Research ArticleUndecimated Lifting Wavelet Packet Transform withBoundary Treatment for Machinery Incipient Fault Diagnosis

Lixiang Duan1 Yangshen Wang1 Jinjiang Wang1 Laibin Zhang1 and Jinglong Chen2

1School of Mechanical and Transportation Engineering China University of Petroleum Beijing 102249 China2School of Material Science and Engineering Xian University of Architecture and Technology Xirsquoan 710055 China

Correspondence should be addressed to Jinjiang Wang jiw09005engruconnedu

Received 26 May 2015 Revised 22 July 2015 Accepted 25 August 2015

Academic Editor Jose V Araujo dos Santos

Copyright copy 2016 Lixiang Duan 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

Effective signal processing in fault detection and diagnosis (FDD) is an important measure to prevent failure and accidents ofmachinery To address the end distortion and frequency aliasing issues in conventional lifting wavelet transform a Volterraseries assisted undecimated lifting wavelet packet transform (ULWPT) is investigated for machinery incipient fault diagnosisUndecimated lifting wavelet packet transform is firstly formulated to eliminate the frequency aliasing issue in traditional liftingwavelet packet transform Next Volterra series as a boundary treatment method is used to preprocess the signal to suppress theend distortion in undecimated lifting wavelet packet transform Finally the decomposed wavelet coefficients are trimmed to theoriginal length as the signal of interest for machinery incipient fault detection Experimental study on a reciprocating compressoris performed to demonstrate the effectiveness of the presented method The results show that the presented method outperformsthe conventional approach by dramatically enhancing the weak defect feature extraction for reciprocating compressor valve faultdiagnosis

1 Introduction

Fault detection and diagnosis play an important role inmachinery condition monitoring to improve product qualityand avoid catastrophic damage or huge production loss [1 2]Increasing demand on system reliability has accelerated theinstallation of sensors to acquire the machinery conditionstatus However the signals caused by incipient fault com-ponents are usually weak and severely drowned out by thestrong noise from machinery vibration and measurementsystem [3] which pose significant challenge on machineryfault diagnosis at early stage

Much effort has been put on developing effective signalprocessing for fault detection and diagnosis during thepast decades Various signal processing techniques includingwavelet transform [4] empirical mode decomposition [5]Wigner-Ville distribution [6] singular value decomposition(SVD) denoising [7 8] and blind source separation [9 10]have been investigated for noise suppression enhanced weakfeature extraction and signal time-frequency decomposition

Among these signal processing methods wavelet transformis the most widely investigated technique Peng and Chuconducted an overview of wavelet transform for machin-ery condition monitoring [11] Yan et al reviewed recentapplications of wavelet transform in rotary machinery faultdiagnosis [12] Discrete wavelet transform was investigatedto extract fault features for gearbox defect diagnosis [13] Awavelet filter-based method with minimal Shannon entropycriterion was investigated for vibration signal denoising inbearing defect prognosis [14] In [15] a multiscale envelopingorder spectrogram based on continuous complex wavelettransform was developed for bearing incipient defect diag-nosis under nonstationary operating conditions A dual-treecomplex wavelet transform based adaptive wavelet shrinkagetechnique was investigated for mechanical vibration signaldenoising [16]

Liftingwavelet transform which is also named as second-generation wavelet transform has attracted considerableattention for machinery fault diagnosis It is implementedthrough lifting scheme by recursive prediction and updating

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 9792807 9 pageshttpdxdoiorg10115520169792807

2 Shock and Vibration

operations to decompose the signal in time domain whichhas the superiorities for example faster implementationbeing independent of Fourier transform and meanwhileretaining all advantages of traditional wavelet transform In[17] a lifting wavelet packet decomposition method waspresented to extract fault features for bearing performancedegradation assessment A redundant lifting wavelet packettransform was applied to diagnose gearbox and engine [18]A multiwavelet lifting scheme to optimize lifting schemewas presented for gearbox diagnosis [19] A combination oflifting wavelet and finite element method was investigatedfor the quantitative identification of pipeline cracks [20] Inthe abovementioned studies decimated lifting scheme usingdownsampling algorithm is commonly used which leads tofrequency aliasing in the decomposition results [21] On theother hand the lifting scheme causes end distortion whichconfuses or misleads the diagnosis Thus an undecimatedlifting wavelet transform with boundary treatment is neededto suppress frequency aliasing and end distortion

Various boundary treatment methods are investigatedto suppress end distortion in lifting wavelet transform In[22 23] the order of predictor in the lifting scheme is reducedby overlapping the edges to suppress end distortion Differentboundary extension methods such as zero-padding exten-sion symmetric extension periodic extension zero-ordersmoothing extension and one-order smoothing extensionmethods are investigated in [24 25] These methods couldsuppress the end distortion to some degree but not up tosatisfaction To address these issues this paper presents anew signal processingmethod namedVolterra series assistedundecimated lifting wavelet packet transform by extendingour prior work [26] First of all Volterra series [27] as aboundary treatment method is used to extend both endsof the signal Then the wavelet coefficients decomposed bythe undecimated lifting wavelet packet are chopped back tooriginal length to serve as the signals of interest formachineryincipient fault detection Finally the effectiveness of thepresented method is demonstrated on valve incipient defectdiagnosis in a reciprocating compressorThus the intellectualmerits of this paper are outlined including the following(1) a Volterra assisted undecimated lifting wavelet packettransformmethod is presented to suppress the end distortionand frequency aliasing issues in conventional lifting wavelettransform and (2) the formula to optimize the number ofextended signal in boundary treatment using Volterra seriesis derived according to the decomposition property of undec-imated lifting wavelet packet transform

The rest of the paper is organized as follows Section 2introduces the theoretical background of lifting wavelettransform and Volterra series Section 3 presents the theoret-ical framework of Volterra series assisted undecimated liftingwavelet packet transform and the equations of the undeci-mated lifting wavelet packet Performance comparison of dif-ferent boundary treatment methods is also discussed herebyAn experimental study of incipient fault detection of recipro-cating compressor valve using the presented method is con-ducted and the analysis results are discussed in Section 4Theconclusions are finally drawn in Section 5

2 Theoretical Background

21 LiftingWavelet Transform Lifting wavelet transform wasfirstly presented by Sweldens in the 1990s [28] Based on lift-ing scheme it calculates the wavelet coefficients using poly-nomial interpolationmethod and constructs scaling functionto obtain the low frequency coefficients of the signal If thescaling function curve is smooth and the ringing artifact ofboundary is reduced adequately an ideal wavelet coefficientcan be acquired using interpolator split schemes

Lifting wavelet transform consists of three steps splitprediction and update [29]

(1) Split Several methods for signal split are available Onemethod could be dividing signals into left and right halvesHowever the result will be unsatisfying due to the lowrelativity degree between the left and the right halves Onemore effective method is to divide the data 119909[119899] into even set119909119890[119899] = 119909[2119899] and odd set 119909

119900[119899] = 119909[2119899 + 1] where 119899 is the

number of data

(2) Prediction The purpose of prediction is to eliminatethe low frequency components of signals and preserve thehigh frequency part The odd set 119909

119900[119899] can be predicted

from the even set 119909119890[119899] and the prediction operator P =

[1199011 1199012 119901

119873] The prediction value is P(119909

119890[119899]) The differ-

ence between the practical value and the prediction value isdefined as 119889[119899]

119889 [119899] = 119909119900[119899] minus P (119909

119890[119899]) (1)

where 119889[119899] also called detailed signal reflects the high fre-quency component of the signals Here119873 is a dual vanishingmoment that determines the smoothness of the interpolationfunction

(3) Update In order to reduce the frequency aliasing effectthe odd set 119909

119890[119899] is updated using detail signal 119889[119899] and

update operatorU = [1199061 1199062 119906

1198731015840]The result of this step is

the approximation signal 119888[119899] that reflects the low frequencycoefficient of the signals which can be written as

119888 [119899] = 119909119890[119899] + U (119889 [119899])P (119909

119890[119899]) (2)

Signals can be decomposed by lifting wavelet transformusing the above iterative operation of approximation signal119888[119899] Prediction coefficients can be calculated by the Lagrangeinterpolation formula As long as the length of the updateoperator is the same as that of the prediction operator theupdate coefficient value will be half of the correspondingprediction coefficient [30] However the downsampling algo-rithm used in the conventional lifting wavelet transform willlead to frequency aliasing because the transformed signalbecomes half of its length in the previous layer The updatealgorithm can reduce but not completely eliminate frequencyaliasing The signal after downsampling algorithm will notmeet the conditions of sampling theorem which leads tounexpected virtual frequency components Thus an undec-imated lifting wavelet transform is outlined and discussed toeliminate frequency aliasing in this study

Shock and Vibration 3

22 Volterra Series Volterra series was initially proposed byan Italian mathematician named Vito Volterra in the 1880sDue to its powerful ability tomodel the behavior of nonlinearsystems the theory has attracted a great deal of attentionand soon gained its applications in many fields If the inputof a nonlinear discrete time system is X(119899) = [119909(119899) 119909(119899 minus

1) 119909(119899 minus 1198731015840

+ 1)] and the output is 119910(119899) =997888

119909

1015840

(119899 + 1) thesystem function can construct nonlinear prediction modelwith the expansion of Volterra series as given by [31]

997888

119909

1015840

(119899 + 1) = ℎ0+

+infin

sum

119894=0

ℎ1(119894) 119909 (119899 minus 119894) +

+infin

sum

119894=0

+infin

sum

119895=0

ℎ2(119894 119895)

sdot 119909 (119899 minus 119894) 119909 (119899 minus 119895) + sdot sdot sdot

+

+infin

sum

119894=0

+infin

sum

119895=0

sdot sdot sdot

+infin

sum

119896=0

ℎ119901(119894 119895 119896) 119909 (119899 minus 119894) 119909 (119899 minus 119895)

sdot sdot sdot 119909 (119899 minus 119896)

(3)

where ℎ119901(119894 119895 119896) is the 119901th order nucleus of Volterra

The expansion of this infinite order series is extremelydifficult in practical applications Generally the second-ordertruncation is employed as follows

997888

119909

1015840

(119899 + 1) = ℎ0+

1198731minus1

sum

119894=0

ℎ1(119894) 119909 (119899 minus 119894)

+

1198732minus1

sum

119894=0

1198732minus1

sum

119895=0

ℎ2(119894 119895) 119909 (119899 minus 119894) 119909 (119899 minus 119895)

(4)

where1198731and119873

2represent the length of filtersTheminimum

embedding dimension119898 of the signal can be obtained usingthe fault near-neighbor method Consequently 119873

1and 119873

2

can be set as119898 [32]Volterra series is used to extend both ends of the signal to

address the end distortion issue in lifting wavelet transformThe integrative approach of Volterra series and lifting wavelettransform is presented in detail in the following sections

The input vector isX(119899) = [1 119909(119899) 119909(119899minus1) 119909(119899minus119898minus

1) 1199092

(119899) 119909(119899)119909(119899 minus 1) 1199092

(119899 minus 119898 + 1)]119879 The prediction

coefficient vector is W(119899) = [ℎ0 ℎ1(0) ℎ1(1) ℎ

1(119898 minus

1) ℎ2(0 0) ℎ

2(0 1) ℎ

2(119898 minus 1119898 minus 1)]

119879 Thus the expres-sion of (4) can be rewritten as [33]

997888

119909

1015840

(119899 + 1) = X119879 (119899)W (119899) (5)

The prediction coefficient vectorW(119899) is calculated usingthe recursive least-squares method (RLS) To elaborate con-sider

119876 (0) = 120575minus1I (6)

where 120575 is a very small normal number and I is the identitymatrix ThusW(0) is set as

W (0) = 0 (7)

andW(119899) is calculated by carrying on the following iterativecomputation

Consider

119866 (119899) =

120583minus1

119876 (119899 minus 1)X (119899)

1 + 120583minus1X119879 (119899) 119876 (119899 minus 1)X (119899)

(8)

where 120583 is a forgetting factorConsider the following

120572 (119899) = 119863 (119899) minusW119879 (119899 minus 1)X (119899) (9)

where119863(119899) is an ideal output signal Thus

W (119899) = W (119899 minus 1) + 119866 (119899) 120572 (119899)

119876 (119899) = 120583minus1

119876 (119899 minus 1) minus 120583minus1

119866 (119899)X119879 (119899) 119876 (119899 minus 1)

(10)

where 120572(119899) and 119866(119899) are intermediate variablesThe details of theoretical knowledge of lifting wavelet

transform and Volterra series model have been discussedabove Volterra series is used to extend both ends of thesignal to address the end distortion issue in lifting wavelettransformThe formulation of Volterra series assisted undec-imated lifting wavelet transform is illustrated below

3 The Proposed Method

A Volterra series-assisted undecimated lifting wavelet packettransform is proposed for machinery incipient defect diag-nosis to eliminate the frequency aliasing and end distortionissues in traditional lifting wavelet transform and its flow-chart is shown in Figure 1

First both ends of the original signal are extended andpredicted with Volterra series model in which the param-eter of extension number is optimized Next the extendedsignal is decomposed by undecimated lifting wavelet packettransform and the subband with the highest energy ratio isselected as the band of interestThe selected subband signal isthen trimmed back to its original length for envelope analysisFinally themachinery status is assessed according to the anal-ysis results The details of undecimated lifting wavelet packettransform with boundary treatment are discussed as below

31 Formulation of Undecimated LWPT with Boundary Treat-ment The undecimated algorithm can eliminate frequencyaliasing because the length of coefficients at each level isequal to that of the raw signal The P and U can be schemedout as initial prediction operator and initial update operatorrespectively and the undecimated algorithm is deduced asfollows

Assuming P = 119901119894 where 119894 = 1 2 119873 the expression

of the undecimated prediction operator of the 119897th level isgiven by [34]

P[119897]119895

=

119901119894 119895 = 2

119897

sdot 119894

0 119895 = 2119897

sdot 119894

119895 = 1 2119897

119873 (11)

4 Shock and Vibration

0 001 002 003 004 005 006

Extension signal

Machinery Volterra series model

Envelope analysis

Status assessment

Data acquisition0 001 002 003 004

0

5

10

15

Parameter optimization

0 005 01 015 02 025

01

X44

0 005 01 015 02 025

010

X43

0 005 01 015 02 025

01

X42

0 005 01 015 02 025

010

X41

Undecimated LWPTData trimming

0 005 01 015 02 0250 001 002 003 004 005 006 007

02468

Prediction coefficient

Extension number

Undecimated prediction operator P

Undecimated update

operator U

minus4

minus2

minus8

minus6

0246

minus4

minus2

minus6

minus1

minus1

minus10

minus10

minus10

minus5

minus15

0

5

10

15

minus10

minus5

minus15

Time t (s) Time t (s)

W(n) nadd

Am

plitu

dea

(mmmiddotsminus

2 )

Am

plitu

dea

(mmmiddotsminus

2 )

Figure 1 Flowchart of the proposed approach for machinery incipient defect diagnosis

According to the principle of equidistance subdivision thepredictor coefficient 119901

119894is calculated by Lagrange interpola-

tion formula as [29]

119901119894=

119873

prod

119896=1

119896 =119894

(119873 + 1) 2 minus 119896

119894 minus 119896

119894 = 1 2 119873 (12)

where 119894 is an index number and119873 is the predictor lengthSimilarly assuming U = 119906

119894 where 119894 = 1 2 119873

1015840 theexpression of the undecimated update operator of the 119897th levelis obtained by [34]

U[119897]119895

=

119906119894 119895 = 2

119897

sdot 119894

0 119895 = 2119897

sdot 119894

119895 = 1 2119897

1198731015840

(13)

where 119906119894is the update coefficient and is usually set as the half

of the predictor coefficient 119901119894

Assuming 119909119897119896is the 119896th frequency band of the 119897th level

signal decomposed from the original signal 119909[119899] 119909119897(119896minus1)

and119909119897119896can be obtained by dividing 119909

(119897minus1)(1198962)

119909119897(119896minus1)

(119899) = 119909(119897minus1)(1198962)

(119899)

minus [119901[119897]

1119909(119897minus1)(1198962)

(119899 minus 2119897minus1

(119873 + 1) + 1)

+ 119901[119897]

2119909(119897minus1)(1198962)

(119899 minus 2119897minus1

(119873 + 1) + 2) + sdot sdot sdot

+ 119901[119897]

2119897119873

119909(119897minus1)(1198962)

(119899 + 2119897minus1

(119873 minus 1))]

119909119897119896(119899) = 119909

(119897minus1)(1198962)(119899)

+ [119906[119897]

1119909119897(119896minus1)

(119899 minus 2119897minus1

(1198731015840

+ 1) + 1)

+ 119906[119897]

2119909119897(119896minus1)

(119899 minus 2119897minus1

(1198731015840

+ 1) + 2) + sdot sdot sdot

+ 119906[119897]

2119897

119909119897(119896minus1)

(119899 + 2119897minus1

(1198731015840

minus 1))]

(14)

where 119896 = 2 4 6 2119897 119901[119897]119894

is the undecimated predictionoperator of 119909

119897(119896minus1) and 119906[119897]

119894is the undecimated update opera-

tor of 119909119897119896

Assuming the undecimated prediction operator P = 119901119894

and the undecimated update operator U = 119906119894 where 119894 =

1 2 4 119909119894is the 119894th data of X The 119894th data of the first level

detailed signal decomposed by undecimated lifting waveletpacket transform is obtained as follows

1198891119894

= 119909119894minus (119909119894minus3

times 1199011+ 119909119894minus2

times 0 + 119909119894minus1

times 1199012+ 119909119894times 0

+ 119909119894+1

times 1199013+ 119909119894+3

times 1199014)

(15)

It can be deduced that three numbersmust be extended tothe left end of 119909 when calculating 119889

11and the same number

to the right when calculating the last data 1198891119899 Similarly for

the second level 6 more numbers should be extended to eachend and so on More generally assuming 119873

119891is the decom-

posed level of undecimated lifting wavelet packet transformthe extension number of the original signal at each end can becalculated as

119899add =

119873119891

sum

119897=1

[2119897minus1

(119873 + 1) minus 1]

+

119873119891

sum

119897=1

[2119897minus1

(1198731015840

+ 1) minus 1]

(16)

According to (16) the length of extended signal inVolterra series model is determined and the signal isextended based on the procedure of Volterra series model inSection 2 The performance of formulated method is dis-cussed below

Shock and Vibration 5

0

01

02

03

End distortion

Characteristic signal

End distortion

minus02

minus01

002 004 006 008 01 0120Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )

(a)

End distortion

Characteristic signal

End distortion

002 004 006 008 01 0120Time t (s)

minus2

minus1

0

1

2

Am

plitu

dea

(mmmiddotsminus

2 )

(b)

Characteristic signalEnd distortion

minus8

minus6

minus4

minus2

0

2

4

6

8

002 004 006 008 01 0120Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )

(c)

Characteristic signal

002 004 006 008 01 0120Time t (s)

minus02

minus01

0

01

02

03

Am

plitu

dea

(mmmiddotsminus

2 )

(d)

Figure 2 One subband in the fourth layer decomposed using wavelet packet and preprocessed by different methods (a) no extension (b)zero-padding extension (c) periodic extension and (d) second-order Volterra series prediction

32 Performance Comparison of Boundary Treatment Meth-ods To evaluate the performance of various boundary treat-ment methods a synthesized signal is constructed as follows

119909 (119905) = 03 exp (minus2120587 times 100119905) sin (2120587 times 400119905)

+ 15 sin(2120587 times 50119905 +

120587

4

)

+ 3 sin(2120587 times 100119905 +

120587

6

)

(17)

The sampling frequency is set as 16000Hz The synthesizedsignal is decomposed into four layers using the undecimatedlifting wavelet transform Figure 2 shows one subband of thefourth layer corresponding to the pulse train signal underdifferent preprocessing scenarios (a) no extension (b) zero-padding extension (c) periodic extension and (d) second-order Volterra series prediction respectively

There are obvious end distortions shown by the arrows inFigures 2(a)ndash2(c)Therefore if the characteristic signal is just

located in the end it will be submerged by the end distor-tions In Figures 2(b)-2(c) the characteristic signal in themiddle is insignificant because the end distortion is strongThe Volterra series based preprocessing method shows theoptimal performance since the end distortions have beenrestrained effectively as shown in Figure 2(d)

4 Experimental Study

A reciprocating compressor (model number 4HOS-6) in apetrochemical plant in China is used as the experimentaltestbed to evaluate the performance of the developedmethodas shown in Figure 3 It is a 4-cylinder natural gas recipro-cating compressor driven by an 8-cylinder gas engine withrated power of 1600 kW The rotating speed of crankshaft is860 rpm which drives the plungers to strike 860 times perminute back and forth The motion of the plungers changesthe volume of the cylinders When the plunger moves downthe increased volume of cylinder opens intake valve and

6 Shock and Vibration

1

2 3

4

(a)

Naturalgas

engineCrankshaft box

4th cylinder 3rd cylinder

2nd cylinder1st cylinderAccelerometer(in the bottom)

(b)

Figure 3 Illustration of experimental setup (a) data acquisition system A control panel B data acquisition box C accelerometer on theexhaust valve lidD exhaust valve and (b) diagram of reciprocating compressor

0 005 01 015 02 025

0

5

Periodic impact Periodic impact minus10

minus5

Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )

Figure 4 The time series vibrational signal of normal valve

closes the exhaust valve When the plunger moves up thecompression of cylinder opens the exhaust valve and closesthe intake valve Valve is composed of lifting limiter springdisc and seat of which spring is the most susceptible tofailure

To diagnose the valve failure in the 2nd cylinder anaccelerometer is placed on the exhaust valve lid A customerdesigned data acquisition system (model number MDES-5)as shown in Figure 3(a) is used to acquire the measurementsIt consists of a laptop and a data acquisition box configuredin a master-slave system The sampling rate is set as 16 kHzin this study Figures 4 and 5 show the time series vibrationalsignals near the exhaust valve of 2nd cylinder under normalcondition and abnormal condition respectively For compar-ison the amplitude of the vibration signal under abnormalcondition is greater than the one of the normal signal Takingthe RMS (root mean square) as the criterion the RMS of theabnormal signal is calculated as 4725msdotsminus2 which is greaterthan that of the normal signal (2227msdotsminus2) However theperiodic impact feature caused by the collision of the valvedisc and seat is difficult to observe due to strong backgroundnoise

For further analysis the proposed method based onVolterra series and undecimated lifting wavelet packet trans-form is used to process the signals The two ends of the orig-inal signal are extended and predicted by the Volterra seriesmodel Then the predicted signal is decomposed into four

0 005 01 015 02 025

05

1015

minus10

minus15

minus5

Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )

Figure 5 The time series vibrational signal of abnormal valve

1 20

50100

Level 1

270

9730

1 2 3 40

50100

Level 2

736 007 23296928

1 2 3 4 5 6 7 80

50100

Level 3

063 426 012 001 1259 10794039 3121

0 2 4 6 8 10 12 14 160

50100

Level 4

084 006 240 144 004 005 001 000 764 407 899 279 2262 1448 2883 575

Figure 6 Energy distribution of four-level subbands

layers by undecimated lifting wavelet packet transformwhere 119873 = 119873

1015840

= 4 From (11)ndash(13) the predictionoperator P and the update operator U can be obtained asP = [minus00625 05625 05625 minus00625] and U = [minus00313

02813 02813 minus00313] Finally the 16 subbands of thefourth level are obtained and the energy distribution of thesubbands at different levels is shown in Figure 6 After undec-imated lifting wavelet packet decomposition the majorityof the energy rests on the high frequency parts which

Shock and Vibration 7

minus8

minus6

minus4

minus2

02468

minus8

minus6

minus4

minus2

02468

003 004 005 006 007002 008Time t (s)

005

(a)

(b)

010 02 025015Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )A

mpl

itude

a(m

mmiddotsminus

2 )

Figure 7 The wavelet coefficient at selected subband and itsenvelope under normal condition

minus6

minus4

minus2

0246

minus8

minus6

minus4

minus2

024

86

003 004 005 006 007002 008Time t (s)

005

(a)

(b)

010 02 025015Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )A

mpl

itude

a(m

mmiddotsminus

2 )

Figure 8 The wavelet coefficient at selected subband and itsenvelope under abnormal condition

correspond to the detailed wavelet coefficients The subbandwith the highest energy ratio is selected as the bandof interestThewavelet coefficients and their envelopes from the selectedsubbands are shown in Figures 7 and 8 corresponding tothe normal condition and abnormal condition respectivelyFromboth figures the periodic impact features are visible andeasily identified

The signal in one period as shown by the red blockin Figures 7(a) and 8(a) is extracted for detailed analysisAccording to the working principle of exhaust valve the front

half curve from 0012 to 0047 sec represents the valve-closephase while the latter one from 0047 to 0082 sec shows theexhaust phase The exhaust valve closes at point A and thevalve opens again at point BUnder normal condition there isno chattering when normal valve closes because the stiffnessof spring is rigid enough to close tightly By contrast valvechatters significantly which is caused by the repeated collisionof valve and seat during the time period 119879 (indicating theaging and stretch decline of spring in the valve) The valvewith aged spring cannot close tightly resulting in gas leak andefficiency degradation When this compressor is stopped andchecked the spring failure is validated by visual inspectionand the exhaust valve is maintained Therefore the pre-sented incipient fault diagnosis method avoids accidents andprevents huge production loss

In addition the energy of the selected band signal isused to evaluate the performance of the presented methodFigure 9(a) shows the energies of the selected band sig-nals under four different scenarios (I) undecimated LWPT(ULWPT) under normal machinery condition (II) undec-imated LWPT (ULWPT) under abnormal machinery con-dition (III) Volterra series assisted undecimated LWPT(V-ULWPT) under normal machinery condition and (IV)Volterra series assisted undecimated LWPT (V-ULWPT)under abnormal machinery condition From the analysisresults it is found that the normal and abnormal machineryconditions can be differentiated according to energy ofselected subband using undecimated lifting wavelet packettransform Since Volterra series assisted ULWPT suppressesthe end distortion the energy of selected subband using V-ULWPT is less than that using ULWPT However the energyratio of abnormal to normal conditions using V-ULWPTis larger than that using ULWPT as shown in Figure 9(b)thus validating the effectiveness of presented Volterra seriesassisted lifting wavelet packet transform method

The computational efficiency of the developed techniquehas also been evaluated on a desktop (Lenovo YangtianT4900d model Lenovo Inc Beijing China) with a 33 GHzCPU and 4GB memory It takes approximately 119 secondsto process a data string containing 4096 data points whichis equivalent to processing 0256 seconds data length under a16 kHz sampling rate Thus the presented method is applica-ble for online machinery diagnosis in practical applications

5 Conclusions

An enhanced weak feature extraction approach that com-bines Volterra series model and undecimated lifting waveletpacket transform has been presented for incipient fault detec-tion The analysis results show that Volterra series assistedundecimated lifting wavelet packet transform eliminates thefrequency aliasing and end distortion issues in conventionallifting wavelet packet transform method thus it is significantto machinery incipient defect diagnosis especially for weakimpact feature extraction From the engineering applicationperspective the weak impact signals caused by the fault ofa valve in reciprocating compressor are analyzed using thepresented method and the early spring failure is detectedfrom the extractedweak features using the presentedmethod

8 Shock and Vibration

Ener

gy (J

)

I normal ULWPT II abnormal ULWPT

I II III IV

III normal V-ULWPT IV abnormal V-ULWPT

600E + 03

550E + 03

500E + 03

450E + 03

400E + 03

350E + 03

300E + 03

250E + 03

200E + 03

150E + 03

100E + 03

500E + 02

000E + 00

428E + 03

136E + 03

475E + 03

219E + 03

(a)

I II

I ULWPT abnormalnormalII V-ULWPT abnormalnormal

216

314

0

05

1

15

2

25

3

35

4

Ener

gy ra

tio

(b)

Figure 9 Energy comparison among different conditions (a) energy comparison chart (b) energy ratio chart

Conflict of Interests

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

Acknowledgments

This research acknowledges the financial support providedby National Science foundation of China (no 51504274)Science Foundation ofChinaUniversity of PetroleumBeijing(nos 2462014YJRC039 and 2462015YQ0403) and BeijingMunicipal Education Commission Cooperative Construc-tion Project Valuable comments from anonymous reviewersare greatly appreciated

References

[1] V Venkatasubramanian R Rengaswamy K Yin and S NKavuri ldquoA review of process fault detection and diagnosis partI quantitativemodel-basedmethodsrdquoComputers and ChemicalEngineering vol 27 no 3 pp 293ndash311 2003

[2] M Liang and I S Bozchalooi ldquoAn energy operator approachto joint application of amplitude and frequency-demodulationsfor bearing fault detectionrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1473ndash1494 2010

[3] Z Zhong Z Jiang Y Long and X Zhan ldquoAnalysis on thenoise for the different gearboxes of the heavy truckrdquo Shock andVibration vol 2015 Article ID 476460 5 pages 2015

[4] W J Wang ldquoWavelet transform in vibration analysis formechanical fault diagnosisrdquo Shock and Vibration vol 3 no 1pp 17ndash26 1996

[5] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[6] N Baydar and A Ball ldquoA comparative study of acoustic andvibration signals in detection of gear failures usingWigner-VilledistributionrdquoMechanical Systems and Signal Processing vol 15no 6 pp 1091ndash1107 2001

[7] W-X Yang and P W Tse ldquoDevelopment of an advanced noisereductionmethod for vibration analysis based on singular valuedecompositionrdquo NDT amp E International vol 36 no 6 pp 419ndash432 2003

[8] H Hassanpour A Zehtabian and S J Sadati ldquoTime domainsignal enhancement based on an optimized singular vectordenoising algorithmrdquoDigital Signal Processing vol 22 no 5 pp786ndash794 2012

[9] J Wang R X Gao and R Yan ldquoIntegration of EEMD and ICAfor wind turbine gearbox diagnosisrdquoWind Energy vol 17 no 5pp 757ndash773 2014

[10] A Sadhu and B Hazra ldquoA novel damage detection algorithmusing time-series analysis-based blind source separationrdquo Shockand Vibration vol 20 no 3 pp 423ndash438 2013

[11] Z K Peng and F L Chu ldquoApplication of the wavelet trans-form in machine condition monitoring and fault diagnosticsa review with bibliographyrdquo Mechanical Systems and SignalProcessing vol 18 no 2 pp 199ndash221 2004

[12] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosis ofrotary machines a review with applicationsrdquo Signal Processingvol 96 pp 1ndash15 2014

[13] N Saravanan and K I Ramachandran ldquoIncipient gear box faultdiagnosis using discrete wavelet transform (DWT) for featureextraction and classification using artificial neural network(ANN)rdquo Expert Systems with Applications vol 37 no 6 pp4168ndash4181 2010

[14] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weaksignature detection method and its application on rolling ele-ment bearing prognosticsrdquo Journal of Sound and Vibration vol289 no 4-5 pp 1066ndash1090 2006

[15] J Wang R X Gao and R Yan ldquoMulti-scale enveloping orderspectrogram for rotatingmachine health diagnosisrdquoMechanicalSystems and Signal Processing vol 46 no 1 pp 28ndash44 2014

Shock and Vibration 9

[16] Y-L Chen P-L Zhang B Li and D-H Wu ldquoDenoising ofmechanical vibration signals using quantum-inspired adaptivewavelet shrinkagerdquo Shock and Vibration vol 2014 Article ID848097 7 pages 2014

[17] Y Pan J Chen and X Li ldquoBearing performance degradationassessment based on lifting wavelet packet decomposition andfuzzy c-meansrdquo Mechanical Systems and Signal Processing vol24 no 2 pp 559ndash566 2010

[18] R ZhouWBaoN Li XHuang andDYu ldquoMechanical equip-ment fault diagnosis based on redundant second generationwavelet packet transformrdquo Digital Signal Processing A ReviewJournal vol 20 no 1 pp 276ndash288 2010

[19] J Yuan Z He and Y Zi ldquoGear fault detection using customizedmultiwavelet lifting schemesrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1509ndash1528 2010

[20] Y He X Chen J Xiang and Z He ldquoAdaptive multiresolutionfinite element method based on second generation waveletsrdquoFinite Elements in Analysis and Design vol 43 no 6-7 pp 566ndash579 2007

[21] W Bao R Zhou J Yang D Yu and N Li ldquoAnti-aliasinglifting scheme formechanical vibration fault feature extractionrdquoMechanical Systems and Signal Processing vol 23 no 5 pp1458ndash1473 2009

[22] R L Claypoole G M Davis W Sweldens and R G BaraniukldquoNonlinear wavelet transforms for image coding via liftingrdquoIEEE Transactions on Image Processing vol 12 no 12 pp 1449ndash1459 2003

[23] R L Claypoole R G Baraniuk and R D Nowak ldquoAdaptivewavelet transforms via liftingrdquo Report ECE TR-9304 1999httpsscholarshipriceeduhandle191119808

[24] B Appleton and H Talbot ldquoRecursive filtering of images withsymmetric extensionrdquo Signal Processing vol 85 no 8 pp 1546ndash1556 2005

[25] B Wohlberg and C M Brislawn ldquoSymmetric extension forlifted filter banks and obstructions to reversible implementa-tionrdquo Signal Processing vol 88 no 1 pp 131ndash145 2008

[26] L Duan L Zhang and J Chen ldquoBoundary treatment of liftingwavelet transform based on Volterra series model and its appli-cationrdquo Chinese Journal of Scientific Instrument vol 33 no 1pp 7ndash12 2012

[27] D Aiordachioaie E Ceanga R de Keyser and Y Naka ldquoDetec-tion and classification of non-linearities based on Volterrakernels processingrdquo Engineering Applications of Artificial Intel-ligence vol 14 no 4 pp 497ndash503 2001

[28] W Sweldens ldquoThe lifting scheme a custom-design constructionof biorthogonal waveletsrdquo Applied and Computational Har-monic Analysis vol 3 no 2 pp 186ndash200 1996

[29] W Sweldens ldquoThe Lifting Scheme a construction of secondgeneration waveletsrdquo SIAM Journal on Mathematical Analysisvol 29 no 2 pp 511ndash546 1998

[30] A Ben and T Hugues ldquoRecursive filtering of images with sym-metric extensionrdquo Signal Processing vol 85 no 8 pp 1546ndash15562005

[31] D Aiordachioaie E Ceanga R DeKeyser andYNaka ldquoDetec-tion and classification of non-linearities based on Volterrakernels processingrdquo Engineering Applications of Artificial Intel-ligence vol 14 no 4 pp 497ndash503 2001

[32] M B Kennel R Brown and H D I Abarbanel ldquoDeterminingembedding dimension for phase-space reconstruction using ageometrical constructionrdquo Physical Review A vol 45 no 6 pp3403ndash3411 1992

[33] J Zhang and X Xiao ldquoPredicting low-dimensional chaotic timeseries using Volterra adaptive filtersrdquo Acta Physica Sinica vol49 no 3 pp 403ndash408 1999

[34] J Hongkai H Zhengjia D Chendong and C Peng ldquoGear-box fault diagnosis using adaptive redundant Lifting SchemerdquoMechanical Systems and Signal Processing vol 20 no 8 pp1992ndash2006 2006

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Control Scienceand Engineering

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International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Electrical and Computer Engineering

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

2 Shock and Vibration

operations to decompose the signal in time domain whichhas the superiorities for example faster implementationbeing independent of Fourier transform and meanwhileretaining all advantages of traditional wavelet transform In[17] a lifting wavelet packet decomposition method waspresented to extract fault features for bearing performancedegradation assessment A redundant lifting wavelet packettransform was applied to diagnose gearbox and engine [18]A multiwavelet lifting scheme to optimize lifting schemewas presented for gearbox diagnosis [19] A combination oflifting wavelet and finite element method was investigatedfor the quantitative identification of pipeline cracks [20] Inthe abovementioned studies decimated lifting scheme usingdownsampling algorithm is commonly used which leads tofrequency aliasing in the decomposition results [21] On theother hand the lifting scheme causes end distortion whichconfuses or misleads the diagnosis Thus an undecimatedlifting wavelet transform with boundary treatment is neededto suppress frequency aliasing and end distortion

Various boundary treatment methods are investigatedto suppress end distortion in lifting wavelet transform In[22 23] the order of predictor in the lifting scheme is reducedby overlapping the edges to suppress end distortion Differentboundary extension methods such as zero-padding exten-sion symmetric extension periodic extension zero-ordersmoothing extension and one-order smoothing extensionmethods are investigated in [24 25] These methods couldsuppress the end distortion to some degree but not up tosatisfaction To address these issues this paper presents anew signal processingmethod namedVolterra series assistedundecimated lifting wavelet packet transform by extendingour prior work [26] First of all Volterra series [27] as aboundary treatment method is used to extend both endsof the signal Then the wavelet coefficients decomposed bythe undecimated lifting wavelet packet are chopped back tooriginal length to serve as the signals of interest formachineryincipient fault detection Finally the effectiveness of thepresented method is demonstrated on valve incipient defectdiagnosis in a reciprocating compressorThus the intellectualmerits of this paper are outlined including the following(1) a Volterra assisted undecimated lifting wavelet packettransformmethod is presented to suppress the end distortionand frequency aliasing issues in conventional lifting wavelettransform and (2) the formula to optimize the number ofextended signal in boundary treatment using Volterra seriesis derived according to the decomposition property of undec-imated lifting wavelet packet transform

The rest of the paper is organized as follows Section 2introduces the theoretical background of lifting wavelettransform and Volterra series Section 3 presents the theoret-ical framework of Volterra series assisted undecimated liftingwavelet packet transform and the equations of the undeci-mated lifting wavelet packet Performance comparison of dif-ferent boundary treatment methods is also discussed herebyAn experimental study of incipient fault detection of recipro-cating compressor valve using the presented method is con-ducted and the analysis results are discussed in Section 4Theconclusions are finally drawn in Section 5

2 Theoretical Background

21 LiftingWavelet Transform Lifting wavelet transform wasfirstly presented by Sweldens in the 1990s [28] Based on lift-ing scheme it calculates the wavelet coefficients using poly-nomial interpolationmethod and constructs scaling functionto obtain the low frequency coefficients of the signal If thescaling function curve is smooth and the ringing artifact ofboundary is reduced adequately an ideal wavelet coefficientcan be acquired using interpolator split schemes

Lifting wavelet transform consists of three steps splitprediction and update [29]

(1) Split Several methods for signal split are available Onemethod could be dividing signals into left and right halvesHowever the result will be unsatisfying due to the lowrelativity degree between the left and the right halves Onemore effective method is to divide the data 119909[119899] into even set119909119890[119899] = 119909[2119899] and odd set 119909

119900[119899] = 119909[2119899 + 1] where 119899 is the

number of data

(2) Prediction The purpose of prediction is to eliminatethe low frequency components of signals and preserve thehigh frequency part The odd set 119909

119900[119899] can be predicted

from the even set 119909119890[119899] and the prediction operator P =

[1199011 1199012 119901

119873] The prediction value is P(119909

119890[119899]) The differ-

ence between the practical value and the prediction value isdefined as 119889[119899]

119889 [119899] = 119909119900[119899] minus P (119909

119890[119899]) (1)

where 119889[119899] also called detailed signal reflects the high fre-quency component of the signals Here119873 is a dual vanishingmoment that determines the smoothness of the interpolationfunction

(3) Update In order to reduce the frequency aliasing effectthe odd set 119909

119890[119899] is updated using detail signal 119889[119899] and

update operatorU = [1199061 1199062 119906

1198731015840]The result of this step is

the approximation signal 119888[119899] that reflects the low frequencycoefficient of the signals which can be written as

119888 [119899] = 119909119890[119899] + U (119889 [119899])P (119909

119890[119899]) (2)

Signals can be decomposed by lifting wavelet transformusing the above iterative operation of approximation signal119888[119899] Prediction coefficients can be calculated by the Lagrangeinterpolation formula As long as the length of the updateoperator is the same as that of the prediction operator theupdate coefficient value will be half of the correspondingprediction coefficient [30] However the downsampling algo-rithm used in the conventional lifting wavelet transform willlead to frequency aliasing because the transformed signalbecomes half of its length in the previous layer The updatealgorithm can reduce but not completely eliminate frequencyaliasing The signal after downsampling algorithm will notmeet the conditions of sampling theorem which leads tounexpected virtual frequency components Thus an undec-imated lifting wavelet transform is outlined and discussed toeliminate frequency aliasing in this study

Shock and Vibration 3

22 Volterra Series Volterra series was initially proposed byan Italian mathematician named Vito Volterra in the 1880sDue to its powerful ability tomodel the behavior of nonlinearsystems the theory has attracted a great deal of attentionand soon gained its applications in many fields If the inputof a nonlinear discrete time system is X(119899) = [119909(119899) 119909(119899 minus

1) 119909(119899 minus 1198731015840

+ 1)] and the output is 119910(119899) =997888

119909

1015840

(119899 + 1) thesystem function can construct nonlinear prediction modelwith the expansion of Volterra series as given by [31]

997888

119909

1015840

(119899 + 1) = ℎ0+

+infin

sum

119894=0

ℎ1(119894) 119909 (119899 minus 119894) +

+infin

sum

119894=0

+infin

sum

119895=0

ℎ2(119894 119895)

sdot 119909 (119899 minus 119894) 119909 (119899 minus 119895) + sdot sdot sdot

+

+infin

sum

119894=0

+infin

sum

119895=0

sdot sdot sdot

+infin

sum

119896=0

ℎ119901(119894 119895 119896) 119909 (119899 minus 119894) 119909 (119899 minus 119895)

sdot sdot sdot 119909 (119899 minus 119896)

(3)

where ℎ119901(119894 119895 119896) is the 119901th order nucleus of Volterra

The expansion of this infinite order series is extremelydifficult in practical applications Generally the second-ordertruncation is employed as follows

997888

119909

1015840

(119899 + 1) = ℎ0+

1198731minus1

sum

119894=0

ℎ1(119894) 119909 (119899 minus 119894)

+

1198732minus1

sum

119894=0

1198732minus1

sum

119895=0

ℎ2(119894 119895) 119909 (119899 minus 119894) 119909 (119899 minus 119895)

(4)

where1198731and119873

2represent the length of filtersTheminimum

embedding dimension119898 of the signal can be obtained usingthe fault near-neighbor method Consequently 119873

1and 119873

2

can be set as119898 [32]Volterra series is used to extend both ends of the signal to

address the end distortion issue in lifting wavelet transformThe integrative approach of Volterra series and lifting wavelettransform is presented in detail in the following sections

The input vector isX(119899) = [1 119909(119899) 119909(119899minus1) 119909(119899minus119898minus

1) 1199092

(119899) 119909(119899)119909(119899 minus 1) 1199092

(119899 minus 119898 + 1)]119879 The prediction

coefficient vector is W(119899) = [ℎ0 ℎ1(0) ℎ1(1) ℎ

1(119898 minus

1) ℎ2(0 0) ℎ

2(0 1) ℎ

2(119898 minus 1119898 minus 1)]

119879 Thus the expres-sion of (4) can be rewritten as [33]

997888

119909

1015840

(119899 + 1) = X119879 (119899)W (119899) (5)

The prediction coefficient vectorW(119899) is calculated usingthe recursive least-squares method (RLS) To elaborate con-sider

119876 (0) = 120575minus1I (6)

where 120575 is a very small normal number and I is the identitymatrix ThusW(0) is set as

W (0) = 0 (7)

andW(119899) is calculated by carrying on the following iterativecomputation

Consider

119866 (119899) =

120583minus1

119876 (119899 minus 1)X (119899)

1 + 120583minus1X119879 (119899) 119876 (119899 minus 1)X (119899)

(8)

where 120583 is a forgetting factorConsider the following

120572 (119899) = 119863 (119899) minusW119879 (119899 minus 1)X (119899) (9)

where119863(119899) is an ideal output signal Thus

W (119899) = W (119899 minus 1) + 119866 (119899) 120572 (119899)

119876 (119899) = 120583minus1

119876 (119899 minus 1) minus 120583minus1

119866 (119899)X119879 (119899) 119876 (119899 minus 1)

(10)

where 120572(119899) and 119866(119899) are intermediate variablesThe details of theoretical knowledge of lifting wavelet

transform and Volterra series model have been discussedabove Volterra series is used to extend both ends of thesignal to address the end distortion issue in lifting wavelettransformThe formulation of Volterra series assisted undec-imated lifting wavelet transform is illustrated below

3 The Proposed Method

A Volterra series-assisted undecimated lifting wavelet packettransform is proposed for machinery incipient defect diag-nosis to eliminate the frequency aliasing and end distortionissues in traditional lifting wavelet transform and its flow-chart is shown in Figure 1

First both ends of the original signal are extended andpredicted with Volterra series model in which the param-eter of extension number is optimized Next the extendedsignal is decomposed by undecimated lifting wavelet packettransform and the subband with the highest energy ratio isselected as the band of interestThe selected subband signal isthen trimmed back to its original length for envelope analysisFinally themachinery status is assessed according to the anal-ysis results The details of undecimated lifting wavelet packettransform with boundary treatment are discussed as below

31 Formulation of Undecimated LWPT with Boundary Treat-ment The undecimated algorithm can eliminate frequencyaliasing because the length of coefficients at each level isequal to that of the raw signal The P and U can be schemedout as initial prediction operator and initial update operatorrespectively and the undecimated algorithm is deduced asfollows

Assuming P = 119901119894 where 119894 = 1 2 119873 the expression

of the undecimated prediction operator of the 119897th level isgiven by [34]

P[119897]119895

=

119901119894 119895 = 2

119897

sdot 119894

0 119895 = 2119897

sdot 119894

119895 = 1 2119897

119873 (11)

4 Shock and Vibration

0 001 002 003 004 005 006

Extension signal

Machinery Volterra series model

Envelope analysis

Status assessment

Data acquisition0 001 002 003 004

0

5

10

15

Parameter optimization

0 005 01 015 02 025

01

X44

0 005 01 015 02 025

010

X43

0 005 01 015 02 025

01

X42

0 005 01 015 02 025

010

X41

Undecimated LWPTData trimming

0 005 01 015 02 0250 001 002 003 004 005 006 007

02468

Prediction coefficient

Extension number

Undecimated prediction operator P

Undecimated update

operator U

minus4

minus2

minus8

minus6

0246

minus4

minus2

minus6

minus1

minus1

minus10

minus10

minus10

minus5

minus15

0

5

10

15

minus10

minus5

minus15

Time t (s) Time t (s)

W(n) nadd

Am

plitu

dea

(mmmiddotsminus

2 )

Am

plitu

dea

(mmmiddotsminus

2 )

Figure 1 Flowchart of the proposed approach for machinery incipient defect diagnosis

According to the principle of equidistance subdivision thepredictor coefficient 119901

119894is calculated by Lagrange interpola-

tion formula as [29]

119901119894=

119873

prod

119896=1

119896 =119894

(119873 + 1) 2 minus 119896

119894 minus 119896

119894 = 1 2 119873 (12)

where 119894 is an index number and119873 is the predictor lengthSimilarly assuming U = 119906

119894 where 119894 = 1 2 119873

1015840 theexpression of the undecimated update operator of the 119897th levelis obtained by [34]

U[119897]119895

=

119906119894 119895 = 2

119897

sdot 119894

0 119895 = 2119897

sdot 119894

119895 = 1 2119897

1198731015840

(13)

where 119906119894is the update coefficient and is usually set as the half

of the predictor coefficient 119901119894

Assuming 119909119897119896is the 119896th frequency band of the 119897th level

signal decomposed from the original signal 119909[119899] 119909119897(119896minus1)

and119909119897119896can be obtained by dividing 119909

(119897minus1)(1198962)

119909119897(119896minus1)

(119899) = 119909(119897minus1)(1198962)

(119899)

minus [119901[119897]

1119909(119897minus1)(1198962)

(119899 minus 2119897minus1

(119873 + 1) + 1)

+ 119901[119897]

2119909(119897minus1)(1198962)

(119899 minus 2119897minus1

(119873 + 1) + 2) + sdot sdot sdot

+ 119901[119897]

2119897119873

119909(119897minus1)(1198962)

(119899 + 2119897minus1

(119873 minus 1))]

119909119897119896(119899) = 119909

(119897minus1)(1198962)(119899)

+ [119906[119897]

1119909119897(119896minus1)

(119899 minus 2119897minus1

(1198731015840

+ 1) + 1)

+ 119906[119897]

2119909119897(119896minus1)

(119899 minus 2119897minus1

(1198731015840

+ 1) + 2) + sdot sdot sdot

+ 119906[119897]

2119897

119909119897(119896minus1)

(119899 + 2119897minus1

(1198731015840

minus 1))]

(14)

where 119896 = 2 4 6 2119897 119901[119897]119894

is the undecimated predictionoperator of 119909

119897(119896minus1) and 119906[119897]

119894is the undecimated update opera-

tor of 119909119897119896

Assuming the undecimated prediction operator P = 119901119894

and the undecimated update operator U = 119906119894 where 119894 =

1 2 4 119909119894is the 119894th data of X The 119894th data of the first level

detailed signal decomposed by undecimated lifting waveletpacket transform is obtained as follows

1198891119894

= 119909119894minus (119909119894minus3

times 1199011+ 119909119894minus2

times 0 + 119909119894minus1

times 1199012+ 119909119894times 0

+ 119909119894+1

times 1199013+ 119909119894+3

times 1199014)

(15)

It can be deduced that three numbersmust be extended tothe left end of 119909 when calculating 119889

11and the same number

to the right when calculating the last data 1198891119899 Similarly for

the second level 6 more numbers should be extended to eachend and so on More generally assuming 119873

119891is the decom-

posed level of undecimated lifting wavelet packet transformthe extension number of the original signal at each end can becalculated as

119899add =

119873119891

sum

119897=1

[2119897minus1

(119873 + 1) minus 1]

+

119873119891

sum

119897=1

[2119897minus1

(1198731015840

+ 1) minus 1]

(16)

According to (16) the length of extended signal inVolterra series model is determined and the signal isextended based on the procedure of Volterra series model inSection 2 The performance of formulated method is dis-cussed below

Shock and Vibration 5

0

01

02

03

End distortion

Characteristic signal

End distortion

minus02

minus01

002 004 006 008 01 0120Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )

(a)

End distortion

Characteristic signal

End distortion

002 004 006 008 01 0120Time t (s)

minus2

minus1

0

1

2

Am

plitu

dea

(mmmiddotsminus

2 )

(b)

Characteristic signalEnd distortion

minus8

minus6

minus4

minus2

0

2

4

6

8

002 004 006 008 01 0120Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )

(c)

Characteristic signal

002 004 006 008 01 0120Time t (s)

minus02

minus01

0

01

02

03

Am

plitu

dea

(mmmiddotsminus

2 )

(d)

Figure 2 One subband in the fourth layer decomposed using wavelet packet and preprocessed by different methods (a) no extension (b)zero-padding extension (c) periodic extension and (d) second-order Volterra series prediction

32 Performance Comparison of Boundary Treatment Meth-ods To evaluate the performance of various boundary treat-ment methods a synthesized signal is constructed as follows

119909 (119905) = 03 exp (minus2120587 times 100119905) sin (2120587 times 400119905)

+ 15 sin(2120587 times 50119905 +

120587

4

)

+ 3 sin(2120587 times 100119905 +

120587

6

)

(17)

The sampling frequency is set as 16000Hz The synthesizedsignal is decomposed into four layers using the undecimatedlifting wavelet transform Figure 2 shows one subband of thefourth layer corresponding to the pulse train signal underdifferent preprocessing scenarios (a) no extension (b) zero-padding extension (c) periodic extension and (d) second-order Volterra series prediction respectively

There are obvious end distortions shown by the arrows inFigures 2(a)ndash2(c)Therefore if the characteristic signal is just

located in the end it will be submerged by the end distor-tions In Figures 2(b)-2(c) the characteristic signal in themiddle is insignificant because the end distortion is strongThe Volterra series based preprocessing method shows theoptimal performance since the end distortions have beenrestrained effectively as shown in Figure 2(d)

4 Experimental Study

A reciprocating compressor (model number 4HOS-6) in apetrochemical plant in China is used as the experimentaltestbed to evaluate the performance of the developedmethodas shown in Figure 3 It is a 4-cylinder natural gas recipro-cating compressor driven by an 8-cylinder gas engine withrated power of 1600 kW The rotating speed of crankshaft is860 rpm which drives the plungers to strike 860 times perminute back and forth The motion of the plungers changesthe volume of the cylinders When the plunger moves downthe increased volume of cylinder opens intake valve and

6 Shock and Vibration

1

2 3

4

(a)

Naturalgas

engineCrankshaft box

4th cylinder 3rd cylinder

2nd cylinder1st cylinderAccelerometer(in the bottom)

(b)

Figure 3 Illustration of experimental setup (a) data acquisition system A control panel B data acquisition box C accelerometer on theexhaust valve lidD exhaust valve and (b) diagram of reciprocating compressor

0 005 01 015 02 025

0

5

Periodic impact Periodic impact minus10

minus5

Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )

Figure 4 The time series vibrational signal of normal valve

closes the exhaust valve When the plunger moves up thecompression of cylinder opens the exhaust valve and closesthe intake valve Valve is composed of lifting limiter springdisc and seat of which spring is the most susceptible tofailure

To diagnose the valve failure in the 2nd cylinder anaccelerometer is placed on the exhaust valve lid A customerdesigned data acquisition system (model number MDES-5)as shown in Figure 3(a) is used to acquire the measurementsIt consists of a laptop and a data acquisition box configuredin a master-slave system The sampling rate is set as 16 kHzin this study Figures 4 and 5 show the time series vibrationalsignals near the exhaust valve of 2nd cylinder under normalcondition and abnormal condition respectively For compar-ison the amplitude of the vibration signal under abnormalcondition is greater than the one of the normal signal Takingthe RMS (root mean square) as the criterion the RMS of theabnormal signal is calculated as 4725msdotsminus2 which is greaterthan that of the normal signal (2227msdotsminus2) However theperiodic impact feature caused by the collision of the valvedisc and seat is difficult to observe due to strong backgroundnoise

For further analysis the proposed method based onVolterra series and undecimated lifting wavelet packet trans-form is used to process the signals The two ends of the orig-inal signal are extended and predicted by the Volterra seriesmodel Then the predicted signal is decomposed into four

0 005 01 015 02 025

05

1015

minus10

minus15

minus5

Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )

Figure 5 The time series vibrational signal of abnormal valve

1 20

50100

Level 1

270

9730

1 2 3 40

50100

Level 2

736 007 23296928

1 2 3 4 5 6 7 80

50100

Level 3

063 426 012 001 1259 10794039 3121

0 2 4 6 8 10 12 14 160

50100

Level 4

084 006 240 144 004 005 001 000 764 407 899 279 2262 1448 2883 575

Figure 6 Energy distribution of four-level subbands

layers by undecimated lifting wavelet packet transformwhere 119873 = 119873

1015840

= 4 From (11)ndash(13) the predictionoperator P and the update operator U can be obtained asP = [minus00625 05625 05625 minus00625] and U = [minus00313

02813 02813 minus00313] Finally the 16 subbands of thefourth level are obtained and the energy distribution of thesubbands at different levels is shown in Figure 6 After undec-imated lifting wavelet packet decomposition the majorityof the energy rests on the high frequency parts which

Shock and Vibration 7

minus8

minus6

minus4

minus2

02468

minus8

minus6

minus4

minus2

02468

003 004 005 006 007002 008Time t (s)

005

(a)

(b)

010 02 025015Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )A

mpl

itude

a(m

mmiddotsminus

2 )

Figure 7 The wavelet coefficient at selected subband and itsenvelope under normal condition

minus6

minus4

minus2

0246

minus8

minus6

minus4

minus2

024

86

003 004 005 006 007002 008Time t (s)

005

(a)

(b)

010 02 025015Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )A

mpl

itude

a(m

mmiddotsminus

2 )

Figure 8 The wavelet coefficient at selected subband and itsenvelope under abnormal condition

correspond to the detailed wavelet coefficients The subbandwith the highest energy ratio is selected as the bandof interestThewavelet coefficients and their envelopes from the selectedsubbands are shown in Figures 7 and 8 corresponding tothe normal condition and abnormal condition respectivelyFromboth figures the periodic impact features are visible andeasily identified

The signal in one period as shown by the red blockin Figures 7(a) and 8(a) is extracted for detailed analysisAccording to the working principle of exhaust valve the front

half curve from 0012 to 0047 sec represents the valve-closephase while the latter one from 0047 to 0082 sec shows theexhaust phase The exhaust valve closes at point A and thevalve opens again at point BUnder normal condition there isno chattering when normal valve closes because the stiffnessof spring is rigid enough to close tightly By contrast valvechatters significantly which is caused by the repeated collisionof valve and seat during the time period 119879 (indicating theaging and stretch decline of spring in the valve) The valvewith aged spring cannot close tightly resulting in gas leak andefficiency degradation When this compressor is stopped andchecked the spring failure is validated by visual inspectionand the exhaust valve is maintained Therefore the pre-sented incipient fault diagnosis method avoids accidents andprevents huge production loss

In addition the energy of the selected band signal isused to evaluate the performance of the presented methodFigure 9(a) shows the energies of the selected band sig-nals under four different scenarios (I) undecimated LWPT(ULWPT) under normal machinery condition (II) undec-imated LWPT (ULWPT) under abnormal machinery con-dition (III) Volterra series assisted undecimated LWPT(V-ULWPT) under normal machinery condition and (IV)Volterra series assisted undecimated LWPT (V-ULWPT)under abnormal machinery condition From the analysisresults it is found that the normal and abnormal machineryconditions can be differentiated according to energy ofselected subband using undecimated lifting wavelet packettransform Since Volterra series assisted ULWPT suppressesthe end distortion the energy of selected subband using V-ULWPT is less than that using ULWPT However the energyratio of abnormal to normal conditions using V-ULWPTis larger than that using ULWPT as shown in Figure 9(b)thus validating the effectiveness of presented Volterra seriesassisted lifting wavelet packet transform method

The computational efficiency of the developed techniquehas also been evaluated on a desktop (Lenovo YangtianT4900d model Lenovo Inc Beijing China) with a 33 GHzCPU and 4GB memory It takes approximately 119 secondsto process a data string containing 4096 data points whichis equivalent to processing 0256 seconds data length under a16 kHz sampling rate Thus the presented method is applica-ble for online machinery diagnosis in practical applications

5 Conclusions

An enhanced weak feature extraction approach that com-bines Volterra series model and undecimated lifting waveletpacket transform has been presented for incipient fault detec-tion The analysis results show that Volterra series assistedundecimated lifting wavelet packet transform eliminates thefrequency aliasing and end distortion issues in conventionallifting wavelet packet transform method thus it is significantto machinery incipient defect diagnosis especially for weakimpact feature extraction From the engineering applicationperspective the weak impact signals caused by the fault ofa valve in reciprocating compressor are analyzed using thepresented method and the early spring failure is detectedfrom the extractedweak features using the presentedmethod

8 Shock and Vibration

Ener

gy (J

)

I normal ULWPT II abnormal ULWPT

I II III IV

III normal V-ULWPT IV abnormal V-ULWPT

600E + 03

550E + 03

500E + 03

450E + 03

400E + 03

350E + 03

300E + 03

250E + 03

200E + 03

150E + 03

100E + 03

500E + 02

000E + 00

428E + 03

136E + 03

475E + 03

219E + 03

(a)

I II

I ULWPT abnormalnormalII V-ULWPT abnormalnormal

216

314

0

05

1

15

2

25

3

35

4

Ener

gy ra

tio

(b)

Figure 9 Energy comparison among different conditions (a) energy comparison chart (b) energy ratio chart

Conflict of Interests

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

Acknowledgments

This research acknowledges the financial support providedby National Science foundation of China (no 51504274)Science Foundation ofChinaUniversity of PetroleumBeijing(nos 2462014YJRC039 and 2462015YQ0403) and BeijingMunicipal Education Commission Cooperative Construc-tion Project Valuable comments from anonymous reviewersare greatly appreciated

References

[1] V Venkatasubramanian R Rengaswamy K Yin and S NKavuri ldquoA review of process fault detection and diagnosis partI quantitativemodel-basedmethodsrdquoComputers and ChemicalEngineering vol 27 no 3 pp 293ndash311 2003

[2] M Liang and I S Bozchalooi ldquoAn energy operator approachto joint application of amplitude and frequency-demodulationsfor bearing fault detectionrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1473ndash1494 2010

[3] Z Zhong Z Jiang Y Long and X Zhan ldquoAnalysis on thenoise for the different gearboxes of the heavy truckrdquo Shock andVibration vol 2015 Article ID 476460 5 pages 2015

[4] W J Wang ldquoWavelet transform in vibration analysis formechanical fault diagnosisrdquo Shock and Vibration vol 3 no 1pp 17ndash26 1996

[5] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[6] N Baydar and A Ball ldquoA comparative study of acoustic andvibration signals in detection of gear failures usingWigner-VilledistributionrdquoMechanical Systems and Signal Processing vol 15no 6 pp 1091ndash1107 2001

[7] W-X Yang and P W Tse ldquoDevelopment of an advanced noisereductionmethod for vibration analysis based on singular valuedecompositionrdquo NDT amp E International vol 36 no 6 pp 419ndash432 2003

[8] H Hassanpour A Zehtabian and S J Sadati ldquoTime domainsignal enhancement based on an optimized singular vectordenoising algorithmrdquoDigital Signal Processing vol 22 no 5 pp786ndash794 2012

[9] J Wang R X Gao and R Yan ldquoIntegration of EEMD and ICAfor wind turbine gearbox diagnosisrdquoWind Energy vol 17 no 5pp 757ndash773 2014

[10] A Sadhu and B Hazra ldquoA novel damage detection algorithmusing time-series analysis-based blind source separationrdquo Shockand Vibration vol 20 no 3 pp 423ndash438 2013

[11] Z K Peng and F L Chu ldquoApplication of the wavelet trans-form in machine condition monitoring and fault diagnosticsa review with bibliographyrdquo Mechanical Systems and SignalProcessing vol 18 no 2 pp 199ndash221 2004

[12] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosis ofrotary machines a review with applicationsrdquo Signal Processingvol 96 pp 1ndash15 2014

[13] N Saravanan and K I Ramachandran ldquoIncipient gear box faultdiagnosis using discrete wavelet transform (DWT) for featureextraction and classification using artificial neural network(ANN)rdquo Expert Systems with Applications vol 37 no 6 pp4168ndash4181 2010

[14] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weaksignature detection method and its application on rolling ele-ment bearing prognosticsrdquo Journal of Sound and Vibration vol289 no 4-5 pp 1066ndash1090 2006

[15] J Wang R X Gao and R Yan ldquoMulti-scale enveloping orderspectrogram for rotatingmachine health diagnosisrdquoMechanicalSystems and Signal Processing vol 46 no 1 pp 28ndash44 2014

Shock and Vibration 9

[16] Y-L Chen P-L Zhang B Li and D-H Wu ldquoDenoising ofmechanical vibration signals using quantum-inspired adaptivewavelet shrinkagerdquo Shock and Vibration vol 2014 Article ID848097 7 pages 2014

[17] Y Pan J Chen and X Li ldquoBearing performance degradationassessment based on lifting wavelet packet decomposition andfuzzy c-meansrdquo Mechanical Systems and Signal Processing vol24 no 2 pp 559ndash566 2010

[18] R ZhouWBaoN Li XHuang andDYu ldquoMechanical equip-ment fault diagnosis based on redundant second generationwavelet packet transformrdquo Digital Signal Processing A ReviewJournal vol 20 no 1 pp 276ndash288 2010

[19] J Yuan Z He and Y Zi ldquoGear fault detection using customizedmultiwavelet lifting schemesrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1509ndash1528 2010

[20] Y He X Chen J Xiang and Z He ldquoAdaptive multiresolutionfinite element method based on second generation waveletsrdquoFinite Elements in Analysis and Design vol 43 no 6-7 pp 566ndash579 2007

[21] W Bao R Zhou J Yang D Yu and N Li ldquoAnti-aliasinglifting scheme formechanical vibration fault feature extractionrdquoMechanical Systems and Signal Processing vol 23 no 5 pp1458ndash1473 2009

[22] R L Claypoole G M Davis W Sweldens and R G BaraniukldquoNonlinear wavelet transforms for image coding via liftingrdquoIEEE Transactions on Image Processing vol 12 no 12 pp 1449ndash1459 2003

[23] R L Claypoole R G Baraniuk and R D Nowak ldquoAdaptivewavelet transforms via liftingrdquo Report ECE TR-9304 1999httpsscholarshipriceeduhandle191119808

[24] B Appleton and H Talbot ldquoRecursive filtering of images withsymmetric extensionrdquo Signal Processing vol 85 no 8 pp 1546ndash1556 2005

[25] B Wohlberg and C M Brislawn ldquoSymmetric extension forlifted filter banks and obstructions to reversible implementa-tionrdquo Signal Processing vol 88 no 1 pp 131ndash145 2008

[26] L Duan L Zhang and J Chen ldquoBoundary treatment of liftingwavelet transform based on Volterra series model and its appli-cationrdquo Chinese Journal of Scientific Instrument vol 33 no 1pp 7ndash12 2012

[27] D Aiordachioaie E Ceanga R de Keyser and Y Naka ldquoDetec-tion and classification of non-linearities based on Volterrakernels processingrdquo Engineering Applications of Artificial Intel-ligence vol 14 no 4 pp 497ndash503 2001

[28] W Sweldens ldquoThe lifting scheme a custom-design constructionof biorthogonal waveletsrdquo Applied and Computational Har-monic Analysis vol 3 no 2 pp 186ndash200 1996

[29] W Sweldens ldquoThe Lifting Scheme a construction of secondgeneration waveletsrdquo SIAM Journal on Mathematical Analysisvol 29 no 2 pp 511ndash546 1998

[30] A Ben and T Hugues ldquoRecursive filtering of images with sym-metric extensionrdquo Signal Processing vol 85 no 8 pp 1546ndash15562005

[31] D Aiordachioaie E Ceanga R DeKeyser andYNaka ldquoDetec-tion and classification of non-linearities based on Volterrakernels processingrdquo Engineering Applications of Artificial Intel-ligence vol 14 no 4 pp 497ndash503 2001

[32] M B Kennel R Brown and H D I Abarbanel ldquoDeterminingembedding dimension for phase-space reconstruction using ageometrical constructionrdquo Physical Review A vol 45 no 6 pp3403ndash3411 1992

[33] J Zhang and X Xiao ldquoPredicting low-dimensional chaotic timeseries using Volterra adaptive filtersrdquo Acta Physica Sinica vol49 no 3 pp 403ndash408 1999

[34] J Hongkai H Zhengjia D Chendong and C Peng ldquoGear-box fault diagnosis using adaptive redundant Lifting SchemerdquoMechanical Systems and Signal Processing vol 20 no 8 pp1992ndash2006 2006

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Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

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Advances inOptoElectronics

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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International Journal of

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

Shock and Vibration 3

22 Volterra Series Volterra series was initially proposed byan Italian mathematician named Vito Volterra in the 1880sDue to its powerful ability tomodel the behavior of nonlinearsystems the theory has attracted a great deal of attentionand soon gained its applications in many fields If the inputof a nonlinear discrete time system is X(119899) = [119909(119899) 119909(119899 minus

1) 119909(119899 minus 1198731015840

+ 1)] and the output is 119910(119899) =997888

119909

1015840

(119899 + 1) thesystem function can construct nonlinear prediction modelwith the expansion of Volterra series as given by [31]

997888

119909

1015840

(119899 + 1) = ℎ0+

+infin

sum

119894=0

ℎ1(119894) 119909 (119899 minus 119894) +

+infin

sum

119894=0

+infin

sum

119895=0

ℎ2(119894 119895)

sdot 119909 (119899 minus 119894) 119909 (119899 minus 119895) + sdot sdot sdot

+

+infin

sum

119894=0

+infin

sum

119895=0

sdot sdot sdot

+infin

sum

119896=0

ℎ119901(119894 119895 119896) 119909 (119899 minus 119894) 119909 (119899 minus 119895)

sdot sdot sdot 119909 (119899 minus 119896)

(3)

where ℎ119901(119894 119895 119896) is the 119901th order nucleus of Volterra

The expansion of this infinite order series is extremelydifficult in practical applications Generally the second-ordertruncation is employed as follows

997888

119909

1015840

(119899 + 1) = ℎ0+

1198731minus1

sum

119894=0

ℎ1(119894) 119909 (119899 minus 119894)

+

1198732minus1

sum

119894=0

1198732minus1

sum

119895=0

ℎ2(119894 119895) 119909 (119899 minus 119894) 119909 (119899 minus 119895)

(4)

where1198731and119873

2represent the length of filtersTheminimum

embedding dimension119898 of the signal can be obtained usingthe fault near-neighbor method Consequently 119873

1and 119873

2

can be set as119898 [32]Volterra series is used to extend both ends of the signal to

address the end distortion issue in lifting wavelet transformThe integrative approach of Volterra series and lifting wavelettransform is presented in detail in the following sections

The input vector isX(119899) = [1 119909(119899) 119909(119899minus1) 119909(119899minus119898minus

1) 1199092

(119899) 119909(119899)119909(119899 minus 1) 1199092

(119899 minus 119898 + 1)]119879 The prediction

coefficient vector is W(119899) = [ℎ0 ℎ1(0) ℎ1(1) ℎ

1(119898 minus

1) ℎ2(0 0) ℎ

2(0 1) ℎ

2(119898 minus 1119898 minus 1)]

119879 Thus the expres-sion of (4) can be rewritten as [33]

997888

119909

1015840

(119899 + 1) = X119879 (119899)W (119899) (5)

The prediction coefficient vectorW(119899) is calculated usingthe recursive least-squares method (RLS) To elaborate con-sider

119876 (0) = 120575minus1I (6)

where 120575 is a very small normal number and I is the identitymatrix ThusW(0) is set as

W (0) = 0 (7)

andW(119899) is calculated by carrying on the following iterativecomputation

Consider

119866 (119899) =

120583minus1

119876 (119899 minus 1)X (119899)

1 + 120583minus1X119879 (119899) 119876 (119899 minus 1)X (119899)

(8)

where 120583 is a forgetting factorConsider the following

120572 (119899) = 119863 (119899) minusW119879 (119899 minus 1)X (119899) (9)

where119863(119899) is an ideal output signal Thus

W (119899) = W (119899 minus 1) + 119866 (119899) 120572 (119899)

119876 (119899) = 120583minus1

119876 (119899 minus 1) minus 120583minus1

119866 (119899)X119879 (119899) 119876 (119899 minus 1)

(10)

where 120572(119899) and 119866(119899) are intermediate variablesThe details of theoretical knowledge of lifting wavelet

transform and Volterra series model have been discussedabove Volterra series is used to extend both ends of thesignal to address the end distortion issue in lifting wavelettransformThe formulation of Volterra series assisted undec-imated lifting wavelet transform is illustrated below

3 The Proposed Method

A Volterra series-assisted undecimated lifting wavelet packettransform is proposed for machinery incipient defect diag-nosis to eliminate the frequency aliasing and end distortionissues in traditional lifting wavelet transform and its flow-chart is shown in Figure 1

First both ends of the original signal are extended andpredicted with Volterra series model in which the param-eter of extension number is optimized Next the extendedsignal is decomposed by undecimated lifting wavelet packettransform and the subband with the highest energy ratio isselected as the band of interestThe selected subband signal isthen trimmed back to its original length for envelope analysisFinally themachinery status is assessed according to the anal-ysis results The details of undecimated lifting wavelet packettransform with boundary treatment are discussed as below

31 Formulation of Undecimated LWPT with Boundary Treat-ment The undecimated algorithm can eliminate frequencyaliasing because the length of coefficients at each level isequal to that of the raw signal The P and U can be schemedout as initial prediction operator and initial update operatorrespectively and the undecimated algorithm is deduced asfollows

Assuming P = 119901119894 where 119894 = 1 2 119873 the expression

of the undecimated prediction operator of the 119897th level isgiven by [34]

P[119897]119895

=

119901119894 119895 = 2

119897

sdot 119894

0 119895 = 2119897

sdot 119894

119895 = 1 2119897

119873 (11)

4 Shock and Vibration

0 001 002 003 004 005 006

Extension signal

Machinery Volterra series model

Envelope analysis

Status assessment

Data acquisition0 001 002 003 004

0

5

10

15

Parameter optimization

0 005 01 015 02 025

01

X44

0 005 01 015 02 025

010

X43

0 005 01 015 02 025

01

X42

0 005 01 015 02 025

010

X41

Undecimated LWPTData trimming

0 005 01 015 02 0250 001 002 003 004 005 006 007

02468

Prediction coefficient

Extension number

Undecimated prediction operator P

Undecimated update

operator U

minus4

minus2

minus8

minus6

0246

minus4

minus2

minus6

minus1

minus1

minus10

minus10

minus10

minus5

minus15

0

5

10

15

minus10

minus5

minus15

Time t (s) Time t (s)

W(n) nadd

Am

plitu

dea

(mmmiddotsminus

2 )

Am

plitu

dea

(mmmiddotsminus

2 )

Figure 1 Flowchart of the proposed approach for machinery incipient defect diagnosis

According to the principle of equidistance subdivision thepredictor coefficient 119901

119894is calculated by Lagrange interpola-

tion formula as [29]

119901119894=

119873

prod

119896=1

119896 =119894

(119873 + 1) 2 minus 119896

119894 minus 119896

119894 = 1 2 119873 (12)

where 119894 is an index number and119873 is the predictor lengthSimilarly assuming U = 119906

119894 where 119894 = 1 2 119873

1015840 theexpression of the undecimated update operator of the 119897th levelis obtained by [34]

U[119897]119895

=

119906119894 119895 = 2

119897

sdot 119894

0 119895 = 2119897

sdot 119894

119895 = 1 2119897

1198731015840

(13)

where 119906119894is the update coefficient and is usually set as the half

of the predictor coefficient 119901119894

Assuming 119909119897119896is the 119896th frequency band of the 119897th level

signal decomposed from the original signal 119909[119899] 119909119897(119896minus1)

and119909119897119896can be obtained by dividing 119909

(119897minus1)(1198962)

119909119897(119896minus1)

(119899) = 119909(119897minus1)(1198962)

(119899)

minus [119901[119897]

1119909(119897minus1)(1198962)

(119899 minus 2119897minus1

(119873 + 1) + 1)

+ 119901[119897]

2119909(119897minus1)(1198962)

(119899 minus 2119897minus1

(119873 + 1) + 2) + sdot sdot sdot

+ 119901[119897]

2119897119873

119909(119897minus1)(1198962)

(119899 + 2119897minus1

(119873 minus 1))]

119909119897119896(119899) = 119909

(119897minus1)(1198962)(119899)

+ [119906[119897]

1119909119897(119896minus1)

(119899 minus 2119897minus1

(1198731015840

+ 1) + 1)

+ 119906[119897]

2119909119897(119896minus1)

(119899 minus 2119897minus1

(1198731015840

+ 1) + 2) + sdot sdot sdot

+ 119906[119897]

2119897

119909119897(119896minus1)

(119899 + 2119897minus1

(1198731015840

minus 1))]

(14)

where 119896 = 2 4 6 2119897 119901[119897]119894

is the undecimated predictionoperator of 119909

119897(119896minus1) and 119906[119897]

119894is the undecimated update opera-

tor of 119909119897119896

Assuming the undecimated prediction operator P = 119901119894

and the undecimated update operator U = 119906119894 where 119894 =

1 2 4 119909119894is the 119894th data of X The 119894th data of the first level

detailed signal decomposed by undecimated lifting waveletpacket transform is obtained as follows

1198891119894

= 119909119894minus (119909119894minus3

times 1199011+ 119909119894minus2

times 0 + 119909119894minus1

times 1199012+ 119909119894times 0

+ 119909119894+1

times 1199013+ 119909119894+3

times 1199014)

(15)

It can be deduced that three numbersmust be extended tothe left end of 119909 when calculating 119889

11and the same number

to the right when calculating the last data 1198891119899 Similarly for

the second level 6 more numbers should be extended to eachend and so on More generally assuming 119873

119891is the decom-

posed level of undecimated lifting wavelet packet transformthe extension number of the original signal at each end can becalculated as

119899add =

119873119891

sum

119897=1

[2119897minus1

(119873 + 1) minus 1]

+

119873119891

sum

119897=1

[2119897minus1

(1198731015840

+ 1) minus 1]

(16)

According to (16) the length of extended signal inVolterra series model is determined and the signal isextended based on the procedure of Volterra series model inSection 2 The performance of formulated method is dis-cussed below

Shock and Vibration 5

0

01

02

03

End distortion

Characteristic signal

End distortion

minus02

minus01

002 004 006 008 01 0120Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )

(a)

End distortion

Characteristic signal

End distortion

002 004 006 008 01 0120Time t (s)

minus2

minus1

0

1

2

Am

plitu

dea

(mmmiddotsminus

2 )

(b)

Characteristic signalEnd distortion

minus8

minus6

minus4

minus2

0

2

4

6

8

002 004 006 008 01 0120Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )

(c)

Characteristic signal

002 004 006 008 01 0120Time t (s)

minus02

minus01

0

01

02

03

Am

plitu

dea

(mmmiddotsminus

2 )

(d)

Figure 2 One subband in the fourth layer decomposed using wavelet packet and preprocessed by different methods (a) no extension (b)zero-padding extension (c) periodic extension and (d) second-order Volterra series prediction

32 Performance Comparison of Boundary Treatment Meth-ods To evaluate the performance of various boundary treat-ment methods a synthesized signal is constructed as follows

119909 (119905) = 03 exp (minus2120587 times 100119905) sin (2120587 times 400119905)

+ 15 sin(2120587 times 50119905 +

120587

4

)

+ 3 sin(2120587 times 100119905 +

120587

6

)

(17)

The sampling frequency is set as 16000Hz The synthesizedsignal is decomposed into four layers using the undecimatedlifting wavelet transform Figure 2 shows one subband of thefourth layer corresponding to the pulse train signal underdifferent preprocessing scenarios (a) no extension (b) zero-padding extension (c) periodic extension and (d) second-order Volterra series prediction respectively

There are obvious end distortions shown by the arrows inFigures 2(a)ndash2(c)Therefore if the characteristic signal is just

located in the end it will be submerged by the end distor-tions In Figures 2(b)-2(c) the characteristic signal in themiddle is insignificant because the end distortion is strongThe Volterra series based preprocessing method shows theoptimal performance since the end distortions have beenrestrained effectively as shown in Figure 2(d)

4 Experimental Study

A reciprocating compressor (model number 4HOS-6) in apetrochemical plant in China is used as the experimentaltestbed to evaluate the performance of the developedmethodas shown in Figure 3 It is a 4-cylinder natural gas recipro-cating compressor driven by an 8-cylinder gas engine withrated power of 1600 kW The rotating speed of crankshaft is860 rpm which drives the plungers to strike 860 times perminute back and forth The motion of the plungers changesthe volume of the cylinders When the plunger moves downthe increased volume of cylinder opens intake valve and

6 Shock and Vibration

1

2 3

4

(a)

Naturalgas

engineCrankshaft box

4th cylinder 3rd cylinder

2nd cylinder1st cylinderAccelerometer(in the bottom)

(b)

Figure 3 Illustration of experimental setup (a) data acquisition system A control panel B data acquisition box C accelerometer on theexhaust valve lidD exhaust valve and (b) diagram of reciprocating compressor

0 005 01 015 02 025

0

5

Periodic impact Periodic impact minus10

minus5

Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )

Figure 4 The time series vibrational signal of normal valve

closes the exhaust valve When the plunger moves up thecompression of cylinder opens the exhaust valve and closesthe intake valve Valve is composed of lifting limiter springdisc and seat of which spring is the most susceptible tofailure

To diagnose the valve failure in the 2nd cylinder anaccelerometer is placed on the exhaust valve lid A customerdesigned data acquisition system (model number MDES-5)as shown in Figure 3(a) is used to acquire the measurementsIt consists of a laptop and a data acquisition box configuredin a master-slave system The sampling rate is set as 16 kHzin this study Figures 4 and 5 show the time series vibrationalsignals near the exhaust valve of 2nd cylinder under normalcondition and abnormal condition respectively For compar-ison the amplitude of the vibration signal under abnormalcondition is greater than the one of the normal signal Takingthe RMS (root mean square) as the criterion the RMS of theabnormal signal is calculated as 4725msdotsminus2 which is greaterthan that of the normal signal (2227msdotsminus2) However theperiodic impact feature caused by the collision of the valvedisc and seat is difficult to observe due to strong backgroundnoise

For further analysis the proposed method based onVolterra series and undecimated lifting wavelet packet trans-form is used to process the signals The two ends of the orig-inal signal are extended and predicted by the Volterra seriesmodel Then the predicted signal is decomposed into four

0 005 01 015 02 025

05

1015

minus10

minus15

minus5

Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )

Figure 5 The time series vibrational signal of abnormal valve

1 20

50100

Level 1

270

9730

1 2 3 40

50100

Level 2

736 007 23296928

1 2 3 4 5 6 7 80

50100

Level 3

063 426 012 001 1259 10794039 3121

0 2 4 6 8 10 12 14 160

50100

Level 4

084 006 240 144 004 005 001 000 764 407 899 279 2262 1448 2883 575

Figure 6 Energy distribution of four-level subbands

layers by undecimated lifting wavelet packet transformwhere 119873 = 119873

1015840

= 4 From (11)ndash(13) the predictionoperator P and the update operator U can be obtained asP = [minus00625 05625 05625 minus00625] and U = [minus00313

02813 02813 minus00313] Finally the 16 subbands of thefourth level are obtained and the energy distribution of thesubbands at different levels is shown in Figure 6 After undec-imated lifting wavelet packet decomposition the majorityof the energy rests on the high frequency parts which

Shock and Vibration 7

minus8

minus6

minus4

minus2

02468

minus8

minus6

minus4

minus2

02468

003 004 005 006 007002 008Time t (s)

005

(a)

(b)

010 02 025015Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )A

mpl

itude

a(m

mmiddotsminus

2 )

Figure 7 The wavelet coefficient at selected subband and itsenvelope under normal condition

minus6

minus4

minus2

0246

minus8

minus6

minus4

minus2

024

86

003 004 005 006 007002 008Time t (s)

005

(a)

(b)

010 02 025015Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )A

mpl

itude

a(m

mmiddotsminus

2 )

Figure 8 The wavelet coefficient at selected subband and itsenvelope under abnormal condition

correspond to the detailed wavelet coefficients The subbandwith the highest energy ratio is selected as the bandof interestThewavelet coefficients and their envelopes from the selectedsubbands are shown in Figures 7 and 8 corresponding tothe normal condition and abnormal condition respectivelyFromboth figures the periodic impact features are visible andeasily identified

The signal in one period as shown by the red blockin Figures 7(a) and 8(a) is extracted for detailed analysisAccording to the working principle of exhaust valve the front

half curve from 0012 to 0047 sec represents the valve-closephase while the latter one from 0047 to 0082 sec shows theexhaust phase The exhaust valve closes at point A and thevalve opens again at point BUnder normal condition there isno chattering when normal valve closes because the stiffnessof spring is rigid enough to close tightly By contrast valvechatters significantly which is caused by the repeated collisionof valve and seat during the time period 119879 (indicating theaging and stretch decline of spring in the valve) The valvewith aged spring cannot close tightly resulting in gas leak andefficiency degradation When this compressor is stopped andchecked the spring failure is validated by visual inspectionand the exhaust valve is maintained Therefore the pre-sented incipient fault diagnosis method avoids accidents andprevents huge production loss

In addition the energy of the selected band signal isused to evaluate the performance of the presented methodFigure 9(a) shows the energies of the selected band sig-nals under four different scenarios (I) undecimated LWPT(ULWPT) under normal machinery condition (II) undec-imated LWPT (ULWPT) under abnormal machinery con-dition (III) Volterra series assisted undecimated LWPT(V-ULWPT) under normal machinery condition and (IV)Volterra series assisted undecimated LWPT (V-ULWPT)under abnormal machinery condition From the analysisresults it is found that the normal and abnormal machineryconditions can be differentiated according to energy ofselected subband using undecimated lifting wavelet packettransform Since Volterra series assisted ULWPT suppressesthe end distortion the energy of selected subband using V-ULWPT is less than that using ULWPT However the energyratio of abnormal to normal conditions using V-ULWPTis larger than that using ULWPT as shown in Figure 9(b)thus validating the effectiveness of presented Volterra seriesassisted lifting wavelet packet transform method

The computational efficiency of the developed techniquehas also been evaluated on a desktop (Lenovo YangtianT4900d model Lenovo Inc Beijing China) with a 33 GHzCPU and 4GB memory It takes approximately 119 secondsto process a data string containing 4096 data points whichis equivalent to processing 0256 seconds data length under a16 kHz sampling rate Thus the presented method is applica-ble for online machinery diagnosis in practical applications

5 Conclusions

An enhanced weak feature extraction approach that com-bines Volterra series model and undecimated lifting waveletpacket transform has been presented for incipient fault detec-tion The analysis results show that Volterra series assistedundecimated lifting wavelet packet transform eliminates thefrequency aliasing and end distortion issues in conventionallifting wavelet packet transform method thus it is significantto machinery incipient defect diagnosis especially for weakimpact feature extraction From the engineering applicationperspective the weak impact signals caused by the fault ofa valve in reciprocating compressor are analyzed using thepresented method and the early spring failure is detectedfrom the extractedweak features using the presentedmethod

8 Shock and Vibration

Ener

gy (J

)

I normal ULWPT II abnormal ULWPT

I II III IV

III normal V-ULWPT IV abnormal V-ULWPT

600E + 03

550E + 03

500E + 03

450E + 03

400E + 03

350E + 03

300E + 03

250E + 03

200E + 03

150E + 03

100E + 03

500E + 02

000E + 00

428E + 03

136E + 03

475E + 03

219E + 03

(a)

I II

I ULWPT abnormalnormalII V-ULWPT abnormalnormal

216

314

0

05

1

15

2

25

3

35

4

Ener

gy ra

tio

(b)

Figure 9 Energy comparison among different conditions (a) energy comparison chart (b) energy ratio chart

Conflict of Interests

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

Acknowledgments

This research acknowledges the financial support providedby National Science foundation of China (no 51504274)Science Foundation ofChinaUniversity of PetroleumBeijing(nos 2462014YJRC039 and 2462015YQ0403) and BeijingMunicipal Education Commission Cooperative Construc-tion Project Valuable comments from anonymous reviewersare greatly appreciated

References

[1] V Venkatasubramanian R Rengaswamy K Yin and S NKavuri ldquoA review of process fault detection and diagnosis partI quantitativemodel-basedmethodsrdquoComputers and ChemicalEngineering vol 27 no 3 pp 293ndash311 2003

[2] M Liang and I S Bozchalooi ldquoAn energy operator approachto joint application of amplitude and frequency-demodulationsfor bearing fault detectionrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1473ndash1494 2010

[3] Z Zhong Z Jiang Y Long and X Zhan ldquoAnalysis on thenoise for the different gearboxes of the heavy truckrdquo Shock andVibration vol 2015 Article ID 476460 5 pages 2015

[4] W J Wang ldquoWavelet transform in vibration analysis formechanical fault diagnosisrdquo Shock and Vibration vol 3 no 1pp 17ndash26 1996

[5] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[6] N Baydar and A Ball ldquoA comparative study of acoustic andvibration signals in detection of gear failures usingWigner-VilledistributionrdquoMechanical Systems and Signal Processing vol 15no 6 pp 1091ndash1107 2001

[7] W-X Yang and P W Tse ldquoDevelopment of an advanced noisereductionmethod for vibration analysis based on singular valuedecompositionrdquo NDT amp E International vol 36 no 6 pp 419ndash432 2003

[8] H Hassanpour A Zehtabian and S J Sadati ldquoTime domainsignal enhancement based on an optimized singular vectordenoising algorithmrdquoDigital Signal Processing vol 22 no 5 pp786ndash794 2012

[9] J Wang R X Gao and R Yan ldquoIntegration of EEMD and ICAfor wind turbine gearbox diagnosisrdquoWind Energy vol 17 no 5pp 757ndash773 2014

[10] A Sadhu and B Hazra ldquoA novel damage detection algorithmusing time-series analysis-based blind source separationrdquo Shockand Vibration vol 20 no 3 pp 423ndash438 2013

[11] Z K Peng and F L Chu ldquoApplication of the wavelet trans-form in machine condition monitoring and fault diagnosticsa review with bibliographyrdquo Mechanical Systems and SignalProcessing vol 18 no 2 pp 199ndash221 2004

[12] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosis ofrotary machines a review with applicationsrdquo Signal Processingvol 96 pp 1ndash15 2014

[13] N Saravanan and K I Ramachandran ldquoIncipient gear box faultdiagnosis using discrete wavelet transform (DWT) for featureextraction and classification using artificial neural network(ANN)rdquo Expert Systems with Applications vol 37 no 6 pp4168ndash4181 2010

[14] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weaksignature detection method and its application on rolling ele-ment bearing prognosticsrdquo Journal of Sound and Vibration vol289 no 4-5 pp 1066ndash1090 2006

[15] J Wang R X Gao and R Yan ldquoMulti-scale enveloping orderspectrogram for rotatingmachine health diagnosisrdquoMechanicalSystems and Signal Processing vol 46 no 1 pp 28ndash44 2014

Shock and Vibration 9

[16] Y-L Chen P-L Zhang B Li and D-H Wu ldquoDenoising ofmechanical vibration signals using quantum-inspired adaptivewavelet shrinkagerdquo Shock and Vibration vol 2014 Article ID848097 7 pages 2014

[17] Y Pan J Chen and X Li ldquoBearing performance degradationassessment based on lifting wavelet packet decomposition andfuzzy c-meansrdquo Mechanical Systems and Signal Processing vol24 no 2 pp 559ndash566 2010

[18] R ZhouWBaoN Li XHuang andDYu ldquoMechanical equip-ment fault diagnosis based on redundant second generationwavelet packet transformrdquo Digital Signal Processing A ReviewJournal vol 20 no 1 pp 276ndash288 2010

[19] J Yuan Z He and Y Zi ldquoGear fault detection using customizedmultiwavelet lifting schemesrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1509ndash1528 2010

[20] Y He X Chen J Xiang and Z He ldquoAdaptive multiresolutionfinite element method based on second generation waveletsrdquoFinite Elements in Analysis and Design vol 43 no 6-7 pp 566ndash579 2007

[21] W Bao R Zhou J Yang D Yu and N Li ldquoAnti-aliasinglifting scheme formechanical vibration fault feature extractionrdquoMechanical Systems and Signal Processing vol 23 no 5 pp1458ndash1473 2009

[22] R L Claypoole G M Davis W Sweldens and R G BaraniukldquoNonlinear wavelet transforms for image coding via liftingrdquoIEEE Transactions on Image Processing vol 12 no 12 pp 1449ndash1459 2003

[23] R L Claypoole R G Baraniuk and R D Nowak ldquoAdaptivewavelet transforms via liftingrdquo Report ECE TR-9304 1999httpsscholarshipriceeduhandle191119808

[24] B Appleton and H Talbot ldquoRecursive filtering of images withsymmetric extensionrdquo Signal Processing vol 85 no 8 pp 1546ndash1556 2005

[25] B Wohlberg and C M Brislawn ldquoSymmetric extension forlifted filter banks and obstructions to reversible implementa-tionrdquo Signal Processing vol 88 no 1 pp 131ndash145 2008

[26] L Duan L Zhang and J Chen ldquoBoundary treatment of liftingwavelet transform based on Volterra series model and its appli-cationrdquo Chinese Journal of Scientific Instrument vol 33 no 1pp 7ndash12 2012

[27] D Aiordachioaie E Ceanga R de Keyser and Y Naka ldquoDetec-tion and classification of non-linearities based on Volterrakernels processingrdquo Engineering Applications of Artificial Intel-ligence vol 14 no 4 pp 497ndash503 2001

[28] W Sweldens ldquoThe lifting scheme a custom-design constructionof biorthogonal waveletsrdquo Applied and Computational Har-monic Analysis vol 3 no 2 pp 186ndash200 1996

[29] W Sweldens ldquoThe Lifting Scheme a construction of secondgeneration waveletsrdquo SIAM Journal on Mathematical Analysisvol 29 no 2 pp 511ndash546 1998

[30] A Ben and T Hugues ldquoRecursive filtering of images with sym-metric extensionrdquo Signal Processing vol 85 no 8 pp 1546ndash15562005

[31] D Aiordachioaie E Ceanga R DeKeyser andYNaka ldquoDetec-tion and classification of non-linearities based on Volterrakernels processingrdquo Engineering Applications of Artificial Intel-ligence vol 14 no 4 pp 497ndash503 2001

[32] M B Kennel R Brown and H D I Abarbanel ldquoDeterminingembedding dimension for phase-space reconstruction using ageometrical constructionrdquo Physical Review A vol 45 no 6 pp3403ndash3411 1992

[33] J Zhang and X Xiao ldquoPredicting low-dimensional chaotic timeseries using Volterra adaptive filtersrdquo Acta Physica Sinica vol49 no 3 pp 403ndash408 1999

[34] J Hongkai H Zhengjia D Chendong and C Peng ldquoGear-box fault diagnosis using adaptive redundant Lifting SchemerdquoMechanical Systems and Signal Processing vol 20 no 8 pp1992ndash2006 2006

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RoboticsJournal of

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

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Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

4 Shock and Vibration

0 001 002 003 004 005 006

Extension signal

Machinery Volterra series model

Envelope analysis

Status assessment

Data acquisition0 001 002 003 004

0

5

10

15

Parameter optimization

0 005 01 015 02 025

01

X44

0 005 01 015 02 025

010

X43

0 005 01 015 02 025

01

X42

0 005 01 015 02 025

010

X41

Undecimated LWPTData trimming

0 005 01 015 02 0250 001 002 003 004 005 006 007

02468

Prediction coefficient

Extension number

Undecimated prediction operator P

Undecimated update

operator U

minus4

minus2

minus8

minus6

0246

minus4

minus2

minus6

minus1

minus1

minus10

minus10

minus10

minus5

minus15

0

5

10

15

minus10

minus5

minus15

Time t (s) Time t (s)

W(n) nadd

Am

plitu

dea

(mmmiddotsminus

2 )

Am

plitu

dea

(mmmiddotsminus

2 )

Figure 1 Flowchart of the proposed approach for machinery incipient defect diagnosis

According to the principle of equidistance subdivision thepredictor coefficient 119901

119894is calculated by Lagrange interpola-

tion formula as [29]

119901119894=

119873

prod

119896=1

119896 =119894

(119873 + 1) 2 minus 119896

119894 minus 119896

119894 = 1 2 119873 (12)

where 119894 is an index number and119873 is the predictor lengthSimilarly assuming U = 119906

119894 where 119894 = 1 2 119873

1015840 theexpression of the undecimated update operator of the 119897th levelis obtained by [34]

U[119897]119895

=

119906119894 119895 = 2

119897

sdot 119894

0 119895 = 2119897

sdot 119894

119895 = 1 2119897

1198731015840

(13)

where 119906119894is the update coefficient and is usually set as the half

of the predictor coefficient 119901119894

Assuming 119909119897119896is the 119896th frequency band of the 119897th level

signal decomposed from the original signal 119909[119899] 119909119897(119896minus1)

and119909119897119896can be obtained by dividing 119909

(119897minus1)(1198962)

119909119897(119896minus1)

(119899) = 119909(119897minus1)(1198962)

(119899)

minus [119901[119897]

1119909(119897minus1)(1198962)

(119899 minus 2119897minus1

(119873 + 1) + 1)

+ 119901[119897]

2119909(119897minus1)(1198962)

(119899 minus 2119897minus1

(119873 + 1) + 2) + sdot sdot sdot

+ 119901[119897]

2119897119873

119909(119897minus1)(1198962)

(119899 + 2119897minus1

(119873 minus 1))]

119909119897119896(119899) = 119909

(119897minus1)(1198962)(119899)

+ [119906[119897]

1119909119897(119896minus1)

(119899 minus 2119897minus1

(1198731015840

+ 1) + 1)

+ 119906[119897]

2119909119897(119896minus1)

(119899 minus 2119897minus1

(1198731015840

+ 1) + 2) + sdot sdot sdot

+ 119906[119897]

2119897

119909119897(119896minus1)

(119899 + 2119897minus1

(1198731015840

minus 1))]

(14)

where 119896 = 2 4 6 2119897 119901[119897]119894

is the undecimated predictionoperator of 119909

119897(119896minus1) and 119906[119897]

119894is the undecimated update opera-

tor of 119909119897119896

Assuming the undecimated prediction operator P = 119901119894

and the undecimated update operator U = 119906119894 where 119894 =

1 2 4 119909119894is the 119894th data of X The 119894th data of the first level

detailed signal decomposed by undecimated lifting waveletpacket transform is obtained as follows

1198891119894

= 119909119894minus (119909119894minus3

times 1199011+ 119909119894minus2

times 0 + 119909119894minus1

times 1199012+ 119909119894times 0

+ 119909119894+1

times 1199013+ 119909119894+3

times 1199014)

(15)

It can be deduced that three numbersmust be extended tothe left end of 119909 when calculating 119889

11and the same number

to the right when calculating the last data 1198891119899 Similarly for

the second level 6 more numbers should be extended to eachend and so on More generally assuming 119873

119891is the decom-

posed level of undecimated lifting wavelet packet transformthe extension number of the original signal at each end can becalculated as

119899add =

119873119891

sum

119897=1

[2119897minus1

(119873 + 1) minus 1]

+

119873119891

sum

119897=1

[2119897minus1

(1198731015840

+ 1) minus 1]

(16)

According to (16) the length of extended signal inVolterra series model is determined and the signal isextended based on the procedure of Volterra series model inSection 2 The performance of formulated method is dis-cussed below

Shock and Vibration 5

0

01

02

03

End distortion

Characteristic signal

End distortion

minus02

minus01

002 004 006 008 01 0120Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )

(a)

End distortion

Characteristic signal

End distortion

002 004 006 008 01 0120Time t (s)

minus2

minus1

0

1

2

Am

plitu

dea

(mmmiddotsminus

2 )

(b)

Characteristic signalEnd distortion

minus8

minus6

minus4

minus2

0

2

4

6

8

002 004 006 008 01 0120Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )

(c)

Characteristic signal

002 004 006 008 01 0120Time t (s)

minus02

minus01

0

01

02

03

Am

plitu

dea

(mmmiddotsminus

2 )

(d)

Figure 2 One subband in the fourth layer decomposed using wavelet packet and preprocessed by different methods (a) no extension (b)zero-padding extension (c) periodic extension and (d) second-order Volterra series prediction

32 Performance Comparison of Boundary Treatment Meth-ods To evaluate the performance of various boundary treat-ment methods a synthesized signal is constructed as follows

119909 (119905) = 03 exp (minus2120587 times 100119905) sin (2120587 times 400119905)

+ 15 sin(2120587 times 50119905 +

120587

4

)

+ 3 sin(2120587 times 100119905 +

120587

6

)

(17)

The sampling frequency is set as 16000Hz The synthesizedsignal is decomposed into four layers using the undecimatedlifting wavelet transform Figure 2 shows one subband of thefourth layer corresponding to the pulse train signal underdifferent preprocessing scenarios (a) no extension (b) zero-padding extension (c) periodic extension and (d) second-order Volterra series prediction respectively

There are obvious end distortions shown by the arrows inFigures 2(a)ndash2(c)Therefore if the characteristic signal is just

located in the end it will be submerged by the end distor-tions In Figures 2(b)-2(c) the characteristic signal in themiddle is insignificant because the end distortion is strongThe Volterra series based preprocessing method shows theoptimal performance since the end distortions have beenrestrained effectively as shown in Figure 2(d)

4 Experimental Study

A reciprocating compressor (model number 4HOS-6) in apetrochemical plant in China is used as the experimentaltestbed to evaluate the performance of the developedmethodas shown in Figure 3 It is a 4-cylinder natural gas recipro-cating compressor driven by an 8-cylinder gas engine withrated power of 1600 kW The rotating speed of crankshaft is860 rpm which drives the plungers to strike 860 times perminute back and forth The motion of the plungers changesthe volume of the cylinders When the plunger moves downthe increased volume of cylinder opens intake valve and

6 Shock and Vibration

1

2 3

4

(a)

Naturalgas

engineCrankshaft box

4th cylinder 3rd cylinder

2nd cylinder1st cylinderAccelerometer(in the bottom)

(b)

Figure 3 Illustration of experimental setup (a) data acquisition system A control panel B data acquisition box C accelerometer on theexhaust valve lidD exhaust valve and (b) diagram of reciprocating compressor

0 005 01 015 02 025

0

5

Periodic impact Periodic impact minus10

minus5

Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )

Figure 4 The time series vibrational signal of normal valve

closes the exhaust valve When the plunger moves up thecompression of cylinder opens the exhaust valve and closesthe intake valve Valve is composed of lifting limiter springdisc and seat of which spring is the most susceptible tofailure

To diagnose the valve failure in the 2nd cylinder anaccelerometer is placed on the exhaust valve lid A customerdesigned data acquisition system (model number MDES-5)as shown in Figure 3(a) is used to acquire the measurementsIt consists of a laptop and a data acquisition box configuredin a master-slave system The sampling rate is set as 16 kHzin this study Figures 4 and 5 show the time series vibrationalsignals near the exhaust valve of 2nd cylinder under normalcondition and abnormal condition respectively For compar-ison the amplitude of the vibration signal under abnormalcondition is greater than the one of the normal signal Takingthe RMS (root mean square) as the criterion the RMS of theabnormal signal is calculated as 4725msdotsminus2 which is greaterthan that of the normal signal (2227msdotsminus2) However theperiodic impact feature caused by the collision of the valvedisc and seat is difficult to observe due to strong backgroundnoise

For further analysis the proposed method based onVolterra series and undecimated lifting wavelet packet trans-form is used to process the signals The two ends of the orig-inal signal are extended and predicted by the Volterra seriesmodel Then the predicted signal is decomposed into four

0 005 01 015 02 025

05

1015

minus10

minus15

minus5

Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )

Figure 5 The time series vibrational signal of abnormal valve

1 20

50100

Level 1

270

9730

1 2 3 40

50100

Level 2

736 007 23296928

1 2 3 4 5 6 7 80

50100

Level 3

063 426 012 001 1259 10794039 3121

0 2 4 6 8 10 12 14 160

50100

Level 4

084 006 240 144 004 005 001 000 764 407 899 279 2262 1448 2883 575

Figure 6 Energy distribution of four-level subbands

layers by undecimated lifting wavelet packet transformwhere 119873 = 119873

1015840

= 4 From (11)ndash(13) the predictionoperator P and the update operator U can be obtained asP = [minus00625 05625 05625 minus00625] and U = [minus00313

02813 02813 minus00313] Finally the 16 subbands of thefourth level are obtained and the energy distribution of thesubbands at different levels is shown in Figure 6 After undec-imated lifting wavelet packet decomposition the majorityof the energy rests on the high frequency parts which

Shock and Vibration 7

minus8

minus6

minus4

minus2

02468

minus8

minus6

minus4

minus2

02468

003 004 005 006 007002 008Time t (s)

005

(a)

(b)

010 02 025015Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )A

mpl

itude

a(m

mmiddotsminus

2 )

Figure 7 The wavelet coefficient at selected subband and itsenvelope under normal condition

minus6

minus4

minus2

0246

minus8

minus6

minus4

minus2

024

86

003 004 005 006 007002 008Time t (s)

005

(a)

(b)

010 02 025015Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )A

mpl

itude

a(m

mmiddotsminus

2 )

Figure 8 The wavelet coefficient at selected subband and itsenvelope under abnormal condition

correspond to the detailed wavelet coefficients The subbandwith the highest energy ratio is selected as the bandof interestThewavelet coefficients and their envelopes from the selectedsubbands are shown in Figures 7 and 8 corresponding tothe normal condition and abnormal condition respectivelyFromboth figures the periodic impact features are visible andeasily identified

The signal in one period as shown by the red blockin Figures 7(a) and 8(a) is extracted for detailed analysisAccording to the working principle of exhaust valve the front

half curve from 0012 to 0047 sec represents the valve-closephase while the latter one from 0047 to 0082 sec shows theexhaust phase The exhaust valve closes at point A and thevalve opens again at point BUnder normal condition there isno chattering when normal valve closes because the stiffnessof spring is rigid enough to close tightly By contrast valvechatters significantly which is caused by the repeated collisionof valve and seat during the time period 119879 (indicating theaging and stretch decline of spring in the valve) The valvewith aged spring cannot close tightly resulting in gas leak andefficiency degradation When this compressor is stopped andchecked the spring failure is validated by visual inspectionand the exhaust valve is maintained Therefore the pre-sented incipient fault diagnosis method avoids accidents andprevents huge production loss

In addition the energy of the selected band signal isused to evaluate the performance of the presented methodFigure 9(a) shows the energies of the selected band sig-nals under four different scenarios (I) undecimated LWPT(ULWPT) under normal machinery condition (II) undec-imated LWPT (ULWPT) under abnormal machinery con-dition (III) Volterra series assisted undecimated LWPT(V-ULWPT) under normal machinery condition and (IV)Volterra series assisted undecimated LWPT (V-ULWPT)under abnormal machinery condition From the analysisresults it is found that the normal and abnormal machineryconditions can be differentiated according to energy ofselected subband using undecimated lifting wavelet packettransform Since Volterra series assisted ULWPT suppressesthe end distortion the energy of selected subband using V-ULWPT is less than that using ULWPT However the energyratio of abnormal to normal conditions using V-ULWPTis larger than that using ULWPT as shown in Figure 9(b)thus validating the effectiveness of presented Volterra seriesassisted lifting wavelet packet transform method

The computational efficiency of the developed techniquehas also been evaluated on a desktop (Lenovo YangtianT4900d model Lenovo Inc Beijing China) with a 33 GHzCPU and 4GB memory It takes approximately 119 secondsto process a data string containing 4096 data points whichis equivalent to processing 0256 seconds data length under a16 kHz sampling rate Thus the presented method is applica-ble for online machinery diagnosis in practical applications

5 Conclusions

An enhanced weak feature extraction approach that com-bines Volterra series model and undecimated lifting waveletpacket transform has been presented for incipient fault detec-tion The analysis results show that Volterra series assistedundecimated lifting wavelet packet transform eliminates thefrequency aliasing and end distortion issues in conventionallifting wavelet packet transform method thus it is significantto machinery incipient defect diagnosis especially for weakimpact feature extraction From the engineering applicationperspective the weak impact signals caused by the fault ofa valve in reciprocating compressor are analyzed using thepresented method and the early spring failure is detectedfrom the extractedweak features using the presentedmethod

8 Shock and Vibration

Ener

gy (J

)

I normal ULWPT II abnormal ULWPT

I II III IV

III normal V-ULWPT IV abnormal V-ULWPT

600E + 03

550E + 03

500E + 03

450E + 03

400E + 03

350E + 03

300E + 03

250E + 03

200E + 03

150E + 03

100E + 03

500E + 02

000E + 00

428E + 03

136E + 03

475E + 03

219E + 03

(a)

I II

I ULWPT abnormalnormalII V-ULWPT abnormalnormal

216

314

0

05

1

15

2

25

3

35

4

Ener

gy ra

tio

(b)

Figure 9 Energy comparison among different conditions (a) energy comparison chart (b) energy ratio chart

Conflict of Interests

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

Acknowledgments

This research acknowledges the financial support providedby National Science foundation of China (no 51504274)Science Foundation ofChinaUniversity of PetroleumBeijing(nos 2462014YJRC039 and 2462015YQ0403) and BeijingMunicipal Education Commission Cooperative Construc-tion Project Valuable comments from anonymous reviewersare greatly appreciated

References

[1] V Venkatasubramanian R Rengaswamy K Yin and S NKavuri ldquoA review of process fault detection and diagnosis partI quantitativemodel-basedmethodsrdquoComputers and ChemicalEngineering vol 27 no 3 pp 293ndash311 2003

[2] M Liang and I S Bozchalooi ldquoAn energy operator approachto joint application of amplitude and frequency-demodulationsfor bearing fault detectionrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1473ndash1494 2010

[3] Z Zhong Z Jiang Y Long and X Zhan ldquoAnalysis on thenoise for the different gearboxes of the heavy truckrdquo Shock andVibration vol 2015 Article ID 476460 5 pages 2015

[4] W J Wang ldquoWavelet transform in vibration analysis formechanical fault diagnosisrdquo Shock and Vibration vol 3 no 1pp 17ndash26 1996

[5] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[6] N Baydar and A Ball ldquoA comparative study of acoustic andvibration signals in detection of gear failures usingWigner-VilledistributionrdquoMechanical Systems and Signal Processing vol 15no 6 pp 1091ndash1107 2001

[7] W-X Yang and P W Tse ldquoDevelopment of an advanced noisereductionmethod for vibration analysis based on singular valuedecompositionrdquo NDT amp E International vol 36 no 6 pp 419ndash432 2003

[8] H Hassanpour A Zehtabian and S J Sadati ldquoTime domainsignal enhancement based on an optimized singular vectordenoising algorithmrdquoDigital Signal Processing vol 22 no 5 pp786ndash794 2012

[9] J Wang R X Gao and R Yan ldquoIntegration of EEMD and ICAfor wind turbine gearbox diagnosisrdquoWind Energy vol 17 no 5pp 757ndash773 2014

[10] A Sadhu and B Hazra ldquoA novel damage detection algorithmusing time-series analysis-based blind source separationrdquo Shockand Vibration vol 20 no 3 pp 423ndash438 2013

[11] Z K Peng and F L Chu ldquoApplication of the wavelet trans-form in machine condition monitoring and fault diagnosticsa review with bibliographyrdquo Mechanical Systems and SignalProcessing vol 18 no 2 pp 199ndash221 2004

[12] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosis ofrotary machines a review with applicationsrdquo Signal Processingvol 96 pp 1ndash15 2014

[13] N Saravanan and K I Ramachandran ldquoIncipient gear box faultdiagnosis using discrete wavelet transform (DWT) for featureextraction and classification using artificial neural network(ANN)rdquo Expert Systems with Applications vol 37 no 6 pp4168ndash4181 2010

[14] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weaksignature detection method and its application on rolling ele-ment bearing prognosticsrdquo Journal of Sound and Vibration vol289 no 4-5 pp 1066ndash1090 2006

[15] J Wang R X Gao and R Yan ldquoMulti-scale enveloping orderspectrogram for rotatingmachine health diagnosisrdquoMechanicalSystems and Signal Processing vol 46 no 1 pp 28ndash44 2014

Shock and Vibration 9

[16] Y-L Chen P-L Zhang B Li and D-H Wu ldquoDenoising ofmechanical vibration signals using quantum-inspired adaptivewavelet shrinkagerdquo Shock and Vibration vol 2014 Article ID848097 7 pages 2014

[17] Y Pan J Chen and X Li ldquoBearing performance degradationassessment based on lifting wavelet packet decomposition andfuzzy c-meansrdquo Mechanical Systems and Signal Processing vol24 no 2 pp 559ndash566 2010

[18] R ZhouWBaoN Li XHuang andDYu ldquoMechanical equip-ment fault diagnosis based on redundant second generationwavelet packet transformrdquo Digital Signal Processing A ReviewJournal vol 20 no 1 pp 276ndash288 2010

[19] J Yuan Z He and Y Zi ldquoGear fault detection using customizedmultiwavelet lifting schemesrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1509ndash1528 2010

[20] Y He X Chen J Xiang and Z He ldquoAdaptive multiresolutionfinite element method based on second generation waveletsrdquoFinite Elements in Analysis and Design vol 43 no 6-7 pp 566ndash579 2007

[21] W Bao R Zhou J Yang D Yu and N Li ldquoAnti-aliasinglifting scheme formechanical vibration fault feature extractionrdquoMechanical Systems and Signal Processing vol 23 no 5 pp1458ndash1473 2009

[22] R L Claypoole G M Davis W Sweldens and R G BaraniukldquoNonlinear wavelet transforms for image coding via liftingrdquoIEEE Transactions on Image Processing vol 12 no 12 pp 1449ndash1459 2003

[23] R L Claypoole R G Baraniuk and R D Nowak ldquoAdaptivewavelet transforms via liftingrdquo Report ECE TR-9304 1999httpsscholarshipriceeduhandle191119808

[24] B Appleton and H Talbot ldquoRecursive filtering of images withsymmetric extensionrdquo Signal Processing vol 85 no 8 pp 1546ndash1556 2005

[25] B Wohlberg and C M Brislawn ldquoSymmetric extension forlifted filter banks and obstructions to reversible implementa-tionrdquo Signal Processing vol 88 no 1 pp 131ndash145 2008

[26] L Duan L Zhang and J Chen ldquoBoundary treatment of liftingwavelet transform based on Volterra series model and its appli-cationrdquo Chinese Journal of Scientific Instrument vol 33 no 1pp 7ndash12 2012

[27] D Aiordachioaie E Ceanga R de Keyser and Y Naka ldquoDetec-tion and classification of non-linearities based on Volterrakernels processingrdquo Engineering Applications of Artificial Intel-ligence vol 14 no 4 pp 497ndash503 2001

[28] W Sweldens ldquoThe lifting scheme a custom-design constructionof biorthogonal waveletsrdquo Applied and Computational Har-monic Analysis vol 3 no 2 pp 186ndash200 1996

[29] W Sweldens ldquoThe Lifting Scheme a construction of secondgeneration waveletsrdquo SIAM Journal on Mathematical Analysisvol 29 no 2 pp 511ndash546 1998

[30] A Ben and T Hugues ldquoRecursive filtering of images with sym-metric extensionrdquo Signal Processing vol 85 no 8 pp 1546ndash15562005

[31] D Aiordachioaie E Ceanga R DeKeyser andYNaka ldquoDetec-tion and classification of non-linearities based on Volterrakernels processingrdquo Engineering Applications of Artificial Intel-ligence vol 14 no 4 pp 497ndash503 2001

[32] M B Kennel R Brown and H D I Abarbanel ldquoDeterminingembedding dimension for phase-space reconstruction using ageometrical constructionrdquo Physical Review A vol 45 no 6 pp3403ndash3411 1992

[33] J Zhang and X Xiao ldquoPredicting low-dimensional chaotic timeseries using Volterra adaptive filtersrdquo Acta Physica Sinica vol49 no 3 pp 403ndash408 1999

[34] J Hongkai H Zhengjia D Chendong and C Peng ldquoGear-box fault diagnosis using adaptive redundant Lifting SchemerdquoMechanical Systems and Signal Processing vol 20 no 8 pp1992ndash2006 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 5

0

01

02

03

End distortion

Characteristic signal

End distortion

minus02

minus01

002 004 006 008 01 0120Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )

(a)

End distortion

Characteristic signal

End distortion

002 004 006 008 01 0120Time t (s)

minus2

minus1

0

1

2

Am

plitu

dea

(mmmiddotsminus

2 )

(b)

Characteristic signalEnd distortion

minus8

minus6

minus4

minus2

0

2

4

6

8

002 004 006 008 01 0120Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )

(c)

Characteristic signal

002 004 006 008 01 0120Time t (s)

minus02

minus01

0

01

02

03

Am

plitu

dea

(mmmiddotsminus

2 )

(d)

Figure 2 One subband in the fourth layer decomposed using wavelet packet and preprocessed by different methods (a) no extension (b)zero-padding extension (c) periodic extension and (d) second-order Volterra series prediction

32 Performance Comparison of Boundary Treatment Meth-ods To evaluate the performance of various boundary treat-ment methods a synthesized signal is constructed as follows

119909 (119905) = 03 exp (minus2120587 times 100119905) sin (2120587 times 400119905)

+ 15 sin(2120587 times 50119905 +

120587

4

)

+ 3 sin(2120587 times 100119905 +

120587

6

)

(17)

The sampling frequency is set as 16000Hz The synthesizedsignal is decomposed into four layers using the undecimatedlifting wavelet transform Figure 2 shows one subband of thefourth layer corresponding to the pulse train signal underdifferent preprocessing scenarios (a) no extension (b) zero-padding extension (c) periodic extension and (d) second-order Volterra series prediction respectively

There are obvious end distortions shown by the arrows inFigures 2(a)ndash2(c)Therefore if the characteristic signal is just

located in the end it will be submerged by the end distor-tions In Figures 2(b)-2(c) the characteristic signal in themiddle is insignificant because the end distortion is strongThe Volterra series based preprocessing method shows theoptimal performance since the end distortions have beenrestrained effectively as shown in Figure 2(d)

4 Experimental Study

A reciprocating compressor (model number 4HOS-6) in apetrochemical plant in China is used as the experimentaltestbed to evaluate the performance of the developedmethodas shown in Figure 3 It is a 4-cylinder natural gas recipro-cating compressor driven by an 8-cylinder gas engine withrated power of 1600 kW The rotating speed of crankshaft is860 rpm which drives the plungers to strike 860 times perminute back and forth The motion of the plungers changesthe volume of the cylinders When the plunger moves downthe increased volume of cylinder opens intake valve and

6 Shock and Vibration

1

2 3

4

(a)

Naturalgas

engineCrankshaft box

4th cylinder 3rd cylinder

2nd cylinder1st cylinderAccelerometer(in the bottom)

(b)

Figure 3 Illustration of experimental setup (a) data acquisition system A control panel B data acquisition box C accelerometer on theexhaust valve lidD exhaust valve and (b) diagram of reciprocating compressor

0 005 01 015 02 025

0

5

Periodic impact Periodic impact minus10

minus5

Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )

Figure 4 The time series vibrational signal of normal valve

closes the exhaust valve When the plunger moves up thecompression of cylinder opens the exhaust valve and closesthe intake valve Valve is composed of lifting limiter springdisc and seat of which spring is the most susceptible tofailure

To diagnose the valve failure in the 2nd cylinder anaccelerometer is placed on the exhaust valve lid A customerdesigned data acquisition system (model number MDES-5)as shown in Figure 3(a) is used to acquire the measurementsIt consists of a laptop and a data acquisition box configuredin a master-slave system The sampling rate is set as 16 kHzin this study Figures 4 and 5 show the time series vibrationalsignals near the exhaust valve of 2nd cylinder under normalcondition and abnormal condition respectively For compar-ison the amplitude of the vibration signal under abnormalcondition is greater than the one of the normal signal Takingthe RMS (root mean square) as the criterion the RMS of theabnormal signal is calculated as 4725msdotsminus2 which is greaterthan that of the normal signal (2227msdotsminus2) However theperiodic impact feature caused by the collision of the valvedisc and seat is difficult to observe due to strong backgroundnoise

For further analysis the proposed method based onVolterra series and undecimated lifting wavelet packet trans-form is used to process the signals The two ends of the orig-inal signal are extended and predicted by the Volterra seriesmodel Then the predicted signal is decomposed into four

0 005 01 015 02 025

05

1015

minus10

minus15

minus5

Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )

Figure 5 The time series vibrational signal of abnormal valve

1 20

50100

Level 1

270

9730

1 2 3 40

50100

Level 2

736 007 23296928

1 2 3 4 5 6 7 80

50100

Level 3

063 426 012 001 1259 10794039 3121

0 2 4 6 8 10 12 14 160

50100

Level 4

084 006 240 144 004 005 001 000 764 407 899 279 2262 1448 2883 575

Figure 6 Energy distribution of four-level subbands

layers by undecimated lifting wavelet packet transformwhere 119873 = 119873

1015840

= 4 From (11)ndash(13) the predictionoperator P and the update operator U can be obtained asP = [minus00625 05625 05625 minus00625] and U = [minus00313

02813 02813 minus00313] Finally the 16 subbands of thefourth level are obtained and the energy distribution of thesubbands at different levels is shown in Figure 6 After undec-imated lifting wavelet packet decomposition the majorityof the energy rests on the high frequency parts which

Shock and Vibration 7

minus8

minus6

minus4

minus2

02468

minus8

minus6

minus4

minus2

02468

003 004 005 006 007002 008Time t (s)

005

(a)

(b)

010 02 025015Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )A

mpl

itude

a(m

mmiddotsminus

2 )

Figure 7 The wavelet coefficient at selected subband and itsenvelope under normal condition

minus6

minus4

minus2

0246

minus8

minus6

minus4

minus2

024

86

003 004 005 006 007002 008Time t (s)

005

(a)

(b)

010 02 025015Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )A

mpl

itude

a(m

mmiddotsminus

2 )

Figure 8 The wavelet coefficient at selected subband and itsenvelope under abnormal condition

correspond to the detailed wavelet coefficients The subbandwith the highest energy ratio is selected as the bandof interestThewavelet coefficients and their envelopes from the selectedsubbands are shown in Figures 7 and 8 corresponding tothe normal condition and abnormal condition respectivelyFromboth figures the periodic impact features are visible andeasily identified

The signal in one period as shown by the red blockin Figures 7(a) and 8(a) is extracted for detailed analysisAccording to the working principle of exhaust valve the front

half curve from 0012 to 0047 sec represents the valve-closephase while the latter one from 0047 to 0082 sec shows theexhaust phase The exhaust valve closes at point A and thevalve opens again at point BUnder normal condition there isno chattering when normal valve closes because the stiffnessof spring is rigid enough to close tightly By contrast valvechatters significantly which is caused by the repeated collisionof valve and seat during the time period 119879 (indicating theaging and stretch decline of spring in the valve) The valvewith aged spring cannot close tightly resulting in gas leak andefficiency degradation When this compressor is stopped andchecked the spring failure is validated by visual inspectionand the exhaust valve is maintained Therefore the pre-sented incipient fault diagnosis method avoids accidents andprevents huge production loss

In addition the energy of the selected band signal isused to evaluate the performance of the presented methodFigure 9(a) shows the energies of the selected band sig-nals under four different scenarios (I) undecimated LWPT(ULWPT) under normal machinery condition (II) undec-imated LWPT (ULWPT) under abnormal machinery con-dition (III) Volterra series assisted undecimated LWPT(V-ULWPT) under normal machinery condition and (IV)Volterra series assisted undecimated LWPT (V-ULWPT)under abnormal machinery condition From the analysisresults it is found that the normal and abnormal machineryconditions can be differentiated according to energy ofselected subband using undecimated lifting wavelet packettransform Since Volterra series assisted ULWPT suppressesthe end distortion the energy of selected subband using V-ULWPT is less than that using ULWPT However the energyratio of abnormal to normal conditions using V-ULWPTis larger than that using ULWPT as shown in Figure 9(b)thus validating the effectiveness of presented Volterra seriesassisted lifting wavelet packet transform method

The computational efficiency of the developed techniquehas also been evaluated on a desktop (Lenovo YangtianT4900d model Lenovo Inc Beijing China) with a 33 GHzCPU and 4GB memory It takes approximately 119 secondsto process a data string containing 4096 data points whichis equivalent to processing 0256 seconds data length under a16 kHz sampling rate Thus the presented method is applica-ble for online machinery diagnosis in practical applications

5 Conclusions

An enhanced weak feature extraction approach that com-bines Volterra series model and undecimated lifting waveletpacket transform has been presented for incipient fault detec-tion The analysis results show that Volterra series assistedundecimated lifting wavelet packet transform eliminates thefrequency aliasing and end distortion issues in conventionallifting wavelet packet transform method thus it is significantto machinery incipient defect diagnosis especially for weakimpact feature extraction From the engineering applicationperspective the weak impact signals caused by the fault ofa valve in reciprocating compressor are analyzed using thepresented method and the early spring failure is detectedfrom the extractedweak features using the presentedmethod

8 Shock and Vibration

Ener

gy (J

)

I normal ULWPT II abnormal ULWPT

I II III IV

III normal V-ULWPT IV abnormal V-ULWPT

600E + 03

550E + 03

500E + 03

450E + 03

400E + 03

350E + 03

300E + 03

250E + 03

200E + 03

150E + 03

100E + 03

500E + 02

000E + 00

428E + 03

136E + 03

475E + 03

219E + 03

(a)

I II

I ULWPT abnormalnormalII V-ULWPT abnormalnormal

216

314

0

05

1

15

2

25

3

35

4

Ener

gy ra

tio

(b)

Figure 9 Energy comparison among different conditions (a) energy comparison chart (b) energy ratio chart

Conflict of Interests

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

Acknowledgments

This research acknowledges the financial support providedby National Science foundation of China (no 51504274)Science Foundation ofChinaUniversity of PetroleumBeijing(nos 2462014YJRC039 and 2462015YQ0403) and BeijingMunicipal Education Commission Cooperative Construc-tion Project Valuable comments from anonymous reviewersare greatly appreciated

References

[1] V Venkatasubramanian R Rengaswamy K Yin and S NKavuri ldquoA review of process fault detection and diagnosis partI quantitativemodel-basedmethodsrdquoComputers and ChemicalEngineering vol 27 no 3 pp 293ndash311 2003

[2] M Liang and I S Bozchalooi ldquoAn energy operator approachto joint application of amplitude and frequency-demodulationsfor bearing fault detectionrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1473ndash1494 2010

[3] Z Zhong Z Jiang Y Long and X Zhan ldquoAnalysis on thenoise for the different gearboxes of the heavy truckrdquo Shock andVibration vol 2015 Article ID 476460 5 pages 2015

[4] W J Wang ldquoWavelet transform in vibration analysis formechanical fault diagnosisrdquo Shock and Vibration vol 3 no 1pp 17ndash26 1996

[5] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[6] N Baydar and A Ball ldquoA comparative study of acoustic andvibration signals in detection of gear failures usingWigner-VilledistributionrdquoMechanical Systems and Signal Processing vol 15no 6 pp 1091ndash1107 2001

[7] W-X Yang and P W Tse ldquoDevelopment of an advanced noisereductionmethod for vibration analysis based on singular valuedecompositionrdquo NDT amp E International vol 36 no 6 pp 419ndash432 2003

[8] H Hassanpour A Zehtabian and S J Sadati ldquoTime domainsignal enhancement based on an optimized singular vectordenoising algorithmrdquoDigital Signal Processing vol 22 no 5 pp786ndash794 2012

[9] J Wang R X Gao and R Yan ldquoIntegration of EEMD and ICAfor wind turbine gearbox diagnosisrdquoWind Energy vol 17 no 5pp 757ndash773 2014

[10] A Sadhu and B Hazra ldquoA novel damage detection algorithmusing time-series analysis-based blind source separationrdquo Shockand Vibration vol 20 no 3 pp 423ndash438 2013

[11] Z K Peng and F L Chu ldquoApplication of the wavelet trans-form in machine condition monitoring and fault diagnosticsa review with bibliographyrdquo Mechanical Systems and SignalProcessing vol 18 no 2 pp 199ndash221 2004

[12] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosis ofrotary machines a review with applicationsrdquo Signal Processingvol 96 pp 1ndash15 2014

[13] N Saravanan and K I Ramachandran ldquoIncipient gear box faultdiagnosis using discrete wavelet transform (DWT) for featureextraction and classification using artificial neural network(ANN)rdquo Expert Systems with Applications vol 37 no 6 pp4168ndash4181 2010

[14] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weaksignature detection method and its application on rolling ele-ment bearing prognosticsrdquo Journal of Sound and Vibration vol289 no 4-5 pp 1066ndash1090 2006

[15] J Wang R X Gao and R Yan ldquoMulti-scale enveloping orderspectrogram for rotatingmachine health diagnosisrdquoMechanicalSystems and Signal Processing vol 46 no 1 pp 28ndash44 2014

Shock and Vibration 9

[16] Y-L Chen P-L Zhang B Li and D-H Wu ldquoDenoising ofmechanical vibration signals using quantum-inspired adaptivewavelet shrinkagerdquo Shock and Vibration vol 2014 Article ID848097 7 pages 2014

[17] Y Pan J Chen and X Li ldquoBearing performance degradationassessment based on lifting wavelet packet decomposition andfuzzy c-meansrdquo Mechanical Systems and Signal Processing vol24 no 2 pp 559ndash566 2010

[18] R ZhouWBaoN Li XHuang andDYu ldquoMechanical equip-ment fault diagnosis based on redundant second generationwavelet packet transformrdquo Digital Signal Processing A ReviewJournal vol 20 no 1 pp 276ndash288 2010

[19] J Yuan Z He and Y Zi ldquoGear fault detection using customizedmultiwavelet lifting schemesrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1509ndash1528 2010

[20] Y He X Chen J Xiang and Z He ldquoAdaptive multiresolutionfinite element method based on second generation waveletsrdquoFinite Elements in Analysis and Design vol 43 no 6-7 pp 566ndash579 2007

[21] W Bao R Zhou J Yang D Yu and N Li ldquoAnti-aliasinglifting scheme formechanical vibration fault feature extractionrdquoMechanical Systems and Signal Processing vol 23 no 5 pp1458ndash1473 2009

[22] R L Claypoole G M Davis W Sweldens and R G BaraniukldquoNonlinear wavelet transforms for image coding via liftingrdquoIEEE Transactions on Image Processing vol 12 no 12 pp 1449ndash1459 2003

[23] R L Claypoole R G Baraniuk and R D Nowak ldquoAdaptivewavelet transforms via liftingrdquo Report ECE TR-9304 1999httpsscholarshipriceeduhandle191119808

[24] B Appleton and H Talbot ldquoRecursive filtering of images withsymmetric extensionrdquo Signal Processing vol 85 no 8 pp 1546ndash1556 2005

[25] B Wohlberg and C M Brislawn ldquoSymmetric extension forlifted filter banks and obstructions to reversible implementa-tionrdquo Signal Processing vol 88 no 1 pp 131ndash145 2008

[26] L Duan L Zhang and J Chen ldquoBoundary treatment of liftingwavelet transform based on Volterra series model and its appli-cationrdquo Chinese Journal of Scientific Instrument vol 33 no 1pp 7ndash12 2012

[27] D Aiordachioaie E Ceanga R de Keyser and Y Naka ldquoDetec-tion and classification of non-linearities based on Volterrakernels processingrdquo Engineering Applications of Artificial Intel-ligence vol 14 no 4 pp 497ndash503 2001

[28] W Sweldens ldquoThe lifting scheme a custom-design constructionof biorthogonal waveletsrdquo Applied and Computational Har-monic Analysis vol 3 no 2 pp 186ndash200 1996

[29] W Sweldens ldquoThe Lifting Scheme a construction of secondgeneration waveletsrdquo SIAM Journal on Mathematical Analysisvol 29 no 2 pp 511ndash546 1998

[30] A Ben and T Hugues ldquoRecursive filtering of images with sym-metric extensionrdquo Signal Processing vol 85 no 8 pp 1546ndash15562005

[31] D Aiordachioaie E Ceanga R DeKeyser andYNaka ldquoDetec-tion and classification of non-linearities based on Volterrakernels processingrdquo Engineering Applications of Artificial Intel-ligence vol 14 no 4 pp 497ndash503 2001

[32] M B Kennel R Brown and H D I Abarbanel ldquoDeterminingembedding dimension for phase-space reconstruction using ageometrical constructionrdquo Physical Review A vol 45 no 6 pp3403ndash3411 1992

[33] J Zhang and X Xiao ldquoPredicting low-dimensional chaotic timeseries using Volterra adaptive filtersrdquo Acta Physica Sinica vol49 no 3 pp 403ndash408 1999

[34] J Hongkai H Zhengjia D Chendong and C Peng ldquoGear-box fault diagnosis using adaptive redundant Lifting SchemerdquoMechanical Systems and Signal Processing vol 20 no 8 pp1992ndash2006 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Shock and Vibration

1

2 3

4

(a)

Naturalgas

engineCrankshaft box

4th cylinder 3rd cylinder

2nd cylinder1st cylinderAccelerometer(in the bottom)

(b)

Figure 3 Illustration of experimental setup (a) data acquisition system A control panel B data acquisition box C accelerometer on theexhaust valve lidD exhaust valve and (b) diagram of reciprocating compressor

0 005 01 015 02 025

0

5

Periodic impact Periodic impact minus10

minus5

Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )

Figure 4 The time series vibrational signal of normal valve

closes the exhaust valve When the plunger moves up thecompression of cylinder opens the exhaust valve and closesthe intake valve Valve is composed of lifting limiter springdisc and seat of which spring is the most susceptible tofailure

To diagnose the valve failure in the 2nd cylinder anaccelerometer is placed on the exhaust valve lid A customerdesigned data acquisition system (model number MDES-5)as shown in Figure 3(a) is used to acquire the measurementsIt consists of a laptop and a data acquisition box configuredin a master-slave system The sampling rate is set as 16 kHzin this study Figures 4 and 5 show the time series vibrationalsignals near the exhaust valve of 2nd cylinder under normalcondition and abnormal condition respectively For compar-ison the amplitude of the vibration signal under abnormalcondition is greater than the one of the normal signal Takingthe RMS (root mean square) as the criterion the RMS of theabnormal signal is calculated as 4725msdotsminus2 which is greaterthan that of the normal signal (2227msdotsminus2) However theperiodic impact feature caused by the collision of the valvedisc and seat is difficult to observe due to strong backgroundnoise

For further analysis the proposed method based onVolterra series and undecimated lifting wavelet packet trans-form is used to process the signals The two ends of the orig-inal signal are extended and predicted by the Volterra seriesmodel Then the predicted signal is decomposed into four

0 005 01 015 02 025

05

1015

minus10

minus15

minus5

Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )

Figure 5 The time series vibrational signal of abnormal valve

1 20

50100

Level 1

270

9730

1 2 3 40

50100

Level 2

736 007 23296928

1 2 3 4 5 6 7 80

50100

Level 3

063 426 012 001 1259 10794039 3121

0 2 4 6 8 10 12 14 160

50100

Level 4

084 006 240 144 004 005 001 000 764 407 899 279 2262 1448 2883 575

Figure 6 Energy distribution of four-level subbands

layers by undecimated lifting wavelet packet transformwhere 119873 = 119873

1015840

= 4 From (11)ndash(13) the predictionoperator P and the update operator U can be obtained asP = [minus00625 05625 05625 minus00625] and U = [minus00313

02813 02813 minus00313] Finally the 16 subbands of thefourth level are obtained and the energy distribution of thesubbands at different levels is shown in Figure 6 After undec-imated lifting wavelet packet decomposition the majorityof the energy rests on the high frequency parts which

Shock and Vibration 7

minus8

minus6

minus4

minus2

02468

minus8

minus6

minus4

minus2

02468

003 004 005 006 007002 008Time t (s)

005

(a)

(b)

010 02 025015Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )A

mpl

itude

a(m

mmiddotsminus

2 )

Figure 7 The wavelet coefficient at selected subband and itsenvelope under normal condition

minus6

minus4

minus2

0246

minus8

minus6

minus4

minus2

024

86

003 004 005 006 007002 008Time t (s)

005

(a)

(b)

010 02 025015Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )A

mpl

itude

a(m

mmiddotsminus

2 )

Figure 8 The wavelet coefficient at selected subband and itsenvelope under abnormal condition

correspond to the detailed wavelet coefficients The subbandwith the highest energy ratio is selected as the bandof interestThewavelet coefficients and their envelopes from the selectedsubbands are shown in Figures 7 and 8 corresponding tothe normal condition and abnormal condition respectivelyFromboth figures the periodic impact features are visible andeasily identified

The signal in one period as shown by the red blockin Figures 7(a) and 8(a) is extracted for detailed analysisAccording to the working principle of exhaust valve the front

half curve from 0012 to 0047 sec represents the valve-closephase while the latter one from 0047 to 0082 sec shows theexhaust phase The exhaust valve closes at point A and thevalve opens again at point BUnder normal condition there isno chattering when normal valve closes because the stiffnessof spring is rigid enough to close tightly By contrast valvechatters significantly which is caused by the repeated collisionof valve and seat during the time period 119879 (indicating theaging and stretch decline of spring in the valve) The valvewith aged spring cannot close tightly resulting in gas leak andefficiency degradation When this compressor is stopped andchecked the spring failure is validated by visual inspectionand the exhaust valve is maintained Therefore the pre-sented incipient fault diagnosis method avoids accidents andprevents huge production loss

In addition the energy of the selected band signal isused to evaluate the performance of the presented methodFigure 9(a) shows the energies of the selected band sig-nals under four different scenarios (I) undecimated LWPT(ULWPT) under normal machinery condition (II) undec-imated LWPT (ULWPT) under abnormal machinery con-dition (III) Volterra series assisted undecimated LWPT(V-ULWPT) under normal machinery condition and (IV)Volterra series assisted undecimated LWPT (V-ULWPT)under abnormal machinery condition From the analysisresults it is found that the normal and abnormal machineryconditions can be differentiated according to energy ofselected subband using undecimated lifting wavelet packettransform Since Volterra series assisted ULWPT suppressesthe end distortion the energy of selected subband using V-ULWPT is less than that using ULWPT However the energyratio of abnormal to normal conditions using V-ULWPTis larger than that using ULWPT as shown in Figure 9(b)thus validating the effectiveness of presented Volterra seriesassisted lifting wavelet packet transform method

The computational efficiency of the developed techniquehas also been evaluated on a desktop (Lenovo YangtianT4900d model Lenovo Inc Beijing China) with a 33 GHzCPU and 4GB memory It takes approximately 119 secondsto process a data string containing 4096 data points whichis equivalent to processing 0256 seconds data length under a16 kHz sampling rate Thus the presented method is applica-ble for online machinery diagnosis in practical applications

5 Conclusions

An enhanced weak feature extraction approach that com-bines Volterra series model and undecimated lifting waveletpacket transform has been presented for incipient fault detec-tion The analysis results show that Volterra series assistedundecimated lifting wavelet packet transform eliminates thefrequency aliasing and end distortion issues in conventionallifting wavelet packet transform method thus it is significantto machinery incipient defect diagnosis especially for weakimpact feature extraction From the engineering applicationperspective the weak impact signals caused by the fault ofa valve in reciprocating compressor are analyzed using thepresented method and the early spring failure is detectedfrom the extractedweak features using the presentedmethod

8 Shock and Vibration

Ener

gy (J

)

I normal ULWPT II abnormal ULWPT

I II III IV

III normal V-ULWPT IV abnormal V-ULWPT

600E + 03

550E + 03

500E + 03

450E + 03

400E + 03

350E + 03

300E + 03

250E + 03

200E + 03

150E + 03

100E + 03

500E + 02

000E + 00

428E + 03

136E + 03

475E + 03

219E + 03

(a)

I II

I ULWPT abnormalnormalII V-ULWPT abnormalnormal

216

314

0

05

1

15

2

25

3

35

4

Ener

gy ra

tio

(b)

Figure 9 Energy comparison among different conditions (a) energy comparison chart (b) energy ratio chart

Conflict of Interests

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

Acknowledgments

This research acknowledges the financial support providedby National Science foundation of China (no 51504274)Science Foundation ofChinaUniversity of PetroleumBeijing(nos 2462014YJRC039 and 2462015YQ0403) and BeijingMunicipal Education Commission Cooperative Construc-tion Project Valuable comments from anonymous reviewersare greatly appreciated

References

[1] V Venkatasubramanian R Rengaswamy K Yin and S NKavuri ldquoA review of process fault detection and diagnosis partI quantitativemodel-basedmethodsrdquoComputers and ChemicalEngineering vol 27 no 3 pp 293ndash311 2003

[2] M Liang and I S Bozchalooi ldquoAn energy operator approachto joint application of amplitude and frequency-demodulationsfor bearing fault detectionrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1473ndash1494 2010

[3] Z Zhong Z Jiang Y Long and X Zhan ldquoAnalysis on thenoise for the different gearboxes of the heavy truckrdquo Shock andVibration vol 2015 Article ID 476460 5 pages 2015

[4] W J Wang ldquoWavelet transform in vibration analysis formechanical fault diagnosisrdquo Shock and Vibration vol 3 no 1pp 17ndash26 1996

[5] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[6] N Baydar and A Ball ldquoA comparative study of acoustic andvibration signals in detection of gear failures usingWigner-VilledistributionrdquoMechanical Systems and Signal Processing vol 15no 6 pp 1091ndash1107 2001

[7] W-X Yang and P W Tse ldquoDevelopment of an advanced noisereductionmethod for vibration analysis based on singular valuedecompositionrdquo NDT amp E International vol 36 no 6 pp 419ndash432 2003

[8] H Hassanpour A Zehtabian and S J Sadati ldquoTime domainsignal enhancement based on an optimized singular vectordenoising algorithmrdquoDigital Signal Processing vol 22 no 5 pp786ndash794 2012

[9] J Wang R X Gao and R Yan ldquoIntegration of EEMD and ICAfor wind turbine gearbox diagnosisrdquoWind Energy vol 17 no 5pp 757ndash773 2014

[10] A Sadhu and B Hazra ldquoA novel damage detection algorithmusing time-series analysis-based blind source separationrdquo Shockand Vibration vol 20 no 3 pp 423ndash438 2013

[11] Z K Peng and F L Chu ldquoApplication of the wavelet trans-form in machine condition monitoring and fault diagnosticsa review with bibliographyrdquo Mechanical Systems and SignalProcessing vol 18 no 2 pp 199ndash221 2004

[12] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosis ofrotary machines a review with applicationsrdquo Signal Processingvol 96 pp 1ndash15 2014

[13] N Saravanan and K I Ramachandran ldquoIncipient gear box faultdiagnosis using discrete wavelet transform (DWT) for featureextraction and classification using artificial neural network(ANN)rdquo Expert Systems with Applications vol 37 no 6 pp4168ndash4181 2010

[14] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weaksignature detection method and its application on rolling ele-ment bearing prognosticsrdquo Journal of Sound and Vibration vol289 no 4-5 pp 1066ndash1090 2006

[15] J Wang R X Gao and R Yan ldquoMulti-scale enveloping orderspectrogram for rotatingmachine health diagnosisrdquoMechanicalSystems and Signal Processing vol 46 no 1 pp 28ndash44 2014

Shock and Vibration 9

[16] Y-L Chen P-L Zhang B Li and D-H Wu ldquoDenoising ofmechanical vibration signals using quantum-inspired adaptivewavelet shrinkagerdquo Shock and Vibration vol 2014 Article ID848097 7 pages 2014

[17] Y Pan J Chen and X Li ldquoBearing performance degradationassessment based on lifting wavelet packet decomposition andfuzzy c-meansrdquo Mechanical Systems and Signal Processing vol24 no 2 pp 559ndash566 2010

[18] R ZhouWBaoN Li XHuang andDYu ldquoMechanical equip-ment fault diagnosis based on redundant second generationwavelet packet transformrdquo Digital Signal Processing A ReviewJournal vol 20 no 1 pp 276ndash288 2010

[19] J Yuan Z He and Y Zi ldquoGear fault detection using customizedmultiwavelet lifting schemesrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1509ndash1528 2010

[20] Y He X Chen J Xiang and Z He ldquoAdaptive multiresolutionfinite element method based on second generation waveletsrdquoFinite Elements in Analysis and Design vol 43 no 6-7 pp 566ndash579 2007

[21] W Bao R Zhou J Yang D Yu and N Li ldquoAnti-aliasinglifting scheme formechanical vibration fault feature extractionrdquoMechanical Systems and Signal Processing vol 23 no 5 pp1458ndash1473 2009

[22] R L Claypoole G M Davis W Sweldens and R G BaraniukldquoNonlinear wavelet transforms for image coding via liftingrdquoIEEE Transactions on Image Processing vol 12 no 12 pp 1449ndash1459 2003

[23] R L Claypoole R G Baraniuk and R D Nowak ldquoAdaptivewavelet transforms via liftingrdquo Report ECE TR-9304 1999httpsscholarshipriceeduhandle191119808

[24] B Appleton and H Talbot ldquoRecursive filtering of images withsymmetric extensionrdquo Signal Processing vol 85 no 8 pp 1546ndash1556 2005

[25] B Wohlberg and C M Brislawn ldquoSymmetric extension forlifted filter banks and obstructions to reversible implementa-tionrdquo Signal Processing vol 88 no 1 pp 131ndash145 2008

[26] L Duan L Zhang and J Chen ldquoBoundary treatment of liftingwavelet transform based on Volterra series model and its appli-cationrdquo Chinese Journal of Scientific Instrument vol 33 no 1pp 7ndash12 2012

[27] D Aiordachioaie E Ceanga R de Keyser and Y Naka ldquoDetec-tion and classification of non-linearities based on Volterrakernels processingrdquo Engineering Applications of Artificial Intel-ligence vol 14 no 4 pp 497ndash503 2001

[28] W Sweldens ldquoThe lifting scheme a custom-design constructionof biorthogonal waveletsrdquo Applied and Computational Har-monic Analysis vol 3 no 2 pp 186ndash200 1996

[29] W Sweldens ldquoThe Lifting Scheme a construction of secondgeneration waveletsrdquo SIAM Journal on Mathematical Analysisvol 29 no 2 pp 511ndash546 1998

[30] A Ben and T Hugues ldquoRecursive filtering of images with sym-metric extensionrdquo Signal Processing vol 85 no 8 pp 1546ndash15562005

[31] D Aiordachioaie E Ceanga R DeKeyser andYNaka ldquoDetec-tion and classification of non-linearities based on Volterrakernels processingrdquo Engineering Applications of Artificial Intel-ligence vol 14 no 4 pp 497ndash503 2001

[32] M B Kennel R Brown and H D I Abarbanel ldquoDeterminingembedding dimension for phase-space reconstruction using ageometrical constructionrdquo Physical Review A vol 45 no 6 pp3403ndash3411 1992

[33] J Zhang and X Xiao ldquoPredicting low-dimensional chaotic timeseries using Volterra adaptive filtersrdquo Acta Physica Sinica vol49 no 3 pp 403ndash408 1999

[34] J Hongkai H Zhengjia D Chendong and C Peng ldquoGear-box fault diagnosis using adaptive redundant Lifting SchemerdquoMechanical Systems and Signal Processing vol 20 no 8 pp1992ndash2006 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 7

minus8

minus6

minus4

minus2

02468

minus8

minus6

minus4

minus2

02468

003 004 005 006 007002 008Time t (s)

005

(a)

(b)

010 02 025015Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )A

mpl

itude

a(m

mmiddotsminus

2 )

Figure 7 The wavelet coefficient at selected subband and itsenvelope under normal condition

minus6

minus4

minus2

0246

minus8

minus6

minus4

minus2

024

86

003 004 005 006 007002 008Time t (s)

005

(a)

(b)

010 02 025015Time t (s)

Am

plitu

dea

(mmmiddotsminus

2 )A

mpl

itude

a(m

mmiddotsminus

2 )

Figure 8 The wavelet coefficient at selected subband and itsenvelope under abnormal condition

correspond to the detailed wavelet coefficients The subbandwith the highest energy ratio is selected as the bandof interestThewavelet coefficients and their envelopes from the selectedsubbands are shown in Figures 7 and 8 corresponding tothe normal condition and abnormal condition respectivelyFromboth figures the periodic impact features are visible andeasily identified

The signal in one period as shown by the red blockin Figures 7(a) and 8(a) is extracted for detailed analysisAccording to the working principle of exhaust valve the front

half curve from 0012 to 0047 sec represents the valve-closephase while the latter one from 0047 to 0082 sec shows theexhaust phase The exhaust valve closes at point A and thevalve opens again at point BUnder normal condition there isno chattering when normal valve closes because the stiffnessof spring is rigid enough to close tightly By contrast valvechatters significantly which is caused by the repeated collisionof valve and seat during the time period 119879 (indicating theaging and stretch decline of spring in the valve) The valvewith aged spring cannot close tightly resulting in gas leak andefficiency degradation When this compressor is stopped andchecked the spring failure is validated by visual inspectionand the exhaust valve is maintained Therefore the pre-sented incipient fault diagnosis method avoids accidents andprevents huge production loss

In addition the energy of the selected band signal isused to evaluate the performance of the presented methodFigure 9(a) shows the energies of the selected band sig-nals under four different scenarios (I) undecimated LWPT(ULWPT) under normal machinery condition (II) undec-imated LWPT (ULWPT) under abnormal machinery con-dition (III) Volterra series assisted undecimated LWPT(V-ULWPT) under normal machinery condition and (IV)Volterra series assisted undecimated LWPT (V-ULWPT)under abnormal machinery condition From the analysisresults it is found that the normal and abnormal machineryconditions can be differentiated according to energy ofselected subband using undecimated lifting wavelet packettransform Since Volterra series assisted ULWPT suppressesthe end distortion the energy of selected subband using V-ULWPT is less than that using ULWPT However the energyratio of abnormal to normal conditions using V-ULWPTis larger than that using ULWPT as shown in Figure 9(b)thus validating the effectiveness of presented Volterra seriesassisted lifting wavelet packet transform method

The computational efficiency of the developed techniquehas also been evaluated on a desktop (Lenovo YangtianT4900d model Lenovo Inc Beijing China) with a 33 GHzCPU and 4GB memory It takes approximately 119 secondsto process a data string containing 4096 data points whichis equivalent to processing 0256 seconds data length under a16 kHz sampling rate Thus the presented method is applica-ble for online machinery diagnosis in practical applications

5 Conclusions

An enhanced weak feature extraction approach that com-bines Volterra series model and undecimated lifting waveletpacket transform has been presented for incipient fault detec-tion The analysis results show that Volterra series assistedundecimated lifting wavelet packet transform eliminates thefrequency aliasing and end distortion issues in conventionallifting wavelet packet transform method thus it is significantto machinery incipient defect diagnosis especially for weakimpact feature extraction From the engineering applicationperspective the weak impact signals caused by the fault ofa valve in reciprocating compressor are analyzed using thepresented method and the early spring failure is detectedfrom the extractedweak features using the presentedmethod

8 Shock and Vibration

Ener

gy (J

)

I normal ULWPT II abnormal ULWPT

I II III IV

III normal V-ULWPT IV abnormal V-ULWPT

600E + 03

550E + 03

500E + 03

450E + 03

400E + 03

350E + 03

300E + 03

250E + 03

200E + 03

150E + 03

100E + 03

500E + 02

000E + 00

428E + 03

136E + 03

475E + 03

219E + 03

(a)

I II

I ULWPT abnormalnormalII V-ULWPT abnormalnormal

216

314

0

05

1

15

2

25

3

35

4

Ener

gy ra

tio

(b)

Figure 9 Energy comparison among different conditions (a) energy comparison chart (b) energy ratio chart

Conflict of Interests

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

Acknowledgments

This research acknowledges the financial support providedby National Science foundation of China (no 51504274)Science Foundation ofChinaUniversity of PetroleumBeijing(nos 2462014YJRC039 and 2462015YQ0403) and BeijingMunicipal Education Commission Cooperative Construc-tion Project Valuable comments from anonymous reviewersare greatly appreciated

References

[1] V Venkatasubramanian R Rengaswamy K Yin and S NKavuri ldquoA review of process fault detection and diagnosis partI quantitativemodel-basedmethodsrdquoComputers and ChemicalEngineering vol 27 no 3 pp 293ndash311 2003

[2] M Liang and I S Bozchalooi ldquoAn energy operator approachto joint application of amplitude and frequency-demodulationsfor bearing fault detectionrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1473ndash1494 2010

[3] Z Zhong Z Jiang Y Long and X Zhan ldquoAnalysis on thenoise for the different gearboxes of the heavy truckrdquo Shock andVibration vol 2015 Article ID 476460 5 pages 2015

[4] W J Wang ldquoWavelet transform in vibration analysis formechanical fault diagnosisrdquo Shock and Vibration vol 3 no 1pp 17ndash26 1996

[5] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[6] N Baydar and A Ball ldquoA comparative study of acoustic andvibration signals in detection of gear failures usingWigner-VilledistributionrdquoMechanical Systems and Signal Processing vol 15no 6 pp 1091ndash1107 2001

[7] W-X Yang and P W Tse ldquoDevelopment of an advanced noisereductionmethod for vibration analysis based on singular valuedecompositionrdquo NDT amp E International vol 36 no 6 pp 419ndash432 2003

[8] H Hassanpour A Zehtabian and S J Sadati ldquoTime domainsignal enhancement based on an optimized singular vectordenoising algorithmrdquoDigital Signal Processing vol 22 no 5 pp786ndash794 2012

[9] J Wang R X Gao and R Yan ldquoIntegration of EEMD and ICAfor wind turbine gearbox diagnosisrdquoWind Energy vol 17 no 5pp 757ndash773 2014

[10] A Sadhu and B Hazra ldquoA novel damage detection algorithmusing time-series analysis-based blind source separationrdquo Shockand Vibration vol 20 no 3 pp 423ndash438 2013

[11] Z K Peng and F L Chu ldquoApplication of the wavelet trans-form in machine condition monitoring and fault diagnosticsa review with bibliographyrdquo Mechanical Systems and SignalProcessing vol 18 no 2 pp 199ndash221 2004

[12] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosis ofrotary machines a review with applicationsrdquo Signal Processingvol 96 pp 1ndash15 2014

[13] N Saravanan and K I Ramachandran ldquoIncipient gear box faultdiagnosis using discrete wavelet transform (DWT) for featureextraction and classification using artificial neural network(ANN)rdquo Expert Systems with Applications vol 37 no 6 pp4168ndash4181 2010

[14] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weaksignature detection method and its application on rolling ele-ment bearing prognosticsrdquo Journal of Sound and Vibration vol289 no 4-5 pp 1066ndash1090 2006

[15] J Wang R X Gao and R Yan ldquoMulti-scale enveloping orderspectrogram for rotatingmachine health diagnosisrdquoMechanicalSystems and Signal Processing vol 46 no 1 pp 28ndash44 2014

Shock and Vibration 9

[16] Y-L Chen P-L Zhang B Li and D-H Wu ldquoDenoising ofmechanical vibration signals using quantum-inspired adaptivewavelet shrinkagerdquo Shock and Vibration vol 2014 Article ID848097 7 pages 2014

[17] Y Pan J Chen and X Li ldquoBearing performance degradationassessment based on lifting wavelet packet decomposition andfuzzy c-meansrdquo Mechanical Systems and Signal Processing vol24 no 2 pp 559ndash566 2010

[18] R ZhouWBaoN Li XHuang andDYu ldquoMechanical equip-ment fault diagnosis based on redundant second generationwavelet packet transformrdquo Digital Signal Processing A ReviewJournal vol 20 no 1 pp 276ndash288 2010

[19] J Yuan Z He and Y Zi ldquoGear fault detection using customizedmultiwavelet lifting schemesrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1509ndash1528 2010

[20] Y He X Chen J Xiang and Z He ldquoAdaptive multiresolutionfinite element method based on second generation waveletsrdquoFinite Elements in Analysis and Design vol 43 no 6-7 pp 566ndash579 2007

[21] W Bao R Zhou J Yang D Yu and N Li ldquoAnti-aliasinglifting scheme formechanical vibration fault feature extractionrdquoMechanical Systems and Signal Processing vol 23 no 5 pp1458ndash1473 2009

[22] R L Claypoole G M Davis W Sweldens and R G BaraniukldquoNonlinear wavelet transforms for image coding via liftingrdquoIEEE Transactions on Image Processing vol 12 no 12 pp 1449ndash1459 2003

[23] R L Claypoole R G Baraniuk and R D Nowak ldquoAdaptivewavelet transforms via liftingrdquo Report ECE TR-9304 1999httpsscholarshipriceeduhandle191119808

[24] B Appleton and H Talbot ldquoRecursive filtering of images withsymmetric extensionrdquo Signal Processing vol 85 no 8 pp 1546ndash1556 2005

[25] B Wohlberg and C M Brislawn ldquoSymmetric extension forlifted filter banks and obstructions to reversible implementa-tionrdquo Signal Processing vol 88 no 1 pp 131ndash145 2008

[26] L Duan L Zhang and J Chen ldquoBoundary treatment of liftingwavelet transform based on Volterra series model and its appli-cationrdquo Chinese Journal of Scientific Instrument vol 33 no 1pp 7ndash12 2012

[27] D Aiordachioaie E Ceanga R de Keyser and Y Naka ldquoDetec-tion and classification of non-linearities based on Volterrakernels processingrdquo Engineering Applications of Artificial Intel-ligence vol 14 no 4 pp 497ndash503 2001

[28] W Sweldens ldquoThe lifting scheme a custom-design constructionof biorthogonal waveletsrdquo Applied and Computational Har-monic Analysis vol 3 no 2 pp 186ndash200 1996

[29] W Sweldens ldquoThe Lifting Scheme a construction of secondgeneration waveletsrdquo SIAM Journal on Mathematical Analysisvol 29 no 2 pp 511ndash546 1998

[30] A Ben and T Hugues ldquoRecursive filtering of images with sym-metric extensionrdquo Signal Processing vol 85 no 8 pp 1546ndash15562005

[31] D Aiordachioaie E Ceanga R DeKeyser andYNaka ldquoDetec-tion and classification of non-linearities based on Volterrakernels processingrdquo Engineering Applications of Artificial Intel-ligence vol 14 no 4 pp 497ndash503 2001

[32] M B Kennel R Brown and H D I Abarbanel ldquoDeterminingembedding dimension for phase-space reconstruction using ageometrical constructionrdquo Physical Review A vol 45 no 6 pp3403ndash3411 1992

[33] J Zhang and X Xiao ldquoPredicting low-dimensional chaotic timeseries using Volterra adaptive filtersrdquo Acta Physica Sinica vol49 no 3 pp 403ndash408 1999

[34] J Hongkai H Zhengjia D Chendong and C Peng ldquoGear-box fault diagnosis using adaptive redundant Lifting SchemerdquoMechanical Systems and Signal Processing vol 20 no 8 pp1992ndash2006 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Shock and Vibration

Ener

gy (J

)

I normal ULWPT II abnormal ULWPT

I II III IV

III normal V-ULWPT IV abnormal V-ULWPT

600E + 03

550E + 03

500E + 03

450E + 03

400E + 03

350E + 03

300E + 03

250E + 03

200E + 03

150E + 03

100E + 03

500E + 02

000E + 00

428E + 03

136E + 03

475E + 03

219E + 03

(a)

I II

I ULWPT abnormalnormalII V-ULWPT abnormalnormal

216

314

0

05

1

15

2

25

3

35

4

Ener

gy ra

tio

(b)

Figure 9 Energy comparison among different conditions (a) energy comparison chart (b) energy ratio chart

Conflict of Interests

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

Acknowledgments

This research acknowledges the financial support providedby National Science foundation of China (no 51504274)Science Foundation ofChinaUniversity of PetroleumBeijing(nos 2462014YJRC039 and 2462015YQ0403) and BeijingMunicipal Education Commission Cooperative Construc-tion Project Valuable comments from anonymous reviewersare greatly appreciated

References

[1] V Venkatasubramanian R Rengaswamy K Yin and S NKavuri ldquoA review of process fault detection and diagnosis partI quantitativemodel-basedmethodsrdquoComputers and ChemicalEngineering vol 27 no 3 pp 293ndash311 2003

[2] M Liang and I S Bozchalooi ldquoAn energy operator approachto joint application of amplitude and frequency-demodulationsfor bearing fault detectionrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1473ndash1494 2010

[3] Z Zhong Z Jiang Y Long and X Zhan ldquoAnalysis on thenoise for the different gearboxes of the heavy truckrdquo Shock andVibration vol 2015 Article ID 476460 5 pages 2015

[4] W J Wang ldquoWavelet transform in vibration analysis formechanical fault diagnosisrdquo Shock and Vibration vol 3 no 1pp 17ndash26 1996

[5] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[6] N Baydar and A Ball ldquoA comparative study of acoustic andvibration signals in detection of gear failures usingWigner-VilledistributionrdquoMechanical Systems and Signal Processing vol 15no 6 pp 1091ndash1107 2001

[7] W-X Yang and P W Tse ldquoDevelopment of an advanced noisereductionmethod for vibration analysis based on singular valuedecompositionrdquo NDT amp E International vol 36 no 6 pp 419ndash432 2003

[8] H Hassanpour A Zehtabian and S J Sadati ldquoTime domainsignal enhancement based on an optimized singular vectordenoising algorithmrdquoDigital Signal Processing vol 22 no 5 pp786ndash794 2012

[9] J Wang R X Gao and R Yan ldquoIntegration of EEMD and ICAfor wind turbine gearbox diagnosisrdquoWind Energy vol 17 no 5pp 757ndash773 2014

[10] A Sadhu and B Hazra ldquoA novel damage detection algorithmusing time-series analysis-based blind source separationrdquo Shockand Vibration vol 20 no 3 pp 423ndash438 2013

[11] Z K Peng and F L Chu ldquoApplication of the wavelet trans-form in machine condition monitoring and fault diagnosticsa review with bibliographyrdquo Mechanical Systems and SignalProcessing vol 18 no 2 pp 199ndash221 2004

[12] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosis ofrotary machines a review with applicationsrdquo Signal Processingvol 96 pp 1ndash15 2014

[13] N Saravanan and K I Ramachandran ldquoIncipient gear box faultdiagnosis using discrete wavelet transform (DWT) for featureextraction and classification using artificial neural network(ANN)rdquo Expert Systems with Applications vol 37 no 6 pp4168ndash4181 2010

[14] H Qiu J Lee J Lin and G Yu ldquoWavelet filter-based weaksignature detection method and its application on rolling ele-ment bearing prognosticsrdquo Journal of Sound and Vibration vol289 no 4-5 pp 1066ndash1090 2006

[15] J Wang R X Gao and R Yan ldquoMulti-scale enveloping orderspectrogram for rotatingmachine health diagnosisrdquoMechanicalSystems and Signal Processing vol 46 no 1 pp 28ndash44 2014

Shock and Vibration 9

[16] Y-L Chen P-L Zhang B Li and D-H Wu ldquoDenoising ofmechanical vibration signals using quantum-inspired adaptivewavelet shrinkagerdquo Shock and Vibration vol 2014 Article ID848097 7 pages 2014

[17] Y Pan J Chen and X Li ldquoBearing performance degradationassessment based on lifting wavelet packet decomposition andfuzzy c-meansrdquo Mechanical Systems and Signal Processing vol24 no 2 pp 559ndash566 2010

[18] R ZhouWBaoN Li XHuang andDYu ldquoMechanical equip-ment fault diagnosis based on redundant second generationwavelet packet transformrdquo Digital Signal Processing A ReviewJournal vol 20 no 1 pp 276ndash288 2010

[19] J Yuan Z He and Y Zi ldquoGear fault detection using customizedmultiwavelet lifting schemesrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1509ndash1528 2010

[20] Y He X Chen J Xiang and Z He ldquoAdaptive multiresolutionfinite element method based on second generation waveletsrdquoFinite Elements in Analysis and Design vol 43 no 6-7 pp 566ndash579 2007

[21] W Bao R Zhou J Yang D Yu and N Li ldquoAnti-aliasinglifting scheme formechanical vibration fault feature extractionrdquoMechanical Systems and Signal Processing vol 23 no 5 pp1458ndash1473 2009

[22] R L Claypoole G M Davis W Sweldens and R G BaraniukldquoNonlinear wavelet transforms for image coding via liftingrdquoIEEE Transactions on Image Processing vol 12 no 12 pp 1449ndash1459 2003

[23] R L Claypoole R G Baraniuk and R D Nowak ldquoAdaptivewavelet transforms via liftingrdquo Report ECE TR-9304 1999httpsscholarshipriceeduhandle191119808

[24] B Appleton and H Talbot ldquoRecursive filtering of images withsymmetric extensionrdquo Signal Processing vol 85 no 8 pp 1546ndash1556 2005

[25] B Wohlberg and C M Brislawn ldquoSymmetric extension forlifted filter banks and obstructions to reversible implementa-tionrdquo Signal Processing vol 88 no 1 pp 131ndash145 2008

[26] L Duan L Zhang and J Chen ldquoBoundary treatment of liftingwavelet transform based on Volterra series model and its appli-cationrdquo Chinese Journal of Scientific Instrument vol 33 no 1pp 7ndash12 2012

[27] D Aiordachioaie E Ceanga R de Keyser and Y Naka ldquoDetec-tion and classification of non-linearities based on Volterrakernels processingrdquo Engineering Applications of Artificial Intel-ligence vol 14 no 4 pp 497ndash503 2001

[28] W Sweldens ldquoThe lifting scheme a custom-design constructionof biorthogonal waveletsrdquo Applied and Computational Har-monic Analysis vol 3 no 2 pp 186ndash200 1996

[29] W Sweldens ldquoThe Lifting Scheme a construction of secondgeneration waveletsrdquo SIAM Journal on Mathematical Analysisvol 29 no 2 pp 511ndash546 1998

[30] A Ben and T Hugues ldquoRecursive filtering of images with sym-metric extensionrdquo Signal Processing vol 85 no 8 pp 1546ndash15562005

[31] D Aiordachioaie E Ceanga R DeKeyser andYNaka ldquoDetec-tion and classification of non-linearities based on Volterrakernels processingrdquo Engineering Applications of Artificial Intel-ligence vol 14 no 4 pp 497ndash503 2001

[32] M B Kennel R Brown and H D I Abarbanel ldquoDeterminingembedding dimension for phase-space reconstruction using ageometrical constructionrdquo Physical Review A vol 45 no 6 pp3403ndash3411 1992

[33] J Zhang and X Xiao ldquoPredicting low-dimensional chaotic timeseries using Volterra adaptive filtersrdquo Acta Physica Sinica vol49 no 3 pp 403ndash408 1999

[34] J Hongkai H Zhengjia D Chendong and C Peng ldquoGear-box fault diagnosis using adaptive redundant Lifting SchemerdquoMechanical Systems and Signal Processing vol 20 no 8 pp1992ndash2006 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 9

[16] Y-L Chen P-L Zhang B Li and D-H Wu ldquoDenoising ofmechanical vibration signals using quantum-inspired adaptivewavelet shrinkagerdquo Shock and Vibration vol 2014 Article ID848097 7 pages 2014

[17] Y Pan J Chen and X Li ldquoBearing performance degradationassessment based on lifting wavelet packet decomposition andfuzzy c-meansrdquo Mechanical Systems and Signal Processing vol24 no 2 pp 559ndash566 2010

[18] R ZhouWBaoN Li XHuang andDYu ldquoMechanical equip-ment fault diagnosis based on redundant second generationwavelet packet transformrdquo Digital Signal Processing A ReviewJournal vol 20 no 1 pp 276ndash288 2010

[19] J Yuan Z He and Y Zi ldquoGear fault detection using customizedmultiwavelet lifting schemesrdquo Mechanical Systems and SignalProcessing vol 24 no 5 pp 1509ndash1528 2010

[20] Y He X Chen J Xiang and Z He ldquoAdaptive multiresolutionfinite element method based on second generation waveletsrdquoFinite Elements in Analysis and Design vol 43 no 6-7 pp 566ndash579 2007

[21] W Bao R Zhou J Yang D Yu and N Li ldquoAnti-aliasinglifting scheme formechanical vibration fault feature extractionrdquoMechanical Systems and Signal Processing vol 23 no 5 pp1458ndash1473 2009

[22] R L Claypoole G M Davis W Sweldens and R G BaraniukldquoNonlinear wavelet transforms for image coding via liftingrdquoIEEE Transactions on Image Processing vol 12 no 12 pp 1449ndash1459 2003

[23] R L Claypoole R G Baraniuk and R D Nowak ldquoAdaptivewavelet transforms via liftingrdquo Report ECE TR-9304 1999httpsscholarshipriceeduhandle191119808

[24] B Appleton and H Talbot ldquoRecursive filtering of images withsymmetric extensionrdquo Signal Processing vol 85 no 8 pp 1546ndash1556 2005

[25] B Wohlberg and C M Brislawn ldquoSymmetric extension forlifted filter banks and obstructions to reversible implementa-tionrdquo Signal Processing vol 88 no 1 pp 131ndash145 2008

[26] L Duan L Zhang and J Chen ldquoBoundary treatment of liftingwavelet transform based on Volterra series model and its appli-cationrdquo Chinese Journal of Scientific Instrument vol 33 no 1pp 7ndash12 2012

[27] D Aiordachioaie E Ceanga R de Keyser and Y Naka ldquoDetec-tion and classification of non-linearities based on Volterrakernels processingrdquo Engineering Applications of Artificial Intel-ligence vol 14 no 4 pp 497ndash503 2001

[28] W Sweldens ldquoThe lifting scheme a custom-design constructionof biorthogonal waveletsrdquo Applied and Computational Har-monic Analysis vol 3 no 2 pp 186ndash200 1996

[29] W Sweldens ldquoThe Lifting Scheme a construction of secondgeneration waveletsrdquo SIAM Journal on Mathematical Analysisvol 29 no 2 pp 511ndash546 1998

[30] A Ben and T Hugues ldquoRecursive filtering of images with sym-metric extensionrdquo Signal Processing vol 85 no 8 pp 1546ndash15562005

[31] D Aiordachioaie E Ceanga R DeKeyser andYNaka ldquoDetec-tion and classification of non-linearities based on Volterrakernels processingrdquo Engineering Applications of Artificial Intel-ligence vol 14 no 4 pp 497ndash503 2001

[32] M B Kennel R Brown and H D I Abarbanel ldquoDeterminingembedding dimension for phase-space reconstruction using ageometrical constructionrdquo Physical Review A vol 45 no 6 pp3403ndash3411 1992

[33] J Zhang and X Xiao ldquoPredicting low-dimensional chaotic timeseries using Volterra adaptive filtersrdquo Acta Physica Sinica vol49 no 3 pp 403ndash408 1999

[34] J Hongkai H Zhengjia D Chendong and C Peng ldquoGear-box fault diagnosis using adaptive redundant Lifting SchemerdquoMechanical Systems and Signal Processing vol 20 no 8 pp1992ndash2006 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of