DANISH CENTER FOR APPLIED

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i I j DANISH CENTER FOR i APPLIED MATHEMATICS AND MECHANICS i

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Poul Andersen Dept, of Naval Architecture and Offshore Engineering Martin P. Bends0e Dept, of Mathematics Esben Byskov Dept, of Structural Engineering and Materials Ove Ditlevsen Dept, of Structural Engineering and Materials J. Buhr Hansen Dept, of Hydrodynamics and Water Resources Ivar G. Jonsson Dept, of Hydrodynamics and Water Resources Wolfhard Kliem Dept, of Mathematics H. Sanstmp Kristensen Dept, of Energy Engineering P. Scheel Larsen Dept, of Energy Engineering Frithiof I. Niordson Dept, of Solid Mechanics Pauli Pedersen Dept, of Solid Mechanics P. Temdrup Pedersen Dept, of Naval Architecture and Offshore Engineering P. Grove Thomsen Dept, of Mathematical Modelling Hans True Dept, of Mathematical Modelling Viggo Tvergaard Dept, of Solid Mechanics

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Pauli Pedersen, Docent, Dr.techn. Department of Solid Mechanics, Building 404 Technical University of Denmark DK-2800 Lyngby, Denmark DISCLAIMER

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FLUID TRANSPORT DUE TO NONLINEAR FLUID-STRUCTURE INTERACTION

by

Jakob Sondergaard Jensen

SOLID MECHANICS

TECHNICAL UNIVERSITY OF DENMARK

FLUID TRANSPORT DUE TO NONLINEAR FLUID-STRUCTURE INTERACTION

Jakob Sondergaard Jensen Department of Solid Mechanics, Technical University of Denmark, DK-2800 Lyngby

Abstract - This work considers nonlinear fluid-structure interaction for a vibrating pipe contain­ ing fluid. Transverse pipe vibrations will force the fluid to move relative to the pipe creating uni ­ directional fluid flow towards the pipe end. The fluid flow induced affects the damping and the stiffness of the pipe. The behavior of the system in response to lateral resonant base excitation is analyzed numerically and by the use of a perturbation method (multiple scales). Exciting the pipe in the fundamental mode of vibration seems to be most effective for high mean fluid speed, whereas higher modes of vibration can be used to transport fluid with the same fluid speed but with smaller magnitude of pipe vibrations. The effect of the nonlinear geometrical terms is ana­ lyzed and these terms are shown to affect the response for higher modes of vibration. Experimen­ tal investigations show good agreement with theoretical predictions.

1. INTRODUCTION

This work deals with fluid flow in a vibrating pipe driven by nonlinear components of the vibra ­ tory forces. A theoretical and experimental investigation of system response to lateral resonant base excitation is carried out When a fluid-filled pipe performs transverse vibrations, transverse forces affect each fluid element. Depending on the pipe slope, components of the forces act tangent to the pipe axis forcing the fluid element to move relative to the pipe. The bulk movement of the fluid is de­ termined by the total contribution of tangential forces on all the fluid elements in the pipe. In absence of external pressure forces, the axial motion of the fluid is governed completely by the nonlinear vibratory forces. The fluid motion induced also nonlinearly affects the dynamic be­ havior of the pipe. Thus the effect to be studied is governed by nonlinear interaction. Previous work has been done on nonlinear dynamic interaction between structures and movable media in contact with the structure. Seemingly paradoxical behavior of vibrating rods with pointmasses free to slide has been described by Blekhman & Malakhova (1986) and Chelomei (1983). Blekhman and Malakhova investigated systems of rigid and flexible rods, carrying collars with holes slightly larger than those of the rods. For certain conditions of exci­ tation the rod could be stabilized in upright position with a collar fixed in ‘flying ’ position along the rod. Linear theory predicts no coupling between collar- and rod-motion and the col ­ lar would be expected to simply fall under the influence of gravity. Also, the above mentioned phenomenon is potentially applicable for passive vibration damping (Babitsky & Veprik 1993, Thomsen 1996). Bajaj et al. (1980) and Rousselet & Herrmann (1981) considered fluid-structure interaction in fluid-conveying pipes. The fluid was affected by a large upstream fluid pressure in addition to the weak nonlinear vibratory forces. Consequently, the effect of the nonlinear interaction forces was to modify the fluid speed on a small-scale level. Lee et al. (1995) investigated non ­ linear interaction between longitudinal, transverse and radial vibrations of a pipe and the speed 2

and pressure of conveyed fluid. In a case study they considered a pinned-pinned pipe convey ­ ing fluid with prescribed speed. Linear studies on structures traversed by moving pointmasses have been studied by e.g. Sadiku & Leipholz (1987). The pointmass speed was externally prescribed, and the effect of the moving mass on the structural behavior was analyzed, while nonlinear interaction was ne­ glected. Pipes conveying fluid have also been the subject of many investigations in the linear domain, (e.g. Benjamin 1961, Pai'doussis & Issid 1974). These studies consider externally pre­ scribed fluid speed neglecting nonlinear interaction. For reviews of work on fluid-conveying pipes see e.g. Pai'doussis & Li (1993) or Semler et al. (1994). Also related to this work are studies on conveyance of solid material in vibrating rigid pipes or on plates. Long et al. (1994) investigated conveyance of a small solid block in a spatially- curved tube by means of vibration. Sliding, non-sliding and flying motion of the block was considered. If careful attention was given to the excitation-amplitude and -frequency the block could be transported through the tube. The motion of the rigid tubes was prescribed and no attention was given to the influence of the motion of the block on the vibration of the tube. Other work on vibrational transportation of solid and granular material can be found in Blekhman (1994). This paper is organized as follows. In section two the model is presented and the equations of motion are derived. Section three presents numerically obtained solutions to the model equations. The results show that resonant pipe vibrations generally create uni-directional fluid flow towards the pipe end. In the fourth section a perturbation analysis is conducted (multiple scales). Results are presented for two specific examples: resonant excitation of the first and fourth mode of vibration, respectively. The effect of key parameters on system response is analyzed, and so is the effect of the geometrical nonlinearities. The geometrical nonlinearities are shown to affect the response for higher modes of vibration. A comparison between the different modes of vibration with respect to fluid transport ability is provided, showing the fun­ damental mode to generally produce the highest mean fluid speed for a given power input. Section five describes experimental results for the first and second modes of vibration, showing good agreement with theoretically predicted results.

2. MODEL EQUATIONS

Figure la shows the model. A flexible cantilever pipe is connected to a fluid-filled reservoir, supplying fluid to the pipe. The fluid speed in the reservoir is vanishing, and the fluid pressure is atmospheric. The pipe is inextensible and uniform with length l, constant mass per length pA, and flexural rigidity El. The pipe performs in-plane transverse vibrations with small but finite rotations in the fixed coordinate system under the influence of lateral periodic base excitation given by the time-dependent displacement fcosQt. The pipe is slender so that Bemoulli-Euler beam theory can be applied and rotary inertia is neglected. External dissipation is included in form of linear viscous damping. Figure 1 a) System model, and b) applied coordinate system.

The fluid is considered incompressible with mass per length m. The flow is described by the time-dependent speed v(z). The fluid speed is taken as a mean of the cross-sectional speed profile and is measured positive towards the pipe end. The fluid flow is affected by internal friction between the fluid and the pipe wall, characterized by the pipe wall stress Figure lb shows the coordinate system. A pipe element is characterized by the position vector r[s,t)=(x(s,t),y(s,t)), where x(s,t) and y(s,r) are the positions of the element along each axis in the fixed coordinate system {%% respectively, and s is a curvelinear coordinate se [0,f]. The position of a fluid element relative to the pipe inlet is given by the curvelinear coordinate «, defined so that u=v. The inextensibility condition provides the relation (x'f+(y'f=l, where primes denote differentiation with respect to s. Hamilton ’s principle is employed for setting up the equations of motion. The kinetic and potential energy density of a combined pipe- and fluid-element is, respectively:

9-=U§| +im = i pA(y2 +x2)+%m(ii2 +y2 +x2 +2u(yy'+ix')), (la)

g=jf.g'-q/)ds = j£ds. (2)

The variation of the action integral added to the virtual work done by the non- conservative forces acting on the system vanishes: (e.g. Sender 1994)

SI+j',&Wt = 0. (3)

The virtual work of non-conservative forces can be divided in two: 8cHf=SQ,t/pi^+8cH/fluid, i.e. the work associated with a virtual displacement of the pipe, and the work associated with a virtual displacement of the fluid. The two contributions are given as: 4

ftWpip' =-mzi^L+ii^-j-5r, =-mu{ylSyl +xtSxl)-mu2(y'tSyt +x'l8xt), (4a)

ftWfiuij = -T»<,u?Sfiuij&‘+-—(Po - ^)&- (4b) P fluid

The virtual work done on the pipe (4a) equals the product of the virtual tip displacement and the fluid momentum. The virtual work on the fluid (4b) originates from the work done by the pipe wall friction and by the difference in fluid pressure force between the pipe inlet and outlet, Po and Pe respectively, for a virtual displacement of a fluid element Su. In (4b), Spun and Pfluid denotes the fluid tube circumference and the fluid density, respectively. External viscous damping is added later directly in modal form. The last term in (4a) is rewritten using the inextensibility condition. The pipe wall friction W/ of (4b) is proportional to the fluid speed for laminar flow, (e.g. White 1991), and to squared fluid speed for turbulent flow, (e.g. Bajaj et.al 1980). That is Wi=ajb where aj is a pipe wall friction coefficient, and 7=1,2 for laminar and turbulent flow respectively. Assum­ ing that the fluid discharges into atmospheric pressure, the Bernoulli equation, (e.g. White 1991), can be used to relate the flow conditions at the pipe inlet to the conditions at the fluid reservoir. The fluid pressure difference can then be written as P«~Pf^Apflunii'. Equations (4a)-(4b) become:

= -mu(yt8yt +*,&,)- £ mu2 |y"(1+/2) - y"£y'y"ds^Syds, (5a)

=-attSI,uUi\uh'\8u-%mu28u. (5b)

The variation of the action integral is:

(6)

By using partial integration, eliminating x by using the inextensibility condition, and requir­ ing (3) to be fulfilled for arbitrary admissible variations in Sy and Su one obtains: 33S '-/(i-H/2)( ddSddS\ A dy ds dy' ds2 By" Adx ds3x’) (7a) +/'(!+i/2)

t‘( 3 3S 3Sy * Ur oft At -imu1 (7b)

Equations (7a)-(7b) are expanded, substituting ii=v. Small but finite pipe rotations imply that (y'f cannot be neglected compared to unity whereas (y'f and higher order terms can. This yields two coupled nondimensional equations governing the in-plane transverse vibrations of the pipe and the speed of the fluid, respectively: 5

w + w"" + [w'jfjjcw'2 + w'w')d^ ] + HwV'r]' + 2/?CAv'(l+w'2) + £/V"(l + w'2)+j$£/(l - £K'(l+fw'2) (8a) + n/'j^P/Ww + tfVw"+\pUwn)d£, = pQ2 cosQz,

t/+l(a >|CZH|+i[Z)C/ = -^[vviv ,-(l-iw,2)jJovV+w,2)<;^+j5:pfi2smj»:jy£f5,

(8b) valid for /M). Dots and primes here denote differentiation with respect to rand 5 respectively, and the nondimensional quantities are:

..._y-/cosf2r f 1 El t «■? M' e ’ P- Tl(pA+m)£4’’ (9) ais&J2 a2^JUad^ 1 m a, = ]j(pA+m)’ l~ -JmEI ’

The first and second term in (8a) represents linear inertia and stiffness respectively and the third and fourth term the nonlinear inertia and stiffness. The remanding terms on the left hand side represent the fluid effect on the pipe vibrations due to the coriolis, centrifugal and angle- acceleration forces. Equation (8a) is similar to equations previously obtained, (e.g. Sender 1994) except for the presence of the external forcing term. The left hand side of the fluid equation (8b) contains the inertia and friction terms. The right hand side describes the vibratory forcing from the transverse pipe motion. The equation agrees with those of Bajaj et.al. (1980) and Rousselet & Herrmann (1981), except that there is no upstream fluid pressure in (8b). Hence, in the system considered fluid motion occurs only through coupling to the transverse pipe motion. A mode-shape expansion is applied in order to recast (8a)-(8b) into a set of ordinary differ­ ential equations. Using the expansion vK|,t)=Xfci9<(' t) ?%(§). where q>,(§) is the orthogonal set of cantilever mode shapes, and q,{t) are the generalized coordinates, yields n ordinary differen­ tial equations for the pipe and a single ordinary equation for the fluid:

q, +2^C0,ql+(02qi +2+ti(Pf,tU+gltU2)qt +2jSU

(10a) + £(Aofo, +%PFiUJJ+GllJJ2)qlqlq„ + + q,qj = p&fi2 cos £2z, kj,ms 1 1 for i=l,...,n, N^l + iu)u = -P%hn{qtqt + qtq,) k,M (10b) ~iP k.l.m.psl 6

In equation (10a) modal viscous damping, characterized by the modal damping ratio Q, has been added. In (10b) the explicit dependence on the base excitation has been neglected, since the base displacement amplitude is assumed to be much smaller than the pipe deflection

{p«q)~ The constants: tit, 1%, kit, fit, ga, hit, Aitim, Fum, Gum, Khm, Xuim and HtMp are com ­ puted from the expansion functions (see appendix). If the forcing frequency £2 is near a natural frequency tit, then a single cantilever mode shape will be dominant and a single-mode approximation (n=1) can be applied. Since the near resonant case yields the largest pipe motion and thus the strongest vibratory forcing of the fluid, this case is of interest and will be considered in the following.

3. NUMERICAL ANALYSIS OF SYSTEM BEHAVIOR

In this section time-series are shown depicting basic fluid and pipe motion. As examples, re­ sults are given for the first and fourth modes of vibration for excitation in sharp resonance, i.e. Q=(i>i, i=l,4. The results are based on numerical integration (fifth- and sixth-order Runge-Kutta-Vemer) of the single-mode approximation of (10a)-(10b). Figure 2a,c depicts the fluid speed as a function of time for the first and fourth mode of vi ­ bration, respectively. A positive mean fluid speed appears, indicating flow in direction towards the pipe end. Small oscillations are seen to overlay the mean speed. The amplitude and fre­ quency of the oscillations increase with the mode number, but the amplitude is always small compared to the mean speed. Figure 2b,d shows the corresponding deflections of the pipe end. The vibrations are seen to be regular periodic with maximum tip deflection around 9% of the pipe length for vibrations in the first mode and around 0.3% for the fourth mode. The signifi ­ cant difference in vibration amplitude is seen even though the corresponding values of the mean fluid speed are comparable. As will appear from section 4.3, the decrease in vibration amplitude happens at expense of higher power consumption. According to the model there is shown to be a mean flow through the pipe when the pipe vibrates. Similar qualitative behavior as in Figure 2 is seen for a wide range of parameters in the case of near resonant forcing. It can be shown also that all modes of vibration create the displayed effect. The shown effect is purely nonlinear and is to be analyzed in further detail in the following. 7

figure 2 Numerical simulation of system behavior for the first and the fourth modes of vibration, a,c) fluid speed U, and b.d) pipe end deflection 2qc, a,b) f2=a)i, p=0.003, c,d) f2=etj, p=0.00015; %=08, /?=0.2, 6=0.01.

4. PERTURBATION ANALYSIS OF RESONANT MOTION

A perturbation analysis of the model equations is performed in order to obtain more insight ; into the behavior of the system, and to analyze the effect of variation of the system parameters, i A multiple scales method, (e.g. Nayfeh & Mook 1979), is employed for obtaining results for ; the near resonant case. Only positive values of the fluid speed are considered, i.e. | U'11 =Z/"'. j To perform the analysis, two different time-scales are introduced. A fast time-scale T0com- | parable with natural- and excitation-frequencies, and a slow time-scale Ti=sT0 describing the ( slow modulation of the pipe-amplitude and -phase and of the mean fluid speed. The smallness ' parameter e is introduced in order to indicate small terms. Analyzing the system response in the near resonant case, a single-mode approximation of 1 (10a)-(10b) is applied:

, q, +Q>)qi + 2E(£iG>i +[ik liU)qi +£(PfuU+ gilU2)qj +s(Ami+\pFmU+GmU2)q2

+2 EfiUKMqfqi+£XBqllq,q, +9?) = Q7i?,f22cosf2T, i ! U+^ajU’-' +iU)U = -£pkliq,q-£p(h,i +iHimqf)(qiqi+qf), (lib)

where £ are used to keep track of orders of magnitude for the different terms. Nonlinear, ’ damping and forcing terms have been considered to be of order £ compared to inertia terms in (11a) and (lib) respectively. In the following, indices indicating the mode number is omitted i for brevity.

i 8

A zero-order approximation to (1 la)-(l lb) is looked for. To obtain this, a first-order ex­ pansion is introduced:

? = %(%,,%)+%(%,%), U = U0(T0,T,)+eUi(T0,T,). (12a,b)

Inserting (12a,b) into (1 la)-(l lb) and equating coefficients of the same powers in e, yields to order d:

D^q0+CO2q0 = 0, D0U0= 0. (13a,b)

And to order d: D2qt +(o2q, = —2D0Dtq0-2(& +PkU0)D0q0-(pfD0U0 +gUl)q0 ~(A+ipFD0U0 +GUl)ql-2PKUaqlD0q0 (14a) “ %9o(9o#o9o + (£>0q0 f)+pdCl2 costir,

DJUX =-DlU0-j(aJUt' +Wo)U0-Pkq 0Dlq0-p(h+iHql)(q0D2q0+(D0q0)2),

(14b) where D'pd'/dTj. The solutions to (13a,b) are:

q0 = Be!dr° + Be1* 7 * , U0 =A, (15a,b) where A and B are real and complex functions of Ti, respectively. A bar denotes the complex conjugate. The unknown functions A and B are to be determined. Substituting (15a,b) into the e'-order equations (14a)-(14b), yields D2q, =-(2ia>B'+2ia(&+PkA)B+gA2B)eid7 °

-(3A+3GA2+2iaPKA-2a2x)B2Be“sr° (16a)

- (A+GA2 +2icopKA - 2co2x)B2eVaT° +%ptiQ2eiar° +cc,

D0Ut =-iA'-^(«^-' +±A)A+pC02(k+HBB)BB 2P (16b) +pm2(Jc+2h + 2HBB)B2e2i°i7 ° + Pco2HBie4:oT‘ +cc, where ()'=d/3Tu and cc denotes the complex conjugates of the preceding terms. Terms proportional to e"“r° and terms independent of To yield secular terms in the solution to (16a) and (16b), respectively. The case of ti=to yields a near resonant term in (16a), which will produce a small divisor term in the solution. These terms are required to vanish. The nearness of the excitation frequency to the natural frequencies can be expressed as

ti = ta + ecr => £2T0=aT0+dT„ (17) 9

where <7 is a so-called detuning parameter. Inserting (17) into (16a)-(16b) and eliminating the resonant and near resonant terms in (16a) and constant terms in (16b) yields, separating real and imaginary part for the pipe equation:

cob' = -

4.1 First vibration mode , linear geometry

The system is considered for t=l, i.e. the forcing frequency is chosen near the fundamental natural frequency (S2=a)i), so that the fundamental mode of vibration is dominant. The stationary behavior of the system is obtained by setting f>'=i/=A'=0 in (18a)-(18c). For low to moderate amplitude of forcing (p<=0.2), the effect of the higher order geometrical non- linearities is inessential. Neglecting these higher order terms, the stationary solution for the modal pipe amplitude bi and the mean fluid speed A become:

!p2t??I24=[, +pkllA)2 +(jguA2-col(Q-cotjf^b 2, (19a)

(

Equation (19b) is used to eliminate A from (19a). Equation (19a) is then solved for b\ using a Newton-Raphson method. The stability of the solutions is determined by computing eigen ­ values of the Jacobian of (18a)-(18c), evaluated at the solution points. Figure 3 shows frequency response curves depicting the mean fluid speed A and the tip amplitude 26| versus the relative forcing frequency Q!cO\. The perturbation solution, based on (19a)-(19b), is compared to results obtained by numerical integration of (lla)-(llb) without the geometrical nonlinearities. The tip amplitude is also compared to the linear case with the fluid fixed inside the pipe, corresponding to setting A=0 in (19a). As appears from Figure 3b, significant motion of the pipe is confined to a frequency band near the linear resonance frequency I2=to,. This is reflected in the fluid behavior, shown in Figure 3a, in that outside the resonant region, the vibrations of the pipe are too small to induce significant fluid motion. The nonlinear coupling to the fluid damps the vibrations of the pipe in the resonant region, as appears from b). Compared to the case where the fluid is fixed inside the pipe, the amplitude of the pipe vibrations is significantly reduced. The fluid flow also slightly increases the stiffness of the pipe. This appears as a slight right bending of the response curves (hardly noticeable in the figure). For other modes of vibration, the fluid flow has a sof ­ tening effect, bending response curves left. In all cases this effect is small compared to the ef­ fect of damping. From comparison with numerical results it appears that the perturbation solu ­ tion adequately describes the qualitative behavior of the system. 10

relative forcing frequency, fl/oi, relative forcing frequency, Cl/a> t Figure 3 a) Mean fluid speed A and b) tip amplitude 2bi for the fundamental mode of vibration (i=l), versus relative forcing frequency Q}C0\ for p=0.01; (—, stable solutions), (O; numerical integration); oN).8, f?=0.2, 51=0.01.

Figure 4 shows the effect of four key system parameters on the mean fluid speed A: forcing amplitude p, mass ratio /?, turbulent wall friction coefficient a 2 and modal viscous damping ratio £1. The response is shown in a forcing frequency domain near the linear resonance fre­ quency £2=Oh. Figure 4a,b shows the effect of the amplitude of base excitation and the amount of viscous damping. The values of these parameters directly influence the amplitude of the pipe vibrations, but only indirectly influence the fluid speed. Increasing p or decreasing £1 results in larger amplitude of vibrations which increases the fluid speed. If the viscous damping is increased the response curves are also seen to flatten. Figure 4c,d depicts the effect of the relative fluid mass and the turbulent wall friction coef ­ ficient. Both parameters directly influence the mean fluid speed. Increasing % restricts fluid flow and decreasing /? decreases the magnitude of the nonlinear pipe-fluid coupling. In both cases the mean fluid speed is decreased. Decreased mean speed lowers the fluid damping on the pipe vibrations, and lower mass ratio will also directly reduce the fluid effect on the pipe motion. Consequently, the amplitude of pipe vibrations will grow. This will affect the fluid, but this secondary effect is small compared to the direct influence. Two limit curves are shown. The case of /?=1 in Figure 4c represents the maximum nonlin ­ ear coupling, i.e. a dense fluid in a thin-walled pipe of light material. Maximum nonlinear coupling maximizes A, i.e. the nondimensional mean fluid speed. Large nonlinear coupling also maximizes the damping of the pipe vibrations, resulting in a smooth response curve. The limit case of %=0 in Figure 4d maximizes the mean fluid speed, since the fluid then flows freely through the pipe. The flow of fluid is however still restricted, due to the stationary fluid in the upstream fluid reservoir which has to be accelerated. 11

relative forcing frequency, O/o), relative forcing frequency, £/o>,

relative forcing frequency, Q/o, relative forcing frequency, fi/cij Figure 4 Mean fluid speed A versus relative forcing frequency C2la>lt a) effect of forcing amplitude p, b) effect of mass ratio /?, c) effect of turbulent friction coefficient % and d) effect of external viscous modal damping coefficient £*; parameter values are (unless explicitly stated): p=0.01, %=0.8, f?=0J2, £i=0.01.

4.2 Fourth vibration mode , nonlinear geometry

The forcing frequency is now assumed to be near the fourth natural frequency (i=4, £l=ra»), so that the fourth mode of vibration is dominant. The third order geometrical nonlinearities will qualitatively affect the system response even for low-amplitude excitation of the base. Includ­ ing these higher order geometrical nonlinearities, the stationary solution to (18a)-(18c) can be found as: IfW =f[(4g«A' +G,*A' (20a) +[

CQf,A+|A)A = +iH^)bl (20b)

Figure 5 shows frequency response curves depicting the mean fluid speed A and the tip amplitude 264 versus the relative forcing frequency 12to 4. The perturbation solution, based on (20a)-(20b), is compared to results obtained by numerical integration of a single mode ap ­ proximation of (lla)-(llb). The tip amplitude is also compared to the case of a fixed fluid, 12

corresponding to letting A=0 in (20a), and to the case where the higher order geometrical non- linearities are neglected in (20a)-(20b). Figure 5b indicates how the nonlinear geometrical terms affect the system response. A left bending of the frequency response curve appears. This effect is due to nonlinear inertia which has a softening effect. It can be shown that nonlinear inertia is the dominant third order geo ­ metrical nonlinearity for second and higher modes of vibration. For the fundamental mode nonlinear curvature is dominant, pulling the response curves slightly to the right. This effect is however negligible as mentioned in section 4.1. Figure 5b shows that the damping effect of the fluid on the vibrations of the pipe is much less prominent compared to vibrations in the first mode. The response curve is seen to nearly match the case of a fixed fluid if the relative fre­ quency of forcing QI(ih is increased slowly (the response jumps to the upper stable solution branch at 43=0.975). If however 42/tot is decreased slowly past the resonance region the re­ sponse is ‘topped off due to the nonlinear coupling to the fluid. At the linear resonance fre­ quency the pipe amplitude nearly matches the response with the fluid being fixed. Figure 5a shows response curves for the mean fluid speed A. The left bending of the re­ sponse curve reduces A with approximately 25% at the linear resonance frequency, compared to the case where the third order geometrical nonlinearities are neglected. The maximum at­ tainable fluid speed is however almost unchanged, but appears to depend on if the forcing fre­ quency is increased or decreased through the resonant region. Figure 6 displays the effect of four key system parameters on the mean fluid speed A: forc ­ ing amplitude p, mass ratio jS2, turbulent wall friction coefficient % and modal viscous damping 6-

tlinear geometry

0.95 1 1.0 0.95 1 1.05 relative forcing frequency, fi/a 4 relative forcing frequency, Figure 5 a) Mean fluid speed A and b) pipe end amplitude 2bt for the fourth vibration mode (i=4), ver­ sus relative forcing frequency for p=0.0007; (—stable, unstable solutions), (O, numerical in­ tegration); c2=0.8, /?=0.2, £,=0.01. 13

Figure 6a,b shows the effect of p and $«. Changing the amplitude of forcing and the amount of modal viscous damping affects the amplitude of vibrations directly. Larger forcing and weaker damping increase the amplitude causing the geometrical nonlinearities to come into effect. This is directly reflected in the fluid response curves as a left bending. The amount of viscous damping is critical, as appears from Figure 6d. If the damping ratio exceeds about 5% the resonant top vanishes with low mean fluid speed as a result Figure 6b,c shows the effect of the mass ratio and the turbulent wall friction a2. Chang ­ ing these parameters affect the fluid speed directly. Increasing the amount of wall friction in­ creases the resistance on the fluid flow and similarly decreasing the mass ratio decreases the nonlinear pipe-fluid coupling. The secondary effect on the fluid, mentioned in section 4.1, is noticeable here. With decreased mean fluid speed, the fluid damping of the pipe vibrations is decreased, which causes the geometrical nonlinearities to come into effect. This is reflected in the fluid response curves, causing left bending.

o —------1------1 J o I—------1------1------L 0.95 1 1.05 0.95 , 1 1.0 relative forcing frequency, fl/o)4 relative forcing frequency.

0.95 1 1.0 0.95 1 1.05 relative forcing frequency. Qfu>4 relative forcing frequency. O/<04 Figure 6 Mean fluid speed A versus relative forcing frequency a) effect of forcing amplitude p% b) effect of mass ratio yS 2, c) effect of turbulent friction coefficient cts and d) effect of external viscous modal damping coefficient (—,•••; stable, unstable solutions); parameter values: (unless explicitly stated in the figures) p=0.0007, a 2=0.S, /f=0.2, &=0.01. 14

4.3 Comparison between vibration modes

The four lowest modes of vibration are compared with respect to the mean fluid speed for a given power input. The power input is measured as a nondimensional RMS-value (Root-Mean- Square) given as p2C2. The comparison is based on the perturbation solution in (19a)-(19b), i.e. the model without the third order geometrical nonlinearities. For the higher order nonline ­ arities to be negligible, also for higher modes of vibration, very low amplitudes of forcing have been chosen. Figure 7 shows the mean fluid speed A versus the power supplied for the four lowest modes of vibration. The four modes are compared for constant modal viscous damping of 1% (5=0.01), and for the case of zero damping (5=0). The relative behavior of the different modes appears to depend on the chosen damping form. With constant damping ratio of 1%, as shown in Figure 7a, the fundamental mode will produce the highest mean fluid speed, and the mean speed will drop significantly with increas ­ ing mode number. With no viscous damping, as shown in Figure 7b, the difference in mean fluid speed is small. In this case the fundamental mode produces the lowest mean fluid speed. The second mode is seen to be the most effective and slightly favorable compared to higher modes (the curves for the third and fourth mode cannot be told apart in the figure). Curves for even higher modes of vibration will coincide with the curves for the third and fourth mode.

5. EXPERIMENTAL INVESTIGATION

Experiments are carried out for horizontal acrylic pipes subjected to lateral resonant excitation. Two different pipes are used to investigate fluid motion in response to first and second mode vibrations. The experiments are set to cover both laminar and turbulent flow for both consid ­ ered modes.

5.1 Experimental setup

Figure 8 shows the experimental setup. The pipe is mounted on a shaker with a steal socket, and acrylic fittings are used to tighten the pipe. Water is enabled to flow to the pipe from a reservoir, through a short silicone rubber tube. The reservoir is placed on a weight, enabling the mass of the reservoir to be determined. A power amplifier feeds the shaker. The frequency and magnitude of the harmonic base displacement can be adjusted on the frequency generator connected to the power amplifier. The acceleration of the base is measured by an accelerome ­ ter mounted on the moving part of the shaker. The signal from the accelerometer is led through a charge amplifier to the signal analyzer. In order to be able to perform the experiments, the pipes have to be inclined slightly. Oth­ erwise fluid will inevitably discharge the pipe, even for a non-vibrating pipe. The inclination causes gravity forces to come into effect and restrict the fluid flow. This is counteracted in the experiments by increasing the water level in the reservoir so it matches the level of the pipe end. 15

mode 1

Figure 7 Mean fluid speed A versus supplied power, a) for £=0.01, and b for £=0, for the four lowest vibration modes; f2=&>„ or2=0.8, /3:=0.2.

The experimental procedure was as follows; The reservoir is filled so that the pipe is com ­ pletely filled with water. The natural frequency of the filled pipe is determined by slowly sweeping the excitation frequency and noting the frequency for maximum tip deflection. After refilling the reservoir, the mass of the reservoir is noted. The pipe is then forced with a fre­ quency matching the natural frequency, and current power is adjusted to match the desired amplitude of forcing. After a forcing period of 20-120 seconds, depending on the actual fluid speed, the mass of the reservoir is noted again, so the amount of discharged fluid can be com ­ puted.

5.2 Material data

Table 1 lists the data of the two pipes and water.

Pipe i 2 Length l fml 0.40 0.60 Inner diameter D, [m] 4.70xl0"3 4.80xl0"3 Outer diameter D, [m] 7.10xl0"3 7.25xl0"3 Young ’s modulus E [N/m2] 2.15x10’ 2.15x10’ Pipe density p„ [kg/m 3l 1.24X103 1.24X103 Fluid density pr [kg/m 3] 998 998 Mass ratio /32 0.39 0.39 Laminar friction coefficient «, 0.10 0.22 Turbulent friction coefficient a 2 2.6 3.9 Viscous damping coefficient 0.04 0.03 Table 1 Material data for the two pipes.

Young ’s modulus E is found by measuring the damped natural frequencies for the empty pipes and correcting for damping. A common mean value for the two pipes has been chosen. The friction coefficients «i, % are determined by measuring the volume flow in a non-vibrating pipe, and the viscous damping coefficient £ is computed by measuring the vibration amplitude for resonant vibrations of an empty pipe. 16

Signal analyzer (Data precision 6000) A

' Power an] Charge amp1ifiet(B&K 2 35)

Accelerometer(B&K 4;

Vibration cxcitcr(B &K 4808) w ,,cr ,c!cr,oir

Figure 8 a) Experimental setup and b) schematics of the setup.

5.3 Experimental results

Experimental results for the two different pipes are shown and compared with theoretical re­ sults. The theoretical results are based on the perturbation solution in (19a)-(19b), i.e. the model with linear geometry. For the given parameter values the third order geometrical non- linearities have no influence on the response. The volume flow V is chosen as base of compari ­ son between the analytical and the experimental results. Figure 9 shows theoretical and experimental values for V, versus the excitation amplitude p for pipe 1 and pipe 2. The volume flow is measured in milliliters per second [ml/s]. A laminar and a turbulent theoretical curve is shown for both pipes. The Reynolds number at the crossing of the two curves are computed to 2250-2300 for both pipes. This matches the theoretical values for pipe flow: 2000-2300, (e.g. White 1991). Thus the experimentally de­ termined values for external damping and pipe wall friction can be expected to be adequately accurate. The results for both pipes show good agreement between predicted and measured volume flow. Maximum deviation is about 15%, but the bulk of the experimental points is seen to lie near the theoretical curves. Experimental results also seem to indicate the predicted shift from laminar flow (/= 1) to turbulent flow (/=2), in that the location of the experimental points drops off to match the lower slope of the turbulent curves for higher values of p.

6. SUMMARY AND CONCLUSIONS

Vibration-induced fluid flow caused by nonlinear fluid-structure interaction has been investi ­ gated. Equations of motion governing the transverse vibrations of a cantilever pipe and the speed of an internal incompressible fluid were derived. The two equations are coupled through non ­ linear terms. The pipe vibrations affect the fluid through nonlinear forcing, and the induced fluid flow affects the vibrations of the pipe by modifying the damping and stiffness. 17

■v

i e 0.005 0.0 0.0005 0.001 0.0015 0.002 0.0025 forcing amplitude, p forcing amplitude, p Figure 9 Experimental and theoretical values for the volume flow V [ml/s] versus the forcing amplitude p for the two lowest modes of vibration; (—, perturbation solution), (□, experimental data); a) pipe 1, Z2=t»i, ai=0.1,01=2.6,5*1=0.04, P'=039, b) pipe 2,42=6%, a t=0.22, cr2=3.9,5*2=0.03, f?=039.

Numerical studies showed that the vibrations of the pipe cause the fluid to obtain a mean speed component towards the pipe end, in addition to a small oscillating component. Resonant excitation of all cantilever vibration modes can be used to obtain this effect. It was noticed that the magnitude of pipe vibrations necessary to obtain a given mean fluid speed is significantly lower for high modes. A perturbation analysis was performed using a multiple scales method. Frequency response curves were obtained for near-resonant excitation. Noticeable fluid motion occurs for near- resonant excitation only, where the nonlinear vibratory forcing is sufficiently large. The fluid damps the amplitude in the resonant region. Two specific examples were considered: resonant excitation of the first and fourth modes of vibration. The results showed qualitative similarities. The response for the fourth mode of vibration is however affected by nonlinear pipe geometry whereas for the fundamental mode, this effect is negligible. The fluid damping effect on the pipe vibrations was seen to be largest for the fundamental mode. For constant viscous damping ratio, the fundamental mode of vibration produces the highest mean fluid speed for a given power input. The mean fluid speed decreases with higher modes. If no external viscous damping is present, the second mode of vibration is favorable in this re­ spect, and higher modes are equally and only slightly less effective. Experimental investigations were performed for resonant excitation of the first and second modes of vibration. The results showed good qualitative agreement with theoretical results for both laminar and turbulent flow. This work has illustrated and quantified a phenomenon associated with nonlinear fluid- structure interaction. The nonlinear coupling explains transport of fluid through flexible canti ­ lever pipes, as seen in experiments. Linear theory predicts no coupling between pipe- and fluid- motion and consequently no transport of fluid. The phenomenon described, poses an alternative to previous suggested vibration convey ­ ance, in that it relies on resonant excitation of the structure. When resonant forcing is applied, the transport was shown to be effective for a wide range of system parameters. Since the driv ­ ing forces involved are nonlinear, they are inherently weak, and the efficiency associated with 18

the fluid transport is poor. There could however be other advantages using this device as a mean of transporting/pumping. The mechanism is extremely simple with no internal moving parts. The device is therefore thought to be suitable for micro purposes, e.g. applications in medicine. Advantages are also possible when dealing with toxic or aggressive fluids, which are not to come in contact with moving parts. Other means of excitation might also be used, e.g. parametric or transient excitation. Fur­ ther work is required in these areas to fully understand this mechanism of transport. Also high frequency excitation may introduce effects not included in the present model.

Acknowledgment - the author wishes to thank Jon Juel Thomsen for suggesting the project and for many valuable discussions.

REFERENCES

Babitsky , V.I. & Veprik , A.M. 1993 Damping of beam forced vibration by a moving washer. Journal of Sound and Vibration, Vol. 166(1), 77-85.

Bajaj , A.K., Sethna , P.R. & Lundgren , T.S. 1980 Hopf bifurkation phenomena in tubes carrying a fluid. SIAM Journal of Applied Mathematics, Vol. 39(2), 213-230.

Benjamin , T.B. 1961 Dynamics of a system of articulated pipes conveying fluid. I. Theory. Proceeding of the Royal Society (London), A 261,457-486. BLEKHMAN, I I. 1994 Vibrational mechanics: Nonlinear dynamic effects, general approach, applications (in russian). Fizmatlit Publishing Company.

BLEKHMAN, I.I. & Malakhova , O.Z. 1986 Quasiequilibrium positions of the Chelomei pen­ dulum. Sov. Phys. Dokl., Vol. 31(3), 229-231. CHELOMEI, V.N. 1983 Mechanical paradoxes caused by vibrations. Sov. Phys. Dokl, Vol. 28(5), 387-390.

Lee, U., Pak , C.H. & HONG, S.C. 1995 The dynamics of a piping system with internal un­ steady flow. Journal of Sound and Vibration, Vol. 180(2), 297-311.

LONG, Y.G., Na GAYA, K & Niwa , H. 1994 Vibration conveyance in spatial-curved tubes. Journal of Vibration and Acoustics, Vol. 116,38-46.

Nayfeh , A.H. & MOOK, D.T. 1979 Nonlinear Vibrations. Wiley-Interscience. PAlDOUSSIS, M.P. & ISSID, N.T. 1974 Dynamic stability of pipes conveying fluid. Journal of Sound and Vibration, Vol. 33(3), 267-294. PAlDOUSSIS, M.P. & Li, G.X. 1993 Pipes conveying fluid: a model dynamical problem. Jour­ nal of Fluids and Structures, Vol. 7,137-204. ROUSSELET, J. & HERRMANN, G. 1981 Dynamical behaviour of continues cantilevered pipes conveying fluid near critical velocities. Journal of Applied Mechanics, Vol. 48, 943-947. SADIKU, S. & LEIPHOLZ, H.H.E. 1987 On the dynamics of elastic systems with moving con ­ centrated masses. Ingenieur-Archiv, Vol. 57,223-242. SEMLER, C., Li, G.X. & PAlDOUSSIS, M.P. 1994 The non-linear equations of motion of pipes conveying fluid. Journal of Sound and Vibration, Vol. 169(5), 577-599. 19

Thomsen , J.J. 1996 Vibration suppresion using self-arranging mass: Effects of adding restor ­ ing force. Journal of Sound and Vibration, (to appear). WHITE, F.M. 1991 Viscous fluid flow. McGRAW-HILL.

APPENDIX

The cantilever mode-shapes are given as

=&,<%. K =-}'a}l

K = j'0

Altta = jfP&ZY,

Gain = jfPiVkVWJZ ~ £

FiUm = ^,9t

Hum = Xm, = j'0

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