Hindawi Publishing CorporationISRNMechanical EngineeringVolume 2013 Article ID 249035 10 pageshttpdxdoiorg1011552013249035

Research ArticleFinite Element Multibody Simulation of a Breathing Crack ina Rotor with a Cohesive Zone Model

Rugerri Toni Liong and Carsten Proppe

Institute of Engineering Mechanics Karlsruhe Institute of Technology Kaiserstraszlige 10 76131 Karlsruhe Germany

Correspondence should be addressed to Rugerri Toni Liong liongitmuni-karlsruhede

Received 17 January 2013 Accepted 6 February 2013

Academic Editors N Anifantis R Brighenti X Deng F Liu and J Seok

Copyright copy 2013 R T Liong and C Proppe This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The breathing mechanism of a transversely cracked shaft and its influence on a rotor system that appears due to shaft weight andinertia forces is studiedThe presence of a crack reduces the stiffness of the rotor system and introduces a stiffness variation duringthe revolution of the shaft Here 3D finite element (FE) model and multibody simulation (MBS) are introduced to predict and toanalyse the breathingmechanism on a transverse cracked shaft It is based on a cohesive zonemodel (CZM) instead of linear-elasticfracture mechanics (LEFM) First the elastic cracked shaft is modelled by 3D FE As a second step the 3D FE model of the shaft istransferred into anMBSmodel in order to analyze the dynamic loads due to the crack and the inertia force acting during rotation atdifferent rotating speeds Finally the vibration responses in the centroid of the shaft obtained fromMBS have been exported into FEmodel in order to observe the breathing mechanism A bilinear crack closure model is proposedThe accuracy of the bilinear crackclosure model and the solution techniques have been demonstrated by a comparison with the corresponding results of previouspublications

1 Introduction

Fatigue cracking of rotor shafts has long been identified asa limiting factor for safe and reliable operation of turbo-machines It can lead to catastrophic failure and great eco-nomic loss if not detected early A crack in the rotor causeslocal changes in stiffness These changes in turn affect thedynamics of the system frequency of the natural vibrationsand the amplitudes of forced vibrations are changed If acracked shaft rotates under external loading the crack opensand closes regularly during the revolution of the shaft itbreathes The breathing mechanism is produced by the stressdistribution around the crack mainly due to the action ofbending moment while the effect of torsion is negligibleUsually shaft cracks breathewhen crack sizes are small runn-ing speeds are low and radial forces are large [1]

The influence of a breathing crack on the vibration ofa rotating shaft has been in the focus of many researchersComprehensive literature survey of various crack modellingtechniques system behaviour of cracked rotor and detectionprocedures to diagnose fracture damage were contributed

Wauer [2] Dimarogonas [3] Sabnavis et al [4] and Kumarand Rastogi [5] More recent studies have been reviewedby Bachschmid et al [6] They noted that the breathingmechanism of cracks in rotating shafts can be accuratelyinvestigated by means of 3D nonlinear finite element modelsBased on these simulations simple approximation of thebreathing mechanism describing the location and extensionof the crack closure line during rotation can be established

The breathing mechanism is the result of the stress andstrain distribution around the cracked area which is dueto static loads like the weight bearing reaction forces anddynamical loads and due to the unbalance- and the vibra-tion-induced inertia force distribution [7] The accuratemodelling of the breathing behaviour that is the graduallyopening and closing of the crack during one rotation ofthe shaft still needs further investigations An originalmethod for calculating the constitutive law of a cracked beamsection under bending has been proposed by Andrieux andVare [8] Based on three-dimensional computations takinginto account the unilateral contact between the lips of thecrack it consists in defining a constitutive relation between

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the bending moment applied to the cracked section and theresulting field of displacements compatible with the beamtheory so that it can be used in rotor-dynamics softwareVare and Andrieux [9] extended this method in order toshow how shear effects can be implemented in the model of acracked section Arem and Maitournam [10] presented stiff-ness variations deduced from 3D FE calculations accountingfor the unilateral contact between the crack lips as originallydeveloped by Andrieux and Vare [8] Dimarogonas andPaipetis [11] introduced the FE models of the rotating shaftcracks They used the fracture mechanics and obtained a full6 times 6 flexibility matrix for a transverse open surface crackon shaft Darpe et al [12] studied the coupling betweenlongitudinal lateral and torsional vibrations together fora rotating cracked shaft By including the axial degree offreedom in their analysis the stiffness matrix formulatedis an extension of the one developed by Ostachowicz andKrawczuk [13] Kulesza and Sawicki [14] used the rigidFEM for modelling a crack in a rotating shaft The crack ispresented as a set of spring-damping elements of variablestiffness connecting two sections of the shaft Bouboulas andAnifantis [15] develop a finite elementmodel in order to studythe vibrational behavior of a beam with a nonpropagatingedge crack According to their model the beam is discretizedinto finite elements while the breathing crack behavior istreated as a full frictional contact problem between the cracksurfaces

The cohesive zone model (CZM) has been widely usedas an alternative to stress intensity factor-based fracturemechanics It can deal with the nonlinear zone ahead ofthe crack tip due to plasticity or microcracking The CZMdescribes material failure on a more phenomenological basis(ie without considering the material microstructure) Thegeneral advantage of this model when compared to classicalfracture mechanics is that the parameters of the respectivemodels depend only on thematerial and not on the geometryThis concept guarantees transferability from specimen tostructure over a wide range of geometries The origin ofthe cohesive zone concept can be traced back to the stripyield model proposed by Dugdale [16] and Barenblatt [17]in which the narrow zone of localized deformation ahead ofthe crack tip was substituted by cohesive traction betweenthe bounding surfaces The constitutive behaviour whichcauses the cohesive elements to open and eventually to failis described by the so-called traction separation law It relatesthe traction vector to the displacement jump across the inter-face and is usually called separation The energy dissipatedby the element until total failure is derived as the integral ofthe traction-separationHowever the traction-separation lawdepends on the stress state which can be characterised bythe triaxiality (ie the hydrostatic stress divided by the vonMises equivalent stress) This issue was first investigated bySiegmund and Brocks [18 19] The approach was extended tosimulation of dynamic ductile crack growth by Anvari et al[20] Scheider [21] Banerjee andManivasagam [22] proposeda versatile CZM to predict ductile fracture at different statesof stress The formulation developed for mode-I plane strainaccounts explicitly for triaxiality of the stress state by usingbasic elastic-plastic constitutive relations combined with two

new model parameters which are independent of the stressstate

The theory of strain energy release rate and SIF combinedwith rotordynamics built the foundation of the dynamic ofcracked rotors based on LEFM [23] This theory has twomajor limitations namely it can only be used if there is aninitial crack and the fracture process zone must be smallcompared with the dimensions of the shaft Due to geometri-cal complexity some simplifications had been made for thecrack profile such as straight-edged circular and ellipticalcrack model to analyze such crack problem Since analyticalSIFs for an edge crack in a rotating cylinder are not availablethe shaft is considered to be a sumof elementary independentrectangular strips (Andrieux andVare [8]) and no interactionbetween them is assumed to take place (Chasalevris andPapadopoulos [24]) The geometric functions that describethe strain energy density are often not accurate enough dueto the fact that the crack passes from stress state caused bythe vertical moment to that of a horizontal moment Thenthe compliance is obtained by integrating along the cracktip If the crack depth exceeds the radius of the shaft thenthe elements of the compliance matrix present a divergenceThis is due to the singularity that the strain energy releaserate method has near the edges of the crack tip givingthus the infinite values It was reported by Papadopoulos[25] that divergence does not reflect reality The crack tip issupposed to be formed by the boundary between the crackedareas and the uncracked areas for the regions in which thebreathing crack is open However the SIF will not appearat the boundary between the closed cracked areas and theopen cracked areas [7] Liong and Proppe [26] proposed amethod for the evaluation of the stiffness losses in a rotorwith a transverse breathing crackTheir method is based on aCZM and accounts explicitly for triaxiality of the stress stateby using constitutive relations

The aim of the present study is to propose a method forthe evaluation of stiffness losses of the cracked shaft based ona CZM The method provides a direct relationship betweenthe material properties of the shaft and the relative crackdepth thus representing the physical phenomenon closelyThis paper introduces a method based on a 3D FE modeland MBS in order to study the breathing mechanism of anelastic cracked shaftTheCZMformulation is implemented ina 3D FEmodelThe elastic cracked shaft with various relativecrack depths is modelled by 3D FE and then transferred intoan MBS model in order to analyze the dynamic loads actingduring rotation at different rotating speeds The vibrationresponses in the centroid of the rotating shaft obtained fromMBS have been exported into FE to observe the breathingmechanism The accuracy of the results is demonstratedthrough comparisons with the results available in the liter-ature

2 The Cohesive Zone Model

21 Basis of the Cohesive Process Zone Model In the CZMfracture nucleates as discontinuity surface able to transmittensile load before opening above a given displacementFormation and extension of this surface require that the

ISRNMechanical Engineering 3

Initialcrack

Crackedzone Cohesive fracture zone

Undamagedligament

120575120590max

120590 = 0120590 = 119891(120575)

120575119888

Figure 1 Cohesive crack process zone

maximum principal stress reaches a given value namely thecohesive strength of thematerialWhen this occurs the cracksurface initiates or grows perpendicularly to the directionof the maximum principal stress The two faces of thesurface exert on each other equal and opposite tensile stresses(cohesive stresses) whose value is a unique function 119891(120575)of the separation 120575 between the faces When the separationreaches another given value (the critical separation 120575

119888) the

cohesive stress vanishes and fracture takes place Fractureconsists of the initiation and propagation of a crack producedby the opening and advance of the cohesive zone (the zonewhere the cohesive stresses act) ahead of the crack tip asshown in Figure 1

The fracture behaviour of each material is described bythe cohesive traction as a function

120590119899= 120590max119891 (120575) (1)

where 120590max is the peak value of traction The function 119891(120575)defines the shape of the traction-separation law and the areaunder the cohesive law curve is the work of separation orcohesive energy 119866

119888

119866119888= int

120575119888

0

120590119899(120575) d120575 (2)

Ideally the model within the cohesive zone should beable to replicate the constitutive behaviour of the undamagedmaterial linear elastic followed by strain hardening till theconditions for the initiation of softening due to damage arereached The softening process representative of increasingmaterial degradation is triggered by rapid growth of voids asconsequence of a highly triaxial state of stress

Since the CZM is a phenomenological model variousformulations for defining the shape of traction-separation lawand the cohesive values are in use [27] A versatile CZM topredict ductile fracture at different states of stress is proposedin [22] The formulation developed for mode-I plane strainaccounts explicitly for triaxiality of the stress state by usingbasic elastic-plastic constitutive relations combined withtwo stress-state-independent new model parameters Theproposed traction-separation law has three distinct regions ofconstitutive behaviour the traction separation law is linear upto the separation limit 120575

119899= 1205751and exhibits strain hardening

Linearregion

120590max

Hardeningregion

Softeningregion

Cohesive energy

0 1205751 1205752 120575c = 120575sep120575119899

120590119899

120590119884

119870119901

Figure 2 Traction separation law for ductile materials

up to 120575119899= 1205752followed by a softening curve The relevant

variables and zones are sketched in Figure 2

120590119899= (1 + radic3120594ef)

2119864

3120575 for 0 lt 120575 lt 120575

1 (3)

The linear behaviour of the cohesive zone exists till theseparation limit defined by the von Mises yield condition isreached

1205751=radic3120590119884

2119864 (4)

Further separation results in strain hardening up to

1205752=radic3

2(119862eminus(32)120594ef + 120590119884

119864) (5)

22 Numerical Representation of Cohesive Zone ModelsWhen fracture proceeds energymust be supplied by externalloads The bounding material undergoes elastoplastic defor-mation involving elastic energy and plastic dissipative energyIn addition to plasticity energy is supplied to the fractureprocess zone in the form of cohesive energy that is dissipatedwithin the cohesive elements The cohesive energy is the sumof the surface energy and all dissipative processes that takeplace within the crack tip regime For the present problem aperfect energy balance between external work119882 and the sumof elastic energy 119864el and cohesive energy 119866

119888[27 28] will be

assumed The energy balance is given by

119882 = 119864el + 119866119888 (6)

119882 = int

119905

0

(int119881

120590 120576el d119881) d119905 + int119905

0

(int119878

120590119899 d119878) d119905 (7)

where 120590 120576el 120590119899 and are nominal stress tensor elasticstrain rate cohesive traction and cohesive separation raterespectively (including the terms for specimen volume119881 andthe internal specimen surface 119878)The external work due to theapplied force is given by

119882 = int

119905

0

(int119878

t sdot v d119878) d119905 + int119905

0

(int119881

b sdot v d119881) d119905 (8)

where v is the velocity field vector t the exterior surfacetraction vector and b the body force vector

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Real

Finite elements

IdealizationContinuum elements

Crack tip

Cohesive elements

Continuum

Crack

Brokencohesiveelements

Separationof cohesiveelements

Cohesivetractions120590119899 = 119891(120575)

Δ119886

Figure 3 Representation of the fracture process using CZM in FE

The cohesive surface contribution representing the crackand the process zone in front of the crack tip is described bythe integral over the internal surface 119878 In this formulation 120590

119899

denotes the cohesive strength that is the maximum tractionvalue that can be sustained within the cohesive zone Thecohesive length 120575 is the value of the displacement jump acrossthe crack surfaces at which the stress carrying capacity of thecohesive elements reaches its maximum value By creatingnew surfaces the traction and the stiffness of the cohesivezone elements connecting these newly created surfaces aremade to vanish but the displacements across them are stillcontinuous During FE analysis the amount of externalwork elastic work and cohesive energy is calculated Theenergy balance given by (7) and (8) is maintained in all FEcomputations

23 Finite Element Implementation The cohesive surfacecontribution as shown in Figure 3 is implemented into anFE code for eight node elements based on CZM conceptsThe fine mesh around the crack tip is shown in Figure 4After making convergence studies the shaft is discretizedwith 4108 plane strains and 8 node quadrilateral elementsThe length of elements is ℓ where the length of shaft is20ℓ Near the crack tip the size of the element is ℓ20A total of 4833 nodes are used to model the geometry asshown in Figure 4 The fracture process zone is modeled by8 node rectangular cohesive elements having zero thickness38 cohesive elements are used One face of cohesive elementsis connected to continuum elements with 4 nodes whilethe other face is given symmetric displacement boundaryconditions Thus an artificial interface is created along thefracture process zone

The maximum cohesive traction is assumed to be thesame as the yield strength of the material of shaft namely120590119899= 250MPa The maximum separation at the end of the

elastic zone is 1205751= 10 120583m The interface stiffness or penalty

stiffness 119870119901can be obtained directly from 119870

119901= 1205901198991205751and

is used to ensure a stiff connection between the surfaces of

Crack tip1

20

Crack tip1

20985747985747

Figure 4 Finite element mesh model near the crack tip

the material discontinuity The values of the penalty stiffnessshould be selected in a way such that the connection betweenthe two elements and numerical stability are guaranteed

3 Investigation of Breathing Mechanism

31 Method and Assumptions The breathing mechanism of asimple cracked rotor on rigid supports has been investigatedThe method of the investigation is summarized in Figure 5Several assumptions are made the location of the crack isassumed to be known at the mid-span of the shaft whichis the same as the maximum deflection point Only a singletransverse crack has been considered in the shaft The planestrain condition is assumed at the crack front due to thegeometry constraint Both ends of the shaft are rigidlysupported The shaft has a uniform circular cross sectionLength ℓ and diameter 119889 of the shaft are assumed 10m and008m respectivelyThematerial of the shaft is considered tobe homogeneous and isotropic Youngrsquos modulus 119864 Poissonrsquosratio 120584 and mass density 120588 are 210GPa 03 and 7850 kgm3respectively Yield strength 120590

119884and ultimate strength of

material 120590max are 250MPa and 400MPa respectively Thecohesive elements along the crack surfaces are implementedas interface elements as shown in Figure 6

ISRNMechanical Engineering 5Ab

aqus

Adam

sAb

aqus

Build FE model of cracked elastic shaft- Define relative crack depth 119886119889- Define crack location- Define interface crack points- Define continuum and cohesive material

Transfer FE model to MBS

- Connect cracked elastic shaft to bearings and rigid disk- Vibration responses due to static (weight) and dynamic

(due to crack and inertia force) during rotation

Transfer output of MBS into FE- Based on the vibration responses the breathing crack

is investigated

loads

mechanism

Figure 5 Principle of the breathing crack simulation using inte-grated FE and MBS

Continuumelements

Cohesiveelements

Continuumelements

Figure 6 FE model of an elastic cracked shaft

32 Breathing Mechanism Simulation The breathing mech-anism generated by the rotating bending load used in theliterature has some limitations Due to the presence ofinertia forces the dynamic behaviour of rotating structuresis different from those of static structures Although in caseof weight dominance the amplitude of the vibration responsedue to inertia forces is smaller than the amplitude due toweight forces elastic forces and presence of a crack (if theshaft is assumed to be balanced) using the inertia force takeninto account will yield more accurate results The 3D FEcalculations allow the breathing mechanism to be predictedaccurately The breathing mechanism is strongly influencedby the weight of the shaft [29] In many publications on thebreathing crack assumptions on the breathing mechanismhave been made For transverse cracks in rotating shafts thevariation of the crack front line perpendicular to the crackfront has been extensively addressed by Darpe et al [12]Thisbreathing crack form is also used by Jun et al [30] and Sinouand Lees [31] An elliptical shape has been proposed by Shihand Chen [32] Based on FE model simulation and somereported experimental results the breathing crack shape ismodelled by a parabolic shape that opens and closes due tobending stresses (Liong and Proppe [26 33 34])

The idea is that the vibration responses in the centroid ofthe shaft obtained from MBS are exported into FE model inorder to analyse the breathing mechanism as schematicallyshown in Figure 7 The opening crack is simulated for onecycle of revolution of the cracked shaft specimen in steady-state condition The breathing mechanism is generated bythe bending due to external load (weight) by increasing theangle by steps of 12058712 rad The breathing (open and closedcrack areas are evaluated in each angular step) is observedby the nodal displacement and the stress distribution (tensileor compressive stress) around the crack The prediction ofthe breathing mechanism was performed by the followingsteps for each angle of rotation stress distribution due to thebending moment is recorded over the cross section Com-pressive (negative) stresses indicate closed region The crackopens where zero or very small positive numerical values ofstresses appear and the contact forces vanish Displacementsand stresses can be observed at the crack surfaces In order toavoid local deformations due to the application of loads thedeflection and stress distribution at each crack element areamust be evaluated as shown in Figure 8

4 Results and Discussion

41 BreathingMechanism Results Figure 9 displays the resultof breathing mechanism during half-revolution of a rotat-ing shaft (from closed to open crack) with relative crackdepth 119886119889 = 01 As can be seen the crack will start to appearand begin to open at 1205876 The crack opens more slowly atthe beginning but increases its opening at 1205873 At 51205876 it isalready completely open Similarly the crack closes again at71205876 and increases its closing at 31205872 The crack is alreadycompletely closed at 111205876 It is interesting to note that theangular positions where the crack opens again (from fullyclosed) are the same as where the crack begins to close

Results of the relative crack depth versus the shaft rotationangle during breathing are shown in Figure 10 Four functionsare used to approximate the relative crack depth namelylinear quadratic harmonic and power functions Table 1gives the approximations for a relative crack depth 119886119889 as afunction of shaft rotation angle 120579 It is shown that harmonicand quadratic functions have the minimal error and thenearest correlation with the simulation results In this sensethe relative crack depth during crack opening should beunderstood not as linear increasing but as harmonic functionor quadratic polynomial

42 Discussion Due to the presence of gravity the upperportion of the cracked rotor at the beginning of a rotationis under compression and the crack is closed As the rotorcontinues to rotate and the gravity direction is constant theupper part now comes in the lower tensile region causingthe crack to open Moreover the accuracy of the results ismodelled by a bilinear model as shown in Figure 11 Thecrack begins opening at shaft rotation angle 120579

0and at crack

front angle 120572 that defines the angle between two crack frontlines The shaft rotates with shaft rotation angle 120579 and thecrack front angle 120572 increases until maximum crack width119887max is reached The crack front angle 120572 and the crack width

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FLEX BODY 1MARKER 1FLEX BODY 1INT NODE

FLEX BODY 1INT NODE

FLEX BODY 1INT NODE 2000FLEX BODY 1MARKER 3

Gravity

119909

119909

119909

119910

119910

119910

119911

119911

119911

1

05

0

minus05

minus10 01 02 03 04 05 06 07 08 09 1

119905 (s)

times10minus5

0 01 02

0

minus05

minus1

minus15

minus2

times10minus5

119911 119910

10000

30000

Am

plitu

de119910

(m)

Am

plitu

de119909

(m)

119899 = 600 rpm (119891 = 10HZ) 119886119889 = 01

Figure 7 The transfer of the vibration responses fromMBS into FE

119887 are nonlinearly increased during rotation from 1205790to 120579max

(Figures 12 and 13) One can obtain the crack front angle 120572and the crack width 119887 as functions of the shaft rotation angle120579 respectively by

120572 =

0 120579 le 1205790

crack still closedminus0018120579

3

+ 00161205792

+035120579 + 24 1205790le 120579 le 120579max

crack opens0 120579 ge 120579max

crack opens completely

119887

119887max

=

0 120579 le 1205790

crack still closed0076120579

4

minus 0541205793

+ 111205792

minus023120579 minus 00056 1205790le 120579 le 120579max

crack opens0 120579 ge 120579max

crack opens completely(9)

For the relative crack depth 119886119889 = 01 the crack beginsto open at 120579

0= 1205876 and 120572 = 51205876 while the crack opens

completely at 120579max = 51205876 and 120572 = 120587 Therefore the crackfront angle and the crack width during crack opening are

120572 = minus00181205793

+ 00161205792

+ 035120579 + 24120587

6le 120579 le

5120587

6

119887

119887max= 0076120579

4

minus 0541205793

+ 111205792

minus 023120579 minus 00056

120587

6le 120579 le

5120587

6

(10)

The relative crack area (ie ratio of the partially opencrack area to the fully open crack area) with respect to theangle of rotation is used as shown in Figure 14 The resultsare similar when the crack closes The relative crack area ofsimulation results is in good agreement with the crack closurestraight linemodel [33] between rotation angles1205873 and 51205876andwith the bilinearmodel in the range from0 to120587 but thereare some relevant differences with respect to the crack closureperpendicular line used byDarpe et al [12]The areamomentof inertia about the rotation axis can be approximated Finallythe stiffness of the cracked shaft can be evaluated (stiffness isfound to vary linearly with the area moment of inertia) Thenormalised stiffness (ratio between stiffness of the crackedshaft and stiffness of the uncracked shaft) is presented asshown in Figure 15 It is shown that the normalized stiffnessbetween breathing mechanism results and the bilinear crackclosure model is very close which validates the accuracy ofbilinear model

The fracture mechanics analysis presupposes the exis-tence of an infinitely sharp crack leading to the singular cracktip fields However in real materials neither the sharpness ofthe crack nor the stress levels near the crack tip region can be

ISRNMechanical Engineering 7

Crack

Tensile stressor zero stress

Displacement+minus

Figure 8 Evaluation of displacements and stresses for state of crack

0∘ 30∘ 45∘

60∘ 75∘ 90∘

105∘ 120∘ 135∘

150∘ 165∘ 180∘

Figure 9 Breathing crack of the rotating shaft relative crack depth119886119889 = 01 at 600 rpm

infinite [27] As an alternative approach to this singularity-driven fracture approach the concept of CZM is proposedCZM has evolved as a preferred method to analyze fractureproblems not only because it avoids the singularity but alsobecause it can be easily implemented in a numerical methodof analysis as in FE method CZM presupposes the presenceof a fracture process zone where the energy is transferred

from external work in the crack regions In this study theexternal work flows as recoverable elastic strain energy andcohesive energy are examined the latter encompassing thework and other energy consuming mechanisms within thefracture process zone Process zone is defined as the regionwithin the separating surfaces where the surface tractionvalues are nonzeroThis also implies that processes occurringwithin the process zone are accounted for only throughthe traction-displacement relations The traction-separationrelations for crack interfaces are demonstrated such thatwith increasing interfacial separation the traction across theinterface reaches amaximum itmeans the elastic limit whereplastic dissipation in the surrounding (bounding) material isnot accounted for in the process zone

5 Conclusions

In this study a simulation process of FE and MBS isemployed An elastic cracked shaft is modelled by FE whichis transferred into MBS model The vibration responses inthe centroid of the shaft obtained from MBS have beentransferred again into an FE model in order to observe thebreathing mechanism and to study the stiffness losses due tobreathing at different rotating speeds

The results have shown that the breathing mechanismis influenced by the vibration due to inertia forces and byrelative crack depth It is shown that the relative crack depthduring crack opening should be understood not as linearincreasing but as harmonic function or quadratic polynomialIt can be noted that as long as the relative crack depth issmall the bilinear crack closure model may be used Themain difference with respect to the straight line crack closuremodel is that the opening crack in the simulation is notconstant at the beginningThe relative crack area of breathingmechanism results is in good agreement with the straight linecrack closuremodel between rotation angles1205873 and1205876 andwith the bilinear crack closuremodel in the range from0 to120587Furthermore the area moment of inertia about the rotationaxis can be approximated From this information the stiffnesslosses of the shaft can be evaluated It has been shown thatthe normalized stiffness between simulation and the bilinearcrack closure model is very close which validates the bilinearmodel

CZM are being increasingly used to simulate crackInstead of an infinitely sharp crack envisaged in fracturemechanics CZM presupposes the presence of a fracture pro-cess zone where the energy is transferred from external workin the crack regions The traction-separation relations forcrack interfaces are demonstrated such that with increasinginterfacial separation the traction across the interface reachesa maximum it means the elastic limit where plastic dissipa-tion in the surrounding (bounding)material is not accountedfor in the process zone

In light of the presented results and the conclusionsseveral subjects can be recommended for future research Animportant point on which the knowledge could be improvedis the prediction of crack propagation on cracked rotor andresidual life estimation The CZM can be easily implementedin FE model to analyze the dynamic behaviour of a cracked

8 ISRNMechanical Engineering

0 120587 2120587

119886119889

01

008

006

004

002

0

Linear model119891(119909) = 119886119909 + 119887

Confidence bounds 95

Goodness of fitSSE = 00005796R-square = 09516

120579 (rad)

119886 = 003887 (003226 004548)

119887 = 000131 (minus001043 001305)

(a)

0 120587 2120587

01

008

006

004

002

0

119886119889

Quadratic polynomial119891(119909) = 1198861199092 + 119887119909 + 119888

Confidence bounds 95

Goodness of fitSSE = 1141119890 minus 005R-square = 09990

120579 (rad)

119886 = minus001187 (minus001325 minus001050)119887 = 007617 (007174 008059)

119888 = minus001985 (minus002289 minus001683)

(b)

0 120587 2120587

01

008

006

004

002

0

119886119889

General model of sinus function

Confidence bounds 95

Goodness of fitSSE = 6158119890 minus 06R-square = 09995

120579 (rad)

119891(119909) = 119886 sin (119887119909 + 119888)

119886 = 009969 (009806 010130)119887 = 059340 (056990 061680)119888 = minus014540 (minus016410 minus012660)

(c)

0 120587 2120587

01

008

006

004

002

0

119886119889

General model of power119891(119909) = 119886119909119887 + 119888

Confidence bounds 95

Goodness of fitSSE = 9952119890 minus 05R-square = 09917

120579 (rad)

119886 = 012310 (004674 019950)119887 = 037300 (015200 059400)119888 = minus007730 (minus015120 minus000336)

(d)

Figure 10 Curve fitting of breathing mechanism results

Table 1 Curve fitting of the simulation results for 119886119889 = 01 120579 in [rad]

Model Mathematical model SSE R-squareLinear function 119886119889 = 0039120579 + 0001 0000 580 0951 6Quadratic polynomial 119886119889 = minus00121205792 + 0076120579minus 0020 0000 011 0999 0Sinus function 119886119889 = 0010sin (0593120579minus 0145) 0000 006 0999 5Power function 119886119889 = 01231205790373minus 0077 0000 099 0991 7

Crack closure line 1205721205720

119886

119887

119887max

Bilinear borderline

120575 gt 100120583m (crack opens)

10120583m le 120575 100120583m

120575 lt 10120583m

120575 = 0 (crack still closed)le

Figure 11 Bilinear crack closure model

ISRNMechanical Engineering 9

0 1205872 120587

120587

51205876

21205873

120572 = minus0084 lowast 1205792 + 051 lowast 120579 + 24

120572 = minus0018 lowast 1205793 + 0016 lowast 1205792 + 035 lowast 120579 + 24

SimulationQuadraticCubic

120579

120572

(rad)

(rad

)

Figure 12 Crack front angle 120572 during rotation of shaft

0 1205872 120587

1

09

08

07

06

05

04

03

02

01

0

119887119887 m

ax

SimulationCubic4th degree

= minus0069 lowast 1205793 + 02 lowast 1205792 + 036 lowast 120579 minus 0066

= 0076 lowast 1205794 minus 054lowast 1205793 + 11 lowast 1205792 minus 023 lowast 120579 minus

120579 (rad)

00056119887119887max

119887119887max

Figure 13 Normalized crack width 119887119887max during rotation of shaft

1

09

08

07

06

05

04

03

02

01

00 1205876 1205873 1205872 21205873 51205876 120587

Breathing mechanismBilinear crack closure model

Straight line crack closure [33]Perpendicular crack closure [12]

119860op

en119860

crac

k

120579 (rad)

Figure 14 Comparison of relative crack area between breathingmechanism results straight line perpendicular and bilinear crackclosure model for a shallow crack 119886119889 = 01

098

096

094

092

09

088

086

084

082

080 1205876 1205873 1205872 21205873 51205876 120587

Breathing mechanismBilinear crack closure model

Straight line crack closure [33]Perpendicular crack closure [12]

120578

120585

119870cr

ack119870

120579 (rad)

Figure 15 Comparison of normalized stiffness between breathingmechanism results straight line perpendicular and bilinear crackclosure model for a shallow crack 119886119889 = 01

shaft It is recommended to use CZM to study differenttypes of cracks such as longitudinal and slant cracks andalso multiple cracks Some other parameters such as internaldamping and thermal transients could be studied to obtainresults for their effect on the breathing mechanism Furtheranalysis on crack morphology is extremely important tounderstand the dynamic behaviour of cracked shaft thiswould include shallow and wide cracks

Acknowledgments

The authors acknowledge the support from Deutsche For-schungsgemeinschaft and Open Access Publishing Fund ofKarlsruhe Institute of Technology

References

[1] S K Georgantzinos and N K Anifantis ldquoAn insight into thebreathing mechanism of a crack in a rotating shaftrdquo Journal ofSound and Vibration vol 318 no 1-2 pp 279ndash295 2008

[2] J Wauer ldquoOn the dynamics of cracked rotors literature surveyrdquoApplied Mechanics Review vol 43 no 1 pp 13ndash17 1990

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

[4] G Sabnavis R G Kirk M Kasarda and D Quinn ldquoCrackedshaft detection and diagnostics a literature reviewrdquo Shock andVibration Digest vol 36 no 4 pp 287ndash296 2004

[5] 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

[6] N Bachschmid P Pennacchi and E Tanzi ldquoSome remarks onbreathing mechanism on non-linear effects and on slant andhelicoidal cracksrdquo Mechanical Systems and Signal Processingvol 22 no 4 pp 879ndash904 2008

[7] N Bachschmid P Pennachi and E Tanzi Cracked RotorsSpringer Berlin Germany 2011

10 ISRNMechanical Engineering

[8] S Andrieux and C Vare ldquoA 3D cracked beam model withunilateral contact Application to rotorsrdquo European Journal ofMechanics A vol 21 no 5 pp 793ndash810 2002

[9] C Vare and S Andrieux ldquoModeling of a cracked beam sectionunder bendingrdquo in Proceedings of the 18th International Confer-ence on Structural Mechanics in Reactor Technolohy (SMiRT 18)pp 281ndash290 Beijing China 2005

[10] S E Arem and HMaitournam ldquoA cracked beam finite elementfor rotating shaft dynamics and stability analysisrdquo Journal ofMechanics of Materials and Structures vol 3 no 5 pp 893ndash9102008

[11] A D Dimarogonas and S A Paipetis Analytical Methods inRotorDynamics Applied Science Publishers LondonUK 1983

[12] A K Darpe K Gupta and A Chawla ldquoCoupled bendinglongitudinal and torsional vibrations of a cracked rotorrdquo Journalof Sound and Vibration vol 269 no 1-2 pp 33ndash60 2004

[13] W M Ostachowicz and M Krawczuk ldquoCoupled torsional andbending vibrations of a rotor with an open crackrdquo Archive ofApplied Mechanics vol 62 no 3 pp 191ndash201 1992

[14] Z Kulesza and J T Sawicki ldquoRigid finite element model ofa cracked rotorrdquo Journal of Sound and Vibration vol 331 pp4145ndash4169 2012

[15] A S Bouboulas and N K Anifantis ldquoFinite element modelingof a vibrating beam with a breathing crack observations oncrack detectionrdquo Structural Health Monitoring vol 10 no 2 pp131ndash145 2011

[16] D S Dugdale ldquoYielding of steel sheets containing slitsrdquo Journalof the Mechanics and Physics of Solids vol 8 no 2 pp 100ndash1041960

[17] G I Barenblatt ldquoThemathematical theory of equilibrium crackin brittle fracturerdquoAdvances in AppliedMechanics vol 7 pp 55ndash129 1962

[18] T Siegmund andW Brocks ldquoTensile decohesion by local failurecriteriardquo Technische Mechanik vol 18 no 4 pp 261ndash270 1998

[19] T Siegmund and W Brocks ldquoA numerical study on thecorrelation between the work of separation and the dissipationrate in ductile fracturerdquo Engineering Fracture Mechanics vol 67no 2 pp 139ndash154 2000

[20] M Anvari I Scheider and CThaulow ldquoSimulation of dynamicductile crack growth using strain-rate and triaxiality-dependentcohesive elementsrdquo Engineering Fracture Mechanics vol 73 no15 pp 2210ndash2228 2006

[21] I Scheider ldquoDerivation of separation laws for cohesive modelsin the course of ductile fracturerdquo Engineering Fracture Mechan-ics vol 76 no 10 pp 1450ndash1459 2009

[22] A Banerjee and R Manivasagam ldquoTriaxiality dependent cohe-sive zone modelrdquo Engineering Fracture Mechanics vol 76 no12 pp 1761ndash1770 2009

[23] A D Dimarogonas and C A Papadopoulos ldquoVibration ofcracked shafts in bendingrdquo Journal of Sound and Vibration vol91 no 4 pp 583ndash593 1983

[24] A C Chasalevris andCA Papadopoulos ldquoA continuousmodelapproach for cross-coupled bending vibrations of a rotor-bearing system with a transverse breathing crackrdquo Mechanismand Machine Theory vol 44 no 6 pp 1176ndash1191 2009

[25] 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

[26] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel to the analysis of a rotor with a transverse crackrdquo Journalof Sound and Vibration vol 332 no 8 pp 2098ndash2110 2013

[27] C Shet andN Chandra ldquoAnalysis of energy balancewhen usingCohesive Zone Models to simulate fracture processesrdquo Journalof EngineeringMaterials andTechnology vol 124 no 4 pp 440ndash450 2002

[28] H Li and N Chandra ldquoAnalysis of crack growth and crack-tip plasticity in ductile materials using cohesive zone modelsrdquoInternational Journal of Plasticity vol 19 no 6 pp 849ndash8822003

[29] N Bachschmid P Pennacchi and E Tanzi ldquoOn the evolutionof vibrations in cracked rotorsrdquo in Proceedings of the 8thIFToMM International Conference on Rotor Dynamics pp 304ndash310 Seoul Korea 2010

[30] O S Jun H J Eun Y Y Earmme and C W Lee ldquoModellingand vibration analysis of a simple rotor with a breathing crackrdquoJournal of Sound andVibration vol 155 no 2 pp 273ndash290 1992

[31] J J Sinou and A W Lees ldquoThe influence of cracks in rotatingshaftsrdquo Journal of Sound and Vibration vol 285 no 4-5 pp1015ndash1037 2005

[32] Y S Shih and J J Chen ldquoAnalysis of fatigue crack growth on acracked shaftrdquo International Journal of Fatigue vol 19 no 6 pp477ndash485 1997

[33] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel for the investigation of the dynamic behavior of arotating shaft with a transverse crackrdquo in Proceedings of the 8thIFToMM International Conference on Rotor Dynamics pp 628ndash636 Seoul Korea September 2010

[34] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel to the analysis of a rotor with a transverse crackrdquo inProceedings of the 8th International Conference on StructuralDynamics pp 3434ndash3442 Leuven Belgium 2011

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International Journal of

2 ISRNMechanical Engineering

the bending moment applied to the cracked section and theresulting field of displacements compatible with the beamtheory so that it can be used in rotor-dynamics softwareVare and Andrieux [9] extended this method in order toshow how shear effects can be implemented in the model of acracked section Arem and Maitournam [10] presented stiff-ness variations deduced from 3D FE calculations accountingfor the unilateral contact between the crack lips as originallydeveloped by Andrieux and Vare [8] Dimarogonas andPaipetis [11] introduced the FE models of the rotating shaftcracks They used the fracture mechanics and obtained a full6 times 6 flexibility matrix for a transverse open surface crackon shaft Darpe et al [12] studied the coupling betweenlongitudinal lateral and torsional vibrations together fora rotating cracked shaft By including the axial degree offreedom in their analysis the stiffness matrix formulatedis an extension of the one developed by Ostachowicz andKrawczuk [13] Kulesza and Sawicki [14] used the rigidFEM for modelling a crack in a rotating shaft The crack ispresented as a set of spring-damping elements of variablestiffness connecting two sections of the shaft Bouboulas andAnifantis [15] develop a finite elementmodel in order to studythe vibrational behavior of a beam with a nonpropagatingedge crack According to their model the beam is discretizedinto finite elements while the breathing crack behavior istreated as a full frictional contact problem between the cracksurfaces

The cohesive zone model (CZM) has been widely usedas an alternative to stress intensity factor-based fracturemechanics It can deal with the nonlinear zone ahead ofthe crack tip due to plasticity or microcracking The CZMdescribes material failure on a more phenomenological basis(ie without considering the material microstructure) Thegeneral advantage of this model when compared to classicalfracture mechanics is that the parameters of the respectivemodels depend only on thematerial and not on the geometryThis concept guarantees transferability from specimen tostructure over a wide range of geometries The origin ofthe cohesive zone concept can be traced back to the stripyield model proposed by Dugdale [16] and Barenblatt [17]in which the narrow zone of localized deformation ahead ofthe crack tip was substituted by cohesive traction betweenthe bounding surfaces The constitutive behaviour whichcauses the cohesive elements to open and eventually to failis described by the so-called traction separation law It relatesthe traction vector to the displacement jump across the inter-face and is usually called separation The energy dissipatedby the element until total failure is derived as the integral ofthe traction-separationHowever the traction-separation lawdepends on the stress state which can be characterised bythe triaxiality (ie the hydrostatic stress divided by the vonMises equivalent stress) This issue was first investigated bySiegmund and Brocks [18 19] The approach was extended tosimulation of dynamic ductile crack growth by Anvari et al[20] Scheider [21] Banerjee andManivasagam [22] proposeda versatile CZM to predict ductile fracture at different statesof stress The formulation developed for mode-I plane strainaccounts explicitly for triaxiality of the stress state by usingbasic elastic-plastic constitutive relations combined with two

new model parameters which are independent of the stressstate

The theory of strain energy release rate and SIF combinedwith rotordynamics built the foundation of the dynamic ofcracked rotors based on LEFM [23] This theory has twomajor limitations namely it can only be used if there is aninitial crack and the fracture process zone must be smallcompared with the dimensions of the shaft Due to geometri-cal complexity some simplifications had been made for thecrack profile such as straight-edged circular and ellipticalcrack model to analyze such crack problem Since analyticalSIFs for an edge crack in a rotating cylinder are not availablethe shaft is considered to be a sumof elementary independentrectangular strips (Andrieux andVare [8]) and no interactionbetween them is assumed to take place (Chasalevris andPapadopoulos [24]) The geometric functions that describethe strain energy density are often not accurate enough dueto the fact that the crack passes from stress state caused bythe vertical moment to that of a horizontal moment Thenthe compliance is obtained by integrating along the cracktip If the crack depth exceeds the radius of the shaft thenthe elements of the compliance matrix present a divergenceThis is due to the singularity that the strain energy releaserate method has near the edges of the crack tip givingthus the infinite values It was reported by Papadopoulos[25] that divergence does not reflect reality The crack tip issupposed to be formed by the boundary between the crackedareas and the uncracked areas for the regions in which thebreathing crack is open However the SIF will not appearat the boundary between the closed cracked areas and theopen cracked areas [7] Liong and Proppe [26] proposed amethod for the evaluation of the stiffness losses in a rotorwith a transverse breathing crackTheir method is based on aCZM and accounts explicitly for triaxiality of the stress stateby using constitutive relations

The aim of the present study is to propose a method forthe evaluation of stiffness losses of the cracked shaft based ona CZM The method provides a direct relationship betweenthe material properties of the shaft and the relative crackdepth thus representing the physical phenomenon closelyThis paper introduces a method based on a 3D FE modeland MBS in order to study the breathing mechanism of anelastic cracked shaftTheCZMformulation is implemented ina 3D FEmodelThe elastic cracked shaft with various relativecrack depths is modelled by 3D FE and then transferred intoan MBS model in order to analyze the dynamic loads actingduring rotation at different rotating speeds The vibrationresponses in the centroid of the rotating shaft obtained fromMBS have been exported into FE to observe the breathingmechanism The accuracy of the results is demonstratedthrough comparisons with the results available in the liter-ature

2 The Cohesive Zone Model

21 Basis of the Cohesive Process Zone Model In the CZMfracture nucleates as discontinuity surface able to transmittensile load before opening above a given displacementFormation and extension of this surface require that the

ISRNMechanical Engineering 3

Initialcrack

Crackedzone Cohesive fracture zone

Undamagedligament

120575120590max

120590 = 0120590 = 119891(120575)

120575119888

Figure 1 Cohesive crack process zone

maximum principal stress reaches a given value namely thecohesive strength of thematerialWhen this occurs the cracksurface initiates or grows perpendicularly to the directionof the maximum principal stress The two faces of thesurface exert on each other equal and opposite tensile stresses(cohesive stresses) whose value is a unique function 119891(120575)of the separation 120575 between the faces When the separationreaches another given value (the critical separation 120575

119888) the

cohesive stress vanishes and fracture takes place Fractureconsists of the initiation and propagation of a crack producedby the opening and advance of the cohesive zone (the zonewhere the cohesive stresses act) ahead of the crack tip asshown in Figure 1

The fracture behaviour of each material is described bythe cohesive traction as a function

120590119899= 120590max119891 (120575) (1)

where 120590max is the peak value of traction The function 119891(120575)defines the shape of the traction-separation law and the areaunder the cohesive law curve is the work of separation orcohesive energy 119866

119888

119866119888= int

120575119888

0

120590119899(120575) d120575 (2)

Ideally the model within the cohesive zone should beable to replicate the constitutive behaviour of the undamagedmaterial linear elastic followed by strain hardening till theconditions for the initiation of softening due to damage arereached The softening process representative of increasingmaterial degradation is triggered by rapid growth of voids asconsequence of a highly triaxial state of stress

Since the CZM is a phenomenological model variousformulations for defining the shape of traction-separation lawand the cohesive values are in use [27] A versatile CZM topredict ductile fracture at different states of stress is proposedin [22] The formulation developed for mode-I plane strainaccounts explicitly for triaxiality of the stress state by usingbasic elastic-plastic constitutive relations combined withtwo stress-state-independent new model parameters Theproposed traction-separation law has three distinct regions ofconstitutive behaviour the traction separation law is linear upto the separation limit 120575

119899= 1205751and exhibits strain hardening

Linearregion

120590max

Hardeningregion

Softeningregion

Cohesive energy

0 1205751 1205752 120575c = 120575sep120575119899

120590119899

120590119884

119870119901

Figure 2 Traction separation law for ductile materials

up to 120575119899= 1205752followed by a softening curve The relevant

variables and zones are sketched in Figure 2

120590119899= (1 + radic3120594ef)

2119864

3120575 for 0 lt 120575 lt 120575

1 (3)

The linear behaviour of the cohesive zone exists till theseparation limit defined by the von Mises yield condition isreached

1205751=radic3120590119884

2119864 (4)

Further separation results in strain hardening up to

1205752=radic3

2(119862eminus(32)120594ef + 120590119884

119864) (5)

22 Numerical Representation of Cohesive Zone ModelsWhen fracture proceeds energymust be supplied by externalloads The bounding material undergoes elastoplastic defor-mation involving elastic energy and plastic dissipative energyIn addition to plasticity energy is supplied to the fractureprocess zone in the form of cohesive energy that is dissipatedwithin the cohesive elements The cohesive energy is the sumof the surface energy and all dissipative processes that takeplace within the crack tip regime For the present problem aperfect energy balance between external work119882 and the sumof elastic energy 119864el and cohesive energy 119866

119888[27 28] will be

assumed The energy balance is given by

119882 = 119864el + 119866119888 (6)

119882 = int

119905

0

(int119881

120590 120576el d119881) d119905 + int119905

0

(int119878

120590119899 d119878) d119905 (7)

where 120590 120576el 120590119899 and are nominal stress tensor elasticstrain rate cohesive traction and cohesive separation raterespectively (including the terms for specimen volume119881 andthe internal specimen surface 119878)The external work due to theapplied force is given by

119882 = int

119905

0

(int119878

t sdot v d119878) d119905 + int119905

0

(int119881

b sdot v d119881) d119905 (8)

where v is the velocity field vector t the exterior surfacetraction vector and b the body force vector

4 ISRNMechanical Engineering

Real

Finite elements

IdealizationContinuum elements

Crack tip

Cohesive elements

Continuum

Crack

Brokencohesiveelements

Separationof cohesiveelements

Cohesivetractions120590119899 = 119891(120575)

Δ119886

Figure 3 Representation of the fracture process using CZM in FE

The cohesive surface contribution representing the crackand the process zone in front of the crack tip is described bythe integral over the internal surface 119878 In this formulation 120590

119899

denotes the cohesive strength that is the maximum tractionvalue that can be sustained within the cohesive zone Thecohesive length 120575 is the value of the displacement jump acrossthe crack surfaces at which the stress carrying capacity of thecohesive elements reaches its maximum value By creatingnew surfaces the traction and the stiffness of the cohesivezone elements connecting these newly created surfaces aremade to vanish but the displacements across them are stillcontinuous During FE analysis the amount of externalwork elastic work and cohesive energy is calculated Theenergy balance given by (7) and (8) is maintained in all FEcomputations

23 Finite Element Implementation The cohesive surfacecontribution as shown in Figure 3 is implemented into anFE code for eight node elements based on CZM conceptsThe fine mesh around the crack tip is shown in Figure 4After making convergence studies the shaft is discretizedwith 4108 plane strains and 8 node quadrilateral elementsThe length of elements is ℓ where the length of shaft is20ℓ Near the crack tip the size of the element is ℓ20A total of 4833 nodes are used to model the geometry asshown in Figure 4 The fracture process zone is modeled by8 node rectangular cohesive elements having zero thickness38 cohesive elements are used One face of cohesive elementsis connected to continuum elements with 4 nodes whilethe other face is given symmetric displacement boundaryconditions Thus an artificial interface is created along thefracture process zone

The maximum cohesive traction is assumed to be thesame as the yield strength of the material of shaft namely120590119899= 250MPa The maximum separation at the end of the

elastic zone is 1205751= 10 120583m The interface stiffness or penalty

stiffness 119870119901can be obtained directly from 119870

119901= 1205901198991205751and

is used to ensure a stiff connection between the surfaces of

Crack tip1

20

Crack tip1

20985747985747

Figure 4 Finite element mesh model near the crack tip

the material discontinuity The values of the penalty stiffnessshould be selected in a way such that the connection betweenthe two elements and numerical stability are guaranteed

3 Investigation of Breathing Mechanism

31 Method and Assumptions The breathing mechanism of asimple cracked rotor on rigid supports has been investigatedThe method of the investigation is summarized in Figure 5Several assumptions are made the location of the crack isassumed to be known at the mid-span of the shaft whichis the same as the maximum deflection point Only a singletransverse crack has been considered in the shaft The planestrain condition is assumed at the crack front due to thegeometry constraint Both ends of the shaft are rigidlysupported The shaft has a uniform circular cross sectionLength ℓ and diameter 119889 of the shaft are assumed 10m and008m respectivelyThematerial of the shaft is considered tobe homogeneous and isotropic Youngrsquos modulus 119864 Poissonrsquosratio 120584 and mass density 120588 are 210GPa 03 and 7850 kgm3respectively Yield strength 120590

119884and ultimate strength of

material 120590max are 250MPa and 400MPa respectively Thecohesive elements along the crack surfaces are implementedas interface elements as shown in Figure 6

ISRNMechanical Engineering 5Ab

aqus

Adam

sAb

aqus

Build FE model of cracked elastic shaft- Define relative crack depth 119886119889- Define crack location- Define interface crack points- Define continuum and cohesive material

Transfer FE model to MBS

- Connect cracked elastic shaft to bearings and rigid disk- Vibration responses due to static (weight) and dynamic

(due to crack and inertia force) during rotation

Transfer output of MBS into FE- Based on the vibration responses the breathing crack

is investigated

loads

mechanism

Figure 5 Principle of the breathing crack simulation using inte-grated FE and MBS

Continuumelements

Cohesiveelements

Continuumelements

Figure 6 FE model of an elastic cracked shaft

32 Breathing Mechanism Simulation The breathing mech-anism generated by the rotating bending load used in theliterature has some limitations Due to the presence ofinertia forces the dynamic behaviour of rotating structuresis different from those of static structures Although in caseof weight dominance the amplitude of the vibration responsedue to inertia forces is smaller than the amplitude due toweight forces elastic forces and presence of a crack (if theshaft is assumed to be balanced) using the inertia force takeninto account will yield more accurate results The 3D FEcalculations allow the breathing mechanism to be predictedaccurately The breathing mechanism is strongly influencedby the weight of the shaft [29] In many publications on thebreathing crack assumptions on the breathing mechanismhave been made For transverse cracks in rotating shafts thevariation of the crack front line perpendicular to the crackfront has been extensively addressed by Darpe et al [12]Thisbreathing crack form is also used by Jun et al [30] and Sinouand Lees [31] An elliptical shape has been proposed by Shihand Chen [32] Based on FE model simulation and somereported experimental results the breathing crack shape ismodelled by a parabolic shape that opens and closes due tobending stresses (Liong and Proppe [26 33 34])

The idea is that the vibration responses in the centroid ofthe shaft obtained from MBS are exported into FE model inorder to analyse the breathing mechanism as schematicallyshown in Figure 7 The opening crack is simulated for onecycle of revolution of the cracked shaft specimen in steady-state condition The breathing mechanism is generated bythe bending due to external load (weight) by increasing theangle by steps of 12058712 rad The breathing (open and closedcrack areas are evaluated in each angular step) is observedby the nodal displacement and the stress distribution (tensileor compressive stress) around the crack The prediction ofthe breathing mechanism was performed by the followingsteps for each angle of rotation stress distribution due to thebending moment is recorded over the cross section Com-pressive (negative) stresses indicate closed region The crackopens where zero or very small positive numerical values ofstresses appear and the contact forces vanish Displacementsand stresses can be observed at the crack surfaces In order toavoid local deformations due to the application of loads thedeflection and stress distribution at each crack element areamust be evaluated as shown in Figure 8

4 Results and Discussion

41 BreathingMechanism Results Figure 9 displays the resultof breathing mechanism during half-revolution of a rotat-ing shaft (from closed to open crack) with relative crackdepth 119886119889 = 01 As can be seen the crack will start to appearand begin to open at 1205876 The crack opens more slowly atthe beginning but increases its opening at 1205873 At 51205876 it isalready completely open Similarly the crack closes again at71205876 and increases its closing at 31205872 The crack is alreadycompletely closed at 111205876 It is interesting to note that theangular positions where the crack opens again (from fullyclosed) are the same as where the crack begins to close

Results of the relative crack depth versus the shaft rotationangle during breathing are shown in Figure 10 Four functionsare used to approximate the relative crack depth namelylinear quadratic harmonic and power functions Table 1gives the approximations for a relative crack depth 119886119889 as afunction of shaft rotation angle 120579 It is shown that harmonicand quadratic functions have the minimal error and thenearest correlation with the simulation results In this sensethe relative crack depth during crack opening should beunderstood not as linear increasing but as harmonic functionor quadratic polynomial

42 Discussion Due to the presence of gravity the upperportion of the cracked rotor at the beginning of a rotationis under compression and the crack is closed As the rotorcontinues to rotate and the gravity direction is constant theupper part now comes in the lower tensile region causingthe crack to open Moreover the accuracy of the results ismodelled by a bilinear model as shown in Figure 11 Thecrack begins opening at shaft rotation angle 120579

0and at crack

front angle 120572 that defines the angle between two crack frontlines The shaft rotates with shaft rotation angle 120579 and thecrack front angle 120572 increases until maximum crack width119887max is reached The crack front angle 120572 and the crack width

6 ISRNMechanical Engineering

FLEX BODY 1MARKER 1FLEX BODY 1INT NODE

FLEX BODY 1INT NODE

FLEX BODY 1INT NODE 2000FLEX BODY 1MARKER 3

Gravity

119909

119909

119909

119910

119910

119910

119911

119911

119911

1

05

0

minus05

minus10 01 02 03 04 05 06 07 08 09 1

119905 (s)

times10minus5

0 01 02

0

minus05

minus1

minus15

minus2

times10minus5

119911 119910

10000

30000

Am

plitu

de119910

(m)

Am

plitu

de119909

(m)

119899 = 600 rpm (119891 = 10HZ) 119886119889 = 01

Figure 7 The transfer of the vibration responses fromMBS into FE

119887 are nonlinearly increased during rotation from 1205790to 120579max

(Figures 12 and 13) One can obtain the crack front angle 120572and the crack width 119887 as functions of the shaft rotation angle120579 respectively by

120572 =

0 120579 le 1205790

crack still closedminus0018120579

3

+ 00161205792

+035120579 + 24 1205790le 120579 le 120579max

crack opens0 120579 ge 120579max

crack opens completely

119887

119887max

=

0 120579 le 1205790

crack still closed0076120579

4

minus 0541205793

+ 111205792

minus023120579 minus 00056 1205790le 120579 le 120579max

crack opens0 120579 ge 120579max

crack opens completely(9)

For the relative crack depth 119886119889 = 01 the crack beginsto open at 120579

0= 1205876 and 120572 = 51205876 while the crack opens

completely at 120579max = 51205876 and 120572 = 120587 Therefore the crackfront angle and the crack width during crack opening are

120572 = minus00181205793

+ 00161205792

+ 035120579 + 24120587

6le 120579 le

5120587

6

119887

119887max= 0076120579

4

minus 0541205793

+ 111205792

minus 023120579 minus 00056

120587

6le 120579 le

5120587

6

(10)

The relative crack area (ie ratio of the partially opencrack area to the fully open crack area) with respect to theangle of rotation is used as shown in Figure 14 The resultsare similar when the crack closes The relative crack area ofsimulation results is in good agreement with the crack closurestraight linemodel [33] between rotation angles1205873 and 51205876andwith the bilinearmodel in the range from0 to120587 but thereare some relevant differences with respect to the crack closureperpendicular line used byDarpe et al [12]The areamomentof inertia about the rotation axis can be approximated Finallythe stiffness of the cracked shaft can be evaluated (stiffness isfound to vary linearly with the area moment of inertia) Thenormalised stiffness (ratio between stiffness of the crackedshaft and stiffness of the uncracked shaft) is presented asshown in Figure 15 It is shown that the normalized stiffnessbetween breathing mechanism results and the bilinear crackclosure model is very close which validates the accuracy ofbilinear model

The fracture mechanics analysis presupposes the exis-tence of an infinitely sharp crack leading to the singular cracktip fields However in real materials neither the sharpness ofthe crack nor the stress levels near the crack tip region can be

ISRNMechanical Engineering 7

Crack

Tensile stressor zero stress

Displacement+minus

Figure 8 Evaluation of displacements and stresses for state of crack

0∘ 30∘ 45∘

60∘ 75∘ 90∘

105∘ 120∘ 135∘

150∘ 165∘ 180∘

Figure 9 Breathing crack of the rotating shaft relative crack depth119886119889 = 01 at 600 rpm

infinite [27] As an alternative approach to this singularity-driven fracture approach the concept of CZM is proposedCZM has evolved as a preferred method to analyze fractureproblems not only because it avoids the singularity but alsobecause it can be easily implemented in a numerical methodof analysis as in FE method CZM presupposes the presenceof a fracture process zone where the energy is transferred

from external work in the crack regions In this study theexternal work flows as recoverable elastic strain energy andcohesive energy are examined the latter encompassing thework and other energy consuming mechanisms within thefracture process zone Process zone is defined as the regionwithin the separating surfaces where the surface tractionvalues are nonzeroThis also implies that processes occurringwithin the process zone are accounted for only throughthe traction-displacement relations The traction-separationrelations for crack interfaces are demonstrated such thatwith increasing interfacial separation the traction across theinterface reaches amaximum itmeans the elastic limit whereplastic dissipation in the surrounding (bounding) material isnot accounted for in the process zone

5 Conclusions

In this study a simulation process of FE and MBS isemployed An elastic cracked shaft is modelled by FE whichis transferred into MBS model The vibration responses inthe centroid of the shaft obtained from MBS have beentransferred again into an FE model in order to observe thebreathing mechanism and to study the stiffness losses due tobreathing at different rotating speeds

The results have shown that the breathing mechanismis influenced by the vibration due to inertia forces and byrelative crack depth It is shown that the relative crack depthduring crack opening should be understood not as linearincreasing but as harmonic function or quadratic polynomialIt can be noted that as long as the relative crack depth issmall the bilinear crack closure model may be used Themain difference with respect to the straight line crack closuremodel is that the opening crack in the simulation is notconstant at the beginningThe relative crack area of breathingmechanism results is in good agreement with the straight linecrack closuremodel between rotation angles1205873 and1205876 andwith the bilinear crack closuremodel in the range from0 to120587Furthermore the area moment of inertia about the rotationaxis can be approximated From this information the stiffnesslosses of the shaft can be evaluated It has been shown thatthe normalized stiffness between simulation and the bilinearcrack closure model is very close which validates the bilinearmodel

CZM are being increasingly used to simulate crackInstead of an infinitely sharp crack envisaged in fracturemechanics CZM presupposes the presence of a fracture pro-cess zone where the energy is transferred from external workin the crack regions The traction-separation relations forcrack interfaces are demonstrated such that with increasinginterfacial separation the traction across the interface reachesa maximum it means the elastic limit where plastic dissipa-tion in the surrounding (bounding)material is not accountedfor in the process zone

In light of the presented results and the conclusionsseveral subjects can be recommended for future research Animportant point on which the knowledge could be improvedis the prediction of crack propagation on cracked rotor andresidual life estimation The CZM can be easily implementedin FE model to analyze the dynamic behaviour of a cracked

8 ISRNMechanical Engineering

0 120587 2120587

119886119889

01

008

006

004

002

0

Linear model119891(119909) = 119886119909 + 119887

Confidence bounds 95

Goodness of fitSSE = 00005796R-square = 09516

120579 (rad)

119886 = 003887 (003226 004548)

119887 = 000131 (minus001043 001305)

(a)

0 120587 2120587

01

008

006

004

002

0

119886119889

Quadratic polynomial119891(119909) = 1198861199092 + 119887119909 + 119888

Confidence bounds 95

Goodness of fitSSE = 1141119890 minus 005R-square = 09990

120579 (rad)

119886 = minus001187 (minus001325 minus001050)119887 = 007617 (007174 008059)

119888 = minus001985 (minus002289 minus001683)

(b)

0 120587 2120587

01

008

006

004

002

0

119886119889

General model of sinus function

Confidence bounds 95

Goodness of fitSSE = 6158119890 minus 06R-square = 09995

120579 (rad)

119891(119909) = 119886 sin (119887119909 + 119888)

119886 = 009969 (009806 010130)119887 = 059340 (056990 061680)119888 = minus014540 (minus016410 minus012660)

(c)

0 120587 2120587

01

008

006

004

002

0

119886119889

General model of power119891(119909) = 119886119909119887 + 119888

Confidence bounds 95

Goodness of fitSSE = 9952119890 minus 05R-square = 09917

120579 (rad)

119886 = 012310 (004674 019950)119887 = 037300 (015200 059400)119888 = minus007730 (minus015120 minus000336)

(d)

Figure 10 Curve fitting of breathing mechanism results

Table 1 Curve fitting of the simulation results for 119886119889 = 01 120579 in [rad]

Model Mathematical model SSE R-squareLinear function 119886119889 = 0039120579 + 0001 0000 580 0951 6Quadratic polynomial 119886119889 = minus00121205792 + 0076120579minus 0020 0000 011 0999 0Sinus function 119886119889 = 0010sin (0593120579minus 0145) 0000 006 0999 5Power function 119886119889 = 01231205790373minus 0077 0000 099 0991 7

Crack closure line 1205721205720

119886

119887

119887max

Bilinear borderline

120575 gt 100120583m (crack opens)

10120583m le 120575 100120583m

120575 lt 10120583m

120575 = 0 (crack still closed)le

Figure 11 Bilinear crack closure model

ISRNMechanical Engineering 9

0 1205872 120587

120587

51205876

21205873

120572 = minus0084 lowast 1205792 + 051 lowast 120579 + 24

120572 = minus0018 lowast 1205793 + 0016 lowast 1205792 + 035 lowast 120579 + 24

SimulationQuadraticCubic

120579

120572

(rad)

(rad

)

Figure 12 Crack front angle 120572 during rotation of shaft

0 1205872 120587

1

09

08

07

06

05

04

03

02

01

0

119887119887 m

ax

SimulationCubic4th degree

= minus0069 lowast 1205793 + 02 lowast 1205792 + 036 lowast 120579 minus 0066

= 0076 lowast 1205794 minus 054lowast 1205793 + 11 lowast 1205792 minus 023 lowast 120579 minus

120579 (rad)

00056119887119887max

119887119887max

Figure 13 Normalized crack width 119887119887max during rotation of shaft

1

09

08

07

06

05

04

03

02

01

00 1205876 1205873 1205872 21205873 51205876 120587

Breathing mechanismBilinear crack closure model

Straight line crack closure [33]Perpendicular crack closure [12]

119860op

en119860

crac

k

120579 (rad)

Figure 14 Comparison of relative crack area between breathingmechanism results straight line perpendicular and bilinear crackclosure model for a shallow crack 119886119889 = 01

098

096

094

092

09

088

086

084

082

080 1205876 1205873 1205872 21205873 51205876 120587

Breathing mechanismBilinear crack closure model

Straight line crack closure [33]Perpendicular crack closure [12]

120578

120585

119870cr

ack119870

120579 (rad)

Figure 15 Comparison of normalized stiffness between breathingmechanism results straight line perpendicular and bilinear crackclosure model for a shallow crack 119886119889 = 01

shaft It is recommended to use CZM to study differenttypes of cracks such as longitudinal and slant cracks andalso multiple cracks Some other parameters such as internaldamping and thermal transients could be studied to obtainresults for their effect on the breathing mechanism Furtheranalysis on crack morphology is extremely important tounderstand the dynamic behaviour of cracked shaft thiswould include shallow and wide cracks

Acknowledgments

The authors acknowledge the support from Deutsche For-schungsgemeinschaft and Open Access Publishing Fund ofKarlsruhe Institute of Technology

References

[1] S K Georgantzinos and N K Anifantis ldquoAn insight into thebreathing mechanism of a crack in a rotating shaftrdquo Journal ofSound and Vibration vol 318 no 1-2 pp 279ndash295 2008

[2] J Wauer ldquoOn the dynamics of cracked rotors literature surveyrdquoApplied Mechanics Review vol 43 no 1 pp 13ndash17 1990

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

[4] G Sabnavis R G Kirk M Kasarda and D Quinn ldquoCrackedshaft detection and diagnostics a literature reviewrdquo Shock andVibration Digest vol 36 no 4 pp 287ndash296 2004

[5] 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

[6] N Bachschmid P Pennacchi and E Tanzi ldquoSome remarks onbreathing mechanism on non-linear effects and on slant andhelicoidal cracksrdquo Mechanical Systems and Signal Processingvol 22 no 4 pp 879ndash904 2008

[7] N Bachschmid P Pennachi and E Tanzi Cracked RotorsSpringer Berlin Germany 2011

10 ISRNMechanical Engineering

[8] S Andrieux and C Vare ldquoA 3D cracked beam model withunilateral contact Application to rotorsrdquo European Journal ofMechanics A vol 21 no 5 pp 793ndash810 2002

[9] C Vare and S Andrieux ldquoModeling of a cracked beam sectionunder bendingrdquo in Proceedings of the 18th International Confer-ence on Structural Mechanics in Reactor Technolohy (SMiRT 18)pp 281ndash290 Beijing China 2005

[10] S E Arem and HMaitournam ldquoA cracked beam finite elementfor rotating shaft dynamics and stability analysisrdquo Journal ofMechanics of Materials and Structures vol 3 no 5 pp 893ndash9102008

[11] A D Dimarogonas and S A Paipetis Analytical Methods inRotorDynamics Applied Science Publishers LondonUK 1983

[12] A K Darpe K Gupta and A Chawla ldquoCoupled bendinglongitudinal and torsional vibrations of a cracked rotorrdquo Journalof Sound and Vibration vol 269 no 1-2 pp 33ndash60 2004

[13] W M Ostachowicz and M Krawczuk ldquoCoupled torsional andbending vibrations of a rotor with an open crackrdquo Archive ofApplied Mechanics vol 62 no 3 pp 191ndash201 1992

[14] Z Kulesza and J T Sawicki ldquoRigid finite element model ofa cracked rotorrdquo Journal of Sound and Vibration vol 331 pp4145ndash4169 2012

[15] A S Bouboulas and N K Anifantis ldquoFinite element modelingof a vibrating beam with a breathing crack observations oncrack detectionrdquo Structural Health Monitoring vol 10 no 2 pp131ndash145 2011

[16] D S Dugdale ldquoYielding of steel sheets containing slitsrdquo Journalof the Mechanics and Physics of Solids vol 8 no 2 pp 100ndash1041960

[17] G I Barenblatt ldquoThemathematical theory of equilibrium crackin brittle fracturerdquoAdvances in AppliedMechanics vol 7 pp 55ndash129 1962

[18] T Siegmund andW Brocks ldquoTensile decohesion by local failurecriteriardquo Technische Mechanik vol 18 no 4 pp 261ndash270 1998

[19] T Siegmund and W Brocks ldquoA numerical study on thecorrelation between the work of separation and the dissipationrate in ductile fracturerdquo Engineering Fracture Mechanics vol 67no 2 pp 139ndash154 2000

[20] M Anvari I Scheider and CThaulow ldquoSimulation of dynamicductile crack growth using strain-rate and triaxiality-dependentcohesive elementsrdquo Engineering Fracture Mechanics vol 73 no15 pp 2210ndash2228 2006

[21] I Scheider ldquoDerivation of separation laws for cohesive modelsin the course of ductile fracturerdquo Engineering Fracture Mechan-ics vol 76 no 10 pp 1450ndash1459 2009

[22] A Banerjee and R Manivasagam ldquoTriaxiality dependent cohe-sive zone modelrdquo Engineering Fracture Mechanics vol 76 no12 pp 1761ndash1770 2009

[23] A D Dimarogonas and C A Papadopoulos ldquoVibration ofcracked shafts in bendingrdquo Journal of Sound and Vibration vol91 no 4 pp 583ndash593 1983

[24] A C Chasalevris andCA Papadopoulos ldquoA continuousmodelapproach for cross-coupled bending vibrations of a rotor-bearing system with a transverse breathing crackrdquo Mechanismand Machine Theory vol 44 no 6 pp 1176ndash1191 2009

[25] 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

[26] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel to the analysis of a rotor with a transverse crackrdquo Journalof Sound and Vibration vol 332 no 8 pp 2098ndash2110 2013

[27] C Shet andN Chandra ldquoAnalysis of energy balancewhen usingCohesive Zone Models to simulate fracture processesrdquo Journalof EngineeringMaterials andTechnology vol 124 no 4 pp 440ndash450 2002

[28] H Li and N Chandra ldquoAnalysis of crack growth and crack-tip plasticity in ductile materials using cohesive zone modelsrdquoInternational Journal of Plasticity vol 19 no 6 pp 849ndash8822003

[29] N Bachschmid P Pennacchi and E Tanzi ldquoOn the evolutionof vibrations in cracked rotorsrdquo in Proceedings of the 8thIFToMM International Conference on Rotor Dynamics pp 304ndash310 Seoul Korea 2010

[30] O S Jun H J Eun Y Y Earmme and C W Lee ldquoModellingand vibration analysis of a simple rotor with a breathing crackrdquoJournal of Sound andVibration vol 155 no 2 pp 273ndash290 1992

[31] J J Sinou and A W Lees ldquoThe influence of cracks in rotatingshaftsrdquo Journal of Sound and Vibration vol 285 no 4-5 pp1015ndash1037 2005

[32] Y S Shih and J J Chen ldquoAnalysis of fatigue crack growth on acracked shaftrdquo International Journal of Fatigue vol 19 no 6 pp477ndash485 1997

[33] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel for the investigation of the dynamic behavior of arotating shaft with a transverse crackrdquo in Proceedings of the 8thIFToMM International Conference on Rotor Dynamics pp 628ndash636 Seoul Korea September 2010

[34] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel to the analysis of a rotor with a transverse crackrdquo inProceedings of the 8th International Conference on StructuralDynamics pp 3434ndash3442 Leuven Belgium 2011

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International Journal of

ISRNMechanical Engineering 3

Initialcrack

Crackedzone Cohesive fracture zone

Undamagedligament

120575120590max

120590 = 0120590 = 119891(120575)

120575119888

Figure 1 Cohesive crack process zone

maximum principal stress reaches a given value namely thecohesive strength of thematerialWhen this occurs the cracksurface initiates or grows perpendicularly to the directionof the maximum principal stress The two faces of thesurface exert on each other equal and opposite tensile stresses(cohesive stresses) whose value is a unique function 119891(120575)of the separation 120575 between the faces When the separationreaches another given value (the critical separation 120575

119888) the

cohesive stress vanishes and fracture takes place Fractureconsists of the initiation and propagation of a crack producedby the opening and advance of the cohesive zone (the zonewhere the cohesive stresses act) ahead of the crack tip asshown in Figure 1

The fracture behaviour of each material is described bythe cohesive traction as a function

120590119899= 120590max119891 (120575) (1)

where 120590max is the peak value of traction The function 119891(120575)defines the shape of the traction-separation law and the areaunder the cohesive law curve is the work of separation orcohesive energy 119866

119888

119866119888= int

120575119888

0

120590119899(120575) d120575 (2)

Ideally the model within the cohesive zone should beable to replicate the constitutive behaviour of the undamagedmaterial linear elastic followed by strain hardening till theconditions for the initiation of softening due to damage arereached The softening process representative of increasingmaterial degradation is triggered by rapid growth of voids asconsequence of a highly triaxial state of stress

Since the CZM is a phenomenological model variousformulations for defining the shape of traction-separation lawand the cohesive values are in use [27] A versatile CZM topredict ductile fracture at different states of stress is proposedin [22] The formulation developed for mode-I plane strainaccounts explicitly for triaxiality of the stress state by usingbasic elastic-plastic constitutive relations combined withtwo stress-state-independent new model parameters Theproposed traction-separation law has three distinct regions ofconstitutive behaviour the traction separation law is linear upto the separation limit 120575

119899= 1205751and exhibits strain hardening

Linearregion

120590max

Hardeningregion

Softeningregion

Cohesive energy

0 1205751 1205752 120575c = 120575sep120575119899

120590119899

120590119884

119870119901

Figure 2 Traction separation law for ductile materials

up to 120575119899= 1205752followed by a softening curve The relevant

variables and zones are sketched in Figure 2

120590119899= (1 + radic3120594ef)

2119864

3120575 for 0 lt 120575 lt 120575

1 (3)

The linear behaviour of the cohesive zone exists till theseparation limit defined by the von Mises yield condition isreached

1205751=radic3120590119884

2119864 (4)

Further separation results in strain hardening up to

1205752=radic3

2(119862eminus(32)120594ef + 120590119884

119864) (5)

22 Numerical Representation of Cohesive Zone ModelsWhen fracture proceeds energymust be supplied by externalloads The bounding material undergoes elastoplastic defor-mation involving elastic energy and plastic dissipative energyIn addition to plasticity energy is supplied to the fractureprocess zone in the form of cohesive energy that is dissipatedwithin the cohesive elements The cohesive energy is the sumof the surface energy and all dissipative processes that takeplace within the crack tip regime For the present problem aperfect energy balance between external work119882 and the sumof elastic energy 119864el and cohesive energy 119866

119888[27 28] will be

assumed The energy balance is given by

119882 = 119864el + 119866119888 (6)

119882 = int

119905

0

(int119881

120590 120576el d119881) d119905 + int119905

0

(int119878

120590119899 d119878) d119905 (7)

where 120590 120576el 120590119899 and are nominal stress tensor elasticstrain rate cohesive traction and cohesive separation raterespectively (including the terms for specimen volume119881 andthe internal specimen surface 119878)The external work due to theapplied force is given by

119882 = int

119905

0

(int119878

t sdot v d119878) d119905 + int119905

0

(int119881

b sdot v d119881) d119905 (8)

where v is the velocity field vector t the exterior surfacetraction vector and b the body force vector

4 ISRNMechanical Engineering

Real

Finite elements

IdealizationContinuum elements

Crack tip

Cohesive elements

Continuum

Crack

Brokencohesiveelements

Separationof cohesiveelements

Cohesivetractions120590119899 = 119891(120575)

Δ119886

Figure 3 Representation of the fracture process using CZM in FE

The cohesive surface contribution representing the crackand the process zone in front of the crack tip is described bythe integral over the internal surface 119878 In this formulation 120590

119899

denotes the cohesive strength that is the maximum tractionvalue that can be sustained within the cohesive zone Thecohesive length 120575 is the value of the displacement jump acrossthe crack surfaces at which the stress carrying capacity of thecohesive elements reaches its maximum value By creatingnew surfaces the traction and the stiffness of the cohesivezone elements connecting these newly created surfaces aremade to vanish but the displacements across them are stillcontinuous During FE analysis the amount of externalwork elastic work and cohesive energy is calculated Theenergy balance given by (7) and (8) is maintained in all FEcomputations

23 Finite Element Implementation The cohesive surfacecontribution as shown in Figure 3 is implemented into anFE code for eight node elements based on CZM conceptsThe fine mesh around the crack tip is shown in Figure 4After making convergence studies the shaft is discretizedwith 4108 plane strains and 8 node quadrilateral elementsThe length of elements is ℓ where the length of shaft is20ℓ Near the crack tip the size of the element is ℓ20A total of 4833 nodes are used to model the geometry asshown in Figure 4 The fracture process zone is modeled by8 node rectangular cohesive elements having zero thickness38 cohesive elements are used One face of cohesive elementsis connected to continuum elements with 4 nodes whilethe other face is given symmetric displacement boundaryconditions Thus an artificial interface is created along thefracture process zone

The maximum cohesive traction is assumed to be thesame as the yield strength of the material of shaft namely120590119899= 250MPa The maximum separation at the end of the

elastic zone is 1205751= 10 120583m The interface stiffness or penalty

stiffness 119870119901can be obtained directly from 119870

119901= 1205901198991205751and

is used to ensure a stiff connection between the surfaces of

Crack tip1

20

Crack tip1

20985747985747

Figure 4 Finite element mesh model near the crack tip

the material discontinuity The values of the penalty stiffnessshould be selected in a way such that the connection betweenthe two elements and numerical stability are guaranteed

3 Investigation of Breathing Mechanism

31 Method and Assumptions The breathing mechanism of asimple cracked rotor on rigid supports has been investigatedThe method of the investigation is summarized in Figure 5Several assumptions are made the location of the crack isassumed to be known at the mid-span of the shaft whichis the same as the maximum deflection point Only a singletransverse crack has been considered in the shaft The planestrain condition is assumed at the crack front due to thegeometry constraint Both ends of the shaft are rigidlysupported The shaft has a uniform circular cross sectionLength ℓ and diameter 119889 of the shaft are assumed 10m and008m respectivelyThematerial of the shaft is considered tobe homogeneous and isotropic Youngrsquos modulus 119864 Poissonrsquosratio 120584 and mass density 120588 are 210GPa 03 and 7850 kgm3respectively Yield strength 120590

119884and ultimate strength of

material 120590max are 250MPa and 400MPa respectively Thecohesive elements along the crack surfaces are implementedas interface elements as shown in Figure 6

ISRNMechanical Engineering 5Ab

aqus

Adam

sAb

aqus

Build FE model of cracked elastic shaft- Define relative crack depth 119886119889- Define crack location- Define interface crack points- Define continuum and cohesive material

Transfer FE model to MBS

- Connect cracked elastic shaft to bearings and rigid disk- Vibration responses due to static (weight) and dynamic

(due to crack and inertia force) during rotation

Transfer output of MBS into FE- Based on the vibration responses the breathing crack

is investigated

loads

mechanism

Figure 5 Principle of the breathing crack simulation using inte-grated FE and MBS

Continuumelements

Cohesiveelements

Continuumelements

Figure 6 FE model of an elastic cracked shaft

32 Breathing Mechanism Simulation The breathing mech-anism generated by the rotating bending load used in theliterature has some limitations Due to the presence ofinertia forces the dynamic behaviour of rotating structuresis different from those of static structures Although in caseof weight dominance the amplitude of the vibration responsedue to inertia forces is smaller than the amplitude due toweight forces elastic forces and presence of a crack (if theshaft is assumed to be balanced) using the inertia force takeninto account will yield more accurate results The 3D FEcalculations allow the breathing mechanism to be predictedaccurately The breathing mechanism is strongly influencedby the weight of the shaft [29] In many publications on thebreathing crack assumptions on the breathing mechanismhave been made For transverse cracks in rotating shafts thevariation of the crack front line perpendicular to the crackfront has been extensively addressed by Darpe et al [12]Thisbreathing crack form is also used by Jun et al [30] and Sinouand Lees [31] An elliptical shape has been proposed by Shihand Chen [32] Based on FE model simulation and somereported experimental results the breathing crack shape ismodelled by a parabolic shape that opens and closes due tobending stresses (Liong and Proppe [26 33 34])

The idea is that the vibration responses in the centroid ofthe shaft obtained from MBS are exported into FE model inorder to analyse the breathing mechanism as schematicallyshown in Figure 7 The opening crack is simulated for onecycle of revolution of the cracked shaft specimen in steady-state condition The breathing mechanism is generated bythe bending due to external load (weight) by increasing theangle by steps of 12058712 rad The breathing (open and closedcrack areas are evaluated in each angular step) is observedby the nodal displacement and the stress distribution (tensileor compressive stress) around the crack The prediction ofthe breathing mechanism was performed by the followingsteps for each angle of rotation stress distribution due to thebending moment is recorded over the cross section Com-pressive (negative) stresses indicate closed region The crackopens where zero or very small positive numerical values ofstresses appear and the contact forces vanish Displacementsand stresses can be observed at the crack surfaces In order toavoid local deformations due to the application of loads thedeflection and stress distribution at each crack element areamust be evaluated as shown in Figure 8

4 Results and Discussion

41 BreathingMechanism Results Figure 9 displays the resultof breathing mechanism during half-revolution of a rotat-ing shaft (from closed to open crack) with relative crackdepth 119886119889 = 01 As can be seen the crack will start to appearand begin to open at 1205876 The crack opens more slowly atthe beginning but increases its opening at 1205873 At 51205876 it isalready completely open Similarly the crack closes again at71205876 and increases its closing at 31205872 The crack is alreadycompletely closed at 111205876 It is interesting to note that theangular positions where the crack opens again (from fullyclosed) are the same as where the crack begins to close

Results of the relative crack depth versus the shaft rotationangle during breathing are shown in Figure 10 Four functionsare used to approximate the relative crack depth namelylinear quadratic harmonic and power functions Table 1gives the approximations for a relative crack depth 119886119889 as afunction of shaft rotation angle 120579 It is shown that harmonicand quadratic functions have the minimal error and thenearest correlation with the simulation results In this sensethe relative crack depth during crack opening should beunderstood not as linear increasing but as harmonic functionor quadratic polynomial

42 Discussion Due to the presence of gravity the upperportion of the cracked rotor at the beginning of a rotationis under compression and the crack is closed As the rotorcontinues to rotate and the gravity direction is constant theupper part now comes in the lower tensile region causingthe crack to open Moreover the accuracy of the results ismodelled by a bilinear model as shown in Figure 11 Thecrack begins opening at shaft rotation angle 120579

0and at crack

front angle 120572 that defines the angle between two crack frontlines The shaft rotates with shaft rotation angle 120579 and thecrack front angle 120572 increases until maximum crack width119887max is reached The crack front angle 120572 and the crack width

6 ISRNMechanical Engineering

FLEX BODY 1MARKER 1FLEX BODY 1INT NODE

FLEX BODY 1INT NODE

FLEX BODY 1INT NODE 2000FLEX BODY 1MARKER 3

Gravity

119909

119909

119909

119910

119910

119910

119911

119911

119911

1

05

0

minus05

minus10 01 02 03 04 05 06 07 08 09 1

119905 (s)

times10minus5

0 01 02

0

minus05

minus1

minus15

minus2

times10minus5

119911 119910

10000

30000

Am

plitu

de119910

(m)

Am

plitu

de119909

(m)

119899 = 600 rpm (119891 = 10HZ) 119886119889 = 01

Figure 7 The transfer of the vibration responses fromMBS into FE

119887 are nonlinearly increased during rotation from 1205790to 120579max

(Figures 12 and 13) One can obtain the crack front angle 120572and the crack width 119887 as functions of the shaft rotation angle120579 respectively by

120572 =

0 120579 le 1205790

crack still closedminus0018120579

3

+ 00161205792

+035120579 + 24 1205790le 120579 le 120579max

crack opens0 120579 ge 120579max

crack opens completely

119887

119887max

=

0 120579 le 1205790

crack still closed0076120579

4

minus 0541205793

+ 111205792

minus023120579 minus 00056 1205790le 120579 le 120579max

crack opens0 120579 ge 120579max

crack opens completely(9)

For the relative crack depth 119886119889 = 01 the crack beginsto open at 120579

0= 1205876 and 120572 = 51205876 while the crack opens

completely at 120579max = 51205876 and 120572 = 120587 Therefore the crackfront angle and the crack width during crack opening are

120572 = minus00181205793

+ 00161205792

+ 035120579 + 24120587

6le 120579 le

5120587

6

119887

119887max= 0076120579

4

minus 0541205793

+ 111205792

minus 023120579 minus 00056

120587

6le 120579 le

5120587

6

(10)

The relative crack area (ie ratio of the partially opencrack area to the fully open crack area) with respect to theangle of rotation is used as shown in Figure 14 The resultsare similar when the crack closes The relative crack area ofsimulation results is in good agreement with the crack closurestraight linemodel [33] between rotation angles1205873 and 51205876andwith the bilinearmodel in the range from0 to120587 but thereare some relevant differences with respect to the crack closureperpendicular line used byDarpe et al [12]The areamomentof inertia about the rotation axis can be approximated Finallythe stiffness of the cracked shaft can be evaluated (stiffness isfound to vary linearly with the area moment of inertia) Thenormalised stiffness (ratio between stiffness of the crackedshaft and stiffness of the uncracked shaft) is presented asshown in Figure 15 It is shown that the normalized stiffnessbetween breathing mechanism results and the bilinear crackclosure model is very close which validates the accuracy ofbilinear model

The fracture mechanics analysis presupposes the exis-tence of an infinitely sharp crack leading to the singular cracktip fields However in real materials neither the sharpness ofthe crack nor the stress levels near the crack tip region can be

ISRNMechanical Engineering 7

Crack

Tensile stressor zero stress

Displacement+minus

Figure 8 Evaluation of displacements and stresses for state of crack

0∘ 30∘ 45∘

60∘ 75∘ 90∘

105∘ 120∘ 135∘

150∘ 165∘ 180∘

Figure 9 Breathing crack of the rotating shaft relative crack depth119886119889 = 01 at 600 rpm

infinite [27] As an alternative approach to this singularity-driven fracture approach the concept of CZM is proposedCZM has evolved as a preferred method to analyze fractureproblems not only because it avoids the singularity but alsobecause it can be easily implemented in a numerical methodof analysis as in FE method CZM presupposes the presenceof a fracture process zone where the energy is transferred

from external work in the crack regions In this study theexternal work flows as recoverable elastic strain energy andcohesive energy are examined the latter encompassing thework and other energy consuming mechanisms within thefracture process zone Process zone is defined as the regionwithin the separating surfaces where the surface tractionvalues are nonzeroThis also implies that processes occurringwithin the process zone are accounted for only throughthe traction-displacement relations The traction-separationrelations for crack interfaces are demonstrated such thatwith increasing interfacial separation the traction across theinterface reaches amaximum itmeans the elastic limit whereplastic dissipation in the surrounding (bounding) material isnot accounted for in the process zone

5 Conclusions

In this study a simulation process of FE and MBS isemployed An elastic cracked shaft is modelled by FE whichis transferred into MBS model The vibration responses inthe centroid of the shaft obtained from MBS have beentransferred again into an FE model in order to observe thebreathing mechanism and to study the stiffness losses due tobreathing at different rotating speeds

The results have shown that the breathing mechanismis influenced by the vibration due to inertia forces and byrelative crack depth It is shown that the relative crack depthduring crack opening should be understood not as linearincreasing but as harmonic function or quadratic polynomialIt can be noted that as long as the relative crack depth issmall the bilinear crack closure model may be used Themain difference with respect to the straight line crack closuremodel is that the opening crack in the simulation is notconstant at the beginningThe relative crack area of breathingmechanism results is in good agreement with the straight linecrack closuremodel between rotation angles1205873 and1205876 andwith the bilinear crack closuremodel in the range from0 to120587Furthermore the area moment of inertia about the rotationaxis can be approximated From this information the stiffnesslosses of the shaft can be evaluated It has been shown thatthe normalized stiffness between simulation and the bilinearcrack closure model is very close which validates the bilinearmodel

CZM are being increasingly used to simulate crackInstead of an infinitely sharp crack envisaged in fracturemechanics CZM presupposes the presence of a fracture pro-cess zone where the energy is transferred from external workin the crack regions The traction-separation relations forcrack interfaces are demonstrated such that with increasinginterfacial separation the traction across the interface reachesa maximum it means the elastic limit where plastic dissipa-tion in the surrounding (bounding)material is not accountedfor in the process zone

In light of the presented results and the conclusionsseveral subjects can be recommended for future research Animportant point on which the knowledge could be improvedis the prediction of crack propagation on cracked rotor andresidual life estimation The CZM can be easily implementedin FE model to analyze the dynamic behaviour of a cracked

8 ISRNMechanical Engineering

0 120587 2120587

119886119889

01

008

006

004

002

0

Linear model119891(119909) = 119886119909 + 119887

Confidence bounds 95

Goodness of fitSSE = 00005796R-square = 09516

120579 (rad)

119886 = 003887 (003226 004548)

119887 = 000131 (minus001043 001305)

(a)

0 120587 2120587

01

008

006

004

002

0

119886119889

Quadratic polynomial119891(119909) = 1198861199092 + 119887119909 + 119888

Confidence bounds 95

Goodness of fitSSE = 1141119890 minus 005R-square = 09990

120579 (rad)

119886 = minus001187 (minus001325 minus001050)119887 = 007617 (007174 008059)

119888 = minus001985 (minus002289 minus001683)

(b)

0 120587 2120587

01

008

006

004

002

0

119886119889

General model of sinus function

Confidence bounds 95

Goodness of fitSSE = 6158119890 minus 06R-square = 09995

120579 (rad)

119891(119909) = 119886 sin (119887119909 + 119888)

119886 = 009969 (009806 010130)119887 = 059340 (056990 061680)119888 = minus014540 (minus016410 minus012660)

(c)

0 120587 2120587

01

008

006

004

002

0

119886119889

General model of power119891(119909) = 119886119909119887 + 119888

Confidence bounds 95

Goodness of fitSSE = 9952119890 minus 05R-square = 09917

120579 (rad)

119886 = 012310 (004674 019950)119887 = 037300 (015200 059400)119888 = minus007730 (minus015120 minus000336)

(d)

Figure 10 Curve fitting of breathing mechanism results

Table 1 Curve fitting of the simulation results for 119886119889 = 01 120579 in [rad]

Model Mathematical model SSE R-squareLinear function 119886119889 = 0039120579 + 0001 0000 580 0951 6Quadratic polynomial 119886119889 = minus00121205792 + 0076120579minus 0020 0000 011 0999 0Sinus function 119886119889 = 0010sin (0593120579minus 0145) 0000 006 0999 5Power function 119886119889 = 01231205790373minus 0077 0000 099 0991 7

Crack closure line 1205721205720

119886

119887

119887max

Bilinear borderline

120575 gt 100120583m (crack opens)

10120583m le 120575 100120583m

120575 lt 10120583m

120575 = 0 (crack still closed)le

Figure 11 Bilinear crack closure model

ISRNMechanical Engineering 9

0 1205872 120587

120587

51205876

21205873

120572 = minus0084 lowast 1205792 + 051 lowast 120579 + 24

120572 = minus0018 lowast 1205793 + 0016 lowast 1205792 + 035 lowast 120579 + 24

SimulationQuadraticCubic

120579

120572

(rad)

(rad

)

Figure 12 Crack front angle 120572 during rotation of shaft

0 1205872 120587

1

09

08

07

06

05

04

03

02

01

0

119887119887 m

ax

SimulationCubic4th degree

= minus0069 lowast 1205793 + 02 lowast 1205792 + 036 lowast 120579 minus 0066

= 0076 lowast 1205794 minus 054lowast 1205793 + 11 lowast 1205792 minus 023 lowast 120579 minus

120579 (rad)

00056119887119887max

119887119887max

Figure 13 Normalized crack width 119887119887max during rotation of shaft

1

09

08

07

06

05

04

03

02

01

00 1205876 1205873 1205872 21205873 51205876 120587

Breathing mechanismBilinear crack closure model

Straight line crack closure [33]Perpendicular crack closure [12]

119860op

en119860

crac

k

120579 (rad)

Figure 14 Comparison of relative crack area between breathingmechanism results straight line perpendicular and bilinear crackclosure model for a shallow crack 119886119889 = 01

098

096

094

092

09

088

086

084

082

080 1205876 1205873 1205872 21205873 51205876 120587

Breathing mechanismBilinear crack closure model

Straight line crack closure [33]Perpendicular crack closure [12]

120578

120585

119870cr

ack119870

120579 (rad)

Figure 15 Comparison of normalized stiffness between breathingmechanism results straight line perpendicular and bilinear crackclosure model for a shallow crack 119886119889 = 01

shaft It is recommended to use CZM to study differenttypes of cracks such as longitudinal and slant cracks andalso multiple cracks Some other parameters such as internaldamping and thermal transients could be studied to obtainresults for their effect on the breathing mechanism Furtheranalysis on crack morphology is extremely important tounderstand the dynamic behaviour of cracked shaft thiswould include shallow and wide cracks

Acknowledgments

The authors acknowledge the support from Deutsche For-schungsgemeinschaft and Open Access Publishing Fund ofKarlsruhe Institute of Technology

References

[1] S K Georgantzinos and N K Anifantis ldquoAn insight into thebreathing mechanism of a crack in a rotating shaftrdquo Journal ofSound and Vibration vol 318 no 1-2 pp 279ndash295 2008

[2] J Wauer ldquoOn the dynamics of cracked rotors literature surveyrdquoApplied Mechanics Review vol 43 no 1 pp 13ndash17 1990

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

[4] G Sabnavis R G Kirk M Kasarda and D Quinn ldquoCrackedshaft detection and diagnostics a literature reviewrdquo Shock andVibration Digest vol 36 no 4 pp 287ndash296 2004

[5] 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

[6] N Bachschmid P Pennacchi and E Tanzi ldquoSome remarks onbreathing mechanism on non-linear effects and on slant andhelicoidal cracksrdquo Mechanical Systems and Signal Processingvol 22 no 4 pp 879ndash904 2008

[7] N Bachschmid P Pennachi and E Tanzi Cracked RotorsSpringer Berlin Germany 2011

10 ISRNMechanical Engineering

[8] S Andrieux and C Vare ldquoA 3D cracked beam model withunilateral contact Application to rotorsrdquo European Journal ofMechanics A vol 21 no 5 pp 793ndash810 2002

[9] C Vare and S Andrieux ldquoModeling of a cracked beam sectionunder bendingrdquo in Proceedings of the 18th International Confer-ence on Structural Mechanics in Reactor Technolohy (SMiRT 18)pp 281ndash290 Beijing China 2005

[10] S E Arem and HMaitournam ldquoA cracked beam finite elementfor rotating shaft dynamics and stability analysisrdquo Journal ofMechanics of Materials and Structures vol 3 no 5 pp 893ndash9102008

[11] A D Dimarogonas and S A Paipetis Analytical Methods inRotorDynamics Applied Science Publishers LondonUK 1983

[12] A K Darpe K Gupta and A Chawla ldquoCoupled bendinglongitudinal and torsional vibrations of a cracked rotorrdquo Journalof Sound and Vibration vol 269 no 1-2 pp 33ndash60 2004

[13] W M Ostachowicz and M Krawczuk ldquoCoupled torsional andbending vibrations of a rotor with an open crackrdquo Archive ofApplied Mechanics vol 62 no 3 pp 191ndash201 1992

[14] Z Kulesza and J T Sawicki ldquoRigid finite element model ofa cracked rotorrdquo Journal of Sound and Vibration vol 331 pp4145ndash4169 2012

[15] A S Bouboulas and N K Anifantis ldquoFinite element modelingof a vibrating beam with a breathing crack observations oncrack detectionrdquo Structural Health Monitoring vol 10 no 2 pp131ndash145 2011

[16] D S Dugdale ldquoYielding of steel sheets containing slitsrdquo Journalof the Mechanics and Physics of Solids vol 8 no 2 pp 100ndash1041960

[17] G I Barenblatt ldquoThemathematical theory of equilibrium crackin brittle fracturerdquoAdvances in AppliedMechanics vol 7 pp 55ndash129 1962

[18] T Siegmund andW Brocks ldquoTensile decohesion by local failurecriteriardquo Technische Mechanik vol 18 no 4 pp 261ndash270 1998

[19] T Siegmund and W Brocks ldquoA numerical study on thecorrelation between the work of separation and the dissipationrate in ductile fracturerdquo Engineering Fracture Mechanics vol 67no 2 pp 139ndash154 2000

[20] M Anvari I Scheider and CThaulow ldquoSimulation of dynamicductile crack growth using strain-rate and triaxiality-dependentcohesive elementsrdquo Engineering Fracture Mechanics vol 73 no15 pp 2210ndash2228 2006

[21] I Scheider ldquoDerivation of separation laws for cohesive modelsin the course of ductile fracturerdquo Engineering Fracture Mechan-ics vol 76 no 10 pp 1450ndash1459 2009

[22] A Banerjee and R Manivasagam ldquoTriaxiality dependent cohe-sive zone modelrdquo Engineering Fracture Mechanics vol 76 no12 pp 1761ndash1770 2009

[23] A D Dimarogonas and C A Papadopoulos ldquoVibration ofcracked shafts in bendingrdquo Journal of Sound and Vibration vol91 no 4 pp 583ndash593 1983

[24] A C Chasalevris andCA Papadopoulos ldquoA continuousmodelapproach for cross-coupled bending vibrations of a rotor-bearing system with a transverse breathing crackrdquo Mechanismand Machine Theory vol 44 no 6 pp 1176ndash1191 2009

[25] 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

[26] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel to the analysis of a rotor with a transverse crackrdquo Journalof Sound and Vibration vol 332 no 8 pp 2098ndash2110 2013

[27] C Shet andN Chandra ldquoAnalysis of energy balancewhen usingCohesive Zone Models to simulate fracture processesrdquo Journalof EngineeringMaterials andTechnology vol 124 no 4 pp 440ndash450 2002

[28] H Li and N Chandra ldquoAnalysis of crack growth and crack-tip plasticity in ductile materials using cohesive zone modelsrdquoInternational Journal of Plasticity vol 19 no 6 pp 849ndash8822003

[29] N Bachschmid P Pennacchi and E Tanzi ldquoOn the evolutionof vibrations in cracked rotorsrdquo in Proceedings of the 8thIFToMM International Conference on Rotor Dynamics pp 304ndash310 Seoul Korea 2010

[30] O S Jun H J Eun Y Y Earmme and C W Lee ldquoModellingand vibration analysis of a simple rotor with a breathing crackrdquoJournal of Sound andVibration vol 155 no 2 pp 273ndash290 1992

[31] J J Sinou and A W Lees ldquoThe influence of cracks in rotatingshaftsrdquo Journal of Sound and Vibration vol 285 no 4-5 pp1015ndash1037 2005

[32] Y S Shih and J J Chen ldquoAnalysis of fatigue crack growth on acracked shaftrdquo International Journal of Fatigue vol 19 no 6 pp477ndash485 1997

[33] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel for the investigation of the dynamic behavior of arotating shaft with a transverse crackrdquo in Proceedings of the 8thIFToMM International Conference on Rotor Dynamics pp 628ndash636 Seoul Korea September 2010

[34] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel to the analysis of a rotor with a transverse crackrdquo inProceedings of the 8th International Conference on StructuralDynamics pp 3434ndash3442 Leuven Belgium 2011

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International Journal of

4 ISRNMechanical Engineering

Real

Finite elements

IdealizationContinuum elements

Crack tip

Cohesive elements

Continuum

Crack

Brokencohesiveelements

Separationof cohesiveelements

Cohesivetractions120590119899 = 119891(120575)

Δ119886

Figure 3 Representation of the fracture process using CZM in FE

The cohesive surface contribution representing the crackand the process zone in front of the crack tip is described bythe integral over the internal surface 119878 In this formulation 120590

119899

denotes the cohesive strength that is the maximum tractionvalue that can be sustained within the cohesive zone Thecohesive length 120575 is the value of the displacement jump acrossthe crack surfaces at which the stress carrying capacity of thecohesive elements reaches its maximum value By creatingnew surfaces the traction and the stiffness of the cohesivezone elements connecting these newly created surfaces aremade to vanish but the displacements across them are stillcontinuous During FE analysis the amount of externalwork elastic work and cohesive energy is calculated Theenergy balance given by (7) and (8) is maintained in all FEcomputations

23 Finite Element Implementation The cohesive surfacecontribution as shown in Figure 3 is implemented into anFE code for eight node elements based on CZM conceptsThe fine mesh around the crack tip is shown in Figure 4After making convergence studies the shaft is discretizedwith 4108 plane strains and 8 node quadrilateral elementsThe length of elements is ℓ where the length of shaft is20ℓ Near the crack tip the size of the element is ℓ20A total of 4833 nodes are used to model the geometry asshown in Figure 4 The fracture process zone is modeled by8 node rectangular cohesive elements having zero thickness38 cohesive elements are used One face of cohesive elementsis connected to continuum elements with 4 nodes whilethe other face is given symmetric displacement boundaryconditions Thus an artificial interface is created along thefracture process zone

The maximum cohesive traction is assumed to be thesame as the yield strength of the material of shaft namely120590119899= 250MPa The maximum separation at the end of the

elastic zone is 1205751= 10 120583m The interface stiffness or penalty

stiffness 119870119901can be obtained directly from 119870

119901= 1205901198991205751and

is used to ensure a stiff connection between the surfaces of

Crack tip1

20

Crack tip1

20985747985747

Figure 4 Finite element mesh model near the crack tip

the material discontinuity The values of the penalty stiffnessshould be selected in a way such that the connection betweenthe two elements and numerical stability are guaranteed

3 Investigation of Breathing Mechanism

31 Method and Assumptions The breathing mechanism of asimple cracked rotor on rigid supports has been investigatedThe method of the investigation is summarized in Figure 5Several assumptions are made the location of the crack isassumed to be known at the mid-span of the shaft whichis the same as the maximum deflection point Only a singletransverse crack has been considered in the shaft The planestrain condition is assumed at the crack front due to thegeometry constraint Both ends of the shaft are rigidlysupported The shaft has a uniform circular cross sectionLength ℓ and diameter 119889 of the shaft are assumed 10m and008m respectivelyThematerial of the shaft is considered tobe homogeneous and isotropic Youngrsquos modulus 119864 Poissonrsquosratio 120584 and mass density 120588 are 210GPa 03 and 7850 kgm3respectively Yield strength 120590

119884and ultimate strength of

material 120590max are 250MPa and 400MPa respectively Thecohesive elements along the crack surfaces are implementedas interface elements as shown in Figure 6

ISRNMechanical Engineering 5Ab

aqus

Adam

sAb

aqus

Build FE model of cracked elastic shaft- Define relative crack depth 119886119889- Define crack location- Define interface crack points- Define continuum and cohesive material

Transfer FE model to MBS

- Connect cracked elastic shaft to bearings and rigid disk- Vibration responses due to static (weight) and dynamic

(due to crack and inertia force) during rotation

Transfer output of MBS into FE- Based on the vibration responses the breathing crack

is investigated

loads

mechanism

Figure 5 Principle of the breathing crack simulation using inte-grated FE and MBS

Continuumelements

Cohesiveelements

Continuumelements

Figure 6 FE model of an elastic cracked shaft

32 Breathing Mechanism Simulation The breathing mech-anism generated by the rotating bending load used in theliterature has some limitations Due to the presence ofinertia forces the dynamic behaviour of rotating structuresis different from those of static structures Although in caseof weight dominance the amplitude of the vibration responsedue to inertia forces is smaller than the amplitude due toweight forces elastic forces and presence of a crack (if theshaft is assumed to be balanced) using the inertia force takeninto account will yield more accurate results The 3D FEcalculations allow the breathing mechanism to be predictedaccurately The breathing mechanism is strongly influencedby the weight of the shaft [29] In many publications on thebreathing crack assumptions on the breathing mechanismhave been made For transverse cracks in rotating shafts thevariation of the crack front line perpendicular to the crackfront has been extensively addressed by Darpe et al [12]Thisbreathing crack form is also used by Jun et al [30] and Sinouand Lees [31] An elliptical shape has been proposed by Shihand Chen [32] Based on FE model simulation and somereported experimental results the breathing crack shape ismodelled by a parabolic shape that opens and closes due tobending stresses (Liong and Proppe [26 33 34])

The idea is that the vibration responses in the centroid ofthe shaft obtained from MBS are exported into FE model inorder to analyse the breathing mechanism as schematicallyshown in Figure 7 The opening crack is simulated for onecycle of revolution of the cracked shaft specimen in steady-state condition The breathing mechanism is generated bythe bending due to external load (weight) by increasing theangle by steps of 12058712 rad The breathing (open and closedcrack areas are evaluated in each angular step) is observedby the nodal displacement and the stress distribution (tensileor compressive stress) around the crack The prediction ofthe breathing mechanism was performed by the followingsteps for each angle of rotation stress distribution due to thebending moment is recorded over the cross section Com-pressive (negative) stresses indicate closed region The crackopens where zero or very small positive numerical values ofstresses appear and the contact forces vanish Displacementsand stresses can be observed at the crack surfaces In order toavoid local deformations due to the application of loads thedeflection and stress distribution at each crack element areamust be evaluated as shown in Figure 8

4 Results and Discussion

41 BreathingMechanism Results Figure 9 displays the resultof breathing mechanism during half-revolution of a rotat-ing shaft (from closed to open crack) with relative crackdepth 119886119889 = 01 As can be seen the crack will start to appearand begin to open at 1205876 The crack opens more slowly atthe beginning but increases its opening at 1205873 At 51205876 it isalready completely open Similarly the crack closes again at71205876 and increases its closing at 31205872 The crack is alreadycompletely closed at 111205876 It is interesting to note that theangular positions where the crack opens again (from fullyclosed) are the same as where the crack begins to close

Results of the relative crack depth versus the shaft rotationangle during breathing are shown in Figure 10 Four functionsare used to approximate the relative crack depth namelylinear quadratic harmonic and power functions Table 1gives the approximations for a relative crack depth 119886119889 as afunction of shaft rotation angle 120579 It is shown that harmonicand quadratic functions have the minimal error and thenearest correlation with the simulation results In this sensethe relative crack depth during crack opening should beunderstood not as linear increasing but as harmonic functionor quadratic polynomial

42 Discussion Due to the presence of gravity the upperportion of the cracked rotor at the beginning of a rotationis under compression and the crack is closed As the rotorcontinues to rotate and the gravity direction is constant theupper part now comes in the lower tensile region causingthe crack to open Moreover the accuracy of the results ismodelled by a bilinear model as shown in Figure 11 Thecrack begins opening at shaft rotation angle 120579

0and at crack

front angle 120572 that defines the angle between two crack frontlines The shaft rotates with shaft rotation angle 120579 and thecrack front angle 120572 increases until maximum crack width119887max is reached The crack front angle 120572 and the crack width

6 ISRNMechanical Engineering

FLEX BODY 1MARKER 1FLEX BODY 1INT NODE

FLEX BODY 1INT NODE

FLEX BODY 1INT NODE 2000FLEX BODY 1MARKER 3

Gravity

119909

119909

119909

119910

119910

119910

119911

119911

119911

1

05

0

minus05

minus10 01 02 03 04 05 06 07 08 09 1

119905 (s)

times10minus5

0 01 02

0

minus05

minus1

minus15

minus2

times10minus5

119911 119910

10000

30000

Am

plitu

de119910

(m)

Am

plitu

de119909

(m)

119899 = 600 rpm (119891 = 10HZ) 119886119889 = 01

Figure 7 The transfer of the vibration responses fromMBS into FE

119887 are nonlinearly increased during rotation from 1205790to 120579max

(Figures 12 and 13) One can obtain the crack front angle 120572and the crack width 119887 as functions of the shaft rotation angle120579 respectively by

120572 =

0 120579 le 1205790

crack still closedminus0018120579

3

+ 00161205792

+035120579 + 24 1205790le 120579 le 120579max

crack opens0 120579 ge 120579max

crack opens completely

119887

119887max

=

0 120579 le 1205790

crack still closed0076120579

4

minus 0541205793

+ 111205792

minus023120579 minus 00056 1205790le 120579 le 120579max

crack opens0 120579 ge 120579max

crack opens completely(9)

For the relative crack depth 119886119889 = 01 the crack beginsto open at 120579

0= 1205876 and 120572 = 51205876 while the crack opens

completely at 120579max = 51205876 and 120572 = 120587 Therefore the crackfront angle and the crack width during crack opening are

120572 = minus00181205793

+ 00161205792

+ 035120579 + 24120587

6le 120579 le

5120587

6

119887

119887max= 0076120579

4

minus 0541205793

+ 111205792

minus 023120579 minus 00056

120587

6le 120579 le

5120587

6

(10)

The relative crack area (ie ratio of the partially opencrack area to the fully open crack area) with respect to theangle of rotation is used as shown in Figure 14 The resultsare similar when the crack closes The relative crack area ofsimulation results is in good agreement with the crack closurestraight linemodel [33] between rotation angles1205873 and 51205876andwith the bilinearmodel in the range from0 to120587 but thereare some relevant differences with respect to the crack closureperpendicular line used byDarpe et al [12]The areamomentof inertia about the rotation axis can be approximated Finallythe stiffness of the cracked shaft can be evaluated (stiffness isfound to vary linearly with the area moment of inertia) Thenormalised stiffness (ratio between stiffness of the crackedshaft and stiffness of the uncracked shaft) is presented asshown in Figure 15 It is shown that the normalized stiffnessbetween breathing mechanism results and the bilinear crackclosure model is very close which validates the accuracy ofbilinear model

The fracture mechanics analysis presupposes the exis-tence of an infinitely sharp crack leading to the singular cracktip fields However in real materials neither the sharpness ofthe crack nor the stress levels near the crack tip region can be

ISRNMechanical Engineering 7

Crack

Tensile stressor zero stress

Displacement+minus

Figure 8 Evaluation of displacements and stresses for state of crack

0∘ 30∘ 45∘

60∘ 75∘ 90∘

105∘ 120∘ 135∘

150∘ 165∘ 180∘

Figure 9 Breathing crack of the rotating shaft relative crack depth119886119889 = 01 at 600 rpm

infinite [27] As an alternative approach to this singularity-driven fracture approach the concept of CZM is proposedCZM has evolved as a preferred method to analyze fractureproblems not only because it avoids the singularity but alsobecause it can be easily implemented in a numerical methodof analysis as in FE method CZM presupposes the presenceof a fracture process zone where the energy is transferred

from external work in the crack regions In this study theexternal work flows as recoverable elastic strain energy andcohesive energy are examined the latter encompassing thework and other energy consuming mechanisms within thefracture process zone Process zone is defined as the regionwithin the separating surfaces where the surface tractionvalues are nonzeroThis also implies that processes occurringwithin the process zone are accounted for only throughthe traction-displacement relations The traction-separationrelations for crack interfaces are demonstrated such thatwith increasing interfacial separation the traction across theinterface reaches amaximum itmeans the elastic limit whereplastic dissipation in the surrounding (bounding) material isnot accounted for in the process zone

5 Conclusions

In this study a simulation process of FE and MBS isemployed An elastic cracked shaft is modelled by FE whichis transferred into MBS model The vibration responses inthe centroid of the shaft obtained from MBS have beentransferred again into an FE model in order to observe thebreathing mechanism and to study the stiffness losses due tobreathing at different rotating speeds

The results have shown that the breathing mechanismis influenced by the vibration due to inertia forces and byrelative crack depth It is shown that the relative crack depthduring crack opening should be understood not as linearincreasing but as harmonic function or quadratic polynomialIt can be noted that as long as the relative crack depth issmall the bilinear crack closure model may be used Themain difference with respect to the straight line crack closuremodel is that the opening crack in the simulation is notconstant at the beginningThe relative crack area of breathingmechanism results is in good agreement with the straight linecrack closuremodel between rotation angles1205873 and1205876 andwith the bilinear crack closuremodel in the range from0 to120587Furthermore the area moment of inertia about the rotationaxis can be approximated From this information the stiffnesslosses of the shaft can be evaluated It has been shown thatthe normalized stiffness between simulation and the bilinearcrack closure model is very close which validates the bilinearmodel

CZM are being increasingly used to simulate crackInstead of an infinitely sharp crack envisaged in fracturemechanics CZM presupposes the presence of a fracture pro-cess zone where the energy is transferred from external workin the crack regions The traction-separation relations forcrack interfaces are demonstrated such that with increasinginterfacial separation the traction across the interface reachesa maximum it means the elastic limit where plastic dissipa-tion in the surrounding (bounding)material is not accountedfor in the process zone

In light of the presented results and the conclusionsseveral subjects can be recommended for future research Animportant point on which the knowledge could be improvedis the prediction of crack propagation on cracked rotor andresidual life estimation The CZM can be easily implementedin FE model to analyze the dynamic behaviour of a cracked

8 ISRNMechanical Engineering

0 120587 2120587

119886119889

01

008

006

004

002

0

Linear model119891(119909) = 119886119909 + 119887

Confidence bounds 95

Goodness of fitSSE = 00005796R-square = 09516

120579 (rad)

119886 = 003887 (003226 004548)

119887 = 000131 (minus001043 001305)

(a)

0 120587 2120587

01

008

006

004

002

0

119886119889

Quadratic polynomial119891(119909) = 1198861199092 + 119887119909 + 119888

Confidence bounds 95

Goodness of fitSSE = 1141119890 minus 005R-square = 09990

120579 (rad)

119886 = minus001187 (minus001325 minus001050)119887 = 007617 (007174 008059)

119888 = minus001985 (minus002289 minus001683)

(b)

0 120587 2120587

01

008

006

004

002

0

119886119889

General model of sinus function

Confidence bounds 95

Goodness of fitSSE = 6158119890 minus 06R-square = 09995

120579 (rad)

119891(119909) = 119886 sin (119887119909 + 119888)

119886 = 009969 (009806 010130)119887 = 059340 (056990 061680)119888 = minus014540 (minus016410 minus012660)

(c)

0 120587 2120587

01

008

006

004

002

0

119886119889

General model of power119891(119909) = 119886119909119887 + 119888

Confidence bounds 95

Goodness of fitSSE = 9952119890 minus 05R-square = 09917

120579 (rad)

119886 = 012310 (004674 019950)119887 = 037300 (015200 059400)119888 = minus007730 (minus015120 minus000336)

(d)

Figure 10 Curve fitting of breathing mechanism results

Table 1 Curve fitting of the simulation results for 119886119889 = 01 120579 in [rad]

Model Mathematical model SSE R-squareLinear function 119886119889 = 0039120579 + 0001 0000 580 0951 6Quadratic polynomial 119886119889 = minus00121205792 + 0076120579minus 0020 0000 011 0999 0Sinus function 119886119889 = 0010sin (0593120579minus 0145) 0000 006 0999 5Power function 119886119889 = 01231205790373minus 0077 0000 099 0991 7

Crack closure line 1205721205720

119886

119887

119887max

Bilinear borderline

120575 gt 100120583m (crack opens)

10120583m le 120575 100120583m

120575 lt 10120583m

120575 = 0 (crack still closed)le

Figure 11 Bilinear crack closure model

ISRNMechanical Engineering 9

0 1205872 120587

120587

51205876

21205873

120572 = minus0084 lowast 1205792 + 051 lowast 120579 + 24

120572 = minus0018 lowast 1205793 + 0016 lowast 1205792 + 035 lowast 120579 + 24

SimulationQuadraticCubic

120579

120572

(rad)

(rad

)

Figure 12 Crack front angle 120572 during rotation of shaft

0 1205872 120587

1

09

08

07

06

05

04

03

02

01

0

119887119887 m

ax

SimulationCubic4th degree

= minus0069 lowast 1205793 + 02 lowast 1205792 + 036 lowast 120579 minus 0066

= 0076 lowast 1205794 minus 054lowast 1205793 + 11 lowast 1205792 minus 023 lowast 120579 minus

120579 (rad)

00056119887119887max

119887119887max

Figure 13 Normalized crack width 119887119887max during rotation of shaft

1

09

08

07

06

05

04

03

02

01

00 1205876 1205873 1205872 21205873 51205876 120587

Breathing mechanismBilinear crack closure model

Straight line crack closure [33]Perpendicular crack closure [12]

119860op

en119860

crac

k

120579 (rad)

Figure 14 Comparison of relative crack area between breathingmechanism results straight line perpendicular and bilinear crackclosure model for a shallow crack 119886119889 = 01

098

096

094

092

09

088

086

084

082

080 1205876 1205873 1205872 21205873 51205876 120587

Breathing mechanismBilinear crack closure model

Straight line crack closure [33]Perpendicular crack closure [12]

120578

120585

119870cr

ack119870

120579 (rad)

Figure 15 Comparison of normalized stiffness between breathingmechanism results straight line perpendicular and bilinear crackclosure model for a shallow crack 119886119889 = 01

shaft It is recommended to use CZM to study differenttypes of cracks such as longitudinal and slant cracks andalso multiple cracks Some other parameters such as internaldamping and thermal transients could be studied to obtainresults for their effect on the breathing mechanism Furtheranalysis on crack morphology is extremely important tounderstand the dynamic behaviour of cracked shaft thiswould include shallow and wide cracks

Acknowledgments

The authors acknowledge the support from Deutsche For-schungsgemeinschaft and Open Access Publishing Fund ofKarlsruhe Institute of Technology

References

[1] S K Georgantzinos and N K Anifantis ldquoAn insight into thebreathing mechanism of a crack in a rotating shaftrdquo Journal ofSound and Vibration vol 318 no 1-2 pp 279ndash295 2008

[2] J Wauer ldquoOn the dynamics of cracked rotors literature surveyrdquoApplied Mechanics Review vol 43 no 1 pp 13ndash17 1990

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

[4] G Sabnavis R G Kirk M Kasarda and D Quinn ldquoCrackedshaft detection and diagnostics a literature reviewrdquo Shock andVibration Digest vol 36 no 4 pp 287ndash296 2004

[5] 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

[6] N Bachschmid P Pennacchi and E Tanzi ldquoSome remarks onbreathing mechanism on non-linear effects and on slant andhelicoidal cracksrdquo Mechanical Systems and Signal Processingvol 22 no 4 pp 879ndash904 2008

[7] N Bachschmid P Pennachi and E Tanzi Cracked RotorsSpringer Berlin Germany 2011

10 ISRNMechanical Engineering

[8] S Andrieux and C Vare ldquoA 3D cracked beam model withunilateral contact Application to rotorsrdquo European Journal ofMechanics A vol 21 no 5 pp 793ndash810 2002

[9] C Vare and S Andrieux ldquoModeling of a cracked beam sectionunder bendingrdquo in Proceedings of the 18th International Confer-ence on Structural Mechanics in Reactor Technolohy (SMiRT 18)pp 281ndash290 Beijing China 2005

[10] S E Arem and HMaitournam ldquoA cracked beam finite elementfor rotating shaft dynamics and stability analysisrdquo Journal ofMechanics of Materials and Structures vol 3 no 5 pp 893ndash9102008

[11] A D Dimarogonas and S A Paipetis Analytical Methods inRotorDynamics Applied Science Publishers LondonUK 1983

[12] A K Darpe K Gupta and A Chawla ldquoCoupled bendinglongitudinal and torsional vibrations of a cracked rotorrdquo Journalof Sound and Vibration vol 269 no 1-2 pp 33ndash60 2004

[13] W M Ostachowicz and M Krawczuk ldquoCoupled torsional andbending vibrations of a rotor with an open crackrdquo Archive ofApplied Mechanics vol 62 no 3 pp 191ndash201 1992

[14] Z Kulesza and J T Sawicki ldquoRigid finite element model ofa cracked rotorrdquo Journal of Sound and Vibration vol 331 pp4145ndash4169 2012

[15] A S Bouboulas and N K Anifantis ldquoFinite element modelingof a vibrating beam with a breathing crack observations oncrack detectionrdquo Structural Health Monitoring vol 10 no 2 pp131ndash145 2011

[16] D S Dugdale ldquoYielding of steel sheets containing slitsrdquo Journalof the Mechanics and Physics of Solids vol 8 no 2 pp 100ndash1041960

[17] G I Barenblatt ldquoThemathematical theory of equilibrium crackin brittle fracturerdquoAdvances in AppliedMechanics vol 7 pp 55ndash129 1962

[18] T Siegmund andW Brocks ldquoTensile decohesion by local failurecriteriardquo Technische Mechanik vol 18 no 4 pp 261ndash270 1998

[19] T Siegmund and W Brocks ldquoA numerical study on thecorrelation between the work of separation and the dissipationrate in ductile fracturerdquo Engineering Fracture Mechanics vol 67no 2 pp 139ndash154 2000

[20] M Anvari I Scheider and CThaulow ldquoSimulation of dynamicductile crack growth using strain-rate and triaxiality-dependentcohesive elementsrdquo Engineering Fracture Mechanics vol 73 no15 pp 2210ndash2228 2006

[21] I Scheider ldquoDerivation of separation laws for cohesive modelsin the course of ductile fracturerdquo Engineering Fracture Mechan-ics vol 76 no 10 pp 1450ndash1459 2009

[22] A Banerjee and R Manivasagam ldquoTriaxiality dependent cohe-sive zone modelrdquo Engineering Fracture Mechanics vol 76 no12 pp 1761ndash1770 2009

[23] A D Dimarogonas and C A Papadopoulos ldquoVibration ofcracked shafts in bendingrdquo Journal of Sound and Vibration vol91 no 4 pp 583ndash593 1983

[24] A C Chasalevris andCA Papadopoulos ldquoA continuousmodelapproach for cross-coupled bending vibrations of a rotor-bearing system with a transverse breathing crackrdquo Mechanismand Machine Theory vol 44 no 6 pp 1176ndash1191 2009

[25] 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

[26] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel to the analysis of a rotor with a transverse crackrdquo Journalof Sound and Vibration vol 332 no 8 pp 2098ndash2110 2013

[27] C Shet andN Chandra ldquoAnalysis of energy balancewhen usingCohesive Zone Models to simulate fracture processesrdquo Journalof EngineeringMaterials andTechnology vol 124 no 4 pp 440ndash450 2002

[28] H Li and N Chandra ldquoAnalysis of crack growth and crack-tip plasticity in ductile materials using cohesive zone modelsrdquoInternational Journal of Plasticity vol 19 no 6 pp 849ndash8822003

[29] N Bachschmid P Pennacchi and E Tanzi ldquoOn the evolutionof vibrations in cracked rotorsrdquo in Proceedings of the 8thIFToMM International Conference on Rotor Dynamics pp 304ndash310 Seoul Korea 2010

[30] O S Jun H J Eun Y Y Earmme and C W Lee ldquoModellingand vibration analysis of a simple rotor with a breathing crackrdquoJournal of Sound andVibration vol 155 no 2 pp 273ndash290 1992

[31] J J Sinou and A W Lees ldquoThe influence of cracks in rotatingshaftsrdquo Journal of Sound and Vibration vol 285 no 4-5 pp1015ndash1037 2005

[32] Y S Shih and J J Chen ldquoAnalysis of fatigue crack growth on acracked shaftrdquo International Journal of Fatigue vol 19 no 6 pp477ndash485 1997

[33] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel for the investigation of the dynamic behavior of arotating shaft with a transverse crackrdquo in Proceedings of the 8thIFToMM International Conference on Rotor Dynamics pp 628ndash636 Seoul Korea September 2010

[34] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel to the analysis of a rotor with a transverse crackrdquo inProceedings of the 8th International Conference on StructuralDynamics pp 3434ndash3442 Leuven Belgium 2011

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DistributedSensor Networks

International Journal of

ISRNMechanical Engineering 5Ab

aqus

Adam

sAb

aqus

Build FE model of cracked elastic shaft- Define relative crack depth 119886119889- Define crack location- Define interface crack points- Define continuum and cohesive material

Transfer FE model to MBS

- Connect cracked elastic shaft to bearings and rigid disk- Vibration responses due to static (weight) and dynamic

(due to crack and inertia force) during rotation

Transfer output of MBS into FE- Based on the vibration responses the breathing crack

is investigated

loads

mechanism

Figure 5 Principle of the breathing crack simulation using inte-grated FE and MBS

Continuumelements

Cohesiveelements

Continuumelements

Figure 6 FE model of an elastic cracked shaft

32 Breathing Mechanism Simulation The breathing mech-anism generated by the rotating bending load used in theliterature has some limitations Due to the presence ofinertia forces the dynamic behaviour of rotating structuresis different from those of static structures Although in caseof weight dominance the amplitude of the vibration responsedue to inertia forces is smaller than the amplitude due toweight forces elastic forces and presence of a crack (if theshaft is assumed to be balanced) using the inertia force takeninto account will yield more accurate results The 3D FEcalculations allow the breathing mechanism to be predictedaccurately The breathing mechanism is strongly influencedby the weight of the shaft [29] In many publications on thebreathing crack assumptions on the breathing mechanismhave been made For transverse cracks in rotating shafts thevariation of the crack front line perpendicular to the crackfront has been extensively addressed by Darpe et al [12]Thisbreathing crack form is also used by Jun et al [30] and Sinouand Lees [31] An elliptical shape has been proposed by Shihand Chen [32] Based on FE model simulation and somereported experimental results the breathing crack shape ismodelled by a parabolic shape that opens and closes due tobending stresses (Liong and Proppe [26 33 34])

The idea is that the vibration responses in the centroid ofthe shaft obtained from MBS are exported into FE model inorder to analyse the breathing mechanism as schematicallyshown in Figure 7 The opening crack is simulated for onecycle of revolution of the cracked shaft specimen in steady-state condition The breathing mechanism is generated bythe bending due to external load (weight) by increasing theangle by steps of 12058712 rad The breathing (open and closedcrack areas are evaluated in each angular step) is observedby the nodal displacement and the stress distribution (tensileor compressive stress) around the crack The prediction ofthe breathing mechanism was performed by the followingsteps for each angle of rotation stress distribution due to thebending moment is recorded over the cross section Com-pressive (negative) stresses indicate closed region The crackopens where zero or very small positive numerical values ofstresses appear and the contact forces vanish Displacementsand stresses can be observed at the crack surfaces In order toavoid local deformations due to the application of loads thedeflection and stress distribution at each crack element areamust be evaluated as shown in Figure 8

4 Results and Discussion

41 BreathingMechanism Results Figure 9 displays the resultof breathing mechanism during half-revolution of a rotat-ing shaft (from closed to open crack) with relative crackdepth 119886119889 = 01 As can be seen the crack will start to appearand begin to open at 1205876 The crack opens more slowly atthe beginning but increases its opening at 1205873 At 51205876 it isalready completely open Similarly the crack closes again at71205876 and increases its closing at 31205872 The crack is alreadycompletely closed at 111205876 It is interesting to note that theangular positions where the crack opens again (from fullyclosed) are the same as where the crack begins to close

Results of the relative crack depth versus the shaft rotationangle during breathing are shown in Figure 10 Four functionsare used to approximate the relative crack depth namelylinear quadratic harmonic and power functions Table 1gives the approximations for a relative crack depth 119886119889 as afunction of shaft rotation angle 120579 It is shown that harmonicand quadratic functions have the minimal error and thenearest correlation with the simulation results In this sensethe relative crack depth during crack opening should beunderstood not as linear increasing but as harmonic functionor quadratic polynomial

42 Discussion Due to the presence of gravity the upperportion of the cracked rotor at the beginning of a rotationis under compression and the crack is closed As the rotorcontinues to rotate and the gravity direction is constant theupper part now comes in the lower tensile region causingthe crack to open Moreover the accuracy of the results ismodelled by a bilinear model as shown in Figure 11 Thecrack begins opening at shaft rotation angle 120579

0and at crack

front angle 120572 that defines the angle between two crack frontlines The shaft rotates with shaft rotation angle 120579 and thecrack front angle 120572 increases until maximum crack width119887max is reached The crack front angle 120572 and the crack width

6 ISRNMechanical Engineering

FLEX BODY 1MARKER 1FLEX BODY 1INT NODE

FLEX BODY 1INT NODE

FLEX BODY 1INT NODE 2000FLEX BODY 1MARKER 3

Gravity

119909

119909

119909

119910

119910

119910

119911

119911

119911

1

05

0

minus05

minus10 01 02 03 04 05 06 07 08 09 1

119905 (s)

times10minus5

0 01 02

0

minus05

minus1

minus15

minus2

times10minus5

119911 119910

10000

30000

Am

plitu

de119910

(m)

Am

plitu

de119909

(m)

119899 = 600 rpm (119891 = 10HZ) 119886119889 = 01

Figure 7 The transfer of the vibration responses fromMBS into FE

119887 are nonlinearly increased during rotation from 1205790to 120579max

(Figures 12 and 13) One can obtain the crack front angle 120572and the crack width 119887 as functions of the shaft rotation angle120579 respectively by

120572 =

0 120579 le 1205790

crack still closedminus0018120579

3

+ 00161205792

+035120579 + 24 1205790le 120579 le 120579max

crack opens0 120579 ge 120579max

crack opens completely

119887

119887max

=

0 120579 le 1205790

crack still closed0076120579

4

minus 0541205793

+ 111205792

minus023120579 minus 00056 1205790le 120579 le 120579max

crack opens0 120579 ge 120579max

crack opens completely(9)

For the relative crack depth 119886119889 = 01 the crack beginsto open at 120579

0= 1205876 and 120572 = 51205876 while the crack opens

completely at 120579max = 51205876 and 120572 = 120587 Therefore the crackfront angle and the crack width during crack opening are

120572 = minus00181205793

+ 00161205792

+ 035120579 + 24120587

6le 120579 le

5120587

6

119887

119887max= 0076120579

4

minus 0541205793

+ 111205792

minus 023120579 minus 00056

120587

6le 120579 le

5120587

6

(10)

The relative crack area (ie ratio of the partially opencrack area to the fully open crack area) with respect to theangle of rotation is used as shown in Figure 14 The resultsare similar when the crack closes The relative crack area ofsimulation results is in good agreement with the crack closurestraight linemodel [33] between rotation angles1205873 and 51205876andwith the bilinearmodel in the range from0 to120587 but thereare some relevant differences with respect to the crack closureperpendicular line used byDarpe et al [12]The areamomentof inertia about the rotation axis can be approximated Finallythe stiffness of the cracked shaft can be evaluated (stiffness isfound to vary linearly with the area moment of inertia) Thenormalised stiffness (ratio between stiffness of the crackedshaft and stiffness of the uncracked shaft) is presented asshown in Figure 15 It is shown that the normalized stiffnessbetween breathing mechanism results and the bilinear crackclosure model is very close which validates the accuracy ofbilinear model

The fracture mechanics analysis presupposes the exis-tence of an infinitely sharp crack leading to the singular cracktip fields However in real materials neither the sharpness ofthe crack nor the stress levels near the crack tip region can be

ISRNMechanical Engineering 7

Crack

Tensile stressor zero stress

Displacement+minus

Figure 8 Evaluation of displacements and stresses for state of crack

0∘ 30∘ 45∘

60∘ 75∘ 90∘

105∘ 120∘ 135∘

150∘ 165∘ 180∘

Figure 9 Breathing crack of the rotating shaft relative crack depth119886119889 = 01 at 600 rpm

infinite [27] As an alternative approach to this singularity-driven fracture approach the concept of CZM is proposedCZM has evolved as a preferred method to analyze fractureproblems not only because it avoids the singularity but alsobecause it can be easily implemented in a numerical methodof analysis as in FE method CZM presupposes the presenceof a fracture process zone where the energy is transferred

from external work in the crack regions In this study theexternal work flows as recoverable elastic strain energy andcohesive energy are examined the latter encompassing thework and other energy consuming mechanisms within thefracture process zone Process zone is defined as the regionwithin the separating surfaces where the surface tractionvalues are nonzeroThis also implies that processes occurringwithin the process zone are accounted for only throughthe traction-displacement relations The traction-separationrelations for crack interfaces are demonstrated such thatwith increasing interfacial separation the traction across theinterface reaches amaximum itmeans the elastic limit whereplastic dissipation in the surrounding (bounding) material isnot accounted for in the process zone

5 Conclusions

In this study a simulation process of FE and MBS isemployed An elastic cracked shaft is modelled by FE whichis transferred into MBS model The vibration responses inthe centroid of the shaft obtained from MBS have beentransferred again into an FE model in order to observe thebreathing mechanism and to study the stiffness losses due tobreathing at different rotating speeds

The results have shown that the breathing mechanismis influenced by the vibration due to inertia forces and byrelative crack depth It is shown that the relative crack depthduring crack opening should be understood not as linearincreasing but as harmonic function or quadratic polynomialIt can be noted that as long as the relative crack depth issmall the bilinear crack closure model may be used Themain difference with respect to the straight line crack closuremodel is that the opening crack in the simulation is notconstant at the beginningThe relative crack area of breathingmechanism results is in good agreement with the straight linecrack closuremodel between rotation angles1205873 and1205876 andwith the bilinear crack closuremodel in the range from0 to120587Furthermore the area moment of inertia about the rotationaxis can be approximated From this information the stiffnesslosses of the shaft can be evaluated It has been shown thatthe normalized stiffness between simulation and the bilinearcrack closure model is very close which validates the bilinearmodel

CZM are being increasingly used to simulate crackInstead of an infinitely sharp crack envisaged in fracturemechanics CZM presupposes the presence of a fracture pro-cess zone where the energy is transferred from external workin the crack regions The traction-separation relations forcrack interfaces are demonstrated such that with increasinginterfacial separation the traction across the interface reachesa maximum it means the elastic limit where plastic dissipa-tion in the surrounding (bounding)material is not accountedfor in the process zone

In light of the presented results and the conclusionsseveral subjects can be recommended for future research Animportant point on which the knowledge could be improvedis the prediction of crack propagation on cracked rotor andresidual life estimation The CZM can be easily implementedin FE model to analyze the dynamic behaviour of a cracked

8 ISRNMechanical Engineering

0 120587 2120587

119886119889

01

008

006

004

002

0

Linear model119891(119909) = 119886119909 + 119887

Confidence bounds 95

Goodness of fitSSE = 00005796R-square = 09516

120579 (rad)

119886 = 003887 (003226 004548)

119887 = 000131 (minus001043 001305)

(a)

0 120587 2120587

01

008

006

004

002

0

119886119889

Quadratic polynomial119891(119909) = 1198861199092 + 119887119909 + 119888

Confidence bounds 95

Goodness of fitSSE = 1141119890 minus 005R-square = 09990

120579 (rad)

119886 = minus001187 (minus001325 minus001050)119887 = 007617 (007174 008059)

119888 = minus001985 (minus002289 minus001683)

(b)

0 120587 2120587

01

008

006

004

002

0

119886119889

General model of sinus function

Confidence bounds 95

Goodness of fitSSE = 6158119890 minus 06R-square = 09995

120579 (rad)

119891(119909) = 119886 sin (119887119909 + 119888)

119886 = 009969 (009806 010130)119887 = 059340 (056990 061680)119888 = minus014540 (minus016410 minus012660)

(c)

0 120587 2120587

01

008

006

004

002

0

119886119889

General model of power119891(119909) = 119886119909119887 + 119888

Confidence bounds 95

Goodness of fitSSE = 9952119890 minus 05R-square = 09917

120579 (rad)

119886 = 012310 (004674 019950)119887 = 037300 (015200 059400)119888 = minus007730 (minus015120 minus000336)

(d)

Figure 10 Curve fitting of breathing mechanism results

Table 1 Curve fitting of the simulation results for 119886119889 = 01 120579 in [rad]

Model Mathematical model SSE R-squareLinear function 119886119889 = 0039120579 + 0001 0000 580 0951 6Quadratic polynomial 119886119889 = minus00121205792 + 0076120579minus 0020 0000 011 0999 0Sinus function 119886119889 = 0010sin (0593120579minus 0145) 0000 006 0999 5Power function 119886119889 = 01231205790373minus 0077 0000 099 0991 7

Crack closure line 1205721205720

119886

119887

119887max

Bilinear borderline

120575 gt 100120583m (crack opens)

10120583m le 120575 100120583m

120575 lt 10120583m

120575 = 0 (crack still closed)le

Figure 11 Bilinear crack closure model

ISRNMechanical Engineering 9

0 1205872 120587

120587

51205876

21205873

120572 = minus0084 lowast 1205792 + 051 lowast 120579 + 24

120572 = minus0018 lowast 1205793 + 0016 lowast 1205792 + 035 lowast 120579 + 24

SimulationQuadraticCubic

120579

120572

(rad)

(rad

)

Figure 12 Crack front angle 120572 during rotation of shaft

0 1205872 120587

1

09

08

07

06

05

04

03

02

01

0

119887119887 m

ax

SimulationCubic4th degree

= minus0069 lowast 1205793 + 02 lowast 1205792 + 036 lowast 120579 minus 0066

= 0076 lowast 1205794 minus 054lowast 1205793 + 11 lowast 1205792 minus 023 lowast 120579 minus

120579 (rad)

00056119887119887max

119887119887max

Figure 13 Normalized crack width 119887119887max during rotation of shaft

1

09

08

07

06

05

04

03

02

01

00 1205876 1205873 1205872 21205873 51205876 120587

Breathing mechanismBilinear crack closure model

Straight line crack closure [33]Perpendicular crack closure [12]

119860op

en119860

crac

k

120579 (rad)

Figure 14 Comparison of relative crack area between breathingmechanism results straight line perpendicular and bilinear crackclosure model for a shallow crack 119886119889 = 01

098

096

094

092

09

088

086

084

082

080 1205876 1205873 1205872 21205873 51205876 120587

Breathing mechanismBilinear crack closure model

Straight line crack closure [33]Perpendicular crack closure [12]

120578

120585

119870cr

ack119870

120579 (rad)

Figure 15 Comparison of normalized stiffness between breathingmechanism results straight line perpendicular and bilinear crackclosure model for a shallow crack 119886119889 = 01

shaft It is recommended to use CZM to study differenttypes of cracks such as longitudinal and slant cracks andalso multiple cracks Some other parameters such as internaldamping and thermal transients could be studied to obtainresults for their effect on the breathing mechanism Furtheranalysis on crack morphology is extremely important tounderstand the dynamic behaviour of cracked shaft thiswould include shallow and wide cracks

Acknowledgments

The authors acknowledge the support from Deutsche For-schungsgemeinschaft and Open Access Publishing Fund ofKarlsruhe Institute of Technology

References

[1] S K Georgantzinos and N K Anifantis ldquoAn insight into thebreathing mechanism of a crack in a rotating shaftrdquo Journal ofSound and Vibration vol 318 no 1-2 pp 279ndash295 2008

[2] J Wauer ldquoOn the dynamics of cracked rotors literature surveyrdquoApplied Mechanics Review vol 43 no 1 pp 13ndash17 1990

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

[4] G Sabnavis R G Kirk M Kasarda and D Quinn ldquoCrackedshaft detection and diagnostics a literature reviewrdquo Shock andVibration Digest vol 36 no 4 pp 287ndash296 2004

[5] 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

[6] N Bachschmid P Pennacchi and E Tanzi ldquoSome remarks onbreathing mechanism on non-linear effects and on slant andhelicoidal cracksrdquo Mechanical Systems and Signal Processingvol 22 no 4 pp 879ndash904 2008

[7] N Bachschmid P Pennachi and E Tanzi Cracked RotorsSpringer Berlin Germany 2011

10 ISRNMechanical Engineering

[8] S Andrieux and C Vare ldquoA 3D cracked beam model withunilateral contact Application to rotorsrdquo European Journal ofMechanics A vol 21 no 5 pp 793ndash810 2002

[9] C Vare and S Andrieux ldquoModeling of a cracked beam sectionunder bendingrdquo in Proceedings of the 18th International Confer-ence on Structural Mechanics in Reactor Technolohy (SMiRT 18)pp 281ndash290 Beijing China 2005

[10] S E Arem and HMaitournam ldquoA cracked beam finite elementfor rotating shaft dynamics and stability analysisrdquo Journal ofMechanics of Materials and Structures vol 3 no 5 pp 893ndash9102008

[11] A D Dimarogonas and S A Paipetis Analytical Methods inRotorDynamics Applied Science Publishers LondonUK 1983

[12] A K Darpe K Gupta and A Chawla ldquoCoupled bendinglongitudinal and torsional vibrations of a cracked rotorrdquo Journalof Sound and Vibration vol 269 no 1-2 pp 33ndash60 2004

[13] W M Ostachowicz and M Krawczuk ldquoCoupled torsional andbending vibrations of a rotor with an open crackrdquo Archive ofApplied Mechanics vol 62 no 3 pp 191ndash201 1992

[14] Z Kulesza and J T Sawicki ldquoRigid finite element model ofa cracked rotorrdquo Journal of Sound and Vibration vol 331 pp4145ndash4169 2012

[15] A S Bouboulas and N K Anifantis ldquoFinite element modelingof a vibrating beam with a breathing crack observations oncrack detectionrdquo Structural Health Monitoring vol 10 no 2 pp131ndash145 2011

[16] D S Dugdale ldquoYielding of steel sheets containing slitsrdquo Journalof the Mechanics and Physics of Solids vol 8 no 2 pp 100ndash1041960

[17] G I Barenblatt ldquoThemathematical theory of equilibrium crackin brittle fracturerdquoAdvances in AppliedMechanics vol 7 pp 55ndash129 1962

[18] T Siegmund andW Brocks ldquoTensile decohesion by local failurecriteriardquo Technische Mechanik vol 18 no 4 pp 261ndash270 1998

[19] T Siegmund and W Brocks ldquoA numerical study on thecorrelation between the work of separation and the dissipationrate in ductile fracturerdquo Engineering Fracture Mechanics vol 67no 2 pp 139ndash154 2000

[20] M Anvari I Scheider and CThaulow ldquoSimulation of dynamicductile crack growth using strain-rate and triaxiality-dependentcohesive elementsrdquo Engineering Fracture Mechanics vol 73 no15 pp 2210ndash2228 2006

[21] I Scheider ldquoDerivation of separation laws for cohesive modelsin the course of ductile fracturerdquo Engineering Fracture Mechan-ics vol 76 no 10 pp 1450ndash1459 2009

[22] A Banerjee and R Manivasagam ldquoTriaxiality dependent cohe-sive zone modelrdquo Engineering Fracture Mechanics vol 76 no12 pp 1761ndash1770 2009

[23] A D Dimarogonas and C A Papadopoulos ldquoVibration ofcracked shafts in bendingrdquo Journal of Sound and Vibration vol91 no 4 pp 583ndash593 1983

[24] A C Chasalevris andCA Papadopoulos ldquoA continuousmodelapproach for cross-coupled bending vibrations of a rotor-bearing system with a transverse breathing crackrdquo Mechanismand Machine Theory vol 44 no 6 pp 1176ndash1191 2009

[25] 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

[26] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel to the analysis of a rotor with a transverse crackrdquo Journalof Sound and Vibration vol 332 no 8 pp 2098ndash2110 2013

[27] C Shet andN Chandra ldquoAnalysis of energy balancewhen usingCohesive Zone Models to simulate fracture processesrdquo Journalof EngineeringMaterials andTechnology vol 124 no 4 pp 440ndash450 2002

[28] H Li and N Chandra ldquoAnalysis of crack growth and crack-tip plasticity in ductile materials using cohesive zone modelsrdquoInternational Journal of Plasticity vol 19 no 6 pp 849ndash8822003

[29] N Bachschmid P Pennacchi and E Tanzi ldquoOn the evolutionof vibrations in cracked rotorsrdquo in Proceedings of the 8thIFToMM International Conference on Rotor Dynamics pp 304ndash310 Seoul Korea 2010

[30] O S Jun H J Eun Y Y Earmme and C W Lee ldquoModellingand vibration analysis of a simple rotor with a breathing crackrdquoJournal of Sound andVibration vol 155 no 2 pp 273ndash290 1992

[31] J J Sinou and A W Lees ldquoThe influence of cracks in rotatingshaftsrdquo Journal of Sound and Vibration vol 285 no 4-5 pp1015ndash1037 2005

[32] Y S Shih and J J Chen ldquoAnalysis of fatigue crack growth on acracked shaftrdquo International Journal of Fatigue vol 19 no 6 pp477ndash485 1997

[33] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel for the investigation of the dynamic behavior of arotating shaft with a transverse crackrdquo in Proceedings of the 8thIFToMM International Conference on Rotor Dynamics pp 628ndash636 Seoul Korea September 2010

[34] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel to the analysis of a rotor with a transverse crackrdquo inProceedings of the 8th International Conference on StructuralDynamics pp 3434ndash3442 Leuven Belgium 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 ISRNMechanical Engineering

FLEX BODY 1MARKER 1FLEX BODY 1INT NODE

FLEX BODY 1INT NODE

FLEX BODY 1INT NODE 2000FLEX BODY 1MARKER 3

Gravity

119909

119909

119909

119910

119910

119910

119911

119911

119911

1

05

0

minus05

minus10 01 02 03 04 05 06 07 08 09 1

119905 (s)

times10minus5

0 01 02

0

minus05

minus1

minus15

minus2

times10minus5

119911 119910

10000

30000

Am

plitu

de119910

(m)

Am

plitu

de119909

(m)

119899 = 600 rpm (119891 = 10HZ) 119886119889 = 01

Figure 7 The transfer of the vibration responses fromMBS into FE

119887 are nonlinearly increased during rotation from 1205790to 120579max

(Figures 12 and 13) One can obtain the crack front angle 120572and the crack width 119887 as functions of the shaft rotation angle120579 respectively by

120572 =

0 120579 le 1205790

crack still closedminus0018120579

3

+ 00161205792

+035120579 + 24 1205790le 120579 le 120579max

crack opens0 120579 ge 120579max

crack opens completely

119887

119887max

=

0 120579 le 1205790

crack still closed0076120579

4

minus 0541205793

+ 111205792

minus023120579 minus 00056 1205790le 120579 le 120579max

crack opens0 120579 ge 120579max

crack opens completely(9)

For the relative crack depth 119886119889 = 01 the crack beginsto open at 120579

0= 1205876 and 120572 = 51205876 while the crack opens

completely at 120579max = 51205876 and 120572 = 120587 Therefore the crackfront angle and the crack width during crack opening are

120572 = minus00181205793

+ 00161205792

+ 035120579 + 24120587

6le 120579 le

5120587

6

119887

119887max= 0076120579

4

minus 0541205793

+ 111205792

minus 023120579 minus 00056

120587

6le 120579 le

5120587

6

(10)

The relative crack area (ie ratio of the partially opencrack area to the fully open crack area) with respect to theangle of rotation is used as shown in Figure 14 The resultsare similar when the crack closes The relative crack area ofsimulation results is in good agreement with the crack closurestraight linemodel [33] between rotation angles1205873 and 51205876andwith the bilinearmodel in the range from0 to120587 but thereare some relevant differences with respect to the crack closureperpendicular line used byDarpe et al [12]The areamomentof inertia about the rotation axis can be approximated Finallythe stiffness of the cracked shaft can be evaluated (stiffness isfound to vary linearly with the area moment of inertia) Thenormalised stiffness (ratio between stiffness of the crackedshaft and stiffness of the uncracked shaft) is presented asshown in Figure 15 It is shown that the normalized stiffnessbetween breathing mechanism results and the bilinear crackclosure model is very close which validates the accuracy ofbilinear model

The fracture mechanics analysis presupposes the exis-tence of an infinitely sharp crack leading to the singular cracktip fields However in real materials neither the sharpness ofthe crack nor the stress levels near the crack tip region can be

ISRNMechanical Engineering 7

Crack

Tensile stressor zero stress

Displacement+minus

Figure 8 Evaluation of displacements and stresses for state of crack

0∘ 30∘ 45∘

60∘ 75∘ 90∘

105∘ 120∘ 135∘

150∘ 165∘ 180∘

Figure 9 Breathing crack of the rotating shaft relative crack depth119886119889 = 01 at 600 rpm

infinite [27] As an alternative approach to this singularity-driven fracture approach the concept of CZM is proposedCZM has evolved as a preferred method to analyze fractureproblems not only because it avoids the singularity but alsobecause it can be easily implemented in a numerical methodof analysis as in FE method CZM presupposes the presenceof a fracture process zone where the energy is transferred

from external work in the crack regions In this study theexternal work flows as recoverable elastic strain energy andcohesive energy are examined the latter encompassing thework and other energy consuming mechanisms within thefracture process zone Process zone is defined as the regionwithin the separating surfaces where the surface tractionvalues are nonzeroThis also implies that processes occurringwithin the process zone are accounted for only throughthe traction-displacement relations The traction-separationrelations for crack interfaces are demonstrated such thatwith increasing interfacial separation the traction across theinterface reaches amaximum itmeans the elastic limit whereplastic dissipation in the surrounding (bounding) material isnot accounted for in the process zone

5 Conclusions

In this study a simulation process of FE and MBS isemployed An elastic cracked shaft is modelled by FE whichis transferred into MBS model The vibration responses inthe centroid of the shaft obtained from MBS have beentransferred again into an FE model in order to observe thebreathing mechanism and to study the stiffness losses due tobreathing at different rotating speeds

The results have shown that the breathing mechanismis influenced by the vibration due to inertia forces and byrelative crack depth It is shown that the relative crack depthduring crack opening should be understood not as linearincreasing but as harmonic function or quadratic polynomialIt can be noted that as long as the relative crack depth issmall the bilinear crack closure model may be used Themain difference with respect to the straight line crack closuremodel is that the opening crack in the simulation is notconstant at the beginningThe relative crack area of breathingmechanism results is in good agreement with the straight linecrack closuremodel between rotation angles1205873 and1205876 andwith the bilinear crack closuremodel in the range from0 to120587Furthermore the area moment of inertia about the rotationaxis can be approximated From this information the stiffnesslosses of the shaft can be evaluated It has been shown thatthe normalized stiffness between simulation and the bilinearcrack closure model is very close which validates the bilinearmodel

CZM are being increasingly used to simulate crackInstead of an infinitely sharp crack envisaged in fracturemechanics CZM presupposes the presence of a fracture pro-cess zone where the energy is transferred from external workin the crack regions The traction-separation relations forcrack interfaces are demonstrated such that with increasinginterfacial separation the traction across the interface reachesa maximum it means the elastic limit where plastic dissipa-tion in the surrounding (bounding)material is not accountedfor in the process zone

In light of the presented results and the conclusionsseveral subjects can be recommended for future research Animportant point on which the knowledge could be improvedis the prediction of crack propagation on cracked rotor andresidual life estimation The CZM can be easily implementedin FE model to analyze the dynamic behaviour of a cracked

8 ISRNMechanical Engineering

0 120587 2120587

119886119889

01

008

006

004

002

0

Linear model119891(119909) = 119886119909 + 119887

Confidence bounds 95

Goodness of fitSSE = 00005796R-square = 09516

120579 (rad)

119886 = 003887 (003226 004548)

119887 = 000131 (minus001043 001305)

(a)

0 120587 2120587

01

008

006

004

002

0

119886119889

Quadratic polynomial119891(119909) = 1198861199092 + 119887119909 + 119888

Confidence bounds 95

Goodness of fitSSE = 1141119890 minus 005R-square = 09990

120579 (rad)

119886 = minus001187 (minus001325 minus001050)119887 = 007617 (007174 008059)

119888 = minus001985 (minus002289 minus001683)

(b)

0 120587 2120587

01

008

006

004

002

0

119886119889

General model of sinus function

Confidence bounds 95

Goodness of fitSSE = 6158119890 minus 06R-square = 09995

120579 (rad)

119891(119909) = 119886 sin (119887119909 + 119888)

119886 = 009969 (009806 010130)119887 = 059340 (056990 061680)119888 = minus014540 (minus016410 minus012660)

(c)

0 120587 2120587

01

008

006

004

002

0

119886119889

General model of power119891(119909) = 119886119909119887 + 119888

Confidence bounds 95

Goodness of fitSSE = 9952119890 minus 05R-square = 09917

120579 (rad)

119886 = 012310 (004674 019950)119887 = 037300 (015200 059400)119888 = minus007730 (minus015120 minus000336)

(d)

Figure 10 Curve fitting of breathing mechanism results

Table 1 Curve fitting of the simulation results for 119886119889 = 01 120579 in [rad]

Model Mathematical model SSE R-squareLinear function 119886119889 = 0039120579 + 0001 0000 580 0951 6Quadratic polynomial 119886119889 = minus00121205792 + 0076120579minus 0020 0000 011 0999 0Sinus function 119886119889 = 0010sin (0593120579minus 0145) 0000 006 0999 5Power function 119886119889 = 01231205790373minus 0077 0000 099 0991 7

Crack closure line 1205721205720

119886

119887

119887max

Bilinear borderline

120575 gt 100120583m (crack opens)

10120583m le 120575 100120583m

120575 lt 10120583m

120575 = 0 (crack still closed)le

Figure 11 Bilinear crack closure model

ISRNMechanical Engineering 9

0 1205872 120587

120587

51205876

21205873

120572 = minus0084 lowast 1205792 + 051 lowast 120579 + 24

120572 = minus0018 lowast 1205793 + 0016 lowast 1205792 + 035 lowast 120579 + 24

SimulationQuadraticCubic

120579

120572

(rad)

(rad

)

Figure 12 Crack front angle 120572 during rotation of shaft

0 1205872 120587

1

09

08

07

06

05

04

03

02

01

0

119887119887 m

ax

SimulationCubic4th degree

= minus0069 lowast 1205793 + 02 lowast 1205792 + 036 lowast 120579 minus 0066

= 0076 lowast 1205794 minus 054lowast 1205793 + 11 lowast 1205792 minus 023 lowast 120579 minus

120579 (rad)

00056119887119887max

119887119887max

Figure 13 Normalized crack width 119887119887max during rotation of shaft

1

09

08

07

06

05

04

03

02

01

00 1205876 1205873 1205872 21205873 51205876 120587

Breathing mechanismBilinear crack closure model

Straight line crack closure [33]Perpendicular crack closure [12]

119860op

en119860

crac

k

120579 (rad)

Figure 14 Comparison of relative crack area between breathingmechanism results straight line perpendicular and bilinear crackclosure model for a shallow crack 119886119889 = 01

098

096

094

092

09

088

086

084

082

080 1205876 1205873 1205872 21205873 51205876 120587

Breathing mechanismBilinear crack closure model

Straight line crack closure [33]Perpendicular crack closure [12]

120578

120585

119870cr

ack119870

120579 (rad)

Figure 15 Comparison of normalized stiffness between breathingmechanism results straight line perpendicular and bilinear crackclosure model for a shallow crack 119886119889 = 01

shaft It is recommended to use CZM to study differenttypes of cracks such as longitudinal and slant cracks andalso multiple cracks Some other parameters such as internaldamping and thermal transients could be studied to obtainresults for their effect on the breathing mechanism Furtheranalysis on crack morphology is extremely important tounderstand the dynamic behaviour of cracked shaft thiswould include shallow and wide cracks

Acknowledgments

The authors acknowledge the support from Deutsche For-schungsgemeinschaft and Open Access Publishing Fund ofKarlsruhe Institute of Technology

References

[1] S K Georgantzinos and N K Anifantis ldquoAn insight into thebreathing mechanism of a crack in a rotating shaftrdquo Journal ofSound and Vibration vol 318 no 1-2 pp 279ndash295 2008

[2] J Wauer ldquoOn the dynamics of cracked rotors literature surveyrdquoApplied Mechanics Review vol 43 no 1 pp 13ndash17 1990

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

[4] G Sabnavis R G Kirk M Kasarda and D Quinn ldquoCrackedshaft detection and diagnostics a literature reviewrdquo Shock andVibration Digest vol 36 no 4 pp 287ndash296 2004

[5] 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

[6] N Bachschmid P Pennacchi and E Tanzi ldquoSome remarks onbreathing mechanism on non-linear effects and on slant andhelicoidal cracksrdquo Mechanical Systems and Signal Processingvol 22 no 4 pp 879ndash904 2008

[7] N Bachschmid P Pennachi and E Tanzi Cracked RotorsSpringer Berlin Germany 2011

10 ISRNMechanical Engineering

[8] S Andrieux and C Vare ldquoA 3D cracked beam model withunilateral contact Application to rotorsrdquo European Journal ofMechanics A vol 21 no 5 pp 793ndash810 2002

[9] C Vare and S Andrieux ldquoModeling of a cracked beam sectionunder bendingrdquo in Proceedings of the 18th International Confer-ence on Structural Mechanics in Reactor Technolohy (SMiRT 18)pp 281ndash290 Beijing China 2005

[10] S E Arem and HMaitournam ldquoA cracked beam finite elementfor rotating shaft dynamics and stability analysisrdquo Journal ofMechanics of Materials and Structures vol 3 no 5 pp 893ndash9102008

[11] A D Dimarogonas and S A Paipetis Analytical Methods inRotorDynamics Applied Science Publishers LondonUK 1983

[12] A K Darpe K Gupta and A Chawla ldquoCoupled bendinglongitudinal and torsional vibrations of a cracked rotorrdquo Journalof Sound and Vibration vol 269 no 1-2 pp 33ndash60 2004

[13] W M Ostachowicz and M Krawczuk ldquoCoupled torsional andbending vibrations of a rotor with an open crackrdquo Archive ofApplied Mechanics vol 62 no 3 pp 191ndash201 1992

[14] Z Kulesza and J T Sawicki ldquoRigid finite element model ofa cracked rotorrdquo Journal of Sound and Vibration vol 331 pp4145ndash4169 2012

[15] A S Bouboulas and N K Anifantis ldquoFinite element modelingof a vibrating beam with a breathing crack observations oncrack detectionrdquo Structural Health Monitoring vol 10 no 2 pp131ndash145 2011

[16] D S Dugdale ldquoYielding of steel sheets containing slitsrdquo Journalof the Mechanics and Physics of Solids vol 8 no 2 pp 100ndash1041960

[17] G I Barenblatt ldquoThemathematical theory of equilibrium crackin brittle fracturerdquoAdvances in AppliedMechanics vol 7 pp 55ndash129 1962

[18] T Siegmund andW Brocks ldquoTensile decohesion by local failurecriteriardquo Technische Mechanik vol 18 no 4 pp 261ndash270 1998

[19] T Siegmund and W Brocks ldquoA numerical study on thecorrelation between the work of separation and the dissipationrate in ductile fracturerdquo Engineering Fracture Mechanics vol 67no 2 pp 139ndash154 2000

[20] M Anvari I Scheider and CThaulow ldquoSimulation of dynamicductile crack growth using strain-rate and triaxiality-dependentcohesive elementsrdquo Engineering Fracture Mechanics vol 73 no15 pp 2210ndash2228 2006

[21] I Scheider ldquoDerivation of separation laws for cohesive modelsin the course of ductile fracturerdquo Engineering Fracture Mechan-ics vol 76 no 10 pp 1450ndash1459 2009

[22] A Banerjee and R Manivasagam ldquoTriaxiality dependent cohe-sive zone modelrdquo Engineering Fracture Mechanics vol 76 no12 pp 1761ndash1770 2009

[23] A D Dimarogonas and C A Papadopoulos ldquoVibration ofcracked shafts in bendingrdquo Journal of Sound and Vibration vol91 no 4 pp 583ndash593 1983

[24] A C Chasalevris andCA Papadopoulos ldquoA continuousmodelapproach for cross-coupled bending vibrations of a rotor-bearing system with a transverse breathing crackrdquo Mechanismand Machine Theory vol 44 no 6 pp 1176ndash1191 2009

[25] 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

[26] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel to the analysis of a rotor with a transverse crackrdquo Journalof Sound and Vibration vol 332 no 8 pp 2098ndash2110 2013

[27] C Shet andN Chandra ldquoAnalysis of energy balancewhen usingCohesive Zone Models to simulate fracture processesrdquo Journalof EngineeringMaterials andTechnology vol 124 no 4 pp 440ndash450 2002

[28] H Li and N Chandra ldquoAnalysis of crack growth and crack-tip plasticity in ductile materials using cohesive zone modelsrdquoInternational Journal of Plasticity vol 19 no 6 pp 849ndash8822003

[29] N Bachschmid P Pennacchi and E Tanzi ldquoOn the evolutionof vibrations in cracked rotorsrdquo in Proceedings of the 8thIFToMM International Conference on Rotor Dynamics pp 304ndash310 Seoul Korea 2010

[30] O S Jun H J Eun Y Y Earmme and C W Lee ldquoModellingand vibration analysis of a simple rotor with a breathing crackrdquoJournal of Sound andVibration vol 155 no 2 pp 273ndash290 1992

[31] J J Sinou and A W Lees ldquoThe influence of cracks in rotatingshaftsrdquo Journal of Sound and Vibration vol 285 no 4-5 pp1015ndash1037 2005

[32] Y S Shih and J J Chen ldquoAnalysis of fatigue crack growth on acracked shaftrdquo International Journal of Fatigue vol 19 no 6 pp477ndash485 1997

[33] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel for the investigation of the dynamic behavior of arotating shaft with a transverse crackrdquo in Proceedings of the 8thIFToMM International Conference on Rotor Dynamics pp 628ndash636 Seoul Korea September 2010

[34] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel to the analysis of a rotor with a transverse crackrdquo inProceedings of the 8th International Conference on StructuralDynamics pp 3434ndash3442 Leuven Belgium 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

ISRNMechanical Engineering 7

Crack

Tensile stressor zero stress

Displacement+minus

Figure 8 Evaluation of displacements and stresses for state of crack

0∘ 30∘ 45∘

60∘ 75∘ 90∘

105∘ 120∘ 135∘

150∘ 165∘ 180∘

Figure 9 Breathing crack of the rotating shaft relative crack depth119886119889 = 01 at 600 rpm

infinite [27] As an alternative approach to this singularity-driven fracture approach the concept of CZM is proposedCZM has evolved as a preferred method to analyze fractureproblems not only because it avoids the singularity but alsobecause it can be easily implemented in a numerical methodof analysis as in FE method CZM presupposes the presenceof a fracture process zone where the energy is transferred

from external work in the crack regions In this study theexternal work flows as recoverable elastic strain energy andcohesive energy are examined the latter encompassing thework and other energy consuming mechanisms within thefracture process zone Process zone is defined as the regionwithin the separating surfaces where the surface tractionvalues are nonzeroThis also implies that processes occurringwithin the process zone are accounted for only throughthe traction-displacement relations The traction-separationrelations for crack interfaces are demonstrated such thatwith increasing interfacial separation the traction across theinterface reaches amaximum itmeans the elastic limit whereplastic dissipation in the surrounding (bounding) material isnot accounted for in the process zone

5 Conclusions

In this study a simulation process of FE and MBS isemployed An elastic cracked shaft is modelled by FE whichis transferred into MBS model The vibration responses inthe centroid of the shaft obtained from MBS have beentransferred again into an FE model in order to observe thebreathing mechanism and to study the stiffness losses due tobreathing at different rotating speeds

The results have shown that the breathing mechanismis influenced by the vibration due to inertia forces and byrelative crack depth It is shown that the relative crack depthduring crack opening should be understood not as linearincreasing but as harmonic function or quadratic polynomialIt can be noted that as long as the relative crack depth issmall the bilinear crack closure model may be used Themain difference with respect to the straight line crack closuremodel is that the opening crack in the simulation is notconstant at the beginningThe relative crack area of breathingmechanism results is in good agreement with the straight linecrack closuremodel between rotation angles1205873 and1205876 andwith the bilinear crack closuremodel in the range from0 to120587Furthermore the area moment of inertia about the rotationaxis can be approximated From this information the stiffnesslosses of the shaft can be evaluated It has been shown thatthe normalized stiffness between simulation and the bilinearcrack closure model is very close which validates the bilinearmodel

CZM are being increasingly used to simulate crackInstead of an infinitely sharp crack envisaged in fracturemechanics CZM presupposes the presence of a fracture pro-cess zone where the energy is transferred from external workin the crack regions The traction-separation relations forcrack interfaces are demonstrated such that with increasinginterfacial separation the traction across the interface reachesa maximum it means the elastic limit where plastic dissipa-tion in the surrounding (bounding)material is not accountedfor in the process zone

In light of the presented results and the conclusionsseveral subjects can be recommended for future research Animportant point on which the knowledge could be improvedis the prediction of crack propagation on cracked rotor andresidual life estimation The CZM can be easily implementedin FE model to analyze the dynamic behaviour of a cracked

8 ISRNMechanical Engineering

0 120587 2120587

119886119889

01

008

006

004

002

0

Linear model119891(119909) = 119886119909 + 119887

Confidence bounds 95

Goodness of fitSSE = 00005796R-square = 09516

120579 (rad)

119886 = 003887 (003226 004548)

119887 = 000131 (minus001043 001305)

(a)

0 120587 2120587

01

008

006

004

002

0

119886119889

Quadratic polynomial119891(119909) = 1198861199092 + 119887119909 + 119888

Confidence bounds 95

Goodness of fitSSE = 1141119890 minus 005R-square = 09990

120579 (rad)

119886 = minus001187 (minus001325 minus001050)119887 = 007617 (007174 008059)

119888 = minus001985 (minus002289 minus001683)

(b)

0 120587 2120587

01

008

006

004

002

0

119886119889

General model of sinus function

Confidence bounds 95

Goodness of fitSSE = 6158119890 minus 06R-square = 09995

120579 (rad)

119891(119909) = 119886 sin (119887119909 + 119888)

119886 = 009969 (009806 010130)119887 = 059340 (056990 061680)119888 = minus014540 (minus016410 minus012660)

(c)

0 120587 2120587

01

008

006

004

002

0

119886119889

General model of power119891(119909) = 119886119909119887 + 119888

Confidence bounds 95

Goodness of fitSSE = 9952119890 minus 05R-square = 09917

120579 (rad)

119886 = 012310 (004674 019950)119887 = 037300 (015200 059400)119888 = minus007730 (minus015120 minus000336)

(d)

Figure 10 Curve fitting of breathing mechanism results

Table 1 Curve fitting of the simulation results for 119886119889 = 01 120579 in [rad]

Model Mathematical model SSE R-squareLinear function 119886119889 = 0039120579 + 0001 0000 580 0951 6Quadratic polynomial 119886119889 = minus00121205792 + 0076120579minus 0020 0000 011 0999 0Sinus function 119886119889 = 0010sin (0593120579minus 0145) 0000 006 0999 5Power function 119886119889 = 01231205790373minus 0077 0000 099 0991 7

Crack closure line 1205721205720

119886

119887

119887max

Bilinear borderline

120575 gt 100120583m (crack opens)

10120583m le 120575 100120583m

120575 lt 10120583m

120575 = 0 (crack still closed)le

Figure 11 Bilinear crack closure model

ISRNMechanical Engineering 9

0 1205872 120587

120587

51205876

21205873

120572 = minus0084 lowast 1205792 + 051 lowast 120579 + 24

120572 = minus0018 lowast 1205793 + 0016 lowast 1205792 + 035 lowast 120579 + 24

SimulationQuadraticCubic

120579

120572

(rad)

(rad

)

Figure 12 Crack front angle 120572 during rotation of shaft

0 1205872 120587

1

09

08

07

06

05

04

03

02

01

0

119887119887 m

ax

SimulationCubic4th degree

= minus0069 lowast 1205793 + 02 lowast 1205792 + 036 lowast 120579 minus 0066

= 0076 lowast 1205794 minus 054lowast 1205793 + 11 lowast 1205792 minus 023 lowast 120579 minus

120579 (rad)

00056119887119887max

119887119887max

Figure 13 Normalized crack width 119887119887max during rotation of shaft

1

09

08

07

06

05

04

03

02

01

00 1205876 1205873 1205872 21205873 51205876 120587

Breathing mechanismBilinear crack closure model

Straight line crack closure [33]Perpendicular crack closure [12]

119860op

en119860

crac

k

120579 (rad)

Figure 14 Comparison of relative crack area between breathingmechanism results straight line perpendicular and bilinear crackclosure model for a shallow crack 119886119889 = 01

098

096

094

092

09

088

086

084

082

080 1205876 1205873 1205872 21205873 51205876 120587

Breathing mechanismBilinear crack closure model

Straight line crack closure [33]Perpendicular crack closure [12]

120578

120585

119870cr

ack119870

120579 (rad)

Figure 15 Comparison of normalized stiffness between breathingmechanism results straight line perpendicular and bilinear crackclosure model for a shallow crack 119886119889 = 01

shaft It is recommended to use CZM to study differenttypes of cracks such as longitudinal and slant cracks andalso multiple cracks Some other parameters such as internaldamping and thermal transients could be studied to obtainresults for their effect on the breathing mechanism Furtheranalysis on crack morphology is extremely important tounderstand the dynamic behaviour of cracked shaft thiswould include shallow and wide cracks

Acknowledgments

The authors acknowledge the support from Deutsche For-schungsgemeinschaft and Open Access Publishing Fund ofKarlsruhe Institute of Technology

References

[1] S K Georgantzinos and N K Anifantis ldquoAn insight into thebreathing mechanism of a crack in a rotating shaftrdquo Journal ofSound and Vibration vol 318 no 1-2 pp 279ndash295 2008

[2] J Wauer ldquoOn the dynamics of cracked rotors literature surveyrdquoApplied Mechanics Review vol 43 no 1 pp 13ndash17 1990

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

[4] G Sabnavis R G Kirk M Kasarda and D Quinn ldquoCrackedshaft detection and diagnostics a literature reviewrdquo Shock andVibration Digest vol 36 no 4 pp 287ndash296 2004

[5] 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

[6] N Bachschmid P Pennacchi and E Tanzi ldquoSome remarks onbreathing mechanism on non-linear effects and on slant andhelicoidal cracksrdquo Mechanical Systems and Signal Processingvol 22 no 4 pp 879ndash904 2008

[7] N Bachschmid P Pennachi and E Tanzi Cracked RotorsSpringer Berlin Germany 2011

10 ISRNMechanical Engineering

[8] S Andrieux and C Vare ldquoA 3D cracked beam model withunilateral contact Application to rotorsrdquo European Journal ofMechanics A vol 21 no 5 pp 793ndash810 2002

[9] C Vare and S Andrieux ldquoModeling of a cracked beam sectionunder bendingrdquo in Proceedings of the 18th International Confer-ence on Structural Mechanics in Reactor Technolohy (SMiRT 18)pp 281ndash290 Beijing China 2005

[10] S E Arem and HMaitournam ldquoA cracked beam finite elementfor rotating shaft dynamics and stability analysisrdquo Journal ofMechanics of Materials and Structures vol 3 no 5 pp 893ndash9102008

[11] A D Dimarogonas and S A Paipetis Analytical Methods inRotorDynamics Applied Science Publishers LondonUK 1983

[12] A K Darpe K Gupta and A Chawla ldquoCoupled bendinglongitudinal and torsional vibrations of a cracked rotorrdquo Journalof Sound and Vibration vol 269 no 1-2 pp 33ndash60 2004

[13] W M Ostachowicz and M Krawczuk ldquoCoupled torsional andbending vibrations of a rotor with an open crackrdquo Archive ofApplied Mechanics vol 62 no 3 pp 191ndash201 1992

[14] Z Kulesza and J T Sawicki ldquoRigid finite element model ofa cracked rotorrdquo Journal of Sound and Vibration vol 331 pp4145ndash4169 2012

[15] A S Bouboulas and N K Anifantis ldquoFinite element modelingof a vibrating beam with a breathing crack observations oncrack detectionrdquo Structural Health Monitoring vol 10 no 2 pp131ndash145 2011

[16] D S Dugdale ldquoYielding of steel sheets containing slitsrdquo Journalof the Mechanics and Physics of Solids vol 8 no 2 pp 100ndash1041960

[17] G I Barenblatt ldquoThemathematical theory of equilibrium crackin brittle fracturerdquoAdvances in AppliedMechanics vol 7 pp 55ndash129 1962

[18] T Siegmund andW Brocks ldquoTensile decohesion by local failurecriteriardquo Technische Mechanik vol 18 no 4 pp 261ndash270 1998

[19] T Siegmund and W Brocks ldquoA numerical study on thecorrelation between the work of separation and the dissipationrate in ductile fracturerdquo Engineering Fracture Mechanics vol 67no 2 pp 139ndash154 2000

[20] M Anvari I Scheider and CThaulow ldquoSimulation of dynamicductile crack growth using strain-rate and triaxiality-dependentcohesive elementsrdquo Engineering Fracture Mechanics vol 73 no15 pp 2210ndash2228 2006

[21] I Scheider ldquoDerivation of separation laws for cohesive modelsin the course of ductile fracturerdquo Engineering Fracture Mechan-ics vol 76 no 10 pp 1450ndash1459 2009

[22] A Banerjee and R Manivasagam ldquoTriaxiality dependent cohe-sive zone modelrdquo Engineering Fracture Mechanics vol 76 no12 pp 1761ndash1770 2009

[23] A D Dimarogonas and C A Papadopoulos ldquoVibration ofcracked shafts in bendingrdquo Journal of Sound and Vibration vol91 no 4 pp 583ndash593 1983

[24] A C Chasalevris andCA Papadopoulos ldquoA continuousmodelapproach for cross-coupled bending vibrations of a rotor-bearing system with a transverse breathing crackrdquo Mechanismand Machine Theory vol 44 no 6 pp 1176ndash1191 2009

[25] 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

[26] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel to the analysis of a rotor with a transverse crackrdquo Journalof Sound and Vibration vol 332 no 8 pp 2098ndash2110 2013

[27] C Shet andN Chandra ldquoAnalysis of energy balancewhen usingCohesive Zone Models to simulate fracture processesrdquo Journalof EngineeringMaterials andTechnology vol 124 no 4 pp 440ndash450 2002

[28] H Li and N Chandra ldquoAnalysis of crack growth and crack-tip plasticity in ductile materials using cohesive zone modelsrdquoInternational Journal of Plasticity vol 19 no 6 pp 849ndash8822003

[29] N Bachschmid P Pennacchi and E Tanzi ldquoOn the evolutionof vibrations in cracked rotorsrdquo in Proceedings of the 8thIFToMM International Conference on Rotor Dynamics pp 304ndash310 Seoul Korea 2010

[30] O S Jun H J Eun Y Y Earmme and C W Lee ldquoModellingand vibration analysis of a simple rotor with a breathing crackrdquoJournal of Sound andVibration vol 155 no 2 pp 273ndash290 1992

[31] J J Sinou and A W Lees ldquoThe influence of cracks in rotatingshaftsrdquo Journal of Sound and Vibration vol 285 no 4-5 pp1015ndash1037 2005

[32] Y S Shih and J J Chen ldquoAnalysis of fatigue crack growth on acracked shaftrdquo International Journal of Fatigue vol 19 no 6 pp477ndash485 1997

[33] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel for the investigation of the dynamic behavior of arotating shaft with a transverse crackrdquo in Proceedings of the 8thIFToMM International Conference on Rotor Dynamics pp 628ndash636 Seoul Korea September 2010

[34] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel to the analysis of a rotor with a transverse crackrdquo inProceedings of the 8th International Conference on StructuralDynamics pp 3434ndash3442 Leuven Belgium 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 ISRNMechanical Engineering

0 120587 2120587

119886119889

01

008

006

004

002

0

Linear model119891(119909) = 119886119909 + 119887

Confidence bounds 95

Goodness of fitSSE = 00005796R-square = 09516

120579 (rad)

119886 = 003887 (003226 004548)

119887 = 000131 (minus001043 001305)

(a)

0 120587 2120587

01

008

006

004

002

0

119886119889

Quadratic polynomial119891(119909) = 1198861199092 + 119887119909 + 119888

Confidence bounds 95

Goodness of fitSSE = 1141119890 minus 005R-square = 09990

120579 (rad)

119886 = minus001187 (minus001325 minus001050)119887 = 007617 (007174 008059)

119888 = minus001985 (minus002289 minus001683)

(b)

0 120587 2120587

01

008

006

004

002

0

119886119889

General model of sinus function

Confidence bounds 95

Goodness of fitSSE = 6158119890 minus 06R-square = 09995

120579 (rad)

119891(119909) = 119886 sin (119887119909 + 119888)

119886 = 009969 (009806 010130)119887 = 059340 (056990 061680)119888 = minus014540 (minus016410 minus012660)

(c)

0 120587 2120587

01

008

006

004

002

0

119886119889

General model of power119891(119909) = 119886119909119887 + 119888

Confidence bounds 95

Goodness of fitSSE = 9952119890 minus 05R-square = 09917

120579 (rad)

119886 = 012310 (004674 019950)119887 = 037300 (015200 059400)119888 = minus007730 (minus015120 minus000336)

(d)

Figure 10 Curve fitting of breathing mechanism results

Table 1 Curve fitting of the simulation results for 119886119889 = 01 120579 in [rad]

Model Mathematical model SSE R-squareLinear function 119886119889 = 0039120579 + 0001 0000 580 0951 6Quadratic polynomial 119886119889 = minus00121205792 + 0076120579minus 0020 0000 011 0999 0Sinus function 119886119889 = 0010sin (0593120579minus 0145) 0000 006 0999 5Power function 119886119889 = 01231205790373minus 0077 0000 099 0991 7

Crack closure line 1205721205720

119886

119887

119887max

Bilinear borderline

120575 gt 100120583m (crack opens)

10120583m le 120575 100120583m

120575 lt 10120583m

120575 = 0 (crack still closed)le

Figure 11 Bilinear crack closure model

ISRNMechanical Engineering 9

0 1205872 120587

120587

51205876

21205873

120572 = minus0084 lowast 1205792 + 051 lowast 120579 + 24

120572 = minus0018 lowast 1205793 + 0016 lowast 1205792 + 035 lowast 120579 + 24

SimulationQuadraticCubic

120579

120572

(rad)

(rad

)

Figure 12 Crack front angle 120572 during rotation of shaft

0 1205872 120587

1

09

08

07

06

05

04

03

02

01

0

119887119887 m

ax

SimulationCubic4th degree

= minus0069 lowast 1205793 + 02 lowast 1205792 + 036 lowast 120579 minus 0066

= 0076 lowast 1205794 minus 054lowast 1205793 + 11 lowast 1205792 minus 023 lowast 120579 minus

120579 (rad)

00056119887119887max

119887119887max

Figure 13 Normalized crack width 119887119887max during rotation of shaft

1

09

08

07

06

05

04

03

02

01

00 1205876 1205873 1205872 21205873 51205876 120587

Breathing mechanismBilinear crack closure model

Straight line crack closure [33]Perpendicular crack closure [12]

119860op

en119860

crac

k

120579 (rad)

Figure 14 Comparison of relative crack area between breathingmechanism results straight line perpendicular and bilinear crackclosure model for a shallow crack 119886119889 = 01

098

096

094

092

09

088

086

084

082

080 1205876 1205873 1205872 21205873 51205876 120587

Breathing mechanismBilinear crack closure model

Straight line crack closure [33]Perpendicular crack closure [12]

120578

120585

119870cr

ack119870

120579 (rad)

Figure 15 Comparison of normalized stiffness between breathingmechanism results straight line perpendicular and bilinear crackclosure model for a shallow crack 119886119889 = 01

shaft It is recommended to use CZM to study differenttypes of cracks such as longitudinal and slant cracks andalso multiple cracks Some other parameters such as internaldamping and thermal transients could be studied to obtainresults for their effect on the breathing mechanism Furtheranalysis on crack morphology is extremely important tounderstand the dynamic behaviour of cracked shaft thiswould include shallow and wide cracks

Acknowledgments

The authors acknowledge the support from Deutsche For-schungsgemeinschaft and Open Access Publishing Fund ofKarlsruhe Institute of Technology

References

[1] S K Georgantzinos and N K Anifantis ldquoAn insight into thebreathing mechanism of a crack in a rotating shaftrdquo Journal ofSound and Vibration vol 318 no 1-2 pp 279ndash295 2008

[2] J Wauer ldquoOn the dynamics of cracked rotors literature surveyrdquoApplied Mechanics Review vol 43 no 1 pp 13ndash17 1990

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

[4] G Sabnavis R G Kirk M Kasarda and D Quinn ldquoCrackedshaft detection and diagnostics a literature reviewrdquo Shock andVibration Digest vol 36 no 4 pp 287ndash296 2004

[5] 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

[6] N Bachschmid P Pennacchi and E Tanzi ldquoSome remarks onbreathing mechanism on non-linear effects and on slant andhelicoidal cracksrdquo Mechanical Systems and Signal Processingvol 22 no 4 pp 879ndash904 2008

[7] N Bachschmid P Pennachi and E Tanzi Cracked RotorsSpringer Berlin Germany 2011

10 ISRNMechanical Engineering

[8] S Andrieux and C Vare ldquoA 3D cracked beam model withunilateral contact Application to rotorsrdquo European Journal ofMechanics A vol 21 no 5 pp 793ndash810 2002

[9] C Vare and S Andrieux ldquoModeling of a cracked beam sectionunder bendingrdquo in Proceedings of the 18th International Confer-ence on Structural Mechanics in Reactor Technolohy (SMiRT 18)pp 281ndash290 Beijing China 2005

[10] S E Arem and HMaitournam ldquoA cracked beam finite elementfor rotating shaft dynamics and stability analysisrdquo Journal ofMechanics of Materials and Structures vol 3 no 5 pp 893ndash9102008

[11] A D Dimarogonas and S A Paipetis Analytical Methods inRotorDynamics Applied Science Publishers LondonUK 1983

[12] A K Darpe K Gupta and A Chawla ldquoCoupled bendinglongitudinal and torsional vibrations of a cracked rotorrdquo Journalof Sound and Vibration vol 269 no 1-2 pp 33ndash60 2004

[13] W M Ostachowicz and M Krawczuk ldquoCoupled torsional andbending vibrations of a rotor with an open crackrdquo Archive ofApplied Mechanics vol 62 no 3 pp 191ndash201 1992

[14] Z Kulesza and J T Sawicki ldquoRigid finite element model ofa cracked rotorrdquo Journal of Sound and Vibration vol 331 pp4145ndash4169 2012

[15] A S Bouboulas and N K Anifantis ldquoFinite element modelingof a vibrating beam with a breathing crack observations oncrack detectionrdquo Structural Health Monitoring vol 10 no 2 pp131ndash145 2011

[16] D S Dugdale ldquoYielding of steel sheets containing slitsrdquo Journalof the Mechanics and Physics of Solids vol 8 no 2 pp 100ndash1041960

[17] G I Barenblatt ldquoThemathematical theory of equilibrium crackin brittle fracturerdquoAdvances in AppliedMechanics vol 7 pp 55ndash129 1962

[18] T Siegmund andW Brocks ldquoTensile decohesion by local failurecriteriardquo Technische Mechanik vol 18 no 4 pp 261ndash270 1998

[19] T Siegmund and W Brocks ldquoA numerical study on thecorrelation between the work of separation and the dissipationrate in ductile fracturerdquo Engineering Fracture Mechanics vol 67no 2 pp 139ndash154 2000

[20] M Anvari I Scheider and CThaulow ldquoSimulation of dynamicductile crack growth using strain-rate and triaxiality-dependentcohesive elementsrdquo Engineering Fracture Mechanics vol 73 no15 pp 2210ndash2228 2006

[21] I Scheider ldquoDerivation of separation laws for cohesive modelsin the course of ductile fracturerdquo Engineering Fracture Mechan-ics vol 76 no 10 pp 1450ndash1459 2009

[22] A Banerjee and R Manivasagam ldquoTriaxiality dependent cohe-sive zone modelrdquo Engineering Fracture Mechanics vol 76 no12 pp 1761ndash1770 2009

[23] A D Dimarogonas and C A Papadopoulos ldquoVibration ofcracked shafts in bendingrdquo Journal of Sound and Vibration vol91 no 4 pp 583ndash593 1983

[24] A C Chasalevris andCA Papadopoulos ldquoA continuousmodelapproach for cross-coupled bending vibrations of a rotor-bearing system with a transverse breathing crackrdquo Mechanismand Machine Theory vol 44 no 6 pp 1176ndash1191 2009

[25] 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

[26] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel to the analysis of a rotor with a transverse crackrdquo Journalof Sound and Vibration vol 332 no 8 pp 2098ndash2110 2013

[27] C Shet andN Chandra ldquoAnalysis of energy balancewhen usingCohesive Zone Models to simulate fracture processesrdquo Journalof EngineeringMaterials andTechnology vol 124 no 4 pp 440ndash450 2002

[28] H Li and N Chandra ldquoAnalysis of crack growth and crack-tip plasticity in ductile materials using cohesive zone modelsrdquoInternational Journal of Plasticity vol 19 no 6 pp 849ndash8822003

[29] N Bachschmid P Pennacchi and E Tanzi ldquoOn the evolutionof vibrations in cracked rotorsrdquo in Proceedings of the 8thIFToMM International Conference on Rotor Dynamics pp 304ndash310 Seoul Korea 2010

[30] O S Jun H J Eun Y Y Earmme and C W Lee ldquoModellingand vibration analysis of a simple rotor with a breathing crackrdquoJournal of Sound andVibration vol 155 no 2 pp 273ndash290 1992

[31] J J Sinou and A W Lees ldquoThe influence of cracks in rotatingshaftsrdquo Journal of Sound and Vibration vol 285 no 4-5 pp1015ndash1037 2005

[32] Y S Shih and J J Chen ldquoAnalysis of fatigue crack growth on acracked shaftrdquo International Journal of Fatigue vol 19 no 6 pp477ndash485 1997

[33] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel for the investigation of the dynamic behavior of arotating shaft with a transverse crackrdquo in Proceedings of the 8thIFToMM International Conference on Rotor Dynamics pp 628ndash636 Seoul Korea September 2010

[34] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel to the analysis of a rotor with a transverse crackrdquo inProceedings of the 8th International Conference on StructuralDynamics pp 3434ndash3442 Leuven Belgium 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

ISRNMechanical Engineering 9

0 1205872 120587

120587

51205876

21205873

120572 = minus0084 lowast 1205792 + 051 lowast 120579 + 24

120572 = minus0018 lowast 1205793 + 0016 lowast 1205792 + 035 lowast 120579 + 24

SimulationQuadraticCubic

120579

120572

(rad)

(rad

)

Figure 12 Crack front angle 120572 during rotation of shaft

0 1205872 120587

1

09

08

07

06

05

04

03

02

01

0

119887119887 m

ax

SimulationCubic4th degree

= minus0069 lowast 1205793 + 02 lowast 1205792 + 036 lowast 120579 minus 0066

= 0076 lowast 1205794 minus 054lowast 1205793 + 11 lowast 1205792 minus 023 lowast 120579 minus

120579 (rad)

00056119887119887max

119887119887max

Figure 13 Normalized crack width 119887119887max during rotation of shaft

1

09

08

07

06

05

04

03

02

01

00 1205876 1205873 1205872 21205873 51205876 120587

Breathing mechanismBilinear crack closure model

Straight line crack closure [33]Perpendicular crack closure [12]

119860op

en119860

crac

k

120579 (rad)

Figure 14 Comparison of relative crack area between breathingmechanism results straight line perpendicular and bilinear crackclosure model for a shallow crack 119886119889 = 01

098

096

094

092

09

088

086

084

082

080 1205876 1205873 1205872 21205873 51205876 120587

Breathing mechanismBilinear crack closure model

Straight line crack closure [33]Perpendicular crack closure [12]

120578

120585

119870cr

ack119870

120579 (rad)

Figure 15 Comparison of normalized stiffness between breathingmechanism results straight line perpendicular and bilinear crackclosure model for a shallow crack 119886119889 = 01

shaft It is recommended to use CZM to study differenttypes of cracks such as longitudinal and slant cracks andalso multiple cracks Some other parameters such as internaldamping and thermal transients could be studied to obtainresults for their effect on the breathing mechanism Furtheranalysis on crack morphology is extremely important tounderstand the dynamic behaviour of cracked shaft thiswould include shallow and wide cracks

Acknowledgments

The authors acknowledge the support from Deutsche For-schungsgemeinschaft and Open Access Publishing Fund ofKarlsruhe Institute of Technology

References

[1] S K Georgantzinos and N K Anifantis ldquoAn insight into thebreathing mechanism of a crack in a rotating shaftrdquo Journal ofSound and Vibration vol 318 no 1-2 pp 279ndash295 2008

[2] J Wauer ldquoOn the dynamics of cracked rotors literature surveyrdquoApplied Mechanics Review vol 43 no 1 pp 13ndash17 1990

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

[4] G Sabnavis R G Kirk M Kasarda and D Quinn ldquoCrackedshaft detection and diagnostics a literature reviewrdquo Shock andVibration Digest vol 36 no 4 pp 287ndash296 2004

[5] 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

[6] N Bachschmid P Pennacchi and E Tanzi ldquoSome remarks onbreathing mechanism on non-linear effects and on slant andhelicoidal cracksrdquo Mechanical Systems and Signal Processingvol 22 no 4 pp 879ndash904 2008

[7] N Bachschmid P Pennachi and E Tanzi Cracked RotorsSpringer Berlin Germany 2011

10 ISRNMechanical Engineering

[8] S Andrieux and C Vare ldquoA 3D cracked beam model withunilateral contact Application to rotorsrdquo European Journal ofMechanics A vol 21 no 5 pp 793ndash810 2002

[9] C Vare and S Andrieux ldquoModeling of a cracked beam sectionunder bendingrdquo in Proceedings of the 18th International Confer-ence on Structural Mechanics in Reactor Technolohy (SMiRT 18)pp 281ndash290 Beijing China 2005

[10] S E Arem and HMaitournam ldquoA cracked beam finite elementfor rotating shaft dynamics and stability analysisrdquo Journal ofMechanics of Materials and Structures vol 3 no 5 pp 893ndash9102008

[11] A D Dimarogonas and S A Paipetis Analytical Methods inRotorDynamics Applied Science Publishers LondonUK 1983

[12] A K Darpe K Gupta and A Chawla ldquoCoupled bendinglongitudinal and torsional vibrations of a cracked rotorrdquo Journalof Sound and Vibration vol 269 no 1-2 pp 33ndash60 2004

[13] W M Ostachowicz and M Krawczuk ldquoCoupled torsional andbending vibrations of a rotor with an open crackrdquo Archive ofApplied Mechanics vol 62 no 3 pp 191ndash201 1992

[14] Z Kulesza and J T Sawicki ldquoRigid finite element model ofa cracked rotorrdquo Journal of Sound and Vibration vol 331 pp4145ndash4169 2012

[15] A S Bouboulas and N K Anifantis ldquoFinite element modelingof a vibrating beam with a breathing crack observations oncrack detectionrdquo Structural Health Monitoring vol 10 no 2 pp131ndash145 2011

[16] D S Dugdale ldquoYielding of steel sheets containing slitsrdquo Journalof the Mechanics and Physics of Solids vol 8 no 2 pp 100ndash1041960

[17] G I Barenblatt ldquoThemathematical theory of equilibrium crackin brittle fracturerdquoAdvances in AppliedMechanics vol 7 pp 55ndash129 1962

[18] T Siegmund andW Brocks ldquoTensile decohesion by local failurecriteriardquo Technische Mechanik vol 18 no 4 pp 261ndash270 1998

[19] T Siegmund and W Brocks ldquoA numerical study on thecorrelation between the work of separation and the dissipationrate in ductile fracturerdquo Engineering Fracture Mechanics vol 67no 2 pp 139ndash154 2000

[20] M Anvari I Scheider and CThaulow ldquoSimulation of dynamicductile crack growth using strain-rate and triaxiality-dependentcohesive elementsrdquo Engineering Fracture Mechanics vol 73 no15 pp 2210ndash2228 2006

[21] I Scheider ldquoDerivation of separation laws for cohesive modelsin the course of ductile fracturerdquo Engineering Fracture Mechan-ics vol 76 no 10 pp 1450ndash1459 2009

[22] A Banerjee and R Manivasagam ldquoTriaxiality dependent cohe-sive zone modelrdquo Engineering Fracture Mechanics vol 76 no12 pp 1761ndash1770 2009

[23] A D Dimarogonas and C A Papadopoulos ldquoVibration ofcracked shafts in bendingrdquo Journal of Sound and Vibration vol91 no 4 pp 583ndash593 1983

[24] A C Chasalevris andCA Papadopoulos ldquoA continuousmodelapproach for cross-coupled bending vibrations of a rotor-bearing system with a transverse breathing crackrdquo Mechanismand Machine Theory vol 44 no 6 pp 1176ndash1191 2009

[25] 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

[26] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel to the analysis of a rotor with a transverse crackrdquo Journalof Sound and Vibration vol 332 no 8 pp 2098ndash2110 2013

[27] C Shet andN Chandra ldquoAnalysis of energy balancewhen usingCohesive Zone Models to simulate fracture processesrdquo Journalof EngineeringMaterials andTechnology vol 124 no 4 pp 440ndash450 2002

[28] H Li and N Chandra ldquoAnalysis of crack growth and crack-tip plasticity in ductile materials using cohesive zone modelsrdquoInternational Journal of Plasticity vol 19 no 6 pp 849ndash8822003

[29] N Bachschmid P Pennacchi and E Tanzi ldquoOn the evolutionof vibrations in cracked rotorsrdquo in Proceedings of the 8thIFToMM International Conference on Rotor Dynamics pp 304ndash310 Seoul Korea 2010

[30] O S Jun H J Eun Y Y Earmme and C W Lee ldquoModellingand vibration analysis of a simple rotor with a breathing crackrdquoJournal of Sound andVibration vol 155 no 2 pp 273ndash290 1992

[31] J J Sinou and A W Lees ldquoThe influence of cracks in rotatingshaftsrdquo Journal of Sound and Vibration vol 285 no 4-5 pp1015ndash1037 2005

[32] Y S Shih and J J Chen ldquoAnalysis of fatigue crack growth on acracked shaftrdquo International Journal of Fatigue vol 19 no 6 pp477ndash485 1997

[33] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel for the investigation of the dynamic behavior of arotating shaft with a transverse crackrdquo in Proceedings of the 8thIFToMM International Conference on Rotor Dynamics pp 628ndash636 Seoul Korea September 2010

[34] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel to the analysis of a rotor with a transverse crackrdquo inProceedings of the 8th International Conference on StructuralDynamics pp 3434ndash3442 Leuven Belgium 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 ISRNMechanical Engineering

[8] S Andrieux and C Vare ldquoA 3D cracked beam model withunilateral contact Application to rotorsrdquo European Journal ofMechanics A vol 21 no 5 pp 793ndash810 2002

[9] C Vare and S Andrieux ldquoModeling of a cracked beam sectionunder bendingrdquo in Proceedings of the 18th International Confer-ence on Structural Mechanics in Reactor Technolohy (SMiRT 18)pp 281ndash290 Beijing China 2005

[10] S E Arem and HMaitournam ldquoA cracked beam finite elementfor rotating shaft dynamics and stability analysisrdquo Journal ofMechanics of Materials and Structures vol 3 no 5 pp 893ndash9102008

[11] A D Dimarogonas and S A Paipetis Analytical Methods inRotorDynamics Applied Science Publishers LondonUK 1983

[12] A K Darpe K Gupta and A Chawla ldquoCoupled bendinglongitudinal and torsional vibrations of a cracked rotorrdquo Journalof Sound and Vibration vol 269 no 1-2 pp 33ndash60 2004

[13] W M Ostachowicz and M Krawczuk ldquoCoupled torsional andbending vibrations of a rotor with an open crackrdquo Archive ofApplied Mechanics vol 62 no 3 pp 191ndash201 1992

[14] Z Kulesza and J T Sawicki ldquoRigid finite element model ofa cracked rotorrdquo Journal of Sound and Vibration vol 331 pp4145ndash4169 2012

[15] A S Bouboulas and N K Anifantis ldquoFinite element modelingof a vibrating beam with a breathing crack observations oncrack detectionrdquo Structural Health Monitoring vol 10 no 2 pp131ndash145 2011

[16] D S Dugdale ldquoYielding of steel sheets containing slitsrdquo Journalof the Mechanics and Physics of Solids vol 8 no 2 pp 100ndash1041960

[17] G I Barenblatt ldquoThemathematical theory of equilibrium crackin brittle fracturerdquoAdvances in AppliedMechanics vol 7 pp 55ndash129 1962

[18] T Siegmund andW Brocks ldquoTensile decohesion by local failurecriteriardquo Technische Mechanik vol 18 no 4 pp 261ndash270 1998

[19] T Siegmund and W Brocks ldquoA numerical study on thecorrelation between the work of separation and the dissipationrate in ductile fracturerdquo Engineering Fracture Mechanics vol 67no 2 pp 139ndash154 2000

[20] M Anvari I Scheider and CThaulow ldquoSimulation of dynamicductile crack growth using strain-rate and triaxiality-dependentcohesive elementsrdquo Engineering Fracture Mechanics vol 73 no15 pp 2210ndash2228 2006

[21] I Scheider ldquoDerivation of separation laws for cohesive modelsin the course of ductile fracturerdquo Engineering Fracture Mechan-ics vol 76 no 10 pp 1450ndash1459 2009

[22] A Banerjee and R Manivasagam ldquoTriaxiality dependent cohe-sive zone modelrdquo Engineering Fracture Mechanics vol 76 no12 pp 1761ndash1770 2009

[23] A D Dimarogonas and C A Papadopoulos ldquoVibration ofcracked shafts in bendingrdquo Journal of Sound and Vibration vol91 no 4 pp 583ndash593 1983

[24] A C Chasalevris andCA Papadopoulos ldquoA continuousmodelapproach for cross-coupled bending vibrations of a rotor-bearing system with a transverse breathing crackrdquo Mechanismand Machine Theory vol 44 no 6 pp 1176ndash1191 2009

[25] 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

[26] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel to the analysis of a rotor with a transverse crackrdquo Journalof Sound and Vibration vol 332 no 8 pp 2098ndash2110 2013

[27] C Shet andN Chandra ldquoAnalysis of energy balancewhen usingCohesive Zone Models to simulate fracture processesrdquo Journalof EngineeringMaterials andTechnology vol 124 no 4 pp 440ndash450 2002

[28] H Li and N Chandra ldquoAnalysis of crack growth and crack-tip plasticity in ductile materials using cohesive zone modelsrdquoInternational Journal of Plasticity vol 19 no 6 pp 849ndash8822003

[29] N Bachschmid P Pennacchi and E Tanzi ldquoOn the evolutionof vibrations in cracked rotorsrdquo in Proceedings of the 8thIFToMM International Conference on Rotor Dynamics pp 304ndash310 Seoul Korea 2010

[30] O S Jun H J Eun Y Y Earmme and C W Lee ldquoModellingand vibration analysis of a simple rotor with a breathing crackrdquoJournal of Sound andVibration vol 155 no 2 pp 273ndash290 1992

[31] J J Sinou and A W Lees ldquoThe influence of cracks in rotatingshaftsrdquo Journal of Sound and Vibration vol 285 no 4-5 pp1015ndash1037 2005

[32] Y S Shih and J J Chen ldquoAnalysis of fatigue crack growth on acracked shaftrdquo International Journal of Fatigue vol 19 no 6 pp477ndash485 1997

[33] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel for the investigation of the dynamic behavior of arotating shaft with a transverse crackrdquo in Proceedings of the 8thIFToMM International Conference on Rotor Dynamics pp 628ndash636 Seoul Korea September 2010

[34] R T Liong and C Proppe ldquoApplication of the cohesive zonemodel to the analysis of a rotor with a transverse crackrdquo inProceedings of the 8th International Conference on StructuralDynamics pp 3434ndash3442 Leuven Belgium 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of