research article multicrack localization in rotors based

Research ArticleMulticrack Localization in Rotors Based on ProperOrthogonal Decomposition Using Fractal Dimension andGapped Smoothing Method

Zhiwen Lu1 Dawei Dong1 Shancheng Cao2 Huajiang Ouyang23 and Chunrong Hua1

1School of Mechanical Engineering Southwest Jiaotong University Chengdu Sichuan 610031 China2School of Engineering University of Liverpool Liverpool L69 3GH UK3The State Key Laboratory of Structural Analysis for Industrial Equipment Dalian University of Technology DalianLiaoning 116023 China

Correspondence should be addressed to Chunrong Hua hcrongswjtucn

Received 30 May 2016 Revised 2 August 2016 Accepted 4 September 2016

Academic Editor Marcello Vanali

Copyright copy 2016 Zhiwen Lu 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

Multicrack localization in operating rotor systems is still a challenge today Focusing on this challenge a new approach basedon proper orthogonal decomposition (POD) is proposed for multicrack localization in rotors A two-disc rotor-bearing systemwith breathing cracks is established by the finite element method and simulated sensors are distributed along the rotor to obtainthe steady-state transverse responses required by POD Based on the discontinuities introduced in the proper orthogonal modes(POMs) at the locations of cracks the characteristic POM (CPOM) which is sensitive to crack locations and robust to noise isselected for cracks localization Instead of using the CPOMdirectly due to its difficulty to localize incipient cracks damage indexesusing fractal dimension (FD) and gapped smoothing method (GSM) are adopted in order to extract the locations more efficientlyThe method proposed in this work is validated to be effective for multicrack localization in rotors by numerical experiments onrotors in different crack configuration cases considering the effects of noise In addition the feasibility of using fewer sensors is alsoinvestigated

1 Introduction

Rotors are one of the most important components of rotatingmachines which are widely used in many engineering fieldssuch as turbines generators and aeroengines Cracks inrotors are the most critical and fundamental damage whichmay lead to a sudden and catastrophic failure of equipmentSo it is of vital significance to identify these cracks in orderto reduce maintenance cost and avoid failure of a rotatingmachine In view of the importance crack identification inrotors has been the focus of many investigations in recentdecades and numerous papers have been published [1ndash9]but crack localization is still a challenge for operating rotorsystems

A brief review of the relevant studies of crack localizationin rotors is given firstly The localization methods can beclassified as model-based and non-model-based methods It

should be noted that model-based methods defined here arethe methods which require a mathematical representationof the system under study for example a partial differentialequation of motion of a rotor as a beam or mass and stiffnessmatrices of a finite element model of a rotor

When it comes to model-based methods in crack iden-tification approaches based on equivalent crack forces willbe mentioned at the first beginning Equivalent crack forcemethods consider the effects of cracks as equivalent forcesapplied in intact systems They have been adopted to iden-tify the location and depth of cracks in rotors by manyresearchers such as Pennacchi et al [10] Lees et al [11]and Sekhar [12] There are also some other model-basedmethods Dong et al [13] presented a method based onintersection of the contour curves of the first three naturalfrequencies obtained from a rotor modelled by wavelet finiteelement method to determine the crack location and depth

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 2375859 17 pageshttpdxdoiorg10115520162375859

2 Shock and Vibration

Seibold andWeinert [14] proposed a method in time domainbased on a bank of Extended Kalman Filters to realize thecrack location and depth identification for an operating rotorMethods based on pattern recognition can be model-basedwhen the samples for training are obtained from mathe-matical models rather than experiments Their main ideasare extracting features sensitive to crack parameters whichthen are trained by artificial intellectual methods or machinelearning methods to obtain the relationship between thecrack parameters and features and then matching the mea-sured features with the established relationship to determinethe crack parameters Zapico-Valle et al [15] adopted the arti-ficial neural network which trained by samples gathered froma finite element model to identify crack location and depth ofrotors Soffker et al [16] compared a modern model-basedtechnique based on a proportional-integral observer witha signal-based technique based on support vector machineusing features extracted from wavelets to identify a crack inan operating rotor Methods based on optimization are alsooften model-based because a large number of iterations arerequired and it is almost impossible if there is no modelGenetic algorithm was adopted by Saridakis et al [17] Xianget al [18] and He et al [19] to minimize the differencebetween real outputs and model outputs to determine thelocation and depth of a crack in a rotor-bearing systemCavalini Jr et al [20] put forward a crack identificationmethodology using external diagnostic forces at certain fre-quencies to obtain the nonlinear combinational resonanceswhich were used as the objective function of a differentialevolution optimization code to determine the crack locationand depth minimizing the difference between the measuredand modelled rotor system

In contrast withmodel-basedmethods some crack local-ization methods do not need a mathematical model of thesystem under study and are thus called non-model-basedmethods which often just need inputs and outputs of thesystem or even outputs only Rubio et al [21] used changesin resonant and antiresonant frequencies to detect cracklocations in a two-cracked torsional shaft Rahman et al [22]utilized the changes in phase angle of frequency responsefunction to identify the location and depth of a rotor with anopen crack Seo et al [23] proposed a method for open cracklocalization by comparing the map of the modal constantsof the reverse directional frequency response functions withthe reference map of the uncracked model These methodsare based on changes of natural dynamic characteristicsthough no mathematical model of the system is neededthey often require information from intact systems whichsometimes is not convenient to obtain There are also somenon-model-based methods which do not need referenceinformation from intact systems ODS measured by sensorsdistributed along rotors was used for crack localization inrotors by Saravanan and Sekhar [24] and Zhang et al [25]Babu and Sekhar [26] proposed a modified ODS methodcalled amplitude deviation curve to identify cracks in rotor-bearing systems A residual ODS based method consideringhigher harmonic components of exciting frequency wasdeveloped to localize cracks in a rotor by Asnaashari andSinha [27] Singh and Tiwari [28] proposed an algorithm

for crack localization based on the fact that cracks causeslope discontinuities in the shaft deflection Due to the lowersensitivity of ODS to incipient cracks some after-treatmenttechniques were developed such as wavelets [29] FD [30 31]and GSM [32]

Both model-based and non-model-based methods havetheir advantages and disadvantages In this paper a non-model-based method based on POD is proposed to realizemulticrack localization for operating rotors The POD is amultivariate statistical method which aims at obtaining acompact representation of vibration data [33 34] GalvanettoandViolaris [35] investigated the feasibility of POD to localizea crack in beams Shane and Jha [36] applied POD to detectdamage in composite beams POD combined with radialbasis functions was proposed by Benaissa et al [37] toidentify a crack in plate structures However to the authorsrsquobest knowledge the POD has not been used for multicracklocalization in operating rotor systems

Due to the breathing phenomenon of cracks duringrotation and the difficulty to generate breathing cracks inrotors a model that reflects the essential behaviour of a crackis vitally important to get the response of cracked rotorsmoreaccurately There are many methods to model a crack Anonlinear 3D finite element method was adopted to modela breathing crack in a rotor in [38] which may be the mostaccurate model but the computation workload is very heavyIn [39] a rigid finite element method was put forward tomodel a cracked rotor which also had good accuracy tomodel a breathing crack Papadopoulosrsquo review paper [4]elaborated the approach for modelling cracks in rotors basedon SERR and showed that the model put forward by Darpe[40] which was also based on SERR could characterise abreathing crack in rotors more accurately and it had theadvantages of allowing general excitations without assumingthat the gravitational force was dominant and the behaviourof the breathing crack was response-dependent instead ofbeing rotation-dependent So considering the complexity incomputation and accuracy in modelling in relation to othermethods Darpersquosmethod is adopted tomodel a cracked rotorin this investigation

In this work the feasibility of multicrack localizationbased on POD using FD and GSM in operating rotor systemsis validated by numerical investigationTheproposedmethodis a kind of non-model-based approach and it does not needthe knowledge of the undamaged rotorsThe rest of the paperis organized as follows In Section 2 the model of a two-discrotor-bearing systemconsidering the static unbalance of discswith response-dependent breathing cracks is established bythe finite element method Section 3 presents the multicracklocalization method based on POD using FD and GSMIn Section 4 numerical experiments are carried out for themulticracked rotor with different crack configuration casesFinally conclusions are drawn

2 Cracked Rotor Modelling

A finite element model of the cracked rotor consideringbending-torsion coupling introduced by static unbalanceis established in this work A generalized breathing crack

Shock and Vibration 3

x

y

z

CCL

P1

P2

P3P4

P5

P6

P7

P8

P9

P10

P11

P12120579

x998400

y998400

z998400

xL

Figure 1 Schematic diagram of cracked shaft element

model which can represent any crack angles and any typesof excitations applied to the rotor is adopted To modelthe cracked rotor the key point is to simulate the crackappropriately and calculate the stiffness matrix of the crackelement After that through assembling the cracked anduncracked elements the finite elementmodel of the rotor canbe obtained

21 Model of a Cracked Shaft Element Figure 1 shows acracked shaft element of length 119897 and radius 119877 1198751ndash11987512 are theloads acting on the 12 degrees of freedom of the two nodesin the element coordinate system 119909-119910-119911 The local coordinatesystem 1199091015840-1199101015840-1199111015840 is defined on the flat crack face to describethe crack cross-section 120579 is the crack angle between the crackface and the shaft centre line (formed by the negative 1199111015840-axis turning to the negative 119909-axis in the counter-clockwisedirection)119909L is the location of the crack centre in the elementcoordinate system CCL is an imaginary line that separatesthe open and closed parts of the crack which will be used tosimulate the breathing of crackThehatched area correspondsto the open area of the crack

The flexibility matrix (G0)6times6 of the uncracked elementand the additional flexibility matrix (Gc)6times6 of the crackedelement can be derived based on SERR theory [4] and thedetailed expressions can be obtained from [40]

According to the assumption of linear elasticity the totalflexibility matrix of the crack element is the sum of G0 andGc

Gce = G0 + Gc (1)

With the flexibility matrix the stiffness matrix can be derivedconsidering static equilibrium of crack element (as shown inFigure 1) and definition of the stiffness

The nodal force vector of an element can be expressed interms 1198751ndash1198756 via transformation matrix T as

1198751 1198752 11987512T = T 1198751 1198752 1198756T (2)

Here

TT =[[[[[[[[[[[[

1 0 0 0 0 0 minus1 0 0 0 0 00 1 0 0 0 0 0 minus1 0 0 0 1198970 0 1 0 0 0 0 0 minus1 0 minus119897 00 0 0 1 0 0 0 0 0 minus1 0 00 0 0 0 1 0 0 0 0 0 minus1 00 0 0 0 0 1 0 0 0 0 0 minus1

]]]]]]]]]]]]

(3)

According to Hookersquos law the left term of (4) is the vector ofdisplacement along 1198751ndash11987512

TT 1199061 1199062 11990612T = Gce 1198751 1198752 1198756T (4)

Thus from (2) and (4)

1198751 1198752 11987512T = T (Gce)minus1 TT 1199061 1199062 11990612T (5)

So the stiffness matrix of the cracked element Kce and thestiffness matrix of the uncracked element Kuce are

Kce = T (Gce)minus1 TTKuce = T (G0)minus1 TT (6)

22 Breathing Crack Model To consider the breathing phe-nomenon what matters the most is to describe the variationof crack section In this paper CCLmethod in [40] is adoptedto model the breathing crack This method assumes thatthe CCL is perpendicular to the crack edge and separatesthe open and closed parts of the crack which can be seenin Figure 1 And the position of CCL is determined bycalculating the opening mode SIF 119870I by (7) which dependson the crack element nodal forces so the crack is response-dependent nonlinear A positive 119870I corresponds to the opencrack state and a negative one to the closed state And theCCL is located at the position where the sign of 119870I changes


Crack 2x

y

z

Crack 1

(a)

y

zo

eCrack 1

Crack 2

120579Ω

120579Ω 120579x

120579phi

120578

120585

m

e

120573

(b)

Figure 2 (a) Schematic diagram of cracked two-disc rotor-bearing system (b) Definition of rotating and stationary coordinates

Once the CCL is ascertained the stiffness matrix of the crackelement can be obtained

119870I =6sum119894=1

119870I119894 (7)

here119870I119894 is the opening mode SIF contributed by 11987511989423 Equations of Motion of Cracked Rotor-Bearing SystemThe rotor-bearing system considered in this work is shown inFigure 2 The rotor is discretized by two-node Timoshenkobeam elements The discs are considered rigid bodies whichhave three translational and three rotational inertias Andthey are added to themassmatrix elements at the correspond-ing degrees of freedom Gyroscopic effect of the two discsis also included The ball bearings are simplified as stiffnessand damping one of which constrains one axial degree offreedomThe torsional degree of freedom in power input endof the rotor is also constrained The rotating frequency ofthe rotor is Ω By assembling the system matrix of crackedelements and uncracked elements the finite element modelcan be established

Denote q119894 as displacement vector of node 119894 having 6degrees of freedom

q119894 = 119909119894 119910119894 119911119894 120579119909119894 120579119910119894 120579119911119894T (8)

The equations of motion in the stationary coordinate systemcan be written as follows

Mq + (D + ΩDg) q + K (119905) q = Fu + Fg + Fexq = q1 q2 q119894 q119899T

(9)

where M is the system mass matrix D = 119886M + 119887K issystem damping matrix considering the Rayleigh dampingDg is systemgyroscopicmatrixK(119905) is system stiffnessmatrixwhich will be updated as the crack breathes Fu is excitationdue to static unbalance of discs Fg is excitation due tothe gravitational force and Fex is external excitation duringoperation

As for the disc located at node 119894 the gravitationalexcitation vector is

Fg119894 = 0 minus119898g 0 0 0 0T (10)

The excitation due to static unbalance is

Fu119894 = 119865u119909119894 119865u119910119894 119865u119911119894119872u119909119894119872u119910119894119872u119911119894T (11)

where the elements in Fu119894 can be expressed as [41]

119865u119909119894 = 0119872u119910119894 = 0119872u119911119894 = 0119865u119910119894 = 119898119890 [(Ω + 119909119894)2 cos (Ω119905 + 120579119909119894 + 120573)

+ 119909119894 sin (Ω119905 + 120579119909119894 + 120573)] 119865u119911119894 = 119898119890 [(Ω + 119909119894)2 sin (Ω119905 + 120579119909119894 + 120573)

+ x119894 cos (Ω119905 + 120579119909119894 + 120573)] 119872u119909119894 = 119898119890 [ sin (Ω119905 + 120579119909119894 + 120573)

minus ( + g) cos (Ω119905 + 120579119909119894 + 120573)]

(12)

As shown in Figure 2(b) 120579Ω(119905) is the relative angular dis-placement between the rotating coordinate and the stationarycoordinate which will be Ω119905 when the rotor is rotating ata constant speed Ω 120579119909(119905) is the torsional angle 120573 is theunbalance orientation angle of the disc

From (12) one can see that the excitation introducedby unbalance is bending-torsion coupled So the equationsof motion of the cracked rotor are response-dependentnonlinear and the excitation term is bending-torsion coupledThe Newmark method [42] is used to solve the equationsnumerically The stiffness and damping matrices and thecoupled excitation term are updated at each integration step


and the next time step will not start until the response inthe current time step reaches convergence which means theincrement of displacement is less than the tolerance

3 POD Based Multicrack Localization Method

31 Theory of POD The mathematical formulation of PODwas reviewed in [34] and will be briefly introduced in thefollowing

Let 120598(119909 119905) be a random field on Π and it can be writtenas

120598 (119909 119905) = 120583 (119909) + 120599 (119909 119905) (13)

where 120583(119909) is the mean value part and 120599(119909 119905) is the timevarying part

The goal of POD is to obtain the most characteristicstructure120593(119909)of an ensemble of snapshots (a snapshot at time119905119896 is defined as 120599119896(119909) = 120599(119909 119905119896)) of 120599(119909 119905) It is equivalent tofind the basis function that maximizes the ensemble averageof the inner products between 120599119896(119909) and 120593(119909)

max120593(119909)

119869 (120593 (119909)) subject to 1003817100381710038171003817120593 (119909)10038171003817100381710038172 = 1

where 119869 (120593 (119909)) = ⟨10038161003816100381610038161003816(120599119896 (119909) 120593 (119909))100381610038161003816100381610038162⟩ (14)

Here (119891(119909) 119892(119909)) = intΠ119891(119909)119892(119909)d119909 denotes the inner prod-

uct inΠ |sdot|denotes themodulus ⟨sdot⟩ is the averaging operator120593(119909) = (120593(119909) 120593(119909))12 denotes the norm of a functionBy introducing Lagrange multiplier the optimization

problem can be expressed as

max120593(119909)

119871 (120593 (119909)) where 119871 (120593 (119909)) = ⟨10038161003816100381610038161003816(120599119896 (119909) 120593 (119909))100381610038161003816100381610038162⟩ minus 120582 (1003817100381710038171003817120593 (119909)10038171003817100381710038172 minus 1) (15)

To reach the maximum the derivative of 119871(120593(119909)) should bezero which is derived as [43]

intΠ⟨120599119896 (119909) 120599119896 (119910)⟩ 120593 (119910) d119910 = 120582120593 (119909) (16)

where ⟨120599119896(119909)120599119896(119910)⟩ is the averaged autocorrelation functionThe optimized solution is given by the orthogonal eigen-

functions 120593119894(119909) of (16) called POMs The correspondingeigenvalues 120582119894 are POVs

The mathematical formulation mentioned above is thecontinuous form of POD however in real practice thedata obtained are discretized in time and space so discreterealization of POD by SVD is used in this work

To start with POD the system response matrix Y whichis measured simultaneously by 119899 sensors at different locationsneeds to be obtained

Y = [y1 y119899] = [[[

11991011 sdot sdot sdot 1199101119899sdot sdot sdot sdot sdot sdot sdot sdot sdot1199101198981 sdot sdot sdot 119910119898119899

]]] (17)

where 119898 is the sample length Y corresponds to the dis-cretized form of field 120598(119909 119905) in (13)

As for the discretized data the averaged autocorrelationfunction is replaced by covariance matrix which can beestimated by the sample covariancematrixCs then the POMsand POVs correspond to the eigenvectors and eigenvalues ofCs respectively In particular if the data have a zeromeanCscan be expressed as

Cs = 1119899YYT (18)

SVD of Y can be written as

Y = USVT (19)

where U119898times119898 is an orthogonal matrix containing the leftsingular vectors S119898times119899 is a pseudo-diagonal matrix withsingular values at the diagonal entries V119899times119899 is an orthogonalmatrix containing the right singular vectors

According to (19) one can get

YYT = USVTVSTUT = US2UT (20)

Then

Cs = U(S2119899 )UT (21)

So one can see that the eigenvectors or POMs of Cs are theleft singular vectors of Y and the eigenvalues or POVs of Csare the squares of singular values of Y divided by 119899

The idea of multicrack localization based on POD is thatthe characteristic structure of measured system response of acracked system will be different from that of an uncrackedsystem and cracks will introduce local discontinuities inthe POMs while the POMs should be continuous for anuncracked system where there is no other factor whichintroduces discontinuity for example a large lumped mass

32 Damage Indexes from FD and GSM When there is nocrack the POMs will be continuous but discontinuities willbe introduced at the locations of cracks In order to amplifythe effect of the discontinuities in localization damageindexes based on FD and GSM are used

321 FD Based on POMs The FD of a curve defined by 119899points (O1 O119899) is estimated by [44]

FD = log10 (119899 minus 1)log10 (119899 minus 1) + log10 (119889119871)

119889 = max2le119894le119899

dist (O1O119894) 119871 = 119899minus1sum119894=1

dist (O119894O119894+1) (22)

Here dist(sdot sdot) denotes the distance between two pointsSo the FD of a specific curve is definite and it is

a measurement of the complexity of a curve Generally


speaking places where discontinuities occur will show highcomplexity which is the main idea to use FD as damageindex of cracks In order to detect discontinuities in a POM asliding window is used to truncate the curveThe FD in everywindow is calculated to represent the complexity of the localsegment falling into thewindow and a proper window chosencan amplify the local discontinuities of the whole curve

Let 119872 be the width of the sliding window and 119904 be thesliding step then the FD in the 119895th window can be expressedas

FD (119895) = log10 (119872 minus 1)log10 (119872 minus 1) + log10 (119889 (119895) 119871 (119895)) (23)

119889 (119895) = max(119895minus1)119904+1le119902le(119895minus1)119904+119872

dist (O(119895minus1)119904+1O119902) (24)

119871 (119895) = (119895minus1)119904+119872minus1sum119902=(119895minus1)119904+1

dist (O119902O119902+1) (25)

During the process of crack localization 119895 responds to thelocation of midpoint in the window

322 GSMBased on POMs TheGSM is a kind of polynomialcurve fitting method It is used to extract the discontinuitiesinduced by cracks in the POMs in this paper Its main ideais to fit the cracked POM using gapped polynomial to obtainthe approximate uncracked POM and then to calculate thedifference function between the actual POM and the fittedPOM Large differences indicate presence of cracks

Generally speaking the order of gapped polynomial ischosen to be three so the gapped polynomial function(GPF3119894 ) at the gapped point O119894(119909119894 119910119894) can be written as [32]

GPF3119894 = 1198860 + 1198861119909119894 + 11988621199092119894 + 11988631199093119894 (26)

where 1198860 1198861 1198862 and 1198863 are determined by O119894minus2 O119894minus1 O119894+1and O119894+2

However for the crack localization in rotors the gappedlinear interpolation is found to be more efficient In this casethe gapped polynomial function (GPF1119894 ) can be expressed as

GPF1119894 = 1198870 + 1198871119909119894 (27)

where 1198870 and 1198871 are determined by O119894minus1 O119894+1Then two damage indexes are put forward as the squared

difference between the gapped polynomial function and thecorresponding value of the actual POM

DI3119894 = (GPF3119894 minus 119910119894)2 DI1119894 = (GPF1119894 minus 119910119894)2

(28)

4 Numerical Investigation

In order to investigate the multicrack localization methodsnumerical experiments are carried out for the rotor-bearingsystem shown in Figure 2 and its detailed parameter valuesare given in Table 1 where 119886 and 119887 are calculated by assuming

Table 1 Parameters of the cracked rotor

Parameter Value (units)Shaft length 056mShaft diameter 001mDisc diameter 0074mDisc thickness 0025mDisc eccentricity 2times 10minus5mUnbalance orientation angle 0Density of steel 78times 103 kgm3

Youngrsquos modulus 211times 1011 PaPoissonrsquos ratio 03Gravitational acceleration 98ms2

Rayleigh damping coefficient (119886) 044Rayleigh damping coefficient (119887) 43times 10minus5

Bearing stiffness 25times 105NmBearing damping 100NsmFirst critical speed 1663 rminRotating speed 540 rmin

modal damping ratios of the first two modes being 0005 and001 And the first critical speed is calculated in the no-crackcondition

The rotor is discretized into 28 equivalent two-nodetwelve-degree-of-freedom Timoshenko beam elements andcracks with different configurations are embedded usingthe cracked shaft elements All the cracks considered aretransverse ones and the cracks are assumed not to propagateduring the short period of excitation while measurement ismade Newmark method is adopted to obtain the responsesin time domain The Newmark constants are 025 and05 respectively the sampling frequency is 5000Hz or theintegration step is 2 times 10minus4 s and the accuracy of convergencefor each step is set to 10minus11

41 Response Characteristics of a Cracked Rotor In order toidentify the crack locations in the rotor response characteris-tics will be studied first Figure 3 gives the vertical steady-stateresponses of the rotor in time and frequency domainswithoutany crack with one crack and with two cracks respectivelymeasured from a single sensor located in the 14th elementof the rotor And 119883 represents the frequency correspondingto the rotating speed The vertical responses of rotor run-upwith angular acceleration 10 rads2 in time domain withoutany crack with one crack and with two cracks are shown inFigure 4 The response characteristics are consistent with theresults in [45] so one can believe that the model establishedand its solution are correct

From Figures 3 and 4 one can see that the presenceof superharmonic components (or subharmonic resonances)generated by nonlinearity introduced by breathing of cracksis a clear indicator of a crack but there is no qualitativedifference between responses of a rotor with a single crackand a rotor with double cracks whether in steady or unsteadystate

Because the crack number cannot be known in advancecrack detection results could be misleading which shows


No crackOne crackTwo cracks

1X

2X 3X

times10minus4

times10minus6

minus39

minus38

minus37

minus36

Vert

ical

resp

onse

(m)

01 02 03 04 05 06 07 08 09 10Time (s)

0

2

4

6A

mpl

itude

5 10 15 20 25 30 350Frequency (Hz)

Figure 3 Steady-state responses comparison of rotors without any crack with one crack and with two cracks (the one crack located in the12th element with depth of 02 and the two cracks located in the 11th and 17th elements both with depth of 02)

the difficulties of multicrack localization by measuring theresponses just from a single sensor since no space informa-tion is produced And it can also be concluded that thosemethods suitable for a single-crack rotor are not alwayssuitable for a multicrack rotor In view of the difficultiesof multicrack localization in rotors using methods withoutspace information POD is introduced for the operating rotorin the same situation as the rotor which was used to getthe responses in Figures 3 and 4 and the first and secondnormalized POMs are shown in Figure 5

As it can be seen from Figure 5 that the one-crack andtwo-crack cases can be identified by POM1 and POM2 whilethe two cases are difficult or impossible to be distinguishedwithout space information as previously shown in Figures 3and 4 And from Figure 5 one can see that the cracks willintroduce discontinuities in POMs Therefore multicracklocalization can be realized by detecting the discontinuitylocations in POMs

42 Localization of a Double-Cracked Rotor Using FD andGSM with the CPOM Focusing on the cases of the rotorwith double cracks of varying depths and relative phaseangles at different locations as shown in Table 2 where therelative phase angle is defined as the angle between thepositive normal lines of the two-crack tips which is shownas 120579phi in Figure 2(b) FD and GSM with the CPOM areused respectively to localize the cracks In order to simulatemeasurement errors white Gaussian noise is added to theoriginal response y so the noise-polluted response yN can beexpressed as

yN = y + NLradicsum(119910119894 minus 120583)2119873 r (29)

where 119873 is the length of y NL is the constant noise levelwithin (0 1) 120583 is the mean value of y r is an N-length vectorof normally distributed randomnumbers with zeromean andvariance equal to 1 Figure 6 is a typical response of a double-cracked rotor without and with noise

421 Localization Results and Robustness of the MethodWithout losing generality case 2 is chosen to determine theCPOM which is the most robust to noise and most sensitiveto cracks All the cases are measured by 29 sensors in thecorresponding nodes except when investigating the effects ofsensor numbers In order to investigate the effect of noise onPOMs a higher noise level of 5 is considered and the POMsfrom the first order to the forth order are compared with thecorresponding unnoised ones in Figure 7

From Figure 7 one can see that the cracks will affect allthe first four POMs but the first two POMs are less sensitiveto noise In addition the discontinuity locations in higherorder POMs are dominated by one of the cracks for examplePOM2 and POM3 are dominated by crack 2 while POM4is mainly influenced by crack 1 Therefore in view of therobustness to noise and sensitivity to cracks POM1 is selectedas the CPOM to identify multicrack locations for variouscases of cracked rotors in Table 2 However it is still not easyto identify crack locations from the CPOM directly so after-treatment methods which can amplify discontinuities in theCPOM are required

In order to amplify discontinuities in the CPOM furthertreatment is performed by GSM and FDWhen GSM is usedthe cubic and linear gapped interpolations are comparedAndthe width of sliding window 119872 for FD in (23) is set to 3which is determined by trial and error From Figures 8(a) and8(b) one can see that multicrack localization using GSM by


No crack

times10minus3

minus15

minus1

minus05

0

05

1

Vert

ical

resp

onse

(m)

0 10 15 20 255Time (s)

(a)

13 subharmonicresonance


One crack

times10minus3

minus15

minus1

minus05

0

05

1

Vert

ical

resp

onse

(m)

5 10 15 20 250Time (s)

(b)



Two cracks

times10minus3

minus2

minus15

minus1

minus05

0

05

1

Vert

ical

resp

onse

(m)

5 10 15 20 250Time (s)

(c)

Figure 4 Run-up responses comparison of rotors without any crack with one crack and with two cracks (a) No crack (b) One crack locatedin the 12th element with depth of 02 (c) Two cracks located in the 11th and 17th elements both with depth of 02

Table 2 Cases of the rotor with cracks of varying depths at different locations

Case Crack 1 Crack 2 Relative phase angle1 01119863 200ndash220mm (the 11th element) 01119863 320ndash340mm (the 17th element) 0∘

2 01119863 200ndash220mm (the 11th element) 02119863 320ndash340mm (the 17th element) 0∘




6 02119863 20ndash40mm (the 3rd element) 02119863 280ndash300mm (the 14th element) 0∘




One crack

Crack

minus1

minus05

0

05

1PO

M1

5 10 15 20 25 300Nodes

5 10 15 20 25 300Nodes

minus1

minus05

0

05

1

POM

2

(a)Two cracks

Crack 1 Crack 2

minus1

minus05

0

05

1

POM

1

5 10 15 20 25 300Nodes

5 10 15 20 25 300Nodes

minus1

minus05

0

05

1

POM

2

(b)

Figure 5 POMs comparison between the rotor with one crack and the rotor with two cracks (a) One crack located in the 12th element withdepth of 02 (b) Two cracks located in the 11th and 17th elements both with depth of 02

No noiseNoise = 5

021 022 023 024minus376

minus374

minus372

minus37

minus368

times10minus4

times10minus4

minus39

minus385

minus38

minus375

minus37

minus365

minus36

Vert

ical

resp

onse

(m)

005 01 015 02 025 03 035 040Time (s)

Figure 6 Typical response of the double-cracked rotor in steady state

cubic gapped interpolation can identify the locations roughlybut the resolution is lower and it is more sensitive to noisecompared with GSMby linear gapped interpolation as shownFigure 8(a) In addition multicrack localization result usingFD is also quite good in Figure 8(c) So in the followingGSM by linear gapped interpolation and FD will be used formulticrack localization (see Figures 9ndash15)

From Figures 8ndash15 one can see that all the double-crackcases are identified correctly and themethod based onCPOMusing GSMwith linear gapped interpolation and FD is robustto noise In Figure 8 though the two cracks are locatedcorrectly there are two more discontinuities apart from thecrack locations which correspond to the locations of thetwo discs but these discontinuities are relatively weak Andfortunately as the crack depth increases the discontinuitiesinduced by discs almost disappear And from Figure 12 onecan see that the method is still reliable even when a crack islocated in the same element as the disc in case 5 So it can be

concluded that crack locations can be identified regardless ofthe disc locations Besides cracks at different locations withdifferent depths can be localized and the deeper the crack thelarger the corresponding magnitude of the damage indexeswhich can be seen in Figures 9 11 and 13 And one can alsosee that even if a crack is near a bearing it can also be localizedcorrectly as shown in Figure 13 From Figures 10 14 and 15one can see that under the same crack depths and locationsthe relative phase angle will change the values of damageindexes Because the relative phase angle between two crackswill definitely influence the response of the rotor thus theCPOM will be different However the localization results arestill quite good which means that the proposed method issuitable for cracks in rotors with any crack phase angles

422 Effects of Sensor Numbers In order to investigate thefeasibility to reduce sensor numbers fewer sensors are usedtomeasure the responses of the cracked rotor in case 3 Fifteen


200 250 300 350minus1

minus08

minus06

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1PO

M1

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

POM

2

minus1

minus05

0

05

1

POM

3

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

POM

4

No noiseNoise = 5

No noiseNoise = 5

Figure 7 The first four POMs of double-cracked rotor in case 2

sensors are used and the location results using GSM and FDare shown in Figure 16

As can be seen from Figure 16 the locations of the twocracks are identified correctly and also insensitive to noisebut with lower resolution As a matter of fact the numberof sensors determines the spatial resolution and thus it willinfluence the accuracy of crack localization So the moresensors are used the more accurate localization is in theoryAs for the minimal number of sensors it can be assumed thatthere are 119873c cracks (this number is unknown) For GSM bylinear gapped interpolation to cover the worst situation thereshould be at least 3119873c + 1 sensors shown as Figure 17(a) forFD method with window width of 3 at least 3119873c + 3 sensorsare required shown as Figure 17(b)

In practice when a crack is localized using 119899 sensorsand if it is suspected that the accuracy is poor all these 119899sensors can be placed around the damage location and thelocal responses are measured again This will lead to a moreaccurate localization

5 Conclusions

Numerical investigation is carried out for multicrack local-ization in rotors based on proper orthogonal decomposition(POD) using fractal dimension (FD) and gapped smooth-ing method (GSM) A two-disc rotor-bearing system withresponse-dependent breathing cracks at different locations ofvarying depths considering the static unbalance of the twodiscs is established by the finite element method Throughcomparing response characteristics of the rotor with a singlecrack and two cracks it is observed that it is very difficult

or impossible to distinguish a multicrack case from a single-crack case just based on the response from one sensor Soproper orthogonal modes (POMs) are extracted by PODfrom the responses ldquomeasuredrdquo from sensors distributedalong the rotor Discontinuities are found to have beenintroduced by cracks at the corresponding locations in thePOMs Considering the sensitivity to cracks and noise thecharacteristic POM (CPOM) is selected Instead of utilizingthe CPOM directly after-treatment techniques of FD andGSM are used to amplify the discontinuities in the CPOMto realize the multicrack localization more effectively All thelocalization results for the rotor with cracks at different loca-tions of varying depths based on CPOM using FD and GSMare quite good And the crack localizationmethod is robust tonoise and fewer sensors are still feasible to successfully locatethe cracks In addition regardless of input excitations onlyresponses are needed by the proposedmethodWhat is moreno prior knowledge about the model is demanded which isof great significance for rotors with complex structures andcomplicated boundaries that are difficult tomodelThereforethe method will be useful in real applications Howevervibration-based damage identification relies heavily on mea-surement technology For some machines working in hostileenvironments such as steam turbines noncontact heat- andhumidity-resistant sensors should be used Without good-quality vibration data the proposed method would not workwell

Abbreviations

CCL Crack closure lineCPOM Characteristic proper orthogonal mode


No noiseNoise = 5

No noiseNoise = 5

Interpolation without noiseInterpolation with noise

GSM

Crack 1 Crack 2

Disc 1 Disc 2

times10minus4

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

1

2

3

4

DI1

100 200 300 400 500 6000Length (mm)

(a)

No noiseNoise = 5

No noiseNoise = 5


GSM

Crack 1

Crack 2

times10minus5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

2

DI3

100 200 300 400 500 6000Length (mm)

(b)

No noiseNoise = 5

Crack 1 Crack 2

Disc 1 Disc 2

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

1

1

10001

10001

FD

No noiseNoise = 5

(c)

Figure 8 Localization results of double-cracked rotor in case 1 (a) Localization using GSM by linear gapped interpolation (b) Localizationusing GSM by cubic gapped interpolation (c) Localization using FD

FD Fractal dimensionGSM Gapped smoothing methodODS Operational deflection shapePOD Proper orthogonal decompositionPOM Proper orthogonal modePOV Proper orthogonal valueSERR Strain energy release rate

SIF Stress intensity factorSVD Singular value decomposition

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper


times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

100 200 300 400 500 6000Length (mm)

No noiseNoise = 5

No noiseNoise = 5


(a)

Crack 1

Crack 2

Disc 1 Disc 2

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD

100 200 300 400 500 6000Length (mm)

No noiseNoise = 5

No noiseNoise = 5

(b)

Figure 9 Localization results of double-cracked rotor in case 2 (a) Localization using GSM by linear gapped interpolation (b) Localizationusing FD

Crack 1 Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

times10minus4

100 200 300 400 500 6000Length (mm)

0

2

4

6

8

DI1

(a)

Crack 1 Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

DI1

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD100 200 300 400 500 6000

Length (mm)

(b)


Crack 1 amp Crack 2disc 1

Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

times10minus3

0

05

1

DI1

(a)

Crack 1 ampdisc 1

Crack 2

Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD100 200 300 400 500 6000

Length (mm)

(b)


Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

100 200 300 400 500 6000Length (mm)

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

2

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10005

1001

FD

(b)


Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

2

4

6

8

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

1001

1002

1003

FD

100 200 300 400 500 6000Length (mm)

(b)

Figure 16 Localization results of double-cracked rotorwith fewer sensors in case 3 (a) Localization usingGSMby linear gapped interpolation(b) Localization using FD


Crack 1

Crack 1

Crack 1

Crack 2

middot middot middot

n = 7

n = 4

n = 3Nc + 1

Crack Nc

(a)

Crack 1

Crack 1

Crack 1

Crack 2


n = 9

n = 6

n = 3Nc + 3

Crack Nc

(b)

Figure 17 Determination of the minimal number of sensors (a) Localization using GSM by linear gapped interpolation (b) Localizationusing FD

Acknowledgments

This study is partly supported by the National NaturalScience Foundation of China (51405399) and the Fun-damental Research Funds for the Central Universities(DUT16RC(3)027) and carried out by the first author duringhis visit to the University of Liverpool sponsored by theChina Scholarship Council

References

[1] A Bovsunovsky and C Surace ldquoNon-linearities in the vibra-tions of elastic structures with a closing crack a state of the artreviewrdquo Mechanical Systems and Signal Processing vol 62 pp129ndash148 2015

[2] W Fan and P Qiao ldquoVibration-based damage identificationmethods a review and comparative studyrdquo Structural HealthMonitoring vol 10 no 1 pp 83ndash111 2011

[3] C Kumar and V Rastogi ldquoA brief review on dynamics of acracked rotorrdquo International Journal of Rotating Machinery vol2009 Article ID 758108 6 pages 2009

[4] C A Papadopoulos ldquoThe strain energy release approach formodeling cracks in rotors a state of the art reviewrdquoMechanicalSystems and Signal Processing vol 22 no 4 pp 763ndash789 2008

[5] Y J Yan L Cheng Z Y Wu and L H Yam ldquoDevelopmentin vibration-based structural damage detection techniquerdquoMechanical Systems and Signal Processing vol 21 no 5 pp2198ndash2211 2007

[6] E P Carden and P Fanning ldquoVibration based conditionmonitoring a reviewrdquo Structural Health Monitoring vol 3 no4 pp 355ndash377 2004

[7] G Sabnavis R G Kirk M Kasarda and D Quinn ldquoCrackedshaft detection and diagnostics a literature reviewrdquo Shock andVibration Digest vol 36 no 4 p 287 2004

[8] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998

[9] A D Dimarogonas ldquoVibration of cracked structures a state ofthe art reviewrdquo Engineering Fracture Mechanics vol 55 no 5pp 831ndash857 1996

[10] P Pennacchi N Bachschmid and A Vania ldquoA model-basedidentification method of transverse cracks in rotating shaftssuitable for industrialmachinesrdquoMechanical Systems and SignalProcessing vol 20 no 8 pp 2112ndash2147 2006

[11] A W Lees J K Sinha and M I Friswell ldquoModel-basedidentification of rotating machinesrdquo Mechanical Systems andSignal Processing vol 23 no 6 pp 1884ndash1893 2009

[12] A S Sekhar ldquoModel-based identification of two cracks in arotor systemrdquoMechanical Systems and Signal Processing vol 18no 4 pp 977ndash983 2004

[13] H B Dong X F Chen B Li K Y Qi and Z J He ldquoRotorcrack detection based on high-precisionmodal parameter iden-tificationmethod andwavelet finite elementmodelrdquoMechanicalSystems and Signal Processing vol 23 no 3 pp 869ndash883 2009

[14] S Seibold and K Weinert ldquoA time domain method for thelocalization of cracks in rotorsrdquo Journal of Sound and Vibrationvol 195 no 1 pp 57ndash73 1996

[15] J L Zapico-Valle E Rodrıguez M Garcıa-Dieguez and J LCortizo ldquoRotor crack identification based on neural networksand modal datardquoMeccanica vol 49 no 2 pp 305ndash324 2014

[16] D Soffker C Wei S Wolff and M-S Saadawia ldquoDetection ofrotor cracks comparison of an old model-based approach witha new signal-based approachrdquo Nonlinear Dynamics vol 83 no3 pp 1153ndash1170 2016

[17] K M Saridakis A C Chasalevris C A Papadopoulos and AJ Dentsoras ldquoApplying neural networks genetic algorithms andfuzzy logic for the identification of cracks in shafts by usingcoupled response measurementsrdquo Computers and Structuresvol 86 no 11-12 pp 1318ndash1338 2008

[18] J W Xiang Y Zhong X F Chen and Z J He ldquoCrack detectionin a shaft by combination of wavelet-based elements and geneticalgorithmrdquo International Journal of Solids and Structures vol45 no 17 pp 4782ndash4795 2008


[19] Y He D Guo and F Chu ldquoUsing genetic algorithms and finiteelementmethods to detect shaft crack for rotor-bearing systemrdquoMathematics and Computers in Simulation vol 57 no 1-2 pp95ndash108 2001

[20] A A Cavalini Jr L Sanches N Bachschmid and V SteffenJr ldquoCrack identification for rotating machines based on anonlinear approachrdquoMechanical Systems and Signal Processingvol 79 pp 72ndash85 2016

[21] L Rubio J Fernandez-Saez and A Morassi ldquoIdentificationof two cracks in a rod by minimal resonant and antiresonantfrequency datardquo Mechanical Systems and Signal Processing vol60 pp 1ndash13 2015

[22] A G A Rahman Z Ismail S Noroozi and O Z Chao ldquoStudyof open crack in rotor shaft using changes in frequency responsefunction phaserdquo International Journal of Damage Mechanicsvol 22 no 6 pp 791ndash807 2013

[23] Y-H Seo C-W Lee and K C Park ldquoCrack identification ina rotating shaft via the reverse directional frequency responsefunctionsrdquo Journal of Vibration and Acoustics vol 131 no 1 p11012 2009

[24] K Saravanan and A S Sekhar ldquoCrack detection in a rotor byoperational deflection shape and kurtosis using laser vibrome-ter measurementsrdquo Journal of Vibration and Control vol 19 no8 pp 1227ndash1239 2012

[25] C L Zhang B Li Z Yang W Xiao and Z He ldquoCracklocation identification of rotating rotor systems using operatingdeflection shape datardquo Science China Technological Sciences vol56 no 7 pp 1723ndash1732 2013

[26] T R Babu and A S Sekhar ldquoDetection of two cracks in arotor-bearing system using amplitude deviation curverdquo Journalof Sound and Vibration vol 314 no 3ndash5 pp 457ndash464 2008

[27] E Asnaashari and J K Sinha ldquoComparative study between theR-ODS and DNDmethods for damage detection in structuresrdquoMeasurement vol 66 pp 80ndash89 2015

[28] S K Singh and R Tiwari ldquoDetection and localisation of mul-tiple cracks in a shaft system an experimental investigationrdquoMeasurement vol 53 pp 182ndash193 2014

[29] S-T Quek Q Wang L Zhang and K-K Ang ldquoSensitivityanalysis of crack detection in beams by wavelet techniquerdquoInternational Journal of Mechanical Sciences vol 43 no 12 pp2899ndash2910 2001

[30] P Z Qiao and M S Cao ldquoWaveform fractal dimension formode shape-based damage identification of beam-type struc-turesrdquo International Journal of Solids and Structures vol 45 no22-23 pp 5946ndash5961 2008

[31] Y-Y Jiang B Li Z-S Zhang and X-F Chen ldquoIdentificationof crack location in beam structures using wavelet transformand fractal dimensionrdquo Shock and Vibration vol 2015 ArticleID 832763 10 pages 2015

[32] M K Yoon D Heider J W Gillespie Jr C P Ratcliffe and RM Crane ldquoLocal damage detection using the two-dimensionalgapped smoothing methodrdquo Journal of Sound and Vibrationvol 279 no 1-2 pp 119ndash139 2005

[33] Y C Liang H P Lee S P Lim W Z Lin K H Lee and C GWu ldquoProper orthogonal decomposition and its applicationsmdashpart I theoryrdquo Journal of Sound and Vibration vol 252 no 3pp 527ndash544 2002

[34] G Kerschen J-C Golinval A F Vakakis and L A BergmanldquoThe method of proper orthogonal decomposition for dynami-cal characterization and order reduction ofmechanical systemsan overviewrdquo Nonlinear Dynamics vol 41 no 1ndash3 pp 147ndash1692005

[35] U Galvanetto and G Violaris ldquoNumerical investigation of anew damage detection method based on proper orthogonaldecompositionrdquoMechanical Systems and Signal Processing vol21 no 3 pp 1346ndash1361 2007

[36] C Shane and R Jha ldquoProper orthogonal decomposition basedalgorithm for detecting damage location and severity in com-posite beamsrdquoMechanical Systems and Signal Processing vol 25no 3 pp 1062ndash1072 2011

[37] B Benaissa N A Hocine I Belaidi A Hamrani and VPettarin ldquoCrack identification using model reduction basedon proper orthogonal decomposition coupled with radial basisfunctionsrdquo Structural and Multidisciplinary Optimization vol54 no 2 pp 265ndash274 2016

[38] G I Giannopoulos S K Georgantzinos and N K AnifantisldquoCoupled vibration response of a shaft with a breathing crackrdquoJournal of Sound and Vibration vol 336 pp 191ndash206 2015

[39] Z Kulesza and J T Sawicki ldquoRigid finite element model of acracked rotorrdquo Journal of Sound and Vibration vol 331 no 18pp 4145ndash4169 2012

[40] A K Darpe ldquoCoupled vibrations of a rotor with slant crackrdquoJournal of Sound and Vibration vol 305 no 1-2 pp 172ndash1932007

[41] Z W Yuan F L Chu and Y L Lin ldquoExternal and internalcoupling effects of rotorrsquos bending and torsional vibrationsunder unbalancesrdquo Journal of Sound and Vibration vol 299 no1-2 pp 339ndash347 2007

[42] N M Newmark ldquoA method of computation for structuraldynamicsrdquo Journal of the Engineering Mechanics Division vol85 no 3 pp 67ndash94 1959

[43] P Holmes J L Lumley and G Berkooz Turbulence Coher-ent Structures Dynamical Systems and Symmetry CambridgeMonographs on Mechanics Cambridge University Press 1996

[44] M J Katz ldquoFractals and the analysis of waveformsrdquo Computersin Biology and Medicine vol 18 no 3 pp 145ndash156 1988

[45] N H Chandra and A S Sekhar ldquoFault detection in rotorbearing systems using time frequency techniquesrdquo MechanicalSystems and Signal Processing vol 72-73 pp 105ndash133 2016

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Seibold andWeinert [14] proposed a method in time domainbased on a bank of Extended Kalman Filters to realize thecrack location and depth identification for an operating rotorMethods based on pattern recognition can be model-basedwhen the samples for training are obtained from mathe-matical models rather than experiments Their main ideasare extracting features sensitive to crack parameters whichthen are trained by artificial intellectual methods or machinelearning methods to obtain the relationship between thecrack parameters and features and then matching the mea-sured features with the established relationship to determinethe crack parameters Zapico-Valle et al [15] adopted the arti-ficial neural network which trained by samples gathered froma finite element model to identify crack location and depth ofrotors Soffker et al [16] compared a modern model-basedtechnique based on a proportional-integral observer witha signal-based technique based on support vector machineusing features extracted from wavelets to identify a crack inan operating rotor Methods based on optimization are alsooften model-based because a large number of iterations arerequired and it is almost impossible if there is no modelGenetic algorithm was adopted by Saridakis et al [17] Xianget al [18] and He et al [19] to minimize the differencebetween real outputs and model outputs to determine thelocation and depth of a crack in a rotor-bearing systemCavalini Jr et al [20] put forward a crack identificationmethodology using external diagnostic forces at certain fre-quencies to obtain the nonlinear combinational resonanceswhich were used as the objective function of a differentialevolution optimization code to determine the crack locationand depth minimizing the difference between the measuredand modelled rotor system

In contrast withmodel-basedmethods some crack local-ization methods do not need a mathematical model of thesystem under study and are thus called non-model-basedmethods which often just need inputs and outputs of thesystem or even outputs only Rubio et al [21] used changesin resonant and antiresonant frequencies to detect cracklocations in a two-cracked torsional shaft Rahman et al [22]utilized the changes in phase angle of frequency responsefunction to identify the location and depth of a rotor with anopen crack Seo et al [23] proposed a method for open cracklocalization by comparing the map of the modal constantsof the reverse directional frequency response functions withthe reference map of the uncracked model These methodsare based on changes of natural dynamic characteristicsthough no mathematical model of the system is neededthey often require information from intact systems whichsometimes is not convenient to obtain There are also somenon-model-based methods which do not need referenceinformation from intact systems ODS measured by sensorsdistributed along rotors was used for crack localization inrotors by Saravanan and Sekhar [24] and Zhang et al [25]Babu and Sekhar [26] proposed a modified ODS methodcalled amplitude deviation curve to identify cracks in rotor-bearing systems A residual ODS based method consideringhigher harmonic components of exciting frequency wasdeveloped to localize cracks in a rotor by Asnaashari andSinha [27] Singh and Tiwari [28] proposed an algorithm

for crack localization based on the fact that cracks causeslope discontinuities in the shaft deflection Due to the lowersensitivity of ODS to incipient cracks some after-treatmenttechniques were developed such as wavelets [29] FD [30 31]and GSM [32]

Both model-based and non-model-based methods havetheir advantages and disadvantages In this paper a non-model-based method based on POD is proposed to realizemulticrack localization for operating rotors The POD is amultivariate statistical method which aims at obtaining acompact representation of vibration data [33 34] GalvanettoandViolaris [35] investigated the feasibility of POD to localizea crack in beams Shane and Jha [36] applied POD to detectdamage in composite beams POD combined with radialbasis functions was proposed by Benaissa et al [37] toidentify a crack in plate structures However to the authorsrsquobest knowledge the POD has not been used for multicracklocalization in operating rotor systems

Due to the breathing phenomenon of cracks duringrotation and the difficulty to generate breathing cracks inrotors a model that reflects the essential behaviour of a crackis vitally important to get the response of cracked rotorsmoreaccurately There are many methods to model a crack Anonlinear 3D finite element method was adopted to modela breathing crack in a rotor in [38] which may be the mostaccurate model but the computation workload is very heavyIn [39] a rigid finite element method was put forward tomodel a cracked rotor which also had good accuracy tomodel a breathing crack Papadopoulosrsquo review paper [4]elaborated the approach for modelling cracks in rotors basedon SERR and showed that the model put forward by Darpe[40] which was also based on SERR could characterise abreathing crack in rotors more accurately and it had theadvantages of allowing general excitations without assumingthat the gravitational force was dominant and the behaviourof the breathing crack was response-dependent instead ofbeing rotation-dependent So considering the complexity incomputation and accuracy in modelling in relation to othermethods Darpersquosmethod is adopted tomodel a cracked rotorin this investigation

In this work the feasibility of multicrack localizationbased on POD using FD and GSM in operating rotor systemsis validated by numerical investigationTheproposedmethodis a kind of non-model-based approach and it does not needthe knowledge of the undamaged rotorsThe rest of the paperis organized as follows In Section 2 the model of a two-discrotor-bearing systemconsidering the static unbalance of discswith response-dependent breathing cracks is established bythe finite element method Section 3 presents the multicracklocalization method based on POD using FD and GSMIn Section 4 numerical experiments are carried out for themulticracked rotor with different crack configuration casesFinally conclusions are drawn

2 Cracked Rotor Modelling

A finite element model of the cracked rotor consideringbending-torsion coupling introduced by static unbalanceis established in this work A generalized breathing crack


x

y

z

CCL

P1

P2

P3P4

P5

P6

P7

P8

P9

P10

P11

P12120579

x998400

y998400

z998400

xL






Gce = G0 + Gc (1)



1198751 1198752 11987512T = T 1198751 1198752 1198756T (2)

Here

TT =[[[[[[[[[[[[


]]]]]]]]]]]]

(3)


TT 1199061 1199062 11990612T = Gce 1198751 1198752 1198756T (4)


1198751 1198752 11987512T = T (Gce)minus1 TT 1199061 1199062 11990612T (5)





Crack 2x

y

z

Crack 1

(a)

y

zo

eCrack 1

Crack 2

120579Ω

120579Ω 120579x

120579phi

120578

120585

m

e

120573

(b)



119870I =6sum119894=1

119870I119894 (7)



q119894 = 119909119894 119910119894 119911119894 120579119909119894 120579119910119894 120579119911119894T (8)



(9)



Fg119894 = 0 minus119898g 0 0 0 0T (10)


Fu119894 = 119865u119909119894 119865u119910119894 119865u119911119894119872u119909119894119872u119910119894119872u119911119894T (11)


119865u119909119894 = 0119872u119910119894 = 0119872u119911119894 = 0119865u119910119894 = 119898119890 [(Ω + 119909119894)2 cos (Ω119905 + 120579119909119894 + 120573)

+ 119909119894 sin (Ω119905 + 120579119909119894 + 120573)] 119865u119911119894 = 119898119890 [(Ω + 119909119894)2 sin (Ω119905 + 120579119909119894 + 120573)

+ x119894 cos (Ω119905 + 120579119909119894 + 120573)] 119872u119909119894 = 119898119890 [ sin (Ω119905 + 120579119909119894 + 120573)

minus ( + g) cos (Ω119905 + 120579119909119894 + 120573)]

(12)








120598 (119909 119905) = 120583 (119909) + 120599 (119909 119905) (13)



max120593(119909)

119869 (120593 (119909)) subject to 1003817100381710038171003817120593 (119909)10038171003817100381710038172 = 1

where 119869 (120593 (119909)) = ⟨10038161003816100381610038161003816(120599119896 (119909) 120593 (119909))100381610038161003816100381610038162⟩ (14)




max120593(119909)



intΠ⟨120599119896 (119909) 120599119896 (119910)⟩ 120593 (119910) d119910 = 120582120593 (119909) (16)





Y = [y1 y119899] = [[[


]]] (17)



Cs = 1119899YYT (18)


Y = USVT (19)




Then

Cs = U(S2119899 )UT (21)






119889 = max2le119894le119899

dist (O1O119894) 119871 = 119899minus1sum119894=1

dist (O119894O119894+1) (22)








dist (O(119895minus1)119904+1O119902) (24)


dist (O119902O119902+1) (25)




GPF3119894 = 1198860 + 1198861119909119894 + 11988621199092119894 + 11988631199093119894 (26)



GPF1119894 = 1198870 + 1198871119909119894 (27)




(28)















1X

2X 3X

times10minus4

times10minus6

minus39

minus38

minus37

minus36

Vert

ical

resp

onse

(m)

01 02 03 04 05 06 07 08 09 10Time (s)

0

2

4

6A

mpl

itude

5 10 15 20 25 30 350Frequency (Hz)











No crack

times10minus3

minus15

minus1

minus05

0

05

1

Vert

ical

resp

onse

(m)

0 10 15 20 255Time (s)

(a)



One crack

times10minus3

minus15

minus1

minus05

0

05

1

Vert

ical

resp

onse

(m)

5 10 15 20 250Time (s)

(b)



Two cracks

times10minus3

minus2

minus15

minus1

minus05

0

05

1

Vert

ical

resp

onse

(m)

5 10 15 20 250Time (s)

(c)












One crack

Crack

minus1

minus05

0

05

1PO

M1

5 10 15 20 25 300Nodes

5 10 15 20 25 300Nodes

minus1

minus05

0

05

1

POM

2

(a)Two cracks

Crack 1 Crack 2

minus1

minus05

0

05

1

POM

1

5 10 15 20 25 300Nodes

5 10 15 20 25 300Nodes

minus1

minus05

0

05

1

POM

2

(b)


No noiseNoise = 5

021 022 023 024minus376

minus374

minus372

minus37

minus368

times10minus4

times10minus4

minus39

minus385

minus38

minus375

minus37

minus365

minus36

Vert

ical

resp

onse

(m)

005 01 015 02 025 03 035 040Time (s)







200 250 300 350minus1

minus08

minus06

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1PO

M1

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

POM

2

minus1

minus05

0

05

1

POM

3

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

POM

4

No noiseNoise = 5

No noiseNoise = 5





5 Conclusions



Abbreviations



No noiseNoise = 5

No noiseNoise = 5


GSM

Crack 1 Crack 2

Disc 1 Disc 2

times10minus4

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

1

2

3

4

DI1

100 200 300 400 500 6000Length (mm)

(a)

No noiseNoise = 5

No noiseNoise = 5


GSM

Crack 1

Crack 2

times10minus5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

2

DI3

100 200 300 400 500 6000Length (mm)

(b)

No noiseNoise = 5

Crack 1 Crack 2

Disc 1 Disc 2

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

1

1

10001

10001

FD

No noiseNoise = 5

(c)




Competing Interests



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

100 200 300 400 500 6000Length (mm)

No noiseNoise = 5

No noiseNoise = 5


(a)

Crack 1

Crack 2

Disc 1 Disc 2

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD

100 200 300 400 500 6000Length (mm)

No noiseNoise = 5

No noiseNoise = 5

(b)


Crack 1 Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

times10minus4

100 200 300 400 500 6000Length (mm)

0

2

4

6

8

DI1

(a)

Crack 1 Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

DI1

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD100 200 300 400 500 6000

Length (mm)

(b)



Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

times10minus3

0

05

1

DI1

(a)

Crack 1 ampdisc 1

Crack 2

Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD100 200 300 400 500 6000

Length (mm)

(b)


Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

100 200 300 400 500 6000Length (mm)

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

2

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10005

1001

FD

(b)


Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

2

4

6

8

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

1001

1002

1003

FD

100 200 300 400 500 6000Length (mm)

(b)



Crack 1

Crack 1

Crack 1

Crack 2


n = 7

n = 4

n = 3Nc + 1

Crack Nc

(a)

Crack 1

Crack 1

Crack 1

Crack 2


n = 9

n = 6

n = 3Nc + 3

Crack Nc

(b)


Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










x

y

z

CCL

P1

P2

P3P4

P5

P6

P7

P8

P9

P10

P11

P12120579

x998400

y998400

z998400

xL






Gce = G0 + Gc (1)



1198751 1198752 11987512T = T 1198751 1198752 1198756T (2)

Here

TT =[[[[[[[[[[[[


]]]]]]]]]]]]

(3)


TT 1199061 1199062 11990612T = Gce 1198751 1198752 1198756T (4)


1198751 1198752 11987512T = T (Gce)minus1 TT 1199061 1199062 11990612T (5)





Crack 2x

y

z

Crack 1

(a)

y

zo

eCrack 1

Crack 2

120579Ω

120579Ω 120579x

120579phi

120578

120585

m

e

120573

(b)



119870I =6sum119894=1

119870I119894 (7)



q119894 = 119909119894 119910119894 119911119894 120579119909119894 120579119910119894 120579119911119894T (8)



(9)



Fg119894 = 0 minus119898g 0 0 0 0T (10)


Fu119894 = 119865u119909119894 119865u119910119894 119865u119911119894119872u119909119894119872u119910119894119872u119911119894T (11)


119865u119909119894 = 0119872u119910119894 = 0119872u119911119894 = 0119865u119910119894 = 119898119890 [(Ω + 119909119894)2 cos (Ω119905 + 120579119909119894 + 120573)

+ 119909119894 sin (Ω119905 + 120579119909119894 + 120573)] 119865u119911119894 = 119898119890 [(Ω + 119909119894)2 sin (Ω119905 + 120579119909119894 + 120573)

+ x119894 cos (Ω119905 + 120579119909119894 + 120573)] 119872u119909119894 = 119898119890 [ sin (Ω119905 + 120579119909119894 + 120573)

minus ( + g) cos (Ω119905 + 120579119909119894 + 120573)]

(12)








120598 (119909 119905) = 120583 (119909) + 120599 (119909 119905) (13)



max120593(119909)

119869 (120593 (119909)) subject to 1003817100381710038171003817120593 (119909)10038171003817100381710038172 = 1

where 119869 (120593 (119909)) = ⟨10038161003816100381610038161003816(120599119896 (119909) 120593 (119909))100381610038161003816100381610038162⟩ (14)




max120593(119909)



intΠ⟨120599119896 (119909) 120599119896 (119910)⟩ 120593 (119910) d119910 = 120582120593 (119909) (16)





Y = [y1 y119899] = [[[


]]] (17)



Cs = 1119899YYT (18)


Y = USVT (19)




Then

Cs = U(S2119899 )UT (21)






119889 = max2le119894le119899

dist (O1O119894) 119871 = 119899minus1sum119894=1

dist (O119894O119894+1) (22)








dist (O(119895minus1)119904+1O119902) (24)


dist (O119902O119902+1) (25)




GPF3119894 = 1198860 + 1198861119909119894 + 11988621199092119894 + 11988631199093119894 (26)



GPF1119894 = 1198870 + 1198871119909119894 (27)




(28)















1X

2X 3X

times10minus4

times10minus6

minus39

minus38

minus37

minus36

Vert

ical

resp

onse

(m)

01 02 03 04 05 06 07 08 09 10Time (s)

0

2

4

6A

mpl

itude

5 10 15 20 25 30 350Frequency (Hz)











No crack

times10minus3

minus15

minus1

minus05

0

05

1

Vert

ical

resp

onse

(m)

0 10 15 20 255Time (s)

(a)



One crack

times10minus3

minus15

minus1

minus05

0

05

1

Vert

ical

resp

onse

(m)

5 10 15 20 250Time (s)

(b)



Two cracks

times10minus3

minus2

minus15

minus1

minus05

0

05

1

Vert

ical

resp

onse

(m)

5 10 15 20 250Time (s)

(c)












One crack

Crack

minus1

minus05

0

05

1PO

M1

5 10 15 20 25 300Nodes

5 10 15 20 25 300Nodes

minus1

minus05

0

05

1

POM

2

(a)Two cracks

Crack 1 Crack 2

minus1

minus05

0

05

1

POM

1

5 10 15 20 25 300Nodes

5 10 15 20 25 300Nodes

minus1

minus05

0

05

1

POM

2

(b)


No noiseNoise = 5

021 022 023 024minus376

minus374

minus372

minus37

minus368

times10minus4

times10minus4

minus39

minus385

minus38

minus375

minus37

minus365

minus36

Vert

ical

resp

onse

(m)

005 01 015 02 025 03 035 040Time (s)







200 250 300 350minus1

minus08

minus06

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1PO

M1

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

POM

2

minus1

minus05

0

05

1

POM

3

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

POM

4

No noiseNoise = 5

No noiseNoise = 5





5 Conclusions



Abbreviations



No noiseNoise = 5

No noiseNoise = 5


GSM

Crack 1 Crack 2

Disc 1 Disc 2

times10minus4

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

1

2

3

4

DI1

100 200 300 400 500 6000Length (mm)

(a)

No noiseNoise = 5

No noiseNoise = 5


GSM

Crack 1

Crack 2

times10minus5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

2

DI3

100 200 300 400 500 6000Length (mm)

(b)

No noiseNoise = 5

Crack 1 Crack 2

Disc 1 Disc 2

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

1

1

10001

10001

FD

No noiseNoise = 5

(c)




Competing Interests



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

100 200 300 400 500 6000Length (mm)

No noiseNoise = 5

No noiseNoise = 5


(a)

Crack 1

Crack 2

Disc 1 Disc 2

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD

100 200 300 400 500 6000Length (mm)

No noiseNoise = 5

No noiseNoise = 5

(b)


Crack 1 Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

times10minus4

100 200 300 400 500 6000Length (mm)

0

2

4

6

8

DI1

(a)

Crack 1 Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

DI1

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD100 200 300 400 500 6000

Length (mm)

(b)



Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

times10minus3

0

05

1

DI1

(a)

Crack 1 ampdisc 1

Crack 2

Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD100 200 300 400 500 6000

Length (mm)

(b)


Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

100 200 300 400 500 6000Length (mm)

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

2

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10005

1001

FD

(b)


Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

2

4

6

8

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

1001

1002

1003

FD

100 200 300 400 500 6000Length (mm)

(b)



Crack 1

Crack 1

Crack 1

Crack 2


n = 7

n = 4

n = 3Nc + 1

Crack Nc

(a)

Crack 1

Crack 1

Crack 1

Crack 2


n = 9

n = 6

n = 3Nc + 3

Crack Nc

(b)


Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Crack 2x

y

z

Crack 1

(a)

y

zo

eCrack 1

Crack 2

120579Ω

120579Ω 120579x

120579phi

120578

120585

m

e

120573

(b)



119870I =6sum119894=1

119870I119894 (7)



q119894 = 119909119894 119910119894 119911119894 120579119909119894 120579119910119894 120579119911119894T (8)



(9)



Fg119894 = 0 minus119898g 0 0 0 0T (10)


Fu119894 = 119865u119909119894 119865u119910119894 119865u119911119894119872u119909119894119872u119910119894119872u119911119894T (11)


119865u119909119894 = 0119872u119910119894 = 0119872u119911119894 = 0119865u119910119894 = 119898119890 [(Ω + 119909119894)2 cos (Ω119905 + 120579119909119894 + 120573)

+ 119909119894 sin (Ω119905 + 120579119909119894 + 120573)] 119865u119911119894 = 119898119890 [(Ω + 119909119894)2 sin (Ω119905 + 120579119909119894 + 120573)

+ x119894 cos (Ω119905 + 120579119909119894 + 120573)] 119872u119909119894 = 119898119890 [ sin (Ω119905 + 120579119909119894 + 120573)

minus ( + g) cos (Ω119905 + 120579119909119894 + 120573)]

(12)








120598 (119909 119905) = 120583 (119909) + 120599 (119909 119905) (13)



max120593(119909)

119869 (120593 (119909)) subject to 1003817100381710038171003817120593 (119909)10038171003817100381710038172 = 1

where 119869 (120593 (119909)) = ⟨10038161003816100381610038161003816(120599119896 (119909) 120593 (119909))100381610038161003816100381610038162⟩ (14)




max120593(119909)



intΠ⟨120599119896 (119909) 120599119896 (119910)⟩ 120593 (119910) d119910 = 120582120593 (119909) (16)





Y = [y1 y119899] = [[[


]]] (17)



Cs = 1119899YYT (18)


Y = USVT (19)




Then

Cs = U(S2119899 )UT (21)






119889 = max2le119894le119899

dist (O1O119894) 119871 = 119899minus1sum119894=1

dist (O119894O119894+1) (22)








dist (O(119895minus1)119904+1O119902) (24)


dist (O119902O119902+1) (25)




GPF3119894 = 1198860 + 1198861119909119894 + 11988621199092119894 + 11988631199093119894 (26)



GPF1119894 = 1198870 + 1198871119909119894 (27)




(28)















1X

2X 3X

times10minus4

times10minus6

minus39

minus38

minus37

minus36

Vert

ical

resp

onse

(m)

01 02 03 04 05 06 07 08 09 10Time (s)

0

2

4

6A

mpl

itude

5 10 15 20 25 30 350Frequency (Hz)











No crack

times10minus3

minus15

minus1

minus05

0

05

1

Vert

ical

resp

onse

(m)

0 10 15 20 255Time (s)

(a)



One crack

times10minus3

minus15

minus1

minus05

0

05

1

Vert

ical

resp

onse

(m)

5 10 15 20 250Time (s)

(b)



Two cracks

times10minus3

minus2

minus15

minus1

minus05

0

05

1

Vert

ical

resp

onse

(m)

5 10 15 20 250Time (s)

(c)












One crack

Crack

minus1

minus05

0

05

1PO

M1

5 10 15 20 25 300Nodes

5 10 15 20 25 300Nodes

minus1

minus05

0

05

1

POM

2

(a)Two cracks

Crack 1 Crack 2

minus1

minus05

0

05

1

POM

1

5 10 15 20 25 300Nodes

5 10 15 20 25 300Nodes

minus1

minus05

0

05

1

POM

2

(b)


No noiseNoise = 5

021 022 023 024minus376

minus374

minus372

minus37

minus368

times10minus4

times10minus4

minus39

minus385

minus38

minus375

minus37

minus365

minus36

Vert

ical

resp

onse

(m)

005 01 015 02 025 03 035 040Time (s)







200 250 300 350minus1

minus08

minus06

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1PO

M1

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

POM

2

minus1

minus05

0

05

1

POM

3

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

POM

4

No noiseNoise = 5

No noiseNoise = 5





5 Conclusions



Abbreviations



No noiseNoise = 5

No noiseNoise = 5


GSM

Crack 1 Crack 2

Disc 1 Disc 2

times10minus4

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

1

2

3

4

DI1

100 200 300 400 500 6000Length (mm)

(a)

No noiseNoise = 5

No noiseNoise = 5


GSM

Crack 1

Crack 2

times10minus5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

2

DI3

100 200 300 400 500 6000Length (mm)

(b)

No noiseNoise = 5

Crack 1 Crack 2

Disc 1 Disc 2

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

1

1

10001

10001

FD

No noiseNoise = 5

(c)




Competing Interests



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

100 200 300 400 500 6000Length (mm)

No noiseNoise = 5

No noiseNoise = 5


(a)

Crack 1

Crack 2

Disc 1 Disc 2

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD

100 200 300 400 500 6000Length (mm)

No noiseNoise = 5

No noiseNoise = 5

(b)


Crack 1 Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

times10minus4

100 200 300 400 500 6000Length (mm)

0

2

4

6

8

DI1

(a)

Crack 1 Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

DI1

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD100 200 300 400 500 6000

Length (mm)

(b)



Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

times10minus3

0

05

1

DI1

(a)

Crack 1 ampdisc 1

Crack 2

Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD100 200 300 400 500 6000

Length (mm)

(b)


Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

100 200 300 400 500 6000Length (mm)

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

2

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10005

1001

FD

(b)


Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

2

4

6

8

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

1001

1002

1003

FD

100 200 300 400 500 6000Length (mm)

(b)



Crack 1

Crack 1

Crack 1

Crack 2


n = 7

n = 4

n = 3Nc + 1

Crack Nc

(a)

Crack 1

Crack 1

Crack 1

Crack 2


n = 9

n = 6

n = 3Nc + 3

Crack Nc

(b)


Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














120598 (119909 119905) = 120583 (119909) + 120599 (119909 119905) (13)



max120593(119909)

119869 (120593 (119909)) subject to 1003817100381710038171003817120593 (119909)10038171003817100381710038172 = 1

where 119869 (120593 (119909)) = ⟨10038161003816100381610038161003816(120599119896 (119909) 120593 (119909))100381610038161003816100381610038162⟩ (14)




max120593(119909)



intΠ⟨120599119896 (119909) 120599119896 (119910)⟩ 120593 (119910) d119910 = 120582120593 (119909) (16)





Y = [y1 y119899] = [[[


]]] (17)



Cs = 1119899YYT (18)


Y = USVT (19)




Then

Cs = U(S2119899 )UT (21)






119889 = max2le119894le119899

dist (O1O119894) 119871 = 119899minus1sum119894=1

dist (O119894O119894+1) (22)








dist (O(119895minus1)119904+1O119902) (24)


dist (O119902O119902+1) (25)




GPF3119894 = 1198860 + 1198861119909119894 + 11988621199092119894 + 11988631199093119894 (26)



GPF1119894 = 1198870 + 1198871119909119894 (27)




(28)















1X

2X 3X

times10minus4

times10minus6

minus39

minus38

minus37

minus36

Vert

ical

resp

onse

(m)

01 02 03 04 05 06 07 08 09 10Time (s)

0

2

4

6A

mpl

itude

5 10 15 20 25 30 350Frequency (Hz)











No crack

times10minus3

minus15

minus1

minus05

0

05

1

Vert

ical

resp

onse

(m)

0 10 15 20 255Time (s)

(a)



One crack

times10minus3

minus15

minus1

minus05

0

05

1

Vert

ical

resp

onse

(m)

5 10 15 20 250Time (s)

(b)



Two cracks

times10minus3

minus2

minus15

minus1

minus05

0

05

1

Vert

ical

resp

onse

(m)

5 10 15 20 250Time (s)

(c)












One crack

Crack

minus1

minus05

0

05

1PO

M1

5 10 15 20 25 300Nodes

5 10 15 20 25 300Nodes

minus1

minus05

0

05

1

POM

2

(a)Two cracks

Crack 1 Crack 2

minus1

minus05

0

05

1

POM

1

5 10 15 20 25 300Nodes

5 10 15 20 25 300Nodes

minus1

minus05

0

05

1

POM

2

(b)


No noiseNoise = 5

021 022 023 024minus376

minus374

minus372

minus37

minus368

times10minus4

times10minus4

minus39

minus385

minus38

minus375

minus37

minus365

minus36

Vert

ical

resp

onse

(m)

005 01 015 02 025 03 035 040Time (s)







200 250 300 350minus1

minus08

minus06

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1PO

M1

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

POM

2

minus1

minus05

0

05

1

POM

3

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

POM

4

No noiseNoise = 5

No noiseNoise = 5





5 Conclusions



Abbreviations



No noiseNoise = 5

No noiseNoise = 5


GSM

Crack 1 Crack 2

Disc 1 Disc 2

times10minus4

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

1

2

3

4

DI1

100 200 300 400 500 6000Length (mm)

(a)

No noiseNoise = 5

No noiseNoise = 5


GSM

Crack 1

Crack 2

times10minus5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

2

DI3

100 200 300 400 500 6000Length (mm)

(b)

No noiseNoise = 5

Crack 1 Crack 2

Disc 1 Disc 2

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

1

1

10001

10001

FD

No noiseNoise = 5

(c)




Competing Interests



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

100 200 300 400 500 6000Length (mm)

No noiseNoise = 5

No noiseNoise = 5


(a)

Crack 1

Crack 2

Disc 1 Disc 2

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD

100 200 300 400 500 6000Length (mm)

No noiseNoise = 5

No noiseNoise = 5

(b)


Crack 1 Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

times10minus4

100 200 300 400 500 6000Length (mm)

0

2

4

6

8

DI1

(a)

Crack 1 Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

DI1

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD100 200 300 400 500 6000

Length (mm)

(b)



Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

times10minus3

0

05

1

DI1

(a)

Crack 1 ampdisc 1

Crack 2

Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD100 200 300 400 500 6000

Length (mm)

(b)


Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

100 200 300 400 500 6000Length (mm)

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

2

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10005

1001

FD

(b)


Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

2

4

6

8

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

1001

1002

1003

FD

100 200 300 400 500 6000Length (mm)

(b)



Crack 1

Crack 1

Crack 1

Crack 2


n = 7

n = 4

n = 3Nc + 1

Crack Nc

(a)

Crack 1

Crack 1

Crack 1

Crack 2


n = 9

n = 6

n = 3Nc + 3

Crack Nc

(b)


Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














dist (O(119895minus1)119904+1O119902) (24)


dist (O119902O119902+1) (25)




GPF3119894 = 1198860 + 1198861119909119894 + 11988621199092119894 + 11988631199093119894 (26)



GPF1119894 = 1198870 + 1198871119909119894 (27)




(28)















1X

2X 3X

times10minus4

times10minus6

minus39

minus38

minus37

minus36

Vert

ical

resp

onse

(m)

01 02 03 04 05 06 07 08 09 10Time (s)

0

2

4

6A

mpl

itude

5 10 15 20 25 30 350Frequency (Hz)











No crack

times10minus3

minus15

minus1

minus05

0

05

1

Vert

ical

resp

onse

(m)

0 10 15 20 255Time (s)

(a)



One crack

times10minus3

minus15

minus1

minus05

0

05

1

Vert

ical

resp

onse

(m)

5 10 15 20 250Time (s)

(b)



Two cracks

times10minus3

minus2

minus15

minus1

minus05

0

05

1

Vert

ical

resp

onse

(m)

5 10 15 20 250Time (s)

(c)












One crack

Crack

minus1

minus05

0

05

1PO

M1

5 10 15 20 25 300Nodes

5 10 15 20 25 300Nodes

minus1

minus05

0

05

1

POM

2

(a)Two cracks

Crack 1 Crack 2

minus1

minus05

0

05

1

POM

1

5 10 15 20 25 300Nodes

5 10 15 20 25 300Nodes

minus1

minus05

0

05

1

POM

2

(b)


No noiseNoise = 5

021 022 023 024minus376

minus374

minus372

minus37

minus368

times10minus4

times10minus4

minus39

minus385

minus38

minus375

minus37

minus365

minus36

Vert

ical

resp

onse

(m)

005 01 015 02 025 03 035 040Time (s)







200 250 300 350minus1

minus08

minus06

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1PO

M1

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

POM

2

minus1

minus05

0

05

1

POM

3

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

POM

4

No noiseNoise = 5

No noiseNoise = 5





5 Conclusions



Abbreviations



No noiseNoise = 5

No noiseNoise = 5


GSM

Crack 1 Crack 2

Disc 1 Disc 2

times10minus4

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

1

2

3

4

DI1

100 200 300 400 500 6000Length (mm)

(a)

No noiseNoise = 5

No noiseNoise = 5


GSM

Crack 1

Crack 2

times10minus5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

2

DI3

100 200 300 400 500 6000Length (mm)

(b)

No noiseNoise = 5

Crack 1 Crack 2

Disc 1 Disc 2

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

1

1

10001

10001

FD

No noiseNoise = 5

(c)




Competing Interests



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

100 200 300 400 500 6000Length (mm)

No noiseNoise = 5

No noiseNoise = 5


(a)

Crack 1

Crack 2

Disc 1 Disc 2

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD

100 200 300 400 500 6000Length (mm)

No noiseNoise = 5

No noiseNoise = 5

(b)


Crack 1 Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

times10minus4

100 200 300 400 500 6000Length (mm)

0

2

4

6

8

DI1

(a)

Crack 1 Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

DI1

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD100 200 300 400 500 6000

Length (mm)

(b)



Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

times10minus3

0

05

1

DI1

(a)

Crack 1 ampdisc 1

Crack 2

Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD100 200 300 400 500 6000

Length (mm)

(b)


Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

100 200 300 400 500 6000Length (mm)

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

2

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10005

1001

FD

(b)


Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

2

4

6

8

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

1001

1002

1003

FD

100 200 300 400 500 6000Length (mm)

(b)



Crack 1

Crack 1

Crack 1

Crack 2


n = 7

n = 4

n = 3Nc + 1

Crack Nc

(a)

Crack 1

Crack 1

Crack 1

Crack 2


n = 9

n = 6

n = 3Nc + 3

Crack Nc

(b)


Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1X

2X 3X

times10minus4

times10minus6

minus39

minus38

minus37

minus36

Vert

ical

resp

onse

(m)

01 02 03 04 05 06 07 08 09 10Time (s)

0

2

4

6A

mpl

itude

5 10 15 20 25 30 350Frequency (Hz)











No crack

times10minus3

minus15

minus1

minus05

0

05

1

Vert

ical

resp

onse

(m)

0 10 15 20 255Time (s)

(a)



One crack

times10minus3

minus15

minus1

minus05

0

05

1

Vert

ical

resp

onse

(m)

5 10 15 20 250Time (s)

(b)



Two cracks

times10minus3

minus2

minus15

minus1

minus05

0

05

1

Vert

ical

resp

onse

(m)

5 10 15 20 250Time (s)

(c)












One crack

Crack

minus1

minus05

0

05

1PO

M1

5 10 15 20 25 300Nodes

5 10 15 20 25 300Nodes

minus1

minus05

0

05

1

POM

2

(a)Two cracks

Crack 1 Crack 2

minus1

minus05

0

05

1

POM

1

5 10 15 20 25 300Nodes

5 10 15 20 25 300Nodes

minus1

minus05

0

05

1

POM

2

(b)


No noiseNoise = 5

021 022 023 024minus376

minus374

minus372

minus37

minus368

times10minus4

times10minus4

minus39

minus385

minus38

minus375

minus37

minus365

minus36

Vert

ical

resp

onse

(m)

005 01 015 02 025 03 035 040Time (s)







200 250 300 350minus1

minus08

minus06

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1PO

M1

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

POM

2

minus1

minus05

0

05

1

POM

3

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

POM

4

No noiseNoise = 5

No noiseNoise = 5





5 Conclusions



Abbreviations



No noiseNoise = 5

No noiseNoise = 5


GSM

Crack 1 Crack 2

Disc 1 Disc 2

times10minus4

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

1

2

3

4

DI1

100 200 300 400 500 6000Length (mm)

(a)

No noiseNoise = 5

No noiseNoise = 5


GSM

Crack 1

Crack 2

times10minus5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

2

DI3

100 200 300 400 500 6000Length (mm)

(b)

No noiseNoise = 5

Crack 1 Crack 2

Disc 1 Disc 2

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

1

1

10001

10001

FD

No noiseNoise = 5

(c)




Competing Interests



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

100 200 300 400 500 6000Length (mm)

No noiseNoise = 5

No noiseNoise = 5


(a)

Crack 1

Crack 2

Disc 1 Disc 2

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD

100 200 300 400 500 6000Length (mm)

No noiseNoise = 5

No noiseNoise = 5

(b)


Crack 1 Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

times10minus4

100 200 300 400 500 6000Length (mm)

0

2

4

6

8

DI1

(a)

Crack 1 Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

DI1

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD100 200 300 400 500 6000

Length (mm)

(b)



Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

times10minus3

0

05

1

DI1

(a)

Crack 1 ampdisc 1

Crack 2

Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD100 200 300 400 500 6000

Length (mm)

(b)


Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

100 200 300 400 500 6000Length (mm)

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

2

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10005

1001

FD

(b)


Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

2

4

6

8

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

1001

1002

1003

FD

100 200 300 400 500 6000Length (mm)

(b)



Crack 1

Crack 1

Crack 1

Crack 2


n = 7

n = 4

n = 3Nc + 1

Crack Nc

(a)

Crack 1

Crack 1

Crack 1

Crack 2


n = 9

n = 6

n = 3Nc + 3

Crack Nc

(b)


Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










No crack

times10minus3

minus15

minus1

minus05

0

05

1

Vert

ical

resp

onse

(m)

0 10 15 20 255Time (s)

(a)



One crack

times10minus3

minus15

minus1

minus05

0

05

1

Vert

ical

resp

onse

(m)

5 10 15 20 250Time (s)

(b)



Two cracks

times10minus3

minus2

minus15

minus1

minus05

0

05

1

Vert

ical

resp

onse

(m)

5 10 15 20 250Time (s)

(c)












One crack

Crack

minus1

minus05

0

05

1PO

M1

5 10 15 20 25 300Nodes

5 10 15 20 25 300Nodes

minus1

minus05

0

05

1

POM

2

(a)Two cracks

Crack 1 Crack 2

minus1

minus05

0

05

1

POM

1

5 10 15 20 25 300Nodes

5 10 15 20 25 300Nodes

minus1

minus05

0

05

1

POM

2

(b)


No noiseNoise = 5

021 022 023 024minus376

minus374

minus372

minus37

minus368

times10minus4

times10minus4

minus39

minus385

minus38

minus375

minus37

minus365

minus36

Vert

ical

resp

onse

(m)

005 01 015 02 025 03 035 040Time (s)







200 250 300 350minus1

minus08

minus06

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1PO

M1

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

POM

2

minus1

minus05

0

05

1

POM

3

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

POM

4

No noiseNoise = 5

No noiseNoise = 5





5 Conclusions



Abbreviations



No noiseNoise = 5

No noiseNoise = 5


GSM

Crack 1 Crack 2

Disc 1 Disc 2

times10minus4

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

1

2

3

4

DI1

100 200 300 400 500 6000Length (mm)

(a)

No noiseNoise = 5

No noiseNoise = 5


GSM

Crack 1

Crack 2

times10minus5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

2

DI3

100 200 300 400 500 6000Length (mm)

(b)

No noiseNoise = 5

Crack 1 Crack 2

Disc 1 Disc 2

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

1

1

10001

10001

FD

No noiseNoise = 5

(c)




Competing Interests



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

100 200 300 400 500 6000Length (mm)

No noiseNoise = 5

No noiseNoise = 5


(a)

Crack 1

Crack 2

Disc 1 Disc 2

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD

100 200 300 400 500 6000Length (mm)

No noiseNoise = 5

No noiseNoise = 5

(b)


Crack 1 Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

times10minus4

100 200 300 400 500 6000Length (mm)

0

2

4

6

8

DI1

(a)

Crack 1 Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

DI1

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD100 200 300 400 500 6000

Length (mm)

(b)



Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

times10minus3

0

05

1

DI1

(a)

Crack 1 ampdisc 1

Crack 2

Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD100 200 300 400 500 6000

Length (mm)

(b)


Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

100 200 300 400 500 6000Length (mm)

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

2

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10005

1001

FD

(b)


Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

2

4

6

8

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

1001

1002

1003

FD

100 200 300 400 500 6000Length (mm)

(b)



Crack 1

Crack 1

Crack 1

Crack 2


n = 7

n = 4

n = 3Nc + 1

Crack Nc

(a)

Crack 1

Crack 1

Crack 1

Crack 2


n = 9

n = 6

n = 3Nc + 3

Crack Nc

(b)


Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










One crack

Crack

minus1

minus05

0

05

1PO

M1

5 10 15 20 25 300Nodes

5 10 15 20 25 300Nodes

minus1

minus05

0

05

1

POM

2

(a)Two cracks

Crack 1 Crack 2

minus1

minus05

0

05

1

POM

1

5 10 15 20 25 300Nodes

5 10 15 20 25 300Nodes

minus1

minus05

0

05

1

POM

2

(b)


No noiseNoise = 5

021 022 023 024minus376

minus374

minus372

minus37

minus368

times10minus4

times10minus4

minus39

minus385

minus38

minus375

minus37

minus365

minus36

Vert

ical

resp

onse

(m)

005 01 015 02 025 03 035 040Time (s)







200 250 300 350minus1

minus08

minus06

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1PO

M1

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

POM

2

minus1

minus05

0

05

1

POM

3

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

POM

4

No noiseNoise = 5

No noiseNoise = 5





5 Conclusions



Abbreviations



No noiseNoise = 5

No noiseNoise = 5


GSM

Crack 1 Crack 2

Disc 1 Disc 2

times10minus4

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

1

2

3

4

DI1

100 200 300 400 500 6000Length (mm)

(a)

No noiseNoise = 5

No noiseNoise = 5


GSM

Crack 1

Crack 2

times10minus5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

2

DI3

100 200 300 400 500 6000Length (mm)

(b)

No noiseNoise = 5

Crack 1 Crack 2

Disc 1 Disc 2

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

1

1

10001

10001

FD

No noiseNoise = 5

(c)




Competing Interests



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

100 200 300 400 500 6000Length (mm)

No noiseNoise = 5

No noiseNoise = 5


(a)

Crack 1

Crack 2

Disc 1 Disc 2

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD

100 200 300 400 500 6000Length (mm)

No noiseNoise = 5

No noiseNoise = 5

(b)


Crack 1 Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

times10minus4

100 200 300 400 500 6000Length (mm)

0

2

4

6

8

DI1

(a)

Crack 1 Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

DI1

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD100 200 300 400 500 6000

Length (mm)

(b)



Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

times10minus3

0

05

1

DI1

(a)

Crack 1 ampdisc 1

Crack 2

Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD100 200 300 400 500 6000

Length (mm)

(b)


Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

100 200 300 400 500 6000Length (mm)

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

2

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10005

1001

FD

(b)


Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

2

4

6

8

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

1001

1002

1003

FD

100 200 300 400 500 6000Length (mm)

(b)



Crack 1

Crack 1

Crack 1

Crack 2


n = 7

n = 4

n = 3Nc + 1

Crack Nc

(a)

Crack 1

Crack 1

Crack 1

Crack 2


n = 9

n = 6

n = 3Nc + 3

Crack Nc

(b)


Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










200 250 300 350minus1

minus08

minus06

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1PO

M1

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

POM

2

minus1

minus05

0

05

1

POM

3

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

POM

4

No noiseNoise = 5

No noiseNoise = 5





5 Conclusions



Abbreviations



No noiseNoise = 5

No noiseNoise = 5


GSM

Crack 1 Crack 2

Disc 1 Disc 2

times10minus4

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

1

2

3

4

DI1

100 200 300 400 500 6000Length (mm)

(a)

No noiseNoise = 5

No noiseNoise = 5


GSM

Crack 1

Crack 2

times10minus5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

2

DI3

100 200 300 400 500 6000Length (mm)

(b)

No noiseNoise = 5

Crack 1 Crack 2

Disc 1 Disc 2

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

1

1

10001

10001

FD

No noiseNoise = 5

(c)




Competing Interests



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

100 200 300 400 500 6000Length (mm)

No noiseNoise = 5

No noiseNoise = 5


(a)

Crack 1

Crack 2

Disc 1 Disc 2

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD

100 200 300 400 500 6000Length (mm)

No noiseNoise = 5

No noiseNoise = 5

(b)


Crack 1 Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

times10minus4

100 200 300 400 500 6000Length (mm)

0

2

4

6

8

DI1

(a)

Crack 1 Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

DI1

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD100 200 300 400 500 6000

Length (mm)

(b)



Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

times10minus3

0

05

1

DI1

(a)

Crack 1 ampdisc 1

Crack 2

Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD100 200 300 400 500 6000

Length (mm)

(b)


Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

100 200 300 400 500 6000Length (mm)

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

2

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10005

1001

FD

(b)


Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

2

4

6

8

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

1001

1002

1003

FD

100 200 300 400 500 6000Length (mm)

(b)



Crack 1

Crack 1

Crack 1

Crack 2


n = 7

n = 4

n = 3Nc + 1

Crack Nc

(a)

Crack 1

Crack 1

Crack 1

Crack 2


n = 9

n = 6

n = 3Nc + 3

Crack Nc

(b)


Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










No noiseNoise = 5

No noiseNoise = 5


GSM

Crack 1 Crack 2

Disc 1 Disc 2

times10minus4

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

1

2

3

4

DI1

100 200 300 400 500 6000Length (mm)

(a)

No noiseNoise = 5

No noiseNoise = 5


GSM

Crack 1

Crack 2

times10minus5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

2

DI3

100 200 300 400 500 6000Length (mm)

(b)

No noiseNoise = 5

Crack 1 Crack 2

Disc 1 Disc 2

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

1

1

10001

10001

FD

No noiseNoise = 5

(c)




Competing Interests



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

100 200 300 400 500 6000Length (mm)

No noiseNoise = 5

No noiseNoise = 5


(a)

Crack 1

Crack 2

Disc 1 Disc 2

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD

100 200 300 400 500 6000Length (mm)

No noiseNoise = 5

No noiseNoise = 5

(b)


Crack 1 Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

times10minus4

100 200 300 400 500 6000Length (mm)

0

2

4

6

8

DI1

(a)

Crack 1 Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

DI1

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD100 200 300 400 500 6000

Length (mm)

(b)



Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

times10minus3

0

05

1

DI1

(a)

Crack 1 ampdisc 1

Crack 2

Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD100 200 300 400 500 6000

Length (mm)

(b)


Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

100 200 300 400 500 6000Length (mm)

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

2

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10005

1001

FD

(b)


Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

2

4

6

8

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

1001

1002

1003

FD

100 200 300 400 500 6000Length (mm)

(b)



Crack 1

Crack 1

Crack 1

Crack 2


n = 7

n = 4

n = 3Nc + 1

Crack Nc

(a)

Crack 1

Crack 1

Crack 1

Crack 2


n = 9

n = 6

n = 3Nc + 3

Crack Nc

(b)


Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

100 200 300 400 500 6000Length (mm)

No noiseNoise = 5

No noiseNoise = 5


(a)

Crack 1

Crack 2

Disc 1 Disc 2

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD

100 200 300 400 500 6000Length (mm)

No noiseNoise = 5

No noiseNoise = 5

(b)


Crack 1 Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

times10minus4

100 200 300 400 500 6000Length (mm)

0

2

4

6

8

DI1

(a)

Crack 1 Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

DI1

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD100 200 300 400 500 6000

Length (mm)

(b)



Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

times10minus3

0

05

1

DI1

(a)

Crack 1 ampdisc 1

Crack 2

Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD100 200 300 400 500 6000

Length (mm)

(b)


Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

100 200 300 400 500 6000Length (mm)

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

2

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10005

1001

FD

(b)


Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

2

4

6

8

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

1001

1002

1003

FD

100 200 300 400 500 6000Length (mm)

(b)



Crack 1

Crack 1

Crack 1

Crack 2


n = 7

n = 4

n = 3Nc + 1

Crack Nc

(a)

Crack 1

Crack 1

Crack 1

Crack 2


n = 9

n = 6

n = 3Nc + 3

Crack Nc

(b)


Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


100 200 300 400 500 6000Length (mm)

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

DI1

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD100 200 300 400 500 6000

Length (mm)

(b)



Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

times10minus3

0

05

1

DI1

(a)

Crack 1 ampdisc 1

Crack 2

Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD100 200 300 400 500 6000

Length (mm)

(b)


Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

100 200 300 400 500 6000Length (mm)

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

2

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10005

1001

FD

(b)


Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

2

4

6

8

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

1001

1002

1003

FD

100 200 300 400 500 6000Length (mm)

(b)



Crack 1

Crack 1

Crack 1

Crack 2


n = 7

n = 4

n = 3Nc + 1

Crack Nc

(a)

Crack 1

Crack 1

Crack 1

Crack 2


n = 9

n = 6

n = 3Nc + 3

Crack Nc

(b)


Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10002

10004

10006

FD100 200 300 400 500 6000

Length (mm)

(b)


Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

0

05

1

DI1

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

10001

10002

10003

10004

FD

100 200 300 400 500 6000Length (mm)

(b)



times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

2

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10005

1001

FD

(b)


Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

2

4

6

8

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

1001

1002

1003

FD

100 200 300 400 500 6000Length (mm)

(b)



Crack 1

Crack 1

Crack 1

Crack 2


n = 7

n = 4

n = 3Nc + 1

Crack Nc

(a)

Crack 1

Crack 1

Crack 1

Crack 2


n = 9

n = 6

n = 3Nc + 3

Crack Nc

(b)


Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










times10minus3

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5


minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

05

1

15

2

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1

Crack 2

Disc 1 Disc 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

100 200 300 400 500 6000Length (mm)

1

10005

1001

FD

(b)


Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5


times10minus3

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

0

2

4

6

8

DI1

100 200 300 400 500 6000Length (mm)

(a)

Crack 1 Crack 2

No noiseNoise = 5

No noiseNoise = 5

minus1

minus05

0

05

1

CPO

M

100 200 300 400 500 6000Length (mm)

1

1001

1002

1003

FD

100 200 300 400 500 6000Length (mm)

(b)



Crack 1

Crack 1

Crack 1

Crack 2


n = 7

n = 4

n = 3Nc + 1

Crack Nc

(a)

Crack 1

Crack 1

Crack 1

Crack 2


n = 9

n = 6

n = 3Nc + 3

Crack Nc

(b)


Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Crack 1

Crack 1

Crack 1

Crack 2


n = 7

n = 4

n = 3Nc + 1

Crack Nc

(a)

Crack 1

Crack 1

Crack 1

Crack 2


n = 9

n = 6

n = 3Nc + 3

Crack Nc

(b)


Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article multicrack localization in rotors based

Documents