Hopf Bif Phenomena Bajaj

7/29/2019 Hopf Bif Phenomena Bajaj

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SIAM J.APPL. MATH. ? 1980 Societyfor ndustrial nd AppliedMathematicsVol. 39, No. 2, October1980 0036-1399/80/3902-0003 $01.00/0

HOPF BIFURCATION PHENOMENA IN TUBES CARRYING A FLUID*A. K. BAJAJ,t . R. SETHNAt AND T. S. LUNDGRENt

Abstract.elf-inducedlanar onlinearscillationsf cantileverube arryingn ncompressibleluidare nvestigated.he systemehaviorsgovernedy nonself-adjointntegrodifferentialquationnd nordinaryifferentialquationwhich re couplednonlinearly.he systemehavior epends n threeparameters,o, he lowelocity,,themass atio f he ube nd he luidnd , a parameterelatedothepressureoss n he ube. or mall o, he ero olutions symptoticallytable. spo s ncreased,or ach 3,a critical alueofPo,Po= Pcr,is reachedwhen he erosolutionecomes nstabley a pairofcomplexconjugateigenvaluesf he inearizedystemrossinghe maginaryxis. he ystematisfiesonditionsorHopf ifurcationnd he ero olutionifurcatesnto eriodicrbits.hebifurcatederiodicolutionsredeterminedsingenter-manifoldnd veragingdeas.In general,he ystemquationsnclude hangesn flow elocityntroducedy thetubemotions.However,ased nthe rderf ,two lasses fproblemsre howno xist.When isveryarge omparedtothe izeofthemotion,t s shownhat luidelocityhangesonot nterhefirstrder heoryndtheresultshow hat n this ase only upercriticalifurcationccurs.n the second ase, either uborsupercriticalolutionsrefoundooccur ependingnthevalue f .Introduction.heproblemsssociated ith hemotionfflexibleubes onveyingafluidave een f nterestor long ime. ecent evelopmentstart ith hework fBenjaminn articulatedubes 1], 2].Much fthework s connected ith he inearplane roblem.his sa nonself-adjointwo oint oundaryalueproblemependenton twoparameters.he physical roblem finterests associatedwith hedeter-minationfthe owest alues f these arameterst which he nitiallytraightubebecomes nstable. hisproblem,venwith he aidofcomputerolutions,resentsconsiderableifficulty.monghe eferencesn thisubjectheworksfGregoryndPaidoussis3], 4] and Paidoussis nd ssid 5] may ementioned.The nonlinearehavior f tubescarrying fluids known obe qualitativelydifferentependingn theboundaryonditions.f the ube ssupportedt both nds(simplyupportedrfixed-)he nstabilityf he ero olutionccurs ue tothe assageof a single igenvaluef the inearizedystemhroughero nto heright alf f thecomplex lane. f, n the ther and, he ube ssupportedt oneendandfree ttheother cantileverase) the instabilitys due to thepassageof a pairof complexeigenvaluesrossingver o therightalf lane. n this asewe will how hat Hopfbifurcationnto periodicmotionccurs.Holmes 6]hasprovenhenonexistencefHopfbifurcationntubes upportedtboth nds.The nonlinearroblemormotionsn a planehasnotbeenwidelytudied.thasbeen tudied yHolmes7],ThurmanndMote 8], ndRousselet ndHerrmann9],[10].Theworkn 8] snot elatedo theproblemfbifurcation.n[7]Holmes oesacompletetudyf he asewhen he ube ssupportedtboth nds yusing Galerkinapproximation.n 10],RousseletndHerrmanntudyhe antileverase.They ivecompleteerivationf he ystemquations hichncludeshe ffectf he luctuations

in he luid elocityaused y hemotionf he ube. heirnalysissbased nmethodsof nalysisor onlinearrdinaryifferentialquationsyKrylovndBogoliubovndtheresults emonstrateome nterestinghysicalhenomenafthe olutions.Wetreat he ame ystemquationserivedyRousseletndHerrmann10].We,however,sea methodf nalysisasedon thework fChow ndMallet-Paret11].*Received y he ditors ay4, 1979.tDepartmentfAerospaceEngineeringnd Mechanics, niversityf Minnesota,Minneapolis,Minnesota5455.Thiswork as upportedy heNational cience oundationnder rant SF-ENG-76-00030.

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214 A. K. BAJAJ, P. R. SETHNA AND T. S. LUNDGRENThis is appropriate or nonlinear wo pointboundary alue problemswhen theequations re a coupled ystem f partial nd ordinary ifferentialquations sdiscussed ere.Notonly s ourmethod f nalysisasically ifferentrom hat n 10]butwe alsoidentifyhecrucial oleofa parameterhat istinguisheswo lasses fsolutionshat requalitativelyifferent.onsequently, anyfour onclusionsiffermarkedlyromhose n 10]. Furthermore,e give esults f a simple riticalxperi-menthat ives ualitativeonfirmationfourresults.Othermethodsf nalysis,uch s onealong he ines fJosephndSattinger12],arepossible.Wehave hosen hepresentmethodecause f tspotentialor eneral-ization or he uturetudy f hemore omplexroblem f hree imensional otionsof tubes. t should e mentionedhat ur results re based on certain ssumptionsregardingheexistence fa centermanifoldndthe moothnessf theflow n themanifold.

System quations. hederivationf the quations fmotions available n theliterature10], 13]. Wewill herefore erely utline hederivationnd define hesystem arameters.We assume hathe ubehasuniformircularrossection nd s ong omparedoitsdiameter.ts enterine s nextensiblend nitiallytraightnd t selastic. hefluidis assumedobe incompressible,ndtheflow elocitys assumed o be parallel o thetube enterine nduniformcross he ross ection.We are nterestednplanemotions.et0 be theorigint the enterineofthetube tthepoint fsupport nd et OZ be along heunderformedenterine.Themotionsre assumed o occur n theX-Z plane.The arc ength along hedeformedtube coincideswith heZ coordinatenthe undeformedtate. f u and w are thedisplacementsf'theenterine n heX andZ directionsespectivelyhen he ositionvector of a point n thecenterine s r= u s, t)i w s, t)kwhere andk are fixedorthogonalnit ectors.The accelerationf a point nthe enterineofthe ube sa2r/at2nd that faparticleffluids(+Uj-I r(s,t)+dU t(s,t)at as ~~dtwhereU(t) isthe elocityf he luid ith espectothe ube nd = ar/as.fmandMare themassesperunitengthf thetube nd fluid espectivelyndQ is the tressresultantna cross ection fa combined ube ndfluid lement,hen orce alancegives aQ a2r(s, ) /Ma + a\2rst)+MdU t(1) ds-m 2 +Mi-+Ud-is. ('t)+M-Ut1) ~~~~as at at aS ' dt

LetM be the ouple esultantna cross ection. henbalance fmomentsives(2) -+txQ=0.as

Thecouple esultant sassumed obe related o hekinematicaluantitiesy heBernoulli-Eulerquation(3) M = EIt x ataswhere I istheflexuraligidityfthe ube. urthermore,ince he ube enterlines



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HOPF BIFURCATION PHENOMENA 215inextensibleehave(4) (-) +( a) =1.

The unknowns f theproblemare u(s, t), w(s,t) and U(t). The equations arederivedor he asewhere he ressuretthe ixed ndof he ubesa constantnd heresistanceo thefluidmotion nthetube s that ppropriateor urbulentlow.Weassumehatt s proportionalothe quare f he luid elocityndtothe ength f hetube, ,with constantfproportionalitywhichependsn the atio f he engthothe quare oot fthe nternalross ectionreaA, the oughnessfthe ube ndtheReynoldsumberftheflow.Theforce alance n a fluid lementnthe angentialirection,henntegratedover he engthfthe ube, ivesL a2w wa 2taud2 2) /awa a=a\ dMa(Uo-U2- |( * -+-d- de-M -L=OJ\at2 aS at2 aSI dt

whereUo is the constant lowvelocitywhichoccurswhenthe tube is not inmotion, = fL/IAwith = +//i/4"frictionactor"sthe onstantfproportionalityandMaU2 representsheconstantressureorce t thefixednds = 0 ofthetube.Duringhe nalysistbecomes lear hat he uantityU = U(t) - Uo instead fU(t) is themore ppropriateariable ndthedimensionlessormsfthevariablesu s,t),w s,t)and AU(t), which re taken obe 7, ', andvrespectively,anbe shown osatisfyhefollowingormidableystemf ntegrodifferentialquations:

2 2 2 a2n 4+2po-+Pa o 2a+ a4a-i2 ax ar aX aXa27 dv 2 a7=aX2 [13(1-x) d + 2pov+v v2]-2,3vax2LPkXdTPO V J axari(5) 3 a [a1{(a2)2 +( a2) 21]2a ax ax axa q1 2 2

anr an 2aLaxJ)~xar 2

where U w M12S77 ';=LSv =(EI L(U - Uo), x=L-~ LPo= ( L = (M+M) L) 2

l M 1/2 Lf(m+M) and Lf-7X



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216 A. K. BAJAJ, P. R. SETHNA AND T. S. LUNDGRENEquation5) arises rom he orce ndmoment alanceawsfor n elementf hetubeandfluid aken ogether,quation6) is theresult f force alanceon a fluidelementnthe angentialirectionhen ntegratedver he engthnd equation7)states he nextensibilityondition4)indimensionlessorm.heboundaryonditionsfor tubefixedt one endandfree t theother an beshownobe ([13])

(8) at1 a2 _ __tx=1.2(8) 7=71-=? atx=O and a 3= 0 atx=l.axax axWethushavetheproblemtatement5), (6), (7) and 8) which sdependentnthree arameterso,/3nd a. Hereporepresentshe tatic low ate, 3 epresentshedensityatio f thefluidnd the ubewhile representshe ombinationffrictionfactornd the lendernessfthe ube.Wewant ostudy onlinear otionsn a small eighborhoodf he ero olution' 0, ;-0, v 0. We thereforentroducehe scalingparameter >0, and let= E"2u (note hathis isdifferentromhatn 1)).With egardov, twill e shownbelow hat he rder fmagnitudefv depends nthe rder fmagnitudefa.Introducing7= E1/2u andeliminatingfrom5), 6) and 7)we have

a2u a2u 2a2u a4u22P -+Po 2+ 4ar axar ax axa2 a2ur dv=-2,8v aa 8(a-x) d+2pov +vJ2

(9) a3 u 3 2u a3u2 [(X2) + x aX2X23a [au 1' rxi j a2u \2ax ax Jx a(ax ar)

2L 2 2+ 4 X X +0ax2PO +2PoX ax r,+7X-4) 2IdxJ+O(8)and dv 2 ~1 x a2u\~2 a 3U a2ua(10) dv+2poa-v a = | [| 2dx - 2 dxd,r eJ[Jlax arl ax ax r ar axi + O(E2),where - a//3 ndtheboundaryonditionsre the ame s in 8).

Estimate fv. Let us first ote that herightideof the ordinaryifferentialequation10),for nybounded (x,r), sa bounded unctionfr.LetrrXj a u~2~ aU a3U1 a2u u(11) b(r)=J [IJ {( d u ) +du d u21 dxv--- du- dx+O(E).axr ~aX aXr2J a axiThen 10) can bewrittens

(12) -=-a (2Po v)v -sE 6b(,r)]Jr a!To obtain nestimatefv r) wecanappealtoMalkin's esult nsystems ith"constantlycting isturbance[14]regarding(r) s thedisturbance.ince 12) is ascalar quation, owever,he esultsclear ymerelyxamininghe ign fdv/drn he



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HOPF BIFURCATION PHENOMENA 217neighborhoodf he rigin 0for ll b r),notinghatbr)I bo or llrwhere o ssomepositive umber.t s clear hat here xists positivelynvariantntervalntheneighborhoodfv 0whichs of0(E/a).We recall hat representshe lendernessf he ube nd hereforesusually otsmall.Wecan hoose = 0(e8), rc 0, nwhich ase t srelated oEorwecan hoosetobe independentfE.Thesecond ase s thephysicallyore ealisticne.Thus wocases rise.(i) Ifr O,then iso E) andthereforenthe irstrder heory here ermsnlyof0(E) are akennto ccount, -r) anbe gnored.orrectionsuetochangesnfluidvelocityhall ccurnhigher rder alculations.(ii) If is ndependentfEorr=0, v r)= 0(E) and hus r)has obe consideredinthe irstrderheorytself.nthis ase,we have coupled ystemf quationsn 9)and 10)with = EV.

Theanalysisiven elow sfor hemore omplicatedase ii)because he esultsfor he case (i) caneasilybe derived rom hosefor ase (ii).The final esultsndconclusionsre, nthe ther and, iven or oth he ases. twill e shown hat heresa qualitativeifferencen theresultsor he wo ases.Equationsnvector orm. heboundaryalueproblemor ,for =0,

82 82 282 842u a2u, 2U a4U- +2po13 .Po ~-2 4-- O,a-P ax r ax aau a2u a uu=-=O atx=O, 2-= =O atx=lax Ax Ax

isnonstandard15].Althought anbe convertedoa standardigenvalueroblemorvectorunctions,here re ome undamentalifficultiesithhis onversionhich ediscussntheAppendix.heformulationiven elow,whichsnot nique,ssures heappropriaterthogonalityfthe igenfunctionsfthe inear peratornd ts djoint.Leta prime enote derivativeith espectox. Let(13) ua=2 and ui =-+2popu

Then 9) and 10) canbe writtennthevector ormau= ) 62(14) --Lu+sN1(u)+EN2(U, V)+O(? )Orand

a-(15) ---2potv + g(u) + 0(?)where

u=(l) 0= -P20po(."()i)N 3;- {( 2 + 2U'2U'2U}+ [U (u1 u2 2poj6u-p o U2-2 }dx2dxl132 2

Upo 2 ( 2Nl u) [{(2)+uuu} ui 0PU 2)d2dij

L ~~~~~~~~0N2(u,&)~[(lx) /u2+2povu 2+Pu2 + 2pof3u'i 4pf32u'-2i5f3(u' -2pof3u')]



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218 A. K. BAJAJ, P. R. SETHNA AND T. S. LUNDGRENand 1 rx , 2u 2 aU 3u u ulg Ju o aolx )+y.X r2Jdx ur2 a J dx.

Linearizedquation. orE= 0, theboundaryalueproblem or reduces o(16) au=LuOrwith

U2(0, t) = U2(0, t) = U2(1, t) = u"'(1, t) =0.Letting = w(x)eAT,we findhat he igenvalueroblemor16) is(17) Aw= Lwwith oundaryonditions

w2(0) = w2 (0) = W2 (1) = w ' (1) = 0.Assuminghe olutionobeoftheform 2(x)= A2eax, weget hat he xponent''a must atisfy

4 22 2=(18) a +pOa2+2po0/Aa+A2=0alongwith r 1 1 1(19) ~~~~detaaa a2 2a3 2a(19) det[a1e a2e2 a3e 3 a2ea44

a3ea1 a3ea2 a3ea3 a3ea4where i(A), i = 1, 2, 3,4 arethe our oots f 18) and 19)resultsromheboundaryconditions.he solutionf 17) then akes heform

4(20) w2(x)= E A2i eaixi=lSince 19) istranscendentalnthe i(A), t sa transcendentalquationnA.Theeigenvalues areknown o be denumerable ith ofiniteccumulationoints 16].Furthermore,incethe coefficientsre real,theeigenvalues,fcomplex, ccur ncomplexonjugateairs.The eigenvalues bviously ependon po and p. For po=0 the problemsself-adjointnd reduces otheproblemfa simple antileveream.Theeigenvalueswhen o= 0 caneasily e showno be imaginaryithA = iwwhereGoatisfies

(21) 1+coshlcocoslco=0.Forpo 0 and mall,ll the igenvaluesre nthe eft alf lane.This anbe seenas follows:Letpobe small. hen heproblem17) takes heform

(22) wij+A2w2=-2,3poAw +0(pg)alongwith heboundaryonditionsw2=w2=0 atx=0 andw2=w =0 atx=1.



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HOPF BIFURCATION PHENOMENA 219Since he olutionsreanalyticntheparametero,we et(23) Z wpW2= W2JPO

p=oand00A= E A1p'o.j=o

Then quatingoefficientsfpowers fpo,wesee that ermsf0(1) give(24) w20+Aw20=0alongwith

w20=w20=O atx=O andw' =w?"=0 atx=1,while,romerms fO(po), onegets(25) wivXA2w =-2,fAowo -2AoAW20where

w21 w21=O atx=0 andw'1= w" =O atx = 1.Similarlyquationsorrespondingohigherrder erms anbe obtained.

Althoughnexplicitxpressionorw20(x) anbecalculated, eget urresultymerelypplyinghenecessaryolvabilityonditionothe roblem25).The"solvabil-ity ondition"tates hatw20(x) must e orthogonalotherightide of 25),thats,1 1w 20w(X)W20(X) dx+A1J 2o(x) dx=0

orAl loW20(X)W20(X) dx

Jo,W20ox) dxHowever,1 1 1

w20(X)w20(x) dx w20 x)I- W2o(x)W2 (x) dxandthus akingheboundaryonditionsnto ccount, eobtain

A o,Ww20x)dxIt then ollowshat or 3>0,A1 sreal andnegativend A= iw+Alpo+O(po).Since hiss true or achrootwof 21), tfollowshat he igenvaluesor o mall ndpositivere nthe eft alf lane.Weare nterestednthenonlinearehaviorf he ystemorarger alues ftheparametershatmake he inearizedystemnstable.nstabilityanoccurwhen orgiven ,po i.e., he low ate) s ncreased-to-ariticalalue twhich neigenvalue rseveral igenvaluesross nto heright alf lane.To study hebehavior fall theeigenvaluesor llpo Ois a very ifficultask.Wemerelyostulatehathenstabilityoccurswhen forsome value of PO Pcr> 0 there is a pair of complex conjugate



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220 A. K. BAJAJ, P. R. SETHNA AND T. S. LUNDGRENeigenvalueshat rosses he maginaryxis romhe eft othe ight alf f he omplexplane. urthermore,t s assumed hat, or < Po Pcr, llthe ther igenvaluesemaininthe eft alf lane.More precisely,here xists pair of complex onjugate igenvalues andA,A A po, 3),such that

A(Pcr,)= iWo, wo0 and dRe(A) #0.dpo PO=PcrTheseconditions,s is wellknown,re the onditionsor Hopfbifurcation16].InFig.1numericalolutions or hebehaviorf he hreeowest igenvaluess afunctionfpo re hown orhe asewhen82 = 0.60. t s clear hat he ehaviorsquitecomplicated. enote hatf he emainingigenvaluesemainn he eft alf lane orallpo, Po Pcr,then henumericalolutionsgreewith urpostulate.We alsonotethathe igenvalueorrespondingothe econd igenvaluef he elf adjointroblem(Po 0) is theonethat oes nto heright alf fthecomplex lane.Thishas beenobserved y thers3], 5] both rom umericalalculationsndby tudyinghe modeshape"oftheperiodic ifurcated otionxperimentally.

-40P?2 =0.6

-2012 r2nd Mode

100 201 2j 0 7

2C20 .'N I - I . I. 1 . 1FIG. 1.Behaviorf igenvaluesorrespondingo he owest odesf heystems a functionf he lowvelocityo.Correspondingoeach igenvalue i heresaneigenfunction").We firsteorderthe igenvalueso that he necorrespondingothe riticaligenvaluecrisA1 Acr.Thecorrespondingeriodic olution,or - po, s

(26) u(7r,) =2 Re (w )(x)eiwoT)wherew(')(x) sa vectorigenfunctionorrespondingothe igenvalue 1.Toobtain he eriodicolutionf henonlinearroblem eneed he djointothelinear roblem16).Adjoint ystem.f (x)andb(x)are omplexector alued unctionsfx,wedefinetheusualL2 inner roduct

(a,b) aT dx0



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HOPF BIFURCATION PHENOMENA 221whereverbarenotesomplexonjugate.heoperatordjointoL isthe perator*forwhich

(La, b) = (a, L*b).Then he igenvalueroblemdjoint o 16) is(27) Aq L*qwhere

q(x) = (qi(x)) L* (_2(jl(.)iV 2po3( )')while l(x) satisfiesheboundaryonditions

ql=q.=0 atx=0 andq'l+pgql=o, q'i +p2q' -2po,8Aq1=0 atx=.IfA saneigenvaluef he riginalroblemhenA san eigenvaluef he djointproblem. husat Po Pcr,theadjointproblem lso has a pairofpure maginaryeigenvaluesicoo.n [13] it s shown hat, iven heeigenfunctionsor heoriginalproblem,necancomputehe igenfunctionsor he djoint roblem.pecificallyfAXisthefthigenvaluehen hefirstomponentigenvectorfthe djoint roblems4

(28) q1 (x) Z ak,A2k;eaki(-x)j =1wherek1 is the oot f 18)for hekth igenvaluend 2ki is as n 20)alsofor hektheigenvalue.The orthogonalityelations etween thvectorw(s) nd therth ectorq(r) oftheadjoint roblemurnutto be(w(s), q(r)) = 5rs

where rss theKroneckerelta.Bifurcationntoperiodic olution.We nowcompute he nonlinear eriodicsolutionor alues fpo n theneighborhoodfPcr. s stated n the ntroduction,hemathematicalasisof the nalysiss along he ines f 11].The actual alculations,however,rea variationdopted or ur ase, nd re nthe piritf hework17]for"monofrequentscillations."Sincewe are interestednthe solutionswhenpois nearPcr, etPo= Pcr 8ALFore = 0,theperiodicolutionfthe inear roblemsgiven y

(29) uO a[w(1)ei" +*(1) e-i+]where a" is the mplitudend kok f or, is thephaseof themotion. orthislinear roblem, and k rearbitraryndtheperiod fthe olutions2r/rlwo.Fore $ 0, etu be decomposedsthe umu= uo ywhere o ies nthe ubspacespannedyw(l) nd w')for ll "a" andkwhile ies nthe ubspace rthogonalotheadjointigenvectors(l)andq(l) For = 0,the eriodicolutionies n manifold (M)spanned yw(l) nd*w')for ll "a" and f.Following11],for $ 0,we assume hat(i) there xists smoothnvariantentermanifold

M(e): y=y*(a, ,k), 0=y*(O, q, eA) aa (0,X,0)



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222 A. K. BAJAJ, P. R. SETHNA AND T. S. LUNDGRENpassing hroughhe rigin = 0, andtangento the a, X, tu) pace;(ii) thedifferentialquation nduces smoothlownM(e);(iii) all orbitsying ear he riginor e (-co, co) ieonM(e).These ssumptions,lthoughlausible,ivenhedistributionf he igenvaluestPo=Pcr, learly equire proof.With heaboveassumptionsnmind ndusing hevariables and f s variables arameterizinghemanifold,et(30) u=uO(a,) s?ul(a,, ++e2u2(a, , ,)+0(e3)and(31) v=vo(a, f)+esj(a, )+0(e2)where he ependencenx andr hasbeen uppressed,hile n themanifold (e), letthe olution egiven y

da= (a)+82A (a) )+ 0 (3),dr(32)=r w00+B(a, ,)+ e2B2(a, ,u)+0(s3).di-

We shallnowdevelopa formal rocedure o obtain xplicit xpressionsorA, B, j=1, 2, andequations overning;', j = 0, 1,2,*Letthe perators,N1 ndN2, lso, eexpandedn ermsf by ntroducingu s:L(po) L(pcr E,) = L(O) + s ot LpoN1(u,po)= N1(u,Pcr+EtA))(u, A)

= Ni(uo, 0) + s- (uo,0) + Niu(uo, )U1+ 0(S 2),dpo(33) pNA(U,v,po)-=N2(U, , u)= N2(uo,vo, )+ sN26(uo,Vo,O)v1 SN2u(uo, o, )u

aN22+s at-(uo, Vo, )+o((2)aPoand

g(u) = g(uo)+ egu(uo)ui 0(e2)where 1u, 2UndN2VreFrecheterivativesf he perators1 ndN2with espectto u andv respectivelyndgu s theFrechet erivativefg(u)with espectou.Theoperator (O)has igenvaluesicoowithorrespondingigenfunctions(l)(x)and*(1)(x).All other igenvaluesfL(O)havenegativeealparts.Substituting30), 31), 33) n 14)and 15),taking32) nto ccount,nd quatingcoefficientsfequal powers f ,wegetfor o(34) coo = L(O)uo,(35) -+ 2po&avog(uo);dir



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HOPF BIFURCATION PHENOMENA 223for i:

au, aLau au(36) Wu-= L(O)ul+/i-duO+N2(uo, vo,O)+Nl(uo, O)-du-A1-uB1i,aq, ~ aPo aa aq

dvi2(37) - + 2poa-v-,gu(uo)ul avo.di-Let us now find he solutions equentially. o,as defined n (29) satisfies34). Theng(uo)is known nd sinceg is quadratic nu, thesolutionfor 35) is oftheform(38) vo(7r) a2(El+ E2 e2 +E2 e2")whereE1 andE2 are constants etermined yg(uo) and the constant -.Now, et usturn urattentionoterms f0(e). SinceNj(uo,0) isperiodicnX withperiod217rnd is cubic nuo, thas a complexFourierrepresentation(39) Nj(uo,0) = '41(a, x) eimqIm=-+1,3where

4lm(a, x) N(uo,0) eim dqi, m= ? 1,3.The vector unctionsDm(a, ) can nowbe expanded neigenfunctions(l)(x) andw)(x), i= 1,2,3, ... , so that

00(Dm(a,X) = ,m(a)wn)(x)-00with q5 (a) = (4V(a, x),q((n))where (n)(x) are theeigenfunctionsftheadjoint problem.Similarly

00(40) N2(uo,10, )= Omkm(a)w(n)(x) eimqIm=-+1,3 -ooand

aL(O) 0 ilWn()(41) uo = a Z [3(n1) '1'+n-') e""]w()(x)aPo -00Let us also expandthe solutionul as

-00(42) ul a,0,x, , u)=,bm(a,/i)w )x 3emM -fi1,3 -oThis content downloaded from 152.66.143.189 on Fri, 29 Mar 2013 13:24:21 PM

All use subject to JSTOR Terms and Conditions


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224 A. K. BAJAJ, P. R. SETHNA AND T. S. LUNDGRENSubstituting39), (40), (41) and (42) in (36) we get

m==?13{ i be(a, x)[imWo-n]W( )(x)} eqimM=41,3 -oo00= E { E [,km(a)+ ma)]w (n)3(x emqm -+1,3 -oo+ a [l )eq iq)ei](n)-00 n ne l

-Al{w(l) ei' + *(i) e }-I iaB l{w(l) e'q' (1) e 1'}.To assure heperiodicityful,we choose

(43) Al +iaBl = ,uasl + +11 +21orAI + iaB1 = af31 + ((D 2',+))

Comparingheothererms, e getb1(a, +) lnA'1) a) + 0k2na) + ,ua/ nbn a, /i)(44) (iwo-An)

b'(a,Au)=O, n=?2,3,* ,and0 3)a 0 23)() n=i1 ,3ln a)+2k (a)(45) b iw=-A n=?1,2,3,-..3iwo-Akn I

Thus, u a@ ) E { E X~ln+?2na+an W(n)( /i a, x, , /i) = Y, k.() ()+a3m w)(x)} eimIm=4?1 -oo,no$l imo -An(46) + c){m)(a)+A2m (a) x)}eM=4-3 -oo lftoM -An

is the olution f 36).Similarly,ecancontinuendconstructigherpproximationsothe olution.Now etus consider43) in omedetail. he terms 1)andk1)aregeneratedythenonlinearerms i(uo, ) andN2(uo,v ,0) respectively.1(uo, ) iscubicnu so that0(1)can be writtens (l) ==a (C31+iD31)whereC31 andD31 are constantseterminedy theprojection fN1(uo,) on theadjoint igenfunction(l).Similarly,2(Uo,Vo, ) is inear nuo ndv-oachwhile -0squadraticn"a". Therefore12) anbe writtens

(12) a3(C32 + iD32)and the mplitudendphase quations32)become(47) Li 8[IaCi + (C31 + C32)a 3]+ 0(82)



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HOPF BIFURCATION PHENOMENA 225and(48) e o=wo+E[LD1 + (D31 D32)a2]+ 0(e2)where

,8(1l) C, + iDl.In thegenericase C31 C32)#0 andfor mall ,the mplitudea" of hebifurcatedperiodic olutions determinedy(49) 2o 1LCa = (C31+ C32)

The stabilityfthis ifurcatedolution epends n the ign f C31 C32).Theperiodic olutions stable ncase (C31 C32) is negativewhile t is unstablewhen(C31 C32) s positive.nthe irstase thebifurcations said o be "supercritical"hile"subcritical" ifurcationefers o the secondcase. In the nongenericase thiscoefficientf 3 vanishesnd then igherrder ermsn 32)have obe evaluatedodeterminehe mplitudeftheperiodicolutionnd ts tability11]. In the aseofsubcriticalifurcation,ince hefirstpproximationbtained ives nstableolutions,they renot he nesphysicallybservedn ny xperiment.f stable eriodicolutiondoesoccur,ts mplitudend tabilityanbeobtainedy valuatingigherrderermsin 32).

Once havingdetermined"a" as above, the correction o the frequency foscillationf he eriodicolutionsgiven y quation48).Thenew requencysgivenby(50) (y) = ojo+-[kDi + (D31+D32)a 0

Resultsnddiscussion. eshallnowdescribehenumericalesults,t s cklarhatthe ystemehavior epends n three arameters3,o, nda with he ssumptionsregarding determininghe wocategoriesfsolutions.fa 0(6 ) and r?0, theeffectfthevariationsntheflow elocity o notenter heresultsndwecan takeC32 =32 = 0. Incase is ndependentfE,C32 andD32 ingeneralrenot ero ndwehave he oupled ystem.Thecriticalalue fpo,namelycr,twhich opf ifurcationccurs, ependsn/8in complicatedanner. graphf cr s. 32 is givennFig.2. t s een hat or ertain,8 herere hree alues f cr.However,fwerestrictur tudyothose aseswhereheflows ncreasedraduallyntil ereach crthenhe art f he cr s. 2 curvehatsof nterestsrestrictedothediscontinuousurve 1,C1,D1, B2, C2,D2. Thus, lltheoreticalurves nddiscussion iven elow hallbe restrictedovalues fPcr hatfollowhediscontinuousurve ndperturbationsromt.InFig.3,resultsfthenonlinearnalysisor he ase i) aregiven. heseresultsapply othecase ofrelativelylender ubes. t shows hevariationfC1,C31 and1C1/31j,which ives he quare f he mplitudef he ifurcatederiodicolutionorA= 1, safunctionf 2 Wenote hat 1 0 for ll/confirminghe act hat he erosolution 0 is unstable orPO Pcr. t is also seen thatC31 s alwaysnegativeirrespectivef he alue f/8nd huswehave he onclusionhat,n he aseof lendertubes, heHopf bifurcationsre supercriticalor ll /3ndthat here xist tableperiodicolutionsor > 0. TheresultsnFig.3 are nmarked ontrast ith hose fRousseletnd Herrmann10].



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226 A. K. BAJAJ, P. R. SETHNA AND T. S. LUNDGREN15-

UNSTABLE Da

t CI~~~~~~~~~D54 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

FIG 2. Criticallow elocitys. 32.

20.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90~~~~~~ 0-4 - 2/ / ~~~~~~~~~5-

-8 - N-A 4C31 ' // ~~~~~~~~~~~~~C10- 1c1ic311C3112 _ 1 6 1-16 ---C3, 15-Cfj ,/IC3,~~~~~~~~~~~IC/C1

- -- Ic,,c20I2~~2

FIG. 3. Cl, C31 and IC1/C311 s.,.

The resultsor he ase ii), .e.,when (ir) 0(e), aregiven nFigs. and 5. Forthis ase,C1 andC3. remainhe ame s for he ase i).However, hevalueofC32 snonzeron his ase nd tdependsna = Lf/IA,where sthe frictionactor" hich,ingeneral,ependsn the oughnessf he ube nd heReynoldsumberf he low.Thisfactors relativelyonstantor widerange fReynolds umbersakingaluesbetween.004 and0.015.Figure shows he ariationf 3 -(C31+ C32), asafunctionof 2, fordifferentalues f a. The values fC3 for = 1.6 and a = 0.8 are alwaysnegativeor ny/3ndthuswe againhave nly upercriticalifurcationsn pite f hefact hat luid elocityhanges avebeen a-kennto ccount.t will e shownnFig.5that his s the aseas long s a > 0.71. For thepurpose fcomparison,e have lsoshown he urve orC31.t s clear hat sa increases,he urve or 3= C31 C32 eryrapidlypproacheshe urve orC31 herebyndicatinghat, erhaps, realisticalue



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HOPF BIFURCATION PHENOMENA 227ofa, abovewhich elocity luctuationseed notbe taken nto ccount ntheanalysis,ssomewhere round2 to 3.Whena < 0.71, for nstance fa = 0.4, C3 has bothnegative nd positivevaluesandthusdepending ponthevalueof/3upercritical rsubcritical ifurcationsccur.FromFig.4 it sseenthat s a isdecreased, heproportion fC3 taking ositive aluesincreases.Thusone expects hat, or ach/3,here s a critical alue ofa, a = acr, suchthat or > ac,rC3 isnegativewhilefor < acr, C3 ispositive. herefore or given/3,fa > acr, thezerosolutionu 0 bifurcatesupercriticallynto stableperiodic olutionwhereas fa < acr, thebifurcations subcritical.

4 -0 1 0.2 4 0 80

C3//-8

'I~~~~~~1 a0.4-12 0X=0.8-X=1.6-16 _ C 31

FIG. 4. C3 VS. 3 for various values of a.The variation f acr as a function f /32 is shown nFig. 5. For a > 0.71, all thebifurcations re supercriticalwhile fora


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228 A. K. BAJAJ,P. R. SETHNA AND T. S. LUNDGREN

1.0 Supercritical0.8 _

a 0.60.4-0.2 Subcritical0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

)322FIG. 5. Criticalslendernessarameter" vs.f3

Appendix. ereweshall iscusshe ifficultyssociated ithhe onversionf henonstandard15]scalar igenvalueroblemnto ector orm.The nonstandardigenvalueroblems(A. 1) wiv+pgw"+2po0fAw'+A w= Owith oundaryonditions(A.2) w=w' =O atx=O and w"= w"'=0 atx=1.The standardhoice, oconverthe inearizedquation9) into ector orms

U = U2, U1 = i2 = iwhich ives he inearystem(A.3) a = Lluwhereu = (uI, U2),(A.4) Li -2po/3( *)' -po2( *)"-_( *)andu2satisfiesheboundaryonditionsA.2).TheeigenvalueroblemorA.3) is(A.5) Aw=L1wwherew2satisfiesA.2).LetA, r= 1,2,***,be theeigenvaluesf A.5) with hecorrespondingigen-functions eingw(r.Using agrange'sormula,heproblemdjoint o A.5) turnsuttobe(A.6) L*q=gq



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HOPF BIFURCATION PHENOMENA 229where(A.7) L*= [2poG( )' 1]If Aj andq(i) are theeigenvalue-eigenvectorair for A.6), then (i) satisfiesheboundaryonditions

qi) =ql() =0 atx=O and(A.8) q (i) +P2qW = qif/i) p2q ")-2p'lArq =0 atx = 1.Now, akingA.4) into ccountndusing(A.9) (Liw(r), q(i)) = (W(r),L*q(i))it can be shown hatwe do notgettheusualorthogonalityelationw(r), q(i))=0 Vr,j,r#j.Thisdifficultyanbe overcomeyredefiningl andu2 in 13). ThischoicewassuggestedyDavidH. Terman. heprocess, owever,snotunique.Analternateormulationhichs not nthe tandard ormutwhich oesgiveweightedrthogonalityonditionsduetoC.S. Hsu. nthis ormulationeconverthescalar roblemoa vectorystemftheformauM-= L3uwhereu=(ut, u2)', u1= u, u2=u,

M=j7 2PoI )] and L3=[O P()1 ()v].Inthis ase the rthogonalityelationsurn ut obe

(Mw(r) q(S)) rsand(L3W(r), q(r)) = Ar, r, = 1,2,

REFERENCES[1] T. B. BENJAMIN, Dynamics of a systemfarticulated ipesconveying luid. Theory, roc.Roy. Soc.LondonSer.A. 261 1961),pp.457-486.[2] ,Dynamicsof systemf rticulatedipes onveyingluid. I. Experiments,bid.,261 (1961), pp.487-499.[3] R. W. GREGORY AND M. P. PAIDOUSSIS, Unstable scillationftubularantileversonveyingluid.Theory,bid.,293 (1966), pp. 512-527.[4] , Unstable scillation ftubular antileversonveyingluid. I. Experiments,bid.,293 (1966), pp.528-542.[5] M. P. PAIDOUSSIS AND T. N. SSID, Dynamic tabilityfpipes onveyingfluid,.Sound andVibrations,33 (1974), no. 6, pp. 267-294.[6] P. J.HOLMES, Pipessupportedt both nds cannot lutter,.Appl. Mech., 45 (1978), pp. 619-622.[7] , Bifurcationsodivergencendflutternflownduced scillations: finite imensionalnalysis,J.SoundandVibrations, 3 (1977), no. 4, pp. 471-503.[8] A. L. THURMAN AND C. D. MOTE, JR, Nonlinear scillationsf cylinderontaining lowingluid,J.Engrg.for ndustry,rans.ASME Ser. E, 91 (1969), pp. 1147-1155.[9] J.ROUSSELET AND G. HERRMANN, Flutter f rticulated ipes t finitemplitude, .Appl. Mech., 44(1977),pp.154-158.



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230 A. K. BAJAJ, P. R. SETHNA AND T. S. LUNDGREN[10] , Dynamic behavior fcontinuous antilever ipes conveying luidnearcritical elocities, anu-script.[11] S. N. CHOW AND J. MALLET-PARET, Integral veraging nd bifurcation,. Differential Equations, 26(1977), pp. 112-159.[12] D. D. JOSEPHAND D. H. SATTINGER, Bifurcatingimeperiodic olutions nd their tability, rch.Rational Mech. Anal., 45 (1972), pp. 79-109.[13] T. S. LUNDGREN, P. R. SETHNA AND A. K. BAJAJ,tability oundaries or low nducedmotions ftubeswith n inclined erminal ozzle,J. Sound and Vibrations, 4 (1979), pp. 553-571.[14] I. G. MALKIN, Stabilityn thecase of constantlyctingdisturbances,rikl.Mat. Mek., 8 (1944), pp.241-245.[15] M. A. NAIMARK, LinearDifferentialperators, art , FrederickUngar,New York, 1968.[16] J. E. MARSDEN AND M. MCCRAKEN, The Hopf Bifurcation nd itsApplications, pringer-Verlag,New York, 1976.[17] N. N. BOGOLIUBOV AND Y. A. MITROPOLSKY, Asymptotic ethods n the Theory f Non-linearOscillations,HindustanPub., Delhi, 1961.

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Hopf Bif Phenomena Bajaj