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Experiments on vertical slender flexible cylinders clamped at both ends and subjected to axial flow


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Experiments on vertical slender flexible cylinders clamped at both ends and subjected to axial flow
Y Modarres

-Sadeghi, M.P Pa?doussis, C Semler and E Grinevich Phil. Trans. R. Soc. A 2008 366, 1275-1296 doi: 10.1098/rsta.2007.2131

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Phil. Trans. R. Soc. A (2008) 366, 1275–1296 doi:10.1098/rsta.2007.2131 Published online 5 November 2007

Experiments on vertical slender ?exible cylinders clamped at both ends and subjected to axial ?ow
¨DOUSSIS *, C. S EMLER B Y Y. M ODARRES -S ADEGHI , M. P. P AI AND E. G RINEVICH

Department of Mechanical Engineering, McGill University, 817 Sherbrooke Street West, Montreal, Quebec, Canada H3A 2K6
Three series of experiments were conducted on vertical clamped–clamped cylinders in order to observe experimentally the dynamical behaviour of the system, and the results are compared with theoretical predictions. In the ?rst series of experiments, the downstream end of the clamped–clamped cylinder was free to slide axially, while in the second, the downstream end was ?xed; the in?uence of externally applied axial compression was also studied in this series of experiments. The third series of experiments was similar to the second, except that a considerably more slender, hollow cylinder was used. In these experiments, the cylinder lost stability by divergence at a suf?ciently high ?ow velocity and the amplitude of buckling increased thereafter. At higher ?ow velocities, the cylinder lost stability by ?utter (attainable only in the third series of experiments), con?rming experimentally the existence of a post-divergence oscillatory instability, which was previously predicted by both linear and nonlinear theory. Good quantitative agreement is obtained between theory and experiment for the amplitude of buckling, and for the critical ?ow velocities.
Keywords: ?exible cylinders; axial ?ow; stability; nonlinear dynamics; post-divergence ?utter; experiments

1. Introduction The dynamical behaviour of cylinders supported at both ends and subjected to axial ?ow was studied very comprehensively using a linear theory by Pa? ¨doussis (1966a, 1973, 2004), supported also by experiments (Pa? ¨doussis 1966b). To analyse the system behaviour using the linear model, the dimensionless complex frequencies for the cylinder were plotted in Argand diagrams with the dimensionless ?ow velocity as the independent parameter (Pa? ¨doussis 1973). It was shown that, for small ?ow velocities, the free motion of the cylinder is damped. However, for suf?ciently high ?ow velocities, the ?rst mode frequency becomes purely imaginary corresponding to the onset of divergence, followed by divergence in the second mode. At slightly
* Author for correspondence (michael.paidoussis@mcgill.ca; mary.?orilli@mcgill.ca). Electronic supplementary material is available at http://dx.doi.org/10.1098/rsta.2007.2131 or via http://www.journals.royalsoc.ac.uk. One contribution of 6 to a Theme Issue ‘Experimental nonlinear dynamics II. Fluids’.

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higher ?ow velocity, the negative branches of the ?rst- and second-mode loci coalesce and leave the imaginary-frequency axis at a point where the damping is negative, indicating the onset of coupled-mode ?utter. The interested reader is also referred to the work of Chen & Wambsganss (1972); see also Chen (1987). For a cantilevered cylinder with a fairly well-streamlined free end, it was shown that the cylinder loses stability by divergence in its ?rst mode, but, as the ?ow velocity is increased further, it regains stability (Pa? ¨doussis 1966a, 1973); subsequently, it loses stability by single- or coupled-mode ?utter, depending on the system parameters. At higher ?ow velocities, the cylinder loses stability in the third mode. Further work was done on pinned–free cylinders and strings in axial ?ow by Triantafyllou & Chryssostomidis (1984, 1985). In addition, some closely related work on towed cylinders in axial ?ow is of interest (Pa? ¨doussis 1968; Ortloff & Ives 1969; Pao 1970; Lee 1981; Kennedy 1987; Dowling 1988a,b). Pa? ¨doussis (1966b) conducted a series of experiments to study the dynamical behaviour of ?exible cylinders in axial ?ow and to measure the limits of stability, and compared them with the theoretical results found using the linear model. In experiments with clamped–free cylinders, a thin metal strip was embedded in the cylinder along its length, in the vertical plane of symmetry, effectively limiting all motion to the horizontal plane and also providing additional support in the vertical plane in cases where the weight of the cylinder was considerably greater, or smaller, than that of the displaced water. Cylinders with both ends supported were not ?tted with a metal strip, so that the development of instability would not be impeded by excessive resistance to axial extension. In all the tests, at small ?ow velocities, small random vibrations were damped. For cantilevered cylinders, tapered at the free end, the system became unstable at suf?ciently high ?ow velocity. The system ?rst buckled and then, as the ?ow velocity was increased, it developed second-mode oscillation (?utter), followed by third-mode oscillation. Similar observations were made with a cylinder pinned at the upstream end and free at the other end. For a very long simply supported cylinder, it was observed that it sagged slightly at its midpoint under its own weight. Increasing the ?ow velocity in such cases exaggerated this sag and slowly shifted it to the horizontal plane, but no distinct threshold of buckling and no oscillatory motions were observed. For shorter cylinders, however, at a suf?ciently high ?ow velocity, a small bow developed just downstream of the midpoint of the cylinder, increasing in amplitude with ?ow; thus the system lost stability by divergence (buckling). This was followed by a spontaneous second-mode oscillation. Lopes et al. (2002) derived a nonlinear equation of motion to describe the dynamics of a slender cantilevered cylinder in axial ?ow, generally terminated by an ogival free end, using the inextensibility assumption, which is reasonable for cantilevered cylinders. Inviscid forces were modelled by an extension of Lighthill’s slender-body theory to third-order accuracy. The viscous, hydrostatic and gravity-related terms were derived separately to the same accuracy. The equation of motion was obtained via Hamilton’s principle. The boundary conditions related to the ogival free end were also derived separately. Using this model, the nonlinear dynamics of the system was studied (Semler et al. 2002). It was found that the cylinder ?rst loses stability by divergence in its ?rst mode, the amplitude of which increases with ?ow. This is gradually transformed to divergence of predominantly second-mode shape, before the
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system is restabilized. At slightly higher ?ow velocities, stability is lost by second-mode ?utter, which at still higher ?ows is succeeded by third-mode ?utter. No chaotic oscillation was observed in this study. One important result, obtained by examining the system from the nonlinear point of view, is that postdivergence ?utter does materialize, as predicted by linear theory, whether there is post-divergence restabilization or not. This theoretical work was also supported by an experimental study (Pa? ¨doussis et al. 2002) using a water tunnel with a vertical test section. The ?exible cylinder was mounted vertically in the middle of the water tunnel, with no central metal strip embedded in it, as it was no longer essential (as it was in the earlier experiments with a horizontal cylinder). Initially, as the ?ow was increased, ?ow-induced damping was generated, but small vibration could be observed in which the cylinder responded to the turbulence-induced ?uctuating pressure ?eld. At higher ?ow velocities, the system developed divergence in its ?rst mode and then regained its equilibrium con?guration, before developing ?utter spontaneously in its second mode. As the ?ow velocity continued to increase, second-mode ?utter was succeeded by third-mode ?utter, and in some cases fourth-mode ?utter. This was the dynamical behaviour for a cylinder with a reasonably well-streamlined ogival end shape at the free end. If, however, the end was completely blunt, then neither static (divergence) nor dynamic (?utter) instabilities materialized; the reasons for this are elucidated in Pa? ¨doussis (2004, §8.3.3). Later, a nonlinear model was derived for a cylinder with both ends supported subjected to axial ?ow (Modarres-Sadeghi et al. 2005). Using this nonlinear model, the behaviour of the system was studied for cylinders with various boundary conditions (Modarres-Sadeghi et al. 2007). Much of the research work referred to in the foregoing was curiosity driven. However, the work on cylinders in axial ?ow was inspired by applications to heat exchanger and nuclear reactor internals (Chen 1987; Pa? ¨doussis 2004). Also, the work on towed cylinders was undertaken for application to the stability of (i) the Dracone barge (Hawthorne 1961), (ii) acoustic arrays used in underwater oil and gas exploration (Dowling 1988a,b; Poddubny et al. 1995; Sudarsan et al. 1997), and (iii) towed underwater pipelines (Sarv & John 2000). Other applications have also emerged, e.g. in ?bre spinning and wire coating (Poddubny & Saltanov 1991; Papanastasiou et al. 2000), and for modelling the dynamics of high-speed trains in tunnels (Sugimoto 1996; Sugimoto & Kugo 2001; Tanaka et al. 2001). The experimental behaviour of a vertical cylinder supported at both ends is discussed in this paper from a nonlinear point of view. According to the nonlinear theory cited above, as the ?ow velocity is increased, the cylinder buckles at a certain ?ow velocity and the amplitude of buckling increases thereafter. At higher ?ow, the statically deformed cylinder undergoes a Hopf bifurcation, which gives rise to periodic oscillations, followed by quasi-periodic and chaotic oscillations. The aim of the experiments to be described was (i) to measure again the critical ?ow velocities for divergence (buckling) and ?utter, (ii) to observe the post-divergence behaviour of the system and measure the corresponding data (the amplitude of buckling, as well as the amplitude and the frequency of oscillations in the case of dynamic motions, if they materialize), and (iii) to compare the experimental observations with the predictions of nonlinear theory, both qualitatively and quantitatively, in order to validate the theoretical model, hopefully by observing experimentally the same sequence of dynamical states predicted by the nonlinear theory.
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2. The nonlinear model, methods of solution and typical theoretical results The weakly nonlinear equations of motion for a slender ?exible cylinder of length L, diameter D and mass per unit length m, subjected to axial ?ow (with ?ow velocity U and added mass per unit length, in uncon?ned ?ow, M ), have been derived by Modarres-Sadeghi et al. (2005) and are given in appendix A. In that derivation, the cylinder centreline is considered to be extensible, and hence two coupled nonlinear equations describe its motion, involving both longitudinal and transverse displacements. The ?uid forces are formulated in terms of several components, for convenience. For high Reynolds number ?ows, the dominant inviscid component is modelled by an extension of Lighthill’s slender-body work; frictional, hydrostatic and pressure-loss forces are then added to the inviscid component. The derivation of the equations of motion is carried out in a Lagrangian framework, and the resultant equations are correct to third order of magnitude, where the transverse displacement of the cylinder is of ?rst order. The resulting nonlinear partial differential equations are then discretized using the Galerkin technique, leading to a set of second-order ordinary differential equations, which can be recast in ?rst-order form. To solve this set of ordinary differential equations, two methods have been used: Houbolt’s ?nite difference method (Semler ?ves 1986), which is a computational package et al. 1996) and AUTO (Doedel & Kerne based on continuation methods. For details on the derivation of the equations of motion and also the methods of solution, see Modarres-Sadeghi (2006). In the equations of motion (see appendix A), z and h are, respectively, the nondimensional displacements in the longitudinal and transverse direction; U is the dimensionless ?ow velocity, used extensively as the independent parameter in  and G  are studying the dynamics of the system; b is a mass ratio; P0, P dimensionless measures of axial ?exibility, pressurization and externally imposed uniform tension, respectively; c n and c t are the coef?cients of frictional forces in the normal and tangential (longitudinal) direction, respectively; cd is the coef?cient of transverse form drag; dZ0 if the downstream end is free to slide axially (or wholly free), and dZ1 if it is axially ?xed; n is the Poisson ratio; g is a gravity coef?cient; cb is the base-drag coef?cient acting in the longitudinal direction at the downstream end of the cylinder when dZ0; 3 is the slenderness ratio; hZD/Dh is a hydrodynamic con?nement coef?cient, Dh being the hydraulic diameter; and c is an added mass coef?cient which increases with increasing con?nement. Figure 1a shows a typical theoretical bifurcation diagram of the system of a clamped–clamped cylinder found by AUTO, where q1 (?rst generalized coordinate in the transverse direction, which is representative of the overall transverse displacement of the cylinder) is plotted versus non-dimensional ?ow velocity, U . The cylinder is at its original equilibrium state at low ?ow velocities. With increasing ?ow, it loses stability via a supercritical pitchfork bifurcation at a nondimensional ?ow velocity U x 2p (BP in ?gure 1a) leading to divergence. The original equilibrium becomes unstable for U O 2p; the dotted line on the horizontal axis corresponds to this unstable state. The amplitude of buckling (q1) increases with the ?ow (U ). The static solution eventually loses stability, and the system develops ?utter via a supercritical Hopf bifurcation at U Z 21:60 (HB in ?gure 1b) giving rise to periodic solutions. This oscillatory motion loses stability via a period-doubling bifurcation at U Z 21:92 and the system develops period-2
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(a) 0.15

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BP

0 5 10 15 20 25 30

(b) 0.100 TR 0.095 PD

q1 0.090

0.085

HB

0.080 21.5

22.0

22.5

Figure 1. (a,b) A typical bifurcation diagram for a clamped–clamped cylinder.

motions afterward (PD in ?gure 1b). Quasi-periodic oscillations are observed for ?ow velocities greater than U Z 22:03 (TR in ?gure 1b), where a torus bifurcation occurs. The quasi-periodic oscillations become chaotic at U x 23 and remain chaotic thereafter for the ?ow range studied. In the theoretical results presented in the following sections, the eigenfunctions of a clamped–clamped cylinder for the transverse deformation are used as basis functions for the transverse displacement. For the case of a clamped–sliding cylinder (§4), the eigenfunctions of a ?xed–sliding bar undergoing axial vibration are used as basis functions for the axial displacement, re?ecting the fact that axial displacement at the lower end of the cylinder can occur freely; however, for the case of a clamped–clamped cylinder with no end-sliding (§§5 and 6), the eigenfunctions of a bar with both ends ?xed have been used. In the computations of §4, only two modes in each direction have been used to discretize the partial differential equation, because the maximum ?ow studied therein is not very high, and this was suf?cient to obtain converged results; in contrast, at least six modes in each direction have been used in the computations of §§5 and 6. Six modes in
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upper support

(b) flexible cylinder ring lower support screw
Figure 2. (a) A schematic, exploded view of the cylinder and its upstream and downstream ends and (b) a schematic view of the downstream end of the system with a ring inserted between the lower end of the cylinder and the lower support (second series of experiments).

each direction were also used in generating the bifurcation diagram of ?gure 1. An appropriate convergence study has been conducted for this purpose (Modarres-Sadeghi 2006). 3. Experimental set-up The experiments were conducted with ?exible cylinders, which were made of silicone rubber by casting in special moulds. Each cylinder was ?tted at its two extremities with metal discs (of the same diameter as the cylinder), which could be screwed onto different support assemblies at the upper and lower ends to provide clamped boundary conditions (see ?gure 2a).1 The upper support was well streamlined. The cylinder was mounted vertically in the test section of the water tunnel, which is shown schematically in ?gure 3. The testsection (‘channel’) diameter is DchZ0.20 m and its length is LchZ0.75 m. Flow straighteners, screens and a large ?ow-area reduction upstream were used to ensure a uniform axial ?ow-stream in the test section. The velocity pro?le is ?at and uniform over the central portion of the test section, covering at least 15 cm in diameter. The highest attainable ?ow velocity in the water tunnel is 5 m sK1 (see Pa? ¨doussis (2004, ch. 8), for a complete description of the water tunnel). To measure the midpoint displacement of the cylinder, two non-contacting optical motion-followers were used, measuring the displacement in two perpendicular directions to guarantee that the plane and the value of the
1

For details about the method of making these cylinders, see Pa? ¨doussis (1998, appendix D).

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vent to drain

heat exchanger water supply manometers turning vanes grids pump test section

venturis
Figure 3. Schematic view of the water tunnel (adapted from Pa? ¨doussis et al. (2002)).

measured by device 1 plane of maximum displacement B measuring device 1 A measuring device 2

measured by device 2
Figure 4. Schematic of the motion-followers (‘measuring devices’) set-up. Distance AB is the maximum measured displacement based on the measurements by the two devices.

maximum displacement were determined. Figure 4 shows schematically how the two motion-followers were placed relative to the cylinder. The ?rst measuring device (‘device 1’) measured displacements of the cylinder perpendicular to the axis of the device, while the second measured those in the direction of its axis. The output of these two motion-followers was saved in a computer for later analysis by the LABVIEW software. The resultant of these two measurements, as shown in ?gure 4, gives the plane and the value of the maximum displacement, which is the value that can be compared with the theoretical predictions. In each experiment, the ?ow velocity was increased gradually from zero, and at each step the midpoint displacement of the cylinder was measured. The dimensional and dimensionless parameters for the cylinders used in the experiments are listed in tables 1 and 2. First, experiments with a cylinder clamped at both ends, but free to slide axially at its downstream end are discussed (§4). Then, the results for clamped– clamped cylinders with no axial sliding are presented (§5); the in?uence of axial compression is also studied experimentally in this case. Because no dynamic instability was observed in these two series of experiments due to ?ow limitations of the water tunnel, another series was conducted in which a more ?exible
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Table 1. Physical parameters of the cylinders used in the three series of experiments.

physical parameter outer cylinder diameter, Do (mm) inner cylinder diameter, Di (mm) length, L (mm) mass per unit length, m (kg mK1) ?exural rigidity, EI (N m2)

series 1 and 2 25.4 0 520 0.577 0.0559

series 3 15.6 9.4 435 0.1445 0.0065

Table 2. Dimensionless parameters used in the theoretical calculations for the three series of experiments, with cylinders the physical parameters of which are given table 1. (The nondimensional parameters are de?ned in equation (A 3).) series 1 and 2 3 P0 6707 9124.2 b 0.47 0.57 g 1.83 K5.78 3 20.47 27.88 h 0.125 0.0768 c 1.032 1.0119 cd 0.0 0.005 ct 0.025 0.02 c n/c t 1 0.5

cylinder was used; these experiments, in which ?utter was observed, are discussed in §6. In all the experiments, graphs of the frequency versus non-dimensional ?ow velocity are presented for pre-buckling turbulence-induced vibrations of the cylinders. Also, in each case, bifurcation diagrams of the system are presented in which the maximum midpoint amplitude of the cylinder is plotted versus nondimensional ?ow velocity. Different methods for the determination of the onset of divergence are discussed. The in?uence of different parameters on the onset of periodic motions is also discussed.

4. First series of experiments: clamped–sliding cylinder In the ?rst series of experiments, the lower end of the cylinder was free to slide axially in its support; the set screw at the lower support (?gure 2) was not tightened. For the parameters of this cylinder and the maximum ?ow rate attainable in the water tunnel, the maximum non-dimensional ?ow velocity for this system was U Z 7. At small ?ow velocities, turbulence-induced damped vibrations were observed about the stretched-straight equilibrium position of the cylinder. With increasing ?ow, the cylinder buckled, essentially in its ?rst beam-mode shape, i.e. its de?ection amplitude at the midpoint increased sharply. This, however, occurred at almost the maximum attainable ?ow; therefore, the cylinder behaviour substantially beyond the onset of divergence could not be observed. The PSD plots of the vibrations of the cylinder were produced at each ?ow velocity to construct the frequency versus ?ow graph and thus obtain the critical value for divergence (as discussed in the next paragraph). Figure 5 shows the PSDs of the vibrations of the cylinder measured at its midpoint at two different ?ow velocities before the onset of divergence. The dominant frequency of vibration is f1Z3.5 Hz
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0 –20 –40 –60 –80 –100 0 5 f 10

15 0

5 f

10

15

Figure 5. PSD plots for the ?rst series of experiments with a clamped–sliding cylinder with parameters given in table 1 (series 1) at (a) U Z 2:2 and (b) U Z 5:6.

for U Z 2:2 (?gure 5a), which can be identi?ed with the ?rst eigenfrequency of the system at this ?ow. At U Z 5:6, two peaks are noticeable in the PSD plot: f1Z2 Hz and f2Z8 Hz, which correspond, respectively, to the ?rst and the second eigenfrequency of the system.2 It is observed that the eigenfrequencies of the system vary with the ?ow velocity. Figure 6a shows how the dominant frequency (u1Z2pf1) varies with the nondimensional ?ow velocity for this system. The asterisks are the experimental results and the dots are the theoretical ones. Theoretically, the ?rst-mode frequency varies with U quasi-parabolically, and at the onset of divergence the ?rst-mode frequency vanishes: u1Z0. The experimental values follow very closely the theoretical parabola, showing good agreement between theory and experiment. To approximate the critical ?ow velocity for divergence, one should ?t a parabola to the experimental results (the continuous line in ?gure 6a) and extend it to cross the ?ow-velocity axis at a point which may be considered as the critical ?ow velocity for divergence. This is called ‘the ?rst method’ for determining the critical point for divergence in this paper. By doing so, it was found that the critical ?ow ?1? velocity for divergence is U BP Z 6:5, as shown in ?gure 6a. Figure 6b shows the experimental (the asterisks) and theoretical (heavy solid line) bifurcation diagrams, for the whole range of attainable ?ow velocities in the water tunnel. Theoretically, the de?ection should be zero, from U Z 0 to the critical ?ow velocity for divergence, U BP , when the amplitude should increase precipitously with ?ow. In the experiments, however, one obtains non-zero h for all 0 ! U ! U BP , as seen in ?gure 6b for the experimental results. In what we refer to as the ?rst zone in the experimental bifurcation diagram, non-zero de?ection amplitudes are observed due to the growth of initial geometric and structural imperfections of the cylinder. The second zone is for U O U BP where h increases much more rapidly with U ; showing the occurrence of divergence. To ?nd the critical ?ow for divergence, one can ?t the results in the two above-mentioned zones by two straight lines and consider the ?ow velocity at their intersection as the critical ?ow for divergence (see ?gure 6b). We call this ‘the second method’
2

Although the deformation was sensed at the midpoint of the cylinder, the second mode was still picked up, since, with ?ow, the midpoint is not a stationary node.

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(1) BP

second zone first zone

0.6 0.4 0.2 0

8

0

2

4

6
(2) BP (3) BP

8

Figure 6. (a) Experimental (asterisk) and theoretical (dot) graphs of frequency versus dimensionless ?ow velocity for a typical series 1 experiment; the continuous line is the parabola ?tted to the experimental results. (b) Theoretical (heavy continuous line) and experimental (asterisk) bifurcation diagrams. Table 3. The experimental critical ?ow velocities for the onset of divergence based on the three methods discussed in §4, together with the corresponding theoretical result for the clamped–sliding cylinder of the ?rst series of experiments with physical parameters given in table 1 and the corresponding error.
?1? ?2? ?3?

?rst method, U BP values error (%) 6.5 5.8

second method, U BP 5.8 15.9

third method, U BP 6.2 10.1

theory, U th BP 6.9 —

for determining the critical ?ow velocity for divergence. In this case, the critical ?2? ?ow velocity for divergence is U BP Z 5:8, as shown in ?gure 6b. Another, pragmatic criterion used to de?ne the critical value for divergence is to de?ne its threshold at hZ0.01 or v/DZ0.2. If the cylinder amplitude exceeds this threshold, the cylinder may be considered buckled. This is ‘the third method’ for determining the critical ?ow velocity for divergence in this paper, and it ?3? yields U BP Z 6:2 for this system (see ?gure 6b). Table 3 lists the critical values for divergence found by the three methods discussed above, together with the corresponding theoretical value. The ?rst method is in very good agreement with the theoretical result, with only 5.8% error; this does not necessarily mean, however, that this method is superior to, say, the second. For the theoretical results, the non-dimensional parameters given in table 2 have been used, where P0, b, g, h, c and 3 are calculated using the physical parameters of the cylinder given in table 1.3 The coef?cient of transverse form drag, c d, can also be calculated, following Pa? ¨doussis (2004, appendix Q)4:
? 2 ? ? 2 ? hZD/Dch and c Z Dch C D 2 = Dch KD 2 , according to Pa? ¨doussis (2004, p. 796). 4 lin In Pa? Q), the following relation is given: cd Z ?4=p??U =U ?CD , where ¨doussis (2004, appendix p??? p???? 2 CD Z ?p=2?DUCd and Cd Z 2 2 S ,???????? in? which S Z UR =n. Now, substituting these relations p???????? ? p?? ?= p backwards, one ?nds Cd Z 4 2=p D???????? U =n, and therefore CD Z 2p 2Un; assuming further that ? lin U =U x O?1?, one obtains cd Z 8 2Un .?????????? It can be shown that the coef?cient used in nonlinear p ? nonlin lin calculations is cd Z cd Z cd =U Z 8 2n=U:
3

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p??????????? cd Z 8 2n=U; where U is the radian frequency and n the kinematic viscosity; for water ?ow, nZ10K6 m2 sK1. One can take c n/c tZ1 and the tangential frictional coef?cient to be c tZ0.025. The effect of the frictional terms on the threshold of divergence is small, and using different values of c n and c t has little effect on the theoretical results. However, as discussed in §6, this is not the case for dynamic instabilities. 5. Second series of experiments: clamped–clamped cylinder with no sliding In this series of experiments, the same cylinder, clamped at both ends was used, but no end-sliding was permitted. Similarly to the ?rst series of experiments, the midpoint displacement of the cylinder was measured at each ?ow velocity and the corresponding PSD plots were produced. The dominant frequency, e.g. at f1Z3 Hz for U Z 3:2, can be identi?ed with the ?rst eigenfrequency of the system. The PSD plot at, for example, U Z 6:1 shows three peaks: at f1Z2 Hz, f2Z8 Hz and f3Z17 Hz (not shown here), which correspond to the ?rst, second and third eigenfrequencies of the buckled system, respectively; it is noted that the dominant ?rst-mode frequency `-vis its value at zero ?ow, but it is not peak has migrated further towards zero vis-a zero. In this regard, it is noted that, in contrast to linear theory, nonlinear theory does not imply zero net rigidity in the buckled state, and the ?rst, second and third mode frequencies are those of the buckled cylinder. Figure 7a,b is of the same nature as ?gure 6a,b. For U ! 5; the experimental values for the frequency of vibrations follow the theoretical curve very closely. For higher ?ow velocities, however, the experimental values deviate from the theoretical ones and it seems that they tend towards a constant value: u1Z10 rad sK1, in contrast to the theoretical curve, which goes towards zero parabolically. This deviation is due to the fact that, in the experiments, because axial sliding is prevented, increased de?ection generates an increase in tension and therefore the frequency never approaches zero (?gure 7b shows the non-zero pre-buckling de?ection of the cylinder). The theory, on the other hand, assumes that there is no pre-buckling transverse de?ection, and therefore that there is no tension in the cylinder before the onset of buckling. It ought to be noted that, in ?tting the parabola to estimate the critical ?ow velocity for divergence by the ?rst method, only the sampling points which follow a parabolic trend have been taken into account and the almost-constant-frequency points have been neglected. The ?tted ?1? parabola shown in ?gure 7a yields the corresponding critical ?ow velocity U BP x 6:8. In ?gure 7b, it is seen that the post-buckling range in the bifurcation diagram is very limited ?6:3 ! U ! 7?, making it dif?cult to reach any major conclusions as to the post-buckling behaviour of the system. However, for the available range, the experimental amplitude of buckling is in good agreement with the theoretical one. The ?rst row in table 4 gives the critical values for the pitchfork bifurcation found using the three methods previously discussed, together with the corresponding theoretical result. Reasonably good agreement is obtained between the theory and the experiment. Because, in both cases discussed so far, divergence occurred at a ?ow velocity very close to the maximum ?ow attainable in the water tunnel, it was not possible to study the post-divergence behaviour of the system. In what follows,
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1

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5
(3) BP

6
(2) BP

7

0

(c) frequency (rad s–1)

25 20 15 10 5 0 1 2 3 4 5 6
(1) BP

(d )

0.06 0.05 0.04 0.03 0.02 0.01

7

0

1

2

3

4

5

6

7

Figure 7. Experimental and theoretical results for the second series of experiments with a clamped– clamped cylinder with no end-sliding and with parameters given in table 1. (a) Experimental (asterisk) and theoretical (dot) graphs of frequency versus dimensionless ?ow velocity for a typical series 2 experiment; (b) comparison between theory (continuous line) and experiment (asterisk) in the form of bifurcation diagrams; (c,d ) same graphs for the cylinder under an externally applied  ZK axial compression of G 20:6.

we discuss the experiments with the same cylinder under axial compression; in this case, the critical non-dimensional ?ow velocity for divergence is lower than that in the present case, making it possible to observe the post-divergence behaviour of the cylinder over a wider ?ow range. To apply an axial compression to the cylinder, rings of varying thickness, with the same outer diameter as the cylinder (Dr,oZDZ0.0254 m) and with the inner diameter of Dr,iZ0.0067 m were inserted between the lower end of the cylinder and the lower support assembly (see ?gure 2b). The inserted ring, depending on its thickness, results in pre-straining (shortening) the cylinder and therefore applies a pre-stress (an axial compression) on the cylinder with no ?ow. To relate the applied axial pre-straining to the non-dimensional compression,  Z sA Z E 3A. Therefore,  2 =EI , where T , we start from the de?nition G  Z TL KG  Z DAL=I , where D is the pre-shortening and A is the cylinder cross-sectional G  Z 1:29 ! 104 D, area. The other quantities are de?ned in table 1. Therefore, G  are negative. where it is understood that in these experiments D and G The procedure, measurements and the nomenclature of the ?gures are the same as previously described. Three different values for axial compression were
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Table 4. The experimental critical ?ow velocities for the onset of divergence based on the three methods discussed in the text, together with the corresponding theoretical results for the second series of experiments, for the clamped–clamped cylinder with the physical parameters as in table 1, for different values of externally applied axial compression. critical value of U BP (K) experiment pre-shortening (mm) 0.0 1.0 1.6 2.3 axial com pression KG 0 12.9 20.6 29.7 ?rst method 6.8 6.1 5.8 5.3 second method 5.8 5.5 5.4 4.2 third method 5.3 5.0 5.0 4.5 theory 6.25 5.12 4.30 3.10

 Z K12:9, K20.6 and K29.7. The graphs of the used in the experiments, namely G frequency of turbulence-induced vibrations of the cylinder versus the ?ow velocity were produced for each case. Figure 7c shows the graph for an axial  ZK compression of G 20:6 as an example (the other cases are not shown). In all these cases, similarly to the case of no axial compression, from a certain ?ow velocity on, the frequencies do not follow the theoretical parabola; they tend to an almost constant asymptote, instead of continuing towards zero. By increasing the axial compression, the deviation of the experimental results from the theoretical curve starts at lower ?ow velocities, due to the fact that the original position of the cylinder in the experiment, even before turning on the ?ow, was not totally straight anymore, as a result of the relatively large applied initial compression. A non-negligible bow existed at zero ?ow, which is not taken into account in the theory; the theory presumes an initially straight cylinder.5 This deviation from the theoretical assumptions affects the pre-buckling behaviour of the system much more than the post-buckling behaviour. The reason is that the theory assumes a straight cylinder, while in the experiment the cylinder is not straight before buckling. After buckling, the cylinder is no longer straight in both theory and experiment. The bifurcation diagrams for the system under axial compression were also produced for all these cases (see ?gure 7d for an example). As the axial compression increases, the range of ?ow velocity in which the cylinder is buckled becomes wider and, therefore, the amplitude of buckling can be compared more meaningfully with the theoretical results. Fairly good quantitative agreement between theory and experiment is observed in ?gure 7d. Similarly to the case of a clamped–sliding or clamped–clamped cylinder with no end-sliding and no axial compression, the main qualitative difference is that in the experimental results the amplitude of the buckled cylinder increases with ?ow exponentially, while theory predicts a parabolic increase in the amplitude. The imperfections and the deformation-induced tension in the cylinder can be thought of as the main source for the difference.
5

The occurrence of the bow of course means that the cylinder has developed buckling due to compression a great deal sooner (for smaller compression) than theoretically predicted, because of imperfections.

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(a) 8 7 6 5
BP

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4 3 2 1 0

(b) 8 7 6 5
BP

4 3 2 1 0 5 10 15 20 G 25 30 35 40 45

Figure 8. Critical ?ow velocities for divergence of the cylinder clamped at both ends and with no end-sliding (series 2 experiments with parameters given in table 1) for different externally applied axial compressions, found by using the three methods discussed in the paper, assuming the cylinder is initially (a) straight and (b) bowed. The continuous line is the corresponding theoretical curve. Filled squares, ?rst method; ?lled triangles, second method; ?lled diamonds, third method.

The critical ?ow velocities for divergence are given in table 4. They are also plotted in ?gure 8a and compared with theory. The theoretical result (continuous line) shows that, as expected, by increasing the axial compression, the critical ?ow velocity for divergence decreases, and at a certain value of axial compression  x 41, which is slightly the cylinder buckles even with no ?ow. This occurs at KG different from the critical value of the Euler problem for a clamped–clamped  Z 4p2 , due to the effect of gravity, which increases the critical value column, KG for divergence. It is noted that, as the axial compression becomes larger, agreement of experiment with theory deteriorates. The main reason for this is that, owing to imperfections, the cylinder bows away from the straight con?guration; and, in doing so, releases some of the compression that would have otherwise been present, had it remained straight. Thus, the actual value of  to which the cylinder is subjected at the beginning of the experiment (for KG U Z 0) is lower than theoretically calculated; meaning that for high enough values  the experimental points should really be moved to the left, thus closer to of KG the theoretical prediction. This, indeed, is what happens, as shown in ?gure 8b. , KG eff , were calculated by a simple In ?gure 8b, the ‘effective’ values of KG model, taking into account the reduction in compression as outlined above, assuming the maximum amplitude due to imperfection-related bowing to be
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10 mm and a half-sinusoid for the bowed cylinder at U Z 0; then, the bowed cylinder length is calculated, and the effective (reduced) value of the ring eff may be deduced from the original KG . thickness calculated, based on which KG As seen in ?gure 8b, the experimental results are felicitously closer to theory than they are if this compression relief is not accounted for. For all the cases studied in this series of experiments, the cylinder did not undergo any dynamic instability before the maximum attainable ?ow velocity was reached, encouraging us to conduct another series of experiments with a more ?exible clamped–clamped cylinder, as discussed in §6, in order to hopefully observe the post-divergence dynamic instability predicted by theory. 6. Third series of experiments: a more ?exible cylinder with no end-sliding In this series of experiments a more slender, hollow ?exible clamped–clamped cylinder was used. The same set-up as for the previous series of experiments was used. Similarly to the previous experiments, at small ?ow velocities the cylinder was straight and the turbulence-induced vibrations were damped (?gure 9a). With increasing ?ow, the cylinder buckled essentially in a ?rst beam-mode shape (?gure 9b) and the amplitude of buckling increased with ?ow (?gure 9c). When the ?ow reached almost the maximum possible, the cylinder started to oscillate in its second mode around its equilibrium position (?gure 9d ). This is also shown in the electronic supplementary material video. In what follows, the observations are discussed quantitatively. Figure 10a shows how the frequency of turbulence-induced vibrations varies with the non-dimensional ?ow velocity. Comments similar to those made for the other experiments can be made here also. The critical ?ow velocity for divergence ?1? based on this graph is U BP x 7:5, which is larger than the theoretical value of th U BP Z 6:25. Figure 10b shows the bifurcation diagram of the system. For small ?ow velocities, the cylinder behaviour is similar to that previously discussed, in the sense that there is a sharp increase in the rate of change of cylinder de?ection amplitude with ?ow, corresponding to the onset of divergence. The critical ?ow ?3? ?2? velocity for divergence is at U BP x 6:2 by the third method; and at U BP x 5:4 by the second method. The cylinder buckles mainly in its ?rst mode and the amplitude of buckling increases with ?ow until U x 10, where a sudden decrease in the experimental midpoint de?ection amplitude of the cylinder is observed for a short range of ?ow, after which the cylinder begins to oscillate, mainly in its second mode, at U x 11:2. As seen in ?gure 10b, the oscillatory motion occurs just before the maximum available ?ow is attained ?U max x 11:5?. Figure 11a –c shows the PSD plots of this cylinder in the pre-buckling ?U Z 4:2?, buckled ?U Z 6:5? and oscillatory ?U Z 11:2? states. The three peaks in the PSD plot for U Z 4:2 (f1Z3, f2Z8 and f3Z16) may be identi?ed with the ?rst, second and third eigenfrequencies of the system. With increasing ?ow, the peaks move to the left; at U Z 6:5, they are at f1Z2, f2Z7 and f3Z15. The two peaks in the PSD plot for U Z 11:2 most probably correspond to the second and the third eigenfrequencies of the system (at fZ1.8 and 3.3), the ?rst mode being the small peak at fZ0.8. This ties in with the fact that the observed oscillations were of predominantly second-mode shape, with a sizable third-mode
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(a)

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(b)

(c)

(d )

Figure 9. The cylinder of the third series of experiments at different states: (a) pre-buckling, (b) small-amplitude buckling, (c) large-amplitude buckling and (d ) oscillatory motions.

contribution. Figure 11d shows the time history of the oscillatory motion of the cylinder at U exp HB Z 11:2, where the displacement measured by the ?rst measuring device is plotted for a time interval of 10 s. The parameters used in the theoretical calculations to compare the results with the experimental observations are given in table 2, corresponding the quantities given in table 1 (series 3). Some of the parameters, however, cannot be determined with certainty, namely those related to the viscous terms (namely c n, c t and c d). These parameters related to the damping and viscous forces can vary over a certain range, depending on the cylinder and character of the ?ow. The in?uence of these parameters on the critical ?ow velocity for the Hopf bifurcation has been carefully studied theoretically. It was found that their effect on the threshold of post-divergence ?utter of this system (which is inherently conservative, bar the dissipative and viscosity-related frictional effects) is quite substantial,6 unlike for the inherently non-conservative
For instance, one would suspect that cd would have an important effect, and it does: varying cd from zero to 0.08 changes U HB from 21.8 to 35.5 approximately (with the other parameters as in table 3). On the other hand, for cnZct , the effect of varying the two together is rather smaller. But, if cns ct , their ratio cn/ ct has an ‘unexpectedly’ important effect: 8:5 ! U HB ! 27 for 0:5 ! cn =ct ! 2, where cn/ctZ0.5 for rough cylinders and cn/ctZ2 for smooth ones (Ortloff & Ives 1969); in the light of ?uid–structure energy-transfer considerations, however, this is hardly surprising (refer to Pa? ¨doussis (2004, §8.3.3)).
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(a) 25 frequency (rads–1) 20 15 10 5 0 1 2 3 4 5 6 7
(1) BP

1291
1.657

(b) 0.06 0.05 0.04 h 0.03 0.02 0.01 8 0 2 4 6 8 10
(2) BP

dynamic oscillations 1.100

0.550

0 12

Figure 10. Comparison between theory and experiment for the third series of experiments with a small-diameter clamped–clamped cylinder with no end-sliding and with the parameters given in table 1. (a) Experimental (asterisk) and theoretical (dot) graphs of frequency versus dimensionless ?ow velocity; (b) comparison between theory (continuous line) and experiment (asterisk) in the form of bifurcation diagrams.

system of a cantilevered cylinder in axial ?ow, for instance. Taking the reasonable values of c tZ0.02, c n/c tZ0.5 (the cylinder is hydrodynamically rough), cdZ0.005 (see Pa? ¨doussis (2004, appendix Q)) and taking into account structural damping (auZavZ1) (see Modarres-Sadeghi 2006), the theoretical critical ?ow velocity for the Hopf bifurcation is found to be U th HB Z 15:7, which is quite different, but not unreasonably far from the experimental value of 11.2. According to the theoretical results, the cylinder oscillates around its buckled state and the frequency of oscillation is f thZ3 Hz; in the experiments, on the other hand, the cylinder oscillates around its original equilibrium position (inexplicably, as predicted by linear theory) with a frequency of f expZ1.8 Hz.7 These two frequencies are of comparable magnitude. The difference in the nature of oscillations (around the buckled state in theory and around the original equilibrium position in experiment) could be due to the fact that the model used for the theoretical predictions is correct to third-order magnitude; perhaps, to capture the relatively large-amplitude oscillations of the system, one needs to use a higher-order model (e.g. ?fth order). To study the in?uence of the externally applied axial compression on the dynamic instability of the system, experiments were conducted with this smalldiameter cylinder, with a ring inserted between the lower end of the cylinder  Z K19:0. and the downstream support, applying an axial compression of G Similarly to the case of the large-diameter cylinder (§5), under axial compression the cylinder buckles at a lower ?ow velocity, and the amplitude of buckling at a given ?ow velocity is larger than that of the cylinder with no axial compression. The critical ?ow velocities for divergence obtained by the ?1? ?2? ?3? three methods discussed before are U BP Z 6:1, U BP Z 5:0, U BP Z 4:3, while
7

It is recalled that linear theory predicts that the onset of ?utter arises by the locus emerging from the imaginary-frequency axis in Argand diagrams, with essentially zero frequency; the frequency increases thereafter as the ?ow velocity is increased.

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(a) power spectrum (dB) 0 – 20 – 40 – 60 – 80 –100

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(b)

0

5

10 f

15

20

(c) power spectrum (dB)

0 –10 – 20 –30 – 40 –50 – 60 –70 0 5 10 f 15 20

(d )

0.04 0.03 0.02 0.01

h

0 –0.01 –0.02 –0.03 –0.04 0 2 4 t 6 8 10

Figure 11. PSD plots for the third series of experiments with a small-diameter clamped–clamped cylinder with no end-sliding and with the parameters given in table 1 at (a) U Z 4:2, (b) U Z 6:5 and (c) U Z 11:2, together with (d ) the time history at U Z 11:2.

U th BP Z 4:5; thus, except for the ?rst method, reasonably good agreement between the theoretical and experimental values is obtained. What is remarkable in this case is that ?utter occurs at almost the same ?ow velocity as with no axial compression! In fact, the cylinder under axial compression buckles at a lower ?ow velocity ?U x 4:5?, and remains buckled for a wider range of ?ow velocities, until it starts oscillating at U x 11. This is in agreement with what the theoretical nonlinear model predicts. The theoretical value for  Z K19 is approximately the same as for G  Z 0 ?U HB x 15:7?.8 By changing the G external axial compression from 0 to K150, the critical ?ow velocity for the Hopf bifurcation increases smoothly from U Z 21:8 to 22.3. The change in  from 0 to K19 is almost negligible according to the critical ?ow for varying G these results. Also, looking at the nonlinear equations of motion of the system  (the (Modarres-Sadeghi et al. 2005), one can see that all the terms involving G non-dimensional axial tension/compression parameter) are associated with static terms, i.e. terms with no time derivatives of the unknowns. This, however, does not imply that there should be no effect, or a negligible effect, on  is varied. the dynamic behaviour of the system as G
8

 from zero to K150 produces a change in U HB of the order of 5% only. Indeed, changing G

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7. Conclusions Three series of experiments have been conducted to study the dynamical behaviour of slender ?exible cylinders clamped at both ends and subjected to axial ?ow. The downstream end of the cylinder could either slide axially or was wholly ?xed. Nonlinear theory predicts that the system loses stability via a pitchfork bifurcation, leading to buckling, and that at higher ?ows a Hopf bifurcation occurs, emanating from the statically divergent solution. The second mode of the system becomes unstable at the Hopf bifurcation and the cylinder oscillates about the buckled state. This periodic oscillation is then followed by quasi-periodic and chaotic ones. The existence of the post-divergence dynamic instability had previously been predicted by Pa? ¨doussis (1966a, 1973) using a linear model and was con?rmed by experiments (Pa? ¨doussis 1966b), but the interest in those experiments was con?ned to the linear realm. Here, in the ?rst series of experiments, the lower end of the cylinder was free to slide axially, while in the second series, axial sliding was prevented. In all the experiments, small static de?ections were observed at small ?ow velocities, re?ecting the growth of the initial material and geometric imperfections. After a certain ?ow velocity, the cylinder amplitude increased more rapidly with ?ow. This ?ow velocity corresponds to the critical value for divergence. When applying axial compression in the case of ?xed ends, it was observed that the critical ?ow velocity for divergence decreased, while for a given ?ow velocity, the amplitude of buckling increased, as one would expect. Good agreement between theoretical and experimental results was observed for both the critical ?ow velocity for divergence and the amplitude of buckling. No dynamic instabilities were observed in these two series of experiments because the cylinder was rather stiff, hence the maximum attainable dimensionless ?ow velocity was insuf?ciently high for that. The third series of experiments were conducted with a more ?exible hollow cylinder, making it possible to reach higher dimensionless ?ow velocities. Similarly to the previous two series of experiments, the cylinder buckled, and the amplitude of buckling increased with ?ow. In this case, however, at higher ?ow, the cylinder developed a dynamic instability (?utter) and oscillated mainly in its second mode. This proves, experimentally, the existence of post-divergence ?utter, previously predicted by linear theory and the nonlinear theoretical model. These observations also con?rm the experimental results of Pa? ¨doussis (1966b). It was observed that the theoretical value for the Hopf bifurcation was largely dependent on the dissipation- and ?uid friction-related system parameters, for which (to the best available knowledge) precise values cannot be assigned. However, based on a separate theoretical parametric study on the effect of these parameters on the ?utter threshold, and using reasonable values, experimental ?utter thresholds were found to be within 30% of the theoretical values. This is not satisfactorily close, indicating that further work needs to be done in this respect. It was also observed, both experimentally and theoretically, that the in?uence of externally applied axial compression on the onset of post-divergence ?utter is negligible for the range of parameters of this study.
The authors gratefully acknowledge the support given to this research by the Natural Sciences and Engineering Research Council of Canada (NSERC).
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Appendix A. Nonlinear equations of motion The nonlinear equations of motion for the system of a slender cylinder subjected to axial ?ow have been derived by Modarres-Sadeghi et al. (2005) and in their non-dimensional form are   p??? 0 0 Kc b h _ h C U 2 h 00 h 0 K P0 ?z 00 C h 0 h 00 ? K?h 00 h 000 C h 0 h?4? ? h 0 C 2 U bh ?1 Kb?z  p???  1 2 1 1 b 0 02 _ Ch _ 2 C 3 U 2 ct ?1 C h ? K 3 U ?c n K c t ? hh C 3c t bh 2 4 2 U # p??? 0    _j Ch _ jh 0 j? b?h jh h02 1 1 2 0 00 0 0 0 ! zK C 1 K d Kx h h K 3 U c d h h jh j C 2 2 U 2
0

"

 ! _ jh _j bh 1 2   C 2 C U c b ?1 Kd? C Gd C ?1 K2n?Pd h 0 h 00 2 U #   h02 1 0 00 Cg z K C 1 K d Kx h h C O?e5 ? Z 0 2 2
0

"

?A 1?

and
    p??? 0 p??? 0 7 02 5 02 2 00 C3 b Uz _ _ 1 K h C cU h 1 K h Kch0 ?bz C 2 c U bh ?1 C ?c K 1?b?h 4 2    p??? 3 p??? 0 p??? 0 2 00 02 0 _ _ C U h00 _ h K c 4 U z C 2 bz C C 2U z ? Kcbh bh bhh 2  ! 1 2 3 0 2 00 00 0 00 00 0   C U cb ?1 Kd? C Gd C ?1K2n?Pd Kh C h z C h z C h h 2 2 p??? 0     0 _ _ _ _ 1 2 j h j C h j h j? b h j h j 3 0 2 00 b ? h 0 0 00 0 0 00 C 3 U cd h jh j C C 2 K P0 z h C z h C h h 2 U 2 U C v ?4? K?8h0 h00 h000 C h0 z?4? C 2h0 h?4? C 2h00 C 2z0 h?4? C 4z00 h000 C 3z000 h00 ?    1 1 3 1 C 3 U 2 c t ?1 C h? K h0 C ?z Kz?1??1 Kd??h00 C 1 K d Kx 2 2 2 !   _ 02 1 h _ 2 h0 hh 3 0 2 00 1 2 00 0 00 00 0 03 0 0 ! Kh C h z C h z C h h C b 2 K 3 U ?c n K c t ? h C h z C b 2 2 2 U U ! p??? p??? 1 1 b _ 1 b3=2 3 b b 0 _K _C 2z _ _C h h C 3 U 2 ?c n C c t h ?h 0 C 3 U 2 c n h zh 2 2 2 U3 U U U &   1 3 1 C g h0 K h0 C ?z Kz?1??1Kd??h00 C 1 K d Kx 2 2  ' 3 2 ? A 2? ! Kh00 C h0 z00 C h00 z0 C h0 h00 C O?e5 ? Z 0; 2
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where the non-dimensional parameters are related to the dimensional ones using the following relations:  1=2  1=2 X u v EI t rA rA xZ ; zZ ; hZ ; tZ ; ; UZ UL; b Z 2 L L L m C rA EI m C rA L ?m KrA?gL3 4 4 4 4 L ; c n Z CN ; c t Z CT ; c d Z CDp ; c b Z C b ; 3 Z ; p p p p D EI 2  2  2 TL D  PAL EAL M Z h Z ; PZ ; G ; P0 Z ; cZ : Dh rA EI EI EI gZ

?A 3?

In equation (A 3), u and v are displacements in x - and y -directions, respectively; L and D are the cylinder length and diameter; C b and CDp are base and form drag coef?cients; CN and C T are frictional coef?cients in the normal and tangential directions, respectively; EI is the ?exural rigidity of the cylinder; rA is the added mass of the ?uid per unit length; m is the mass per unit length of the  and P  are externally applied tension and pressurization, cylinder; and T respectively.

References
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