research article correlation coefficient of simplified...

Research ArticleCorrelation Coefficient of Simplified Neutrosophic Setsfor Bearing Fault Diagnosis

Lilian Shi

Department of Electrical and Information Engineering Shaoxing University 508 Huancheng West RoadShaoxing Zhejiang Province 312000 China

Correspondence should be addressed to Lilian Shi slliansinacom

Received 18 May 2016 Revised 26 September 2016 Accepted 3 October 2016

Academic Editor Mariano Artes

Copyright copy 2016 Lilian Shi This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In order to process the vagueness in vibration fault diagnosis of rolling bearing a new correlation coefficient of simplifiedneutrosophic sets (SNSs) is proposed Vibration signals of rolling bearings are acquired by an acceleration sensor and amorphological filter is used to reduce the noise effect Wavelet packet is applied to decompose the vibration signals into eightsubfrequency bands and the eigenvectors associated with energy eigenvalue of each frequency are extracted for fault features TheSNSs of each fault types are established according to energy eigenvectors Finally a correlation coefficient of two SNSs is proposedto diagnose the bearing fault types The experimental results show that the proposed method can effectively diagnose the bearingfaults

1 Introduction

A rolling bearing is an important rotating part in a mechan-ical equipment and its quality decides the operation per-formance of the equipment A faulty bearing may cause thewhole equipment to operate abnormally Bearing faults mustbe effectively diagnosed to avoid catastrophic mechanicalfailures and significant economic losses

The vibration signals of rolling bearings often indicatesome fault information When the fault occurs in rotatingbearings different characteristic frequencies of vibrationsignals can be generated periodically [1] Actually for theoriginal vibration signals many useful fault features areusually hidden in noise and the relationship between faultsymptoms and causations is very complex so it is difficultto make accurate and quantitative analysis for fault typesIn recent years many studies have been devoted to the faultdiagnosis of rolling bearing There are two critical issues fordiagnosing bearing faults from vibration signals One issue ishow to extract fault features from vibration signals Anotherone is how to analyze fault features and recognize fault typesaccording to these features

In order to extract useful fault features from vibrationsignals many techniques such as time domain frequency

domain and time-frequency domainmethods are extensivelyinvestigated [2] In the time domain method key parameterscan be extracted directly from the original vibration signalssuch as root mean square (RMS) crest factor peak andprobability density function [3] In addition time domainsignals can be transformed into frequency domain by Fouriertransform However Fourier analysis may cause informationloss during the transformation particularly for nonstationarysignals The vibration of a rolling bearing is typically non-stationary so it is difficult to extract accurate and completefault features when adopting the traditional analysis only inthe time or frequency domain In time-frequency domainthewavelet can revealmore complete information for nonsta-tionary signals [4] Many research results show that a waveletpacket is an effective tool to extract features from vibrationsignals for bearing fault diagnosis [5ndash8]

The next key issue is to recognize fault types of bearingsaccording to the extracted fault features from vibrationsignals To solve this problem various approaches such asexpert systems [9 10] neural networks [3 11 12] and fuzzyapproaches [13ndash15] have been developed for fault diagnosisover the past few years Fuzzy theory has attracted increasingattention in bearing fault diagnosis and many researches

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 5414361 11 pageshttpdxdoiorg10115520165414361

2 Shock and Vibration

show that fuzzy theory is an effective tool to diagnose bearingfaults

Fuzzy sets (FSs) theory was proposed by Zadeh (1965)for handling uncertain information using single membershipdegree function [16] The fuzzy sets were extended to intu-itionistic fuzzy sets (IFSs) [17] and interval valued intuitionis-tic fuzzy sets (IVIFSs) [18] by usingmembership degree func-tion non-membership degree function and degree functionof hesitation simultaneously FSs IFSs and IVIFSs have beenwidely applied in various fields However FSs IFSs andIVIFSs cannot deal with some types of uncertainties such asthe indeterminate information and inconsistent informationin real physical problems Furthermore Smarandache [19]proposed neutrosophy theory from philosophical point ofview Neutrosophic sets (NSs) are characterized by a truth-membership function an indeterminacy-membership func-tion and a falsity-membership functionThe functions ofNSstake the value from real standard or nonstandard subsets of]minus0 1+[ [19] andNSs are difficult to be applied in engineeringareas For the real engineering applications neutrosophicsets (NSs) can be described as simplified neutrosophic sets(SNSs) [20] with the normal standard real unit interval[0 1] One major advantage of SNSs is the ability to performanalysis problems involving imprecise undetermined andinconsistent data Recently SNSs have been applied in manydifferent fuzzy problems such as medical diagnosis problems[21 22] decision making problems [20 23] and imageprocessing [24]

For vibrational fault diagnosis of rolling bearing thereis no direct accurate and quantitative relationship betweenfault vibration characteristics and fault types Therefore thefault-diagnosis process has certain vagueness This papermainly focuses on the fault diagnosis of rolling bearingsbased on vibration signals and SNSs In this work a morpho-logical filter and wavelet packet decomposition are appliedto preprocess the original vibration signals and the SNSsof each fault type will be established according to energyeigenvectors The fault types will be diagnosed using a newcorrelation coefficient of SNSs

The rest of the paper is organized as follows Section 2gives the experimental system Section 3 gives the datapreprocessing techniques including morphological opening-closing operation and wavelet packet decomposition InSection 4 some basic concepts of SNSs and a new correlationcoefficient are introduced firstly and then the fault-diagnosismethod is presented based on SNSs Conclusions of this workare summarized in Section 5

2 Experimental Setup

This study was carried out with the experimental apparatusshown in Figure 1 The principal axis is driven by an ACmotor and the vibration signals of bearings are acquired byan acceleration sensor and a data acquiring card NI USB-6251The vibration signals will be processed using a computerand displayed by an oscilloscope Some vibration signals areacquired by the experimental device of Jiliang University inChina [25] shown in Figure 2 In this experiment the type

(1)

(2)(3)(4)

(5)

(6)

(7)

(8)

(9)

Figure 1 Diagram of experimental system (1)Motor (2) driver (3)principal axis (4) bearing (5) core axes (6) acceleration sensor (7)acquisition data card (8) oscilloscope and (9) computer

Figure 2 Experimental device

Table 1 Bearing parameters

Parameters Values (mm)Outer race diameter 35Inner race diameter 15Ball diameter d 75Thickness 11Number of balls 7Pitch diameter D 25Contact angle 120572 0

of bearings is NSK 6202 deep groove ball bearing whosespecifications are listed in Table 1

In order to diagnose the fault of bearings four types ofbearings are used healthy outer race fault inner race faultand ball fault bearingsThe core axis is driven at the rotationalspeed of 25Hz NI Labview Signal Express will be applied fordata acquisition with 10KHz sampling frequency and 02 ssample time

When a fault exists in a bearing vibration impulses willhappen at a specific frequency Theoretically when a bearing

Shock and Vibration 3

Vibration signals acquisition

Data preprocessing bymorphological filtering

Decompose the vibration signalsinto energy spectrum

Extract energy spectrum featuresof vibration signals

Built the neutrosophic sets offour types of bearings according

to energy spectrum features

Calculate the correlation coefficientbetween testing bearing and four

types of bearings

Diagnose fault types of bearingsaccording to maximumcorrelation coefficient

Figure 3 Block diagram of diagnosis for fault bearing usingneutrosophic sets

rotates at a constant speed the fault frequencies can becalculated by the following [26]

119865119874 = 1198731198612 (1 minus 119889119863 sdot cos120572) sdot 119865119903

119865119868 = 1198731198612 (1 + 119889119863 sdot cos120572) sdot 119865119903

119865119861 = 119863119889 [1 minus ( 119889

119863 sdot cos120572)2] sdot 119865119903

(1)

where 119889 is the diameter of the rolling elements119863 is the pitchdiameter 119865119903 is the rotational speed of the shaft 119873119861 is thenumber of rolling elements and 119865119874 119865119868 and 119865119861 represent thefault frequencies of outer race fault inner race fault and ballfault of a bearing respectively

According to (1) we can calculate the fault frequencies119865119874 = 6125Hz 119865119868 = 11375Hz and 119865119861 = 7583Hz in thisexperiment

3 Vibration Signal Data Preprocessing

The framework of diagnosing process is shown in Figure 3The original vibration signals of bearings are usually

ridden with noise It is difficult to extract the fault featuresdirectly from original vibration signals In order to removethe strong noises and detect the effective signals for bearingfaults diagnosis data processing algorithms are necessary

to be performed In this experiment a morphological filteris used to remove high frequencies noise from the originalvibration signals firstly and then wavelet packet is applied todecompose the signals into the individual frequencies

31 Morphological Filter A morphological filter is a nonlin-ear signal processing and analysis tool in time domain andit can be composed of several morphological operations [27]The basic morphological operators include dilation erosionopening and closing Assume that 119909(119899) and a structuralelement 119910(119899) are discrete signals defined in119883 = 0 1 119873minus1 and119861 = 0 1 119872minus1 respectively and119873 ge 119872 the fourbasic operators of 119909(119899) on 119910(119899) are defined as follows

Dilation

119909 oplus 119910 = max119898=01119872minus1

119909 (119899 + 119898) + 119910 (119898) (119899 = 0 1 119873 + 119872)

(2)

Erosion

119909 ⊖ 119910 = min119898=01119872minus1

119909 (119899 + 119898) minus 119910 (119898) (119899 = 0 1 119873 + 119872) (3)

Opening

119909 ∘ 119910 = (119909 ⊖ 119910 oplus 119910) (4)

Closing

119909 ∙ 119910 = (119909 oplus 119910 ⊖ 119910) (5)

Morphological opening-closing filter as follows

119865oc = (119909 ∘ 119910 ∙ 119910) (6)

In this experiment the morphological opening-closingfilter 119865oc was used to remove the strong noises

Figures 4ndash6 show the signals of rolling element bearingswith outer race inner race and ball fault respectively [25]In these figures the fault signals have distinguishing peakvalue features at the fault frequencies The results in Figures4ndash6 indicate thatmorphological filter is an effective denoisingtechnique for vibration signals of ball bearings

32 Wavelet Packet Decomposition According to the struc-ture of wavelet decomposition the input vibration signal canbe decomposed into low-frequency and high-frequency partsfor each stepThe selection of a suitable level for the hierarchydepends on the signal experience and actual needs [4 7]The wavelet packet was applied to decompose the vibrationsignals into eight subfrequency bands in the practical appli-cation [2 6] Based on review of earlier researcher in thiswork 3-level wavelet packet decomposition is considered forbearing fault diagnosis and experimental results show thatthe bearing fault feature can be extracted effectively from thedecomposed signals


0 005 01 015 02minus1

0

1

Time (s)

Am

plitu

de (m

s2)

(a) Original vibration in time domain

0 500 1000 1500 20000

100200300400500600700

Frequency (Hz)

Am

plitu

de (m

s2) FO

2 lowast FO

(b) Fast Fourier transform after morphological filter

Figure 4 Signals of rolling bearing in outer race fault

0 005 01 015 02minus05

0

05

Time (s)

Am

plitu

de (m

s2)


0 200 400 600 800 10000

100

200

300

Frequency (Hz)

Am

plitu

de (m

s2)

FI

2 lowast FI


Figure 5 Signals of rolling bearing in inner race fault

In this experiment the vibration signals of bearing arepreprocessed firstly by a morphological filter and then aredecomposed using 3-level wavelet packet

Assume that 119909(119905) is a vibration signal 119871(sdot) and 119867(sdot) arequadrature mirror filters representing low-pass and high-pass wavelet filters respectively These filters associate withthe scaling function and wavelet function and satisfy thecondition 119867(119899) = (minus1)119899119871(1 minus 119899) Then the signal 119909(119905)can be decomposed into a set of high- and low-frequencycomponents by the following recursive relationships

1199091198952119896 = sum119899

119871 (119899) sdot 119909119895minus11198961199091198952119896+1 = sum

119899

119867(119899) sdot 119909119895minus1119896(7)

where 1199091198952119896 denotes the wavelet coefficients at the119895th level and2119896th subbandThe diagram of 3-level wavelet packet decomposition is

shown in Figure 7 In Figure 7 the frequency intervals of eachband can be computed by ((119896 minus 1)11989111990424 11989611989111990424] where 119891119904 issampling frequency In this work 119891119904 = 10 kHz and 11989111990424 =625Hz The frequency intervals are given in Table 2

The vibration signal 119909(119905) can be expressed as follows

119909 (119905) = 23minus1sum119896=0

1199093119896 (119905) 119896 = 0 1 7 (8)

where 119896 represents eight subfrequency bands and1199093119896(119905) is thewavelet coefficient at the 3-level and 119896th subfrequency band

Table 2 Frequency intervals of eight subfrequency bands

Signals Frequency (Hz)11990930 (0 625]11990931 (625 1250]11990932 (1250 1875]11990933 (1875 2500]11990934 (2500 3125]11990935 (3125 3750]11990936 (3750 4375]11990937 (4375 5000]

After the decomposition the energy in each subfrequencyband can be defined as

1198641198963 = int 10038161003816100381610038161199093119896 (119905)10038161003816100381610038162 119889119905 =119873sum119894=0

10038161003816100381610038161199093119896 (119894)10038161003816100381610038162 119896 = 0 1 7 (9)

where 1199093119896(119894) is the 119894th discrete point amplitude of waveletcoefficient (1199093119896(119905)) and119873 is its discrete point number in eachsubfrequency

The faults of rolling bearings will greatly influence thewavelet packet energy of vibration signals so it is veryuseful to extract the energy eigenvalue for diagnosing bearingfaults In this experiment an eigenvector based on energyeigenvalue of each frequency can be constructed as follows

119879 = 11986403 11986413 11986423 11986433 11986443 11986453 11986463 11986473 (10)


0 005 01 015 02minus1

0

1

Time (s)

Am

plitu

de (m

s2)


0 200 400 600 800 10000

100

200

300

400

500

Frequency (Hz)

Am

plitu

de (m

s2)

FB

2 lowast FB


Figure 6 Signals of rolling bearing in ball fault

Table 3 Energy interval ranges at the eight subfrequency bands

Fault types Energy in each frequency band119864lowast03 119864lowast13 119864lowast23 119864lowast33 119864lowast43 119864lowast53 119864lowast63 119864lowast731198601 (healthy) [076 080] [095 100] [011 015] [064 071] [004 006] [000 002] [001 005] [000 003]

1198602 (outer race fault) [100 100] [024 039] [002 003] [013 022] [000 001] [000 001] [001 001] [001 001]1198603 (ball fault) [082 093] [100 100] [011 016] [065 074] [006 008] [000 004] [002 006] [000 001]1198604 (inner race fault) [100 100] [049 055] [006 010] [020 024] [000 000] [002 003] [003 005] [003 006]

HL

H HL

H H H HL L L L

L

Vibration signal x(t)

x10

x20

x30 x31 x32 x33 x34 x35 x36 x37

x21 x22 x23

x11

Figure 7 Diagram of 3-level wavelet packet decomposition (L low-pass filter H high-pass filter)

Furthermore assume that 1198643max is the maximum value ofthe energy eigenvalue in the 3-level subfrequency band andthen the eigenvalues can be normalized as follows

119864lowast1198963 = 11986411989631198643max 119896 = 0 1 7 (11)

By the above normalization the energy eigenvalue of thewavelet packet energy of vibration signals was bounded to

[0 1] and then the normalized eigenvector can be describedas follows

119879lowast = 119864lowast03 119864lowast13 119864lowast23 119864lowast33 119864lowast43 119864lowast53 119864lowast63 119864lowast73 (12)

The normalized energy eigenvalues of vibration signalsare shown in Figure 8

For diffident type faults of bearing the eigenvalue ofthe wavelet packet energy has the distinguishing distributionat the individual subfrequency band According to a lot ofexperimentation and data comparison we extract the lowerbound and upper bound of the energy eigenvalues for typicalfaults of bearing and establish the energy interval ranges asshown in Table 3 and the energy interval ranges can be usedto diagnose fault types of rolling bearings in the next step

4 Fault Diagnosis of Rolling BearingBased on SNSs

In this section we briefly introduce basic concepts of simpli-fied neutrosophic sets (SNSs) and propose a new correlationcoefficient of two SNSs which will be needed in the followinganalysisThen we establish the fault SNSs of bearings accord-ing to energy features Finally we present the method forfault diagnosis of rolling bearing according to the correlationcoefficient of SNSs

41 Simplified Neutrosophic Sets (SNSs)

Definition 1 (see [19]) Let119880 be a universe of discourse thenthe neutrosophic set (NS) 119860 is defined by

119860 = ⟨119909 119879119860 (119909) 119868119860 (119909) 119865119860 (119909)⟩ 119909 isin 119880 (13)


1 2 3 4 5 6 7 80

01

02

03

04

05

06

07

08

09

1N

orm

aliz

ed en

ergy

eige

nval

ue

Subfrequency band

(a) Healthy rolling bearing

1 2 3 4 5 6 7 80

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

ener

gy ei

genv

alue

Subfrequency band

(b) Rolling bearing in outer race fault

1 2 3 4 5 6 7 80

01

02

03

04

05

06

07

08

09

1

Subfrequency band

Nor

mal

ized

ener

gy ei

genv

alue

(c) Rolling bearing in inner race fault

1 2 3 4 5 6 7 80

01

02

03

04

05

06

07

08

09

1

Subfrequency band

Nor

mal

ized

ener

gy ei

genv

alue

(d) Rolling bearing in ball fault

Figure 8 Energy histogram of rolling bearing

where the functions 119879119860(119909) 119868119860(119909) and 119865119860(119909) represent atruth-membership function an indeterminacy-membershipfunction and a falsity-membership function of the element119909 isin 119880 to the set 119860 respectively with the conditions119879119860(119909) 119868119860(119909) 119865119860(119909) 119880 rarr]minus0 1+[ and minus0 le sup119879119860(119909) +sup 119868119860(119909) + sup119865119860(119909119894) le 3+

The above concept of a neutrosophic set (NS) is presentedfrom philosophical point of view and it takes the value fromreal standard or nonstandard subsets of ]minus0 1+[ It will bedifficult to apply ]minus0 1+[ in scientific and engineering areasFor the real applications a simplified neutrosophic set (SNS)is introduced by Ye [20] as the following definition

Definition 2 (see [20]) Let 119880 be a space of points (objects)with generic elements in 119880 denoted by 119909 A simplifiedneutrosophic set (SNS) 119860 in 119880 is characterized by a truth-membership function 119879119860(119909) an indeterminacy-membershipfunction 119868119860(119909) and a falsity-membership function119865119860(119909) Foreach point 119909 in 119880 119879119860(119909) 119868119860(119909) and 119865119860(119909) are singleton

subintervalssubsets in the real standard [0 1] such that119879119860(119909) 119868119860(119909) 119865119860(119909) isin [0 1] Then a simplified neutrosophicset (SNS) is denoted by

119860 = ⟨119909 119879119860 (119909) 119868119860 (119909) 119865119860 (119909)⟩ | 119909 isin 119880 (14)

Obviously a simplified neutrosophic set (SNS) is a sub-class of the neutrosophic set (NS) and satisfies the conditions119879119860(119909) 119868119860(119909) 119865119860(119909) 119880 rarr [0 1] and 0 le 119879119860(119909) + 119868119860(119909) +119865119860(119909) le 342 Correlation Coefficient for SNSs Correlation coefficientis an important tool for determining the correlation degreebetween fuzzy sets Therefore a new correlation coefficientof two SNSs is proposed by the following definition

Definition 3 Assume that there are two SNSs 119860 = ⟨119909119894119879119860(119909119894) 119868119860(119909119894) 119865119860(119909119894)⟩ | 119909119894 isin 119880 and 119861 = ⟨119909119894 119879119861(119909119894) 119868119861(119909119894)119865119861(119909119894)⟩ | 119909119894 isin 119880 in the universe of discourse 119880 = 1199091


1199092 119909119899 119909119894 isin 119880 A correlation coefficient between SNSs isdefined as follows

119872SNS (119860 119861) = 1119899119899sum119894=1

min [119879119860 (119909119894) 119879119861 (119909119894)] +min [119868119860 (119909119894) 119868119861 (119909119894)] +min [119865119860 (119909119894) 119865119861 (119909119894)]radic119879119860 (119909119894) 119879119861 (119909119894) + radic119868119860 (119909119894) 119868119861 (119909119894) + radic119865119860 (119909119894) 119865119861 (119909119894)

(15)

where the symbol ldquominrdquo is the minimum operation

According to the above definition the correlation coeffi-cient of SNSs 119860 and 119861 satisfies the following properties

(P1) 0 le 119872SNS(119860 119861) le 1

(P2) 119872SNS(119860 119861) = 119872SNS(119861 119860)(P3) 119872SNS(119860 119861) = 1 if and only if 119860 = 119861If we consider the weights of 119909119894 a weighted correlation

coefficient between SNSs 119860 and 119861 is proposed as follows

119872SNS (119860 119861) = 119899sum119894=1

119908119894min [119879119860 (119909119894) 119879119861 (119909119894)] +min [119868119860 (119909119894) 119868119861 (119909119894)] +min [119865119860 (119909119894) 119865119861 (119909119894)]radic119879119860 (119909119894) 119879119861 (119909119894) + radic119868119860 (119909119894) 119868119861 (119909119894) + radic119865119860 (119909119894) 119865119861 (119909119894)

(16)

where 119908119894 isin [0 1] and sum119899119894=1 119908119894 = 1 for 119894 = 1 2 11989943 Bearings Neutrosophic SetsModels Based on Energy Eigen-vectors The SNSs models of rolling bearings can be builtaccording to the energy intervals of the eight subfrequencybands as shown in Table 3

Assume that a set of bearing faults is 119860 = 1198601 (healthy)1198602 (outer race fault) 1198603 (ball fault) 1198604 (inner race fault)and a set of energy eigenvector is 119864 = 1198901(119864lowast03 ) 1198902(119864lowast13 )1198903(119864lowast23 ) 1198904(119864lowast33 ) 1198905(119864lowast43 ) 1198906(119864lowast53 ) 1198907(119864lowast63 ) 1198908(119864lowast73 ) InTable 3let 119879119860119896(119890119894) and 119880119860119896(119890119894) (119896 = 1 2 3 4 119894 = 1 2 8) bethe lower bound and upper bound of the characteristic value119890119894 for 119860119896 respectively then the characteristic intervals ofrolling bearing can be represented by

119860119896 = (1198901 [119879119860119896 (1199091) 119880119860119896 (1198901)]) (1198902 [119879119860119896 (1198902) 119880119860119896 (1198902)]) (1198903 [119879119860119896 (1198903) 119880119860119896 (1198903)]) (1198904 [119879119860119896 (1198904) 119880119860119896 (1198904)]) (1198905 [119879119860119896 (1198905) 119880119860119896 (1198905)]) (1198906 [119879119860119896 (1198906) 119880119860119896 (1198906)]) (1198907 [119879119860119896 (1198907) 119880119860119896 (1198907)]) (1198908 [119879119860119896 (1198908) 119880119860119896 (1198908)]) (119896 = 1 2 3 4)

(17)

Let 119880119860119896(119890119894) = 1 minus 119865119860119896(119890119894) and 119868119860119896(119890119894) = 119880119860119896(119890119894) minus 119879119860119896(119890119894)for 119896 = 1 4 and 119894 = 1 8 If 119880119860119896(119890119894) minus 119879119860119896(119890119894) le 001then let 119868119860119896(119890119894) = 001 In this case the sets 119860119896 can beextended to simplified neutrosophic sets (SNSs) and can berewritten as

119860119896 = ⟨1198901 119879119860119896 (1198901) 119868119860119896 (1198901) 119865119860119896 (1198901)⟩ ⟨1198902 119879119860119896 (1198902) 119868119860119896 (1198902) 119865119860119896 (1198902)⟩ ⟨1198903 119879119860119896 (1198903) 119868119860119896 (1198903) 119865119860119896 (1198903)⟩

⟨1198904 119879119860119896 (1198904) 119868119860119896 (1198904) 119865119860119896 (1198904)⟩ ⟨1198905 119879119860119896 (1198905) 119868119860119896 (1198905) 119865119860119896 (1198905)⟩ ⟨1198906 119879119860119896 (1198906) 119868119860119896 (1198906) 119865119860119896 (1198906)⟩ ⟨1198907 119879119860119896 (1198907) 119868119860119896 (1198907) 119865119860119896 (1198907)⟩ ⟨1198908 119879119860119896 (1198908) 119868119860119896 (1198908) 119865119860119896 (1198908)⟩

(18)

where 119879119860119896(119890119894) 119880 rarr [0 1] 119868119860119896(119890119894) 119880 rarr [0 1] 119865119860119896(119890119894) 119880 rarr[0 1] and 0 ge 119879119860119896(119890119894) + 119868119860119896(119890119894) + 119865119860119896(119890119894) le 3 for 119896 = 1 4and 119894 = 1 8

According to the definition of neutrosophic sets thenumbers 119879119860119896(119890119894) 119868119860119896(119890119894) and 119865119860119896(119890119894) represent a truth-membership an indeterminacy-membership and a falsity-membership respectively The neutrosophic sets of bearingfault types are shown in Table 4 Here 1198601 1198602 1198603 and 1198604are healthy outer race fault ball fault and inner race faultbearings respectively

44 Rolling Bearing Fault Diagnosis Using Correlation Coeffi-cient In this section we apply the correlation coefficient ofSNSs to diagnose rolling bearing faults Assume that 119860119896 (119896 =1 2 3 4) are SNSs models of rolling bearing faults and 119860 119905 isa testing rolling bearing signal expressed by a SNS Then wecan calculate the correlation coefficient value 119872SNS(119860119896 119860 119905)(119896 = 1 2 3 4) using (16) Finally the fault-diagnosis orderof the fault-testing sample can be ranked according to thecorrelation coefficient value and the proper diagnosis119860119896lowast forthe bearing fault 119860 119905 is derived by

119896lowast = arg max1le119896le4

119872SNS (119860119896 119860 119905) (19)

This paper considers the same importance of the energyvalues in each frequency band therefore the weights of 119908119894(119894 = 1 2 8) are 119908119894 = 18


Table4En

ergy

values

ofbearingfaulttypes

representedby

theform

ofSN

S

Faulttypes

Energy

ineach

frequ

ency

band

119864lowast0 3119864lowast1 3




119860 1⟨119890 1

0760

14020

⟩⟨119890 20

95005

000⟩

⟨119890 301

1004

085⟩⟨

119890 4064

00702

9⟩⟨119890 5

0040

02094

⟩⟨119890 60

00002

098⟩

⟨119890 700

1005

095⟩⟨

119890 8000

00309

7⟩119860 2

⟨119890 110

0001

000⟩⟨

119890 2024

01506

1 ⟩⟨119890 3

0020

01097

⟩⟨119890 40

13009

078 ⟩

⟨119890 500

0001

100⟩⟨

119890 6000

00109

9 ⟩⟨119890 7

0010

01099

⟩⟨119890 80

01001

099 ⟩

119860 3⟨119890 1

0820

11007

⟩⟨119890 21

00001

000 ⟩

⟨119890 301

1005

084⟩⟨

119890 4065

01002

5 ⟩⟨119890 5

0060

03092

⟩⟨119890 60

00004

096 ⟩

⟨119890 700

2004

094⟩⟨

119890 8000

00110

0 ⟩119860 4

⟨119890 110

0001

000⟩⟨

119890 2049

00704

5 ⟩⟨119890 3

0060

04090

⟩⟨119890 40

20004

076 ⟩

⟨119890 500

0001

100⟩⟨

119890 6002

00109

7 ⟩⟨119890 7

0030

03095

⟩⟨119890 80

03003

094 ⟩


Table 5 Fault diagnosis results based on the correlation coefficient of SNSs and SVM

Fault type Method Test sample Diagnosis result Diagnosis accuracyrate ()Healthy Outer race fault Ball fault Inner race fault

Healthy SNSsSVM

30 2827

23

93390

Outer race fault SNSsSVM 30 28

2822

933933

Ball fault SNSsSVM 30 5

72523

833767

Inner race fault SNSsSVM 30 5

3025

100833

45 Results andDiscussions Todemonstrate the effectivenessof the new diagnosis method we now provide two examplesfor fault diagnosis of bearings Let us consider two testingbearing samples 1198611 and 1198612 described as neutrosophic sets

1198611 = ⟨1198901 100 001 000⟩ ⟨1198902 051 001 049⟩ ⟨1198903 008 001 092⟩ ⟨1198904 024 001 076⟩ ⟨1198905 000 001 100⟩ ⟨1198906 003 001 097⟩ ⟨1198907 005 001 095⟩ ⟨1198908 006 001 094⟩

1198612 = ⟨1198901 100 001 000⟩ ⟨1198902 039 001 061⟩ ⟨1198903 003 001 097⟩ ⟨1198904 022 001 078⟩ ⟨1198905 000 001 100⟩ ⟨1198906 001 001 100⟩ ⟨1198907 001 001 100⟩ ⟨1198908 001 001 100⟩

(20)

The correlation coefficient values between SNSs 119861119895 (119895 =1 2) and 119860119896 (119896 = 1 2 3 4) can calculated by (17) as follows

119872SNS (1198601 1198611) = 08787119872SNS (1198602 1198611) = 09483119872SNS (1198603 1198611) = 08746119872SNS (1198604 1198611) = 09819119872SNS (1198601 1198612) = 08587119872SNS (1198602 1198612) = 09714119872SNS (1198603 1198612) = 08566119872SNS (1198604 1198612) = 09590

(21)

For the fault-testing sample 1198611 119872SNS(1198604 1198611) is themaximum correlation coefficient and 119872SNS(1198602 1198611) is thesecond correlation coefficient According to the principle ofcorrelation coefficient the fault-diagnosis order is as follows

1198604 997888rarr 1198602 997888rarr 1198601 997888rarr 1198603 (22)

Therefore we can determine that the testing bearing isan inner race fault bearing By actual observing the testing

bearing inner race was covered with scratches and thereforethe diagnosis result is correct

Similarly for the fault-testing sample 1198612 the fault-diagnosis order is as follows

1198602 997888rarr 1198604 997888rarr 1198601 997888rarr 1198603 (23)

By actual checking the fault of the bearing firstly resultedfrom damage of outer race and then inner race So thediagnosis results are consistent with the actual situation

In the experiment 120 rolling bearings were used fortesting samples In order to verify the effectiveness of thefault-diagnosis method proposed in this paper we extractedthe energy eigenvalues of bearing vibration signals firstly andthen diagnosed the bearing faults using the correlation coef-ficient of SNSs and the support vector machine (SVM) [25]respectivelyThe fault-diagnosis results of rolling bearings areshown in Table 5 By comparing the diagnosis results shownin Table 5 it is clear that the diagnosis accuracy rate based onthe correlation coefficient of SNSs is much higher than theaccuracy rate based on SVM

For further comparison Table 6 lists the diagnosis resultsbased on the correlation coefficient of SNSs SVM BP andGA-BP [28] methods respectively Obviously the methodbased on the correlation coefficient of SNSs can achieve theaverage accuracy rate of 925 and it is higher than the onesbased on the other methods

The above comparisons demonstrate that the proposedmethod in this paper is effective in the bearing fault diagnosis

5 Conclusion

To diagnose rolling bearing faults a new fault-diagnosismethod was developed by combining correlation coefficientof SNSs with wavelet packet decomposition A series ofexperiments were conducted to diagnose rolling bearingfaults and the experimental results demonstrated that theproposed method can effectively identify the bearing faultsFor the novel fault-diagnosis method there exist two keyissues (1) extracting useful fault features by wavelet packetdecomposition (2) building the accurate SNSs models ofbearing faults In the future work the two issues will befurther improved based on the analysis of a large amount of


Table 6 Fault diagnosis results based on the correlation coefficient of SNSs SVM BP and GA-BP

Diagnosis method Diagnosis accuracy rate () Average accuracy ()Healthy Outer race fault Ball fault Inner race fault

SNS 933 933 833 100 925

SVM 90 933 767 833 858

GA-BP 100 80 70 90 85

BP 90 70 50 90 75

experimental data since they will influence the accuracy offault-diagnosis results

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] A S Raj and N Murali ldquoA novel application of Lucy-Richardson deconvolution bearing fault diagnosisrdquo Journal ofVibration and Control vol 21 no 6 pp 1055ndash1067 2015

[2] J Lin and L Qu ldquoFeature extraction based on morlet waveletand its application for mechanical fault diagnosisrdquo Journal ofSound and Vibration vol 234 no 1 pp 135ndash148 2000

[3] P K Kankar S C Sharma and S P Harsha ldquoFault diagnosis ofball bearings using machine learning methodsrdquo Expert Systemswith Applications vol 38 no 3 pp 1876ndash1886 2011

[4] D H Pandya S H Upadhyay and S P Harsha ldquoFault diagnosisof rolling element bearing by using multinomial logistic regres-sion and wavelet packet transformrdquo Soft Computing vol 18 no2 pp 255ndash266 2014

[5] R Yan M Shan J Cui and Y Wu ldquoMutual information-assisted wavelet function selection for enhanced rolling bearingfault diagnosisrdquo Shock and Vibration vol 2015 Article ID794921 9 pages 2015

[6] P H Nguyen and J-M Kim ldquoMultifault diagnosis of rolling ele-ment bearings using a wavelet kurtogram and vector median-based feature analysisrdquo Shock and Vibration vol 2015 ArticleID 320508 14 pages 2015

[7] C Rajeswari B Sathiyabhama S Devendiran andKManivan-nan ldquoBearing fault diagnosis using wavelet packet transformhybrid PSO and support vector machinerdquo Procedia Engineeringvol 97 pp 1772ndash1783 2014

[8] B Vishwash P S Pai N Sriram R Ahmed H Kumar and GVijay ldquoMultiscale slope feature extraction for gear and bearingfault diagnosis using wavelet transformrdquo Procedia MaterialsScience vol 5 pp 1650ndash1659 2014

[9] L-Y Zhao L Wang and R-Q Yan ldquoRolling bearing faultdiagnosis based on wavelet packet decomposition and multi-scale permutation entropyrdquo Entropy vol 17 no 9 pp 6447ndash6461 2015

[10] P Jayaswal S N Verma and A KWadhwani ldquoDevelopment ofEBP-Artificial neural network expert system for rolling elementbearing fault diagnosisrdquo Journal of Vibration and Control vol17 no 8 pp 1131ndash1148 2011

[11] L-L Jiang H-K Yin X-J Li and S-W Tang ldquoFault diagnosisof rotatingmachinery based onmultisensor information fusion

using SVMand time-domain featuresrdquo Shock andVibration vol2014 Article ID 418178 8 pages 2014

[12] A Moosavian H Ahmadi A Tabatabaeefar and M Khaz-aee ldquoComparison of two classifiers K-nearest neighbor andartificial neural network for fault diagnosis on a main enginejournal-bearingrdquo Shock and Vibration vol 20 no 2 pp 263ndash272 2013

[13] V Sugumaran and K I Ramachandran ldquoFault diagnosis ofroller bearing using fuzzy classifier and histogram featureswith focus on automatic rule learningrdquo Expert Systems withApplications vol 38 no 5 pp 4901ndash4907 2011

[14] W Y Liu and J G Han ldquoRolling element bearing faultrecognition approach based on fuzzy clustering bispectrumestimationrdquo Shock and Vibration vol 20 no 2 pp 213ndash2252013

[15] F Harrouche and A Felkaoui ldquoAutomation of fault diagnosisof bearing by application of fuzzy inference system (FIS)rdquoMechanics and Industry vol 15 no 6 pp 477ndash485 2014

[16] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[17] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[18] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

[19] F Smarandache A Unifying Field in Logics Neutrosophy Neu-trophic Probability Set and Logic American Research PressRehoboth Mass USA 1999

[20] J Ye ldquoVector similarity measures of simplified neutrosophicsets and their application in multicriteria decision makingrdquoInternational Journal of Fuzzy Systems vol 16 no 2 pp 204ndash211 2014

[21] J Ye ldquoImproved cosine similarity measures of simplified neu-trosophic sets for medical diagnosesrdquo Artificial Intelligence inMedicine vol 63 no 3 pp 171ndash179 2015

[22] A Q Ansari R Biswas and S Aggarwal ldquoProposal for appli-cability of neutrosophic set theory in medical AIrdquo InternationalJournal of Computer Applications vol 27 no 5 pp 5ndash11 2011

[23] J Ye ldquoMulticriteria decision-making method using the cor-relation coefficient under single-valued neutrosophic environ-mentrdquo International Journal of General Systems vol 42 no 4pp 386ndash394 2013

[24] Y Guo andH D Cheng ldquoNew neutrosophic approach to imagesegmentationrdquo Pattern Recognition vol 42 no 5 pp 587ndash5952009

[25] Y P Sang Research on production test and intelligent diagnosisof rolling element bearing [MS thesis] China Jiliang University2015


[26] R Li P Sopon and D He ldquoFault features extraction for bearingprognosticsrdquo Journal of Intelligent Manufacturing vol 23 no 2pp 313ndash321 2012

[27] Y Q Wang X P Wang and L C Wang ldquoPitch detectionmethod based on morphological filteringrdquo Applied Mechanicsand Materials vol 596 pp 433ndash436 2014

[28] B S Huang W Xu and X F Zhou ldquoRolling bearing diagnosisbased on LMD and neural networkrdquo International Journal ofComputer Science vol 10 no 3 pp 304ndash309 2013


show that fuzzy theory is an effective tool to diagnose bearingfaults

Fuzzy sets (FSs) theory was proposed by Zadeh (1965)for handling uncertain information using single membershipdegree function [16] The fuzzy sets were extended to intu-itionistic fuzzy sets (IFSs) [17] and interval valued intuitionis-tic fuzzy sets (IVIFSs) [18] by usingmembership degree func-tion non-membership degree function and degree functionof hesitation simultaneously FSs IFSs and IVIFSs have beenwidely applied in various fields However FSs IFSs andIVIFSs cannot deal with some types of uncertainties such asthe indeterminate information and inconsistent informationin real physical problems Furthermore Smarandache [19]proposed neutrosophy theory from philosophical point ofview Neutrosophic sets (NSs) are characterized by a truth-membership function an indeterminacy-membership func-tion and a falsity-membership functionThe functions ofNSstake the value from real standard or nonstandard subsets of]minus0 1+[ [19] andNSs are difficult to be applied in engineeringareas For the real engineering applications neutrosophicsets (NSs) can be described as simplified neutrosophic sets(SNSs) [20] with the normal standard real unit interval[0 1] One major advantage of SNSs is the ability to performanalysis problems involving imprecise undetermined andinconsistent data Recently SNSs have been applied in manydifferent fuzzy problems such as medical diagnosis problems[21 22] decision making problems [20 23] and imageprocessing [24]

For vibrational fault diagnosis of rolling bearing thereis no direct accurate and quantitative relationship betweenfault vibration characteristics and fault types Therefore thefault-diagnosis process has certain vagueness This papermainly focuses on the fault diagnosis of rolling bearingsbased on vibration signals and SNSs In this work a morpho-logical filter and wavelet packet decomposition are appliedto preprocess the original vibration signals and the SNSsof each fault type will be established according to energyeigenvectors The fault types will be diagnosed using a newcorrelation coefficient of SNSs

The rest of the paper is organized as follows Section 2gives the experimental system Section 3 gives the datapreprocessing techniques including morphological opening-closing operation and wavelet packet decomposition InSection 4 some basic concepts of SNSs and a new correlationcoefficient are introduced firstly and then the fault-diagnosismethod is presented based on SNSs Conclusions of this workare summarized in Section 5

2 Experimental Setup

This study was carried out with the experimental apparatusshown in Figure 1 The principal axis is driven by an ACmotor and the vibration signals of bearings are acquired byan acceleration sensor and a data acquiring card NI USB-6251The vibration signals will be processed using a computerand displayed by an oscilloscope Some vibration signals areacquired by the experimental device of Jiliang University inChina [25] shown in Figure 2 In this experiment the type

(1)

(2)(3)(4)

(5)

(6)

(7)

(8)

(9)

Figure 1 Diagram of experimental system (1)Motor (2) driver (3)principal axis (4) bearing (5) core axes (6) acceleration sensor (7)acquisition data card (8) oscilloscope and (9) computer

Figure 2 Experimental device

Table 1 Bearing parameters

Parameters Values (mm)Outer race diameter 35Inner race diameter 15Ball diameter d 75Thickness 11Number of balls 7Pitch diameter D 25Contact angle 120572 0

of bearings is NSK 6202 deep groove ball bearing whosespecifications are listed in Table 1

In order to diagnose the fault of bearings four types ofbearings are used healthy outer race fault inner race faultand ball fault bearingsThe core axis is driven at the rotationalspeed of 25Hz NI Labview Signal Express will be applied fordata acquisition with 10KHz sampling frequency and 02 ssample time

When a fault exists in a bearing vibration impulses willhappen at a specific frequency Theoretically when a bearing









types of bearings





119865119868 = 1198731198612 (1 + 119889119863 sdot cos120572) sdot 119865119903

119865119861 = 119863119889 [1 minus ( 119889

119863 sdot cos120572)2] sdot 119865119903

(1)








Dilation

119909 oplus 119910 = max119898=01119872minus1

119909 (119899 + 119898) + 119910 (119898) (119899 = 0 1 119873 + 119872)

(2)

Erosion

119909 ⊖ 119910 = min119898=01119872minus1

119909 (119899 + 119898) minus 119910 (119898) (119899 = 0 1 119873 + 119872) (3)

Opening

119909 ∘ 119910 = (119909 ⊖ 119910 oplus 119910) (4)

Closing

119909 ∙ 119910 = (119909 oplus 119910 ⊖ 119910) (5)


119865oc = (119909 ∘ 119910 ∙ 119910) (6)





0 005 01 015 02minus1

0

1

Time (s)

Am

plitu

de (m

s2)


0 500 1000 1500 20000

100200300400500600700

Frequency (Hz)

Am

plitu

de (m

s2) FO

2 lowast FO



0 005 01 015 02minus05

0

05

Time (s)

Am

plitu

de (m

s2)


0 200 400 600 800 10000

100

200

300

Frequency (Hz)

Am

plitu

de (m

s2)

FI

2 lowast FI





1199091198952119896 = sum119899

119871 (119899) sdot 119909119895minus11198961199091198952119896+1 = sum

119899

119867(119899) sdot 119909119895minus1119896(7)




119909 (119905) = 23minus1sum119896=0

1199093119896 (119905) 119896 = 0 1 7 (8)





1198641198963 = int 10038161003816100381610038161199093119896 (119905)10038161003816100381610038162 119889119905 =119873sum119894=0

10038161003816100381610038161199093119896 (119894)10038161003816100381610038162 119896 = 0 1 7 (9)



119879 = 11986403 11986413 11986423 11986433 11986443 11986453 11986463 11986473 (10)


0 005 01 015 02minus1

0

1

Time (s)

Am

plitu

de (m

s2)


0 200 400 600 800 10000

100

200

300

400

500

Frequency (Hz)

Am

plitu

de (m

s2)

FB

2 lowast FB






HL

H HL

H H H HL L L L

L


x10

x20

x30 x31 x32 x33 x34 x35 x36 x37

x21 x22 x23

x11



119864lowast1198963 = 11986411989631198643max 119896 = 0 1 7 (11)










119860 = ⟨119909 119879119860 (119909) 119868119860 (119909) 119865119860 (119909)⟩ 119909 isin 119880 (13)


1 2 3 4 5 6 7 80

01

02

03

04

05

06

07

08

09

1N

orm

aliz

ed en

ergy

eige

nval

ue

Subfrequency band


1 2 3 4 5 6 7 80

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

ener

gy ei

genv

alue

Subfrequency band


1 2 3 4 5 6 7 80

01

02

03

04

05

06

07

08

09

1

Subfrequency band

Nor

mal

ized

ener

gy ei

genv

alue


1 2 3 4 5 6 7 80

01

02

03

04

05

06

07

08

09

1

Subfrequency band

Nor

mal

ized

ener

gy ei

genv

alue







119860 = ⟨119909 119879119860 (119909) 119868119860 (119909) 119865119860 (119909)⟩ | 119909 isin 119880 (14)





119872SNS (119860 119861) = 1119899119899sum119894=1


(15)



(P1) 0 le 119872SNS(119860 119861) le 1



119872SNS (119860 119861) = 119899sum119894=1


(16)



119860119896 = (1198901 [119879119860119896 (1199091) 119880119860119896 (1198901)]) (1198902 [119879119860119896 (1198902) 119880119860119896 (1198902)]) (1198903 [119879119860119896 (1198903) 119880119860119896 (1198903)]) (1198904 [119879119860119896 (1198904) 119880119860119896 (1198904)]) (1198905 [119879119860119896 (1198905) 119880119860119896 (1198905)]) (1198906 [119879119860119896 (1198906) 119880119860119896 (1198906)]) (1198907 [119879119860119896 (1198907) 119880119860119896 (1198907)]) (1198908 [119879119860119896 (1198908) 119880119860119896 (1198908)]) (119896 = 1 2 3 4)

(17)


119860119896 = ⟨1198901 119879119860119896 (1198901) 119868119860119896 (1198901) 119865119860119896 (1198901)⟩ ⟨1198902 119879119860119896 (1198902) 119868119860119896 (1198902) 119865119860119896 (1198902)⟩ ⟨1198903 119879119860119896 (1198903) 119868119860119896 (1198903) 119865119860119896 (1198903)⟩

⟨1198904 119879119860119896 (1198904) 119868119860119896 (1198904) 119865119860119896 (1198904)⟩ ⟨1198905 119879119860119896 (1198905) 119868119860119896 (1198905) 119865119860119896 (1198905)⟩ ⟨1198906 119879119860119896 (1198906) 119868119860119896 (1198906) 119865119860119896 (1198906)⟩ ⟨1198907 119879119860119896 (1198907) 119868119860119896 (1198907) 119865119860119896 (1198907)⟩ ⟨1198908 119879119860119896 (1198908) 119868119860119896 (1198908) 119865119860119896 (1198908)⟩

(18)





119872SNS (119860119896 119860 119905) (19)



Table4En

ergy

values

ofbearingfaulttypes

representedby

theform

ofSN

S

Faulttypes

Energy

ineach

frequ

ency

band





119860 1⟨119890 1

0760

14020

⟩⟨119890 20

95005

000⟩

⟨119890 301

1004

085⟩⟨

119890 4064

00702

9⟩⟨119890 5

0040

02094

⟩⟨119890 60

00002

098⟩

⟨119890 700

1005

095⟩⟨

119890 8000

00309

7⟩119860 2

⟨119890 110

0001

000⟩⟨

119890 2024

01506

1 ⟩⟨119890 3

0020

01097

⟩⟨119890 40

13009

078 ⟩

⟨119890 500

0001

100⟩⟨

119890 6000

00109

9 ⟩⟨119890 7

0010

01099

⟩⟨119890 80

01001

099 ⟩

119860 3⟨119890 1

0820

11007

⟩⟨119890 21

00001

000 ⟩

⟨119890 301

1005

084⟩⟨

119890 4065

01002

5 ⟩⟨119890 5

0060

03092

⟩⟨119890 60

00004

096 ⟩

⟨119890 700

2004

094⟩⟨

119890 8000

00110

0 ⟩119860 4

⟨119890 110

0001

000⟩⟨

119890 2049

00704

5 ⟩⟨119890 3

0060

04090

⟩⟨119890 40

20004

076 ⟩

⟨119890 500

0001

100⟩⟨

119890 6002

00109

7 ⟩⟨119890 7

0030

03095

⟩⟨119890 80

03003

094 ⟩




Healthy SNSsSVM

30 2827

23

93390


2822

933933


72523

833767


3025

100833


1198611 = ⟨1198901 100 001 000⟩ ⟨1198902 051 001 049⟩ ⟨1198903 008 001 092⟩ ⟨1198904 024 001 076⟩ ⟨1198905 000 001 100⟩ ⟨1198906 003 001 097⟩ ⟨1198907 005 001 095⟩ ⟨1198908 006 001 094⟩

1198612 = ⟨1198901 100 001 000⟩ ⟨1198902 039 001 061⟩ ⟨1198903 003 001 097⟩ ⟨1198904 022 001 078⟩ ⟨1198905 000 001 100⟩ ⟨1198906 001 001 100⟩ ⟨1198907 001 001 100⟩ ⟨1198908 001 001 100⟩

(20)



(21)


1198604 997888rarr 1198602 997888rarr 1198601 997888rarr 1198603 (22)




1198602 997888rarr 1198604 997888rarr 1198601 997888rarr 1198603 (23)





5 Conclusion





SNS 933 933 833 100 925

SVM 90 933 767 833 858

GA-BP 100 80 70 90 85

BP 90 70 50 90 75


Competing Interests


References







































types of bearings





119865119868 = 1198731198612 (1 + 119889119863 sdot cos120572) sdot 119865119903

119865119861 = 119863119889 [1 minus ( 119889

119863 sdot cos120572)2] sdot 119865119903

(1)








Dilation

119909 oplus 119910 = max119898=01119872minus1

119909 (119899 + 119898) + 119910 (119898) (119899 = 0 1 119873 + 119872)

(2)

Erosion

119909 ⊖ 119910 = min119898=01119872minus1

119909 (119899 + 119898) minus 119910 (119898) (119899 = 0 1 119873 + 119872) (3)

Opening

119909 ∘ 119910 = (119909 ⊖ 119910 oplus 119910) (4)

Closing

119909 ∙ 119910 = (119909 oplus 119910 ⊖ 119910) (5)


119865oc = (119909 ∘ 119910 ∙ 119910) (6)





0 005 01 015 02minus1

0

1

Time (s)

Am

plitu

de (m

s2)


0 500 1000 1500 20000

100200300400500600700

Frequency (Hz)

Am

plitu

de (m

s2) FO

2 lowast FO



0 005 01 015 02minus05

0

05

Time (s)

Am

plitu

de (m

s2)


0 200 400 600 800 10000

100

200

300

Frequency (Hz)

Am

plitu

de (m

s2)

FI

2 lowast FI





1199091198952119896 = sum119899

119871 (119899) sdot 119909119895minus11198961199091198952119896+1 = sum

119899

119867(119899) sdot 119909119895minus1119896(7)




119909 (119905) = 23minus1sum119896=0

1199093119896 (119905) 119896 = 0 1 7 (8)





1198641198963 = int 10038161003816100381610038161199093119896 (119905)10038161003816100381610038162 119889119905 =119873sum119894=0

10038161003816100381610038161199093119896 (119894)10038161003816100381610038162 119896 = 0 1 7 (9)



119879 = 11986403 11986413 11986423 11986433 11986443 11986453 11986463 11986473 (10)


0 005 01 015 02minus1

0

1

Time (s)

Am

plitu

de (m

s2)


0 200 400 600 800 10000

100

200

300

400

500

Frequency (Hz)

Am

plitu

de (m

s2)

FB

2 lowast FB






HL

H HL

H H H HL L L L

L


x10

x20

x30 x31 x32 x33 x34 x35 x36 x37

x21 x22 x23

x11



119864lowast1198963 = 11986411989631198643max 119896 = 0 1 7 (11)










119860 = ⟨119909 119879119860 (119909) 119868119860 (119909) 119865119860 (119909)⟩ 119909 isin 119880 (13)


1 2 3 4 5 6 7 80

01

02

03

04

05

06

07

08

09

1N

orm

aliz

ed en

ergy

eige

nval

ue

Subfrequency band


1 2 3 4 5 6 7 80

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

ener

gy ei

genv

alue

Subfrequency band


1 2 3 4 5 6 7 80

01

02

03

04

05

06

07

08

09

1

Subfrequency band

Nor

mal

ized

ener

gy ei

genv

alue


1 2 3 4 5 6 7 80

01

02

03

04

05

06

07

08

09

1

Subfrequency band

Nor

mal

ized

ener

gy ei

genv

alue







119860 = ⟨119909 119879119860 (119909) 119868119860 (119909) 119865119860 (119909)⟩ | 119909 isin 119880 (14)





119872SNS (119860 119861) = 1119899119899sum119894=1


(15)



(P1) 0 le 119872SNS(119860 119861) le 1



119872SNS (119860 119861) = 119899sum119894=1


(16)



119860119896 = (1198901 [119879119860119896 (1199091) 119880119860119896 (1198901)]) (1198902 [119879119860119896 (1198902) 119880119860119896 (1198902)]) (1198903 [119879119860119896 (1198903) 119880119860119896 (1198903)]) (1198904 [119879119860119896 (1198904) 119880119860119896 (1198904)]) (1198905 [119879119860119896 (1198905) 119880119860119896 (1198905)]) (1198906 [119879119860119896 (1198906) 119880119860119896 (1198906)]) (1198907 [119879119860119896 (1198907) 119880119860119896 (1198907)]) (1198908 [119879119860119896 (1198908) 119880119860119896 (1198908)]) (119896 = 1 2 3 4)

(17)


119860119896 = ⟨1198901 119879119860119896 (1198901) 119868119860119896 (1198901) 119865119860119896 (1198901)⟩ ⟨1198902 119879119860119896 (1198902) 119868119860119896 (1198902) 119865119860119896 (1198902)⟩ ⟨1198903 119879119860119896 (1198903) 119868119860119896 (1198903) 119865119860119896 (1198903)⟩

⟨1198904 119879119860119896 (1198904) 119868119860119896 (1198904) 119865119860119896 (1198904)⟩ ⟨1198905 119879119860119896 (1198905) 119868119860119896 (1198905) 119865119860119896 (1198905)⟩ ⟨1198906 119879119860119896 (1198906) 119868119860119896 (1198906) 119865119860119896 (1198906)⟩ ⟨1198907 119879119860119896 (1198907) 119868119860119896 (1198907) 119865119860119896 (1198907)⟩ ⟨1198908 119879119860119896 (1198908) 119868119860119896 (1198908) 119865119860119896 (1198908)⟩

(18)





119872SNS (119860119896 119860 119905) (19)



Table4En

ergy

values

ofbearingfaulttypes

representedby

theform

ofSN

S

Faulttypes

Energy

ineach

frequ

ency

band





119860 1⟨119890 1

0760

14020

⟩⟨119890 20

95005

000⟩

⟨119890 301

1004

085⟩⟨

119890 4064

00702

9⟩⟨119890 5

0040

02094

⟩⟨119890 60

00002

098⟩

⟨119890 700

1005

095⟩⟨

119890 8000

00309

7⟩119860 2

⟨119890 110

0001

000⟩⟨

119890 2024

01506

1 ⟩⟨119890 3

0020

01097

⟩⟨119890 40

13009

078 ⟩

⟨119890 500

0001

100⟩⟨

119890 6000

00109

9 ⟩⟨119890 7

0010

01099

⟩⟨119890 80

01001

099 ⟩

119860 3⟨119890 1

0820

11007

⟩⟨119890 21

00001

000 ⟩

⟨119890 301

1005

084⟩⟨

119890 4065

01002

5 ⟩⟨119890 5

0060

03092

⟩⟨119890 60

00004

096 ⟩

⟨119890 700

2004

094⟩⟨

119890 8000

00110

0 ⟩119860 4

⟨119890 110

0001

000⟩⟨

119890 2049

00704

5 ⟩⟨119890 3

0060

04090

⟩⟨119890 40

20004

076 ⟩

⟨119890 500

0001

100⟩⟨

119890 6002

00109

7 ⟩⟨119890 7

0030

03095

⟩⟨119890 80

03003

094 ⟩




Healthy SNSsSVM

30 2827

23

93390


2822

933933


72523

833767


3025

100833


1198611 = ⟨1198901 100 001 000⟩ ⟨1198902 051 001 049⟩ ⟨1198903 008 001 092⟩ ⟨1198904 024 001 076⟩ ⟨1198905 000 001 100⟩ ⟨1198906 003 001 097⟩ ⟨1198907 005 001 095⟩ ⟨1198908 006 001 094⟩

1198612 = ⟨1198901 100 001 000⟩ ⟨1198902 039 001 061⟩ ⟨1198903 003 001 097⟩ ⟨1198904 022 001 078⟩ ⟨1198905 000 001 100⟩ ⟨1198906 001 001 100⟩ ⟨1198907 001 001 100⟩ ⟨1198908 001 001 100⟩

(20)



(21)


1198604 997888rarr 1198602 997888rarr 1198601 997888rarr 1198603 (22)




1198602 997888rarr 1198604 997888rarr 1198601 997888rarr 1198603 (23)





5 Conclusion





SNS 933 933 833 100 925

SVM 90 933 767 833 858

GA-BP 100 80 70 90 85

BP 90 70 50 90 75


Competing Interests


References
































0 005 01 015 02minus1

0

1

Time (s)

Am

plitu

de (m

s2)


0 500 1000 1500 20000

100200300400500600700

Frequency (Hz)

Am

plitu

de (m

s2) FO

2 lowast FO



0 005 01 015 02minus05

0

05

Time (s)

Am

plitu

de (m

s2)


0 200 400 600 800 10000

100

200

300

Frequency (Hz)

Am

plitu

de (m

s2)

FI

2 lowast FI





1199091198952119896 = sum119899

119871 (119899) sdot 119909119895minus11198961199091198952119896+1 = sum

119899

119867(119899) sdot 119909119895minus1119896(7)




119909 (119905) = 23minus1sum119896=0

1199093119896 (119905) 119896 = 0 1 7 (8)





1198641198963 = int 10038161003816100381610038161199093119896 (119905)10038161003816100381610038162 119889119905 =119873sum119894=0

10038161003816100381610038161199093119896 (119894)10038161003816100381610038162 119896 = 0 1 7 (9)



119879 = 11986403 11986413 11986423 11986433 11986443 11986453 11986463 11986473 (10)


0 005 01 015 02minus1

0

1

Time (s)

Am

plitu

de (m

s2)


0 200 400 600 800 10000

100

200

300

400

500

Frequency (Hz)

Am

plitu

de (m

s2)

FB

2 lowast FB






HL

H HL

H H H HL L L L

L


x10

x20

x30 x31 x32 x33 x34 x35 x36 x37

x21 x22 x23

x11



119864lowast1198963 = 11986411989631198643max 119896 = 0 1 7 (11)










119860 = ⟨119909 119879119860 (119909) 119868119860 (119909) 119865119860 (119909)⟩ 119909 isin 119880 (13)


1 2 3 4 5 6 7 80

01

02

03

04

05

06

07

08

09

1N

orm

aliz

ed en

ergy

eige

nval

ue

Subfrequency band


1 2 3 4 5 6 7 80

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

ener

gy ei

genv

alue

Subfrequency band


1 2 3 4 5 6 7 80

01

02

03

04

05

06

07

08

09

1

Subfrequency band

Nor

mal

ized

ener

gy ei

genv

alue


1 2 3 4 5 6 7 80

01

02

03

04

05

06

07

08

09

1

Subfrequency band

Nor

mal

ized

ener

gy ei

genv

alue







119860 = ⟨119909 119879119860 (119909) 119868119860 (119909) 119865119860 (119909)⟩ | 119909 isin 119880 (14)





119872SNS (119860 119861) = 1119899119899sum119894=1


(15)



(P1) 0 le 119872SNS(119860 119861) le 1



119872SNS (119860 119861) = 119899sum119894=1


(16)



119860119896 = (1198901 [119879119860119896 (1199091) 119880119860119896 (1198901)]) (1198902 [119879119860119896 (1198902) 119880119860119896 (1198902)]) (1198903 [119879119860119896 (1198903) 119880119860119896 (1198903)]) (1198904 [119879119860119896 (1198904) 119880119860119896 (1198904)]) (1198905 [119879119860119896 (1198905) 119880119860119896 (1198905)]) (1198906 [119879119860119896 (1198906) 119880119860119896 (1198906)]) (1198907 [119879119860119896 (1198907) 119880119860119896 (1198907)]) (1198908 [119879119860119896 (1198908) 119880119860119896 (1198908)]) (119896 = 1 2 3 4)

(17)


119860119896 = ⟨1198901 119879119860119896 (1198901) 119868119860119896 (1198901) 119865119860119896 (1198901)⟩ ⟨1198902 119879119860119896 (1198902) 119868119860119896 (1198902) 119865119860119896 (1198902)⟩ ⟨1198903 119879119860119896 (1198903) 119868119860119896 (1198903) 119865119860119896 (1198903)⟩

⟨1198904 119879119860119896 (1198904) 119868119860119896 (1198904) 119865119860119896 (1198904)⟩ ⟨1198905 119879119860119896 (1198905) 119868119860119896 (1198905) 119865119860119896 (1198905)⟩ ⟨1198906 119879119860119896 (1198906) 119868119860119896 (1198906) 119865119860119896 (1198906)⟩ ⟨1198907 119879119860119896 (1198907) 119868119860119896 (1198907) 119865119860119896 (1198907)⟩ ⟨1198908 119879119860119896 (1198908) 119868119860119896 (1198908) 119865119860119896 (1198908)⟩

(18)





119872SNS (119860119896 119860 119905) (19)



Table4En

ergy

values

ofbearingfaulttypes

representedby

theform

ofSN

S

Faulttypes

Energy

ineach

frequ

ency

band





119860 1⟨119890 1

0760

14020

⟩⟨119890 20

95005

000⟩

⟨119890 301

1004

085⟩⟨

119890 4064

00702

9⟩⟨119890 5

0040

02094

⟩⟨119890 60

00002

098⟩

⟨119890 700

1005

095⟩⟨

119890 8000

00309

7⟩119860 2

⟨119890 110

0001

000⟩⟨

119890 2024

01506

1 ⟩⟨119890 3

0020

01097

⟩⟨119890 40

13009

078 ⟩

⟨119890 500

0001

100⟩⟨

119890 6000

00109

9 ⟩⟨119890 7

0010

01099

⟩⟨119890 80

01001

099 ⟩

119860 3⟨119890 1

0820

11007

⟩⟨119890 21

00001

000 ⟩

⟨119890 301

1005

084⟩⟨

119890 4065

01002

5 ⟩⟨119890 5

0060

03092

⟩⟨119890 60

00004

096 ⟩

⟨119890 700

2004

094⟩⟨

119890 8000

00110

0 ⟩119860 4

⟨119890 110

0001

000⟩⟨

119890 2049

00704

5 ⟩⟨119890 3

0060

04090

⟩⟨119890 40

20004

076 ⟩

⟨119890 500

0001

100⟩⟨

119890 6002

00109

7 ⟩⟨119890 7

0030

03095

⟩⟨119890 80

03003

094 ⟩




Healthy SNSsSVM

30 2827

23

93390


2822

933933


72523

833767


3025

100833


1198611 = ⟨1198901 100 001 000⟩ ⟨1198902 051 001 049⟩ ⟨1198903 008 001 092⟩ ⟨1198904 024 001 076⟩ ⟨1198905 000 001 100⟩ ⟨1198906 003 001 097⟩ ⟨1198907 005 001 095⟩ ⟨1198908 006 001 094⟩

1198612 = ⟨1198901 100 001 000⟩ ⟨1198902 039 001 061⟩ ⟨1198903 003 001 097⟩ ⟨1198904 022 001 078⟩ ⟨1198905 000 001 100⟩ ⟨1198906 001 001 100⟩ ⟨1198907 001 001 100⟩ ⟨1198908 001 001 100⟩

(20)



(21)


1198604 997888rarr 1198602 997888rarr 1198601 997888rarr 1198603 (22)




1198602 997888rarr 1198604 997888rarr 1198601 997888rarr 1198603 (23)





5 Conclusion





SNS 933 933 833 100 925

SVM 90 933 767 833 858

GA-BP 100 80 70 90 85

BP 90 70 50 90 75


Competing Interests


References
































0 005 01 015 02minus1

0

1

Time (s)

Am

plitu

de (m

s2)


0 200 400 600 800 10000

100

200

300

400

500

Frequency (Hz)

Am

plitu

de (m

s2)

FB

2 lowast FB






HL

H HL

H H H HL L L L

L


x10

x20

x30 x31 x32 x33 x34 x35 x36 x37

x21 x22 x23

x11



119864lowast1198963 = 11986411989631198643max 119896 = 0 1 7 (11)










119860 = ⟨119909 119879119860 (119909) 119868119860 (119909) 119865119860 (119909)⟩ 119909 isin 119880 (13)


1 2 3 4 5 6 7 80

01

02

03

04

05

06

07

08

09

1N

orm

aliz

ed en

ergy

eige

nval

ue

Subfrequency band


1 2 3 4 5 6 7 80

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

ener

gy ei

genv

alue

Subfrequency band


1 2 3 4 5 6 7 80

01

02

03

04

05

06

07

08

09

1

Subfrequency band

Nor

mal

ized

ener

gy ei

genv

alue


1 2 3 4 5 6 7 80

01

02

03

04

05

06

07

08

09

1

Subfrequency band

Nor

mal

ized

ener

gy ei

genv

alue







119860 = ⟨119909 119879119860 (119909) 119868119860 (119909) 119865119860 (119909)⟩ | 119909 isin 119880 (14)





119872SNS (119860 119861) = 1119899119899sum119894=1


(15)



(P1) 0 le 119872SNS(119860 119861) le 1



119872SNS (119860 119861) = 119899sum119894=1


(16)



119860119896 = (1198901 [119879119860119896 (1199091) 119880119860119896 (1198901)]) (1198902 [119879119860119896 (1198902) 119880119860119896 (1198902)]) (1198903 [119879119860119896 (1198903) 119880119860119896 (1198903)]) (1198904 [119879119860119896 (1198904) 119880119860119896 (1198904)]) (1198905 [119879119860119896 (1198905) 119880119860119896 (1198905)]) (1198906 [119879119860119896 (1198906) 119880119860119896 (1198906)]) (1198907 [119879119860119896 (1198907) 119880119860119896 (1198907)]) (1198908 [119879119860119896 (1198908) 119880119860119896 (1198908)]) (119896 = 1 2 3 4)

(17)


119860119896 = ⟨1198901 119879119860119896 (1198901) 119868119860119896 (1198901) 119865119860119896 (1198901)⟩ ⟨1198902 119879119860119896 (1198902) 119868119860119896 (1198902) 119865119860119896 (1198902)⟩ ⟨1198903 119879119860119896 (1198903) 119868119860119896 (1198903) 119865119860119896 (1198903)⟩

⟨1198904 119879119860119896 (1198904) 119868119860119896 (1198904) 119865119860119896 (1198904)⟩ ⟨1198905 119879119860119896 (1198905) 119868119860119896 (1198905) 119865119860119896 (1198905)⟩ ⟨1198906 119879119860119896 (1198906) 119868119860119896 (1198906) 119865119860119896 (1198906)⟩ ⟨1198907 119879119860119896 (1198907) 119868119860119896 (1198907) 119865119860119896 (1198907)⟩ ⟨1198908 119879119860119896 (1198908) 119868119860119896 (1198908) 119865119860119896 (1198908)⟩

(18)





119872SNS (119860119896 119860 119905) (19)



Table4En

ergy

values

ofbearingfaulttypes

representedby

theform

ofSN

S

Faulttypes

Energy

ineach

frequ

ency

band





119860 1⟨119890 1

0760

14020

⟩⟨119890 20

95005

000⟩

⟨119890 301

1004

085⟩⟨

119890 4064

00702

9⟩⟨119890 5

0040

02094

⟩⟨119890 60

00002

098⟩

⟨119890 700

1005

095⟩⟨

119890 8000

00309

7⟩119860 2

⟨119890 110

0001

000⟩⟨

119890 2024

01506

1 ⟩⟨119890 3

0020

01097

⟩⟨119890 40

13009

078 ⟩

⟨119890 500

0001

100⟩⟨

119890 6000

00109

9 ⟩⟨119890 7

0010

01099

⟩⟨119890 80

01001

099 ⟩

119860 3⟨119890 1

0820

11007

⟩⟨119890 21

00001

000 ⟩

⟨119890 301

1005

084⟩⟨

119890 4065

01002

5 ⟩⟨119890 5

0060

03092

⟩⟨119890 60

00004

096 ⟩

⟨119890 700

2004

094⟩⟨

119890 8000

00110

0 ⟩119860 4

⟨119890 110

0001

000⟩⟨

119890 2049

00704

5 ⟩⟨119890 3

0060

04090

⟩⟨119890 40

20004

076 ⟩

⟨119890 500

0001

100⟩⟨

119890 6002

00109

7 ⟩⟨119890 7

0030

03095

⟩⟨119890 80

03003

094 ⟩




Healthy SNSsSVM

30 2827

23

93390


2822

933933


72523

833767


3025

100833


1198611 = ⟨1198901 100 001 000⟩ ⟨1198902 051 001 049⟩ ⟨1198903 008 001 092⟩ ⟨1198904 024 001 076⟩ ⟨1198905 000 001 100⟩ ⟨1198906 003 001 097⟩ ⟨1198907 005 001 095⟩ ⟨1198908 006 001 094⟩

1198612 = ⟨1198901 100 001 000⟩ ⟨1198902 039 001 061⟩ ⟨1198903 003 001 097⟩ ⟨1198904 022 001 078⟩ ⟨1198905 000 001 100⟩ ⟨1198906 001 001 100⟩ ⟨1198907 001 001 100⟩ ⟨1198908 001 001 100⟩

(20)



(21)


1198604 997888rarr 1198602 997888rarr 1198601 997888rarr 1198603 (22)




1198602 997888rarr 1198604 997888rarr 1198601 997888rarr 1198603 (23)





5 Conclusion





SNS 933 933 833 100 925

SVM 90 933 767 833 858

GA-BP 100 80 70 90 85

BP 90 70 50 90 75


Competing Interests


References
































1 2 3 4 5 6 7 80

01

02

03

04

05

06

07

08

09

1N

orm

aliz

ed en

ergy

eige

nval

ue

Subfrequency band


1 2 3 4 5 6 7 80

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

ener

gy ei

genv

alue

Subfrequency band


1 2 3 4 5 6 7 80

01

02

03

04

05

06

07

08

09

1

Subfrequency band

Nor

mal

ized

ener

gy ei

genv

alue


1 2 3 4 5 6 7 80

01

02

03

04

05

06

07

08

09

1

Subfrequency band

Nor

mal

ized

ener

gy ei

genv

alue







119860 = ⟨119909 119879119860 (119909) 119868119860 (119909) 119865119860 (119909)⟩ | 119909 isin 119880 (14)





119872SNS (119860 119861) = 1119899119899sum119894=1


(15)



(P1) 0 le 119872SNS(119860 119861) le 1



119872SNS (119860 119861) = 119899sum119894=1


(16)



119860119896 = (1198901 [119879119860119896 (1199091) 119880119860119896 (1198901)]) (1198902 [119879119860119896 (1198902) 119880119860119896 (1198902)]) (1198903 [119879119860119896 (1198903) 119880119860119896 (1198903)]) (1198904 [119879119860119896 (1198904) 119880119860119896 (1198904)]) (1198905 [119879119860119896 (1198905) 119880119860119896 (1198905)]) (1198906 [119879119860119896 (1198906) 119880119860119896 (1198906)]) (1198907 [119879119860119896 (1198907) 119880119860119896 (1198907)]) (1198908 [119879119860119896 (1198908) 119880119860119896 (1198908)]) (119896 = 1 2 3 4)

(17)


119860119896 = ⟨1198901 119879119860119896 (1198901) 119868119860119896 (1198901) 119865119860119896 (1198901)⟩ ⟨1198902 119879119860119896 (1198902) 119868119860119896 (1198902) 119865119860119896 (1198902)⟩ ⟨1198903 119879119860119896 (1198903) 119868119860119896 (1198903) 119865119860119896 (1198903)⟩

⟨1198904 119879119860119896 (1198904) 119868119860119896 (1198904) 119865119860119896 (1198904)⟩ ⟨1198905 119879119860119896 (1198905) 119868119860119896 (1198905) 119865119860119896 (1198905)⟩ ⟨1198906 119879119860119896 (1198906) 119868119860119896 (1198906) 119865119860119896 (1198906)⟩ ⟨1198907 119879119860119896 (1198907) 119868119860119896 (1198907) 119865119860119896 (1198907)⟩ ⟨1198908 119879119860119896 (1198908) 119868119860119896 (1198908) 119865119860119896 (1198908)⟩

(18)





119872SNS (119860119896 119860 119905) (19)



Table4En

ergy

values

ofbearingfaulttypes

representedby

theform

ofSN

S

Faulttypes

Energy

ineach

frequ

ency

band





119860 1⟨119890 1

0760

14020

⟩⟨119890 20

95005

000⟩

⟨119890 301

1004

085⟩⟨

119890 4064

00702

9⟩⟨119890 5

0040

02094

⟩⟨119890 60

00002

098⟩

⟨119890 700

1005

095⟩⟨

119890 8000

00309

7⟩119860 2

⟨119890 110

0001

000⟩⟨

119890 2024

01506

1 ⟩⟨119890 3

0020

01097

⟩⟨119890 40

13009

078 ⟩

⟨119890 500

0001

100⟩⟨

119890 6000

00109

9 ⟩⟨119890 7

0010

01099

⟩⟨119890 80

01001

099 ⟩

119860 3⟨119890 1

0820

11007

⟩⟨119890 21

00001

000 ⟩

⟨119890 301

1005

084⟩⟨

119890 4065

01002

5 ⟩⟨119890 5

0060

03092

⟩⟨119890 60

00004

096 ⟩

⟨119890 700

2004

094⟩⟨

119890 8000

00110

0 ⟩119860 4

⟨119890 110

0001

000⟩⟨

119890 2049

00704

5 ⟩⟨119890 3

0060

04090

⟩⟨119890 40

20004

076 ⟩

⟨119890 500

0001

100⟩⟨

119890 6002

00109

7 ⟩⟨119890 7

0030

03095

⟩⟨119890 80

03003

094 ⟩




Healthy SNSsSVM

30 2827

23

93390


2822

933933


72523

833767


3025

100833


1198611 = ⟨1198901 100 001 000⟩ ⟨1198902 051 001 049⟩ ⟨1198903 008 001 092⟩ ⟨1198904 024 001 076⟩ ⟨1198905 000 001 100⟩ ⟨1198906 003 001 097⟩ ⟨1198907 005 001 095⟩ ⟨1198908 006 001 094⟩

1198612 = ⟨1198901 100 001 000⟩ ⟨1198902 039 001 061⟩ ⟨1198903 003 001 097⟩ ⟨1198904 022 001 078⟩ ⟨1198905 000 001 100⟩ ⟨1198906 001 001 100⟩ ⟨1198907 001 001 100⟩ ⟨1198908 001 001 100⟩

(20)



(21)


1198604 997888rarr 1198602 997888rarr 1198601 997888rarr 1198603 (22)




1198602 997888rarr 1198604 997888rarr 1198601 997888rarr 1198603 (23)





5 Conclusion





SNS 933 933 833 100 925

SVM 90 933 767 833 858

GA-BP 100 80 70 90 85

BP 90 70 50 90 75


Competing Interests


References

































119872SNS (119860 119861) = 1119899119899sum119894=1


(15)



(P1) 0 le 119872SNS(119860 119861) le 1



119872SNS (119860 119861) = 119899sum119894=1


(16)



119860119896 = (1198901 [119879119860119896 (1199091) 119880119860119896 (1198901)]) (1198902 [119879119860119896 (1198902) 119880119860119896 (1198902)]) (1198903 [119879119860119896 (1198903) 119880119860119896 (1198903)]) (1198904 [119879119860119896 (1198904) 119880119860119896 (1198904)]) (1198905 [119879119860119896 (1198905) 119880119860119896 (1198905)]) (1198906 [119879119860119896 (1198906) 119880119860119896 (1198906)]) (1198907 [119879119860119896 (1198907) 119880119860119896 (1198907)]) (1198908 [119879119860119896 (1198908) 119880119860119896 (1198908)]) (119896 = 1 2 3 4)

(17)


119860119896 = ⟨1198901 119879119860119896 (1198901) 119868119860119896 (1198901) 119865119860119896 (1198901)⟩ ⟨1198902 119879119860119896 (1198902) 119868119860119896 (1198902) 119865119860119896 (1198902)⟩ ⟨1198903 119879119860119896 (1198903) 119868119860119896 (1198903) 119865119860119896 (1198903)⟩

⟨1198904 119879119860119896 (1198904) 119868119860119896 (1198904) 119865119860119896 (1198904)⟩ ⟨1198905 119879119860119896 (1198905) 119868119860119896 (1198905) 119865119860119896 (1198905)⟩ ⟨1198906 119879119860119896 (1198906) 119868119860119896 (1198906) 119865119860119896 (1198906)⟩ ⟨1198907 119879119860119896 (1198907) 119868119860119896 (1198907) 119865119860119896 (1198907)⟩ ⟨1198908 119879119860119896 (1198908) 119868119860119896 (1198908) 119865119860119896 (1198908)⟩

(18)





119872SNS (119860119896 119860 119905) (19)



Table4En

ergy

values

ofbearingfaulttypes

representedby

theform

ofSN

S

Faulttypes

Energy

ineach

frequ

ency

band





119860 1⟨119890 1

0760

14020

⟩⟨119890 20

95005

000⟩

⟨119890 301

1004

085⟩⟨

119890 4064

00702

9⟩⟨119890 5

0040

02094

⟩⟨119890 60

00002

098⟩

⟨119890 700

1005

095⟩⟨

119890 8000

00309

7⟩119860 2

⟨119890 110

0001

000⟩⟨

119890 2024

01506

1 ⟩⟨119890 3

0020

01097

⟩⟨119890 40

13009

078 ⟩

⟨119890 500

0001

100⟩⟨

119890 6000

00109

9 ⟩⟨119890 7

0010

01099

⟩⟨119890 80

01001

099 ⟩

119860 3⟨119890 1

0820

11007

⟩⟨119890 21

00001

000 ⟩

⟨119890 301

1005

084⟩⟨

119890 4065

01002

5 ⟩⟨119890 5

0060

03092

⟩⟨119890 60

00004

096 ⟩

⟨119890 700

2004

094⟩⟨

119890 8000

00110

0 ⟩119860 4

⟨119890 110

0001

000⟩⟨

119890 2049

00704

5 ⟩⟨119890 3

0060

04090

⟩⟨119890 40

20004

076 ⟩

⟨119890 500

0001

100⟩⟨

119890 6002

00109

7 ⟩⟨119890 7

0030

03095

⟩⟨119890 80

03003

094 ⟩




Healthy SNSsSVM

30 2827

23

93390


2822

933933


72523

833767


3025

100833


1198611 = ⟨1198901 100 001 000⟩ ⟨1198902 051 001 049⟩ ⟨1198903 008 001 092⟩ ⟨1198904 024 001 076⟩ ⟨1198905 000 001 100⟩ ⟨1198906 003 001 097⟩ ⟨1198907 005 001 095⟩ ⟨1198908 006 001 094⟩

1198612 = ⟨1198901 100 001 000⟩ ⟨1198902 039 001 061⟩ ⟨1198903 003 001 097⟩ ⟨1198904 022 001 078⟩ ⟨1198905 000 001 100⟩ ⟨1198906 001 001 100⟩ ⟨1198907 001 001 100⟩ ⟨1198908 001 001 100⟩

(20)



(21)


1198604 997888rarr 1198602 997888rarr 1198601 997888rarr 1198603 (22)




1198602 997888rarr 1198604 997888rarr 1198601 997888rarr 1198603 (23)





5 Conclusion





SNS 933 933 833 100 925

SVM 90 933 767 833 858

GA-BP 100 80 70 90 85

BP 90 70 50 90 75


Competing Interests


References
































Table4En

ergy

values

ofbearingfaulttypes

representedby

theform

ofSN

S

Faulttypes

Energy

ineach

frequ

ency

band





119860 1⟨119890 1

0760

14020

⟩⟨119890 20

95005

000⟩

⟨119890 301

1004

085⟩⟨

119890 4064

00702

9⟩⟨119890 5

0040

02094

⟩⟨119890 60

00002

098⟩

⟨119890 700

1005

095⟩⟨

119890 8000

00309

7⟩119860 2

⟨119890 110

0001

000⟩⟨

119890 2024

01506

1 ⟩⟨119890 3

0020

01097

⟩⟨119890 40

13009

078 ⟩

⟨119890 500

0001

100⟩⟨

119890 6000

00109

9 ⟩⟨119890 7

0010

01099

⟩⟨119890 80

01001

099 ⟩

119860 3⟨119890 1

0820

11007

⟩⟨119890 21

00001

000 ⟩

⟨119890 301

1005

084⟩⟨

119890 4065

01002

5 ⟩⟨119890 5

0060

03092

⟩⟨119890 60

00004

096 ⟩

⟨119890 700

2004

094⟩⟨

119890 8000

00110

0 ⟩119860 4

⟨119890 110

0001

000⟩⟨

119890 2049

00704

5 ⟩⟨119890 3

0060

04090

⟩⟨119890 40

20004

076 ⟩

⟨119890 500

0001

100⟩⟨

119890 6002

00109

7 ⟩⟨119890 7

0030

03095

⟩⟨119890 80

03003

094 ⟩




Healthy SNSsSVM

30 2827

23

93390


2822

933933


72523

833767


3025

100833


1198611 = ⟨1198901 100 001 000⟩ ⟨1198902 051 001 049⟩ ⟨1198903 008 001 092⟩ ⟨1198904 024 001 076⟩ ⟨1198905 000 001 100⟩ ⟨1198906 003 001 097⟩ ⟨1198907 005 001 095⟩ ⟨1198908 006 001 094⟩

1198612 = ⟨1198901 100 001 000⟩ ⟨1198902 039 001 061⟩ ⟨1198903 003 001 097⟩ ⟨1198904 022 001 078⟩ ⟨1198905 000 001 100⟩ ⟨1198906 001 001 100⟩ ⟨1198907 001 001 100⟩ ⟨1198908 001 001 100⟩

(20)



(21)


1198604 997888rarr 1198602 997888rarr 1198601 997888rarr 1198603 (22)




1198602 997888rarr 1198604 997888rarr 1198601 997888rarr 1198603 (23)





5 Conclusion





SNS 933 933 833 100 925

SVM 90 933 767 833 858

GA-BP 100 80 70 90 85

BP 90 70 50 90 75


Competing Interests


References


































Healthy SNSsSVM

30 2827

23

93390


2822

933933


72523

833767


3025

100833


1198611 = ⟨1198901 100 001 000⟩ ⟨1198902 051 001 049⟩ ⟨1198903 008 001 092⟩ ⟨1198904 024 001 076⟩ ⟨1198905 000 001 100⟩ ⟨1198906 003 001 097⟩ ⟨1198907 005 001 095⟩ ⟨1198908 006 001 094⟩

1198612 = ⟨1198901 100 001 000⟩ ⟨1198902 039 001 061⟩ ⟨1198903 003 001 097⟩ ⟨1198904 022 001 078⟩ ⟨1198905 000 001 100⟩ ⟨1198906 001 001 100⟩ ⟨1198907 001 001 100⟩ ⟨1198908 001 001 100⟩

(20)



(21)


1198604 997888rarr 1198602 997888rarr 1198601 997888rarr 1198603 (22)




1198602 997888rarr 1198604 997888rarr 1198601 997888rarr 1198603 (23)





5 Conclusion





SNS 933 933 833 100 925

SVM 90 933 767 833 858

GA-BP 100 80 70 90 85

BP 90 70 50 90 75


Competing Interests


References


































SNS 933 933 833 100 925

SVM 90 933 767 833 858

GA-BP 100 80 70 90 85

BP 90 70 50 90 75


Competing Interests


References































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