

1GΔZ(t)1113960 1113961

32 ΔZ(t)ge 0

0 ΔZ(t)lt 0

⎧⎪⎨

⎪⎩(1)

where G is the wheel-rail contact constant and ΔZ(t) is theelastic compression deformation between the wheel and rail

-e elastic compression between wheel and rail is de-termined by the displacement of wheel and rail at the wheel-rail contact point expressed as

ΔZ(t) ywi(t)minusyr xwi t( 1113857minusy0(t) i 1sim4 (2)

where ywi is the displacement of the wheel yr is the dis-placement of the rail and y0 is the track irregularity which isthe main excitation of vehicle-track coupling system -epower spectral density of ballastless track irregularities ofChina high-speed railway [15] is as follows

S(f)

10544 times 10minus5 times fminus33891 fle 00187

35588 times 10minus3 times fminus19271 00187ltfle 00474

19784 times 10minus3 times fminus13643 00474ltfle 01533

39488 times 10minus3 times fminus34516 fgt 01555

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(3)

where f is the spatial frequency-e random track irregularities in spatial domain yirr can

be calculated with IFFT-e dynamic equation of the vehicle-track coupling sys-

tem is obtained by combining the vehicle vibration equationwith the track vibration equation which is expressed as

2 Shock and Vibration

[M] A +[C] V +[K] X P (4)

where [M] [C] and [K] are the mass damping andstiffness matrices of the vehicle-track coupling model re-spectively A V and X are the acceleration velocityand displacement vectors of the coupling model re-spectively and P is the load vector of the coupling model

Vehicle-track coupling model is a system of nonlineardifferential equations which can be solved by the numericalintegration method

24 Rail Corrugation Model Rail corrugation is a kind ofharmonic irregularity which is expressed by a cosinefunction

ycor(t) 12

a 1minus cos2πvt

λ1113874 11138751113876 1113877 0le tle

L

v1113874 1113875 (5)

where a is the depth of rail corrugation λ is the wavelengthof rail corrugation v is the train speed and L is the length ofrail corrugation

In this paper a 130-meter track is simulated with a 10-meter rail corrugation at the middle position from 60degm to70degm of which the wavelength is from 100mm to 150mmand the depth is from 001mm to 01mm Figure 2 shows asection of rail corrugation of which the wavelength is150mm and the depth is 001mm

-erefore the integrated track irregularity of the vehicle-track system can be expressed as

y0(t) yirr(t) + ycor(t) (6)

Obviously rail corrugation will influence the vibrationresponse of wheels through wheel-rail interaction Railcorrugation is a kind of short-wavelength irregularity whilethe wavelength of random track irregularity is much longer-erefore high-frequency vibration will appear in the vi-bration response of wheels if rail corrugation exists and themain frequency of high-frequency band matches thewavelength of rail corrugation -is is the basic principle ofdetecting rail corrugation using wheel dynamic responses

25 Model Validation It is necessary to validate the modelaccuracy because wheel responses used in the followingdetection algorithm are acquired from the model Differentconditions with different wavelength and depth of railcorrugation are simulated and the results are compared withWangrsquos study [16] -e effect of rail corrugation wavelengthon wheel-rail force is shown in Figure 3 in which thedepth of rail corrugation is 01mm and the train speed is300 kmh -e results show that the influence of wavelengthon wheel-rail force is nonlinear and the maximum wheel-rail force is at 120mm wavelength Sensitive wavelength is

RailRail fastener yr (x t)

yw4yw3

yt2

βt2

yw2yw1

yt1

v

βt1

yc

βc

ys (x t)SlabCA mortarFoundation

Car body

Bogic

Wheelsetx

Figure 1 -e vehicle-track system model of high-speed railway

Table 1 Parameters for Chinese high-speed train CRH3 [13]

Parameter Value UnitMass of car body 40000 kgMass of bogie 3200 kgMass of wheelset 2400 kgPitch inertia of car body 547000 kgmiddotm2

Pitch inertia of bogie 6800 kgmiddotm2

Stiffness of primary suspension system 208times106 NmStiffness of secondary suspension system 8times105 NmDamping of primary suspension system 1times 105 NmiddotsmDamping of secondary suspension system 12times105 NmiddotsmSemilongitudinal distance between bogies 86875 mSemilongitudinal distance betweenwheelsets in bogie 125 m

Radius of wheel 046 m

Table 2 Parameters of the CRTS II slab track [13]

Parameter Value UnitElastic modulus of rail 21times 1011 Nm2

Inertia of rail 3217times10minus5 m4

Mass of rail 60 kgmStiffness of rail fastener 6times107 NmDamping of rail fastener 477times104 NmiddotsmElastic modulus of slab 39times1010 Nm2

Inertia of slab 85times10minus5 m4

Mass of slab 1275 kgmLength of single slab 65 mStiffness of CA mortar 9times108 NmDamping of CA mortar 83times104 Nmiddotsm

Shock and Vibration 3

the wavelength when rail corrugation has the greatest effecton wheel-rail interaction -e sensitive wavelength of thesimulation model is 120mm In addition the main fre-quency of wheel-rail force is related to the wavelength anddecreases with the increase of wavelength -e effect of railcorrugation depth on maximum wheel-rail force and railvibration acceleration is shown in Figure 4 -e wavelengthof rail corrugation is determined to be 100mm and the trainruns at 300 kmh With the increase of rail corrugationdepth the values of wheel-rail force and rail vibration ac-celeration gradually increase -e simulation results showgood consistence with the conclusion of Wang

3 Detection Method

A detection method for rail corrugation is proposed bywhich the wavelength can be estimated through analyzingmain frequency of wheel vibration responses based onEEMD and the depth can be classified with bispectrumfeatures and SVM -e detection flow chart is shown inFigure 5 LIBSVM [17] toolbox and particle swarm opti-mization (PSO) are used to optimize the parameters of SVM

31 Wavelength Estimation -e process of wavelength es-timation can be described as 4 steps

(1) Decompose wheel acceleration signal using EEMDmethod

(2) Eliminate the false intrinsic modal functions (IMFs)according to the correlation analysis between theIMFs and the original signal

(3) Select IMFs with high-frequency components andobtain their main frequency using FFT

(4) Estimate the wavelength of rail corrugation based onmain frequency and train speed

311 Background Huang et al [18] defined the signal sat-isfying the following two conditions as intrinsicmodal function(IMF) (1) the number of extreme points and zero-crossingpoints is equal or different by one (2) the mean value of theupper envelope formed by the local maximum points and thelower envelope formed by the local minimum points is zero

-e EMD process for signal x(t) is as follows

(1) Find all local maximum points and local minimumpoints of the signal x(t)

(2) Fit the upper and lower envelope and obtain theaverage m(t) of the upper and lower envelope andthen calculate

h(t) x(t)minusm(t) (7)

(3) If h(t) satisfies the above two conditions then h(t) isan IMF denoted as c(t) otherwise h(t) is treated asthe original signal repeat the above two steps

(4) Calculate the residual signal

r(t) x(t)minus c(t) (8)

(5) Regard r(t) as the original data and repeat the abovesteps until the residual signal is a monotonefunction

(6) -e original signal x(t) can be expressed as the sumof all IMFs and residual signal r(t)

x(t) 1113944n

i1ci(t) + r(t) (9)

Whe

el-r

ail v

ertic

al fo

rce (

kN)

500

400

300

200

100

Mai

n fre

quen

cy (H

z)

2000

1500

1000

500

050 100 150

Wavelength of corrugation (mm)

Wheel-rail vertical forceMain frequency

200

Figure 3 Effect of rail corrugation wavelength on wheel-railinteraction

Whe

el-r

ail v

ertic

al fo

rce (

kN)

400

300

200

Rail

acce

lera

tion

(g)

300

200

100

Wheel-rail vertical forceRail acceleration

005 01 015Depth of corrugation (mm)

02

Figure 4 Effect of rail corrugation depth on wheel-rail interaction

times10ndash3

y cor

(mm

)

5

0

ndash5

ndash10

ndash1558 60 62 64 66 68

Distance (m)70 72

Figure 2 Rail corrugation

4 Shock and Vibration

-e orthogonality of IMFs is used to evaluate the de-composition results -e overall index of orthogonality isdefined as

IO 1113944T

t0

1113936n+1j1 1113936

n+1k1 cj(t)ck(t)

x2(t)⎛⎝ ⎞⎠ (jne k) (10)

-e mode mixing problem occurs when the signal isanalyzed by IMF sifting method because there are multiplefrequency components in an IMF To avoid this problemWu and Huang [19] proposed EEMD based on noise-assisted data analysis method

-e EEMD process for signal x(t) is as follows

(1) Add a white noise with amplitude α to the signal x(t)(2) -e signal with noise is analyzed by EMD(3) Repeat the above steps N times and average the

results of EMD

-e difference between the input signal and the corre-sponding IMF is called the final standard deviation of errorwhich is expressed by e -e added white noise satisfies thefollowing rule [19]

e αN

radic (11)

where α is the amplitude of the added white noise and N isthe number of ensemble members

312 Selection of EEMDKey Parameters If the amplitude ofthe white noise is too small the mixing of the modes cannotbe weakened On the contrary the final standard deviationof error will be large If the number of ensemble members istoo small the influence of added white noise cannot be

eliminated in contrast the calculation cost will be increased-erefore it is very important to select suitable amplitude ofthe white noise and number of ensemble members whenEEMD is employed to analyze signals -e amplitude ofwhite noise is defined as

α k middot σ (12)

where σ is the standard deviation of the signal and k is acoefficient

-e purpose of added white noise is to improve thedistribution of extreme points of signal -e extreme pointdistribution index is defined as

SI

S21 + S22 + S23 + S24

1113969

(13)

where S1 is the standard deviation of amplitude interval ofmaximum points S2 is the standard deviation of coordinateinterval of maximum points S3 is the standard deviation ofamplitude interval of minimum points and S4 is the stan-dard deviation of coordinate interval of minimum points

-e vibration acceleration of wheel with the condition of10-meter rail corrugation is shown in Figure 6 White noisewith different amplitudes is added to the signal and theextreme point distribution index SI is calculated It can beseen from Figure 7 that the extreme point distribution indexSI tends to be stable when the amplitude of white noiseincreases to a certain value

In this paper the extreme point distribution index SI isused to determine the optimal range of white noise am-plitude -en the added white noise amplitude α and thenumber N of ensemble members are selected respectivelyaccording to the orthogonal coefficient IO and equation (11)-e parameter selection algorithm of EEMD is shown in

Vibration acceleration of

wheel

EEMD

Selection of IMFs

High frequency componentMain frequency

Wavelength of corrugation

Reconstructed signal

Bispectrum features

SVM

Depth of corrugation

Train speed

Figure 5 Rail corrugation detection method

Shock and Vibration 5

Figure 8 Firstly the optimum range of white noise am-plitude is determined by the stability value of SI In order toensure that the amplitude of white noise can improve thedistribution of signal extreme points the minimum value ofk is 0001-emaximum value of k is set to be 001 so that theoptimization range is not too small In order to select theappropriate amplitude of white noise it is necessary todetermine the number of ensemble members in advance-e numberN of ensemble members should not be too largeotherwise the calculation time is very long If N is too smallEEMD will be invalid After many trials N is initially set tobe 10 EEMD analyses of the signal are carried out in theoptimal range of white noise amplitude and the de-composition results are evaluated by the orthogonal co-efficients -e white noise amplitude corresponding to theoptimal decomposition result is the best white noise am-plitude α In this paper the error less than 1 can be ac-ceptable -e final standard deviation of error is set to be001 -en the bestN is determined according to the rules ofthe final standard deviation of error and the white noiseamplitude Meanwhile the minimum value of N is 10 toavoid the EEMD invalidation caused by too small N

-e vibration acceleration signals of wheel in Figure 6 areanalyzed by EMD and EEMD respectively Figure 9 shows thefirst four IMFs after decomposition IMF components arearranged from high frequency to low frequency so the first

IMF (IMF1) is the high-frequency component of the signal Itcan be seen from Figure 9(a) that IMF1 of EMD analysis haslow-frequency signals at other locations besides high-frequency signals caused by rail corrugation which in-dicates that there is mode mixing in EMD analysis Appar-ently the mode mixing will influence the detection results InFigure 9(b) there is only the high-frequency componentsrelated with rail corrugation and no low frequency in IMF1-erefore EEMD analysis is an effectivemethod to extract thepart of the signal caused by rail corrugation

313 Wavelength Calculation After the IMFs are obtainedby EEMD analysis the IMFs are selected to eliminate theeffects of noise and the random irregularity of the track IMFselection and signal reconstruction algorithm are shown inFigure 10 -e false components are eliminated by corre-lation analysis and the high-frequency components arereconstructed into a high-frequency signal which is causedby rail corrugation

Estimation of wavelength of rail corrugation by mainfrequency of the reconstructed signal yr is expressed as

λ v

fm (14)

where v is train speed and fm is the main frequency of thereconstructed signal which can be obtained by FFT

32 Features Extraction of Corrugation Depth -e vibrationacceleration signals of wheel under rail corrugation arenonlinear nonstationary and non-Gaussian Higher-orderspectra are useful in analyzing nonlinearity and non-Gaussianity of the signal as it provides high noise immu-nity [20] -e frequency domain representation of third-order cumulants of signals is called bispectrum which isexpressed as

B f1 f2( 1113857 E X f1( 1113857X f2( 1113857Xlowast

f1 + f2( 11138571113858 1113859 (15)

Whe

etse

t acc

eler

atio

n (g

)

2

1

0

ndash1

ndash230 40 50 60

Distance (m)70 80 90 100

Figure 6 Vibration acceleration signal of wheel

SI

80

60

40

20

00 005

Amplitude coefficient k of white noise01 015 02

Figure 7 -e extreme point distribution index SI

Calculate standard deviation σ of signal

Find the k when SI is stable defined as kmaxIf kmax lt 001 let kmax = 001

k = 0001~kmax N = 10

Find the k when the IO is minimal defined as ko

α = komiddotσ

N = (αe)2 e = 001if N lt 10 let N = 10

Figure 8 Selection algorithm of EEMD parameters

6 Shock and Vibration

2 IMF1

30 40 50 60 70 80 90 100

0

ndash2

2 IMF2

30 40 50 60 70 80 90 100

0

ndash2

05 IMF3

30 40 50 60 70 80 90 100

0

ndash05

05 IMF4

30 40 50 60 70 80 90 100

0

ndash05

(a)

05 IMF1

30 40 50 60 70 80 90 100

0

ndash05

1 IMF2

30 40 50 60 70 80 90 100

0

ndash1

2 IMF3

30 40 50 60 70 80 90 100

0

ndash2

1 IMF4

30 40 50 60 70 80 90 100

0

ndash1

(b)

Figure 9 -e first four IMFs of the signal (a) EMD (b) EEMD

Ci (i = 1 2 n) are n IMFs of the signal

CORi (i = 1 2 n) are the correlation coefficients of IMF and original signal

respectively

CORthreshold = max CORi (i = 1 2 n)10If CORi lt CORthreshold eliminate ci

fi (i = 1 2 ) are the main frequency of ci respectively

yr = sum (ci) i = 1 2

If fi gt 400Hz reserve ci

Figure 10 IMFs selection and signal reconstruction algorithm

Shock and Vibration 7

where X(f) is the Fourier transform of the signal lowast denotescomplex conjugate E[] denotes the expectation operationand the frequency f may be normalized by the Nyquistfrequency to be between 0 and 1

-e discrete bispectrum matrix B is obtained by the highorder spectral analysis of the reconstructed signal yr(t) -esize of the bispectrum matrix B is QtimesQ In order to rec-ognize the depth of rail corrugation easily bispectrumfeatures need to be extracted

-ree amplitude features of bispectrum [21]

Amp1 1

Q2 1113944Ω

|B(i j)|

Amp2 1113944Ωlog(|B(i j)|)

Amp3 1113944

Q

i1log(|B(i i)|)

(16)

-ree moment features of bispectrum [22]

Mom1 1113944

Q

i1i middot log(|B(i i)|)

Mom2 1113944

Q

i1(iminusM)

2middot log(|B(i i)|)

Mom3 1113944Ω

i2 + j21113969

middot |B(i j)|

(17)

-ree entropy features of bispectrum [23]

Ent1 minus1113944Ω

p1(i j) middot log p1(i j)( 1113857 (18)

where p1(i j) |B(i j)|1113936Ω|B(i j)|

Ent2 minus1113944Ω

p2(i j) middot log p2(i j)( 1113857 (19)

where p2(i j) |B(i j)|21113936Ω|B(i j)|2

Ent3 minus1113944Ω

p3(i j) middot log p3(i j)( 1113857 (20)

where p3(i j) |B(i j)|31113936Ω|B(i j)|3-e above nine bispectrum features have different am-

plitudes at different rail corrugation depths as shown inFigure 11 -ree entropy features of bispectrum changenonlinearly with the increase of rail corrugation depth whilethe other features of bispectrum increase with the increase ofrail corrugation depth -e bispectrum features can wellreflect the depth of rail corrugation

-e train speed and the wavelength of the rail corru-gation also affect the vibration response of the wheel-erefore the train speed v and the estimated wavelength λare also taken as the feature parameters -e extractedfeatures are constructed into a features vector and classifiedusing the SVM model

4 Detection Result

41 Data Preparation In the simulation model the trainspeeds are 250 kmh to 350 kmh the wavelengths of rail

corrugation are 100mm to 150mm and the depths of railcorrugation are 001mm to 01mm -ere are 660 signalsAfter repeated 10 simulations 6600 sets of vibration ac-celeration signals of wheel are obtained -e signal length is60m including 25m before the wheel enters the rail cor-rugation and 25m after it leaves the rail corrugation

In order to verify the antinoise ability of the detectionmethod a Gaussian noise signal is added to the wheel ac-celeration signal to simulate the measurement noise

xmeasurement xsimulation + Ep times N times var xsimulation( 1113857 (21)

where Ep is the noise level N is a standard normal dis-tribution vector with zero mean and unit standard deviationand var() denotes the standard deviation of signal

42 Wavelength Estimation of Rail Corrugation -e esti-mation results of rail corrugation wavelength are shown inFigure 12 It can be seen from Figure 12 that the wavelengthof rail corrugation can be estimated at different corrugationdepths and train speeds and the noise has no effect on theestimation algorithm

-e relative error between the estimated valueof wavelength and the true value of wavelength is definedas

error 1n

1113944

n

i1

λestimationi minus λtruei

11138681113868111386811138681113868111386811138681113868

λtruei

(22)

where n is the number of data samplesIn order to analyze the influence of corrugation depth on

wavelength estimation depths of 001mm 005mm and01mm rail corrugation are selected respectively -ewavelength estimation results with 0 1 3 and 5noise levels are shown in Table 3 -e wavelength estimationerror of rail wavelength is almost unchanged under differentdepths and the maximum value of wavelength estimationerror is not more than 02 -e wavelength estimationerrors are very small which shows that the estimation of railwavelength is completely reliable Moreover the simulationresults show that the proposed method can estimate the

001 002 003 004 005 006 007 008 009 01Depth of corrugation (mm)

0

02

04

06

08

1

Nor

mal

ized

ampl

itude

Amp1Amp2Amp3

Mom1Mom2Mom3

Ent1Ent2Ent3

Figure 11 Bispectrum features at different depths of corrugation(train speed is 250 kmh and wavelength is 100mm)

8 Shock and Vibration

wavelength well under the conditions of different depths andmeasurement noises

-e train speeds are 250 kmh 300 kmh and 350 kmhand the wavelength of rail corrugation is estimated re-spectively -e wavelength estimation results with 0 13 and 5 noise levels are shown in Table 4 -e

wavelength estimation error of rail wavelength is almostunchanged at different depths and the maximum value ofwavelength estimation error is not more than 025 -ewavelength estimation error is also very small which in-dicates that the wavelength estimation effect of rail corru-gation is very good Meanwhile the simulation results

9999

100

10001

Wav

elen

gth

(mm

)

True value ofwavelength0 noise

1 noise3 noise5 noise

002 004 006 008 01

110

11005

002 004 006 008 0111995

120

12005

002 004 006 008 011298

1299

130

002 004 006 008 011395

140

1405

002 004 006 008 01e depth of rail corrugation (mm)

1495

150

1505

002 004 006 008 01

(a)

True value ofwavelength0 noise

1 noise3 noise5 noise

9998

100

10002

Wav

elen

gth

(mm

)

250 300 350

250 300 350

1098

110

1102

250 300 3501198

120

1202

250 300 3501298

1299

130

1301

250 300 3501395

140

1405

250 300 350Train speed (kmh)

1495

150

1505

(b)

Figure 12 -e estimation results of rail corrugation wavelength (a) At different depths of rail corrugation (b) At different train speeds

Table 3 Wavelength estimation error at different depths of rail corrugation

CaseDepth 001mm Depth 005mm Depth 01mm

Nil 1 3 5 Nil 1 3 5 Nil 1 3 5100mm 00045 00045 00045 00045 00045 00045 00045 00045 00045 00045 00045 00045110mm 00900 00897 00908 00910 00902 00900 00900 00899 00899 00899 00900 00899120mm 00205 00045 00045 00063 00367 00276 00330 00275 00313 00349 00367 00330130mm 00968 00968 00980 00994 00968 00968 00968 00968 00968 00968 00968 00968140mm 01252 01124 01001 01100 01347 01286 01230 01260 01360 01357 01329 01358150mm 00376 00045 00045 00087 01453 00623 00543 00601 01563 01293 01091 01068

Shock and Vibration 9

illustrate that the proposed method can estimate thewavelength excellent under the conditions of different trainspeeds and measurement noises

43 Depth Classification of Rail Corrugation According tothe values of depth of rail corrugations three depth levels aredivided as shown in Table 5

Bispectrum features estimated wavelength and trainspeed are taken as inputs of SVM and depth level is takenas SVM output -e 80 of all simulation data is used totrain the SVM model and the remaining 20 is used fortesting

-e classification results of rail corrugation depth levelwith different levels of noise are shown in Figure 13

-e reliability of the proposedmethod is evaluated by theaccuracy of the test results -e ratio between the number ofcorrect classification results and the number of total testsamples is defined as the detection accuracy which can beexpressed as

accuray Numcorrect

Numtotaltimes 100 (23)

-e detection results of depth level with 0 1 3 and5 noise levels are shown in Table 6 -e detection accuracyis more than 99 indicating that the extracted corrugationdepth feature is effective -e proposed detection methodhas strong antinoise ability and still has high accuracy underdifferent measurement noises

5 Conclusion

Rail corrugation is a short-wavelength track irregularityexcitation and accordingly causes high-frequency vibrationresponse of wheel A rail detection algorithm based onEEMD and bispectrum features is proposed using wheelvibration acceleration Different frequency components ofwheel vibration response are extracted with EEMD of wheelvibration acceleration signal -e parameters α and N ofEEMD are optimized using the orthogonal coefficient IOand the extreme point distribution index SI which reducesthe mode mixing and calculation time of the de-composition -e main frequency is found after EEMD tocalculate the wavelength-e depth detection is regarded asa classification problem with SVM -e high-frequencysignal of rail corrugation is reconstructed through theselection of IMFs by which not only the interference oftrack random irregularity is avoided but also noise influ-ence Bispectrum features which are extracted from the

Table 4 Wavelength estimation error at different train speeds

CaseTrain speed 250 kmh Train speed 300 kmh Train speed 350 kmh

Nil 1 3 5 Nil 1 3 5 Nil 1 3 5100mm 0 0 0 0 0 0 0 0 00093 00093 00093 00093110mm 00999 00999 00999 00997 00834 00834 00834 00836 00927 00927 00927 00927120mm 01657 01198 00938 01058 0 0 0 0 00093 00093 00093 00093130mm 00999 00999 00999 00999 00999 00999 00999 01001 00907 00907 00910 00910140mm 01234 01070 01016 01107 01325 01275 01217 01265 01428 01412 01391 01386150mm 00399 0 0 00100 01421 00524 00474 00050 02124 01778 01616 01639

Table 5 Definition of depth levels of rail corrugation

Depth levels Depths of rail corrugation (mm)Level 1 a≦ 004Level 2 004lt a≦ 007Level 3 agt 007

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 3Level 2

(a)

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 2 Level 3

(b)

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 2 Level 3

(c)

Figure 13 Continued

10 Shock and Vibration

reconstructed signal combining with train speed and thecorrugation wavelength are input into SVM -e nu-merical simulation results show the proposed detectionalgorithm can accurately identify the existence of railcorrugation -e error between the estimated wavelengthand the real value is small and the accuracy of corrugationdepth level classification is high And the results alsorepresent robustness under different train speeds andmeasurement noises -e proposed rail corrugation de-tection method provides a feasible scheme for the in-service vehicle test in the future

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 they have no conflicts of interest

Acknowledgments

-is work was supported by the National Key Researchand Development Program of China under Grant no2016YFB1200401

References

[1] K H Oostermeijer ldquoReview on short pitch rail corrugationstudiesrdquo Wear vol 265 no 9-10 pp 1231ndash1237 2008

[2] Z Q Jiang D L Si W Li et al ldquoOn rail corrugation of highspeed railway (in Chinese)rdquo China Railway Science vol 35no 4 pp 9ndash14 2014

[3] B M Hopkins and S Taheri ldquoTrack health monitoring usingwaveletrdquo in Proceedings of ASME 2010 Rail TransportationDivision Fall Technical Conference pp 9ndash15 Roanoke VAUSA October 2010

[4] M Molodova Z Li and R Dollevoet ldquoAxle box accelerationmeasurement and simulation for detection of short trackdefectsrdquo Wear vol 271 no 1-2 pp 349ndash356 2011

[5] W Huang W Zhang Y Du et al ldquoDetection of rail cor-rugation based on fiber laser accelerometersrdquo MeasurementScience and Technology vol 24 no 9 article 094014 2013

[6] S Kaewunruen ldquoMonitoring of rail corrugation growth onsharp curves for track maintenance prioritisationrdquo In-ternational Journal of Acoustics and Vibration vol 23 no 1pp 35ndash43 2018

[7] P Salvador V Naranjo R Insa and P Teixeira ldquoAxleboxaccelerations their acquisition and time-frequency charac-terisation for railway track monitoring purposesrdquo Measure-ment vol 82 pp 301ndash312 2016

[8] Z Shen Q Wang Y Shen et al ldquoAccent extraction ofemotional speech based on modified ensemble empiricalmode decompositionrdquo in Proceedings of Instrumentation andMeasurement Technology Conference Austin TX USA May2010

[9] W Guo and P W Tse ldquoA novel signal compression methodbased on optimal ensemble empirical mode decompositionfor bearing vibration signalsrdquo Journal of Sound and Vibrationvol 332 no 2 pp 423ndash441 2013

[10] X Xue J Zhou Y Xu W Zhu and C Li ldquoAn adaptively fastensemble empirical mode decomposition method and itsapplications to rolling element bearing fault diagnosisrdquoMechanical Systems and Signal Processing vol 62-63pp 444ndash459 2015

[11] M Kedadouche M -omas and A Tahan ldquoA comparativestudy between empirical wavelet transforms and empiricalmode decomposition methods application to bearing defectdiagnosisrdquo Mechanical Systems and Signal Processing vol 81pp 88ndash107 2016

[12] W Zhai and X Sun ldquoA detailed model for investigatingvertical interaction between railway vehicle and trackrdquo Ve-hicle System Dynamics vol 23 no 1 pp 603ndash615 1994

[13] X Lei and J Wang ldquoDynamic analysis of the train and slabtrack coupling system with finite elements in a moving frameof referencerdquo Journal of Vibration and Control vol 20 no 9pp 1301ndash1317 2013

[14] W Zhai KWang and C Cai ldquoFundamentals of vehicle-trackcoupled dynamicsrdquo Vehicle System Dynamics vol 47 no 11pp 1349ndash1376 2009

[15] X Kang X B Liu H Y LI et al ldquoPSD of ballastless trackirregularities of high-speed railway (in Chinese)rdquo ScientiaSinica Technologica vol 44 no 7 pp 687ndash696 2014

[16] K Wang P Liu W Zhai C Huang Z Chen and J GaoldquoWheelrail dynamic interaction due to excitation of railcorrugation in high-speed railwayrdquo Science China Techno-logical Sciences vol 58 no 2 pp 226ndash235 2015

[17] C C Chang and C J Lin ldquoLLIBSVM a library for supportvector machinesrdquo ACM Transactions on Intelligent Systemsand Technology vol 2 no 3 pp 1ndash27 2011

[18] 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 the RoyalSociety A-Mathematical Physical and Engineering Sciencesvol 454 no 1971 pp 909ndash995 1998

Table 6 -e accuracy of depth level classification results at dif-ferent measurement noises

Noise level Nil 1 3 5Accuracy of level 1 () 9981 100 100 100Accuracy of level 2 () 9949 9975 9975 9949Accuracy of level 3 () 9949 9975 100 100Total accuracy () 9962 9985 9992 9985

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 2 Level 3

(d)

Figure 13 -e classification results of rail corrugation depth level(a) 0 noise (b) 1 noise (c) 3 noise (d) 5 noise

Shock and Vibration 11

[19] Z Wu and N E Huang ldquoEnsemble empirical mode de-composition a noise-assisted data analysis methodrdquo Ad-vances in Adaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[20] LW Jian and T-C Lim ldquoAutomated detection of diabetes bymeans of higher order spectral features obtained from heartrate signalsrdquo Journal of Medical Imaging and Health In-formatics vol 3 no 3 pp 440ndash447 2013

[21] R Yuvaraj M Murugappan N Mohamed Ibrahim et alldquoDetection of emotions in Parkinsonrsquos disease using higherorder spectral features from brainrsquos electrical activityrdquo Bio-medical Signal Processing and Control vol 14 pp 108ndash1162014

[22] CKY M Hariharan R Ngadiran A H Adom S Yaacoband K Polat ldquoHybrid BBO_PSO and higher order spectralfeatures for emotion and stress recognition from naturalspeechrdquo Applied Soft Computing vol 56 pp 217ndash232 2017

[23] L Saidi J Ben Ali F Fnaiech et al ldquoApplication of higherorder spectral features and support vector machines forbearing faults classificationrdquo ISA Transactions vol 54pp 193ndash206 2015

12 Shock and Vibration

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[M] A +[C] V +[K] X P (4)

where [M] [C] and [K] are the mass damping andstiffness matrices of the vehicle-track coupling model re-spectively A V and X are the acceleration velocityand displacement vectors of the coupling model re-spectively and P is the load vector of the coupling model

Vehicle-track coupling model is a system of nonlineardifferential equations which can be solved by the numericalintegration method

24 Rail Corrugation Model Rail corrugation is a kind ofharmonic irregularity which is expressed by a cosinefunction

ycor(t) 12

a 1minus cos2πvt

λ1113874 11138751113876 1113877 0le tle

L

v1113874 1113875 (5)

where a is the depth of rail corrugation λ is the wavelengthof rail corrugation v is the train speed and L is the length ofrail corrugation

In this paper a 130-meter track is simulated with a 10-meter rail corrugation at the middle position from 60degm to70degm of which the wavelength is from 100mm to 150mmand the depth is from 001mm to 01mm Figure 2 shows asection of rail corrugation of which the wavelength is150mm and the depth is 001mm

-erefore the integrated track irregularity of the vehicle-track system can be expressed as

y0(t) yirr(t) + ycor(t) (6)

Obviously rail corrugation will influence the vibrationresponse of wheels through wheel-rail interaction Railcorrugation is a kind of short-wavelength irregularity whilethe wavelength of random track irregularity is much longer-erefore high-frequency vibration will appear in the vi-bration response of wheels if rail corrugation exists and themain frequency of high-frequency band matches thewavelength of rail corrugation -is is the basic principle ofdetecting rail corrugation using wheel dynamic responses

25 Model Validation It is necessary to validate the modelaccuracy because wheel responses used in the followingdetection algorithm are acquired from the model Differentconditions with different wavelength and depth of railcorrugation are simulated and the results are compared withWangrsquos study [16] -e effect of rail corrugation wavelengthon wheel-rail force is shown in Figure 3 in which thedepth of rail corrugation is 01mm and the train speed is300 kmh -e results show that the influence of wavelengthon wheel-rail force is nonlinear and the maximum wheel-rail force is at 120mm wavelength Sensitive wavelength is

RailRail fastener yr (x t)

yw4yw3

yt2

βt2

yw2yw1

yt1

v

βt1

yc

βc

ys (x t)SlabCA mortarFoundation

Car body

Bogic

Wheelsetx

Figure 1 -e vehicle-track system model of high-speed railway

Table 1 Parameters for Chinese high-speed train CRH3 [13]

Parameter Value UnitMass of car body 40000 kgMass of bogie 3200 kgMass of wheelset 2400 kgPitch inertia of car body 547000 kgmiddotm2

Pitch inertia of bogie 6800 kgmiddotm2

Stiffness of primary suspension system 208times106 NmStiffness of secondary suspension system 8times105 NmDamping of primary suspension system 1times 105 NmiddotsmDamping of secondary suspension system 12times105 NmiddotsmSemilongitudinal distance between bogies 86875 mSemilongitudinal distance betweenwheelsets in bogie 125 m

Radius of wheel 046 m

Table 2 Parameters of the CRTS II slab track [13]

Parameter Value UnitElastic modulus of rail 21times 1011 Nm2

Inertia of rail 3217times10minus5 m4

Mass of rail 60 kgmStiffness of rail fastener 6times107 NmDamping of rail fastener 477times104 NmiddotsmElastic modulus of slab 39times1010 Nm2

Inertia of slab 85times10minus5 m4

Mass of slab 1275 kgmLength of single slab 65 mStiffness of CA mortar 9times108 NmDamping of CA mortar 83times104 Nmiddotsm

Shock and Vibration 3

the wavelength when rail corrugation has the greatest effecton wheel-rail interaction -e sensitive wavelength of thesimulation model is 120mm In addition the main fre-quency of wheel-rail force is related to the wavelength anddecreases with the increase of wavelength -e effect of railcorrugation depth on maximum wheel-rail force and railvibration acceleration is shown in Figure 4 -e wavelengthof rail corrugation is determined to be 100mm and the trainruns at 300 kmh With the increase of rail corrugationdepth the values of wheel-rail force and rail vibration ac-celeration gradually increase -e simulation results showgood consistence with the conclusion of Wang

3 Detection Method

A detection method for rail corrugation is proposed bywhich the wavelength can be estimated through analyzingmain frequency of wheel vibration responses based onEEMD and the depth can be classified with bispectrumfeatures and SVM -e detection flow chart is shown inFigure 5 LIBSVM [17] toolbox and particle swarm opti-mization (PSO) are used to optimize the parameters of SVM

31 Wavelength Estimation -e process of wavelength es-timation can be described as 4 steps

(1) Decompose wheel acceleration signal using EEMDmethod

(2) Eliminate the false intrinsic modal functions (IMFs)according to the correlation analysis between theIMFs and the original signal

(3) Select IMFs with high-frequency components andobtain their main frequency using FFT

(4) Estimate the wavelength of rail corrugation based onmain frequency and train speed

311 Background Huang et al [18] defined the signal sat-isfying the following two conditions as intrinsicmodal function(IMF) (1) the number of extreme points and zero-crossingpoints is equal or different by one (2) the mean value of theupper envelope formed by the local maximum points and thelower envelope formed by the local minimum points is zero

-e EMD process for signal x(t) is as follows

(1) Find all local maximum points and local minimumpoints of the signal x(t)

(2) Fit the upper and lower envelope and obtain theaverage m(t) of the upper and lower envelope andthen calculate

h(t) x(t)minusm(t) (7)

(3) If h(t) satisfies the above two conditions then h(t) isan IMF denoted as c(t) otherwise h(t) is treated asthe original signal repeat the above two steps

(4) Calculate the residual signal

r(t) x(t)minus c(t) (8)

(5) Regard r(t) as the original data and repeat the abovesteps until the residual signal is a monotonefunction

(6) -e original signal x(t) can be expressed as the sumof all IMFs and residual signal r(t)

x(t) 1113944n

i1ci(t) + r(t) (9)

Whe

el-r

ail v

ertic

al fo

rce (

kN)

500

400

300

200

100

Mai

n fre

quen

cy (H

z)

2000

1500

1000

500

050 100 150

Wavelength of corrugation (mm)

Wheel-rail vertical forceMain frequency

200

Figure 3 Effect of rail corrugation wavelength on wheel-railinteraction

Whe

el-r

ail v

ertic

al fo

rce (

kN)

400

300

200

Rail

acce

lera

tion

(g)

300

200

100

Wheel-rail vertical forceRail acceleration

005 01 015Depth of corrugation (mm)

02

Figure 4 Effect of rail corrugation depth on wheel-rail interaction

times10ndash3

y cor

(mm

)

5

0

ndash5

ndash10

ndash1558 60 62 64 66 68

Distance (m)70 72

Figure 2 Rail corrugation

4 Shock and Vibration

-e orthogonality of IMFs is used to evaluate the de-composition results -e overall index of orthogonality isdefined as

IO 1113944T

t0

1113936n+1j1 1113936

n+1k1 cj(t)ck(t)

x2(t)⎛⎝ ⎞⎠ (jne k) (10)

-e mode mixing problem occurs when the signal isanalyzed by IMF sifting method because there are multiplefrequency components in an IMF To avoid this problemWu and Huang [19] proposed EEMD based on noise-assisted data analysis method

-e EEMD process for signal x(t) is as follows

(1) Add a white noise with amplitude α to the signal x(t)(2) -e signal with noise is analyzed by EMD(3) Repeat the above steps N times and average the

results of EMD

-e difference between the input signal and the corre-sponding IMF is called the final standard deviation of errorwhich is expressed by e -e added white noise satisfies thefollowing rule [19]

e αN

radic (11)

where α is the amplitude of the added white noise and N isthe number of ensemble members

312 Selection of EEMDKey Parameters If the amplitude ofthe white noise is too small the mixing of the modes cannotbe weakened On the contrary the final standard deviationof error will be large If the number of ensemble members istoo small the influence of added white noise cannot be

eliminated in contrast the calculation cost will be increased-erefore it is very important to select suitable amplitude ofthe white noise and number of ensemble members whenEEMD is employed to analyze signals -e amplitude ofwhite noise is defined as

α k middot σ (12)

where σ is the standard deviation of the signal and k is acoefficient

-e purpose of added white noise is to improve thedistribution of extreme points of signal -e extreme pointdistribution index is defined as

SI

S21 + S22 + S23 + S24

1113969

(13)

where S1 is the standard deviation of amplitude interval ofmaximum points S2 is the standard deviation of coordinateinterval of maximum points S3 is the standard deviation ofamplitude interval of minimum points and S4 is the stan-dard deviation of coordinate interval of minimum points

-e vibration acceleration of wheel with the condition of10-meter rail corrugation is shown in Figure 6 White noisewith different amplitudes is added to the signal and theextreme point distribution index SI is calculated It can beseen from Figure 7 that the extreme point distribution indexSI tends to be stable when the amplitude of white noiseincreases to a certain value

In this paper the extreme point distribution index SI isused to determine the optimal range of white noise am-plitude -en the added white noise amplitude α and thenumber N of ensemble members are selected respectivelyaccording to the orthogonal coefficient IO and equation (11)-e parameter selection algorithm of EEMD is shown in

Vibration acceleration of

wheel

EEMD

Selection of IMFs

High frequency componentMain frequency

Wavelength of corrugation

Reconstructed signal

Bispectrum features

SVM

Depth of corrugation

Train speed

Figure 5 Rail corrugation detection method

Shock and Vibration 5

Figure 8 Firstly the optimum range of white noise am-plitude is determined by the stability value of SI In order toensure that the amplitude of white noise can improve thedistribution of signal extreme points the minimum value ofk is 0001-emaximum value of k is set to be 001 so that theoptimization range is not too small In order to select theappropriate amplitude of white noise it is necessary todetermine the number of ensemble members in advance-e numberN of ensemble members should not be too largeotherwise the calculation time is very long If N is too smallEEMD will be invalid After many trials N is initially set tobe 10 EEMD analyses of the signal are carried out in theoptimal range of white noise amplitude and the de-composition results are evaluated by the orthogonal co-efficients -e white noise amplitude corresponding to theoptimal decomposition result is the best white noise am-plitude α In this paper the error less than 1 can be ac-ceptable -e final standard deviation of error is set to be001 -en the bestN is determined according to the rules ofthe final standard deviation of error and the white noiseamplitude Meanwhile the minimum value of N is 10 toavoid the EEMD invalidation caused by too small N

-e vibration acceleration signals of wheel in Figure 6 areanalyzed by EMD and EEMD respectively Figure 9 shows thefirst four IMFs after decomposition IMF components arearranged from high frequency to low frequency so the first

IMF (IMF1) is the high-frequency component of the signal Itcan be seen from Figure 9(a) that IMF1 of EMD analysis haslow-frequency signals at other locations besides high-frequency signals caused by rail corrugation which in-dicates that there is mode mixing in EMD analysis Appar-ently the mode mixing will influence the detection results InFigure 9(b) there is only the high-frequency componentsrelated with rail corrugation and no low frequency in IMF1-erefore EEMD analysis is an effectivemethod to extract thepart of the signal caused by rail corrugation

313 Wavelength Calculation After the IMFs are obtainedby EEMD analysis the IMFs are selected to eliminate theeffects of noise and the random irregularity of the track IMFselection and signal reconstruction algorithm are shown inFigure 10 -e false components are eliminated by corre-lation analysis and the high-frequency components arereconstructed into a high-frequency signal which is causedby rail corrugation

Estimation of wavelength of rail corrugation by mainfrequency of the reconstructed signal yr is expressed as

λ v

fm (14)

where v is train speed and fm is the main frequency of thereconstructed signal which can be obtained by FFT

32 Features Extraction of Corrugation Depth -e vibrationacceleration signals of wheel under rail corrugation arenonlinear nonstationary and non-Gaussian Higher-orderspectra are useful in analyzing nonlinearity and non-Gaussianity of the signal as it provides high noise immu-nity [20] -e frequency domain representation of third-order cumulants of signals is called bispectrum which isexpressed as

B f1 f2( 1113857 E X f1( 1113857X f2( 1113857Xlowast

f1 + f2( 11138571113858 1113859 (15)

Whe

etse

t acc

eler

atio

n (g

)

2

1

0

ndash1

ndash230 40 50 60

Distance (m)70 80 90 100

Figure 6 Vibration acceleration signal of wheel

SI

80

60

40

20

00 005

Amplitude coefficient k of white noise01 015 02

Figure 7 -e extreme point distribution index SI

Calculate standard deviation σ of signal

Find the k when SI is stable defined as kmaxIf kmax lt 001 let kmax = 001

k = 0001~kmax N = 10

Find the k when the IO is minimal defined as ko

α = komiddotσ

N = (αe)2 e = 001if N lt 10 let N = 10

Figure 8 Selection algorithm of EEMD parameters

6 Shock and Vibration

2 IMF1

30 40 50 60 70 80 90 100

0

ndash2

2 IMF2

30 40 50 60 70 80 90 100

0

ndash2

05 IMF3

30 40 50 60 70 80 90 100

0

ndash05

05 IMF4

30 40 50 60 70 80 90 100

0

ndash05

(a)

05 IMF1

30 40 50 60 70 80 90 100

0

ndash05

1 IMF2

30 40 50 60 70 80 90 100

0

ndash1

2 IMF3

30 40 50 60 70 80 90 100

0

ndash2

1 IMF4

30 40 50 60 70 80 90 100

0

ndash1

(b)

Figure 9 -e first four IMFs of the signal (a) EMD (b) EEMD

Ci (i = 1 2 n) are n IMFs of the signal

CORi (i = 1 2 n) are the correlation coefficients of IMF and original signal

respectively

CORthreshold = max CORi (i = 1 2 n)10If CORi lt CORthreshold eliminate ci

fi (i = 1 2 ) are the main frequency of ci respectively

yr = sum (ci) i = 1 2

If fi gt 400Hz reserve ci

Figure 10 IMFs selection and signal reconstruction algorithm

Shock and Vibration 7

where X(f) is the Fourier transform of the signal lowast denotescomplex conjugate E[] denotes the expectation operationand the frequency f may be normalized by the Nyquistfrequency to be between 0 and 1

-e discrete bispectrum matrix B is obtained by the highorder spectral analysis of the reconstructed signal yr(t) -esize of the bispectrum matrix B is QtimesQ In order to rec-ognize the depth of rail corrugation easily bispectrumfeatures need to be extracted

-ree amplitude features of bispectrum [21]

Amp1 1

Q2 1113944Ω

|B(i j)|

Amp2 1113944Ωlog(|B(i j)|)

Amp3 1113944

Q

i1log(|B(i i)|)

(16)

-ree moment features of bispectrum [22]

Mom1 1113944

Q

i1i middot log(|B(i i)|)

Mom2 1113944

Q

i1(iminusM)

2middot log(|B(i i)|)

Mom3 1113944Ω

i2 + j21113969

middot |B(i j)|

(17)

-ree entropy features of bispectrum [23]

Ent1 minus1113944Ω

p1(i j) middot log p1(i j)( 1113857 (18)

where p1(i j) |B(i j)|1113936Ω|B(i j)|

Ent2 minus1113944Ω

p2(i j) middot log p2(i j)( 1113857 (19)

where p2(i j) |B(i j)|21113936Ω|B(i j)|2

Ent3 minus1113944Ω

p3(i j) middot log p3(i j)( 1113857 (20)

where p3(i j) |B(i j)|31113936Ω|B(i j)|3-e above nine bispectrum features have different am-

plitudes at different rail corrugation depths as shown inFigure 11 -ree entropy features of bispectrum changenonlinearly with the increase of rail corrugation depth whilethe other features of bispectrum increase with the increase ofrail corrugation depth -e bispectrum features can wellreflect the depth of rail corrugation

-e train speed and the wavelength of the rail corru-gation also affect the vibration response of the wheel-erefore the train speed v and the estimated wavelength λare also taken as the feature parameters -e extractedfeatures are constructed into a features vector and classifiedusing the SVM model

4 Detection Result

41 Data Preparation In the simulation model the trainspeeds are 250 kmh to 350 kmh the wavelengths of rail

corrugation are 100mm to 150mm and the depths of railcorrugation are 001mm to 01mm -ere are 660 signalsAfter repeated 10 simulations 6600 sets of vibration ac-celeration signals of wheel are obtained -e signal length is60m including 25m before the wheel enters the rail cor-rugation and 25m after it leaves the rail corrugation

In order to verify the antinoise ability of the detectionmethod a Gaussian noise signal is added to the wheel ac-celeration signal to simulate the measurement noise

xmeasurement xsimulation + Ep times N times var xsimulation( 1113857 (21)

where Ep is the noise level N is a standard normal dis-tribution vector with zero mean and unit standard deviationand var() denotes the standard deviation of signal

42 Wavelength Estimation of Rail Corrugation -e esti-mation results of rail corrugation wavelength are shown inFigure 12 It can be seen from Figure 12 that the wavelengthof rail corrugation can be estimated at different corrugationdepths and train speeds and the noise has no effect on theestimation algorithm

-e relative error between the estimated valueof wavelength and the true value of wavelength is definedas

error 1n

1113944

n

i1

λestimationi minus λtruei

11138681113868111386811138681113868111386811138681113868

λtruei

(22)

where n is the number of data samplesIn order to analyze the influence of corrugation depth on

wavelength estimation depths of 001mm 005mm and01mm rail corrugation are selected respectively -ewavelength estimation results with 0 1 3 and 5noise levels are shown in Table 3 -e wavelength estimationerror of rail wavelength is almost unchanged under differentdepths and the maximum value of wavelength estimationerror is not more than 02 -e wavelength estimationerrors are very small which shows that the estimation of railwavelength is completely reliable Moreover the simulationresults show that the proposed method can estimate the

001 002 003 004 005 006 007 008 009 01Depth of corrugation (mm)

0

02

04

06

08

1

Nor

mal

ized

ampl

itude

Amp1Amp2Amp3

Mom1Mom2Mom3

Ent1Ent2Ent3

Figure 11 Bispectrum features at different depths of corrugation(train speed is 250 kmh and wavelength is 100mm)

8 Shock and Vibration

wavelength well under the conditions of different depths andmeasurement noises

-e train speeds are 250 kmh 300 kmh and 350 kmhand the wavelength of rail corrugation is estimated re-spectively -e wavelength estimation results with 0 13 and 5 noise levels are shown in Table 4 -e

wavelength estimation error of rail wavelength is almostunchanged at different depths and the maximum value ofwavelength estimation error is not more than 025 -ewavelength estimation error is also very small which in-dicates that the wavelength estimation effect of rail corru-gation is very good Meanwhile the simulation results

9999

100

10001

Wav

elen

gth

(mm

)

True value ofwavelength0 noise

1 noise3 noise5 noise

002 004 006 008 01

110

11005

002 004 006 008 0111995

120

12005

002 004 006 008 011298

1299

130

002 004 006 008 011395

140

1405

002 004 006 008 01e depth of rail corrugation (mm)

1495

150

1505

002 004 006 008 01

(a)

True value ofwavelength0 noise

1 noise3 noise5 noise

9998

100

10002

Wav

elen

gth

(mm

)

250 300 350

250 300 350

1098

110

1102

250 300 3501198

120

1202

250 300 3501298

1299

130

1301

250 300 3501395

140

1405

250 300 350Train speed (kmh)

1495

150

1505

(b)

Figure 12 -e estimation results of rail corrugation wavelength (a) At different depths of rail corrugation (b) At different train speeds

Table 3 Wavelength estimation error at different depths of rail corrugation

CaseDepth 001mm Depth 005mm Depth 01mm

Nil 1 3 5 Nil 1 3 5 Nil 1 3 5100mm 00045 00045 00045 00045 00045 00045 00045 00045 00045 00045 00045 00045110mm 00900 00897 00908 00910 00902 00900 00900 00899 00899 00899 00900 00899120mm 00205 00045 00045 00063 00367 00276 00330 00275 00313 00349 00367 00330130mm 00968 00968 00980 00994 00968 00968 00968 00968 00968 00968 00968 00968140mm 01252 01124 01001 01100 01347 01286 01230 01260 01360 01357 01329 01358150mm 00376 00045 00045 00087 01453 00623 00543 00601 01563 01293 01091 01068

Shock and Vibration 9

illustrate that the proposed method can estimate thewavelength excellent under the conditions of different trainspeeds and measurement noises

43 Depth Classification of Rail Corrugation According tothe values of depth of rail corrugations three depth levels aredivided as shown in Table 5

Bispectrum features estimated wavelength and trainspeed are taken as inputs of SVM and depth level is takenas SVM output -e 80 of all simulation data is used totrain the SVM model and the remaining 20 is used fortesting

-e classification results of rail corrugation depth levelwith different levels of noise are shown in Figure 13

-e reliability of the proposedmethod is evaluated by theaccuracy of the test results -e ratio between the number ofcorrect classification results and the number of total testsamples is defined as the detection accuracy which can beexpressed as

accuray Numcorrect

Numtotaltimes 100 (23)

-e detection results of depth level with 0 1 3 and5 noise levels are shown in Table 6 -e detection accuracyis more than 99 indicating that the extracted corrugationdepth feature is effective -e proposed detection methodhas strong antinoise ability and still has high accuracy underdifferent measurement noises

5 Conclusion

Rail corrugation is a short-wavelength track irregularityexcitation and accordingly causes high-frequency vibrationresponse of wheel A rail detection algorithm based onEEMD and bispectrum features is proposed using wheelvibration acceleration Different frequency components ofwheel vibration response are extracted with EEMD of wheelvibration acceleration signal -e parameters α and N ofEEMD are optimized using the orthogonal coefficient IOand the extreme point distribution index SI which reducesthe mode mixing and calculation time of the de-composition -e main frequency is found after EEMD tocalculate the wavelength-e depth detection is regarded asa classification problem with SVM -e high-frequencysignal of rail corrugation is reconstructed through theselection of IMFs by which not only the interference oftrack random irregularity is avoided but also noise influ-ence Bispectrum features which are extracted from the

Table 4 Wavelength estimation error at different train speeds

CaseTrain speed 250 kmh Train speed 300 kmh Train speed 350 kmh

Nil 1 3 5 Nil 1 3 5 Nil 1 3 5100mm 0 0 0 0 0 0 0 0 00093 00093 00093 00093110mm 00999 00999 00999 00997 00834 00834 00834 00836 00927 00927 00927 00927120mm 01657 01198 00938 01058 0 0 0 0 00093 00093 00093 00093130mm 00999 00999 00999 00999 00999 00999 00999 01001 00907 00907 00910 00910140mm 01234 01070 01016 01107 01325 01275 01217 01265 01428 01412 01391 01386150mm 00399 0 0 00100 01421 00524 00474 00050 02124 01778 01616 01639

Table 5 Definition of depth levels of rail corrugation

Depth levels Depths of rail corrugation (mm)Level 1 a≦ 004Level 2 004lt a≦ 007Level 3 agt 007

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 3Level 2

(a)

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 2 Level 3

(b)

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 2 Level 3

(c)

Figure 13 Continued

10 Shock and Vibration

reconstructed signal combining with train speed and thecorrugation wavelength are input into SVM -e nu-merical simulation results show the proposed detectionalgorithm can accurately identify the existence of railcorrugation -e error between the estimated wavelengthand the real value is small and the accuracy of corrugationdepth level classification is high And the results alsorepresent robustness under different train speeds andmeasurement noises -e proposed rail corrugation de-tection method provides a feasible scheme for the in-service vehicle test in the future

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 they have no conflicts of interest

Acknowledgments

-is work was supported by the National Key Researchand Development Program of China under Grant no2016YFB1200401

References

[1] K H Oostermeijer ldquoReview on short pitch rail corrugationstudiesrdquo Wear vol 265 no 9-10 pp 1231ndash1237 2008

[2] Z Q Jiang D L Si W Li et al ldquoOn rail corrugation of highspeed railway (in Chinese)rdquo China Railway Science vol 35no 4 pp 9ndash14 2014

[3] B M Hopkins and S Taheri ldquoTrack health monitoring usingwaveletrdquo in Proceedings of ASME 2010 Rail TransportationDivision Fall Technical Conference pp 9ndash15 Roanoke VAUSA October 2010

[4] M Molodova Z Li and R Dollevoet ldquoAxle box accelerationmeasurement and simulation for detection of short trackdefectsrdquo Wear vol 271 no 1-2 pp 349ndash356 2011

[5] W Huang W Zhang Y Du et al ldquoDetection of rail cor-rugation based on fiber laser accelerometersrdquo MeasurementScience and Technology vol 24 no 9 article 094014 2013

[6] S Kaewunruen ldquoMonitoring of rail corrugation growth onsharp curves for track maintenance prioritisationrdquo In-ternational Journal of Acoustics and Vibration vol 23 no 1pp 35ndash43 2018

[7] P Salvador V Naranjo R Insa and P Teixeira ldquoAxleboxaccelerations their acquisition and time-frequency charac-terisation for railway track monitoring purposesrdquo Measure-ment vol 82 pp 301ndash312 2016

[8] Z Shen Q Wang Y Shen et al ldquoAccent extraction ofemotional speech based on modified ensemble empiricalmode decompositionrdquo in Proceedings of Instrumentation andMeasurement Technology Conference Austin TX USA May2010

[9] W Guo and P W Tse ldquoA novel signal compression methodbased on optimal ensemble empirical mode decompositionfor bearing vibration signalsrdquo Journal of Sound and Vibrationvol 332 no 2 pp 423ndash441 2013

[10] X Xue J Zhou Y Xu W Zhu and C Li ldquoAn adaptively fastensemble empirical mode decomposition method and itsapplications to rolling element bearing fault diagnosisrdquoMechanical Systems and Signal Processing vol 62-63pp 444ndash459 2015

[11] M Kedadouche M -omas and A Tahan ldquoA comparativestudy between empirical wavelet transforms and empiricalmode decomposition methods application to bearing defectdiagnosisrdquo Mechanical Systems and Signal Processing vol 81pp 88ndash107 2016

[12] W Zhai and X Sun ldquoA detailed model for investigatingvertical interaction between railway vehicle and trackrdquo Ve-hicle System Dynamics vol 23 no 1 pp 603ndash615 1994

[13] X Lei and J Wang ldquoDynamic analysis of the train and slabtrack coupling system with finite elements in a moving frameof referencerdquo Journal of Vibration and Control vol 20 no 9pp 1301ndash1317 2013

[14] W Zhai KWang and C Cai ldquoFundamentals of vehicle-trackcoupled dynamicsrdquo Vehicle System Dynamics vol 47 no 11pp 1349ndash1376 2009

[15] X Kang X B Liu H Y LI et al ldquoPSD of ballastless trackirregularities of high-speed railway (in Chinese)rdquo ScientiaSinica Technologica vol 44 no 7 pp 687ndash696 2014

[16] K Wang P Liu W Zhai C Huang Z Chen and J GaoldquoWheelrail dynamic interaction due to excitation of railcorrugation in high-speed railwayrdquo Science China Techno-logical Sciences vol 58 no 2 pp 226ndash235 2015

[17] C C Chang and C J Lin ldquoLLIBSVM a library for supportvector machinesrdquo ACM Transactions on Intelligent Systemsand Technology vol 2 no 3 pp 1ndash27 2011

[18] 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 the RoyalSociety A-Mathematical Physical and Engineering Sciencesvol 454 no 1971 pp 909ndash995 1998

Table 6 -e accuracy of depth level classification results at dif-ferent measurement noises

Noise level Nil 1 3 5Accuracy of level 1 () 9981 100 100 100Accuracy of level 2 () 9949 9975 9975 9949Accuracy of level 3 () 9949 9975 100 100Total accuracy () 9962 9985 9992 9985

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 2 Level 3

(d)

Figure 13 -e classification results of rail corrugation depth level(a) 0 noise (b) 1 noise (c) 3 noise (d) 5 noise

Shock and Vibration 11

[19] Z Wu and N E Huang ldquoEnsemble empirical mode de-composition a noise-assisted data analysis methodrdquo Ad-vances in Adaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[20] LW Jian and T-C Lim ldquoAutomated detection of diabetes bymeans of higher order spectral features obtained from heartrate signalsrdquo Journal of Medical Imaging and Health In-formatics vol 3 no 3 pp 440ndash447 2013

[21] R Yuvaraj M Murugappan N Mohamed Ibrahim et alldquoDetection of emotions in Parkinsonrsquos disease using higherorder spectral features from brainrsquos electrical activityrdquo Bio-medical Signal Processing and Control vol 14 pp 108ndash1162014

[22] CKY M Hariharan R Ngadiran A H Adom S Yaacoband K Polat ldquoHybrid BBO_PSO and higher order spectralfeatures for emotion and stress recognition from naturalspeechrdquo Applied Soft Computing vol 56 pp 217ndash232 2017

[23] L Saidi J Ben Ali F Fnaiech et al ldquoApplication of higherorder spectral features and support vector machines forbearing faults classificationrdquo ISA Transactions vol 54pp 193ndash206 2015

12 Shock and Vibration

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Submit your manuscripts atwwwhindawicom

the wavelength when rail corrugation has the greatest effecton wheel-rail interaction -e sensitive wavelength of thesimulation model is 120mm In addition the main fre-quency of wheel-rail force is related to the wavelength anddecreases with the increase of wavelength -e effect of railcorrugation depth on maximum wheel-rail force and railvibration acceleration is shown in Figure 4 -e wavelengthof rail corrugation is determined to be 100mm and the trainruns at 300 kmh With the increase of rail corrugationdepth the values of wheel-rail force and rail vibration ac-celeration gradually increase -e simulation results showgood consistence with the conclusion of Wang

3 Detection Method

A detection method for rail corrugation is proposed bywhich the wavelength can be estimated through analyzingmain frequency of wheel vibration responses based onEEMD and the depth can be classified with bispectrumfeatures and SVM -e detection flow chart is shown inFigure 5 LIBSVM [17] toolbox and particle swarm opti-mization (PSO) are used to optimize the parameters of SVM

31 Wavelength Estimation -e process of wavelength es-timation can be described as 4 steps

(1) Decompose wheel acceleration signal using EEMDmethod

(2) Eliminate the false intrinsic modal functions (IMFs)according to the correlation analysis between theIMFs and the original signal

(3) Select IMFs with high-frequency components andobtain their main frequency using FFT

(4) Estimate the wavelength of rail corrugation based onmain frequency and train speed

311 Background Huang et al [18] defined the signal sat-isfying the following two conditions as intrinsicmodal function(IMF) (1) the number of extreme points and zero-crossingpoints is equal or different by one (2) the mean value of theupper envelope formed by the local maximum points and thelower envelope formed by the local minimum points is zero

-e EMD process for signal x(t) is as follows

(1) Find all local maximum points and local minimumpoints of the signal x(t)

(2) Fit the upper and lower envelope and obtain theaverage m(t) of the upper and lower envelope andthen calculate

h(t) x(t)minusm(t) (7)

(3) If h(t) satisfies the above two conditions then h(t) isan IMF denoted as c(t) otherwise h(t) is treated asthe original signal repeat the above two steps

(4) Calculate the residual signal

r(t) x(t)minus c(t) (8)

(5) Regard r(t) as the original data and repeat the abovesteps until the residual signal is a monotonefunction

(6) -e original signal x(t) can be expressed as the sumof all IMFs and residual signal r(t)

x(t) 1113944n

i1ci(t) + r(t) (9)

Whe

el-r

ail v

ertic

al fo

rce (

kN)

500

400

300

200

100

Mai

n fre

quen

cy (H

z)

2000

1500

1000

500

050 100 150

Wavelength of corrugation (mm)

Wheel-rail vertical forceMain frequency

200

Figure 3 Effect of rail corrugation wavelength on wheel-railinteraction

Whe

el-r

ail v

ertic

al fo

rce (

kN)

400

300

200

Rail

acce

lera

tion

(g)

300

200

100

Wheel-rail vertical forceRail acceleration

005 01 015Depth of corrugation (mm)

02

Figure 4 Effect of rail corrugation depth on wheel-rail interaction

times10ndash3

y cor

(mm

)

5

0

ndash5

ndash10

ndash1558 60 62 64 66 68

Distance (m)70 72

Figure 2 Rail corrugation

4 Shock and Vibration

-e orthogonality of IMFs is used to evaluate the de-composition results -e overall index of orthogonality isdefined as

IO 1113944T

t0

1113936n+1j1 1113936

n+1k1 cj(t)ck(t)

x2(t)⎛⎝ ⎞⎠ (jne k) (10)

-e mode mixing problem occurs when the signal isanalyzed by IMF sifting method because there are multiplefrequency components in an IMF To avoid this problemWu and Huang [19] proposed EEMD based on noise-assisted data analysis method

-e EEMD process for signal x(t) is as follows

(1) Add a white noise with amplitude α to the signal x(t)(2) -e signal with noise is analyzed by EMD(3) Repeat the above steps N times and average the

results of EMD

-e difference between the input signal and the corre-sponding IMF is called the final standard deviation of errorwhich is expressed by e -e added white noise satisfies thefollowing rule [19]

e αN

radic (11)

where α is the amplitude of the added white noise and N isthe number of ensemble members

312 Selection of EEMDKey Parameters If the amplitude ofthe white noise is too small the mixing of the modes cannotbe weakened On the contrary the final standard deviationof error will be large If the number of ensemble members istoo small the influence of added white noise cannot be

eliminated in contrast the calculation cost will be increased-erefore it is very important to select suitable amplitude ofthe white noise and number of ensemble members whenEEMD is employed to analyze signals -e amplitude ofwhite noise is defined as

α k middot σ (12)

where σ is the standard deviation of the signal and k is acoefficient

-e purpose of added white noise is to improve thedistribution of extreme points of signal -e extreme pointdistribution index is defined as

SI

S21 + S22 + S23 + S24

1113969

(13)

where S1 is the standard deviation of amplitude interval ofmaximum points S2 is the standard deviation of coordinateinterval of maximum points S3 is the standard deviation ofamplitude interval of minimum points and S4 is the stan-dard deviation of coordinate interval of minimum points

-e vibration acceleration of wheel with the condition of10-meter rail corrugation is shown in Figure 6 White noisewith different amplitudes is added to the signal and theextreme point distribution index SI is calculated It can beseen from Figure 7 that the extreme point distribution indexSI tends to be stable when the amplitude of white noiseincreases to a certain value

In this paper the extreme point distribution index SI isused to determine the optimal range of white noise am-plitude -en the added white noise amplitude α and thenumber N of ensemble members are selected respectivelyaccording to the orthogonal coefficient IO and equation (11)-e parameter selection algorithm of EEMD is shown in

Vibration acceleration of

wheel

EEMD

Selection of IMFs

High frequency componentMain frequency

Wavelength of corrugation

Reconstructed signal

Bispectrum features

SVM

Depth of corrugation

Train speed

Figure 5 Rail corrugation detection method

Shock and Vibration 5

Figure 8 Firstly the optimum range of white noise am-plitude is determined by the stability value of SI In order toensure that the amplitude of white noise can improve thedistribution of signal extreme points the minimum value ofk is 0001-emaximum value of k is set to be 001 so that theoptimization range is not too small In order to select theappropriate amplitude of white noise it is necessary todetermine the number of ensemble members in advance-e numberN of ensemble members should not be too largeotherwise the calculation time is very long If N is too smallEEMD will be invalid After many trials N is initially set tobe 10 EEMD analyses of the signal are carried out in theoptimal range of white noise amplitude and the de-composition results are evaluated by the orthogonal co-efficients -e white noise amplitude corresponding to theoptimal decomposition result is the best white noise am-plitude α In this paper the error less than 1 can be ac-ceptable -e final standard deviation of error is set to be001 -en the bestN is determined according to the rules ofthe final standard deviation of error and the white noiseamplitude Meanwhile the minimum value of N is 10 toavoid the EEMD invalidation caused by too small N

-e vibration acceleration signals of wheel in Figure 6 areanalyzed by EMD and EEMD respectively Figure 9 shows thefirst four IMFs after decomposition IMF components arearranged from high frequency to low frequency so the first

IMF (IMF1) is the high-frequency component of the signal Itcan be seen from Figure 9(a) that IMF1 of EMD analysis haslow-frequency signals at other locations besides high-frequency signals caused by rail corrugation which in-dicates that there is mode mixing in EMD analysis Appar-ently the mode mixing will influence the detection results InFigure 9(b) there is only the high-frequency componentsrelated with rail corrugation and no low frequency in IMF1-erefore EEMD analysis is an effectivemethod to extract thepart of the signal caused by rail corrugation

313 Wavelength Calculation After the IMFs are obtainedby EEMD analysis the IMFs are selected to eliminate theeffects of noise and the random irregularity of the track IMFselection and signal reconstruction algorithm are shown inFigure 10 -e false components are eliminated by corre-lation analysis and the high-frequency components arereconstructed into a high-frequency signal which is causedby rail corrugation

Estimation of wavelength of rail corrugation by mainfrequency of the reconstructed signal yr is expressed as

λ v

fm (14)

where v is train speed and fm is the main frequency of thereconstructed signal which can be obtained by FFT

32 Features Extraction of Corrugation Depth -e vibrationacceleration signals of wheel under rail corrugation arenonlinear nonstationary and non-Gaussian Higher-orderspectra are useful in analyzing nonlinearity and non-Gaussianity of the signal as it provides high noise immu-nity [20] -e frequency domain representation of third-order cumulants of signals is called bispectrum which isexpressed as

B f1 f2( 1113857 E X f1( 1113857X f2( 1113857Xlowast

f1 + f2( 11138571113858 1113859 (15)

Whe

etse

t acc

eler

atio

n (g

)

2

1

0

ndash1

ndash230 40 50 60

Distance (m)70 80 90 100

Figure 6 Vibration acceleration signal of wheel

SI

80

60

40

20

00 005

Amplitude coefficient k of white noise01 015 02

Figure 7 -e extreme point distribution index SI

Calculate standard deviation σ of signal

Find the k when SI is stable defined as kmaxIf kmax lt 001 let kmax = 001

k = 0001~kmax N = 10

Find the k when the IO is minimal defined as ko

α = komiddotσ

N = (αe)2 e = 001if N lt 10 let N = 10

Figure 8 Selection algorithm of EEMD parameters

6 Shock and Vibration

2 IMF1

30 40 50 60 70 80 90 100

0

ndash2

2 IMF2

30 40 50 60 70 80 90 100

0

ndash2

05 IMF3

30 40 50 60 70 80 90 100

0

ndash05

05 IMF4

30 40 50 60 70 80 90 100

0

ndash05

(a)

05 IMF1

30 40 50 60 70 80 90 100

0

ndash05

1 IMF2

30 40 50 60 70 80 90 100

0

ndash1

2 IMF3

30 40 50 60 70 80 90 100

0

ndash2

1 IMF4

30 40 50 60 70 80 90 100

0

ndash1

(b)

Figure 9 -e first four IMFs of the signal (a) EMD (b) EEMD

Ci (i = 1 2 n) are n IMFs of the signal

CORi (i = 1 2 n) are the correlation coefficients of IMF and original signal

respectively

CORthreshold = max CORi (i = 1 2 n)10If CORi lt CORthreshold eliminate ci

fi (i = 1 2 ) are the main frequency of ci respectively

yr = sum (ci) i = 1 2

If fi gt 400Hz reserve ci

Figure 10 IMFs selection and signal reconstruction algorithm

Shock and Vibration 7

where X(f) is the Fourier transform of the signal lowast denotescomplex conjugate E[] denotes the expectation operationand the frequency f may be normalized by the Nyquistfrequency to be between 0 and 1

-e discrete bispectrum matrix B is obtained by the highorder spectral analysis of the reconstructed signal yr(t) -esize of the bispectrum matrix B is QtimesQ In order to rec-ognize the depth of rail corrugation easily bispectrumfeatures need to be extracted

-ree amplitude features of bispectrum [21]

Amp1 1

Q2 1113944Ω

|B(i j)|

Amp2 1113944Ωlog(|B(i j)|)

Amp3 1113944

Q

i1log(|B(i i)|)

(16)

-ree moment features of bispectrum [22]

Mom1 1113944

Q

i1i middot log(|B(i i)|)

Mom2 1113944

Q

i1(iminusM)

2middot log(|B(i i)|)

Mom3 1113944Ω

i2 + j21113969

middot |B(i j)|

(17)

-ree entropy features of bispectrum [23]

Ent1 minus1113944Ω

p1(i j) middot log p1(i j)( 1113857 (18)

where p1(i j) |B(i j)|1113936Ω|B(i j)|

Ent2 minus1113944Ω

p2(i j) middot log p2(i j)( 1113857 (19)

where p2(i j) |B(i j)|21113936Ω|B(i j)|2

Ent3 minus1113944Ω

p3(i j) middot log p3(i j)( 1113857 (20)

where p3(i j) |B(i j)|31113936Ω|B(i j)|3-e above nine bispectrum features have different am-

plitudes at different rail corrugation depths as shown inFigure 11 -ree entropy features of bispectrum changenonlinearly with the increase of rail corrugation depth whilethe other features of bispectrum increase with the increase ofrail corrugation depth -e bispectrum features can wellreflect the depth of rail corrugation

-e train speed and the wavelength of the rail corru-gation also affect the vibration response of the wheel-erefore the train speed v and the estimated wavelength λare also taken as the feature parameters -e extractedfeatures are constructed into a features vector and classifiedusing the SVM model

4 Detection Result

41 Data Preparation In the simulation model the trainspeeds are 250 kmh to 350 kmh the wavelengths of rail

corrugation are 100mm to 150mm and the depths of railcorrugation are 001mm to 01mm -ere are 660 signalsAfter repeated 10 simulations 6600 sets of vibration ac-celeration signals of wheel are obtained -e signal length is60m including 25m before the wheel enters the rail cor-rugation and 25m after it leaves the rail corrugation

In order to verify the antinoise ability of the detectionmethod a Gaussian noise signal is added to the wheel ac-celeration signal to simulate the measurement noise

xmeasurement xsimulation + Ep times N times var xsimulation( 1113857 (21)

where Ep is the noise level N is a standard normal dis-tribution vector with zero mean and unit standard deviationand var() denotes the standard deviation of signal

42 Wavelength Estimation of Rail Corrugation -e esti-mation results of rail corrugation wavelength are shown inFigure 12 It can be seen from Figure 12 that the wavelengthof rail corrugation can be estimated at different corrugationdepths and train speeds and the noise has no effect on theestimation algorithm

-e relative error between the estimated valueof wavelength and the true value of wavelength is definedas

error 1n

1113944

n

i1

λestimationi minus λtruei

11138681113868111386811138681113868111386811138681113868

λtruei

(22)

where n is the number of data samplesIn order to analyze the influence of corrugation depth on

wavelength estimation depths of 001mm 005mm and01mm rail corrugation are selected respectively -ewavelength estimation results with 0 1 3 and 5noise levels are shown in Table 3 -e wavelength estimationerror of rail wavelength is almost unchanged under differentdepths and the maximum value of wavelength estimationerror is not more than 02 -e wavelength estimationerrors are very small which shows that the estimation of railwavelength is completely reliable Moreover the simulationresults show that the proposed method can estimate the

001 002 003 004 005 006 007 008 009 01Depth of corrugation (mm)

0

02

04

06

08

1

Nor

mal

ized

ampl

itude

Amp1Amp2Amp3

Mom1Mom2Mom3

Ent1Ent2Ent3

Figure 11 Bispectrum features at different depths of corrugation(train speed is 250 kmh and wavelength is 100mm)

8 Shock and Vibration

wavelength well under the conditions of different depths andmeasurement noises

-e train speeds are 250 kmh 300 kmh and 350 kmhand the wavelength of rail corrugation is estimated re-spectively -e wavelength estimation results with 0 13 and 5 noise levels are shown in Table 4 -e

wavelength estimation error of rail wavelength is almostunchanged at different depths and the maximum value ofwavelength estimation error is not more than 025 -ewavelength estimation error is also very small which in-dicates that the wavelength estimation effect of rail corru-gation is very good Meanwhile the simulation results

9999

100

10001

Wav

elen

gth

(mm

)

True value ofwavelength0 noise

1 noise3 noise5 noise

002 004 006 008 01

110

11005

002 004 006 008 0111995

120

12005

002 004 006 008 011298

1299

130

002 004 006 008 011395

140

1405

002 004 006 008 01e depth of rail corrugation (mm)

1495

150

1505

002 004 006 008 01

(a)

True value ofwavelength0 noise

1 noise3 noise5 noise

9998

100

10002

Wav

elen

gth

(mm

)

250 300 350

250 300 350

1098

110

1102

250 300 3501198

120

1202

250 300 3501298

1299

130

1301

250 300 3501395

140

1405

250 300 350Train speed (kmh)

1495

150

1505

(b)

Figure 12 -e estimation results of rail corrugation wavelength (a) At different depths of rail corrugation (b) At different train speeds

Table 3 Wavelength estimation error at different depths of rail corrugation

CaseDepth 001mm Depth 005mm Depth 01mm

Nil 1 3 5 Nil 1 3 5 Nil 1 3 5100mm 00045 00045 00045 00045 00045 00045 00045 00045 00045 00045 00045 00045110mm 00900 00897 00908 00910 00902 00900 00900 00899 00899 00899 00900 00899120mm 00205 00045 00045 00063 00367 00276 00330 00275 00313 00349 00367 00330130mm 00968 00968 00980 00994 00968 00968 00968 00968 00968 00968 00968 00968140mm 01252 01124 01001 01100 01347 01286 01230 01260 01360 01357 01329 01358150mm 00376 00045 00045 00087 01453 00623 00543 00601 01563 01293 01091 01068

Shock and Vibration 9

illustrate that the proposed method can estimate thewavelength excellent under the conditions of different trainspeeds and measurement noises

43 Depth Classification of Rail Corrugation According tothe values of depth of rail corrugations three depth levels aredivided as shown in Table 5

Bispectrum features estimated wavelength and trainspeed are taken as inputs of SVM and depth level is takenas SVM output -e 80 of all simulation data is used totrain the SVM model and the remaining 20 is used fortesting

-e classification results of rail corrugation depth levelwith different levels of noise are shown in Figure 13

-e reliability of the proposedmethod is evaluated by theaccuracy of the test results -e ratio between the number ofcorrect classification results and the number of total testsamples is defined as the detection accuracy which can beexpressed as

accuray Numcorrect

Numtotaltimes 100 (23)

-e detection results of depth level with 0 1 3 and5 noise levels are shown in Table 6 -e detection accuracyis more than 99 indicating that the extracted corrugationdepth feature is effective -e proposed detection methodhas strong antinoise ability and still has high accuracy underdifferent measurement noises

5 Conclusion

Rail corrugation is a short-wavelength track irregularityexcitation and accordingly causes high-frequency vibrationresponse of wheel A rail detection algorithm based onEEMD and bispectrum features is proposed using wheelvibration acceleration Different frequency components ofwheel vibration response are extracted with EEMD of wheelvibration acceleration signal -e parameters α and N ofEEMD are optimized using the orthogonal coefficient IOand the extreme point distribution index SI which reducesthe mode mixing and calculation time of the de-composition -e main frequency is found after EEMD tocalculate the wavelength-e depth detection is regarded asa classification problem with SVM -e high-frequencysignal of rail corrugation is reconstructed through theselection of IMFs by which not only the interference oftrack random irregularity is avoided but also noise influ-ence Bispectrum features which are extracted from the

Table 4 Wavelength estimation error at different train speeds

CaseTrain speed 250 kmh Train speed 300 kmh Train speed 350 kmh

Nil 1 3 5 Nil 1 3 5 Nil 1 3 5100mm 0 0 0 0 0 0 0 0 00093 00093 00093 00093110mm 00999 00999 00999 00997 00834 00834 00834 00836 00927 00927 00927 00927120mm 01657 01198 00938 01058 0 0 0 0 00093 00093 00093 00093130mm 00999 00999 00999 00999 00999 00999 00999 01001 00907 00907 00910 00910140mm 01234 01070 01016 01107 01325 01275 01217 01265 01428 01412 01391 01386150mm 00399 0 0 00100 01421 00524 00474 00050 02124 01778 01616 01639

Table 5 Definition of depth levels of rail corrugation

Depth levels Depths of rail corrugation (mm)Level 1 a≦ 004Level 2 004lt a≦ 007Level 3 agt 007

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 3Level 2

(a)

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 2 Level 3

(b)

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 2 Level 3

(c)

Figure 13 Continued

10 Shock and Vibration

reconstructed signal combining with train speed and thecorrugation wavelength are input into SVM -e nu-merical simulation results show the proposed detectionalgorithm can accurately identify the existence of railcorrugation -e error between the estimated wavelengthand the real value is small and the accuracy of corrugationdepth level classification is high And the results alsorepresent robustness under different train speeds andmeasurement noises -e proposed rail corrugation de-tection method provides a feasible scheme for the in-service vehicle test in the future

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 they have no conflicts of interest

Acknowledgments

-is work was supported by the National Key Researchand Development Program of China under Grant no2016YFB1200401

References

[1] K H Oostermeijer ldquoReview on short pitch rail corrugationstudiesrdquo Wear vol 265 no 9-10 pp 1231ndash1237 2008

[2] Z Q Jiang D L Si W Li et al ldquoOn rail corrugation of highspeed railway (in Chinese)rdquo China Railway Science vol 35no 4 pp 9ndash14 2014

[3] B M Hopkins and S Taheri ldquoTrack health monitoring usingwaveletrdquo in Proceedings of ASME 2010 Rail TransportationDivision Fall Technical Conference pp 9ndash15 Roanoke VAUSA October 2010

[4] M Molodova Z Li and R Dollevoet ldquoAxle box accelerationmeasurement and simulation for detection of short trackdefectsrdquo Wear vol 271 no 1-2 pp 349ndash356 2011

[5] W Huang W Zhang Y Du et al ldquoDetection of rail cor-rugation based on fiber laser accelerometersrdquo MeasurementScience and Technology vol 24 no 9 article 094014 2013

[6] S Kaewunruen ldquoMonitoring of rail corrugation growth onsharp curves for track maintenance prioritisationrdquo In-ternational Journal of Acoustics and Vibration vol 23 no 1pp 35ndash43 2018

[7] P Salvador V Naranjo R Insa and P Teixeira ldquoAxleboxaccelerations their acquisition and time-frequency charac-terisation for railway track monitoring purposesrdquo Measure-ment vol 82 pp 301ndash312 2016

[8] Z Shen Q Wang Y Shen et al ldquoAccent extraction ofemotional speech based on modified ensemble empiricalmode decompositionrdquo in Proceedings of Instrumentation andMeasurement Technology Conference Austin TX USA May2010

[9] W Guo and P W Tse ldquoA novel signal compression methodbased on optimal ensemble empirical mode decompositionfor bearing vibration signalsrdquo Journal of Sound and Vibrationvol 332 no 2 pp 423ndash441 2013

[10] X Xue J Zhou Y Xu W Zhu and C Li ldquoAn adaptively fastensemble empirical mode decomposition method and itsapplications to rolling element bearing fault diagnosisrdquoMechanical Systems and Signal Processing vol 62-63pp 444ndash459 2015

[11] M Kedadouche M -omas and A Tahan ldquoA comparativestudy between empirical wavelet transforms and empiricalmode decomposition methods application to bearing defectdiagnosisrdquo Mechanical Systems and Signal Processing vol 81pp 88ndash107 2016

[12] W Zhai and X Sun ldquoA detailed model for investigatingvertical interaction between railway vehicle and trackrdquo Ve-hicle System Dynamics vol 23 no 1 pp 603ndash615 1994

[13] X Lei and J Wang ldquoDynamic analysis of the train and slabtrack coupling system with finite elements in a moving frameof referencerdquo Journal of Vibration and Control vol 20 no 9pp 1301ndash1317 2013

[14] W Zhai KWang and C Cai ldquoFundamentals of vehicle-trackcoupled dynamicsrdquo Vehicle System Dynamics vol 47 no 11pp 1349ndash1376 2009

[15] X Kang X B Liu H Y LI et al ldquoPSD of ballastless trackirregularities of high-speed railway (in Chinese)rdquo ScientiaSinica Technologica vol 44 no 7 pp 687ndash696 2014

[16] K Wang P Liu W Zhai C Huang Z Chen and J GaoldquoWheelrail dynamic interaction due to excitation of railcorrugation in high-speed railwayrdquo Science China Techno-logical Sciences vol 58 no 2 pp 226ndash235 2015

[17] C C Chang and C J Lin ldquoLLIBSVM a library for supportvector machinesrdquo ACM Transactions on Intelligent Systemsand Technology vol 2 no 3 pp 1ndash27 2011

[18] 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 the RoyalSociety A-Mathematical Physical and Engineering Sciencesvol 454 no 1971 pp 909ndash995 1998

Table 6 -e accuracy of depth level classification results at dif-ferent measurement noises

Noise level Nil 1 3 5Accuracy of level 1 () 9981 100 100 100Accuracy of level 2 () 9949 9975 9975 9949Accuracy of level 3 () 9949 9975 100 100Total accuracy () 9962 9985 9992 9985

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 2 Level 3

(d)

Figure 13 -e classification results of rail corrugation depth level(a) 0 noise (b) 1 noise (c) 3 noise (d) 5 noise

Shock and Vibration 11

[19] Z Wu and N E Huang ldquoEnsemble empirical mode de-composition a noise-assisted data analysis methodrdquo Ad-vances in Adaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[20] LW Jian and T-C Lim ldquoAutomated detection of diabetes bymeans of higher order spectral features obtained from heartrate signalsrdquo Journal of Medical Imaging and Health In-formatics vol 3 no 3 pp 440ndash447 2013

[21] R Yuvaraj M Murugappan N Mohamed Ibrahim et alldquoDetection of emotions in Parkinsonrsquos disease using higherorder spectral features from brainrsquos electrical activityrdquo Bio-medical Signal Processing and Control vol 14 pp 108ndash1162014

[22] CKY M Hariharan R Ngadiran A H Adom S Yaacoband K Polat ldquoHybrid BBO_PSO and higher order spectralfeatures for emotion and stress recognition from naturalspeechrdquo Applied Soft Computing vol 56 pp 217ndash232 2017

[23] L Saidi J Ben Ali F Fnaiech et al ldquoApplication of higherorder spectral features and support vector machines forbearing faults classificationrdquo ISA Transactions vol 54pp 193ndash206 2015

12 Shock and Vibration

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Submit your manuscripts atwwwhindawicom

-e orthogonality of IMFs is used to evaluate the de-composition results -e overall index of orthogonality isdefined as

IO 1113944T

t0

1113936n+1j1 1113936

n+1k1 cj(t)ck(t)

x2(t)⎛⎝ ⎞⎠ (jne k) (10)

-e mode mixing problem occurs when the signal isanalyzed by IMF sifting method because there are multiplefrequency components in an IMF To avoid this problemWu and Huang [19] proposed EEMD based on noise-assisted data analysis method

-e EEMD process for signal x(t) is as follows

(1) Add a white noise with amplitude α to the signal x(t)(2) -e signal with noise is analyzed by EMD(3) Repeat the above steps N times and average the

results of EMD

-e difference between the input signal and the corre-sponding IMF is called the final standard deviation of errorwhich is expressed by e -e added white noise satisfies thefollowing rule [19]

e αN

radic (11)

where α is the amplitude of the added white noise and N isthe number of ensemble members

312 Selection of EEMDKey Parameters If the amplitude ofthe white noise is too small the mixing of the modes cannotbe weakened On the contrary the final standard deviationof error will be large If the number of ensemble members istoo small the influence of added white noise cannot be

eliminated in contrast the calculation cost will be increased-erefore it is very important to select suitable amplitude ofthe white noise and number of ensemble members whenEEMD is employed to analyze signals -e amplitude ofwhite noise is defined as

α k middot σ (12)

where σ is the standard deviation of the signal and k is acoefficient

-e purpose of added white noise is to improve thedistribution of extreme points of signal -e extreme pointdistribution index is defined as

SI

S21 + S22 + S23 + S24

1113969

(13)

where S1 is the standard deviation of amplitude interval ofmaximum points S2 is the standard deviation of coordinateinterval of maximum points S3 is the standard deviation ofamplitude interval of minimum points and S4 is the stan-dard deviation of coordinate interval of minimum points

-e vibration acceleration of wheel with the condition of10-meter rail corrugation is shown in Figure 6 White noisewith different amplitudes is added to the signal and theextreme point distribution index SI is calculated It can beseen from Figure 7 that the extreme point distribution indexSI tends to be stable when the amplitude of white noiseincreases to a certain value

In this paper the extreme point distribution index SI isused to determine the optimal range of white noise am-plitude -en the added white noise amplitude α and thenumber N of ensemble members are selected respectivelyaccording to the orthogonal coefficient IO and equation (11)-e parameter selection algorithm of EEMD is shown in

Vibration acceleration of

wheel

EEMD

Selection of IMFs

High frequency componentMain frequency

Wavelength of corrugation

Reconstructed signal

Bispectrum features

SVM

Depth of corrugation

Train speed

Figure 5 Rail corrugation detection method

Shock and Vibration 5

Figure 8 Firstly the optimum range of white noise am-plitude is determined by the stability value of SI In order toensure that the amplitude of white noise can improve thedistribution of signal extreme points the minimum value ofk is 0001-emaximum value of k is set to be 001 so that theoptimization range is not too small In order to select theappropriate amplitude of white noise it is necessary todetermine the number of ensemble members in advance-e numberN of ensemble members should not be too largeotherwise the calculation time is very long If N is too smallEEMD will be invalid After many trials N is initially set tobe 10 EEMD analyses of the signal are carried out in theoptimal range of white noise amplitude and the de-composition results are evaluated by the orthogonal co-efficients -e white noise amplitude corresponding to theoptimal decomposition result is the best white noise am-plitude α In this paper the error less than 1 can be ac-ceptable -e final standard deviation of error is set to be001 -en the bestN is determined according to the rules ofthe final standard deviation of error and the white noiseamplitude Meanwhile the minimum value of N is 10 toavoid the EEMD invalidation caused by too small N

-e vibration acceleration signals of wheel in Figure 6 areanalyzed by EMD and EEMD respectively Figure 9 shows thefirst four IMFs after decomposition IMF components arearranged from high frequency to low frequency so the first

IMF (IMF1) is the high-frequency component of the signal Itcan be seen from Figure 9(a) that IMF1 of EMD analysis haslow-frequency signals at other locations besides high-frequency signals caused by rail corrugation which in-dicates that there is mode mixing in EMD analysis Appar-ently the mode mixing will influence the detection results InFigure 9(b) there is only the high-frequency componentsrelated with rail corrugation and no low frequency in IMF1-erefore EEMD analysis is an effectivemethod to extract thepart of the signal caused by rail corrugation

313 Wavelength Calculation After the IMFs are obtainedby EEMD analysis the IMFs are selected to eliminate theeffects of noise and the random irregularity of the track IMFselection and signal reconstruction algorithm are shown inFigure 10 -e false components are eliminated by corre-lation analysis and the high-frequency components arereconstructed into a high-frequency signal which is causedby rail corrugation

Estimation of wavelength of rail corrugation by mainfrequency of the reconstructed signal yr is expressed as

λ v

fm (14)

where v is train speed and fm is the main frequency of thereconstructed signal which can be obtained by FFT

32 Features Extraction of Corrugation Depth -e vibrationacceleration signals of wheel under rail corrugation arenonlinear nonstationary and non-Gaussian Higher-orderspectra are useful in analyzing nonlinearity and non-Gaussianity of the signal as it provides high noise immu-nity [20] -e frequency domain representation of third-order cumulants of signals is called bispectrum which isexpressed as

B f1 f2( 1113857 E X f1( 1113857X f2( 1113857Xlowast

f1 + f2( 11138571113858 1113859 (15)

Whe

etse

t acc

eler

atio

n (g

)

2

1

0

ndash1

ndash230 40 50 60

Distance (m)70 80 90 100

Figure 6 Vibration acceleration signal of wheel

SI

80

60

40

20

00 005

Amplitude coefficient k of white noise01 015 02

Figure 7 -e extreme point distribution index SI

Calculate standard deviation σ of signal

Find the k when SI is stable defined as kmaxIf kmax lt 001 let kmax = 001

k = 0001~kmax N = 10

Find the k when the IO is minimal defined as ko

α = komiddotσ

N = (αe)2 e = 001if N lt 10 let N = 10

Figure 8 Selection algorithm of EEMD parameters

6 Shock and Vibration

2 IMF1

30 40 50 60 70 80 90 100

0

ndash2

2 IMF2

30 40 50 60 70 80 90 100

0

ndash2

05 IMF3

30 40 50 60 70 80 90 100

0

ndash05

05 IMF4

30 40 50 60 70 80 90 100

0

ndash05

(a)

05 IMF1

30 40 50 60 70 80 90 100

0

ndash05

1 IMF2

30 40 50 60 70 80 90 100

0

ndash1

2 IMF3

30 40 50 60 70 80 90 100

0

ndash2

1 IMF4

30 40 50 60 70 80 90 100

0

ndash1

(b)

Figure 9 -e first four IMFs of the signal (a) EMD (b) EEMD

Ci (i = 1 2 n) are n IMFs of the signal

CORi (i = 1 2 n) are the correlation coefficients of IMF and original signal

respectively

CORthreshold = max CORi (i = 1 2 n)10If CORi lt CORthreshold eliminate ci

fi (i = 1 2 ) are the main frequency of ci respectively

yr = sum (ci) i = 1 2

If fi gt 400Hz reserve ci

Figure 10 IMFs selection and signal reconstruction algorithm

Shock and Vibration 7

where X(f) is the Fourier transform of the signal lowast denotescomplex conjugate E[] denotes the expectation operationand the frequency f may be normalized by the Nyquistfrequency to be between 0 and 1

-e discrete bispectrum matrix B is obtained by the highorder spectral analysis of the reconstructed signal yr(t) -esize of the bispectrum matrix B is QtimesQ In order to rec-ognize the depth of rail corrugation easily bispectrumfeatures need to be extracted

-ree amplitude features of bispectrum [21]

Amp1 1

Q2 1113944Ω

|B(i j)|

Amp2 1113944Ωlog(|B(i j)|)

Amp3 1113944

Q

i1log(|B(i i)|)

(16)

-ree moment features of bispectrum [22]

Mom1 1113944

Q

i1i middot log(|B(i i)|)

Mom2 1113944

Q

i1(iminusM)

2middot log(|B(i i)|)

Mom3 1113944Ω

i2 + j21113969

middot |B(i j)|

(17)

-ree entropy features of bispectrum [23]

Ent1 minus1113944Ω

p1(i j) middot log p1(i j)( 1113857 (18)

where p1(i j) |B(i j)|1113936Ω|B(i j)|

Ent2 minus1113944Ω

p2(i j) middot log p2(i j)( 1113857 (19)

where p2(i j) |B(i j)|21113936Ω|B(i j)|2

Ent3 minus1113944Ω

p3(i j) middot log p3(i j)( 1113857 (20)

where p3(i j) |B(i j)|31113936Ω|B(i j)|3-e above nine bispectrum features have different am-

plitudes at different rail corrugation depths as shown inFigure 11 -ree entropy features of bispectrum changenonlinearly with the increase of rail corrugation depth whilethe other features of bispectrum increase with the increase ofrail corrugation depth -e bispectrum features can wellreflect the depth of rail corrugation

-e train speed and the wavelength of the rail corru-gation also affect the vibration response of the wheel-erefore the train speed v and the estimated wavelength λare also taken as the feature parameters -e extractedfeatures are constructed into a features vector and classifiedusing the SVM model

4 Detection Result

41 Data Preparation In the simulation model the trainspeeds are 250 kmh to 350 kmh the wavelengths of rail

corrugation are 100mm to 150mm and the depths of railcorrugation are 001mm to 01mm -ere are 660 signalsAfter repeated 10 simulations 6600 sets of vibration ac-celeration signals of wheel are obtained -e signal length is60m including 25m before the wheel enters the rail cor-rugation and 25m after it leaves the rail corrugation

In order to verify the antinoise ability of the detectionmethod a Gaussian noise signal is added to the wheel ac-celeration signal to simulate the measurement noise

xmeasurement xsimulation + Ep times N times var xsimulation( 1113857 (21)

where Ep is the noise level N is a standard normal dis-tribution vector with zero mean and unit standard deviationand var() denotes the standard deviation of signal

42 Wavelength Estimation of Rail Corrugation -e esti-mation results of rail corrugation wavelength are shown inFigure 12 It can be seen from Figure 12 that the wavelengthof rail corrugation can be estimated at different corrugationdepths and train speeds and the noise has no effect on theestimation algorithm

-e relative error between the estimated valueof wavelength and the true value of wavelength is definedas

error 1n

1113944

n

i1

λestimationi minus λtruei

11138681113868111386811138681113868111386811138681113868

λtruei

(22)

where n is the number of data samplesIn order to analyze the influence of corrugation depth on

wavelength estimation depths of 001mm 005mm and01mm rail corrugation are selected respectively -ewavelength estimation results with 0 1 3 and 5noise levels are shown in Table 3 -e wavelength estimationerror of rail wavelength is almost unchanged under differentdepths and the maximum value of wavelength estimationerror is not more than 02 -e wavelength estimationerrors are very small which shows that the estimation of railwavelength is completely reliable Moreover the simulationresults show that the proposed method can estimate the

001 002 003 004 005 006 007 008 009 01Depth of corrugation (mm)

0

02

04

06

08

1

Nor

mal

ized

ampl

itude

Amp1Amp2Amp3

Mom1Mom2Mom3

Ent1Ent2Ent3

Figure 11 Bispectrum features at different depths of corrugation(train speed is 250 kmh and wavelength is 100mm)

8 Shock and Vibration

wavelength well under the conditions of different depths andmeasurement noises

-e train speeds are 250 kmh 300 kmh and 350 kmhand the wavelength of rail corrugation is estimated re-spectively -e wavelength estimation results with 0 13 and 5 noise levels are shown in Table 4 -e

wavelength estimation error of rail wavelength is almostunchanged at different depths and the maximum value ofwavelength estimation error is not more than 025 -ewavelength estimation error is also very small which in-dicates that the wavelength estimation effect of rail corru-gation is very good Meanwhile the simulation results

9999

100

10001

Wav

elen

gth

(mm

)

True value ofwavelength0 noise

1 noise3 noise5 noise

002 004 006 008 01

110

11005

002 004 006 008 0111995

120

12005

002 004 006 008 011298

1299

130

002 004 006 008 011395

140

1405

002 004 006 008 01e depth of rail corrugation (mm)

1495

150

1505

002 004 006 008 01

(a)

True value ofwavelength0 noise

1 noise3 noise5 noise

9998

100

10002

Wav

elen

gth

(mm

)

250 300 350

250 300 350

1098

110

1102

250 300 3501198

120

1202

250 300 3501298

1299

130

1301

250 300 3501395

140

1405

250 300 350Train speed (kmh)

1495

150

1505

(b)

Figure 12 -e estimation results of rail corrugation wavelength (a) At different depths of rail corrugation (b) At different train speeds

Table 3 Wavelength estimation error at different depths of rail corrugation

CaseDepth 001mm Depth 005mm Depth 01mm

Nil 1 3 5 Nil 1 3 5 Nil 1 3 5100mm 00045 00045 00045 00045 00045 00045 00045 00045 00045 00045 00045 00045110mm 00900 00897 00908 00910 00902 00900 00900 00899 00899 00899 00900 00899120mm 00205 00045 00045 00063 00367 00276 00330 00275 00313 00349 00367 00330130mm 00968 00968 00980 00994 00968 00968 00968 00968 00968 00968 00968 00968140mm 01252 01124 01001 01100 01347 01286 01230 01260 01360 01357 01329 01358150mm 00376 00045 00045 00087 01453 00623 00543 00601 01563 01293 01091 01068

Shock and Vibration 9

illustrate that the proposed method can estimate thewavelength excellent under the conditions of different trainspeeds and measurement noises

43 Depth Classification of Rail Corrugation According tothe values of depth of rail corrugations three depth levels aredivided as shown in Table 5

Bispectrum features estimated wavelength and trainspeed are taken as inputs of SVM and depth level is takenas SVM output -e 80 of all simulation data is used totrain the SVM model and the remaining 20 is used fortesting

-e classification results of rail corrugation depth levelwith different levels of noise are shown in Figure 13

-e reliability of the proposedmethod is evaluated by theaccuracy of the test results -e ratio between the number ofcorrect classification results and the number of total testsamples is defined as the detection accuracy which can beexpressed as

accuray Numcorrect

Numtotaltimes 100 (23)

-e detection results of depth level with 0 1 3 and5 noise levels are shown in Table 6 -e detection accuracyis more than 99 indicating that the extracted corrugationdepth feature is effective -e proposed detection methodhas strong antinoise ability and still has high accuracy underdifferent measurement noises

5 Conclusion

Rail corrugation is a short-wavelength track irregularityexcitation and accordingly causes high-frequency vibrationresponse of wheel A rail detection algorithm based onEEMD and bispectrum features is proposed using wheelvibration acceleration Different frequency components ofwheel vibration response are extracted with EEMD of wheelvibration acceleration signal -e parameters α and N ofEEMD are optimized using the orthogonal coefficient IOand the extreme point distribution index SI which reducesthe mode mixing and calculation time of the de-composition -e main frequency is found after EEMD tocalculate the wavelength-e depth detection is regarded asa classification problem with SVM -e high-frequencysignal of rail corrugation is reconstructed through theselection of IMFs by which not only the interference oftrack random irregularity is avoided but also noise influ-ence Bispectrum features which are extracted from the

Table 4 Wavelength estimation error at different train speeds

CaseTrain speed 250 kmh Train speed 300 kmh Train speed 350 kmh

Nil 1 3 5 Nil 1 3 5 Nil 1 3 5100mm 0 0 0 0 0 0 0 0 00093 00093 00093 00093110mm 00999 00999 00999 00997 00834 00834 00834 00836 00927 00927 00927 00927120mm 01657 01198 00938 01058 0 0 0 0 00093 00093 00093 00093130mm 00999 00999 00999 00999 00999 00999 00999 01001 00907 00907 00910 00910140mm 01234 01070 01016 01107 01325 01275 01217 01265 01428 01412 01391 01386150mm 00399 0 0 00100 01421 00524 00474 00050 02124 01778 01616 01639

Table 5 Definition of depth levels of rail corrugation

Depth levels Depths of rail corrugation (mm)Level 1 a≦ 004Level 2 004lt a≦ 007Level 3 agt 007

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 3Level 2

(a)

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 2 Level 3

(b)

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 2 Level 3

(c)

Figure 13 Continued

10 Shock and Vibration

reconstructed signal combining with train speed and thecorrugation wavelength are input into SVM -e nu-merical simulation results show the proposed detectionalgorithm can accurately identify the existence of railcorrugation -e error between the estimated wavelengthand the real value is small and the accuracy of corrugationdepth level classification is high And the results alsorepresent robustness under different train speeds andmeasurement noises -e proposed rail corrugation de-tection method provides a feasible scheme for the in-service vehicle test in the future

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 they have no conflicts of interest

Acknowledgments

-is work was supported by the National Key Researchand Development Program of China under Grant no2016YFB1200401

References

[1] K H Oostermeijer ldquoReview on short pitch rail corrugationstudiesrdquo Wear vol 265 no 9-10 pp 1231ndash1237 2008

[2] Z Q Jiang D L Si W Li et al ldquoOn rail corrugation of highspeed railway (in Chinese)rdquo China Railway Science vol 35no 4 pp 9ndash14 2014

[3] B M Hopkins and S Taheri ldquoTrack health monitoring usingwaveletrdquo in Proceedings of ASME 2010 Rail TransportationDivision Fall Technical Conference pp 9ndash15 Roanoke VAUSA October 2010

[4] M Molodova Z Li and R Dollevoet ldquoAxle box accelerationmeasurement and simulation for detection of short trackdefectsrdquo Wear vol 271 no 1-2 pp 349ndash356 2011

[5] W Huang W Zhang Y Du et al ldquoDetection of rail cor-rugation based on fiber laser accelerometersrdquo MeasurementScience and Technology vol 24 no 9 article 094014 2013

[6] S Kaewunruen ldquoMonitoring of rail corrugation growth onsharp curves for track maintenance prioritisationrdquo In-ternational Journal of Acoustics and Vibration vol 23 no 1pp 35ndash43 2018

[7] P Salvador V Naranjo R Insa and P Teixeira ldquoAxleboxaccelerations their acquisition and time-frequency charac-terisation for railway track monitoring purposesrdquo Measure-ment vol 82 pp 301ndash312 2016

[8] Z Shen Q Wang Y Shen et al ldquoAccent extraction ofemotional speech based on modified ensemble empiricalmode decompositionrdquo in Proceedings of Instrumentation andMeasurement Technology Conference Austin TX USA May2010

[9] W Guo and P W Tse ldquoA novel signal compression methodbased on optimal ensemble empirical mode decompositionfor bearing vibration signalsrdquo Journal of Sound and Vibrationvol 332 no 2 pp 423ndash441 2013

[10] X Xue J Zhou Y Xu W Zhu and C Li ldquoAn adaptively fastensemble empirical mode decomposition method and itsapplications to rolling element bearing fault diagnosisrdquoMechanical Systems and Signal Processing vol 62-63pp 444ndash459 2015

[11] M Kedadouche M -omas and A Tahan ldquoA comparativestudy between empirical wavelet transforms and empiricalmode decomposition methods application to bearing defectdiagnosisrdquo Mechanical Systems and Signal Processing vol 81pp 88ndash107 2016

[12] W Zhai and X Sun ldquoA detailed model for investigatingvertical interaction between railway vehicle and trackrdquo Ve-hicle System Dynamics vol 23 no 1 pp 603ndash615 1994

[13] X Lei and J Wang ldquoDynamic analysis of the train and slabtrack coupling system with finite elements in a moving frameof referencerdquo Journal of Vibration and Control vol 20 no 9pp 1301ndash1317 2013

[14] W Zhai KWang and C Cai ldquoFundamentals of vehicle-trackcoupled dynamicsrdquo Vehicle System Dynamics vol 47 no 11pp 1349ndash1376 2009

[15] X Kang X B Liu H Y LI et al ldquoPSD of ballastless trackirregularities of high-speed railway (in Chinese)rdquo ScientiaSinica Technologica vol 44 no 7 pp 687ndash696 2014

[16] K Wang P Liu W Zhai C Huang Z Chen and J GaoldquoWheelrail dynamic interaction due to excitation of railcorrugation in high-speed railwayrdquo Science China Techno-logical Sciences vol 58 no 2 pp 226ndash235 2015

[17] C C Chang and C J Lin ldquoLLIBSVM a library for supportvector machinesrdquo ACM Transactions on Intelligent Systemsand Technology vol 2 no 3 pp 1ndash27 2011

[18] 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 the RoyalSociety A-Mathematical Physical and Engineering Sciencesvol 454 no 1971 pp 909ndash995 1998

Table 6 -e accuracy of depth level classification results at dif-ferent measurement noises

Noise level Nil 1 3 5Accuracy of level 1 () 9981 100 100 100Accuracy of level 2 () 9949 9975 9975 9949Accuracy of level 3 () 9949 9975 100 100Total accuracy () 9962 9985 9992 9985

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 2 Level 3

(d)

Figure 13 -e classification results of rail corrugation depth level(a) 0 noise (b) 1 noise (c) 3 noise (d) 5 noise

Shock and Vibration 11

[19] Z Wu and N E Huang ldquoEnsemble empirical mode de-composition a noise-assisted data analysis methodrdquo Ad-vances in Adaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[20] LW Jian and T-C Lim ldquoAutomated detection of diabetes bymeans of higher order spectral features obtained from heartrate signalsrdquo Journal of Medical Imaging and Health In-formatics vol 3 no 3 pp 440ndash447 2013

[21] R Yuvaraj M Murugappan N Mohamed Ibrahim et alldquoDetection of emotions in Parkinsonrsquos disease using higherorder spectral features from brainrsquos electrical activityrdquo Bio-medical Signal Processing and Control vol 14 pp 108ndash1162014

[22] CKY M Hariharan R Ngadiran A H Adom S Yaacoband K Polat ldquoHybrid BBO_PSO and higher order spectralfeatures for emotion and stress recognition from naturalspeechrdquo Applied Soft Computing vol 56 pp 217ndash232 2017

[23] L Saidi J Ben Ali F Fnaiech et al ldquoApplication of higherorder spectral features and support vector machines forbearing faults classificationrdquo ISA Transactions vol 54pp 193ndash206 2015

12 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Figure 8 Firstly the optimum range of white noise am-plitude is determined by the stability value of SI In order toensure that the amplitude of white noise can improve thedistribution of signal extreme points the minimum value ofk is 0001-emaximum value of k is set to be 001 so that theoptimization range is not too small In order to select theappropriate amplitude of white noise it is necessary todetermine the number of ensemble members in advance-e numberN of ensemble members should not be too largeotherwise the calculation time is very long If N is too smallEEMD will be invalid After many trials N is initially set tobe 10 EEMD analyses of the signal are carried out in theoptimal range of white noise amplitude and the de-composition results are evaluated by the orthogonal co-efficients -e white noise amplitude corresponding to theoptimal decomposition result is the best white noise am-plitude α In this paper the error less than 1 can be ac-ceptable -e final standard deviation of error is set to be001 -en the bestN is determined according to the rules ofthe final standard deviation of error and the white noiseamplitude Meanwhile the minimum value of N is 10 toavoid the EEMD invalidation caused by too small N

-e vibration acceleration signals of wheel in Figure 6 areanalyzed by EMD and EEMD respectively Figure 9 shows thefirst four IMFs after decomposition IMF components arearranged from high frequency to low frequency so the first

IMF (IMF1) is the high-frequency component of the signal Itcan be seen from Figure 9(a) that IMF1 of EMD analysis haslow-frequency signals at other locations besides high-frequency signals caused by rail corrugation which in-dicates that there is mode mixing in EMD analysis Appar-ently the mode mixing will influence the detection results InFigure 9(b) there is only the high-frequency componentsrelated with rail corrugation and no low frequency in IMF1-erefore EEMD analysis is an effectivemethod to extract thepart of the signal caused by rail corrugation

313 Wavelength Calculation After the IMFs are obtainedby EEMD analysis the IMFs are selected to eliminate theeffects of noise and the random irregularity of the track IMFselection and signal reconstruction algorithm are shown inFigure 10 -e false components are eliminated by corre-lation analysis and the high-frequency components arereconstructed into a high-frequency signal which is causedby rail corrugation

Estimation of wavelength of rail corrugation by mainfrequency of the reconstructed signal yr is expressed as

λ v

fm (14)

where v is train speed and fm is the main frequency of thereconstructed signal which can be obtained by FFT

32 Features Extraction of Corrugation Depth -e vibrationacceleration signals of wheel under rail corrugation arenonlinear nonstationary and non-Gaussian Higher-orderspectra are useful in analyzing nonlinearity and non-Gaussianity of the signal as it provides high noise immu-nity [20] -e frequency domain representation of third-order cumulants of signals is called bispectrum which isexpressed as

B f1 f2( 1113857 E X f1( 1113857X f2( 1113857Xlowast

f1 + f2( 11138571113858 1113859 (15)

Whe

etse

t acc

eler

atio

n (g

)

2

1

0

ndash1

ndash230 40 50 60

Distance (m)70 80 90 100

Figure 6 Vibration acceleration signal of wheel

SI

80

60

40

20

00 005

Amplitude coefficient k of white noise01 015 02

Figure 7 -e extreme point distribution index SI

Calculate standard deviation σ of signal

Find the k when SI is stable defined as kmaxIf kmax lt 001 let kmax = 001

k = 0001~kmax N = 10

Find the k when the IO is minimal defined as ko

α = komiddotσ

N = (αe)2 e = 001if N lt 10 let N = 10

Figure 8 Selection algorithm of EEMD parameters

6 Shock and Vibration

2 IMF1

30 40 50 60 70 80 90 100

0

ndash2

2 IMF2

30 40 50 60 70 80 90 100

0

ndash2

05 IMF3

30 40 50 60 70 80 90 100

0

ndash05

05 IMF4

30 40 50 60 70 80 90 100

0

ndash05

(a)

05 IMF1

30 40 50 60 70 80 90 100

0

ndash05

1 IMF2

30 40 50 60 70 80 90 100

0

ndash1

2 IMF3

30 40 50 60 70 80 90 100

0

ndash2

1 IMF4

30 40 50 60 70 80 90 100

0

ndash1

(b)

Figure 9 -e first four IMFs of the signal (a) EMD (b) EEMD

Ci (i = 1 2 n) are n IMFs of the signal

CORi (i = 1 2 n) are the correlation coefficients of IMF and original signal

respectively

CORthreshold = max CORi (i = 1 2 n)10If CORi lt CORthreshold eliminate ci

fi (i = 1 2 ) are the main frequency of ci respectively

yr = sum (ci) i = 1 2

If fi gt 400Hz reserve ci

Figure 10 IMFs selection and signal reconstruction algorithm

Shock and Vibration 7

where X(f) is the Fourier transform of the signal lowast denotescomplex conjugate E[] denotes the expectation operationand the frequency f may be normalized by the Nyquistfrequency to be between 0 and 1

-e discrete bispectrum matrix B is obtained by the highorder spectral analysis of the reconstructed signal yr(t) -esize of the bispectrum matrix B is QtimesQ In order to rec-ognize the depth of rail corrugation easily bispectrumfeatures need to be extracted

-ree amplitude features of bispectrum [21]

Amp1 1

Q2 1113944Ω

|B(i j)|

Amp2 1113944Ωlog(|B(i j)|)

Amp3 1113944

Q

i1log(|B(i i)|)

(16)

-ree moment features of bispectrum [22]

Mom1 1113944

Q

i1i middot log(|B(i i)|)

Mom2 1113944

Q

i1(iminusM)

2middot log(|B(i i)|)

Mom3 1113944Ω

i2 + j21113969

middot |B(i j)|

(17)

-ree entropy features of bispectrum [23]

Ent1 minus1113944Ω

p1(i j) middot log p1(i j)( 1113857 (18)

where p1(i j) |B(i j)|1113936Ω|B(i j)|

Ent2 minus1113944Ω

p2(i j) middot log p2(i j)( 1113857 (19)

where p2(i j) |B(i j)|21113936Ω|B(i j)|2

Ent3 minus1113944Ω

p3(i j) middot log p3(i j)( 1113857 (20)

where p3(i j) |B(i j)|31113936Ω|B(i j)|3-e above nine bispectrum features have different am-

plitudes at different rail corrugation depths as shown inFigure 11 -ree entropy features of bispectrum changenonlinearly with the increase of rail corrugation depth whilethe other features of bispectrum increase with the increase ofrail corrugation depth -e bispectrum features can wellreflect the depth of rail corrugation

-e train speed and the wavelength of the rail corru-gation also affect the vibration response of the wheel-erefore the train speed v and the estimated wavelength λare also taken as the feature parameters -e extractedfeatures are constructed into a features vector and classifiedusing the SVM model

4 Detection Result

41 Data Preparation In the simulation model the trainspeeds are 250 kmh to 350 kmh the wavelengths of rail

corrugation are 100mm to 150mm and the depths of railcorrugation are 001mm to 01mm -ere are 660 signalsAfter repeated 10 simulations 6600 sets of vibration ac-celeration signals of wheel are obtained -e signal length is60m including 25m before the wheel enters the rail cor-rugation and 25m after it leaves the rail corrugation

In order to verify the antinoise ability of the detectionmethod a Gaussian noise signal is added to the wheel ac-celeration signal to simulate the measurement noise

xmeasurement xsimulation + Ep times N times var xsimulation( 1113857 (21)

where Ep is the noise level N is a standard normal dis-tribution vector with zero mean and unit standard deviationand var() denotes the standard deviation of signal

42 Wavelength Estimation of Rail Corrugation -e esti-mation results of rail corrugation wavelength are shown inFigure 12 It can be seen from Figure 12 that the wavelengthof rail corrugation can be estimated at different corrugationdepths and train speeds and the noise has no effect on theestimation algorithm

-e relative error between the estimated valueof wavelength and the true value of wavelength is definedas

error 1n

1113944

n

i1

λestimationi minus λtruei

11138681113868111386811138681113868111386811138681113868

λtruei

(22)

where n is the number of data samplesIn order to analyze the influence of corrugation depth on

wavelength estimation depths of 001mm 005mm and01mm rail corrugation are selected respectively -ewavelength estimation results with 0 1 3 and 5noise levels are shown in Table 3 -e wavelength estimationerror of rail wavelength is almost unchanged under differentdepths and the maximum value of wavelength estimationerror is not more than 02 -e wavelength estimationerrors are very small which shows that the estimation of railwavelength is completely reliable Moreover the simulationresults show that the proposed method can estimate the

001 002 003 004 005 006 007 008 009 01Depth of corrugation (mm)

0

02

04

06

08

1

Nor

mal

ized

ampl

itude

Amp1Amp2Amp3

Mom1Mom2Mom3

Ent1Ent2Ent3

Figure 11 Bispectrum features at different depths of corrugation(train speed is 250 kmh and wavelength is 100mm)

8 Shock and Vibration

wavelength well under the conditions of different depths andmeasurement noises

-e train speeds are 250 kmh 300 kmh and 350 kmhand the wavelength of rail corrugation is estimated re-spectively -e wavelength estimation results with 0 13 and 5 noise levels are shown in Table 4 -e

wavelength estimation error of rail wavelength is almostunchanged at different depths and the maximum value ofwavelength estimation error is not more than 025 -ewavelength estimation error is also very small which in-dicates that the wavelength estimation effect of rail corru-gation is very good Meanwhile the simulation results

9999

100

10001

Wav

elen

gth

(mm

)

True value ofwavelength0 noise

1 noise3 noise5 noise

002 004 006 008 01

110

11005

002 004 006 008 0111995

120

12005

002 004 006 008 011298

1299

130

002 004 006 008 011395

140

1405

002 004 006 008 01e depth of rail corrugation (mm)

1495

150

1505

002 004 006 008 01

(a)

True value ofwavelength0 noise

1 noise3 noise5 noise

9998

100

10002

Wav

elen

gth

(mm

)

250 300 350

250 300 350

1098

110

1102

250 300 3501198

120

1202

250 300 3501298

1299

130

1301

250 300 3501395

140

1405

250 300 350Train speed (kmh)

1495

150

1505

(b)

Figure 12 -e estimation results of rail corrugation wavelength (a) At different depths of rail corrugation (b) At different train speeds

Table 3 Wavelength estimation error at different depths of rail corrugation

CaseDepth 001mm Depth 005mm Depth 01mm

Nil 1 3 5 Nil 1 3 5 Nil 1 3 5100mm 00045 00045 00045 00045 00045 00045 00045 00045 00045 00045 00045 00045110mm 00900 00897 00908 00910 00902 00900 00900 00899 00899 00899 00900 00899120mm 00205 00045 00045 00063 00367 00276 00330 00275 00313 00349 00367 00330130mm 00968 00968 00980 00994 00968 00968 00968 00968 00968 00968 00968 00968140mm 01252 01124 01001 01100 01347 01286 01230 01260 01360 01357 01329 01358150mm 00376 00045 00045 00087 01453 00623 00543 00601 01563 01293 01091 01068

Shock and Vibration 9

illustrate that the proposed method can estimate thewavelength excellent under the conditions of different trainspeeds and measurement noises

43 Depth Classification of Rail Corrugation According tothe values of depth of rail corrugations three depth levels aredivided as shown in Table 5

Bispectrum features estimated wavelength and trainspeed are taken as inputs of SVM and depth level is takenas SVM output -e 80 of all simulation data is used totrain the SVM model and the remaining 20 is used fortesting

-e classification results of rail corrugation depth levelwith different levels of noise are shown in Figure 13

-e reliability of the proposedmethod is evaluated by theaccuracy of the test results -e ratio between the number ofcorrect classification results and the number of total testsamples is defined as the detection accuracy which can beexpressed as

accuray Numcorrect

Numtotaltimes 100 (23)

-e detection results of depth level with 0 1 3 and5 noise levels are shown in Table 6 -e detection accuracyis more than 99 indicating that the extracted corrugationdepth feature is effective -e proposed detection methodhas strong antinoise ability and still has high accuracy underdifferent measurement noises

5 Conclusion

Rail corrugation is a short-wavelength track irregularityexcitation and accordingly causes high-frequency vibrationresponse of wheel A rail detection algorithm based onEEMD and bispectrum features is proposed using wheelvibration acceleration Different frequency components ofwheel vibration response are extracted with EEMD of wheelvibration acceleration signal -e parameters α and N ofEEMD are optimized using the orthogonal coefficient IOand the extreme point distribution index SI which reducesthe mode mixing and calculation time of the de-composition -e main frequency is found after EEMD tocalculate the wavelength-e depth detection is regarded asa classification problem with SVM -e high-frequencysignal of rail corrugation is reconstructed through theselection of IMFs by which not only the interference oftrack random irregularity is avoided but also noise influ-ence Bispectrum features which are extracted from the

Table 4 Wavelength estimation error at different train speeds

CaseTrain speed 250 kmh Train speed 300 kmh Train speed 350 kmh

Nil 1 3 5 Nil 1 3 5 Nil 1 3 5100mm 0 0 0 0 0 0 0 0 00093 00093 00093 00093110mm 00999 00999 00999 00997 00834 00834 00834 00836 00927 00927 00927 00927120mm 01657 01198 00938 01058 0 0 0 0 00093 00093 00093 00093130mm 00999 00999 00999 00999 00999 00999 00999 01001 00907 00907 00910 00910140mm 01234 01070 01016 01107 01325 01275 01217 01265 01428 01412 01391 01386150mm 00399 0 0 00100 01421 00524 00474 00050 02124 01778 01616 01639

Table 5 Definition of depth levels of rail corrugation

Depth levels Depths of rail corrugation (mm)Level 1 a≦ 004Level 2 004lt a≦ 007Level 3 agt 007

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 3Level 2

(a)

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 2 Level 3

(b)

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 2 Level 3

(c)

Figure 13 Continued

10 Shock and Vibration

reconstructed signal combining with train speed and thecorrugation wavelength are input into SVM -e nu-merical simulation results show the proposed detectionalgorithm can accurately identify the existence of railcorrugation -e error between the estimated wavelengthand the real value is small and the accuracy of corrugationdepth level classification is high And the results alsorepresent robustness under different train speeds andmeasurement noises -e proposed rail corrugation de-tection method provides a feasible scheme for the in-service vehicle test in the future

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 they have no conflicts of interest

Acknowledgments

-is work was supported by the National Key Researchand Development Program of China under Grant no2016YFB1200401

References

[1] K H Oostermeijer ldquoReview on short pitch rail corrugationstudiesrdquo Wear vol 265 no 9-10 pp 1231ndash1237 2008

[2] Z Q Jiang D L Si W Li et al ldquoOn rail corrugation of highspeed railway (in Chinese)rdquo China Railway Science vol 35no 4 pp 9ndash14 2014

[3] B M Hopkins and S Taheri ldquoTrack health monitoring usingwaveletrdquo in Proceedings of ASME 2010 Rail TransportationDivision Fall Technical Conference pp 9ndash15 Roanoke VAUSA October 2010

[4] M Molodova Z Li and R Dollevoet ldquoAxle box accelerationmeasurement and simulation for detection of short trackdefectsrdquo Wear vol 271 no 1-2 pp 349ndash356 2011

[5] W Huang W Zhang Y Du et al ldquoDetection of rail cor-rugation based on fiber laser accelerometersrdquo MeasurementScience and Technology vol 24 no 9 article 094014 2013

[6] S Kaewunruen ldquoMonitoring of rail corrugation growth onsharp curves for track maintenance prioritisationrdquo In-ternational Journal of Acoustics and Vibration vol 23 no 1pp 35ndash43 2018

[7] P Salvador V Naranjo R Insa and P Teixeira ldquoAxleboxaccelerations their acquisition and time-frequency charac-terisation for railway track monitoring purposesrdquo Measure-ment vol 82 pp 301ndash312 2016

[8] Z Shen Q Wang Y Shen et al ldquoAccent extraction ofemotional speech based on modified ensemble empiricalmode decompositionrdquo in Proceedings of Instrumentation andMeasurement Technology Conference Austin TX USA May2010

[9] W Guo and P W Tse ldquoA novel signal compression methodbased on optimal ensemble empirical mode decompositionfor bearing vibration signalsrdquo Journal of Sound and Vibrationvol 332 no 2 pp 423ndash441 2013

[10] X Xue J Zhou Y Xu W Zhu and C Li ldquoAn adaptively fastensemble empirical mode decomposition method and itsapplications to rolling element bearing fault diagnosisrdquoMechanical Systems and Signal Processing vol 62-63pp 444ndash459 2015

[11] M Kedadouche M -omas and A Tahan ldquoA comparativestudy between empirical wavelet transforms and empiricalmode decomposition methods application to bearing defectdiagnosisrdquo Mechanical Systems and Signal Processing vol 81pp 88ndash107 2016

[12] W Zhai and X Sun ldquoA detailed model for investigatingvertical interaction between railway vehicle and trackrdquo Ve-hicle System Dynamics vol 23 no 1 pp 603ndash615 1994

[13] X Lei and J Wang ldquoDynamic analysis of the train and slabtrack coupling system with finite elements in a moving frameof referencerdquo Journal of Vibration and Control vol 20 no 9pp 1301ndash1317 2013

[14] W Zhai KWang and C Cai ldquoFundamentals of vehicle-trackcoupled dynamicsrdquo Vehicle System Dynamics vol 47 no 11pp 1349ndash1376 2009

[15] X Kang X B Liu H Y LI et al ldquoPSD of ballastless trackirregularities of high-speed railway (in Chinese)rdquo ScientiaSinica Technologica vol 44 no 7 pp 687ndash696 2014

[16] K Wang P Liu W Zhai C Huang Z Chen and J GaoldquoWheelrail dynamic interaction due to excitation of railcorrugation in high-speed railwayrdquo Science China Techno-logical Sciences vol 58 no 2 pp 226ndash235 2015

[17] C C Chang and C J Lin ldquoLLIBSVM a library for supportvector machinesrdquo ACM Transactions on Intelligent Systemsand Technology vol 2 no 3 pp 1ndash27 2011

[18] 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 the RoyalSociety A-Mathematical Physical and Engineering Sciencesvol 454 no 1971 pp 909ndash995 1998

Table 6 -e accuracy of depth level classification results at dif-ferent measurement noises

Noise level Nil 1 3 5Accuracy of level 1 () 9981 100 100 100Accuracy of level 2 () 9949 9975 9975 9949Accuracy of level 3 () 9949 9975 100 100Total accuracy () 9962 9985 9992 9985

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 2 Level 3

(d)

Figure 13 -e classification results of rail corrugation depth level(a) 0 noise (b) 1 noise (c) 3 noise (d) 5 noise

Shock and Vibration 11

[19] Z Wu and N E Huang ldquoEnsemble empirical mode de-composition a noise-assisted data analysis methodrdquo Ad-vances in Adaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[20] LW Jian and T-C Lim ldquoAutomated detection of diabetes bymeans of higher order spectral features obtained from heartrate signalsrdquo Journal of Medical Imaging and Health In-formatics vol 3 no 3 pp 440ndash447 2013

[21] R Yuvaraj M Murugappan N Mohamed Ibrahim et alldquoDetection of emotions in Parkinsonrsquos disease using higherorder spectral features from brainrsquos electrical activityrdquo Bio-medical Signal Processing and Control vol 14 pp 108ndash1162014

[22] CKY M Hariharan R Ngadiran A H Adom S Yaacoband K Polat ldquoHybrid BBO_PSO and higher order spectralfeatures for emotion and stress recognition from naturalspeechrdquo Applied Soft Computing vol 56 pp 217ndash232 2017

[23] L Saidi J Ben Ali F Fnaiech et al ldquoApplication of higherorder spectral features and support vector machines forbearing faults classificationrdquo ISA Transactions vol 54pp 193ndash206 2015

12 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

2 IMF1

30 40 50 60 70 80 90 100

0

ndash2

2 IMF2

30 40 50 60 70 80 90 100

0

ndash2

05 IMF3

30 40 50 60 70 80 90 100

0

ndash05

05 IMF4

30 40 50 60 70 80 90 100

0

ndash05

(a)

05 IMF1

30 40 50 60 70 80 90 100

0

ndash05

1 IMF2

30 40 50 60 70 80 90 100

0

ndash1

2 IMF3

30 40 50 60 70 80 90 100

0

ndash2

1 IMF4

30 40 50 60 70 80 90 100

0

ndash1

(b)

Figure 9 -e first four IMFs of the signal (a) EMD (b) EEMD

Ci (i = 1 2 n) are n IMFs of the signal

CORi (i = 1 2 n) are the correlation coefficients of IMF and original signal

respectively

CORthreshold = max CORi (i = 1 2 n)10If CORi lt CORthreshold eliminate ci

fi (i = 1 2 ) are the main frequency of ci respectively

yr = sum (ci) i = 1 2

If fi gt 400Hz reserve ci

Figure 10 IMFs selection and signal reconstruction algorithm

Shock and Vibration 7

where X(f) is the Fourier transform of the signal lowast denotescomplex conjugate E[] denotes the expectation operationand the frequency f may be normalized by the Nyquistfrequency to be between 0 and 1

-e discrete bispectrum matrix B is obtained by the highorder spectral analysis of the reconstructed signal yr(t) -esize of the bispectrum matrix B is QtimesQ In order to rec-ognize the depth of rail corrugation easily bispectrumfeatures need to be extracted

-ree amplitude features of bispectrum [21]

Amp1 1

Q2 1113944Ω

|B(i j)|

Amp2 1113944Ωlog(|B(i j)|)

Amp3 1113944

Q

i1log(|B(i i)|)

(16)

-ree moment features of bispectrum [22]

Mom1 1113944

Q

i1i middot log(|B(i i)|)

Mom2 1113944

Q

i1(iminusM)

2middot log(|B(i i)|)

Mom3 1113944Ω

i2 + j21113969

middot |B(i j)|

(17)

-ree entropy features of bispectrum [23]

Ent1 minus1113944Ω

p1(i j) middot log p1(i j)( 1113857 (18)

where p1(i j) |B(i j)|1113936Ω|B(i j)|

Ent2 minus1113944Ω

p2(i j) middot log p2(i j)( 1113857 (19)

where p2(i j) |B(i j)|21113936Ω|B(i j)|2

Ent3 minus1113944Ω

p3(i j) middot log p3(i j)( 1113857 (20)

where p3(i j) |B(i j)|31113936Ω|B(i j)|3-e above nine bispectrum features have different am-

plitudes at different rail corrugation depths as shown inFigure 11 -ree entropy features of bispectrum changenonlinearly with the increase of rail corrugation depth whilethe other features of bispectrum increase with the increase ofrail corrugation depth -e bispectrum features can wellreflect the depth of rail corrugation

-e train speed and the wavelength of the rail corru-gation also affect the vibration response of the wheel-erefore the train speed v and the estimated wavelength λare also taken as the feature parameters -e extractedfeatures are constructed into a features vector and classifiedusing the SVM model

4 Detection Result

41 Data Preparation In the simulation model the trainspeeds are 250 kmh to 350 kmh the wavelengths of rail

corrugation are 100mm to 150mm and the depths of railcorrugation are 001mm to 01mm -ere are 660 signalsAfter repeated 10 simulations 6600 sets of vibration ac-celeration signals of wheel are obtained -e signal length is60m including 25m before the wheel enters the rail cor-rugation and 25m after it leaves the rail corrugation

In order to verify the antinoise ability of the detectionmethod a Gaussian noise signal is added to the wheel ac-celeration signal to simulate the measurement noise

xmeasurement xsimulation + Ep times N times var xsimulation( 1113857 (21)

where Ep is the noise level N is a standard normal dis-tribution vector with zero mean and unit standard deviationand var() denotes the standard deviation of signal

42 Wavelength Estimation of Rail Corrugation -e esti-mation results of rail corrugation wavelength are shown inFigure 12 It can be seen from Figure 12 that the wavelengthof rail corrugation can be estimated at different corrugationdepths and train speeds and the noise has no effect on theestimation algorithm

-e relative error between the estimated valueof wavelength and the true value of wavelength is definedas

error 1n

1113944

n

i1

λestimationi minus λtruei

11138681113868111386811138681113868111386811138681113868

λtruei

(22)

where n is the number of data samplesIn order to analyze the influence of corrugation depth on

wavelength estimation depths of 001mm 005mm and01mm rail corrugation are selected respectively -ewavelength estimation results with 0 1 3 and 5noise levels are shown in Table 3 -e wavelength estimationerror of rail wavelength is almost unchanged under differentdepths and the maximum value of wavelength estimationerror is not more than 02 -e wavelength estimationerrors are very small which shows that the estimation of railwavelength is completely reliable Moreover the simulationresults show that the proposed method can estimate the

001 002 003 004 005 006 007 008 009 01Depth of corrugation (mm)

0

02

04

06

08

1

Nor

mal

ized

ampl

itude

Amp1Amp2Amp3

Mom1Mom2Mom3

Ent1Ent2Ent3

Figure 11 Bispectrum features at different depths of corrugation(train speed is 250 kmh and wavelength is 100mm)

8 Shock and Vibration

wavelength well under the conditions of different depths andmeasurement noises

-e train speeds are 250 kmh 300 kmh and 350 kmhand the wavelength of rail corrugation is estimated re-spectively -e wavelength estimation results with 0 13 and 5 noise levels are shown in Table 4 -e

wavelength estimation error of rail wavelength is almostunchanged at different depths and the maximum value ofwavelength estimation error is not more than 025 -ewavelength estimation error is also very small which in-dicates that the wavelength estimation effect of rail corru-gation is very good Meanwhile the simulation results

9999

100

10001

Wav

elen

gth

(mm

)

True value ofwavelength0 noise

1 noise3 noise5 noise

002 004 006 008 01

110

11005

002 004 006 008 0111995

120

12005

002 004 006 008 011298

1299

130

002 004 006 008 011395

140

1405

002 004 006 008 01e depth of rail corrugation (mm)

1495

150

1505

002 004 006 008 01

(a)

True value ofwavelength0 noise

1 noise3 noise5 noise

9998

100

10002

Wav

elen

gth

(mm

)

250 300 350

250 300 350

1098

110

1102

250 300 3501198

120

1202

250 300 3501298

1299

130

1301

250 300 3501395

140

1405

250 300 350Train speed (kmh)

1495

150

1505

(b)

Figure 12 -e estimation results of rail corrugation wavelength (a) At different depths of rail corrugation (b) At different train speeds

Table 3 Wavelength estimation error at different depths of rail corrugation

CaseDepth 001mm Depth 005mm Depth 01mm

Nil 1 3 5 Nil 1 3 5 Nil 1 3 5100mm 00045 00045 00045 00045 00045 00045 00045 00045 00045 00045 00045 00045110mm 00900 00897 00908 00910 00902 00900 00900 00899 00899 00899 00900 00899120mm 00205 00045 00045 00063 00367 00276 00330 00275 00313 00349 00367 00330130mm 00968 00968 00980 00994 00968 00968 00968 00968 00968 00968 00968 00968140mm 01252 01124 01001 01100 01347 01286 01230 01260 01360 01357 01329 01358150mm 00376 00045 00045 00087 01453 00623 00543 00601 01563 01293 01091 01068

Shock and Vibration 9

illustrate that the proposed method can estimate thewavelength excellent under the conditions of different trainspeeds and measurement noises

43 Depth Classification of Rail Corrugation According tothe values of depth of rail corrugations three depth levels aredivided as shown in Table 5

Bispectrum features estimated wavelength and trainspeed are taken as inputs of SVM and depth level is takenas SVM output -e 80 of all simulation data is used totrain the SVM model and the remaining 20 is used fortesting

-e classification results of rail corrugation depth levelwith different levels of noise are shown in Figure 13

-e reliability of the proposedmethod is evaluated by theaccuracy of the test results -e ratio between the number ofcorrect classification results and the number of total testsamples is defined as the detection accuracy which can beexpressed as

accuray Numcorrect

Numtotaltimes 100 (23)

-e detection results of depth level with 0 1 3 and5 noise levels are shown in Table 6 -e detection accuracyis more than 99 indicating that the extracted corrugationdepth feature is effective -e proposed detection methodhas strong antinoise ability and still has high accuracy underdifferent measurement noises

5 Conclusion

Rail corrugation is a short-wavelength track irregularityexcitation and accordingly causes high-frequency vibrationresponse of wheel A rail detection algorithm based onEEMD and bispectrum features is proposed using wheelvibration acceleration Different frequency components ofwheel vibration response are extracted with EEMD of wheelvibration acceleration signal -e parameters α and N ofEEMD are optimized using the orthogonal coefficient IOand the extreme point distribution index SI which reducesthe mode mixing and calculation time of the de-composition -e main frequency is found after EEMD tocalculate the wavelength-e depth detection is regarded asa classification problem with SVM -e high-frequencysignal of rail corrugation is reconstructed through theselection of IMFs by which not only the interference oftrack random irregularity is avoided but also noise influ-ence Bispectrum features which are extracted from the

Table 4 Wavelength estimation error at different train speeds

CaseTrain speed 250 kmh Train speed 300 kmh Train speed 350 kmh

Nil 1 3 5 Nil 1 3 5 Nil 1 3 5100mm 0 0 0 0 0 0 0 0 00093 00093 00093 00093110mm 00999 00999 00999 00997 00834 00834 00834 00836 00927 00927 00927 00927120mm 01657 01198 00938 01058 0 0 0 0 00093 00093 00093 00093130mm 00999 00999 00999 00999 00999 00999 00999 01001 00907 00907 00910 00910140mm 01234 01070 01016 01107 01325 01275 01217 01265 01428 01412 01391 01386150mm 00399 0 0 00100 01421 00524 00474 00050 02124 01778 01616 01639

Table 5 Definition of depth levels of rail corrugation

Depth levels Depths of rail corrugation (mm)Level 1 a≦ 004Level 2 004lt a≦ 007Level 3 agt 007

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 3Level 2

(a)

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 2 Level 3

(b)

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 2 Level 3

(c)

Figure 13 Continued

10 Shock and Vibration

reconstructed signal combining with train speed and thecorrugation wavelength are input into SVM -e nu-merical simulation results show the proposed detectionalgorithm can accurately identify the existence of railcorrugation -e error between the estimated wavelengthand the real value is small and the accuracy of corrugationdepth level classification is high And the results alsorepresent robustness under different train speeds andmeasurement noises -e proposed rail corrugation de-tection method provides a feasible scheme for the in-service vehicle test in the future

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 they have no conflicts of interest

Acknowledgments

-is work was supported by the National Key Researchand Development Program of China under Grant no2016YFB1200401

References

[1] K H Oostermeijer ldquoReview on short pitch rail corrugationstudiesrdquo Wear vol 265 no 9-10 pp 1231ndash1237 2008

[2] Z Q Jiang D L Si W Li et al ldquoOn rail corrugation of highspeed railway (in Chinese)rdquo China Railway Science vol 35no 4 pp 9ndash14 2014

[3] B M Hopkins and S Taheri ldquoTrack health monitoring usingwaveletrdquo in Proceedings of ASME 2010 Rail TransportationDivision Fall Technical Conference pp 9ndash15 Roanoke VAUSA October 2010

[4] M Molodova Z Li and R Dollevoet ldquoAxle box accelerationmeasurement and simulation for detection of short trackdefectsrdquo Wear vol 271 no 1-2 pp 349ndash356 2011

[5] W Huang W Zhang Y Du et al ldquoDetection of rail cor-rugation based on fiber laser accelerometersrdquo MeasurementScience and Technology vol 24 no 9 article 094014 2013

[6] S Kaewunruen ldquoMonitoring of rail corrugation growth onsharp curves for track maintenance prioritisationrdquo In-ternational Journal of Acoustics and Vibration vol 23 no 1pp 35ndash43 2018

[7] P Salvador V Naranjo R Insa and P Teixeira ldquoAxleboxaccelerations their acquisition and time-frequency charac-terisation for railway track monitoring purposesrdquo Measure-ment vol 82 pp 301ndash312 2016

[8] Z Shen Q Wang Y Shen et al ldquoAccent extraction ofemotional speech based on modified ensemble empiricalmode decompositionrdquo in Proceedings of Instrumentation andMeasurement Technology Conference Austin TX USA May2010

[9] W Guo and P W Tse ldquoA novel signal compression methodbased on optimal ensemble empirical mode decompositionfor bearing vibration signalsrdquo Journal of Sound and Vibrationvol 332 no 2 pp 423ndash441 2013

[10] X Xue J Zhou Y Xu W Zhu and C Li ldquoAn adaptively fastensemble empirical mode decomposition method and itsapplications to rolling element bearing fault diagnosisrdquoMechanical Systems and Signal Processing vol 62-63pp 444ndash459 2015

[11] M Kedadouche M -omas and A Tahan ldquoA comparativestudy between empirical wavelet transforms and empiricalmode decomposition methods application to bearing defectdiagnosisrdquo Mechanical Systems and Signal Processing vol 81pp 88ndash107 2016

[12] W Zhai and X Sun ldquoA detailed model for investigatingvertical interaction between railway vehicle and trackrdquo Ve-hicle System Dynamics vol 23 no 1 pp 603ndash615 1994

[13] X Lei and J Wang ldquoDynamic analysis of the train and slabtrack coupling system with finite elements in a moving frameof referencerdquo Journal of Vibration and Control vol 20 no 9pp 1301ndash1317 2013

[14] W Zhai KWang and C Cai ldquoFundamentals of vehicle-trackcoupled dynamicsrdquo Vehicle System Dynamics vol 47 no 11pp 1349ndash1376 2009

[15] X Kang X B Liu H Y LI et al ldquoPSD of ballastless trackirregularities of high-speed railway (in Chinese)rdquo ScientiaSinica Technologica vol 44 no 7 pp 687ndash696 2014

[16] K Wang P Liu W Zhai C Huang Z Chen and J GaoldquoWheelrail dynamic interaction due to excitation of railcorrugation in high-speed railwayrdquo Science China Techno-logical Sciences vol 58 no 2 pp 226ndash235 2015

[17] C C Chang and C J Lin ldquoLLIBSVM a library for supportvector machinesrdquo ACM Transactions on Intelligent Systemsand Technology vol 2 no 3 pp 1ndash27 2011

[18] 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 the RoyalSociety A-Mathematical Physical and Engineering Sciencesvol 454 no 1971 pp 909ndash995 1998

Table 6 -e accuracy of depth level classification results at dif-ferent measurement noises

Noise level Nil 1 3 5Accuracy of level 1 () 9981 100 100 100Accuracy of level 2 () 9949 9975 9975 9949Accuracy of level 3 () 9949 9975 100 100Total accuracy () 9962 9985 9992 9985

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 2 Level 3

(d)

Figure 13 -e classification results of rail corrugation depth level(a) 0 noise (b) 1 noise (c) 3 noise (d) 5 noise

Shock and Vibration 11

[19] Z Wu and N E Huang ldquoEnsemble empirical mode de-composition a noise-assisted data analysis methodrdquo Ad-vances in Adaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[20] LW Jian and T-C Lim ldquoAutomated detection of diabetes bymeans of higher order spectral features obtained from heartrate signalsrdquo Journal of Medical Imaging and Health In-formatics vol 3 no 3 pp 440ndash447 2013

[21] R Yuvaraj M Murugappan N Mohamed Ibrahim et alldquoDetection of emotions in Parkinsonrsquos disease using higherorder spectral features from brainrsquos electrical activityrdquo Bio-medical Signal Processing and Control vol 14 pp 108ndash1162014

[22] CKY M Hariharan R Ngadiran A H Adom S Yaacoband K Polat ldquoHybrid BBO_PSO and higher order spectralfeatures for emotion and stress recognition from naturalspeechrdquo Applied Soft Computing vol 56 pp 217ndash232 2017

[23] L Saidi J Ben Ali F Fnaiech et al ldquoApplication of higherorder spectral features and support vector machines forbearing faults classificationrdquo ISA Transactions vol 54pp 193ndash206 2015

12 Shock and Vibration

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Submit your manuscripts atwwwhindawicom

where X(f) is the Fourier transform of the signal lowast denotescomplex conjugate E[] denotes the expectation operationand the frequency f may be normalized by the Nyquistfrequency to be between 0 and 1

-e discrete bispectrum matrix B is obtained by the highorder spectral analysis of the reconstructed signal yr(t) -esize of the bispectrum matrix B is QtimesQ In order to rec-ognize the depth of rail corrugation easily bispectrumfeatures need to be extracted

-ree amplitude features of bispectrum [21]

Amp1 1

Q2 1113944Ω

|B(i j)|

Amp2 1113944Ωlog(|B(i j)|)

Amp3 1113944

Q

i1log(|B(i i)|)

(16)

-ree moment features of bispectrum [22]

Mom1 1113944

Q

i1i middot log(|B(i i)|)

Mom2 1113944

Q

i1(iminusM)

2middot log(|B(i i)|)

Mom3 1113944Ω

i2 + j21113969

middot |B(i j)|

(17)

-ree entropy features of bispectrum [23]

Ent1 minus1113944Ω

p1(i j) middot log p1(i j)( 1113857 (18)

where p1(i j) |B(i j)|1113936Ω|B(i j)|

Ent2 minus1113944Ω

p2(i j) middot log p2(i j)( 1113857 (19)

where p2(i j) |B(i j)|21113936Ω|B(i j)|2

Ent3 minus1113944Ω

p3(i j) middot log p3(i j)( 1113857 (20)

where p3(i j) |B(i j)|31113936Ω|B(i j)|3-e above nine bispectrum features have different am-

plitudes at different rail corrugation depths as shown inFigure 11 -ree entropy features of bispectrum changenonlinearly with the increase of rail corrugation depth whilethe other features of bispectrum increase with the increase ofrail corrugation depth -e bispectrum features can wellreflect the depth of rail corrugation

-e train speed and the wavelength of the rail corru-gation also affect the vibration response of the wheel-erefore the train speed v and the estimated wavelength λare also taken as the feature parameters -e extractedfeatures are constructed into a features vector and classifiedusing the SVM model

4 Detection Result

41 Data Preparation In the simulation model the trainspeeds are 250 kmh to 350 kmh the wavelengths of rail

corrugation are 100mm to 150mm and the depths of railcorrugation are 001mm to 01mm -ere are 660 signalsAfter repeated 10 simulations 6600 sets of vibration ac-celeration signals of wheel are obtained -e signal length is60m including 25m before the wheel enters the rail cor-rugation and 25m after it leaves the rail corrugation

In order to verify the antinoise ability of the detectionmethod a Gaussian noise signal is added to the wheel ac-celeration signal to simulate the measurement noise

xmeasurement xsimulation + Ep times N times var xsimulation( 1113857 (21)

where Ep is the noise level N is a standard normal dis-tribution vector with zero mean and unit standard deviationand var() denotes the standard deviation of signal

42 Wavelength Estimation of Rail Corrugation -e esti-mation results of rail corrugation wavelength are shown inFigure 12 It can be seen from Figure 12 that the wavelengthof rail corrugation can be estimated at different corrugationdepths and train speeds and the noise has no effect on theestimation algorithm

-e relative error between the estimated valueof wavelength and the true value of wavelength is definedas

error 1n

1113944

n

i1

λestimationi minus λtruei

11138681113868111386811138681113868111386811138681113868

λtruei

(22)

where n is the number of data samplesIn order to analyze the influence of corrugation depth on

wavelength estimation depths of 001mm 005mm and01mm rail corrugation are selected respectively -ewavelength estimation results with 0 1 3 and 5noise levels are shown in Table 3 -e wavelength estimationerror of rail wavelength is almost unchanged under differentdepths and the maximum value of wavelength estimationerror is not more than 02 -e wavelength estimationerrors are very small which shows that the estimation of railwavelength is completely reliable Moreover the simulationresults show that the proposed method can estimate the

001 002 003 004 005 006 007 008 009 01Depth of corrugation (mm)

0

02

04

06

08

1

Nor

mal

ized

ampl

itude

Amp1Amp2Amp3

Mom1Mom2Mom3

Ent1Ent2Ent3

Figure 11 Bispectrum features at different depths of corrugation(train speed is 250 kmh and wavelength is 100mm)

8 Shock and Vibration

wavelength well under the conditions of different depths andmeasurement noises

-e train speeds are 250 kmh 300 kmh and 350 kmhand the wavelength of rail corrugation is estimated re-spectively -e wavelength estimation results with 0 13 and 5 noise levels are shown in Table 4 -e

wavelength estimation error of rail wavelength is almostunchanged at different depths and the maximum value ofwavelength estimation error is not more than 025 -ewavelength estimation error is also very small which in-dicates that the wavelength estimation effect of rail corru-gation is very good Meanwhile the simulation results

9999

100

10001

Wav

elen

gth

(mm

)

True value ofwavelength0 noise

1 noise3 noise5 noise

002 004 006 008 01

110

11005

002 004 006 008 0111995

120

12005

002 004 006 008 011298

1299

130

002 004 006 008 011395

140

1405

002 004 006 008 01e depth of rail corrugation (mm)

1495

150

1505

002 004 006 008 01

(a)

True value ofwavelength0 noise

1 noise3 noise5 noise

9998

100

10002

Wav

elen

gth

(mm

)

250 300 350

250 300 350

1098

110

1102

250 300 3501198

120

1202

250 300 3501298

1299

130

1301

250 300 3501395

140

1405

250 300 350Train speed (kmh)

1495

150

1505

(b)

Figure 12 -e estimation results of rail corrugation wavelength (a) At different depths of rail corrugation (b) At different train speeds

Table 3 Wavelength estimation error at different depths of rail corrugation

CaseDepth 001mm Depth 005mm Depth 01mm

Nil 1 3 5 Nil 1 3 5 Nil 1 3 5100mm 00045 00045 00045 00045 00045 00045 00045 00045 00045 00045 00045 00045110mm 00900 00897 00908 00910 00902 00900 00900 00899 00899 00899 00900 00899120mm 00205 00045 00045 00063 00367 00276 00330 00275 00313 00349 00367 00330130mm 00968 00968 00980 00994 00968 00968 00968 00968 00968 00968 00968 00968140mm 01252 01124 01001 01100 01347 01286 01230 01260 01360 01357 01329 01358150mm 00376 00045 00045 00087 01453 00623 00543 00601 01563 01293 01091 01068

Shock and Vibration 9

illustrate that the proposed method can estimate thewavelength excellent under the conditions of different trainspeeds and measurement noises

43 Depth Classification of Rail Corrugation According tothe values of depth of rail corrugations three depth levels aredivided as shown in Table 5

Bispectrum features estimated wavelength and trainspeed are taken as inputs of SVM and depth level is takenas SVM output -e 80 of all simulation data is used totrain the SVM model and the remaining 20 is used fortesting

-e classification results of rail corrugation depth levelwith different levels of noise are shown in Figure 13

-e reliability of the proposedmethod is evaluated by theaccuracy of the test results -e ratio between the number ofcorrect classification results and the number of total testsamples is defined as the detection accuracy which can beexpressed as

accuray Numcorrect

Numtotaltimes 100 (23)

-e detection results of depth level with 0 1 3 and5 noise levels are shown in Table 6 -e detection accuracyis more than 99 indicating that the extracted corrugationdepth feature is effective -e proposed detection methodhas strong antinoise ability and still has high accuracy underdifferent measurement noises

5 Conclusion

Rail corrugation is a short-wavelength track irregularityexcitation and accordingly causes high-frequency vibrationresponse of wheel A rail detection algorithm based onEEMD and bispectrum features is proposed using wheelvibration acceleration Different frequency components ofwheel vibration response are extracted with EEMD of wheelvibration acceleration signal -e parameters α and N ofEEMD are optimized using the orthogonal coefficient IOand the extreme point distribution index SI which reducesthe mode mixing and calculation time of the de-composition -e main frequency is found after EEMD tocalculate the wavelength-e depth detection is regarded asa classification problem with SVM -e high-frequencysignal of rail corrugation is reconstructed through theselection of IMFs by which not only the interference oftrack random irregularity is avoided but also noise influ-ence Bispectrum features which are extracted from the

Table 4 Wavelength estimation error at different train speeds

CaseTrain speed 250 kmh Train speed 300 kmh Train speed 350 kmh

Nil 1 3 5 Nil 1 3 5 Nil 1 3 5100mm 0 0 0 0 0 0 0 0 00093 00093 00093 00093110mm 00999 00999 00999 00997 00834 00834 00834 00836 00927 00927 00927 00927120mm 01657 01198 00938 01058 0 0 0 0 00093 00093 00093 00093130mm 00999 00999 00999 00999 00999 00999 00999 01001 00907 00907 00910 00910140mm 01234 01070 01016 01107 01325 01275 01217 01265 01428 01412 01391 01386150mm 00399 0 0 00100 01421 00524 00474 00050 02124 01778 01616 01639

Table 5 Definition of depth levels of rail corrugation

Depth levels Depths of rail corrugation (mm)Level 1 a≦ 004Level 2 004lt a≦ 007Level 3 agt 007

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 3Level 2

(a)

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 2 Level 3

(b)

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 2 Level 3

(c)

Figure 13 Continued

10 Shock and Vibration

reconstructed signal combining with train speed and thecorrugation wavelength are input into SVM -e nu-merical simulation results show the proposed detectionalgorithm can accurately identify the existence of railcorrugation -e error between the estimated wavelengthand the real value is small and the accuracy of corrugationdepth level classification is high And the results alsorepresent robustness under different train speeds andmeasurement noises -e proposed rail corrugation de-tection method provides a feasible scheme for the in-service vehicle test in the future

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 they have no conflicts of interest

Acknowledgments

-is work was supported by the National Key Researchand Development Program of China under Grant no2016YFB1200401

References

[1] K H Oostermeijer ldquoReview on short pitch rail corrugationstudiesrdquo Wear vol 265 no 9-10 pp 1231ndash1237 2008

[2] Z Q Jiang D L Si W Li et al ldquoOn rail corrugation of highspeed railway (in Chinese)rdquo China Railway Science vol 35no 4 pp 9ndash14 2014

[3] B M Hopkins and S Taheri ldquoTrack health monitoring usingwaveletrdquo in Proceedings of ASME 2010 Rail TransportationDivision Fall Technical Conference pp 9ndash15 Roanoke VAUSA October 2010

[4] M Molodova Z Li and R Dollevoet ldquoAxle box accelerationmeasurement and simulation for detection of short trackdefectsrdquo Wear vol 271 no 1-2 pp 349ndash356 2011

[5] W Huang W Zhang Y Du et al ldquoDetection of rail cor-rugation based on fiber laser accelerometersrdquo MeasurementScience and Technology vol 24 no 9 article 094014 2013

[6] S Kaewunruen ldquoMonitoring of rail corrugation growth onsharp curves for track maintenance prioritisationrdquo In-ternational Journal of Acoustics and Vibration vol 23 no 1pp 35ndash43 2018

[7] P Salvador V Naranjo R Insa and P Teixeira ldquoAxleboxaccelerations their acquisition and time-frequency charac-terisation for railway track monitoring purposesrdquo Measure-ment vol 82 pp 301ndash312 2016

[8] Z Shen Q Wang Y Shen et al ldquoAccent extraction ofemotional speech based on modified ensemble empiricalmode decompositionrdquo in Proceedings of Instrumentation andMeasurement Technology Conference Austin TX USA May2010

[9] W Guo and P W Tse ldquoA novel signal compression methodbased on optimal ensemble empirical mode decompositionfor bearing vibration signalsrdquo Journal of Sound and Vibrationvol 332 no 2 pp 423ndash441 2013

[10] X Xue J Zhou Y Xu W Zhu and C Li ldquoAn adaptively fastensemble empirical mode decomposition method and itsapplications to rolling element bearing fault diagnosisrdquoMechanical Systems and Signal Processing vol 62-63pp 444ndash459 2015

[11] M Kedadouche M -omas and A Tahan ldquoA comparativestudy between empirical wavelet transforms and empiricalmode decomposition methods application to bearing defectdiagnosisrdquo Mechanical Systems and Signal Processing vol 81pp 88ndash107 2016

[12] W Zhai and X Sun ldquoA detailed model for investigatingvertical interaction between railway vehicle and trackrdquo Ve-hicle System Dynamics vol 23 no 1 pp 603ndash615 1994

[13] X Lei and J Wang ldquoDynamic analysis of the train and slabtrack coupling system with finite elements in a moving frameof referencerdquo Journal of Vibration and Control vol 20 no 9pp 1301ndash1317 2013

[14] W Zhai KWang and C Cai ldquoFundamentals of vehicle-trackcoupled dynamicsrdquo Vehicle System Dynamics vol 47 no 11pp 1349ndash1376 2009

[15] X Kang X B Liu H Y LI et al ldquoPSD of ballastless trackirregularities of high-speed railway (in Chinese)rdquo ScientiaSinica Technologica vol 44 no 7 pp 687ndash696 2014

[16] K Wang P Liu W Zhai C Huang Z Chen and J GaoldquoWheelrail dynamic interaction due to excitation of railcorrugation in high-speed railwayrdquo Science China Techno-logical Sciences vol 58 no 2 pp 226ndash235 2015

[17] C C Chang and C J Lin ldquoLLIBSVM a library for supportvector machinesrdquo ACM Transactions on Intelligent Systemsand Technology vol 2 no 3 pp 1ndash27 2011

[18] 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 the RoyalSociety A-Mathematical Physical and Engineering Sciencesvol 454 no 1971 pp 909ndash995 1998

Table 6 -e accuracy of depth level classification results at dif-ferent measurement noises

Noise level Nil 1 3 5Accuracy of level 1 () 9981 100 100 100Accuracy of level 2 () 9949 9975 9975 9949Accuracy of level 3 () 9949 9975 100 100Total accuracy () 9962 9985 9992 9985

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 2 Level 3

(d)

Figure 13 -e classification results of rail corrugation depth level(a) 0 noise (b) 1 noise (c) 3 noise (d) 5 noise

Shock and Vibration 11

[19] Z Wu and N E Huang ldquoEnsemble empirical mode de-composition a noise-assisted data analysis methodrdquo Ad-vances in Adaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[20] LW Jian and T-C Lim ldquoAutomated detection of diabetes bymeans of higher order spectral features obtained from heartrate signalsrdquo Journal of Medical Imaging and Health In-formatics vol 3 no 3 pp 440ndash447 2013

[21] R Yuvaraj M Murugappan N Mohamed Ibrahim et alldquoDetection of emotions in Parkinsonrsquos disease using higherorder spectral features from brainrsquos electrical activityrdquo Bio-medical Signal Processing and Control vol 14 pp 108ndash1162014

[22] CKY M Hariharan R Ngadiran A H Adom S Yaacoband K Polat ldquoHybrid BBO_PSO and higher order spectralfeatures for emotion and stress recognition from naturalspeechrdquo Applied Soft Computing vol 56 pp 217ndash232 2017

[23] L Saidi J Ben Ali F Fnaiech et al ldquoApplication of higherorder spectral features and support vector machines forbearing faults classificationrdquo ISA Transactions vol 54pp 193ndash206 2015

12 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

wavelength well under the conditions of different depths andmeasurement noises

-e train speeds are 250 kmh 300 kmh and 350 kmhand the wavelength of rail corrugation is estimated re-spectively -e wavelength estimation results with 0 13 and 5 noise levels are shown in Table 4 -e

wavelength estimation error of rail wavelength is almostunchanged at different depths and the maximum value ofwavelength estimation error is not more than 025 -ewavelength estimation error is also very small which in-dicates that the wavelength estimation effect of rail corru-gation is very good Meanwhile the simulation results

9999

100

10001

Wav

elen

gth

(mm

)

True value ofwavelength0 noise

1 noise3 noise5 noise

002 004 006 008 01

110

11005

002 004 006 008 0111995

120

12005

002 004 006 008 011298

1299

130

002 004 006 008 011395

140

1405

002 004 006 008 01e depth of rail corrugation (mm)

1495

150

1505

002 004 006 008 01

(a)

True value ofwavelength0 noise

1 noise3 noise5 noise

9998

100

10002

Wav

elen

gth

(mm

)

250 300 350

250 300 350

1098

110

1102

250 300 3501198

120

1202

250 300 3501298

1299

130

1301

250 300 3501395

140

1405

250 300 350Train speed (kmh)

1495

150

1505

(b)

Figure 12 -e estimation results of rail corrugation wavelength (a) At different depths of rail corrugation (b) At different train speeds

Table 3 Wavelength estimation error at different depths of rail corrugation

CaseDepth 001mm Depth 005mm Depth 01mm

Nil 1 3 5 Nil 1 3 5 Nil 1 3 5100mm 00045 00045 00045 00045 00045 00045 00045 00045 00045 00045 00045 00045110mm 00900 00897 00908 00910 00902 00900 00900 00899 00899 00899 00900 00899120mm 00205 00045 00045 00063 00367 00276 00330 00275 00313 00349 00367 00330130mm 00968 00968 00980 00994 00968 00968 00968 00968 00968 00968 00968 00968140mm 01252 01124 01001 01100 01347 01286 01230 01260 01360 01357 01329 01358150mm 00376 00045 00045 00087 01453 00623 00543 00601 01563 01293 01091 01068

Shock and Vibration 9

illustrate that the proposed method can estimate thewavelength excellent under the conditions of different trainspeeds and measurement noises

43 Depth Classification of Rail Corrugation According tothe values of depth of rail corrugations three depth levels aredivided as shown in Table 5

Bispectrum features estimated wavelength and trainspeed are taken as inputs of SVM and depth level is takenas SVM output -e 80 of all simulation data is used totrain the SVM model and the remaining 20 is used fortesting

-e classification results of rail corrugation depth levelwith different levels of noise are shown in Figure 13

-e reliability of the proposedmethod is evaluated by theaccuracy of the test results -e ratio between the number ofcorrect classification results and the number of total testsamples is defined as the detection accuracy which can beexpressed as

accuray Numcorrect

Numtotaltimes 100 (23)

-e detection results of depth level with 0 1 3 and5 noise levels are shown in Table 6 -e detection accuracyis more than 99 indicating that the extracted corrugationdepth feature is effective -e proposed detection methodhas strong antinoise ability and still has high accuracy underdifferent measurement noises

5 Conclusion

Rail corrugation is a short-wavelength track irregularityexcitation and accordingly causes high-frequency vibrationresponse of wheel A rail detection algorithm based onEEMD and bispectrum features is proposed using wheelvibration acceleration Different frequency components ofwheel vibration response are extracted with EEMD of wheelvibration acceleration signal -e parameters α and N ofEEMD are optimized using the orthogonal coefficient IOand the extreme point distribution index SI which reducesthe mode mixing and calculation time of the de-composition -e main frequency is found after EEMD tocalculate the wavelength-e depth detection is regarded asa classification problem with SVM -e high-frequencysignal of rail corrugation is reconstructed through theselection of IMFs by which not only the interference oftrack random irregularity is avoided but also noise influ-ence Bispectrum features which are extracted from the

Table 4 Wavelength estimation error at different train speeds

CaseTrain speed 250 kmh Train speed 300 kmh Train speed 350 kmh

Nil 1 3 5 Nil 1 3 5 Nil 1 3 5100mm 0 0 0 0 0 0 0 0 00093 00093 00093 00093110mm 00999 00999 00999 00997 00834 00834 00834 00836 00927 00927 00927 00927120mm 01657 01198 00938 01058 0 0 0 0 00093 00093 00093 00093130mm 00999 00999 00999 00999 00999 00999 00999 01001 00907 00907 00910 00910140mm 01234 01070 01016 01107 01325 01275 01217 01265 01428 01412 01391 01386150mm 00399 0 0 00100 01421 00524 00474 00050 02124 01778 01616 01639

Table 5 Definition of depth levels of rail corrugation

Depth levels Depths of rail corrugation (mm)Level 1 a≦ 004Level 2 004lt a≦ 007Level 3 agt 007

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 3Level 2

(a)

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 2 Level 3

(b)

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 2 Level 3

(c)

Figure 13 Continued

10 Shock and Vibration

reconstructed signal combining with train speed and thecorrugation wavelength are input into SVM -e nu-merical simulation results show the proposed detectionalgorithm can accurately identify the existence of railcorrugation -e error between the estimated wavelengthand the real value is small and the accuracy of corrugationdepth level classification is high And the results alsorepresent robustness under different train speeds andmeasurement noises -e proposed rail corrugation de-tection method provides a feasible scheme for the in-service vehicle test in the future

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 they have no conflicts of interest

Acknowledgments

-is work was supported by the National Key Researchand Development Program of China under Grant no2016YFB1200401

References

[1] K H Oostermeijer ldquoReview on short pitch rail corrugationstudiesrdquo Wear vol 265 no 9-10 pp 1231ndash1237 2008

[2] Z Q Jiang D L Si W Li et al ldquoOn rail corrugation of highspeed railway (in Chinese)rdquo China Railway Science vol 35no 4 pp 9ndash14 2014

[3] B M Hopkins and S Taheri ldquoTrack health monitoring usingwaveletrdquo in Proceedings of ASME 2010 Rail TransportationDivision Fall Technical Conference pp 9ndash15 Roanoke VAUSA October 2010

[4] M Molodova Z Li and R Dollevoet ldquoAxle box accelerationmeasurement and simulation for detection of short trackdefectsrdquo Wear vol 271 no 1-2 pp 349ndash356 2011

[5] W Huang W Zhang Y Du et al ldquoDetection of rail cor-rugation based on fiber laser accelerometersrdquo MeasurementScience and Technology vol 24 no 9 article 094014 2013

[6] S Kaewunruen ldquoMonitoring of rail corrugation growth onsharp curves for track maintenance prioritisationrdquo In-ternational Journal of Acoustics and Vibration vol 23 no 1pp 35ndash43 2018

[7] P Salvador V Naranjo R Insa and P Teixeira ldquoAxleboxaccelerations their acquisition and time-frequency charac-terisation for railway track monitoring purposesrdquo Measure-ment vol 82 pp 301ndash312 2016

[8] Z Shen Q Wang Y Shen et al ldquoAccent extraction ofemotional speech based on modified ensemble empiricalmode decompositionrdquo in Proceedings of Instrumentation andMeasurement Technology Conference Austin TX USA May2010

[9] W Guo and P W Tse ldquoA novel signal compression methodbased on optimal ensemble empirical mode decompositionfor bearing vibration signalsrdquo Journal of Sound and Vibrationvol 332 no 2 pp 423ndash441 2013

[10] X Xue J Zhou Y Xu W Zhu and C Li ldquoAn adaptively fastensemble empirical mode decomposition method and itsapplications to rolling element bearing fault diagnosisrdquoMechanical Systems and Signal Processing vol 62-63pp 444ndash459 2015

[11] M Kedadouche M -omas and A Tahan ldquoA comparativestudy between empirical wavelet transforms and empiricalmode decomposition methods application to bearing defectdiagnosisrdquo Mechanical Systems and Signal Processing vol 81pp 88ndash107 2016

[12] W Zhai and X Sun ldquoA detailed model for investigatingvertical interaction between railway vehicle and trackrdquo Ve-hicle System Dynamics vol 23 no 1 pp 603ndash615 1994

[13] X Lei and J Wang ldquoDynamic analysis of the train and slabtrack coupling system with finite elements in a moving frameof referencerdquo Journal of Vibration and Control vol 20 no 9pp 1301ndash1317 2013

[14] W Zhai KWang and C Cai ldquoFundamentals of vehicle-trackcoupled dynamicsrdquo Vehicle System Dynamics vol 47 no 11pp 1349ndash1376 2009

[15] X Kang X B Liu H Y LI et al ldquoPSD of ballastless trackirregularities of high-speed railway (in Chinese)rdquo ScientiaSinica Technologica vol 44 no 7 pp 687ndash696 2014

[16] K Wang P Liu W Zhai C Huang Z Chen and J GaoldquoWheelrail dynamic interaction due to excitation of railcorrugation in high-speed railwayrdquo Science China Techno-logical Sciences vol 58 no 2 pp 226ndash235 2015

[17] C C Chang and C J Lin ldquoLLIBSVM a library for supportvector machinesrdquo ACM Transactions on Intelligent Systemsand Technology vol 2 no 3 pp 1ndash27 2011

[18] 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 the RoyalSociety A-Mathematical Physical and Engineering Sciencesvol 454 no 1971 pp 909ndash995 1998

Table 6 -e accuracy of depth level classification results at dif-ferent measurement noises

Noise level Nil 1 3 5Accuracy of level 1 () 9981 100 100 100Accuracy of level 2 () 9949 9975 9975 9949Accuracy of level 3 () 9949 9975 100 100Total accuracy () 9962 9985 9992 9985

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 2 Level 3

(d)

Figure 13 -e classification results of rail corrugation depth level(a) 0 noise (b) 1 noise (c) 3 noise (d) 5 noise

Shock and Vibration 11

[19] Z Wu and N E Huang ldquoEnsemble empirical mode de-composition a noise-assisted data analysis methodrdquo Ad-vances in Adaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[20] LW Jian and T-C Lim ldquoAutomated detection of diabetes bymeans of higher order spectral features obtained from heartrate signalsrdquo Journal of Medical Imaging and Health In-formatics vol 3 no 3 pp 440ndash447 2013

[21] R Yuvaraj M Murugappan N Mohamed Ibrahim et alldquoDetection of emotions in Parkinsonrsquos disease using higherorder spectral features from brainrsquos electrical activityrdquo Bio-medical Signal Processing and Control vol 14 pp 108ndash1162014

[22] CKY M Hariharan R Ngadiran A H Adom S Yaacoband K Polat ldquoHybrid BBO_PSO and higher order spectralfeatures for emotion and stress recognition from naturalspeechrdquo Applied Soft Computing vol 56 pp 217ndash232 2017

[23] L Saidi J Ben Ali F Fnaiech et al ldquoApplication of higherorder spectral features and support vector machines forbearing faults classificationrdquo ISA Transactions vol 54pp 193ndash206 2015

12 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

illustrate that the proposed method can estimate thewavelength excellent under the conditions of different trainspeeds and measurement noises

43 Depth Classification of Rail Corrugation According tothe values of depth of rail corrugations three depth levels aredivided as shown in Table 5

Bispectrum features estimated wavelength and trainspeed are taken as inputs of SVM and depth level is takenas SVM output -e 80 of all simulation data is used totrain the SVM model and the remaining 20 is used fortesting

-e classification results of rail corrugation depth levelwith different levels of noise are shown in Figure 13

-e reliability of the proposedmethod is evaluated by theaccuracy of the test results -e ratio between the number ofcorrect classification results and the number of total testsamples is defined as the detection accuracy which can beexpressed as

accuray Numcorrect

Numtotaltimes 100 (23)

-e detection results of depth level with 0 1 3 and5 noise levels are shown in Table 6 -e detection accuracyis more than 99 indicating that the extracted corrugationdepth feature is effective -e proposed detection methodhas strong antinoise ability and still has high accuracy underdifferent measurement noises

5 Conclusion

Rail corrugation is a short-wavelength track irregularityexcitation and accordingly causes high-frequency vibrationresponse of wheel A rail detection algorithm based onEEMD and bispectrum features is proposed using wheelvibration acceleration Different frequency components ofwheel vibration response are extracted with EEMD of wheelvibration acceleration signal -e parameters α and N ofEEMD are optimized using the orthogonal coefficient IOand the extreme point distribution index SI which reducesthe mode mixing and calculation time of the de-composition -e main frequency is found after EEMD tocalculate the wavelength-e depth detection is regarded asa classification problem with SVM -e high-frequencysignal of rail corrugation is reconstructed through theselection of IMFs by which not only the interference oftrack random irregularity is avoided but also noise influ-ence Bispectrum features which are extracted from the

Table 4 Wavelength estimation error at different train speeds

CaseTrain speed 250 kmh Train speed 300 kmh Train speed 350 kmh

Nil 1 3 5 Nil 1 3 5 Nil 1 3 5100mm 0 0 0 0 0 0 0 0 00093 00093 00093 00093110mm 00999 00999 00999 00997 00834 00834 00834 00836 00927 00927 00927 00927120mm 01657 01198 00938 01058 0 0 0 0 00093 00093 00093 00093130mm 00999 00999 00999 00999 00999 00999 00999 01001 00907 00907 00910 00910140mm 01234 01070 01016 01107 01325 01275 01217 01265 01428 01412 01391 01386150mm 00399 0 0 00100 01421 00524 00474 00050 02124 01778 01616 01639

Table 5 Definition of depth levels of rail corrugation

Depth levels Depths of rail corrugation (mm)Level 1 a≦ 004Level 2 004lt a≦ 007Level 3 agt 007

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 3Level 2

(a)

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 2 Level 3

(b)

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 2 Level 3

(c)

Figure 13 Continued

10 Shock and Vibration

reconstructed signal combining with train speed and thecorrugation wavelength are input into SVM -e nu-merical simulation results show the proposed detectionalgorithm can accurately identify the existence of railcorrugation -e error between the estimated wavelengthand the real value is small and the accuracy of corrugationdepth level classification is high And the results alsorepresent robustness under different train speeds andmeasurement noises -e proposed rail corrugation de-tection method provides a feasible scheme for the in-service vehicle test in the future

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 they have no conflicts of interest

Acknowledgments

-is work was supported by the National Key Researchand Development Program of China under Grant no2016YFB1200401

References

[1] K H Oostermeijer ldquoReview on short pitch rail corrugationstudiesrdquo Wear vol 265 no 9-10 pp 1231ndash1237 2008

[2] Z Q Jiang D L Si W Li et al ldquoOn rail corrugation of highspeed railway (in Chinese)rdquo China Railway Science vol 35no 4 pp 9ndash14 2014

[3] B M Hopkins and S Taheri ldquoTrack health monitoring usingwaveletrdquo in Proceedings of ASME 2010 Rail TransportationDivision Fall Technical Conference pp 9ndash15 Roanoke VAUSA October 2010

[4] M Molodova Z Li and R Dollevoet ldquoAxle box accelerationmeasurement and simulation for detection of short trackdefectsrdquo Wear vol 271 no 1-2 pp 349ndash356 2011

[5] W Huang W Zhang Y Du et al ldquoDetection of rail cor-rugation based on fiber laser accelerometersrdquo MeasurementScience and Technology vol 24 no 9 article 094014 2013

[6] S Kaewunruen ldquoMonitoring of rail corrugation growth onsharp curves for track maintenance prioritisationrdquo In-ternational Journal of Acoustics and Vibration vol 23 no 1pp 35ndash43 2018

[7] P Salvador V Naranjo R Insa and P Teixeira ldquoAxleboxaccelerations their acquisition and time-frequency charac-terisation for railway track monitoring purposesrdquo Measure-ment vol 82 pp 301ndash312 2016

[8] Z Shen Q Wang Y Shen et al ldquoAccent extraction ofemotional speech based on modified ensemble empiricalmode decompositionrdquo in Proceedings of Instrumentation andMeasurement Technology Conference Austin TX USA May2010

[9] W Guo and P W Tse ldquoA novel signal compression methodbased on optimal ensemble empirical mode decompositionfor bearing vibration signalsrdquo Journal of Sound and Vibrationvol 332 no 2 pp 423ndash441 2013

[10] X Xue J Zhou Y Xu W Zhu and C Li ldquoAn adaptively fastensemble empirical mode decomposition method and itsapplications to rolling element bearing fault diagnosisrdquoMechanical Systems and Signal Processing vol 62-63pp 444ndash459 2015

[11] M Kedadouche M -omas and A Tahan ldquoA comparativestudy between empirical wavelet transforms and empiricalmode decomposition methods application to bearing defectdiagnosisrdquo Mechanical Systems and Signal Processing vol 81pp 88ndash107 2016

[12] W Zhai and X Sun ldquoA detailed model for investigatingvertical interaction between railway vehicle and trackrdquo Ve-hicle System Dynamics vol 23 no 1 pp 603ndash615 1994

[13] X Lei and J Wang ldquoDynamic analysis of the train and slabtrack coupling system with finite elements in a moving frameof referencerdquo Journal of Vibration and Control vol 20 no 9pp 1301ndash1317 2013

[14] W Zhai KWang and C Cai ldquoFundamentals of vehicle-trackcoupled dynamicsrdquo Vehicle System Dynamics vol 47 no 11pp 1349ndash1376 2009

[15] X Kang X B Liu H Y LI et al ldquoPSD of ballastless trackirregularities of high-speed railway (in Chinese)rdquo ScientiaSinica Technologica vol 44 no 7 pp 687ndash696 2014

[16] K Wang P Liu W Zhai C Huang Z Chen and J GaoldquoWheelrail dynamic interaction due to excitation of railcorrugation in high-speed railwayrdquo Science China Techno-logical Sciences vol 58 no 2 pp 226ndash235 2015

[17] C C Chang and C J Lin ldquoLLIBSVM a library for supportvector machinesrdquo ACM Transactions on Intelligent Systemsand Technology vol 2 no 3 pp 1ndash27 2011

[18] 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 the RoyalSociety A-Mathematical Physical and Engineering Sciencesvol 454 no 1971 pp 909ndash995 1998

Table 6 -e accuracy of depth level classification results at dif-ferent measurement noises

Noise level Nil 1 3 5Accuracy of level 1 () 9981 100 100 100Accuracy of level 2 () 9949 9975 9975 9949Accuracy of level 3 () 9949 9975 100 100Total accuracy () 9962 9985 9992 9985

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 2 Level 3

(d)

Figure 13 -e classification results of rail corrugation depth level(a) 0 noise (b) 1 noise (c) 3 noise (d) 5 noise

Shock and Vibration 11

[19] Z Wu and N E Huang ldquoEnsemble empirical mode de-composition a noise-assisted data analysis methodrdquo Ad-vances in Adaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[20] LW Jian and T-C Lim ldquoAutomated detection of diabetes bymeans of higher order spectral features obtained from heartrate signalsrdquo Journal of Medical Imaging and Health In-formatics vol 3 no 3 pp 440ndash447 2013

[21] R Yuvaraj M Murugappan N Mohamed Ibrahim et alldquoDetection of emotions in Parkinsonrsquos disease using higherorder spectral features from brainrsquos electrical activityrdquo Bio-medical Signal Processing and Control vol 14 pp 108ndash1162014

[22] CKY M Hariharan R Ngadiran A H Adom S Yaacoband K Polat ldquoHybrid BBO_PSO and higher order spectralfeatures for emotion and stress recognition from naturalspeechrdquo Applied Soft Computing vol 56 pp 217ndash232 2017

[23] L Saidi J Ben Ali F Fnaiech et al ldquoApplication of higherorder spectral features and support vector machines forbearing faults classificationrdquo ISA Transactions vol 54pp 193ndash206 2015

12 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

reconstructed signal combining with train speed and thecorrugation wavelength are input into SVM -e nu-merical simulation results show the proposed detectionalgorithm can accurately identify the existence of railcorrugation -e error between the estimated wavelengthand the real value is small and the accuracy of corrugationdepth level classification is high And the results alsorepresent robustness under different train speeds andmeasurement noises -e proposed rail corrugation de-tection method provides a feasible scheme for the in-service vehicle test in the future

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 they have no conflicts of interest

Acknowledgments

-is work was supported by the National Key Researchand Development Program of China under Grant no2016YFB1200401

References

[1] K H Oostermeijer ldquoReview on short pitch rail corrugationstudiesrdquo Wear vol 265 no 9-10 pp 1231ndash1237 2008

[2] Z Q Jiang D L Si W Li et al ldquoOn rail corrugation of highspeed railway (in Chinese)rdquo China Railway Science vol 35no 4 pp 9ndash14 2014

[3] B M Hopkins and S Taheri ldquoTrack health monitoring usingwaveletrdquo in Proceedings of ASME 2010 Rail TransportationDivision Fall Technical Conference pp 9ndash15 Roanoke VAUSA October 2010

[4] M Molodova Z Li and R Dollevoet ldquoAxle box accelerationmeasurement and simulation for detection of short trackdefectsrdquo Wear vol 271 no 1-2 pp 349ndash356 2011

[5] W Huang W Zhang Y Du et al ldquoDetection of rail cor-rugation based on fiber laser accelerometersrdquo MeasurementScience and Technology vol 24 no 9 article 094014 2013

[6] S Kaewunruen ldquoMonitoring of rail corrugation growth onsharp curves for track maintenance prioritisationrdquo In-ternational Journal of Acoustics and Vibration vol 23 no 1pp 35ndash43 2018

[7] P Salvador V Naranjo R Insa and P Teixeira ldquoAxleboxaccelerations their acquisition and time-frequency charac-terisation for railway track monitoring purposesrdquo Measure-ment vol 82 pp 301ndash312 2016

[8] Z Shen Q Wang Y Shen et al ldquoAccent extraction ofemotional speech based on modified ensemble empiricalmode decompositionrdquo in Proceedings of Instrumentation andMeasurement Technology Conference Austin TX USA May2010

[9] W Guo and P W Tse ldquoA novel signal compression methodbased on optimal ensemble empirical mode decompositionfor bearing vibration signalsrdquo Journal of Sound and Vibrationvol 332 no 2 pp 423ndash441 2013

[10] X Xue J Zhou Y Xu W Zhu and C Li ldquoAn adaptively fastensemble empirical mode decomposition method and itsapplications to rolling element bearing fault diagnosisrdquoMechanical Systems and Signal Processing vol 62-63pp 444ndash459 2015

[11] M Kedadouche M -omas and A Tahan ldquoA comparativestudy between empirical wavelet transforms and empiricalmode decomposition methods application to bearing defectdiagnosisrdquo Mechanical Systems and Signal Processing vol 81pp 88ndash107 2016

[12] W Zhai and X Sun ldquoA detailed model for investigatingvertical interaction between railway vehicle and trackrdquo Ve-hicle System Dynamics vol 23 no 1 pp 603ndash615 1994

[13] X Lei and J Wang ldquoDynamic analysis of the train and slabtrack coupling system with finite elements in a moving frameof referencerdquo Journal of Vibration and Control vol 20 no 9pp 1301ndash1317 2013

[14] W Zhai KWang and C Cai ldquoFundamentals of vehicle-trackcoupled dynamicsrdquo Vehicle System Dynamics vol 47 no 11pp 1349ndash1376 2009

[15] X Kang X B Liu H Y LI et al ldquoPSD of ballastless trackirregularities of high-speed railway (in Chinese)rdquo ScientiaSinica Technologica vol 44 no 7 pp 687ndash696 2014

[16] K Wang P Liu W Zhai C Huang Z Chen and J GaoldquoWheelrail dynamic interaction due to excitation of railcorrugation in high-speed railwayrdquo Science China Techno-logical Sciences vol 58 no 2 pp 226ndash235 2015

[17] C C Chang and C J Lin ldquoLLIBSVM a library for supportvector machinesrdquo ACM Transactions on Intelligent Systemsand Technology vol 2 no 3 pp 1ndash27 2011

[18] 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 the RoyalSociety A-Mathematical Physical and Engineering Sciencesvol 454 no 1971 pp 909ndash995 1998

Table 6 -e accuracy of depth level classification results at dif-ferent measurement noises

Noise level Nil 1 3 5Accuracy of level 1 () 9981 100 100 100Accuracy of level 2 () 9949 9975 9975 9949Accuracy of level 3 () 9949 9975 100 100Total accuracy () 9962 9985 9992 9985

0 200 400 600 800 1000 1200 1400Testing set

1

2

3

Dep

th le

vel

Level 1 Level 2 Level 3

(d)

Figure 13 -e classification results of rail corrugation depth level(a) 0 noise (b) 1 noise (c) 3 noise (d) 5 noise

Shock and Vibration 11

[19] Z Wu and N E Huang ldquoEnsemble empirical mode de-composition a noise-assisted data analysis methodrdquo Ad-vances in Adaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[20] LW Jian and T-C Lim ldquoAutomated detection of diabetes bymeans of higher order spectral features obtained from heartrate signalsrdquo Journal of Medical Imaging and Health In-formatics vol 3 no 3 pp 440ndash447 2013

[21] R Yuvaraj M Murugappan N Mohamed Ibrahim et alldquoDetection of emotions in Parkinsonrsquos disease using higherorder spectral features from brainrsquos electrical activityrdquo Bio-medical Signal Processing and Control vol 14 pp 108ndash1162014

[22] CKY M Hariharan R Ngadiran A H Adom S Yaacoband K Polat ldquoHybrid BBO_PSO and higher order spectralfeatures for emotion and stress recognition from naturalspeechrdquo Applied Soft Computing vol 56 pp 217ndash232 2017

[23] L Saidi J Ben Ali F Fnaiech et al ldquoApplication of higherorder spectral features and support vector machines forbearing faults classificationrdquo ISA Transactions vol 54pp 193ndash206 2015

12 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

[19] Z Wu and N E Huang ldquoEnsemble empirical mode de-composition a noise-assisted data analysis methodrdquo Ad-vances in Adaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[20] LW Jian and T-C Lim ldquoAutomated detection of diabetes bymeans of higher order spectral features obtained from heartrate signalsrdquo Journal of Medical Imaging and Health In-formatics vol 3 no 3 pp 440ndash447 2013

[21] R Yuvaraj M Murugappan N Mohamed Ibrahim et alldquoDetection of emotions in Parkinsonrsquos disease using higherorder spectral features from brainrsquos electrical activityrdquo Bio-medical Signal Processing and Control vol 14 pp 108ndash1162014

[22] CKY M Hariharan R Ngadiran A H Adom S Yaacoband K Polat ldquoHybrid BBO_PSO and higher order spectralfeatures for emotion and stress recognition from naturalspeechrdquo Applied Soft Computing vol 56 pp 217ndash232 2017

[23] L Saidi J Ben Ali F Fnaiech et al ldquoApplication of higherorder spectral features and support vector machines forbearing faults classificationrdquo ISA Transactions vol 54pp 193ndash206 2015

12 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom