Research ArticleModeling and Optimization for Fault Diagnosis ofElectromechanical Systems Based on Zero Crossing Algorithm

Qing Chen 1 Tao Liu 1 Xing Wu12 and Hua Li1

1Key Laboratory for Advanced Equipment Intelligent Manufacturing Technology of Yunnan ProvinceKunming University of Science and Technology Kunming 650500 China2Yunnan Vocational College of Mechanical and Electrical Technology Kunming 650203 China

Correspondence should be addressed to Tao Liu kmliutaoaliyuncom

Received 24 April 2020 Accepted 30 June 2020 Published 27 July 2020

Guest Editor Jian Huang

Copyright copy 2020 Qing Chen et al (is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

(e demand of system security and reliability in the modern industrial process is ever-increasing and fault diagnosis technologyhas always been a crucial research direction in the control field Due to the complexity nonlinearity and coupling of multi-tudinous control systems precise system modeling for fault diagnosis is attracting more attention In this paper we propose animproved method of electromechanical systems fault diagnosis based on zero-crossing (ZC) algorithm which can present thecalculation model of zero-crossing rate (ZCR) and optimize the parameters of ZC algorithm by establishing a criterion functionmodel to improve the diagnosis accuracy and robustness of ZC characteristic model (e simulation validates the influence ofdifferent signal-to-noise ratio (SNR) on ZC feature recognition ability and indicates that the within-between distance model iseffective to enhance the diagnose accuracy of ZC feature Finally the method is applied to the diagnosis of motor fault bearingwhich confirms the necessity and effectiveness of the model improvement and parameter optimization and verifies the robustnessto the load

1 Introduction

(e electromechanical system is a vital part in the process ofmodern industrial production and manufacturing (e re-search of fault diagnosis and prediction methods which canbe applied to nonlinear electromechanical systems and re-alize the real-time monitoring diagnosis prediction andstate-based maintenance of electromechanical system hasan important significance for improving the safety and re-liability of electromechanical system As a crucial compo-nent of electromechanical systems the stability andreliability of motor operation directly affect the normaloperation of electromechanical systems especially in thefields of CNC machine tools [1] robots [2] and aircraft [3]which require relatively high production accuracy effi-ciency and reliability In particular rolling element bearings(REBs) are the most commonly machine elements used inalmost all rotary machinery and is also the vulnerablecomponent of rotating machinery [4 5] (e health status of

REBs directly affects the functioning of the motor whichaccounts for almost 40ndash50 of motor fault [6 7] As aconsequence the fault diagnosis of motor REBs plays acrucial role to the smooth reliable and safe handling of thewhole electromechanical system

Mathematical modeling and analysis are useful to designand study control systems [8ndash10] and also can be used toanalyze the behavior of the real systems under differentoperating conditions test signal processing methods andnew detection and prognosis techniques Over the last fewdecades the dynamic modeling of rolling bearings has beenextensively studied McFadden and Smith established amodel to illustrate the vibration characteristics of a singlepoint defect in the inner race of REB under the same radialload (e model comprehensively considers the impacts ofbearing geometry bearing size distribution transfer func-tion and vibration index attenuation By comparing theradial vibration spectra predicted with measured the ex-cellent performance of the model is validated [11] Sawalhi

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 9267838 13 pageshttpsdoiorg10115520209267838

and Randall proposed a combination of gear and bearingdynamic model In this model the outspread faults of theinner and outer rings of REBs in the presence of gear in-teractions can be studied [12] Due to the distinct superiorityof U-model in nonlinear control the development ofU-model based gear and bearing state prediction is prom-ising [13 14] For the sake of research of the nonlineardynamic characteristics of the REB system containing sur-face defect a theoretical model is presented by Rafsanjaniet al In order to investigate the linear stability of the de-fective bearing rotor system with changes in the parametersof the conversion system the classic Floquet theorem wasembedded in this model In the solution of this model thepeak-to-peak frequency response of the system was obtainedin some cases and the basic paths of offset collimation andchaotic motion for different internal radial clearance weredetermined [15] Based on the research of the dynamicmodel on REB the fault diagnosis using the measurementsignal in the system has been developed rapidly In theprocess of system operation the symptoms of failure will bereflected in the detection information (measurement signal)the resultful features of the signal will be extracted by meansof analysis and then the diagnosis scheme will be establishedbased upon the fault symptom analysis and the priorknowledge of the healthy system Nevertheless the quality offeatures extracted from signals is always a thorny problemSince the vibration data of a faulty rolling bearing is usuallyunstable and nonlinear and contains relatively weak faultfeatures it has always attracted wide attention from re-searchers Over the past decades a mass of approaches offeature extraction for REB fault diagnosing have beenproposed which can be roughly summarized as the fol-lowing categories according to the distinction of principlesand properties Firstly feature extraction based on thetraditional time domain parameters for example crestfactor peak-to-peak value kurtosis root-mean squareshape factor and standard deviation etc [16ndash18] secondlyfrequency domain parameters such as power spectraldensity power spectrum [19] thirdly analysis based ontime-frequency domain methods for instance spectrogramand wavelet transform [20ndash22] finally based on multipleparameters the mixed feature extraction realized for ex-ample the method of [23 24] have extracted blended pa-rameter features of time domain frequency domain andtime-frequency

It is found that in most former feature extractiontechniques a large number of complex calculations areneeded for vibration signal data for improving the accuracyof fault identification Compared with these traditionalmethods as ZC features generate directly from the count ofthe ZC interval in the time domain ZC feature algorithm hasan obvious advantage which can well reflect the change ofsignal frequency with fast convergence and is easy to cal-culate (erefore ZC algorithm has been successfully ap-plied to speech recognition [25] vehicle classification [26]and biomedical applications [27 28] Moreover the ZCmethod is also commonly used in signal processing andmechanical fault diagnosis William and Hoffman showedthe ZC method combined with an ANN is effective in early

detection and diagnosis of bearing failures [29] Liu et alproposed a bearing performance degradation estimationmethod based on the ZC characteristics and coupled hiddenMarkov model and proved that the ZC features can detectthe early degradation stage of the bearing [30] Ukil et alpresented a feature extraction method of current ZC mo-ment to detect short-circuit fault of stator winding of seriesasynchronous induction motors [31] Gonzalez and Kinsnershowed that ZC could be used to identify different parts ofthe composite signal and the advantage of the ZC com-puting immediately in the time domain is attractive to realtime implementation [32] Waghmare et al presented amethodology based on piecewise energy and correspondingmean of signal ZC in environmental noise which has abroad application prospect in dealing with the underwatertarget-radiated noise [33] For the first time Nayana andGeethanjali used time domain parameters waveform pa-rameters slope sign changes (SSC) simple sign integral andWilson amplitude to established mean absolute value andZC to identify failures of motors and by comparing withconventional features they proved the proposed featuresperform better [34]

Literatures [29 30] demonstrated the advantages of ZCfeatures in bearing fault diagnosis such as small compu-tation and fast speed However by reason of no stationaryand nonlinearity of REB fault signals the recognition rate ofthe extracted ZC features in REB fault recognition is lowespecially in the case of low signal-to-noise ratio (SNR) Forsolving this problem this paper firstly analyzes the recog-nition ability of ZC features under different SNR conditionsthen proposes an adaptive method to enhance the capable ofrecognition fault and verifies the recognition rate throughneural network (e improved fault diagnosis method forelectromechanical system which optimized the calculationmodel of ZCR and optimized the parameters of ZC algo-rithm by establishing a criterion function model greatlyimproves the fault recognition rate and is robust to the load

(e remaining of this article is organized as follows InSection 2 the theory of ZC feature algorithm is brieflyreviewed and the improved model of ZC calculation and themethod of parameters optimized through modeling a cri-terion function with within-between distance are intro-duced and Section 3 simulation signals are used to verify theinfluence of ZC feature recognition ability at disparate SNRand verifies the validation of the presented method InSection 4 the experiment is analyzed to verify the methodand the conclusions are given in the Section 5

2 Theoretical Background

21 ZC Characteristic Feature Algorithm (e measureswidely used to represent the characteristics of ZC include themean ZCR density of the time interval between continuousZCs and excess threshold measure According to the ex-perience of literature [33] compared with other ZC mea-surements the ZC feature of excess threshold measurementis more conducive to fault diagnosis at relatively high SNR(erefore in this paper the ZC feature of excess thresholdmeasurement is used to method improvement and

2 Mathematical Problems in Engineering

parameter optimization for achieving the more accuracy offault diagnosis at low SNR

In order to describe the short-time waveform the timedomain signal collected should be divided into a diminutiveobservation window firstly In addition the measure of theobservation window should be greater than the maximumexpected duration between continuous ZCs TL is defined asthe measure of the observation window TQ represents themaximum expected duration between the continuous ZCsand TQ le TL (en divide the duration range [0 TQ] into Qintervals by the threshold of Q+ 1 and define Ti as the ithinterval Let Ci represent the count of ZC intervals withduration in the range of Tjminus1 and Tj and x(n) is the short-time waveform with the data sequence N then Ci can beobtained by

Ci 12

1113944

N

n2[sgn[x(n)] minus sgn[x(n minus 1)]] (1)

where the sgn[x] is sign function and expressed as

sgn[x(n)] 1(xge 0)

minus1(xlt 0)1113896 (2)

(en the eigenvector of ZC counts is normalized by theobservation window and defined as

Fcount C

TL

1

TL

C1C2 CQ1113872 1113873 (3)

and the function of excess thresholdmeasurement can becalculated by the following formula

f Ti( 1113857 12

1113944

Qminus1

kj

Ck+1 Tk + Tk+1( 1113857 (4)

where 12Ck+1(Tk + Tk+1) is used to approximately calculatethe sum of ZC durations in the range Tk ltT≪Tk+1

(ereby the eigenvector of excess threshold measure-ment can be obtained by

Fduration 1

TL

f T0( 1113857 f T0( 1113857 f TQminus11113872 11138731113960 1113961 (5)

22 Criterion Function Modeling (e purpose of featureextraction is to acquire the most available information relatedto the equipment condition so as to realize fault identification(e classification ability of feature vectors directly determinesthe ability of fault recognition Based on the idea of Fishercriterion [35] to calculate the distance between categories andwithin categories the recognition ability of features can beimproved by finding a functional relationship between theintraclass distance and interclass distance of features In otherwords the smaller the intraclass distance and the larger theinterclass distance of the feature clustering results the higherthe discrimination degree of the classification results will be(erefore we can construct a criterion function that reflectsthe within-between distance

Assume that the feature set (x1 x2 xN) to beclassified is classified as class C x

(j)i j 1 2 C i 1113966

1 2 nj (e data quantity of class j is nj (e intraclassdistance of class j can be expressed as follows

S(j)w

1nj

1113944

nj

i1x

(j)i minus mj1113872 1113873 x

(j)i minus mj1113872 1113873

T (6)

where mj is the average of the samples of class j andmj 1nj 1113936

nj

i1 x(j)i (j 1 2 C)

Let S(k)w and mk denote the intraclass distance and the

samplesmean of class k( kne j and kleC) respectively the inter-class distance between class j and k can be expressed as follows

SB 1113944 mj minus mk1113872 1113873 mj minus mk1113872 1113873T (7)

(en the function represents the ability to recognizebetween classes can be expressed as follows

I SB

S(j)w + S

(k)w

(8)

On the premise that the highest recognition capacity ofthe two categories is not weakened let us assume that thediscrepancy of I value between other categories is thesmallest and the recognition rate is the best and then thecriterion function can be expressed by the following relation

minϕ Imax Imin( 1113857 ϕ Imax minus Imin (9)

where Imax and Imin represent the maximum and minimumI values of all categories respectively

23 Parameter-Optimized ZC Based on Within-BetweenDistanceModels According to formula (1) we know that theZC count has nothing to do with the magnitude of signalamplitude and is only related to the positive and negative signsof signals However ZC count is sensitive to the presence ofnoise as random noises repeatedly cross the coordinate axis inthe background a large number of false ZCR will be generatedand affect the recognition result Equation (1) is suitable fordiscrete calculation but not conducive to the analysis of relatedsignal waveform (erefore formula (1) is firstly transformedinto the following equation

Ci 12

1113944

N

n2[1 minus sgn[x(n) middot x(n minus 1)]] (10)

Consider further the influence of random noise increasea threshold ε to offset part of false ZCR generated by noiserepeatedly crossing the coordinate axis (erefore the zero-crossing calculation formula can be revised into the form offormula (11)

Ci 12

1113944

N

n2[1 minus sgn[x(n) middot x(n minus 1) + ε]] (11)

(e subsequent work is the optimization of threshold ε(e method of extracting ZC features from ZC duration isdescribed in Section 21 It is noteworthy that the length ofthe observation window must greater than the maximumexpected time interval between the continuous ZC intervalsso the ZC feature should ensure that the minimum

Mathematical Problems in Engineering 3

frequency of interest can be discovered In order to improvedata utilization and maintain data continuity 50 overlaprate is set When the maximum expected time is divided intoQ intervals by the threshold of Q+ 1 Q group eigenvectorsare generated We know that the extracted features aredifferent on various time scales therefore the eigenvector isaffected by the length and number of observation windowLiterature [36] has discussed the optimization method ofthese two thresholds here the optimized window data areused directly Keep the optimized window data unchangedand change the value of ε to extract the ZC feature re-spectively (e ε must be changed by the specified step sizeand adjusted for ZC features changes Calculate the inter-class distance and intraclass distance of each ZC feature(en values of I with diverse values of ε are calculatedaccording to formula (9) If the critical point is found theoptimized ε value is obtained (e flow chart of improvedmethod of ZC features extraction is manifested in Figure 1

Detailed descriptions of the flow chart in Figure 1

Step 1 Input original vibration signalsStep 2 Calculate the peak-to-peak value as the basis forcalculating the search step size and search scopeStep 3 Set the search range and step size of ε(e searchstep size needs to be based on the experience of makingmultiple attempts at the actual signal In the simulationone-thousandth of the peak value is used as the steplength and 20 steps are used as the search rangeStep 4 Extract ZC feature vectors of every signal atdifferent ε values (e number and length of observationwindow were optimized before feature extraction andthe data overlapped by 50Step 5 Calculate the value of I between different statecategories according to formula (8) Suppose there are 4kinds of state signals and then 6 values of I are formedone for every 2 kinds of signals ε value is searched 20times and then a 6times 20 matrix is formedStep 6 Calculate the maximum and minimum of I foreach search guaranteed the highest recognition ca-pacity of the two categories is not weakened and in-crease the value of I between the two classes that aremost difficult to recognize So when formula (9) issatisfied the optimization is achieved According to thestep size of each signal and the number of steps theoptimized ε value of each signal can be obtainedStep 7 According to the optimized ε value of eachsignal extract the optimized ZC featuresStep 8 Bring the optimized features into neural net-work for fault diagnosis and verification and the resultsare analyzed

3 Simulation Analysis

For the sake of verifying the necessity and effectiveness of thepresented method four simulated signals of normal innerring fault rolling element fault and outer ring fault of rolling

bearing are used for analysis (e simulation signals areobtained from the following formula

x(t) 1113936M

i1Ais t minus iT minus τi( 1113857 + n(i)

Ai A0 cos 2πQt + φA( 1113857 + CA

s(t) eminus Bt sin 2πfnt + φw( 1113857

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

(12)

where the system sampling frequency fs 12000Hz theresonant frequency fn 3000Hz the inner ring fault fre-quency fi 162Hz the rolling element fault frequencyfb 142Hz the outer ring fault frequency fi 142Hz therotational frequency fr 30Hz and the damping ratioB 400 Four simulated signals are a group a total of threegroups (ree random noises with a SNR of 30 dB 5 dB andminus5 dB are added separately to three groups signal (egrouping of simulation signals is shown in Table 1

31 Analysis of ZC Feature Noise Resistance In orderto make the extracted features represent the running stateinformation of the bearing effectively the length of the datasegment calculated with ZC feature should contain at least onecomplete rotation period (e rotational frequency is 30Hz sothe length of an observation window must greater than 130secondswhich contains 360 points at the sampling frequency of12K Follow the method in Section 23 above the data frag-ment length is 512 and the number of fragment is 8 Asdescribed earlier the data overlap rate is set to 50 whichcontains 256 points at the sampling frequency of 12K Eachstate data are divided into 40 samples and contain data of 05seconds (6000 points) Figure 2 shows the ZC features(FVduration) of the four state signals extracted from each set ofsignals when SNR 30 5 and minus5 From the left to the right ofthe figure is the distribution of FVduration of four state signalswhen SNR is 30 5 and minus5 (e graphs above the figure are thedistribution of all samples under a certain feature and thebelow ones are the differentiation of all eigenvectors under acertain sample It can be seen from the three groups of graphsthat the ZC feature is very sensitive to random noise and therecognition ability between ZC features of various state signalsis decreasing with random noise increasing When SNR 30FVduration still has fairish recognition ability however whenSNR minus5 recognition ability of FVduration begins to fail

32 Analysis of ε Value On the basis of the method de-scription in Section 23 extract the ZC features and calculatethe various values of I with different values of ε(en calculatethe values of ϕ according to formula (9) and find theminimumvalue of ϕ (e relationship between ϕ and ε is shown inFigures 3ndash5 under different kinds of SNRs(e figures indicatethat with the increase of the ε value the capacity of distinguishfault is improved but they will not continue indefinitely Inother words specifically value of ϕwould not always decreasebut has a critical value such as the values marked with arrowsin three figures and then with the value of ϕ increasing therecognition capacity of certain two types will decrease(erefore the optimized values of ε can be obtained in adifferent kind of SNR (e value of ε is represented by the

4 Mathematical Problems in Engineering

number of steps and the step sizes of the four state signalsunder different kind SNR are shown in Table 2

It can be seen from Figures 3ndash5 that the optimal numberof steps correspond to the minimum ϕ combined with thestep size in Table 2 and the optimized ε value can be cal-culated as shown in Table 3

(en the ZC features extracted from above optimized εvalues are shown in Figure 6

FromFigure 6 we can clearly see that all samples of signals indiverse states in the same feature can be separated from each

other and similarly all features of signals in diverse states underthe same sample can be separated fromeach other Consequentlythe ZC characteristics of the four states can be recognizedCompared with the identification result before optimization inFigure 2 the effectiveness of the method is proved

33 Optimization Validation In order to further prove theavailability of themethod the optimized ZC eigenvectors areidentified by three-layer feed-forward neural network andthe result of verification is shown in Table 4 which is the

099

098

1

097

096

094

093

095

0 5 10 15 20Samples

Nor

mal

ized

dur

atio

n

S11S12

S13S14

25 30 35 40

(a)

1

098

096

094

092

Nor

mal

ized

dur

atio

n

S21S22

S23S24

Samples0 5 10 15 20 25 30 35 40

(b)

S31S32

S33S34

1

098

096

094

092

Nor

mal

ized

dur

atio

nSamples

0 5 10 15 20 25 30 35 40

(c)1

08

06

04

021 2 3 4 5 6 7 8

Feature

Nor

mal

ized

dur

atio

n

S11S12

S13S14

(d)

1

08

06

04

021 2 3 4 5 6 7 8

Feature

Nor

mal

ized

dur

atio

n

S21S22

S23S24

(e)

1

08

06

04

021 2 3 4 5 6 7 8

Feature

Nor

mal

ized

dur

atio

n

S31S32

S33S34

(f )

Figure 2 ZC features of four state signals (from left to right SNR is 30 5 and minus5 respectively)

Neural networkfault diagnosis

Extract ZC feature vectors ofevery signal at different ε values

Calculate the I value of differentstate categories according to

formula (8)

Calculate the peak-to-peak value

Set the search rangeand step size of ε

Find the step size that satisfiesformula (9) and

obtain the ε value of each signal

Extract the optimized ZCfeature according to the

optimized ε value

Input originalvibration signalStart

End

Figure 1 Flow chart of the optimized parameter

Table 1 (e grouping of simulation signals

Index SNR 30 SNR 5 SNR minus5Simulated normalno fault S11 S21 S31Simulated inner ring fault S12 S22 S32Simulated rolling element fault S13 S23 S33Simulated outer ring fault S14 S24 S34

Mathematical Problems in Engineering 5

average recognition rate of 40 replicates It can be dem-onstrated from Table 4 that this method has a high faultidentification rate for REB simulation signals

4 Experiment Validation

(e necessity and effectiveness of the improved model andparameter optimization are verified by simulation experi-ments In practical applications the identification of rollingbearing faults usually requires the identification method tobe robust for load so the experiment uses the bearing datafrom the Case Western Reserve University Bearing DataCenter to verify whether the proposed method is effective

41 Experiment Data and Schemes (e bearing status dataset is derived from [37] (e type of test bearing is deepgroove bearing and the data are sampled by the acceler-ometer at 12KHz from the drive end of the motor Single-point failure diameter is 0007 inches (e outer ring failureis located at 6 orsquoclock For the motor loads 0 1 2 and 3motor speeds are from 1797 to 1730 RPM Table 5 details theload and speed

(e whole experiment consists of three parts (e firstexperiment verifies the identification effect of four bearingstates under load 0 HP and compares with the traditional ZCmethods (en the bearing state data at load 0 are used asthe training set to test the fault identification effect of thebearing state data under other loads Finally the bearingstate data under any load are used as the training set to testthe fault identification effect of the bearing state data underother loads (e first part of the experiment is to verify theeffect of the method in practical application and theremaining two parts are used to test the robustness of themethod

Step 6ϕ 02835

ϕ

11

1

09

08

07

06

05

04

03

02

Steps2 4 6 8 10 12 14 16 18 20

Figure 3 SNR 30 ϕ and steps trend

Step 10ϕ 0495

ϕ

18

16

14

12

1

08

06

04

Steps2 4 6 8 10 12 14 16 18 20

Figure 4 SNR 5 ϕ and steps trend

Step 12ϕ 07836

ϕ

22

2

18

16

14

12

1

08

06

Steps2 4 6 8 10 12 14 16 18 20

Figure 5 SNR minus5 ϕ and steps trend

Table 2 (e step sizes of the four state signals under differentSNRs

SNR 30 S11 S12 S13 S14Step size 0001 0008 0012 0020SNR 5 S21 S22 S23 S34Step size 0002 0007 0011 0020SNR minus5 S31 S32 S33 S34Step size 0003 0009 0012 0020

Table 3 (e optimized ε value of the four state signals underdifferent SNRs

SNR 30 S11 S12 S13 S14ε 0006 0048 0072 012SNR 5 S21 S22 S23 S34ε 002 007 011 020SNR minus5 S31 S32 S33 S34ε 0036 0108 0144 024

6 Mathematical Problems in Engineering

42 Results andAnalysis Firstly the spectrum and envelopespectrum analysis of bearing state data under load 0 test arecarried out to understand the influence of noise on faultdiagnosis Figures 7 and 8 show the frequency spectrum andenvelope spectrum of normal state inner ring fault rollingelement fault and outer ring fault of REB Due to themodulation of the noise signal the characteristic frequencycannot be detected from the frequency spectrum In Fig-ure 8 the characteristic frequency of outer ring failure andthe frequency doubling can be observed but the faultcharacteristic frequency of the inner ring is not obvious andthe characteristic frequency of rolling element is unavailableSo the fault of rolling element cannot be identified effec-tively as the influence of strong noise and signal modulation

and the random signal is spread throughout the frequencydomain It can be concluded from the above analysis that inpractice random noise abundant is an important factorhindering the accuracy of fault identification Due to theexistence of high frequency noise it is necessary to improvethe ZC method model and optimize the parameters so as toimprove the accuracy of fault diagnosis and make the ad-vantages of fast calculation of ZC method useful

(e rotational speed of load 0 is 1797 rpm from Table 5therefore the magnitude of the observation window shouldbe greater than 0034 seconds which contains 360 points atthe sampling frequency of 12K Follow the method inSection 31 the data fragment length is 720 and the numberof fragment is 8 (en each length of data collected from thefour bearing states is 10 seconds which is divided into 40segments with 50 overlap rate According to the method inthis paper optimization parameters in the improved modelrespectively are calculated and shown in Table 6 (e ratioshows the ratio of the optimized ε to the value of peak-to-peak Figure 9 shows the comparison of ZC characteristics offour bearing states before and after the model improvementand parameter optimization

From Figures 9(a) and 9(c) we can see that before themodel improvement parameter optimization the faultcharacteristics of partial samples of rolling element andouter ring are similar and difficult to distinguish which willlead to misjudgment From Figures 9(b) and 9(d) we can seethat the sample characteristics of rolling element fault andthe outer ring fault are completely distinguished through

1

08

06

04

02

0

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

S11S12

S13S14

(a)

1

08

06

04

02

0

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

S21S22

S23S24

(b)

1

08

06

04

02

0

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

S31S32

S33S34

(c)1

08

06

04

02

0

Feature

Nor

mal

ized

dur

atio

n

1 2 3 4 5 6 7 8

S11S12

S13S14

(d)

1

08

06

04

02

0

Feature

Nor

mal

ized

dur

atio

n

1 2 3 4 5 6 7 8

S21S22

S23S24

(e)

1

08

06

04

02

0

Feature

Nor

mal

ized

dur

atio

n

1 2 3 4 5 6 7 8

S31S32

S33S34

(f )

Figure 6 Optimized ZC features of four state signals (from left to right SNR is 30 5 and minus5 respectively)

Table 4 (e average identification rate of simulation fault

Index SNR 30 SNR 5 SNR minus5Classification rate () FVduration FVduration FVdurationNormal 100 100 100Inner ring defect 100 100 9985Rolling element defect 100 100 9985Outer ring defect 100 100 100

Table 5 (e motor load and corresponding speed

Index Load 0 Load 1 Load 2 Load 3Motor speedRPM 1797 1772 1750 1730

Mathematical Problems in Engineering 7

Normal frequency spectrum01

005

0

Frequency (Hz)

Am

plitu

de (V

)

0 1000 2000 3000 4000 5000 6000

(a)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Inner frequency spectrum01

005

0Am

plitu

de (V

)

(b)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Ball frequency spectrum004

002

0Am

plitu

de (V

)

(c)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Outer frequency spectrum02

01

0Am

plitu

de (V

)

(d)

Figure 7 Frequency spectrum of four bearing states

Normal envelope spectrum004

002

0

Frequency (Hz)

Am

plitu

de (V

)

0 1000 2000 3000 4000 5000 6000

(a)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Inner envelope spectrum02

01

0Am

plitu

de (V

)

(b)

Figure 8 Continued

8 Mathematical Problems in Engineering

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Ball envelope spectrum002

001

0Am

plitu

de (V

)

(c)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Outer envelope spectrum1

05

0Am

plitu

de (V

)

(d)

Figure 8 Envelope spectrum of four bearing states

Table 6 Optimized values of ε and theirs ratios to peak-to-peak

Index Normal Inner race fault Ball fault Outer race faultε 0000598 0003109 0001211 0002041Ratio () 01 01 01 0029

1

08

06

04

02

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

NormalInner race defect

Ball defectOuter race defect

(a)

1

08

06

04

0

02

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

NormalInner race defect

Ball defectOuter race defect

(b)

Feature1 2 3 4 5 6 7 8

1

08

06

04

02

0

Nor

mal

ized

dur

atio

n

NormalInner race defect

Ball defectOuter race defect

(c)

Feature1 2 3 4 5 6 7 8

1

08

06

04

0

02Nor

mal

ized

dur

atio

n

NormalInner race defect

Ball defectOuter race defect

(d)

Figure 9 ZC duration features before (a and c) and after (b and d) the model improvement and parameter optimization

Mathematical Problems in Engineering 9

adopting the method of the model improvement and pa-rameter optimization

In order to compare the recognition rate all the ei-genvectors are taken to a three-layer neural network whose

weights and thresholds are optimized by genetic algorithmto improve the diagnostic efficiency and accuracy (enumber of neurons in the hidden layer is 5 and the totalnumber of samples is 160 including training set sample 120and test set sample 40 Each bearing state contains 30training set samples and 10 test set samples arranged inorder according to normal inner ring fault rolling elementfault and outer ring fault (e result of verification is shownin Figure 10 which is the average recognition rate of 40replicates (e recognition rate of both training prior sta-tistics and test set is always 100 and proves that purposedmethod can identify bearing faults accurately

As a comparison the recognition result before the modelimprovement and parameter optimization is shown inFigure 11

(e results in the Figure 11 represent the average rec-ognition accuracy of 40 repeated experiments from which itcan be intuitively seen that the recognition rate of normalstate and inner ring fault is 100 but the recognition rate ofrolling element fault and outer ring fault is not satisfactory(e partial misjudgment between rolling element fault andouter ring fault has affected the fault recognition rate andthe comparison of the average recognition rate of 40 ex-periments is shown in Table 7

Clas

s4

35

3

25

2

15

1

Train sample0 20 40 60 80 100 120

(a)Cl

ass

4

35

3

25

2

15

1

Test sample0 5 10 2015 25 3530 40

(b)

Figure 10 Identification results of adopting the method of the model improvement and parameter optimization

Table 7 Comparison of mean recognition rate before and afteroptimization

Index Training set Test setAccuracy of before optimization () 9333 9225Accuracy of after optimization () 100 100

Table 8 Optimized ε values of various bearing states under threeloads

Index Normal Inner ringfault

Rollingelement fault

Outer ringfault

ε underload 1 0000521 0002938 0001299 0001776

ε underload 2 0000635 0003065 0001171 0001793

ε underload 3 0000548 0003188 0001321 0001809

Table 9 Recognition rate of various bearing states under threeloads

Index Normal Inner ringfault

Rollingelement fault

Outer ringfault

Accuracy ofload 1 () 100 100 9917 9917

Accuracy ofload 2 () 100 975 95 925

Accuracy ofload 3 () 9917 9917 9833 9833

Table 10 Recognition rate of different load HPs as training sets

Index Test set recognition rateTraining set Load 0 Load 1 Load 2 Load 3Load 0 100 9972 9683 9912Load 1 9813 9985 9672 9783Load 2 9775 9728 9883 9752Load 3 9815 9723 9695 9917

10 Mathematical Problems in Engineering

(e experimental results show that the method of themodel improvement and parameter optimization is effectivefor bearing fault identification under the same load

Next the experiment will verify whether the method inthis paper is robust for the load (e experimental data ofload 0 is still used as the training set according to ratios of εto peak-to-peak in Table 6 calculate the optimization pa-rameters under the other load and extract the ZC features asthe test set (e length and number of observation windowsshould be consistent with that under load 0 when extractingthe ZC features under other load Table 8 shows the opti-mized ε values of various bearing states and under threeloads

(e same neural network is applied to identify thebearing fault and the results are shown in Table 9

According to the recognition results in Table 9 althoughthe fault recognition rate at load 2 is lower the overallrecognition rate of other loads still achieves the expectedeffect (e analysis of the error identification samples showsthat these samples are quite distinct from the other samplesof the same state so the validity of themethod is not affected(e experiment proves that the method is applicable to otherloads when the load information is known

Finally the experiment verifies the test effect under anyload and the recognition effect of unknown load has beentested Similarly the method proposed in this paper isemployed for obtaining characteristic samples the samplesunder each load are used as training sets and the samples ofother load are test sets (e overall recognition rate mea-sured is shown in Table 10 and each result is the averagerecognition rate of 40 tests as well

According to the analysis of the results presented inTable 10 the recognition accuracy of samples under thesame load as the training set and the test set is relatively highWhen samples under different loads are used as training setsto test other load samples the recognition accuracy isslightly lower because a few samples deviate from the

characteristics of similar samples In general the experi-mental results demonstrate that the method has good loadrobustness so as to accurately identify the bearing fault evenif the bearing load is not known

5 Conclusion

In this paper an improved fault diagnosis method forelectromechanical system based on ZC algorithm is pro-posed which optimized the calculation model of ZCR andthe parameters of ZC algorithm by establishing a criterionfunction model (e fault diagnosis experiment of motorREB proves that the method successfully overcomes theshortcoming of ZC feature sensitive to noise and can realizeaccurate fault diagnosis without knowing the bearing loadCombined with the merits of saving computing time in thecase of big data the method is expected to monitor thecrucial parts of electromechanical system in real time so asto provide security and reliability guaranty for accuratecontrol of the electromechanical system

Data Availability

(e data used to support the study are available from thecorresponding author upon request

Conflicts of Interest

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

Acknowledgments

(is work was supported by the National Key Research andDevelopment plan of China under Grant 2018YFB1306103the National Natural Science Foundation of China underGrant 51875272 and the Provincial School Education

Clas

s4

35

3

25

2

15

1

Train sample0 20 40 60 80 100 120

(a)

Clas

s

4

35

3

25

2

15

1

Test sample0 5 10 2015 25 3530 40

(b)

Figure 11 Identification results before the model improvement and parameter optimization

Mathematical Problems in Engineering 11

Cooperation Project of Yunnan Province under GrantKKDA202001003

References

[1] M R Khoshdarregi S Tappe and Y Altintas ldquoIntegratedfive-Axis trajectory shaping and contour error compensationfor high-speed CNC machine toolsrdquo IEEEASME Transac-tions on Mechatronics vol 19 no 6 pp 1859ndash1871 2014

[2] R Colombo I Cusmano I Sterpi A Mazzone C Delconteand F Pisano ldquoTest-retest reliability of robotic assessmentmeasures for the evaluation of upper limb recoveryrdquo IEEETransactions on Neural Systems and Rehabilitation Engi-neering vol 22 no 5 pp 1020ndash1029 2014

[3] B Xu D Wang Y Zhang and Z Shi ldquoDOB-based neuralcontrol of flexible hypersonic flight vehicle considering windeffectsrdquo IEEE Transactions on Industrial Electronics vol 64no 11 pp 8676ndash8685 2017

[4] X Ding Q He and N Luo ldquoA fusion feature and its im-provement based on locality preserving projections for rollingelement bearing fault classificationrdquo Journal of Sound andVibration vol 335 pp 367ndash383 2015

[5] Y Zhang and R B Randall ldquoRolling element bearing faultdiagnosis based on the combination of genetic algorithms andfast kurtogramrdquo Mechanical Systems and Signal Processingvol 23 no 5 pp 1509ndash1517 2009

[6] J Benali M Sayadi F Fnaiech B Morello and N ZerhounildquoImportance of the fourth and fifth intrinsic mode functionsfor bearing fault diagnosisrdquo in Proceedings of the 14th In-ternational Conference on Sciences and Techniques of Auto-matic Control amp Computer EngineeringmdashSTArsquo2013 SousseTunisia December 2013

[7] S Nandi H A Toliyat and X Li ldquoCondition monitoring andfault diagnosis of electrical motors-A reviewrdquo IEEE Trans-actions on Energy Conversion vol 20 no 4 pp 719ndash729 2005

[8] J Huang S Ri T Fukuda and Y Wang ldquoA disturbanceobserver based sliding mode control for a class of under-actuated robotic system with mismatched uncertaintiesrdquoIEEE Transactions on Automatic Control vol 64 no 6pp 2480ndash2487 2019

[9] J Huang M Zhang S Ri C Xiong Z Li and Y Kang ldquoHigh-order disturbance observer based sliding mode control formobile wheeled inverted pendulum systemsrdquo IEEE Trans-actions on Industrial Electronics vol 67 no 3 pp 2030ndash20412020

[10] H Jian M H Ri DWu and S Ri ldquoInterval type-2 fuzzy logicmodeling and control of a mobile two-wheeled invertedpendulumrdquo IEEE Transactions on Fuzzy Systems vol 26no 4 pp 2030ndash2038 2018

[11] P D Mcfadden and J D Smith ldquoModel for the vibrationproduced by a single point defect in a rolling elementbearingrdquo Journal of Sound and Vibration vol 96 no 1pp 69ndash82 1984

[12] N Sawalhi and R B Randall ldquoSimulating gear and bearinginteractions in the presence of faultsrdquoMechanical Systems andSignal Processing vol 22 no 8 pp 1924ndash1951 2008

[13] Q Zhu L Liu S Li and W Zhang ldquoControl of complexnonlinear dynamic rational systemsrdquo Complexity vol 2018Article ID 8953035 12 pages 2018

[14] Q M Zhu D Y Zhao and J Zhang ldquoA general U-blockmodel-based design procedure for nonlinear polynomialcontrol systemsrdquo International Journal of Systems Sciencevol 47 no 14 pp 3465ndash3475 2015

[15] A Rafsanjani S Abbasion A Farshidianfar andH Moeenfard ldquoNonlinear dynamic modeling of surfacedefects in rolling element bearing systemsrdquo Journal of Soundamp Vibration vol 319 no 3-5 pp 1150ndash1174 2009

[16] T W Rauber F de Assis Boldt and F M Varejao ldquoHet-erogeneous feature models and feature selection applied tobearing fault diagnosisrdquo IEEE Transactions on IndustrialElectronics vol 62 no 1 pp 637ndash646 2015

[17] J Chebil M Hrairi and N Abushikhah ldquoSignal analysis ofvibration measurements for condition monitoring of bear-ingsrdquo Australian Journal of Basic amp Applied Sciences vol 5no 1 p 70 2011

[18] M D Prieto G Cirrincione A G Espinosa J A Ortega andH Henao ldquobearing fault detection by a novel condition-monitoring scheme based on statistical-time features andneural networksrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 pp 3398ndash3407 2013

[19] A Garcia-Perez R d J Romero-Troncoso E Cabal-Yepezand R A Osornio-Rios ldquo(e application of high-resolutionspectral analysis for identifying multiple combined faults ininduction motorsrdquo IEEE Transactions on Industrial Elec-tronics vol 58 no 5 pp 2002ndash2010 2011

[20] Z K Peng and F L Chu ldquoApplication of the wavelettransform in machine condition monitoring and fault diag-nostics a review with bibliographyrdquo Mechanical Systems andSignal Processing vol 18 no 2 pp 199ndash221 2004

[21] J Seshadrinath B Singh and B K Panigrahi ldquoInvestigationof vibration signatures for multiple fault diagnosis in variablefrequency drives using complex waveletsrdquo IEEE Transactionson Power Electronics vol 29 no 2 pp 936ndash945 2014

[22] W Su F Wang H Zhu Z Zhang and Z Guo ldquoRollingelement bearing faults diagnosis based on optimal Morletwavelet filter and autocorrelation enhancementrdquo MechanicalSystems and Signal Processing vol 24 no 5 pp 1458ndash14722010

[23] X Jin M Zhao T W S Chow andM Pecht ldquoMotor bearingfault diagnosis using trace ratio linear discriminant analysisrdquoIEEE Transactions on Industrial Electronics vol 61 no 5pp 2441ndash2451 2014

[24] M Zhao X Jin Z Zhang and B Li ldquoFault diagnosis of rollingelement bearings via discriminative subspace learning visu-alization and classificationrdquo Expert Systems with Applicationsvol 41 no 7 pp 3391ndash3401 2014

[25] R Niederjohn ldquoA mathematical formulation and comparisonof zero-crossing analysis techniques which have been appliedto automatioc speech recognitionrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 23 no 4pp 373ndash380 1975

[26] P William andM Hoffman ldquoEfficient sensor network vehicleclassification using peak harmonics of acoustic emissionsrdquo inProceedings of the SPIE Defense and Security Symposium Orlando FL USA March 2008

[27] A Shahidi Zandi R Tafreshi M Javidan and G A DumontldquoPredicting epileptic seizures in scalp EEG based on a vari-ational bayesian Gaussian mixture model of zero-crossingintervalsrdquo IEEE Transactions on Biomedical Engineeringvol 60 no 5 pp 1401ndash1413 2013

[28] C-H Wu H-C Chang P-L Lee et al ldquoFrequency recog-nition in an SSVEP-based brain computer interface usingempirical mode decomposition and refined generalized zero-crossingrdquo Journal of Neuroscience Methods vol 196 no 1pp 170ndash181 2011

12 Mathematical Problems in Engineering

[29] P E William and M W Hoffman ldquoIdentification of bearingfaults using time domain zero-crossingsrdquoMechanical Systemsand Signal Processing vol 25 no 8 pp 3078ndash3088 2011

[30] T Liu J Chen and G Dong ldquoZero crossing and coupledhidden Markov model for a rolling bearing performancedegradation assessmentrdquo Journal of Vibration and Controlvol 20 no 16 pp 2487ndash2500 2013

[31] A Ukil S Chen and A Andenna ldquoDetection of stator shortcircuit faults in three-phase induction motors using motorcurrent zero crossing instantsrdquo Electric Power Systems Re-search vol 81 no 4 pp 1036ndash1044 2011

[32] J D T Gonzalez and W Kinsner ldquoZero-crossing analysis ofLevy walks for real-time feature extractionrdquo in Proceedings ofthe IEEE International Conference on Electro InformationTechnology EIT Grand Forks ND USA May 2016

[33] R G Waghmare S L Nalbalwar and A Das ldquoTransientsignal detection on the basis of energy and zero crossingdetectorsrdquo Procedia Engineering vol 30 pp 129ndash134 2012

[34] B R Nayana and P Geethanjali ldquoAnalysis of statistical time-domain features effectiveness in identification of bearingfaults from vibration signalrdquo IEEE Sensors Journal vol 17no 17 pp 5618ndash5625 2017

[35] C Fisher Encyclopedia of Biometrics p 549 Springer USBoston MA USA 2009

[36] Q Chen W Xing L Tao and L Hua ldquoZero-crossing featureextraction based on threshold optimization for rolling ele-ment bearingrdquo in Lecture Notes in Electrical Engineeringpp 409ndash425 Springer Berlin Germany 2018

[37] Case Western Reserve University Bearing Data CenterWebsite accessed on Mar 23 2015 [Online]Available httpcsegroupscaseedubearingdatacenterpagesdownload-data-file

Mathematical Problems in Engineering 13

parameter optimization for achieving the more accuracy offault diagnosis at low SNR

In order to describe the short-time waveform the timedomain signal collected should be divided into a diminutiveobservation window firstly In addition the measure of theobservation window should be greater than the maximumexpected duration between continuous ZCs TL is defined asthe measure of the observation window TQ represents themaximum expected duration between the continuous ZCsand TQ le TL (en divide the duration range [0 TQ] into Qintervals by the threshold of Q+ 1 and define Ti as the ithinterval Let Ci represent the count of ZC intervals withduration in the range of Tjminus1 and Tj and x(n) is the short-time waveform with the data sequence N then Ci can beobtained by

Ci 12

1113944

N

n2[sgn[x(n)] minus sgn[x(n minus 1)]] (1)

where the sgn[x] is sign function and expressed as

sgn[x(n)] 1(xge 0)

minus1(xlt 0)1113896 (2)

(en the eigenvector of ZC counts is normalized by theobservation window and defined as

Fcount C

TL

1

TL

C1C2 CQ1113872 1113873 (3)

and the function of excess thresholdmeasurement can becalculated by the following formula

f Ti( 1113857 12

1113944

Qminus1

kj

Ck+1 Tk + Tk+1( 1113857 (4)

where 12Ck+1(Tk + Tk+1) is used to approximately calculatethe sum of ZC durations in the range Tk ltT≪Tk+1

(ereby the eigenvector of excess threshold measure-ment can be obtained by

Fduration 1

TL

f T0( 1113857 f T0( 1113857 f TQminus11113872 11138731113960 1113961 (5)

22 Criterion Function Modeling (e purpose of featureextraction is to acquire the most available information relatedto the equipment condition so as to realize fault identification(e classification ability of feature vectors directly determinesthe ability of fault recognition Based on the idea of Fishercriterion [35] to calculate the distance between categories andwithin categories the recognition ability of features can beimproved by finding a functional relationship between theintraclass distance and interclass distance of features In otherwords the smaller the intraclass distance and the larger theinterclass distance of the feature clustering results the higherthe discrimination degree of the classification results will be(erefore we can construct a criterion function that reflectsthe within-between distance

Assume that the feature set (x1 x2 xN) to beclassified is classified as class C x

(j)i j 1 2 C i 1113966

1 2 nj (e data quantity of class j is nj (e intraclassdistance of class j can be expressed as follows

S(j)w

1nj

1113944

nj

i1x

(j)i minus mj1113872 1113873 x

(j)i minus mj1113872 1113873

T (6)

where mj is the average of the samples of class j andmj 1nj 1113936

nj

i1 x(j)i (j 1 2 C)

Let S(k)w and mk denote the intraclass distance and the

samplesmean of class k( kne j and kleC) respectively the inter-class distance between class j and k can be expressed as follows

SB 1113944 mj minus mk1113872 1113873 mj minus mk1113872 1113873T (7)

(en the function represents the ability to recognizebetween classes can be expressed as follows

I SB

S(j)w + S

(k)w

(8)

On the premise that the highest recognition capacity ofthe two categories is not weakened let us assume that thediscrepancy of I value between other categories is thesmallest and the recognition rate is the best and then thecriterion function can be expressed by the following relation

minϕ Imax Imin( 1113857 ϕ Imax minus Imin (9)

where Imax and Imin represent the maximum and minimumI values of all categories respectively

23 Parameter-Optimized ZC Based on Within-BetweenDistanceModels According to formula (1) we know that theZC count has nothing to do with the magnitude of signalamplitude and is only related to the positive and negative signsof signals However ZC count is sensitive to the presence ofnoise as random noises repeatedly cross the coordinate axis inthe background a large number of false ZCR will be generatedand affect the recognition result Equation (1) is suitable fordiscrete calculation but not conducive to the analysis of relatedsignal waveform (erefore formula (1) is firstly transformedinto the following equation

Ci 12

1113944

N

n2[1 minus sgn[x(n) middot x(n minus 1)]] (10)

Consider further the influence of random noise increasea threshold ε to offset part of false ZCR generated by noiserepeatedly crossing the coordinate axis (erefore the zero-crossing calculation formula can be revised into the form offormula (11)

Ci 12

1113944

N

n2[1 minus sgn[x(n) middot x(n minus 1) + ε]] (11)

(e subsequent work is the optimization of threshold ε(e method of extracting ZC features from ZC duration isdescribed in Section 21 It is noteworthy that the length ofthe observation window must greater than the maximumexpected time interval between the continuous ZC intervalsso the ZC feature should ensure that the minimum

Mathematical Problems in Engineering 3

frequency of interest can be discovered In order to improvedata utilization and maintain data continuity 50 overlaprate is set When the maximum expected time is divided intoQ intervals by the threshold of Q+ 1 Q group eigenvectorsare generated We know that the extracted features aredifferent on various time scales therefore the eigenvector isaffected by the length and number of observation windowLiterature [36] has discussed the optimization method ofthese two thresholds here the optimized window data areused directly Keep the optimized window data unchangedand change the value of ε to extract the ZC feature re-spectively (e ε must be changed by the specified step sizeand adjusted for ZC features changes Calculate the inter-class distance and intraclass distance of each ZC feature(en values of I with diverse values of ε are calculatedaccording to formula (9) If the critical point is found theoptimized ε value is obtained (e flow chart of improvedmethod of ZC features extraction is manifested in Figure 1

Detailed descriptions of the flow chart in Figure 1

Step 1 Input original vibration signalsStep 2 Calculate the peak-to-peak value as the basis forcalculating the search step size and search scopeStep 3 Set the search range and step size of ε(e searchstep size needs to be based on the experience of makingmultiple attempts at the actual signal In the simulationone-thousandth of the peak value is used as the steplength and 20 steps are used as the search rangeStep 4 Extract ZC feature vectors of every signal atdifferent ε values (e number and length of observationwindow were optimized before feature extraction andthe data overlapped by 50Step 5 Calculate the value of I between different statecategories according to formula (8) Suppose there are 4kinds of state signals and then 6 values of I are formedone for every 2 kinds of signals ε value is searched 20times and then a 6times 20 matrix is formedStep 6 Calculate the maximum and minimum of I foreach search guaranteed the highest recognition ca-pacity of the two categories is not weakened and in-crease the value of I between the two classes that aremost difficult to recognize So when formula (9) issatisfied the optimization is achieved According to thestep size of each signal and the number of steps theoptimized ε value of each signal can be obtainedStep 7 According to the optimized ε value of eachsignal extract the optimized ZC featuresStep 8 Bring the optimized features into neural net-work for fault diagnosis and verification and the resultsare analyzed

3 Simulation Analysis

For the sake of verifying the necessity and effectiveness of thepresented method four simulated signals of normal innerring fault rolling element fault and outer ring fault of rolling

bearing are used for analysis (e simulation signals areobtained from the following formula

x(t) 1113936M

i1Ais t minus iT minus τi( 1113857 + n(i)

Ai A0 cos 2πQt + φA( 1113857 + CA

s(t) eminus Bt sin 2πfnt + φw( 1113857

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

(12)

where the system sampling frequency fs 12000Hz theresonant frequency fn 3000Hz the inner ring fault fre-quency fi 162Hz the rolling element fault frequencyfb 142Hz the outer ring fault frequency fi 142Hz therotational frequency fr 30Hz and the damping ratioB 400 Four simulated signals are a group a total of threegroups (ree random noises with a SNR of 30 dB 5 dB andminus5 dB are added separately to three groups signal (egrouping of simulation signals is shown in Table 1

31 Analysis of ZC Feature Noise Resistance In orderto make the extracted features represent the running stateinformation of the bearing effectively the length of the datasegment calculated with ZC feature should contain at least onecomplete rotation period (e rotational frequency is 30Hz sothe length of an observation window must greater than 130secondswhich contains 360 points at the sampling frequency of12K Follow the method in Section 23 above the data frag-ment length is 512 and the number of fragment is 8 Asdescribed earlier the data overlap rate is set to 50 whichcontains 256 points at the sampling frequency of 12K Eachstate data are divided into 40 samples and contain data of 05seconds (6000 points) Figure 2 shows the ZC features(FVduration) of the four state signals extracted from each set ofsignals when SNR 30 5 and minus5 From the left to the right ofthe figure is the distribution of FVduration of four state signalswhen SNR is 30 5 and minus5 (e graphs above the figure are thedistribution of all samples under a certain feature and thebelow ones are the differentiation of all eigenvectors under acertain sample It can be seen from the three groups of graphsthat the ZC feature is very sensitive to random noise and therecognition ability between ZC features of various state signalsis decreasing with random noise increasing When SNR 30FVduration still has fairish recognition ability however whenSNR minus5 recognition ability of FVduration begins to fail

32 Analysis of ε Value On the basis of the method de-scription in Section 23 extract the ZC features and calculatethe various values of I with different values of ε(en calculatethe values of ϕ according to formula (9) and find theminimumvalue of ϕ (e relationship between ϕ and ε is shown inFigures 3ndash5 under different kinds of SNRs(e figures indicatethat with the increase of the ε value the capacity of distinguishfault is improved but they will not continue indefinitely Inother words specifically value of ϕwould not always decreasebut has a critical value such as the values marked with arrowsin three figures and then with the value of ϕ increasing therecognition capacity of certain two types will decrease(erefore the optimized values of ε can be obtained in adifferent kind of SNR (e value of ε is represented by the

4 Mathematical Problems in Engineering

number of steps and the step sizes of the four state signalsunder different kind SNR are shown in Table 2

It can be seen from Figures 3ndash5 that the optimal numberof steps correspond to the minimum ϕ combined with thestep size in Table 2 and the optimized ε value can be cal-culated as shown in Table 3

(en the ZC features extracted from above optimized εvalues are shown in Figure 6

FromFigure 6 we can clearly see that all samples of signals indiverse states in the same feature can be separated from each

other and similarly all features of signals in diverse states underthe same sample can be separated fromeach other Consequentlythe ZC characteristics of the four states can be recognizedCompared with the identification result before optimization inFigure 2 the effectiveness of the method is proved

33 Optimization Validation In order to further prove theavailability of themethod the optimized ZC eigenvectors areidentified by three-layer feed-forward neural network andthe result of verification is shown in Table 4 which is the

099

098

1

097

096

094

093

095

0 5 10 15 20Samples

Nor

mal

ized

dur

atio

n

S11S12

S13S14

25 30 35 40

(a)

1

098

096

094

092

Nor

mal

ized

dur

atio

n

S21S22

S23S24

Samples0 5 10 15 20 25 30 35 40

(b)

S31S32

S33S34

1

098

096

094

092

Nor

mal

ized

dur

atio

nSamples

0 5 10 15 20 25 30 35 40

(c)1

08

06

04

021 2 3 4 5 6 7 8

Feature

Nor

mal

ized

dur

atio

n

S11S12

S13S14

(d)

1

08

06

04

021 2 3 4 5 6 7 8

Feature

Nor

mal

ized

dur

atio

n

S21S22

S23S24

(e)

1

08

06

04

021 2 3 4 5 6 7 8

Feature

Nor

mal

ized

dur

atio

n

S31S32

S33S34

(f )

Figure 2 ZC features of four state signals (from left to right SNR is 30 5 and minus5 respectively)

Neural networkfault diagnosis

Extract ZC feature vectors ofevery signal at different ε values

Calculate the I value of differentstate categories according to

formula (8)

Calculate the peak-to-peak value

Set the search rangeand step size of ε

Find the step size that satisfiesformula (9) and

obtain the ε value of each signal

Extract the optimized ZCfeature according to the

optimized ε value

Input originalvibration signalStart

End

Figure 1 Flow chart of the optimized parameter

Table 1 (e grouping of simulation signals

Index SNR 30 SNR 5 SNR minus5Simulated normalno fault S11 S21 S31Simulated inner ring fault S12 S22 S32Simulated rolling element fault S13 S23 S33Simulated outer ring fault S14 S24 S34

Mathematical Problems in Engineering 5

average recognition rate of 40 replicates It can be dem-onstrated from Table 4 that this method has a high faultidentification rate for REB simulation signals

4 Experiment Validation

(e necessity and effectiveness of the improved model andparameter optimization are verified by simulation experi-ments In practical applications the identification of rollingbearing faults usually requires the identification method tobe robust for load so the experiment uses the bearing datafrom the Case Western Reserve University Bearing DataCenter to verify whether the proposed method is effective

41 Experiment Data and Schemes (e bearing status dataset is derived from [37] (e type of test bearing is deepgroove bearing and the data are sampled by the acceler-ometer at 12KHz from the drive end of the motor Single-point failure diameter is 0007 inches (e outer ring failureis located at 6 orsquoclock For the motor loads 0 1 2 and 3motor speeds are from 1797 to 1730 RPM Table 5 details theload and speed

(e whole experiment consists of three parts (e firstexperiment verifies the identification effect of four bearingstates under load 0 HP and compares with the traditional ZCmethods (en the bearing state data at load 0 are used asthe training set to test the fault identification effect of thebearing state data under other loads Finally the bearingstate data under any load are used as the training set to testthe fault identification effect of the bearing state data underother loads (e first part of the experiment is to verify theeffect of the method in practical application and theremaining two parts are used to test the robustness of themethod

Step 6ϕ 02835

ϕ

11

1

09

08

07

06

05

04

03

02

Steps2 4 6 8 10 12 14 16 18 20

Figure 3 SNR 30 ϕ and steps trend

Step 10ϕ 0495

ϕ

18

16

14

12

1

08

06

04

Steps2 4 6 8 10 12 14 16 18 20

Figure 4 SNR 5 ϕ and steps trend

Step 12ϕ 07836

ϕ

22

2

18

16

14

12

1

08

06

Steps2 4 6 8 10 12 14 16 18 20

Figure 5 SNR minus5 ϕ and steps trend

Table 2 (e step sizes of the four state signals under differentSNRs

SNR 30 S11 S12 S13 S14Step size 0001 0008 0012 0020SNR 5 S21 S22 S23 S34Step size 0002 0007 0011 0020SNR minus5 S31 S32 S33 S34Step size 0003 0009 0012 0020

Table 3 (e optimized ε value of the four state signals underdifferent SNRs

SNR 30 S11 S12 S13 S14ε 0006 0048 0072 012SNR 5 S21 S22 S23 S34ε 002 007 011 020SNR minus5 S31 S32 S33 S34ε 0036 0108 0144 024

6 Mathematical Problems in Engineering

42 Results andAnalysis Firstly the spectrum and envelopespectrum analysis of bearing state data under load 0 test arecarried out to understand the influence of noise on faultdiagnosis Figures 7 and 8 show the frequency spectrum andenvelope spectrum of normal state inner ring fault rollingelement fault and outer ring fault of REB Due to themodulation of the noise signal the characteristic frequencycannot be detected from the frequency spectrum In Fig-ure 8 the characteristic frequency of outer ring failure andthe frequency doubling can be observed but the faultcharacteristic frequency of the inner ring is not obvious andthe characteristic frequency of rolling element is unavailableSo the fault of rolling element cannot be identified effec-tively as the influence of strong noise and signal modulation

and the random signal is spread throughout the frequencydomain It can be concluded from the above analysis that inpractice random noise abundant is an important factorhindering the accuracy of fault identification Due to theexistence of high frequency noise it is necessary to improvethe ZC method model and optimize the parameters so as toimprove the accuracy of fault diagnosis and make the ad-vantages of fast calculation of ZC method useful

(e rotational speed of load 0 is 1797 rpm from Table 5therefore the magnitude of the observation window shouldbe greater than 0034 seconds which contains 360 points atthe sampling frequency of 12K Follow the method inSection 31 the data fragment length is 720 and the numberof fragment is 8 (en each length of data collected from thefour bearing states is 10 seconds which is divided into 40segments with 50 overlap rate According to the method inthis paper optimization parameters in the improved modelrespectively are calculated and shown in Table 6 (e ratioshows the ratio of the optimized ε to the value of peak-to-peak Figure 9 shows the comparison of ZC characteristics offour bearing states before and after the model improvementand parameter optimization

From Figures 9(a) and 9(c) we can see that before themodel improvement parameter optimization the faultcharacteristics of partial samples of rolling element andouter ring are similar and difficult to distinguish which willlead to misjudgment From Figures 9(b) and 9(d) we can seethat the sample characteristics of rolling element fault andthe outer ring fault are completely distinguished through

1

08

06

04

02

0

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

S11S12

S13S14

(a)

1

08

06

04

02

0

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

S21S22

S23S24

(b)

1

08

06

04

02

0

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

S31S32

S33S34

(c)1

08

06

04

02

0

Feature

Nor

mal

ized

dur

atio

n

1 2 3 4 5 6 7 8

S11S12

S13S14

(d)

1

08

06

04

02

0

Feature

Nor

mal

ized

dur

atio

n

1 2 3 4 5 6 7 8

S21S22

S23S24

(e)

1

08

06

04

02

0

Feature

Nor

mal

ized

dur

atio

n

1 2 3 4 5 6 7 8

S31S32

S33S34

(f )

Figure 6 Optimized ZC features of four state signals (from left to right SNR is 30 5 and minus5 respectively)

Table 4 (e average identification rate of simulation fault

Index SNR 30 SNR 5 SNR minus5Classification rate () FVduration FVduration FVdurationNormal 100 100 100Inner ring defect 100 100 9985Rolling element defect 100 100 9985Outer ring defect 100 100 100

Table 5 (e motor load and corresponding speed

Index Load 0 Load 1 Load 2 Load 3Motor speedRPM 1797 1772 1750 1730

Mathematical Problems in Engineering 7

Normal frequency spectrum01

005

0

Frequency (Hz)

Am

plitu

de (V

)

0 1000 2000 3000 4000 5000 6000

(a)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Inner frequency spectrum01

005

0Am

plitu

de (V

)

(b)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Ball frequency spectrum004

002

0Am

plitu

de (V

)

(c)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Outer frequency spectrum02

01

0Am

plitu

de (V

)

(d)

Figure 7 Frequency spectrum of four bearing states

Normal envelope spectrum004

002

0

Frequency (Hz)

Am

plitu

de (V

)

0 1000 2000 3000 4000 5000 6000

(a)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Inner envelope spectrum02

01

0Am

plitu

de (V

)

(b)

Figure 8 Continued

8 Mathematical Problems in Engineering

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Ball envelope spectrum002

001

0Am

plitu

de (V

)

(c)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Outer envelope spectrum1

05

0Am

plitu

de (V

)

(d)

Figure 8 Envelope spectrum of four bearing states

Table 6 Optimized values of ε and theirs ratios to peak-to-peak

Index Normal Inner race fault Ball fault Outer race faultε 0000598 0003109 0001211 0002041Ratio () 01 01 01 0029

1

08

06

04

02

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

NormalInner race defect

Ball defectOuter race defect

(a)

1

08

06

04

0

02

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

NormalInner race defect

Ball defectOuter race defect

(b)

Feature1 2 3 4 5 6 7 8

1

08

06

04

02

0

Nor

mal

ized

dur

atio

n

NormalInner race defect

Ball defectOuter race defect

(c)

Feature1 2 3 4 5 6 7 8

1

08

06

04

0

02Nor

mal

ized

dur

atio

n

NormalInner race defect

Ball defectOuter race defect

(d)

Figure 9 ZC duration features before (a and c) and after (b and d) the model improvement and parameter optimization

Mathematical Problems in Engineering 9

adopting the method of the model improvement and pa-rameter optimization

In order to compare the recognition rate all the ei-genvectors are taken to a three-layer neural network whose

weights and thresholds are optimized by genetic algorithmto improve the diagnostic efficiency and accuracy (enumber of neurons in the hidden layer is 5 and the totalnumber of samples is 160 including training set sample 120and test set sample 40 Each bearing state contains 30training set samples and 10 test set samples arranged inorder according to normal inner ring fault rolling elementfault and outer ring fault (e result of verification is shownin Figure 10 which is the average recognition rate of 40replicates (e recognition rate of both training prior sta-tistics and test set is always 100 and proves that purposedmethod can identify bearing faults accurately

As a comparison the recognition result before the modelimprovement and parameter optimization is shown inFigure 11

(e results in the Figure 11 represent the average rec-ognition accuracy of 40 repeated experiments from which itcan be intuitively seen that the recognition rate of normalstate and inner ring fault is 100 but the recognition rate ofrolling element fault and outer ring fault is not satisfactory(e partial misjudgment between rolling element fault andouter ring fault has affected the fault recognition rate andthe comparison of the average recognition rate of 40 ex-periments is shown in Table 7

Clas

s4

35

3

25

2

15

1

Train sample0 20 40 60 80 100 120

(a)Cl

ass

4

35

3

25

2

15

1

Test sample0 5 10 2015 25 3530 40

(b)

Figure 10 Identification results of adopting the method of the model improvement and parameter optimization

Table 7 Comparison of mean recognition rate before and afteroptimization

Index Training set Test setAccuracy of before optimization () 9333 9225Accuracy of after optimization () 100 100

Table 8 Optimized ε values of various bearing states under threeloads

Index Normal Inner ringfault

Rollingelement fault

Outer ringfault

ε underload 1 0000521 0002938 0001299 0001776

ε underload 2 0000635 0003065 0001171 0001793

ε underload 3 0000548 0003188 0001321 0001809

Table 9 Recognition rate of various bearing states under threeloads

Index Normal Inner ringfault

Rollingelement fault

Outer ringfault

Accuracy ofload 1 () 100 100 9917 9917

Accuracy ofload 2 () 100 975 95 925

Accuracy ofload 3 () 9917 9917 9833 9833

Table 10 Recognition rate of different load HPs as training sets

Index Test set recognition rateTraining set Load 0 Load 1 Load 2 Load 3Load 0 100 9972 9683 9912Load 1 9813 9985 9672 9783Load 2 9775 9728 9883 9752Load 3 9815 9723 9695 9917

10 Mathematical Problems in Engineering

(e experimental results show that the method of themodel improvement and parameter optimization is effectivefor bearing fault identification under the same load

Next the experiment will verify whether the method inthis paper is robust for the load (e experimental data ofload 0 is still used as the training set according to ratios of εto peak-to-peak in Table 6 calculate the optimization pa-rameters under the other load and extract the ZC features asthe test set (e length and number of observation windowsshould be consistent with that under load 0 when extractingthe ZC features under other load Table 8 shows the opti-mized ε values of various bearing states and under threeloads

(e same neural network is applied to identify thebearing fault and the results are shown in Table 9

According to the recognition results in Table 9 althoughthe fault recognition rate at load 2 is lower the overallrecognition rate of other loads still achieves the expectedeffect (e analysis of the error identification samples showsthat these samples are quite distinct from the other samplesof the same state so the validity of themethod is not affected(e experiment proves that the method is applicable to otherloads when the load information is known

Finally the experiment verifies the test effect under anyload and the recognition effect of unknown load has beentested Similarly the method proposed in this paper isemployed for obtaining characteristic samples the samplesunder each load are used as training sets and the samples ofother load are test sets (e overall recognition rate mea-sured is shown in Table 10 and each result is the averagerecognition rate of 40 tests as well

According to the analysis of the results presented inTable 10 the recognition accuracy of samples under thesame load as the training set and the test set is relatively highWhen samples under different loads are used as training setsto test other load samples the recognition accuracy isslightly lower because a few samples deviate from the

characteristics of similar samples In general the experi-mental results demonstrate that the method has good loadrobustness so as to accurately identify the bearing fault evenif the bearing load is not known

5 Conclusion

In this paper an improved fault diagnosis method forelectromechanical system based on ZC algorithm is pro-posed which optimized the calculation model of ZCR andthe parameters of ZC algorithm by establishing a criterionfunction model (e fault diagnosis experiment of motorREB proves that the method successfully overcomes theshortcoming of ZC feature sensitive to noise and can realizeaccurate fault diagnosis without knowing the bearing loadCombined with the merits of saving computing time in thecase of big data the method is expected to monitor thecrucial parts of electromechanical system in real time so asto provide security and reliability guaranty for accuratecontrol of the electromechanical system

Data Availability

(e data used to support the study are available from thecorresponding author upon request

Conflicts of Interest

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

Acknowledgments

(is work was supported by the National Key Research andDevelopment plan of China under Grant 2018YFB1306103the National Natural Science Foundation of China underGrant 51875272 and the Provincial School Education

Clas

s4

35

3

25

2

15

1

Train sample0 20 40 60 80 100 120

(a)

Clas

s

4

35

3

25

2

15

1

Test sample0 5 10 2015 25 3530 40

(b)

Figure 11 Identification results before the model improvement and parameter optimization

Mathematical Problems in Engineering 11

Cooperation Project of Yunnan Province under GrantKKDA202001003

References

[1] M R Khoshdarregi S Tappe and Y Altintas ldquoIntegratedfive-Axis trajectory shaping and contour error compensationfor high-speed CNC machine toolsrdquo IEEEASME Transac-tions on Mechatronics vol 19 no 6 pp 1859ndash1871 2014

[2] R Colombo I Cusmano I Sterpi A Mazzone C Delconteand F Pisano ldquoTest-retest reliability of robotic assessmentmeasures for the evaluation of upper limb recoveryrdquo IEEETransactions on Neural Systems and Rehabilitation Engi-neering vol 22 no 5 pp 1020ndash1029 2014

[3] B Xu D Wang Y Zhang and Z Shi ldquoDOB-based neuralcontrol of flexible hypersonic flight vehicle considering windeffectsrdquo IEEE Transactions on Industrial Electronics vol 64no 11 pp 8676ndash8685 2017

[4] X Ding Q He and N Luo ldquoA fusion feature and its im-provement based on locality preserving projections for rollingelement bearing fault classificationrdquo Journal of Sound andVibration vol 335 pp 367ndash383 2015

[5] Y Zhang and R B Randall ldquoRolling element bearing faultdiagnosis based on the combination of genetic algorithms andfast kurtogramrdquo Mechanical Systems and Signal Processingvol 23 no 5 pp 1509ndash1517 2009

[6] J Benali M Sayadi F Fnaiech B Morello and N ZerhounildquoImportance of the fourth and fifth intrinsic mode functionsfor bearing fault diagnosisrdquo in Proceedings of the 14th In-ternational Conference on Sciences and Techniques of Auto-matic Control amp Computer EngineeringmdashSTArsquo2013 SousseTunisia December 2013

[7] S Nandi H A Toliyat and X Li ldquoCondition monitoring andfault diagnosis of electrical motors-A reviewrdquo IEEE Trans-actions on Energy Conversion vol 20 no 4 pp 719ndash729 2005

[8] J Huang S Ri T Fukuda and Y Wang ldquoA disturbanceobserver based sliding mode control for a class of under-actuated robotic system with mismatched uncertaintiesrdquoIEEE Transactions on Automatic Control vol 64 no 6pp 2480ndash2487 2019

[9] J Huang M Zhang S Ri C Xiong Z Li and Y Kang ldquoHigh-order disturbance observer based sliding mode control formobile wheeled inverted pendulum systemsrdquo IEEE Trans-actions on Industrial Electronics vol 67 no 3 pp 2030ndash20412020

[10] H Jian M H Ri DWu and S Ri ldquoInterval type-2 fuzzy logicmodeling and control of a mobile two-wheeled invertedpendulumrdquo IEEE Transactions on Fuzzy Systems vol 26no 4 pp 2030ndash2038 2018

[11] P D Mcfadden and J D Smith ldquoModel for the vibrationproduced by a single point defect in a rolling elementbearingrdquo Journal of Sound and Vibration vol 96 no 1pp 69ndash82 1984

[12] N Sawalhi and R B Randall ldquoSimulating gear and bearinginteractions in the presence of faultsrdquoMechanical Systems andSignal Processing vol 22 no 8 pp 1924ndash1951 2008

[13] Q Zhu L Liu S Li and W Zhang ldquoControl of complexnonlinear dynamic rational systemsrdquo Complexity vol 2018Article ID 8953035 12 pages 2018

[14] Q M Zhu D Y Zhao and J Zhang ldquoA general U-blockmodel-based design procedure for nonlinear polynomialcontrol systemsrdquo International Journal of Systems Sciencevol 47 no 14 pp 3465ndash3475 2015

[15] A Rafsanjani S Abbasion A Farshidianfar andH Moeenfard ldquoNonlinear dynamic modeling of surfacedefects in rolling element bearing systemsrdquo Journal of Soundamp Vibration vol 319 no 3-5 pp 1150ndash1174 2009

[16] T W Rauber F de Assis Boldt and F M Varejao ldquoHet-erogeneous feature models and feature selection applied tobearing fault diagnosisrdquo IEEE Transactions on IndustrialElectronics vol 62 no 1 pp 637ndash646 2015

[17] J Chebil M Hrairi and N Abushikhah ldquoSignal analysis ofvibration measurements for condition monitoring of bear-ingsrdquo Australian Journal of Basic amp Applied Sciences vol 5no 1 p 70 2011

[18] M D Prieto G Cirrincione A G Espinosa J A Ortega andH Henao ldquobearing fault detection by a novel condition-monitoring scheme based on statistical-time features andneural networksrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 pp 3398ndash3407 2013

[19] A Garcia-Perez R d J Romero-Troncoso E Cabal-Yepezand R A Osornio-Rios ldquo(e application of high-resolutionspectral analysis for identifying multiple combined faults ininduction motorsrdquo IEEE Transactions on Industrial Elec-tronics vol 58 no 5 pp 2002ndash2010 2011

[20] Z K Peng and F L Chu ldquoApplication of the wavelettransform in machine condition monitoring and fault diag-nostics a review with bibliographyrdquo Mechanical Systems andSignal Processing vol 18 no 2 pp 199ndash221 2004

[21] J Seshadrinath B Singh and B K Panigrahi ldquoInvestigationof vibration signatures for multiple fault diagnosis in variablefrequency drives using complex waveletsrdquo IEEE Transactionson Power Electronics vol 29 no 2 pp 936ndash945 2014

[22] W Su F Wang H Zhu Z Zhang and Z Guo ldquoRollingelement bearing faults diagnosis based on optimal Morletwavelet filter and autocorrelation enhancementrdquo MechanicalSystems and Signal Processing vol 24 no 5 pp 1458ndash14722010

[23] X Jin M Zhao T W S Chow andM Pecht ldquoMotor bearingfault diagnosis using trace ratio linear discriminant analysisrdquoIEEE Transactions on Industrial Electronics vol 61 no 5pp 2441ndash2451 2014

[24] M Zhao X Jin Z Zhang and B Li ldquoFault diagnosis of rollingelement bearings via discriminative subspace learning visu-alization and classificationrdquo Expert Systems with Applicationsvol 41 no 7 pp 3391ndash3401 2014

[25] R Niederjohn ldquoA mathematical formulation and comparisonof zero-crossing analysis techniques which have been appliedto automatioc speech recognitionrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 23 no 4pp 373ndash380 1975

[26] P William andM Hoffman ldquoEfficient sensor network vehicleclassification using peak harmonics of acoustic emissionsrdquo inProceedings of the SPIE Defense and Security Symposium Orlando FL USA March 2008

[27] A Shahidi Zandi R Tafreshi M Javidan and G A DumontldquoPredicting epileptic seizures in scalp EEG based on a vari-ational bayesian Gaussian mixture model of zero-crossingintervalsrdquo IEEE Transactions on Biomedical Engineeringvol 60 no 5 pp 1401ndash1413 2013

[28] C-H Wu H-C Chang P-L Lee et al ldquoFrequency recog-nition in an SSVEP-based brain computer interface usingempirical mode decomposition and refined generalized zero-crossingrdquo Journal of Neuroscience Methods vol 196 no 1pp 170ndash181 2011

12 Mathematical Problems in Engineering

[29] P E William and M W Hoffman ldquoIdentification of bearingfaults using time domain zero-crossingsrdquoMechanical Systemsand Signal Processing vol 25 no 8 pp 3078ndash3088 2011

[30] T Liu J Chen and G Dong ldquoZero crossing and coupledhidden Markov model for a rolling bearing performancedegradation assessmentrdquo Journal of Vibration and Controlvol 20 no 16 pp 2487ndash2500 2013

[31] A Ukil S Chen and A Andenna ldquoDetection of stator shortcircuit faults in three-phase induction motors using motorcurrent zero crossing instantsrdquo Electric Power Systems Re-search vol 81 no 4 pp 1036ndash1044 2011

[32] J D T Gonzalez and W Kinsner ldquoZero-crossing analysis ofLevy walks for real-time feature extractionrdquo in Proceedings ofthe IEEE International Conference on Electro InformationTechnology EIT Grand Forks ND USA May 2016

[33] R G Waghmare S L Nalbalwar and A Das ldquoTransientsignal detection on the basis of energy and zero crossingdetectorsrdquo Procedia Engineering vol 30 pp 129ndash134 2012

[34] B R Nayana and P Geethanjali ldquoAnalysis of statistical time-domain features effectiveness in identification of bearingfaults from vibration signalrdquo IEEE Sensors Journal vol 17no 17 pp 5618ndash5625 2017

[35] C Fisher Encyclopedia of Biometrics p 549 Springer USBoston MA USA 2009

[36] Q Chen W Xing L Tao and L Hua ldquoZero-crossing featureextraction based on threshold optimization for rolling ele-ment bearingrdquo in Lecture Notes in Electrical Engineeringpp 409ndash425 Springer Berlin Germany 2018

[37] Case Western Reserve University Bearing Data CenterWebsite accessed on Mar 23 2015 [Online]Available httpcsegroupscaseedubearingdatacenterpagesdownload-data-file

Mathematical Problems in Engineering 13

frequency of interest can be discovered In order to improvedata utilization and maintain data continuity 50 overlaprate is set When the maximum expected time is divided intoQ intervals by the threshold of Q+ 1 Q group eigenvectorsare generated We know that the extracted features aredifferent on various time scales therefore the eigenvector isaffected by the length and number of observation windowLiterature [36] has discussed the optimization method ofthese two thresholds here the optimized window data areused directly Keep the optimized window data unchangedand change the value of ε to extract the ZC feature re-spectively (e ε must be changed by the specified step sizeand adjusted for ZC features changes Calculate the inter-class distance and intraclass distance of each ZC feature(en values of I with diverse values of ε are calculatedaccording to formula (9) If the critical point is found theoptimized ε value is obtained (e flow chart of improvedmethod of ZC features extraction is manifested in Figure 1

Detailed descriptions of the flow chart in Figure 1

Step 1 Input original vibration signalsStep 2 Calculate the peak-to-peak value as the basis forcalculating the search step size and search scopeStep 3 Set the search range and step size of ε(e searchstep size needs to be based on the experience of makingmultiple attempts at the actual signal In the simulationone-thousandth of the peak value is used as the steplength and 20 steps are used as the search rangeStep 4 Extract ZC feature vectors of every signal atdifferent ε values (e number and length of observationwindow were optimized before feature extraction andthe data overlapped by 50Step 5 Calculate the value of I between different statecategories according to formula (8) Suppose there are 4kinds of state signals and then 6 values of I are formedone for every 2 kinds of signals ε value is searched 20times and then a 6times 20 matrix is formedStep 6 Calculate the maximum and minimum of I foreach search guaranteed the highest recognition ca-pacity of the two categories is not weakened and in-crease the value of I between the two classes that aremost difficult to recognize So when formula (9) issatisfied the optimization is achieved According to thestep size of each signal and the number of steps theoptimized ε value of each signal can be obtainedStep 7 According to the optimized ε value of eachsignal extract the optimized ZC featuresStep 8 Bring the optimized features into neural net-work for fault diagnosis and verification and the resultsare analyzed

3 Simulation Analysis

For the sake of verifying the necessity and effectiveness of thepresented method four simulated signals of normal innerring fault rolling element fault and outer ring fault of rolling

bearing are used for analysis (e simulation signals areobtained from the following formula

x(t) 1113936M

i1Ais t minus iT minus τi( 1113857 + n(i)

Ai A0 cos 2πQt + φA( 1113857 + CA

s(t) eminus Bt sin 2πfnt + φw( 1113857

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

(12)

where the system sampling frequency fs 12000Hz theresonant frequency fn 3000Hz the inner ring fault fre-quency fi 162Hz the rolling element fault frequencyfb 142Hz the outer ring fault frequency fi 142Hz therotational frequency fr 30Hz and the damping ratioB 400 Four simulated signals are a group a total of threegroups (ree random noises with a SNR of 30 dB 5 dB andminus5 dB are added separately to three groups signal (egrouping of simulation signals is shown in Table 1

31 Analysis of ZC Feature Noise Resistance In orderto make the extracted features represent the running stateinformation of the bearing effectively the length of the datasegment calculated with ZC feature should contain at least onecomplete rotation period (e rotational frequency is 30Hz sothe length of an observation window must greater than 130secondswhich contains 360 points at the sampling frequency of12K Follow the method in Section 23 above the data frag-ment length is 512 and the number of fragment is 8 Asdescribed earlier the data overlap rate is set to 50 whichcontains 256 points at the sampling frequency of 12K Eachstate data are divided into 40 samples and contain data of 05seconds (6000 points) Figure 2 shows the ZC features(FVduration) of the four state signals extracted from each set ofsignals when SNR 30 5 and minus5 From the left to the right ofthe figure is the distribution of FVduration of four state signalswhen SNR is 30 5 and minus5 (e graphs above the figure are thedistribution of all samples under a certain feature and thebelow ones are the differentiation of all eigenvectors under acertain sample It can be seen from the three groups of graphsthat the ZC feature is very sensitive to random noise and therecognition ability between ZC features of various state signalsis decreasing with random noise increasing When SNR 30FVduration still has fairish recognition ability however whenSNR minus5 recognition ability of FVduration begins to fail

32 Analysis of ε Value On the basis of the method de-scription in Section 23 extract the ZC features and calculatethe various values of I with different values of ε(en calculatethe values of ϕ according to formula (9) and find theminimumvalue of ϕ (e relationship between ϕ and ε is shown inFigures 3ndash5 under different kinds of SNRs(e figures indicatethat with the increase of the ε value the capacity of distinguishfault is improved but they will not continue indefinitely Inother words specifically value of ϕwould not always decreasebut has a critical value such as the values marked with arrowsin three figures and then with the value of ϕ increasing therecognition capacity of certain two types will decrease(erefore the optimized values of ε can be obtained in adifferent kind of SNR (e value of ε is represented by the

4 Mathematical Problems in Engineering

number of steps and the step sizes of the four state signalsunder different kind SNR are shown in Table 2

It can be seen from Figures 3ndash5 that the optimal numberof steps correspond to the minimum ϕ combined with thestep size in Table 2 and the optimized ε value can be cal-culated as shown in Table 3

(en the ZC features extracted from above optimized εvalues are shown in Figure 6

FromFigure 6 we can clearly see that all samples of signals indiverse states in the same feature can be separated from each

other and similarly all features of signals in diverse states underthe same sample can be separated fromeach other Consequentlythe ZC characteristics of the four states can be recognizedCompared with the identification result before optimization inFigure 2 the effectiveness of the method is proved

33 Optimization Validation In order to further prove theavailability of themethod the optimized ZC eigenvectors areidentified by three-layer feed-forward neural network andthe result of verification is shown in Table 4 which is the

099

098

1

097

096

094

093

095

0 5 10 15 20Samples

Nor

mal

ized

dur

atio

n

S11S12

S13S14

25 30 35 40

(a)

1

098

096

094

092

Nor

mal

ized

dur

atio

n

S21S22

S23S24

Samples0 5 10 15 20 25 30 35 40

(b)

S31S32

S33S34

1

098

096

094

092

Nor

mal

ized

dur

atio

nSamples

0 5 10 15 20 25 30 35 40

(c)1

08

06

04

021 2 3 4 5 6 7 8

Feature

Nor

mal

ized

dur

atio

n

S11S12

S13S14

(d)

1

08

06

04

021 2 3 4 5 6 7 8

Feature

Nor

mal

ized

dur

atio

n

S21S22

S23S24

(e)

1

08

06

04

021 2 3 4 5 6 7 8

Feature

Nor

mal

ized

dur

atio

n

S31S32

S33S34

(f )

Figure 2 ZC features of four state signals (from left to right SNR is 30 5 and minus5 respectively)

Neural networkfault diagnosis

Extract ZC feature vectors ofevery signal at different ε values

Calculate the I value of differentstate categories according to

formula (8)

Calculate the peak-to-peak value

Set the search rangeand step size of ε

Find the step size that satisfiesformula (9) and

obtain the ε value of each signal

Extract the optimized ZCfeature according to the

optimized ε value

Input originalvibration signalStart

End

Figure 1 Flow chart of the optimized parameter

Table 1 (e grouping of simulation signals

Index SNR 30 SNR 5 SNR minus5Simulated normalno fault S11 S21 S31Simulated inner ring fault S12 S22 S32Simulated rolling element fault S13 S23 S33Simulated outer ring fault S14 S24 S34

Mathematical Problems in Engineering 5

average recognition rate of 40 replicates It can be dem-onstrated from Table 4 that this method has a high faultidentification rate for REB simulation signals

4 Experiment Validation

(e necessity and effectiveness of the improved model andparameter optimization are verified by simulation experi-ments In practical applications the identification of rollingbearing faults usually requires the identification method tobe robust for load so the experiment uses the bearing datafrom the Case Western Reserve University Bearing DataCenter to verify whether the proposed method is effective

41 Experiment Data and Schemes (e bearing status dataset is derived from [37] (e type of test bearing is deepgroove bearing and the data are sampled by the acceler-ometer at 12KHz from the drive end of the motor Single-point failure diameter is 0007 inches (e outer ring failureis located at 6 orsquoclock For the motor loads 0 1 2 and 3motor speeds are from 1797 to 1730 RPM Table 5 details theload and speed

(e whole experiment consists of three parts (e firstexperiment verifies the identification effect of four bearingstates under load 0 HP and compares with the traditional ZCmethods (en the bearing state data at load 0 are used asthe training set to test the fault identification effect of thebearing state data under other loads Finally the bearingstate data under any load are used as the training set to testthe fault identification effect of the bearing state data underother loads (e first part of the experiment is to verify theeffect of the method in practical application and theremaining two parts are used to test the robustness of themethod

Step 6ϕ 02835

ϕ

11

1

09

08

07

06

05

04

03

02

Steps2 4 6 8 10 12 14 16 18 20

Figure 3 SNR 30 ϕ and steps trend

Step 10ϕ 0495

ϕ

18

16

14

12

1

08

06

04

Steps2 4 6 8 10 12 14 16 18 20

Figure 4 SNR 5 ϕ and steps trend

Step 12ϕ 07836

ϕ

22

2

18

16

14

12

1

08

06

Steps2 4 6 8 10 12 14 16 18 20

Figure 5 SNR minus5 ϕ and steps trend

Table 2 (e step sizes of the four state signals under differentSNRs

SNR 30 S11 S12 S13 S14Step size 0001 0008 0012 0020SNR 5 S21 S22 S23 S34Step size 0002 0007 0011 0020SNR minus5 S31 S32 S33 S34Step size 0003 0009 0012 0020

Table 3 (e optimized ε value of the four state signals underdifferent SNRs

SNR 30 S11 S12 S13 S14ε 0006 0048 0072 012SNR 5 S21 S22 S23 S34ε 002 007 011 020SNR minus5 S31 S32 S33 S34ε 0036 0108 0144 024

6 Mathematical Problems in Engineering

42 Results andAnalysis Firstly the spectrum and envelopespectrum analysis of bearing state data under load 0 test arecarried out to understand the influence of noise on faultdiagnosis Figures 7 and 8 show the frequency spectrum andenvelope spectrum of normal state inner ring fault rollingelement fault and outer ring fault of REB Due to themodulation of the noise signal the characteristic frequencycannot be detected from the frequency spectrum In Fig-ure 8 the characteristic frequency of outer ring failure andthe frequency doubling can be observed but the faultcharacteristic frequency of the inner ring is not obvious andthe characteristic frequency of rolling element is unavailableSo the fault of rolling element cannot be identified effec-tively as the influence of strong noise and signal modulation

and the random signal is spread throughout the frequencydomain It can be concluded from the above analysis that inpractice random noise abundant is an important factorhindering the accuracy of fault identification Due to theexistence of high frequency noise it is necessary to improvethe ZC method model and optimize the parameters so as toimprove the accuracy of fault diagnosis and make the ad-vantages of fast calculation of ZC method useful

(e rotational speed of load 0 is 1797 rpm from Table 5therefore the magnitude of the observation window shouldbe greater than 0034 seconds which contains 360 points atthe sampling frequency of 12K Follow the method inSection 31 the data fragment length is 720 and the numberof fragment is 8 (en each length of data collected from thefour bearing states is 10 seconds which is divided into 40segments with 50 overlap rate According to the method inthis paper optimization parameters in the improved modelrespectively are calculated and shown in Table 6 (e ratioshows the ratio of the optimized ε to the value of peak-to-peak Figure 9 shows the comparison of ZC characteristics offour bearing states before and after the model improvementand parameter optimization

From Figures 9(a) and 9(c) we can see that before themodel improvement parameter optimization the faultcharacteristics of partial samples of rolling element andouter ring are similar and difficult to distinguish which willlead to misjudgment From Figures 9(b) and 9(d) we can seethat the sample characteristics of rolling element fault andthe outer ring fault are completely distinguished through

1

08

06

04

02

0

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

S11S12

S13S14

(a)

1

08

06

04

02

0

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

S21S22

S23S24

(b)

1

08

06

04

02

0

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

S31S32

S33S34

(c)1

08

06

04

02

0

Feature

Nor

mal

ized

dur

atio

n

1 2 3 4 5 6 7 8

S11S12

S13S14

(d)

1

08

06

04

02

0

Feature

Nor

mal

ized

dur

atio

n

1 2 3 4 5 6 7 8

S21S22

S23S24

(e)

1

08

06

04

02

0

Feature

Nor

mal

ized

dur

atio

n

1 2 3 4 5 6 7 8

S31S32

S33S34

(f )

Figure 6 Optimized ZC features of four state signals (from left to right SNR is 30 5 and minus5 respectively)

Table 4 (e average identification rate of simulation fault

Index SNR 30 SNR 5 SNR minus5Classification rate () FVduration FVduration FVdurationNormal 100 100 100Inner ring defect 100 100 9985Rolling element defect 100 100 9985Outer ring defect 100 100 100

Table 5 (e motor load and corresponding speed

Index Load 0 Load 1 Load 2 Load 3Motor speedRPM 1797 1772 1750 1730

Mathematical Problems in Engineering 7

Normal frequency spectrum01

005

0

Frequency (Hz)

Am

plitu

de (V

)

0 1000 2000 3000 4000 5000 6000

(a)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Inner frequency spectrum01

005

0Am

plitu

de (V

)

(b)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Ball frequency spectrum004

002

0Am

plitu

de (V

)

(c)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Outer frequency spectrum02

01

0Am

plitu

de (V

)

(d)

Figure 7 Frequency spectrum of four bearing states

Normal envelope spectrum004

002

0

Frequency (Hz)

Am

plitu

de (V

)

0 1000 2000 3000 4000 5000 6000

(a)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Inner envelope spectrum02

01

0Am

plitu

de (V

)

(b)

Figure 8 Continued

8 Mathematical Problems in Engineering

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Ball envelope spectrum002

001

0Am

plitu

de (V

)

(c)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Outer envelope spectrum1

05

0Am

plitu

de (V

)

(d)

Figure 8 Envelope spectrum of four bearing states

Table 6 Optimized values of ε and theirs ratios to peak-to-peak

Index Normal Inner race fault Ball fault Outer race faultε 0000598 0003109 0001211 0002041Ratio () 01 01 01 0029

1

08

06

04

02

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

NormalInner race defect

Ball defectOuter race defect

(a)

1

08

06

04

0

02

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

NormalInner race defect

Ball defectOuter race defect

(b)

Feature1 2 3 4 5 6 7 8

1

08

06

04

02

0

Nor

mal

ized

dur

atio

n

NormalInner race defect

Ball defectOuter race defect

(c)

Feature1 2 3 4 5 6 7 8

1

08

06

04

0

02Nor

mal

ized

dur

atio

n

NormalInner race defect

Ball defectOuter race defect

(d)

Figure 9 ZC duration features before (a and c) and after (b and d) the model improvement and parameter optimization

Mathematical Problems in Engineering 9

adopting the method of the model improvement and pa-rameter optimization

In order to compare the recognition rate all the ei-genvectors are taken to a three-layer neural network whose

weights and thresholds are optimized by genetic algorithmto improve the diagnostic efficiency and accuracy (enumber of neurons in the hidden layer is 5 and the totalnumber of samples is 160 including training set sample 120and test set sample 40 Each bearing state contains 30training set samples and 10 test set samples arranged inorder according to normal inner ring fault rolling elementfault and outer ring fault (e result of verification is shownin Figure 10 which is the average recognition rate of 40replicates (e recognition rate of both training prior sta-tistics and test set is always 100 and proves that purposedmethod can identify bearing faults accurately

As a comparison the recognition result before the modelimprovement and parameter optimization is shown inFigure 11

(e results in the Figure 11 represent the average rec-ognition accuracy of 40 repeated experiments from which itcan be intuitively seen that the recognition rate of normalstate and inner ring fault is 100 but the recognition rate ofrolling element fault and outer ring fault is not satisfactory(e partial misjudgment between rolling element fault andouter ring fault has affected the fault recognition rate andthe comparison of the average recognition rate of 40 ex-periments is shown in Table 7

Clas

s4

35

3

25

2

15

1

Train sample0 20 40 60 80 100 120

(a)Cl

ass

4

35

3

25

2

15

1

Test sample0 5 10 2015 25 3530 40

(b)

Figure 10 Identification results of adopting the method of the model improvement and parameter optimization

Table 7 Comparison of mean recognition rate before and afteroptimization

Index Training set Test setAccuracy of before optimization () 9333 9225Accuracy of after optimization () 100 100

Table 8 Optimized ε values of various bearing states under threeloads

Index Normal Inner ringfault

Rollingelement fault

Outer ringfault

ε underload 1 0000521 0002938 0001299 0001776

ε underload 2 0000635 0003065 0001171 0001793

ε underload 3 0000548 0003188 0001321 0001809

Table 9 Recognition rate of various bearing states under threeloads

Index Normal Inner ringfault

Rollingelement fault

Outer ringfault

Accuracy ofload 1 () 100 100 9917 9917

Accuracy ofload 2 () 100 975 95 925

Accuracy ofload 3 () 9917 9917 9833 9833

Table 10 Recognition rate of different load HPs as training sets

Index Test set recognition rateTraining set Load 0 Load 1 Load 2 Load 3Load 0 100 9972 9683 9912Load 1 9813 9985 9672 9783Load 2 9775 9728 9883 9752Load 3 9815 9723 9695 9917

10 Mathematical Problems in Engineering

(e experimental results show that the method of themodel improvement and parameter optimization is effectivefor bearing fault identification under the same load

Next the experiment will verify whether the method inthis paper is robust for the load (e experimental data ofload 0 is still used as the training set according to ratios of εto peak-to-peak in Table 6 calculate the optimization pa-rameters under the other load and extract the ZC features asthe test set (e length and number of observation windowsshould be consistent with that under load 0 when extractingthe ZC features under other load Table 8 shows the opti-mized ε values of various bearing states and under threeloads

(e same neural network is applied to identify thebearing fault and the results are shown in Table 9

According to the recognition results in Table 9 althoughthe fault recognition rate at load 2 is lower the overallrecognition rate of other loads still achieves the expectedeffect (e analysis of the error identification samples showsthat these samples are quite distinct from the other samplesof the same state so the validity of themethod is not affected(e experiment proves that the method is applicable to otherloads when the load information is known

Finally the experiment verifies the test effect under anyload and the recognition effect of unknown load has beentested Similarly the method proposed in this paper isemployed for obtaining characteristic samples the samplesunder each load are used as training sets and the samples ofother load are test sets (e overall recognition rate mea-sured is shown in Table 10 and each result is the averagerecognition rate of 40 tests as well

According to the analysis of the results presented inTable 10 the recognition accuracy of samples under thesame load as the training set and the test set is relatively highWhen samples under different loads are used as training setsto test other load samples the recognition accuracy isslightly lower because a few samples deviate from the

characteristics of similar samples In general the experi-mental results demonstrate that the method has good loadrobustness so as to accurately identify the bearing fault evenif the bearing load is not known

5 Conclusion

In this paper an improved fault diagnosis method forelectromechanical system based on ZC algorithm is pro-posed which optimized the calculation model of ZCR andthe parameters of ZC algorithm by establishing a criterionfunction model (e fault diagnosis experiment of motorREB proves that the method successfully overcomes theshortcoming of ZC feature sensitive to noise and can realizeaccurate fault diagnosis without knowing the bearing loadCombined with the merits of saving computing time in thecase of big data the method is expected to monitor thecrucial parts of electromechanical system in real time so asto provide security and reliability guaranty for accuratecontrol of the electromechanical system

Data Availability

(e data used to support the study are available from thecorresponding author upon request

Conflicts of Interest

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

Acknowledgments

(is work was supported by the National Key Research andDevelopment plan of China under Grant 2018YFB1306103the National Natural Science Foundation of China underGrant 51875272 and the Provincial School Education

Clas

s4

35

3

25

2

15

1

Train sample0 20 40 60 80 100 120

(a)

Clas

s

4

35

3

25

2

15

1

Test sample0 5 10 2015 25 3530 40

(b)

Figure 11 Identification results before the model improvement and parameter optimization

Mathematical Problems in Engineering 11

Cooperation Project of Yunnan Province under GrantKKDA202001003

References

[1] M R Khoshdarregi S Tappe and Y Altintas ldquoIntegratedfive-Axis trajectory shaping and contour error compensationfor high-speed CNC machine toolsrdquo IEEEASME Transac-tions on Mechatronics vol 19 no 6 pp 1859ndash1871 2014

[2] R Colombo I Cusmano I Sterpi A Mazzone C Delconteand F Pisano ldquoTest-retest reliability of robotic assessmentmeasures for the evaluation of upper limb recoveryrdquo IEEETransactions on Neural Systems and Rehabilitation Engi-neering vol 22 no 5 pp 1020ndash1029 2014

[3] B Xu D Wang Y Zhang and Z Shi ldquoDOB-based neuralcontrol of flexible hypersonic flight vehicle considering windeffectsrdquo IEEE Transactions on Industrial Electronics vol 64no 11 pp 8676ndash8685 2017

[4] X Ding Q He and N Luo ldquoA fusion feature and its im-provement based on locality preserving projections for rollingelement bearing fault classificationrdquo Journal of Sound andVibration vol 335 pp 367ndash383 2015

[5] Y Zhang and R B Randall ldquoRolling element bearing faultdiagnosis based on the combination of genetic algorithms andfast kurtogramrdquo Mechanical Systems and Signal Processingvol 23 no 5 pp 1509ndash1517 2009

[6] J Benali M Sayadi F Fnaiech B Morello and N ZerhounildquoImportance of the fourth and fifth intrinsic mode functionsfor bearing fault diagnosisrdquo in Proceedings of the 14th In-ternational Conference on Sciences and Techniques of Auto-matic Control amp Computer EngineeringmdashSTArsquo2013 SousseTunisia December 2013

[7] S Nandi H A Toliyat and X Li ldquoCondition monitoring andfault diagnosis of electrical motors-A reviewrdquo IEEE Trans-actions on Energy Conversion vol 20 no 4 pp 719ndash729 2005

[8] J Huang S Ri T Fukuda and Y Wang ldquoA disturbanceobserver based sliding mode control for a class of under-actuated robotic system with mismatched uncertaintiesrdquoIEEE Transactions on Automatic Control vol 64 no 6pp 2480ndash2487 2019

[9] J Huang M Zhang S Ri C Xiong Z Li and Y Kang ldquoHigh-order disturbance observer based sliding mode control formobile wheeled inverted pendulum systemsrdquo IEEE Trans-actions on Industrial Electronics vol 67 no 3 pp 2030ndash20412020

[10] H Jian M H Ri DWu and S Ri ldquoInterval type-2 fuzzy logicmodeling and control of a mobile two-wheeled invertedpendulumrdquo IEEE Transactions on Fuzzy Systems vol 26no 4 pp 2030ndash2038 2018

[11] P D Mcfadden and J D Smith ldquoModel for the vibrationproduced by a single point defect in a rolling elementbearingrdquo Journal of Sound and Vibration vol 96 no 1pp 69ndash82 1984

[12] N Sawalhi and R B Randall ldquoSimulating gear and bearinginteractions in the presence of faultsrdquoMechanical Systems andSignal Processing vol 22 no 8 pp 1924ndash1951 2008

[13] Q Zhu L Liu S Li and W Zhang ldquoControl of complexnonlinear dynamic rational systemsrdquo Complexity vol 2018Article ID 8953035 12 pages 2018

[14] Q M Zhu D Y Zhao and J Zhang ldquoA general U-blockmodel-based design procedure for nonlinear polynomialcontrol systemsrdquo International Journal of Systems Sciencevol 47 no 14 pp 3465ndash3475 2015

[15] A Rafsanjani S Abbasion A Farshidianfar andH Moeenfard ldquoNonlinear dynamic modeling of surfacedefects in rolling element bearing systemsrdquo Journal of Soundamp Vibration vol 319 no 3-5 pp 1150ndash1174 2009

[16] T W Rauber F de Assis Boldt and F M Varejao ldquoHet-erogeneous feature models and feature selection applied tobearing fault diagnosisrdquo IEEE Transactions on IndustrialElectronics vol 62 no 1 pp 637ndash646 2015

[17] J Chebil M Hrairi and N Abushikhah ldquoSignal analysis ofvibration measurements for condition monitoring of bear-ingsrdquo Australian Journal of Basic amp Applied Sciences vol 5no 1 p 70 2011

[18] M D Prieto G Cirrincione A G Espinosa J A Ortega andH Henao ldquobearing fault detection by a novel condition-monitoring scheme based on statistical-time features andneural networksrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 pp 3398ndash3407 2013

[19] A Garcia-Perez R d J Romero-Troncoso E Cabal-Yepezand R A Osornio-Rios ldquo(e application of high-resolutionspectral analysis for identifying multiple combined faults ininduction motorsrdquo IEEE Transactions on Industrial Elec-tronics vol 58 no 5 pp 2002ndash2010 2011

[20] Z K Peng and F L Chu ldquoApplication of the wavelettransform in machine condition monitoring and fault diag-nostics a review with bibliographyrdquo Mechanical Systems andSignal Processing vol 18 no 2 pp 199ndash221 2004

[21] J Seshadrinath B Singh and B K Panigrahi ldquoInvestigationof vibration signatures for multiple fault diagnosis in variablefrequency drives using complex waveletsrdquo IEEE Transactionson Power Electronics vol 29 no 2 pp 936ndash945 2014

[22] W Su F Wang H Zhu Z Zhang and Z Guo ldquoRollingelement bearing faults diagnosis based on optimal Morletwavelet filter and autocorrelation enhancementrdquo MechanicalSystems and Signal Processing vol 24 no 5 pp 1458ndash14722010

[23] X Jin M Zhao T W S Chow andM Pecht ldquoMotor bearingfault diagnosis using trace ratio linear discriminant analysisrdquoIEEE Transactions on Industrial Electronics vol 61 no 5pp 2441ndash2451 2014

[24] M Zhao X Jin Z Zhang and B Li ldquoFault diagnosis of rollingelement bearings via discriminative subspace learning visu-alization and classificationrdquo Expert Systems with Applicationsvol 41 no 7 pp 3391ndash3401 2014

[25] R Niederjohn ldquoA mathematical formulation and comparisonof zero-crossing analysis techniques which have been appliedto automatioc speech recognitionrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 23 no 4pp 373ndash380 1975

[26] P William andM Hoffman ldquoEfficient sensor network vehicleclassification using peak harmonics of acoustic emissionsrdquo inProceedings of the SPIE Defense and Security Symposium Orlando FL USA March 2008

[27] A Shahidi Zandi R Tafreshi M Javidan and G A DumontldquoPredicting epileptic seizures in scalp EEG based on a vari-ational bayesian Gaussian mixture model of zero-crossingintervalsrdquo IEEE Transactions on Biomedical Engineeringvol 60 no 5 pp 1401ndash1413 2013

[28] C-H Wu H-C Chang P-L Lee et al ldquoFrequency recog-nition in an SSVEP-based brain computer interface usingempirical mode decomposition and refined generalized zero-crossingrdquo Journal of Neuroscience Methods vol 196 no 1pp 170ndash181 2011

12 Mathematical Problems in Engineering

[29] P E William and M W Hoffman ldquoIdentification of bearingfaults using time domain zero-crossingsrdquoMechanical Systemsand Signal Processing vol 25 no 8 pp 3078ndash3088 2011

[30] T Liu J Chen and G Dong ldquoZero crossing and coupledhidden Markov model for a rolling bearing performancedegradation assessmentrdquo Journal of Vibration and Controlvol 20 no 16 pp 2487ndash2500 2013

[31] A Ukil S Chen and A Andenna ldquoDetection of stator shortcircuit faults in three-phase induction motors using motorcurrent zero crossing instantsrdquo Electric Power Systems Re-search vol 81 no 4 pp 1036ndash1044 2011

[32] J D T Gonzalez and W Kinsner ldquoZero-crossing analysis ofLevy walks for real-time feature extractionrdquo in Proceedings ofthe IEEE International Conference on Electro InformationTechnology EIT Grand Forks ND USA May 2016

[33] R G Waghmare S L Nalbalwar and A Das ldquoTransientsignal detection on the basis of energy and zero crossingdetectorsrdquo Procedia Engineering vol 30 pp 129ndash134 2012

[34] B R Nayana and P Geethanjali ldquoAnalysis of statistical time-domain features effectiveness in identification of bearingfaults from vibration signalrdquo IEEE Sensors Journal vol 17no 17 pp 5618ndash5625 2017

[35] C Fisher Encyclopedia of Biometrics p 549 Springer USBoston MA USA 2009

[36] Q Chen W Xing L Tao and L Hua ldquoZero-crossing featureextraction based on threshold optimization for rolling ele-ment bearingrdquo in Lecture Notes in Electrical Engineeringpp 409ndash425 Springer Berlin Germany 2018

[37] Case Western Reserve University Bearing Data CenterWebsite accessed on Mar 23 2015 [Online]Available httpcsegroupscaseedubearingdatacenterpagesdownload-data-file

Mathematical Problems in Engineering 13

number of steps and the step sizes of the four state signalsunder different kind SNR are shown in Table 2

It can be seen from Figures 3ndash5 that the optimal numberof steps correspond to the minimum ϕ combined with thestep size in Table 2 and the optimized ε value can be cal-culated as shown in Table 3

(en the ZC features extracted from above optimized εvalues are shown in Figure 6

FromFigure 6 we can clearly see that all samples of signals indiverse states in the same feature can be separated from each

other and similarly all features of signals in diverse states underthe same sample can be separated fromeach other Consequentlythe ZC characteristics of the four states can be recognizedCompared with the identification result before optimization inFigure 2 the effectiveness of the method is proved

33 Optimization Validation In order to further prove theavailability of themethod the optimized ZC eigenvectors areidentified by three-layer feed-forward neural network andthe result of verification is shown in Table 4 which is the

099

098

1

097

096

094

093

095

0 5 10 15 20Samples

Nor

mal

ized

dur

atio

n

S11S12

S13S14

25 30 35 40

(a)

1

098

096

094

092

Nor

mal

ized

dur

atio

n

S21S22

S23S24

Samples0 5 10 15 20 25 30 35 40

(b)

S31S32

S33S34

1

098

096

094

092

Nor

mal

ized

dur

atio

nSamples

0 5 10 15 20 25 30 35 40

(c)1

08

06

04

021 2 3 4 5 6 7 8

Feature

Nor

mal

ized

dur

atio

n

S11S12

S13S14

(d)

1

08

06

04

021 2 3 4 5 6 7 8

Feature

Nor

mal

ized

dur

atio

n

S21S22

S23S24

(e)

1

08

06

04

021 2 3 4 5 6 7 8

Feature

Nor

mal

ized

dur

atio

n

S31S32

S33S34

(f )

Figure 2 ZC features of four state signals (from left to right SNR is 30 5 and minus5 respectively)

Neural networkfault diagnosis

Extract ZC feature vectors ofevery signal at different ε values

Calculate the I value of differentstate categories according to

formula (8)

Calculate the peak-to-peak value

Set the search rangeand step size of ε

Find the step size that satisfiesformula (9) and

obtain the ε value of each signal

Extract the optimized ZCfeature according to the

optimized ε value

Input originalvibration signalStart

End

Figure 1 Flow chart of the optimized parameter

Table 1 (e grouping of simulation signals

Index SNR 30 SNR 5 SNR minus5Simulated normalno fault S11 S21 S31Simulated inner ring fault S12 S22 S32Simulated rolling element fault S13 S23 S33Simulated outer ring fault S14 S24 S34

Mathematical Problems in Engineering 5

average recognition rate of 40 replicates It can be dem-onstrated from Table 4 that this method has a high faultidentification rate for REB simulation signals

4 Experiment Validation

(e necessity and effectiveness of the improved model andparameter optimization are verified by simulation experi-ments In practical applications the identification of rollingbearing faults usually requires the identification method tobe robust for load so the experiment uses the bearing datafrom the Case Western Reserve University Bearing DataCenter to verify whether the proposed method is effective

41 Experiment Data and Schemes (e bearing status dataset is derived from [37] (e type of test bearing is deepgroove bearing and the data are sampled by the acceler-ometer at 12KHz from the drive end of the motor Single-point failure diameter is 0007 inches (e outer ring failureis located at 6 orsquoclock For the motor loads 0 1 2 and 3motor speeds are from 1797 to 1730 RPM Table 5 details theload and speed

(e whole experiment consists of three parts (e firstexperiment verifies the identification effect of four bearingstates under load 0 HP and compares with the traditional ZCmethods (en the bearing state data at load 0 are used asthe training set to test the fault identification effect of thebearing state data under other loads Finally the bearingstate data under any load are used as the training set to testthe fault identification effect of the bearing state data underother loads (e first part of the experiment is to verify theeffect of the method in practical application and theremaining two parts are used to test the robustness of themethod

Step 6ϕ 02835

ϕ

11

1

09

08

07

06

05

04

03

02

Steps2 4 6 8 10 12 14 16 18 20

Figure 3 SNR 30 ϕ and steps trend

Step 10ϕ 0495

ϕ

18

16

14

12

1

08

06

04

Steps2 4 6 8 10 12 14 16 18 20

Figure 4 SNR 5 ϕ and steps trend

Step 12ϕ 07836

ϕ

22

2

18

16

14

12

1

08

06

Steps2 4 6 8 10 12 14 16 18 20

Figure 5 SNR minus5 ϕ and steps trend

Table 2 (e step sizes of the four state signals under differentSNRs

SNR 30 S11 S12 S13 S14Step size 0001 0008 0012 0020SNR 5 S21 S22 S23 S34Step size 0002 0007 0011 0020SNR minus5 S31 S32 S33 S34Step size 0003 0009 0012 0020

Table 3 (e optimized ε value of the four state signals underdifferent SNRs

SNR 30 S11 S12 S13 S14ε 0006 0048 0072 012SNR 5 S21 S22 S23 S34ε 002 007 011 020SNR minus5 S31 S32 S33 S34ε 0036 0108 0144 024

6 Mathematical Problems in Engineering

42 Results andAnalysis Firstly the spectrum and envelopespectrum analysis of bearing state data under load 0 test arecarried out to understand the influence of noise on faultdiagnosis Figures 7 and 8 show the frequency spectrum andenvelope spectrum of normal state inner ring fault rollingelement fault and outer ring fault of REB Due to themodulation of the noise signal the characteristic frequencycannot be detected from the frequency spectrum In Fig-ure 8 the characteristic frequency of outer ring failure andthe frequency doubling can be observed but the faultcharacteristic frequency of the inner ring is not obvious andthe characteristic frequency of rolling element is unavailableSo the fault of rolling element cannot be identified effec-tively as the influence of strong noise and signal modulation

and the random signal is spread throughout the frequencydomain It can be concluded from the above analysis that inpractice random noise abundant is an important factorhindering the accuracy of fault identification Due to theexistence of high frequency noise it is necessary to improvethe ZC method model and optimize the parameters so as toimprove the accuracy of fault diagnosis and make the ad-vantages of fast calculation of ZC method useful

(e rotational speed of load 0 is 1797 rpm from Table 5therefore the magnitude of the observation window shouldbe greater than 0034 seconds which contains 360 points atthe sampling frequency of 12K Follow the method inSection 31 the data fragment length is 720 and the numberof fragment is 8 (en each length of data collected from thefour bearing states is 10 seconds which is divided into 40segments with 50 overlap rate According to the method inthis paper optimization parameters in the improved modelrespectively are calculated and shown in Table 6 (e ratioshows the ratio of the optimized ε to the value of peak-to-peak Figure 9 shows the comparison of ZC characteristics offour bearing states before and after the model improvementand parameter optimization

From Figures 9(a) and 9(c) we can see that before themodel improvement parameter optimization the faultcharacteristics of partial samples of rolling element andouter ring are similar and difficult to distinguish which willlead to misjudgment From Figures 9(b) and 9(d) we can seethat the sample characteristics of rolling element fault andthe outer ring fault are completely distinguished through

1

08

06

04

02

0

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

S11S12

S13S14

(a)

1

08

06

04

02

0

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

S21S22

S23S24

(b)

1

08

06

04

02

0

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

S31S32

S33S34

(c)1

08

06

04

02

0

Feature

Nor

mal

ized

dur

atio

n

1 2 3 4 5 6 7 8

S11S12

S13S14

(d)

1

08

06

04

02

0

Feature

Nor

mal

ized

dur

atio

n

1 2 3 4 5 6 7 8

S21S22

S23S24

(e)

1

08

06

04

02

0

Feature

Nor

mal

ized

dur

atio

n

1 2 3 4 5 6 7 8

S31S32

S33S34

(f )

Figure 6 Optimized ZC features of four state signals (from left to right SNR is 30 5 and minus5 respectively)

Table 4 (e average identification rate of simulation fault

Index SNR 30 SNR 5 SNR minus5Classification rate () FVduration FVduration FVdurationNormal 100 100 100Inner ring defect 100 100 9985Rolling element defect 100 100 9985Outer ring defect 100 100 100

Table 5 (e motor load and corresponding speed

Index Load 0 Load 1 Load 2 Load 3Motor speedRPM 1797 1772 1750 1730

Mathematical Problems in Engineering 7

Normal frequency spectrum01

005

0

Frequency (Hz)

Am

plitu

de (V

)

0 1000 2000 3000 4000 5000 6000

(a)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Inner frequency spectrum01

005

0Am

plitu

de (V

)

(b)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Ball frequency spectrum004

002

0Am

plitu

de (V

)

(c)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Outer frequency spectrum02

01

0Am

plitu

de (V

)

(d)

Figure 7 Frequency spectrum of four bearing states

Normal envelope spectrum004

002

0

Frequency (Hz)

Am

plitu

de (V

)

0 1000 2000 3000 4000 5000 6000

(a)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Inner envelope spectrum02

01

0Am

plitu

de (V

)

(b)

Figure 8 Continued

8 Mathematical Problems in Engineering

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Ball envelope spectrum002

001

0Am

plitu

de (V

)

(c)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Outer envelope spectrum1

05

0Am

plitu

de (V

)

(d)

Figure 8 Envelope spectrum of four bearing states

Table 6 Optimized values of ε and theirs ratios to peak-to-peak

Index Normal Inner race fault Ball fault Outer race faultε 0000598 0003109 0001211 0002041Ratio () 01 01 01 0029

1

08

06

04

02

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

NormalInner race defect

Ball defectOuter race defect

(a)

1

08

06

04

0

02

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

NormalInner race defect

Ball defectOuter race defect

(b)

Feature1 2 3 4 5 6 7 8

1

08

06

04

02

0

Nor

mal

ized

dur

atio

n

NormalInner race defect

Ball defectOuter race defect

(c)

Feature1 2 3 4 5 6 7 8

1

08

06

04

0

02Nor

mal

ized

dur

atio

n

NormalInner race defect

Ball defectOuter race defect

(d)

Figure 9 ZC duration features before (a and c) and after (b and d) the model improvement and parameter optimization

Mathematical Problems in Engineering 9

adopting the method of the model improvement and pa-rameter optimization

In order to compare the recognition rate all the ei-genvectors are taken to a three-layer neural network whose

weights and thresholds are optimized by genetic algorithmto improve the diagnostic efficiency and accuracy (enumber of neurons in the hidden layer is 5 and the totalnumber of samples is 160 including training set sample 120and test set sample 40 Each bearing state contains 30training set samples and 10 test set samples arranged inorder according to normal inner ring fault rolling elementfault and outer ring fault (e result of verification is shownin Figure 10 which is the average recognition rate of 40replicates (e recognition rate of both training prior sta-tistics and test set is always 100 and proves that purposedmethod can identify bearing faults accurately

As a comparison the recognition result before the modelimprovement and parameter optimization is shown inFigure 11

(e results in the Figure 11 represent the average rec-ognition accuracy of 40 repeated experiments from which itcan be intuitively seen that the recognition rate of normalstate and inner ring fault is 100 but the recognition rate ofrolling element fault and outer ring fault is not satisfactory(e partial misjudgment between rolling element fault andouter ring fault has affected the fault recognition rate andthe comparison of the average recognition rate of 40 ex-periments is shown in Table 7

Clas

s4

35

3

25

2

15

1

Train sample0 20 40 60 80 100 120

(a)Cl

ass

4

35

3

25

2

15

1

Test sample0 5 10 2015 25 3530 40

(b)

Figure 10 Identification results of adopting the method of the model improvement and parameter optimization

Table 7 Comparison of mean recognition rate before and afteroptimization

Index Training set Test setAccuracy of before optimization () 9333 9225Accuracy of after optimization () 100 100

Table 8 Optimized ε values of various bearing states under threeloads

Index Normal Inner ringfault

Rollingelement fault

Outer ringfault

ε underload 1 0000521 0002938 0001299 0001776

ε underload 2 0000635 0003065 0001171 0001793

ε underload 3 0000548 0003188 0001321 0001809

Table 9 Recognition rate of various bearing states under threeloads

Index Normal Inner ringfault

Rollingelement fault

Outer ringfault

Accuracy ofload 1 () 100 100 9917 9917

Accuracy ofload 2 () 100 975 95 925

Accuracy ofload 3 () 9917 9917 9833 9833

Table 10 Recognition rate of different load HPs as training sets

Index Test set recognition rateTraining set Load 0 Load 1 Load 2 Load 3Load 0 100 9972 9683 9912Load 1 9813 9985 9672 9783Load 2 9775 9728 9883 9752Load 3 9815 9723 9695 9917

10 Mathematical Problems in Engineering

(e experimental results show that the method of themodel improvement and parameter optimization is effectivefor bearing fault identification under the same load

Next the experiment will verify whether the method inthis paper is robust for the load (e experimental data ofload 0 is still used as the training set according to ratios of εto peak-to-peak in Table 6 calculate the optimization pa-rameters under the other load and extract the ZC features asthe test set (e length and number of observation windowsshould be consistent with that under load 0 when extractingthe ZC features under other load Table 8 shows the opti-mized ε values of various bearing states and under threeloads

(e same neural network is applied to identify thebearing fault and the results are shown in Table 9

According to the recognition results in Table 9 althoughthe fault recognition rate at load 2 is lower the overallrecognition rate of other loads still achieves the expectedeffect (e analysis of the error identification samples showsthat these samples are quite distinct from the other samplesof the same state so the validity of themethod is not affected(e experiment proves that the method is applicable to otherloads when the load information is known

Finally the experiment verifies the test effect under anyload and the recognition effect of unknown load has beentested Similarly the method proposed in this paper isemployed for obtaining characteristic samples the samplesunder each load are used as training sets and the samples ofother load are test sets (e overall recognition rate mea-sured is shown in Table 10 and each result is the averagerecognition rate of 40 tests as well

According to the analysis of the results presented inTable 10 the recognition accuracy of samples under thesame load as the training set and the test set is relatively highWhen samples under different loads are used as training setsto test other load samples the recognition accuracy isslightly lower because a few samples deviate from the

characteristics of similar samples In general the experi-mental results demonstrate that the method has good loadrobustness so as to accurately identify the bearing fault evenif the bearing load is not known

5 Conclusion

In this paper an improved fault diagnosis method forelectromechanical system based on ZC algorithm is pro-posed which optimized the calculation model of ZCR andthe parameters of ZC algorithm by establishing a criterionfunction model (e fault diagnosis experiment of motorREB proves that the method successfully overcomes theshortcoming of ZC feature sensitive to noise and can realizeaccurate fault diagnosis without knowing the bearing loadCombined with the merits of saving computing time in thecase of big data the method is expected to monitor thecrucial parts of electromechanical system in real time so asto provide security and reliability guaranty for accuratecontrol of the electromechanical system

Data Availability

(e data used to support the study are available from thecorresponding author upon request

Conflicts of Interest

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

Acknowledgments

(is work was supported by the National Key Research andDevelopment plan of China under Grant 2018YFB1306103the National Natural Science Foundation of China underGrant 51875272 and the Provincial School Education

Clas

s4

35

3

25

2

15

1

Train sample0 20 40 60 80 100 120

(a)

Clas

s

4

35

3

25

2

15

1

Test sample0 5 10 2015 25 3530 40

(b)

Figure 11 Identification results before the model improvement and parameter optimization

Mathematical Problems in Engineering 11

Cooperation Project of Yunnan Province under GrantKKDA202001003

References

[1] M R Khoshdarregi S Tappe and Y Altintas ldquoIntegratedfive-Axis trajectory shaping and contour error compensationfor high-speed CNC machine toolsrdquo IEEEASME Transac-tions on Mechatronics vol 19 no 6 pp 1859ndash1871 2014

[2] R Colombo I Cusmano I Sterpi A Mazzone C Delconteand F Pisano ldquoTest-retest reliability of robotic assessmentmeasures for the evaluation of upper limb recoveryrdquo IEEETransactions on Neural Systems and Rehabilitation Engi-neering vol 22 no 5 pp 1020ndash1029 2014

[3] B Xu D Wang Y Zhang and Z Shi ldquoDOB-based neuralcontrol of flexible hypersonic flight vehicle considering windeffectsrdquo IEEE Transactions on Industrial Electronics vol 64no 11 pp 8676ndash8685 2017

[4] X Ding Q He and N Luo ldquoA fusion feature and its im-provement based on locality preserving projections for rollingelement bearing fault classificationrdquo Journal of Sound andVibration vol 335 pp 367ndash383 2015

[5] Y Zhang and R B Randall ldquoRolling element bearing faultdiagnosis based on the combination of genetic algorithms andfast kurtogramrdquo Mechanical Systems and Signal Processingvol 23 no 5 pp 1509ndash1517 2009

[6] J Benali M Sayadi F Fnaiech B Morello and N ZerhounildquoImportance of the fourth and fifth intrinsic mode functionsfor bearing fault diagnosisrdquo in Proceedings of the 14th In-ternational Conference on Sciences and Techniques of Auto-matic Control amp Computer EngineeringmdashSTArsquo2013 SousseTunisia December 2013

[7] S Nandi H A Toliyat and X Li ldquoCondition monitoring andfault diagnosis of electrical motors-A reviewrdquo IEEE Trans-actions on Energy Conversion vol 20 no 4 pp 719ndash729 2005

[8] J Huang S Ri T Fukuda and Y Wang ldquoA disturbanceobserver based sliding mode control for a class of under-actuated robotic system with mismatched uncertaintiesrdquoIEEE Transactions on Automatic Control vol 64 no 6pp 2480ndash2487 2019

[9] J Huang M Zhang S Ri C Xiong Z Li and Y Kang ldquoHigh-order disturbance observer based sliding mode control formobile wheeled inverted pendulum systemsrdquo IEEE Trans-actions on Industrial Electronics vol 67 no 3 pp 2030ndash20412020

[10] H Jian M H Ri DWu and S Ri ldquoInterval type-2 fuzzy logicmodeling and control of a mobile two-wheeled invertedpendulumrdquo IEEE Transactions on Fuzzy Systems vol 26no 4 pp 2030ndash2038 2018

[11] P D Mcfadden and J D Smith ldquoModel for the vibrationproduced by a single point defect in a rolling elementbearingrdquo Journal of Sound and Vibration vol 96 no 1pp 69ndash82 1984

[12] N Sawalhi and R B Randall ldquoSimulating gear and bearinginteractions in the presence of faultsrdquoMechanical Systems andSignal Processing vol 22 no 8 pp 1924ndash1951 2008

[13] Q Zhu L Liu S Li and W Zhang ldquoControl of complexnonlinear dynamic rational systemsrdquo Complexity vol 2018Article ID 8953035 12 pages 2018

[14] Q M Zhu D Y Zhao and J Zhang ldquoA general U-blockmodel-based design procedure for nonlinear polynomialcontrol systemsrdquo International Journal of Systems Sciencevol 47 no 14 pp 3465ndash3475 2015

[15] A Rafsanjani S Abbasion A Farshidianfar andH Moeenfard ldquoNonlinear dynamic modeling of surfacedefects in rolling element bearing systemsrdquo Journal of Soundamp Vibration vol 319 no 3-5 pp 1150ndash1174 2009

[16] T W Rauber F de Assis Boldt and F M Varejao ldquoHet-erogeneous feature models and feature selection applied tobearing fault diagnosisrdquo IEEE Transactions on IndustrialElectronics vol 62 no 1 pp 637ndash646 2015

[17] J Chebil M Hrairi and N Abushikhah ldquoSignal analysis ofvibration measurements for condition monitoring of bear-ingsrdquo Australian Journal of Basic amp Applied Sciences vol 5no 1 p 70 2011

[18] M D Prieto G Cirrincione A G Espinosa J A Ortega andH Henao ldquobearing fault detection by a novel condition-monitoring scheme based on statistical-time features andneural networksrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 pp 3398ndash3407 2013

[19] A Garcia-Perez R d J Romero-Troncoso E Cabal-Yepezand R A Osornio-Rios ldquo(e application of high-resolutionspectral analysis for identifying multiple combined faults ininduction motorsrdquo IEEE Transactions on Industrial Elec-tronics vol 58 no 5 pp 2002ndash2010 2011

[20] Z K Peng and F L Chu ldquoApplication of the wavelettransform in machine condition monitoring and fault diag-nostics a review with bibliographyrdquo Mechanical Systems andSignal Processing vol 18 no 2 pp 199ndash221 2004

[21] J Seshadrinath B Singh and B K Panigrahi ldquoInvestigationof vibration signatures for multiple fault diagnosis in variablefrequency drives using complex waveletsrdquo IEEE Transactionson Power Electronics vol 29 no 2 pp 936ndash945 2014

[22] W Su F Wang H Zhu Z Zhang and Z Guo ldquoRollingelement bearing faults diagnosis based on optimal Morletwavelet filter and autocorrelation enhancementrdquo MechanicalSystems and Signal Processing vol 24 no 5 pp 1458ndash14722010

[23] X Jin M Zhao T W S Chow andM Pecht ldquoMotor bearingfault diagnosis using trace ratio linear discriminant analysisrdquoIEEE Transactions on Industrial Electronics vol 61 no 5pp 2441ndash2451 2014

[24] M Zhao X Jin Z Zhang and B Li ldquoFault diagnosis of rollingelement bearings via discriminative subspace learning visu-alization and classificationrdquo Expert Systems with Applicationsvol 41 no 7 pp 3391ndash3401 2014

[25] R Niederjohn ldquoA mathematical formulation and comparisonof zero-crossing analysis techniques which have been appliedto automatioc speech recognitionrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 23 no 4pp 373ndash380 1975

[26] P William andM Hoffman ldquoEfficient sensor network vehicleclassification using peak harmonics of acoustic emissionsrdquo inProceedings of the SPIE Defense and Security Symposium Orlando FL USA March 2008

[27] A Shahidi Zandi R Tafreshi M Javidan and G A DumontldquoPredicting epileptic seizures in scalp EEG based on a vari-ational bayesian Gaussian mixture model of zero-crossingintervalsrdquo IEEE Transactions on Biomedical Engineeringvol 60 no 5 pp 1401ndash1413 2013

[28] C-H Wu H-C Chang P-L Lee et al ldquoFrequency recog-nition in an SSVEP-based brain computer interface usingempirical mode decomposition and refined generalized zero-crossingrdquo Journal of Neuroscience Methods vol 196 no 1pp 170ndash181 2011

12 Mathematical Problems in Engineering

[29] P E William and M W Hoffman ldquoIdentification of bearingfaults using time domain zero-crossingsrdquoMechanical Systemsand Signal Processing vol 25 no 8 pp 3078ndash3088 2011

[30] T Liu J Chen and G Dong ldquoZero crossing and coupledhidden Markov model for a rolling bearing performancedegradation assessmentrdquo Journal of Vibration and Controlvol 20 no 16 pp 2487ndash2500 2013

[31] A Ukil S Chen and A Andenna ldquoDetection of stator shortcircuit faults in three-phase induction motors using motorcurrent zero crossing instantsrdquo Electric Power Systems Re-search vol 81 no 4 pp 1036ndash1044 2011

[32] J D T Gonzalez and W Kinsner ldquoZero-crossing analysis ofLevy walks for real-time feature extractionrdquo in Proceedings ofthe IEEE International Conference on Electro InformationTechnology EIT Grand Forks ND USA May 2016

[33] R G Waghmare S L Nalbalwar and A Das ldquoTransientsignal detection on the basis of energy and zero crossingdetectorsrdquo Procedia Engineering vol 30 pp 129ndash134 2012

[34] B R Nayana and P Geethanjali ldquoAnalysis of statistical time-domain features effectiveness in identification of bearingfaults from vibration signalrdquo IEEE Sensors Journal vol 17no 17 pp 5618ndash5625 2017

[35] C Fisher Encyclopedia of Biometrics p 549 Springer USBoston MA USA 2009

[36] Q Chen W Xing L Tao and L Hua ldquoZero-crossing featureextraction based on threshold optimization for rolling ele-ment bearingrdquo in Lecture Notes in Electrical Engineeringpp 409ndash425 Springer Berlin Germany 2018

[37] Case Western Reserve University Bearing Data CenterWebsite accessed on Mar 23 2015 [Online]Available httpcsegroupscaseedubearingdatacenterpagesdownload-data-file

Mathematical Problems in Engineering 13

average recognition rate of 40 replicates It can be dem-onstrated from Table 4 that this method has a high faultidentification rate for REB simulation signals

4 Experiment Validation

(e necessity and effectiveness of the improved model andparameter optimization are verified by simulation experi-ments In practical applications the identification of rollingbearing faults usually requires the identification method tobe robust for load so the experiment uses the bearing datafrom the Case Western Reserve University Bearing DataCenter to verify whether the proposed method is effective

41 Experiment Data and Schemes (e bearing status dataset is derived from [37] (e type of test bearing is deepgroove bearing and the data are sampled by the acceler-ometer at 12KHz from the drive end of the motor Single-point failure diameter is 0007 inches (e outer ring failureis located at 6 orsquoclock For the motor loads 0 1 2 and 3motor speeds are from 1797 to 1730 RPM Table 5 details theload and speed

(e whole experiment consists of three parts (e firstexperiment verifies the identification effect of four bearingstates under load 0 HP and compares with the traditional ZCmethods (en the bearing state data at load 0 are used asthe training set to test the fault identification effect of thebearing state data under other loads Finally the bearingstate data under any load are used as the training set to testthe fault identification effect of the bearing state data underother loads (e first part of the experiment is to verify theeffect of the method in practical application and theremaining two parts are used to test the robustness of themethod

Step 6ϕ 02835

ϕ

11

1

09

08

07

06

05

04

03

02

Steps2 4 6 8 10 12 14 16 18 20

Figure 3 SNR 30 ϕ and steps trend

Step 10ϕ 0495

ϕ

18

16

14

12

1

08

06

04

Steps2 4 6 8 10 12 14 16 18 20

Figure 4 SNR 5 ϕ and steps trend

Step 12ϕ 07836

ϕ

22

2

18

16

14

12

1

08

06

Steps2 4 6 8 10 12 14 16 18 20

Figure 5 SNR minus5 ϕ and steps trend

Table 2 (e step sizes of the four state signals under differentSNRs

SNR 30 S11 S12 S13 S14Step size 0001 0008 0012 0020SNR 5 S21 S22 S23 S34Step size 0002 0007 0011 0020SNR minus5 S31 S32 S33 S34Step size 0003 0009 0012 0020

Table 3 (e optimized ε value of the four state signals underdifferent SNRs

SNR 30 S11 S12 S13 S14ε 0006 0048 0072 012SNR 5 S21 S22 S23 S34ε 002 007 011 020SNR minus5 S31 S32 S33 S34ε 0036 0108 0144 024

6 Mathematical Problems in Engineering

42 Results andAnalysis Firstly the spectrum and envelopespectrum analysis of bearing state data under load 0 test arecarried out to understand the influence of noise on faultdiagnosis Figures 7 and 8 show the frequency spectrum andenvelope spectrum of normal state inner ring fault rollingelement fault and outer ring fault of REB Due to themodulation of the noise signal the characteristic frequencycannot be detected from the frequency spectrum In Fig-ure 8 the characteristic frequency of outer ring failure andthe frequency doubling can be observed but the faultcharacteristic frequency of the inner ring is not obvious andthe characteristic frequency of rolling element is unavailableSo the fault of rolling element cannot be identified effec-tively as the influence of strong noise and signal modulation

and the random signal is spread throughout the frequencydomain It can be concluded from the above analysis that inpractice random noise abundant is an important factorhindering the accuracy of fault identification Due to theexistence of high frequency noise it is necessary to improvethe ZC method model and optimize the parameters so as toimprove the accuracy of fault diagnosis and make the ad-vantages of fast calculation of ZC method useful

(e rotational speed of load 0 is 1797 rpm from Table 5therefore the magnitude of the observation window shouldbe greater than 0034 seconds which contains 360 points atthe sampling frequency of 12K Follow the method inSection 31 the data fragment length is 720 and the numberof fragment is 8 (en each length of data collected from thefour bearing states is 10 seconds which is divided into 40segments with 50 overlap rate According to the method inthis paper optimization parameters in the improved modelrespectively are calculated and shown in Table 6 (e ratioshows the ratio of the optimized ε to the value of peak-to-peak Figure 9 shows the comparison of ZC characteristics offour bearing states before and after the model improvementand parameter optimization

From Figures 9(a) and 9(c) we can see that before themodel improvement parameter optimization the faultcharacteristics of partial samples of rolling element andouter ring are similar and difficult to distinguish which willlead to misjudgment From Figures 9(b) and 9(d) we can seethat the sample characteristics of rolling element fault andthe outer ring fault are completely distinguished through

1

08

06

04

02

0

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

S11S12

S13S14

(a)

1

08

06

04

02

0

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

S21S22

S23S24

(b)

1

08

06

04

02

0

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

S31S32

S33S34

(c)1

08

06

04

02

0

Feature

Nor

mal

ized

dur

atio

n

1 2 3 4 5 6 7 8

S11S12

S13S14

(d)

1

08

06

04

02

0

Feature

Nor

mal

ized

dur

atio

n

1 2 3 4 5 6 7 8

S21S22

S23S24

(e)

1

08

06

04

02

0

Feature

Nor

mal

ized

dur

atio

n

1 2 3 4 5 6 7 8

S31S32

S33S34

(f )

Figure 6 Optimized ZC features of four state signals (from left to right SNR is 30 5 and minus5 respectively)

Table 4 (e average identification rate of simulation fault

Index SNR 30 SNR 5 SNR minus5Classification rate () FVduration FVduration FVdurationNormal 100 100 100Inner ring defect 100 100 9985Rolling element defect 100 100 9985Outer ring defect 100 100 100

Table 5 (e motor load and corresponding speed

Index Load 0 Load 1 Load 2 Load 3Motor speedRPM 1797 1772 1750 1730

Mathematical Problems in Engineering 7

Normal frequency spectrum01

005

0

Frequency (Hz)

Am

plitu

de (V

)

0 1000 2000 3000 4000 5000 6000

(a)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Inner frequency spectrum01

005

0Am

plitu

de (V

)

(b)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Ball frequency spectrum004

002

0Am

plitu

de (V

)

(c)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Outer frequency spectrum02

01

0Am

plitu

de (V

)

(d)

Figure 7 Frequency spectrum of four bearing states

Normal envelope spectrum004

002

0

Frequency (Hz)

Am

plitu

de (V

)

0 1000 2000 3000 4000 5000 6000

(a)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Inner envelope spectrum02

01

0Am

plitu

de (V

)

(b)

Figure 8 Continued

8 Mathematical Problems in Engineering

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Ball envelope spectrum002

001

0Am

plitu

de (V

)

(c)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Outer envelope spectrum1

05

0Am

plitu

de (V

)

(d)

Figure 8 Envelope spectrum of four bearing states

Table 6 Optimized values of ε and theirs ratios to peak-to-peak

Index Normal Inner race fault Ball fault Outer race faultε 0000598 0003109 0001211 0002041Ratio () 01 01 01 0029

1

08

06

04

02

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

NormalInner race defect

Ball defectOuter race defect

(a)

1

08

06

04

0

02

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

NormalInner race defect

Ball defectOuter race defect

(b)

Feature1 2 3 4 5 6 7 8

1

08

06

04

02

0

Nor

mal

ized

dur

atio

n

NormalInner race defect

Ball defectOuter race defect

(c)

Feature1 2 3 4 5 6 7 8

1

08

06

04

0

02Nor

mal

ized

dur

atio

n

NormalInner race defect

Ball defectOuter race defect

(d)

Figure 9 ZC duration features before (a and c) and after (b and d) the model improvement and parameter optimization

Mathematical Problems in Engineering 9

adopting the method of the model improvement and pa-rameter optimization

In order to compare the recognition rate all the ei-genvectors are taken to a three-layer neural network whose

weights and thresholds are optimized by genetic algorithmto improve the diagnostic efficiency and accuracy (enumber of neurons in the hidden layer is 5 and the totalnumber of samples is 160 including training set sample 120and test set sample 40 Each bearing state contains 30training set samples and 10 test set samples arranged inorder according to normal inner ring fault rolling elementfault and outer ring fault (e result of verification is shownin Figure 10 which is the average recognition rate of 40replicates (e recognition rate of both training prior sta-tistics and test set is always 100 and proves that purposedmethod can identify bearing faults accurately

As a comparison the recognition result before the modelimprovement and parameter optimization is shown inFigure 11

(e results in the Figure 11 represent the average rec-ognition accuracy of 40 repeated experiments from which itcan be intuitively seen that the recognition rate of normalstate and inner ring fault is 100 but the recognition rate ofrolling element fault and outer ring fault is not satisfactory(e partial misjudgment between rolling element fault andouter ring fault has affected the fault recognition rate andthe comparison of the average recognition rate of 40 ex-periments is shown in Table 7

Clas

s4

35

3

25

2

15

1

Train sample0 20 40 60 80 100 120

(a)Cl

ass

4

35

3

25

2

15

1

Test sample0 5 10 2015 25 3530 40

(b)

Figure 10 Identification results of adopting the method of the model improvement and parameter optimization

Table 7 Comparison of mean recognition rate before and afteroptimization

Index Training set Test setAccuracy of before optimization () 9333 9225Accuracy of after optimization () 100 100

Table 8 Optimized ε values of various bearing states under threeloads

Index Normal Inner ringfault

Rollingelement fault

Outer ringfault

ε underload 1 0000521 0002938 0001299 0001776

ε underload 2 0000635 0003065 0001171 0001793

ε underload 3 0000548 0003188 0001321 0001809

Table 9 Recognition rate of various bearing states under threeloads

Index Normal Inner ringfault

Rollingelement fault

Outer ringfault

Accuracy ofload 1 () 100 100 9917 9917

Accuracy ofload 2 () 100 975 95 925

Accuracy ofload 3 () 9917 9917 9833 9833

Table 10 Recognition rate of different load HPs as training sets

Index Test set recognition rateTraining set Load 0 Load 1 Load 2 Load 3Load 0 100 9972 9683 9912Load 1 9813 9985 9672 9783Load 2 9775 9728 9883 9752Load 3 9815 9723 9695 9917

10 Mathematical Problems in Engineering

(e experimental results show that the method of themodel improvement and parameter optimization is effectivefor bearing fault identification under the same load

Next the experiment will verify whether the method inthis paper is robust for the load (e experimental data ofload 0 is still used as the training set according to ratios of εto peak-to-peak in Table 6 calculate the optimization pa-rameters under the other load and extract the ZC features asthe test set (e length and number of observation windowsshould be consistent with that under load 0 when extractingthe ZC features under other load Table 8 shows the opti-mized ε values of various bearing states and under threeloads

(e same neural network is applied to identify thebearing fault and the results are shown in Table 9

According to the recognition results in Table 9 althoughthe fault recognition rate at load 2 is lower the overallrecognition rate of other loads still achieves the expectedeffect (e analysis of the error identification samples showsthat these samples are quite distinct from the other samplesof the same state so the validity of themethod is not affected(e experiment proves that the method is applicable to otherloads when the load information is known

Finally the experiment verifies the test effect under anyload and the recognition effect of unknown load has beentested Similarly the method proposed in this paper isemployed for obtaining characteristic samples the samplesunder each load are used as training sets and the samples ofother load are test sets (e overall recognition rate mea-sured is shown in Table 10 and each result is the averagerecognition rate of 40 tests as well

According to the analysis of the results presented inTable 10 the recognition accuracy of samples under thesame load as the training set and the test set is relatively highWhen samples under different loads are used as training setsto test other load samples the recognition accuracy isslightly lower because a few samples deviate from the

characteristics of similar samples In general the experi-mental results demonstrate that the method has good loadrobustness so as to accurately identify the bearing fault evenif the bearing load is not known

5 Conclusion

In this paper an improved fault diagnosis method forelectromechanical system based on ZC algorithm is pro-posed which optimized the calculation model of ZCR andthe parameters of ZC algorithm by establishing a criterionfunction model (e fault diagnosis experiment of motorREB proves that the method successfully overcomes theshortcoming of ZC feature sensitive to noise and can realizeaccurate fault diagnosis without knowing the bearing loadCombined with the merits of saving computing time in thecase of big data the method is expected to monitor thecrucial parts of electromechanical system in real time so asto provide security and reliability guaranty for accuratecontrol of the electromechanical system

Data Availability

(e data used to support the study are available from thecorresponding author upon request

Conflicts of Interest

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

Acknowledgments

(is work was supported by the National Key Research andDevelopment plan of China under Grant 2018YFB1306103the National Natural Science Foundation of China underGrant 51875272 and the Provincial School Education

Clas

s4

35

3

25

2

15

1

Train sample0 20 40 60 80 100 120

(a)

Clas

s

4

35

3

25

2

15

1

Test sample0 5 10 2015 25 3530 40

(b)

Figure 11 Identification results before the model improvement and parameter optimization

Mathematical Problems in Engineering 11

Cooperation Project of Yunnan Province under GrantKKDA202001003

References

[1] M R Khoshdarregi S Tappe and Y Altintas ldquoIntegratedfive-Axis trajectory shaping and contour error compensationfor high-speed CNC machine toolsrdquo IEEEASME Transac-tions on Mechatronics vol 19 no 6 pp 1859ndash1871 2014

[2] R Colombo I Cusmano I Sterpi A Mazzone C Delconteand F Pisano ldquoTest-retest reliability of robotic assessmentmeasures for the evaluation of upper limb recoveryrdquo IEEETransactions on Neural Systems and Rehabilitation Engi-neering vol 22 no 5 pp 1020ndash1029 2014

[3] B Xu D Wang Y Zhang and Z Shi ldquoDOB-based neuralcontrol of flexible hypersonic flight vehicle considering windeffectsrdquo IEEE Transactions on Industrial Electronics vol 64no 11 pp 8676ndash8685 2017

[4] X Ding Q He and N Luo ldquoA fusion feature and its im-provement based on locality preserving projections for rollingelement bearing fault classificationrdquo Journal of Sound andVibration vol 335 pp 367ndash383 2015

[5] Y Zhang and R B Randall ldquoRolling element bearing faultdiagnosis based on the combination of genetic algorithms andfast kurtogramrdquo Mechanical Systems and Signal Processingvol 23 no 5 pp 1509ndash1517 2009

[6] J Benali M Sayadi F Fnaiech B Morello and N ZerhounildquoImportance of the fourth and fifth intrinsic mode functionsfor bearing fault diagnosisrdquo in Proceedings of the 14th In-ternational Conference on Sciences and Techniques of Auto-matic Control amp Computer EngineeringmdashSTArsquo2013 SousseTunisia December 2013

[7] S Nandi H A Toliyat and X Li ldquoCondition monitoring andfault diagnosis of electrical motors-A reviewrdquo IEEE Trans-actions on Energy Conversion vol 20 no 4 pp 719ndash729 2005

[8] J Huang S Ri T Fukuda and Y Wang ldquoA disturbanceobserver based sliding mode control for a class of under-actuated robotic system with mismatched uncertaintiesrdquoIEEE Transactions on Automatic Control vol 64 no 6pp 2480ndash2487 2019

[9] J Huang M Zhang S Ri C Xiong Z Li and Y Kang ldquoHigh-order disturbance observer based sliding mode control formobile wheeled inverted pendulum systemsrdquo IEEE Trans-actions on Industrial Electronics vol 67 no 3 pp 2030ndash20412020

[10] H Jian M H Ri DWu and S Ri ldquoInterval type-2 fuzzy logicmodeling and control of a mobile two-wheeled invertedpendulumrdquo IEEE Transactions on Fuzzy Systems vol 26no 4 pp 2030ndash2038 2018

[11] P D Mcfadden and J D Smith ldquoModel for the vibrationproduced by a single point defect in a rolling elementbearingrdquo Journal of Sound and Vibration vol 96 no 1pp 69ndash82 1984

[12] N Sawalhi and R B Randall ldquoSimulating gear and bearinginteractions in the presence of faultsrdquoMechanical Systems andSignal Processing vol 22 no 8 pp 1924ndash1951 2008

[13] Q Zhu L Liu S Li and W Zhang ldquoControl of complexnonlinear dynamic rational systemsrdquo Complexity vol 2018Article ID 8953035 12 pages 2018

[14] Q M Zhu D Y Zhao and J Zhang ldquoA general U-blockmodel-based design procedure for nonlinear polynomialcontrol systemsrdquo International Journal of Systems Sciencevol 47 no 14 pp 3465ndash3475 2015

[15] A Rafsanjani S Abbasion A Farshidianfar andH Moeenfard ldquoNonlinear dynamic modeling of surfacedefects in rolling element bearing systemsrdquo Journal of Soundamp Vibration vol 319 no 3-5 pp 1150ndash1174 2009

[16] T W Rauber F de Assis Boldt and F M Varejao ldquoHet-erogeneous feature models and feature selection applied tobearing fault diagnosisrdquo IEEE Transactions on IndustrialElectronics vol 62 no 1 pp 637ndash646 2015

[17] J Chebil M Hrairi and N Abushikhah ldquoSignal analysis ofvibration measurements for condition monitoring of bear-ingsrdquo Australian Journal of Basic amp Applied Sciences vol 5no 1 p 70 2011

[18] M D Prieto G Cirrincione A G Espinosa J A Ortega andH Henao ldquobearing fault detection by a novel condition-monitoring scheme based on statistical-time features andneural networksrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 pp 3398ndash3407 2013

[19] A Garcia-Perez R d J Romero-Troncoso E Cabal-Yepezand R A Osornio-Rios ldquo(e application of high-resolutionspectral analysis for identifying multiple combined faults ininduction motorsrdquo IEEE Transactions on Industrial Elec-tronics vol 58 no 5 pp 2002ndash2010 2011

[20] Z K Peng and F L Chu ldquoApplication of the wavelettransform in machine condition monitoring and fault diag-nostics a review with bibliographyrdquo Mechanical Systems andSignal Processing vol 18 no 2 pp 199ndash221 2004

[21] J Seshadrinath B Singh and B K Panigrahi ldquoInvestigationof vibration signatures for multiple fault diagnosis in variablefrequency drives using complex waveletsrdquo IEEE Transactionson Power Electronics vol 29 no 2 pp 936ndash945 2014

[22] W Su F Wang H Zhu Z Zhang and Z Guo ldquoRollingelement bearing faults diagnosis based on optimal Morletwavelet filter and autocorrelation enhancementrdquo MechanicalSystems and Signal Processing vol 24 no 5 pp 1458ndash14722010

[23] X Jin M Zhao T W S Chow andM Pecht ldquoMotor bearingfault diagnosis using trace ratio linear discriminant analysisrdquoIEEE Transactions on Industrial Electronics vol 61 no 5pp 2441ndash2451 2014

[24] M Zhao X Jin Z Zhang and B Li ldquoFault diagnosis of rollingelement bearings via discriminative subspace learning visu-alization and classificationrdquo Expert Systems with Applicationsvol 41 no 7 pp 3391ndash3401 2014

[25] R Niederjohn ldquoA mathematical formulation and comparisonof zero-crossing analysis techniques which have been appliedto automatioc speech recognitionrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 23 no 4pp 373ndash380 1975

[26] P William andM Hoffman ldquoEfficient sensor network vehicleclassification using peak harmonics of acoustic emissionsrdquo inProceedings of the SPIE Defense and Security Symposium Orlando FL USA March 2008

[27] A Shahidi Zandi R Tafreshi M Javidan and G A DumontldquoPredicting epileptic seizures in scalp EEG based on a vari-ational bayesian Gaussian mixture model of zero-crossingintervalsrdquo IEEE Transactions on Biomedical Engineeringvol 60 no 5 pp 1401ndash1413 2013

[28] C-H Wu H-C Chang P-L Lee et al ldquoFrequency recog-nition in an SSVEP-based brain computer interface usingempirical mode decomposition and refined generalized zero-crossingrdquo Journal of Neuroscience Methods vol 196 no 1pp 170ndash181 2011

12 Mathematical Problems in Engineering

[29] P E William and M W Hoffman ldquoIdentification of bearingfaults using time domain zero-crossingsrdquoMechanical Systemsand Signal Processing vol 25 no 8 pp 3078ndash3088 2011

[30] T Liu J Chen and G Dong ldquoZero crossing and coupledhidden Markov model for a rolling bearing performancedegradation assessmentrdquo Journal of Vibration and Controlvol 20 no 16 pp 2487ndash2500 2013

[31] A Ukil S Chen and A Andenna ldquoDetection of stator shortcircuit faults in three-phase induction motors using motorcurrent zero crossing instantsrdquo Electric Power Systems Re-search vol 81 no 4 pp 1036ndash1044 2011

[32] J D T Gonzalez and W Kinsner ldquoZero-crossing analysis ofLevy walks for real-time feature extractionrdquo in Proceedings ofthe IEEE International Conference on Electro InformationTechnology EIT Grand Forks ND USA May 2016

[33] R G Waghmare S L Nalbalwar and A Das ldquoTransientsignal detection on the basis of energy and zero crossingdetectorsrdquo Procedia Engineering vol 30 pp 129ndash134 2012

[34] B R Nayana and P Geethanjali ldquoAnalysis of statistical time-domain features effectiveness in identification of bearingfaults from vibration signalrdquo IEEE Sensors Journal vol 17no 17 pp 5618ndash5625 2017

[35] C Fisher Encyclopedia of Biometrics p 549 Springer USBoston MA USA 2009

[36] Q Chen W Xing L Tao and L Hua ldquoZero-crossing featureextraction based on threshold optimization for rolling ele-ment bearingrdquo in Lecture Notes in Electrical Engineeringpp 409ndash425 Springer Berlin Germany 2018

[37] Case Western Reserve University Bearing Data CenterWebsite accessed on Mar 23 2015 [Online]Available httpcsegroupscaseedubearingdatacenterpagesdownload-data-file

Mathematical Problems in Engineering 13

42 Results andAnalysis Firstly the spectrum and envelopespectrum analysis of bearing state data under load 0 test arecarried out to understand the influence of noise on faultdiagnosis Figures 7 and 8 show the frequency spectrum andenvelope spectrum of normal state inner ring fault rollingelement fault and outer ring fault of REB Due to themodulation of the noise signal the characteristic frequencycannot be detected from the frequency spectrum In Fig-ure 8 the characteristic frequency of outer ring failure andthe frequency doubling can be observed but the faultcharacteristic frequency of the inner ring is not obvious andthe characteristic frequency of rolling element is unavailableSo the fault of rolling element cannot be identified effec-tively as the influence of strong noise and signal modulation

and the random signal is spread throughout the frequencydomain It can be concluded from the above analysis that inpractice random noise abundant is an important factorhindering the accuracy of fault identification Due to theexistence of high frequency noise it is necessary to improvethe ZC method model and optimize the parameters so as toimprove the accuracy of fault diagnosis and make the ad-vantages of fast calculation of ZC method useful

(e rotational speed of load 0 is 1797 rpm from Table 5therefore the magnitude of the observation window shouldbe greater than 0034 seconds which contains 360 points atthe sampling frequency of 12K Follow the method inSection 31 the data fragment length is 720 and the numberof fragment is 8 (en each length of data collected from thefour bearing states is 10 seconds which is divided into 40segments with 50 overlap rate According to the method inthis paper optimization parameters in the improved modelrespectively are calculated and shown in Table 6 (e ratioshows the ratio of the optimized ε to the value of peak-to-peak Figure 9 shows the comparison of ZC characteristics offour bearing states before and after the model improvementand parameter optimization

From Figures 9(a) and 9(c) we can see that before themodel improvement parameter optimization the faultcharacteristics of partial samples of rolling element andouter ring are similar and difficult to distinguish which willlead to misjudgment From Figures 9(b) and 9(d) we can seethat the sample characteristics of rolling element fault andthe outer ring fault are completely distinguished through

1

08

06

04

02

0

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

S11S12

S13S14

(a)

1

08

06

04

02

0

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

S21S22

S23S24

(b)

1

08

06

04

02

0

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

S31S32

S33S34

(c)1

08

06

04

02

0

Feature

Nor

mal

ized

dur

atio

n

1 2 3 4 5 6 7 8

S11S12

S13S14

(d)

1

08

06

04

02

0

Feature

Nor

mal

ized

dur

atio

n

1 2 3 4 5 6 7 8

S21S22

S23S24

(e)

1

08

06

04

02

0

Feature

Nor

mal

ized

dur

atio

n

1 2 3 4 5 6 7 8

S31S32

S33S34

(f )

Figure 6 Optimized ZC features of four state signals (from left to right SNR is 30 5 and minus5 respectively)

Table 4 (e average identification rate of simulation fault

Index SNR 30 SNR 5 SNR minus5Classification rate () FVduration FVduration FVdurationNormal 100 100 100Inner ring defect 100 100 9985Rolling element defect 100 100 9985Outer ring defect 100 100 100

Table 5 (e motor load and corresponding speed

Index Load 0 Load 1 Load 2 Load 3Motor speedRPM 1797 1772 1750 1730

Mathematical Problems in Engineering 7

Normal frequency spectrum01

005

0

Frequency (Hz)

Am

plitu

de (V

)

0 1000 2000 3000 4000 5000 6000

(a)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Inner frequency spectrum01

005

0Am

plitu

de (V

)

(b)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Ball frequency spectrum004

002

0Am

plitu

de (V

)

(c)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Outer frequency spectrum02

01

0Am

plitu

de (V

)

(d)

Figure 7 Frequency spectrum of four bearing states

Normal envelope spectrum004

002

0

Frequency (Hz)

Am

plitu

de (V

)

0 1000 2000 3000 4000 5000 6000

(a)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Inner envelope spectrum02

01

0Am

plitu

de (V

)

(b)

Figure 8 Continued

8 Mathematical Problems in Engineering

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Ball envelope spectrum002

001

0Am

plitu

de (V

)

(c)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Outer envelope spectrum1

05

0Am

plitu

de (V

)

(d)

Figure 8 Envelope spectrum of four bearing states

Table 6 Optimized values of ε and theirs ratios to peak-to-peak

Index Normal Inner race fault Ball fault Outer race faultε 0000598 0003109 0001211 0002041Ratio () 01 01 01 0029

1

08

06

04

02

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

NormalInner race defect

Ball defectOuter race defect

(a)

1

08

06

04

0

02

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

NormalInner race defect

Ball defectOuter race defect

(b)

Feature1 2 3 4 5 6 7 8

1

08

06

04

02

0

Nor

mal

ized

dur

atio

n

NormalInner race defect

Ball defectOuter race defect

(c)

Feature1 2 3 4 5 6 7 8

1

08

06

04

0

02Nor

mal

ized

dur

atio

n

NormalInner race defect

Ball defectOuter race defect

(d)

Figure 9 ZC duration features before (a and c) and after (b and d) the model improvement and parameter optimization

Mathematical Problems in Engineering 9

adopting the method of the model improvement and pa-rameter optimization

In order to compare the recognition rate all the ei-genvectors are taken to a three-layer neural network whose

weights and thresholds are optimized by genetic algorithmto improve the diagnostic efficiency and accuracy (enumber of neurons in the hidden layer is 5 and the totalnumber of samples is 160 including training set sample 120and test set sample 40 Each bearing state contains 30training set samples and 10 test set samples arranged inorder according to normal inner ring fault rolling elementfault and outer ring fault (e result of verification is shownin Figure 10 which is the average recognition rate of 40replicates (e recognition rate of both training prior sta-tistics and test set is always 100 and proves that purposedmethod can identify bearing faults accurately

As a comparison the recognition result before the modelimprovement and parameter optimization is shown inFigure 11

(e results in the Figure 11 represent the average rec-ognition accuracy of 40 repeated experiments from which itcan be intuitively seen that the recognition rate of normalstate and inner ring fault is 100 but the recognition rate ofrolling element fault and outer ring fault is not satisfactory(e partial misjudgment between rolling element fault andouter ring fault has affected the fault recognition rate andthe comparison of the average recognition rate of 40 ex-periments is shown in Table 7

Clas

s4

35

3

25

2

15

1

Train sample0 20 40 60 80 100 120

(a)Cl

ass

4

35

3

25

2

15

1

Test sample0 5 10 2015 25 3530 40

(b)

Figure 10 Identification results of adopting the method of the model improvement and parameter optimization

Table 7 Comparison of mean recognition rate before and afteroptimization

Index Training set Test setAccuracy of before optimization () 9333 9225Accuracy of after optimization () 100 100

Table 8 Optimized ε values of various bearing states under threeloads

Index Normal Inner ringfault

Rollingelement fault

Outer ringfault

ε underload 1 0000521 0002938 0001299 0001776

ε underload 2 0000635 0003065 0001171 0001793

ε underload 3 0000548 0003188 0001321 0001809

Table 9 Recognition rate of various bearing states under threeloads

Index Normal Inner ringfault

Rollingelement fault

Outer ringfault

Accuracy ofload 1 () 100 100 9917 9917

Accuracy ofload 2 () 100 975 95 925

Accuracy ofload 3 () 9917 9917 9833 9833

Table 10 Recognition rate of different load HPs as training sets

Index Test set recognition rateTraining set Load 0 Load 1 Load 2 Load 3Load 0 100 9972 9683 9912Load 1 9813 9985 9672 9783Load 2 9775 9728 9883 9752Load 3 9815 9723 9695 9917

10 Mathematical Problems in Engineering

(e experimental results show that the method of themodel improvement and parameter optimization is effectivefor bearing fault identification under the same load

Next the experiment will verify whether the method inthis paper is robust for the load (e experimental data ofload 0 is still used as the training set according to ratios of εto peak-to-peak in Table 6 calculate the optimization pa-rameters under the other load and extract the ZC features asthe test set (e length and number of observation windowsshould be consistent with that under load 0 when extractingthe ZC features under other load Table 8 shows the opti-mized ε values of various bearing states and under threeloads

(e same neural network is applied to identify thebearing fault and the results are shown in Table 9

According to the recognition results in Table 9 althoughthe fault recognition rate at load 2 is lower the overallrecognition rate of other loads still achieves the expectedeffect (e analysis of the error identification samples showsthat these samples are quite distinct from the other samplesof the same state so the validity of themethod is not affected(e experiment proves that the method is applicable to otherloads when the load information is known

Finally the experiment verifies the test effect under anyload and the recognition effect of unknown load has beentested Similarly the method proposed in this paper isemployed for obtaining characteristic samples the samplesunder each load are used as training sets and the samples ofother load are test sets (e overall recognition rate mea-sured is shown in Table 10 and each result is the averagerecognition rate of 40 tests as well

According to the analysis of the results presented inTable 10 the recognition accuracy of samples under thesame load as the training set and the test set is relatively highWhen samples under different loads are used as training setsto test other load samples the recognition accuracy isslightly lower because a few samples deviate from the

characteristics of similar samples In general the experi-mental results demonstrate that the method has good loadrobustness so as to accurately identify the bearing fault evenif the bearing load is not known

5 Conclusion

In this paper an improved fault diagnosis method forelectromechanical system based on ZC algorithm is pro-posed which optimized the calculation model of ZCR andthe parameters of ZC algorithm by establishing a criterionfunction model (e fault diagnosis experiment of motorREB proves that the method successfully overcomes theshortcoming of ZC feature sensitive to noise and can realizeaccurate fault diagnosis without knowing the bearing loadCombined with the merits of saving computing time in thecase of big data the method is expected to monitor thecrucial parts of electromechanical system in real time so asto provide security and reliability guaranty for accuratecontrol of the electromechanical system

Data Availability

(e data used to support the study are available from thecorresponding author upon request

Conflicts of Interest

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

Acknowledgments

(is work was supported by the National Key Research andDevelopment plan of China under Grant 2018YFB1306103the National Natural Science Foundation of China underGrant 51875272 and the Provincial School Education

Clas

s4

35

3

25

2

15

1

Train sample0 20 40 60 80 100 120

(a)

Clas

s

4

35

3

25

2

15

1

Test sample0 5 10 2015 25 3530 40

(b)

Figure 11 Identification results before the model improvement and parameter optimization

Mathematical Problems in Engineering 11

Cooperation Project of Yunnan Province under GrantKKDA202001003

References

[1] M R Khoshdarregi S Tappe and Y Altintas ldquoIntegratedfive-Axis trajectory shaping and contour error compensationfor high-speed CNC machine toolsrdquo IEEEASME Transac-tions on Mechatronics vol 19 no 6 pp 1859ndash1871 2014

[2] R Colombo I Cusmano I Sterpi A Mazzone C Delconteand F Pisano ldquoTest-retest reliability of robotic assessmentmeasures for the evaluation of upper limb recoveryrdquo IEEETransactions on Neural Systems and Rehabilitation Engi-neering vol 22 no 5 pp 1020ndash1029 2014

[3] B Xu D Wang Y Zhang and Z Shi ldquoDOB-based neuralcontrol of flexible hypersonic flight vehicle considering windeffectsrdquo IEEE Transactions on Industrial Electronics vol 64no 11 pp 8676ndash8685 2017

[4] X Ding Q He and N Luo ldquoA fusion feature and its im-provement based on locality preserving projections for rollingelement bearing fault classificationrdquo Journal of Sound andVibration vol 335 pp 367ndash383 2015

[5] Y Zhang and R B Randall ldquoRolling element bearing faultdiagnosis based on the combination of genetic algorithms andfast kurtogramrdquo Mechanical Systems and Signal Processingvol 23 no 5 pp 1509ndash1517 2009

[6] J Benali M Sayadi F Fnaiech B Morello and N ZerhounildquoImportance of the fourth and fifth intrinsic mode functionsfor bearing fault diagnosisrdquo in Proceedings of the 14th In-ternational Conference on Sciences and Techniques of Auto-matic Control amp Computer EngineeringmdashSTArsquo2013 SousseTunisia December 2013

[7] S Nandi H A Toliyat and X Li ldquoCondition monitoring andfault diagnosis of electrical motors-A reviewrdquo IEEE Trans-actions on Energy Conversion vol 20 no 4 pp 719ndash729 2005

[8] J Huang S Ri T Fukuda and Y Wang ldquoA disturbanceobserver based sliding mode control for a class of under-actuated robotic system with mismatched uncertaintiesrdquoIEEE Transactions on Automatic Control vol 64 no 6pp 2480ndash2487 2019

[9] J Huang M Zhang S Ri C Xiong Z Li and Y Kang ldquoHigh-order disturbance observer based sliding mode control formobile wheeled inverted pendulum systemsrdquo IEEE Trans-actions on Industrial Electronics vol 67 no 3 pp 2030ndash20412020

[10] H Jian M H Ri DWu and S Ri ldquoInterval type-2 fuzzy logicmodeling and control of a mobile two-wheeled invertedpendulumrdquo IEEE Transactions on Fuzzy Systems vol 26no 4 pp 2030ndash2038 2018

[11] P D Mcfadden and J D Smith ldquoModel for the vibrationproduced by a single point defect in a rolling elementbearingrdquo Journal of Sound and Vibration vol 96 no 1pp 69ndash82 1984

[12] N Sawalhi and R B Randall ldquoSimulating gear and bearinginteractions in the presence of faultsrdquoMechanical Systems andSignal Processing vol 22 no 8 pp 1924ndash1951 2008

[13] Q Zhu L Liu S Li and W Zhang ldquoControl of complexnonlinear dynamic rational systemsrdquo Complexity vol 2018Article ID 8953035 12 pages 2018

[14] Q M Zhu D Y Zhao and J Zhang ldquoA general U-blockmodel-based design procedure for nonlinear polynomialcontrol systemsrdquo International Journal of Systems Sciencevol 47 no 14 pp 3465ndash3475 2015

[15] A Rafsanjani S Abbasion A Farshidianfar andH Moeenfard ldquoNonlinear dynamic modeling of surfacedefects in rolling element bearing systemsrdquo Journal of Soundamp Vibration vol 319 no 3-5 pp 1150ndash1174 2009

[16] T W Rauber F de Assis Boldt and F M Varejao ldquoHet-erogeneous feature models and feature selection applied tobearing fault diagnosisrdquo IEEE Transactions on IndustrialElectronics vol 62 no 1 pp 637ndash646 2015

[17] J Chebil M Hrairi and N Abushikhah ldquoSignal analysis ofvibration measurements for condition monitoring of bear-ingsrdquo Australian Journal of Basic amp Applied Sciences vol 5no 1 p 70 2011

[18] M D Prieto G Cirrincione A G Espinosa J A Ortega andH Henao ldquobearing fault detection by a novel condition-monitoring scheme based on statistical-time features andneural networksrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 pp 3398ndash3407 2013

[19] A Garcia-Perez R d J Romero-Troncoso E Cabal-Yepezand R A Osornio-Rios ldquo(e application of high-resolutionspectral analysis for identifying multiple combined faults ininduction motorsrdquo IEEE Transactions on Industrial Elec-tronics vol 58 no 5 pp 2002ndash2010 2011

[20] Z K Peng and F L Chu ldquoApplication of the wavelettransform in machine condition monitoring and fault diag-nostics a review with bibliographyrdquo Mechanical Systems andSignal Processing vol 18 no 2 pp 199ndash221 2004

[21] J Seshadrinath B Singh and B K Panigrahi ldquoInvestigationof vibration signatures for multiple fault diagnosis in variablefrequency drives using complex waveletsrdquo IEEE Transactionson Power Electronics vol 29 no 2 pp 936ndash945 2014

[22] W Su F Wang H Zhu Z Zhang and Z Guo ldquoRollingelement bearing faults diagnosis based on optimal Morletwavelet filter and autocorrelation enhancementrdquo MechanicalSystems and Signal Processing vol 24 no 5 pp 1458ndash14722010

[23] X Jin M Zhao T W S Chow andM Pecht ldquoMotor bearingfault diagnosis using trace ratio linear discriminant analysisrdquoIEEE Transactions on Industrial Electronics vol 61 no 5pp 2441ndash2451 2014

[24] M Zhao X Jin Z Zhang and B Li ldquoFault diagnosis of rollingelement bearings via discriminative subspace learning visu-alization and classificationrdquo Expert Systems with Applicationsvol 41 no 7 pp 3391ndash3401 2014

[25] R Niederjohn ldquoA mathematical formulation and comparisonof zero-crossing analysis techniques which have been appliedto automatioc speech recognitionrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 23 no 4pp 373ndash380 1975

[26] P William andM Hoffman ldquoEfficient sensor network vehicleclassification using peak harmonics of acoustic emissionsrdquo inProceedings of the SPIE Defense and Security Symposium Orlando FL USA March 2008

[27] A Shahidi Zandi R Tafreshi M Javidan and G A DumontldquoPredicting epileptic seizures in scalp EEG based on a vari-ational bayesian Gaussian mixture model of zero-crossingintervalsrdquo IEEE Transactions on Biomedical Engineeringvol 60 no 5 pp 1401ndash1413 2013

[28] C-H Wu H-C Chang P-L Lee et al ldquoFrequency recog-nition in an SSVEP-based brain computer interface usingempirical mode decomposition and refined generalized zero-crossingrdquo Journal of Neuroscience Methods vol 196 no 1pp 170ndash181 2011

12 Mathematical Problems in Engineering

[29] P E William and M W Hoffman ldquoIdentification of bearingfaults using time domain zero-crossingsrdquoMechanical Systemsand Signal Processing vol 25 no 8 pp 3078ndash3088 2011

[30] T Liu J Chen and G Dong ldquoZero crossing and coupledhidden Markov model for a rolling bearing performancedegradation assessmentrdquo Journal of Vibration and Controlvol 20 no 16 pp 2487ndash2500 2013

[31] A Ukil S Chen and A Andenna ldquoDetection of stator shortcircuit faults in three-phase induction motors using motorcurrent zero crossing instantsrdquo Electric Power Systems Re-search vol 81 no 4 pp 1036ndash1044 2011

[32] J D T Gonzalez and W Kinsner ldquoZero-crossing analysis ofLevy walks for real-time feature extractionrdquo in Proceedings ofthe IEEE International Conference on Electro InformationTechnology EIT Grand Forks ND USA May 2016

[33] R G Waghmare S L Nalbalwar and A Das ldquoTransientsignal detection on the basis of energy and zero crossingdetectorsrdquo Procedia Engineering vol 30 pp 129ndash134 2012

[34] B R Nayana and P Geethanjali ldquoAnalysis of statistical time-domain features effectiveness in identification of bearingfaults from vibration signalrdquo IEEE Sensors Journal vol 17no 17 pp 5618ndash5625 2017

[35] C Fisher Encyclopedia of Biometrics p 549 Springer USBoston MA USA 2009

[36] Q Chen W Xing L Tao and L Hua ldquoZero-crossing featureextraction based on threshold optimization for rolling ele-ment bearingrdquo in Lecture Notes in Electrical Engineeringpp 409ndash425 Springer Berlin Germany 2018

[37] Case Western Reserve University Bearing Data CenterWebsite accessed on Mar 23 2015 [Online]Available httpcsegroupscaseedubearingdatacenterpagesdownload-data-file

Mathematical Problems in Engineering 13

Normal frequency spectrum01

005

0

Frequency (Hz)

Am

plitu

de (V

)

0 1000 2000 3000 4000 5000 6000

(a)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Inner frequency spectrum01

005

0Am

plitu

de (V

)

(b)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Ball frequency spectrum004

002

0Am

plitu

de (V

)

(c)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Outer frequency spectrum02

01

0Am

plitu

de (V

)

(d)

Figure 7 Frequency spectrum of four bearing states

Normal envelope spectrum004

002

0

Frequency (Hz)

Am

plitu

de (V

)

0 1000 2000 3000 4000 5000 6000

(a)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Inner envelope spectrum02

01

0Am

plitu

de (V

)

(b)

Figure 8 Continued

8 Mathematical Problems in Engineering

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Ball envelope spectrum002

001

0Am

plitu

de (V

)

(c)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Outer envelope spectrum1

05

0Am

plitu

de (V

)

(d)

Figure 8 Envelope spectrum of four bearing states

Table 6 Optimized values of ε and theirs ratios to peak-to-peak

Index Normal Inner race fault Ball fault Outer race faultε 0000598 0003109 0001211 0002041Ratio () 01 01 01 0029

1

08

06

04

02

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

NormalInner race defect

Ball defectOuter race defect

(a)

1

08

06

04

0

02

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

NormalInner race defect

Ball defectOuter race defect

(b)

Feature1 2 3 4 5 6 7 8

1

08

06

04

02

0

Nor

mal

ized

dur

atio

n

NormalInner race defect

Ball defectOuter race defect

(c)

Feature1 2 3 4 5 6 7 8

1

08

06

04

0

02Nor

mal

ized

dur

atio

n

NormalInner race defect

Ball defectOuter race defect

(d)

Figure 9 ZC duration features before (a and c) and after (b and d) the model improvement and parameter optimization

Mathematical Problems in Engineering 9

adopting the method of the model improvement and pa-rameter optimization

In order to compare the recognition rate all the ei-genvectors are taken to a three-layer neural network whose

weights and thresholds are optimized by genetic algorithmto improve the diagnostic efficiency and accuracy (enumber of neurons in the hidden layer is 5 and the totalnumber of samples is 160 including training set sample 120and test set sample 40 Each bearing state contains 30training set samples and 10 test set samples arranged inorder according to normal inner ring fault rolling elementfault and outer ring fault (e result of verification is shownin Figure 10 which is the average recognition rate of 40replicates (e recognition rate of both training prior sta-tistics and test set is always 100 and proves that purposedmethod can identify bearing faults accurately

As a comparison the recognition result before the modelimprovement and parameter optimization is shown inFigure 11

(e results in the Figure 11 represent the average rec-ognition accuracy of 40 repeated experiments from which itcan be intuitively seen that the recognition rate of normalstate and inner ring fault is 100 but the recognition rate ofrolling element fault and outer ring fault is not satisfactory(e partial misjudgment between rolling element fault andouter ring fault has affected the fault recognition rate andthe comparison of the average recognition rate of 40 ex-periments is shown in Table 7

Clas

s4

35

3

25

2

15

1

Train sample0 20 40 60 80 100 120

(a)Cl

ass

4

35

3

25

2

15

1

Test sample0 5 10 2015 25 3530 40

(b)

Figure 10 Identification results of adopting the method of the model improvement and parameter optimization

Table 7 Comparison of mean recognition rate before and afteroptimization

Index Training set Test setAccuracy of before optimization () 9333 9225Accuracy of after optimization () 100 100

Table 8 Optimized ε values of various bearing states under threeloads

Index Normal Inner ringfault

Rollingelement fault

Outer ringfault

ε underload 1 0000521 0002938 0001299 0001776

ε underload 2 0000635 0003065 0001171 0001793

ε underload 3 0000548 0003188 0001321 0001809

Table 9 Recognition rate of various bearing states under threeloads

Index Normal Inner ringfault

Rollingelement fault

Outer ringfault

Accuracy ofload 1 () 100 100 9917 9917

Accuracy ofload 2 () 100 975 95 925

Accuracy ofload 3 () 9917 9917 9833 9833

Table 10 Recognition rate of different load HPs as training sets

Index Test set recognition rateTraining set Load 0 Load 1 Load 2 Load 3Load 0 100 9972 9683 9912Load 1 9813 9985 9672 9783Load 2 9775 9728 9883 9752Load 3 9815 9723 9695 9917

10 Mathematical Problems in Engineering

(e experimental results show that the method of themodel improvement and parameter optimization is effectivefor bearing fault identification under the same load

Next the experiment will verify whether the method inthis paper is robust for the load (e experimental data ofload 0 is still used as the training set according to ratios of εto peak-to-peak in Table 6 calculate the optimization pa-rameters under the other load and extract the ZC features asthe test set (e length and number of observation windowsshould be consistent with that under load 0 when extractingthe ZC features under other load Table 8 shows the opti-mized ε values of various bearing states and under threeloads

(e same neural network is applied to identify thebearing fault and the results are shown in Table 9

According to the recognition results in Table 9 althoughthe fault recognition rate at load 2 is lower the overallrecognition rate of other loads still achieves the expectedeffect (e analysis of the error identification samples showsthat these samples are quite distinct from the other samplesof the same state so the validity of themethod is not affected(e experiment proves that the method is applicable to otherloads when the load information is known

Finally the experiment verifies the test effect under anyload and the recognition effect of unknown load has beentested Similarly the method proposed in this paper isemployed for obtaining characteristic samples the samplesunder each load are used as training sets and the samples ofother load are test sets (e overall recognition rate mea-sured is shown in Table 10 and each result is the averagerecognition rate of 40 tests as well

According to the analysis of the results presented inTable 10 the recognition accuracy of samples under thesame load as the training set and the test set is relatively highWhen samples under different loads are used as training setsto test other load samples the recognition accuracy isslightly lower because a few samples deviate from the

characteristics of similar samples In general the experi-mental results demonstrate that the method has good loadrobustness so as to accurately identify the bearing fault evenif the bearing load is not known

5 Conclusion

In this paper an improved fault diagnosis method forelectromechanical system based on ZC algorithm is pro-posed which optimized the calculation model of ZCR andthe parameters of ZC algorithm by establishing a criterionfunction model (e fault diagnosis experiment of motorREB proves that the method successfully overcomes theshortcoming of ZC feature sensitive to noise and can realizeaccurate fault diagnosis without knowing the bearing loadCombined with the merits of saving computing time in thecase of big data the method is expected to monitor thecrucial parts of electromechanical system in real time so asto provide security and reliability guaranty for accuratecontrol of the electromechanical system

Data Availability

(e data used to support the study are available from thecorresponding author upon request

Conflicts of Interest

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

Acknowledgments

(is work was supported by the National Key Research andDevelopment plan of China under Grant 2018YFB1306103the National Natural Science Foundation of China underGrant 51875272 and the Provincial School Education

Clas

s4

35

3

25

2

15

1

Train sample0 20 40 60 80 100 120

(a)

Clas

s

4

35

3

25

2

15

1

Test sample0 5 10 2015 25 3530 40

(b)

Figure 11 Identification results before the model improvement and parameter optimization

Mathematical Problems in Engineering 11

Cooperation Project of Yunnan Province under GrantKKDA202001003

References

[1] M R Khoshdarregi S Tappe and Y Altintas ldquoIntegratedfive-Axis trajectory shaping and contour error compensationfor high-speed CNC machine toolsrdquo IEEEASME Transac-tions on Mechatronics vol 19 no 6 pp 1859ndash1871 2014

[2] R Colombo I Cusmano I Sterpi A Mazzone C Delconteand F Pisano ldquoTest-retest reliability of robotic assessmentmeasures for the evaluation of upper limb recoveryrdquo IEEETransactions on Neural Systems and Rehabilitation Engi-neering vol 22 no 5 pp 1020ndash1029 2014

[3] B Xu D Wang Y Zhang and Z Shi ldquoDOB-based neuralcontrol of flexible hypersonic flight vehicle considering windeffectsrdquo IEEE Transactions on Industrial Electronics vol 64no 11 pp 8676ndash8685 2017

[4] X Ding Q He and N Luo ldquoA fusion feature and its im-provement based on locality preserving projections for rollingelement bearing fault classificationrdquo Journal of Sound andVibration vol 335 pp 367ndash383 2015

[5] Y Zhang and R B Randall ldquoRolling element bearing faultdiagnosis based on the combination of genetic algorithms andfast kurtogramrdquo Mechanical Systems and Signal Processingvol 23 no 5 pp 1509ndash1517 2009

[6] J Benali M Sayadi F Fnaiech B Morello and N ZerhounildquoImportance of the fourth and fifth intrinsic mode functionsfor bearing fault diagnosisrdquo in Proceedings of the 14th In-ternational Conference on Sciences and Techniques of Auto-matic Control amp Computer EngineeringmdashSTArsquo2013 SousseTunisia December 2013

[7] S Nandi H A Toliyat and X Li ldquoCondition monitoring andfault diagnosis of electrical motors-A reviewrdquo IEEE Trans-actions on Energy Conversion vol 20 no 4 pp 719ndash729 2005

[8] J Huang S Ri T Fukuda and Y Wang ldquoA disturbanceobserver based sliding mode control for a class of under-actuated robotic system with mismatched uncertaintiesrdquoIEEE Transactions on Automatic Control vol 64 no 6pp 2480ndash2487 2019

[9] J Huang M Zhang S Ri C Xiong Z Li and Y Kang ldquoHigh-order disturbance observer based sliding mode control formobile wheeled inverted pendulum systemsrdquo IEEE Trans-actions on Industrial Electronics vol 67 no 3 pp 2030ndash20412020

[10] H Jian M H Ri DWu and S Ri ldquoInterval type-2 fuzzy logicmodeling and control of a mobile two-wheeled invertedpendulumrdquo IEEE Transactions on Fuzzy Systems vol 26no 4 pp 2030ndash2038 2018

[11] P D Mcfadden and J D Smith ldquoModel for the vibrationproduced by a single point defect in a rolling elementbearingrdquo Journal of Sound and Vibration vol 96 no 1pp 69ndash82 1984

[12] N Sawalhi and R B Randall ldquoSimulating gear and bearinginteractions in the presence of faultsrdquoMechanical Systems andSignal Processing vol 22 no 8 pp 1924ndash1951 2008

[13] Q Zhu L Liu S Li and W Zhang ldquoControl of complexnonlinear dynamic rational systemsrdquo Complexity vol 2018Article ID 8953035 12 pages 2018

[14] Q M Zhu D Y Zhao and J Zhang ldquoA general U-blockmodel-based design procedure for nonlinear polynomialcontrol systemsrdquo International Journal of Systems Sciencevol 47 no 14 pp 3465ndash3475 2015

[15] A Rafsanjani S Abbasion A Farshidianfar andH Moeenfard ldquoNonlinear dynamic modeling of surfacedefects in rolling element bearing systemsrdquo Journal of Soundamp Vibration vol 319 no 3-5 pp 1150ndash1174 2009

[16] T W Rauber F de Assis Boldt and F M Varejao ldquoHet-erogeneous feature models and feature selection applied tobearing fault diagnosisrdquo IEEE Transactions on IndustrialElectronics vol 62 no 1 pp 637ndash646 2015

[17] J Chebil M Hrairi and N Abushikhah ldquoSignal analysis ofvibration measurements for condition monitoring of bear-ingsrdquo Australian Journal of Basic amp Applied Sciences vol 5no 1 p 70 2011

[18] M D Prieto G Cirrincione A G Espinosa J A Ortega andH Henao ldquobearing fault detection by a novel condition-monitoring scheme based on statistical-time features andneural networksrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 pp 3398ndash3407 2013

[19] A Garcia-Perez R d J Romero-Troncoso E Cabal-Yepezand R A Osornio-Rios ldquo(e application of high-resolutionspectral analysis for identifying multiple combined faults ininduction motorsrdquo IEEE Transactions on Industrial Elec-tronics vol 58 no 5 pp 2002ndash2010 2011

[20] Z K Peng and F L Chu ldquoApplication of the wavelettransform in machine condition monitoring and fault diag-nostics a review with bibliographyrdquo Mechanical Systems andSignal Processing vol 18 no 2 pp 199ndash221 2004

[21] J Seshadrinath B Singh and B K Panigrahi ldquoInvestigationof vibration signatures for multiple fault diagnosis in variablefrequency drives using complex waveletsrdquo IEEE Transactionson Power Electronics vol 29 no 2 pp 936ndash945 2014

[22] W Su F Wang H Zhu Z Zhang and Z Guo ldquoRollingelement bearing faults diagnosis based on optimal Morletwavelet filter and autocorrelation enhancementrdquo MechanicalSystems and Signal Processing vol 24 no 5 pp 1458ndash14722010

[23] X Jin M Zhao T W S Chow andM Pecht ldquoMotor bearingfault diagnosis using trace ratio linear discriminant analysisrdquoIEEE Transactions on Industrial Electronics vol 61 no 5pp 2441ndash2451 2014

[24] M Zhao X Jin Z Zhang and B Li ldquoFault diagnosis of rollingelement bearings via discriminative subspace learning visu-alization and classificationrdquo Expert Systems with Applicationsvol 41 no 7 pp 3391ndash3401 2014

[25] R Niederjohn ldquoA mathematical formulation and comparisonof zero-crossing analysis techniques which have been appliedto automatioc speech recognitionrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 23 no 4pp 373ndash380 1975

[26] P William andM Hoffman ldquoEfficient sensor network vehicleclassification using peak harmonics of acoustic emissionsrdquo inProceedings of the SPIE Defense and Security Symposium Orlando FL USA March 2008

[27] A Shahidi Zandi R Tafreshi M Javidan and G A DumontldquoPredicting epileptic seizures in scalp EEG based on a vari-ational bayesian Gaussian mixture model of zero-crossingintervalsrdquo IEEE Transactions on Biomedical Engineeringvol 60 no 5 pp 1401ndash1413 2013

[28] C-H Wu H-C Chang P-L Lee et al ldquoFrequency recog-nition in an SSVEP-based brain computer interface usingempirical mode decomposition and refined generalized zero-crossingrdquo Journal of Neuroscience Methods vol 196 no 1pp 170ndash181 2011

12 Mathematical Problems in Engineering

[29] P E William and M W Hoffman ldquoIdentification of bearingfaults using time domain zero-crossingsrdquoMechanical Systemsand Signal Processing vol 25 no 8 pp 3078ndash3088 2011

[30] T Liu J Chen and G Dong ldquoZero crossing and coupledhidden Markov model for a rolling bearing performancedegradation assessmentrdquo Journal of Vibration and Controlvol 20 no 16 pp 2487ndash2500 2013

[31] A Ukil S Chen and A Andenna ldquoDetection of stator shortcircuit faults in three-phase induction motors using motorcurrent zero crossing instantsrdquo Electric Power Systems Re-search vol 81 no 4 pp 1036ndash1044 2011

[32] J D T Gonzalez and W Kinsner ldquoZero-crossing analysis ofLevy walks for real-time feature extractionrdquo in Proceedings ofthe IEEE International Conference on Electro InformationTechnology EIT Grand Forks ND USA May 2016

[33] R G Waghmare S L Nalbalwar and A Das ldquoTransientsignal detection on the basis of energy and zero crossingdetectorsrdquo Procedia Engineering vol 30 pp 129ndash134 2012

[34] B R Nayana and P Geethanjali ldquoAnalysis of statistical time-domain features effectiveness in identification of bearingfaults from vibration signalrdquo IEEE Sensors Journal vol 17no 17 pp 5618ndash5625 2017

[35] C Fisher Encyclopedia of Biometrics p 549 Springer USBoston MA USA 2009

[36] Q Chen W Xing L Tao and L Hua ldquoZero-crossing featureextraction based on threshold optimization for rolling ele-ment bearingrdquo in Lecture Notes in Electrical Engineeringpp 409ndash425 Springer Berlin Germany 2018

[37] Case Western Reserve University Bearing Data CenterWebsite accessed on Mar 23 2015 [Online]Available httpcsegroupscaseedubearingdatacenterpagesdownload-data-file

Mathematical Problems in Engineering 13

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Ball envelope spectrum002

001

0Am

plitu

de (V

)

(c)

Frequency (Hz)0 1000 2000 3000 4000 5000 6000

Outer envelope spectrum1

05

0Am

plitu

de (V

)

(d)

Figure 8 Envelope spectrum of four bearing states

Table 6 Optimized values of ε and theirs ratios to peak-to-peak

Index Normal Inner race fault Ball fault Outer race faultε 0000598 0003109 0001211 0002041Ratio () 01 01 01 0029

1

08

06

04

02

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

NormalInner race defect

Ball defectOuter race defect

(a)

1

08

06

04

0

02

Samples

Nor

mal

ized

dur

atio

n

0 5 10 15 20 25 30 35 40

NormalInner race defect

Ball defectOuter race defect

(b)

Feature1 2 3 4 5 6 7 8

1

08

06

04

02

0

Nor

mal

ized

dur

atio

n

NormalInner race defect

Ball defectOuter race defect

(c)

Feature1 2 3 4 5 6 7 8

1

08

06

04

0

02Nor

mal

ized

dur

atio

n

NormalInner race defect

Ball defectOuter race defect

(d)

Figure 9 ZC duration features before (a and c) and after (b and d) the model improvement and parameter optimization

Mathematical Problems in Engineering 9

adopting the method of the model improvement and pa-rameter optimization

In order to compare the recognition rate all the ei-genvectors are taken to a three-layer neural network whose

weights and thresholds are optimized by genetic algorithmto improve the diagnostic efficiency and accuracy (enumber of neurons in the hidden layer is 5 and the totalnumber of samples is 160 including training set sample 120and test set sample 40 Each bearing state contains 30training set samples and 10 test set samples arranged inorder according to normal inner ring fault rolling elementfault and outer ring fault (e result of verification is shownin Figure 10 which is the average recognition rate of 40replicates (e recognition rate of both training prior sta-tistics and test set is always 100 and proves that purposedmethod can identify bearing faults accurately

As a comparison the recognition result before the modelimprovement and parameter optimization is shown inFigure 11

(e results in the Figure 11 represent the average rec-ognition accuracy of 40 repeated experiments from which itcan be intuitively seen that the recognition rate of normalstate and inner ring fault is 100 but the recognition rate ofrolling element fault and outer ring fault is not satisfactory(e partial misjudgment between rolling element fault andouter ring fault has affected the fault recognition rate andthe comparison of the average recognition rate of 40 ex-periments is shown in Table 7

Clas

s4

35

3

25

2

15

1

Train sample0 20 40 60 80 100 120

(a)Cl

ass

4

35

3

25

2

15

1

Test sample0 5 10 2015 25 3530 40

(b)

Figure 10 Identification results of adopting the method of the model improvement and parameter optimization

Table 7 Comparison of mean recognition rate before and afteroptimization

Index Training set Test setAccuracy of before optimization () 9333 9225Accuracy of after optimization () 100 100

Table 8 Optimized ε values of various bearing states under threeloads

Index Normal Inner ringfault

Rollingelement fault

Outer ringfault

ε underload 1 0000521 0002938 0001299 0001776

ε underload 2 0000635 0003065 0001171 0001793

ε underload 3 0000548 0003188 0001321 0001809

Table 9 Recognition rate of various bearing states under threeloads

Index Normal Inner ringfault

Rollingelement fault

Outer ringfault

Accuracy ofload 1 () 100 100 9917 9917

Accuracy ofload 2 () 100 975 95 925

Accuracy ofload 3 () 9917 9917 9833 9833

Table 10 Recognition rate of different load HPs as training sets

Index Test set recognition rateTraining set Load 0 Load 1 Load 2 Load 3Load 0 100 9972 9683 9912Load 1 9813 9985 9672 9783Load 2 9775 9728 9883 9752Load 3 9815 9723 9695 9917

10 Mathematical Problems in Engineering

(e experimental results show that the method of themodel improvement and parameter optimization is effectivefor bearing fault identification under the same load

Next the experiment will verify whether the method inthis paper is robust for the load (e experimental data ofload 0 is still used as the training set according to ratios of εto peak-to-peak in Table 6 calculate the optimization pa-rameters under the other load and extract the ZC features asthe test set (e length and number of observation windowsshould be consistent with that under load 0 when extractingthe ZC features under other load Table 8 shows the opti-mized ε values of various bearing states and under threeloads

(e same neural network is applied to identify thebearing fault and the results are shown in Table 9

According to the recognition results in Table 9 althoughthe fault recognition rate at load 2 is lower the overallrecognition rate of other loads still achieves the expectedeffect (e analysis of the error identification samples showsthat these samples are quite distinct from the other samplesof the same state so the validity of themethod is not affected(e experiment proves that the method is applicable to otherloads when the load information is known

Finally the experiment verifies the test effect under anyload and the recognition effect of unknown load has beentested Similarly the method proposed in this paper isemployed for obtaining characteristic samples the samplesunder each load are used as training sets and the samples ofother load are test sets (e overall recognition rate mea-sured is shown in Table 10 and each result is the averagerecognition rate of 40 tests as well

According to the analysis of the results presented inTable 10 the recognition accuracy of samples under thesame load as the training set and the test set is relatively highWhen samples under different loads are used as training setsto test other load samples the recognition accuracy isslightly lower because a few samples deviate from the

characteristics of similar samples In general the experi-mental results demonstrate that the method has good loadrobustness so as to accurately identify the bearing fault evenif the bearing load is not known

5 Conclusion

In this paper an improved fault diagnosis method forelectromechanical system based on ZC algorithm is pro-posed which optimized the calculation model of ZCR andthe parameters of ZC algorithm by establishing a criterionfunction model (e fault diagnosis experiment of motorREB proves that the method successfully overcomes theshortcoming of ZC feature sensitive to noise and can realizeaccurate fault diagnosis without knowing the bearing loadCombined with the merits of saving computing time in thecase of big data the method is expected to monitor thecrucial parts of electromechanical system in real time so asto provide security and reliability guaranty for accuratecontrol of the electromechanical system

Data Availability

(e data used to support the study are available from thecorresponding author upon request

Conflicts of Interest

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

Acknowledgments

(is work was supported by the National Key Research andDevelopment plan of China under Grant 2018YFB1306103the National Natural Science Foundation of China underGrant 51875272 and the Provincial School Education

Clas

s4

35

3

25

2

15

1

Train sample0 20 40 60 80 100 120

(a)

Clas

s

4

35

3

25

2

15

1

Test sample0 5 10 2015 25 3530 40

(b)

Figure 11 Identification results before the model improvement and parameter optimization

Mathematical Problems in Engineering 11

Cooperation Project of Yunnan Province under GrantKKDA202001003

References

[1] M R Khoshdarregi S Tappe and Y Altintas ldquoIntegratedfive-Axis trajectory shaping and contour error compensationfor high-speed CNC machine toolsrdquo IEEEASME Transac-tions on Mechatronics vol 19 no 6 pp 1859ndash1871 2014

[2] R Colombo I Cusmano I Sterpi A Mazzone C Delconteand F Pisano ldquoTest-retest reliability of robotic assessmentmeasures for the evaluation of upper limb recoveryrdquo IEEETransactions on Neural Systems and Rehabilitation Engi-neering vol 22 no 5 pp 1020ndash1029 2014

[3] B Xu D Wang Y Zhang and Z Shi ldquoDOB-based neuralcontrol of flexible hypersonic flight vehicle considering windeffectsrdquo IEEE Transactions on Industrial Electronics vol 64no 11 pp 8676ndash8685 2017

[4] X Ding Q He and N Luo ldquoA fusion feature and its im-provement based on locality preserving projections for rollingelement bearing fault classificationrdquo Journal of Sound andVibration vol 335 pp 367ndash383 2015

[5] Y Zhang and R B Randall ldquoRolling element bearing faultdiagnosis based on the combination of genetic algorithms andfast kurtogramrdquo Mechanical Systems and Signal Processingvol 23 no 5 pp 1509ndash1517 2009

[6] J Benali M Sayadi F Fnaiech B Morello and N ZerhounildquoImportance of the fourth and fifth intrinsic mode functionsfor bearing fault diagnosisrdquo in Proceedings of the 14th In-ternational Conference on Sciences and Techniques of Auto-matic Control amp Computer EngineeringmdashSTArsquo2013 SousseTunisia December 2013

[7] S Nandi H A Toliyat and X Li ldquoCondition monitoring andfault diagnosis of electrical motors-A reviewrdquo IEEE Trans-actions on Energy Conversion vol 20 no 4 pp 719ndash729 2005

[8] J Huang S Ri T Fukuda and Y Wang ldquoA disturbanceobserver based sliding mode control for a class of under-actuated robotic system with mismatched uncertaintiesrdquoIEEE Transactions on Automatic Control vol 64 no 6pp 2480ndash2487 2019

[9] J Huang M Zhang S Ri C Xiong Z Li and Y Kang ldquoHigh-order disturbance observer based sliding mode control formobile wheeled inverted pendulum systemsrdquo IEEE Trans-actions on Industrial Electronics vol 67 no 3 pp 2030ndash20412020

[10] H Jian M H Ri DWu and S Ri ldquoInterval type-2 fuzzy logicmodeling and control of a mobile two-wheeled invertedpendulumrdquo IEEE Transactions on Fuzzy Systems vol 26no 4 pp 2030ndash2038 2018

[11] P D Mcfadden and J D Smith ldquoModel for the vibrationproduced by a single point defect in a rolling elementbearingrdquo Journal of Sound and Vibration vol 96 no 1pp 69ndash82 1984

[12] N Sawalhi and R B Randall ldquoSimulating gear and bearinginteractions in the presence of faultsrdquoMechanical Systems andSignal Processing vol 22 no 8 pp 1924ndash1951 2008

[13] Q Zhu L Liu S Li and W Zhang ldquoControl of complexnonlinear dynamic rational systemsrdquo Complexity vol 2018Article ID 8953035 12 pages 2018

[14] Q M Zhu D Y Zhao and J Zhang ldquoA general U-blockmodel-based design procedure for nonlinear polynomialcontrol systemsrdquo International Journal of Systems Sciencevol 47 no 14 pp 3465ndash3475 2015

[15] A Rafsanjani S Abbasion A Farshidianfar andH Moeenfard ldquoNonlinear dynamic modeling of surfacedefects in rolling element bearing systemsrdquo Journal of Soundamp Vibration vol 319 no 3-5 pp 1150ndash1174 2009

[16] T W Rauber F de Assis Boldt and F M Varejao ldquoHet-erogeneous feature models and feature selection applied tobearing fault diagnosisrdquo IEEE Transactions on IndustrialElectronics vol 62 no 1 pp 637ndash646 2015

[17] J Chebil M Hrairi and N Abushikhah ldquoSignal analysis ofvibration measurements for condition monitoring of bear-ingsrdquo Australian Journal of Basic amp Applied Sciences vol 5no 1 p 70 2011

[18] M D Prieto G Cirrincione A G Espinosa J A Ortega andH Henao ldquobearing fault detection by a novel condition-monitoring scheme based on statistical-time features andneural networksrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 pp 3398ndash3407 2013

[19] A Garcia-Perez R d J Romero-Troncoso E Cabal-Yepezand R A Osornio-Rios ldquo(e application of high-resolutionspectral analysis for identifying multiple combined faults ininduction motorsrdquo IEEE Transactions on Industrial Elec-tronics vol 58 no 5 pp 2002ndash2010 2011

[20] Z K Peng and F L Chu ldquoApplication of the wavelettransform in machine condition monitoring and fault diag-nostics a review with bibliographyrdquo Mechanical Systems andSignal Processing vol 18 no 2 pp 199ndash221 2004

[21] J Seshadrinath B Singh and B K Panigrahi ldquoInvestigationof vibration signatures for multiple fault diagnosis in variablefrequency drives using complex waveletsrdquo IEEE Transactionson Power Electronics vol 29 no 2 pp 936ndash945 2014

[22] W Su F Wang H Zhu Z Zhang and Z Guo ldquoRollingelement bearing faults diagnosis based on optimal Morletwavelet filter and autocorrelation enhancementrdquo MechanicalSystems and Signal Processing vol 24 no 5 pp 1458ndash14722010

[23] X Jin M Zhao T W S Chow andM Pecht ldquoMotor bearingfault diagnosis using trace ratio linear discriminant analysisrdquoIEEE Transactions on Industrial Electronics vol 61 no 5pp 2441ndash2451 2014

[24] M Zhao X Jin Z Zhang and B Li ldquoFault diagnosis of rollingelement bearings via discriminative subspace learning visu-alization and classificationrdquo Expert Systems with Applicationsvol 41 no 7 pp 3391ndash3401 2014

[25] R Niederjohn ldquoA mathematical formulation and comparisonof zero-crossing analysis techniques which have been appliedto automatioc speech recognitionrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 23 no 4pp 373ndash380 1975

[26] P William andM Hoffman ldquoEfficient sensor network vehicleclassification using peak harmonics of acoustic emissionsrdquo inProceedings of the SPIE Defense and Security Symposium Orlando FL USA March 2008

[27] A Shahidi Zandi R Tafreshi M Javidan and G A DumontldquoPredicting epileptic seizures in scalp EEG based on a vari-ational bayesian Gaussian mixture model of zero-crossingintervalsrdquo IEEE Transactions on Biomedical Engineeringvol 60 no 5 pp 1401ndash1413 2013

[28] C-H Wu H-C Chang P-L Lee et al ldquoFrequency recog-nition in an SSVEP-based brain computer interface usingempirical mode decomposition and refined generalized zero-crossingrdquo Journal of Neuroscience Methods vol 196 no 1pp 170ndash181 2011

12 Mathematical Problems in Engineering

[29] P E William and M W Hoffman ldquoIdentification of bearingfaults using time domain zero-crossingsrdquoMechanical Systemsand Signal Processing vol 25 no 8 pp 3078ndash3088 2011

[30] T Liu J Chen and G Dong ldquoZero crossing and coupledhidden Markov model for a rolling bearing performancedegradation assessmentrdquo Journal of Vibration and Controlvol 20 no 16 pp 2487ndash2500 2013

[31] A Ukil S Chen and A Andenna ldquoDetection of stator shortcircuit faults in three-phase induction motors using motorcurrent zero crossing instantsrdquo Electric Power Systems Re-search vol 81 no 4 pp 1036ndash1044 2011

[32] J D T Gonzalez and W Kinsner ldquoZero-crossing analysis ofLevy walks for real-time feature extractionrdquo in Proceedings ofthe IEEE International Conference on Electro InformationTechnology EIT Grand Forks ND USA May 2016

[33] R G Waghmare S L Nalbalwar and A Das ldquoTransientsignal detection on the basis of energy and zero crossingdetectorsrdquo Procedia Engineering vol 30 pp 129ndash134 2012

[34] B R Nayana and P Geethanjali ldquoAnalysis of statistical time-domain features effectiveness in identification of bearingfaults from vibration signalrdquo IEEE Sensors Journal vol 17no 17 pp 5618ndash5625 2017

[35] C Fisher Encyclopedia of Biometrics p 549 Springer USBoston MA USA 2009

[36] Q Chen W Xing L Tao and L Hua ldquoZero-crossing featureextraction based on threshold optimization for rolling ele-ment bearingrdquo in Lecture Notes in Electrical Engineeringpp 409ndash425 Springer Berlin Germany 2018

[37] Case Western Reserve University Bearing Data CenterWebsite accessed on Mar 23 2015 [Online]Available httpcsegroupscaseedubearingdatacenterpagesdownload-data-file

Mathematical Problems in Engineering 13

adopting the method of the model improvement and pa-rameter optimization

In order to compare the recognition rate all the ei-genvectors are taken to a three-layer neural network whose

weights and thresholds are optimized by genetic algorithmto improve the diagnostic efficiency and accuracy (enumber of neurons in the hidden layer is 5 and the totalnumber of samples is 160 including training set sample 120and test set sample 40 Each bearing state contains 30training set samples and 10 test set samples arranged inorder according to normal inner ring fault rolling elementfault and outer ring fault (e result of verification is shownin Figure 10 which is the average recognition rate of 40replicates (e recognition rate of both training prior sta-tistics and test set is always 100 and proves that purposedmethod can identify bearing faults accurately

As a comparison the recognition result before the modelimprovement and parameter optimization is shown inFigure 11

(e results in the Figure 11 represent the average rec-ognition accuracy of 40 repeated experiments from which itcan be intuitively seen that the recognition rate of normalstate and inner ring fault is 100 but the recognition rate ofrolling element fault and outer ring fault is not satisfactory(e partial misjudgment between rolling element fault andouter ring fault has affected the fault recognition rate andthe comparison of the average recognition rate of 40 ex-periments is shown in Table 7

Clas

s4

35

3

25

2

15

1

Train sample0 20 40 60 80 100 120

(a)Cl

ass

4

35

3

25

2

15

1

Test sample0 5 10 2015 25 3530 40

(b)

Figure 10 Identification results of adopting the method of the model improvement and parameter optimization

Table 7 Comparison of mean recognition rate before and afteroptimization

Index Training set Test setAccuracy of before optimization () 9333 9225Accuracy of after optimization () 100 100

Table 8 Optimized ε values of various bearing states under threeloads

Index Normal Inner ringfault

Rollingelement fault

Outer ringfault

ε underload 1 0000521 0002938 0001299 0001776

ε underload 2 0000635 0003065 0001171 0001793

ε underload 3 0000548 0003188 0001321 0001809

Table 9 Recognition rate of various bearing states under threeloads

Index Normal Inner ringfault

Rollingelement fault

Outer ringfault

Accuracy ofload 1 () 100 100 9917 9917

Accuracy ofload 2 () 100 975 95 925

Accuracy ofload 3 () 9917 9917 9833 9833

Table 10 Recognition rate of different load HPs as training sets

Index Test set recognition rateTraining set Load 0 Load 1 Load 2 Load 3Load 0 100 9972 9683 9912Load 1 9813 9985 9672 9783Load 2 9775 9728 9883 9752Load 3 9815 9723 9695 9917

10 Mathematical Problems in Engineering

(e experimental results show that the method of themodel improvement and parameter optimization is effectivefor bearing fault identification under the same load

Next the experiment will verify whether the method inthis paper is robust for the load (e experimental data ofload 0 is still used as the training set according to ratios of εto peak-to-peak in Table 6 calculate the optimization pa-rameters under the other load and extract the ZC features asthe test set (e length and number of observation windowsshould be consistent with that under load 0 when extractingthe ZC features under other load Table 8 shows the opti-mized ε values of various bearing states and under threeloads

(e same neural network is applied to identify thebearing fault and the results are shown in Table 9

According to the recognition results in Table 9 althoughthe fault recognition rate at load 2 is lower the overallrecognition rate of other loads still achieves the expectedeffect (e analysis of the error identification samples showsthat these samples are quite distinct from the other samplesof the same state so the validity of themethod is not affected(e experiment proves that the method is applicable to otherloads when the load information is known

Finally the experiment verifies the test effect under anyload and the recognition effect of unknown load has beentested Similarly the method proposed in this paper isemployed for obtaining characteristic samples the samplesunder each load are used as training sets and the samples ofother load are test sets (e overall recognition rate mea-sured is shown in Table 10 and each result is the averagerecognition rate of 40 tests as well

According to the analysis of the results presented inTable 10 the recognition accuracy of samples under thesame load as the training set and the test set is relatively highWhen samples under different loads are used as training setsto test other load samples the recognition accuracy isslightly lower because a few samples deviate from the

characteristics of similar samples In general the experi-mental results demonstrate that the method has good loadrobustness so as to accurately identify the bearing fault evenif the bearing load is not known

5 Conclusion

In this paper an improved fault diagnosis method forelectromechanical system based on ZC algorithm is pro-posed which optimized the calculation model of ZCR andthe parameters of ZC algorithm by establishing a criterionfunction model (e fault diagnosis experiment of motorREB proves that the method successfully overcomes theshortcoming of ZC feature sensitive to noise and can realizeaccurate fault diagnosis without knowing the bearing loadCombined with the merits of saving computing time in thecase of big data the method is expected to monitor thecrucial parts of electromechanical system in real time so asto provide security and reliability guaranty for accuratecontrol of the electromechanical system

Data Availability

(e data used to support the study are available from thecorresponding author upon request

Conflicts of Interest

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

Acknowledgments

(is work was supported by the National Key Research andDevelopment plan of China under Grant 2018YFB1306103the National Natural Science Foundation of China underGrant 51875272 and the Provincial School Education

Clas

s4

35

3

25

2

15

1

Train sample0 20 40 60 80 100 120

(a)

Clas

s

4

35

3

25

2

15

1

Test sample0 5 10 2015 25 3530 40

(b)

Figure 11 Identification results before the model improvement and parameter optimization

Mathematical Problems in Engineering 11

Cooperation Project of Yunnan Province under GrantKKDA202001003

References

[1] M R Khoshdarregi S Tappe and Y Altintas ldquoIntegratedfive-Axis trajectory shaping and contour error compensationfor high-speed CNC machine toolsrdquo IEEEASME Transac-tions on Mechatronics vol 19 no 6 pp 1859ndash1871 2014

[2] R Colombo I Cusmano I Sterpi A Mazzone C Delconteand F Pisano ldquoTest-retest reliability of robotic assessmentmeasures for the evaluation of upper limb recoveryrdquo IEEETransactions on Neural Systems and Rehabilitation Engi-neering vol 22 no 5 pp 1020ndash1029 2014

[3] B Xu D Wang Y Zhang and Z Shi ldquoDOB-based neuralcontrol of flexible hypersonic flight vehicle considering windeffectsrdquo IEEE Transactions on Industrial Electronics vol 64no 11 pp 8676ndash8685 2017

[4] X Ding Q He and N Luo ldquoA fusion feature and its im-provement based on locality preserving projections for rollingelement bearing fault classificationrdquo Journal of Sound andVibration vol 335 pp 367ndash383 2015

[5] Y Zhang and R B Randall ldquoRolling element bearing faultdiagnosis based on the combination of genetic algorithms andfast kurtogramrdquo Mechanical Systems and Signal Processingvol 23 no 5 pp 1509ndash1517 2009

[6] J Benali M Sayadi F Fnaiech B Morello and N ZerhounildquoImportance of the fourth and fifth intrinsic mode functionsfor bearing fault diagnosisrdquo in Proceedings of the 14th In-ternational Conference on Sciences and Techniques of Auto-matic Control amp Computer EngineeringmdashSTArsquo2013 SousseTunisia December 2013

[7] S Nandi H A Toliyat and X Li ldquoCondition monitoring andfault diagnosis of electrical motors-A reviewrdquo IEEE Trans-actions on Energy Conversion vol 20 no 4 pp 719ndash729 2005

[8] J Huang S Ri T Fukuda and Y Wang ldquoA disturbanceobserver based sliding mode control for a class of under-actuated robotic system with mismatched uncertaintiesrdquoIEEE Transactions on Automatic Control vol 64 no 6pp 2480ndash2487 2019

[9] J Huang M Zhang S Ri C Xiong Z Li and Y Kang ldquoHigh-order disturbance observer based sliding mode control formobile wheeled inverted pendulum systemsrdquo IEEE Trans-actions on Industrial Electronics vol 67 no 3 pp 2030ndash20412020

[10] H Jian M H Ri DWu and S Ri ldquoInterval type-2 fuzzy logicmodeling and control of a mobile two-wheeled invertedpendulumrdquo IEEE Transactions on Fuzzy Systems vol 26no 4 pp 2030ndash2038 2018

[11] P D Mcfadden and J D Smith ldquoModel for the vibrationproduced by a single point defect in a rolling elementbearingrdquo Journal of Sound and Vibration vol 96 no 1pp 69ndash82 1984

[12] N Sawalhi and R B Randall ldquoSimulating gear and bearinginteractions in the presence of faultsrdquoMechanical Systems andSignal Processing vol 22 no 8 pp 1924ndash1951 2008

[13] Q Zhu L Liu S Li and W Zhang ldquoControl of complexnonlinear dynamic rational systemsrdquo Complexity vol 2018Article ID 8953035 12 pages 2018

[14] Q M Zhu D Y Zhao and J Zhang ldquoA general U-blockmodel-based design procedure for nonlinear polynomialcontrol systemsrdquo International Journal of Systems Sciencevol 47 no 14 pp 3465ndash3475 2015

[15] A Rafsanjani S Abbasion A Farshidianfar andH Moeenfard ldquoNonlinear dynamic modeling of surfacedefects in rolling element bearing systemsrdquo Journal of Soundamp Vibration vol 319 no 3-5 pp 1150ndash1174 2009

[16] T W Rauber F de Assis Boldt and F M Varejao ldquoHet-erogeneous feature models and feature selection applied tobearing fault diagnosisrdquo IEEE Transactions on IndustrialElectronics vol 62 no 1 pp 637ndash646 2015

[17] J Chebil M Hrairi and N Abushikhah ldquoSignal analysis ofvibration measurements for condition monitoring of bear-ingsrdquo Australian Journal of Basic amp Applied Sciences vol 5no 1 p 70 2011

[18] M D Prieto G Cirrincione A G Espinosa J A Ortega andH Henao ldquobearing fault detection by a novel condition-monitoring scheme based on statistical-time features andneural networksrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 pp 3398ndash3407 2013

[19] A Garcia-Perez R d J Romero-Troncoso E Cabal-Yepezand R A Osornio-Rios ldquo(e application of high-resolutionspectral analysis for identifying multiple combined faults ininduction motorsrdquo IEEE Transactions on Industrial Elec-tronics vol 58 no 5 pp 2002ndash2010 2011

[20] Z K Peng and F L Chu ldquoApplication of the wavelettransform in machine condition monitoring and fault diag-nostics a review with bibliographyrdquo Mechanical Systems andSignal Processing vol 18 no 2 pp 199ndash221 2004

[21] J Seshadrinath B Singh and B K Panigrahi ldquoInvestigationof vibration signatures for multiple fault diagnosis in variablefrequency drives using complex waveletsrdquo IEEE Transactionson Power Electronics vol 29 no 2 pp 936ndash945 2014

[22] W Su F Wang H Zhu Z Zhang and Z Guo ldquoRollingelement bearing faults diagnosis based on optimal Morletwavelet filter and autocorrelation enhancementrdquo MechanicalSystems and Signal Processing vol 24 no 5 pp 1458ndash14722010

[23] X Jin M Zhao T W S Chow andM Pecht ldquoMotor bearingfault diagnosis using trace ratio linear discriminant analysisrdquoIEEE Transactions on Industrial Electronics vol 61 no 5pp 2441ndash2451 2014

[24] M Zhao X Jin Z Zhang and B Li ldquoFault diagnosis of rollingelement bearings via discriminative subspace learning visu-alization and classificationrdquo Expert Systems with Applicationsvol 41 no 7 pp 3391ndash3401 2014

[25] R Niederjohn ldquoA mathematical formulation and comparisonof zero-crossing analysis techniques which have been appliedto automatioc speech recognitionrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 23 no 4pp 373ndash380 1975

[26] P William andM Hoffman ldquoEfficient sensor network vehicleclassification using peak harmonics of acoustic emissionsrdquo inProceedings of the SPIE Defense and Security Symposium Orlando FL USA March 2008

[27] A Shahidi Zandi R Tafreshi M Javidan and G A DumontldquoPredicting epileptic seizures in scalp EEG based on a vari-ational bayesian Gaussian mixture model of zero-crossingintervalsrdquo IEEE Transactions on Biomedical Engineeringvol 60 no 5 pp 1401ndash1413 2013

[28] C-H Wu H-C Chang P-L Lee et al ldquoFrequency recog-nition in an SSVEP-based brain computer interface usingempirical mode decomposition and refined generalized zero-crossingrdquo Journal of Neuroscience Methods vol 196 no 1pp 170ndash181 2011

12 Mathematical Problems in Engineering

[29] P E William and M W Hoffman ldquoIdentification of bearingfaults using time domain zero-crossingsrdquoMechanical Systemsand Signal Processing vol 25 no 8 pp 3078ndash3088 2011

[30] T Liu J Chen and G Dong ldquoZero crossing and coupledhidden Markov model for a rolling bearing performancedegradation assessmentrdquo Journal of Vibration and Controlvol 20 no 16 pp 2487ndash2500 2013

[31] A Ukil S Chen and A Andenna ldquoDetection of stator shortcircuit faults in three-phase induction motors using motorcurrent zero crossing instantsrdquo Electric Power Systems Re-search vol 81 no 4 pp 1036ndash1044 2011

[32] J D T Gonzalez and W Kinsner ldquoZero-crossing analysis ofLevy walks for real-time feature extractionrdquo in Proceedings ofthe IEEE International Conference on Electro InformationTechnology EIT Grand Forks ND USA May 2016

[33] R G Waghmare S L Nalbalwar and A Das ldquoTransientsignal detection on the basis of energy and zero crossingdetectorsrdquo Procedia Engineering vol 30 pp 129ndash134 2012

[34] B R Nayana and P Geethanjali ldquoAnalysis of statistical time-domain features effectiveness in identification of bearingfaults from vibration signalrdquo IEEE Sensors Journal vol 17no 17 pp 5618ndash5625 2017

[35] C Fisher Encyclopedia of Biometrics p 549 Springer USBoston MA USA 2009

[36] Q Chen W Xing L Tao and L Hua ldquoZero-crossing featureextraction based on threshold optimization for rolling ele-ment bearingrdquo in Lecture Notes in Electrical Engineeringpp 409ndash425 Springer Berlin Germany 2018

[37] Case Western Reserve University Bearing Data CenterWebsite accessed on Mar 23 2015 [Online]Available httpcsegroupscaseedubearingdatacenterpagesdownload-data-file

Mathematical Problems in Engineering 13

(e experimental results show that the method of themodel improvement and parameter optimization is effectivefor bearing fault identification under the same load

Next the experiment will verify whether the method inthis paper is robust for the load (e experimental data ofload 0 is still used as the training set according to ratios of εto peak-to-peak in Table 6 calculate the optimization pa-rameters under the other load and extract the ZC features asthe test set (e length and number of observation windowsshould be consistent with that under load 0 when extractingthe ZC features under other load Table 8 shows the opti-mized ε values of various bearing states and under threeloads

(e same neural network is applied to identify thebearing fault and the results are shown in Table 9

According to the recognition results in Table 9 althoughthe fault recognition rate at load 2 is lower the overallrecognition rate of other loads still achieves the expectedeffect (e analysis of the error identification samples showsthat these samples are quite distinct from the other samplesof the same state so the validity of themethod is not affected(e experiment proves that the method is applicable to otherloads when the load information is known

Finally the experiment verifies the test effect under anyload and the recognition effect of unknown load has beentested Similarly the method proposed in this paper isemployed for obtaining characteristic samples the samplesunder each load are used as training sets and the samples ofother load are test sets (e overall recognition rate mea-sured is shown in Table 10 and each result is the averagerecognition rate of 40 tests as well

According to the analysis of the results presented inTable 10 the recognition accuracy of samples under thesame load as the training set and the test set is relatively highWhen samples under different loads are used as training setsto test other load samples the recognition accuracy isslightly lower because a few samples deviate from the

characteristics of similar samples In general the experi-mental results demonstrate that the method has good loadrobustness so as to accurately identify the bearing fault evenif the bearing load is not known

5 Conclusion

In this paper an improved fault diagnosis method forelectromechanical system based on ZC algorithm is pro-posed which optimized the calculation model of ZCR andthe parameters of ZC algorithm by establishing a criterionfunction model (e fault diagnosis experiment of motorREB proves that the method successfully overcomes theshortcoming of ZC feature sensitive to noise and can realizeaccurate fault diagnosis without knowing the bearing loadCombined with the merits of saving computing time in thecase of big data the method is expected to monitor thecrucial parts of electromechanical system in real time so asto provide security and reliability guaranty for accuratecontrol of the electromechanical system

Data Availability

(e data used to support the study are available from thecorresponding author upon request

Conflicts of Interest

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

Acknowledgments

(is work was supported by the National Key Research andDevelopment plan of China under Grant 2018YFB1306103the National Natural Science Foundation of China underGrant 51875272 and the Provincial School Education

Clas

s4

35

3

25

2

15

1

Train sample0 20 40 60 80 100 120

(a)

Clas

s

4

35

3

25

2

15

1

Test sample0 5 10 2015 25 3530 40

(b)

Figure 11 Identification results before the model improvement and parameter optimization

Mathematical Problems in Engineering 11

Cooperation Project of Yunnan Province under GrantKKDA202001003

References

[1] M R Khoshdarregi S Tappe and Y Altintas ldquoIntegratedfive-Axis trajectory shaping and contour error compensationfor high-speed CNC machine toolsrdquo IEEEASME Transac-tions on Mechatronics vol 19 no 6 pp 1859ndash1871 2014

[2] R Colombo I Cusmano I Sterpi A Mazzone C Delconteand F Pisano ldquoTest-retest reliability of robotic assessmentmeasures for the evaluation of upper limb recoveryrdquo IEEETransactions on Neural Systems and Rehabilitation Engi-neering vol 22 no 5 pp 1020ndash1029 2014

[3] B Xu D Wang Y Zhang and Z Shi ldquoDOB-based neuralcontrol of flexible hypersonic flight vehicle considering windeffectsrdquo IEEE Transactions on Industrial Electronics vol 64no 11 pp 8676ndash8685 2017

[4] X Ding Q He and N Luo ldquoA fusion feature and its im-provement based on locality preserving projections for rollingelement bearing fault classificationrdquo Journal of Sound andVibration vol 335 pp 367ndash383 2015

[5] Y Zhang and R B Randall ldquoRolling element bearing faultdiagnosis based on the combination of genetic algorithms andfast kurtogramrdquo Mechanical Systems and Signal Processingvol 23 no 5 pp 1509ndash1517 2009

[6] J Benali M Sayadi F Fnaiech B Morello and N ZerhounildquoImportance of the fourth and fifth intrinsic mode functionsfor bearing fault diagnosisrdquo in Proceedings of the 14th In-ternational Conference on Sciences and Techniques of Auto-matic Control amp Computer EngineeringmdashSTArsquo2013 SousseTunisia December 2013

[7] S Nandi H A Toliyat and X Li ldquoCondition monitoring andfault diagnosis of electrical motors-A reviewrdquo IEEE Trans-actions on Energy Conversion vol 20 no 4 pp 719ndash729 2005

[8] J Huang S Ri T Fukuda and Y Wang ldquoA disturbanceobserver based sliding mode control for a class of under-actuated robotic system with mismatched uncertaintiesrdquoIEEE Transactions on Automatic Control vol 64 no 6pp 2480ndash2487 2019

[9] J Huang M Zhang S Ri C Xiong Z Li and Y Kang ldquoHigh-order disturbance observer based sliding mode control formobile wheeled inverted pendulum systemsrdquo IEEE Trans-actions on Industrial Electronics vol 67 no 3 pp 2030ndash20412020

[10] H Jian M H Ri DWu and S Ri ldquoInterval type-2 fuzzy logicmodeling and control of a mobile two-wheeled invertedpendulumrdquo IEEE Transactions on Fuzzy Systems vol 26no 4 pp 2030ndash2038 2018

[11] P D Mcfadden and J D Smith ldquoModel for the vibrationproduced by a single point defect in a rolling elementbearingrdquo Journal of Sound and Vibration vol 96 no 1pp 69ndash82 1984

[12] N Sawalhi and R B Randall ldquoSimulating gear and bearinginteractions in the presence of faultsrdquoMechanical Systems andSignal Processing vol 22 no 8 pp 1924ndash1951 2008

[13] Q Zhu L Liu S Li and W Zhang ldquoControl of complexnonlinear dynamic rational systemsrdquo Complexity vol 2018Article ID 8953035 12 pages 2018

[14] Q M Zhu D Y Zhao and J Zhang ldquoA general U-blockmodel-based design procedure for nonlinear polynomialcontrol systemsrdquo International Journal of Systems Sciencevol 47 no 14 pp 3465ndash3475 2015

[15] A Rafsanjani S Abbasion A Farshidianfar andH Moeenfard ldquoNonlinear dynamic modeling of surfacedefects in rolling element bearing systemsrdquo Journal of Soundamp Vibration vol 319 no 3-5 pp 1150ndash1174 2009

[16] T W Rauber F de Assis Boldt and F M Varejao ldquoHet-erogeneous feature models and feature selection applied tobearing fault diagnosisrdquo IEEE Transactions on IndustrialElectronics vol 62 no 1 pp 637ndash646 2015

[17] J Chebil M Hrairi and N Abushikhah ldquoSignal analysis ofvibration measurements for condition monitoring of bear-ingsrdquo Australian Journal of Basic amp Applied Sciences vol 5no 1 p 70 2011

[18] M D Prieto G Cirrincione A G Espinosa J A Ortega andH Henao ldquobearing fault detection by a novel condition-monitoring scheme based on statistical-time features andneural networksrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 pp 3398ndash3407 2013

[19] A Garcia-Perez R d J Romero-Troncoso E Cabal-Yepezand R A Osornio-Rios ldquo(e application of high-resolutionspectral analysis for identifying multiple combined faults ininduction motorsrdquo IEEE Transactions on Industrial Elec-tronics vol 58 no 5 pp 2002ndash2010 2011

[20] Z K Peng and F L Chu ldquoApplication of the wavelettransform in machine condition monitoring and fault diag-nostics a review with bibliographyrdquo Mechanical Systems andSignal Processing vol 18 no 2 pp 199ndash221 2004

[21] J Seshadrinath B Singh and B K Panigrahi ldquoInvestigationof vibration signatures for multiple fault diagnosis in variablefrequency drives using complex waveletsrdquo IEEE Transactionson Power Electronics vol 29 no 2 pp 936ndash945 2014

[22] W Su F Wang H Zhu Z Zhang and Z Guo ldquoRollingelement bearing faults diagnosis based on optimal Morletwavelet filter and autocorrelation enhancementrdquo MechanicalSystems and Signal Processing vol 24 no 5 pp 1458ndash14722010

[23] X Jin M Zhao T W S Chow andM Pecht ldquoMotor bearingfault diagnosis using trace ratio linear discriminant analysisrdquoIEEE Transactions on Industrial Electronics vol 61 no 5pp 2441ndash2451 2014

[24] M Zhao X Jin Z Zhang and B Li ldquoFault diagnosis of rollingelement bearings via discriminative subspace learning visu-alization and classificationrdquo Expert Systems with Applicationsvol 41 no 7 pp 3391ndash3401 2014

[25] R Niederjohn ldquoA mathematical formulation and comparisonof zero-crossing analysis techniques which have been appliedto automatioc speech recognitionrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 23 no 4pp 373ndash380 1975

[26] P William andM Hoffman ldquoEfficient sensor network vehicleclassification using peak harmonics of acoustic emissionsrdquo inProceedings of the SPIE Defense and Security Symposium Orlando FL USA March 2008

[27] A Shahidi Zandi R Tafreshi M Javidan and G A DumontldquoPredicting epileptic seizures in scalp EEG based on a vari-ational bayesian Gaussian mixture model of zero-crossingintervalsrdquo IEEE Transactions on Biomedical Engineeringvol 60 no 5 pp 1401ndash1413 2013

[28] C-H Wu H-C Chang P-L Lee et al ldquoFrequency recog-nition in an SSVEP-based brain computer interface usingempirical mode decomposition and refined generalized zero-crossingrdquo Journal of Neuroscience Methods vol 196 no 1pp 170ndash181 2011

12 Mathematical Problems in Engineering

[29] P E William and M W Hoffman ldquoIdentification of bearingfaults using time domain zero-crossingsrdquoMechanical Systemsand Signal Processing vol 25 no 8 pp 3078ndash3088 2011

[30] T Liu J Chen and G Dong ldquoZero crossing and coupledhidden Markov model for a rolling bearing performancedegradation assessmentrdquo Journal of Vibration and Controlvol 20 no 16 pp 2487ndash2500 2013

[31] A Ukil S Chen and A Andenna ldquoDetection of stator shortcircuit faults in three-phase induction motors using motorcurrent zero crossing instantsrdquo Electric Power Systems Re-search vol 81 no 4 pp 1036ndash1044 2011

[32] J D T Gonzalez and W Kinsner ldquoZero-crossing analysis ofLevy walks for real-time feature extractionrdquo in Proceedings ofthe IEEE International Conference on Electro InformationTechnology EIT Grand Forks ND USA May 2016

[33] R G Waghmare S L Nalbalwar and A Das ldquoTransientsignal detection on the basis of energy and zero crossingdetectorsrdquo Procedia Engineering vol 30 pp 129ndash134 2012

[34] B R Nayana and P Geethanjali ldquoAnalysis of statistical time-domain features effectiveness in identification of bearingfaults from vibration signalrdquo IEEE Sensors Journal vol 17no 17 pp 5618ndash5625 2017

[35] C Fisher Encyclopedia of Biometrics p 549 Springer USBoston MA USA 2009

[36] Q Chen W Xing L Tao and L Hua ldquoZero-crossing featureextraction based on threshold optimization for rolling ele-ment bearingrdquo in Lecture Notes in Electrical Engineeringpp 409ndash425 Springer Berlin Germany 2018

[37] Case Western Reserve University Bearing Data CenterWebsite accessed on Mar 23 2015 [Online]Available httpcsegroupscaseedubearingdatacenterpagesdownload-data-file

Mathematical Problems in Engineering 13

Cooperation Project of Yunnan Province under GrantKKDA202001003

References

[1] M R Khoshdarregi S Tappe and Y Altintas ldquoIntegratedfive-Axis trajectory shaping and contour error compensationfor high-speed CNC machine toolsrdquo IEEEASME Transac-tions on Mechatronics vol 19 no 6 pp 1859ndash1871 2014

[2] R Colombo I Cusmano I Sterpi A Mazzone C Delconteand F Pisano ldquoTest-retest reliability of robotic assessmentmeasures for the evaluation of upper limb recoveryrdquo IEEETransactions on Neural Systems and Rehabilitation Engi-neering vol 22 no 5 pp 1020ndash1029 2014

[3] B Xu D Wang Y Zhang and Z Shi ldquoDOB-based neuralcontrol of flexible hypersonic flight vehicle considering windeffectsrdquo IEEE Transactions on Industrial Electronics vol 64no 11 pp 8676ndash8685 2017

[4] X Ding Q He and N Luo ldquoA fusion feature and its im-provement based on locality preserving projections for rollingelement bearing fault classificationrdquo Journal of Sound andVibration vol 335 pp 367ndash383 2015

[5] Y Zhang and R B Randall ldquoRolling element bearing faultdiagnosis based on the combination of genetic algorithms andfast kurtogramrdquo Mechanical Systems and Signal Processingvol 23 no 5 pp 1509ndash1517 2009

[6] J Benali M Sayadi F Fnaiech B Morello and N ZerhounildquoImportance of the fourth and fifth intrinsic mode functionsfor bearing fault diagnosisrdquo in Proceedings of the 14th In-ternational Conference on Sciences and Techniques of Auto-matic Control amp Computer EngineeringmdashSTArsquo2013 SousseTunisia December 2013

[7] S Nandi H A Toliyat and X Li ldquoCondition monitoring andfault diagnosis of electrical motors-A reviewrdquo IEEE Trans-actions on Energy Conversion vol 20 no 4 pp 719ndash729 2005

[8] J Huang S Ri T Fukuda and Y Wang ldquoA disturbanceobserver based sliding mode control for a class of under-actuated robotic system with mismatched uncertaintiesrdquoIEEE Transactions on Automatic Control vol 64 no 6pp 2480ndash2487 2019

[9] J Huang M Zhang S Ri C Xiong Z Li and Y Kang ldquoHigh-order disturbance observer based sliding mode control formobile wheeled inverted pendulum systemsrdquo IEEE Trans-actions on Industrial Electronics vol 67 no 3 pp 2030ndash20412020

[10] H Jian M H Ri DWu and S Ri ldquoInterval type-2 fuzzy logicmodeling and control of a mobile two-wheeled invertedpendulumrdquo IEEE Transactions on Fuzzy Systems vol 26no 4 pp 2030ndash2038 2018

[11] P D Mcfadden and J D Smith ldquoModel for the vibrationproduced by a single point defect in a rolling elementbearingrdquo Journal of Sound and Vibration vol 96 no 1pp 69ndash82 1984

[12] N Sawalhi and R B Randall ldquoSimulating gear and bearinginteractions in the presence of faultsrdquoMechanical Systems andSignal Processing vol 22 no 8 pp 1924ndash1951 2008

[13] Q Zhu L Liu S Li and W Zhang ldquoControl of complexnonlinear dynamic rational systemsrdquo Complexity vol 2018Article ID 8953035 12 pages 2018

[14] Q M Zhu D Y Zhao and J Zhang ldquoA general U-blockmodel-based design procedure for nonlinear polynomialcontrol systemsrdquo International Journal of Systems Sciencevol 47 no 14 pp 3465ndash3475 2015

[15] A Rafsanjani S Abbasion A Farshidianfar andH Moeenfard ldquoNonlinear dynamic modeling of surfacedefects in rolling element bearing systemsrdquo Journal of Soundamp Vibration vol 319 no 3-5 pp 1150ndash1174 2009

[16] T W Rauber F de Assis Boldt and F M Varejao ldquoHet-erogeneous feature models and feature selection applied tobearing fault diagnosisrdquo IEEE Transactions on IndustrialElectronics vol 62 no 1 pp 637ndash646 2015

[17] J Chebil M Hrairi and N Abushikhah ldquoSignal analysis ofvibration measurements for condition monitoring of bear-ingsrdquo Australian Journal of Basic amp Applied Sciences vol 5no 1 p 70 2011

[18] M D Prieto G Cirrincione A G Espinosa J A Ortega andH Henao ldquobearing fault detection by a novel condition-monitoring scheme based on statistical-time features andneural networksrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 pp 3398ndash3407 2013

[19] A Garcia-Perez R d J Romero-Troncoso E Cabal-Yepezand R A Osornio-Rios ldquo(e application of high-resolutionspectral analysis for identifying multiple combined faults ininduction motorsrdquo IEEE Transactions on Industrial Elec-tronics vol 58 no 5 pp 2002ndash2010 2011

[20] Z K Peng and F L Chu ldquoApplication of the wavelettransform in machine condition monitoring and fault diag-nostics a review with bibliographyrdquo Mechanical Systems andSignal Processing vol 18 no 2 pp 199ndash221 2004

[21] J Seshadrinath B Singh and B K Panigrahi ldquoInvestigationof vibration signatures for multiple fault diagnosis in variablefrequency drives using complex waveletsrdquo IEEE Transactionson Power Electronics vol 29 no 2 pp 936ndash945 2014

[22] W Su F Wang H Zhu Z Zhang and Z Guo ldquoRollingelement bearing faults diagnosis based on optimal Morletwavelet filter and autocorrelation enhancementrdquo MechanicalSystems and Signal Processing vol 24 no 5 pp 1458ndash14722010

[23] X Jin M Zhao T W S Chow andM Pecht ldquoMotor bearingfault diagnosis using trace ratio linear discriminant analysisrdquoIEEE Transactions on Industrial Electronics vol 61 no 5pp 2441ndash2451 2014

[24] M Zhao X Jin Z Zhang and B Li ldquoFault diagnosis of rollingelement bearings via discriminative subspace learning visu-alization and classificationrdquo Expert Systems with Applicationsvol 41 no 7 pp 3391ndash3401 2014

[25] R Niederjohn ldquoA mathematical formulation and comparisonof zero-crossing analysis techniques which have been appliedto automatioc speech recognitionrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 23 no 4pp 373ndash380 1975

[26] P William andM Hoffman ldquoEfficient sensor network vehicleclassification using peak harmonics of acoustic emissionsrdquo inProceedings of the SPIE Defense and Security Symposium Orlando FL USA March 2008

[27] A Shahidi Zandi R Tafreshi M Javidan and G A DumontldquoPredicting epileptic seizures in scalp EEG based on a vari-ational bayesian Gaussian mixture model of zero-crossingintervalsrdquo IEEE Transactions on Biomedical Engineeringvol 60 no 5 pp 1401ndash1413 2013

[28] C-H Wu H-C Chang P-L Lee et al ldquoFrequency recog-nition in an SSVEP-based brain computer interface usingempirical mode decomposition and refined generalized zero-crossingrdquo Journal of Neuroscience Methods vol 196 no 1pp 170ndash181 2011

12 Mathematical Problems in Engineering

[29] P E William and M W Hoffman ldquoIdentification of bearingfaults using time domain zero-crossingsrdquoMechanical Systemsand Signal Processing vol 25 no 8 pp 3078ndash3088 2011

[30] T Liu J Chen and G Dong ldquoZero crossing and coupledhidden Markov model for a rolling bearing performancedegradation assessmentrdquo Journal of Vibration and Controlvol 20 no 16 pp 2487ndash2500 2013

[31] A Ukil S Chen and A Andenna ldquoDetection of stator shortcircuit faults in three-phase induction motors using motorcurrent zero crossing instantsrdquo Electric Power Systems Re-search vol 81 no 4 pp 1036ndash1044 2011

[32] J D T Gonzalez and W Kinsner ldquoZero-crossing analysis ofLevy walks for real-time feature extractionrdquo in Proceedings ofthe IEEE International Conference on Electro InformationTechnology EIT Grand Forks ND USA May 2016

[33] R G Waghmare S L Nalbalwar and A Das ldquoTransientsignal detection on the basis of energy and zero crossingdetectorsrdquo Procedia Engineering vol 30 pp 129ndash134 2012

[34] B R Nayana and P Geethanjali ldquoAnalysis of statistical time-domain features effectiveness in identification of bearingfaults from vibration signalrdquo IEEE Sensors Journal vol 17no 17 pp 5618ndash5625 2017

[35] C Fisher Encyclopedia of Biometrics p 549 Springer USBoston MA USA 2009

[36] Q Chen W Xing L Tao and L Hua ldquoZero-crossing featureextraction based on threshold optimization for rolling ele-ment bearingrdquo in Lecture Notes in Electrical Engineeringpp 409ndash425 Springer Berlin Germany 2018

[37] Case Western Reserve University Bearing Data CenterWebsite accessed on Mar 23 2015 [Online]Available httpcsegroupscaseedubearingdatacenterpagesdownload-data-file

Mathematical Problems in Engineering 13

[29] P E William and M W Hoffman ldquoIdentification of bearingfaults using time domain zero-crossingsrdquoMechanical Systemsand Signal Processing vol 25 no 8 pp 3078ndash3088 2011

[30] T Liu J Chen and G Dong ldquoZero crossing and coupledhidden Markov model for a rolling bearing performancedegradation assessmentrdquo Journal of Vibration and Controlvol 20 no 16 pp 2487ndash2500 2013

[31] A Ukil S Chen and A Andenna ldquoDetection of stator shortcircuit faults in three-phase induction motors using motorcurrent zero crossing instantsrdquo Electric Power Systems Re-search vol 81 no 4 pp 1036ndash1044 2011

[32] J D T Gonzalez and W Kinsner ldquoZero-crossing analysis ofLevy walks for real-time feature extractionrdquo in Proceedings ofthe IEEE International Conference on Electro InformationTechnology EIT Grand Forks ND USA May 2016

[33] R G Waghmare S L Nalbalwar and A Das ldquoTransientsignal detection on the basis of energy and zero crossingdetectorsrdquo Procedia Engineering vol 30 pp 129ndash134 2012

[34] B R Nayana and P Geethanjali ldquoAnalysis of statistical time-domain features effectiveness in identification of bearingfaults from vibration signalrdquo IEEE Sensors Journal vol 17no 17 pp 5618ndash5625 2017

[35] C Fisher Encyclopedia of Biometrics p 549 Springer USBoston MA USA 2009

[36] Q Chen W Xing L Tao and L Hua ldquoZero-crossing featureextraction based on threshold optimization for rolling ele-ment bearingrdquo in Lecture Notes in Electrical Engineeringpp 409ndash425 Springer Berlin Germany 2018

[37] Case Western Reserve University Bearing Data CenterWebsite accessed on Mar 23 2015 [Online]Available httpcsegroupscaseedubearingdatacenterpagesdownload-data-file

Mathematical Problems in Engineering 13