AN EXPERIMENTAL STUDY OF CRACK DETECTION IN A ROTATING SHAFT

AN EXPERIMENTAL STUDY OF CRACK DETECTION IN A ROTATING SHAFT

BY

CYINGMINGXIAO.B.ENG.

A THESIS SUBMITIED TO TIlE SCHOOLOF GRADUATE STUDIES IN PARTIAL FULFlLLMENT OF TIlE

REQUIREMENTS FOR TIlE DEGREE OF MASTEROF ENGINEERING

FACULTY OF ENGINEERING AND APPLIED SCIENCE MEMORIAL UNIVERSITY OF NEWFOUNDLAND

JULy1995

ST• .x>HN'S NEWFOUNDLAND CANADA

."'.

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Abstract

This thesis investigatedthe dynamic characteristicsof a rotatingshaft by using modal experimen talmethod andan experimentalmodel has beende velopedforcrack detection.

Inthisstudy. animitationcrac k.whichsimulatesthe characteristic ofatransverse crack.in the middle spanofarotatingshaft,is designed and therotating shaftexperimentsetupismade up.Experi ments arecarri ed outindifferentcrack parametersand differentsize rotorsinforward and backward rotation.

It Isfound that thenaturalfrequencyofthe rotating shaftdecreasesand frequencyresponse amplitude increaseswithincreasi ngcrack parameter.The ma ximumdifference ofnatural frequencybetweencracked anduncrucked shaftis14.7%.andthe maximum differenceof frequencyrespon se amplitudewithacrackis4 timesashighasthat without crack.Itisalso foundthat the rotatingshaft inbackward. rotationhas lIlesame vibration responseas thatin forwardrotation.

Theexperimentresults showthatthecriticalspeedof the rotatingshaftis sensitivetocracks.

Therefore.themethod maybeusedinindustrialapplicationfordetectioncrack in rotating shafts.

Acknowledgements

The author wouldliketoappreciate the support and encouragement from individualsin the program of Masterof Engineering.Particular thanks are due:

(a)Dr.M.R.Haddar a,Professorof ocean engineering.for hisguidance and financialsupport throughoutthisprogram.

(b)Dr.J.J.Sharp, Dr.Swamidas. Dr.R.Seshadri.Dr.G.Sabin and Dr.L M.Lyefor their courses and help.

(d)Mr.H.Dye,Mr.A.Bursey and other staff in the Machine Shop andStructure Lab.for their technical servicesand suggestionsin manufacturing thesetup and processingthe experiment.

(e) Finally,his family members, especially his wife, daughter and parents. for their understanding and encouragement.

iii

Contents

Abstract Adalowledceme nts Table orContl:nts Ust orFiCUfts UstorTable.

ListorSymbo ls 1 IntroduCtioD 2 UcemblftReview

2.1AnalyticalApproach. . .

2.2ExperimentalApproadl. . . ... . . ... . .. . . ... ....••••... 2.3Summary•• • • ••• • •.••.•• • • • • • •.••.••.• •••.• • ••••.• • • • l ModlllAaalr sis1beOI7

3.1AnalyticalApproach 3.1.1Theoretical Method

3.1.2FiniteElementMethod.•.•...•... ..•.

iii

vii viii

8 8 8 21 3.1.3FiniteElement Model ofExperimentalSetup. .... . . .... . 31 3.2Experimental Modal Approach .... .. . • .. .. • . • • . .. . .. •... 34 3.2.1Mod.a.I AnalysisTheory ... • .. . . . • ...•... .... ... ).4

3.2.2Experimental Modal AnalysisMethods 3.23 ModalData Acquisition .... 3.2.4 Mo dalParam eter Estimate 3.2.5Modal Data Presentati on•. 3.2.6Summary

.. £SperimeatalSiudy

4.1 Experimentalsetup . 4.2Transducer.

4.3lnstrument 4.4Procedur e . 4.5Data Analysis

4.5.1Freq uency 4.5.2Damp ing

!IiIUsuiaandDiscussioD S.1Static Deflection..

5.2Unc raclcedShaft .

S.3ended Shaft .

5.4 Forward and Back ward Rotation . S.SComparison betw een Uncracked and Cracked Shaft . S.6 Summary

6 Conclusion

40 42 46 47 47

4 .

49 SO

sa

63 69 ... . ... 69 69 74 74 77 71 . . . ... .. 78 78 80 81 88

List of Figures

3.1 Ca>Uniformsimplysupportedbeamwith massincentre...

(b)Uniform simply supported beam (e) Movementandshearforce 3.2 Element. node and basic element 3.3 Deflection of ashaft... 3.4 Experi mentalsetup

3.5 (a)Simplification of experimentalsetup (b)FEM modelof experimental setup .• ..

3.6 Timeand frequency domain .

3.7 DiscreteFourier transform concept .

10 24 25 32 33 33 J8 44 . .... ... ... .. . 52 53 54 Prin ciple of theproxi mityprobeinoperati on . 4.4 Experimental setup andinstrum ent... . . .•. . . . •. 61 4.1 Natural «)Ordinatey-z and rotating coordinate'11-~

4.2 Simulation crackwithvariable depth .... ... . . . .. . . . •• • • •.

4.3

4.5 Mountedproximity probes

4.6 (a) Size of experi men talsetup .

(b)Rotor .

(e)Disk and Oange . 4.7 Bandwidthmethod .

62 66 67 68 70 4.8 Rotation speed and frequencyVIamplitude(e-O.case 1) ...•.••••••• 71

vi 4.9 Rotationspeedandfrequenq vs amplitude(C-O,case2) 72

4.10 Amplitud e vsrelativefreq uency(case 1). 13

4.11Amplitude\ISrelativefrequency(case 2) 73

5.1 Verticaldeflectionvsphase angle ... 76

S.2 Horizontal deflection vspbase angle 76

5.3 Speed vs amplitud e(C""O.OO83) •.... . 83

5.4 SpeedY$amplitude(e-O.1833) •.•• . 83

5.5 CalSpeed\ISamplitude (c:O.2041) .. ... . . • . . . .. 84 (b)Speed vsamplitu de (backward rotation. c-O.2041)... ... 84

5.6 SpeedV$amplitude (c-Q.3333) •.•. . ... 85

5.7 Ca>Speed vsamplitude(c=O.0083.case2).. 86

(b) Speedvsamplitu de (e-O.1833,case 2) 86

5.8 Speed vsamplitude (e:-O.2041. case2) 87

5.9 Speed vsamplitude ( c:=O.33J3.case2)... ..•. . ... ... • . .. 87 5.tO Comparisonof experimental results (case1) ..•••.. . . . .. .... . .. 88 5.11 Comp arison of experimental results (ease 2) . . . __. . ... .... . 89

5.12 Natural frequency vscrackparameter 90

5.13 Amplitud e vs creekparameter .... 91

vii

List of Tables

4.1 (a) Characteristics of accelerometer [B&:K 4378]

(b) Characteristicsof accelerometer[B&:K 4379]

4.2 Characteristicsofproximityprobe.. 4.3 List of instrument . 4.4 Crack parameter .

55 56 57 . .. ..•. . .... 60 63

n

79 5.1 Natural frequen cyof the shaft system .

5.2 Resultscomparison (easeI) .

5.3 Results comparison(ease2) 80

List of symbols

x, Y.z-natural coordinates y

,11.

~

•

rotating coordinates t -timein seconds g - gravity acceleration.m1s2 E -elastic modulus.N/m2 p -mass density,kg/m 3 m-mass, kg/m2

Iz - moment of inertia about z-axis A -cross-sectional area, m2 I-length,m

Co) -natural frequency,Hz

e -strain o • stress

~- damping factor [M] -n x n mass matrix [C]-n x n damping matrix [KJ •n x n stiffness matrix {x(t)} - n x1 vector displacements

viii

{f(tH - nK1vector of forces b(t) - impulse-response function H((0) -frequency response function R(s) • transfer function F«(o) •forcing function

"J,qr - residue

~qr - complex:conjugateof~qr At •systempole

At· -complex:conjugateof "'r N • blockslee (power of 2) T-N~t

4t •time spacing 4£ - frequencyspacing k - O.I.2•...N·[

f_k•kU

c - crack parameter d •damping factor W -thickness of disk

ix

Chapter 1 Introduction

Arotatingshaftisanimport an tpaninrotatingmachinery.suchas theshaftina gene rato r.Duetomanufacturingflaws orcyclic loadin g.cracksfrequ entl yde velopin a rot a ting shaft.Ifcrackspropagatein theshaft. catastrophicfailurewilloccur.Theref ore . wetryto detectthese cracksas earlyas possible.

Thereare differentmethods10inspect creeks.Moda l analysis method s.as oneof non- destructive inspec tion(NO I)methods.hasrecentlybecomeofgre aterimpona nce. InNDI meth ods. ultrasonicwave.X-ray.magnetic-particle. permeationand electric potent ial differe nce CIC.canbe:usedfordetecting crackunde rSialiccond ition.Theacoustic - emissionmethod(Ivanov.(19 84 ).Simmons et 01..[19 84 J and Grabamctal.,(19 82))can beusedforinspect ing cracksonrotatingshafts.Yeta setof specificequipmen t isneed ed (Zhao.M.andLuc,Z. H.•(1989)).Modalanalysis methodhas theadvantagesof simp licityininstru mentationand convenience intesting.Thus.the method iswidely em p loye d in crackdetection inrotating machin ery.Modal analysis isdefinedasthe process of characterizing mod alparameters (mod e shapes. damping fact ors and frequc ncies ) in asystemeitherthrough analytical or experimental approach.The

experimenta lmodalapproachwillbeusedintheexperiment mentionedin thisthesis.

Dueto crac ks. natural frequenciesdec rease.mode shape salterdependingonthe damage mag nitude,unitresponseamplitude smal"increase ordecrease depe nding on the positionofcrack. anddamping alsochanges.Bycomparin gthe diffe re ncesofmodal parameters betwee na crackedand an uncrackedstructure.wecandetect cracks in the testedstructure.

Thepurpose of this study is10carryoutexperimental modal analysisontheunc rac ked and crackedrotat ing shafts supported onelasticbearings and toinvestigatethe relationsh ipbetweennatural frequenciesandcrackparameter,This studywill alsoprovide someuse fu lresultsfor crackdetect ioninindustrialapplications.

In thisstudy.an experimentalsetup forsimulating a crackinarotat ing shaft has been designed.Disp laceme ntandaccele rationsensorsarefixedforcollectingvibration respo nds.Modalparame ters wereestimatedand used to formulaterelationship s among modalparameters:namely.natu ralfrequenciesand dampingfactors.crac k parametersand rotating speed,

Cbapter2

Literature Review

Fatigue cracking in a rotating shaftis one of the main reasons of failure.

Craa

detection inrotating shaftshas receivedmuch an ention since middle1970's.Since a crack influences the stiffness of&shaftand the stiffness influencesthe dynami c behavio ur of theshaft.,modalanalysis orvibration monitoringcan be employedtodetect crack's initiationand growth.

2.1 Analytical Approach

Kolzow (1974]firstpointed. out that the vibrati o n monitoringcouldbe usefulwayin detecting cracksand Shatoshowed the same ideaalittle later (Wauer[1990

an.

The work donebyDimaroaonas[1970] andPatelias(l974 ] introducedthebending stiffness descriptionof a rotor crack whichis determined fromcompliance measurements.The incorporation ofthestiffnesschange causedby •crack into theequation of motionhas been dealtwithin sev.raI papers:Gasch[1976J.Henry[1976J• Mayes [1976Jand

Dimarogonas[1976 J.

Gasch[1976J developed a hingemodel f'orLavalrotors whichis a simplysupponed., massless shaft carryin, a rilid disk inthe middle.inwhich hereplacedtheendc:

mech anismbyan additionalflexib ility andswitchedit on andoffaccordingtowhether theen d::was closed oropen.Hediscov ere dthat resonanceswould occur as the ratarional speedreach ed112,113,ete.,of' theshaft bendingfrequencies..

Henry andOkah- Ava e [1976]employedthe equati onsof'motionwith ashaft section inertia unequalto thatof the crackedshaft. and conclude dthatthere wouldberesonances dueto che crack when the rotationalspeed is equal tolInof thefirst critical speed. They also found chat thevibration response due tothecrackwas hardly detectable whenthe rotationalspeedexceededthefimcritical speed..

MayesandDavi es[1976Jperformed adetailed analytical investigatio nofa turbine shaftwith crack s.Theyderivedaroug h analytical estimation of'thecrack compliance based onenergy principle.Althoughthey consideredthe non-li nearequatio nfera simple rotor.theyobtai ned analytical solution by consideringanopen crack which led toashaft withdissimilarmoments of'inerti aintwoperpendieuJar directions.

GrabowskietaI[1912] argued that in a shaft ofpraaital interest the shaft deflection duetoits ownweightis orders of magnitude grea1er than the vibntionamplitude.

Therefore.hesuggested chatnon-lineari ty doesnoCaffecttheshaftresponse sinc ethe crack opensand closes regularlywith the rotation.

Usingche concep tchat a transverse crackinastructuralmemb er introd uces local flexib ilitydue to thestrain energyconcentration inthevicinity of thecrac:k tipWlder

load.Dimorogonas[1983] and Papadopoulo seeat[1987)derived the complete local flexibilitymatrixof a erecked,rotating shaft They observed the loul flexibility of the shaftand developed an analytical expressionfor the effect of the crack on the dynamic response of the rotor.

Oirr and SchmalhofSt [1987] described the crack more accurately than ethers by a 3- dimensional finiteelement analysisandsuccessfullysimulated the vibrationof. cracked testrotor onthe basis of measu red crack shapes.

Qian et al.[1990] derivedthe element stiffn ess matrixof a beam withacraclc from an integration of stress intensity factorsand then established a finiteelement model of a cracked beam.

Wauer [1990b) and CollinsetaI.[199 1] developeda beam-likerotating substructure inwhich,the stiffness and dampingprope rtiesofasingle crack areaccurately modelled.

Theyalso presented arotatingTimoshenkoshaftwitha single transversecrack in open and closesituation. The governingequation of motionisderived usingthe principles of fracture mechanicsand Galerkin'smethod.The equationsare nonlinear when the crack is open and linear when the crackis closed.When thereis atransition from an open crack toaclosed crack,orviceversa, some of thestiffness and dampingcoefficientschange. The analyticalapproach hasadvantagesforsimulating the dynamical behaviour of a cracked and rotating shaft However,the approach requires enormous amounts of computational time and effort. Furthermore, itis not easy to obtain an accurate measure of the cnck effects usingan analyticalapproach.Besides,.the validityof the results obtained by the analytical approach can be appropriatelyassessedonlybyexperimental

stUdy.Therefore.experimental approach hasbeen widely usedinthe investigation of the addressedproblems.

2.2 Experimental Approach

Mayes and navies[1976]introduced some simplification .especially in relation to time- periodicstep functionand showed the firstexp erim entalresults onthe dynamicsof a cracked shaft

Ziebal'1h and Baumgartner [198 I}estab lished theircrack:model on thebasis of detailed experimentalinvestigati on.They consequen dyformulated the equations of motionin stationarycoordinates andappliedthem to practic:al turbine rotors..They.then compared the anaJytic:al result with the resultsofamodel test. As practical indicators. they suggested significantpeaks invibration amplitudes.,shifting ofnatural frequencies.

unstab le vibrationsandchanges in dccble-Irequencyvibrati oncomponent ImameeaI.[19SS} introduceda HistogramsSignature Analysis Techniqueto reduce the background noiseand eliminatethe harmonic:s whichexistin an unereeked case.

Basedona J.<fimensionalfiniteelementcrack:model and an on-linemodel,aprototype on-line rotorcrack.monitoring system was developedand experimentally tested. The systemwuinstalled and has been operating on aturbin~8eneruorsince 1989.

Wen etal.[1992)tesled a simpleexperimentalmodel systemin different erack paramet ets. He found. that the principal critical speedwilldecreasewiththeincreaseof crack:depth andthere isunstabl e regionin the vicinity of theprincipal eritic:aJspeed. His experimentalresults arein agreementwiththatobtai nedbyperturbation and numerical

methods.

Tamuraetal.(1988],and Liao and Gasch(1992Jbuilltheirexperimentalsetup co simulate the traCk's openineand dosin e.Tamurinvestigat ed theregion of unstable vibrationof a rotor with•transverse crackandfo undthat there is an instability near the rotationalspeedof2IJof critiw speed in luec crack case.Liao measured accelerations ofthe system inst ead of displacementsin the supercritical speedrange.

2.3 Summ ary

Previousstudiesrevi ewed abovehaveinvestiaated thedynam ic chara cteristi csofshafts instaticand rotationalstate.Most of these studies havestudied the changesin natural frequencies.mode shapesand dampingratiosof theshaft due co cracks using analytical method.The study presentedin this thesisfocu seson the changes of natural frequencies using an experimental method.

Chapter 3

Modal Anal ysis Theory

Modalanalysisis a process ofdetermining medal parameters of a systemthrough analyticalor experimental approach.Analytical approach is accomplishedbyusing theoreticalmethodorFinite ElementMethod (FEM).The modal parameters.suchas frequencies and modeshapes canbe predict edusing theseapproaches.Inthe subsequent sections, bothmethods,theoreticaland finiteetemenrmethods.for modal analytical approach.willbe reviewed, thenmodalexperimentalapproach will be addressed.

3.1 Analytical Appro ach

3.1.1TheoreticalMctbod

Inthisstudy,a shaft systemis simplifiedas anuniform simplysuppo nc dbeam,with mass in thecentre asshown inFigurel.ILInorderto show thoanalytical approach simply.at first, we consider onlythe uniformsimply supportedbeunshown inFiaure 3.1b(ShobUl" 1990).

Figure3.1a:Uniform simplysupported beamwithmassincentre

Figure 3.1b:Uniformsimply S\,Ipportedbeam

y

,) _~~ ^{• X}

F(x,t) 6 x

jill

₎

_{~H ~~}

_6X

M ~r I ^{v + ss.} _ax ^{6 x}

- 8x-

Figure3.1c:Momentsand shear forces

10

11 a) Partial Dift"ereadalEquation

Inorderto determin ethe differen tial equationfor thetransversevibntionin theshaft or beam.We consider aninfinitesimal volumeat •~cexfromthe endof the shaft as showuinFipro3.1c. The lenith ofthisinfinitesimal volumeisassumed mbeb.Let v,

v.

M.F(x"t)denotes the transversedisplacement of the beam,. the shear force.

bendinl moment, and the loading per unitlength oftheshaft. respectively. Negledina the rotaryin erti a.thesum ofthemomen ts about the leftend of thesection yields

(3.1)

Takingthe limitISOx approacheszero, the precedingequationleadsto

(3.2)

The dyn amic equilibri um conditionfor the transverse vibrationof theshaftisobtai ned byapplyingNewton 's secondlaw as

(3.3)

Equation3.3can be rewrittenafter simplifica tionas

12

(3.4)

Substituting Equation 3.2intoEquation 3.4yields

The mo ment can be eliminatedfrom thisequation byusingmemoment displacement relationship.To this end,mo ment.M=EIzY-,issubstituted into Equation3.S.This leads to

(3.6)

IfE and~are assumedtobeconstant. Equation3.6becomes

(3.7)

Inthe cue of free vibration,F(x,t )•0 and accordingly

(3.8)

wherecis a constan t defined as

13

(l.9)

b) SeparadOd of Variahlu

Equation3.8is • fourth-order partial differential equation that governs the free transverse vibration of the shaft. The solution of this equation can be obtainedbyusins:the tedmiqueof theseparatio n of variables.Inthis case, we assume a solutionin the Conn

(l.IO) where +(x)is a space--dependent function,and q(t)is•functionthatdependsonly on time.Equation 3.10leadsto

(l.II)

(l.12)

Substituting these equationinto Equationl.lO,we obtain

(l.l3 ) which implies that

(l.l4)

14 whereColisaconstant10be determined.Equation3.14 Iuds 10 thefollowingtwo equations:

(3.IS )

Thesolutionof Equation 3.15is givenby

q-Slsm(,)t+ Broswt forEquation3.16,weassume a solution intheform

t-A.e"'"' Substitutingthisassumed solutioninto Equation3.16yields

(3.16)

(3.17)

(3.18)

whichcan bewrittenas

(J.19) where

(J.20)

Therootsor Equation3.19arc

wherei~- l.therefore,the seneraJ solution of Equation 3.16canbewritten as

(3.21)

whichcan berewritten&5

(J.22)

where AI-(AS +A6JI2.

A3-(A.~A1)12·

A2-(A••A

s)12

A4-(AS+iA7)f2

Equation 3,22canthenberewritten.usinl Euler'sformula ofthecomplexvariables.

(3.23)

16 Substituting3. 11and 3.23 into Equation3.10yields

*).(A,sinh'JZ.A,.<:osh~".A,sin~"A,<:os1l.<)x(Blsin"'t.B,cos"'t) ().24) q Bouudary Condidops

The naturalfrequenciesDCtheshaft.dependon theboundary condition.Intheease.the shaftissimplysupportedatbothendsandtheboundaryconditions are

v(0,t) -0,

y(I,t)- O.

whichimply that ol>(0 )-0, ol>(I)- 0,

v"(O,I)-O v"(I,t)-0

Itis clear thacinthis case,there are two geometricboundary conditionsthatspecify the displacementsat thetwoendsof the shaftand therearc twonaturalboundary conditio nsthat specify the momentsatthe endsoelheshaft..Substitutin g these condition s into Equation3.23 yields

A6+AS⁼0, A6- AS-0

A,sinh~I.A,.<:osh~I-A,sin~I-A,conII"() A,sinh~I.A,.<:osh~I.A""~I.A,conII"()

(3.2l)

(3.26)

11

TherootsofEquation3.26 are

(3.21)

where n-I,2, 3,...

Therefore,weget thenaturalfrequencies

(3.28)

andthe corresponding models ofvibration are

(3.29) When 0=1.wecan rewri teEquation (3.29) as

(3.30)

whereAOis a constant,

The solution for the free vibration of the simply supported shaft can thenbe written

>'(%;)-E

.-1

~.q.-E(C"smwrD.<aSw.t)si.n".r

.-1

^{(3.3 1)}

Whena mass is addedat the centre of the shaft shownill Figure 3.1a.Rayleigh's method canbeused to determinethefundamen talnaturalfrequencyofthesystem.The sum of the kineticenergyandstrain metBY'of. modelremainsconstantand equaltothe

18 maximumkinetic energy.Consequently.theconservationofenergy leadsto

(3032)

whereT· and U are.respectively.themaximu mkinetic and strain enerKies.

The strain energyis

(3033)

where Q.C.BandIare. respectively,the mess, main, volumeandlength oCthe shaft.

UsingHooke'slaw,the stressis given

o-eE ^(3.34)

(3035)

where¢l{x)islateral deflectionofsim ply supportedshaft shown in Equation(3.30).

Substituti ng0andItinto Equation(3033),we have

(3.36)

where I-fyI dA.A is areaofcross section.

Therefore.the strain energy is

(3.37)

19 The kinetic: energy is

(3.38 )

whereTsand Tmis. respedive.ly,the kineticenergy olthe shaft and ithaddedmass..

(3.39)

where"is the massdensity.

(3.40)

where mi is thejthaddedmass and«xi)is thevalueofthe amplitu deat thelocationof

(3.4 1)

Equating themaxium strainenergytothe totalmaxium kineticenergy of theshaft and added

masses.

we have

the equation of freq ue ncybecomes

,

Elf(~H"lJ'd<

y . 0

.e i +'C<)dz.i; ^m#f,%J

g . J-l

Ulin. Equa!ion (3.30) • the terms in Equation (3.43)becomes

10

lUll

Because there is only one mass added in the centre or the shaftinthecase.we have

Substitvtina theseIe.nnS.Equation (3.43)becomes

"."

21

(3.44)

wherern,is shaft'smass. m isaddedmassand

k "M

^{1· -2mm,}

3.1.2 FiniteElementMethod

(3.4S)

Generally, thesolutio nof the vibration problems ofcontinuoussystemsis based onthe assumption that thedeforma tion ofthesystem canbedescribedbyasetofassumed functi ons (Bidkfo rd,1990 ).Byusingthisapproac h. thevibrationof the continuou s system which has an infinitenumberof degrees of freedomisdescribedbya finite numberof ordinary differe ntial equations.This approachcanbeusedinthecaseofgeometrical shape structure elemen ts.Ho we ver, ina largescalesystemwithco mple xshapestructure.

difficultiesmaybeencountered in defining theassumedshape function. Inorderto solvethese problems, FiniteElemen tMeth od(FEM)hasbeenwidelyused inthedynami c analysis.InFEM.atfirst.thestructure is dividedto relativelysmall

22 regionseelled elements which are rigidJyinten::onnecredIJsele<:tednodalpoints.The deformationwithineachelement can thenbedescri be dbyinterpo latingpolyno mials.The coefficients of' thesepolyno mialsaredefined intennsof physical coordinatescalledthe element nodalcoordinatesthat describethe displacem ents andslopes ofselected nodal points on theelement Therefore" the disp lacem ent ofthe element can be expressed by usin.the separation ofvariablesastheproduaof splU'-dependent MetiORS andtime-- dependent nodalcoordinates.Byusing the connectivity between elements.the assumed disp lacement field canbe written intermsofthe elementshape functionandnodal coordinates of the Sb"Uetureor shaft. Usingtheassumed displacementfield. the kineti c:IUId strain energy of each element canbedeveloped. thus definingthe finiteelement mass and stiffness matrix.The energy expressions oftheshaft can be obtainedbysummingthe energyexpressio nsof itselements (Shah an a,1990).In the case of determ iningmodal parameters.such as frequencies.whenwe getthemass andstiffness matrices.itis easy tocalculate the natura!frequencies by FEM.~canbe consideredtoconsistof four steps:discretization,interpolation,elemental fonnulation ,and assemblyandsolution of thefiniteelementequations.

(1) Discfttizatioll

This isthefirststepinthe finiteelementpro ced ure. where thebodyunder examination isdividedinto elementsin suchawaythatthe W'Iknownfield variableis adequately represented throughthebody.Theshaftis dividedinto 11 elementsISshown inFigure 3.2.

(Z)IIllerpo lnoa

23 The finiteelementmethod works by assumingagivendistributionoftheunknown variables through each elements.Theequationsof defining theapproximating distribution is called interpola tion.andcan takeanymathematical form.(0pra ctice.itis usually polynomials.Polynomialsarepopularbecause theyarceasy to formulate andco mpute.

andinparticular their differentiationand integratio nis straightforwardto implement on theco mputer.

Consider a section of theshaft showninFigure 3.3.There is no deformationalong the shaft axis,and planes normal[0the axis before loading remain norm al after deformation.

Theslo pe of the shaft at anysectionisgivenbyav/dx,and thedisplacementin the x direction is found from

U'-Y~

Therefore. the bendingstrain inthe shaftis

(3.46)

(3.47)

Element

t 2 3 4 5 6 7 8 9 10 11

Node

)~!

x

Vj

Figure3.2:Element.node and basic element

+

24

y

x

2S

Fiaure3.3:Deflectionor.shaft

26

TaJrinaone elementshowninFigure 3.2.itisdeveloped with twonodes.and requires a cubic interpolationfunction fortworeasons:Fimly. the element basfoUlboundary c:onditions (the four depees of freedom).whichdemands foW' coefficients in the

~proximatinlf\lnaion..hiaddition.however,toenswethai:tho element satisfies the necessary continuity conditions.thai: isdeflectionand slope continuityat the nodes., the interpolation function must be of third order.The function is givenby

and it must satisfythe boundary conditions:

&lX-O. v-vj and e- iJvl&· oi'

at,,-!.. V-Vj ande-avl&c. -Oj.

The resultingfout equations can easilybesolved to yield

•3("["J _(28,.8)

.. L' L

(3.48)

(l.49)

When equations 3.49are substituted back into the original interpolationfunction,

27 equation 3.48can berearran gedand expresse d as

(3.>0) (3) I1emeabll FormuiatioD

By means of'the shape function,we can get tho element mass and stiffitess matrix.

The basic method of finiteelement formulation isRjuand Galerkinmethod. Galerkin method attacks theweak fonn(weightedResidual)altho differential equation directlyand Ritzmethoduses the energy orfunctional(calculusofvariation) associatedwiththe differentialequation asthebasisforthedevelop mentof the finiteelementmodel.Inthis study.Ritz energy methodis employed.

The kineticenergy ofthe element e isdefined as

(3.H)

whereVandpare the volume and massdensityofthe element and

(3.>2)

(3.53)

Thus.equation3.51canberewrittenas

(3.54)

Furthermore. we get tho element mass matrix.

The strain energy is givenby

wherea '" Exc

Substituting equation 3.47into 3.56.we have

where I '"

I

y2 dA,and the strain energyis

From equation 3.$2we can get

Substituting equation 3.59into equation 3.5S,

28

(3.55)

(3.56)

(3.57)

(3.58)

(3.59)

29

(3.6 0 )

Minimization of the potentialenergythengives thenow familiarformofthe stiffness matrix

I

[K'!. ![N'1[D][N'1^Td;c

whcre [ D]=E I

(4)Assembl y andSolutio nor theFiniteEle mentEquations

(3.61)

Byusingshape function,clement nodal coordinates can be transfonn aed natural coordinate systembecause complex integrationcanbetransformedas simple numerical integration.suchas Gauss quadrature whichcan getahighdegree of accuracy for approac hing the solution.under the natural coordinate system.

Assemblytheseelement matrix,wehave globalmass andstiffnessmatrix

(MJ-E(MJ'

.

rl

(3.62)

Thefree undampedvibratio nis

[K]=~[K]'

[Mlvl'+[K}V=O

30

(3 .63)

(3.64 )

Asinthecaseofmulti-degreeoffreedomsyste ms we assume asolution to Equation 3.64intheform

(3.65)

whereB is thevecto r of amplitudes.tistime.00 isthe frequency, and$is the phase angle.

SubstitutingEquation3.65 into Equation3.64

{[K]-,,'lMlIB=O (3.66 )

Fo rthis systemofequations to havea notrivial solution. thede tcnninan t of the coefficientmatrixmustbeequalto zero, thatis

[K]-,,'lMl=O (3.6 7 )

Sofar.weget theequation 3.67and knowmatrix[K] and[M]. When the boundary conditionsaresubstitutedinto theequations.themodalparameters,such as frequenc ies.

can be caculated.

31 Asmentioned earlier.finiteelement method is ananalyticaltool whic hisbeingused topred ictthe modal parameters.In thepresentedstudy.thegeneral purposefinite elementprogram ABAQUS. VersionS.4.developedbyHibbin.Karl sso n&Soren sen.Inc.

isused foranalyzing modalparameters.

3. 1.3 Finite ElementModel of the Experiment Setup

The experimentsetupinthe study,whichwillbedescribeddewilyinchapte r 4,shown inFigure3.4.consist sof auniform simply suppo rtedshaftand a rotorin the centre of theshaft.

Theshaftcanbedesc ribedby beamele me nts in ABAQUS sinc ethis is a effectiveand easy way forFEM analysis.andtherotor can be simplified asamassinthe shafc center.

Figure3.5 illustrates the FEM modelof the experimentsetupinthestud>:in which the shaft is dividedinto 11elementsby23 nodes.Acoord ingtothemodel.aprogram ron inABAQUSiscompliled.The result of caculation is compared andshown in chapter S.

••••••11' •• • •••

~ .

,=;:;~

_{"n. I ..}

j~

Figure3.4:Experimental setup

1t- ^-& _II

I

_I

----. ^{.u. -}

^{t -}

~ ..

Fi&\lre3.5a: Simplification of experimental setup

JJ

21 23

®

figure3.5b:FE.\t mo delofexperimentalsetup

34

3.2 Ex pe rime ntal Mo dal Ap proac h

Experimentalmodal approachisthe:processof determiningthe modalparameters ofa linear.time-invarian tsystem.One of the commo nrea.sonsofexperimentalmodalapproac h isto validatethe resultofthe analyticalappro ach.Ifan analyticalmodeldoes not exist.

themodal para metersdeterminedexperimentally serveasthe modelforfutureevalua tio ns suchas structural modificat ions.Predominantly. experimentalmodalapproac hisused to explai nadynam ics proble m.vibration oracoustic.thatis not obviousfro m intuitio n.

analyticalmodels, or similar experience(Al bert, 1993).

The process of determiningmodal parametersfrom experimentaldata involvesthe followi ngphases: Modalanalysistheory;Experimentalmodal analysismethod;Modal data acquisitio nand Moda lparameter estimation.

3.2.1 ModalAnalysisTheory

Moda lanalysis theorydealswiththedynami cs ofa structure system.But, die syste m satisfies threeassumptio ns:thefirst.the systemis linearandthat its dynamicresponse can be representedby a seriesofsecondorder differential equations.The second. the systemis time-invariant.whichmeansthatthesyste mparameterssuch as theequivalent mass.stiffnessand damp ing ratio are constantsinstea doffunctions of time.The las t.the systemfollows Maxwell-Betti's reciprocalrela tio nsh ips.

Based ontheaboveassumptio ns.thedynamicsofthesystemis described bytransform relationshi pbetwee n different domains.whic h aretime.freque ncy (Fourier).andLaplace domain.Inorderto understan dmodalanalysistheory easy.the three domains ofasystem

3S arebriefl yreviewed.

(1) TIme Domai n(ImpulseResponseFunc tion )

The generalmathematicalrepresentation ofa single degreeof freedom(SOOF) system is expressed

(3.68 )

whereMisthemass.

Fromdifferentialequationtheory,thetransient responseofthe SDOFsystemtoa tran sient forceinConn of atheore ticalimpulse.can be assumedtobein thefollowing form:

(3.69)

The characteristicfrequencies inthissolution.).,and)., aredeterm ined fromthe differentialequationof Equation 3.68.This yields charac teristic frequenciesof the foll owingform:

(3.70)

Thetransientrespo nseof SDOFcanbedetermined from Equation3.68.assumingthat theinitialconditions acezeroandthatthesystemexcitatio n(I)isaunitimpulse.The responseofthesyste mX(l)tosuch a unitimpulse isknown asthe Impulse-respo nse functionh(t)of the syste m.Therefore:

36

(3.1 1)

(3.12)

where:ar=dampingfactor;Wr=dampednaturalfrequency.

(1:) F~quenc)'Domain (Fftquenl:y. Res ponseFunc tio n)

Equation3.68isthe time-domainrepresentationof the system.Anequivale nteq uationof motionmaybedetermined forthe Fourier of frequency(OJ)domain.This is accomplished by taking the Fourier transformof Equation3.68.Thus, Equation 3.68becomes:

(3.1 3)

Equation3.73is anequivalentrepresentatio nof Equation3.68in the Fourier do main.

Ifthe system forcing functionF(w) andits responsex(00) are known.the system characteri sticH(ro)canbecalculated.Thatis thefrequency-re spon sefunction:

(3.14)

TheIrequeecy-respcnse functionH(w)can be rewritten as afunctionof the complex polesbyusin gthefactored form ofthe polynomial equation as follows:

H(.. ). 11M

(j..-I.,W..-I.;) ^(3.1S)

Figure 3.6is anexample toillustrate the domain transfonnation(from lime to frequencydomain).

37 (3)I.aP-eDoauiD. ( Tnasf'er F\mction)

The equivalentinformation of equation 3.68 can be presentedin the Laplace domainby wayof' Laplacetnnsform.The onlysigni fican t difference in the development(:ORcemS

the(actthat Fourier tnnsfonnisdefinedfrom negativeinf'uti tyCO positiveinfinitywhil.

me Laplacetransformisdefinedfrom zerotopositiv einfinity withinitialconditiOCL

n.

developmentusing Laplacetransform beginsby takingtho Laplace transform of equ&lion 3.68.Thus equation3.68becom es,assumingzero initialconditions;

(3.76) Therefore.tho transferfunction canbedefined just as the frequency·response function thatwasdefinedearlier.

(3.77)

Thequantity H(s) is define das the transferfun ction ofthesystem.Inother words.a transferfunctionrelates tho Laplacetransformof thesysteminputto the Laplace transform of thesystem respon se .Thotransferfunction can alsobe written

(3.78)

The transferfunctionH{s)can berewrittenas

(3.79)

38

0.'5r-~---r-~-~-,....--~---,---r----,---,

0.1

II

I 0.05 '

00 10 0 200 300 400 SOD 600 700 80 0 900 10 0[

Time(1J1000Secl power spectraldensity

35 40

t !

1S 20 ZS 30

Fequeney(Hz) to

'5 f

'0

J ^o '---:--~____::'::______ ~---::---c

^~

Figure].6:Example of domain transformation(caseI. c-O.IOIl3speeds900rpm)

19 (4) Tnaderm RtJadoasbipi or Muld·Dq;rteo( Ftftdoaa Sr_IDS ( MDOt') Just uin the precedingcase (or SDOF.the equivalentinfOnnaDon canbepresented. for MDOF.The equation or motiontora MDOFsystem., usinS matrix notation.isas Collows:

Impulse-respe ese function:

(3.10)

Frequency.responsefun ctio n:

Transferfunction:

N A ...to

H

Gs)·r; = •

....!!!..

J1'II ...1.r-A., s-),,;

where t- time variable

5"'"Laplace variable

COl•frequency variabl e

p-measureddegreeof freedom (respo nse)

(3.1 1)

(3.12)

(l.U)

40 q-mcasu.red degree of freedom (in put)

r-modal vector number Apq r·residue

1,. -

system pole

N-number ofmodalfrequenci es Apqr•- complex conj ugateofApqr A._{t •}-com ple xco nj ugateofAt

3.2.2 Experimental Modal Analy'i'Methods

There arefour general categoriesof experimen talanalysis metho ds:Sinusoidal input- output model;Frequeney-respcn se function; Damped complex exponential respo nseand Generalinput-outp utmodel(Albert, 1993).

Thefrequency-r esponsefunction methodisthe most com mo nlyusedapproach forthe estimationofmodalparameters.Thismethodoriginated asa testingtechniqueas aresult of theuseoffreq uency-response functionsinthe fo rced normal mode excitatio n method to determ in enatural freq uenciesandeffectivenumberof degr ees offreedom.With the adventoCme computer.thefreq uency-respo nse function method became a separate viable techn ique.

rn

thismethod. frequen cyresponsefuncrions are measur edusingexcitationat single or multiplepoints.The relationshipsbetweentheinputF andtherepones X or bothsingle and multipleinputs are shownin equations3.84 through3.8'.

Sing le-inpu trelationshi p:

41

(J .B4)

(3.8S)

Multiple inputrelationsh ip:

The frequency-response functionsare used asinput datatoalgorithms thatestimate modalparameters usinga freque ncy-domainmodel. Throughtheuseofthe fast Fourier transform,.the Fouriertransformofthe frequency-do main mode , theimpulse-r esponse functioncan be calculated for usein modalparameter estimation algorithmsinvolving time-domainmodels.

Sinc:e the Prequency -respc nseFunction Method has theadvantages mentioned above.

themethodisused in the experimentalstudy.

42 3.2.3 Modal Dati. Acquisition

Acquisition of'datadeal ,withtheconvertingofanalog signalsinto acolTeSpOndin a sequen ce ofdieitalvalues that accuratelydescribe the time-v aryin adtaracteri stics of' inputs to and responsesfrom a system.Inthe present study.a Keithley570data acquisitionsystemisused..Once the datais availablein digital form.themostcommon approachistotransfo rmitfrom the time domain tothe frequencydomain usinl •fast Fouri er transform.

Dlsc~teFowierTrwasfonn

The Discrete Fourier Transform (OF[)is the basisforthe formulationofanyfrequency ..

demmn functionindataacquisition systems.Interms ofanintegral Fourier transform.for a function tobeevaluat ed.it museexistforall timein a continuoussense.For the realisticmeasurementsitu ation. data are availablein discretesenseovera limitedperiod..

Figure 3.7representstheDiscrete FourierTransform(OFT) concept Integ ralFouriertransform:

(l.87)

Inverse Fourier transform:

(l.88)

Discrete Fourier transform:

43

Inverse Fourier tnnsfonn:

where

N- btc ekslee (power of 2)

.u'""time spacing

J=frequencyspacingIrr k=O.1.2 N-I

X(f~'

N-' I; ... ;«' ,.)e

-p . <WNl

N -'

>ltJ= I; ...

X(f~.-P'<WNl

(l.B9)

(3.90)

(3.91)

fk-k.r

On the basisof OFT.theperiodicityandsymmetryofthe complex factar e-j(2 xJN)kn can be exploited toincre asetheeffici encyofOFTcomputati ons.ItiscaJled the Fast FourierTransfonn(FFI1.The FIT algorithmforcomputing the OFT ofasequenceis the workhorseofdigital signal processing (Wo wk, (991).Inthisstudy,thcdata collected by Keithley5570in digital form.is transferred as frequency-responsefunctionby FFf usingthe computer sofiware MATLAB,versio n 4.1,which is a technical computing environmentforhigh-performancenumeric: computatio n andvisualization.

Figure3.7:DiscreteFourier transform concept

44

Emo..

The accuratemeasuremen toCfrequency- respons efunctiondepends on thereductionof errorsstemmingfrom the digitalsignal processing.Totake CulladvantA&oDCexperimental data in the ovUuationof experimental proceduresand verification of theoretical approaches. errors. in measurem ent. generallydesignat ednoise. must be reduc:ed10 acceptablelevels.Thegeneral errors arcleaka ge andaHassi ng error.

uabce Envr

Leakageerror isbasicallyduo toaviolation ofanassumptionof thefast Fourier transformalgorithm;namely that the truesignal isperiodic:withinthesample periodused toobserve the samplefunction.Whenbothinput andoutput are totallyobservable (transientinputwithcompletelyobserveddecay outputwithin thesampleperiod)orare harmonicfunctionsol thcrimeperiodofobservationT. therewill benocontri butionto the bias error duetoleakage.Leakageisprobably the most commonsignal-prceessi ng error.Theeffects ofleakagecan beonlyreduced.not c:ompletelyeliminated- Theerror can be reducedbymethod orwindowingor weightingfunctions.

Wmdowi ac

Windowingis aprocess or multiplying sign alsbysome sort of weightingfun ction.By applyingweightfunctionorawindow.leaka&e error canbereduced. There are many windows.such as Rectan gular.BlackmanandHanning incomputer softwareMAl1.AB.

Inthisstudy.a Hanningwindowisused.

AvenICinC

4 '

Averaai nSalso reducesleakage error. It is onevery usefulfeatureof FFT spectrum analysis. Avera gi ngis the abilitytocombine time records (blocks of data)withprevious datatosmoothout the display.Sincevibration dataare not usually stable;namely,it is always variable,we use the signalaveragingto smooth thespectru m and reduce random signals.

Avuaginl can be divided into overlapand summationavenging.For periodic monitorin g, itis strongly recom mended to do summatio naven ging (Wawk. 199 1). Eight 10 sixteen averaging arcall that isnecessarytoget a stable spectrum.In thisstUdy.

summation avera ging is usedto reduceleak ageand noise signals.

Ali_siDIEn'Or

Ifthatfrequency components larger than half ofthesampling fre qu encyeecurinthe analogtime histo ry .high-frequency signal can fonn false peaksin thefrequency domain.

ThisiscalledaHassing.Thisis abyproduct of thedigitizing process.The solutionof reducing the erro risthatsamplingfreque ncyis tworimes as highas the muimum naturalfrequencyof thesystem;Namely,Fsampling 1::2 Fmax'Inthis study,The sampling frequencyis IOOO(Hz)and thenatural frequency is lowerthan IOO(Hz).Therefore.this sampling frequency is high enoughtoreducea1iassina errorandsatisfy the requirement of the data acquisition.

3.2.4 Modal Parameters Estimation

Modal parameter estimation is theesrimaue nof naruralfrequencies, damping factorsand modeshapesfro mthe measured data. Themeasured data can bein relativelyrawfonn

41 in terms of force andrespon.se datainthe arne or frequencydomain.orin&processed (onnsuch as frequency-responsefuncti ons.Modalparameters estimationcarriedoutin this studyisbased upon themeasured databeingin the frequenc:y-.rcsponse function Conn.Thecomputer software usedto perform the estimationis MATLAB (Sianal Analysis)Version 4.1.

3.2.5 Modal Data. PresentatioD

Modalparameters dataobtained fro m experimentand predict edbyfiniteeleme ntmeth od arepresentedin tabularandgraphicalforms.Therefore,the relationship amongtime, frequencyand amplitude versus frequency are shownclearly , and the peakvalues of frequencyresponse functioncanbe compared under the differentcrack parameters.

3.2.6 Summary

Modalanalysisisthe process ofcharacterizing modal parametersusingeith er analytical app ro ac:hor experi men talapproach.Inthischapter,the theoreticalbasisof modalanaly sis methodsisrevi ewed.andexperimentalanalysismethods arediscussed.Inthisstudy.

beamelements providedby software package ABAQUS. SAversion . areused tosimulate the shaft.Intheexperimental partof thisstudy,frequencyresponsefuncti ons obtained usinga softwar e packageMA TLAB,4.1version, is employe d toobtain the modal parametersotthe rotating shaft system.Other digi tal signal processingmethods are appliedindata acquisition inorder toeliminatenoise signal and errol'S,suchasaveraging andwindowing.

48 The result of the analytical study isreported and experimental results are presented in section 5.2.Funhermore. the comparison of modal parametersis givenin section 5.5.

49

Chapter 4

Experimental Study

Experimenta lstudyisan important linkofmodal analysis.andit is a bridge frommodal theoretica lanalysis 10 industria lapplicationsfor crack detection inrotating shafts . Experimenta lstudymakes it possible to gaina direc tinsight into problems whose analyticalsolutionis difficult[0obtain.

In this chapter.an imitationcrack. which simulatesthecharacteristic of practical transversecrack.is designed and a rotating shaftsystemhaving such as crack.ismade up_ Inorder to measure the vibration signal of the system .transducers and data acquisitioninstrumentsare chosen. The general procedures of the experiment is introduced.

4.1 Experiment Setup

Atransverse crack opensandcloses.which is alsocalled breath ing,during rotation inthe case ofhorizontal shaft, because the shaftis deflected by gravity.When thecrack

50 direc tion inthe directionof thegravitydirectio n. saythe crackisopen. whenthe crack directionis in the oppositedirectionofthe gravity,thecrackis closedshown in Figure 4.1.Except for the bothcasesmentioned above. there are both open and closed sections continuouslyvaries duringtherotation .Itis conside red thatthe fle xura lrigidity ofthe shaftalsovarie s.

Inorderto inves tiga tethe relationshi p between modal andcrack para me ters ; namely frequencyand crack depth.weneed a crack of variable depth that we can open andclose on purpose.However.thetransversecrac ks ofshafts inlaborato ries are genera llysmall because theshafts are generallyslender.Also itisdifficultto makedifferentcrackdepth

andthaterrors dueto makingthe crack will influence the crack characteristics. [0thisexperiment, alargecrack is imitatedby the rotorshowninFigure4.2.Itis possible to vary the crackdepth upto a large value andthe shaft can be safelyoperated in rotationbecausethe crack is not propagated .

The rotorconsistsof onemassdisk.tWOflang es andone shaft.The two flangesare symmetrically fixed on bot hsides of themass diskby eightbolts.Ifsomeboltsare remo ved symmetricallyon the both sides, the removed disk sectionopens and closeswith therotationof the rotorsi milarto thebehaviourof a crack. The sha ft is supported by ball bearings andis coupled to amotorby a rubber joint.The experiment setupis shown inFigu re 3.4.

4.2 Transducer

A transducerisade vice forconverti ng the mechan ical motionof vibrationinto electric

51 signal. Thereare three kindsof transducers:displacement.velocity and acceleration.

The mostcommontype of displac ementtransducer istheprox imit yprobeshownin Figure4.3.whichope rates on theeddycurre nt principle.Itsetsup a high-fr equency electric fieldin thegap between the endof probe and the metal surface that is moving.

The proximityprobe senses the change inthegap and thereforemeasurestherelative distance,or displacementbetween theprobe lip and the surface. The proximity probe measuresrelativedisplacement. Yet.anaccelerometer measureabsolutely displacement.

The most common accelerationtransducer is the piezoelectricaccelerometer.

[0theexperi ment, accelerometers[B&K 4378and 4379,1areused.The characteristic s of theaccelerometersis shown in Table 4.1.The untouchedproximityprobes[MStype 924-30] arealsoused .The characteristicsof the proximityprobes areshown in Table 4.2.

Figure4.la:Naturalcoordinatey-z androtatingcoordinateil -{

52

"

One boltremoved

Figure 4.2:Simulationcrackwith variable depth

I

Eddy Current

Figure4.3:Principleof the proximityprobeinoperation S4

Table ".11l:Charaa.cristicsof Accelerometer[BAK4311)

P.,...eler Value

Motorial SteelA!SI 316

SensinaElement PieZDeIectricMaterialPZ2l

Weipt 115irani

Tempe ratu reRange -50to +2500 C

Reference Sensitivityat 159.2Hz 23° C

ChuaeSensitivity 31lP ClS

Volta geSensitivity 258mVlI

Typ ical Undamped NaturalFrequency 13KHz Capac itance (Incl.cabl e) 1221PF Max.TransverseSensiti vity at J O Hz 1.4%

Max. Shock Acceleratio n 20Kms-2peak

Tabl,4.1b: Clwaaeri.stiaoC Accelerometer(&tX4379]

56

P...- Val...

Material Sleel AlSI 316

SensinaElement PiezoelectricMaterialPZ23

Weip t

a "

&RJn

Tempenrvn

llan"

^·50to+Z3O"C

Ilef'ercnceSensitivityIt'0Hz 2JoC

Charg e Sensitivity 310PCI,

VoltageSensitivi ty 248mVla:

Typ icalUndamp ed Natural Frequency 13KHz Capacitance(Incl.ctble) 1230 PF Max.Transverse Sensitivity at30Hz 0.'%

Max. Shoc kAccelention 20Kms ·2pcak

Table 4.2.:OwacterisI:ics oCProximityProbe[MS924-30 )

P... Valuo

SupplyVoltag e 13.5 to JOVDC

LoadCurrent 12mAmax

Output Voltage Range 1 to 9VDC

Outp ut Impedance SOohmmax

LoadResistance.min IKto22Kohms

SlewRate IV/msec

Temperatu reDrift: %2mVOClmm

Linearity :t:O.2SVDCbetweenI and 9 VDC

TemperatureRange

o

to60° C

Diameter 30mm

Protection Class lP67

sa

4.3 Instrument

Inthis experimental setup shown in Figure4.4,the DC motor [Eic:or 4020·22Jdrivesthe shaft:for rotation.Inordertosupply the direct: current,.28 volt:q;es and 30amperestothe motor, aDCpowersupply madebyE&:ELab.ofMUNisused.Mean while,.inorderto control thespeed.ofthemotor,aDC variable autotransformeris used to adjustthemotor's rotational speed.A speed meter [tachmeter 8931]is usedtomeasure thespeed..There were

so·

70rpmsdifferences between measuredspeedandrealspeed of rotation. Thus.

a stroboscope[IRDS 17]isusedtomeasurethespeed ofrotarion.Inthiscase, the difference isonly2·5rpmbetweenmeasured and real speed.whichis accurateenough forthis study.Accelerometersare mounted onthetopof bearinghousing andproxi mity pro besarefixed near the middleofro torin horizontal andvertical directiontopick up vibratio n signalas showninFigure 4.5.The ISvoltageswork ingvolts neededby proximity probes is suppl ied byDC power sup ply (Model10611- Measured signals are fed tothe amplifi ersby cables. andamplified by theamplifiers [B&K 2626and Sundstrand504E) .Afterthat.the amplifiedsignalsare fedtoKeithleydata acquisi o n system [KEITlU..EYS570)to be converted intoadigitalsignal form.Finally.the vibrationsignal sindigitalform are analyzed in Unix comp uter worksta tionbyusing computer software MA TLAB 4.1.

intheexpe rimentalsetup.an oscillos cope [TEKTRONIX 5441] isem ploy ed asa signal monito rin g.

All ofinstru ment and devicesmention edaboveare listedin Table 4.3 S9

Tabl. 4.3: Listof Instrument

60

No. Name of InstNment Typo

1 ProximityProbe MSTypo 92' • 30

2 AcuJerometer BA:K Type .31.

3 Accel erometer B~Type 4]79

,

AIDConverter Keithley Modd 570

5 StorageOseillc sccpe TektronixModel 5441

6 Amplifier Soundmand Typo S04E

7 Amplifier B.tK Type 2626

I Medtanalysis IRD Model 880

9 Stroboscope IRD Model 571

10 Tachometer EMSType 893 I

II DC Power Supply Model 106\

12 Autotransformer Powemat EMI712

13

De

Motor Eicor Model 4]679

I' Computer EM Pac Model 486

15 Monitor Microscan4G1ADI

61

Figure4.4:Experimental setup andinstrument

Fisure4.5Mountedproximityprobes

62

6J 4.4 Pmcedun:

In thisa:periment" a tnnsversecrack.whichopens anddosesduringrotation.isimitated byremovina those bolts toMeenng the diskandflanges u shown Figur.4.2. One.

two,three and fourbolts areremovedsymmetrically from everysid. ofthe disk intho experiment" respectively. TheCBCk depthis expressed by crack parametercolTespondina toth. numberof bolts removed. The enc:k parameteris •non-d.imensionalparameter.Th.

crack depth duotoremovingdifferent boltsis dividedbythe diameteroCche disk and the relationbetweenthe crack parameterand the number ofremoved bolts isshown in Tabl.

4.4

Table4.4: Crac kparam eter

Removedbolts Crackparameter(crackdepth!diameter)

0 0

I 0.0833

2 0.1083

J 0.204 1

4 O.JJJJ

64 The procedureof this experime ntincludesfourste ps.Thefirst.thestatic deflectiondue to the self-weig htoftherotoris measured.Second ly,thenaturalfrequencyof the static syste m ismeasured using a hammer excitationmethod. Third. the uncracked shaft is tested.Next.thecrackedshaftis tested indifferentcrackparametersbetween0.083and 0.333. finally. the crackedshaftin backward rotatio nistested.

The first stepistotest sta tic deflection of therotor.Thestaticdeflectio noftherotor ismeasu red forthe horizon tal andverticaldirect ion respective lybyproxim ityprobes.

show nin Figure 4.5.forcne revolutionperiodic variation from0"to 360".

The secondStep.asingleimpulse.byhittingthe rotorby a rubberhammer.excites the static shaft systemto obtainits natural frequency.

Thethird step.the rotorisdrivenbya Direct Current(DC)electric motor.Then.the DC is adjustedto reach the speed that we require.At the same time.thestroboscopeis trun ed onmentio nedonchapte r4.2. and aimed thenashlight ofstrobeat therotor.After that.the nash ratesare changedslightly.oncethe flash freezes themotion of therotor.

thespe eddisplayedonthe readoutofthestrobe is thespeed recordedin theexperiment.

SinceDCisused tochangespee d of rotation.it have to makesure that speed displayed on readou tdoes notchange; namely.thecurrent isstable.Then. thevibratio nsignalis measured by usingaccelerometersat the speed.Arterthat, direct current isincreasedby adjustingauto transforme r toincrease rotating speed,andmeasure ahighe r speed than before.Therange of tested speed isfrom300rpm102000 rpm.whichcove r thecritical speed ofthe shaftsystem.sincethe themeter's powe risnothigh enoughover 2000rpm.

The founhstep.the procedureis the same as thethirdstep. But. we would remove

6S someboltsandadjust crack parametersbefore the measureme nt.The bolts are removed symmeui caJly onbomsides of the disk.

Intheexperime nt. In ordertoinvesti gate the modal parameter accurately. simula ting crack rotoris testedin differentsizes.which areW=S8(rom) incaseI.and W=90(rom)

in case 2,shownin Figure 4.6b.

The laststep.testincrackedshaftin backward rotation.the procedure isthesameas the thirdstepexceptthat theforward rotation isreplaced bybackward rotation.

Figure4.6a: Sizeof experimentsetup(unit:mm)

66

I ~ . I

- E_L-'>

A ._ _{_.~2s>~} w1ffit:~4-- - -

figure4.6b: rotor

67

.~

¹¹¹²⁴⁵

.200 _ '

.- " ' . *'

1112 00 '5

"

Figure4.6c:Disk andflange

61

69

4.5 Data Analysis

Asment ioned before. thevibra tio n response signal s fromthe transducer s are con vened toa digital form byKeithley5570 .Afte r the digitalformdatainare collected ,they are sent toUnixcomputerworks tationforfurther analysis.

4.5.1Frequency

Thedata analysisconsists of threestages.Firstly,thesedigitalsignalsarerun througha progr am whichaverages thedata, muchofthe noise in thesignalare remo ved.

Meanwhile .a banning weight functi on isused to reduce the leakage error.Then.the vibration response digital signals are transferredfrom time domain10 frequency domain byfast fouriertransformwhich is shownin Figure 3.4. Finally,themeasuredrotation speed andfrequency responseamplitudeve rsus thespeed arerecorded. Repeatingthe procedureuntilall thosemeasuredsignalsordata setsat different rotationalspeed are analyzed. these figuresofvibration response amplitudeversus these rotationalspeedand versus the rotational frequenciescan be drawn asshownin Figure4.8 and 4.9.Fromthese graphs.we can find the naturalfreque ncyofthe shaftsystem in the experimentsetup is 30Hz incase 1.and28.33Hz incase2.

4.5.2 Damping

Basedon thesegraphsFig4.8band Fig 4.9b.the othergraphs. frequency response curves can be developed.bychanging the rotational frequencyaxis torelativerotational

<OJ s.o!-

•.0,..

i i :

... J.o~

70 frequencyaxis

n

Iw(where:

n

isrotatio nfrequencyand cc is natura lfrequency).shown in Figure 4.10and Figure 4.11.

Takingr=O:!was horizo ntalaxisand Amplitude

JJ

as vertical axis,band w idth method (Shabana), one ofexperi me ntalmethods for damping evaluation,can beused to dete rminedda mping ofthe system.Figu re4.7shows the use of this method in evaluating the dampingfactorl;.Using the frequency respon se curve,we can drawahorizon talline at distanceP =(lN2)P..1fromIher-exis.This horizontalline inte rsec tsthefrequency response curve attwo points whichde fine Ihe frequenciesf.andft .The damping factor 1;canbedetermined by equation:~=(fl-rJl2.By means of the bandwidth method.

abtain damping factor l; =0.032in the case I.and dampingfactor l;

=

0.033 in case 2are abtained.

:~

^0.0

:j:

^{1.0 :.0 ).0}

Figure 4.7Bandwidthmethod

Crackparamete r~

i _. ~~

0.1_~

~

_{n nn n n}

Sp••d(rpm )

Figure 4.8a:Speedvs amplitude

Cr.aelcparametere=Q S0.015

.,

i

^0.06

t ."

p~,- ... ====... ~

! : ~

Figure4.8b: FrequencyV$amplitude

71

11

r-~

y - -

4""" ._~y -

....

~~""

~. ^-/(~l

~-- - ^{~2~ \-~~}

~T

llos~.

I ^~

^~l

O~

^§

~

^§

~ ~ ~ ~ ~

^E

~

Speed(rpm)

Fig.4.9a: SpeedVSamplitude(case2)

Cradl:par.m.",e-O

_0.7o.e .e0.5

i

0.'

J"'

^02~._o

!! n n n~ ~

ADIMlonrrequeflq

Fig.4.9b:Frequency vs amplitu de(case2)

caO (ca. 1)

73

;

~

:.---:

-3' ,'t ~:

, i t"

L - ': ~'\~

~~ ~.-,

^...j

~7

~.

S05

I:: l ~ ,

~ ~ ~ ! ~ ~

Ala.rwfrequency(Qt_)

Figure4.10:Amplitude vsrelative frequency (case I)

j ~'"

f

^c...

~02

!~:~S!§!

AI~fr..-r(Qt..)

Figure 4.11:Amplitude vs relative frequency (case 2)

74

Chapter 5

Results and Discussion

lnthis chapter.firstly,the relationship betweenstaticdeflectionof therotor and phase angleinvertical andhorizontaldirectionis discussed. Then.the modalparame ter, frequency and da mpi ng.is conside redin theuncrackedshaftsystem.Afterthat. the experiment results dealing with crackparameter are represented.Next.the vibration response of forward and backward rotation of the shaft areshown.Finally,the modal parametersof thesystem with.and without cracks. are compared.

5.1 S tatic Deflection

The deflection amplitudeof theshaft due to self-weight varies withthevariationof transverse crack:opening andclosing inrotation.The staticdeflection is measuredin horizontaland verticaldirectionrespectively.Themeasuredvaluerepresentsthe static deflectionof theuncrackedand cracked rotorinonerevo lution periodicvariation. which is shownin Figure5.1. lt can be seen thatthe static deflectionincreaseswith increasing

7S or the crack parameter.Inthecaseof verticaldeflection graph.thede flectio nat phase angle 180"isthelargest because ofthe fullyopenedcrack section.and the deflectio nat

cr

isdiesmallest due [0the fullyclosedcrac k section.The resultisin agreement with resultsobtained by GrabowskiandMahrenhol tz and Ziebarth erat.0978.)

I 0'

0.'

I..

g';1i~i~

phAHang'i;

Figure5.1:Vertical Deflection

Horizontaldeflection

Figure5.2:HorizontalDeflection

76

77

S.2 Uncracked Shaf t

in order to determine the naturalfrequency of the shaft system in the experimentalsetup.

a theoreticalcalculation.a finite element method approach and a experimentalexciting evaluationare applied. The result is shown inTable5.1.

Table 5.1: Naturalfrequency of the shaft system

Natural frequency (00)Hz

Method Case I Case 2

Theoretical calculation 32.3 30

FEM calculation 33 3\

Exciting evaluation 30.5 28.5

Experimental measurement 30 28.33

Then.allboltsace tightentedon the rotorandthevibration response is tested at different rotational speeds.The amplitudeversusrotation al speed curve is shownin Figure 5.2.It is notedthat thereis a strong vibration,reson ance,at 30 Hz. This means thatthe natural frequencyof the system is 30 Hz. There is a good agreementwiththe results obtained byoth.erways mentioned above.

18

5.3 Cracked Shaft

We measurevibration responseateachcrack parameter shownin Figure4.1.Tho results areshowninFigure 5.3to5.5,incaseI,andinFigure 5.6to5.9inease 2. Itis found thatthelarg er thecrackpar ameter,the lower thenaturalfrequency is and thebigerthe amplitud e versus the frequencyis.

5.4 Forward and Backward Rotation

B~thattherearcforward and backward rotatingshaftsinmachinery,thevibnrion response is measured in both rota tional direction.forwar d and backward.But we findthat therespo nse in backwardrotation issimi larto the forwardrotation.Thus, we showonly the amplitude versusbackward rotationalspeed curve showninFigure 5.4b(c^aO.2041in caseI).

5.5 Comparison betwe en Uncracked and Crac ked Shaft

We showthevibration response of crackedand untrackedshaftsystemin Figure 4.1to 5.9.and TableS.JandTable 5.4.Whenwe put ihesefigures together. we getFigure 5.10 in caseI.andFigure S.l lin case 2.As a result .we can comparethe experimental results of crackedshaft with that ofuncrackedshaft. We findthat the natural frequency decre asesand amplitude versus thefreque ncy increases withincreasi ngc:ra~kparameter whichare illustra ted in Figures~U2and5.13.There are78% amplitudedifferen~eof