Hindawi Publishing CorporationAdvances in Acoustics and VibrationVolume 2009, Article ID 432340, 19 pagesdoi:10.1155/2009/432340Review ArticleVibration of Slender Structures Subjected to Axial FloworAxially Towed in Quiescent FluidL. Wang and Q. NiDepartment of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, ChinaCorrespondence should be addressed to L. Wang, wangliniping@sohu.comReceived 9 November 2008; Accepted 16 April 2009Recommended by Rama BhatThe vibrations and stability of slender structures subjected to axial ow or axially towed in quiescent uid are discussed in thispaper. Aselective reviewof the research undertaken on it is presented. It is endeavoured to showthat slender structures subjected toaxial ow or axially towed in quiescent uid are capable of displaying rich dynamical behavior. The basic dynamics of straight andcurved pipes conveying uid (with or without motion constraints), carbon nanotubes conveying uid, tubular beams subjectedto both internal and external ows in axial direction, slender structures in axial ow or axially towed in quiescent uid, cylindricalshells conveying or immersed in axial ow, solitary plate or parallel-plate assembly in axial ow; linear, nonlinear, and chaoticdynamics; these and many more are some of the aspects of the problem considered.Copyright 2009 L. Wang and Q. Ni. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.1. IntroductionThe study of ow-induced vibrations of slender structureshas been intensied in the past decades. This may bepartly due to the increased need for stability and reliability,especially in the power generating industry where repeatedequipment failures have evidenced the inadequate state-of-the-art. Thus, it has now become increasingly important totry to understand and to be able to predict the dynamicalbehaviour of slender structures in ow, such as what mightbe found in mechanical equipments, nuclear reactors, heatexchangers, steamgenerators, ocean mining pipes anddrill-strings [13]. Unlike the case of vibrations of slenderstructures inducedbycross-ow, thestudyof dynamicalbehaviours of suchstructures inducedbyaxial owis arelatively newphenomenon, beginning seriously in the1960s [4, 5]. Althoughmost failures are associatedwiththecases of cross-ow, theconditions of axial owhavealso been shown to be of importance. Moreover, quiteapartfrompractical considerations, theseproblemsareofsucient intrinsic interest, inthe realmof dynamics ofvarious dynamical systems subjectedtogyroscopicforces(e.g., axially accelerating belts), to merit study for their ownsake.Motivated by the quest for a fundamental understandingof uid-structure interactions as well as by applications inseveral areas of engineering, the vibrations of axially movingslenderbeamsorcylindersinuidhasalsoattractedtheattention of several investigators [68]. As can be expected,thedynamics of slender structures axiallytowedinuidshould be dierent from that of slender structures conveyinguid or immersed in axial ows.The current paper attempts to present a selective reviewof thepublishedliterature, emphasizingtheworkdealingwithslender structures subjectedtoaxial owor axiallytowed in uid, specically the vibrations and stabilities of (i)pipes conveying uid, both straight and curved (ii) carbonnanotubes conveying uid(iii) tubular beams subjectedboth internal and external axial ows (iv) cylindrical shellsconveyinguid(v) plates inaxial owand(vi) slenderstructures axially towed in quiescent uid. The main purposeof this article is toreviewthe recent literature whichisrelevant to all aspects and ramications of slender structuressubjectedtoaxial oworaxiallytowedinuid. Inorderto display some new results reported on this problem, therelated background theory and some of the early work willalso be discussed.2 Advances in Acoustics and Vibration2. Dynamics of Straight Pipes Conveying FluidSince the dynamics of straight and curved pipes, cantileveredand supported pipes, are fundamentally dierent, they willbe treated separately. In each case the linear dynamics will betreated rst.2.1. Linear Dynamics of Straight Pipes Conveying Fluid2.1.1. The Simplest Equation of Motion. Consider a straightpipe conveying uid (Figure 1). If externally imposed ten-sion, internal damping, gravity, a possible elastic foundationand pressurization eects are either neglected or absent, thelinearequationofmotioncanbewritteninaparticularlysimple form [9]EI4wx4+ MU22wx2+ 2MU2wxt + (M + m)2wt2= 0, (1)where EI is theeectiveexural rigidityof thepipe, misthemassofthepipeperunitlength, Misthemassofuidperunit length, owingwithasteadyowvelocityU, andwisthelateraldeectionofthepipe; xandtarethe axial coordinate and time, respectively. The uid forcesaremodeledintermsof aplugowmodel, whichisthesimplest possible form of the equation of motion for straightpipes conveying uid. Amore detailed treatment of the linearequation of motion was given in [9].Thevarioustermsin(1)maybedened, sequentially,as the exural restoring force, a centrifugal force, a Coriolisforce, and the inertia force.2.1.2. Straight Pipes with Supported Ends. The linear dynam-ics of the system for a pipe simply supported at both ends isvery clear now. After a Galerkin discretization, the equationof motion, asgivenby(1), canbesolvedbyconsideringacharacteristiceigenvalueproblem. Theeigenfrequencies(denoted by in this paper) of the pipe system are generallycomplex. For the sake of convenience, we may dene U as thedimensionless ow velocity, that is, U = (M/EI)1/2LU. Notethat Re() is the oscillation frequency, while Im() is relatedtothedamping, thedampingratiobeingIm()/Re(). Ithas been found that divergence in the rst mode occurs atUc = and in the second at Uc = 2. Linear theory predictsa coupled-mode utter for U> 6.375.In[9], a similar result was obtainedfor a clamped-clamped pipe. In this case, the rst-mode divergence occursat Uc= 2; thenthesystemis restabilizedat U 9.Again, still according to linear theory, another formofpostdivergence (coupled-mode) utter was predicted at U 9.3.However, thephysical existenceof thispostdivergenceutter instability is questionable, since the linear equation ofmotion cannot be used to provide reliable information oncethedisplacementsbecomelarge. Therefore, thispostdiver-gence utter would have to be veried by nonlinear theory,as will be discussed later.2.1.3. Straight Pipes with Clamped-Free Ends. Unlike thestraight pipes with both ends supported, the linear dynamicsLxUFigure 1: Schematic of a uid-conveying straight pipe with bothends supported.foracantileveredsystemmayshowsignicant dierence[10]. Uptonow, it has beenreportedthat theeect ofincreasing U, provided it remains small, is to generate ow-induced damping in the system. For a relatively small owvelocity, it was shown that the value of Im() remains pos-itive. For increasing values of U (e.g., U 4), however, thisdamping begins to be attenuated. The damping eventuallyvanishes (at U5.6) andthenbecomes negative. Thisimplies that a single-degree-of-freedom (DOF) utter via aHopf bifurcation occurs. The presence of nonzero dampingat U = 5.6, therefore, merely postpones the onset of utter.Inthelast twodecades, averticallycantileveredpipeconveying uid upwards was also studied by several inves-tigators[1114]. Itoughttoberecalledthatthisproblemis not whollyacademic. One important applicationmaybe inthe oceanmining industry. The oceanmining ofmanganese nodules, by essentially vacuuming the sea oorfrom a surface vessel, thus involves a exible, long pipe. Inthis case, the vertical pipe is always aspirating uid; thus, thedynamics may be dierent from that of the system conveyinguid downwards, as discussed in the foregoing.Upto now, inseveral experiments, it was observedthat a uid-aspirating pipe is stable for small ow velocity.However, theoretical results [14] haveshownthat uttermight occur even for pipes aspirating uid with sucientlylow-owvelocity. In2005, areappraisalofwhyaspiratingpipes do not utter at innitesimal ow is made byPadoussis et al. [14]. Inthatpaper, thelinearequationofmotion of a cantilevered pipe aspirating uid was written asEI_1 + t_4wx4+_MU2_T pA__2wx2+ 2MU2wxt + cwt+ (M + m + Ma)2wt2= 0(2)in which the added mass per unit length Maof theambient uid has been included; cis the viscous dampingcoecient, A is the cross-section-area of the uid, is thecoecient of Kelvin-Voigt internal dissipation, p is the globalpressurization, and T the externally imposed tension.From (2), if the dissipation is absent or neglected, it canbe easily veried that instability is possible for small internalow velocity. If, however, a realistic amount of dissipation istaken into account, the straight pipe was found to remainstableuptoowvelocitiescoveringtherangeofpracticalinterest (for the ocean mining application, e.g.). Thus, it hasnowbecomeimperativetotrytounderstandthestabilitymechanism of pipes aspirating uid. However, this questionremains unresolved. In[14], it has beensuggestedthatAdvances in Acoustics and Vibration 3more precise assumptions made on the intake ow structureshouldbedevelopedinthenearfuture.Wastheobservedstability only due to dissipation? Or was it because the owat the intake is such as to make utter impossible? To help getto the bottom of things, a CFD study of the ow eld wouldbe helpful, as mentioned in [14].2.2. Nonlinear Dynamics of Straight Pipes Conveying Fluid2.2.1. Straight Pipes withSupportedEnds. Presumingtheexistenceof periodicmotions, therateof workdonebythe uid on the pipe over a period of oscillationTmay beobtained from (1) [13]W = MU_T0__wt_2+ U_wt__wx__L0dt. (3)Clearly, ifbothendsofthestraightpipearepositivelysupported, then (w/t) = 0 at both ends, andW = 0. (4)This implies that self-excited oscillatory motion (utter) isnot possible for pipes with both ends supported. However,this does not mean that the system remains stable even forsuciently high U. The +MU2(2w/x2) term can be viewedas an eective compression associated with the exiting uidmomentumatthedownstreamend. Thismightbelinkedtoaslendercolumnsubjectedtoacompressiveforce. Forhigh enough U, therefore, the system would lose stability bybuckling (static divergence).In fact, even in the context of linear theory, the existenceof utter is problematic for a pipe with positively supportedends. This delicatequestionwas rst addressedbyDoneandSimpson[15]. Thequestionoftheexistenceofpost-divergence utter (coupled-utter) in this system has beenansweredbyHolmesandhiscoworker[1618]. Reference[17] is categorically entitled Pipes supported at both endscannot utterfor pipes positively supported at both ends,that is, where axial sliding is not permitted. Obviously, thisimportant conclusion was based on analyzing the nonlineardynamics of uidelastic systems.All theworkmentionedinthelast paragraphrelatestotheoretical work. Infact, thepostdivergenceutterhasnever been observed experimentally in pipes with both endssupported, though the loss of stability by divergence is easilyobservable. Perhaps this is the most potent evidence ofnonoccurrence of postdivergence utter in pipes with bothends supported.Moreover, parametric resonances may occur if the owin the pipe is not wholly steady but contains periodicpulsations. Recently, quasiperiodic and chaotic motions weredetected in pipes conveying uid with both ends supported;see, for example, [19, 20].2.2.2. Straight Pipes withClamped-Free Ends. For acan-tileveredpipesystem, itisassumedthatthefree-endisatx = L. Then, one obtains [13]W = MU_T0__wt_2+ U_wt__wx__L0dt/=0. (5)By analyzing the above equation, it is clear that W< 0 forU> 0 and small, and free motions of the pipe are damped.If, however, Uissucientlyhigh, whileovermostofthecycle (w/x)Land (w/t)Lhave opposite signs, then onecanobtainW>0. Thisimpliesthatthepipewill gainenergy from the owing uid, and hence free motions willbe amplied.The nonlinear dynamics of straight pipes with clamped-free ends is concordant. If we denote the onset of utter asa Hopf bifurcation, the Hopf bifurcation gives rise to limitcycle motions, which is also what is observed experimentally.Inthis case, however, the dynamics canbe muchmorecomplex. TheHopfbifurcationcanbeeithersupercriticalorsubcritical, dependingontheparameterof massratio(=M/(M+m))andaparameterinvolvingthefrictioncoecient and the slenderness of the pipe [21]. Moreover,the limit cycle motion can be either two or three dimensional[22], again depending on .Thechaoticdynamicsofcantileveredsystemswasalsoinvestigatedextensivelyinthepast twodecades(see, e.g.,[2332]). Inthe rst suchstudy, TangandDowell [23]considered a cantilevered pipe with an inset steel strip andtwo equispaced permanent magnets on either side of the free-end, thus exerting nonlinear forces on the strip and bucklingit into one of the two potential wells on either side of thepipe. If the ow velocity is increased suciently above thecritical value for utter about the buckled state, numericalresults will show that the cantilevered system might displaychaotic motions. This autonomous system was studied onlybrieyandtheoretically. Amoreextensivetheoretical andexperimental study was made of a pipe system with externalexcitation. Intheexperiments, thepipewasexcitedbyashaker. Once again, chaotic motions were detected when theamplitude of the external excitation was suciently higherthan the threshold value of this force for chaos. The chaoticmotions, strongly inuenced by the ow velocity, have alsobeen observed experimentally.At about the same time, Padoussis andMoon[24]undertookacombinedtheoreticalandexperimentalstudyof the autonomous system of a cantilevered pipe conveyinguid. Inthis case, however, the pipe is interactingwithmotion constraints somewhere along the length of the pipe.Theonlynonlinearityconsideredinthesystemwas duetothe nonlinear constraints, modeledby cubic springs.The experiments for a cantilevered pipe, with either air orwater internal ow, showedthat, whentheowvelocitywas sucientlybeyondtheonset of utter for thepipe,the pipe wouldimpact onthe motionconstraints, thusintroducing nonlinear force. The systembecame chaoticthroughthe route of perioddoubling bifurcations. Theanalytical model, after Galerkin discretization to two DOFs.,exhibitedasimilarbehaviour. Thesameanalytical modelwas furtherstudiedandsomenewresults wereobtainedby Padoussis et al. [25]. It was shown that, after the Hopfbifurcation, a symmetry-breaking transcritical-like pitchforkbifurcation occurs, followed by a sequence of period-doublingbifurcations, leadingtochaoticmotions. Sampleresults of phase portraits based on the equation of motiongiven in [25] are shown in Figure 2.4 Advances in Acoustics and VibrationThis same system was restudied with higher dimensionalmodels (upto7 d.o.f.) by Padoussis et al. [26]. Moresignicantly, the impact models for the motion constraintswere improved. Inthis case, for the exact experimentalparameters intheanalytical model, excellent quantitativeagreement (within 510%) was obtained.Finally, by using the full nonlinear equations of motion[27], the same cantilevered system was re-examined, com-pleting the circle of studies on this system. The post-Hopfdynamics predicted with lower d.o.f. (e.g., 3 d.o.f.) was quitedierent from that predicted with higher d.o.f (e.g., 4 d.o.f.).It was found that, the degree of agreement with experimentbecomes excellent for at least 4 d.o.f..Also, ifthecantileveredsystemisstanding, interestingdynamics (e.g., the pipe regains stability after utter instabil-ity as the ow velocity is increased) may arise [28, 29]. How-ever, in the study of [29], the only nonlinearity considered isthe nonlinear force induced by the motion constraints.Moreover, chaoticmotionsmayalsooccurforacan-tilevered system with an intermediate linear spring support,or witha mass addedat the free-end[3032]. In[31,32], the pipe is allowedtovibrate ina 3Dspace. Theintermediate spring supports were disposed in symmetricalfashion with respect to two axes. For the pipe with a four-spring array placed somewhere along the length of the pipe(but not close to the free-end), it was found that the systemlosesstabilityviaaHopfbifurcation(i.e., similarlytotheplanarmotionofacantileveredpipe). Again, apitchfork,or symmetry breaking bifurcation was detected at a higherowvelocity. However, thepipewaspredictedtooscillateasymmetrically; and symmetry is regained for a higher owvelocity. Interestingly, quasiperiodic oscillations, followed bychaoticoscillations, havebeenfoundinsuchadynamicalsystem with suciently high-ow velocity. However, if thearray of four springs is positioned closer to the free-end andthetotal stinessofthespringsismuchlarger, theinitialinstability was predicted to be a pitchfork bifurcation.3. Dynamics Of Curved Pipes Conveying FluidOne might have thought that a similar analysis of straightpipes conveying uid can be extended to curved pipes. This isnot so, however. Actually, in contrast to the systems reportedso far, until 1988, there remained considerable confusion anduncertainty as to the vibrations and stability of curved pipesconveying uid.Workonthe vibrationandstability of curvedpipesconveying uid appears to have started in the 1960s. Someof thekeycontributionsinthisrealmweremadeby, butnot limited to, Svetlitskii [3335], Chen [3638], Doll andMote Jr. [39, 40], Hill and Davis [41], Dupuis and Rousselet[42], Misraet al. [4345], Ni et al. [46], Qiaoet al. [47,48], and Jung and Chung [49]. The systems studied rangefromcurvedpipesshapedascirculararcs, L-orS-shapedcongurations, analyzedbynite-elementtechniques[3941, 4345], transfer-matrix technique [42, 50] or dierentialquadrature method (DQM) [4648, 51, 52].Unlike the straight pipes, motions of curved pipes gener-ally require four displacement variables (three displacementsof the centerline and a twist angle) and hence at least fourequations to govern the motions. For the circular-centerlinepipes, depending on the initial shape and state of the systemas well as assumptions made, it is often possible to decouplethe motions into in-plane and out-of-plane motions.Therearethreemaintheoriesavailableforpredictingthe stability and vibrations of curved pipes conveying uid:the so-called inextensible theories [36, 37, 43, 4648], themodied inextensible theory [44], and the complete exten-sible theories [33, 34, 4042, 44, 45, 49]. The inextensibletheoriesassumethatthecircularcenterlineof thepipeisessentially inextensible and all steady-state stress resultantsareabsentorneglected. Thecompleteextensibletheories,however, donotmakethisassumptionandgenerallytakeinto account the changes in formwith increasing owvelocity, aswell astheforcesgeneratedthereby; thus, thesteady-state axial tension-pressure force has been considered.Themodiedinextensibletheories takeintoaccount theinitial stresses due to owing uid in a curved pipe.The inextensible theories predict that curved pipeswith both ends supported are subject to divergence atsucientlyhigh-owvelocities, similar tostraight pipes.However, instability was predicted to be impossible by usingthemodiedinextensibleandtheextensibletheories.Oneinteresting practical result is that the modied inextensibletheory gives results very close tothose of the completeextensible theory. For the cantilevered system, however, boththe inextensible andextensible theories predict that theform of utter instability might occur with suciently high-ow velocity. The reason may be that the steady-state axialtension-pressureforcehasalesspronouncedeectonthedynamics of cantilever system [48].Here, it shouldbepointedout that, theliteratureonthenonlineardynamicsofcurvedpipesconveyinguidisverylimited. DupuisandRousselet [53]haveundertakenacareful derivationofthenonlinearequationsofmotionbytheNewtonianapproach; however, theirequationsarenoteasytosolvebecauseoftheircomplexity. In2005, byusingtheinextensibletheory, Ni et al. [46] developedacantilevermodel, inwhichacurvedpipeisembeddedinnonlinearfoundations. BasedonDQMdiscretization, thein-planevibrationsofthesystemwerediscussed, showingthat chaotictransients couldoccur. Recently, Qiaoet al.[47] investigated another cantilevered curved pipe conveyinguid with motion constraints (Figure 3) and explored someinteresting dynamics. Again, the inextensible theory wasutilized for the cantilevered system. As can be expected, thecurved pipe would impact on the motion constraints whenthe deection of the pipe becomes reasonably large due toincreasing uid velocity.The analytical model, after DQMdiscretization, exhibitedvarious behaviors (seeFigure4).The route tochaos was showntobe via period-doublebifurcations. This curved pipe model was further analyzedby Lin et al. [48] by applying an external excitation at thefree-end of the curved pipe. In the forced pipe system, theroutes to chaos were found to be via either period-doublebifurcations or quasiperiodic motions. Therefore, the forcedsystem can also display rich dynamics.Advances in Acoustics and Vibration 51 0.5 0 0.5 1Displacement15105051015Velocity(a)1 0.5 0 0.5 1Displacement15105051015Velocity(b)1 0.5 0 0.5 1Displacement15105051015Velocity(c)1 0.5 0 0.5 1Displacement15105051015Velocity(d)1 0.5 0 0.5 1Displacement15105051015Velocity(e)1.5 1 0.5 0 0.5 1 1.5Displacement15105051015Velocity(f)Figure 2: Theoretical phase portraits of the free-end of the pipe, with motion constraints modeled by a cubic spring and 2 d.o.f., for dierentow velocities.Before closing this section, it ought to be noted that theuidvelocityowingincurvedpipeswasassumedtobesteady, in all the related work cited in the foregoing. Moresignicantly, thegeometricnonlinearities inducedbythedeformation of curved pipes have been neglected in [4648].If, however, the uid velocity is not steady and the geometricnonlinearities are considered, the nonlinear dynamics of thecurved pipe may be much richer.6 Advances in Acoustics and VibrationFlow inFigure 3: Schematic of a uid-conveying curved pipe with motionconstraints.4. Vibrations of Nanotubes Conveying FluidAfter the invention of carbon nanotubes (CNTs) by Iijima[54], it has beenshownthat CNTs have goodelectricaland mechanical properties and so they have potentialapplications in design for nanoelectronics, nanodevices,nanocomposites, and so forth, [55]. Because of perfecthollow cylindrical geometry and high mechanical strength,CNTsholdsubstantial promiseasnanocontainersforgasstorage, and as nanopipes for conveying uid (such as gas orwater) [5658]. It is not surprising, therefore, uid owinginside CNTs has become an attractive research topic [5962].Inanattempt tounderstandandbe able topredictthe uid-structure interactions inCNTs conveyinguid,Tuzun et al. [63] developed molecular dynamics simulationsof uids owing through CNTs. It was found that in a uidconveyingCNTsystem, themotionof theCNTs plays asignicant role in the uid ow. For example, a uid owingthrough the CNTs tends to straighten out the CNTas it exes,and simultaneously excites longitudinal vibration modes ofthe CNTs.Sincemolecular dynamicsimulations aredicult forlarge-scale systems, continuum mechanics models have beenutilized to investigate the vibrational behavior of CNTsconveying uid [6470]. Natsuki et al. [64] studied the wavepropagation in single- or double-walled carbon nanotubes(DWCNTs)lledwithinternal owinguidsbyusinganelastic shell model.Ontheotherhand, thetransversevibrationsofuid-conveyingCNTs byusingEuler beamtheory, have beenstudied recently. Yoon et al. [65, 66] have developed a single-elastic Euler beammodel for vibratingCNTs containingowing uids, both for the cantilevered and supportedsystems. Itwasfoundthattheeectofuidowvelocityon the resonant frequencies of CNTs is signicant. Structuralinstability of the CNTs could occur at a critical ow velocity.The critical ow velocity could cover the range of practicalinterest. However, the eects of ow velocity on the resonantfrequencies and the instability of CNTs would be mitigatedwhen a CNT is embedded in a surrounding elastic medium(suchaspolymermatrix). Aspointedout byYoonet al.[65], the available data in the literature showed that the owvelocity inside CNTs might range from 400 m/s to 2000 m/s,or even up to 50000 m/s, in spite of the fact that the availabledataforowvelocityofwaterinsideCNTs(ofverysmallinnermost diameter) are much lower than this value. In 2007,Reddy et al. [67] investigated the eect of uid ow on thefree vibration and instability of uid-conveying single-walledcarbonnanotubes (SWCNTs), using bothatomistic andEuler beam models. Wang et al. [68] reported some results ofan investigation into the inuence of internal owing uidonthecouplingvibrationof uid-lledCNTs; again, theEulerbeammodel wasused. Recently, WangandNi [69]further analyzed the uid-conveying CNT model developedby Yoon et al. [65] and explored the possible postdivergenceutter existing in the same dynamical model. Of course, thepostdivergenceutterwas predictedbythelineartheory.More recently, Wang et al. [70] considered the thermal eecton the vibration and instability of a uid-conveying CNT.Based on the Euler beam theory, it was concluded that at lowor room temperature the critical uid velocity for nanotubeincluding the thermal eect is larger than that excluding thethermal eect and increases with the increase of temperaturechange. At high temperature the critical uid velocity for thenanotubeincludingthethermaleectissmallerthanthatwithout considering the thermal eect and decreases with theincrease of temperature change.It is noted that [6570] only considered the vibrations ofsingle-walled carbon nanotubes conveying uid. Continuingthis line of investigation, the ow-inducedvibrations ofDWCNTs (Figure 5) or MWCNTs conveying uidwerealso examined by using Euler beam theory [7174]. Basedonthe multiple-elastic beammodel, the vander Waalsinteraction between tubes has been accounted for, showingsome important results.Finally, it ought tobe stressedthat, in[6574], thevibrationandstabilityof nanotubesconveyinguidweredescribedbytheEulerbeammodel, whichalsohasbeenutilized to predict the dynamics of straight pipe conveyinguid, as discussed in Section 2. Therefore, the eect resultingfrom the small (nano-) scale on the vibrational propertiesof nanotubes conveying uid has not been included so far.Although the Euler beam model and several other classicalcontinuum models are relevant to some extent, the lengthscales associated with nanotechnology are often sucientlysmall tocall theapplicabilityof continuummodels intoquestion. Themainreasonis that at small lengthscalesthe material microstructure (such as lattice spacing betweenindividual atoms) becomes increasingly important. Theeects of these small length scales can no longer be ignored.This has raised a major challenge to the classical continuummechanics. Therefore, a possible solution is to develop somenew fundamental theories based on the classical continuummodels. Suchnewtheories wouldaccount for the smalllengthscales byincorporatinginformationregardingthebehavior of material microstructure.5. Dynamics of Tubular Beams Subjected toBoth Internal and External Axial FlowsBecauseof important applications andacademicrequire-ments, the vibration and stability of a tubular beam system,subjectedtobothinternal andexternal axial ows, haveAdvances in Acoustics and Vibration 70.02 0.01 0 0.01Dimensionless displacement0.0200.02Dimensionlessvelocity(a)0.012 0.008Dimensionless displacement0.00400.004Dimensionlessvelocity(b)0.04 0.02 0Dimensionless displacement0.0500.05Dimensionlessvelocity(c)0.04 0.02 0 0.02Dimensionless displacement0.10.0500.050.1Dimensionlessvelocity(d)0.04 0.02 0 0.02Dimensionless displacement0.10.0500.050.10.15Dimensionlessvelocity(e)0.04 0.02 0 0.02Dimensionless displacement0.0500.050.1Dimensionlessvelocity(f)0.04 0 0.04Dimensionless displacement0.20.100.10.2Dimensionlessvelocity(g)0.04 0.02 0Dimensionless displacement0.10.0500.050.1Dimensionlessvelocity(h)Figure 4: Theoretical phase portraits of the free-end of the curved pipe, with motion constraints modeled by a cubic spring, for dierentow velocities; from [47].8 Advances in Acoustics and VibrationFluid outFluid inh hR1R2Figure 5: Schematic of a DWCNTs with internal uid ow.UoUiUo(a)UoUiUo(b)Figure6: (a) Schematic of a tubular beamsubjectedtobothinternal and external axial ows; (b) the system considered in [83].been studied by many investigators in the past decades. InFigures 6(a) and 6(b), two typical tubular beam systems aregiven. The conguration of Figure 6(b) thus resembles thatofadrill-stringwithaoatinguid-powereddrill-bit;forexample, several related models developed in [3, 7577].Itisnotedthat, thetubularbeamsshowninFigure6are subjectedconcurrentlytointernal andexternal axialows. In fact, the problem of a tubular beam subjected tobothinternal andexternal owshasbeenstudiedbefore,by many investigators. Cesari and Curioni [78] have studiedthe buckling instability in tubular beams subject to internaland external axial ows. Hannoyer and Padoussis [79]combined theory and experiments on the linear vibrationsandstabilityof atubularbeamwithtwosupportedendsorcantilevered,subject tobothinternalandexternalaxialows; they found multiple divergence and utter instabilities.Theory and experiments were in quite good agreement. Atabout the same time, Grigoriew [80] considered a drill beamwith an initial curvature in the axial stream and analyzed itsstability. Another notable work by Padoussis and Besancon[81] discussed various aspects of the vibrations and stabilityof clusters of tubular beams conveying internal uid and sur-rounded by a conned external axial ow. By calculating theeigenfrequencies ofthetubularbeamsystemandstudyingtheir evolution with various ow velocities of either internalor external uid, the free vibrations were investigated. Wangand Bloom [82] formulated a mathematical model to studythe dynamics of a submerged and inclined concentric tubularbeam system with internal and external ows, the resonantfrequencies of that system obtained and analyzed.Recently, Padoussis et al. [83] reported some interestingresults on this problem. The basis of that work is Luus thesis[84]. Luu [84] and Padoussis et al. [83] diered from thework of [7882] in two signicant ways: rst, the externaland axial ows are countercurrent, and second the two owsare not independent of eachother (see Figure 6(b)). Inthe study by Padoussis et al. [83], a theoretical model wasdeveloped for the dynamics of a hanging tubular cantilever,centrally located in a cylindrical container, with uid owingdownwards insidethecantilever. Theinternal uid, afterexiting from the free-end, is deected at the bottom of thecontainer, and thereafter ows upwards in the annular spacebetweenthecantileverandcontainer. It isnotedthat theconguration developed in Padoussis et al. [83] was inspiredby the geometry of a drill-string with a drill-bit at the lowerend.The drill-string-like system considered in [83] consists ofa uniform tubular beam of length L, external cross-sectionalareaAo, exural rigidityEI, and mass per unit lengthMt,conveying downwards incompressible internal uid of massper unit length Mf, owing axially with constant velocity Ui.The internal uid leaving the lower end of the tubular beamthen ows upwards with velocityUowithin an outer rigidchannel. The linear equation of motion for this system canbe written as [83]EI4wx4+ Mt2wt2+ Mf_2wt2+ 2Ui2wxt + U2i2wx2_+ fAo_2wt22Uo2wxt + U2o2wx2__(T Af pi + Aopo)L +_Mt + Mf fAo_g(L x)12CffDoU2o_1 +DoDh_(L x)_2wx2+__Mt + Mf fAo_g 12CffDoU2o_1 +DoDh__wx+ 12CffDoUowt+ kwt= 0,(6)wherew(x, t)isthelateral deectionofthetubularbeamandtistime; fisthemassdensityoftheuid; gistheacceleration due to gravity; Cfand k are the viscous dampingcoecients; TLis theaxial tension, inducedbytheuidAdvances in Acoustics and Vibration 9LwuvhaxFigure 7: A cylindrical shell, showing some key dimensions.xyUUFixed endFigure 8: Schematic of a solitary plate in axial ow.pressureatthelowerend; Doistheouterdiameterofthetubular beam and Dh is the hydraulic diameter of the annularchannel ow. The denitions ofpoL,piL and can be foundin [83].Basedonnumerical calculations, Padoussisetal. [83]have analyzed the evolution of the eigenfrequencies. It wasfoundthat, if theannularspaceiswide, thedynamicsisdominatedby the inside ow(i.e., the owwithinthedrill-string), forlow-owvelocities, theowincreasesthedamping associated with the presence of the annular uid;if the annular space is narrow, however, the annular ow isdominant,tendingtodestabilizethesystem,givingrisetoutter at remarkably low-ow velocities.From the viewpoint of string-drill dynamics these resultsare interesting for the following reason: it was shown that,even if the drill-bit never makes mechanical contact with thedrill-string, the system experiences utter type of instability;andhencethestringwouldsoontouchthesurroundingwalls. In a real system, however, eective contact between thedrill-bit and the drill-string is inevitable, and so the dynamicsis more likely to resemble those of a pipe with clamped orpinned ends [83].6. Dynamics of Cylindrical ShellsSubjected to Axial FlowOf academic and practical interest is the dynamics of thin-wall pipesconveyinguid. Forverythinpipesconveyinguid, Padoussis and Denise [85, 86] accidentally found thatthis typeof pipesystems arenot onlysubject tobeam-type, but also to shell-type instabilities. For suciently high-ow velocities, if the thin pipe is relatively short, then theinstability observed is not one of lateral motions of the pipe(n =1, where n is the circumferential mode number), butratherinvolvesdeformationofthepipecross-section;thatis, theinstabilityisassociatedwithashell-typebreathingmode(typicallyn=2or3). Whenthediameterof themiddlesurfaceofthethin-wallpipeisrelativelylarge,theproblem should be analyzed by using elastic shell theory. Inthe past decades, the subject was studied theoretically andexperimentally, both for clamped-clamped and cantileveredshells.Consider acylindrical shell either conveyinguidorimmersedinanaxial ow, asshowninFigure7. Inthiscase, however, the internal ow can no longer be treated asa plug ow, but rather as a three-dimensional one. The linearequations of motion may be written as [86]

1(u, v, w) = 2ut2 , 1(u, v, w) = 2vt2 ,

1(u, v, w) = _2wt2qrsh_,(7)in which i (i = 1, 2, 3) denote linear dierential operators ofthe axial coordinate x and the circumferential angle;u,vand w are dened, respectively, as the axial, circumferentialand radial displacements of the middle surface of the shell;qris the radial surface loading per unit area, which can bewritten asqr =pi pe. (8)Inthe above equation, piand peare the internal andexternal pressures exerted on the shell. The uid is assumedtobeinviscidandincompressibleforsimplicity; theowis irrotational. Moreover, piand peare supposedtobecomposed of mean steady components and the perturbationcomponents.If theeect of thesteadycomponentsisignored, theperturbation components may be obtained via potential owtheory [86] and can be written aspi = ian + In+1()/In()_ t + Uix_2w, (9)pe = ean Kn+1()/Kn()_ t + Uex_2w, (10)at r =a 0+and r =a + 0+, respectively, for h/a 1(hand a are dened in Figure 7). In the above two equations,i and e are the uid densities of the internal and externalows, respectively; UiandUeare, respectively, denedastheinternal andexternal owvelocities; andnaretheaxial wavenumber and the circumferential wavenumber,respectively. For the particular case of the internal or externaluid being quiescent, (9) and (10) still apply but with Ui = 0or Uo = 0.10 Advances in Acoustics and VibrationItcanbeseenthatthetermsarisingfromthesquare-brackets operator in (9) and (10) can be written as 2w/t2+U22w/x2+ 2U2w/xt. Thesevarious terms areasso-ciated, respectively, withtheinertiaof theuid, andthecentrifugal andCoriolisforcesofthemovinguid. Thus,the uideect is wholly analogous tothat acting onastraightpipeconveyinguid. Itisnotsurprisingthatthemechanismof underlyinginstabilitiesappearstobequitesimilar to that of beam-like instabilities of pipes conveyinguid. If the shell is supported at both ends, correspondingtoaconservativesystem, itlosesstabilitybydivergenceatacertainowvelocity. Foraslightlyhigherowvelocity,the shell may subject to a coupled-mode utter. Once again,the reliability of the postdivergence utter predicted bymeans of linear theory is questionable and hence needs tobe re-examined by means of nonlinear theory, as discussedlater.In the case of a cantilevered shell conveying internal uid,correspondingtoanonconservativesystem, theinstabilityis intheformof single-modeutter, similar tothecaseofthickerpipes. Thedynamicswithexternalowisquitesimilar to that with internal ow. This form of single-modeutter is what was observed experimentally [85, 86].Experiments with elastomer shells and air-ow showedthat cantilevered shells lose stability by utter, as predictedby theory [86, 87]. Inthe case of shells withclampedends, however, the dynamics observed in experiments withinternal and external ows were quite dierent: with externalowthe systemlost stability by divergence, while withinternal owthesystemlost stabilitybyutter [86, 88].Therefore, thedynamicsofshellsconveyinginternal uidpredicted by linear theory did not agree with that observedexperimentally, sincedivergencehasnot beendetectedinexperiments. It would seemthat re-examination of thedynamics was necessary.Clearly,this re-examinationmustinvolve both nonlinear theory and further experiments. Thiswas discussed in greater detail by Karagiozis et al. [8991]and Padoussis [13].The question of nonoccurrence of divergence in exper-iments withshells conveying internal owwas resolved[13, 92] byconductingexperiments withstier shells. Itwas found that the stier shells did indeed lose stability bydivergence, which is what is predicted by theory.Based on the nonlinear theory, Amabili et al. [93] furtheranalyzed a shell with simply supported ends conveyinguid. Theypredictedthattheshellwouldlosestabilitybydivergenceviaastronglysubcritical pitchforkbifurcation.However, they did not develop coupled-model utter. As theexperiments [89] were always done with shells with clampedends (for experimental convenience), a new nonlinear the-oretical model was developed for shells with clamped ends[90, 91]. Again, the postdivergence utter was not detected.Theory and experiments are in reasonable agreement witheach other quantitatively.7. Dynamics of Plates in Axial FlowVibration of exible plates due to axial ow is an importantissue, for instance, in paper manufacturing and paperprinting[94, 95], alsoinparallel-plateassembliesusedascore elements in some research and power nuclear reactors.7.1. Solitary Plate in Axial Flow. First, consider a twodimensional plate of exural rigidity = Eh3/[12(1 2)], E and being the elastic modulus and Poisson ratio,respectively,h being the plate thickness andpthe density.The plate is subjected to a ow-related perturbation pressurep. The schematic of the analytical systemis showninFigure 8. The linear equation of motion can be written as4wx4+ Cdwt+ Ph2wt2= p, (11)where Cdis viscous damping coecient. It is assumed thatthe ow is inviscid, the complete solution for phas beenobtained by Kornecki et al. [96] and is given byp =L__10_L22wt2+ 2UL 2wt + U22w2_lnx d R(x)_,(12)whereR(x) = U2__w

(1) + (L/U) w(1)_ln(1 x)_w

(0) + (L/U) w(0)_lnx_,(13)inwhichxandaredenedbyx=x/Land =u/L,respectively, u being a dummy variable; t is dimensional time; is the uid density.Theterms giveninthesquare-brackets ontheright-hand side of (12) have the similar functional form as in (1),andthefunctionln|x |maybeviewedastheeectofspatial memory. ThedierenceinthetheoriesmentionedinSections2, 3, 4, 5, and6andthatfortheplatehereissignicant. For pipes, tubular beams and shells of the localuid forces depend only on the local displacement. For theplate problem, however, the local uid forces depend on theglobal ow eld.Byusinglineartheory, it hasbeenfoundthat aplatewith supported ends loses stability by divergence and thenby coupled-model utter subsequently. These theoreticalndings were broadly supported by experiments conductedby Dugundji et al. [97]. It is recalled that, for uid-conveyingpipes with both ends supported, the coupled-model utterinstability, which is predicted by the linear theory, has notbeen observed experimentally. In the case of supported platessubjected to axial ow, however, this coupled-model utterwas indeed observed in experiments.For a cantilevered plate subjected to axial ow, thedynamics is much clearer; the system loses stability by utter,as predicted by theory. Also, this form of instability has beenobserved experimentally. The similarity in the form of theuttering plate to a uttering pipe, in all its features, is quiteremarkable. In the case of a cantilevered plate, however, thepredictionis muchmorecomplicated, sincethevorticityshed by the apping plate into the wake should be taken intoaccount [97100].Advances in Acoustics and Vibration 11Morerecently, TangandPadoussis[101103]furtherinvestigated the instability and nonlinear vibrations of two-dimensional cantilevered plates in axial ow. In [101, 102],a nonlinear equation of motion has been utilized, assumingthemiddleplaneof theplatetobeinextensible, togetherwith the unsteady lumped-vortex model for calculating theunsteadyuidloads. Theutter boundaryobtainedwascomparedwithavailableexperimental data. It was foundthat, when the plate is long, the theoretical predictions arein very good agreement with measurements from dierentexperiments. Incontrast, agreement withexperiments israther poor for short plates. In another recent work reportedbyTangandPadoussis[103], thenonlinearvibrationofthesamedynamical systemdevelopedin[101, 102] wasfurtherstudied. However, anadditional springsupportofeither linear or cubic type was installed at various locationsontheplate. Whentheowvelocityis sucientlyhigh,theplatewas predictedtoexhibit chaoticmotions viaaperiod-doubling route. However, these interesting dynamicalbehaviours should be examined experimentally.7.2. Parallel-Plate Assembly. A parallel-plate assembly, gener-ally, consists of many thin plates stacked in parallel; betweenthese parallel plates there are narrow channels to let coolantow through. The main problem in this type of fuel systemis the static and/or dynamic instabilities due to the owinguid. In some practical tests, large deections and/or utterwereobservedwhentheowvelocitybecamesucientlyhigh(see, e.g., [104107]). The large deections and/orutter might lead to failure in practice.Many attempts have been made to theoretically analyzethe vibrations and study instability. Perhaps the rst study onthis problem was by Miller [108], who presented a theory forpredicting the critical velocity for static divergence (collapse)of parallel-plateassembly. Inhisanalysis, basedonwide-beamtheoryandBernoullistheorem, thecritical velocitywas obtained by equating pressure dierences between chan-nels to the elastic restoring force of a plate. Millers theorywas further improved by Johansson [109], who included theeects of uidfrictionandowredistribution. Scavuzzo[110]andWambsganss[111]madefurtherimprovementsuponMillers andJohanssons model byconsideringthenonlinearity caused by large deections (i.e., geometricnonlinearity). RosenbergandYoungdahl [112]formulateda dynamical model and obtained the same critical velocitybyusingatwo-dimensional mode. YangandZhang[113]developed a multispan elastic beammodel to imitate a typicalsubstructure of a parallel-plate structure. In their analyticalmodel, there exists anarrowchannel betweenthe lowersurface of the wide beamand the upper surface of the bottomplateofthewatertrough. Byusingtheaddedwatermassanddampingcoecients, thefreevibrational frequenciesof the system were analyzed. Yang and Zhang [114] furtherinvestigated a parallel at plate-type structure in rigid watertrough or rigid rectangular tube.Morerecently, GuoandPadoussis[115]developedamore accurate and general theoretical analysis for parallel-plate assembly system. Intheir analysis, the plates weretreated as two dimensional, with a nite length, and the oweld is taken to be inviscid, three dimensional. In [115], theequation of motion of an elastic plate is given by_4wx4+4wx2y2+4wy4_+ Ph2wt2+ P = 0, (14)where P =P(x, y, t) may be viewed as the net load perunit area on the plate, equal to the dierence between theperturbation pressures on the upper and lower surfaces ofthe plate caused by its deection. Because of antisymmetrywith respect to the plate, the perturbation pressures on theupper and lower surfaces must be equal in magnitude, butopposite in sign.Based on (14), several important conclusions were [115](i)single-modedivergence, mostlyintherstmode, andcoupled-mode utter involving adjacent modes were found;(ii) the frequencies at a given ow velocity and the criticalvelocities increase as the aspect ratio decreases; (iii) in thecase of large aspect ratios andsmall channel-height-to-plate-widthratios, the plates lose stability by rst-modedivergence, however, very short plates usually lose stabilitybycoupled-model utter intherst andsecondmodes;(iv) critical velocities for bothdivergenceandutter areinsensitive to changes in damping coecients.Beforeclosingthissection, itshouldberemarkedthatmost studies discussed in the foregoing were based onlinear theories for parallel-plate assembly. Unfortunately,theliteratureonthenonlineardynamicsof suchtypeofstructures is verylimited. If the essential nonlinearityisaccounted for, the dynamical behaviour may be much richer.8. Dynamics of Slender Structures in AxialFlowor Axially Towed in Quiescent Fluid8.1. Slender Structures inAxial Flow. As indicatedintheIntroduction, althoughmost failuresof slenderstructures(mostly cylinders or rods) are associated with the conditionsof cross-ow, the cases of axial ow have also been showntobeofsignicance. Intherstsuchstudy, motivatedbyapplicationtothevibrationof ssilefuel rodsinnuclearreactors, Padoussis [116] has investigated the dynamics ofcylinders or rods inaxial ow. Inhis twolater papers,Padoussis [117, 118] led to a still-used semi-empiricalrelation for predicting the turbulence-induced vibrationlevels in such systems.Further research, however, was mainly driven by curios-ity. Nevertheless, many applications can be found in practice.Theseapplications shouldinclude, but not limitedto(i)dynamics of rods and reactivity monitors in nuclear reactors;(ii) the vibration in closely spaced clusters of cylinders; (iii)the turbulence-induced vibration of tube (or cylinder) arraysin heat exchangers.8.1.1. SolitaryCylinders or Rods. Theschematicof acan-tileveredcylinder inaxial owis showninFigure9. Asdeveloped in [119], the simplest form of the linear equation12 Advances in Acoustics and VibrationR0RgUFlexiblecylinderFigure 9: Schematic of a hanging cylinder in axial ow.x yUFigure 10: Idealized system of a towed cylinder with noncylindricalnose and tail segments.of motion of a cylinder in axial ow may be written asEI4wx4+ MU22wx2+ 2MU2wxt + (m + M)2wt2_12DU2CT(L x) + 12D2U2Cb_2wx2+ 12DUCN_wt+ U wx_+ 12DCDwt= 0,(15)where M = A is the virtual, or added, mass of the uid perunit length for unconned ow, A being the cross-sectionalareaofthecylinderandtheuiddensity, w(x, t)isthelateral deection, CTandCNare viscous force coecientsin the longitudinal and normal direction, respectively, CD isthelinearizedzero-owviscousdragcoecientforlateralmotions, Cbis the base drag coecient, Dis the diameterof the cylinder, and the other symbols are the same as forinternal ow.Ifbothendsaresupported, theequationofmotionisslightlymorecomplicated,dependingalsoonwhetherthedownstream end is free to slide axially, pressurization of theexternal uid, and so on.Surrounding fluidxwyL(t)Rigid wallL(t)(a)xyVLSurrounding fluidK1 K2P0P0(b)Figure 11: The cantilevered and supported axially moving beams.(a) The system considered in [7]. (b) The system considered in [8].Inthecaseof acantileveredcylinder, itwasgenerallysupposed that the free-end is ogive-shaped. The simplest ofthe boundary conditions areEI2wx2= 0,EI3wx3+f MU_wt+ U wx__m +f M_xe2wt2= 0,(16)where xe = (1/A)_LLlA(x)dx, l being the length of the shapedend; f is a parameter rst introduced by Hawthorne [120],equal tounityforatrulystreamlinedend. However, f isgenerally smaller because of 3D ow over the free-end andboundary-layer eects.From(15), it is immediately seenthat its rst lineis identical tothe equationof motionof straight pipesconveying uid (see (1)). Examining (15), it can be foundthat its second line is associated with the viscosity of the uid.In fact, (1) and (15) dier only because of the viscous termsconstituting in (15). Generally, the viscous terms are smallcompared to the rst four terms in (15). Therefore, it maybe expected that the dynamics, for cylinders with supportedends at least, is similar to that of the uid-conveying pipe.This similarity is conrmed by Padoussis [119].For a cylinder with simply supported ends, as discussedin [119], divergence can be predicted at a nondimensionalowvelocityonlyslightlyhigherthanU=intherstmode, followedat U2bydivergenceinthesecondmode. Coupled-utterispredictedat U6.48. Indeed,postdivergence (couple-mode) utter has been observed inexperiments [121]. Infact, recent calculations bymeansofnonlineartheoryhaveconrmedtheexistenceofpost-divergence utter [122], andmore recently reconrmedexperimentally [123]. In the study of [122], it was found thata Hopf bifurcation arises from loss of stability of the trivialequilibriumstate. Moreinterestingly, thesystemdisplaysAdvances in Acoustics and Vibration 130 0.5 1 1.5 2 2.5 3Moving speed020406080Im()First modeSecond modeThird mode(a)0 0.5 1 1.5 2 2.5 3Moving speed4020020Re()First modeSecond modeThird mode(b)Figure 12: The imaginary and real components of the dimensionless frequency, , as functions of the moving speed, V, for the lowest threemodes of a pinned-pinned beam; from [8].0 1 2 3 4Moving speed020406080100120Im()First modeSecond modeThird mode(a)0 1 2 3 4Moving speed4020020Re()First modeSecond modeThird mode(b)Figure 13: The imaginary and real components of the dimensionless frequency, , as functions of the moving speed, V, for the lowest threemodes of a clamped-clamped beam; from [8].quasiperiodic and chaotic motions at higher ows, of course,predicted by means of nonlinear theory.Thedynamicsof acantileveredsystemshouldbedis-cussed here, since it is not similar to that of a cantileveredpipeconveyinguid. Asreportedin[119], for f =0.8,the cantilevered system rst loses stability by divergence atU 2.04, and then by single-mode utter at U 5.16, andafter restabilization by utter in the third mode at U 8.17.The reader may be surprised by the fact that the cantileveredsystem rst loses stability by divergence at low-ow velocity.The reason is that the divergence is related to the presenceof the tapered free-end. It is recalled that the case f =0.8suggests a fairly well-streamlined end. If f = 0, however, theend is blunt, and hence divergence is not possible.The dynamics of cantilevered system predicted by lineartheory has beenre-examinedtheoretically, by means ofnonlinear theory, andexperimentally [124, 125]. It wasfound that the essential dynamical behaviour is as predictedby linear theory. However, the bifurcations do not arise in thesame way.8.1.2. Clustered Cylinders. The dynamics of clustered cylin-ders in axial ow has received considerable attention [126,14 Advances in Acoustics and Vibration127], because such systems exist in many engineeringapplications, as discussed in the foregoing.Thevibrations of suchcylinders comparedwiththatof isolatedcylinder arecharacterizedby(i) theeect ofproximityoftheothercylindersbeingimportant, causingvarious instabilities to occur at lower ow velocities, and (ii)the eect of intercylindermotioncoupling,decreasing thecritical ow velocities. Therefore, predicting the critical owvelocities for instability requires one very important piece ofdata: the cluster geometry and the intercylinder separation.For more details on this topic, one can refer to [127].8.2. Slender Structures TowedinQuiescent Fluid. Acon-siderableamountofnotableworkhasbeenconductedontowed cylinders, rods, or tubular beams. Many applicationsof this systemhaveemergedas follows: (i) vibrations ofextruding metal and plastic rods in uid [6, 7]; (ii) stabilityandvibrationsof extremelylongseismicarrays,mostlytowedbehindboats andusedinmineral explorationinthedeepseas; (iii) thevibrations of towedpipelines foreasyrelocationmostlyusedinocean; (iv) thevibrationsof articulated submarine transporters; (v) high-speed trainstraveling in narrow tunnels.A sketch of towed slender structures shown in Figure 10has received considerable attention [128, 129]. For thisdynamical model, it was found that the dynamics with lowtowingspeedis dominatedbyrigid-bodyinstabilities. Athigher towing speeds exural instabilities might arise, muchas for a cantilevered cylinder. Recently, in a work by Langreet al. [130], utter was indeed found to exist.Another typical system of a exible cylinder or a tubularbeamaxially towed inuid is showninFigure 11. InFigure 11(a), a slender cantilevered beam is extending axiallyin the horizontal direction at a known rate, while immersedin a dense incompressible uid. This tubular beam systemhasbeenstudiedbyTalebandMisra[6]andGosselinetal. [7]. InthestudybyGosselinet al., theuid-dynamicforces obtained by Taleb and Misra were perceived tobenot correctlyaccountedfor. Thus, Gosselinet al. [7]re-examinedtheuid-dynamicforces. It was foundthat,inthe case of lowconstant extensionrates, the systemdisplays a phase of oscillation with increasing amplitude anddecreasingfrequencyuntil themotionisstronglydampedand later becomes statically unstable. For faster deploymentrates, thebeamhasashortutterphaseatthebeginningof thedeployment, followedbyabrief phaseof dampedoscillation until it exhibits static divergence. For fast enoughdeployment rates, the system is unstable from the beginningand never stabilizes. It should be mentioned that the eectivelength of the towed beam is increased with time, since theaxially moving beam is clamped-free. It was also found thatthe axial added mass coecient plays signicant role in thestability of the system.More recently, an axially towed system in uid, showninFigure 11(b), was investigatedby LinandQiao[8].Compared with the system in Gosselin et al. [7], this beamsystem has hybrid supports at both ends. It is worth notingthat theformulaof thetotal axial tension(T(x)) inthesupportedbeamisdierentfromthatinthecantileveredsystem. In the study of Gosselin et al. [7], a nonzero valueofT(L) arises from drag-induced compression at the free-end. Therefore, the nal equation of motion of a supportedsystemdoes dier fromthat of a cantilevered system.Compared with the cantilevered system, the eective lengthof the supported beam system keeps constant with time.Figure 12 [8] shows the eect of moving speed on thevariation of the lowest three eigenvalues (1, 2, and 3) ofthe moving beam with pinned-pined supports. It was foundthat therst moderst becomes unstablebydivergenceinstability when the moving speed becomes equal or largerthan the lowest critical moving speed (nondimensional) V 1.06.However, utter instabilitywas predictedto occur ata higher moving speed (V=2.27) in the rst mode. Thecritical moving speeds at which divergence may occur in thesecond and third modes were both higher than V = 2.27.It is of special interest to see the case of clamped-clampedmovingbeam. Inthis case, typical results are showninFigure 13 [8]. It is obviously seen that divergence in the rstmode occurs atV=2.072 and in the third atV=4.18.However, the second mode was predicted to be stable in therange 0 < V< 4.3.9. ConclusionsThe knowledge base associatedwiththe basic dynamicsof slender structures subjectedtoaxial owor towedinquiescent uid has expanded greatly in recent years, as thenumberofapplicationscontinuestogrow. Obviously, theliterature survey is not exhaustive. For example, in Beckerssurvey[131]associatedwiththetopicof pipesconveyinguid, 223 references were cited; in a more recent review byPadoussisandLi [132], again, morethan200referencesassociated with pipes conveying uid were cited. Thispaper, therefore, presents a selective review of the researchundertaken on the vibrations of slender structures subjectedtoaxialoworaxiallytowedinuid. Undoubtedlymanyimportantcontributionshavebeenmissed. Otheraspects,whichhavebeencoveredinarecent bookbyPadoussis[133], also have not been fully discussed here.Oneitemof particular interest tothereaders is thatthe fundamental understanding and experience gained andthe methodology employed in the studies of the dynamicalmodel of uid-conveying pipes has proved to be very usefulin the study of several other dynamical models, particularlyshells conveying or immersed in axial ow, nanotubesconveyinguid, tubularbeamssubjectedtobothinternaland external axial ows, cylinders and plates in axial ow,and tubular beams or cylinders axially towed in uid.Asreportedin[132], it canbenowunderstoodthat thedynamics of thin shells containing or surrounded by annularows, with applications to aircraft engines and some typesofnuclearreactorsutilizingthermal shields. Similarly, theunderstanding of the dynamics of heat exchanger tube arraysand nuclear reactor fuel clusters subjected to external axialows owes a great deal to the understanding gained in thestudyof pipes conveyinginternal ow. Furthermore, theAdvances in Acoustics and Vibration 15dynamics of various cylindrical beam-like components sub-jected to internal or annular ows (e.g., solar thermal powerplant chimney subjected to internal and external ows) canbe understood in terms of what has been presented in theuid-conveying thin pipe system. Interestingly, however, asindicated in [132], all these mostly unexpected applicationscame 1040 years after the basic research on the topic hadalready been done.From the history of investigating the slender structuressubjectedtoaxial oworaxiallytowedinuid, itcanbeseen that, various important issues, such as, but not limitedto, vibrations of nanotubes conveying uid when accountingfor small-length eect, instabilityofpipes aspiratinguid,andnonlinear vibrations of tubular beams concurrentlysubjectedtointernal andexternal axial ows, remainnotwhollyresolved. Totheauthors knowledge, several itemsmaybe of future interest onanalyzingthe stabilityandvibrationsofslenderstructuresassociatedwithaxial-ow-forces.(i) With the development of computer techniques,utilizingadvancednumerical approaches tosimulate thedynamical behaviour of slender structures subjected to axialowor towed in quiescent uid becomes realistic. To-day, the simulationresults may play animportant role,intuitively showing the dynamical behaviour of such systems.To advance in this direction, numerical (CFD) studies usingANSYS or other procedures should be initiated, which wouldhelp reveal the dynamics closer to the truth. The CFD resultsmay meet the requirement for better coupling between thesolidanduidmodels, andmaking uidmodels morerealistic.(ii) The eect resulting fromthe nanoscale on thevibration of nanotubes conveying uid has not been includedsofar. Atsmall lengthscalesthematerial micro-structurebecomes increasingly important and its eect can no longerbe ignored. Thus, the direct use of classic continuumapproach to small length scales may be questionable. It is apossible solution to extend the classic continuum approachto smaller length scales by accounting for the informationregardingthebehaviorof material microstructure. There-fore, some new theoretical models should be developed toresolve such an important issue.(iii) In practice, the axial ow is always with stochasticvelocity or has a stochastic component superposed onsteady ow. Therefore, the study on stochastic dynamics ofslender structures subjected to axial ow has more practicalapplication in this area. However, the literature on this topicis quite limited.(iv) Nonlinear problems of various slender structuresdiscussed in Sections 2, 3, 4, 5, 6, 7, and 8 are not whollyresolved. For example, the nonlinear equation of motion ofparallel-plate assembly in axial ow, slender structures axiallytowedinquiescentuid, tubularbeamssubjectedtobothinternal and external axial ows (may not be independent),and pipes aspirating uid, have not been derived out, andhence the corresponding nonlinear dynamics has not beenexplored yet. Therefore, much attention may be concentratedonthenonlinearaspectsofthoseslenderstructuresmen-tioned in the foregoing.(v)Inthepast decades, variousmethodsof vibrationcontrol mostlyconsideredthelinearequationsof motionfor slender structures subjected to axial ow. More impor-tantly, in a vibration control system, time-delayed feedbackunavoidablyexists. As reportedbyXuandChung[134],time delayed feedback may change the stability and dynamicsof dynamical systems, leading to much more complexdynamical behaviors. To suppress the amplied oscillations,therefore, the methods of nonlinear control for slenderstructures, subjected to axial ows or towed in uid, shouldalso be developed by considering time delayed feedback.AcknowledgmentThis workis supportedbythe National Natural ScienceFoundationof China (10772071and10802031) andtheScientic Research Foundation of HUST (2006Q003B).References[1]M. P. 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