A wave-based numerical approach for the description of the vibroacoustic behavior of fluid-filled pipes

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Ajit Bhuddi

To cite this version:

Ajit Bhuddi. A wave-based numerical approach for the description of the vibroacoustic behavior of

fluid-filled pipes. Vibrations [physics.class-ph]. Université François Rabelais de Tours, 2015. English.

�tel-01339338�

É ole Do toraleEMSTU

Laboratoire de Mé aniqueetRhéologie, E.A. 2640

THÈSE présentéepar :

Ajit BHUDDI

soutenan e prévue le : 25 novembre 2015

pour obtenir legradede : Do teurde l'UniversitéFrançois - Rabelais deTours

Dis ipline/ Spé ialité : GénieMé anique et Produ tique

Appro he ondulatoire pour la des ription numérique du

omportement vibroa oustique large bande des onduites ave

uide interne

THÈSE dirigée par :

MENCIK Jean-Mathieu Professeur,INSA Centre Valde Loire,Blois

RAPPORTEURS :

DEUJean-François Professeur,ConservatoireNational desArts etMétiers, Paris

DUHAMEL Denis Professeur,E oledesPonts ParisTe h,Marne-la-Vallée

EXAMINATEURS :

BALMESEtienne Professeur,E oleNationale Supérieured'ArtsetMétiers, Paris

MEOStéphane Professeur,UniversitéFrançois Rabelais, Tours

Les onduites ylindriquesutiliséesdansletransportdegazoudeliquidessontlesiège

de phénomènes d'intera tion uide-stru ture qui peuvent onduire à des niveaux

vibra-toiresoua oustiquesex essifs,menaçant l'intégritédelastru tureousour esdenuisan es

sonores. Ave ledur issementdesnormesenmatièred'émissionsa oustiquesetl'apparition

denouveauxmatériaux,l'enjeu estàprésentde développerdesoutilsnumériques e a es

apablesde prédire le omportement detels systèmes ouplés à moindre oût.

Ce travailde thèsevise àappliquer laméthodedes élémentsnis ondulatoiresWave

FiniteElement (WFE) en vuedeprédirele omportement vibroa oustiquede onduites

élastiques axisymétriques ontenant unuide interne. La méthode WFE s'applique à des

guides d'onde homogènes ou périodiques, et repose sur le al ul d'une base d'ondes à

partirdumodèleélémentsnisd'unsous-systèmereprésentatifdusystème omplet. Cette

appro he implique un nombre restreint de degrés de liberté (DDLs), e qui lui permet

d'orir desgains de temps de al ul onsidérables par rapportà laméthode des éléments

nis lassique,laquelle né essiteunmaillage du système omplet.

Dans un premier temps, on s'intéresse au as d'une onduite élastique axisymétrique

ave uideinterne, ennégligeanttouteintera tionave lemilieuambiant. Lesous-système

représentatifesti i onstituéd'untronçon de onduitede trèsfaiblelongueur,etduuide

qu'il ontient. La méthodologie visant à obtenir les modes d'onde à partir de la matri e

de rigidité dynamiquedu sous-système est exposéeet les résultats analysés sous laforme

de ourbes de dispersion. Deux exemples de onduites, omposées respe tivement d'un

matériau homogène et d'une stru ture multi- ou hes onstituée d'un ÷ur souple inséré

entredeux peaux rigides,viennent illustrer l'appro he. Dans unse ond temps, labasede

modesd'ondeest utiliséeenvuede déterminerlaréponsefor ée des onduite delongueur

nie. La onfrontation des résultatsobtenus en termes depressions dansle uideinterne

et de dépla ements de la stru ture élastique ave des résultats issus de la méthode des

élémentsnis permetde validerl'appro he etde mettreen éviden elesgains de tempsde

al ulfavoriséspar l'appro he WFE.

des onduitesélastiquesave uideinterne,immergéesdansunuidea oustiqueextérieur.

Dans e adre, la ondition de Sommerfeld est prise en ompte en tronquant le domaine

uide externe, et en l'entourant d'une ou he d'éléments absorbants  Perfe tly Mat hed

Layer(PML) dans laquelleles ondesa oustiques in identes sont progressivement

amor-ties. Le sous-système représentatifest àprésent onstituéd'untronçon dela onduite, du

uideinternequiyest ontenu,ainsiqueduuideexterneetduPMLquil'entourent. Une

appro he baséesurdesguides d'onde purement a oustiques onne tésà haqueextrémité

dela onduiteprin ipale estparailleursdéveloppée andeprendreen omptela ondition

de rayonnement dans la dire tion longitudinale. Les hamps de pression rayonnée issus

de la méthode WFE sont analysés pour les mêmes onduites mono- etmulti- ou hes que

pré édemment, immergéesdans unuide léger ommedansun uidelourd. Lesrésultats

mettentnotammentenéviden el'impa tdelaprésen eduuidelourdsurlesvibrationsde

la onduiteetlesniveauxdepressionrayonnée, ainsiquel'eetd'atténuation obtenudans

le as de la onduite àstru ture sandwi h. L'e a ité de laméthode WFE esti i en ore

démontrée par rapportà l'appro he lassiquedesélémentsnis, en termesde pré ision et

de gainsde temps de al uls.

Mots lés : méthode WFE ; intera tion uide-stru ture ; rayonnement a oustique ;

Pressure u tuationsandme hani al vibrationsinpipes an ause ex essivenoiseand

even damage thepipeor the ma hinery. Dueto thestri t regulationsonhumanexposure

to noise emission and vibrations from engineering stru tures, theneed for e ient design

toolsisevolving. Theissueistoover omelargeCPUtimesinvolvedbythestandardFinite

Element (FE) method when a large number of degrees of freedom (DOFs) are involved.

Thisthesisinvestigatesthee ien yofthewaveniteelement(WFE)methodtoassessthe

vibro-a ousti behaviorofaxisymmetri elasti pipesintera tingwithinternalandexternal

a ousti uids. TheWFE methoduses numeri al wave basesof smallsizes to apture the

dynami s of waveguides that exhibit uniform or periodi ross-se tions. The sizes ofsu h

basesarelinkedtothenumberofDOFsusedtodes ribethewaveguide ross-se tion,whi h

issmall ompared to thenumberof DOFs ofthe wholesystem.

In the rst instan e, the wave modes hara terizing an innite uid-lled pipe are

omputed from a representative subsystem of small length. These wave modes are then

used to ompute the dispersion urves. The resulting dispersion urves are plotted and

analyzed to highlight the uid-stru ture oupling between the pipe vibrations and the

dynami s of the internal uid. The wave modes are then used as a representative basis

to ompute thefrequen y response fun tions(FRFs)ofnite lengthuid-lled pipeswith

axisymmetri boundary onditions. The method is illustrated intwo ases of single- and

multi-layereduid-lledpipes. Thee ien yoftheWFEmethodintermsofa ura yand

omputational time savingsishighlighted in omparison withthestandard FEanalyzes.

In the se ond part, theproblem of a ousti radiation of uid-lled pipesimmersed in

an external uid is ta kled by onsidering perfe tly mat hed layers (PMLs) to model the

Sommerfeld radiation ondition. Thesele tion riteria ofthePMLarethoroughly dened

for the urrent ase. The WFE method involves the omputation of wave modes along

an axisymmetri multi-physi s waveguide that in orporates a pipe, internal and external

uids, as well as a PML. Those modes are again used to ompute the FRFs of the

mul-tiphysi s system in the form of radiated pressure eld in the external uid. Within the

dire tionatsomedistan efromtheelasti pipeisa hievedby onne tingadditional

a ous-ti waveguides - involving the external uid and the PML - to the left and right ends of

themulti-physi s waveguide. A wave-based matrix formulation isproposed whi h enables

thedispla ement eldof the pipe as well asthepressure elds at anylo ation within the

internal andexternal uidsto be omputed. Numeri alexperimentsare arriedout whi h

involve the same single- and multi-layered uid-lled pipes aspreviously,immersed in an

external uid. The WFE method is again seen apable of predi ting a urately the

vi-broa ousti behaviorof the multi-physi s systems,inparti ular theuidloading ee t on

thepipe vibrations. The ability of the multi-layered pipe to redu e thepressure levels in

the exterior uiddomain is also analyzed. Comparisons with FE results again bring out

interesting omputational time savingsout usingthe WFEmethod.

Keywords: Waveniteelementmethod;uid-stru tureintera tion;a ousti radiation;

Introdu tion générale 18

1 Introdu tion 19

1.1 Obje tivesofthestudy . . . 23

1.2 Reviewof waveguide approa hes . . . 23

1.2.1 Analyti almethods . . . 23

1.2.2 Numeri almethods . . . 36

1.3 Numeri alresults anddis ussion . . . 57

1.3.1 Dispersion urves for3Dpipe . . . 57

1.3.2 Dispersion urves for2Daxisymmetri pipe . . . 61

1.3.3 For edresponseof a3Dand a2D axisymmetri pipe . . . 62

2 Wave Finite Element method for elasto-a ousti waveguides 67 2.1 Elasto-a ousti waveguide . . . 68

2.2 FEmodelof anelasto-a ousti waveguide (uid-lled pipe) . . . 70

2.3 WFE modeling . . . 72

2.3.1 Wave propagation . . . 72

2.3.2 For edresponse omputation . . . 76

2.4 Numeri alexamples . . . 77

2.4.1 Sele tionof FEmesh fora 2Daxisymmetri elasto-a ousti waveguide 77 2.4.2 Dispersion urves . . . 78

2.4.3 For ed response omputation of elasto-a ousti waveguides of nite length . . . 84

3 A ousti radiation of uid-lledpipes 95

3.1 Elasto-a ousti waveguide inan external uid . . . 97

3.2 Review ofnumeri al methods to model aSommerfeld radiation ondition. . 99

3.2.1 BoundaryElement Method (BEM) . . . 99

3.2.2 Prin iple ofInnite Elements . . . 102

3.2.3 Diri hlet-to-Neumann (DtN) Method . . . 103

3.2.4 Perfe tly Mat hed Layers (PMLs) . . . 105

3.3 Axisymmetri PML. . . 108

3.3.1 Coordinate transformation . . . 108

3.3.2 FEformulation . . . 109

3.3.3 PML sele tion . . . 110

3.4 Wavepropagationalongmulti-physi swaveguidesinvolvinguid-lledpipes andexternal uids . . . 113

3.5 A ousti radiationof nite uid-lledpipes . . . 115

3.6 Numeri alresults . . . 120

3.6.1 A ousti radiation ofa single-layered uid-lledpipe . . . 120

3.6.2 A ousti radiation ofa multi-layereduid-lled pipe . . . 127

Con lusion 133

Con lusion générale 135

Appendi es 139

A Flügge's shell equations of motion 139

B Indire t and variational boundary integral equations 141

C Newton-Raphson method 143

D Craig-Bampton method 145

1.1 Someappli ations ofuid-lled pipes. . . 21

1.2 Dispersion urvesfor a ir ular ylindri al pipe. . . 26

1.3 Longitudinal ex itationof abeambyatimeharmoni for e

F e

iωt

. . . 27

1.4 Ex itation of a beam in the transverse dire tion by a time harmoni for e

F e

iωt

anda moment

M

0

e

iωt

.. . . 28

1.5 Co-ordinatesystemand modalshapesof shells. . . 29

1.6 Dispersion urves for a ir ular ylindri al pipe (

n = 0

˜

), Donnell's shell theory(), Flügge'sshelltheory(x). . . 31

1.7 Dispersion urves for a ir ular ylindri al pipe (

n = 1

˜

), Donnell's shell theory(), Flügge'sshelltheory(x). . . 32

1.8 Elasti waveguide (a)s hemati view(b) FEmodel. . . 37

1.9 As hemati representation of waveguide andsubsystems. . . 41

1.10 Atwo levelAMLS partition tree. . . 44

1.11 FEmodelof the pipeusing the SAFEmethod. . . 45

1.12 Axisymmetri modelof a pipe (a) typi al subsystem (b)s hemati view of fullwaveguide. . . 48

1.13 3Dmodelofapipe(a)typi alsubsystem(b)s hemati viewoffullwaveguide. 49 1.14 Dispersion urvesfora3Dpipe using()Donnellshelltheory,(

· · ·

)SAFE method, (thi k lines)Timoshenko theoryand ( ) WFE method. . . 58

1.15 WFE method algorithm . . . 59

1.16 Spatialrepresentationofseveral ross-se tionwavemodeshapesforthepipe (blue olor)at (a)

5, 000

Hz and (b)

10, 000

Hz; (i) longitudinal mode; (ii) shear mode; (iii) bending mode and (iv) mid-frequen y mode, ompared withtheundeformed ross-se tion(red olor). . . 60

1.17 Dispersion urvesfora2Daxisymmetri pipeusing()Donnellshelltheory,

(

· · ·

) SAFEmethodand ( ) WFE method. . . 61

1.18 FRFsof the 3Dpipeobtained usingtheFEmethod,theWFE methodand

theanalyti al method (using longitudinalwavenumber). . . 62

1.19 Relative errorsbetweenthe (a)FEandWFEmethod(b)FEandanalyti al

FRF. . . 64

1.20 FRFsofthe2Daxisymmetri pipeobtainedusingtheFEmethod,the

Craig-Bampton method (CMS), the WFE method and analyti al method (using

longitudinalwavenumber). . . 64

1.21 Relativeerrorsbetweenthe(a)FEandWFEmethod(b)FEandCBmethod

with798 modes( )FEandCBmethodwith100modes(d)FEand

analyt-i al for edresponse. . . 65

2.1 S hemati view ofan elasto-a ousti waveguide. . . 69

2.2 Axisymmetri modelof an elasto-a ousti waveguide: (a)typi al

substru -ture(b)s hemati viewof thefullwaveguide. . . 73

2.3 Pressure FRF over [

1

kHz;

1.5

kHz℄ for a elasto-a ousti waveguide with

various meshes(a)elasti part: 1 quadrati element;a ousti part: () 10

quadrati elements(

· · ·

)5quadrati elements(

− · −

)5linearelements; (b) a ousti part: 5 quadrati elements; elasti part: () 5quadrati elements

(

· · ·

) 1quadrati element (

− −

)10 linearelements (

− · −

) 1linearelement. 79

2.4 WFE method algorithm for elasto-a ousti waveguides . . . 80

2.5 Variation of the uidloading term with respe t to the radial wavenumber

inawater-lled steelshell(analyti al results for

Ω = 1

),

n

˜

=

0

. . . 81

2.6 Dispersion urves for a 2D axisymmetri water-lled steel pipe omputed

using the WFE method (dashed lines) and the analyti al method (solid

lines),

n

˜

=

0

. . . 81

2.7 3D FE model of an elasto-a ousti waveguide: (a) typi al subsystem (b)

s hemati view of the fullwaveguide. . . 83

2.8 Dispersion urves fora 3Dwater-lledsteel pipe omputed using theWFE

method (dashedlines) andthe analyti almethod(solid lines),

˜

n

=

0

and

˜

n

=

1

. . . 83

2.9 Axisymmetri model of rigid-walled a ousti waveguide: (a) typi al

2.10 FRFsoftherigid-walleda ousti waveguide intermsofinternalpressure, as

obtainedwiththeWFE method(dottedline)andtheFEmethod(solidline). 86

2.11 Relativeerrorsbetween theWFE solution andtheFEsolution. . . 86

2.12 FRFs of the 2D axisymmetri elasto-a ousti waveguide, as obtained with

theWFE method(dotted line)and theFEmethod(solidline): (a)internal

pressure; (b)radialdispla ement. . . 88

2.13 RelativeerrorsbetweentheWFEsolutionandtheFEsolution: (a)internal

pressure; (b)radialdispla ement. . . 89

2.14 FRFsofthe3Delasto-a ousti waveguide,asobtainedwiththeWFEmethod

(dottedline)andtheFEmethod(solidline): (a)internalpressure;(b)radial

displa ement. . . 90

2.15 RelativeerrorsbetweentheWFEsolutionandtheFEsolution: (a)internal

pressure; (b)radialdispla ement. . . 90

2.16 Axisymmetri modelofa typi alsubsystemof amultilayered uid-lled pipe. 91

2.17 Dispersion urvesfor a multilayered water-lled steelshell omputed using

the WFE method. . . 92

2.18 FRFs of the multilayered uid-lled pipe in-va uo, as obtained with the

WFE method (dotted line) and the FE method (solid line): (a) internal

pressure; (b)radialdispla ement. . . 92

2.19 RelativeerrorsbetweentheWFEsolutionandtheFEsolution: (a)internal

pressure; (b)radialdispla ement. . . 93

3.1 S hemati view of an elasto-a ousti waveguide keptinan external uidof

innite extent. . . 98

3.2 Geometri al illustration of (a) Finite elements (b) Boundary elements ( )

Inniteelements(d)Absorbinglayer(PML)forana ousti radiationproblem.100

3.3 Divisionof two domainsbyan arti ial boundary(2D DtNproblem). . . . 104

3.4 S hemati viewof a vibroa ousti systemsurrounded bya PML. . . 106

3.5 S hemati viewoftheexternal a ousti partandPML partofthesubsystem.108

3.6 Frequen y evolutions of the sound pressure radiated by the single-layered

uid-lled pipewithdierentnumber ofelements inthePMLmesh. . . 111

3.7 Relativeerrorsbetweenradiatedpressureswithdierentmeshes: (solidline)

10

and

15

elements, (dotted line)

12

and

15

elements. . . 111 3.8 Frequen y evolutions of the sound pressure radiated by the single-layered

3.9 RelativeerrorsbetweenradiatedpressureswithdierentPMLwidths: (solid

line)

0.2

mand

0.3

m, (dottedline)

0.25

mand

0.3

m. . . 112

3.10 Axisymmetri modelofauid-lled pipewithanexternaluidandaPML:

(a)FEmodelof a subsystem;(b)FEmodelof thefullwaveguide.. . . 114

3.11 Axisymmetri modelofthe mainmulti-physi swaveguide onne tedto two

purelya ousti extra waveguides. . . 116

3.12 (solid lines) Dispersion urves for the single-layered uid-lled pipe

sur-rounded by (a) air and (b) water; (dotted lines) dispersion urves for the

uid-lled pipein-va uo;

Ω = ω a/c

L

.. . . 121 3.13 Frequen y evolutions of the sound pressure radiated by the single-layered

uid-lled pipe in (a) air, (b) water, ( ) air (zoom of (a)) and (d) water

(zoomof(b)), at the lo ation(

r

=

0.05

m,

z

=

0.5

m): WFEsolution (dotted line);FEsolution (solid line). . . 123

3.14 Relative errors between the radiated pressures omputed using the WFE

andthe FE methods (a)inair and (b)inwater. . . 124

3.15 Spatial distribution of the sound pressure radiated by the single-layered

uid-lled pipeinair at

5, 000

Hz: (a) WFEsolution; (b) FEsolution. . . . 125 3.16 Spatial distribution of the sound pressure radiated by the single-layered

uid-lled pipeinair at

10, 000

Hz: (a) WFE solution; (b)FEsolution. . . 125 3.17 Spatial distribution of the sound pressure radiated by the single-layered

uid-lled pipeinwater at

5, 000

Hz: (a)WFE solution; (b)FEsolution. . 126 3.18 Spatial distribution of the sound pressure radiated by the single-layered

uid-lled pipeinwater at

10, 000

Hz: (a)WFE solution; (b)FEsolution. . 126 3.19 Radial displa ement of the pipe with air as external uid (solid line) and

water asexternaluid(dotted line). . . 127

3.20 S hemati viewofa subsystemfor the aseof amulti-layereduid-lled pipe.128

3.21 Frequen y evolutions of the sound pressure radiated by the multi-layered

uid-lledpipein(a)airand(b)water,atthelo ation(

r

=

0.0615

m,

z

=

0.5

m): WFE solution (dotted line);FEsolution (solidline). . . 129

3.22 Relative errors between the radiated pressures for the multi-layered pipe

omputed usingthe WFEand theFEmethods (a)inairand (b)inwater. . 130

3.23 Spatialdistributionofthesoundpressureradiatedbythemulti-layered

uid-lled pipeinair at

5, 000

Hz: (a)WFE solution; (b)FE solution. . . 131 3.24 Spatialdistributionofthesoundpressureradiatedbythemulti-layered

3.25 Spatialdistributionofthesoundpressureradiatedbythemulti-layered

uid-lled pipeinwater at

5, 000

Hz: (a) WFEsolution; (b)FEsolution. . . 132 3.26 Spatialdistributionofthesoundpressureradiatedbythemulti-layered

uid-lled pipeinwater at

10, 000

Hz: (a)WFE solution; (b)FEsolution. . . 132 C.1 Geometri alillustrationof the Newton-Raphsonmethod.. . . 144

Contexte et obje tifs

Laprédi tiondu omportementvibroa oustiquede onduitesutiliséesdansletransport

de gaz ou de liquides est un sujet de re her hes ré urrent depuis de nombreuses années.

Detelssystèmesseren ontrent fréquemment dansdesdomainesindustriels variéstels que

l'industrie automobile, le génie ivil, l'aérospatiale ou en ore l'ingénierie oshore. Leur

étude répond à des enjeux é onomiques, de sé urité ou environnementaux. En eet, es

systèmes oupléssontlesiègedephénomènesd'intera tionuide-stru ture omplexesdont

les eets peuvent dégrader l'état de la stru ture, omme engendrer des niveaux sonores

ex essifs. La prédi tion de leur omportement pourrait ainsi permettre la mise en pla e

d'outilsdemaintenan epréventive ou orre tive,visant parexempleàsurveillerl'intégrité

stru turelle de pipelines, ou à lo aliser des défauts sur des onduites immergées. Par

ailleurs, leniveau d'exigen e on ernant les performan es a oustiques de es systèmes est

aujourd'hui ontinuellement a ru et a ompagne le dur issement des normes en matière

d'émissionsa oustiquesetde niveauxvibratoires. Ainsi,laquestion du onforta oustique

o upe désormaisune pla e entrale dansla on eptionde véhi ules terrestres ouaériens.

Une attention parti ulière est par exemple portée aux pots d'é happement et silen ieux

qui équipent les véhi ules à moteur thermiques, et qui jouent un rle primordial sur les

niveaux sonores émis. Des solutions innovantes visant à réduire les niveaux a oustiques

ommen ent à émerger, parmi lesquelles l'utilisation de matériaux multi- ou hes omme

les stru turessandwi h, onstituéesd'un ÷ur souple inséréentredes peaux rigides; leur

e a ité résideraitdansla apa ité du ÷ur àabsorber les vibrationsdespeaux.

Dans e ontexte,ilapparaîtdon né essairededévelopperdesoutilsnumériques apables

deprédire e a ement le omportement vibroa oustiquede systèmes omplexes, qui

peu-ventêtremultiphysiquesetmulti-é helles,en prenant en ompteles diversphénomènesde

ouplage, y ompris l'intera tion ave le milieu a oustique ambiant. L'intégration de tels

outils dans des pro essus d'optimisation suppose par ailleurs des temps de al ul réduits

Dans ette thèse, laméthode des éléments nis ondulatoires  Wave Finite Elements

(WFE)  est appliquée à l'étude de onduites ylindriques omportant un uide interne.

La méthode WFE, qui s'applique à des guides d'ondes homogènes ou périodiques,

re-posesurle al uld'une base d'ondes à partir du modèle élémentsnis d'un sous-système

représentatif du système omplet. Cette base peut ensuite être exploitée pour

déter-miner les réponses for ées du système omplet, de longueur nie, soumis à un

harge-ment harmonique. Elle implique un nombre de degrés de liberté restreint (seul le

sous-système est maillé) et don des temps de al uls réduits par rapport à d'autres

méth-odes né essitant le maillage du système omplet. De nombreux travaux ont ainsi pu

démontrer son e a ité dans l'analyse de stru tures périodiques variées telles que des

poutres, onduites ou plaques homogènes ou raidies [Men ik, 2010 , Duhamel etal.,2006,

Men ik and Gobert, 2012, Men ik andDuhamel, 2015℄, des stru tures sandwi h ou

lam-inées [Men ik, 2010 , Renno andMa e, 2010℄, ou en ore omposées de matériaux dont les

propriétés varient en fon tion de la fréquen e [Wakietal.,2009 ℄, pour n'en iter que

quelques-unes. L'analyse de systèmes multiphysiques tels que des onduites ave uide

interne aétéinitiée dans[Men ik andI h hou,2007 ℄.

Le présent travail viseà implémenter laméthode WFE dansle asde systèmes

multi-physiques onstituésd'unestru tureélastique oupléeave unplusieursuidesa oustiques.

L'obje tifest dedévelopperdesformulations matri iellese a es permettant notamment

le al ul des réponses for ées de es systèmes à moindre oût. L'une des originalités de

l'étude onsiste à prendre en ompte dansla formulation WFE le ouplage ave lemilieu

uide ambient, non borné, an d'évaluer ave pré ision le hamp a oustique rayonné par

le système. Dans ette étude, les onduites ylindriques seront supposées élastiques et

dissipatives, de se tion homogène ou multi- ou hes. Le uide est quant à lui onsidéré

parfait eta oustique(i.e. barotrope et ompressible). L'étudedu rayonnement a oustique

du système oupléest par ailleursrestreinteàdes onduites axisymétriques.

Organisation du manus rit

Le manus rit se omposede trois hapitres prin ipaux.

Aprèsunrappeldu ontexteetdesmotivationsdel'étude, lepremier hapitreprésente

un état de l'art sur les appro hes ondulatoires analytiques et numériques appli ables à

l'étude de onduites ave et sans uide interne. Diverses formulations analytiques visant

proposées et illustrées par le tra é de ourbes de dispersion. Une formulation basée sur

leséquations de Donnell-Mushtari pour les oques ylindriques[Leissa,1993 ℄ etqui prend

en ompte le ouplage uide-stru ture est en parti ulier détaillée pour les onduites ave

uide interne. Elle sera utilisée ommesolution de référen edans le hapitre 2. La suite

du hapitre 1 est onsa rée à laprésentation de méthodesnumériques, parmi lesquellesla

méthode WFE tient une pla e prépondérante. Le prin ipe de la méthode etson

appli a-tion à la ara térisation des ondes dansune onduite ylindrique sans uide interne sont

exposés en détails, avant d'aborder le al ulde réponses for ées de onduites de longueur

niesous hargement harmonique. Cesdeuxaspe ts sontillustrés dansles asdemodèles

3D puis 2D d'une onduite axisymétrique. La prise en ompte d'un uide interne sera

quant à elle développée dansle hapitre 2. Pour nir, d'autresméthodesnumériquessont

rappeléesetappliquéesà lamêmestru ture envuede onfronterlesrésultatsobtenus par

laméthodeWFE,notammentlaméthodeSAFE(SemiAnalyti alFiniteElementMethod)

pour les ourbesde dispersion[Gavri , 1994,Gavri , 1995 ,Hayashi etal.,2003 ℄,ainsique

laméthodedesélémentsnis onventionnelle[Petyt, 2003 ,Bathe,1996 ℄puislaméthodede

Craig-Bampton[Craigand Bampton, 1968 ,Maess andGaul, 2006 ,Bennighof etal., 1997 ℄

pour les réponses for ées.

Le hapitre 2 est dévolu à l'analyse du omportement vibroa oustique de onduites

ylindriquesave uideinterneparlaméthodeWFE.Dans e hapitre onnégligetoute

in-tera tionentreles onduitesetlemilieuambiantextérieur. Laméthodologieestdétailléeet

illustrée omme pré édemment pour desmodèles 3Dpuis2Dde onduitesaxisymétriques

dese tionhomogène. L'e a itédelaméthodeentermesdepré ision etde tempsde

al- ulestdémontrée en onfrontant lesrésultatsà euxobtenus par laméthode deséléments

nis lassique, ainsi qu'à la formulation analytique pré isée dans le hapitre 1. Le as

d'une onduite multi- ou hes omposée d'une stru ture sandwi h (un ÷ur souple inséré

entre deux peaux rigides) est ensuite traité, l'e a ité établie de la méthode permettant

d'envisager sonintégration dansdespro essusd'optimisation.

Dansle hapitre3ons'intéresseaurayonnementa oustiquede onduitesaxisymétriques

ave uide interne, immergées dans un uide externe au repos. La méthode hoisie

pour répondreà etteproblématique reposesurl'utilisation d'unPerfe tly Mat hedLayer

(PML), 'est-à-direune ou he d'éléments absorbants quivient eindre ledomaine

a ous-tiqueextérieurentourant la onduite,etdanslaquellelesondesa oustiquesin identessont

progressivement amorties. Le hoix duPML en termesde dimensionsetde maillageainsi

quesonimplémentation dansle adrede laméthode WFEsont détaillésdans e hapitre.

axisymétriques immergées respe tivement dans un uide léger puis un uide lourd. De

plus, pour ha une de es ongurations sont traités les as d'une onduite de stru ture

homogèneetd'une onduitemulti- ou hesàstru turesandwi h. L'e a itéde l'appro he

proposée estnalement soulignéepar des omparaisons ave des résultatsissus de al uls

Introdu tion

Contents

1.1 Obje tives ofthe study . . . 23

1.2 Review ofwaveguide approa hes . . . 23

1.2.1 Analyti almethods. . . 23

1.2.2 Numeri almethods. . . 36

1.3 Numeri al resultsand dis ussion . . . 57

1.3.1 Dispersion urvesfor3Dpipe . . . 57

1.3.2 Dispersion urvesfor2Daxisymmetri pipe . . . 61

1.3.3 For edresponseofa3Danda2Daxisymmetri pipe. . . 62

Résumé du hapitre

Ce hapitre ommen e par évoquer le ontexte a tuel dans lequel se pla e l'étude et expose

lesmotivationsàl'originedusujetdethèse. Laproblématiquevisée dans etravail on erne

le développement d'un outil numérique e a e apable de prédire le omportement

vibroa- oustique ainsi que le rayonnement a oustique de systèmes périodiques et multiphysiques.

Le hoix de la méthode des éléments nis ondulatoires  Wave Finite Elements (WFE) 

poury répondre est ensuitejustié enprésentantunétat del'artdes appro hesanalytiques

etnumériques existantesappli ablesà l'étude de onduites ylindriques,ave et sansuide

interne.

Les appro hes analytiques visantà ara tériser lesondes d'une onduite inniesans uide

interne sontd'abord passéesenrevue (se tion 1.2.1). Lesformulations issues dela théorie

des poutres d'Euler-Bernoulli et de Timoshenko, puis des oques ylindriques de Donnell

et de Flügge, sont notamment détaillées et illustrées par le tra é de ourbes de dispersion

orrespondant à une onduite ylindrique homogène innie derayon etépaisseur xés. La

méthode des éléments nis  Finite Elements (FE)  est rappelé avant d'aborder diverses

méthodes de sous-stru turation dynamique telles que la méthode de Craig-Bampton (CB)

et la méthodeAMLS (Automated Multi-Level Substru turing). La synthèse se terminepar

la méthodedes éléments nis semi-analytiques  Semi-Analyti al FiniteElements (SAFE)

 puis la méthodeWFE qui est dé rite en détails.

La dernière partie (se tion 1.3) reprend l'exemple de la onduite ylindrique homogène

abordée dans la se tion 1.2.1 envue d'illustrer et de omparer l'e a ité de es méthodes.

Les ourbes dedispersion obtenues ave les méthodes SAFE et WFE sontainsi omparées

à elles issues de la théorie de Donnell. Dans unse ond temps on onsidère une onduite

delongueur niesoumise àun hargement harmoniqueand'analyserles réponses for ées

obtenues par les méthodes FE, WFE et CB. Les omparaisons sont ee tuées pour deux

modélisations 3Det 2Daxisymétrique de la onduite. L'e a ité de la méthodeWFE, en

termes de pré ision etde temps de al ul, est démontrée.

Thisthesisinvestigatesthevibroa ousti behaviorofaxisymmetri elasti pipesofnite

length, oupled with internal and external a ousti uids using the Wave Finite Element

(WFE)method. The pipesstudied hereareassumedtobehomogenousalongtheir length

and exhibit spatial omplex dynami s within the hosen frequen y range. The pipes an

beeithersingle-layeredor multi-layered.

This hapter is on erned with the ba kground and obje tives of the thesis and also

des ribesseveral appli ationsof this work. Also, areview oftheanalyti al and numeri al

methods usedinthe modeling ofuid-lled pipesis proposed.

Context

Predi ting the vibroa ousti behaviorofuid-lledpipesusing wave-based approa hes

has been a resear h topi for many years. These systems are en ountered frequently in

industrialappli ations su h asautomotive, ivil, aerospa e,oshore engineering,et . The

demand for improved a ousti performan es of su h stru tures is growing along withthe

tightening of the legal regulations on human exposure to noise emissions and vibrations.

Forexample,thea ousti omforthasbe omeanimportantaspe tin ommer ialvehi les.

Inthisframework,the studyofthe vibroa ousti behaviorofuid-lled pipeshasre eived

inthedesignofmuers(seeFigure1). Wheneverauidisen losedinanelasti stru ture,

theresultingstru tural vibrationsandthesoundradiationareinuen edbythe

stru ture-a ousti sintera tion. Therefore,thenoiseproblemsinreallifeappli ationsinvolvetreating

the oupled vibroa ousti problemrather than usinga purelya ousti approa h.

Most of the stru tures whi h onvey uid are uniform in one-dire tion and an be

treated as one-dimensional waveguides where the waves travel along the length of the

waveguide. Examples of su h stru tures in lude beams, elasti shells, and so on. The

waveguidesare onsidereduniform,i.e.,thematerialandgeometri alpropertiesremainthe

same along the axisof the waveguide. A number of methods based on wave propagation

have been used to des ribe the dynami behavior of uid- arrying waveguides, su h as

analyti aland numeri al methods.

Figure1.1: Someappli ations ofuid-lled pipes.

Analyti almethodsareusedformodelingsimplewaveguides,su hasrods,beams,du ts

and uid-lled pipes. Theyare basedon a ertain numberof assumptions whi h simplify

the equations of motion. For example, onsider the bending motion of a beam; if the

beamisthin enough ompared to thewavelength, theEuler-Bernoullitheory an beused,

where itisassumedthat the ross-se tionremainsplane andperpendi ularto theneutral

axis during bending, ignoring rotary inertia and shearing ee ts. This simplies many

terms and yields a se ond order partial dierential equation whi h an be easily solved.

The only problemwith this assumption is thatit annotbe validatedat high frequen ies

when thewavelength be omes omparable to the thi kness of waveguides. Other theories

su h asthe one proposed by Timoshenko for homogenous beams an thenbe usedwhi h

in lude shearing ee ts and rotary inertia into the model. But as the stru ture be omes

onsidering oupling onditions at the interfa e between thetwo media. There are many

theories proposed for modeling axisymmetri shells su h as Donnell-Mushtari,

Kir ho-Love,Flügge-Byrne-Lurye, andsoon[Leissa,1993 ℄. Theanalyti al solutionforuid-lled

pipe usingDonnell-Mushtari shell equationsis des ribed inSe tion 1.2.1.2.

In summary, analyti alsolutions involve ertainapproximations andthe solutions are

not a urate at higher frequen ies; it is very di ult to apply analyti al theories for

waveguides with omplex ross-se tions and boundary onditions. Numeri al methods

have been developed to over ome su h issues. Among them, the Finite Element (FE)

method [Petyt, 2003 , Bathe,1996 ℄ has been proved e ient to model ompli ated

stru -tures. Thismethodusesadis retisationofthestru tureintosmallelementsinter onne ted

at nodes with two or more degrees of freedom (DOFs). The DOFs may des ribe the

dis-pla ement/rotationsofanodeinastru turalanalysis. On etheDOFsaredened forea h

element, then the elements are assembled to form the omplete stru ture. This pro ess

results in solving a matrix equation, whose size represents the number of DOFs of the

model. Various analysis su hasstati and dynami an then be arriedout to solve su h

equationsand todeterminethevalues oftheDOFs (su hasdispla ement/rotation).

How-evertheFEmethod suersfromdi ulties regardinglargesize modelsdue toan in rease

inmemory requirement and omputational time. For example,to apture thewavelength

of the stru tural/uid waves at all frequen ies a large number of elements are needed to

dis retiseastru turee iently. Asaruleofthumb,atleast6to8elementsperwavelength

(atthehighestfrequen y of interest)aregenerally re ommended,whi h makesthesize of

theFEmodels onsiderably large.

Thus analysis of long and omplex stru tures via onventional FE method may

re-quire ex essive omputational times. To over ome this,the Wave Finite Element (WFE)

methodhas proved more ee tive to model omplex but periodi stru tures. The present

work on erns the use of the WFE method for modeling the dynami and vibroa ousti

behaviorofapipeandamulti-physi swaveguideaswillbedetailedinSe tions2.3and3.4

respe tively. The WFE method uses a small sli e of a waveguide whi h ismodeled using

the FE method. The mass and stiness matri es are then extra ted and post-pro essed

using periodi ity onditions to form an eigenvalue problem. The eigenvalues and

eigen-ve torsobtained represent thefreewave hara teristi softhesystem. Wave-based matrix

formulations arethenderived to al ulate the freewave propagationand for edresponses

of a one-dimensional waveguide (for example a uid-lled pipe) with various boundary

A reviewof analyti al methods for des ribing the dynami behavior of elasti

waveg-uidesisaddressedinSe tion1.2followedbyananalysisofthenumeri al methods (Se tion

1.2.2). InSe tion1.3,the omparisonofthesemethodsaredis ussedandtheadvantagesof

theWFE method arethoroughly explained forthe omputation of wavesinelasti pipes.

1.1 Obje tives of the study

In this thesis, the vibroa ousti behavior of uid-lled pipe is studied using theWave

Finite Element (WFE) method. Theobje tives arethefollowing:

•

to modelthe dynami behaviorof single-layered and multi-layered uid-lled pipes;

the results will be ompared withthose obtained from thestandard FE method in

orderto validate the modeland highlight thebenets ofthe WFEmethod interms

of omputational times.

•

to model the problem of sound radiation of uid-lled pipes in an exterior

do-mainoftheoreti allyinniteextentusingappropriatenumeri alte hniques(Perfe tly

Mat hed Layers).

1.2 Review of waveguide approa hes

In this se tion, analysis methods for the dynami s of waveguides are reviewed. In

Se tion 1.2.1the lassi alanalyti almethods arereviewedwhi h on erns Euler-Bernoulli

and Timoshenko beams, ir ular ylindri al elasti shells and uid-lled pipes. Su h

ap-proa hes are in general appli able only to simple waveguides. Numeri al methods are

needed to investigate waves in general waveguides. Subse tion 1.2.2 on erns the review

of numeri al methods for omputing thedynami s of waveguides su h astheFEmethod,

omponent mode synthesis (CMS), automated multi-level-substru tuting (AMLS), semi

analyti al methods (for example semi analyti al nite element (SAFE)) and nally the

WFEmethod. The omparisonsofthesemethods aredoneregardingthedispersion urves

and frequen yresponsefun tions.

1.2.1 Analyti al methods

Analyti al methods basedonwavetheoryusewavestodes ribethebehaviorofbeams

and pipes. The omplex motioninthe solids is des ribed inthe form ofparti ular waves

su hasbending,longitudinal, shear,andsoon. Theanalyti al solutionsforthe

In thefollowing se tion, Euler-Bernoulli andTimoshenko beams are reviewed followed by

thetheories ofshells and uid-lledpipes.

1.2.1.1 Euler-Bernoulliand Timoshenko beams

For a thin beam where theEuler Bernoulli theoryapplies, thegoverning equation for

freevibration is given by[Fahyand Gardonio, 2007℄

EI

∂

4

_w

∂x

4

+ ρA

∂

2

w

∂t

2

= 0

(1.1)

where

w

isthe transversedispla ement,

x

thedire tionalongtheaxisofthebeam,

t

isthe time,

E

the Young's modulus,

I

the moment of inertia,

A

the ross-se tional area and

ρ

thedensityof the material.

Assuming time- andspa e- harmoni motion,thedispla ement isgivenby

w(x, t) = ae

(−iβ

b

x+iωt)

_.

(1.2)

where

β

b

is the wavenumber,

ω

is the angular frequen y,

a

is the wave amplitude and

i

=

p

₍₋₁₎

.

Substituting

w(x, t)

inEq. (1.1), we obtain the following dispersion relation of thebeam as

EIβ

_b

4

_{− ρAω}

2

= 0.

(1.3)

This dispersion relation has four solutions whi h denote four freely propagating waves

withtheir wavenumbers

±β

b

and

±iβ

b

,where

β

b

= (ρA/EI)

(1/4)

√

_ω

denotes thebending

wavenumber. Thersttwowavenumbers(whi harereal)representpropagatingwavesand

theother two (imaginary) depi t neareldor evanes ent waves.

Evanes ent waves de ayexponentiallywith distan e, theiramplitudes areonly signi ant

neardis ontinuitiesinthebeamorneartheex itationsour es;theydonottransmitenergy

[Fahyand Gardonio, 2007℄.

The omplete solution ofEq. (1.1) isgivenby

w(x, t) = (a

+

e

−iβ

b

x

_{+ a}

−

_e

iβ

b

x

_{+ a}

+

n

e

−β

b

x

+ a

−

n

e

β

b

x

)e

iωt

(1.4) where

a

+

_{, a}

−

_{, a}

+

n

and

a

−

n

arethe wave amplitudes ofthe right-goingand left-going propa-gating wavesand the orrespondingevanes ent waves.

It should be noted that, in the Euler Bernoulli theory, the rotary inertia and shear

remain plane and normalto the neutral bre during motion. To in lude theshear

defor-mation, the Timoshenko theory [Gra,1991 ℄ for beams an be used. In this framework

rossse tion still remain plane but are no longer normal to the neutral bre. In that

framework,the dispersionrelation isgivenby

 EI

ρA



β

_b

4

₋

I

A



1 +

E

Gk

′



β

_b

2

ω

2

_{− ω}

2

+

ρI

GAk

′

ω

4

_{= 0}

(1.5) where

k

′

isa shear orre tion fa tor ( onstant) whi h dependson the ross se tion, for a

re tangular ross-se tion

k

′

=

0.833

and for a ir ular se tion

k

′

=

0.88

.

G

is the shear

modulus,

G

=

E

2(1+ν)

,with

ν

thePoisson'sratio.

Solving Eq. (1.5), weget

β

b

= ±

v

u

u

u

u

t

I

A

1 +

Gk

E

′

ω

2

_±

r

I

A

1 +

Gk

E

′

ω

2

2

₋

4EI

ρA



−ω

2

₊

ρI

GAk

′

ω

4



2EI

ρA

.

(1.6)

Eq. (1.6) givesfour values of

β

b

,i.e. wavenumbers orresponding to right-going and left-goingpropagating waves and evanes ent wavesat theboundaries.

Longitudinal waves

The longitudinal wavenumbers an be omputed for the axial vibrations of a beam, and

theyare given by[Fahy,2005℄

β

l

= ω

r

ρ

E

.

(1.7)

Shear Waves

Thewavenumbers obtained for shear wavesareof theform[Fahy, 2005 ℄:

β

s

= ω

r

ρ

G

.

(1.8)

As an illustration, we onsider waves appearing in a pipe of innite length. Plotting

the dispersion urves enables to understand the dynami s of the beam and also the

limi-tationsofthe EulerBernoullitheory. Dispersion urves(Figure 1.2)areplottedfor apipe

withinternal radius =

0.0475

mand external radius =

0.05

m, density

ρ

=

7800

kg/m

3

,

Young's modulus

E

=

2 × 10

11

Pa, for a frequen y range of

[10

Hz;

15, 000]

Hz with a frequen y step of

10

Hz. The dispersion urves(real and imaginary part) obtained using Eqs. below (1.3), (1.6), (1.7) and (1.8) areshown inFigure 1.2. Bran h

1

orrespondsto

0

10

20

30

40

ℜ

(

β

)

0

2000

4000

6000

8000

10000

12000

14000

−40

−30

−20

−10

0

Frequency [Hz]

ℑ

(

β

)

1

2

3

4

5

2’

1’

5’

Figure1.2: Dispersion urves fora ir ular ylindri al pipe.

Euler-Bernoulli theory. The dieren e between the bending wavenumbers obtained from

theEuler-Bernoullitheoryand the Timoshenko theoryis learly visibleafter

2000

Hz and

is in reasing with the in rease of the frequen y. Bran h

3

depi ts the shear mode and

bran h

4

,thelongitudinalmode. Bran h

1

′

and

5

′

arethe evanes entmodesobtainedfrom

Timoshenko theory and bran h

2

′

is obtained from Euler-Bernoulli. Bran h

5

uts-on at

1370

Hz and be omespropagating.

Waves ree tion and transmission

Waveswhi h propagate along awaveguide might en ountera dis ontinuity intheform of

aboundaryor hangeinse tion,andsoon. Thein identwaves anberee tedat

bound-aries and may be ree ted and transmitted at dis ontinuities. Assume that an in ident

wave of amplitude

a

i

produ es a ree ted wave of amplitude

a

r

and a transmitted wave

of amplitude

a

i

, then the ree tion oe ient is dened as

r =

a

r

a

i

and the transmission

oe ient isgiven by

t =

a

t

a

i

.

Consider a ase when a propagative wave in a beam impinges at a boundary. The

displa ement at the boundary an bewritten as

w(x, t) = (a

i

e

−iβx

+ a

rp

e

iβx

+ a

rn

e

βx

)e

iωt

(1.9)

where

a

i

,

a

rp

and

a

rn

aretheamplitudesofthein ident,ree tedpropagatingandree ted evanes ent waves, respe tively. The amplitudes of these wavesare determined by the

in- ident and ree tion oe ients. In this asethe ree tion oe ient is given by

a

rp

a

i

and

a

rn

a

i

. Forafreeboundary ondition,

a

rp

a

i

= −i

and

a

rn

a

i

= 1 −i

. Whenawaveimpingesona

dis ontinuity su haspoint mass,both ree tedand transmittedwavesmaybegenerated.

The ontinuity of displa ement and for e equilibrium at the point mass determines the

ree tionand transmission oe ients.

Ex itation of longitudinalwaves

Consider the ase of a semi-innite beam. The left end of thebeamis ex ited by a time

harmoni for e

F e

iωt

inthelongitudinaldire tion. Itgenerateswave

a

+

_e

−iβ

l

x

travelingin

thepositive

x−

dire tionasshown inFigure1.3. Displa ement for x

≥

0is givenby

x

Figure1.3: Longitudinal ex itation ofa beam bya timeharmoni for e

F e

iωt

.

u(x, t) = a

+

e

(iωt−iβ

l

x)

(1.10)

Considering the dynami equilibrium equation of aninnitesimal element of thebeamat

theend

x = 0

F = −EA

∂u

_∂x

,

i.e.

F = iβ

l

EAa

+

(1.11)

Amplitude ofthe ex ited wave isgiven by

a

+

_{= −}

i

β

l

EA

F

(1.12) Displa ement for

x > 0

u(x, t) = −

i

β

l

EA

F e

(iωt−iβ

l

x)

(1.13)

Thefrequen y responseintermsofre eptan e isgiven by(at

x = 0

)

X =

u

0

F

=

1

iβ

l

EA

;

u

0

= u(0)

(1.14)

Ex itation of bending waves

Considernowtheex itationofbendingwavesinbeams(seeFigure1.4). Thesameprin iple

(displa ement and rotation), two internal for es (shear for e and bending moment), two

wave omponents in ea h dire tion (propagating and evanes ent or near eld) and two

possibleex itations(for eandmoment). Thusthe ontinuityequationinvolvedispla ement

and rotation and the equilibrium onditions relate for es and moments. Suppose a time

x

Figure1.4: Ex itationofabeaminthetransversedire tionbyatimeharmoni for e

F e

iωt

and amoment

M

0

e

iωt

.

harmoni for e

F

and/oramoment

M

0

a tontheendofasemi-innitebeam. Itgenerates an ex itedwave

a

+

_e

−iβx

and anear eldwave

a

+

n

e

−βx

.

Thedispla ement for

x ≥ 0

is given by

w(x, t) = a

+

e

−iβx

+ a

+

_n

e

−βx

(1.15)

Considering equilibriumofan innitesimalelement oftherod attheend

x = 0

,

•

for eequilibrium at

x = 0

F = EI

∂

3

_w

∂x

3

(1.16)

•

moment equilibriumat x=0

M

0

= −EI

∂

2

_w

∂x

2

(1.17)

Substituting the displa ement eld(Eq. 1.15) inthefor e/moment equilibriumequations

givesthewave amplitudes.

Similar approa h isusedfor nitelength beams forbending/longitudinal ex itation.

1.2.1.2 Cir ular ylindri al shells

The behavior of an in-va uo pipe is des ribed by onsidering the waves in the axial,

radial and ir umferential (tangential) dire tions (See Figure 1.5). The axial, tangential

to the ir umferential mode order

n

˜

by[Fahy andGardonio, 2007 ℄

[u, v, w] = [U (z), V (z), W (z)]cos(˜

nθ + φ)

0 6 ˜

_{n 6 ∞.}

(1.18)

Foreveryvalueof

n

˜

threeformsofwavespropagatealongthein-va uopipe hara terizedby theirwaveamplitudes[

U, V, W

℄. TheFlügge'stheoryisbasedonKir ho-Lovehypotheses

a

w

u

v

Breathing mode

Bending mode

=0

=1

Figure 1.5: Co-ordinate systemandmodalshapesofshells.

forthinelasti shells[Leissa,1993 ℄. Byusingthistheory,thestrain-displa ement relations

and hanges of urvature of the middle surfa e of a ylindri al shell an be obtained. In

this thesis the analyti al formulations that will be used to validate our numeri al results

are provided by the Donnell's equations, whi h are obtained by negle ting few terms in

Flügge'sequations. See Appendix Afor details ofFlügge's equations.

Equation of motion

Theassumptions onsideredin theDonnell's shelltheoryare[Pri e etal., 1998℄:

•

thethi kness ofthe shellis small ompared to other dimensions;

•

thedispla ements oftheshellaresmall;

•

thetransverse normalstress isnegligible;

•

normals to the referen e surfa e of the shellremain normalsand theshellthi kness

remains un hanged.

These assumptions allow to treat a thin shell of small urvature similar to the ase of a

thinplate. Theresulting equations ofmotion ofDonnell'sshellaregiven by[Leissa, 1993,

Pri e etal.,1998 ℄:

∂

2

w

∂z

2

+

K

0

a

2

∂

2

w

∂θ

2

+

K

0

a

′

_∂

2

_v

∂θ∂z

+

ν

a

∂u

∂z

=

1

c

2

_L

w

¨

(1.19)

K

₀

′

a

∂

2

w

∂θ∂z

+ K

0

∂

2

v

∂z

2

+

1

a

2

∂

2

v

∂θ

2

+

1

a

2

∂w

∂θ

=

1

c

2

_L

v

¨

(1.20)

−

ν

_a

∂v

_∂z

−

_a

1

2

∂v

∂θ

−

ε

a

2

∇

4

zθ

u −

u

a

2

=

1

c

2

L

¨

u +

1

a

q

n

K

(1.21)

where

u

,

v

and

w

are the displa ements in the radial (

r

), azimuthal (

θ

) and axial (

z

) dire tionsrespe tively;

K

0

= (1−ν)/2

,

K

′

0

= (1+ν)/2

,

K = Eh/(1−ν

2

₎

istheextensional

rigidityoftheshell,

a

istheradiusofthemidsurfa eoftheshell,

q

n

representstheexternal for es a tingnormalto thesurfa eof theshell,

∇

2

zθ

= a

2 ∂

2

∂z

2

+

∂

2

∂θ

2

,

c

2

L

= E/[ρ

E

(1 − ν

2

)]

is thesquareof the phasespeed ofa exuralwave propagatingina thinplate and

ε =

h

2

12a

2

. The normalmode shapesintermsof displa ements oftheshellarewritten as

u = Ue

−iβ

z

r

_cos(˜

_nθ)e

iωt

(1.22)

v = Ve

−iβ

z

r

_sin(˜

_nθ)e

iωt

(1.23)

w = We

−iβ

z

r

_cos(˜

_nθ)e

iωt

(1.24)

where

U

,

V

and

W

arethemodal amplitudesinthe

r

,

θ

and

z

dire tionsrespe tively. Theequation ofmotion forthe thin ylindri al pipe an be written inmatrix form as







L

11

L

12

L

13

L

21

L

22

L

23

L

31

L

32

L

33







|

{z

}

L







W

V

U





 =







0

0

0







(1.25) where

L

11

= −Ω

2

+ (β

z

a)

2

+

1

2

(1 − ν)˜n

2

_{, L}

12

=

1

₂

(1 + ν)˜

n(β

z

a), L

13

= ν(β

z

a)

L

21

= L

12

, L

22

= −Ω

2

+

1

₂

(1 − ν)(β

z

a)

2

+ ˜

n

2

, L

23

= ˜

n

L

31

= L

13

, L

32

= L

23

, L

33

= −Ω

2

+ 1 + ε

2

[(β

z

a)

2

+ ˜

n

2

]

2

where

ε

isthe shell thi knessparameter,

Ω

isthe non dimensionalfrequen y

Ω = ωa/c

L

. Thedispersion urvesareobtainedbysolvingthe hara teristi equationdet

(L) = 0

at ea h frequen y. Thisresults inan eight order equationwhose rootsare thewavenumbers

(

β

z

) for the in-va uo pipe. The roots an be omputed numeri ally using the

Newton-Raphson algorithm. They ome in pairs and are of the form

±α

,

±iγ

and

±(Ψ ± iΦ)

.

Thereal(

±α

)andimaginary(

±iγ

)rootsrepresentthepurelypropagatingandevanes ent waves,respe tively,whereasthe omplex solutions(

±(Ψ ± iΦ)

) representpropagatingand

de ayingwaves. Newton-Raphsonmethodisaniterativepro essbasedupontheknowledge

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

50

100

150

ℜ

(

β

z

)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−150

−100

−50

0

Normalized frequency

Ω

ℑ

(

β

z

)

Figure1.6: Dispersion urvesfor a ir ular ylindri al pipe(

n = 0

˜

),Donnell'sshelltheory (), Flügge'sshelltheory(x).

of an initial guess of the sear hed root and the tangent to the urve near that root. For

example, suppose we want to nd the root of an equation

f (x) = 0

, assuming

f (x)

is

dierentiable. Let

r

betheroot ofthisequation. We startwithanestimate

x

0

of

r

. From

x

0

,anewestimate

x

1

is omputedas

x

0

−

f (x

0

)

f

′

_(x

0

)

. Ase ondestimate

x

2

issimilarly omputed from

x

1

,and this pro ess is repeated until the solution rea hes

r

. The Newton-Raphson methodworkswellif

x

0

is loseto

r

,butthe omputationaltimes anbeextremelylargeif theinitial guessistotallydierent. Generalizingthemethod,if

x

n

isthe urrentestimate, thenthenext estimate(

x

n+1

) isgiven by

x

n+1

= x

n

−

f (x

n

)

f

′

_(x

n

)

TheNewton-Raphson method isfullydetailed inAppendixC.

TheNewton-Raphson method is usedhere to ompute thedispersion urves for Donnell

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

50

100

150

ℜ

(

β

z

)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−150

−100

−50

0

Normalized frequency

Ω

ℑ

(

β

z

)

Figure1.7: Dispersion urvesfora ir ular ylindri al pipe(

n = 1

˜

),Donnell'sshelltheory (), Flügge'sshelltheory(x).

as well as Flügge's shell [Leissa,1993 ℄ for

n = 0

˜

and

n = 1

˜

, respe tively for a frequen y range of

[10

Hz;

15, 000

Hz

]

at a frequen y step of

10

Hz. Flügge's shell equations are onsidered to be a urate for des ribing thedynami s of shells, whereas while omputing

theDonnell'sshellequationsafewterms(asdes ribedabove)arenegle tedandthislimits

the use of Donnell's shell theory. However, this theory is onsidered to be su ient to

des ribe thedynami s of thin shells. The resulting dispersion urves from Donnell's and

Flügge'sequations are ompared inFigure1.6 (for

n = 0

˜

) and Figure 1.7(for

n = 1

˜

) and

shows a good agreement for the sele ted frequen y range, sin e the transverse shearing

ee ts play negligible role until

Ω

=

0.9

. In the further se tions, the dispersion urves obtained fromDonnell's equationswill beusedto ompare the numeri al results.

1.2.1.3 Fluid-lled ir ular ylindri al shells

Inthelastfewde ades,greatattentionhasbeenpaidtotheproblemsinvolving oupled

uid-stru ture intera tion whi h requirestreatment of solid and the uid parts together.

Earlierworkinthe analysisofthe dynami sofuid-lledelasti shellsbymeansof

disper-sion urveswasformulatedby[Lin andMorgan, 1956 ℄. [Fuller andFahy,1982 ℄proposeda

travel-inginauid-lledelasti pipe;theee t ofuidloading wasalsoexplained. Itwasfound

thatat low frequen ies, most of theenergy islo atedinthepipe for stru tural ex itation

and intheuid for a ousti ex itation; however, at highfrequen ies, the energy

distribu-tion was found to vary signi antly. Later on, [Fuller, 1983 ℄ extended this approa h and

aptured the mobility of auid-lled shellex ited by an external line for eapplied along

the ir umferen e at the left boundary of pipe. The input mobility was then ompared

withtheinputmobilityofanin-va uo shell. Itwasfound thatatverylowfrequen iesthe

real part of input mobility of the uid-lled pipe is low and lose to the in-va uo result.

As the frequen y is in reased, the amplitude of the input mobility be omes larger than

thein-va uo pipe mobilitymainly due tothestrong uid-stru ture oupling whi hresults

inin reasedradialvibrations. At veryhigh frequen ies,themobilityagain be omes lower

and lose to the in-va uo result. The energy distribution was also explained with radial

for eex itation. [Pavi , 1990 ℄simpliedtheform(asgivenby[Fuller and Fahy, 1982℄)and

investigated the vibrational energy ow forin-va uo and uid-lled pipes.

Free vibrations of ring-stiened ylindri al shells under initial hydrostati pressure

were analyzed by [Gan etal.,2009 ℄ with dierent sets of boundary onditions and the

natural frequen ies were ompared with experimental results. [Zhang etal.,2001 ℄

inves-tigated the vibrations of in-va uo shells using the wave propagation approa h and

om-pared these results with Finite Element analyses. For the oupled uid-stru ture

analy-sis, the analyti al results for dierent boundary onditions were ompared with

numer-i al results obtained using the FE method (to model the shell part) and the Boundary

Element method (to model the uid part). The surfa e variables of the Boundary

Ele-ment model(i.e. pressureand normal parti le velo ity) were solved using boundary

inte-gral method and then the normal velo ity at the surfa e was related with the stru tural

displa ement to model uid-stru ture intera tion [Zhangetal.,2001 ℄. It was found that

the oupled natural frequen y redu es to almost half of its un oupled value for this shell

(with Clamped-Clamped boundary onditions), whi h highlights the importan e of uid

ee ts on the shell vibrations. [Finneveden,1997a ℄ derived simplied equations of

mo-tion for the shell and the uid in uid-lled pipes. Later [Finneveden,1997b ℄ derived

expressions for the modal density in uid-lled pipes and omputed the input power at

point sour es. [Zhu,1995 ℄ studied the oupling between uid and stru ture using the

Rayleigh-Ritz method. [Brevartand Fuller, 1993℄ studied the ee ts of internal ow on

thedistribution of vibrational energy ina uid-lled ylindri al shell. It was shown that

theee t of ow is greatest near oin iden e or ut-on frequen iesof higher order waves.

[Aristegui etal., 2001 ℄ investigated the wave propagation hara teristi s of a pipe with

both internal andexternal uids. The dispersion hara teristi s were omputed and

Analyti al model

Treating the problem of uid-lled pipes is usually done by using the dynami

equa-tions for the pipe (usually using the shell equations), the equation of motion for the

uid, and nally taking into a ount the oupling at the interfa e between the pipe

and the uid. For omputing the sound radiation and for ed response of uid-lled

pipes, it is only ne essary to onsider the form of waves in whi h radial displa ement

is dominant [Fahyand Gardonio, 2007 ℄. Other forms of oupling in lude fri tion

ou-pling and Poisson's oupling. Fri tion oupling represents an axial intera tion aused

by the fri tion between the uid and the pipe whi h is out of the s ope of this thesis

[Hanssonand Sandberg,2001 ℄. Poisson's oupling involves the intera tion between the

pressure in uid and axial stresses in thepipe due to radial ontra tion/expansion. The

details ofthetheoryusedby[FullerandFahy, 1982 ℄isexplained below.

Donnell's shell theory is used to model the solid part whi h has been des ribed in

Se -tion1.2.1.2.

Pressure equation in a rigid-walled ylindri al pipe

TheHelmholtz equationin ylindri al oordinates isgivenby

∂

2

p

∂r

2

+

1

r

∂p

∂r

+

1

r

2

∂

2

p

∂θ

2

+

∂

2

p

∂z

2

+ β

2

_{p = 0.}

(1.27)

Theboundary onditions forthe uidinsidethepipe(of length

L

),whi h is onsideredas rigid-walled, are

 ∂p

∂z



z=0

=

 ∂p

∂z



z=L

=

 ∂p

∂r



r=a

= 0

(1.28)

Thesolution isassumed to be timeharmoni inall thethreedire tions andis given by

p(r, θ, z, t) = P

n

˜

R(r)Θ(θ)Z(z)e

iωt

.

(1.29)

Onsubstituting

p

and theboundary onditions intheHelmholtzequation, we get

d

2

Z

dz

2

= −β

2

˜

n

Z

(1.30)

d

2

_Θ

dθ

2

= −˜n

2

_Θ

(1.31)

r

2

d

2

_R

dr

2

+ r

dR

dr

+ (β

2

˜

n

r

2

− ˜n

2

)R = 0.

(1.32)

Solving these equations we obtain

Z = cos(β

n

˜

z)

,

Θ = cos(˜

nθ)

, and

R = J

n

˜

(β

r

r)

where

˜

n = 0, 1, 2...

is the ir umferential order,

β

n

˜

is the ir umferential wavenumber,

β

r

is the wavenumberinthe radial dire tion,

J

n

˜

istheBessel's fun tion oforder

˜

n

.

Therefore the pressure eld in the ontained uid whi h satises the a ousti wave

equationis given by[Fahyand Gardonio, 2007 ℄

p =

∞

X

˜

n=0

P

˜

n

cos(˜

nθ)J

n

˜

(β

r

r)e

(iωt−ik

n

˜

z

).

(1.33)

Theterm

P

n

˜

isan amplitude oe ient determined bytheboundary onditions,theterm

J

n

˜

(β

r

r)

isa Besselfun tion oftherstkind whi hrepresentsstanding wavesintheradial dire tion andtheterm

cos(˜

nθ)

representsstanding waves inthe ir umferential dire tion.

Coupling onditions

At theuid-stru ture interfa e, to ensure that theuid remains in onta t with thepipe

wall, the radial vibrational velo ity of the uid and the radial velo ity of the shell wall

mustbeequal. Theradialvelo ityof theuid an easily be omputed using theequation

ofmomentum onservation i.e.,

v

r

= −

1

iρ

A

ω

∂p

∂r

(1.34) where

ρ

A

isthe densityofthe internal uid. Thus for aparti ular ir umferential mode

n

˜

theradialvelo ityat the interfa e withtheshellwall is given by

v

r=a

= −

β

r

J

n

˜

′

(β

r

a)

iρ

A

ω

P

n

˜

cos(˜

nθ)e

(iωt−ik

˜

n

z)

.

(1.35)

Theequationsoffreevibrationsofthe oupleduid-lledpipe,representedinmatrixform,

are given by [Fullerand Fahy,1982 ℄. These equations are almost similar to Eq. (1.25)

ex eptthe uidloading term (

F L

) isadded to

L

33

,i.e.

L

33

= −Ω

2

+ 1 + ε

2

[(β

z

a)

2

+ ˜

n

2

]

2

− F L

(1.36)

The uidloading term

F L

isdened by

F L = Ω

2

 ρ

A

ρ

E

  h

a



−1

_{ 1}

β

r

a

  J

˜

n

(β

r

a)

J

′

˜

n

(β

r

a)



.

(1.37)

The radialwavenumber is given by

β

r

= ±

1

a

[Ω

2

(c

L

/c

f

) − (β

z

)

2

]

1/2

,with

c

f

the freewave

speed inthe uid. The wavenumbers are obtained by using det(

L

)=0. Due to the