research article intelligent diagnosis method for...

Research ArticleIntelligent Diagnosis Method for Centrifugal Pump SystemUsing Vibration Signal and Support Vector Machine

Hongtao Xue1 Zhongxing Li1 Huaqing Wang2 and Peng Chen3

1 School of Automotive and Traffic Engineering Jiangsu University 301 Xuefu Road Zhenjiang Jiangsu 212013 China2 Beijing University of Chemical Technology 15 Beisanhuan East Road Chaoyang Beijing 100029 China3 Graduate School of Bioresources Mie University 1577 Kurimamachiya-cho Tsu Mie 514-8507 Japan

Correspondence should be addressed to Huaqing Wang wanghq bucthotmailcom and Peng Chen chenbiomie-uacjp

Received 27 May 2014 Revised 28 August 2014 Accepted 28 August 2014 Published 4 November 2014

Academic Editor Didier Remond

Copyright copy 2014 Hongtao Xue et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper proposed an intelligent diagnosis method for a centrifugal pump system using statistic filter support vector machine(SVM) possibility theory and Dempster-Shafer theory (DST) on the basis of the vibration signals to diagnose frequent faultsin the centrifugal pump at an early stage such as cavitation impeller unbalance and shaft misalignment Firstly statistic filter isused to extract the feature signals of pump faults from the measured vibration signals across an optimum frequency region andnondimensional symptom parameters (NSPs) are defined to represent the feature signals for distinguishing fault types Secondlythe optimal classification hyperplane for distinguishing two states is obtained by SVM and NSPs and its function is defined assynthetic symptom parameter (SSP) in order to increase the diagnosisrsquo sensitivity Finally the possibility functions of the SSP areused to construct a sequential fuzzy diagnosis for fault detection and fault-type identification by possibility theory and DST Theproposed method has been applied to detect the faults of the centrifugal pump and the efficiency of the method has been verifiedusing practical examples

1 Introduction

Pumps come in several types such as centrifugal turbo pro-peller and positive displacement Irrigation pumping plantsare usually of the centrifugal type [1] A centrifugal pumpplays an important role in industries However faults ofpump can cause a high rate of energy loss associated withperformance degradation even the breakdown of a wholesystem and then lead to substantial economic losses There-fore condition diagnosis of the pump system at an early stageis very important

Different approaches have been used for fault detection ofcentrifugal pumps Studies in [2ndash4] adopted indirect param-eters or interrelated parameters to identify the faults but didnot consider the features of a vibration signature Howeverit is gradually being understood that vibration signature isthe most revealing information reflecting the condition ofrotating machinery [5ndash7] Vibration signals were employedfor fault detection and condition monitoring in [8ndash21] so it

is important that fault signal should be sensitively extractedfrom the measured signal when fault occurs However it isdifficult since fault signal is so weak that it is often buriedin strong noise especially at an early stage Studies in [8ndash14] presented that wavelet analysis could detect effectivelycondition change However different wavelet basis functionswere adopted Moreover the features of vibration signalswere captured in different frequency areas [14] Thereforethe line between the application of wavelet analysis and theconstruction of an automatic system is drawn

For noise cancelling namely extracting fault signalmanymethods have been proposed For example band-pass filter[22] adaptive filter [23] Wiener filter [24] and Kalman filter[25] and so forth However in the field of machinery diagno-sis these methods cannot always be applied to failure signalextraction due to the following adverse conditions Firstly inthe case of the band-pass filter the wide band noise cannotbe cancelled Secondly when applying the adaptive filter fornoise cancelling the reference noise must be simultaneously

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 407570 14 pageshttpdxdoiorg1011552014407570

2 Shock and Vibration

measured with the signal Simultaneous measurement of thereference noise is not easily realized in most cases of faultdiagnosis Finally the noise cannot be effectively removedby the Wiener filter and Kalman filter if noise and signal donot follow the normal distribution In order to overcome theproblems above we used the statistic filter to extract the faultsignal from the vibration signal measured in the abnormalstate of a machine

Fault diagnosis can hardly do without intelligent systemssuch as neural networks (NN) and support vector machine(SVM) Studies in [14ndash16] have been carried out to investigatethe use of NN for detecting fault and identifying fault typesHowever the conventional NN cannot reflect the possibilityof ambiguous diagnosis problems and will never convergewhen the first layer symptom parameters have the same val-ues in different states [26] Studies in [17ndash20] have employedSVMs to perform fault diagnosis and the capability ofSVMswhich can efficiently perform a nonlinear classificationwith kernel function has been confirmed However it isdifficult to determine the parameters for a given value ofthe regularization and kernel parameters and to choose anappropriate kernel function

For the above reasons in order to process the uncertainrelationship between symptom parameters (SPs) and ma-chinery conditions improve the efficiency and accuracy offault diagnosis at an early stage and achieve an automaticsystem from measured signals the authors propose an intel-ligent diagnosis for a pump system using support vectormachine (SVM) possibility theory and Dempster-Shafertheory (DST) on the basis of the vibration signals to detectfaults and identify fault types at an early stage as is shown inthe flowchart in Figure 1 The statistic filter is used to detectand extract the failure signals of pump faults across optimumfrequency regions and the failure signals extracted arereflected by nondimensional symptom parameters (NSPs) Inorder to increase the diagnosisrsquo sensitivity a synthetic symp-tom parameter (SSP) is defined with the function of the opti-mal classification hyperplane obtained using SVM Then thepossibility distributions of the SSPs are used to detect faultsand identify fault types by possibility theory andDST Finallythe proposedmethod is evaluated using practical examples ofcentrifugal pump which shows a good performance

2 Experimental Centrifugal Pump System forCondition Diagnosis

Figure 2 shows the testing bed consisting of a pump a motorand a close-loop water piping system The SF-JRO seriesmotor is employed to drive the pump through a couplingand the rotating speed can be controlled via control panelThe experimental pump is a centrifugal pump described asfollows HONDA Pump Type HAS (Volute Pump Horizon-tal Type) Head 40 Output 37 kW and Capacity 75m3hThe capacity of the tank is based on the maximum flow ratewhich can also be controlled by the valve control system Fiveaccelerometers are adopted to acquire the vibration signalswith a sampling frequency of 50 kHz One is mounted onthe pump housing two in the piping direction and vertical

Measure vibration signals in each state

Obtain optimal classification hyperplane by SVM and NSPs

Calculate NSPs in each state

Calculate SSP and construct possibility distributions of SSP

Diagnose using possibility theory and DST

Extract feature signals by statistic filter

Figure 1 Flowchart of intelligent diagnosis

Figure 2 Photograph of the experiment centrifugal pump system

direction of the pump inlet respectively the rest in the pipingdirection and horizontal direction of the pump outlet asshown in Figure 3

Cavitation phenomenon is one of the sources of instabil-ity in a centrifugal pump Bubbles collapse abruptly leadingto damage of the pump and generation of crackling soundand vibration [27] Other faults such as impeller unbalanceand shaftmisalignment between themotor and the pump arediscriminated easily but often occur in pump system Thesefaults can cause serious machine accidents and bring greatproduction losses Therefore these states with normal statewill be discussed to examine whether incipient faults of apump were able to be detected using the proposed methodin a later sectionWhen the pump speed was at 3500 rpm andthe valves in the suction and discharge lines were fully openthe pumpwas in the best situation of operationThe conditionwas seen as normal state Then the conditions of the speedand the valve in the discharge line remained unchanged thevalve in the suction line was slowly turned down until littlebubbles appeared At present the condition was the cavitation

Shock and Vibration 3

Accelerometer

Figure 3 The location of sensors

state with the mild abnormal degree Sequentially the valvein suction line was turned down the amounts of bubbleswere used to decide the cavitation state with the medium andsevere abnormal degree In the case of impeller unbalance dueto impeller damage the damage areas from three impellerswere 25mm2 100mm2 and 225mm2 respectively Then thestates that used three impellers were defined as impellerunbalance with the mild medium and severe abnormaldegree Similarly the states that the pump side of the shaftwas installed to deviate 01mm 05mm and 10mm beyondits specified limit were defined as shaft misalignment withdifferent abnormal degrees

3 Feature Extraction Using Statistic Filter

31 Statistic Filter Statistic filter is a filter which takes in apacket and decides whether to accept or reject it on the basisof the statistical approach [28] In the field ofmachinery diag-nosis studies in [29] have investigated that the filter basedon means of statistical tests of spectrums was convenient andeffective for extracting pure failure signalsHere statistic filterbased ondistinction index (DI) of spectrums betweennormalsignal andmeasured signal is proposed as shown in Figure 4

Measured normal signal and diagnosis signal can bedivided into119873 parts Spectrum analysis is performed on eachpart which generates one spectrum content Going throughall signal parts the spectrum content119865

119894119895(119891119896) of the part 119895 (119895 =

1 2 119873) at the frequency 119891119896(119891119896= 119896 sdot 119891max119872 119896 =

1 2 119872) can be obtained where 119894 is the style of signalsuch as normal signal (119899) and diagnosis signal (119889) 119891max and119872 are the maximal frequency and the number of spectrumanalysis Thus the spectrum components 119865

119899(119891119896) and 119865

119889(119891119896)

of 119873 signal parts under normal and diagnosis states at thefrequency 119891

119896can be denoted in (1) and it is verified that

119865119899(119891119896) and 119865

119889(119891119896) conform to normal distribution as shown

in Figure 5 Consider

119865119899(119891119896) = 119865

1198991(119891119896) 1198651198992(119891119896) 119865

119899119873(119891119896)

119865119889(119891119896) = 119865

1198891(119891119896) 1198651198892(119891119896) 119865

119889119873(119891119896)

(1)

In order to determine whether there is significant differ-ence between the spectrums 119865

119899(119891119896) and 119865

119889(119891119896) of normal and

diagnosis states at the frequency 119891119896 distinction index (DI)

is defined as follows as an index of the quality to evaluatethe spectrumdifference between normal and diagnosis statesSuppose that119909

1and1199092are the values of119865

119899(119891119896) and119865

119889(119891119896) and

conform to the normal distributions119873(1205831 1205901) and119873(120583

2 1205902)

respectively Here 1205831 1205901and 120583

2 1205902are the mean values and

standard deviations of normal and diagnosis states at thefrequency 119891

119896 The larger the value of |119909

1minus 1199092| is the higher

the difference of two states will be Because 119911 = 1199092minus 1199091

is also distributed 119873(1205832minus 1205831 1205902

1+ 1205902

1) we can denote the

density function of 119911 in (2a)When 1205832ge 1205831 the probability of

119911 = 1199092minus1199091(1199092lt 1199091) can be calculated by (2b) of course we

can obtain the same conclusion when 1205832le 1205831 Standardizing

the value of 119911 namely 120583 = (119911 minus (1205832minus1205831))radic120590

2

1+ 1205902

1 (2b) can

be transformed into (2c) Consider

119891 (119911) =1

radic2120587 (1205902

1+ 1205902

2)

exp(minus119911 minus (120583

2minus 1205831)2

2 (1205902

1+ 1205902

2)

) (2a)

119901 = int

0

minusinfin

119891 (119911) 119889119911 (2b)

119901 =1

radic2120587int

minusDI

minusinfin

exp(minus1205832

2)119889120583 (2c)

HereDI = (1205832minus1205831)radic1205902

1+ 1205902

1 Considering the condition

of 1205832le 1205831 the DI value is calculated by the following

DI =10038161003816100381610038161205832 minus 1205831

1003816100381610038161003816

radic1205902

1+ 1205902

2

(2d)

According to the nature of the normal distribution theprobability of 119911 = 119909

2minus 1199091(1199092ge 1199091) is also 119901 then the

probability of 119911 is 2119901 However 119911 shows the overlapping partof normal and diagnosis states the distinction probability ofnormal state or diagnosis state is 1 minus 2119901 then the distinctionrate (DR) is defined as follows

DR = 1 minus 2119901 (2e)

In [30ndash32] DI has been used to judge the sensitivityof a symptom parameter (SP) and it also has been provedthat the larger the value of the DI the higher the DR valuewill be and then the higher the sensitivity of the SP will beMoreover when the DI value is more than 128 the DR valueis more than 09 but the probability of 119911 is less than 01In statistics significance levels such as 01 005 or 001 areused depending on the field of study Here 01 level is usedto determine whether there is a difference between 119865

119899(119891119896)

and 119865119889(119891119896) and then the DI value of 128 is watershed in the

statistic filter based on distinction index of spectrums Whenthe DI value of 119865

119899(119891119896) and 119865

119889(119891119896) is smaller than 128 there

is smaller difference between 119865119899(119891119896) and 119865

119889(119891119896) and then it

is assumed no difference Therefore the spectrum 119865119889(119891119896) of

diagnosis state at the frequency 119891119896will be removed as noise

by a filter Otherwise 119865119889(119891119896) are left with no change For

example the DI value between 119865119899(1198911) and 119865

119889(1198911) as shown


Estimated fault spectrum(when fault occurs)

Statistical comparison between spectrumsof normal signal and diagnosis signal

Estimated noise spectrum

The signal measured for diagnosis

Normal signalmeasured in advance

IFFT

Estimated fault time signal

Distinction indexof spectrum

Figure 4 The principle of statistic filter based on distinction index (DI) of spectrums

f1 f2 f3 middot middot middot

(Hz)

0

1 Part 1 Part 1

Part 2 Part 2

Part N Part N

Spec

trum

0

1

Spec

trum

0

1

Spec

trum

0

1

Spec

trum

f1 f2 f3 middot middot middot Fk

Fk f1 f2 f3 middot middot middot Fk



(Hz)


(Hz) (Hz)

(Hz)

(Hz)

0

1

Spec

trum

0

1

Spec

trum

Normal signal

Fn(f1)

05 08 F(f1)

F(f2)

F(fk)

Distributions

Distributions

Distributions

Fn(f1)

Fn(f2)Fn(f2)

Fn(fk)Fn(fk)

Fd(f1)

Fd(f2)Fd(f2)

Fd(fk)Fd(fk)

Fd(f1)

042 048

001 012

Diagnosis signal

(A1)

(A2)

(Ak)

Figure 5 Spectrums of each part between normal and diagnosis signals and the normal distributions of 119865119899(119891119896) and 119865

119889(119891119896) at the frequencies

1198911and 119891

2

in Figure 5(A1) is 187 then 119865119889(1198911) is left However the DI

value between 119865119899(1198912) and 119865

119889(1198912) as shown in Figure 5(A2)

is 079 then 119865119889(1198912) is removed from original spectrumThus

a filtered 119865119889(119891119896) can be obtained which is controlled by DI A

simple example is introduced to explain a statistic filter basedon distinction index (DI) of spectrum as shown in Figure 6Here black squares are the spectrum values of normal signaland black dots are the spectrum values of diagnosis signalIn these points the DI values of spectrums between normal

and diagnosis signals at 1198912 1198914 1198916 1198918 119891

119872are smaller than

128 theDI values at1198911 1198913 1198915 1198917 aremore than 128Thus

the filtered diagnosis spectrums which are indicated by blacktriangle are estimated as the fault components in frequencydomain and the fault signal in time domain can be obtainedby using inverse fast Fourier transform (IFFT)

Moreover we have tried other values of DI to extract thefaults signal by statistic filter for the condition diagnosis of thecentrifugal pump system and the optimum value is still 128


Table 1 DI values and standard deviation of DI values of 3 NSPs between two states from the signals estimated by statistic filter

Two states In mild step In medium step In severe step DIrsquos standard deviation1198751

1198752

1198753

1198751

1198752

1198753

1198751

1198752

1198753

1198751

1198752

1198753

Normal state and abnormal state 054 061 073 083 079 092 129 116 131 038 028 030Cavitation and impeller unbalance 061 055 080 082 065 088 127 104 094 034 026 007Cavitation and shaft misalignment 058 064 072 076 090 087 103 098 126 023 018 028Impeller unbalance and shaft misalignment 068 077 079 093 084 103 111 109 117 022 017 019

Table 2 DI values and standard deviation of DI values of 3 NSPs between two states from original signals


1198752

1198753

1198751

1198752

1198753

1198751

1198752

1198753

1198751

1198752

1198753


0f1f2f3f4f5

f6f7f8middot middot middot

fM

Spec

trum

Spectrum (Hz)Normal signal Diagnosis signal Signal filtered

Figure 6 Explanation of statistic filter based on distinction index(DI) of spectrums

32 Analysis of Experimental Signals Using Statistic Filter Tomake the signals comparable regardless of differences inmag-nitude the signals of each state are normalized by the follow-ing formula

119909119894=1199091015840

119894minus 1205831015840

1205901015840 (3)

where 1199091015840119894(119894 = 1 2 119873) are the original signal series 1205831015840

and 1205901015840 are the mean and standard deviation of 1199091015840119894 and 119909

119894(119894 =

1 2 119873) are the normalization signal seriesIn present work the normalized signal of each state has

been divided into 128 parts respectively Then statistic filterbased on distinction index of spectrum is performed toobtain the estimated signals under different states Figure 7shows the estimated signal in time domain of shaft misalign-ment when the failure is sever It is obvious that informationrepresenting shaft misalignment has been extracted

4 Nondimensional Symptom Parameters forIntelligent Diagnosis

For intelligent diagnosis symptom parameters (SPs) arerequired and can sensitively detect the occurrence of fault anddistinguish the fault types A large set of symptomparametershas been defined in the pattern recognition field [31] Here

the nondimensional symptom parameters (NSPs) in timedomain commonly used for the fault diagnosis of plantmachinery are considered

When given the digital data 119909119894(119894 = 1 2 119873) of the

vibration signal where 119873 is the number of the signal the 3NSPs in the time domain are defined as follows

1198751=120590

|119909|

1198752=sum119873

119894=1(119909119894minus 119909)3

119873 sdot 1205903

1198753=sum119873

119894=1(119909119894minus 119909)4

119873 sdot 1205904

(4)

where 119909 and 120590 are the mean value and standard deviationof the vibration signal 119909

119894 |119909119894| is the absolute value of the

vibration signal 119909119894and |119909| is the mean value of |119909

119894|

Then the estimated signals obtained in Section 3 usingstatistic filter are performed to calculate 3 NSPs In this paper3 typical faults have been investigatedwith different abnormaldegrees in experiment and then these fault states with thesame abnormal degree are called abnormal state with thesame abnormal degree for purposes of discussion Thus 3NSPs of these fault states with the same abnormal degree areunited to 3 NSPs of abnormal state with the same abnormaldegree Therefore the DI values of 3 NSPs between any twostates in the same abnormal degree are calculated as shownin Table 1 Moreover original signals are also done with thesame work and DI values of which are shown in Table 2

It is obvious that the DI values of 3 NSPs from the signalsestimated by statistic filter are larger than from originalsignals However the DI values of these SPs are yet lowespecially at an incipient stage of a fault Thus the choice ofan effective SP will expend time and effort and the differenti-ation degree of the SP may or may not be high In this studya newmethod in which some SPs integrate an effective SP fordiagnosis by introducingweight coefficient is proposedHere


Time signal of normal state (N)

0 005 01

(s)

(s)

(s)

0 005 01

0 005 01

minus2

0

2A

mpl

itude

(V)

minus2

0

2

Am

plitu

de (V

)

minus2

0

2

Am

plitu

de (V

)

Time signal of shaft misalignment (M)

Estimated signal of misalignment (M)

Standard of statistic filter (SF)

FFT

Spectrum of normal state (N)

Spec

trum

0 5 10 15 20 25

(kHz)

(kHz)

(kHz)

0 5 10 15 20 25

0 5 10 15 20 25

0

8

Spec

trum

0

8

Spec

trum

0

8

Spectrum of shaft misalignment (M)

Spectrum filtered by SF

SF

IFFT

Figure 7 Estimated signal of shaft misalignment in sever step using statistic filter based on distinction index of spectrum

3NSPs are used to integrate an effective SP and this integratedSP is called ldquosynthetic symptom parameter (SSP)rdquo

5 Application of SVM for SyntheticSymptom Parameter

51 Support Vector Machine (SVM) SVM is a relatively newcomputational learningmethod based on the statistical learn-ing theory and the basic idea is to create an optimal classifica-tion hyperplane between the two classes and ensure that thedistance between the boundary and the nearest data point ineach class ismaximized [33] SVMcan efficiently performnotonly a linear classification but also a nonlinear classificationusing kernel function In general a classification hyperplanecan be expressed as follows

120596 sdot 119909 + 119887 = 0 (5)

where 119909 is the sample vector 120596 is a vector that representedthe weight coefficients of 119909 and 119887 is a classification threshold

Recently almost all of practical applications of SVMshaveemployed different kernel functions such as polynomial ker-nel and radial basis function (RBF) kernel to achieve linearclassifications in high-dimensional feature spaces Howeveras said in introduction section it is difficult to determine theparameters for a given value of the regularization and kernelparameters and to choose an appropriate kernel function Butsoft margin SVM was proposed to increase the enchantmentof SVM [34] Soft margin SVM is a modified maximum

x1

x2

x3

Figure 8 The optimal classification hyperplane for identifying twostates using soft margin SVM

margin idea that permits minimum error and relaxes thecondition for the optimal classification hyperplane In orderto explain the soft margin SVM an example is shownin Figure 8 with 3-dimensional situation The white circlesrepresent state 1 and the black points represent state 2 119909

1 1199092

and 1199093represent 3-dimensional sample vector The colorized

plane is the optimal classification hyperplaneSoft margin SVM not only constructs the optimal clas-

sification hyperplane when nonlinear data is separated butalso has a strong generalization capacity when only a smallamount of training samples are available

In order to distinguish states of machinery as precisely aspossible the excellent symptom parameters must be definedfor the automatic diagnosis The symptom parameters mustbe able to sensitively reflect the characteristics of states


However there is not an acceptable method for extractingsymptom parameters from signals measured in each state[35 36] The larger the number of states there is the moredifficult the extraction of symptom parameters is In manycases this has been done merely by trial and errorThe aboveprocess not only is time-consuming and labor intensive butalso cannot always ensure that excellent symptomparameterswill be found

In order to resolve this problem we propose the newmethod be called ldquosequential diagnosis using synthetic symp-tom parameters generated by soft margin SVMrdquo By thismethod the optimum synthetic symptom parameters for dis-tinguishing all states can be sequentially and quickly searchedout Because the synthetic symptom parameters generated bythe soft margin SVM can only be used to distinguish towstates we will sequentially and precisely diagnose states asshown Figure 11

52 Synthetic Symptom Parameter The paper proposes softmargin SVM to achieve the optimization of integratingprimitive SPs in the field of condition diagnosis The optimalclassification hyperplane is shown in (5) where 120596 is theoptimal vector of the weight coefficients and determinesthe diagnostic sensitivity of the SSP Then SSP is defined asfollows

SSP = 120596 sdot [1198751 1198752 sdot sdot sdot 119875119899]1015840

+ 119887 (6)

In order to enhance the sensitivity of extracted features tofailures especially at the incipient stage of a fault 3 NSPs ofeach state are input SVMs to train the classifiers For examplein the first step the diagnosis goal is to detect whether thereis fault so these two classes are normal and abnormal statesIn a normal state 3 NSPs 119909

119894= 1198751119894 1198752119894 1198753119894 (119894 = 1 2 119868)

are extracted to associate with labels 119910119894= 1 (119894 = 1 2 119868)

and to recombine the training data (119909119894 119910119894) (119894 = 1 2 119868) for

a normal state Similarly the training data of abnormal state(119909119895 119910119895) (119895 = 1 2 119869) are obtained to associate with labels

119910119895= minus1 (119895 = 1 2 119869) Here 119868 and 119869 are the class number

of 3 NSPs from normal and abnormal statesWhen these dataare input into SVMs to train for the optimal classificationhyperplane shown as Figure 8 weight coefficient120596 = 120596

1 1205962

1205963 and classification threshold 119887 are obtained and then

the SSPs are obtained shown in Table 3 Then new SSPsbetween two faults are calculated based on the estimatedsignals and the DI values of new SSPs are also calculated asshown in Table 4TheDI values of SSPs are larger than the DIvalues of 3 NSPs In addition the viewpoints about trainingperformance are as follows

(1) Soft margin SVM is a modified maximum marginidea that permits minimum error and relaxes thecondition for the optimal classification hyperplaneTherefore it is not important to give the training per-formance

(2) Training data are input into soft margin SVM onlyto build the optimal classification hyperplane In thispaper the goal is only to define synthetic symptomparameter (SSP)

Table 3 Synthetic symptom parameters (SSPs) between two states

Two states Synthetic symptom parameter (SSP)Normal state andabnormal state SSP

1= minus3587119875

1+ 6109119875

2minus 2234119875

3+ 10006

Cavitation andimpeller unbalance SSP

2= 0164119875

1+ 0897119875

2minus 3220119875

3minus 0107

Cavitation andshaft misalignment SSP

3= minus0593119875

1+ 0031119875

2+ 1071119875

3+ 1934

Impeller unbalanceand shaftmisalignment

SSP4= 1462119875

1minus 2843119875

2minus 4566119875

3+ 2006

Table 4 DI values of new SSPs in the same abnormal degree fromthe estimated signals

Two states In mildstep

In mediumstep

In severestep

Normal state and abnormalstate 128 157 172

Cavitation and impellerunbalance 109 126 154

Cavitation and shaftmisalignment 098 129 147

Impeller unbalance and shaftmisalignment 117 138 156

6 Sequential Fuzzy DiagnosisUsing Possibility Theory andDempster-Shafer Theory

61 Possibility Theory Possibility theory is a mathematicaltheory for dealing with certain types of uncertainty and analternative to probability theory Professor Zadeh first intro-duced possibility theory in 1978 as an extension of his theoryof fuzzy sets and fuzzy logic [37] Dubois and Prade furthercontributed to its development [38] Recently possibilitytheory has been used for fault diagnosis [14 39 40] In [1439 40] possibility theory was applied to condition diagnosisin rotating machinery under varying rotating speeds toprocess the uncertain relationship between the symptomsand fault types In the paper for increasing the diagnosisrsquosensitivity and improving the identification of ambiguousstate possibility theory is used to construct the system ofsequential fuzzy diagnosis

For fuzzy inference the membership function of a SP isnecessary [14 39 40] which can be obtained fromprobabilitydensity functions of the SP using possibility theory In thispaper synthetic symptom parameter (SSP) is selected as thefeatures for fault identification and it is verified that a SSPfollows the Weibull distribution Then probability densityfunctions of a SSP (119891(119909)) can be changed to possibilityfunction (119901(119909)) by the following formulae

119901 (119909119894) =

119873

sum

119896=1

min 120582119894 120582119896 (119894 119896 = 1 2 119873) (7)


120582119894and 120582

119896can be calculated as follows

120582119894= int

119909119894

119909119894minus1

119898

120578sdot (119909 minus 1199090

120578)

119898minus1

expminus(119909 minus 119909

0)119898

120578119889119909

119909119894= 119909 +

6119894 minus 3119873

119873sdot 120590

120582119896= int

119909119896

119909119896minus1

119898

120578sdot (119909 minus 1199090

120578)

119898minus1


0)119898

120578119889119909

119909119896= 119909 +

6119896 minus 3119873

119873sdot 120590

(8)

where 119873 is the division number of the domain of the SSPnamely [119909 minus 3120590 119909 + 3120590] 119909 and 120590 are the mean and thestandard deviation of the SSP respectively119898 120578 and119909

0are the

parameters of the shape scale and location Figure 9 showsan illustration of the possibility function and the probabilitydensity function

To identify machinery condition the membership func-tion is constructed based on the possibility function of theSSP for diagnosis using possibility theory Figure 10 showsthe matching examples of possibility function The commonarea between the possibility functions (119901

119894(119909) 119901119895(119909)) ofmodel

states 119894 119895 and the possibility function (119901119905(119909)) of diagnostic

state 119905 is calculated by the following formula

119878119896= int

119909119905+3120590119905

119909119905minus3120590119905

120595119896 (119909) 119889119909

120595119896 (119909) = min 119901

119896 (119909) 119901119905 (119909)

(119896 = 119894 119895)

(9)

where 119909119905and 120590

119905are the mean and the standard deviation

of the SSP in diagnostic state Moreover the unknown stateexcept state 119894 and state 119895 is also model state UN its possibilityfunction (119901UN(119909)) is calculated by the following formula(10) And the common area between 119901UN(119909) and 119901

119905(119909) is

calculated by the following formula (11) Consider

119901UN (119909) = max 1 minus [119901119894(119909) + 119901

119895(119909)] 0 (10)

119878UN = int119909119905+3120590119905

119909119905minus3120590119905

120595UN (119909) 119889119909

120595UN (119909) = min 119901UN (119909) 119901119905 (119909)

(11)

Therefore the probability that should be diagnosed withmodel states 119894 119895 and UN respectively can be calculated bythe following formulas Then it is judged that the diagnosticstate is the model state with the largest possibility Consider

120596119896=1

120576sdot

119878119896

int119901119896(119909) 119889119909

(119896 = 119894 119895UN)

120576 =119878119894

int119901119894(119909) 119889119909

+

119878119895

int119901119895(119909) 119889119909

+119878UN

int119901UN (119909) 119889119909

(12)

0

02

04

Value of synthetic symptom parameter (x)0

1

05

Prob

abili

ty d

ensit

y fu

nctio

n (f

(x))

f(x)

p(x)

Poss

ibili

ty fu

nctio

n (p(x))

Figure 9 Probability density function (119891(119909)) and possibility func-tion (119901(119909))

minus6 minus5 minus4 minus3 minus2 minus1 0 1 2 3 4 50

p(x)

1

Synthetic symptom parameter (x)

pj(x)pi(x)

pt(x)

Sj

Si

pUN(x)

SUN

Figure 10 Matching example of possibility functions betweenmodel states and diagnostic state

62 Sequential Condition Diagnosis Approach Symptomparameters (SPS) can represent the features of vibrationsignals However they are always sensitive to one or severalfailure types so it is very difficult to find one or more SPsthat simultaneously identify all states but in contrast it iseasy that one or some SPs are found to identify two statesThen a sequential diagnosis method addressing the multiplefailure type identification as shown in Figure 11 is proposedfor condition diagnosis of a centrifugal pump

The first step of the diagnostic system is simple diagnosisIts performance is to detect whether there is fault and thennormal state (119873) and abnormal state (119860) are model states Inorder to generalize the capability of condition diagnosis allSSP1s of each abnormal state with three abnormal degrees

(mild medium and severe) are combined to establish themodel possibility function Since there are only three typicalfaults of the pump in this paper the unknown state exceptnormal state and abnormal state is also a model state If thepossibility grades of normal state (119873) abnormal state (119860)and the unknown state (UN) are expressed as 120596(119873) 120596(119860)and 120596(UN) respectively and 120596(119873) is the largest then thestate is diagnosed as ldquonormal state (119873)rdquo ending with thestop of the diagnosis system Otherwise there is fault thediagnostic system will come into the next step of precisediagnosis

Precise diagnosis is performed to identify the fault typesSince there are three types of faults in the experiment threepatterns are piled up in the precise diagnosis phase In eachpattern two fault types are selected as diagnosis target


Measure vibration signal and extract feature signal

Simple diagnosisin the first step

Normal (N)Abnormal (A)

Normal stateN

A

Calculate NSPs and obtain optimal classification hyperplane by SVM and NSPs


Precision diagnosisin the second step

Cavitation (C)Shaft misalignment (M)Unknown state (UN)

Fault identification

Impeller unbalance (U)Shaft misalignment (M)Unknown state (UN)

Cavitation (C)Impeller unbalance (U)Unknown state (UN)

Fault occurrenceCavitation (C)

Shaft misalignment (M)

Combine the possibilities from three patterns by DST

For example impeller unbalance (U)

Figure 11 Flowchart of sequential diagnosis for a pump system

and then the unknown state except the two selected statesis also regarded as diagnosis target Thus there are threestates in each pattern For example in the first pattern threemodel states are cavitation (119862) impeller unbalance (119880)and unknown state (UN) and then these possibility gradesof all SSP

2s are expressed as 120596

1(119862) 120596

1(119880) and 120596

1(UN)

respectively In the second pattern cavitation (119862) shaftmisalignment (119872) and unknown state (UN) are three modelstates and these possibility grades of all SSP

3s are expressed

as 1205962(119862) 120596

2(119872) and 120596

2(UN) respectively In the third

pattern impeller unbalance (119880) shaft misalignment (119872)and unknown state (UN) are three model states and thesepossibility grades of all SSP


3(119880) 120596

3(119872)

and 1205963(UN) respectively

When the training data synthetic symptom parameters(SSPs) between two states calculated using the proposedmethod are trained to determine the membership functionof fuzzy inference the system of sequential fuzzy diagnosisas shown in Figure 11 has been built

63 Fuzzy Inference Using Dempster-Shafer Theory Demp-ster-Shafer theory (DST) is an effectivemethod that combines

accumulative evidences [41 42] If 119873 evidences are used toidentify119872 classes (119860

1 1198602 119860

119872) and 120596

119894is the possibility

assignment function from the 119894th (119894 = 1 2 119873) evidencethe combining possibility function 120596(119860) can be obtained byDST given as follows

120596 (119860) =1

119896sum

cap119873

119894=1119860119894=119860

119873

prod

119894=1

120596119894(119860119894)

119896 = sum

cap119873

119894=1119860119894 =0

119873

prod

119894=1

120596119894(119860119894)

(13)

DST is widely used in the classification to addressthe uncertainty problem [43ndash45] In the Section 62 threepatterns are simultaneously performed in the second stepObviously the inferences are independent in three patternsand each pattern offers one evidence for obtaining the finalidentification result Then Dempster-Shafer theory (DST) isemployed to combine the possibility of each fault state Ifthe possibility grades of cavitation (119862) impeller unbalance(119880) shaftmisalignment (119872) and unknown state (UN) in thesecond step are 120596(119862) 120596(119880) 120596(119872) and 120596(UN) respectively


then the possibility grade of each fault state is combined byDST as follows

120596 (119862) =1

119896sdot 1205961 (119862) 1205962 (119862) 1205963 (UN)

120596 (119880) =1

119896sdot 1205961(119880) 1205962(UN) 120596

3(119880)

120596 (119872) =1

119896sdot 1205961(UN) 120596

2(119872)120596

3(119872)

120596 (UN) = 1

119896sdot 1205961(UN) 120596

2(UN) 120596

3(UN)

(14)

where 119896 is a normalizing constant 119896 = 1205961(119862)1205962(119862)1205963(UN) +

1205961(119880)1205962(UN)120596

3(119880) + 120596

1(UN)120596

2(119872)120596

3(119872) + 120596

1(UN)120596

2

(UN)1205963(UN) If max120596(119862) 120596(119880) 120596(119872) 120596(UN) = 120596(119862)

then the diagnosed state is judged as ldquocavitation (119862)rdquo In otherwords the state that the possibility grade is the biggest is thediagnosed state

7 Diagnosis and Verification

71 Diagnosis by the ProposedMethod To verify the proposeddiagnostic method the test data sets are again acquired inaccordancewith three typical faults of a centrifugal pump andthe defect degrees of mild medium and severe Moreovereach data set is processed using the proposed methods asinput of the diagnosis system Here two practical diagnosticexamples are shown in Figures 12 and 13

In the first example possibility functions between teststate and model states in the first step are shown in Figure 12119875119873(119909) and 119875

119860(119909) express the possibility functions of normal

state and abnormal state (it includes only cavitation impellerunbalance and shaft misalignment in this paper) 119875UN(119909)expresses the possibility function of the unknown stateexcept normal state and abnormal state 119875

119905(119909) expresses the

possibility function of the test state 119878119873 119878119860 and 119878UN express

the common area between 119875119905(119909) and 119875

119873(119909) 119875

119860(119909) and

119875UN(119909) respectively 120596(119873) 120596(119860) and 120596(UN) the possibilityof normal state abnormal state and the unknown staterespectively are calculated as 08558 00724 and 00718Thusthe diagnosis process stops in the first step and the outcomeof diagnosis system is normal state The diagnosis state isentirely consistent with the original state

For the second example the outcome of the first step asshown in Figure 13(a) is abnormal state and then the diag-nostic system comes into the second step of precision diag-nosis In the second step three patterns are simultaneouslyperformed as shown in Figures 13(b) 13(c) and 13(d) Thediagnosis possibilities are shown in Table 5 Moreover thediagnosis system intelligently employs DST to combine thepossibility grade of each fault state Then 120596(119862) 120596(119880) 120596(119872)and 120596(UN) the possibility of the cavitation (119862) impellerunbalance (119880) shaft misalignment (119872) and unknown state(UN) respectively are calculated as 09649 00001 00098and 00252 So the final outcome is cavitation state Bycomparing the original state the diagnosis result is correct

Similarly 10 test data sets have been performed in thesame procedure and the diagnosis results are shown in

minus3 0 3 6 9

p(x)

1

PN(x)

PA(x)

Pt(x)

SN

SASUN

PUN (x)


Figure 12The first diagnostic example using the sequential diagno-sis proposed

Table 6 The experimental result has shown that all test stateshave been diagnosed correctly but even the faults with milddefect at the early stage have been detected sensitively andidentified rightly

72 Diagnosis by Fuzzy Neural Network Fuzzy neural net-work is often employed to detect fault and identify fault types[14ndash16] This section shows the performance of the diagnosissystem based on fuzzy neural network In general there aretwo methods of direct diagnosis and sequential diagnosis asshown in Figures 14 and 15 Certainly considering the exper-iment in the paper state 119894 means normal state cavitationimpeller unbalance or shaft misalignment sequentially

To compare the proposed diagnosis with the diagnosismethods based on fuzzy neural network the experiment datadescribed in Section 2 are used to train the diagnosis systemsbased on fuzzy neural network as shown in Figures 14 and15 the measured data described in Section 7 are used totest The diagnosis results are shown in Table 7 The directdiagnosis based on fuzzy neural network could hardly detectthe faults with mild degree and 2 test data sets with mediumdegree have been also identified mistakenly For sequentialdiagnosis 6 test data sets with medium and severe degreeshave been diagnosed correctly but the performance that thefaults withmild degreewill be detected and identified is weak

8 Conclusions

To effectively detect faults and correctly identify fault typesfor plant machinery at an early stage an intelligent diagnosismethod using statistic filter SVM possibility theory andDST on the basis of the vibration signals was proposed for acentrifugal pump system and the effectiveness was demon-strated experimentally The superiority of the method pro-posed in this paper can be explained in the following points

(1) Statistic filter is a method of signal processing thatfailure signal is extracted by statistical tests of spec-trums between normal signal and fault signal Theapplication of statistic filter is very effective even ifat an early stage statistic filter can extract effectivelypure feature signal


minus3 0 3 6 9

p(x)

1

PN(x)

PA(x)

Pt(x)

SN

SA

SUN

PUN (x)


(a)

minus3minus6 00

3 6 9

p(x)

1

Pt(x)

PUN (x)


PC(x)

PU(x)

SC

SUSUN

(b)

0

p(x)

1

Pt(x)

PUN (x)

minus3minus6 0 3 6 9


PC(x)

SC

PM(x)

SMSUN

(c)

0

p(x)

1

Pt(x)

PUN (x)



PU(x)

SUN

PM(x)

SM

(d)

Figure 13 The second diagnostic example using the sequential diagnosis proposed

Table 5 The second diagnostic possibilities using the sequential diagnosis proposed

Each step or pattern Possibility of model stateModel state 1 Model state 2 Model state 3

In the first step 120596(119873) = 03200 120596(119860) = 06134 120596(UN) = 00666In the second step

First pattern 120596(119862) = 07486 120596(119880) = 01399 120596(UN) = 01115Second pattern 120596(119862) = 07909 120596(119872) = 00703 120596(UN) = 01389Third pattern 120596(119880) = 00013 120596(119872) = 04328 120596(UN) = 05659

Table 6 The diagnostic results of various states with the defect degrees by the proposed method

Test state Defect degree Simple diagnosis (first step) Precision diagnosis (second step) Judge state120596(119873) 120596(119860) 120596(UN) 120596(119862) 120596(119880) 120596(119872) 120596(UN)

Normal (119873) 08558 00724 00718 119873

Cavitation (119862)Mild 03529 06038 00433 07382 00579 01763 00276 119862

Medium 02311 07621 00068 09218 00053 00204 00525 119862

Severe 03200 06134 00666 09649 00001 00098 00252 119862

Impellerunbalance (119880)

Mild 02026 06813 01161 00153 07468 01934 00445 119880

Medium 01897 07385 00718 00118 08216 01430 00236 119880

Severe 01365 07027 01608 00092 07604 02259 00045 119880

Shaftmisalignment(119872)

Mild 03415 06435 00150 01138 02126 06072 00664 119872

Medium 02071 07446 00483 00158 01901 06831 01110 119872

Severe 03007 06756 00237 00576 01573 06315 01536 119872


Table 7 The diagnostic results of various states with the defect degrees by fuzzy neural network

Test state Defect degree Judge state by fuzzy neural networkDirect diagnosis Sequential diagnosis

Normal (119873) 119873 119873

Cavitation (119862)Mild 119873 119873

Medium 119873 119862

Severe 119862 119862

Impeller unbalance (119880)Mild 119873 119880

Medium 119880 119880

Severe 119880 119880

Shaft misalignment (119872)Mild 119873 119862

Medium 119862 119872

Severe 119872 119872

P1

P2

Sym

ptom

par

amet

ers

Input layer

P3Shaft misalignment

Normal state

Impeller unbalance

Cavitation state

Unknown state

Middle layer Output layer

Figure 14 Direct diagnosis method based on fuzzy neural network

P1

P2

Sym

ptom

par

amet

ers

Input layer

P3


State i

Other state

Unknown state

Figure 15 Sequential diagnosis method based on fuzzy neuralnetwork

(2) Softmargin SVM is used to define synthetic symptomparameter (SSP) and then to raise the discernmentsensitivity of original SPs

(3) The efficiency of intelligent fuzzy diagnosis proposedhere has been verified by applying it to a practicaldiagnosis for incipient faults of the centrifugal pumpwith a performance In fact the perfect performanceof intelligent diagnosis using possibility functions ofthe SSPs is attributed primarily to effective feature

extraction by statistic filter SVMsrsquo generalizationcapability SSPsrsquo high sensitivity andDSTrsquos combiningevidence

In the near future themethod proposed in this paper willbe applied to condition diagnosis in various types of rotatingmachinery in a real plant

Conflict of Interests

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

Acknowledgments

This project is supported by the National Natural ScienceFoundation of China (Grant no 51375037) National Programon Key Basic Research Project (Grant no 2012CB026000)and Program forNewCentury Excellent Talents inUniversity(NCET-12-0759)

References

[1] DMoshou I Gravalos D Kateris N Sawalhi and S LoutridisldquoCondition monitoring in centrifugal irrigation pumps withself-organizing feature visualisationrdquo in Proceedings of theEFITAWCCA Conference pp 116ndash124 2011

[2] S Perovic P J Unsworth and E H Higham ldquoFuzzy logicsystem to detect pump faults from motor current spectrardquo inIndustry Applications Conference vol 1 pp 274ndash280 ChicagoIll USA 2001

[3] C S Kallesoslashe V Cocquempot and R Izadi-ZamanabadildquoModel based fault detection in a centrifugal pump applicationrdquoIEEE Transactions on Control Systems Technology vol 14 no 2pp 204ndash215 2006

[4] S Rajakarunakaran P Venkumar D Devaraj and K S P RaoldquoArtificial neural network approach for fault detection in rotarysystemrdquo Applied Soft Computing Journal vol 8 no 1 pp 740ndash748 2008

[5] P Jayaswal A KWadhwani and K B Mulchandani ldquoMachinefault signature analysisrdquo International Journal of RotatingMachinery vol 2008 Article ID 583982 10 pages 2008


[6] P Brian ldquoGraney pump vibration analysisrdquo Pumps amp Systemsvol 11 pp 24ndash28 2011

[7] J T Renwick and P E Babson ldquoVibration analysis a proventechniques as a productions maintenance toolrdquo IEEE Transac-tions on Industry Applications vol 21 no 2 pp 324ndash332 1985

[8] B Liu and S-F Ling ldquoOn the selection of informative waveletsfor machinery diagnosisrdquo Mechanical Systems and Signal Pro-cessing vol 13 no 1 pp 145ndash162 1999

[9] 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

[10] Q B Zhu ldquoGear fault diagnosis system based on wavelet neuralnetworksrdquo Dynamics of Continuous Discrete and ImpulsiveSystems Series A Mathematical Analysis vol 13 part 2 pp 671ndash673 2006

[11] H Wang P Chen and S Wang ldquoIntelligent diagnosis methodsfor plant machineryrdquo Frontiers of Mechanical Engineering inChina vol 5 no 1 pp 118ndash124 2010

[12] H Wang and P Chen ldquoFuzzy diagnosis method for rotatingmachinery in variable rotating speedrdquo IEEE Sensors Journal vol11 no 1 pp 23ndash34 2011

[13] A Albraik F Althobiani F Gu and A Ball ldquoDiagnosis ofcentrifugal pump faults using vibration methodsrdquo Journal ofPhysics Conference Series vol 364 no 1 Article ID 012139 2012

[14] H Wang and P Chen ldquoIntelligent diagnosis method for acentrifugal pump using features of vibration signalsrdquo NeuralComputing and Applications vol 18 no 4 pp 397ndash405 2009

[15] K Li P Chen and S Wang ldquoAn intelligent diagnosis methodfor rotating machinery using least squares mapping and a fuzzyneural networkrdquo Sensors vol 12 no 5 pp 5919ndash5939 2012

[16] H Wang and P Chen ldquoIntelligent diagnosis method for rollingelement bearing faults using possibility theory and neuralnetworkrdquo Computers and Industrial Engineering vol 60 no 4pp 511ndash518 2011

[17] A Widodo and B-S Yang ldquoApplication of nonlinear featureextraction and support vector machines for fault diagnosis ofinductionmotorsrdquo Expert Systems with Applications vol 33 no1 pp 241ndash250 2007

[18] H T Xue ldquoStructural fault diagnosis of rotating machinerybased on distinctive frequency components and support vectormachinesrdquo in Advanced Intelligent Computing Theories andApplications With Aspects of Artificial Intelligence vol 6839 ofLecture Notes in Computer Science pp 341ndash348 2012

[19] AWidodo andB-S Yang ldquoSupport vectormachine inmachinecondition monitoring and fault diagnosisrdquo Mechanical Systemsand Signal Processing vol 21 no 6 pp 2560ndash2574 2007

[20] H Xue H Wang P Chen K Li and L Song ldquoAutomaticdiagnosis method for structural fault of rotating machinerybased on distinctive frequency components and support vectormachines under varied operating conditionsrdquoNeurocomputingvol 116 pp 326ndash335 2013

[21] H Xue K Li H Wang and P Chen ldquoSequential diagnosismethod for rotating machinery using support vector machinesand possibility theoryrdquo in Intelligent Computing Theories vol7995 of Lecture Notes in Computer Science pp 315ndash324 2013

[22] J C Principe and T Yoon ldquoA new algorithm for the detection oftool breakage inmillingrdquo International Journal of Machine Toolsand Manufacture vol 31 no 4 pp 443ndash454 1991

[23] M Karimi-Ghartemani and M R Iravani ldquoA nonlinear adap-tive filter for online signal analysis in power systems applica-tionsrdquo IEEE Transactions on Power Delivery vol 17 no 2 pp617ndash622 2002

[24] W Zhou B Lu T GHabetler andRGHarley ldquoIncipient bear-ing fault detection via motor stator current noise cancellationusing wiener filterrdquo IEEE Transactions on Industry Applicationsvol 45 no 4 pp 1309ndash1317 2009

[25] A J Volponi H DePold R Ganguli and C Daguang ldquoTheuse of kalman filter and neural network methodologies in gasturbine performance diagnostics a comparative studyrdquo Journalof Engineering for Gas Turbines and Power vol 125 no 4 pp917ndash924 2003

[26] C M Bishop Neural Networks for Pattern Recognition OxfordUniversity Pres New York NY USA 1995

[27] M Cudina ldquoDetection of cavitation phenomenon in a centrifu-gal pump using audible soundrdquo Mechanical Systems and SignalProcessing vol 17 no 6 pp 1335ndash1347 2003

[28] L Feinstein D Schnackenberg R Balupari and D KindredldquoStatistical approaches to DDoS attack detection and responserdquoin Proceedings of the DARPA Information Survivability Confer-ence and Exposition vol 1 pp 303ndash314 2003

[29] T Toyota P Chen and T Mizota ldquoDetection and distinctionof failure signal by means of statistical tests of spectrumrdquo TheJournal of the Japan Society for Precision Engineering vol 58 no6 pp 1041ndash1046 1992

[30] P Chen T Toyota and Z He ldquoAutomated function generationof symptom parameters and application to fault diagnosis ofmachinery under variable operating conditionsrdquo IEEE Trans-actions on Systems Man and Cybernetics Part ASystems andHumans vol 31 no 6 pp 775ndash781 2001

[31] K Li and P Chen ldquoIntelligent method for diagnosing structuralfaults of rotating machinery using ant colony optimizationrdquoSensors vol 11 no 4 pp 4009ndash4029 2011

[32] S G Mallat ldquoTheory for multiresolution signal decompositionthe wavelet representationrdquo IEEE Transactions on Pattern Anal-ysis and Machine Intelligence vol 11 no 7 pp 674ndash693 1989

[33] S R Gunn ldquoSupport vector machines for classification andregressionrdquo Tech Rep University of Southampton 1998

[34] C Cortes and V Vapnik ldquoSupport-vector networksrdquo MachineLearning vol 20 no 3 pp 273ndash297 1995

[35] H Matuyama ldquoDiagnosis algorithmrdquo Journal of JSPE vol 75no 3 pp 35ndash37 1991

[36] J J Richardson Artificial Intelligence in Maintenance NoyesPublications New Jersey NJ USA 1985

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

[38] D Dubois and H Prade Possibility Theory An Approach toComputerized Processing of Uncertainty Plenum Press NewYork NY USA 1988

[39] P Chen and T Toyota ldquoMethod of failure detection by possi-bility theory and dempster amp shafer probability theoryrdquo Journalof KitakyushuMedical and Engineering Cooperative Associationvol 9 pp 1ndash4 1998

[40] P Chen and T Toyota ldquoSequential fuzzy diagnosis for plantmachineryrdquo JSME International Journal Series C MechanicalSystems Machine Elements and Manufacturing vol 46 no 3pp 1121ndash1129 2003

[41] A P Dempster ldquoUpper and lower probabilities induced by amultivalued mappingrdquo The Annals of Mathematical Statisticsvol 38 pp 325ndash339 1967

[42] G Shafer A Mathematical Theory of Evidence Princeton Uni-versity Press 1976


[43] P Chen Foundation and Application of Condition DiagnosisTechnology for Rotating Machinery Sankeisha Press NagoyaJapan 2009

[44] C R Parikh M J Pont and N Barrie Jones ldquoApplication ofDempster-Shafer theory in condition monitoring applicationsa case studyrdquo Pattern Recognition Letters vol 22 no 6-7 pp777ndash785 2001

[45] S Schocken and R A Hummel ldquoOn the use of the DempsterShafer model in information indexing and retrieval applica-tionsrdquo International Journal ofMan-Machine Studies vol 39 no5 pp 843ndash879 1993

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



measured with the signal Simultaneous measurement of thereference noise is not easily realized in most cases of faultdiagnosis Finally the noise cannot be effectively removedby the Wiener filter and Kalman filter if noise and signal donot follow the normal distribution In order to overcome theproblems above we used the statistic filter to extract the faultsignal from the vibration signal measured in the abnormalstate of a machine

Fault diagnosis can hardly do without intelligent systemssuch as neural networks (NN) and support vector machine(SVM) Studies in [14ndash16] have been carried out to investigatethe use of NN for detecting fault and identifying fault typesHowever the conventional NN cannot reflect the possibilityof ambiguous diagnosis problems and will never convergewhen the first layer symptom parameters have the same val-ues in different states [26] Studies in [17ndash20] have employedSVMs to perform fault diagnosis and the capability ofSVMswhich can efficiently perform a nonlinear classificationwith kernel function has been confirmed However it isdifficult to determine the parameters for a given value ofthe regularization and kernel parameters and to choose anappropriate kernel function

For the above reasons in order to process the uncertainrelationship between symptom parameters (SPs) and ma-chinery conditions improve the efficiency and accuracy offault diagnosis at an early stage and achieve an automaticsystem from measured signals the authors propose an intel-ligent diagnosis for a pump system using support vectormachine (SVM) possibility theory and Dempster-Shafertheory (DST) on the basis of the vibration signals to detectfaults and identify fault types at an early stage as is shown inthe flowchart in Figure 1 The statistic filter is used to detectand extract the failure signals of pump faults across optimumfrequency regions and the failure signals extracted arereflected by nondimensional symptom parameters (NSPs) Inorder to increase the diagnosisrsquo sensitivity a synthetic symp-tom parameter (SSP) is defined with the function of the opti-mal classification hyperplane obtained using SVM Then thepossibility distributions of the SSPs are used to detect faultsand identify fault types by possibility theory andDST Finallythe proposedmethod is evaluated using practical examples ofcentrifugal pump which shows a good performance

2 Experimental Centrifugal Pump System forCondition Diagnosis

Figure 2 shows the testing bed consisting of a pump a motorand a close-loop water piping system The SF-JRO seriesmotor is employed to drive the pump through a couplingand the rotating speed can be controlled via control panelThe experimental pump is a centrifugal pump described asfollows HONDA Pump Type HAS (Volute Pump Horizon-tal Type) Head 40 Output 37 kW and Capacity 75m3hThe capacity of the tank is based on the maximum flow ratewhich can also be controlled by the valve control system Fiveaccelerometers are adopted to acquire the vibration signalswith a sampling frequency of 50 kHz One is mounted onthe pump housing two in the piping direction and vertical

Measure vibration signals in each state

Obtain optimal classification hyperplane by SVM and NSPs

Calculate NSPs in each state


Diagnose using possibility theory and DST

Extract feature signals by statistic filter

Figure 1 Flowchart of intelligent diagnosis

Figure 2 Photograph of the experiment centrifugal pump system

direction of the pump inlet respectively the rest in the pipingdirection and horizontal direction of the pump outlet asshown in Figure 3

Cavitation phenomenon is one of the sources of instabil-ity in a centrifugal pump Bubbles collapse abruptly leadingto damage of the pump and generation of crackling soundand vibration [27] Other faults such as impeller unbalanceand shaftmisalignment between themotor and the pump arediscriminated easily but often occur in pump system Thesefaults can cause serious machine accidents and bring greatproduction losses Therefore these states with normal statewill be discussed to examine whether incipient faults of apump were able to be detected using the proposed methodin a later sectionWhen the pump speed was at 3500 rpm andthe valves in the suction and discharge lines were fully openthe pumpwas in the best situation of operationThe conditionwas seen as normal state Then the conditions of the speedand the valve in the discharge line remained unchanged thevalve in the suction line was slowly turned down until littlebubbles appeared At present the condition was the cavitation


Accelerometer






119894119895(119891119896) of the part 119895 (119895 =



119899(119891119896) and 119865

119889(119891119896)



119865119899(119891119896) and 119865



119865119899(119891119896) = 119865

1198991(119891119896) 1198651198992(119891119896) 119865

119899119873(119891119896)

119865119889(119891119896) = 119865

1198891(119891119896) 1198651198892(119891119896) 119865

119889119873(119891119896)

(1)


119899(119891119896) and 119865

119889(119891119896) of normal and




119899(119891119896) and119865

119889(119891119896) and


2 1205902)








1+ 1205902






2

1+ 1205902

1 (2b) can


119891 (119911) =1

radic2120587 (1205902

1+ 1205902

2)


2minus 1205831)2

2 (1205902

1+ 1205902

2)

) (2a)

119901 = int

0

minusinfin

119891 (119911) 119889119911 (2b)

119901 =1

radic2120587int

minusDI

minusinfin

exp(minus1205832

2)119889120583 (2c)


1+ 1205902



DI =10038161003816100381610038161205832 minus 1205831

1003816100381610038161003816

radic1205902

1+ 1205902

2

(2d)




DR = 1 minus 2119901 (2e)


119899(119891119896)



119899(119891119896) and 119865



119889(119891119896) and then it





119889(1198911) as shown







IFFT





(Hz)

0

1 Part 1 Part 1

Part 2 Part 2

Part N Part N

Spec

trum

0

1

Spec

trum

0

1

Spec

trum

0

1

Spec

trum





(Hz)


(Hz) (Hz)

(Hz)

(Hz)

0

1

Spec

trum

0

1

Spec

trum

Normal signal

Fn(f1)

05 08 F(f1)

F(f2)

F(fk)

Distributions

Distributions

Distributions

Fn(f1)

Fn(f2)Fn(f2)

Fn(fk)Fn(fk)

Fd(f1)

Fd(f2)Fd(f2)

Fd(fk)Fd(fk)

Fd(f1)

042 048

001 012

Diagnosis signal

(A1)

(A2)

(Ak)



1198911and 119891

2















1198752

1198753

1198751

1198752

1198753

1198751

1198752

1198753

1198751

1198752

1198753




1198752

1198753

1198751

1198752

1198753

1198751

1198752

1198753

1198751

1198752

1198753


0f1f2f3f4f5


fM

Spec

trum




119909119894=1199091015840

119894minus 1205831015840

1205901015840 (3)



119894(119894 =








1198751=120590

|119909|

1198752=sum119873

119894=1(119909119894minus 119909)3

119873 sdot 1205903

1198753=sum119873

119894=1(119909119894minus 119909)4

119873 sdot 1205904

(4)




119894|





0 005 01

(s)

(s)

(s)

0 005 01

0 005 01

minus2

0

2A

mpl

itude

(V)

minus2

0

2

Am

plitu

de (V

)

minus2

0

2

Am

plitu

de (V

)




FFT


Spec

trum

0 5 10 15 20 25

(kHz)

(kHz)

(kHz)

0 5 10 15 20 25

0 5 10 15 20 25

0

8

Spec

trum

0

8

Spec

trum

0

8



SF

IFFT





120596 sdot 119909 + 119887 = 0 (5)



x1

x2

x3



1 1199092










+ 119887 (6)


119894= 1198751119894 1198752119894 1198753119894 (119894 = 1 2 119868)






1 1205962







1= minus3587119875

1+ 6109119875

2minus 2234119875

3+ 10006


2= 0164119875

1+ 0897119875

2minus 3220119875

3minus 0107


3= minus0593119875

1+ 0031119875

2+ 1071119875

3+ 1934


SSP4= 1462119875

1minus 2843119875

2minus 4566119875

3+ 2006



In mediumstep

In severestep








119901 (119909119894) =

119873

sum

119896=1

min 120582119894 120582119896 (119894 119896 = 1 2 119873) (7)


120582119894and 120582


120582119894= int

119909119894

119909119894minus1

119898

120578sdot (119909 minus 1199090

120578)

119898minus1


0)119898

120578119889119909

119909119894= 119909 +

6119894 minus 3119873

119873sdot 120590

120582119896= int

119909119896

119909119896minus1

119898

120578sdot (119909 minus 1199090

120578)

119898minus1


0)119898

120578119889119909

119909119896= 119909 +

6119896 minus 3119873

119873sdot 120590

(8)


0are the



119894(119909) 119901119895(119909)) ofmodel



119878119896= int

119909119905+3120590119905

119909119905minus3120590119905

120595119896 (119909) 119889119909

120595119896 (119909) = min 119901

119896 (119909) 119901119905 (119909)

(119896 = 119894 119895)

(9)

where 119909119905and 120590



119905(119909) is


119901UN (119909) = max 1 minus [119901119894(119909) + 119901

119895(119909)] 0 (10)

119878UN = int119909119905+3120590119905

119909119905minus3120590119905

120595UN (119909) 119889119909

120595UN (119909) = min 119901UN (119909) 119901119905 (119909)

(11)


120596119896=1

120576sdot

119878119896

int119901119896(119909) 119889119909

(119896 = 119894 119895UN)

120576 =119878119894

int119901119894(119909) 119889119909

+

119878119895

int119901119895(119909) 119889119909

+119878UN

int119901UN (119909) 119889119909

(12)

0

02

04


1

05

Prob

abili

ty d

ensit

y fu

nctio

n (f

(x))

f(x)

p(x)

Poss

ibili

ty fu

nctio

n (p(x))



p(x)

1


pj(x)pi(x)

pt(x)

Sj

Si

pUN(x)

SUN










Normal stateN

A















1(119862) 120596

1(119880) and 120596

1(UN)


3s are expressed

as 1205962(119862) 120596

2(119872) and 120596




3(119880) 120596

3(119872)





1 1198602 119860

119872) and 120596



120596 (119860) =1

119896sum

cap119873

119894=1119860119894=119860

119873

prod

119894=1

120596119894(119860119894)

119896 = sum

cap119873

119894=1119860119894 =0

119873

prod

119894=1

120596119894(119860119894)

(13)




120596 (119862) =1

119896sdot 1205961 (119862) 1205962 (119862) 1205963 (UN)

120596 (119880) =1

119896sdot 1205961(119880) 1205962(UN) 120596

3(119880)

120596 (119872) =1

119896sdot 1205961(UN) 120596

2(119872)120596

3(119872)

120596 (UN) = 1

119896sdot 1205961(UN) 120596

2(UN) 120596

3(UN)

(14)


1205961(119880)1205962(UN)120596

3(119880) + 120596

1(UN)120596

2(119872)120596

3(119872) + 120596

1(UN)120596

2

(UN)1205963(UN) If max120596(119862) 120596(119880) 120596(119872) 120596(UN) = 120596(119862)










119873(119909) 119875

119860(119909) and




minus3 0 3 6 9

p(x)

1

PN(x)

PA(x)

Pt(x)

SN

SASUN

PUN (x)






8 Conclusions




minus3 0 3 6 9

p(x)

1

PN(x)

PA(x)

Pt(x)

SN

SA

SUN

PUN (x)


(a)

minus3minus6 00

3 6 9

p(x)

1

Pt(x)

PUN (x)


PC(x)

PU(x)

SC

SUSUN

(b)

0

p(x)

1

Pt(x)

PUN (x)



PC(x)

SC

PM(x)

SMSUN

(c)

0

p(x)

1

Pt(x)

PUN (x)



PU(x)

SUN

PM(x)

SM

(d)








Normal (119873) 08558 00724 00718 119873

Cavitation (119862)Mild 03529 06038 00433 07382 00579 01763 00276 119862

Medium 02311 07621 00068 09218 00053 00204 00525 119862

Severe 03200 06134 00666 09649 00001 00098 00252 119862


Mild 02026 06813 01161 00153 07468 01934 00445 119880

Medium 01897 07385 00718 00118 08216 01430 00236 119880

Severe 01365 07027 01608 00092 07604 02259 00045 119880


Mild 03415 06435 00150 01138 02126 06072 00664 119872

Medium 02071 07446 00483 00158 01901 06831 01110 119872

Severe 03007 06756 00237 00576 01573 06315 01536 119872




Normal (119873) 119873 119873

Cavitation (119862)Mild 119873 119873

Medium 119873 119862

Severe 119862 119862


Medium 119880 119880

Severe 119880 119880


Medium 119862 119872

Severe 119872 119872

P1

P2

Sym

ptom

par

amet

ers

Input layer


Normal state

Impeller unbalance

Cavitation state

Unknown state



P1

P2

Sym

ptom

par

amet

ers

Input layer

P3


State i

Other state

Unknown state








Acknowledgments


References


















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Accelerometer






119894119895(119891119896) of the part 119895 (119895 =



119899(119891119896) and 119865

119889(119891119896)



119865119899(119891119896) and 119865



119865119899(119891119896) = 119865

1198991(119891119896) 1198651198992(119891119896) 119865

119899119873(119891119896)

119865119889(119891119896) = 119865

1198891(119891119896) 1198651198892(119891119896) 119865

119889119873(119891119896)

(1)


119899(119891119896) and 119865

119889(119891119896) of normal and




119899(119891119896) and119865

119889(119891119896) and


2 1205902)








1+ 1205902






2

1+ 1205902

1 (2b) can


119891 (119911) =1

radic2120587 (1205902

1+ 1205902

2)


2minus 1205831)2

2 (1205902

1+ 1205902

2)

) (2a)

119901 = int

0

minusinfin

119891 (119911) 119889119911 (2b)

119901 =1

radic2120587int

minusDI

minusinfin

exp(minus1205832

2)119889120583 (2c)


1+ 1205902



DI =10038161003816100381610038161205832 minus 1205831

1003816100381610038161003816

radic1205902

1+ 1205902

2

(2d)




DR = 1 minus 2119901 (2e)


119899(119891119896)



119899(119891119896) and 119865



119889(119891119896) and then it





119889(1198911) as shown







IFFT





(Hz)

0

1 Part 1 Part 1

Part 2 Part 2

Part N Part N

Spec

trum

0

1

Spec

trum

0

1

Spec

trum

0

1

Spec

trum





(Hz)


(Hz) (Hz)

(Hz)

(Hz)

0

1

Spec

trum

0

1

Spec

trum

Normal signal

Fn(f1)

05 08 F(f1)

F(f2)

F(fk)

Distributions

Distributions

Distributions

Fn(f1)

Fn(f2)Fn(f2)

Fn(fk)Fn(fk)

Fd(f1)

Fd(f2)Fd(f2)

Fd(fk)Fd(fk)

Fd(f1)

042 048

001 012

Diagnosis signal

(A1)

(A2)

(Ak)



1198911and 119891

2















1198752

1198753

1198751

1198752

1198753

1198751

1198752

1198753

1198751

1198752

1198753




1198752

1198753

1198751

1198752

1198753

1198751

1198752

1198753

1198751

1198752

1198753


0f1f2f3f4f5


fM

Spec

trum




119909119894=1199091015840

119894minus 1205831015840

1205901015840 (3)



119894(119894 =








1198751=120590

|119909|

1198752=sum119873

119894=1(119909119894minus 119909)3

119873 sdot 1205903

1198753=sum119873

119894=1(119909119894minus 119909)4

119873 sdot 1205904

(4)




119894|





0 005 01

(s)

(s)

(s)

0 005 01

0 005 01

minus2

0

2A

mpl

itude

(V)

minus2

0

2

Am

plitu

de (V

)

minus2

0

2

Am

plitu

de (V

)




FFT


Spec

trum

0 5 10 15 20 25

(kHz)

(kHz)

(kHz)

0 5 10 15 20 25

0 5 10 15 20 25

0

8

Spec

trum

0

8

Spec

trum

0

8



SF

IFFT





120596 sdot 119909 + 119887 = 0 (5)



x1

x2

x3



1 1199092










+ 119887 (6)


119894= 1198751119894 1198752119894 1198753119894 (119894 = 1 2 119868)






1 1205962







1= minus3587119875

1+ 6109119875

2minus 2234119875

3+ 10006


2= 0164119875

1+ 0897119875

2minus 3220119875

3minus 0107


3= minus0593119875

1+ 0031119875

2+ 1071119875

3+ 1934


SSP4= 1462119875

1minus 2843119875

2minus 4566119875

3+ 2006



In mediumstep

In severestep








119901 (119909119894) =

119873

sum

119896=1

min 120582119894 120582119896 (119894 119896 = 1 2 119873) (7)


120582119894and 120582


120582119894= int

119909119894

119909119894minus1

119898

120578sdot (119909 minus 1199090

120578)

119898minus1


0)119898

120578119889119909

119909119894= 119909 +

6119894 minus 3119873

119873sdot 120590

120582119896= int

119909119896

119909119896minus1

119898

120578sdot (119909 minus 1199090

120578)

119898minus1


0)119898

120578119889119909

119909119896= 119909 +

6119896 minus 3119873

119873sdot 120590

(8)


0are the



119894(119909) 119901119895(119909)) ofmodel



119878119896= int

119909119905+3120590119905

119909119905minus3120590119905

120595119896 (119909) 119889119909

120595119896 (119909) = min 119901

119896 (119909) 119901119905 (119909)

(119896 = 119894 119895)

(9)

where 119909119905and 120590



119905(119909) is


119901UN (119909) = max 1 minus [119901119894(119909) + 119901

119895(119909)] 0 (10)

119878UN = int119909119905+3120590119905

119909119905minus3120590119905

120595UN (119909) 119889119909

120595UN (119909) = min 119901UN (119909) 119901119905 (119909)

(11)


120596119896=1

120576sdot

119878119896

int119901119896(119909) 119889119909

(119896 = 119894 119895UN)

120576 =119878119894

int119901119894(119909) 119889119909

+

119878119895

int119901119895(119909) 119889119909

+119878UN

int119901UN (119909) 119889119909

(12)

0

02

04


1

05

Prob

abili

ty d

ensit

y fu

nctio

n (f

(x))

f(x)

p(x)

Poss

ibili

ty fu

nctio

n (p(x))



p(x)

1


pj(x)pi(x)

pt(x)

Sj

Si

pUN(x)

SUN










Normal stateN

A















1(119862) 120596

1(119880) and 120596

1(UN)


3s are expressed

as 1205962(119862) 120596

2(119872) and 120596




3(119880) 120596

3(119872)





1 1198602 119860

119872) and 120596



120596 (119860) =1

119896sum

cap119873

119894=1119860119894=119860

119873

prod

119894=1

120596119894(119860119894)

119896 = sum

cap119873

119894=1119860119894 =0

119873

prod

119894=1

120596119894(119860119894)

(13)




120596 (119862) =1

119896sdot 1205961 (119862) 1205962 (119862) 1205963 (UN)

120596 (119880) =1

119896sdot 1205961(119880) 1205962(UN) 120596

3(119880)

120596 (119872) =1

119896sdot 1205961(UN) 120596

2(119872)120596

3(119872)

120596 (UN) = 1

119896sdot 1205961(UN) 120596

2(UN) 120596

3(UN)

(14)


1205961(119880)1205962(UN)120596

3(119880) + 120596

1(UN)120596

2(119872)120596

3(119872) + 120596

1(UN)120596

2

(UN)1205963(UN) If max120596(119862) 120596(119880) 120596(119872) 120596(UN) = 120596(119862)










119873(119909) 119875

119860(119909) and




minus3 0 3 6 9

p(x)

1

PN(x)

PA(x)

Pt(x)

SN

SASUN

PUN (x)






8 Conclusions




minus3 0 3 6 9

p(x)

1

PN(x)

PA(x)

Pt(x)

SN

SA

SUN

PUN (x)


(a)

minus3minus6 00

3 6 9

p(x)

1

Pt(x)

PUN (x)


PC(x)

PU(x)

SC

SUSUN

(b)

0

p(x)

1

Pt(x)

PUN (x)



PC(x)

SC

PM(x)

SMSUN

(c)

0

p(x)

1

Pt(x)

PUN (x)



PU(x)

SUN

PM(x)

SM

(d)








Normal (119873) 08558 00724 00718 119873

Cavitation (119862)Mild 03529 06038 00433 07382 00579 01763 00276 119862

Medium 02311 07621 00068 09218 00053 00204 00525 119862

Severe 03200 06134 00666 09649 00001 00098 00252 119862


Mild 02026 06813 01161 00153 07468 01934 00445 119880

Medium 01897 07385 00718 00118 08216 01430 00236 119880

Severe 01365 07027 01608 00092 07604 02259 00045 119880


Mild 03415 06435 00150 01138 02126 06072 00664 119872

Medium 02071 07446 00483 00158 01901 06831 01110 119872

Severe 03007 06756 00237 00576 01573 06315 01536 119872




Normal (119873) 119873 119873

Cavitation (119862)Mild 119873 119873

Medium 119873 119862

Severe 119862 119862


Medium 119880 119880

Severe 119880 119880


Medium 119862 119872

Severe 119872 119872

P1

P2

Sym

ptom

par

amet

ers

Input layer


Normal state

Impeller unbalance

Cavitation state

Unknown state



P1

P2

Sym

ptom

par

amet

ers

Input layer

P3


State i

Other state

Unknown state








Acknowledgments


References


















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















IFFT





(Hz)

0

1 Part 1 Part 1

Part 2 Part 2

Part N Part N

Spec

trum

0

1

Spec

trum

0

1

Spec

trum

0

1

Spec

trum





(Hz)


(Hz) (Hz)

(Hz)

(Hz)

0

1

Spec

trum

0

1

Spec

trum

Normal signal

Fn(f1)

05 08 F(f1)

F(f2)

F(fk)

Distributions

Distributions

Distributions

Fn(f1)

Fn(f2)Fn(f2)

Fn(fk)Fn(fk)

Fd(f1)

Fd(f2)Fd(f2)

Fd(fk)Fd(fk)

Fd(f1)

042 048

001 012

Diagnosis signal

(A1)

(A2)

(Ak)



1198911and 119891

2















1198752

1198753

1198751

1198752

1198753

1198751

1198752

1198753

1198751

1198752

1198753




1198752

1198753

1198751

1198752

1198753

1198751

1198752

1198753

1198751

1198752

1198753


0f1f2f3f4f5


fM

Spec

trum




119909119894=1199091015840

119894minus 1205831015840

1205901015840 (3)



119894(119894 =








1198751=120590

|119909|

1198752=sum119873

119894=1(119909119894minus 119909)3

119873 sdot 1205903

1198753=sum119873

119894=1(119909119894minus 119909)4

119873 sdot 1205904

(4)




119894|





0 005 01

(s)

(s)

(s)

0 005 01

0 005 01

minus2

0

2A

mpl

itude

(V)

minus2

0

2

Am

plitu

de (V

)

minus2

0

2

Am

plitu

de (V

)




FFT


Spec

trum

0 5 10 15 20 25

(kHz)

(kHz)

(kHz)

0 5 10 15 20 25

0 5 10 15 20 25

0

8

Spec

trum

0

8

Spec

trum

0

8



SF

IFFT





120596 sdot 119909 + 119887 = 0 (5)



x1

x2

x3



1 1199092










+ 119887 (6)


119894= 1198751119894 1198752119894 1198753119894 (119894 = 1 2 119868)






1 1205962







1= minus3587119875

1+ 6109119875

2minus 2234119875

3+ 10006


2= 0164119875

1+ 0897119875

2minus 3220119875

3minus 0107


3= minus0593119875

1+ 0031119875

2+ 1071119875

3+ 1934


SSP4= 1462119875

1minus 2843119875

2minus 4566119875

3+ 2006



In mediumstep

In severestep








119901 (119909119894) =

119873

sum

119896=1

min 120582119894 120582119896 (119894 119896 = 1 2 119873) (7)


120582119894and 120582


120582119894= int

119909119894

119909119894minus1

119898

120578sdot (119909 minus 1199090

120578)

119898minus1


0)119898

120578119889119909

119909119894= 119909 +

6119894 minus 3119873

119873sdot 120590

120582119896= int

119909119896

119909119896minus1

119898

120578sdot (119909 minus 1199090

120578)

119898minus1


0)119898

120578119889119909

119909119896= 119909 +

6119896 minus 3119873

119873sdot 120590

(8)


0are the



119894(119909) 119901119895(119909)) ofmodel



119878119896= int

119909119905+3120590119905

119909119905minus3120590119905

120595119896 (119909) 119889119909

120595119896 (119909) = min 119901

119896 (119909) 119901119905 (119909)

(119896 = 119894 119895)

(9)

where 119909119905and 120590



119905(119909) is


119901UN (119909) = max 1 minus [119901119894(119909) + 119901

119895(119909)] 0 (10)

119878UN = int119909119905+3120590119905

119909119905minus3120590119905

120595UN (119909) 119889119909

120595UN (119909) = min 119901UN (119909) 119901119905 (119909)

(11)


120596119896=1

120576sdot

119878119896

int119901119896(119909) 119889119909

(119896 = 119894 119895UN)

120576 =119878119894

int119901119894(119909) 119889119909

+

119878119895

int119901119895(119909) 119889119909

+119878UN

int119901UN (119909) 119889119909

(12)

0

02

04


1

05

Prob

abili

ty d

ensit

y fu

nctio

n (f

(x))

f(x)

p(x)

Poss

ibili

ty fu

nctio

n (p(x))



p(x)

1


pj(x)pi(x)

pt(x)

Sj

Si

pUN(x)

SUN










Normal stateN

A















1(119862) 120596

1(119880) and 120596

1(UN)


3s are expressed

as 1205962(119862) 120596

2(119872) and 120596




3(119880) 120596

3(119872)





1 1198602 119860

119872) and 120596



120596 (119860) =1

119896sum

cap119873

119894=1119860119894=119860

119873

prod

119894=1

120596119894(119860119894)

119896 = sum

cap119873

119894=1119860119894 =0

119873

prod

119894=1

120596119894(119860119894)

(13)




120596 (119862) =1

119896sdot 1205961 (119862) 1205962 (119862) 1205963 (UN)

120596 (119880) =1

119896sdot 1205961(119880) 1205962(UN) 120596

3(119880)

120596 (119872) =1

119896sdot 1205961(UN) 120596

2(119872)120596

3(119872)

120596 (UN) = 1

119896sdot 1205961(UN) 120596

2(UN) 120596

3(UN)

(14)


1205961(119880)1205962(UN)120596

3(119880) + 120596

1(UN)120596

2(119872)120596

3(119872) + 120596

1(UN)120596

2

(UN)1205963(UN) If max120596(119862) 120596(119880) 120596(119872) 120596(UN) = 120596(119862)










119873(119909) 119875

119860(119909) and




minus3 0 3 6 9

p(x)

1

PN(x)

PA(x)

Pt(x)

SN

SASUN

PUN (x)






8 Conclusions




minus3 0 3 6 9

p(x)

1

PN(x)

PA(x)

Pt(x)

SN

SA

SUN

PUN (x)


(a)

minus3minus6 00

3 6 9

p(x)

1

Pt(x)

PUN (x)


PC(x)

PU(x)

SC

SUSUN

(b)

0

p(x)

1

Pt(x)

PUN (x)



PC(x)

SC

PM(x)

SMSUN

(c)

0

p(x)

1

Pt(x)

PUN (x)



PU(x)

SUN

PM(x)

SM

(d)








Normal (119873) 08558 00724 00718 119873

Cavitation (119862)Mild 03529 06038 00433 07382 00579 01763 00276 119862

Medium 02311 07621 00068 09218 00053 00204 00525 119862

Severe 03200 06134 00666 09649 00001 00098 00252 119862


Mild 02026 06813 01161 00153 07468 01934 00445 119880

Medium 01897 07385 00718 00118 08216 01430 00236 119880

Severe 01365 07027 01608 00092 07604 02259 00045 119880


Mild 03415 06435 00150 01138 02126 06072 00664 119872

Medium 02071 07446 00483 00158 01901 06831 01110 119872

Severe 03007 06756 00237 00576 01573 06315 01536 119872




Normal (119873) 119873 119873

Cavitation (119862)Mild 119873 119873

Medium 119873 119862

Severe 119862 119862


Medium 119880 119880

Severe 119880 119880


Medium 119862 119872

Severe 119872 119872

P1

P2

Sym

ptom

par

amet

ers

Input layer


Normal state

Impeller unbalance

Cavitation state

Unknown state



P1

P2

Sym

ptom

par

amet

ers

Input layer

P3


State i

Other state

Unknown state








Acknowledgments


References


















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












1198752

1198753

1198751

1198752

1198753

1198751

1198752

1198753

1198751

1198752

1198753




1198752

1198753

1198751

1198752

1198753

1198751

1198752

1198753

1198751

1198752

1198753


0f1f2f3f4f5


fM

Spec

trum




119909119894=1199091015840

119894minus 1205831015840

1205901015840 (3)



119894(119894 =








1198751=120590

|119909|

1198752=sum119873

119894=1(119909119894minus 119909)3

119873 sdot 1205903

1198753=sum119873

119894=1(119909119894minus 119909)4

119873 sdot 1205904

(4)




119894|





0 005 01

(s)

(s)

(s)

0 005 01

0 005 01

minus2

0

2A

mpl

itude

(V)

minus2

0

2

Am

plitu

de (V

)

minus2

0

2

Am

plitu

de (V

)




FFT


Spec

trum

0 5 10 15 20 25

(kHz)

(kHz)

(kHz)

0 5 10 15 20 25

0 5 10 15 20 25

0

8

Spec

trum

0

8

Spec

trum

0

8



SF

IFFT





120596 sdot 119909 + 119887 = 0 (5)



x1

x2

x3



1 1199092










+ 119887 (6)


119894= 1198751119894 1198752119894 1198753119894 (119894 = 1 2 119868)






1 1205962







1= minus3587119875

1+ 6109119875

2minus 2234119875

3+ 10006


2= 0164119875

1+ 0897119875

2minus 3220119875

3minus 0107


3= minus0593119875

1+ 0031119875

2+ 1071119875

3+ 1934


SSP4= 1462119875

1minus 2843119875

2minus 4566119875

3+ 2006



In mediumstep

In severestep








119901 (119909119894) =

119873

sum

119896=1

min 120582119894 120582119896 (119894 119896 = 1 2 119873) (7)


120582119894and 120582


120582119894= int

119909119894

119909119894minus1

119898

120578sdot (119909 minus 1199090

120578)

119898minus1


0)119898

120578119889119909

119909119894= 119909 +

6119894 minus 3119873

119873sdot 120590

120582119896= int

119909119896

119909119896minus1

119898

120578sdot (119909 minus 1199090

120578)

119898minus1


0)119898

120578119889119909

119909119896= 119909 +

6119896 minus 3119873

119873sdot 120590

(8)


0are the



119894(119909) 119901119895(119909)) ofmodel



119878119896= int

119909119905+3120590119905

119909119905minus3120590119905

120595119896 (119909) 119889119909

120595119896 (119909) = min 119901

119896 (119909) 119901119905 (119909)

(119896 = 119894 119895)

(9)

where 119909119905and 120590



119905(119909) is


119901UN (119909) = max 1 minus [119901119894(119909) + 119901

119895(119909)] 0 (10)

119878UN = int119909119905+3120590119905

119909119905minus3120590119905

120595UN (119909) 119889119909

120595UN (119909) = min 119901UN (119909) 119901119905 (119909)

(11)


120596119896=1

120576sdot

119878119896

int119901119896(119909) 119889119909

(119896 = 119894 119895UN)

120576 =119878119894

int119901119894(119909) 119889119909

+

119878119895

int119901119895(119909) 119889119909

+119878UN

int119901UN (119909) 119889119909

(12)

0

02

04


1

05

Prob

abili

ty d

ensit

y fu

nctio

n (f

(x))

f(x)

p(x)

Poss

ibili

ty fu

nctio

n (p(x))



p(x)

1


pj(x)pi(x)

pt(x)

Sj

Si

pUN(x)

SUN










Normal stateN

A















1(119862) 120596

1(119880) and 120596

1(UN)


3s are expressed

as 1205962(119862) 120596

2(119872) and 120596




3(119880) 120596

3(119872)





1 1198602 119860

119872) and 120596



120596 (119860) =1

119896sum

cap119873

119894=1119860119894=119860

119873

prod

119894=1

120596119894(119860119894)

119896 = sum

cap119873

119894=1119860119894 =0

119873

prod

119894=1

120596119894(119860119894)

(13)




120596 (119862) =1

119896sdot 1205961 (119862) 1205962 (119862) 1205963 (UN)

120596 (119880) =1

119896sdot 1205961(119880) 1205962(UN) 120596

3(119880)

120596 (119872) =1

119896sdot 1205961(UN) 120596

2(119872)120596

3(119872)

120596 (UN) = 1

119896sdot 1205961(UN) 120596

2(UN) 120596

3(UN)

(14)


1205961(119880)1205962(UN)120596

3(119880) + 120596

1(UN)120596

2(119872)120596

3(119872) + 120596

1(UN)120596

2

(UN)1205963(UN) If max120596(119862) 120596(119880) 120596(119872) 120596(UN) = 120596(119862)










119873(119909) 119875

119860(119909) and




minus3 0 3 6 9

p(x)

1

PN(x)

PA(x)

Pt(x)

SN

SASUN

PUN (x)






8 Conclusions




minus3 0 3 6 9

p(x)

1

PN(x)

PA(x)

Pt(x)

SN

SA

SUN

PUN (x)


(a)

minus3minus6 00

3 6 9

p(x)

1

Pt(x)

PUN (x)


PC(x)

PU(x)

SC

SUSUN

(b)

0

p(x)

1

Pt(x)

PUN (x)



PC(x)

SC

PM(x)

SMSUN

(c)

0

p(x)

1

Pt(x)

PUN (x)



PU(x)

SUN

PM(x)

SM

(d)








Normal (119873) 08558 00724 00718 119873

Cavitation (119862)Mild 03529 06038 00433 07382 00579 01763 00276 119862

Medium 02311 07621 00068 09218 00053 00204 00525 119862

Severe 03200 06134 00666 09649 00001 00098 00252 119862


Mild 02026 06813 01161 00153 07468 01934 00445 119880

Medium 01897 07385 00718 00118 08216 01430 00236 119880

Severe 01365 07027 01608 00092 07604 02259 00045 119880


Mild 03415 06435 00150 01138 02126 06072 00664 119872

Medium 02071 07446 00483 00158 01901 06831 01110 119872

Severe 03007 06756 00237 00576 01573 06315 01536 119872




Normal (119873) 119873 119873

Cavitation (119862)Mild 119873 119873

Medium 119873 119862

Severe 119862 119862


Medium 119880 119880

Severe 119880 119880


Medium 119862 119872

Severe 119872 119872

P1

P2

Sym

ptom

par

amet

ers

Input layer


Normal state

Impeller unbalance

Cavitation state

Unknown state



P1

P2

Sym

ptom

par

amet

ers

Input layer

P3


State i

Other state

Unknown state








Acknowledgments


References


















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0 005 01

(s)

(s)

(s)

0 005 01

0 005 01

minus2

0

2A

mpl

itude

(V)

minus2

0

2

Am

plitu

de (V

)

minus2

0

2

Am

plitu

de (V

)




FFT


Spec

trum

0 5 10 15 20 25

(kHz)

(kHz)

(kHz)

0 5 10 15 20 25

0 5 10 15 20 25

0

8

Spec

trum

0

8

Spec

trum

0

8



SF

IFFT





120596 sdot 119909 + 119887 = 0 (5)



x1

x2

x3



1 1199092










+ 119887 (6)


119894= 1198751119894 1198752119894 1198753119894 (119894 = 1 2 119868)






1 1205962







1= minus3587119875

1+ 6109119875

2minus 2234119875

3+ 10006


2= 0164119875

1+ 0897119875

2minus 3220119875

3minus 0107


3= minus0593119875

1+ 0031119875

2+ 1071119875

3+ 1934


SSP4= 1462119875

1minus 2843119875

2minus 4566119875

3+ 2006



In mediumstep

In severestep








119901 (119909119894) =

119873

sum

119896=1

min 120582119894 120582119896 (119894 119896 = 1 2 119873) (7)


120582119894and 120582


120582119894= int

119909119894

119909119894minus1

119898

120578sdot (119909 minus 1199090

120578)

119898minus1


0)119898

120578119889119909

119909119894= 119909 +

6119894 minus 3119873

119873sdot 120590

120582119896= int

119909119896

119909119896minus1

119898

120578sdot (119909 minus 1199090

120578)

119898minus1


0)119898

120578119889119909

119909119896= 119909 +

6119896 minus 3119873

119873sdot 120590

(8)


0are the



119894(119909) 119901119895(119909)) ofmodel



119878119896= int

119909119905+3120590119905

119909119905minus3120590119905

120595119896 (119909) 119889119909

120595119896 (119909) = min 119901

119896 (119909) 119901119905 (119909)

(119896 = 119894 119895)

(9)

where 119909119905and 120590



119905(119909) is


119901UN (119909) = max 1 minus [119901119894(119909) + 119901

119895(119909)] 0 (10)

119878UN = int119909119905+3120590119905

119909119905minus3120590119905

120595UN (119909) 119889119909

120595UN (119909) = min 119901UN (119909) 119901119905 (119909)

(11)


120596119896=1

120576sdot

119878119896

int119901119896(119909) 119889119909

(119896 = 119894 119895UN)

120576 =119878119894

int119901119894(119909) 119889119909

+

119878119895

int119901119895(119909) 119889119909

+119878UN

int119901UN (119909) 119889119909

(12)

0

02

04


1

05

Prob

abili

ty d

ensit

y fu

nctio

n (f

(x))

f(x)

p(x)

Poss

ibili

ty fu

nctio

n (p(x))



p(x)

1


pj(x)pi(x)

pt(x)

Sj

Si

pUN(x)

SUN










Normal stateN

A















1(119862) 120596

1(119880) and 120596

1(UN)


3s are expressed

as 1205962(119862) 120596

2(119872) and 120596




3(119880) 120596

3(119872)





1 1198602 119860

119872) and 120596



120596 (119860) =1

119896sum

cap119873

119894=1119860119894=119860

119873

prod

119894=1

120596119894(119860119894)

119896 = sum

cap119873

119894=1119860119894 =0

119873

prod

119894=1

120596119894(119860119894)

(13)




120596 (119862) =1

119896sdot 1205961 (119862) 1205962 (119862) 1205963 (UN)

120596 (119880) =1

119896sdot 1205961(119880) 1205962(UN) 120596

3(119880)

120596 (119872) =1

119896sdot 1205961(UN) 120596

2(119872)120596

3(119872)

120596 (UN) = 1

119896sdot 1205961(UN) 120596

2(UN) 120596

3(UN)

(14)


1205961(119880)1205962(UN)120596

3(119880) + 120596

1(UN)120596

2(119872)120596

3(119872) + 120596

1(UN)120596

2

(UN)1205963(UN) If max120596(119862) 120596(119880) 120596(119872) 120596(UN) = 120596(119862)










119873(119909) 119875

119860(119909) and




minus3 0 3 6 9

p(x)

1

PN(x)

PA(x)

Pt(x)

SN

SASUN

PUN (x)






8 Conclusions




minus3 0 3 6 9

p(x)

1

PN(x)

PA(x)

Pt(x)

SN

SA

SUN

PUN (x)


(a)

minus3minus6 00

3 6 9

p(x)

1

Pt(x)

PUN (x)


PC(x)

PU(x)

SC

SUSUN

(b)

0

p(x)

1

Pt(x)

PUN (x)



PC(x)

SC

PM(x)

SMSUN

(c)

0

p(x)

1

Pt(x)

PUN (x)



PU(x)

SUN

PM(x)

SM

(d)








Normal (119873) 08558 00724 00718 119873

Cavitation (119862)Mild 03529 06038 00433 07382 00579 01763 00276 119862

Medium 02311 07621 00068 09218 00053 00204 00525 119862

Severe 03200 06134 00666 09649 00001 00098 00252 119862


Mild 02026 06813 01161 00153 07468 01934 00445 119880

Medium 01897 07385 00718 00118 08216 01430 00236 119880

Severe 01365 07027 01608 00092 07604 02259 00045 119880


Mild 03415 06435 00150 01138 02126 06072 00664 119872

Medium 02071 07446 00483 00158 01901 06831 01110 119872

Severe 03007 06756 00237 00576 01573 06315 01536 119872




Normal (119873) 119873 119873

Cavitation (119862)Mild 119873 119873

Medium 119873 119862

Severe 119862 119862


Medium 119880 119880

Severe 119880 119880


Medium 119862 119872

Severe 119872 119872

P1

P2

Sym

ptom

par

amet

ers

Input layer


Normal state

Impeller unbalance

Cavitation state

Unknown state



P1

P2

Sym

ptom

par

amet

ers

Input layer

P3


State i

Other state

Unknown state








Acknowledgments


References


















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














+ 119887 (6)


119894= 1198751119894 1198752119894 1198753119894 (119894 = 1 2 119868)






1 1205962







1= minus3587119875

1+ 6109119875

2minus 2234119875

3+ 10006


2= 0164119875

1+ 0897119875

2minus 3220119875

3minus 0107


3= minus0593119875

1+ 0031119875

2+ 1071119875

3+ 1934


SSP4= 1462119875

1minus 2843119875

2minus 4566119875

3+ 2006



In mediumstep

In severestep








119901 (119909119894) =

119873

sum

119896=1

min 120582119894 120582119896 (119894 119896 = 1 2 119873) (7)


120582119894and 120582


120582119894= int

119909119894

119909119894minus1

119898

120578sdot (119909 minus 1199090

120578)

119898minus1


0)119898

120578119889119909

119909119894= 119909 +

6119894 minus 3119873

119873sdot 120590

120582119896= int

119909119896

119909119896minus1

119898

120578sdot (119909 minus 1199090

120578)

119898minus1


0)119898

120578119889119909

119909119896= 119909 +

6119896 minus 3119873

119873sdot 120590

(8)


0are the



119894(119909) 119901119895(119909)) ofmodel



119878119896= int

119909119905+3120590119905

119909119905minus3120590119905

120595119896 (119909) 119889119909

120595119896 (119909) = min 119901

119896 (119909) 119901119905 (119909)

(119896 = 119894 119895)

(9)

where 119909119905and 120590



119905(119909) is


119901UN (119909) = max 1 minus [119901119894(119909) + 119901

119895(119909)] 0 (10)

119878UN = int119909119905+3120590119905

119909119905minus3120590119905

120595UN (119909) 119889119909

120595UN (119909) = min 119901UN (119909) 119901119905 (119909)

(11)


120596119896=1

120576sdot

119878119896

int119901119896(119909) 119889119909

(119896 = 119894 119895UN)

120576 =119878119894

int119901119894(119909) 119889119909

+

119878119895

int119901119895(119909) 119889119909

+119878UN

int119901UN (119909) 119889119909

(12)

0

02

04


1

05

Prob

abili

ty d

ensit

y fu

nctio

n (f

(x))

f(x)

p(x)

Poss

ibili

ty fu

nctio

n (p(x))



p(x)

1


pj(x)pi(x)

pt(x)

Sj

Si

pUN(x)

SUN










Normal stateN

A















1(119862) 120596

1(119880) and 120596

1(UN)


3s are expressed

as 1205962(119862) 120596

2(119872) and 120596




3(119880) 120596

3(119872)





1 1198602 119860

119872) and 120596



120596 (119860) =1

119896sum

cap119873

119894=1119860119894=119860

119873

prod

119894=1

120596119894(119860119894)

119896 = sum

cap119873

119894=1119860119894 =0

119873

prod

119894=1

120596119894(119860119894)

(13)




120596 (119862) =1

119896sdot 1205961 (119862) 1205962 (119862) 1205963 (UN)

120596 (119880) =1

119896sdot 1205961(119880) 1205962(UN) 120596

3(119880)

120596 (119872) =1

119896sdot 1205961(UN) 120596

2(119872)120596

3(119872)

120596 (UN) = 1

119896sdot 1205961(UN) 120596

2(UN) 120596

3(UN)

(14)


1205961(119880)1205962(UN)120596

3(119880) + 120596

1(UN)120596

2(119872)120596

3(119872) + 120596

1(UN)120596

2

(UN)1205963(UN) If max120596(119862) 120596(119880) 120596(119872) 120596(UN) = 120596(119862)










119873(119909) 119875

119860(119909) and




minus3 0 3 6 9

p(x)

1

PN(x)

PA(x)

Pt(x)

SN

SASUN

PUN (x)






8 Conclusions




minus3 0 3 6 9

p(x)

1

PN(x)

PA(x)

Pt(x)

SN

SA

SUN

PUN (x)


(a)

minus3minus6 00

3 6 9

p(x)

1

Pt(x)

PUN (x)


PC(x)

PU(x)

SC

SUSUN

(b)

0

p(x)

1

Pt(x)

PUN (x)



PC(x)

SC

PM(x)

SMSUN

(c)

0

p(x)

1

Pt(x)

PUN (x)



PU(x)

SUN

PM(x)

SM

(d)








Normal (119873) 08558 00724 00718 119873

Cavitation (119862)Mild 03529 06038 00433 07382 00579 01763 00276 119862

Medium 02311 07621 00068 09218 00053 00204 00525 119862

Severe 03200 06134 00666 09649 00001 00098 00252 119862


Mild 02026 06813 01161 00153 07468 01934 00445 119880

Medium 01897 07385 00718 00118 08216 01430 00236 119880

Severe 01365 07027 01608 00092 07604 02259 00045 119880


Mild 03415 06435 00150 01138 02126 06072 00664 119872

Medium 02071 07446 00483 00158 01901 06831 01110 119872

Severe 03007 06756 00237 00576 01573 06315 01536 119872




Normal (119873) 119873 119873

Cavitation (119862)Mild 119873 119873

Medium 119873 119862

Severe 119862 119862


Medium 119880 119880

Severe 119880 119880


Medium 119862 119872

Severe 119872 119872

P1

P2

Sym

ptom

par

amet

ers

Input layer


Normal state

Impeller unbalance

Cavitation state

Unknown state



P1

P2

Sym

ptom

par

amet

ers

Input layer

P3


State i

Other state

Unknown state








Acknowledgments


References


















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










120582119894and 120582


120582119894= int

119909119894

119909119894minus1

119898

120578sdot (119909 minus 1199090

120578)

119898minus1


0)119898

120578119889119909

119909119894= 119909 +

6119894 minus 3119873

119873sdot 120590

120582119896= int

119909119896

119909119896minus1

119898

120578sdot (119909 minus 1199090

120578)

119898minus1


0)119898

120578119889119909

119909119896= 119909 +

6119896 minus 3119873

119873sdot 120590

(8)


0are the



119894(119909) 119901119895(119909)) ofmodel



119878119896= int

119909119905+3120590119905

119909119905minus3120590119905

120595119896 (119909) 119889119909

120595119896 (119909) = min 119901

119896 (119909) 119901119905 (119909)

(119896 = 119894 119895)

(9)

where 119909119905and 120590



119905(119909) is


119901UN (119909) = max 1 minus [119901119894(119909) + 119901

119895(119909)] 0 (10)

119878UN = int119909119905+3120590119905

119909119905minus3120590119905

120595UN (119909) 119889119909

120595UN (119909) = min 119901UN (119909) 119901119905 (119909)

(11)


120596119896=1

120576sdot

119878119896

int119901119896(119909) 119889119909

(119896 = 119894 119895UN)

120576 =119878119894

int119901119894(119909) 119889119909

+

119878119895

int119901119895(119909) 119889119909

+119878UN

int119901UN (119909) 119889119909

(12)

0

02

04


1

05

Prob

abili

ty d

ensit

y fu

nctio

n (f

(x))

f(x)

p(x)

Poss

ibili

ty fu

nctio

n (p(x))



p(x)

1


pj(x)pi(x)

pt(x)

Sj

Si

pUN(x)

SUN










Normal stateN

A















1(119862) 120596

1(119880) and 120596

1(UN)


3s are expressed

as 1205962(119862) 120596

2(119872) and 120596




3(119880) 120596

3(119872)





1 1198602 119860

119872) and 120596



120596 (119860) =1

119896sum

cap119873

119894=1119860119894=119860

119873

prod

119894=1

120596119894(119860119894)

119896 = sum

cap119873

119894=1119860119894 =0

119873

prod

119894=1

120596119894(119860119894)

(13)




120596 (119862) =1

119896sdot 1205961 (119862) 1205962 (119862) 1205963 (UN)

120596 (119880) =1

119896sdot 1205961(119880) 1205962(UN) 120596

3(119880)

120596 (119872) =1

119896sdot 1205961(UN) 120596

2(119872)120596

3(119872)

120596 (UN) = 1

119896sdot 1205961(UN) 120596

2(UN) 120596

3(UN)

(14)


1205961(119880)1205962(UN)120596

3(119880) + 120596

1(UN)120596

2(119872)120596

3(119872) + 120596

1(UN)120596

2

(UN)1205963(UN) If max120596(119862) 120596(119880) 120596(119872) 120596(UN) = 120596(119862)










119873(119909) 119875

119860(119909) and




minus3 0 3 6 9

p(x)

1

PN(x)

PA(x)

Pt(x)

SN

SASUN

PUN (x)






8 Conclusions




minus3 0 3 6 9

p(x)

1

PN(x)

PA(x)

Pt(x)

SN

SA

SUN

PUN (x)


(a)

minus3minus6 00

3 6 9

p(x)

1

Pt(x)

PUN (x)


PC(x)

PU(x)

SC

SUSUN

(b)

0

p(x)

1

Pt(x)

PUN (x)



PC(x)

SC

PM(x)

SMSUN

(c)

0

p(x)

1

Pt(x)

PUN (x)



PU(x)

SUN

PM(x)

SM

(d)








Normal (119873) 08558 00724 00718 119873

Cavitation (119862)Mild 03529 06038 00433 07382 00579 01763 00276 119862

Medium 02311 07621 00068 09218 00053 00204 00525 119862

Severe 03200 06134 00666 09649 00001 00098 00252 119862


Mild 02026 06813 01161 00153 07468 01934 00445 119880

Medium 01897 07385 00718 00118 08216 01430 00236 119880

Severe 01365 07027 01608 00092 07604 02259 00045 119880


Mild 03415 06435 00150 01138 02126 06072 00664 119872

Medium 02071 07446 00483 00158 01901 06831 01110 119872

Severe 03007 06756 00237 00576 01573 06315 01536 119872




Normal (119873) 119873 119873

Cavitation (119862)Mild 119873 119873

Medium 119873 119862

Severe 119862 119862


Medium 119880 119880

Severe 119880 119880


Medium 119862 119872

Severe 119872 119872

P1

P2

Sym

ptom

par

amet

ers

Input layer


Normal state

Impeller unbalance

Cavitation state

Unknown state



P1

P2

Sym

ptom

par

amet

ers

Input layer

P3


State i

Other state

Unknown state








Acknowledgments


References


















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













Normal stateN

A















1(119862) 120596

1(119880) and 120596

1(UN)


3s are expressed

as 1205962(119862) 120596

2(119872) and 120596




3(119880) 120596

3(119872)





1 1198602 119860

119872) and 120596



120596 (119860) =1

119896sum

cap119873

119894=1119860119894=119860

119873

prod

119894=1

120596119894(119860119894)

119896 = sum

cap119873

119894=1119860119894 =0

119873

prod

119894=1

120596119894(119860119894)

(13)




120596 (119862) =1

119896sdot 1205961 (119862) 1205962 (119862) 1205963 (UN)

120596 (119880) =1

119896sdot 1205961(119880) 1205962(UN) 120596

3(119880)

120596 (119872) =1

119896sdot 1205961(UN) 120596

2(119872)120596

3(119872)

120596 (UN) = 1

119896sdot 1205961(UN) 120596

2(UN) 120596

3(UN)

(14)


1205961(119880)1205962(UN)120596

3(119880) + 120596

1(UN)120596

2(119872)120596

3(119872) + 120596

1(UN)120596

2

(UN)1205963(UN) If max120596(119862) 120596(119880) 120596(119872) 120596(UN) = 120596(119862)










119873(119909) 119875

119860(119909) and




minus3 0 3 6 9

p(x)

1

PN(x)

PA(x)

Pt(x)

SN

SASUN

PUN (x)






8 Conclusions




minus3 0 3 6 9

p(x)

1

PN(x)

PA(x)

Pt(x)

SN

SA

SUN

PUN (x)


(a)

minus3minus6 00

3 6 9

p(x)

1

Pt(x)

PUN (x)


PC(x)

PU(x)

SC

SUSUN

(b)

0

p(x)

1

Pt(x)

PUN (x)



PC(x)

SC

PM(x)

SMSUN

(c)

0

p(x)

1

Pt(x)

PUN (x)



PU(x)

SUN

PM(x)

SM

(d)








Normal (119873) 08558 00724 00718 119873

Cavitation (119862)Mild 03529 06038 00433 07382 00579 01763 00276 119862

Medium 02311 07621 00068 09218 00053 00204 00525 119862

Severe 03200 06134 00666 09649 00001 00098 00252 119862


Mild 02026 06813 01161 00153 07468 01934 00445 119880

Medium 01897 07385 00718 00118 08216 01430 00236 119880

Severe 01365 07027 01608 00092 07604 02259 00045 119880


Mild 03415 06435 00150 01138 02126 06072 00664 119872

Medium 02071 07446 00483 00158 01901 06831 01110 119872

Severe 03007 06756 00237 00576 01573 06315 01536 119872




Normal (119873) 119873 119873

Cavitation (119862)Mild 119873 119873

Medium 119873 119862

Severe 119862 119862


Medium 119880 119880

Severe 119880 119880


Medium 119862 119872

Severe 119872 119872

P1

P2

Sym

ptom

par

amet

ers

Input layer


Normal state

Impeller unbalance

Cavitation state

Unknown state



P1

P2

Sym

ptom

par

amet

ers

Input layer

P3


State i

Other state

Unknown state








Acknowledgments


References


















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120596 (119862) =1

119896sdot 1205961 (119862) 1205962 (119862) 1205963 (UN)

120596 (119880) =1

119896sdot 1205961(119880) 1205962(UN) 120596

3(119880)

120596 (119872) =1

119896sdot 1205961(UN) 120596

2(119872)120596

3(119872)

120596 (UN) = 1

119896sdot 1205961(UN) 120596

2(UN) 120596

3(UN)

(14)


1205961(119880)1205962(UN)120596

3(119880) + 120596

1(UN)120596

2(119872)120596

3(119872) + 120596

1(UN)120596

2

(UN)1205963(UN) If max120596(119862) 120596(119880) 120596(119872) 120596(UN) = 120596(119862)










119873(119909) 119875

119860(119909) and




minus3 0 3 6 9

p(x)

1

PN(x)

PA(x)

Pt(x)

SN

SASUN

PUN (x)






8 Conclusions




minus3 0 3 6 9

p(x)

1

PN(x)

PA(x)

Pt(x)

SN

SA

SUN

PUN (x)


(a)

minus3minus6 00

3 6 9

p(x)

1

Pt(x)

PUN (x)


PC(x)

PU(x)

SC

SUSUN

(b)

0

p(x)

1

Pt(x)

PUN (x)



PC(x)

SC

PM(x)

SMSUN

(c)

0

p(x)

1

Pt(x)

PUN (x)



PU(x)

SUN

PM(x)

SM

(d)








Normal (119873) 08558 00724 00718 119873

Cavitation (119862)Mild 03529 06038 00433 07382 00579 01763 00276 119862

Medium 02311 07621 00068 09218 00053 00204 00525 119862

Severe 03200 06134 00666 09649 00001 00098 00252 119862


Mild 02026 06813 01161 00153 07468 01934 00445 119880

Medium 01897 07385 00718 00118 08216 01430 00236 119880

Severe 01365 07027 01608 00092 07604 02259 00045 119880


Mild 03415 06435 00150 01138 02126 06072 00664 119872

Medium 02071 07446 00483 00158 01901 06831 01110 119872

Severe 03007 06756 00237 00576 01573 06315 01536 119872




Normal (119873) 119873 119873

Cavitation (119862)Mild 119873 119873

Medium 119873 119862

Severe 119862 119862


Medium 119880 119880

Severe 119880 119880


Medium 119862 119872

Severe 119872 119872

P1

P2

Sym

ptom

par

amet

ers

Input layer


Normal state

Impeller unbalance

Cavitation state

Unknown state



P1

P2

Sym

ptom

par

amet

ers

Input layer

P3


State i

Other state

Unknown state








Acknowledgments


References


















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus3 0 3 6 9

p(x)

1

PN(x)

PA(x)

Pt(x)

SN

SA

SUN

PUN (x)


(a)

minus3minus6 00

3 6 9

p(x)

1

Pt(x)

PUN (x)


PC(x)

PU(x)

SC

SUSUN

(b)

0

p(x)

1

Pt(x)

PUN (x)



PC(x)

SC

PM(x)

SMSUN

(c)

0

p(x)

1

Pt(x)

PUN (x)



PU(x)

SUN

PM(x)

SM

(d)








Normal (119873) 08558 00724 00718 119873

Cavitation (119862)Mild 03529 06038 00433 07382 00579 01763 00276 119862

Medium 02311 07621 00068 09218 00053 00204 00525 119862

Severe 03200 06134 00666 09649 00001 00098 00252 119862


Mild 02026 06813 01161 00153 07468 01934 00445 119880

Medium 01897 07385 00718 00118 08216 01430 00236 119880

Severe 01365 07027 01608 00092 07604 02259 00045 119880


Mild 03415 06435 00150 01138 02126 06072 00664 119872

Medium 02071 07446 00483 00158 01901 06831 01110 119872

Severe 03007 06756 00237 00576 01573 06315 01536 119872




Normal (119873) 119873 119873

Cavitation (119862)Mild 119873 119873

Medium 119873 119862

Severe 119862 119862


Medium 119880 119880

Severe 119880 119880


Medium 119862 119872

Severe 119872 119872

P1

P2

Sym

ptom

par

amet

ers

Input layer


Normal state

Impeller unbalance

Cavitation state

Unknown state



P1

P2

Sym

ptom

par

amet

ers

Input layer

P3


State i

Other state

Unknown state








Acknowledgments


References


















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Normal (119873) 119873 119873

Cavitation (119862)Mild 119873 119873

Medium 119873 119862

Severe 119862 119862


Medium 119880 119880

Severe 119880 119880


Medium 119862 119872

Severe 119872 119872

P1

P2

Sym

ptom

par

amet

ers

Input layer


Normal state

Impeller unbalance

Cavitation state

Unknown state



P1

P2

Sym

ptom

par

amet

ers

Input layer

P3


State i

Other state

Unknown state








Acknowledgments


References


















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation





















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article intelligent diagnosis method for...

Documents