Cross flow and in-line damping measurements from forced excitations of a flexible cylinder in a uniform flow

Transcription

1 Proceedings of OMAE04 23rd International Conference on Offshore Mechanics and Arctic Engineering June 20-25, 2004, Vancouver, British Columbia, Canada OM AE Cross flow and in-line damping measurements from forced excitations of a flexible cylinder in a uniform flow Madan Venugopal, ARMA Group, 804 Betlin Ave, Cupertino, CA 95014, USA. madan@alum.mit.edu J Kim Vandiver, MIT, Room 7-133, 77 Massachusetts Ave, Cambridge, MA USA. kimv@mit.edu Keywords: Damping, flow induced vibration, flexible cylinders, response prediction, forced oscillation. ABSTRACT Tensioned flexible cylindrical structures are important in many ocean engineering applications such as moorings for buoys and platforms, marine risers and towing cables. Modeling the vibration of these structures is complicated because these are complex three-dimensional, unsymmetrical, fluid structure interaction problems. Damping is an important, but poorly understood, component of the response prediction models developed for modeling such systems. In particular, there is a scarcity of good data on damping of flexible cylinders vibrating in uniform and non-uniform external flow. This is, in part, due to the difficulty of measuring fluid damping on a vibrating cylinder in a flow. Results are presented here which address some of these limitations. Forced vibration tests were performed on two 13 ft long tensioned flexible cylinders (an ABS tube and a steel wire) in a current tank to determine in air and still water damping as well as cross flow and in-line damping in a uniform flow. The experimental methodology is described and results are presented for a range of reduced velocities. The results show an increase in fluid damping with increased reduced velocities for small amplitudes of vibration. INTRODUCTION The response of flexible cylinders vibrating in a fluid is controlled significantly by the damping, especially at near resonance frequencies. The damping usually has a structural component and a fluid dynamic component. For vibration of a flexible cylinder in air, the fluid damping is of the same order of magnitude as the structural damping. For flexible cylinders vibrating in water, such as marine risers, mooring cables, subsea pipelines etc, the hydrodynamic damping is usually more significant than the structural damping and in air damping. This paper will therefore focus more on hydrodynamic damping, though the analysis is equally valid for in air damping. For a flexible cylinder in an ocean environment subject to fluid loading from waves and currents, the flow effect can be an excitation or damping of the structure, depending on the phase between the velocity of the structural motion and the force. On a flexible cylinder in a sheared flow, different regions of the same cylinder can be either excited or damped by the flow based on the relative flow velocity. Discussion of fluid damping in the literature is interchangeably used with negative excitation. For purposes of building a good response prediction model, a good estimate of the damping force reflecting the dissipated energy is necessary. It is usually difficult to separate the negative excitation from the dissipated energy by measurements on a flexible cylinder in a flow. The method used for measuring damping as described in this paper reflects the dissipated energy, and hence makes it possible to isolate the fluid damping force. There can be two distinct cases for fluid damping one where the cylinder is oscillating in still water and the other where the cylinder is oscillating in an external flow. One is actually a limiting case of the other, but there are significant differences between the flow conditions. Experimental results are presented for fluid damping in still water, oscillation in the direction in-line with the flow and oscillation normal to the direction of the flow (cross flow). In general, there is a scarcity of measured fluid damping data for flexible cylinders in the literature. Venugopal [1] presents a survey of the literature on fluid damping on cylinders. Using the damping results presented here as well as other experimental data, Venugopal [1] also developed a damping and response prediction model for a flexible cylinder vibrating in a sheared flow. Vikestad et al [2] analyze the validity of the damping model in comparison with further experiments. Chen et al [3], Skop et al [6] and Sarpkaya s [7] still water damping measurements and damping models are discussed in [1]. Low reduced velocity in-line and cross flow damping measurements and models by Chen et al [4], cross flow damping measurements 1 Copyright 2004 by ASME

2 and a model by Vandiver and Chung [8], high reduced velocity models by Gopalakrishnan [9], Triantafyllou et al [10], Chen et al [5] and damping measurements and models from Moe and Verley [11, 12] as well as Griffin and Koopmann, Sarpkaya, Mercier, Staubli and Gopalakrishnan [13, 7, 14, 15, 9] are also discussed in [1]. Most of the measurements are from forced oscillation measurements on rigid cylinders, while a few such as Chen et al and King [3, 4, 5 and 16] are from flexible cylinders. The limitations of the available data for development of a response prediction model are also considered in [1]. In developing a damping model for use in response prediction of flexible cylinders in sheared flows, damping at reduced velocities corresponding to off-resonance conditions are particularly important. MEASUREMENTS OF CROSS FLOW AND IN-LINE DAMPING OF A TENSIONED CYLINDER IN A FLOW As discussed above, damping data is scarce when the goal is to build a damping model for flexible cylinder vibration in sheared flows. To address this problem. some tests were designed and conducted at the Shell Westhollow Research Center in Houston, TX. The methodology and results from these tests will be discussed below. Forced vibration tests have been used to identify structural parameters by Ibanez, et al [17], Smith and Matthiesen [18] and other researchers. This was the technique chosen for the current tests. A 13 ft, long cylinder was mounted vertically in a current tank with bearings at both ends approximating pinned-end supports. The current tank was 20 ft. long by 7 ft. wide by 12 ft. deep and generated a uniform current over the cross section, driven by a ship s propeller. The propeller RPM was directly proportional to the mean current speed in the test section and was pre-calibrated. Figure 1 from Allen, et al [1992] shows a sketch of the current tank. The cylinder was tensioned by adjusting a turnbuckle above the top bearing. The tension was measured by a force transducer mounted below the turnbuckle. Two biaxial accelerometers were mounted at 0.27 L and 0.88 L below the top bearing, where L is the length of the cylinder. Figure 2 shows a schematic of the test setup. The signals from the accelerometers and force transducer were passed through low pass filters and charge amplifiers before reaching the data acquisition systems. A PC-based Tektronix four channel spectrum analyzer and another PC-based data acquisition system using LabTech Notebook software were used to simultaneously record the signals. The results presented here are based on the data acquired using the Tektronix system. A small DC motor was mounted inches below the top bearing and was powered by a 3V DC battery pack in series with a 25 Ohm potentiometer. The motor speed could thus be adjusted by adjusting the potentiometer. An eccentric weight mounted on the motor shaft caused a sinusoidal oscillating force to be generated in the plane normal to the motor shaft. By arranging the plane normal to the motor shaft to be parallel to the longitudinal axis of the test cylinder, an approximation to a point harmonic excitation could be applied to the cylinder at the point of the motor mount. Consider the system shown in Figure 3. A mass, m, rotating at a frequency, ω, mounted on a rigid base is shown. M n is the modal mass of the cable, motor, mount and eccentric mass for mode n. The centrifugal force due to m is meω 2, where e is the eccentricity of the rotating mass. The force has a vertical component and a horizontal component as shown. For the tests conducted, the vertical excitation was arranged to act along the axis of the flexible cylinder tested, where it could only excite longitudinal vibration. Since the longitudinal vibration frequencies were well outside the range of frequencies excited by the motor, longitudinal vibration response was low. Two different motors and rotating masses were used during the tests. For the ABS tube tests, the total mass of the motor, mount and rotating mass was slugs ( grams). The rotating mass alone had a mass of slugs (30.8 grams). The eccentricity of the rotating mass was 0.33 inch. For the steel wire tests the rotating mass had a mass of slugs (40.6grams) and the total mass of the motor, mount and rotating mass was slugs (185.6 grams). The eccentricity, e, of the rotating mass was inch. If the angular position of the mass is measured from a plane parallel to the base, the magnitude of the excitation force, F e, is given as follows [19]. F e (t) = me ω 2 sin(ωt) (1) where m is the mass of the eccentric weight, e is the eccentricity and ω is the circular frequency of rotation. In modal terms, for the motor mounted on a cable, at a location x = x m the equation of motion for the nth mode of vibration is: M n q... (t) + R n q (t) + K n q n (t) = me ω 2 sin(ωt)y n (x m ) (2) n n where M n is the modal mass and is given by: L M n = Y () x 2 n 0 [ρ s + ( M + m) δ ( x - x m )] dx = M n,cable + ( M + m) 2 Y n ( x m ) (3) M + m is the mass of the motor, mount and rotating mass at location x = x m on the cable. ρ s is the mass per unit length of the cable including the added mass. Y n (x m ) is the mode shape at the motor location. 2 Copyright 2004 by ASME

3 R n is the modal damping and is given by, K n is the modal stiffness and is given by, L R n = Y () x 2 0 n K n = - L R(x)dx (4) T Y n (x)y n (x) dx (5) 0 The cable modal mass M n,cable is the dominant part of the modal mass M n, of the combined motor and cable system. The effect of motion of the center of gravity of the motor and cable combination is thus included in the response calculation from Equation 2. However, an exact measure of the exciting force is not required for damping measurements by the methods described below. By adjusting the frequency of rotation of the motor, particular modes of the cylinder could be excited, typically ranging from the second to the fifth. First, the damping of the cylinder was measured in air by means of pluck tests, impulse tests and forced decay tests. In a forced decay test, the cylinder is excited in a particular mode by the rotating mass excitation, and while it is oscillating at a steady state, the motor is stopped. The resulting decaying motion of the cylinder contains only one frequency component. This method was found to be the most reliable for damping measurement. An average damping was then determined from the resulting logarithmic decrement derived from the decaying signals recorded by the biaxial accelerometers. The total resultant acceleration, obtained by appropriately adding the two components recorded by the in-line and cross flow accelerometers, was used to estimate the damping. The tank was then filled with water to a depth of 12 ft. and the damping of the cylinder vibrating in still water was estimated using the same methods as for the in air tests. It should be noted that onlv 12 ft. of the 13 ft. long cylinder was immersed in water. The remaining 1 ft. of the cylinder was in air, thus enabling the motor to be mounted above water and pluck tests also to be performed. The forced decay tests in air and in still water were performed with the excitation applied parallel to the length of the tank and normal to the length of the tank to verify symmetry of the structural damping. The difference in damping in both directions was within 5% of each other and hence an average damping was used. Amplitude dependence of the still water and in air damping was not investigated. This was thought to be acceptable because the literature indicates that for amplitudes less than 0.4 diameters, the still fluid damping is relatively amplitude independent. The modal response amplitudes for the tests conducted were under 0.2 diameters. The steady state amplitudes of oscillation for the chosen mode of the cylinder were recorded for the in air tests and still water tests. With the eccentric weight rotating at the RPM corresponding to the chosen modal frequency of the cylinder, the flow was started. Measurements of the steady state acceleration response in uniform flow were done at regular intervals of flow velocity between 0.5 and 4 ft./sec. Separate sets of tests were done at the same flow velocities with the motor exciting the cylinder in-line and cross flow. The acceleration response was also measured with only flow excitation and no motor excitation. From the acceleration spectra, the RMS amplitudes of acceleration centered on each modal frequency could be estimated. By subtracting the mean square acceleration due to the flow at the desired modal frequency from the mean square acceleration due to the flow plus the motor excitation, the RMS acceleration due to the motor excitation could be determined for each flow velocity. The mean square acceleration for a small frequency band (+ or - 5% of the peak frequency) around the peak frequency excited was used in computing the mean square acceleration. This was done in the in-line and cross flow directions. The quality factor, or dynamic amplification, Q, for a modal response is given by: d 1 Q = ---- = (6) δ st 2 ζ n where, d is the modal dynamic displacement and δ st is the static displacement for α given modal excitation. ζ n is the modal damping ratio for mode n. Equation 6 is valid for damping ratios of below 0.05 and gives only a small error for damping ratios of up to 0.20, which is the range covered by the present tests. Equation 6 implies that the displacement is inversely proportional to the damping for small damping ratios. In these tests, the cylinder with flow was driven at a constant amplitude by the eccentric weight. At various flow speeds, it was assumed that the parameter which changed most significantly was the fluid damping. Hence, any change in the displacement caused by the motor was due to the change in the fluid damping. Since the still water damping and RMS response with the eccentric weight were measured earlier from the same excitation to the cylinder in a flow, the resulting RMS response was assumed to be inverselv proportional to the damping ratio. Considering the equation of motion of the system given by equation 2, the response amplitude, q n, is given by: M n q n r = (7) me Y n (x m ) [(1 r 2 ) 2 + (2ζ n r) 2 ] ½ 3 Copyright 2004 by ASME

4 where, r = ω / ω n, ζ n is the modal damping ratio, ω is the vibration frequency and ω n is the modal natural frequency of the system. At resonance, r = 1 and equation 7 reduces to, M n q n = (8) me Y n (x m ) 2ζ n The modal displacement at the accelerometer location, x = x a, is: X(x = x a ) = Y n (x a ) q n (9) Hence, m e Y n (x a ) Y n (x m ) 1 X(x = x a ) = (10) M n 2ζ n Let X sw, be the response amplitude in still water and X v be the response amplitude with flow velocity > 0, at the accelerometer location x = x a. m e Y n (x a ) Y n (x m ) 1 X sw = (11) M n 2ζ sw m e Y n (x a ) Y n (x m ) 1 X v = (12) M n 2ζ v Therefore, at a given f l o w velocity the modal fluid damping ratio, ζ v is given by X sw d sw ζ v = ζ sw ---- = ζ sw (13) X v d v where ζ is the modal damping ratio and d is the RMS displacement. The suffixes sw and v indicate still water and with a flow velocity, respectively. The displacement is computed from the acceleration by assuming the response is sinusoidal with the natural frequency of the mode. An RMS value of the acceleration computed from the spectrum is used below. Then, a v d v = ---- (14) ω 2 where a v, is the RMS acceleration at the frequency ω. Equation 13 makes the assumption that the damping is inversely proportional to the response for constant excitation at a modal resonance, as discussed earlier. The fluid added mass, which is frequency and flow velocity dependent, is unaccounted for in the above analysis. The result of a change in the added mass is a change in the natural frequency of the cylinder. Fortunately, however, the DC motor tends to synchronize naturally with the resonant frequency. Since, the frequency could be accurately estimated from the spectrum, and changed with flow velocity, the added mass effect was implicitly accounted for. However, the frequency of rotation of the nιotor was not exactly the same for the still water case as with flow. This resulted in the need for a correction factor to be applied to the damping ratio to account for a change in the magnitude of the excitation force (since this was dependent on the motor RPM). This correction was: 4 Copyright 2004 by ASME

5 2 ω sw corr = (15) 2 ω v where ω is the frequency and v and sw are suffixes for flow velocity and still water respectively. The damping ratio with flow is multiplied by the above correction factor. Reduced velocity, Vr, is defined as: V Vr = (16) f e D where V is the flow velocity, f e is the excitation frequency and D is the diameter. From the above analysis, the modal damping ratio in the in-line and cross flow direction was determined as a function of reduced velocity. Tests were done on two separate cylinders - an AΒS tube inch in diameter and a steel wire inch in diameter. In both cases, the flow velocity ranged from 0.5 to 4 ft./sec. The biaxial acclerometers were mounted within the cable for the AΒS tube and on the surface of the steel wire. The AΒS tube was excited in the fifth mode in water, at approximately 23 Hz, using the motor. The tension was maintained at 100 lbf at the top. For the steel wire the tension was maintained constant at 200 lbf and the second mode was excited in water using the motor at approximately 14.2 Hz. Tests were done at a finer grid of velocities for the steel wire than for the AΒS tube. The resulting maximum reduced velocity for the AΒS tube was approximately 3.5 and about 12 for the steel wire for both in-line and cross flow damping results. The results of the damping measurements are discussed in the next section. PRESENTATION OF THE RESULTS The still water, cross flow and in-line damping results are discussed below. The intent of the paper is to focus only on the experimental methodology and results derived from the tests discussed above. Comparisons with a proposed damping model and other experimental results are presented in [1]. Still water damping The in-air damping wasdominated by the structural damping and was measured by decay tests. Still water damping measurements are averages of measurements in two orthogonal directions. The still water total damping includes fluid and structural damping (approximated here by the in-air damping). The predicted values in the following table are from a still water damping model from [1]. Table I Measured and predicted modal damping ratio, ζ, in still water. Flow condition Damping ABS Tube Steel wire In air Measured Still water Measured, total Still water Predicted, total Table I shows the measured still water damping for the ABS tube and the steel wire. Amplitude to diameter ratio, A/D was 0.1 for the ABS tube, 5th mode, and 0.13 for the steel wire, 2nd mode. The structural damping was approximated to be the same as the in air damping which was measured by a forced decay test. The predictions of still water damping are made using a model for the fluid damping developed in [1] with the measured structural (in-air) damping added to it. In the comparison of the modal damping ratios in Table I the flow was assumed to act over the whole 13 feet of the cylinders tested. The resulting error for the damping ratio in making this assumption was less than 0.2% of the value of the damping ratio for the case of the ABS tube and less than 0.05% of the value of the damping ratio for the steel wire. 5 Copyright 2004 by ASME

6 In-line damping The in-line measured damping results for the ABS tube are shown in Figure 4. The experimental results were the average of measurements from the top and bottom accelerometer pairs. Figure 6 shows the experimental results from the steel wire in-line damping tests. The damping at zero reduced velocity is the same as the still water damping results presented earlier. Discrete results for the tests are presented in the plots as opposed to an average shown in Table I. The in-line damping ratios for the ABS tube and the steel wire range from about 1% to 3% for reduced velocities below 3 and appear to linearly increase with reduced velocity in this range. The steel wire tests show results for a larger range of reduced velocities. The in-line damping ratios show two peaks at reduced velocities of 4 and around 6. Further tests are necessary to understand these results. In this range of reduced velocities, the in-line damping ratio is about 2.5% and also appears higher than the cross flow damping ratio. The damping ratios in the reduced velocity range of 6 to 9 are lower and in the order of 1%. For reduced velocities above 3, the ABS tube results show significant scatter. Similarly, the steel wire results show significant scatter for reduced velocities above 10. This is due to the measured value of the displacement, d, in Equations 13 and 14 being small for high damping. Hence, small fluctuations in the displacement result in large fluctuations in the damping, since the damping is inversely proportional to the displacement as shown by Equation 6. Cross flow damping Figures 5 shows the cross flow damping for the ABS tube. The data was the average of measurements from the top and bottom accelerometer pairs. The measured steel wire modal damping ratios are shown in Figure 7. The measured damping in the Shell WRC tests showed considerable experimental scatter at the higher reduced velocities, as with the in-line damping measurements. Damping results for zero reduced velocity represent still water damping results, also as discussed earlier. The damping ratios for the ABS tube from cross flow and in-line excitation show very similar behavior up to reduced velocities below 3. This behavior is observed by Chen et al [15] also. The damping ratio for the ABS tube ranges from 1% to 3% and shows a linear increase with reduced velocity. The cross flow and in-line damping ratios for the steel wire also show similar values for low reduced velocities up to about 3. In general, they are lower than the ABS tube results and are about 1% in this range of reduced velocities. The cross flow damping results for the steel wire stay around 1% in the reduced velocity range between 3 and 6. They do not show the higher peaks as with the in-line results. Around a reduced velocity of 6, the damping ratio shows a marked drop to about 0.7% or lower. For reduced velocities above 8, the damping ratio shows significant increase with reduced velocity and shows values of 6% or higher for reduced velocities around 12. But, the scatter discussed earlier makes it difficult to draw definite conclusions based on the data. SUMMARY AND CONCLUSIONS A survey of available results related to fluid damping on flexible cylinders showed limited available results [1]. A methodology for deriving damping data from forced oscillation tests of a flexible cylinder in a flow is described. Measurements of damping of two flexible cylinders were made in air, in still water and in uniform flow. Based on this data, results are presented here for cross flow and in-line damping for a range of reduced velocities. Because of scatter in the data at high reduced velocities, data from rigid cylinder tests and other sources had to be used to build the appropriate damping models in [1]. However more data on flexible cylinders is needed to validate the in-line high reduced velocity damping model. The data presented here is for a Reynolds number of approximately 5000 to and for smooth cylinders. The effects of Reynolds number and roughness need to be quantified further with experiments. Studies similar to Bearman's [20] on the effects of roughness on drag coefficients in oscillatory flow are needed to understand the effect of roughness on damping ratios in the cross flow and in-line directions. An interim approach may be to model roughness as an effective increase in Reynolds number. The data indicates that the damping ratio in both the in-line and cross flow directions is linearly related to the flow velocity for low reduced velocities (below 3). The still water damping (corresponding to vibration in still water) has an amplitude dependence and is well supported by experimental results. The amplitude dependence in the low reduced velocity model proposed in [1] derives from the use of the still water damping as the limiting zero reduced velocity fluid damping. The test results presented here do not explore the amplitude dependence explictly. The damping at lock-in, can be treated as coming purely from the structural damping, except in the case of a sheared flow. In a sheared flow, when multiple modes are excited but only one is dominant, the non-excited modes will have a damping which may be assumed to be due to the corresponding low reduced velocity model. This will possibly overestimate the damping. By assuming the damping at lock-in to be due to the structural damping, the fluid damping will be underestimated. There will be a dynamic component arising from the fact that the fluid excites the cylinder more at the anti-nodes and less at the nodes and hence there are some points on the cylinder where the fluid force opposes the motion. There is a range of reduced velocity from 4 to 10 where the oscillatory drag coefficient shows considerable dependence on the amplitude of oscillation and reduced velocity. The results presented here are not adequate in this range. With these limitations, the data presented here was used, in part, to develop a damping model which is directly useful for building a good prediction model for the response of flexible cylinders in sheared flows. The response prediction model is 6 Copyright 2004 by ASME

7 implemented in a program SHEAR7 which is described in [1]. Response predictions using this model are discussed in [1] with comparisons to response measurements. The damping model presented in [1] is also validated by Vikestad et al [2,] who conducted extensive damping measurements using a different methodology. Vikestad et al found the damping model in [1], developed from the results presented here, to be conservative. A methodology is presented here to measure fluid damping on a flexible cylinder using forced excitation. Some results for cross flow and in-line fluid damping of a flexible cylinder vibrating in a flow are presented. These results were useful in developing a 3 dimensional response prediction model for a flexible cylinder vibrating in a flow. Additional measurements are needed to validate the higher reduced velocity results presented here. A closer investigation of the in-line damping in the reduced velocity range of 3 to 6 would also be useful. Dependence of the damping ratio on amplitude of vibration, Reynolds number and possibly roughness, needs to be explored further ACKNOWLEDGEMENTS The authors wish to thank the Shell Development Company and Dr.Don Allen for providing the tank time at Shell's Westhollow Research Center. We also wish to thank Joe Haws of Shell for the technical support in carrying out the experiments. The data presented in this paper come from the experiments conducted at the Westhollow Research Center. REFERENCES [1] Venugopal, M., 1996, Damping and response prediction of a flexible cylinder in a current, Ph.D. dissertation, Massachusetts Institute of Technology. [2] Vikestad, K., Larsen, C.M., and Vandiver, J.K., 2000, Norwegian Deepwater Program: Damping of Vortex Induced Vibrations, OTC Offshore Technology Conference [3] Chen, S.S., Wambsgnass, M.W.and Jendrzecjczyk, J.A., 1976, Added mass and damping of a vibrating rod in confined viscous fluids. ASME Journal of Applied Mechanics, 43 pp [4] Chen, S.S., Wambsgnass, M.W., and Jendrzecjczyk, J.A., 1979, Dynamic response of circular tubes subjected to liquid cross flow, Argonne National Laboratory Technical Memorandum ANL-CT [5] Chen, S.S., Zhu, S. Cai, Y., 1995, An unsteady flow theory for vortex-induced vibration, Journal of Sound and Vibration, 184 (1), pp [6] Skop, R.A., Ramberg, S.E. and Ferrer, K.M., 1976, Added Mass and Damping forces on circular cylinders, Naval Research Laboratory Report 7970, Washington D.C. [7] Sarpkaya, T., 1979, Vortex-Induced Oscillations: A selective review, Journal of Applied Mechanics, 46, n. 2. [8] Vandiver, J.K and Chung, T.Y., 1998, Predicted and measured response of flexible cylinders in sheared flow. Proc. ASME Winter Annual Meeting, Symposium on Flow induced Vibrations December [9] Gopalkrishnan, R., 1993, Vortex-Induced Forces on oscillating bluff cylinders, D.Sc. thesis in Oceanographic Engineering, Massachusetts Institute of Technology and Woods Hole Oceanographic Institution. [10] Triantafyllou, M.S., Gopalkrishnan, R. and Grosenbaugh, M.A., 1994, Vortex-induced vibrations in a sheared flow: A new predictive method. Hydroelasticity in Marine Technology, Faltinsen et al.(eds). [11] Moe, G. and Verley, R.L.P.,1980, Hydrodynamic damping of offshore structures in waves and currents, Offshore Technology Conference paper OTC 3798, pp [12] Verley, R.L.P. and Moe, G., 1978, An investigation into the hydrodynamic damping of cylinders oscillated in steady currents of various velocities, SINTEF Report No. STF60 A [13] Griffin, O.M., and Koopmann, G.H., 1977, The vortex-excited lift and reaction forces on resonantly vibrating cylinders, Journal of Sound and Vibration 54 (3) pp [14] Mercier, J., 1980, Large amplitude oscillations of a circular cylinder in a low-speed stream, Ph.D thesis, Stevens Institute of Technology. [15] Staubli, T., 1983, Calculation of the vibration of an elastically mounted cylinder using experimental data from forced oscillation, Journal of Fluids Engineering 105, pp [16] King, R., 1977, A review of vortex shedding research and its application, Ocean Engineering 4, pp [17] Ibanez, P., Spencer, R.B., and Smith, C.B., 1973, Forced vibration tests on electrical distribution equipment, Nuclear Engineering and Design 25 pp [18] Smith, C.B., and Mathiesen, R.B., 1973, Vibration testing of full-scale structures, Nuclear Engineering and Design 25 pp [19] Rao, S.S., 1986, Mechanical Vibrations Addison-Wesley. [20] Bearman, P.W., 1992, Fluid loading of cylinders with application to risers: Results from model tests, Proc. NSF Workshop on Riser Dynamics September 29, 30 and October 1, 1992, University of Michigan, Ann Arbor, MI. 7 Copyright 2004 by ASME

8 Figure 1 Schematic of current tank at Westhollow Research Center 8 Copyright 2004 by ASME

9 Figure 2 Schematic of test setup for damping measurements 9 Copyright 2004 by ASME

10 Figure 3 Schematic of forced excitation mechanism 10 Copyright 2004 by ASME

11 Figure 4 Modal damping ratio inferred from in-line excitation for ABS tube 11 Copyright 2004 by ASME

12 Figure 5 Modal damping ratio inferred from cross flow excitation for ABS tube 12 Copyright 2004 by ASME

13 Figure 6 Modal damping ratio inferred from in-line excitation for steel wire 13 Copyright 2004 by ASME

14 Figure 7 Modal damping ratio inferred from cross flow excitation for steel wire 14 Copyright 2004 by ASME