Dynamic Behaviour Of A Cracked Shaft On Elastic Support.

Dynamic Behaviour Of A Cracked Shaft On Elastic Support.

by

@Liqian g Liu, B.Eng.,M.Eng.

A thesissub mitted to the School of Gradua.te Studies inp&rtialfulfillment of the requir ements Corthe degreeof Masterof Engineering

Facultyof Engineeringand Applied Science Memorial UniversityofNewfou ndland

June1995

St.John's Newfoundland Canada

.+.

NationalUbrary olOM""

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=-~S-

K1A CN4

~bI~ue nalionale DirectiOndesacquisitionsel desservices bticgraptiques

... - -

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crNE DOIVENTETRElMPRIMES OU AUTREMENTREPRQDUlTS SANSSON AlJTORJSATION.

IS8N0-612-06130-2

Canada

To

MyparentsJinkai Liu

ana

Yulan Jia and

MywifeMiao

I

Abstract

This is a simple but comprehensivestudy ofthe dynamic behavior of a shaft withacrackon elasticsupports. The analysisis restricted to the single span shaft with uniformcircularcross-section. The natural frequency and modes of vibrationof a shaft having a transversecrack are investigated using the finite element method. Thelocal flexibility duetothe crack is evaluatedusingthe theory of fracture mechanics.The effect of crack depth on the natural behavior is discussed. The results show that an increasein the depth of the of crecx magnifies the response amplitude and decreases thenaturalfrequencies. The effect of elastic supports on thedynamic behaviorof the shaftispresented throughcomputation.

The range of maximum effect is given.

The clement stiffnessmatrixof a cracked shaft considering thelongitudinal translation and axialrotation is first presented. Thismakes it possible to enely- size the dynamicresponse of a practical shaftbyFEM.A Fortran-77programis developedwhich can beused to calculate the two and threedimensional vibration of a shaftcontainingmore than one transverse crack,concentratedmass andelas- tic foundation.Itcan~JBObe usedin multi-spa nshaft with different cross-section and appliedto some loads.

Acknowledgements

Inmypun uit oftheMuterdegreein Engineering,Ihave received valuable adviceandassistan ce from anumberof people,this isvery much appreciated . In part icular ,Iwould"liketo thank:

<a}Dr.M.R.Hedder e,mysupervisor,for hisguidanceandsupport through- outthe program, and forhis advice andelucidationofthe variousproblemsasso- ciated.with this study.

(b)Dr.J.J.Sharp,Dr. A. S.J.Swamid asandDr.G.Sabinfortheir courses and help.Mrs.MoyaCrocker for her helpduringthecourseof my stu dy.

(c) The SchoolofGraduateStudiesfor providing funding and teaching assls- tantships.

(d)Mr.DavidPressand his staff &1the CentreCor Computer Aided En- gineering fortheir helpin overcomingso many difficulties relatedto computer work.

(e) Finally, Iwouldlike to thankmy dearwife,MiaoLi,forherpatience and understand ing.

Contents

Abstract Acknowledgements List of Tables List ofFigu res ListofSymbols

1 Introductionand LiteratureSurvey 1.1 Litera tureSurvey .. .. •. .. .. .. 1.2 Objective

1.3 Methodology ...

vi vii ix

2 StiffnessMatrixDerivationofSpaceBeam Elementwitha Crack li

2.1 Introduction , .

2.2 Crack-TipStress FieldsforLinear-ElasticBodies . 2.2.1 Crack Tip Stres9IntensityFactors..., , .

2.2.2 EvaluationofKr,KII and K111 of TheSingleEdge Notch 13 2.3 LocalFlexibility . .. .• .. .• . .. . . •... ... .•. 21

iii

2.4 Stiffness Matrixof the Cracked Element

iv

29

3 Test of theProgram to Solve theBea mVibration(NoCr ack ) 34 4 TheEffectof Elasti cSupp ort s andProp elle r Iner t ia011theDy-

namicBehaviour 4,1 KI=K~. • . . . .. 4.2 K1 :f:.K~..

4..3 EffectofThe PropellerInertia ...•.•. •.

4.4 Discussion of The Results . 5 TheEffectof ACr ack on the DynamicBeh aviour

5.1 Calculation Results ..

5.2 Conclusions ... .•....

6 StiffnessMat rixDeriv ation of SpaceBeamElement wit ha Crac k Consideringthe AxisTranslationand Rotation

6.1 Local Flexibility .

6.2 Stiffness Matrix of the Cracked Element 7 Concl usions

7.1 Conclusions 7.2 Recommendat ions.

Refer ences Append ices

38 40 44 48 49

. ,

51 61

82

"

₆₃

68 68 69 70 73

A Free Vibration of•Beam

• •

A.I BendingVibrat ion Equationofa.BeamSubjectedtoan AxialForce 74

B MassandStiffn es sMatrice sDerivationof SpaceBeamElement 78

CFlow Cha rt ofPro gr a m

D Computer Programin Fortran·77

.5

ss

List of Tables

3.1 Comparisonof NaturalFrequencies.. ... ... . ... . 37 4.1 FirstThreeFrequencies..

4.2 First ThreeFrequencies..

, .

"

5.1 FirstThree FrequenciesCorrespondingto Different Crack Depth. 53

List of Figures

2.1 The BasicModesofCrackSurface Displacements.

2.2 Coordinates MeasuredCrom theLeading Edge ofIICrack andthe Stress Components in the CrackTipStress Field •.. 2.3 TheSingleEdge Notch Test SpecimenUnderTension Load... 14 2.4 TheSingleEdgeNotchTest Specimen Under Bending Load ... 16 2.5 TheSingleEdge NotchTest Specimen Under Shear andTorsion Load18 2.6 TheSingle Edge Notch Test Specime nUnder Torsion Load . ." 20 2,7 (a)Acrackedshaftelement in general loading; (b) thecrack section

ofthe shaft .. ... . ....• •..•.. .... .. . •.. 22 2.8 Dimensionlesscompliancesversuscrack depth.(a)en,cu,C6S;(b)

ca,

c••.

C.u; (e)Cu,C3S,Coo;(d)(;22,C33•• • •••• 2.9 Simplysupported shaftwith a crackedelement ,.

28 31 3.1 Flow Chartof theProgram ... .. . . • •..• . ..• . . ... 35 3.2 (a) Simplesupported; (b)Free;(c) Fixed; (d)Cantilever;(e)Propped 36 4.1 (a) shaft; (b)Mesh of elements ... .... . . . .•. . . 39 4.2 thecurve between first frequencyandK . .•... . . . .. .•. 41 4.3 the curvebetween second frequency andK .•...•.• •. . .. 42

vii

viii

4.4 the curvebetweenthird Irequency andK _ 43

4.5 the curve betweenlint frequency andK 45

4.6 thecurvebetweenlIt'COndfrequency andK.• • •.. .• •.•.. -16 4.7 the curve betweenthirdfrequency andK•••".•• •• • •• • •• 47 4.8 Diagramofatailed ,hart... .... .. ...• •. ... .. . .. 49' 5.1 (a>shaftwit h acrack;(b)Meahofelements . 52 5.2 Variat ionsof firstfrequencywithdifferentcrackdept h • . . 54 5,3 Variationsof~eeondfrequencywith differentcrack depth.. . . 55 5.4 Variationsofthird frequencywith differentcrackdepth... 56 5.5 Var iat ions ofnorm alizedchangeinfintfrequencywith crack depth 57 5.6 Var iations ofn~a1izedchangein second frequencywithcrackdepth!i8 5.7 Variations or normalizedchange in thi rdfrequencywith crackdepth59 5.8 FirstModeShapeorShan withDiffere nt CrackDepth 60 6.1 ShaftwithCrackedE1emen\•. . •• • •... • ... .•... &1 A.I (a)abeaminbendin&;(b)free body diagram of&l1element . . . 75 8.1 Spaceframememb er:(a)localdirect ions; (b)globaldirec\ions. . 79 8.2 Beamelementwit h 8 degreee of freedom . . .••. . .. 83

x,y,ll x ,y ,':

M(x,.) V(x,.) f(x,. ) A(x) P P

E G

D R

Li st of Symbols

Coordinates in globalsystem Coordinates in localsystem time

Bendingmomentof theshaft Shear forceoftheshaft External forceper unitlengthofa shaft areaof cross-sectionofa shaft Mass density Externalload

Anglebetween axialandho~izontaldirections Young'selasticitymodulus

Shear modulusofelasticity Passionratio'

Inertia moment of cross-sectionof shaft Diameter ofshaft

Radiusofshaft

Polar momentofinertia ofshaft per unitlength Tension stress

Shear stress Depth ofa crack

ix

h,b

U;

J(u) [J() 1M ]

the gemetricsize

or

^{asam ple}

Three kinds ofstress intensityfactors displacementof crack tip Dimensionlesscompliance Local flexibilitymatrix Stiffnessof elasticsupports J-integral

Stiffness matrix Mass matrix

Chapter 1

Introduction and Literature Survey

Apropeller shaftisaDimportant par tofshippropulsion.Shaf t vibra t ionmon- itoring has beenreceivingincreasing attention in recentyears. Thefailures of shafts due tofatigue cracks makes it imperativ eto have an accurate estimationof shaft natural vibrat ioncharacterist icsin thedesign stage.Vibration monitoring has the greatest pote ntialin crack detection sinceitcanbecarried out withou t dismantling any partof the machine and be done usuallyevenunderoperating condition.

1.1 Liter at ure Su rvey

Fati guecrackingin a shaft isone of themain causesofcatastroph icfailu rewhich isdescribedbyJack and Patterso n (1976). Sincea crackchanges the stiffness that influences thedynamicbehavior ofthe shaft,vibrationmonitoringcouldbeused as ameans ofdetectingcrack initiation and growth. Kolzow(1974) first pointed outthat the vibration monito ring could beusefulindetectingcrackinitiation r.nd.rowtb ,TberefDJ'tIa det aUllld.tudyofthe vibrat iDnalbehavio rof,haftwith

trecsve rsecracks is necessary.

Since themiddle 1970••manyresearchers have realizedthe importanceof this problem.The first workdoneby Dimarogollu (1970)andPaleliu (1914)intro- duced the bending stiffnessdescriptioncf arotorcrack whichis dete rminedfrom compliancem~asurements.Theincorpora tionofthestiffness change causedbya crack intothe equation ofmotion was dealt with in theliteraturebyDimuogo Du (1976).

Gasch (1976, 1993) developedahinge modelfor Lavalrotors(massless shaft).

in which he replacedthe crack mechanismby an additionlLlcrackflexibilityand switched it on and off accordingto whetherthecrack wasclosed oropen.He discoveredthatresonanceswould occurastherotationreach ed

i , i.

ete.,of the shaftbendingfrequencies:

Henry and Okab·Avae (1976)employedthe equationsofmotion witha shalt section interia unequalto that of thecracked shaft, andconcludedthat t.here wouldberesonances due to thecrack when therotational speed equal to

!

^{of the}

fintcriti cal speed whereDis an odd integer.They alsofound thatthevibra- tion response dueto the crack was hard lydet ectablewhen the rotationalspeed exceeded thefirst criticalspeed.

Mayes and Davies (1976)Mayes (1977) performed adet ailedanalyticaland experimental investigat ion ofturb ine shaftswithcracks.They deri ved arough analytical estimationof the crack compliancebased on the energyprinciple.AI·

tbough theyconsideredthenonlinea r equation forasimplerotor, theyobt ained analytical solutions byconsidering anopen crackwhichledto a.shaftwith dis- simila rmoments ofinert iaintwo perpendiculardirection•.

Grabowskiand Mahrenholtz(1982; 1980)arguedthat in a shaft of practical interest the shaftdeflection dueto its twn weight illorders of magnit udegreater than thevibra tionamp lit ude. Ther efore besuggeststhatnon-linear itydoes not affecttheshaftresponse sincetbecrackopens and closes regularly with the rota- tion.

Using the conceptthata tra nsversecrackin a structuralmemberintroduces local flexibility due tothe strain energy concentrationin thevicinityofthe crack tip under load,Dimorogonas and Papadop oulos (1983), Dimorogonas and Paipetis (1983) and Papadopoulosand 'Dimarogo nas( 1987)derivedthe com plete local flexibility matrix of acracked, l\)t ating shaltandverified itexperimental ly. They observed the localflexibility of the shaft due tothecrackand developedan an- alyticalexpression forthecrack'ccel flexibility inrelationtothe crackdepth.

Theyalso showedtheinfluenceofthe crack onthe dynamicreapcnae oftherotor.

Ziebarthand Baumgartner (1981)esta blished their crackmodel on the basis ofdetai led(but quasistat ic)experimentalinvest igation.Theyconsequently for- mulate dthe equations of motion in stationarycoordinatesandapplied themto practica l turbinerotors.Then theycomparedtheanalyticalresultswith the re- sultsof modeltest.As practicalcrackindicators, theysuggested significantpeaks invib rationamplit udes,shiftingofnaturalfrequencies,unstablevib rations,and changes in thedouble-frequency vibratio ncomponent.

Dirrand Schmalhorsts ( 1987) describedthecrack more accuratelythan others bya3-dimensionalfinite elemen t analysisandsuccessfullysimulat edthe vibra- tionsofacrackedtestrotor onthe basis ofmeasuredcrack shapes.

Qian etaJ(1990) derivedtheelement stiffnessmatrixof abeam with acrack

from an integrationof thestress intensityfactorsand thenestablished a finite elementmode(FEM) of a crackedbeam.

Most of the investigatorsconcentrated onthestiffnesschangesdue to a crack, nnd these researchers only considered thecase that the-crackis perpendicular to the axisof shaft.

1.2 Objective

Inthis study,a finite elementmodel is employedto analyzethe dynamic behaviour ofa shafthavinga crack andsupport edon elasticbearings. Through the inves- tigation,some relationsh ipsbetweennat unlfrequenciesof shaftandcrackdepth andstiffnessof eleetlc supports shouldbe found. This work will abo providesome usefulresults for experiment al invest igat ion in thenextstage.

1.3 Methodology

In this study, the firststep istcgiveatheoreticallydescription of the free vibration

o r

^abeam.Furthermo re, afiniteelementmodel is formulated toanalyzethe effect of elastic supports on the dynamicbehaviourof a non-crack shaft and give an approximate evaluationof propellereffect.

Inordertoderive the stiffnesl!'matrixof cracked element,a fracture mechanics approachis usedto studythe effect of the preeeaceofacrack on the dynamic characteristicsof theshaft.

Atlast ,a Fortran-11computer programwas developed.

Chapter 2

St iffn ess Matrix Derivation of Space B eam Element with a Crack

2.1 Introduction

Theelement etiffneeemat rix ofabeamwithacrack was derivedfromaninte- grationofthestressintensi ty factorsendthen afiniteelementmodel(FEM ) ofa cracked beamwasesta blishedbyQianet

ale

^{1990).SekharandPrahhu(1992)}

also presented a similar approach .

2.2 C rack-Tip Stress' Fields for Linear-Elastic Bodies

2.2.1 CrackTipStressInt en sit yFacto rs

Fracturestudiesofstruct ural elementshavebeen revolutionizedinthe recent twenty year s bythe analysisof theirsensitivitytofh.wsor cracklikedefects.

Within these.tudies an essentialingredientisreasonableand proper stress anal- ysisincluding especiallythe flaw withits high local elevationsofstresses from

which fractureprogresses throughvariouscrackpropagationmecbeniemetstress corrosion, fatigue,etc.).

Fullstudies of fracture behavior coverbot hthe stress analysisaspectsandthe material behavior in terms of resistancetothe stresses imposed.The redistri- butionof stressin abody duetotheint roductionof a crackornotchmaybe begun bymethods oflinear-elastic stressanalysis.Of coursethe greatestatten- tion should bepaid totbe highlevel ofetreeseeat orsurroundingthe cracktip which willusuallybe accompaniedbyatleast some plasticityandother non-linear effects.Neverthelesslinear-elasticstressanalysis properlyforms thebasis ofmost current fractu reanalysis foratleast"small scaleyielding" where all substantial non-linearityis confined within alinear-elastic fieldsurroundingthecracktip.

Consequently, the characterand significant parametersof lineer-elaatic cracktip fieldswillhe given firstattention.

The surface of a crack has the dominati ng influenceonthe distributionof stressesnear andaround the crack tip. Otherremote boundaries andloading forcesaffectonlytheintensity of the local stress field at the tip.

The stressfieldsnear cracktips can be dividedinto threebasictypes, each associatedwit ha local modeofdeformati on as illustrated in Figure2.1.(Tadaet al, 1973j

Modell

ModelIl

Flgure 2.1:TheBasic ModesorCrackSurfaceDisplacements .

ModeI is the openins mode which isassociated with local displacementin whichthecracksurfaces move directlyapart(syrr.metril::with respecttothe x- yand x-a planes).ModeIIi. the edge-elidingmode, whichis characterized by displacementsin whichthecrack surfaces slideeveroneanotherperpendicula r totbe leadingedge of the crack (symmet ric with respeet totbe x-y planeand skew-symmetric with respecttothe x-zplane).ModeIIIis tearing mode,finds the crack surfaceslidingwith respectto one anotherparalleltotheleadingedge (skew-symmetri cwith respectto thex-yplane andx-zplane).Tbe superposition ofthese three modes is sufficient todescribethemost general3-dimensionalcase of localcrack-tipdeformation and etreeefields.(Tadaetal,1973)

Themootdirectapproachto determinati onofthe strcSRand displacementfields associated witheachmodefollowsin themannerofIrwin( 1957),based on the methodofWesteI gaard(1939).ModesI and IIcanbe analyzed as 2-dimensional plane-extensionalproblemsofthe theory ofelastici\.ywhich ace subdividedas symmetricand skew-symmetric,respectively,withrespectto thecrackplane.

ModeIIIcanberegardedAIthe 2-dimer.l!lionalpureeheer (ortorsion)problem.

Referringto Figure 2.2for Dotati on,the resultingst ress anddisplacementfields are given below:

Figure2.2; CoordinatesMeasuredfrom the LeadingEdge ofa Crack andthe StressCom ponent!!inthe CrackTip StressField

10

Model Forplane stress

o,=

~cos~,[l - sin~sin~J

+t1dl'+O(rt) (2.1)

{2lrr}f 2 2

(Til

= AcoS _,8

^[1

⁺

^sin_,f)sin:!...,9]

+

O(rt) (2.2) (2'll'r)!f

s , e. e

38 I

'T.,~

=

(27l'r)tcos2'slnicos2"

+

OCr) (2.3)

and for planestrai n (with higher order terms omitted)

Tn =0 (2.5)

Till=0 (2.6)

w=O ('. 9)

Modell

11 For plene etreee

1711

= ..!S.!.!- 1!in~,co3~,cO$~,8 +

O(rt) (2.11) (2J'r )

'l'qI

=

~( KII ^cos_,811_sin-,8Jin!..,O]+O(rt ) (2.12)

21l'r).

andforplanestrain (witb hig herordertermsomitted)

r..~0 ('.14)

TV'=0 (2.15)

w~0 ('.18)

ModeIII Forplaoe 'tr e u

12

.,.~.

""- K11I

t "in~

+'1'",...+O(rl) (2.19)

(2'11"r) 2

T•••

~c:o.t;+O(ri)

_(2'1'r) ^(2.20)

tr:o=O (2.21)

q~=0 (2.22)

q.=O (2.23)

Tq= O (2.2')

11= 0 (2.20)

v=o

^(2.26)

10=

K;'[~Jl"in;

^(2.27)

Equationsfor ModeIand ModeIIhavebeen written(orthe case

o r

plane strain(that is,w= O ) butcanbechangedtoplanestresseMilyby ta.king17.=0 andreplacing Poisson'sratio,II,in the displacements witbanapprop riate value, trl:;r.

13 In equationsfor modesItIIand Ill, higher orderterms such as uniform stresses parallelto the crack,u~andTz zo•and terms cf the crderof square root of r,O(rl), are as indicated.However, normallythese terms are omittedsince as r becomes smallcompared to planar dimensions (in the x-y plane) these higher order terms become negligible comparedto the leadingj;term. Therefore these leading terms are the linear-elasticcracktipSt~9(and displacement)fields.

The parametersKit Ku and KIll in theseequations are called crack tip stress (field)intensityfactorsforthe correspond ingthree modes. SinceKI.KuandKtu are not functions of the coordinates, r and 9, they representthe strength ofthe stress fields surroundingthe cracktip.Alternatelythey may be mathematically viewedas the strengths of the -j;stress singularities at the crac ktip. Their values are determinedby other boundaries of the body and theloads imposed, consequently formulasfor their evaluationcome from a complete stress analysis of a given configuration and loading.

2.2.2 EvaluationofK],](11 andKIllof TheSin gleEdge Notch

FromH. Tada, etal (1913),KI, KuandKIllcan be evaluatedforthe single edge notch specimen byfollowing formulas:

1./{/

Theloading conditionand sizeare shownin Figure2.3

h

h

14

Figure2.3:The Single:Edge Notch Test Specimen Under Tension Load

15

(2.28)

The numerical values ofF(i)canbe calculated by following empiricalFormu- I~.

F(~)~^1.12-0.231(~)

+

_10.55(~)'-21.72(~)'+30.39(~)· ^(2.29) The accuracy is 0.5%for

i

less than0.6.

F(~)~0.265(1-~)'

+

0.857+ 0',65; (2.30)

b b (1-;)

The accurac yis better than 1% for

r

less than 0.2 and 0.5%for

i

greaterthan orequal0.2.

F(!!b>

=

~tanll'a(0.752

+

2.02(i )+0.37(1-8in~)3 (2.31)

1fa 2b

cos;:

The accuracy is better than 0.5%(or any~ For the loading condition shownin Figure2.4

(2.32)

NumericalvaluesofF(i)can be obtained byfollowing empiricalformulae.

- 1- ,

DY~

Figure2.4:The SingleEdgeNotchTestSpecimen UnderBendingLoad 16

17

Theaccuracy is 0.2%for

i

less than orequal 0.6.

F(~b)

=

~tan^lI',a

b(0.923+O.199!! -sin if )4 (2.34)

lI'a cos~~

The accuracy is betterthan0.5%for anyt 2.[( /landK111

Forloading conditionshownin Figure2.5

Fu(!)

=

1.3- O.65(i )+O.37( i )2+O.28(i)3

b

R

a

I""'.'

F1ll{i/=~~

TheaccuracyofFaisbett er than 1% forany

I

Put^{is exact,}

('.35)

('.36)

('.37)

('.38)

18

Q

- 1- ,

Figure2.5:TheSingleEdgeNotch TestSpecimen UnderShearandTorsion Load

19 Forloadingcondition shown inFigur e 2.6

KIll=1'jv;i=iiFru(~) (2.40)

F (~)_ 1.l 22-0.56( V + O.085( i F +O, 1 8(U3 (2.41)

"b -

/i- I

F l/l(i>

⁼

~j:an~

^(2.42)

The eccurecy ofFuis better than2% forany

f

FUJis exact.

- 1- ,

Figure 2.6:TheSingle Edge NotchTest Specimen UnderTOnlion Load 20

21

2.3 Local Flexibility

Consider a shaftwith given stiffness properties,radiusR=D/2,whereDis the diameterofthe shaft,and a transversecrack of depth a ,shown inFigure2.7(a) and (b).The shaftis loaded withaxial.forcePI'shearforcesp~andPa, bend- ing momentsP.and Pr;and torsional momentPs•The dimensionof the local flexibilitymatr ixdependson the numberof degree of freedom,hereit is 6x 6.

H.Tada'i equation(Tada etel,1973) gives theadditionaldispl acement u, due to acrackof depth a, in thei direction,as

Uj

= t& ^fotJ

^{J(a)da (2.43)}

where J(a}is the Strain Energy DeneityFunction (SEDF) andPiisthecorre- spondingload.The SEDFis (Dimarogonaaand Paipetis,1983)

1 6 8 8

J -E'I(~Knl'

+

(~Kml'

+

m(~Kun)'J (2.44) WhereE'=E orE/ (1-1/')forplane stress andplane strain respectiv ely, E isthemodulusofelasticity,m =1

+

II,IIisthe Poissonratio(II=0.3forsteel ) andJ(;jare the Crack StressIntensity Factors(SIF)for thei

=

I,II,IJJmodes and forj=1,2,... ,6,the load index .

22

p, [

;~'.'

~

(b)

Figure2.7: (a) A cracked shaft elementin general loading; (b)the cracksection of the shaft.

23

Thelocalflexibility dueto the crackperunit widthis,by definition ( Dimarogonas and Paipetis,1983 )

au,

C;j

= 8Pj

That is

Cii=fJ;;Pj

[L

^J(A)dA)

or, afterintegratingalongthewidth 2bof the crack,

8'

1 '['

eij

=

8P;oPj[_~JoJ(a)dadz]

(2.45)

(2.46)

(2.47)

The valueof SIFin equation(2.44)arewell knownfromthe literature ( Tada et al,1973)forastrip of unit thickness with atran sverse crack.Sincethe energy densityis a scalar,itis permissibleto integratealongthetip of the crackit beingassumedthat tbe crackdepth is variableand thatthe stressintensityfact or is given forthe elementarystrip. It is knownthatthisapproximationyields acceptable results for engineering accuracy(Dimarogonas and Palpetis,1983 ). Fromreference (Tada et al,1973 )

P,

O'"t=;]i2

(2.48)

(2.49)

(2.50)

0'. =- ,p.

'll'

KI5=0'1,;;:DF2(X)

a . = :Z

^(R2-

.:t2)~

Kn =KI3=K,s=O

Km -0'3.;;:«Fu(X) kp,

(13=,,"IV

Ku.

=

O'm.;'iOFII(I ) 2Pe%

(1fJl1= rR' KIll=Km =K1I4:= K".=O

K,m :=

0'1Viii

Fm(I) kP,

(12=-

,11' KlI/t

=

O'llm..fiOFm(X)

2P,(R2_ %2)1

0"111

= -- .- R'- -

KUII

=

KlIl3=Kim=Km~

=

⁰

24

(2.SI) (2.S2) (2.SJ)

(2.54)

(2.55) (2.56) (2.S7)

(2.58) (2.S9) (2.60) (2.61)

25

where

Herek=6(1+1/)/ (7+6/1)is a shape coefficientfor circularcrosssection, Combiningrelations(2.44),(2.47)and (2.48)-(2.65)yieldsthedimensionless termsof the compliancematrix:

Cl l

= t~:2Cll

^=4-

L a fo'

fFl(h)didy (2.66) (2.67)

(2.68) (2.69)

26

(2.70) (2.71)

(2.72) (2.73)

(2.7<) (2.7')

(2.77)

.ER

j,' r.

'"

^{; -,--v,C3:l}

^="

^{0 DfFJ,(A)dzdy (2.78)}

(2.79)

.ER

j,' r.

'"

^{-,- 2l:n}

^="

gF1Il(~)tUdy (2.80)

-v 0 0

(2.61)

27

c62

~~~:~l =

8

loa l ~iiFllI(h)didii

^(2.82)

(2.83)

c63

;~~:C63

⁼

8 f l

xyFlt(h)dfdg (2.84) (2.85)

cU

:~~ctl6

⁼¹⁶

loa l lA

^{I+mA2]di dy (2.86)}

(2.87) WhereAI

=

f2yFll h),A2=(1-f2)iiFll1(h)andf =xlR,g=ylR,Ii= ylh ,b=bIR.

The dimensionless com pliance matrixit,then.

'"

0 c"⁰

'

0

..

^CIS0 c"⁰

c= '"

_cn⁰0 i'"⁰⁰0

'"

⁰0 ^CH

'

⁰

.. ",

^{Cu0 c"00 12.88)}

'"

^{0 0}

'"

The elementsof this matrixare computedandplottedin Figure2.8.

(b) 28

d".~ ab~~

00.8 008

0.7 07 Cit

0.6 06 _

0.5 0.5 C44

0.4 04

0.3 03

~:i ~ ~ ~~

0.0 00

10-³10-²1O~110° 10¹ J0210³ 10-³10-²10-¹10⁰WI 10²10³ (a)

1. [ll 1.[Zj

~O. aD.

00.8 C31 DO.8

0.1 07

0.6

c. \

^0:6 en

0.5 0.5

0.4 0.4

0.3 0.3

~:~

^C2S

~:~

^C»

0.0 0.0

10-²10-²10-¹100 1QI 10² 1()3 10-³10-²10-¹10010¹ 10² 10'

«)

^(d)

Figure 2.8:Dimensionless compliances versus crack depth.(a )elltcu.C55;(b) Cl4,~4.C45;(c)~e,C38,~;(d)f22,~.

29 Thenthe local flexibility mat rixduetothe crack equations(2.66)-(2.87) and equation(2.88)yields

cuR 0 0 Cl4 Cu

0 C2 2R 0 0 0 C26

CIH=*

^{0 0 ,,,R 0 0}

'.,

(2.89)

C41 0 0 Ctt/R CtsjR 0

'"

⁰ c"^{0 0 CS4/Rcu/R 0}

'"

^{0 0 ... /R}

wherec;j(ij=1,2,..,6)arethe dimensionlesscomplian ce coefficientsand Fa=1fE R²/(1-v^'l) .

Whenneglecti ngtheaxialtrans lationandrotati on,the loca lflexibilit y matrix becomes

(2.90)

2.4 Stiffness Matrix of the Cracked Element

Accordingtothe principleofSaint-Venan t, the stress fieldis affectedonlyin the region adjacent to the crack.Therefore,theelement stiffnessmatrix, exceptfor thecracked element,may beregard ed as unchangedunder a certain limit at ion of element size( Qianet al,1990).The additionalstressenergy of a crack has been studiedthoroughlyin fracture mechanics andthe 8exibilitycoefficient,expressed by a stressintensityfactor,canbe easily derived by means of Castigliano'stheorem in the linear-elastic range.

Consideringa shaft divided intoeleme ntsas shown inFigure 2.9,Thebehavior of theelementssituated to theright ofthecracked elementmay be regarde das

30

external forces appliedtothe cracked element ,whilethebehaviourof elements situatedtoits left mayberegardedas constra.ints(Qi&Jlet aJ,l990iSekhuand Prabhu,1992).Thus,the 8exibility matrixofa crackede~emeDtwithconstra.ints maybecalculated.

31

I I

¹

~ f 'T-rtjj)

~~ I· I

q.

Figure2.9: Simplysupportedsbaft with a.crackedelement

(2.91)

"

Withtheshearingactionneglected,andby usingthestrainenergy, the flexi- bilitycoefficien tsforanelement withou ta crack(seeFigure2.9)canbederived inthe form

C'- 6~1 [2f ~,

_{3t 0}

+

₀

~l

₆

Here EI is the bending,tiffneSli and I is the element length.

Theadditionalflexibilityma.trixdue to thecrackisshewninequation(2.90) Thetotalflexibility mat rixforthecrackedelement isgivenas

(C)⁼(C,]

+

[C,~J From the equilibrium conditions(Figure2.9)

92= -98 /h= -qr+lqe

(2.92)

wherethe transformation matrixIT]is

33

(2.93)

[T)=

-1 0 0 0

o

-1 0 0

o

f -1 0 -I 0 0 -1

1 0 0 0

o

1 0 0

o

^{0 1 0}

o

0 0 1 Sothe st iffness matrixof thecracke deleme nt canbewritt en as

[K,)

=

[T] [C]- '(TjT (2.94)

Chapter 3

Test of the Program to Solve the Beam Vibration(No Crack)

According to the model described in Appendix A,/I,FORTRAN-77program is written. The program flow chartis shown in Figure3.1. To check theprogram, acomparison with the analyticalsolu tionforbeamshavingdifferent boundary condit ions(WeaverandJohnston,1987)ismade.The comparison isshown in Table3.1.

Five cuesare considered.Thesearesimpleeuppc r t,free,fixed, cantileverand proppedbeams whichareshown inFigure 3.2.The beam is divided into Icur elements,each of which bas thesame propertiesE,l,pand A.

Figure 3.1: FlowChart of the Program

35

A _IE ^{• •} A

"'I

(a)

• • •

(b)

~ ^{• •} ~

(e)

~ ^{• •}

(d)

~ ^•

(e)

A

36

Figure 3.2: (a) Simplesupportedj(b) Free; (c) Fixed; (d) Cantilever;(e) Propped

37

Table3.1:ComparisonofNatural Frequencies Structure ModeExact Solution Solutionof the Program

1 2.560E6 2.563E6

Simple 2 4,100E7 4.130E7

3 2.050E8 2.150E8

1 1.316 E7 1.318 E7

F~ 2 9.999 7 1.013E8

3 3.843 3.843

1 1.316E7 1.311 E7

Fixed 2 9.999 E7 1.013E8

3 3.843 E8 4.009E8

1 3.250E5 3.256ES

Cantilever 2 1.276E7 1.279t;7

3 1.001E8 1.016E8

1 6.252E6 6.258E6

Propped 2 6.656E7 6.646E7

3 2.855 8 2.9861:;8

From the Table,itisfound that thereis averygood agreement between enelytlceleolutionandcalculatedresults.

Chapter 4

The Effect of Elastic Supports and Propeller Inertia on the Dynamic Behaviour

Intbe dynamiccalculat ion of a propeller shaft, thebearing supports can be con- sidered as elas ti c supports.The differenceof the stiffness ofbear ings andtheir distributionmayaffectthedynam ic behaviourofshaftgreatly.This isimportant forthe designertooptim izethe alignmentof the shaft.

Figu re4.1showsaone span of shaft with elasticsupports at two ends.The boundarysupports areexpressedbytwo springs intwo directlonspendicular to eachother.Inthe figure,K1andK2representthe stilfnessesof theelastic springs at the two ends.

38

(.)

(I) (2) (3) (4) (5)

(b)

Figure4.1: (a) shaft; (b) Mesh of elements

39

40

Table4.1:FirstThree Frequencies

K w,

w, w ,

rigid a.M01E6O.1033E8O.5375E8 5E12 O.6395E6 O.1027E8 O.5316E8 SEll 0.6329£6 O.9807E8 0.4761ES SElO 0.5066E6 O.6284E7 O.1926E8 5E09O.2617E6 O.1l38E7O.5042E7 5E06 O.3869E5 O.1217E6 O.3463E7 5EO! O.4637E4 O.1l85E5 O.3314E7

When the stiffnesseB ofthe springs atthe two ends oftheshaft arethesame u K,the differentvalueofthe stiffnesseshave great effect onthe naturalbehaviour of theshaft. The resultsare shewninTable4.1.In thecalculation.the shaft is divided intofour elements.Thevalu e of stiffnesses variesfrom finitevalue to a infinite value(rigid).Figure 4.2-Figure4.4showthe curvesbetween thevalueof K andfirstthreenaturalfrequ enciesWI,W2andW3.

41

~o'

^{10' lJ 10'2}

Slillness of Supp ortK(NJm )

Figur e 4.2:thecurve betwee n firstfrequencyand K

'2

I(10'

12 r'----~---~---~~~--~---~

10

c~

.

u,

. .

<J>

10' 10'0 1012

Slillnu s01SupportKIN /5 )

Figure4.3:thecurvebetween seco ndfrequency and K

J10'

·r'--~~~~~~~~~~.---~~~--~

~O'

^{10' 1010 10'1}

Stilln essofSuppor t K(N/m)

10"

Figure 4.4:thecurve bet ween third frequencyandK

44

Table 4.2: First Three Frequencies K1fK2 WI/Well W2!W02 W3!W03 1 1.000 1.000 1.000 2 1.030 1.1143 1.2113 3 1.040 1.1523 1.2965 4 1.0454 1.1711 1.3390 5 1.0482 1.1822 1.3645 6 1.0508 1.1895 1.3806 10 1.05489 1.2040 1.4123

In a practicalengineeri ng problem,the bearings atthetwo ends of theshaft aredifferent .Sotheeffectdue to thedifferentvaluesofsprings on thenatural behaviour shouldheconsidered.In this part,Iusethe valueof~to representthe differencebetweenK1andK2 •Theresultsare shown in Table 4.2.In thetable, WOI, W02andWooarethe first,second and thirdfrequencies respectivelywhenK1 equal to[(2'Figure 4.5-Figure 4.7show theC::Ul"VelIbetween:;-and~,~and

f;,

and,;;; and

t·

.,

1.06r--~--~-~--~-~--~-~--~-~

1.05

1.04

~ 1.03 i

5 6

K1/K2

Figure4.5:the curve between first frequencyand K

10

1.25

1.2

1.15

~

~ 1.1

5 6

K11K2

Figure4.6:the curvebetween second{requen~ya.ndK

••

10

47

1." .--~--~-~--~-~--~-~--~---,

1.35

S

1.25

~

• 1.2

5 5

KlIK2

Figure4.7:the curve between thirdfrequencyand K

10

48

4.3 Effect of The Propeller Inertia

The effectofprop elleriner tiawillbeconsideredin theboundary conditi ons to thepropeller.

ElfJ4~~~,t) +PA 82~:,t)=0 Forasha ftshownin Figure4.8,the boundary conditions become (1)x= O,y= Oand M=O

(2)z='hY=O.~=9~~_

and

(3)x~1

and

Where

(4.1)

Jis themesspolar momen tof iner tia of propell er.

The natur al Crequenciesof followingexample iscarriedout.

the lumped massis32500kg.andlum ped inerti amoment is 16300kgm²•The diameter of shaftis.25m .

The resultsare:

(1)No lum ped mass and inertiamomen t wl=O.190 8xl0⁸(rad j s)

w2=O.8366xlO^ll(rad/ s)

Figure4.8:Diagramofa.tailed shaft

WI=O.4363xlO'(rad/_) (2)Onlyconsiderthe lumpedmaM wl=O.ll38x l()4(rad/ .}

w,= 0,4923xlO'(rad/.}

10]=O.3971xIO^t(rad/ s)

(3) Bothlumped mus&Dd inertiamomentareconsidered wl= O.6604xlo'(rad/l )

w2=O.1376xl 0⁵(rad/s) wl= O.5202xlo' (rad/ l)

4.4 Discussion of The Results

Results of the calculat ion showthat :

49

A.Results of the calculationsshown inFigures 4.2to 4.4 show that fora certain range ofthe values of the bearingstiffness, the natura.!frequenciesof the shaft arevery sensitive to variationsin the bearingstiffness.Within that range the natural frequencies increaserapidly asthe stiffnessincreases. For valuesof bearing stiffness outside that rangethenat uralfrequenciesremain almost un- changed as the stiffness changes. When the bearing stiffness is below a certain range, the bearingbecomes as a "simple"support, while above thatrange,the bearing behaves as "fixed" support.

B. Whenthe stiffnesses ofelasticsupportsatthe twoendsof shaft arenot

1. Withtheincrease of the value of K1/K2 ,the naturalfrequencies also increase.However,the effect on lowermodefrequenciesis less than highermode frequencies.

2. When thevalue ofK,/K2islarger thana certainnumber(forexample, larger than 6 or 7), with the increaseofKl/K2,the natural frequencies have very little change.

C. Considerat ionof theinertia of thepropeller decreasesthe naturalfrequen- cies of the system. From theresults, it can be foundthatthe frequencies will decrease by consideringoflumped mass and inertia moment.

Chapter 5

The Effect of A Crack on the Dynamic Behaviour

5.1 Calcu lation Results

Accordingto the finite element model described in Chapter 4,a.programis written to calculatethe naturaldynamic behaviourof a crackedsbaft.

Whenthe crackis assumedto affectonlystiffness, the naturalfrequencies are obtainedby solvingthe eigenvalueproblem[K]-w³IMI:::O.

Take a one span or beam with a crackat the middleofthe beam. The diameter ofbeam isD,and the depthof crack is a. The meshofelements are shown in Figure5.1

"

ITIv-rTI

~ ~ I I

52

(1) (2) (3) (4) (5)

Figure5.1:(a) shaft witha crack; (b) Mesh ofelements

53

Table5.1:First ThreeFrequenc iesCorrespo ndi ngtoDifferen t CrackDepth

aD w, w, w,

0.0 O.6406E6 0.1033E80.5377E8 0.1 O.6276E6 0.1032E8 0.5371E8 0.2 O.5350E6 O.1029ES O.531OES 0.3 0.41561>6 0.9918E 7 O.3387E8 0.4 O.2935E6 0.1012E8 O.2813E8 0.5 0.1624E6 0.1000E8 0.2365E8

The result sare shown in Table5.1,Figure5.2- Figure5.7. In thetable and figures,WI 'w~andW3are thefirst,second and thirdfrequ enciesrespectively ,Wot.

Wo2andWo3arethe first,secondand thirdfreque nciesrespectivelywhenthe depth orcrack iszero, deltaWItdeltaW2and deltaW3areWl -WohW2-WtnandW3-WQ3.

The first mode shapes cooresponding to differentcrack depthare shown in Figure 5.8.

0.'

0.8

0.7

~O.6

..

0.5.

0.' 0.3

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

alO

Figure5.2:Variations or firatfrequencywithdifferentcrack depth

0.995

0.99

0.975

0,97

55

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

aiD

Figure5.3: Variationsofsecond frequencywithdifferent crackdepth

56

0.'

0.8 ee

!O.7

~ 0.8

0.5

0.4o:--;;-~----;;';--;;-7;---;:':---;;-::---;:':---;;-::---;:'c--_:'c---l0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 aiD

Figure5.4: Variationsofthird frequency withdifferentcrackdepth

57

O.035 r--~-~-~--"---'--~-~--~-~-~

0.03

0.025

!

^0.02

..

Jl

~O.D15

0.01

0.005

O:-~~=:":=---::-7:--:'::---:-=---:~--,-'----'-~--'---.J

_{o 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5}

aiD

Figure 5.5:Variat ionsof normalizedchange in firstfrequencywithcrack depth

58

O.035r--~-~--,.--r---~-~-~--r---~-~

0.03

0.025

0.02

0.015

0.01

e.ccs

0.0 5 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.4 5 0.5

.,0

Figure5.6:Variations of normalized changeinsecondfrequency with crackdepth

59

D.'r--~-~-~--'--"""'-~--~-~-~-~

0.5

0.4

0.3

0.2

0.1

o!---;;-:;----;;'-;-=::::====:':--~:____:':___:c'_:___:'-:---'-:__--,Jo 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 .10

Figure5.7:Variations ofnormalizedchange inthird frequency withcrackdepth

60

a/O.. O.3 a/O.O.4

3.5 alD_a.O

••• •• a/D ..O.1

... a/O=0.2

1.5 2 2.5

Axial Dirsctionof Shal t 0.5

,Q

"

.;";"

.I'~/' .II,

'"

./ I I

.,.;"'"/

,;'

" "

: / ,'I

;-'f.,'

.j"I~

i',;' '"

i""

) 'I

.;'//,/

r, tt.

iI, 'J'II

"

h W v 0.9

la. ..

"' .

-g0.3

"

~O.8 Q

g0.7 i3

~^0.6

~~O.5

..

Figure5.8:First Mode Shapeof Shaft withDifferentCrackDepth

61

5.2 Conclusions

From theresults ,we can getconclusionsas follows:

1.Asexpected, the natural frequencies decrease whenthecrackoccurs,and the maximumamplitudesof the modesha pes become larger.

2. As thecrack depth becomeslarger,theamplit udesofthe mode shapes become large r,and the valuesof nat uralfrequenci esbecome smaller. The general trendof the decreasein naturalfrequencies withthe increasein crack depthis also observed at higher frequencies.

3. When thecrack occursclosetothe middleof theshaft,themax imum ampli tude ofthemodeshape occurs.

Chapter 6

Stiffness Matrix Derivation of Space Beam Element with a Crack Considering the Axis Translation and Rotation

In practical engineering,the shaftisrota ting under the normaloperation at some rotation speed.Thereforeitisnecessary to studythe thecrackeffect on theshaft torsionalvibration.Figure 2.7depicts&typicalcracked shaft in generalloading.

6.1 Local Flexibility

Considera shaft with given stiffness prope rties,radius R=D /2 ,where D is the diameter of the shaft,and a transversecrack of deptha,shown in Figure2.7(a) and(b). The shaft is loaded with axial forceP"s hea rforcesP2andPz,Bendi ng momentP4andPsand torsiona lmomentPs.Thedimension of the localflexibility matrixdependson the number ofdegrees offreedom,here6x6.

From Chapter2,the dimensionless localcompliance matrixisthen.

62

63

'n

₀ ^{0 0 Cl4 Ell 0}

'n

^{0 0 0}

'"

c=

",

^{0 00}

'"

⁰

...

⁰

...

⁰

^"-

^{0 (6.1)}

'"

^{0 0 e..}

...

⁰

0

.., ...

^{0 0}

_'"

The values ofelementsofthis matri xare comp utedaccordingtoequat ion (2.66)·(2.87).

Then thelocalflexibilitymatrixdueto the crackis she wninfollowingequat ion.

cuR 0 0 Cl 4 eu

0 cnR 0 0 0

C ,.

CI",,=

1.

^{0 0 "'R 0 0}

^{e ,.}

(6.2) F,

'"

^{Ell 00 00 "cH./RIR"./RculR 00}

0

,- ,

'"

^{0 0 "'/R}

where CZj(ij

=

1,2,••.•6)are thedimensionlesscompliance coefficients and

Fo=rER"/CI -,,2).

6.2 Stiffness M a t r ix of the Cracked Element

Conside r asbaft dividedinto element sa.lIebowain Figure6.1•

Withthe shearing action neglected,and by usingthestrai n energy, the flexi- bilitycoefficients foranelementwithout ."crack cen be derivedin theform

q.

..

I I ^.l

. 1TT'ITI[o"(f})

~ I I

qn q"

q,

Figure6.1:ShaftwithCrackedElement

(6.3) 65

I _~:& ,

^{0 0}0 ⁰0 ⁰0

&

⁰

Co=

~ ~ t ^Ja -fr ~

o 0

-&1

0

-h

0

O-& 0 0 0 - h

Here EIis thebend ingstiffness, G is torsionalshear modulus,J is torsional inertia momentandI isthe elem ent length .

Theadditionalloca l flexibilitymatrixdue~othe crac kis shownin eque- tion(6.2).

The totalfle x ibilitymatrix forthe cracked element is givenas

[C)=[C,l

+

[0,:]

Fromthe equilibriumconditions(Figure6.1)

(6.4)

Thatis

(Qh Q2,...,qn)T=(TJ(Qr,Q8,..,Qn)T wherethe transformation matrixITIis

-1 0 0 0 0 0

0 -1 0 0 0 0

0 0 -1 0 0 0

0 0 0 -1 0 0

0 0 I 0 -1 0

[TJ =

⁰₁ ^-I₀ ⁰₀ ⁰₀ ⁰₀ ^-1₀

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

So thestiffness matrix ofthecracked elementcan be written8.9

[K.]

=

[TJler ' [Tl^T Whenwithoutcrack

where[Co]-1is

66

(6.5)

(6.6)

(6.7)

67

,~{ ^"W

⁰0

,.,

^{00 ~}~⁰⁰₀⁰

_'" 0 'f'

₀⁰⁰⁰

^"P-

⁰⁰⁰₀

- r]

^(6.8)

- .,,- '¥

Sothesti ffness matrixofelemen t is (KI=(TJlc.r'( T)T=

'f'-

^{0 0 0 0 0}

- 'i'-

^{0 0 0 0} 0

0

"W

^{0 0 0} ~ ⁰ -Ijfl 0 0 0 ~

0 0 1.jfl 0

-'fil

^{0 0} 0

- '1f'

⁰

-'fil

⁰

0 0 0

'f'-

^{0 G 0 0 0}

- "I'

^{0 0}

0 0

-w

⁰

^'¥

^{0 0 0} ~ ⁰

'¥

⁰

0 ~ ^{0 0 0}

'¥

⁰

-w

^{0 0 0}

^'f'

- 'i'-

^{0 0 0 0 0}

'i'-

^{0 0 0 0 0}

0

-...

^nEI 0 0 0 -~ ⁰ ~ ^{0 0 0}

-!51

0 0 -1.jfl 0

,.,

."... _{0 0 0}

_up

₀ OEI-jr- 0

0 0 0

-,

^JG 0 0 0 0 0

lf2

^{0 0}

0 0

..c'fil

0 !H 0 0 0

"Y-

^O ~ ⁰

0 ^-p-OEI 0 0

0 lfl

⁰

- <p

0 0 0

'f'

(6.9) Tbi.isthe lleneraJelemen tItiffneSamatrixorbeamwitho utcrack.

Chapter 7 Conclusions

7.1 Conclusions

The st iffness of elastic supportsof theshaft bee great effect onthe natural be- haviourofthe sh aft. In the case thatthe stiffnesses of th e elaetic supportsat the twoends ofshaft are thesame.(a) withthe increase ofthe stiffness, the natural frequ encies alsoincrease; (b)whenthe stiffnessofthe elastic su pports islerger than a value(which dependsonthe mode),the naturalfrequenciesare almost const antandapproachthe naturalfrequencies whenthe supports are rigid.(e) fora shaft with similar elastic8upp~rts.the naturalfrequenciesvary rapidly when thestiffness iswith inacertain range.This phenomenonshould beconsidered in alignme ntof ashaft. Whenthestiffnesses of elasticsu pports at thetwo ends of shaft are not the same.(a) with the increase ofstiffness differencebetween two supports,thenat ural frequencies also increase;and theeffectonlower mode frequenciesis lessthan higher modefrequencies. (b) whenthestiffness differ- encebetweentwosupp orts isbigenough,the natural frequencieshaveverylittle change.

For a shaft witha crack, the crackeffecton vhe naturalbehavio urof the shaft

68

69 is shown inthefollowing aspects.

I,As expected, the naturalfreq uencies decreaseswhenthecrack occurs,and the maximumamplitud esofthe mode shapesbecome larger .

2, As the crack depthbecomes larger, theamplitudesof themode shapes becomelarger,and the valuesofna tural frequenci esbecome smaller.The gene ral trend of the decreaseinnatural frequencieswith theincreasein crackdepth is also observedathigherfreque ncies.

3, Whenthecrackoccurs closeto the middleofthe shaft,the maximum amplitud eof the mode sha peoccurs.

Inprac tica l eng ineering,measur ing the changes inan adequateDumberof the na t ural frequenci escan be usedtodetect thecrack.Itis importantforanengineer to discover the crackas earlyaspossible and prevent damage of theshaftdue to the presenceofa crack.

7.2 R ecommendations

Thisst udy carries out thecalculat ionresults obtainedby fin iteelementmethod, However,further studiesshould be doneinfollowingtopics:

1.Experimentsshouldbe done inorder tocomparewithcalculation results. 2.In practicalshafts, thecracks may occurinany direction,how the crack affectthedynamic characteristicsshouldbe studiedfurther.

70 REFERENCES

Dim aro gon as , A.D.I1970, "Dynamic Response of CrackedRotors",General ElectricCompany,Schenectady NY

Dimar ogona s,A.D.•1976,"Vibration Engineering",WestPublishers,St.

Paul

Dimar ogonas,A.D.,and Peip efi s,S.A. ,1983, "AnalyticalMethodsinRo- tor Dynamics"IAppliedSciencePublishers

Dimarogonas,A.D.and Papadopoulos,C.A.,1983,"VibrationofCracked Shafts inBending",J.of Sound and Vibration,91,pp 583-593.

Dir r,B. O. an d Schm alh orsts,B.K.,1987. "CrackDepth Analysisof a ro- tating Shaftby Vibration Measurement",RotatingMachineryDynamics, vol 2, 11th BienniulConference on Mechanical Vibrat ionand Noise,Boston, DE vel 2,ASME,pp607-614

Gasch ,R., 1976,"Dynamic Behavior ofa SimpleRotorwith a Cross-sectional crack ",Vibrati onin Rotating Machi nery,PaperNo CI78/76,Institu tion of Mechanical enginee rsConferencePublication,

Gas ch ,R. ,1993,"A Survey ofthe Dynamic Behaviorof a Simple RotatingShaft with a TransverseCrack",J. of Sound and Vibration,160(2), pp 313-332

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Appendices

73

Appendix A

Free Vibration of a Beam

A .I Bending Vibration Equation of a Beam Sub- jected to an Axial Force

For abeam withdifferent boundarycond it ions,the derivationof thevibrat ion eq uat ion is given below.

Considerthefree bodydiagram ofan element of a beam shown in FigureA.I where M(x,t )is the bendingmoment, V(x,t )is the shear force,and((x,t)is the externalforce per unitlength of the beam.

Sincetheinert ia forceacting on the elementof the beamis pA(x )dzW-(z ,t)8'w

74

(A.I)

(a)

f(x,t ),dx

dx

(b)

Figure A.I:(ala beam in bending; (b) free bodydiagram of an elemen t 75

7.

Thentheforceequat ionofmotion inthey direction gives

-(V+

~dZ)+fd%+

^V

+

(P+ dP)Sin(8 +dO) -P$i n8

= PAdX~t~

^(A.2)

Wherepisthemassdensity,A(x)isthecross-sectional areaofthe beamand0 is theangle betweenthe force Pand the x-axi s.Themomentequat ionofmotion aboutapoint0is, (neglectingrotaryinertia)

(M+dM)-(V+ dVl dr+fdr ;:- M : O (A.3)

By writing

dV

= ~d:t

^{and dM}

= ~dX

and neglecting higherorder terms .Equations (A.2)and(A.3)can bewritt en

-

8V~:.t)dX +

fdx

+

(P

+

dP )Si n (8 +de)-PS inO=

PA(X,t)~t~

^ds:(AA)

For small deflection

Sin(6

+

d8);;,8

+

dO= ()

(A.5)

(A

.e )

Fromthe elementary theoryof ben ding ofbeams ,therelati onshipbetwee n bending moment anddeflect ioncanbeexpressedas

M(x ,t )

= EI(X)82~~~, t)

^(A.7)

Where EisYoung'smodulusandI(x)is themomentofinert iaofthe beam crosssectional areaabouttheneutr al axis.Substitutingequation (A.1) int o equa- tion (A.4) and (A.5),we obtain the differential equat ion of motio nfor the forced latera lvib rati onof anonunifor mbeam.

~[EJ(Z:)(j2~~,^t)J+pA(z:)82~~~,t)_p{j2~~~. f)^{=f(z:,t) (A.8)}

77

For the freevibration of a uniformbeam,equation(A.S)reduces to

El4~~~, t)+PA a2~~~. t) _pa2~~~,t)^{=0 (A.9)}

Appendix B

Mass and Stiffness Matrices Derivation of Space Beam Element

Figure B.l( a)depicts a, typical memberiofa.space frame.Eachendofthe memberhas six degrees of freedoms, three translationdegreesand threerota- tionaldegrees. Theprincipalplanes of bending are thez'-y' planeand r'-z' plane. Six numbereddisplacementsindicated at each end ofthe member,consist of translationsand rotationsin the:l',y',Z'direction.With a prismaticmem- ber,the12x 12 stiffnessmatrix for localaxesisccmpceed ofthe following 6x6 submat rices .( Weaver and Johnston, 1987 ).

78

(b)

FigureB.l:Spaceframemember:(a) local directions; (b)globaldirections.

79

80

rti. 0 0 0 0 0

0 121. 0 0 0 6LI.

[K;jJ=~

⁰0 ⁰0 ^121~0 r2L2I~^{0 -6LI~}O. ⁰0

0 0 -6LI~ 0 4L'lI~ 0

0 6LI. 0 0 0 4L2I.

-rll. 0 0 0 0 0

0 -121. 0 0 0 -6LI.

[K~jl = ~

⁰0 ⁰0 ^-12I~0 -r2L2I~^{0 6LI~}0 ⁰0

0 0 -6LI~ 0 2L2I~ 0

0 OLI. 0 0 0 2L2I.

rd. 0 0 0 0 0

0 121. 0 0 0 -OLI.

[K~kl = -ffa

⁰0 ⁰0 ^12I~0 r'lL2I~^{0 6LI~}0 ⁰0

0 0 6LI, 0 4L21, 0

0 -OLI. 0 0 0 4L2I.

(B.1)

(B.2)

(8.3)

Wherepis the mass densityofelement, Aisthe areaofthe cross secti on of beam,Listhelengthofelement ,!...isthe polar momentofinertia of the cross secti on.I~I.are itssecond moment sof area about they'andz'axisrespectiv ely.

raisGI.:/El,l•rlisAL2/1••

For the circular crosssect ion

Dis the diameter ofthe shaft G is the shear modulusof elasticit y

G=2(I~V)

The stiffnessmatrixof element is

81

IK 'J = [K j; K j.]

(B.')

K"i KH

Similarly,the12 )( 12conliatent·m&S$matrixAtfor local directionscontains thefour 6 x 611ubmatrices,

140 0 0 0 0 0

0 156 0 0 0 22L

1

M ;; ) =':";

⁰₀ ⁰0 ¹⁵⁶0 140r;^{0 -22}0^{L 0}0 ^(B.5)

0 0 -22£ 0 'L' 0

0 22L 0 0 0 4L'

7r: 0 0 0 0 0

0 54 0 0 0 13L

rM~jl

⁼

~~~

⁰0 ⁰0 ⁵⁴0 7Or;^{0 -13L}0 ⁰0 ^(B.B)

0 0 13L 0 -3L' 0

0 -13£ 0 0 0 ^_3L1

140 0 0 0 0 0

0 156 0 0 0 -22L

1M;,,]= ~-;:

⁰0 ⁰0 ¹⁵⁶0 14⁰0r: ^22L0 ⁰0 ^(B.7)

0 0 22L .0 'L' 0

0 -22£ 0 0 0 4L'

Wherer; isJ/ A •theradiusof gyrationsquared,Jis the mass polarmoment ofinertia ofshaft per unitlength.

J=",Ir 32 TheCcnslstent.massmatrixM'is

IM 'I=[

M

M j;

lrj

M.. M j. ]

(B.8) Forthe lateral(transverse)vibration of ashaft,itisreasonable to neglect the translationandrot ationin theaxialdirect ion.Therefore thestiffness matrix and consietent.mass matrix of anelementcanbe expressed asfollows: