EXPERIMENTAL INVESTIGATIONS ON VORTEX-INDUCED VIBRATIONS WITH A LONG FLEXIBLE CYLINDER. PART I: MODAL-AMPLITUDE ANALYSIS WITH A VERTICAL CONFIGURATION

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1 EXPERIMENTAL INVESTIGATIONS ON VORTEX-INDUCED VIBRATIONS WITH A LONG FLEXIBLE CYLINDER. PART I: MODAL-AMPLITUDE ANALYSIS WITH A VERTICAL CONFIGURATION Guilherme R. Franzini Celso P. Pesce Rodolfo T. Gonçalves Offshore Mechanics Laborator (LMO), Escola Politécnica (EPUSP), Universit of São Paulo, Brazil. André L. C. Fujarra Department of Mobilit Engineering Federal Universit of Santa Catarina, Brazil. Pedro Mendes Petrobras Resarch Center (CENPES/PDEP/TDUT), Brazil. ABSTRACT An experimental investigation of Vortex-Induced Vibration (VIV) on a long and vertical flexible clinder is presented and some results are analzed. Optical instrumentation allows to directl measure the cartesian coordinates of 43 targets placed along the model. At each instant, modal decomposition based on Galerkin s scheme is applied and, then, modal-amplitude time-histories are obtained. The modal-amplitude time-histories are analzed, allowing to identif similarities in the modal response at the first and the second natural modes lockin. Jumps and phase-shifts transitions are also obtained from the analsis of the modal-amplitude time-histories. Finall, it is also found that modal response amplitudes, plotted as functions of the corresponding, reasonable collapses onto the same curve, providing an experimental evidence that the modal response are the same for different excited modes. Kewords:Vortex-Induced Vibrations, flexible clinder, experiments, modal analsis. INTRODUCTION Vortex-Induced Vibrations (VIV) is a non-linear, selfexcited and self-limited fluid-structure interaction phenomenon which ma be of crucial importance in structural integrit studies of risers and other slender structures commonl found in the offshore engineering scenario. A great deal of investigations have been carried out in the last five decades, mostl dedicated to fundamental Corresponding author. gfranzini@usp.br Formerl at Universit of São Paulo. aspects of VIV of elasticall supported rigid clinders, as thoroughl described in earlier reviews such as, for example, references [], [2] and [3]. Flexible clinders are, however, a much closer representation of risers dnamics. In this context, some aspects such as presence of travelling waves, modal jumps, multimodal excitations, are intrinsic to the dnamics of long flexible clinders; see, for instance, [4], [5], [6], [7] and [8]. Reference [9] presents a review of flexible clinders VIV. Recentl, a comprehensive research project on nonlinear dnamics of risers was conducted at Escola Politécnica of Universit of São Paulo. Three groups of experiments with a long and immersed flexible clinder were carried out for two arrangements - vertical and catenar configurations; namel, (i) pure top-motion (see [] for results concerning the vertical configuration), (ii) pure VIV, (iii) concomitant VIV and top-motion; details ma be found in []. Part of the results from the second group of experiments with the vertical configuration are herein presented. Results from the same group, but for a catenar configuration are presented in Part III, [2], of this series of papers, while Part II (see [3]) focuses on results of group (iii) for the vertical configuration. Instead of a standard VIV analsis (i.e., statistical and spectral analses for each measurement point along the model), this paper focuses on analzing the modalamplitude time-histories obtained after the application of a Galerkin s projection scheme. The objective is to present some aspects enlightened b the modal-amplitude time-histories and examples of analsis that can be carried out with them. It is also discussed on the paper the complimentar character of the modal analsis with re-

2 spect to the standard analsis. The paper is structured as follows: The next Section presents details regarding the experimental arrangement and the analsis methodolog. In the following Section, some results are presented and discussed. Finall, final remarks and perspectives of further works are presented. EXPERIMENTAL ARRANGEMENT AND ANALYSIS METHODOLOGY The experiments were carried out at IPT (Institute for Technological Research of the State of São Paulo) towing tank facilit. Renolds number ranged from 7 to 6. The clinder diameter is D = 22.2mm with unstretched length L = 2552mm. The mass ratio parameter, the immersed to total length and the immersed length to diameter ratios were, respectivel, 3.48, 8% and. Fig. presents a schematic representation of the experimental arrangement. Main properties of the flexible clinder are presented in Tab.. (a) Sketch of the side view. TABLE : Flexible clinder properties and characteristics of the vertical configuration. External diameter D Axial stiffness EA Clinder properties 22.2 mm.2 kn Bending stiffness EI.56 Nm 2 Linear mass m l Immersed weight γ.9 kg/m 7.88 N/m Characteristics of the vertical configuration Unstretched length L o Stretched length L Immersed length L i 2552 mm 262 mm 2257 mm Mass ratio parameter m 3.48 Aspect ratio L i /D 2 L/D 7 Static tension at the top T t 4 N Natural frequencies and modal damping ratios mode n f air N,n ζ n f N,n. Hz.42%.84 Hz Hz.63%.68 Hz 3 3. Hz.89% 2.52 Hz (b) Sketch of the back view. FIGURE : Schematic representations of the experimental arrangement. Carriage speed is from left to right. Displacements of 43 points along the model were measured b an underwater optical tracking sstem and the vertical component of the force at the top was measured b a load cell. Data were sampled at f sp = 6Hz along 3s of stead-state regime. The flexible clinder was tensioned b appling a static axial force at the top of 4N. Free-decaing tests carried out with the immersed model at rest indicated that the natural frequenc of the n-mode is given b f N,n = n.84hz at null towing speed conditions. As presented in [4], the natural frequencies in air roughl follows the relation 2 fn,n air = nhz and the modal structural damping ratio (w.r.t. the critical value) is lower than % for the first three natural modes. In experimental investigations of flexible clinders VIV, it is ver common the use of strain-gages ( [4], [6, 7] or accelerometers ( [8]). These experimental Damping ratios refer to experiments carried out in air, with the same corresponding modal stiffness (regulated b tension) as in water [4]. 2 Actuall, eigen modes are Bessel-like functions [5], [6], making the natural frequenc ratio deviate from a natural number sequence. This effect is a little more pronounced in air than in water.

3 z/l V R, =9.7 Measured Nm = Nm = 2 Nm = 3 [5]), or Bessel-like ones, which incorporate both effects through an ingenious integral averaging procedure; see [6]. Nonetheless, sinusoidal functions, besides simpler, are here considered as sufficient to exemplif the analsis methodolog. Consistentl, the modal reduced velocit can be defined b normalizing the free-stream velocit U b the product between the diameter and the n natural frequenc f N,n. Hence: (t ) V R,n = U f N,n D (3) FIGURE 2: Example of reconstruction of measured data at a particular instant. approaches are indirect was to measure displacements, since the depend either on assumptions regarding the structural modeling or on double integration with respect to time. The large number of points and the direct measurement method allow a richer analsis of the flexible clinder VIV dnamics. In fact, from the directl measured displacements, the analsis methodolog consists in appling the Galerkin s scheme, i.e., to project the deformed configuration of the flexible clinder, at an arbitrar instant t j, onto the space composed b modal functions ψ(z). Being X (z,t j ) and Y (z,t j ) the in-line and cross-wise displacements 3 with respect to the towing direction measured at the point of spanwise coordinate z at t j, the respective modal amplitudes corresponding to a given n mode ma be written, respectivel: ã x n(t j ) = ã n(t j ) = L X (z,t j )ψ n (z)dz L (ψ n(z)) 2 dz L Y (z,t j )ψ n (z)dz L (ψ n(z)) 2 dz () (2) Fig. 2 presents an example of reconstruction of the measured cross-wise displacement at a certain instant t j. As expected, the increase of the number of modes, N m, used in the reconstruction leads to a a better representation of the measured data. Similarl to [6], [7] and [9], the mode functions considered herein are sinusoidal ones ψ n (z) = sin(nπz/l ). Evidentl, much more representative mode functions could be used, as the classic Bessel s, which neglect bending and axial stiffness effects (see, e.g., 3 All displacement amplitudes are normalized b the diameter D. Another aspect of potential interest in the stud of the flexible clinder VIV is the snchronization between in-line and cross-wise oscillation. The modal amplitude time-histories allow not onl the stud of snchronization between natural modes in orthogonal directions but also natural modes in the same direction, such as, for example, snchronization between the first and second modes in the cross-wise direction. In this paper, snchronization is investigated b considering the time-histor of phase-shift between two modal amplitude time-histories. Following the definition presented in [2], snchronization is characterized b a time-invariant phase-shift between two signals. In the VIV context, snchronization analsis was carried out b [9] in for flexible clinder VIV (not considering modal-amplitude time-histories) and b [2] addressing of phase-shift between force and displacement in 2-dof VIV. Let w(t) be a signal and H(w) its Hilbert Transform. The Hilbert Transform allows representing the instantaneous phase (φ w (t)) and amplitude a w (t) of w(t), in the form: H(w) = a w (t)e jφ w(t) (4) The phase-shift between two signals is then given b m n φ q p(t) = βφ p n (t) + φ q m(t) (5) where n and m refer to the mode number, p and q indicate the direction (in-line or cross-wise) and β is the ratio between the dominant frequenc of the two signals. Although Eqn. 5 can be used both for modes in orthogonal directions as well as in the same direction, snchronization analses between modes in the same direction is not herein carried out for the sake of conciseness of this paper. This is left for a further work.

4 RESULTS AND DISCUSSIONS Firstl, it will be discussed how the modal analsis proposed in the previous section ma help as a complimentar tool for the standard analsis. Fig. 3 shows a standard analsis of the cross-wise displacements measured at z/l =.22. Fig. 3(a) presents the characteristic oscillation amplitude (obtained b averageing the % of the highest peaks). For reduced velocities V R, = U / f N, D < 8, there is a qualitative agreement with the results obtained with a rigid clinder elasticall supported. However, due to the onset of the second natural mode lock-in, a classical lower branch for V R, > 8 is not observed. The non-dimensional amplitude spectra presented in 3(b) reveals the presence of lock-in with distinct natural modes depending on the value of V R,. Although interesting, both graphics are not sufficient to discuss the contribution of the distinct natural modes in the flexible clinder VIV. The modal decomposition appears as a useful scheme for the mentioned analsis and examples of use will be addressed below. The first example of use of the modal analsis technique refers to the case in which the lock-in is observed in the first natural mode; in the present experiments at V R, = U / f N, D = Fig. 4(a) presents the modal amplitude time-histor corresponding to the first vibration mode, i.e., n = in Eqns. and 2. As expected the modal-amplitude time-histor ã (t) is practicall monochromatic at the first natural frequenc. It is well known from the literature addressing rigid clinders mounted on two degrees-of-freedom elastic supports (see, for example, [22]) that the in-line vibration takes place at twice the cross-wise vibration frequenc. Hence, it seems natural that, for the first mode lock-in, the second natural mode might be excited in the in-line direction. This fact is confirmed in Fig. 4(b), showing a well defined eight-shaped figure in the ã (t) ãx 2 (t) modal amplitude plane; a neat experimental example of dual resonance (see [23]). Still considering the first-mode lock-in, it is interesting to investigate the snchronization between ã (t ) and ã x 2 (t ). Fig. 5 presents the temporal evolution of the phase-shift 2 φ x and the corresponding histogram normalized b the total number of occurrences. Notice that 2 φ x is practicall time-invariant and close to 5. Hence, the first mode lock-in is accompanied b a ver well-defined snchronization between first natural mode oscillations in the cross-wise direction and second natural mode oscillations in the in-line direction. Consider now the second mode lock-in and, for this, a test condition in which the modal reduced velocit cor- A V R, = U /f N, D (a) Characteristic cross-wise oscillation amplitude at Z/L =.22. (b) Non-dimensional cross-wise amplitude spectra. FIGURE 3: z/l =.22 Results for cross-wise displacements at responding to the second natural mode is V R,2 = Fig. 6 presents both the cross-wise modal amplitude timehistor and the projection in the plane ã 2 ãx 4. Fig. 6(a) clearl reveals two regimes. In the first one (t = t f N, < 5), the characteristic amplitude is similar to that observed in ã (t) when V R, = 5.68 (see Fig. 4(a)), i.e., at practicall the same modal reduced velocit. The second regime is characterized b an increase in the characteristic amplitude. The trajectories in the ã 2 (t) ãx 4 (t) plane, presented in Fig. 6(b), reveal that the increase of amplitude in ã 2 (t) is accompanied b a decrease in ã x 4 (t). In the first regime, a classic eight-shaped trajector in the ã 2 (t) ãx 4 (t) is also shown in red, similar to what appeared in the first mode lock-in. Nonetheless, such regime quickl jumps to another one, where relative phases change and a C-shaped trajector prevails. This 4 Notice that the value of this modal reduced velocit is ver similar to that previousl investigate.

5 VR, = U /fn,d =5.63 is for sure an interesting phenomenon, possibl revealing the co-existence of two dnamic atractors. 2 x φ (t )[ ] 2 VR, = U /fn,d =5.63 a 2 (t ) N 3 t = tfn, 5 5 t = tfn, φx [ ] S 2.3 FIGURE 5: Phase-shift between a (t ) and a x2 (t ). VR, = f /fn, (a) Modal-amplitude time-histor a (t). VR,2 = U /fn,2d =5.78 VR, = U /fn,d = a 2 (t ) t = tfn, a (t ) S f /fn, (a) Modal-amplitude time-histor a 2 (t). a x2 (t ) (b) Trajector in the a (t) a x2 (t) plane. VR,2 = U /fn,2d = FIGURE 4: Modal-amplitude time-histories. VR, = a 2 (t ) It remains of interest the discussion regarding the snchronization between modal amplitude time-histories in cross-wise and in-line directions. For this, consider Fig. 7. In this plot, it is clearl noticeable a sudden phase jump from 42 φx 23 to 42 φx 45 at t 5. Furthermore, considering t > 5, the phase-shift time-histor is much more oscillator, compared to the case corresponding to VR, = 5.63, presented in Fig. 5. Hence, despite the first and second natural modes lock-in indicate similar qualitative behavior of the phase-shifts, the phase modulation is more pronounced on the latter case, indicating a less defined snchronization. The last result herein presented consists in plotting the characteristic values, A xn and A n, as functions of the modal reduced velocit VR,n = U / fn,n D. The characteristic values were computed b averaging the % of the a x4 (t ) (b) Trajector in the a 2 (t) a x4 (t) plane. FIGURE 6: Modal-amplitude time-histories. VR,2 = highest modal-amplitude time-histories peaks. As can be seen in Fig. 8, the values of amplitude A i follow the same trend for the first four natural modes. Furthermore, the re-

6 4 2 φ x (t )[ ] N 2 V R,2 = U /f N,2 D = t = tfn, φ x [ ] FIGURE 7: Phase-shift between ã 2 (t ) and ã x 4 (t ). V R,2 = sponse is remarkabl similar to those usuall encountered for elasticall supported rigid clinders. à n à x n n = n = 2 n = 3 n = V R,n =U /f N,nD (a) Cross-wise direction. n = n = 2 n = 3 n = V R,n =U /f N,nD (b) In-line direction. FIGURE 8: Characteristic values of modal-amplitudes as a function of modal reduced velocit. FINAL REMARKS The present paper presented experimental results of a long, vertical and flexible clinder subjected to Vortex- Induced Vibrations (VIV). The cartesian coordinates of 43 reflexive targets placed along the model were directl measured b an optical tracking sstem. Such optical measuring technique, along with a Galerkin s decomposition scheme, formed an innovative experimental analsis procedure, leading to a direct evaluation of modalamplitude time-histories in both, in-line and cross-wise directions. Some interesting aspects were enlightened through the analsis of the modal-amplitude time-histories. The first-mode lock-in is characterized b a stead-state response with a remarkable snchronization between the modal-amplitude time-histories associated to the first mode in cross-wise direction and the second mode in inline direction. On the other hand, considering the secondmode lock-in, two regimes in the response were clearl identified. The first regime has a remarkable similarit with the one observed for the first-mode lock-in. The second one is characterized b an increase in the crosswise oscillations and a decrease in the in-line amplitudes. These two distinct regime also present slightl different spectral distributions. The characteristic values of the modal-amplitude time-histories were also evaluated. Plotting these results as functions of the modal reduced velocit (i.e, the reduced velocit considering the natural frequenc of each mode), the same trend for all the modes was obtained. Ongoing researches include a deeper analsis of the modal-amplitude time-histories, such as the use of timefrequenc domain signal analsis techniques in order to better identif mode transitions and jumps in the response. Further analsis of snchronization between modes in orthogonal directions and snchronization between modes in the same direction are being carried out. Part I took care of the vertical configuration subjected to pure VIV, pertaining to experimental group (ii). In this same conference, Part II, addresses experimental group (iii), studing the effect of axial motion excitation in the VIV of the same vertical configuration; whereas Part III takes care of pure VIV in a catenar configuration, pertaining to group (ii). ACKNOWLEDGMENT The results presented in the paper were obtained from a comprehensive research project on non-linear dnamics of risers sponsored b Petrobras and carried out in First and third authors ac-

7 knowledge FAPESP (São Paulo Research Foundation) for their post-doctoral scholarship, grants 23/982-2 and 24/243-. The National Council for Research, CNPq, is acknowledged b first, second and fourth authors for the grants 33595/25-, 3899/24-5 and 3959/23-9. Special thanks to the IPT towing tank technical staff. REFERENCES [] Bearman, P. W., 984. Vortex shedding from oscillating bluff bodies. Annual Review of Fluid Mechanics, 6, Jan, pp [2] Williamson, C. H. K., and Govardhan, R. N., 24. Vortex-induced vibrations. Annual Review of Fluids Mechanics, 36, pp [3] Sarpkaa, T., 24. A critical review of the intrinsic nature of vortex-induced vibrations. Journal of Fluids and Structures, 9, pp [4] Pesce, C. P., and Fujarra, A. L. C., 2. Vortexinduced vibrations and jump phenomenon: Experiments with a clamped flexible clinder in water. International Journal of Offshore and Polar Engineering,, pp [5] Facchinetti, M. L., de Langre, E., and Biolle, F., 24. Vortex-induced traveling waves along a cable. European Journal of Mechanics B/Fluids, 23, pp [6] Chaplin, J. R., Bearman, P. W., Huera Huarte, F. J., and Pattenden, R. J., 25. Laborator measurements of Vortex-Induced Vibrations of a vertical tension riser in a stepped current. Jounal of Fluids and Structures, 2, pp [7] Swithenbank, S., 27. Dnamics of long flexible clinders at high-mode number in uniform and sheared flows. PhD thesis, Massachusetts Institute of Technolog. [8] Vandiver, J. K., Jaiswal, V., and Jhingran, V., 29. Insights on vortex-induced, traveling waves on long risers.. Jounal of Fluids and Structures, 25, pp [9] Wu, X., Ge, F., and Hong, Y., 22. A review of recent studies on vortex-induced vibrations of long slender clinders. Journal of Fluids and Structures, 28, pp [] Franzini, G. R., Pesce, C. P., Salles, R., Gonçalves, R. T., Fujarra, A. L. C., and Mendes, P., 25. Experimental investigation with a vertical and flexible clinder in water: response to top motion excitation and parametric resonance. Journal of Vibration and Acoustics, 37 (3), pp [] Pesce, C. P., 23. Riser dnamics: experiments with small scale models. LabOceano - Ten-Years Anniversar Celebration Workshop, April [2] Rateiro, F., Fujarra, A. L. C., Pesce, C. P., Gonçalves, R. T., Franzini, G. R., and Mendes, P., 26. Experimental investigations on vortexinduced vibrations with a long flexible clinder. Part III: modal-amplitude analsis with a catenar configuration. In Proceedings of the th International Conference on Flow-Induced Vibration. [3] Franzini, G. R., Pesce, C. P., Gonçalves, R. T., and Mendes, P., 26. Experimental investigations on Vortex-Induced Vibrations with a long flexible clinder. Part II: effect of axial motion excitation in a vertical configuration.. In Proceedings of the th International Conference on Flow-Induced Vibration. [4] Salles, R., 26. Experimental analsis of fluidstructure interaction phenomena on a vertical flexible clinder: modal coefficients and parametric resonance. Master s thesis, Escola Politécnica da Universidade de São Paulo. [5] Pesce, C. P., Fujarra, A. L. C., Simos, A. N., and Tannuri, E. A., 999. Analtical and closed form solutions for deep water riser-like eigenvalue problem. In Proceedings of the Ninth (9th) International Offshore and Polar Engineering Conference. [6] Mazzilli, C. E. N., Lenci, S., and Demeio, L., 24. Non-linear free vibrations of tensioned vertical risers. In Proceedings of the 8th European Nonlinear Dnamics Conference - ENOC24. [7] Chaplin, J. R., Bearman, P. W., Cheng, Y., Fontaine, E., Graham, J. M. R., Herfjord, K., Huera Huarte, F. J., Isherwood, M., Lambracos, K., Larsen, C. M., Meneghini, J. R., Moe, G., Pattenden, R., Triantafllou, M. S., and Willden, R. H. J., 25. Blind predictions of laborator measurements of Vortex- Induced Vibrations of a tension riser. Jounal of Fluids and Structures, 2, pp [8] Huang, S., Khorasanchi, M., and Herfjord, K., 2. Drag amplification of long flexible riser models undergoing multi-mode VIV in uniform currents. Journal of Fluids and Structures, 27, pp [9] Huera-Huarte, F. J., and Bearman, P. W., 29. Wake structures and vortex-induced vibrations of a long flexible clinder - part :dnamic response. Journal of Fluids and Structures, 25, pp [2] Pikovsk, A., Rosenblum, M., and Kurths, J., 2. Snchronization - A Universal Concept in Nonlinear Sciences. Cambridge Universit Press. [2] Franzini, G. R., Gonçalves, R. T., Meneghini, J. R.,

8 and Fujarra, A. L. C., 23. One and two degreesof-freedom vortex-induced vibration experiments with awed clinders. Journal of Fluids and Structures, 42, pp [22] Jauvtis, N., and Williamson, C. H. K., 24. The effect of two degrees of freedom on vortex-induced vibration at low mass and damping. Journal of Fluid Mechanics, 59, pp [23] Dahl, J. M., Hover, F. S., Triantafllou, M. S., and Oakle, O. H., 2. Dual ressonance in vortexinduced vibrations at subcritical and supercritical Renolds numbers. Journal of Fluid Mechanics, 643, pp