Research Article A Diagnosis Method for Rotation Machinery ...downloads.hindawi.com/journals/mpe/2015/563954.pdf · diagnosis of the rotatingmachinery fault signal is uncertain. In

Research ArticleA Diagnosis Method for Rotation MachineryFaults Based on Dimensionless Indexes Combined with119870-Nearest Neighbor Algorithm

Jianbin Xiong12 Qinghua Zhang12 Zhiping Peng2 Guoxi Sun2

Weichao Xu3 and Qi Wang2

1Guangdong Provincial Key Laboratory of Petrochemical Equipment Fault Diagnosis Maoming 525000 China2School of Computer and Electronic Information Guangdong University of Petrochemical Technology Maoming 525000 China3School of Automation Guangdong University of Technology Guangzhou 510006 China

Correspondence should be addressed to Guoxi Sun 158011382qqcom

Received 30 October 2014 Revised 3 February 2015 Accepted 3 February 2015

Academic Editor Gang Li

Copyright copy 2015 Jianbin Xiong et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

It is difficult to well distinguish the dimensionless indexes between normal petrochemical rotatingmachinery equipment and thosewith complex faultsWhen the conflict of evidence is too big it will result in uncertainty of diagnosisThis paper presents a diagnosismethod for rotation machinery fault based on dimensionless indexes combined with 119870-nearest neighbor (KNN) algorithm Thismethod uses a KNN algorithm and an evidence fusion theoretical formula to process fuzzy data incomplete data and accuratedata This method can transfer the signals from the petrochemical rotating machinery sensors to the reliability manners usingdimensionless indexes and KNN algorithm The input information is further integrated by an evidence synthesis formula to getthe final data The type of fault will be decided based on these data The experimental results show that the proposed method canintegrate data to provide a more reliable and reasonable result thereby reducing the decision risk

1 Introduction

Large rotating machinery and equipment (such as steam tur-bines rotary bearings fans and compressors) are key equip-ment in petroleum chemical metallurgy machinery manu-facturing aerospace and other important engineering fieldsSuch equipment requires high safety and reliability [1] so thestudy of fault diagnosis methods for these types of equipmenthas been a hot topic in this field Vibrationmonitoring signalshave lots of nonlinear random and nonergodic informationwhich causes many complications in the fault signal analysiswhen rotating machinery does not work [2] The time-domain signal of vibration is the most basic and originalsignal References [1 2] extracted failure characteristicsdirectly from the time-domain signal and analyzed the faultdiagnosis showing that maintaining the basic characteristicsof the signal will be very beneficial References [2ndash4] used theprobability density function of the vibration signal to derive

dimensional indexes (the mean and root mean square valuesetc) and dimensionless indexes (waveform margin indexpulse etc) in the amplitude domain In practice althougha dimensional index is sensitive to the fault characteristicsits value will increase with the development of the fault Inaddition as theworking conditions (load speed etc) changethese are easily affected by interference which reduces theirperformance as diagnostic measures [3] By contrast thedimensionless indexes are not sensitive to the disturbanceof the vibration monitoring signal and the performance isstable In particular these dimensionless indexes are sensitiveto neither the changes in amplitude nor the frequency of thesignal That is they have little relationship to the particularworking conditions of the machine [1ndash3] Dimensionlessindexes have been widely used in the fault diagnosis ofrotating machinery For the dimensionless indexes pulseindex and kurtosis index are more sensitive to impact typefault especially in the early fault the large amplitude of

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 563954 9 pageshttpdxdoiorg1011552015563954

2 Mathematical Problems in Engineering

the pulse is less Compare to other parameters kurtosis indexand pulse index rise faster so that fault of the range is largerit is difficult to determine fault type [3 4]

Solving the above problems requires the use of aneffective method which can process uncertain informationrationally systematically and flexibly [5] Evidence theorycan effectively express and deal with uncertain and impre-cise information and other problems [6] However in theactual information fusion system the interference due tothe natural environment or human disturbances often leadsto conflicting reports by the sensors [7] Traditional D-Sevidence theory cannot effectively deal with the integrationof conflicting information When evidence conflict existsbetween global or local information using D-S combinationrules for fusion leads to fusion results which are contrary tothe real value [1 8] Therefore when a high degree of conflictexists between the evidence achieving effective integrationbetween the evidence is an urgent problem that needs to besolved At present there are three major ways for treatingevidence conflict [7] (1) in the improvement method basedon D-S combination rules there is a high degree of conflictbetween the evidence and the D-S combination rules forfusion are used directly This often produces unreasonablefusion results This kind of methods has been presented byYager [9] Dubois and Prade [10] and Feng et al [7] (2)In the second method which is based on modifying theoriginal sources of evidence [8 11] the conflict evidence ispreprocessed first and then evidence combination rules areused for fusing evidence (3) The third method is to modifythe model from the sources based on a known model

These three methods can solve some practical applicationproblems Which method is used depends on the actualsituation and the need In the absence of an effective com-bination rule evaluation criteria it is difficult to determinewhich combination rule is ldquothe most excellentrdquo Smets et alput forward a TBMmodel and believed that the reliability ofthe evidence conflict which resulted from incompleteness ofrecognition framework should be assigned to the empty setMurphy provided an evidence average combination methodThis method is based on the modified model and has fasterconvergence The deficiency of this method is that it uses theunfused evidence simply without considering the intercon-nectedness between them Reference [12] used a modifiedEuclidean distance to determine the correlation betweenthe evidence and to obtain the weight of evidence beforemodifying the original evidence andfinallymaking the fusiondecisions Reference [13] pointed out that the use of a conflictcoefficient to measure the conflict between the evidence wasnot sufficient the conflict between the evidence is also relatedto the pignistic probability distance so these two factorsshould be used together to measure conflict degree betweenthe evidence Reference [7] presented a conflicting intervalevidence fusion method based on the evidence similaritymeasure This method defines the probability conversionrules of the extended pignistic probability and converts theinterval evidence into interval pignistic probability and usesthe fuzzy interval to normalize the Euclidean distance andgets the similarity between interval pignistic thus deter-mining the similarity matrix between any two evidences

and obtaining confidence of the interval evidence Differentsituations of evidence conflict are classified and discussedaccording to the size of the two values However Reference[13] did not consider the distance of the evidence body sothe conflict representation model is still not comprehensiveMore importantly there is no analysis of the conflict andthe relationship leading to the conflict and it has not beendeveloped to give a further approach to the process conflict

Given the above problems we propose an evidencesynthesis method based on dimensionless indexes combinedwith KNN to improve the reliability and the rationality ofthe results This paper is organized as follows Sections 2 3and 4 introduce the problem statement theory and rotatingmachinery fault diagnosis experiment respectively Finally aconclusion is drawn in Section 5

2 Problem Statement

Vibration monitoring signals from rotating machinery in theevent of a fault include nonlinear random and nonergodicinformation which leads to great difficulty in fault signalanalysis Although a dimensional index is sensitive to the faultcharacteristics its value will increase with the developmentof the fault when the working conditions (load speedetc) change it is easy to be affected resulting in unstableperformance The dimensionless index is not sensitive todisturbance of the vibration monitoring signal performanceis stable Of dimensionless indexes the pulse index andthe kurtosis index are more sensitive to impact type faultespecially in the early stage of fault since there is less pulseand no significant increase in other parameter values whilethe kurtosis index and pulse index rise fastlyTherefore thesetwo indexes are more sensitive in the early stage of fault inrotatingmachinery resulting in increased fault interval rangeso it is difficult to distinguish

Definition 1 (see [2 3]) A dimensionless index is made up ofthe ratio of two amounts with the same dimension Whendescribing a particular system it has a certain physical mean-ing In fault diagnosis for dimensionless parameter indexes

Δ120578119909=

[int+infin

minusinfin

|119909|119879

120585 (119909) 119889119909]

1119879

[int+infin

minusinfin

|119909|119898

120585 (119909) 119889119909]

1119898

(1)

where 119909 is the vibration amplitude and 120585(119909) is the probabilitydensity function of vibration amplitude Multiple historicalmonitoring data of a single fault can be calculated using thisfunction

Our Aim We attempted to answer the question regardinghow to integrate reliable and reasonable results when the faultdiagnosis of the rotating machinery fault signal is uncertain

In this paper we addressed the diagnoses of rotatingmachinery fault for large petrochemical enterprises Sensorscollected many kinds of fault data by mechanical oper-ation in real time online and the distance between thisdata and the known training samples was calculated using

Mathematical Problems in Engineering 3

Petrochemicaldevice

Sensor 1

Sensor set

Sensor 2 Sensor 3

Online testing data

parameterCompute dimensionless

Bearingwear

Bearing crack outsider

Bearing crack insider Curved shaft Lack of

bearing

Evidence collection

Compute dimensionless parameter weight

Fusion with D-S

Make a decision

Sensor n

Fusion result mi(i = 1 2 n)

middot middot middot

Figure 1 Flowchart of rotating machinery fault evidence synthesis diagnosis

the KNN algorithm After obtaining the distance betweenthe test samples and the known training samples we tookthe reciprocal value of the distance as the probability thatthe tested sample is the kind of training sample We fusedevidence using the D-S evidence theory synthesis methodto make a final decision about the fault The specific flow isshown in Figure 1

3 Theory

31 Calculation of Dimensionless Indexes and Determinationof the Fault Zone In this paper we processed vibrationmon-itoring signal using the method of dimensionless calculation[14]

Hypothesis 1 (see [1 2 5]) Under Definition 1 and 119879 = 1119898 = 1 then the waveform index

119878119891=

[int+infin

minusinfin

|119909|2

120585 (119909) 119889119909]

12

[int+infin

minusinfin

|119909| 120585 (119909) 119889119909]

=

radic119864 (|119909|2

)

radic119864 (|119909|)

(2)

Similarly (1) when 119879 = infin 119898 = 1 pulse index 119868119891is

defined as

119868119891= lim119879rarrinfin

[int+infin

minusinfin

|119909|119879

120585 (119909) 119889119909]

1119879

[int+infin

minusinfin

|119909| 120585 (119909) 119889119909]

= lim119879rarrinfin

radic119864(|119909|2

)

radic119864 (|119909|)

(3)

(2) when 119879 = infin 119898 = 12 margin index 119862119871119891is defined

as

119862119871119891= lim119879rarrinfin

[int+infin

minusinfin

|119909|119879

120585 (119909) 119889119909]

1119879

[int+infin

minusinfin

|119909|12

120585 (119909) 119889119909]

2= lim119879rarrinfin

119879radic119864(|119909|

119879

)

[radic119864 (|119909|)]2

(4)

(3) when 119879 = infin119898 = 2 peak index 119862119891is defined as

119862119891= lim119879rarrinfin

[int+infin

minusinfin

|119909|119879

120585 (119909) 119889119909]

1119879

[int+infin

minusinfin

|119909|2

120585 (119909) 119889119909]

12


119879radic119864(|119909|

119879

)

radic119864 (|119909|2

)

(5)

(4) when 119879 = infin119898 = 4 kurtosis index119870V is defined as

119870V = lim119879rarrinfin

[int+infin

minusinfin

|119909|4

120585 (119909) 119889119909]

[int+infin

minusinfin

1199092120585 (119909) 119889119909]

2=

119864 (|119909|4

)

[119864 (|119911|2

)]2 (6)

A dimensionless index is made up of the ratio of twoamounts with the same dimension In this paper we moni-tored signals based on the probability density function of themonitoring signal This dimensionless index is a ratio whichis not affected by themagnitude of the signal and the correla-tions between the sensitivity of vibration detector amplifierand themagnification are not large so themonitoring systemwithout calibration can be used in the actual equipment faultdiagnosis [1 14]


To use the dimensionless index in the study of fault diag-nosis we began with petrochemical core units We collecteddata online in real time and calculated the normal state ofthe rotation unit and many kinds of dimensionless indexparameters when each fault happens Then we calculated themaximum value and minimum value of each dimensionlessindex for each of core units in the normal state and all kindsof fault states

Hypothesis 2 119873 monitoring data of vibration data 120576 werecollected under the single fault and119873 is relatively large

Conclusion 1 Under the condition of Definition 1 Hypothe-sis 1 and Hypothesis 2 the expectation of the dimensionlessindex can be approximate

120576minus119879

= 119864 (|120576|119879

) =1

119873

119873

sum

119894=1

10038161003816100381610038161205761198941003816100381610038161003816

119879

(7)

So dimensionless index Δ120578119909approximate

Δ120578119909=

119879radic120576minus119879

119898radic120576minus119898

(8)

if 119879 = infin then 119879radic120576minus119879 asymp max119895=12119873

|120576119895|

Conclusion 2 Under the condition of Definition 1 Hypothe-sis 1 and Hypothesis 2 sets of vibration monitoring data ofa single fault history 120576

1 1205762 120576

119873119896 where the value range is

120576minus119879

isin [120572119879 120573119879] of 120576minus119879 can be calculated then the dimension-

less index fault interval is

Δ120578119909=

[int+infin

minusinfin

|119909|119879

120585 (119909) 119889119909]

1119879

[int+infin

minusinfin

|119909|119898

120585 (119909) 119889119909]

1119898

=

119879radic120576minus119879

119898radic120576minus119898

isin [119888120576 119889120576]

=

[ 119879radic120572119879119879radic120573119879]

[ 119898radic120572119898119898radic120573119898]

isin [

119879radic120572119879

119898radic120573119898

119879radic120573119879

119898radic120572119898

]

(9)

32 KNNAlgorithm Cover andHart proposed the119870-nearestneighbor algorithm (KNN) in 1968 [15] The idea behind thealgorithm is to calculate the distance between tested samplesand known training samples based on a distance functionSelect 119896-nearest sample values and choose an unknownsample according to the 119896-nearest sample valuesThismethodis widely used in fault diagnosis text classification datamining machine learning pattern recognition and imageprocessing and other domains This paper has 119873 faultsamples distributed to 119888 classes 119878

119891119862119891 119868119891119862119871119891 and119870V Each

class has 119873119894samples 119894 = 1 2 119888 We found 119896-nearest

neighbors in all fault samples 119896119894represents the number

of 119896-nearest neighbors distributed to 119888 class The 119896-nearestneighbor judgment function is

119892119894(119909) = 119896

119894 (119894 = 1 2 119888) (10)

33 Evidence Theory

331 Classic D-SAlgorithm D-S evidence theory is an uncer-tainty reasoning method also known as belief functiontheory It is widely used in intelligent data processing infor-mation fusion expert systems data mining fault diagnosistarget identification decision analysis and other domainsThis theory provides useful evidence combination rules tofuse and update evidence information in order to solve theproblem of processing uncertain information

Evidence synthesis is the core of the evidence theoryIt fuses independent evidence information coming fromdifferent information sources in order to produce morereliable evidence information However D-S evidence syn-thesis is limited in different degrees in practical applica-tion especially when evidence conflicts highly or fully Inthese cases D-S evidence synthesis loses efficacy and soresearchers at home and abroad in the field have proposedmany improvements from their different perspectives Atpresent Chinarsquos fault diagnosis technology is widely used inmilitary aerospace chemicals shipbuilding and so forthThere are many theories and methods of fault diagnosis andevidential reasoning has a great significance in fault researchIt contains uncertainty information processing the effectiveintegration of information determinations of the credibilityof the fault indicators formation and decision-making Inthis paper we use the idea of evidence theory combinedwith the dimensionless index to solve such uncertainty prob-lems Through multifeature fusion recognition analysis weimprove the recognition performance and accuracy of faultdiagnosis using effective appropriate diagnosticmethods anddetermine the root cause of failure quickly [9]

In a large crew equipment we can install the sensorsin different parts of large crew of the equipment to achieveequipment testingThe information from sensors provides allthe fault information from each part that needs monitoringand forms a body of evidence Different evidence bodiescorrespond to different credits functions Through analyzingcredit functions we can obtain the corresponding credit andfuse each credit function using certain D-S combinationprinciples to determine the fault eventually

(1) Basic Probability Assignment In the recognition frame-work Θ the basic probability assignment (BPA) is a 2Θ rarr

[0 1] function119898 called the mass function This satisfies

119898(0) = 0

sum

119860subeΘ

119898(119860) = 1(11)

where the 119898(119860) which makes 119898(119860) gt 0 is called a focalelement for 119860

(2) Trust Function Trust function is also known as belieffunction In recognition framework Θ based on BPA trustfunction definition of119898 is

Bel (119861) = sum

119861sube119860

119898(119861) (12)


(3) Likelihood Function Likelihood function is also known asplausibility function In recognition framework Θ based onBPA likelihood function definition of119898 is

119875119897 (119860) = sum

119861cap119860 =0

119898(119861) (13)

(4) Confidence Zone In evidence theory hypothetical 119860 is inrecognition framework Θ

119898 (Θ) = sum

119861⋂119860=Φ

119898(119861)

119898 (119860) =

sumcap119860119894=119860

prod119898

119894=1119898119894(119860119894)

1 minus sumcap119860119894=Φ

prod119898

119894=1119898119894(119860119894)=

sumcap119860119894=119860

prod119898

119894=1119898119894(119860119894)

1 minus 119870

(14)

This equation is a classic synthetic formula from D-Sevidence theory where the size of the 119870 value which rep-resents the conflict between all the evidence is called thenormalization factor The role of 1 minus 119870 is not to assign thenonzero probability values to the empty set in the process ofevidence synthesis [16]

In the classical D-S evidence theory synthesis formulaespecially for the case of a completed conflict (ie 119870 = 1)the results obtained from (13) above are usually not consistentwith the actual situation and the formula loses efficacy Peoplebegan to modify this method on the basis of the originalformulaThere are twomainways in which it can bemodified[8]

(1) Based on Modification Rules [9 17]The key to improvingsynthesis results is how tomanage conflictThe new synthesisrules need to efficiently determine how to allocate conflictand this problem also contains two small problems whichsubsets should the conflict be reassigned to and after deter-mining the subset in what proportion should the conflict beallocated

(2) Based on Modification Evidence Source Modification [17]This presumes that the D-S synthesis rules for evidence the-ory are not themselves wrong When the evidence conflictshighly evidence should be pretreated first and then the D-S evidence theory synthesis rules should be used For thoseevidence sources in which conflicts are great and unreliablewe can use the discount factor and other methods [18] toprocess the evidence source without modifying the synthesisrule

332 Improved D-S Algorithm Since the classic D-S theorycan not manage conflict effectively when evidence conflictshighly the results using the D-S evidence synthesis rule isdifferent from the actual situation Many people in Chinahave proposed various modifications to D-S evidence theoryYe et al proposed an evidence combinationmethod based onthe weight coefficients and the confliction probability distri-bution [16] After calculating the weighting coefficients foreach piece of evidence the following evidence combinationis used for information fusion The steps are as follows [16]

(1) Calculate the degree of confliction 119870119894119895between evi-

dence 119870119894in evidence set 119870 and other evidence 119864

119895(119894 =

1 2 119894 minus 1 119894 + 1 119899) and form confliction vector of 119870119894

[16]

119870119894= (1198961198941 1198961198942 119896

119894119894minus1 119896119894119894 119896

119894119899) (15)

wherein

119870119894119895= sum

119860119894cap119860119895=Φ119860119894isin119864119894 119860119895isin119864119895

119898119894(119860119894)119898119895(119860119895)

(119894 = 1 2 119899)

(16)

(2) Process confliction vector 119870119894(119894 = 1 2 119899) with

normalization

119870119873

119894=(1198961198941 1198961198942 119896

119894119894minus1 119896119894119894 119896

119894119899)

sum119895=1119895 =119894

119896119894119895

= (119870119873

1198941 119870119873

1198942 119870

119873

119894119894minus1 119870119873

119894119894 119870

119873

119894119899)

(17)

(3) After normalization calculate the entropy of conflic-tion vector119870119873

119894

119867119894= sum

119895=1119895 =119894

119896119873

119894ln 119896119873119894

(119894 = 1 2 119899) (18)

(4) Get countdown of entropy119867119894

119867minus1

119894=

1

119867119894

(19)

(5) Calculate the weight coefficient of evidence 119864119894

119908119894=

1198671

119894

sum119899

119895=119894119867minus1

119895

(20)

After calculating the weighting coefficients of each evi-dence the following evidence combination is used for infor-mation fusion steps are as follow [16]

(1) Allocate the probability value to the proposition in theframework according to the evidence provided by evidencesource and establish the weight vector of the evidence source

119882 = (1199081 1199082 119908

119899) (21)

(2) Assume 119882max = max(1199081 1199082 119908

119899) relative weight

vector is available 119882 = (1199081 1199082 119908

119899)119882max then we can

determine the ldquodiscount raterdquo of the basic probability assign-ment value of the evidence Using the ldquodiscount raterdquo to adjustbasic probability assignment value of all the propositionin each recognition framework according to the followingmethod the basic probability assignment value after beingadjusted is [16]

119898lowast

119894(119860119896) = 120572119894119898119894(119860119894) (22)

wherein the discount rate is

119886119894=

119908119894

119908max (23)


Table 1 Various faults intervals (acceleration)

Fault index Bearing wear Bearing crackoutsider

Bearing crackinsider Curved shaft Lack of bearing

Waveform index [1284 1447] [119 1327] [1172 1327] [1172 1312] [119 1494]

Peak index [3140 5959] [1886 4818] [2215 4818] [2026 3686] [2503 6105]

Pulse index [4068 8625] [2320 6044] [2664 6044] [2460 4590] [3098 8602]

Margin index [4935 1076] [2683 7346] [3044 7346] [2832 5647] [3647 1072]

Margin index [3027 1079] [2531 4051] [2167 385] [2167 385] [2531 8378]

(3) By substituting the probability value of all proposi-tions after adjusting into [11] formula we can get a new syn-thetic formula

119898(Φ) = 0

119898 (119860) = 119901 (119860) + 119896 sdot 119902 (119860) 119860 = Φ

119901 (119860) = sum

119860isin119864119894cap119899

119894=1119860119894=119860

119898lowast

1119860 (1)119898

lowast

2119860 (2) sdot sdot sdot 119898

lowast

119899119860 (119899)

119896 = sum

119860isin119864119894cap119899

119894=1119860119894=119860

119898lowast

1119860 (1)119898

lowast

2119860 (2) sdot sdot sdot 119898

lowast

119899119860 (119899)

119902 (119860) =1

119899

119899

sum

119894=1

119898lowast

1(119860)

119898 (Φ) = 1 minus sum

119860subΘ

119898(119860)

(24)

New synthetic formula fully considers the importance ofthe fusion evidence that comes from different data andmakessynthetic results more realistic Moreover except the aboveimproved D-S evidence combination formula in detail thereare many other methods such as D-S evidence combinationformula based on credibility proposed by Sun et al [19] aneffective evidence theory synthesis formula proposed by Li etal [11] This not only reduces the confliction effectively butalso makes the results of synthetic realistic

34 Improved D-S Algorithm Application in the RotatingMachinery Fault Diagnosis We have realized the rotatingmachinery fault diagnosis for large petrochemical enter-prises Through sensors collect all kinds of fault data bymechanical operation in real time online and calculate thedistance between this data and the known training samplesusing KNN algorithm After obtaining the distance betweenthe tested samples and the known training samples take thereciprocal value of the distance as the test sample probabilityand training sample probability The specific flow of fusionevidences using D-S evidence theory synthesis method andmaking a final decision is shown in Figure 1 The implemen-tation steps are as follows

Step 1 Fault data can be collected from petrochemical rota-tion real time

Step 2 Based on the collected failure data dimensionlessindexes can be calculated and fault zone (the maximum and

minimum range in 10 indices) can be set up Use (2) (3)(4) (5) and (6) to calculate the waveform indices peakindicators pulse index margin index and the kurtosis indexrange faults

Step 3 According to the KNN algorithm the number ofnearest fault points 119896 can be found and the distribution canbe derived

Step 4 Use the improved D-S algorithm in (Section 332) tocalculate degree of conflict (119896

119894119895) and then conflict vector (119870

119894)

can be obtained

Step 5 Normalize the conflict vector (119870119894(119894 = 1 2 3 119899))

and calculate the (119870119873119894) using (16)

Step 6 Calculate the entropy119867119894in conflict vector (119870119873

119894) after

normalizing Meanwhile the weighing values 119908119894of 119864119894can be

calculated based on (18)

Step 7 Use (23) to correct D-S fusion data

Step 8 Make final decisions after correction

4 Rotating Machinery FaultDiagnosis Experiment

This experiment was conducted on large rotating machin-ery fault diagnosis experiment platform in petrochemicalequipment fault diagnosis key laboratories of Guangdongprovince Real time data collection of many kinds of faulttypes at Guangdong University of Petrochemical Technology(GDUPT) are shown in Figure 2There are five causes of bear-ing fault in petrochemical rotary setsThere are bearing wearbearing outer crack bearing inner crack bent axle and lack ofbearingThe rotatingmachinery vibration acceleration signalin the process of operation was detected and calculated usinga linear operation to get the waveform indicator 119878

119891 peak

metric 119862119891 pulse index 119868

119904 clearance factor 119862119871

119891 and kurtosis

value119870V for each kind of faultIn order to make the experimental data more accurate

we have collected 1024 fault points for every kind of faultand used them as the training samples Five indexes of thetraining samples were obtained by linear operations and theminimum and maximum values of the five indexes wereselected to confirm the range of the indicators as shownin Table 1 In Table 1 it can be seen in the sensitivity of


(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 2 Rotating machinery fault diagnosis real experiment condition (a) The developed real test bed (b) Fault diagnosis rotatingmachinery test bed (c) Normal bearing (d) Wearing ball bearing (e) Outer ring crack bearing (f) Inner ring crack bearing (g) Bend shaft(h) Lacking ball bearing

various indicators that the waveform index because its scopeis very small is the least sensitive By contrast the sensitiveof the margin index to the jamming signal is much higherIn addition under the same kind of dimensionless index theoverlap of five kinds of faults is significant that is they arehighly conflicted For example for bearing outer cracks andbearing inner cracks the dimensionless index values rangefor the five kinds of indexes is generally low

Choosing a group of data randomly from all the realtime acquired data for example we can choose a bearingcrack value of 3950 and use all the collected 1024 externalbearing crack data to produce an array 119878 and we can get119878(5) = 1 119878(12) = 5 Then the data value of 3950 wassubjected to a linear operation and the fault data valueswere obtained and used in KNN arithmetic First takethe middle values of the five dimensionless indexes as thecentral values of the scope and then calculate the distancefrom the fault value to each central value Here we will get

25 groups of distance values Then convert distance to aprobability value usingKNNalgorithmThewaywe choose isto directly take the reciprocals of those 25 groups of distancevalues and obtain their corresponding probability valuesTheguiding ideology is that when a test samples is closer to atraining sample it has a higher probability to share the samecategory of that training sample In order to make it meet thebasic probability equation (1) a probability value normalizedprocessing was performed in each index and the resultsare shown in Figure 3 Figure 3 lists various fault probabilityvalues under the five indexes Each indicator provides faultprobability values for five kinds of faults including bearingwear bearing outer crack bearing inside crack bent axleand lack of bearing We named each indicator to be a basicprobability distribution function which is also called theevidence collection Five sets of evidence were formed byKNN algorithm and the information from the 5 groups ofevidence collection was fused using D-S evidence theory


Waveform index

Peak index

Pulse index

Margin index

Kurtosis index

0

005

01

015

02

025

03

Five kinds of dimensionless index

Prob

abili

ty v

alue

Bearing wearBearing crack outsiderBearing crack insider

Curved shaftLack of bearing

Figure 3The results ofKNN(Thenumber of faults is 3950 119878(5) = 1119878(12) = 5)

We used classic D-S evidence theory and variousimprovements to D-S evidence theory to match informationfusion and the results are shown in Figure 4 From Figure 4we can see that the evidence collection processing is notstrong enough when it meets the classic D-S evidence theoryespecially the classic source of D-S evidence theory considersall of the evidences are equally important it leads us to theeven wrong conclusion with this situation [20] In view of theabove reasons we used the improved D-S evidence theoryadding different weight coefficients to different evidenceThethree methods in Figure 4 are based on the weight coefficientof the D-S evidence theory synthesis method It can beseen that in comparing the three kinds of synthesis methodsto the classical D-S evidence theory that when evidencewas highly conflicted the other methods increase reliabilityand rationality of the results of synthesis The tested datahowever were from an external bearing crack Despite usingimproved D-S evidence theory the correct diagnosis of thefault was still not obtained

We can see from Figure 4 that from the various sourcesof evidence the probability value for the external bearingcrack fault is not the largest In other words before fusingthe evidence each source of evidence does not think that itis the bearing outer crack that broke down so the final fusionresults are also incorrect

5 Conclusion

There are some problems of identifying complex faults inpetrochemical rotating machinery First the correspondingzone of the dimensionless index is difficult to determine

Bear

ing

wea

r

Bear

ing

crac

k ou

tside

r

Bear

ing

crac

k in

sider

Curv

ed

shaft

Lack

of

bear

ing0

01

02

03

04

05

Five types in petrochemical rotary sets of bearing failurePr

obab

ility

val

ueD-S evidence theory synthetic formulaDirect weighted synthetic formulaSynthesis formula of [16]The synthesis formula of this paper

Figure 4 Evidence theoretical probability comparison table (thenumber of faults is 3950 119878(5) = 1 119878(12) = 5)

Second when the data is transferred from the scene to aremote server it is disturbed by various factors which causetransmission errors Fluctuations in the calculation of therotating machinery fault dimensionless indexes are largeresulting in difficulties with correct fault diagnosis In thispaper we used a rotating machinery fault evidence synthesisdiagnosismethod combining dimensionless indexwithKNNto achieve fault evidence synthesis diagnosis of the rotatingmachinery to make the fusion result more reasonable andreliableThe increased reliability of the results will reduce therisk of decisions based on incorrect information

Conflict of Interests

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

Acknowledgments

The authors would like to thank the Associate Editor Pro-fessor Gang Li and Dr X S Si giving them the opportunityto publish this correspondence paper They would like tosincerely thank and acknowledge the wit-outputs and thetremendous work performed by the Associate Editor and thetwo anonymous reviewers for their insightful suggestions andthorough review which greatly improve this correspondencepaper The authors are also grateful to Professor W XuProfessor L Cai Dr H Y Wu and Dr Z Zhang for their


help to improve writing qualityThis work is supported by theNational Natural Science Foundation of China under Grantnos 61473331 61271380 61174113 and 61272382 the NaturalScience Foundation of Guangdong Province of China (noS2012010009870) the National Natural Science FoundationofGuangdongPetrochemical Equipment FaultDiagnosisKeyLaboratory under Grant no 643513 and the GuangdongUniversity of Petrochemical Technologyrsquos Internal Projectnos 204341 314004

References

[1] Q H Zhang Fault Diagnosis in Unit Based on Artificial ImmuneDetectors System China Petrochemical Press 2008

[2] A S Qing Q H Zhang T Y Li and Q Hu ldquoThe applicationof a compound dimensionless parameter for fault classifying ofrotating machineryrdquoModern Manufacturing Engineering no 4pp 10ndash14 2013

[3] Q H Zhang and Y Z Fu ldquoResearch of adaptive immune net-work intrusion detection modelrdquo International Journal of Sys-tems Control and Communications vol 3 no 3 pp 280ndash2862011

[4] X-S Si C-H Hu J-B Yang and Q Zhang ldquoOn the dynamicevidential reasoning algorithm for fault predictionrdquo ExpertSystems with Applications vol 38 no 5 pp 5061ndash5080 2011

[5] X S Si C H Hu and Z J Zhou ldquoFault predictionmodel basedon evidential reasoning approachrdquo Science in China Series FInformation Sciences vol 53 no 10 pp 2032ndash2046 2010

[6] L Zhang J-W Liu R-C Wang and H-Y Wang ldquoTrustevaluation model based on improved D-S evidence theoryrdquoJournal on Communications vol 34 no 7 pp 167ndash173 2013

[7] H-S Feng X-B Xu and C-L Wen ldquoA new fusion method ofconflicting interval evidence based on the similarity measureof evidencerdquo Journal of Electronics and Information Technologyvol 34 no 4 pp 851ndash857 2012

[8] H-W Guo W-K Shi Q-K Liu and Y Deng ldquoNew combina-tion rule of evidencerdquo Journal of Shanghai Jiaotong Universityvol 40 no 11 pp 1895ndash1902 2006

[9] R R Yager ldquoOn the Dempster-Shafer framework and newcombination rulesrdquo Information Sciences vol 41 no 2 pp 93ndash137 1987

[10] D Dubois and H Prade ldquoRepresentation and combinationof uncertainty with belief functions and possibility measuresrdquoComputational Intelligence vol 4 no 3 pp 244ndash264 1988

[11] B C Li B Wang J Wei C B Qian and Y Q Huang ldquoEffi-cient combination rule of evidence theoryrdquo Journal of DataAcquisition and Processing vol 17 no 1 pp 33ndash36 2002

[12] D Yong S WenKang Z ZhenFu and L Qi ldquoCombining belieffunctions based on distance of evidencerdquo Decision SupportSystems vol 38 no 3 pp 489ndash493 2004

[13] W Liu ldquoAnalyzing the degree of conflict among belief func-tionsrdquo Artificial Intelligence vol 170 no 11 pp 909ndash924 2006

[14] J B Xiong Q H Zhang G X Sun Z P Peng and Q LiangldquoFusion of the dimensionless parameters and filtering methodsin rotating machinery fault diagnosisrdquo Journal of Networks vol9 no 5 pp 1201ndash1207 2014

[15] Y Wang Study on text categorization based on decision tree andK nearest neighbors [PhD thesis] Tientsin University 2006

[16] Q Ye X-PWu andY-X Song ldquoEvidence combinationmethodbased on the weight coefficients and the confliction probability

distributionrdquo Systems Engineering and Electronics vol 28 no 7pp 1014ndash1081 2006

[17] E Lefevre O Colot and P Vannoorenberghe ldquoBelief functioncombination and conflict managementrdquo Information Fusionvol 3 no 2 pp 149ndash162 2002

[18] B He and H L Hu ldquoMulti-level DS evidence combinationstrategyrdquo Computer Engineering and Applications vol 10 pp87ndash90 2004

[19] Q Sun X Ye and W K Gu ldquoA new combination rules of evi-dence theoryrdquo Acta Electronica Sinica vol 28 no 8 pp 117ndash1192000

[20] J B Xiong Intelligence data fusion and its applications in shipdynamic positioning Guangdong university of technology [PhDthesis] Guangdong University of Technology 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the pulse is less Compare to other parameters kurtosis indexand pulse index rise faster so that fault of the range is largerit is difficult to determine fault type [3 4]

Solving the above problems requires the use of aneffective method which can process uncertain informationrationally systematically and flexibly [5] Evidence theorycan effectively express and deal with uncertain and impre-cise information and other problems [6] However in theactual information fusion system the interference due tothe natural environment or human disturbances often leadsto conflicting reports by the sensors [7] Traditional D-Sevidence theory cannot effectively deal with the integrationof conflicting information When evidence conflict existsbetween global or local information using D-S combinationrules for fusion leads to fusion results which are contrary tothe real value [1 8] Therefore when a high degree of conflictexists between the evidence achieving effective integrationbetween the evidence is an urgent problem that needs to besolved At present there are three major ways for treatingevidence conflict [7] (1) in the improvement method basedon D-S combination rules there is a high degree of conflictbetween the evidence and the D-S combination rules forfusion are used directly This often produces unreasonablefusion results This kind of methods has been presented byYager [9] Dubois and Prade [10] and Feng et al [7] (2)In the second method which is based on modifying theoriginal sources of evidence [8 11] the conflict evidence ispreprocessed first and then evidence combination rules areused for fusing evidence (3) The third method is to modifythe model from the sources based on a known model

These three methods can solve some practical applicationproblems Which method is used depends on the actualsituation and the need In the absence of an effective com-bination rule evaluation criteria it is difficult to determinewhich combination rule is ldquothe most excellentrdquo Smets et alput forward a TBMmodel and believed that the reliability ofthe evidence conflict which resulted from incompleteness ofrecognition framework should be assigned to the empty setMurphy provided an evidence average combination methodThis method is based on the modified model and has fasterconvergence The deficiency of this method is that it uses theunfused evidence simply without considering the intercon-nectedness between them Reference [12] used a modifiedEuclidean distance to determine the correlation betweenthe evidence and to obtain the weight of evidence beforemodifying the original evidence andfinallymaking the fusiondecisions Reference [13] pointed out that the use of a conflictcoefficient to measure the conflict between the evidence wasnot sufficient the conflict between the evidence is also relatedto the pignistic probability distance so these two factorsshould be used together to measure conflict degree betweenthe evidence Reference [7] presented a conflicting intervalevidence fusion method based on the evidence similaritymeasure This method defines the probability conversionrules of the extended pignistic probability and converts theinterval evidence into interval pignistic probability and usesthe fuzzy interval to normalize the Euclidean distance andgets the similarity between interval pignistic thus deter-mining the similarity matrix between any two evidences

and obtaining confidence of the interval evidence Differentsituations of evidence conflict are classified and discussedaccording to the size of the two values However Reference[13] did not consider the distance of the evidence body sothe conflict representation model is still not comprehensiveMore importantly there is no analysis of the conflict andthe relationship leading to the conflict and it has not beendeveloped to give a further approach to the process conflict

Given the above problems we propose an evidencesynthesis method based on dimensionless indexes combinedwith KNN to improve the reliability and the rationality ofthe results This paper is organized as follows Sections 2 3and 4 introduce the problem statement theory and rotatingmachinery fault diagnosis experiment respectively Finally aconclusion is drawn in Section 5

2 Problem Statement

Vibration monitoring signals from rotating machinery in theevent of a fault include nonlinear random and nonergodicinformation which leads to great difficulty in fault signalanalysis Although a dimensional index is sensitive to the faultcharacteristics its value will increase with the developmentof the fault when the working conditions (load speedetc) change it is easy to be affected resulting in unstableperformance The dimensionless index is not sensitive todisturbance of the vibration monitoring signal performanceis stable Of dimensionless indexes the pulse index andthe kurtosis index are more sensitive to impact type faultespecially in the early stage of fault since there is less pulseand no significant increase in other parameter values whilethe kurtosis index and pulse index rise fastlyTherefore thesetwo indexes are more sensitive in the early stage of fault inrotatingmachinery resulting in increased fault interval rangeso it is difficult to distinguish

Definition 1 (see [2 3]) A dimensionless index is made up ofthe ratio of two amounts with the same dimension Whendescribing a particular system it has a certain physical mean-ing In fault diagnosis for dimensionless parameter indexes

Δ120578119909=

[int+infin

minusinfin

|119909|119879

120585 (119909) 119889119909]

1119879

[int+infin

minusinfin

|119909|119898

120585 (119909) 119889119909]

1119898

(1)

where 119909 is the vibration amplitude and 120585(119909) is the probabilitydensity function of vibration amplitude Multiple historicalmonitoring data of a single fault can be calculated using thisfunction

Our Aim We attempted to answer the question regardinghow to integrate reliable and reasonable results when the faultdiagnosis of the rotating machinery fault signal is uncertain

In this paper we addressed the diagnoses of rotatingmachinery fault for large petrochemical enterprises Sensorscollected many kinds of fault data by mechanical oper-ation in real time online and the distance between thisdata and the known training samples was calculated using


Petrochemicaldevice

Sensor 1

Sensor set

Sensor 2 Sensor 3

Online testing data


Bearingwear



bearing

Evidence collection


Fusion with D-S

Make a decision

Sensor n





3 Theory



119878119891=

[int+infin

minusinfin

|119909|2

120585 (119909) 119889119909]

12

[int+infin

minusinfin

|119909| 120585 (119909) 119889119909]

=

radic119864 (|119909|2

)

radic119864 (|119909|)

(2)


defined as

119868119891= lim119879rarrinfin

[int+infin

minusinfin

|119909|119879

120585 (119909) 119889119909]

1119879

[int+infin

minusinfin

|119909| 120585 (119909) 119889119909]


radic119864(|119909|2

)

radic119864 (|119909|)

(3)


as

119862119871119891= lim119879rarrinfin

[int+infin

minusinfin

|119909|119879

120585 (119909) 119889119909]

1119879

[int+infin

minusinfin

|119909|12

120585 (119909) 119889119909]


119879radic119864(|119909|

119879

)

[radic119864 (|119909|)]2

(4)


119862119891= lim119879rarrinfin

[int+infin

minusinfin

|119909|119879

120585 (119909) 119889119909]

1119879

[int+infin

minusinfin

|119909|2

120585 (119909) 119889119909]

12


119879radic119864(|119909|

119879

)

radic119864 (|119909|2

)

(5)



[int+infin

minusinfin

|119909|4

120585 (119909) 119889119909]

[int+infin

minusinfin

1199092120585 (119909) 119889119909]

2=

119864 (|119909|4

)

[119864 (|119911|2

)]2 (6)






120576minus119879

= 119864 (|120576|119879

) =1

119873

119873

sum

119894=1

10038161003816100381610038161205761198941003816100381610038161003816

119879

(7)


Δ120578119909=

119879radic120576minus119879

119898radic120576minus119898

(8)


|120576119895|


1 1205762 120576


120576minus119879



Δ120578119909=

[int+infin

minusinfin

|119909|119879

120585 (119909) 119889119909]

1119879

[int+infin

minusinfin

|119909|119898

120585 (119909) 119889119909]

1119898

=

119879radic120576minus119879

119898radic120576minus119898

isin [119888120576 119889120576]

=

[ 119879radic120572119879119879radic120573119879]

[ 119898radic120572119898119898radic120573119898]

isin [

119879radic120572119879

119898radic120573119898

119879radic120573119879

119898radic120572119898

]

(9)


119891119862119891 119868119891119862119871119891 and119870V Each




119892119894(119909) = 119896

119894 (119894 = 1 2 119888) (10)

33 Evidence Theory






119898(0) = 0

sum

119860subeΘ

119898(119860) = 1(11)



Bel (119861) = sum

119861sube119860

119898(119861) (12)



119875119897 (119860) = sum

119861cap119860 =0

119898(119861) (13)


119898 (Θ) = sum

119861⋂119860=Φ

119898(119861)

119898 (119860) =

sumcap119860119894=119860

prod119898

119894=1119898119894(119860119894)


prod119898

119894=1119898119894(119860119894)=

sumcap119860119894=119860

prod119898

119894=1119898119894(119860119894)

1 minus 119870

(14)








119895(119894 =


[16]

119870119894= (1198961198941 1198961198942 119896

119894119894minus1 119896119894119894 119896

119894119899) (15)

wherein

119870119894119895= sum

119860119894cap119860119895=Φ119860119894isin119864119894 119860119895isin119864119895

119898119894(119860119894)119898119895(119860119895)

(119894 = 1 2 119899)

(16)


normalization

119870119873

119894=(1198961198941 1198961198942 119896

119894119894minus1 119896119894119894 119896

119894119899)

sum119895=1119895 =119894

119896119894119895

= (119870119873

1198941 119870119873

1198942 119870

119873

119894119894minus1 119870119873

119894119894 119870

119873

119894119899)

(17)


119894

119867119894= sum

119895=1119895 =119894

119896119873

119894ln 119896119873119894

(119894 = 1 2 119899) (18)


119867minus1

119894=

1

119867119894

(19)


119908119894=

1198671

119894

sum119899

119895=119894119867minus1

119895

(20)



119882 = (1199081 1199082 119908

119899) (21)

(2) Assume 119882max = max(1199081 1199082 119908





119898lowast

119894(119860119896) = 120572119894119898119894(119860119894) (22)


119886119894=

119908119894

119908max (23)





Waveform index [1284 1447] [119 1327] [1172 1327] [1172 1312] [119 1494]

Peak index [3140 5959] [1886 4818] [2215 4818] [2026 3686] [2503 6105]

Pulse index [4068 8625] [2320 6044] [2664 6044] [2460 4590] [3098 8602]

Margin index [4935 1076] [2683 7346] [3044 7346] [2832 5647] [3647 1072]

Margin index [3027 1079] [2531 4051] [2167 385] [2167 385] [2531 8378]


119898(Φ) = 0

119898 (119860) = 119901 (119860) + 119896 sdot 119902 (119860) 119860 = Φ

119901 (119860) = sum

119860isin119864119894cap119899

119894=1119860119894=119860

119898lowast

1119860 (1)119898

lowast

2119860 (2) sdot sdot sdot 119898

lowast

119899119860 (119899)

119896 = sum

119860isin119864119894cap119899

119894=1119860119894=119860

119898lowast

1119860 (1)119898

lowast

2119860 (2) sdot sdot sdot 119898

lowast

119899119860 (119899)

119902 (119860) =1

119899

119899

sum

119894=1

119898lowast

1(119860)

119898 (Φ) = 1 minus sum

119860subΘ

119898(119860)

(24)









119894)

can be obtained




119894) after







119891 peak



119891 and kurtosis




(a) (b)

(c) (d)

(e) (f)

(g) (h)






Waveform index

Peak index

Pulse index

Margin index

Kurtosis index

0

005

01

015

02

025

03


Prob

abili

ty v

alue






5 Conclusion


Bear

ing

wea

r

Bear

ing

crac

k ou

tside

r

Bear

ing

crac

k in

sider

Curv

ed

shaft

Lack

of

bear

ing0

01

02

03

04

05


obab

ility

val






Acknowledgments




References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Petrochemicaldevice

Sensor 1

Sensor set

Sensor 2 Sensor 3

Online testing data


Bearingwear



bearing

Evidence collection


Fusion with D-S

Make a decision

Sensor n





3 Theory



119878119891=

[int+infin

minusinfin

|119909|2

120585 (119909) 119889119909]

12

[int+infin

minusinfin

|119909| 120585 (119909) 119889119909]

=

radic119864 (|119909|2

)

radic119864 (|119909|)

(2)


defined as

119868119891= lim119879rarrinfin

[int+infin

minusinfin

|119909|119879

120585 (119909) 119889119909]

1119879

[int+infin

minusinfin

|119909| 120585 (119909) 119889119909]


radic119864(|119909|2

)

radic119864 (|119909|)

(3)


as

119862119871119891= lim119879rarrinfin

[int+infin

minusinfin

|119909|119879

120585 (119909) 119889119909]

1119879

[int+infin

minusinfin

|119909|12

120585 (119909) 119889119909]


119879radic119864(|119909|

119879

)

[radic119864 (|119909|)]2

(4)


119862119891= lim119879rarrinfin

[int+infin

minusinfin

|119909|119879

120585 (119909) 119889119909]

1119879

[int+infin

minusinfin

|119909|2

120585 (119909) 119889119909]

12


119879radic119864(|119909|

119879

)

radic119864 (|119909|2

)

(5)



[int+infin

minusinfin

|119909|4

120585 (119909) 119889119909]

[int+infin

minusinfin

1199092120585 (119909) 119889119909]

2=

119864 (|119909|4

)

[119864 (|119911|2

)]2 (6)






120576minus119879

= 119864 (|120576|119879

) =1

119873

119873

sum

119894=1

10038161003816100381610038161205761198941003816100381610038161003816

119879

(7)


Δ120578119909=

119879radic120576minus119879

119898radic120576minus119898

(8)


|120576119895|


1 1205762 120576


120576minus119879



Δ120578119909=

[int+infin

minusinfin

|119909|119879

120585 (119909) 119889119909]

1119879

[int+infin

minusinfin

|119909|119898

120585 (119909) 119889119909]

1119898

=

119879radic120576minus119879

119898radic120576minus119898

isin [119888120576 119889120576]

=

[ 119879radic120572119879119879radic120573119879]

[ 119898radic120572119898119898radic120573119898]

isin [

119879radic120572119879

119898radic120573119898

119879radic120573119879

119898radic120572119898

]

(9)


119891119862119891 119868119891119862119871119891 and119870V Each




119892119894(119909) = 119896

119894 (119894 = 1 2 119888) (10)

33 Evidence Theory






119898(0) = 0

sum

119860subeΘ

119898(119860) = 1(11)



Bel (119861) = sum

119861sube119860

119898(119861) (12)



119875119897 (119860) = sum

119861cap119860 =0

119898(119861) (13)


119898 (Θ) = sum

119861⋂119860=Φ

119898(119861)

119898 (119860) =

sumcap119860119894=119860

prod119898

119894=1119898119894(119860119894)


prod119898

119894=1119898119894(119860119894)=

sumcap119860119894=119860

prod119898

119894=1119898119894(119860119894)

1 minus 119870

(14)








119895(119894 =


[16]

119870119894= (1198961198941 1198961198942 119896

119894119894minus1 119896119894119894 119896

119894119899) (15)

wherein

119870119894119895= sum

119860119894cap119860119895=Φ119860119894isin119864119894 119860119895isin119864119895

119898119894(119860119894)119898119895(119860119895)

(119894 = 1 2 119899)

(16)


normalization

119870119873

119894=(1198961198941 1198961198942 119896

119894119894minus1 119896119894119894 119896

119894119899)

sum119895=1119895 =119894

119896119894119895

= (119870119873

1198941 119870119873

1198942 119870

119873

119894119894minus1 119870119873

119894119894 119870

119873

119894119899)

(17)


119894

119867119894= sum

119895=1119895 =119894

119896119873

119894ln 119896119873119894

(119894 = 1 2 119899) (18)


119867minus1

119894=

1

119867119894

(19)


119908119894=

1198671

119894

sum119899

119895=119894119867minus1

119895

(20)



119882 = (1199081 1199082 119908

119899) (21)

(2) Assume 119882max = max(1199081 1199082 119908





119898lowast

119894(119860119896) = 120572119894119898119894(119860119894) (22)


119886119894=

119908119894

119908max (23)





Waveform index [1284 1447] [119 1327] [1172 1327] [1172 1312] [119 1494]

Peak index [3140 5959] [1886 4818] [2215 4818] [2026 3686] [2503 6105]

Pulse index [4068 8625] [2320 6044] [2664 6044] [2460 4590] [3098 8602]

Margin index [4935 1076] [2683 7346] [3044 7346] [2832 5647] [3647 1072]

Margin index [3027 1079] [2531 4051] [2167 385] [2167 385] [2531 8378]


119898(Φ) = 0

119898 (119860) = 119901 (119860) + 119896 sdot 119902 (119860) 119860 = Φ

119901 (119860) = sum

119860isin119864119894cap119899

119894=1119860119894=119860

119898lowast

1119860 (1)119898

lowast

2119860 (2) sdot sdot sdot 119898

lowast

119899119860 (119899)

119896 = sum

119860isin119864119894cap119899

119894=1119860119894=119860

119898lowast

1119860 (1)119898

lowast

2119860 (2) sdot sdot sdot 119898

lowast

119899119860 (119899)

119902 (119860) =1

119899

119899

sum

119894=1

119898lowast

1(119860)

119898 (Φ) = 1 minus sum

119860subΘ

119898(119860)

(24)









119894)

can be obtained




119894) after







119891 peak



119891 and kurtosis




(a) (b)

(c) (d)

(e) (f)

(g) (h)






Waveform index

Peak index

Pulse index

Margin index

Kurtosis index

0

005

01

015

02

025

03


Prob

abili

ty v

alue






5 Conclusion


Bear

ing

wea

r

Bear

ing

crac

k ou

tside

r

Bear

ing

crac

k in

sider

Curv

ed

shaft

Lack

of

bear

ing0

01

02

03

04

05


obab

ility

val






Acknowledgments




References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













120576minus119879

= 119864 (|120576|119879

) =1

119873

119873

sum

119894=1

10038161003816100381610038161205761198941003816100381610038161003816

119879

(7)


Δ120578119909=

119879radic120576minus119879

119898radic120576minus119898

(8)


|120576119895|


1 1205762 120576


120576minus119879



Δ120578119909=

[int+infin

minusinfin

|119909|119879

120585 (119909) 119889119909]

1119879

[int+infin

minusinfin

|119909|119898

120585 (119909) 119889119909]

1119898

=

119879radic120576minus119879

119898radic120576minus119898

isin [119888120576 119889120576]

=

[ 119879radic120572119879119879radic120573119879]

[ 119898radic120572119898119898radic120573119898]

isin [

119879radic120572119879

119898radic120573119898

119879radic120573119879

119898radic120572119898

]

(9)


119891119862119891 119868119891119862119871119891 and119870V Each




119892119894(119909) = 119896

119894 (119894 = 1 2 119888) (10)

33 Evidence Theory






119898(0) = 0

sum

119860subeΘ

119898(119860) = 1(11)



Bel (119861) = sum

119861sube119860

119898(119861) (12)



119875119897 (119860) = sum

119861cap119860 =0

119898(119861) (13)


119898 (Θ) = sum

119861⋂119860=Φ

119898(119861)

119898 (119860) =

sumcap119860119894=119860

prod119898

119894=1119898119894(119860119894)


prod119898

119894=1119898119894(119860119894)=

sumcap119860119894=119860

prod119898

119894=1119898119894(119860119894)

1 minus 119870

(14)








119895(119894 =


[16]

119870119894= (1198961198941 1198961198942 119896

119894119894minus1 119896119894119894 119896

119894119899) (15)

wherein

119870119894119895= sum

119860119894cap119860119895=Φ119860119894isin119864119894 119860119895isin119864119895

119898119894(119860119894)119898119895(119860119895)

(119894 = 1 2 119899)

(16)


normalization

119870119873

119894=(1198961198941 1198961198942 119896

119894119894minus1 119896119894119894 119896

119894119899)

sum119895=1119895 =119894

119896119894119895

= (119870119873

1198941 119870119873

1198942 119870

119873

119894119894minus1 119870119873

119894119894 119870

119873

119894119899)

(17)


119894

119867119894= sum

119895=1119895 =119894

119896119873

119894ln 119896119873119894

(119894 = 1 2 119899) (18)


119867minus1

119894=

1

119867119894

(19)


119908119894=

1198671

119894

sum119899

119895=119894119867minus1

119895

(20)



119882 = (1199081 1199082 119908

119899) (21)

(2) Assume 119882max = max(1199081 1199082 119908





119898lowast

119894(119860119896) = 120572119894119898119894(119860119894) (22)


119886119894=

119908119894

119908max (23)





Waveform index [1284 1447] [119 1327] [1172 1327] [1172 1312] [119 1494]

Peak index [3140 5959] [1886 4818] [2215 4818] [2026 3686] [2503 6105]

Pulse index [4068 8625] [2320 6044] [2664 6044] [2460 4590] [3098 8602]

Margin index [4935 1076] [2683 7346] [3044 7346] [2832 5647] [3647 1072]

Margin index [3027 1079] [2531 4051] [2167 385] [2167 385] [2531 8378]


119898(Φ) = 0

119898 (119860) = 119901 (119860) + 119896 sdot 119902 (119860) 119860 = Φ

119901 (119860) = sum

119860isin119864119894cap119899

119894=1119860119894=119860

119898lowast

1119860 (1)119898

lowast

2119860 (2) sdot sdot sdot 119898

lowast

119899119860 (119899)

119896 = sum

119860isin119864119894cap119899

119894=1119860119894=119860

119898lowast

1119860 (1)119898

lowast

2119860 (2) sdot sdot sdot 119898

lowast

119899119860 (119899)

119902 (119860) =1

119899

119899

sum

119894=1

119898lowast

1(119860)

119898 (Φ) = 1 minus sum

119860subΘ

119898(119860)

(24)









119894)

can be obtained




119894) after







119891 peak



119891 and kurtosis




(a) (b)

(c) (d)

(e) (f)

(g) (h)






Waveform index

Peak index

Pulse index

Margin index

Kurtosis index

0

005

01

015

02

025

03


Prob

abili

ty v

alue






5 Conclusion


Bear

ing

wea

r

Bear

ing

crac

k ou

tside

r

Bear

ing

crac

k in

sider

Curv

ed

shaft

Lack

of

bear

ing0

01

02

03

04

05


obab

ility

val






Acknowledgments




References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119875119897 (119860) = sum

119861cap119860 =0

119898(119861) (13)


119898 (Θ) = sum

119861⋂119860=Φ

119898(119861)

119898 (119860) =

sumcap119860119894=119860

prod119898

119894=1119898119894(119860119894)


prod119898

119894=1119898119894(119860119894)=

sumcap119860119894=119860

prod119898

119894=1119898119894(119860119894)

1 minus 119870

(14)








119895(119894 =


[16]

119870119894= (1198961198941 1198961198942 119896

119894119894minus1 119896119894119894 119896

119894119899) (15)

wherein

119870119894119895= sum

119860119894cap119860119895=Φ119860119894isin119864119894 119860119895isin119864119895

119898119894(119860119894)119898119895(119860119895)

(119894 = 1 2 119899)

(16)


normalization

119870119873

119894=(1198961198941 1198961198942 119896

119894119894minus1 119896119894119894 119896

119894119899)

sum119895=1119895 =119894

119896119894119895

= (119870119873

1198941 119870119873

1198942 119870

119873

119894119894minus1 119870119873

119894119894 119870

119873

119894119899)

(17)


119894

119867119894= sum

119895=1119895 =119894

119896119873

119894ln 119896119873119894

(119894 = 1 2 119899) (18)


119867minus1

119894=

1

119867119894

(19)


119908119894=

1198671

119894

sum119899

119895=119894119867minus1

119895

(20)



119882 = (1199081 1199082 119908

119899) (21)

(2) Assume 119882max = max(1199081 1199082 119908





119898lowast

119894(119860119896) = 120572119894119898119894(119860119894) (22)


119886119894=

119908119894

119908max (23)





Waveform index [1284 1447] [119 1327] [1172 1327] [1172 1312] [119 1494]

Peak index [3140 5959] [1886 4818] [2215 4818] [2026 3686] [2503 6105]

Pulse index [4068 8625] [2320 6044] [2664 6044] [2460 4590] [3098 8602]

Margin index [4935 1076] [2683 7346] [3044 7346] [2832 5647] [3647 1072]

Margin index [3027 1079] [2531 4051] [2167 385] [2167 385] [2531 8378]


119898(Φ) = 0

119898 (119860) = 119901 (119860) + 119896 sdot 119902 (119860) 119860 = Φ

119901 (119860) = sum

119860isin119864119894cap119899

119894=1119860119894=119860

119898lowast

1119860 (1)119898

lowast

2119860 (2) sdot sdot sdot 119898

lowast

119899119860 (119899)

119896 = sum

119860isin119864119894cap119899

119894=1119860119894=119860

119898lowast

1119860 (1)119898

lowast

2119860 (2) sdot sdot sdot 119898

lowast

119899119860 (119899)

119902 (119860) =1

119899

119899

sum

119894=1

119898lowast

1(119860)

119898 (Φ) = 1 minus sum

119860subΘ

119898(119860)

(24)









119894)

can be obtained




119894) after







119891 peak



119891 and kurtosis




(a) (b)

(c) (d)

(e) (f)

(g) (h)






Waveform index

Peak index

Pulse index

Margin index

Kurtosis index

0

005

01

015

02

025

03


Prob

abili

ty v

alue






5 Conclusion


Bear

ing

wea

r

Bear

ing

crac

k ou

tside

r

Bear

ing

crac

k in

sider

Curv

ed

shaft

Lack

of

bear

ing0

01

02

03

04

05


obab

ility

val






Acknowledgments




References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Waveform index [1284 1447] [119 1327] [1172 1327] [1172 1312] [119 1494]

Peak index [3140 5959] [1886 4818] [2215 4818] [2026 3686] [2503 6105]

Pulse index [4068 8625] [2320 6044] [2664 6044] [2460 4590] [3098 8602]

Margin index [4935 1076] [2683 7346] [3044 7346] [2832 5647] [3647 1072]

Margin index [3027 1079] [2531 4051] [2167 385] [2167 385] [2531 8378]


119898(Φ) = 0

119898 (119860) = 119901 (119860) + 119896 sdot 119902 (119860) 119860 = Φ

119901 (119860) = sum

119860isin119864119894cap119899

119894=1119860119894=119860

119898lowast

1119860 (1)119898

lowast

2119860 (2) sdot sdot sdot 119898

lowast

119899119860 (119899)

119896 = sum

119860isin119864119894cap119899

119894=1119860119894=119860

119898lowast

1119860 (1)119898

lowast

2119860 (2) sdot sdot sdot 119898

lowast

119899119860 (119899)

119902 (119860) =1

119899

119899

sum

119894=1

119898lowast

1(119860)

119898 (Φ) = 1 minus sum

119860subΘ

119898(119860)

(24)









119894)

can be obtained




119894) after







119891 peak



119891 and kurtosis




(a) (b)

(c) (d)

(e) (f)

(g) (h)






Waveform index

Peak index

Pulse index

Margin index

Kurtosis index

0

005

01

015

02

025

03


Prob

abili

ty v

alue






5 Conclusion


Bear

ing

wea

r

Bear

ing

crac

k ou

tside

r

Bear

ing

crac

k in

sider

Curv

ed

shaft

Lack

of

bear

ing0

01

02

03

04

05


obab

ility

val






Acknowledgments




References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)

(c) (d)

(e) (f)

(g) (h)






Waveform index

Peak index

Pulse index

Margin index

Kurtosis index

0

005

01

015

02

025

03


Prob

abili

ty v

alue






5 Conclusion


Bear

ing

wea

r

Bear

ing

crac

k ou

tside

r

Bear

ing

crac

k in

sider

Curv

ed

shaft

Lack

of

bear

ing0

01

02

03

04

05


obab

ility

val






Acknowledgments




References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Waveform index

Peak index

Pulse index

Margin index

Kurtosis index

0

005

01

015

02

025

03


Prob

abili

ty v

alue






5 Conclusion


Bear

ing

wea

r

Bear

ing

crac

k ou

tside

r

Bear

ing

crac

k in

sider

Curv

ed

shaft

Lack

of

bear

ing0

01

02

03

04

05


obab

ility

val






Acknowledgments




References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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Research Article A Diagnosis Method for Rotation Machinery ...downloads.hindawi.com/journals/mpe/2015/563954.pdf · diagnosis of the rotatingmachinery fault signal is uncertain. In