Integral transform solutions of dynamic response of a clamped–clamped pipe conveying fluid

ContentslistsavailableatSciVerseScienceDirect

Nuclear

Engineering

and

Design

jo u r n al h om ep a g e :w w w . e l s e v i e r . c o m / l o c a t e / n u c e n g d e s

Integral

transform

solutions

of

dynamic

response

of

a

clamped–clamped

pipe

conveying

fluid

Jijun

Gu

a

_,

_Chen

_An

a

_,

_Menglan

_Duan

c

_,

_Carlos

_Levi

a

_,

_Jian

_Su

b,∗

a_{OceanEngineeringProgram,COPPE,UniversidadeFederaldoRiodeJaneiro,CP68508,RiodeJaneiro21941-972,Brazil} b_{NuclearEngineeringProgram,COPPE,UniversidadeFederaldoRiodeJaneiro,CP68509,RiodeJaneiro21941-972,Brazil} c_{OffshoreOil/GasResearchCenter,ChinaUniversityofPetroleum,Beijing102249,China}

h

i

g

h

l

i

g

h

t

s

 Dynamicresponseofpipeconveyingfluidwasstudiednumerically.

 Thegeneralizedintegraltransformtechnique(GITT)wasapplied.

 Numericalsolutionswithautomaticglobalaccuracycontrolwereobtained.

 Excellentconvergencebehaviorwasshown.

 Modalseparationanalysiswascarriedoutandtheinfluenceofmassratiowasanalyzed.

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received4January2012 Receivedinrevisedform 11September2012 Accepted30September2012 Keywords:

Pipeconveyingfluid Fluid–structureinteraction Internalfluid Clamped–clampedpipe Integraltransform Modalanalysis

a

b

s

t

r

a

c

t

Analysisofdynamicresponseofpipeconveyingfluidisanimportantaspectinnuclearpowerplant

design.Inthepresentpaper,dynamicresponseofaclamped–clampedpipeconveyingfluidwassolved

bythegeneralizedintegraltransformtechnique(GITT).Thegoverningpartialdifferentialequationwas

transformedintoasetofsecond-orderordinarydifferentialequationswhichisthennumericallysolvedby

makinguseofthesubroutineDIVPAGfromIMSLLibrary.Athoroughconvergenceanalysiswasperformed

toyieldsetsofreferenceresultsofthetransversedeflectionatdifferenttimeandspanwiseposition.We

foundgoodagreementbetweenthecomputednaturalfrequenciesatmode1–3andthoseobtainedby

previoustheoreticalstudy.Besides,modalseparationanalysiswascarriedoutandtheinfluenceofmass

ratioondeflectionandnaturalfrequencieswasqualitativelyandquantitativelyassessed.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Pipesconveyingfluidsareessentialpartsofa nuclearpower plant.Flow-inducedvibrationofsuchpipesisamajorconcernfor design,operation, maintenance,andsafetyanalysisofanuclear powerplant.Afluidflowinginapipemayexertforcesonthepipe wall, deflect thepipe, causevibration, and eventually result in long-termfatiguefailures.Analysisofdynamicresponseofapipe excitedbyinternalflowingfluidisanimportantaspectinthedesign ofnuclearpowerplants,sincevibrationsrelatedtofluid–structure interactionmayleadtopiperuptureandthuspossibleaccidents. Themainconsiderationmightfocusonthenaturalfrequencyand modeshapes of pipe, as wellas the critical fluid velocity. The

∗ Correspondingauthor.Tel.:+552125628448;fax:+552125628444 E-mailaddresses:[email protected],[email protected](J.Su).

accuratepredictionofnaturalfrequenciesandmodeshapesofa pipe willhelpselect thepropersupportlocations,which could reducetheoccurrenceoffatigue-relatedpipefailures.Meanwhile, acriticalfluidvelocitywillcausesysteminstability,whichwould inducethepipesfailsuddenlyinashorttimeduration.

Variousaspectsofthedynamiccharacteristicsofpipes convey-ingfluidhavebeenextensivelystudied(Ibrahim,2010,2011),as wellascantileveredpipeconveyingfluid(Païdoussisetal.,2002, 2007;Lopesetal.,2002;Semleretal.,2002;Wadham-Gagnonetal., 2007;Modarres-Sadeghietal.,2007).Inthenuclearengineering fields,Kang(2010)studiedtheeffectsofrotatyinertiaof concen-tratedmassesonthenaturalvibrationsofafluid-conveyingpipe bytheoreticalmodelingand numericalcalculations.Sinhaetal. (2005)derived acombinationof alinearand anon-linear opti-mizationmethodsforthepredictionoftheflow-inducedexcitation forcesandthestructuralresponseateachmeasuredfrequencyall alongthepipelength.Huangetal.(2010)investigatedthenatural 0029-5493/$–seefrontmatter © 2012 Elsevier B.V. All rights reserved.

Nomenclature

Aij integraltransformationcoefficient

Bij integraltransformationcoefficient

EI flexuralstiffness

f dimensionlessnaturalfrequency

i index

j index

L lengthofthepipe mf massofthefluid

mp massofthepipe

N truncationorder Ni normalizationintegral

t dimensionlesstime

T time

u dimensionlessflowvelocity ucd dimensionlesscriticalflowvelocity

U flowvelocity

y transformedlateraldeflectionofthepipe y dimensionlesslateraldeflectionofthepipe Y lateraldeflectionofthepipe

z dimensionlessaxialcoordinate Z axialcoordinate Greeksymbols ˇ massratio ı coefficient  eigenvalue  eigenfunction ˜  normalizedeigenfunction ω naturalangularfrequency Acronyms

DTM DifferentialTransformationMethod DQM DifferentialQuadratureMethod FDM FiniteDifferenceMethod FFT FastFourierTransform

GITT GeneralizedIntegralTransformTechnique HPM HomotopyPerturbationMethod

IMSL InternationalMathematicsandStatisticsLibrary

frequencyofpipeconveyingfluidwithdifferentboundary condi-tions.Lietal.(2011)analyzedfreevibrationofmulti-spanpipe conveyingfluidbydynamic stiffnessmethod.Zhaiet al.(2011) analyzedthedynamicresponseofpipelineconveyingfluidunder randomexcitation.

Fornumerical simulationof dynamic responseof a beamor pipe,manydifferentmethodshavebeenapplied.Galerkinmethod isusedwidelytodealwiththislineardifferentialequation prob-lems, see Lopes et al. (2002), Wadham-Gagnon et al. (2007), Modarres-Sadeghietal.(2007)andHuangetal.(2010).Besides, many other numerical methods have been proposed, such as differential transformation method (DTM) implemented by Ni etal. (2011)toanalyzevibration ofpipesconveyingfluid, pre-ciseintegrationmethod(PIM)performedbyLiuandXuan(2010) toanalyzesupportedpipesconveyingpulsatingfluid,aswellas homotopyperturbationmethod(HPM)(Xuetal.,2010)and dif-ferentialquadraturemethod(DQM)(Linetal.,2007;Qianetal., 2009).However,therearenopreviousstudyendeavoredto per-form vibration analysis of a clamped–clamped pipe conveying fluid based on generalized integral transform technique (GITT) approach.Thisapproach,withitsintrinsiccharacteristicoffinding solutionswithautomaticglobalerrorcontrol(Cotta,1993,1998; CottaandMikhailov,1997),openedupanalternativeperspective

Y U y z L o

Fig.1. Pipeconveyingfluidwithclamped–clampedboundarycondition. inbenchmarkingandcovalidationforsuchdynamicresponse prob-lems.Maetal.(2006)appliedGITTtosolveatransversevibration problemofanaxialmovingstring,AnandSu(2011)performedGITT toanalyzedynamicresponseofclampedaxiallymovingbeams,and convergencebehaviorofintegraltransformsolutionwasexamined. At the present stage, GITT methodology was already success-fullyemployed in thesolution of bendingproblemof clamped orthotropicrectangularplates(Anetal.,2011).Theresulting sys-temoftransformedfunctionsthenofferedanordinarydifferential equationsystemasaboundaryvalueproblem,duetotheparabolic nature of the modelling equations. Following thesame line of research,the present workis thus aimedat utilizing the ideas inGITTmethodologytoconstructahybridanalytical–numerical solution,nowfordynamicresponseofaclamped–clampedpipe conveyingfluid. By adopting thesame fourth-order eigenvalue problem, thetransformed functionssystemresultsin an initial valueproblemtobenumericallysolvedalongtheaxialcoordinate. Themainpurposesare:(1)computationofdeflectionand natu-ralfrequenciesatseveralmodesforvariousvaluesofinternalflow velocityandmassratio,followingathoroughconvergenceanalysis ofeigenfunctionexpansions;(2)criticalcomparisonswithearlier publishedworksforacovalidationagainstthepresentGITTresults. Thiscontributionwillbenefitthedynamicanalysisofflow-induced vibrationsonthedesignandmaintenanceofnuclearpowerplants. 2. Analysis

Consideranelasticpipeconveyingfluidsofaconstantvelocity U,asshowninFig.1.Ifgravitationalforces,internaldamping, exter-nallyimposedtensionandpressurizationeffectsareneglected,the linearequationoftransversemotionY(Z,T)isgivenbyLeeandMote (1997a,b) (mf+mp)

∂

2 Y

∂

T2 +2mf

∂

2Y

∂

Z

∂

T +mfU 2

∂

2 Y

∂

Z2 +EI

∂

4Y

∂

Z4 =0, Z∈(0,L), (1)

where mf and mp aremass densitiesof thefluid andpipe, Lis

thelengthofthepipe,andEIistheflexuralstiffnessofthepipe. Introducingthefollowingdimensionlessvariables

y=Y L, z= Z L, t= EI (mf+mp) T L2, u=



mf EIUL, ˇ = mf mf+mp. (2) Eq.(1)canbewrittenindimensionlessformas

∂

2y

∂

t2 +2



ˇu

∂

2 y

∂

z

∂

t+u 2

∂

2 y

∂

z2 +

∂

4y

∂

z4 =0, z∈(0,1). (3)

Thedimensionlessboundaryconditionsaregivenasfollows: y(0,t)=0, y(1,t)=0,

∂

y(0,t)

∂

z =0,

∂

y(1,t)

∂

z =0 (4)

withinitialconditionsdefinedasfollows: y(z,0)=0,

∂

y(z,0)

∂

t =O(10

−3_{) random noise, (5)}

whereO(10−3)representsarandomnoise.Theseinitialconditions areusuallyadoptedtoanalyzedynamictransversevibrationofa slenderpipe,seeVioletteetal.(2007).

3. Integraltransformsolution

FollowingtheideasinGITT,thenextstepisthatofselecting eigenvalueproblemsandproposingeigenfunctionexpansions.The eigenvalueproblemfortransversedisplacementy(z,t)ischosenas expression(6): d4_ i(z) dz4 = 4 ii(z), 0<z<1 (6)

withthefollowingboundaryconditions i(0)=0, i(1)=0, di

(0) dz =0,

di(1)

dz =0, (7)

where i(z) is the eigenfunction of problem (6), i is the

cor-respondingeigenvalue, theysatisfy thefollowing orthogonality property



1 0

i(z)j(z)dz=ıijNi (8)

withıij=0fori /= j,andıij=1fori=j.Thenorm,ornormalization

integral,iswrittenas Ni=



1 0

_i2(z)dz. (9)

Theselectionofeigenfunctioniscomputedbasedonthe aux-iliaryeigenvalueproblemwiththeboundaryconditionsatisfied, evenforsomecomplexcases(Su,2006).Thenproblem(6)isreadily solvedanalyticallytoyieldtheeigenfunction

i(z)=

⎧

⎪

⎨

⎪

⎩

cos[i(x−0.5)] cos(i/2) − cosh[i(x−0.5)] cosh(i/2) foriodd, sin[i(x−0.5)] sin(i/2) − sinh[i(x−0.5)] sinh(i/2) forieven, (10a,b)

wheretheeigenvaluesareobtainedfromthetranscendental equa-tions:

tanh(i/2)=

−tan(i/2) foriodd,

tan(i/2) forieven.

(11)

Theeigenvalueproblem(6)allowsdefinitionofthefollowing inte-graltransformpair

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

yi(t)=



1 0 ˜ i(z)y(z,t)dz, transform y(z,t)= ∞

i=1 ˜ i(z)yi(t), inversion, (12a,b)

where ˜i(z)isthenormalizedeigenfunction

˜

i(z)= i(z)

N_i1/2.

(13) Now,thenextstepisthustoaccomplishtheintegral transfor-mationoftheoriginalpartialdifferentialsystem.Forthispurpose, Eq.(6),followedbytheboundaryconditionsgivenbyEq.(7),are multipliedby



₀1˜i(z)dz,integratedoverthedomaininz[0,1],and

theinverseformulagivenbyEq.(12b)isemployed.Aftertheusual manipulations,thefollowingcoupledordinarydifferentialsystem results,forthecalculationofthetransformedy(t):

d2_y i(t) dt2 +2



ˇu ∞

j=1 Aijdyi(t) dt +u 2 ∞

j=1 Bijyi(t)+4iyi(t)=0, i=1,2,3,..., (14)

wherethecoefficientsoftheordinarydifferentialsystemaregiven bythefollowingexpressions:

Aij=



1 0 ˜ i(z) d ˜j(z) dz dz, Bij=



1 0 ˜ i(z) d2_˜ j(z) dz2 dz. (15)

Theboundaryconditionsarenaturallysatisfiedbythe eigen-functions.Theinitialconditionsarealsointegraltransformedto eliminatethespatialcoordinate,yielding

yi(0)=0, dyi (0) dt =



1 0 ˜ i(z)

∂

y(z, 0)

∂

t dz, i=1,2,3,... (16) System(14)isnowinanappropriateformfornumericalsolution throughdedicatedsubroutinesforinitialvalueproblems.The sub-routineDIVPAGfromIMSLLibrary(IMSL,2003)iswelltestedand capableofhandlingsuchproblems,offeringautomaticaccuracy control,andforthisproblemtheerrortolerance10−6isselected. Forcomputationalpurpose,theexpansionsarethentruncatedto Norders,soastoreachtheuserestablishedaccuracytargetinfinal solution.TherelatedcoefficientsgivenbyEq.(15)arealsohandled throughIMSLLibrary.Onceyi(t)havebeennumericallyevaluated,

theanalyticalinversion formula(12b)isrecalled torecoverthe dimensionlessfunction ˜y(z,t).

4. Resultsanddiscussion

Thetypicaltimehistoriesandresponsefrequenciesareshownin Fig.2,withdimensionlessparametersu=4.5,ˇ=1.0,andtruncation orderN=64.Thefirstcolumnisdimensionlesstimehistoryofyof thewholerunat5equidistantspaceintervalalongthepipe,the secondcolumnisdimensionlesstimehistoryinaintervalt∈[15, 17],thethirdcolumnisthespectralanalysisofsecondcolumn.As showninthefirstcolumn,thevibrationofthedisplacementkeeps inlong-termstability,andthesecondcolumndemonstratedmore clearlyafairlystablevibration.Severalpeaksofvibration dimen-sionlessfrequencyappearedinspectralanalysis,whichindicates themodeshapehavemulti-modecontributions.

The convergence behavior of integral transform solution is examinedforincreasingtruncationordersN=4,8,16,32and64,the instantaneousdeflectionsatdimensionlesstimet=11areshownin

−3.5 0 3.5 y (10−3) z =0.84 y (10−3) 0 0.2 0.4 PSD −3.5 0 3.5 z =0.67 0 1 2 −3.5 0 3.5 z =0.5 0 1 2 −3.5 0 3.5 z =0.33 0 1 2 0 10 20 30 40 50 −3.5 0 3.5 (a) t z =0.16 15 16 17 (b) t 0 10 20 30 40 50 0 0.5 (c) f

Fig.2.TimehistoryofsimulationatN=64,u=4.5,ˇ=1.0.Firstcolumn:dimensionlesstimehistoryoftransversedimensionlessdeflectionofthewholerunat5equidistant spaceintervalalongthepipe;secondcolumn:dimensionlesstimehistoryinadimensionlesstimeintervalt∈[15,17];thirdcolumn:spectralanalysisofsecondcolumn.

-3 -2 -1 0 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 -3 -2 -1 0

(b)

(c)

u

= 1.5 β = 0.5

u

= 4.5 β = 0.5

z

y

|

_{t= 11}

(10

-3

)

u

= 4.5 β

= 1.0

(a)

y

|

_{t = 11}

(10

-3

)

N = 4 N = 8 N = 16 N = 32 N = 64

y

|

_{t= 11}

(10

-3

)

Fig.3.GITTsolutionswithdifferenttruncationordersNforthedimensionlesstransversedeflectiony(z,t)atdimensionlesstimet=11:(a)u=4.5,ˇ=1.0;(b)u=4.5,ˇ=0.5; (c)u=1.5,ˇ=0.5.

13.0 12.5 12.0 11.5 11.0 10.5 10.0 -5 -4 -3 -2 -1 0 1 2 3 4 5 u =4.5 β =1.0 y (10 -3 ) t N = 4 N= 8 N = 16 N= 32 N= 64

Fig.4.GITTsolutionswithdifferenttruncationordersNfordimensionlesstime historyy(z,t)atu=4.5,ˇ=1.0.

Fig.3withacombinationofdimensionlessflowvelocityu=1.5,4.5 andmassratioˇ=0.5,1.0.ItcanbeobservedthatwhenN16, GITTsolutionspresentaperfectconvergenceintheplotswhere theselinesarenearlyoverlapped.Figs.4–6illustratethe conver-genceofdimensionlesstimetraceinanintervalt∈[10,13]witha combinationofu=1.5,4.5andˇ=0.5,1.0.Itisnoticedthatthe con-vergenceisquitefavorable.However,itshouldbepointedoutthat verylowtruncationordersmaynotcaptureactualdisplacement, suchastheoneinFig.3withN=4and8.

From FFTanalysis shown in the third column of Fig. 2,the multi-modecontributioniscaptured.Itisconvenienttoperform modalanalysisbyGITTsincethesolutionisinformof eigenfunc-tionexpansions.Fig.7showsthemodecontributionfrom1to5 atu=4.5, ˇ=1.0,withtruncationorderN=64.It is clearlyseen thatthe maximumdeflectionsdecreaseswhen themode num-berincreases,andthedeflectionatmode1hasasamevibration amplitude scale withthe original one, which means the origi-naldeflectionsaredominatedbymode1,confirmedalsobythe powerenergyofFFTanalysisshowninFig.2.Actually,the vibra-tionamplitudeapproachesto0whentruncationtermNapproaches toinfinity,illustratingtheexcellentconvergentbehavior.

Figs.8and9presentmodalseparationofdimensionlesstime traceinanintervalt∈[15,17]andu=4.5,ˇ=1.0,truncationorder N=64,atspanwisepositionz=0.5and0.16,respectively.Itdepicts

13.0 12.5 12.0 11.5 11.0 10.5 10.0 -5 -4 -3 -2 -1 0 1 2 3 4 5 u =4.5 β =0.5 y (10 -3 ) t N = 4 N= 8 N = 16 N = 32 N= 64

Fig.5.GITTsolutionswithdifferenttruncationordersNfordimensionlesstime historyy(z,t)atu=4.5,ˇ=0.5. 13.0 12.5 12.0 11.5 11.0 10.5 10.0 -5 -4 -3 -2 -1 0 1 2 3 4 5 u=1.5 β =0.5 y (10 -3 ) t N = 4 N = 8 N= 16 N = 32 N = 64

Fig.6.GITTsolutionswithdifferenttruncationordersNfordimensionlesstime historyy(z,t)atu=1.5,ˇ=0.5.

thatmode2and4havezeroamplitudeswhenz=0.5,whichcanbe verifiedintheFig.8,sincez=0.5isthemidpointwhichkeepsstill atmode2and4.Thesecondcolumnshowsthecleardominated dimensionlessfrequenciesofeachmode.Whenz=0.16,spectral analysispresentsclearlythemulti-modecontributions,andnatural frequencyateachmodecanbeeasilyobtained.

Wenowproceedtowardsthevalidationofproposedapproach againstpreviouslyreportedresultsinliterature.Here,wedefined dimensionlessnaturalangularfrequencyω=2f,wherefis dimen-sionlessnaturalfrequency.Acomparisonofvariationsofωatmode 1–3withincreasingubetweenGITTandtheory(Païdoussis,1998) areillustratedbyFig.10,whichshowsthatωdecreaseswiththe increasingofu,andωatmode1isapproaching0asuisapproaching 2.Thevalue2isdefinedasacriticalinternalflowvelocityucdof

thepipewithaclamped–clampedboundaryconditionby theoret-icalanalysis(Païdoussis,1998),whenu<ucd,theeigenfrequencies

areallpurelyreal,whilstforu>ucdthoseassociatedwiththefirst

modearepurelyimaginary,whichmeansthesystemloss stabil-ity.OurGITTsolutionpresentsadivergencewhenu>ucd,which

verifiesthecriticalvelocity.Besides,itcanbeseenthatnumerical resultsobtainedinpresentworkagreeverywellwiththeoretical predictionsbyPaïdoussis(1998).

Themaximumvibrationdeflectionincreasesastheflow veloc-ityincreases,whichisverifiedbyexperimentGuoandLou(2008). However,thevariationofdeflectionversusflowvelocitywasnot qualitativelyandquantitativelyevaluated.GITTisusedheresince itis convenienttoanalyzedynamicresponseof pipeconveying fluidbothatfrequencyandtimedomain.Fig.11presentsdeflection ratioyM/y∗_Mversusuatˇ=0.1,whereyMandy∗_Mdenotemaximum

deflectionswhenu /= 0andu=0,respectively.Itisseenthatwhen uincreasesfrom0to5,yM/y∗Mincreasesslowlyfrom1to1.5,but

whenitincreasesfrom5andapproachestocriticalvelocity2, yM/y∗_M increasesexponentiallyfrom1.5 to6.Thisphenomenon

depictsthattheinfluenceofinternalflowvelocityhasadramatical increasewhenuapproachesthecriticalflowvelocity,whichshould becarefulinpipedesign.

Finally,weconsidertheinfluenceofmassratioˇondynamic responses of pipe conveyingfluid. Païdoussis (1998) evaluated theinfluenceofmassratioonthecriticalvelocitytoyieldamap of differentkinds of instabilities predictedby linear theoryfor clamped–clampedpipes.However,themaximumvibration deflec-tionandnaturalangularfrequenciesunderdifferentmassratios arealsovaluabletoengineers.Fig.12presentsyM/y∗Mversusˇ,at

u=4.5.ItisobservedthattheyM/y∗Mdecreaseslinearlyfrom1to

−4 0 4 0 0.2 0.4 0.6 0.8 1.0 y (10−3) z Original −4 0 4 y (10−3) mode 1 −1 0 1 y (10−3) mode 2 −5 0 5 y (10−4) mode 3 −2 0 2 y (10−4) mode 4 −5 0 5 y (10−5) mode 5

Fig.7.Modeseparationofinstantaneousdeflectionswithdimensionlesstimeintervalt=1atu=4.5,ˇ=1.0,truncationorderN=64.

naturalangularfrequenciesatmode1–5(ω1,ω2,ω3,ω4,ω5)versus

massratioisshowninTable1.Itisinterestingtonoticethat dimen-sionlessnatural angular frequencyat firstmode(ω1)decreases

monotonicallywhenˇincreasesfrom0to1,whilethoseatmode 2–5(ω2,ω3,ω4,ω5)increasemonotonicallywhenˇincreasesfrom

0to1.

Fromtheexcellentagreementofthepresentresultswith previ-ouslytheoryshownabove,thesolutionofGITTcouldbeconsidered astheexactsolution.Toclarifytheso-called“exactsolution”,itis necessarytomakefinalremarksaboutthenatureoftheproposed approach.Eachtermof ˜i(z)andyi(t) satisfiesboundary

condi-tions,besides,thereisnoapproximationinvolvedintheanalytical

−5 0 5 Original y (10 −3 ) 0 1 2 −5 0 5 mode 1 y (10 −3 ) 0 1 2 −1 0 1 mode 2 _{y (10} −4 ) −1 0 1 −5 0 5 mode 3 _{y (10} −4 ) 0 1 2 −1 0 1 mode 4 _{y (10} −4 ) −1 0 1 15 16 17 −1 0 1 mode 5 y (10 −4 ) (a) t 0 10 20 30 40 50 0 0.2 0.4 (b) f

−1 0 1 Original y (10 −3 ) 0 0.5 −1 0 1 mode 1 _{y (10} −3 ) 0 0.5 −5 0 5 mode 2 _{y (10} −4 ) 0 1 2 −5 0 5 mode 3 _{y (10} −4 ) 0 1 2 −2 0 2 mode 4 y (10 −4 ) 0 0.5 1 15 16 17 −1 0 1 mode 5 y (10 −4 ) (a) t 0 10 20 30 40 50 0 0.2 0.4 (b) f

Fig.9. Modeseparationofdimensionlesstimehistoryy(z,t)inadimensionlesstimeintervalt∈[15,17]atu=4.5,ˇ=1.0,truncationorderN=64,spanwisepositionz=0.16.

derivationofGITTapproach,theproblemsolutionrepresentedby thesummationofarapidlyconverginginfiniteseries(12b) simul-taneouslysatisfiesthegoverningdifferentialequation(3)andthe boundaryconditions (4). Inreal calculations,thederived series solutionneedstobetruncatedsomewhereforcomputational eval-uation.Inotherwords,theproblemcanonlybedealtwiththrough finite series instead of the infinite range in the analytical for-mulation.Therefore,theexactsolutioncanbecharacterizedasa truncatedseriessolution withthedesiredprecision,which can beachievedbycontrollingeverynumericalstepwithprescribed accuracy.Thisautomaticandstraightforwardglobalerrorcontrol

6 5 4 3 2 1 0 0 20 40 60 80 100 120 ω u mode 1 (GITT) mode 2 (GITT) mode 3 (GITT) mode 1 (Paidoussis 1998) mode 2 (Paidoussis 1998) mode 3 (Paidoussis 1998)

Fig.10.Dimensionlessnaturalangularfrequencyωatmode1–3versus dimension-lessflowvelocityu,atˇ=0.1.

proceduremakesGITTparticularlysuitableforbenchmarking pur-posesofsuchdynamicanalysisofstructures.

However,therearesomeothergoodmethods,suchasthetime stepintegrationmethod.Therehavebeentremendousresearches on the method of direct integration algorithms, including the Runge-Kuttamethod,theHouboltmethod,theNewmarkmethod, theWilsonmethod,thecentraldifferencemethod,etc.Theyare allFDM(FiniteDifferenceMethod)approaches,butthefinite dif-ferenceapproximationisalwaysaccompanyingwitherror,andhas alsonumericalproblemincludingstiff,stabilityetc.,notsoideal.For instance,agoodimplementationofaRunge-Kuttamethodrequires anefficientstepsizecontrol.Inprinciple,onetriestochoosethe stepsizeineachstepsuchthatthenewlocalerrordoesnotexceeda

6 5 4 3 2 1 0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 yM / y* M u GITT

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 yM / y* M β GITT

Fig.12.DeflectionratioyM/yM∗ versusmassratioˇ,atu=4.5.

Table1

Dimensionlessnaturalangularfrequenciesω1,ω2,ω3,ω4,ω5versusmassratioˇ,

atu=4.5. ˇ ω1 ω2 ω3 ω4 ω5 0.05 15.6182 53.6764 112.4690 191.0986 289.6548 0.10 15.4387 53.7661 112.5588 191.2781 289.8344 0.15 15.3489 53.9456 112.7383 191.4576 290.0139 0.20 15.1694 54.0354 112.9178 191.6372 290.1934 0.25 15.0796 54.1252 113.0076 191.8167 290.3729 0.30 14.9899 54.2149 113.1871 191.9064 290.5524 0.35 14.8104 54.3944 113.2769 192.0860 290.7320 0.40 14.7206 54.4842 113.4564 192.2655 290.9115 0.45 14.6308 54.5740 113.5461 192.4450 291.0910 0.50 14.4513 54.5740 113.7257 192.5347 291.1808 0.55 14.3616 54.6637 113.8154 192.7143 291.3603 0.60 14.2718 54.7535 113.9052 192.8938 291.5398 0.65 14.1820 54.8432 114.0847 193.0733 291.7193 0.70 14.0923 54.9330 114.1745 193.1631 291.8988 0.75 14.0025 54.9330 114.2642 193.3426 292.0784 0.80 13.9128 55.0228 114.4437 193.5221 292.2579 0.85 13.7332 55.0228 114.5335 193.6119 292.4374 0.90 13.6435 55.1125 114.6233 193.7914 292.5272 0.95 13.5537 55.1125 114.8028 193.8811 292.7067 1.00 13.4640 55.2023 114.8925 194.0607 292.8862

prescribedtolerance.Unfortunately,explicitRunge-Kuttamethods

aregenerallyunsuitableforthesolutionofstiffequationsbecause

theirregionofabsolutestabilityissmall.Thisissueisespecially

importantinthesolutionofpartialdifferentialequations.The

insta-bilityofexplicitRunge-Kuttamethodsmotivatesthedevelopment

ofimplicitmethods,butthecomputationalcostswouldbe

signifi-cantlyincreased.

5. Conclusions

The integral transform method was applied to obtain a

hybridnumerical–analytical solution of dynamic response of a

clamped–clampedpipe conveyingfluid.Theanalysisof

conver-gence behavior and comparison with theoretical results were

performed.Excellentagreementofthepresentresultswith

pre-viously theory demonstrates the convenience and exceptional

computationalperformanceoftheproposedmethodology.

Numer-icalresultsillustratethedynamicresponseisdominatedbymode

1throughmodalseparationanalysis.Thedeflectionratioincreases

asthe flow velocity increases,and decreasesas themass ratio

increases. Meanwhile, the natural angular frequency decreases

monotonicallyatfirstmodebutincreasesmonotonicallyatmode

2–5whenthemassratioincreases.

Acknowledgments

TheauthorsacknowledgegratefullyCNPq,CAPESandFAPERJof

Brazil,theNationalBasicResearchProgramofChina(973Program)

GrantNo.2011CB013702andtheNaturalScienceFoundationof

ChinaGrantNo.50979113forthefinancialsupporttothisresearch.

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