ANALYSIS OF PR OP ELLER SHAFT T RANSVERSE VIBRATIONS

ANALYSISOF

PROP ELLE R SHAFT TRANSVERSE VIBRATIONS

By

@Raman Warikoo, Il.Eng. (Hons.)

A thesis submitted to the School of Graduate Studies inpartialfulfillmentofthe requirements for the degree of

MasterofEnglneerlng

Faculty of Engineering andApillied Science MemorialUniversity ofNewfoundland

Augustrose

St. John's Newfoundland Canada

.+.

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L'auteura accoroe unelicence irrevocable et non exclusiveoermettent

a

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distribuer au vencredescopiesde sathese dequelquemenlere etsousquelqueforme quece soilpourmeltredes exemplairesde cettethese a la dispositiondes personnes lnteressees.

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ISBN 0- 31 5-59 2 21-4

Theaimor thisprojN tWilli tofind"simple,pract ical "nilsufficiently accurate me-thodforfindilll{thenatura lfreq ut'nc~of"propl'llt'rsh dtlU\!i("mbly. Tile-needfor such method s is ft'll atthl"Nrly designsta£C$when suffieientdata aboutthesystem is netavallable andthe nt"C"dtorestrict thecostofanalysis isofi lllPOrlancc.Dueto thesefactsit iii clear thatthemerhodsh"liM 011 the di5cf('lizatiollof rhc conueunm, which can estimatetheuaturalIeequcneics accuratelymay nothe edvisahlc10 go in fo rin the init ial lilages,

III theclassof approximat e methods,towhich thi.'\work 1Jl'long.'\,inv('sti~at ions do ne so far have Cf111II']ymodelled thepro pellrr.whichwas avoidedin the present work.Thepropellerwasconsideredtobea Rexil.l('rotormountedwlth I>\adf>ll .

Tosolve the problem ani\lysiswas carr ied 0111ill three parh . first the blade was stud ied forthe naturalfreqllenci l'!i.mod e llhapn,stat icdeflect io ns ancl the slead}'st atestresses.The bla dewas assumed to he a cant ii<'\'erandthl;'res ultswere cross-checked bya prdimilla ry finit el'km('ntanalYliiliusing beam1'I('IIIf'nts. The UlSumptionwasfoundtob... quite-validafter doing ti,efiniteclementana lysis lIliing

3·n

Iscparamerri c 20nodedclcm(,llts.fortill.'airrotl blad l' effects01 rotat ion.dl('ar deflectjon ,ro tar y inertia. pitch sett ing angleW('~taken into considera t ion.Second theshart-rot orsystem wall stndiccl andlile equationfortilerreneveese vlbra tio ns wasderivedinthecom plex plane whichwassolved for thenaturalIreqncucics, Effectsof fo rwa rd and reve rsewhir l,hxedandsimplysllllporl('(1fo rward bearing end conclition s, tailllha ftle ngt h variatio n,rot ary illl'rli.,andsheardoflocrionwere invest iga ted . Lastlythenaturalfrequ encies oftill"bladeslindthe shan-r otorsystem were ccupl.dto glvether('sulta ntnaturalfrequency orthe propellershaftass('mh ly.

Thehydrodynam iceff('cthasbl'C1Itakenint oconsidera tion byincorporal ingthe addedmass ('ffecta forthepropeller.1'\\'0 appro xlmarjonsforthe natllralfrCtII:('ncy

oftheassembly were arrived at,the zeroth orde randthe Ilrst orde r.Thetrue value ofthefrequency issupposed\.0lie in between the twolimits .Erlectsdue to change in thenumberofblades,blad e geometry andthe shaftpara metersonthepropeller shaftassemblywerecheckedfor.Agreement bet weentheresult sobtained fromthe present workand with thoseavailabl e inthe literature wasfoundto he excellent, Finiteelement analysts comp arisonsalso were verysatisfactor y.

iii

ACKNO\VI EDGEt\IENTS

Iamgn.teful toDr.~L11.lI:ulda r;l for hillI1;lLill'lnceand eeeoueegemeatduring thecourseofmyresearchwork.

I feelindebtedto deangrad uateschool

rcr

providin gmewit hfellowehlpand NSERCCanada for providingme withfinanrialasslstanceduringthe period ofmy st udies.Iam tha nkfulto Dr.G.R.Pe ters,deanohngineeringandDr.T.R.Chari.

associate deanofengineeringfor extendingtheircooperationand neceuar y help duringmy masters program.

A wordof praise for MervinGoodyeu for typing the thesisefficiently.

ToVidyasagarom(Ocean orKnowledge)

"d MyParents

Contents

1 IN T RO DUCT IO N ANDLITE RATUREREVIEW 1.1 Scop('ofllirIll\"('l;lig a tio n ..

1.2 LitN all!T<,SntVl')'.• . 1.:.1.I I'TOp<,ll(,fShaft Analysis. 1.2.1 nllld l"Analysi!l 1.3 [koHr iptionOfTilehn'('5ti~lI liOIl

2 BLADEANALYSI S

2.2.3 Non-Dimensioeahxafi on 2.2.2 MlIlhNllllt iralfo rmulat ion

2.2.·1Solutionofl li<.> Equadous • 2.2 Equa:iotIofMotion•• .

2.2.1 Introduction

10

,. ,

I·'

rs

"

rn

Sample Problem 23 2.2.5

2.1 Ac[('knl ionAnalysis 2.1.1 Int ro<ltlcl ion _,_

2.1.2 Auc!trationofitPoint on the Shaft C1.'lIt('rLine ••. . 2.1.3 Accf"ll'ulionofa PoiulonlIlt'mil"!'. .. .

2.2.6 lIydrodynllrnic Effttt.<i 2.2.i CompariMllls 2.3 Mode Shapes

2..1Prl'liminary CheekUsingFiniter.lc-M·nt Fotrnllh,tio"

.13

33

"

as

2..1.1 Element$clKtioli •.•...••.••. 2....2 [!emf-ntPro~rt i es•.•.. •.. • • • • • •

3.\

2.5 BladeIknectionand Stresses•. .• •.•• •• •..•. .• •.••• 3i 2.5.1 Introduct ion •.• ..••. • . •••••••• .

205.2 formula ti on.• • •

3 ISOPARAMETRICFINITEELEME NTFORM ULATIO NOF THE

BLA D E 50

3.1 Int ro dllction... . .. . 50

3.2 ElementSelection. 50

3.3 Mathe maticalFormulation... 5\

3.4 FonnutatlonofINIMatrix. ... .... .;.\

3.5 Formulat ion ofStrainDisplacemen tMatrix[llJ 55 3.5.1 Str.unVector

3.5.2 JacobianMatri x•. 3.5.3 {I' } Vector•.. . .• .

3.6 Formn lation of ElementStilfn~Ma trixfl..-"). •• • •• ••••.. .• 58 3.6.1 Intqra l ion SCheme.•..• .•.••••• • •••

3.7 FormulationofElementMassMa tr ix(11I"1..• •.• 59 3.8 Formll lolt ion of ElementStres s Stilfn<'SsUlilrix(.l"~1••. 60

3.8.1

[,1

Matr lx... 62

3.9 SolutionofTilt' EigenvalueProble m 63

3.9.1 Dynamic: Condcns arlo n •. 6,\

4 FR E QUENCYANALYSIS OFSHAFT·ROTORSYSTE M 67 ,1.1 Introd ur-tion.• . . . •.

4.2 Formulation ofthe Rotor Equation.

"

"

·t 2.1 AngularMomentum ofthe Rotor.•• . • •. 69

".2.2 Equa tion of~Iotiol\in z= and,=Plant":'!.• •• . iO

vii

.1.:1 EquavionsforIll('T<lilSha ft aud ,III.'O\'l'rhilugPo rtion. i·'

,1..1Nondime usicnalix arlon

,5

·1....' Solution oftheEquation. ,8

,1..').1 Applle ationol theBo ulidaryFondltloas, IiO

Ii COUPLI NGOFBLADEAND ROTOR·SHAFTFREQUEN CIES85

.'j. 1 [ntrodue tion 8.')

."•.2 Forruulatio u of theSt'ri"sSynrlll' tkM('lh O(I. 1:'.", .').:\ Cntcnonfor the Ajlplir<llionofSl'ril'NTyp(' Solufion . Hi

1)..' i\lnddlillgoftheSystem. !)2

.'j..'i DeternunarlonoftheCritkalSpeed 9i

6 CONCLUSIO N S 102

li.l SummaryofthePresent Work 102

6.2 Coneluslons 10:1

G.J Limitatio nsandRccommen datlons . 10·'

References 105

AArrangement ofCoo rd ina te Axes 108

B Tran sformationof CoordinateAxes 110

CVelocity of1\Pointon theBlade 112

D Applicati on of BoundaryConditions 116

E ExplicitExpressionsforpk and pk' 117

F Simplified Eigen valueEquation 119

GEx pr ess ion forArea Inte g ral 122

vili

H Boundary Co nditions Appliedto theThirdTer m on theMU!iISide US

I Exp r"lio n forCen tr ifuga lForc eExer t ed on aRot a ting Cantilever128 J Gau S!Qua d ra t ur eInt e gr a t io n Scheme 131

K Rela t ion ofSt~..Stiffne!il!il Matrix with Nonline a rStra i n!il 132

L StressMatrixI 135

M Charo.c1er uticEquationfor the Shaft·Ro torSyllte m 137

N Couplingof Detl ectionFunctions 140

o

Compu terPrOgrllmforFindingtheNatuj-alFreque nciesand Mod e Sha pes ofaRot at ingPro pell erBla de

PCo m p uterProgramfor Findi ng theNatura l Freq ue ncies of3.Ro- tating PropellerBlade Using FiniteEle m en tAnalysill 153 Q Com puter Pro gr a mfor Com puting the Naturalhequency of a

Rotor ·Sh!'.RSYltem

ix

110

List of Figures

2.1 Blade coor dina te system.• 2.2 Free body dlsgram

or

the blade element 2.3 Bladeprofile atvadons section sNACA 16-018 2..1Bladeprofile atvartoussectionsNACA633-018.•.

11 16 24 25 2.5 Variationofnaturalfrequency forNACA16·018, pitchangle=0 dog. 27 2.6 Variationof natu ralfrequency for NACA}(j·U18,pitchangle=30deg.28 2.7 Variationof naturalfrequency for NACA 16·018,pitchangle

=

45 deg. 29 2.8 Variationof naturalfrequencyforNACA 6:h-018,pitch angle=0

dcg.

2.9 Variation ofna t ural frequencyforNACA 633 -018, pitchangle

=

30 deg.

2.10Variationofna t ural frequencyfor NACA633-018, pitchangle

=

015 . 10

31

~. ~

2.11Comparison

or

modeshapeswiththe casvta ken in Storti,1987. 36

2.12J·D taperedbeam element. 38

2.13Comparisonof naturalfrequencieswithfinite elementanalysisusing taper edbeamelements .

2.14Comp ar isonof modeshapes withfinite elementanalysisusingtapered beam elements

2.15Variationofstaticblade deflections ••

2.16Variationofsteady stale stresses 2.17Le ad variationofthe blade

39

2.18Comparisonofstat ic bladedeflectionswiththe casetakenin Conolly, 1960.

3.1 3-0,20no ded lscparamctrlcelemen t•

3.2 compar iso n ofnatural frequencies,againstthoseobt ai ned using3-D finiteelementanalysis.

4.1 Propelle rshaft schematicdiagram 4.2 FrC('bodydiagramofthe rotor dement

4.3 Variationof natliraIfrequency withrespectto thechange illtheshaft fixity. •.

48

52

6'

82 4.4 Natural frequencyvariationdue' thevaryingtailshaft length. . 84

5.1 Synthesis ofiner t iaand restoringelements . . 86

5.2 Modellingofthepropellershaft assembly 93

5.3 All comb inat ions of inertiaand restoring demen ts 9.1

5.4 Determinationofcrit icalspeed 98

5.5 Effectofblade numberonthe nat uralfrequencyofthe propellershaft 99 5.6 Comparisonsofnat uralfrequency withthe case studiedill Tomsand

Martin1972.

Rotating cantilc\'er.

K.l Transverseforceon an axiallyloaded tr uss member.

100

129

133

List of T ables

2.1 Cordlengthsat various bladesections...•

2.2 Comparisonof nat ural frequencies withthecas etakeninSubra h- manyam,1982 ..

26

34 2.3 Com parison ofnatural frequencieswiththecuetaken in Storti,1987 34 2.4 Comparisonofstaticbladestresseswiththe casetaken in ConoUy,

1960. ..••.•..•• .... ..•• •. .. .• .• .•.••. 49

5.1 Com parisons ofcriticalspeeds (RPM)withthecase studiedin Way.

towich, 1979... . . . ..•. . .. . . . ... .. . 101

NOHENCLAT !lRE

01 angula rvelocity of coordinate systemzyz

u. deflectionof theshaftcenterline alongxax ledueto vibratio ns v. deflectionof the share center line along 11axis due to vibrations

slope of the deflecrlonalong y axis slopeof thedeflectionalong a axis

coord inatesof apoint on the vib ratingshaftcenter line, inanon -rotating coordinateframe

coordi na tesofapoint on the vibratingshaftcenterline, in aro t ating cccrdlnatc frame

XY Z globalcoordinate frame

;ellt frame fixedto theshan centerline.

Tp coordi n atesof apointontheblad e To coordi natesofa.point on the blad eroot

rp.,

coordinates ofapoint on the blade withrespectto itsroot u~ deflection oftheshaftcenterline along ;r;axis, in a rotating

coordinatefr:t.mll,tovibrations

v: deflect ion ofthe shaftCl'lItl'rlinealongyaxis,ina rotating coordinateframe.duetovibration s

angula rvelocity ofcoordinatesystem:z;~y~ z~

I,J,!( unit vectors alongX, 1',Zaxes i,j,k unitvecto rs alongx,y,zaxes i2,h,k~ unitvectors alongXl,1/2,:2axes

t4 deflectionofthebladeatollt~axis due tovibrat io ns W6 deflect ionof the blade along( axisdue tovibratio ns

R hubradiu s

pitchangle

xiii

ii( accelerati onof a point ontheblad e along~axis u~ acceleration of a point on thebla de alongn axis fiC acceler at io nofapointonthe bladealong (axis

shearforce

N centrif ugalforce

slopeduetoboth bendillgmomentand shea r force

At beudingmoment

kN nondimensional factorforceni riful;al force bladelengt h

K shapefa ct or constant area of cross-section

E elastlemodulus

G shear modulus

density

nondim enslonal time paramete r nondimensional constant

{2 nondimensio nalconsta nt nondimensio nalarea

nondimensfcnalareamornentof inert ia nondimensiona leigenvalue

IJk assumedsolut ion for tra nsversedtsplaeomont PIo: assumedsolutionfor slope or thebending moment

eigenvalueof theblade vibrati ons error incorpo ra t ed due to assume dsolutions PKA,Pl,B explicit exp ressions for]J1o:

PKIA.,PA-1B explicit expressionsfor11k eigelwa lueoft hesys t em /h efgenvalne lndex

xiv

/(x) spatialfunctionof theforce on tile blade 9(1) timefunction of the force on theblade [11.'] global stiffnessmat rix ofa uniformcantilever [.\1] globalmass matrix of a uniformcantilever

[li"] global stiffness mat rix oftheblade

[M} global massmat rixofthe blade

{F} force vecto r

{q} displacementvector

OP!r rthnondim ensional mode shape of the blade

T bladethickness

II,V,w components ofgeneralizeddisplacement vector

[N] shapefunction matrix

{e} nodalcoord inatevector (mO] elementmass matrix

{k"1 elomenrstjffnessmat rlx

[~J element stress stiffness matrix [B] straindisplacementmatrix

{tI} derivativeof thenodal displacementvector,w.r.t.x,y,z

{ti'} derivative ofthe nodaldisplacementvector,w.r .t.{,'l,'

Ike] element stressstiffness matrix

[sl stressma tri x

pcisscnsra tio

p] jacobian mat rix

[I'[ inverse of jacobianmatrix

Wi Gauss weights

{cz} linearstrai ns

{fNL} nonlinearstrains

{UNL} strain~ergydueto nonlinear- strOlins [T] transformAtion ma tr ix {orcondenseddisplu em en u lh"nJ red ucedstiffnessmatrixof thesystem (Mnl reduccd mas slllotlrix ofthesyst em II angularmom entumoftherotor

l",z,I'>I"I...z X,!I, :componeats ofthemassmomentofiner tiaoftherotor V_z,V, shear forcealong2:,yaxisof the rotor

Mz,A/, bendi ng mo mentalongx, yaxis o{ t llf"rotor rotortransv erse displacementillthe compktxplane overhangtea nsversedisplacement inthecomple xplane tai lshart tra nsve rsedlsplacemee r in thercmple xplane rotor,ov('r ha ng. tailshaftlengthsrespee tively constantsfo rtheroto r andtails1l4ft w.

constants{or thedisplacementfunctionof the rot or constantsforthe dls plaeementfunct ionof theoverha ng constantsforthe displa remeu tfunctio n ofthetailshaft fixityoftheforwardendbearing

P, natu ralfrequt'llcy of anlsclatedsyste mcomposed of mass

",

a,b, c

and oneofthe restorinl;:ele ments

~ii perturbationparamet er

~. sumoftheperturhationparameterfor\'ariouscombinat ions of inertiaandresto ring elemeeu

non dimenslc nalra di us ofthehub ncndlmenslonalco efflclentsofthearea polynomial

ni

Chapter 1

INTRODUCTION AND LITERATURE REVIEW

Propeller shafeis an importantpart of the ship propuls ionsystem and therecent failures of thepro peller shafts due to fatigue make it imperative to have an accurate estimate of the propeller shaft natural frequencies early in the design stage to avoid resonance,whichof course is one of the prime factors of fatigue failures.This can prove usefulinthe selectionofthe enginespeed and reduction ratios. Medal analysis not onlyhelpsinpredicting criticalspeeds butalso in assessing the dynamic stresses in the propeller blade, which are of equalimportance,especially at the blade root . For the propeller shaft the mainexcitingforce comes fromthe bladeexcitations, whlch arecaused by the non-uniformwakearound the ship propeller.Thereca n be otherexciting forcesdue to massimbalance,hysteresisdamping, fluid friction in thebearings etc.,but sucheffectshavenot been found to be of predominant effect.It isimpor t ant to note thatthepropeller exciting forces can be expressedir, a series,whose frequenciesbeara ratio of multiplesofNtotherotation al speed of the propellershaft,whereNisthe number ofthe blades,

1.1 Scop e of the Invest igat ion

The main aim of tllisprojectwasto lilillasill/pIe,pract icaland sufficientlyaccurate method for finding thenat ur alFrequencies of the propellershaft,which can prove

quiterelia ble and handy in the earlystages ofdesign andallows forthestudyof the e(fectofpropeller shaft geometryon thenatural frequencies.Theanalysiswas car riedantintwo parts : firstthe bladewas analyzedandthentileshaft-roto r system.Thefundam ental frequency ofthe propeller shaftassemblywas estimated by suitab lycouplingthoseobtai ned fortheblade and tha t forthe sha ft-rotorsys'em.

Thenatur alfrequencies ofthetra nsversevibrationofa propulsion shaftingsys- tem can beaccura telypredicted using numericalmethodssuchasfiniteclement analysisas in Kuo,(1965),Holzer's scheme or Myklesta d-Pr oblmethodas inToms and Martin,(1972).Theuse ofsuchmethods usuallyrequiresdetailed informa- lionwhich mayno tbe availableat theearlystagesofdesign process.Inaddition, the costofusingsuchprogra ms doesnetwa rrant theirusewhen wejust need an approximateestimate of thena tu ralfrequencies. Thisdearlyhighlight stheim- porta nce oftheapproximatemethods which are hynomeanscomplicate dandstill arenotacompromise in theiraccuracy.Tilepresent workfallsin this category and satisfies theabovementionedqua litiesbesidestakinginto considerationsome otherimportantcharacteristicsof the propeller shaftwhichare no t availablein the existing literat ure,like theairfoil cross-sectionalarea oftheblades,conaidcrlng the propellerto be aflexible hub mountedwithblades.

1.2 Literature Su r vey 1.2_1Propeller ShaftAnalysis

Inthe early fiftiestwo well-knownapproximate meth ods for de termini ngcritical propellershaft speedswithrespect to lateral vibrationswerede veloped.The first one wasformulate dbyPanagopulos(HI50)whereinthemoment um oftherotating propeller isignored , whereas Jasper (1952) ignores the shaftmassandconsiders the rotor as a thin disc; therotoris anapproximate modelling forthe propeller.These tworesear chersare considered tobethepioneersin thefield ofdeterm ining the natural frequencies of the propeller shaftby such approximate approaches.

Iloek (1976)in early seventiesimp roved the existingmet hodsby consideringtile effectsofpropellermoment andiner tiaforce, however, the propellerwas treatedas a disc.He considersa non-rotatingshafton two supports with au overhung mass.

Startingfromthedifferentialequatio ngoverningthe dynamicbehaviourof abeam , propellermoment and inertiaforce are applied asboundary conditionsat the end of theoverhungbeam.

In auattempt to solvethe problem clflnding thenat ural Irequencle aofa pro- pellershaft Woyto wich(1979)employedDunkerley' s method forthispurpose.The propeller is approximatedas a thindiscattached to a masslesselasticshaft.An equation of motion,derivedb)'DenHartog (1956) givingthe whirlingnaturalfre- quenciesofa disc on arotating shaftis used to get the whirlingnaturalfrequencies of theoverhung portionofthe propellershaft.The frequencies of the excitingforces bear theratioNto the shaft rotation speed,whereNistilt!number of propeller blades.For thetallshaft , transversenaturalfrequencies are calculatedfrom the conventional Euler-Bernoulliequation. Dunkerley'sformulais used tocouple the two natural frequencies to gettheco mbined naturalfrequencyof the systemslm- ilar work canalso befound in Yagodaand Kerchman(HJ82). Inthisfcrmularlon Den-Hartog'sequationgivestwo valuesfor critical speeds oneforfor wa rd whirland anotherfor reverse whirl;italso accountsfor thepropeller shaftforwa rd bearing end conditionswhichcanbe assumedto be fixed orsimplysupported. The forwar d and reversewhirlcorrespondeespccr ivelytotheconditionswherethe propellershaft excitation frequency equals plus orminus times the product of blade number and ro ta tional speed.

lIaddara (1988) treatsthe whole propellershaftsimultaneously andfindsthe coupledtransversevibrationof theshaftingin the horizontalandverticalplanes.

The propeller isappropria tely modeledas a rotor.The shaftisconsideredinthree parts:firsttherotor is takenwhereint!le coupled naturalfrequenciesare obtained by takingiuto consideration the rotaryinertia.The othertwo par ts are the overhung

shaftandthetailshaft.The three segmentsarelinkedby theboundaryconditions.

In thisworkthe effect of propeller mass on thewhirlingnat ural frequenciesof the propeller shaftcanbe clearly seen;for theshaftshaving small tail shaftsthe values of for wardandreverse whirlare found tobe distinctunlikefor tile shafts withlong tailshafts,wherein thetail shaftmass has an overridingeffect.

1.2. 2 Blade Analysis

As stated before theproblemorfindingnaturalfrequenciesofpropellershaftwas solved by couplingthe corresponding nat ural frequenciesofthe blades andtheshaft.

Theblades assumed for thepresentwork areofairfoil section.Thisproblem has not been dealtwithin theexisting literat ure. Subrahma nyamet.al.(1982)used the Reissnerandthepotentialenergy methodfor11manalysisof lateralvibrations of a rotatingblade taking sheardeflectionandrotaryinertia intoaccount; this study has beendone onlyfor beamsof uniformcross-sectionalarea.

Fox(1985)inhis work on rotatingca ntileversconsidersageneralcase ofcompact beamswhereintheflexuralstiffnessin thetwo perpendiculardirectionsarc bot h finiteandofthesameorder so that the assumptionthatthe flexural vibration is confined toasingle plane is nolonger valid.Thisleads to a pairof coupled eq ua tionsofmotionwhichcanbe deeoupledfor some specialcases of pitchangleor cross-section;theresultsobta ined forthese specialcases have beencompa redwlth thoseavaila blein tileliterat ure.The solutionoftheequa tfoushas been carriedout using Galerkin'stechnique.

Morerecently,the bending vibrationsof a class of rotatlng bea ms hasbI'C1l dealtwithby Stortiet.al.(1987).Themethoddependson Iactcrtng

urc

fourl h- orderdifferential operato r,whichappears in the equationof motion,intoa pair of commutingsecond-orderope rators.These twosecond-order differential equations arcthensolved intermsof hypergeometricIuucuons . Thefactorizatio n ofthe fo ur t h order operatorcanbe achievedani)'inCa.5{'of selectedbladeprofilegeometry.

Moreover, the effectsofshear deflection and rotaryinertia have not beentaken lute consideratio n.

1.3 D escr ipti on or The Inves t igation

The workofthi~investisation was carr iedout instar;~which serve115in stu d )'ing thediifert'DtlIISJl«Uof the Iyslem.Tile sta&e5 are givenas:

1. Themodalanillysil of the prope lle r bla de.The blade has beenideal ised as a/Ion uniformcantile\'er!Hoamwithan airfoilsection. Thet"ifed s ofbladerot ation , bladeplrchanglesellin,;,sheardeuecucnandrotary inert ia havebeentakenint oconsldera tlcawhilesolving for thenat ural frequencies andthe modesha pes oftheblade. Themethod em ployed tosolve the problem is based onallimp rovisat ionover Galerkl n'a tech- nique.The assumeddisplacementfllllct iollisadju stedin lter arlo ns ttll the extr act edeigenvaluesbeco mefairly consta nt in theirmagnitudes.As a ptclimiuary cheek finiteelementanal ysisusing taperedbeamelem ent s wascarriedout, whichgaveveryclose comparisons.

2.The anuysi.of the bladestat icdeflectionsand Itrt'!>5t'!l.

3.Tile3-Dfiniteelement analysisof thebladeto check thevalid ityofthe assumption of blade as a ca ntileve rbeam. 2O'noded isoparamal('ric brick elemenh have been employedfor tile! ana!)'.i •.

4.The modalanal)'sis of the sha ft-rot orsystem.Tht"shaft wascousid ered toconsist ofa rotor as anoverh ang withlheforwardendbeari ngofvarl- a'Jle fixity.The eff«ls of rotary iut'r tiaandshea r dt'lI« t ionrOttilerotor ha vebeenconsidere d.In thistypeofa syste mIht' trans versevib ratjon s takeplaceintwoperpendicular pla nesandthl'equationof moti o nI!:OV' emlng themhashcon obt ainedinthe complexplane.The criticalspeeds obtainedthus corres pond tothe forward and reversewhirlofthelihafl.

S.After havingevalua ted theres pecuvcfreqllt'ncieRofHIt' twoJlUb-5y stems ofthepropellerlihaft, nam elytheh1a d('l andthesh,, [l- rolorsystem, an extendedSouthweU-Du nkt'rII'YmethodRivenby Endc and Taniguchi ,

(1976)is employed toestimate thenaturalfrequencies of theassembl y.

Foremploying thismetboda criterionIlMIbeen satisfied,whichjustiflcs the useofthisappro ximation foroursystem.

6.Fora marinepropellerit isnecessaryto consldoraddodmassandadded moment ofine rtiain thedeterminationofthe natur al freq uencies. How.

ever,the solution of the hydr od ynam ic problem is beyond the scopeof t.his work.Recourse to empirica lvalueswasmadeas given byHaddara (1988),Bro oks(1!J80)and Toms(1072).Asensitivity analysiswas car- ried out10lnvestlgat etheImporta nce ofcll!c,.: ating the addedmassof thepropelleraccurate ly.

Chapter 2

BL ADE ANALYSIS

2.1 Accel er aelonAnal ysis 2.1.1 Intr oducti on

Inthis sectionthe expressionsfOTacceleration of a pointon the shaft centerline and on the blade, in the rotatingcoordinatesystemhave been derived. The effects of shaSt bending and shaft vibrat ionshave beenta ken intoaccount.The coordina.te systems fixed to the shaftcenterline and blade have been described in appendix A

2.1.2 Accelera t ion ofaPoint ontheSha ftCenter Line

The angularvelocity of the coordinatesystemxyz, fixed tothe shaft center line.

withrespect to the fixedreference frameXYZ ~sgivenas:(tefer appendix A)

(2.1)

The coordinatesof a pointonthevibrat in g shaftcenter line, which is alignedwith za.xis aregive n as:

i=u.I+v.J (2.2)

where:

avo (2.3)

'l'=-7);

8=

7;;

^(2.')

u.=deflectionof theshalt centerlinealongXaxis,dueto vibrat ions v.:=deflectionof theshaft centerlinealongYaxis,due to vibrations dot s represent differentiat ionw.r.t time

Afterthe transform ationofcoordinateaxes andunderthe assumption:

cosfJ::;"COS,p::l }

sin6R: fJ,s in,p :>::,p

we get:( refer toappendix D)

f:=u, i tv,l

Thevelocity of a pointon theshaftcenterlineas in Torby (1984)isgivenas:

The last ter m e-ppeers sincetheshaftcenter line hasan angular velocity01

I

^{i j}

^k I

111Xf:=

J

8 0 '"(v.¢ -u.9)k u. v, 0

~o (productof twosmall quan titiesisnegligible).

f:=

u ,i+v,i

(2.' )

(2.6)

u.itii,l (2.7)

2.1.3 Acceler at ionof a Point on theBlad e

As given in thefig2.1, coordinates ofa point onthe propeller bladecan be written

(2.8)

From equation(2.5) thecoordinatesofa pointon the shaft center line are:

r=u,i+

v.i

In the rotatingcoordinat e systemit takesthe form?=U~i2

+

V~2,such that

{ .: _u. } [

_{= -}

00'",

_{sin n! cosnt}

^'inn, 1{

^{u, }}_tl.

where:

u~=shaft deflectionsin the %2direct ionin the rotatingframe%2 Y 2Z2 v~

=

shaft deflectionsintheY2direction in the rotat ingframe%2Y2Z2

n

=rotational speed.;>(coordinate frame %2'1/22;2

The coordinatesofapoint at the bladerootare givenas:

(2.9)

(2.10)

Considera pointP011the blade ata.distan ceTffrom the blade root,as shownin fig.2.1.ThepositionvectorofPintheblade localcoo rdinate systemcanbe given

usingequa ti on(2.10) and (2.11)inequation(2.8)we obtain:

(2.11)

in

o

FiglHe2.1:Bladecoordinatesystem

11

where:

R=hub radius

u~

=

transverse deflectionof point P in { direction w~=transverse deflection of pointPin ( direction

e"e2,e3=unit vectorsofthebladelocal coordin ate frame along {, 1], ( respectively.

Equation( 2.12) gives the absolute displacemuJ\tof a point onthe blade in thero- tating coordinatesystem,whit-Itwhen transfcrmodto bladelocal coordinate fra me, takes theform:(see appcndlx III

where!{Jis the pitch settingangle of the blade on thehub.

The angularvelocity of the referenceframeX2Y2Z2rotatingat

n

is given as: ( refer appendix A)

Sincethe coordinate fra me fixed to theroot of blade has time invariant transforrna- tionsR and!{JfromZ2Y 2Z2reference frame,the angular velocityof {1]( coordinate frame is givenas:(see appendixB fordetails)

l'h ¢i+8j+nk

CI

[¢

cos111cos

v

+8sinOtc.os¢- I1sin¢]

f-ea[-~sinnt+8c.osnll

+e3[~cosntsin¢+8sinntsin¢+ncOS\liJ (2.14)

12

from eqution (2.13),wehave

;, (tiI~cos,""

+

ti, )el

+

V~i1

+

(u~^sin

"+

W,)i3+

l'h

x

r,

+[V,+n{u~sin!b+v',lsinT/1+n( u~ cos !JI +tlb)cosIb] lh

+(u~sin"

+ w,1+

(Ii+"llcf,('osu!

+

9sinl1tlcostP-11(v~

+

R+'1)

sin1/l]

i3 from the aboveexpress jce ,f~isgiven as:

i,

== (ii~ c05" +ii.)-(R+ '1)(~roI f!l +jhin nt)sin., -(R+'1)(-¢"innt

+

9cosnt)O,in ,,-nv~cos "lil +(~~+fi{u~,in "+w.)sin"+n( 1i~

COl "

+t4)cos

,, 1

i1 +l(ii~sin"

+

iii, )+(R+'1)(~cosnl

+

BsinfU)cosw +(R +'1){-¢sinnt +9cosn!lncos ¢-Ilt1~sin ¢li3

+n

3x

t,

(14-n1(w" in t/lcOll"+UbCOS1¢)+(v~-2nt1~-n1tl~) cos .p -(R +'1}(4lCOlllt+B'innl)sin TP

+2fi(R+'1)(~sinfit -

6

cosnl) sin

"lit

+

I~:

+ u:

+n(2 wl sin¢+21i, C05,,+ 1i~-

v :-

^R-11)!i 1

+ lw, -

n 1( ut, in" c05¢

+

WISi1l1,,)+(ii:-2n~-n1,,~) .in"

+(R+'1)(~COIllt+isinllt )cos ljt

where:

-20(R

+

1J)(¢liin fit - 8 cosUl) cos1/.l!t3 (2.15) (2.16)

l{=accelerationof a pointontheblad e along

t

axis inthe~I]{local coordinate systemoftheblade

ii"

=

accelerationofa point on tbebladealong71 axisintbetq(localcoordinate roystf'm oftheblad e

13

ii(

=

acceleration ofaflointon the hladc along ' L'lisinthee",~coordin;a.te syste m ofth('hhulr

(for detailsrefertoappendixC )

Equation(2.16)r;iVe.ltheexrressklll forll.bsoluteaccelerat ion of a pointonthe bladeata distance'1from the bladeroot.

Using equll.tions(203J,{2.4)and(2.9)we have:

4ain Ot-8cos Ot 8'

OZ{)I I-".si nfl t -u. cos l1tJ {)2 U'

-Dl{)~

(2.17)

(2.18)

using eQ\lations(2.17) and (2.18)inequation(2.16)wehave:

2.2 Equa t ion of Motion 2.2.1 Intro d u ct ion

(2.19)

Inthis sectionthe equationof motion ofa.rotat ing blade,havinga.ncn-unlfeem cross-sectionalarea and moment ofinertia,hasbeenderiv ed using Newtonian ap- proach.Theeffects ofcentriful~alforce due torotalion ,pitch sett ing angle,shear defor mationand rotaryilil'r li ~havebeentakeninto considera tion.The. propeller bladehasbeen treatedat;:~f>lnlil l·...l·rwithIwndingstiffness a.boutthe majorprin- cipleaxissufficiently largeto ignorebending about this a.xis.The accelerationof the blade is considered purelydueto bladevilJrationsand excitatklnsdue to the

1·'

shaft vibrationshave been ignored since weare interested intheisolated natural frequenciesoftheblade.Thebladeis assu med to have a vanishingthickness at thetip (which simplifiesthetreat ment of boundaryconditions )and a coincident mMS and elastic axes(doubly symmetric),so thatthe torsionalcouplingcan be eliminated.

2.2.2 MathematicalFormulation

Consider thefree-bodydiag ram of a blade element in(1]plane ,as giveninfig.2.2, where in thecoordinate system (1/(is rotat ingwiththe bladeandhasits origin at the blade root.The acceleration of theelement isobtainedfromequat ion ( 2.19) after ignoringtheshaft displacement s duetovibration s,bladedeflect ionalong

e

axis being verysmall in magnitudehasbeen neglected.

Equilibrium of forcesin the( direction yields:

v

cos ~-(V +dV) cos rP+(N +dN ) sinO-NsinO

=

(md,/)( (2.20)

m(=-~ +~(N SinO)

^(2.21)

Equilbrium of momentsin the '/- ( planeyields:

where:

(M

+

dM)-(V

+

dV)d1]- M

pi¢,

EI~

15

(2.22)

(2.23)

(2.24)

(2.25)

(2.26)

1..- '7

Figurl! 2.2:Free body diagram of the bladeelement

IG

V shearforce

~ slope due to bending alone

N centrifugalforce,(referto appendixI )

e

slope due to bendlng and shear tot.lvirt ualmUfperunit lengthoftheblade

M bendingmoment

p density

J area.moment ofinertia E modulusof elast icity

w .

transversedisplacementalong ( u1s fI hubrotationrate

dotsoverthe variable denote differentiationwithreepeetto time.

AscanbeseeninRIlO(1986)the shear forceVdecreases the slopeof thedeflec tion cur ve by anangle1. Nowtheslo pe ofthe deneencncurveduetobot hbending and shear isgiven byq,-1.hence:

(2.27)

shear angle1is givenu :

V 1=KAG V=

(t1J -~)J;AG

where:

1 alleu angle

K shapeconstant,depending on the ctcse-secricnofthe element A crcss-secrlc nal area

G shear modulus

(romequat ion(2.21 )wehave:

(2.28)

(2.29)

elnceBis small

sinB :;,;tan(}:::::~

11

whereadash over the variable repr esents differentiationw.r.t.1/.

Substituting(2,24 ) and( 2.29) into( 2.23), we get:

2,2.3 Non-Dimensionalization

('.3')

Fornon -dlmea aion aliaatio n w.r.t .spaceand timevariables,we definethe fonowing non-dlruenaicnal parameters:

w,

Z

= T

kN

b1'A('1+

^RJd1/

=

^JiG

"

^p1Vl2

=

^E

"

^plVP

A _ii

^A

I where:

F

I bladelengt h R

=

hubradius

Equations(2.31 ) and( 2.32 ) can be written in the followingnon-dimensional form:

wheretheboundarycond itionsare given as:

18

Z,t'

=

0 at

v

»0

j,A,kN =

0 at y

=

1

Equa tions( 2.34) and(2.33)are two coupledpartialdifferentialequa tions in two variab leszand~. Th ecoefficientsof the equa tions arefunctio ns of the independ ent variabley.Since the exact solutionoftheproblem isdifficult to obtain, a numericaltechn iquebasedOilGaler kin's meth od as explainedinMeirovltch (1967) basbeen used.

2.2.4 Solutionofthe Equat ions

\Ve assumea seriessolution for the variableszand~in termsofthe comparison functions'lk and Pkrespect ively, whereIlk andPk satis fy theboundar y conditio ns ofthe system .

~={;Pke;"'~ '

usingthe above expressions inequations ( 2.33 ) and ( 2.3·')we get:

f: {-w~

^j

Pk}

=

f: ^{{i 2 { i Pkl' - il A}

^{(Pk-q/,)} (2.36 )}

k=1 k=l

Galerk in's meth od of weightedresidualsstates thatthe erroriincorporatedin theabovetwoequat ions,due to theassumedsolutionintegratedovertbe domain shouldvanish.

ie,

on'

^{for r=1. 2,3· · ·"}

(2~7)

(2.39) (2.38)

(2.41) wherethe dom ain 0 for tile11l'i'lWllleaseisthC'lelll;thof the blade varying{rom 0 1<>1.

FOtequation(2.35)

f::

.. t ^{, {-4o.l: A} q~- A

'1/rsin2

"'+£ 11.4

(p", -

'1llY- (kNlltY}

For equation(2,36)

(:: i: . . ^, {-l<JH

^p",-

Ct<i

pi,J'

+

^(I

A

(Pt -

qU }

usingequation(2.37)we get:

J t .. ^{, {-loll A}

^'1"-A.'1",t.in' l/)+{1[,4(Pt-Ilk)}'-

(kNq~)I}

q.d1l'"0 (2.4.0)

and forr

=

1,2,3 ' " ''

J"'

^L.-

.. ^, {

^-1oI

"

^",[p",- t,(Ipu

. '"

+£.A

.

(P/r-q,,)

.}

~d.,=O forr=1,2,3"'11 Boundary conditionsfot the blade are :

.,

.,=0 q,.=O .1

(2.42) On the applicationof boun darycond itions,theabove equa tions taketheIcuow.

Ingform:(asexplainedinappendix D)

-wrt r

^{Aqrq/rdY -Sill'w t}

II

Aq. q",dll

talJo b.

10

20

forr= 1,2,3···n

equa tions( 2.43) and(2.44) reduce to the form:

forr

=

l,2,3· · ·n

wI t L

^jPkp,.dy=(2

t ^t'

^j

p~p:'dy ⁺

^II

t

^II

A

^Pr(Pk-

q~)dy

^(2.46)

k;;10 b110 k=t10

fcrrel,2, 3..·n

addingthe above two equations,we get:

=-( 1

t { ^A [q~(Pk

^-91) - Vr(Pk-qDdy k=1"

+(~ t l i 11~1)~rlg ⁺ t l kNq~tJl.dy

k::1U k::l o

=

⁽¹

t [

^j

p~Pkdy + t r

k""q;fJJ.dy

k::10 k::l1 o

t il

t ^A

^{[(PI.-qD(llr-q:Jdll (2.47)}

21

forr:::l, 2, 3· · ·n

fromequation(2.35),we have :(seeappendix E )

• f: .,P . 2.:-tl Aqkdll+ ·~n2 "'tt Aq.dy

(IAk::l 0 A(I b,t0

(2.48)

where:

PK B :::

si~2.p t A qkdy + qk + ~kNqk

^(2.49)

All 0 A(I

simUllI'lywe have:(referappendix E)

'f p~

^=>.

~

^PKIA

⁺ ~

^{PKIB (2.50)}

where:

.'

PKlA ~^-

4-r,,; ;\.

" A

(2.51)

Equation ( 2.47)representsn equationsfornVaJllesofT,on adding thesen equatiou. ,weget:

22

=

(2

t t

^[Ij

p~p~d, ⁺ t t t,\·q.q~J,

...1 ••l

lD

>:1i= 1

+ "t t ^I' .4 11,,-,,)1,.-,:11',

(252) . :l bl}O

On furth ersimplificationandapplicat iollof boundaryconditions theabo\"('equa- tions as glve ninWa ri\.:oo a ndlIaddara. (1989)canbe written as:[see appendix Pl.

=

t t ^{[ _ si n}

^J,",(I

^{, i}

'/r'fl dY + (2[

i

PRlIJPl\tBdy+

l

^kNq:qk(!g

.=1*", \

1 0

^{0 II}

+

^(I

l ..i.

⁽P""SPRB - PRBfJ:'- Pnr/.

+':'1~ld'l

^(2.53)

'1M )

=

1-)

(2.M)

Bo undary conditionsare appiledto thE'third te rm on the musside,whleh state Ihati = Oal, =1 and,. =0 ilL,=O (det ails9ve1l inappendixII).

2.2.5 Sam plePr oblem

Thesoftwa redevelo ped ',uused tofind thena turalfrequl'ncie5 andmode shapes oftwoblade shllvingaitlOiluctio ns NACA 16-0 18 and'IACAG3J -018as shown infig.2,:)andfig.2.'1.wherein thecord ratios for the blade s are givenin table2.1.

Thebladeshavebeenassumed to beorlength 2.8mmountedonahubof diameter 1.4m for a.7mdiame tertail~lLart.Fig. 2.5 tofig. 2.7andfig.2.8tofig.2.10show thevariationof naturalIrequeedeawlth respecttothehub rotation speed,forthese two bladesfor thepitchangll'Sl'ttings of0.30,45 degret'Srespectively.Thedecreaw inthenaturalfrl'fluenriesdue toshear defor matjon androta ryinertia(anbedearly

I.I .---~---..,

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

o.li L _ _L-_ _L - _---J_ _

- - l_ _ --'-_ _ - J

Figure2.3;made profileatvarious sect io nsNACA16-018

1.1...- - - , - - - -- - - ,

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

O.I L._ _...L_ _-l. L..._ _- ' -_ _--'-_ _---l

Figure 2.4:Bladeprofile at varioussectionsNACA633-018

Table 2.1:Cord lengthsat various bladesections

Bladelength

o

.125 .25 .375 .5 .625 .75 .815 .937 1.

Cord lenlA/Diameter .325 .35 .375 .387 .39 .375 .333 .275

o.

.2

490.00

1jI .

O·

...---

420.00 MOOE

m

350.00

LEGENO

...-...

WITHOUTSO/RI

iii a:

280.00 WITHSD/R I

~ 0

UJ a:u.

210.00 to'

'"

z

MODE

n ;=

14 0.00

:=

70.00 MODEI

5.0 0 \0.0 0 15.0 0 20.00 HUB ROTATIONIR.P.S)

1I •

25.00

Figllre2.5:Vuii.tioll ofnaturalfrequencyfor NACA 16-018, pitchangle

=

0 deg.

1;

490. 00

0/ :

30'

42 0.0 0 ^MOOE

^m ---

350.0 0

LEGEND

in

....-.. WITHOUTSD/RI

0: 280.00

- - WITHSD/RI

~ d

'"

'"

u, 210.00

..:

<t

Z _MODEIf

; =

140.0 0

70.00

MODEI _{M 8}

I 25.0 0 I

20.00 15.0 0 10.00 5.00

O.OOL _ _ -.l..._ _--''--_ _

-'-_._-1.,.,-_...,.-:'-:-:

0.00

HU8ROTATION1R.P.SI

Figure2.6:Variation of naturalfrequency forNACA16·018, pitchangle

=

30deg.

490.001, - - - -

420.00 MODE m

350,00

LEGENO

210,00 (;j

a; 2BO.00

a ~

"'

0:u,

~ z

...WITHOUTSDIRI ...WITH SD/R[

MODEl[

140,00

70,00

MODEI II 8

25,00 20,00 15,00 10,00 5,00

OOgoLO- _ - l, -_ - ll.-_ _L-_ _. L_ _ ...l-

HUB ROTATION IR.P.Sl

Figure 2.7:Variationof natural frequency forNACA 16-018, pitch anglee45 deg.

490.00

0/'

O·

420.00

...---

MOOE

m

350.00

en

^LEGEND'

a:

~WITHOUTSOIRI

~

^280.00

0

...-... WITHSOI RI

UJ

a:

"'

>-' 210.00

'"

z

MODE 1I 140.00

70.00 MODE I

5.00 10.00 15.00 20.00 HUB ROTATION (R.P.SI

25.0

Figure 2.8:Variationof naturalIrequaneyforNACA633-018, pitch angle=0dcg.

30

490.00 '1" 30·

4Z0.00

...---..--

MODE

m

350.00

LEGEND:

(j) ... WITHOUT501RI

a:

_280.00

Ii - WITH SO/RI

0

w

0:

U.

~ ZIO.OO

'"

z

MODEII

14 0.00

::!!=='

70.00 MODE1

25.0 10.0 0 15.00 20.00 HUB ROTATION tR.P.Sl 5.00

0.001L..._ _-L_ _-:l-:-:-_-::'=_-=-:-::-~=

0.00

Figure2.9:Variationofnatural frequencyforNACA 63₃-018,pitchangle

=

30

'<g.

.11

490.00~---·---

420.00

350.0 0

;;;

~

^260.00 d

"'

II:

U.

'"' 210.00

'"

z

140.00

70.00 MOOE

m

MODE

n

MODEI

---~.---

LEGENO:

l<--*WITHOUT SD/RIEFFECTS

- WITHSD/RIEFFECTS

HUB ROTATION IRP.S)

Figure2.10:Variationof naturalfrequencyforNACA 633- 018, pitch angle

=

45 deg,

2.2.6 Hydrodynamic Effects

Asmentionedbefore,for marineapplicat ions ofthepropellershart, hydrodyn amic effect shave been takeninto consideratio n by takinginto account the added mas s and added moment ofinertia effec ts ofthepropeller.Theeffects were incorporated byincreasing the massofthe propellerblades by a factor of25% ascan beseen illHadda ra (1988).Itiswort hmentioni ngheretha t the formulationpresent edfor findingthe nat uralfrequenciesofthe propellerbladecantakeintoconsiderat ion anyempiricalfactorassumed fora particularcase.Tocheckthe sensitivity ofthis factoritwasobserved that a10%changeillthisfactor results only in a1%ehange in thevaluesofthenaturalfrequencies.Thisimplies that the factor assumedgives fairlyclose valuesforthenat uralfrequenciesofpropellerblades in wa ter.

2.2.7 Comparisons

Theeffectsofro tary inertia ands:lear defl(.'(:tionon thena t ura!frequencies ofa rota tinguniform beamhave been dealt ...lt h bySubrahmanyam,et.aI.(1982).On checkingthe exam plegiveninthe paperasaspecialcasefor ourst udy,theresults matc hed quiteclose asarc givenintable2.2. However,theeffects of pitchangle on thena t uralfrcqnencles asgiveninthereferencewere seen to contradictthe results obtainedfromthl1analysis ofthe presentwork, whichwasconfirmedfrom the findingsofStorti(lD87).

Stort i stud iedthebeamsofanon-uniform cress-sectionbut theme thodof solu- tic nhasbeenappliedonlyto aparticula r class of cress-sections foritgets compli- cated in thecase of arbita ry beam profiles. Table 2.3 givesthecom pariso nof tile frequencies with the example taken byStorti,whereinthe discrepancyforthefirst modeisnot morethan.5%,which highlightstile accuracyofthe methodInvesu- gated in thiswork,in estim atingthe naturalfrequenciesandmode sha pesof beams.

havingnon-uniform cross-sectiona l Mea.

Table 2.2:Comparison of naturalfrequencieswlththecase taken in Subrahmanyam, 1982

Hub Rotation

=

MO.35rad/ Jcc WithoutS.D&R./ lVithS.D&R./

Mode No. cos7/J Refer ence Calculated Reference Calculated

1 0 5800.04 5776.75 5576.83 5710.79

1I 0 35565.0 35573.87 33832 32618.07

III 0 99365. 6 99391 89677.6 82754.56

I .45 5788.73 5788.16 5746.09 5721.6

II .45 35563.3 35576.43 33832.0 32619.98

III .45 99365.6 99393.49 89677.6 82755.04

seemstobeerroneous

Table2.3: Ccrnparlsonofnatur al freq uencies with the case taken inStor t i,1987

HubRot. 18/,IJ,,<lr (I1Z) 2nd/tfode(II Z)

(HZ Reference Calcu la ted %error Ilofercncs Calculated %error

5 241.5 2'12.7G .5 783 793.62 1.3

10 242 243.29 .5 7R5.6 794.18 1.0

15 243 244.1 .4 7~7 795.24 1.0

20 244 245.4 .5 'j'89.2 796.7

.s

25 247.5 246.97 .2 795.5 708.58 .3

34

2.3 Mo de Sh apes

The ith modeshape for the propellerbladeis givenby:

fori

=

1'"4

where:

a;~(k

=

1- 4}

q.

u .

PI

AI I

l'igen\'ector{or ... givenvalueof;

coslJI<1/-COShJ3k'

+

u. (sinh1J.,1-sinlJ~Y) coetJ.l +coshlJkl

IinlJk1

+

li nhJ3l:1 A1w'

EI

averagetotalvir tual massperunitlengt h ofthe blade average areamomentof inertiaofthe blade natural frequency of theblade

TIlC~mode shapescbmncdusing thepresenttechnique werechecked fo r the case considered b)'Storti(1987)andthefig. 2.11 shows their close agf'('(!ml'nt.

2.4 P r eli minary C heck Using Finite Element Formu- lat ion

Forcheckingthe accuracy of themetho dpresented earlierinthisch..pte r,finite cleme nt analysisof the bladeusingbeamelements""as earned out.Theanalysis Walldone usingthefinil~cle ment analysispackage ANSYS.

2.4.1 ElementSelect ion

To accomodatethencn-unllormarea of the blade,3·nta pered unsymmetrical beam clementswereused.The bla de"'liSdiscrerlzed lnte~Osuchelements.

35

oo

- x-

~ - !J.-

C

--<!>- - { ! } -

o

'"

.;

o

'"

y

/>IOOE II IN REF.

/>lODE II />IOOE I IN REF.

/>IOOE I

o~

0.80

0.20 0.40 0.6 0

BLADE LE NGTH

y . +-:-,--- r-:-:- --.-- - ---.- - .---- --I

0.00 1. 0 0

Figure2.11:Comparison of mode shapeswith the casetakenin Storti,1987

:16

2.4.2 Elemen tProp er ti es

Thebea m clement is defined by two nodesI.Jascan be seen inthefig.2.12.I andJrepresentthe nodesIlt the ends ofthe element.Six degreesof freedom.three rotationalandthree translational.are definedat nodesIandJ.

Otherpara mete rs that are defined for each dl'mentare :

I.Areaat nodesIandJ

il.Area momentofinert ia at nodesIand J

iii.SllI'ardeflectionconsta ntdependingon thecross-sect ionof theele-

The compute rprogra misrunintwostag es, inthefirs tstage itgenera testill' stressstiffnessmatrixfromtilenodalforces exerteddue the bladerot ation.Inthe second stagethe strl'SsstilTne!iS mat rixisusedto findtile eigenvalues and elgenvee- torsofthe system .

The comparisonofthe naturalfrequencies computedwit hthe met hod presented in this workwiththeseobtainedIrom beam finile elemen tanalysisis shownin Fig.

2.13.The mode shapes giveninfig. 2.14also much well.

2.5 BladeDeflectionandStresses 2.5. 1 Intro d uct ion

The propellerbladesshould satisfythl' stren gthrequiremen tstowithstandlong periodsof arduousscrvlcewithoutsufferingfailureor permanentdistortions; also thedeflect ionsunder theloadshouldnot alterthe geometric sha peof the blade.At the sametime.propellersshouldnot be excessively strongincurringhighWeight and costand possiblyprejudicinggoodhydrodyna micd~ign .Inviewofthisit becomes impera tive to dete rmineblad ed~n t'Ctionsandst resses. The methodpresentedin thls secti onlor determiningthe stat icblade delloetjonsIIllltlstudystalestressesis basedon tilemodesum ma tionprocedure. For givinga,ufficientlyaccurate vallie

U

z

Figure 2.12: 3-0tnpe rcdh eam clement

y

•

420.0 0 . - - - -

MODE

:m:

350.00 ^{ill' iI It}

=

LEGEND'

...--.PRESENT WORK

~F.E.M .

iii

280.00

Q;

! o

UJ 0: 210.00 u,

>'

"

^Z

MODElL

140.00 ^{II ? lie}

II

--=

70.00

MODE I

25.0C 10.00 15.0 0 20.00 HUB ROTA TIONIR.P.Sl 5.00

0.00L _ _

l.-_-:l--_.L..---:-1.:-::---::'-:-:

0.00

Figure2.13: Comparisonof naturaJIrcquencleswith finite eleme ntanalysisusing tapered beam elements

3.00

F.E.M.

z

PRESENT WORK

Q

f- 2.00

l.l

'"

..J

...

'"

0

..J 1.00

'"

z

0 in

z

'"

"

a

z

0

z

-1.00

-2.00

0.80 1.00 0.60 0,40

-3.00':-=---,:-=-::---0.20 --,...1.",...- -:-':-0---:-'=-=----,-' 0.00

NON OIMENSIONAL8LAOE LENGTH

Figure2.14: Comparison

or

llIoMshapeswithlillll('dementanalysisusinglapl'[t-.\

beamelements

10

forthe displacement the firstfourmode shapesobtained fro m themodalanalysis weretakenfortilesumma tio n.

2.5.2 Formulation

We considerthe case of a cantileverloaded by a distributed forceP(.r,tl,whose equationofmotiondescrlhlug tit", tra nsversedisplace mentis:

(Ely"(z,t »)"

+

m(z),ij(z,l)::: F(z,1j

=

Pf(z)g{t) (2.55) where:

E modulusof elasticity [ area mome nt of inertia Y transversedisplacement

x spatialcoordinateaJong the lengtll of the blade m total virtualmass per unit lengt h of theblade

, and' representdlfferentla ricn w.r.t.spaceand timevariablesresp ecti vely f(x) spatia lfunction oftheforce

f(t) time functi on oftheforce

P a constant

Weassumeyto bethesumoftilefirstIImodes.

Y(X.t)::: ~¢i(Z)qi(t)

where:

,pi uhmode shapefo r the blade qi :::: timefun ction associatedwithilllmode shape

Using in equatlon]2.5.) ) we have :

multiplylngbyrPjamiintegrating

·11

(2.M)

for

i

=1""n

usingthe bounduy condit ion forthe cantilever'fie have:

t

^"EJ _:f j9,dz+

i: £ ~;~jti;d~ = £ PI(z~jg{t)tlz

^(2.57)

i"110 i_I0 0

forj

=

I- n

Thenormalmodes~jand~jareknown to~atisfy theconditionofortho gonality given by:

10 '

m~itPjd:t =

{ °

,.^[or/^it'.ⁱ .

c mIl orJ=I

"

EI~'.'~/~dz ={

0lorj

~

^i.

10 ' J kj•for}

=

^I

using equat lc es{ 2.58 land(2.59 ).equatio n(2.57)lakestheform (2.58)

12.59)

. : ]{7 } +[T

^{mO" ...}

k.... q.. 0

(k){, )+('iJl;)

=

IF) where

Ikj stlfness mat rixofthe can rilevee (A1J massmatri xof the cantilever {P} forcevector {q} displacementvector

° ]{" } {"}

~. 1 ^{= j:}

(2.GO)

(2.61)

In the cueof prc pellerblade,idealized a.sacantilever ofnon-unifor m area,the normal modeswe arrive atareapproximateand hence theequat ions (2.58)and {2.59IarenotsatisfiedfuUr.

Thisisdearfrom equationI 2.&1 )which sta tt'5that:

IKj= 'IAtj

where(h.' Jand[AI)are thestiffnessandthe mass mat rices ofthesystem respectively bavingnonzero off,dia gonalterms.

Ignori ng tIleoff.d ias:onal clements of1/'-)and1M)matrict' s,whichare tM'gligible as comparedto thediago llaJ clements,vve can writeane-qua t io nrc r the propeller Made similarto equationI'.l.60). Takingthe first rour mod esinto consideration equat ion( 2.60)red ucesto the following nondlmeaslon al form.

where :

m~ _o

_m~~

l _{] { ~}

_q~

_{}= {~:}

_/~

_}

(2.62 )

t.

(39(0

f

/,o;.dll ::::B;

p •

pU'J3

rtb nondi mensio nal modesha peor blade non-dimensionalforcedistribut ion along thebladelength

Upon solving thefoureqeaue ns givenby(2.62) the stead ystate den ectionun be givenas :

where w;istheil helgcnvalue.

Usingin equati on(2.56 ) thestead)'statedlsplaeemcntor the blade can UI'!

givenas:

(:l.G3)

The maximum bendingstressil.tiI.sectionin thebladeisgiveniI.S:

....here:

II ([J9i+~412 +tI344+q~dI:

T blade thicknesaI.tthe section

Fig.2.15andlig.:1.16show the variationofthe staticdeflectionsand steady

state stres sesforthe bladeNM'A Hl-Olf!ta kenasasampleexample in thiswork.

The pressuredistributionM~u l H e dforIh!;'studylegivenin Fig.2.17.

The for mulat ion derived for theblade deflccricne and stresseswasusedtocheck these parametersfor a10feet diameterpropellerspinningat300r.p.m. with ah.p.

or18,000giveninConoUy(19GO).Tileexperimental V&!uu te therefereeeewere predictedfrom the mealUremfllt5^aDamodel. Dlade deflect iolll matchedquiteclose asgiveninF~.2.18, whileas blade suesse eat rnicf.sectionscompared "'ell andare giveninTable 2.4,.

o~

O!. - ~

g .,;

G

~

.:

oo

..;

o~

0.80

0.20 0. 40 0.60

BLADE LENGTH

oo

o;-I-=- ---,-- - - .,.---...,--- ---,c---1

0.00 1.0 0

Figure 2.15:Vuiation

o r

$tatic blade deflections

'"

~-r-'- - - -- - - ,

(fl W'"

( f l o (fl W

...

0:

(fl

'"

~

'"

N

0.00 0.20 0.40 0.60

BLADE LENGTH

Figure2.11;:\':Lri;\1iOILofst(~:l(l)'statestresses

-m

0.60 1.00

'"

~,.---..,

'"

o

..

0.00 0.20 0.40 0.60 0,8 0 1.00

NON D IMENSIONAL BLADE LENGTH

Figure 2.17:Load variation of the blade

'"

o..;;~---...,

oo ..; --<'>-

-I!l-

o

'"

.;

REFERENCE PRESENT/IORK

0.00 0.20 0.40 0.60

BLADE LENGTH

^0.80

Figure2.18:Comparisonofstaticbladedeflections withthecasetaken in Conolly, 1960.

Table2.4:Comparisonofstati cblade stresses withtil('casetakenillConolly.1900.

OIadtfwglh Slrus(Pa) Reference Calculated .315 1.115x~~'

1 1.399

^{x 10"}

.625 6.81x107 4.621X107 .815 2.23 X107 2.2xlOT

·19

Chapter 3

ISOPARAMETRIC FINITE ELEMENT FORMULATION OF THE BLADE

3.1 Introduction

IIIthe preceeding chap terformulat ion forfindingthena t ural frequencyofthepro- pellerbladewas based011the assumpt ion that thebladecan be consideredasa.

cant ilever of vary ing cross section.Sinceweare prima rily interested inlindingthe fundament al modeofvibrationtheassumptionis sup posedto be quite valid. To checkthis,finiteelementanalysis oftileblade was carried outusing 20 noded, serendipity ,solid isoparametric elements.Theanalysistakes intoaccountthe in- crease inthestiffnessof thestructuredueto rotat ionaswell as theadded mas s of thepropeller.For theeigenvalueext ract ion,dyn amiccondensat ionwas carriedalit toreducethe C.P.V.time consumed.

3.2 Element Sel ect ion

Bladegeometrybeing quitecomplicated, selectionoflsoparamctrlc clementsWM thoughttobe appropriate,forthefactthatin such elementsthe generalizedcoor- dinatesandthegeneralizeddisplacementsarcrelatedto thenodalcoordinatesand nodal displacements respectively bythe sameshapeInnctlcr•.Twentynoded

.;0

elementwasehoscn10givea.no nlinear dlspla eemcntfield,whichin turn lakMl care ofthe110nlillei..I::romelr)' ascan beseeninuahrec (1987).Tltt!dC'gree'l of freedom at thenod esof11t('element aretrans la t ionalinnature0110 01::all thetllrt'(' exes, bcn ee the eleme ntis a CO cont inuity elemcer.D)'choosingthe dC'g1l:'('ll of freedom for dis placC'ml'nlS attill'nodt'!l alongallth(' threeUMI, We arC'able10~('t the-ot!lt"rmodesof vibra t ionalso, bcside!ithatfor Ihl'tra nsve rsedf'Re<:lioo .obtained withtilebeamfOfllllllalion. Tileclcmeutcharaclerislksare glven in Fig.:1.1

3.3 M a t hematica l Formulation

Thechosenelement isassocia te dwith a loca lcoordinatesysteme,/((shown in Fig 3.1),whereinthe vahwsof

e,

1/ and (va ryfromvlto1 in thecleme nt.TII(' co o rdina te system

xrz

isIIIl' glohalcartesian coordinat e s}1;If'IIi.

TheJisplac C'11\('1I1vectorat apointinIheeleme nt , M inReo (1982) ,isgiv('nIJ)':

{; }=(Nlld)

Theglobal eocrd lnatevecto roCa pointinthe element is glvenas :

{ ~ ^}=INIl'}

where :

(3.1)

(3.2)

gC'IIl'r alizN(lisp ]lIcemf'1l1alongglohalX •axis generalizeddisplar(,11lentalollgglobal )"•axls gell('rali"Ndisplace men talongglobal l •axi~

gene rali zed coord inatesalongglobalX·a.xis genet allzedeoordlnatcsalong glo bal)'.axis gencmllzcdroordinatesalong globa lZ•axis d = 1 .Jdal displace m en tvec tor

N = s1lapcfunr lion matrix r nodal coordinatevector

Forgettingthe<,igel1\'alllejland 111<'('igen\"('('tol'5cfthcsystemWI"111'<-'(110get thestiflnessaed tIll'maNomatricesofthe system.

51

,',",!

The stlffuess ofthesyst omconsistsof twomaulee s :

l.!k<]the clement srifluessmatrix depending 011the materialproper tie s andglveuas:

(3,3)

2.Ik~1theelementstillne ss matrixresultingduoto 1herotationof the blado. Thls mat rixiscalledthost ressstiffnessma-rlsandlsIndcpendcntofthe mater-ial propertics.Thismatrixisgiven>\5 :

[' II "]

[':1

⁼j[G)T 0 , II[Gld,.

v 0D "

Thcclomontmassmatrlxol thosystemis gin' ll as

where :

In] stra indisplacement matrix volume cl rheelcmeru dvust t y ol thc matcri al

1'1

[G]{ " )

{~} ^{II,rIl,~"" V,rt',~t'"w»//I,yII',,}

[D] nrutc r-inlproper ty matrix whoseoxpro ssionisgivenas:

(3..1)

(3 . f~

E(I -I' ) (l+v)(I- 'lv )

1-V I_II

~Yllllll(/rir

i- ' ll' 2(1 -1')

1-2/, '2(1-v)

1"1+ 1".1 = ' [.III

where :

[Ill convoutlcna! globalslilfn('Ssmat rixolthesyste m [I\~J globalstress stiffncss mat rix of the S,\·st l' lli [M] globalmassmat rixof tll('syst e m A isthoeigenvalue.

3.4 Form ulationof

p rj

Matrix

(:I.Ii)

'l'beshape fUllct iouortheinl t'T!mla li()lIfnnrtlnnmatr ixfortho'lllll()(I('disoparn- metricsolideleme ntisgivonas:

mont,andillth.. nailiralroordiua! r- s)'stl'llIare~iI'('lIlIS:(rl'f(·rtoCook,I!J~7 )

.v;~ ~( I

^{+«i)( 1}

+

1/1/0)( 1

+

«,)( {{,+ IJI/i+

«;

-'2)foraror nerumh- alld

s.

^:=~~ I-

ell: +

^{1111i)( 1}

+ « il

foralypk a lmiclsid{'lind,·Il;h"('uh~' {i=0,11;=-1.( ;=- 1.

From (3.11and( 3.2I,we have

0 , 0 ,

w ,

U} = _{[ ~'}

^{N,00 N,00 N,00 N,00 N001}

^...

^{N"00 N"00}

JJ ^{0 "" '" ,}

^(3.7)

0"

"'"

"",

"

r,

z

{: } =[ ~' ^s,

^{00 N,00 N001 N,00 N,00}

^{... .. .}

^{N"00 N"00}

JJ ^{" " "}

^(3.81

'"

'"

3.5 Formulationof Strain DisplacementMat rix

[B]

Ma.trix

r BJ

relat esthe ur.l.inilltheelement to

ll lt'

nodal displacements.

(.) =

^{(B/{d ) (3.91}

3.5 .1 Stra inVect or

Thestrainvector{e}fora3Delementisgiven as :

5,