Research Papers

Two-Dimensional Nonlinear Analysis of an Untethered Spherical Buoy Due to Wave Loading

[+] Author and Article Information
Zach Ballard

e-mail: zcb@duke.edu

Brian P. Mann

e-mail: bpm4@duke.edu
Dynamical Systems Laboratory,
Department of Mechanical Engineering and Materials Science,
Duke University,
Durham, NC 27705

Contributed by the Design Engineering Division of ASME for publication in the Journal of Computational and Nonlinear Dynamics. Manuscript received September 22, 2011; final manuscript received June 23, 2013; published online July 29, 2013. Assoc. Editor: Hiroshi Yabuno.

J. Comput. Nonlinear Dynam 8(4), 041019 (Jul 29, 2013) (12 pages) Paper No: CND-11-1157; doi: 10.1115/1.4024887 History: Received September 22, 2011; Revised June 23, 2013

The horizontal and vertical motions of a nonlinear spherical buoy, excited by synthetic ocean waves within a wave flume, is numerically and experimentally investigated. First, fluid motion in the wave tank is described using Airy's theory, and the forces on the buoy are determined using a modified form of Morison's equation. The system is then studied statically in order to determine the effects of varying system parameters. Numerical simulations then use the governing equations to compare predicted motions with experimentally observed behavior. Additionally, a commonly used linear formulation is shown to be insufficient in predicting buoy motion, while the nonlinear formulation presented is shown to be accurate.

Copyright © 2013 by ASME
Topics: Waves , Buoys , Fluids
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Wilson, J. F., 2003, Dynamics of Offshore Structures, John Wiley, New York.
Bastien, S., and Sepe, R., 2009, “Ocean Wave Energy Harvesting for Buoy Sensors,” Energy Conversion Congress and Exposition, pp. 3718–3725.
Carpenter, E., Leonard, J., and Yim, S., 1995, “Experimental and Numerical Investigations of Tethered Spar and Sphere Buoys in Irregular Waves,” Ocean Eng., 22, pp. 765–784. [CrossRef]
Lin, H., and Yim, S. C., 2006, “Coupled Surge-Heavemotions of a Moored System. 1: Model Calibration and Parametric Study,” J. Eng. Mech., 132(6), pp. 671–680. [CrossRef]
Ogihara, K., 1980, “Theoretical Analysis on the Transverse Motion of a Buoy by a Surface Wave,” Appl. Ocean Res., 2(2), pp. 51–56. [CrossRef]
Longuet-Higgins, M., 1953, “Mass Transport in Water Waves,” Philos. Trans. R. Soc. London Ser. A, 245(903), pp. 535–581. [CrossRef]
Craik, A. D., 2004, “The Origins of Water Wave Theory,” Annu. Rev. Fluid Mech., 36, pp. 1–28. [CrossRef]
Shaughnessy, E. J., and Katz, I. M., 2005, Introduction to Fluid Mechanics, Oxford University Press, Oxford.
Wade, B., and Dwyer, M., 1978, “Application of Morison's Equation to Fixed Offshore Platforms,” J. Pet. Technol., 30(3), pp. 447–454.
Gottlieb, O., and Yim, S. C., 1997, “Nonlinear Dynamics of a Coupled Surge-Heave Small-Body Ocean Mooring System,” Ocean Eng., 24, pp. 479–495. [CrossRef]
Dean, R. G., 1991, Water Wave Mechanics for Engineers and Scientists, World Scientific, Singapore.
Worden, K., Stansby, P. K., Tomlinson, G. R., and Billings, S. A., 1994, “Identification of Nonlinear Wave Forces,” J. Fluids Struct., 8(1), pp. 19–71. [CrossRef]
Milne-Thompson, L., 1962, Theoretical Hydrodynamics, Macmillan, New York.
Wolfram, J., 1999, “On Alternative Approaches to Linearization and Morison's Equation for Wave Forces,” Proc. Math. Phys. Eng. Sci., 455(1988), pp. 2957–2974. [CrossRef]


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Fig. 4

Oscillating horizontal fluid velocity U(x, z, t) throughout finite depth wave tank using values from Table 1

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Fig. 5

Linear wave theory confirmation: vertical fluid velocity (V) at 0.5 Hz for numerical (solid black) and experimental (dashed blue)

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Fig. 6

Linear wave theory confirmation: vertical fluid velocity (V) at 0.75 Hz for numerical (solid black) and experimental (dashed blue)

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Fig. 3

Diagram of a spherical buoy in a finite depth wave tank

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Fig. 2

Diagram (side-view) of the plunging wavemaker

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Fig. 1

Photo of the wavemaker used for all experiments. Dimensions: 6.7 m L × 0.5 m H × 0.5 m D.

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Fig. 7

Dimensional relations for partially submerged buoy due to wave loading

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Fig. 12

Amplitude ratios versus buoy mass ratio, where x¯ = x-motion amplitude normalized by first mass

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Fig. 10

ωn versus buoy mass: Comparisons of the predicted natural frequency from Eq. (24) (solid) and numerical simulation (circle) using values from Table 1

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Fig. 11

Time series for both horizontal and vertical positions and velocities with mb = 0.1

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Fig. 18

Snapshot of image processing software used to determine the experimental natural frequency of the buoy, as well as the drag coefficient CD

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Fig. 8

Buoyancy force versus dimensionless depth

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Fig. 9

Projected area in the z direction versus dimensionless depth

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Fig. 13

Velocity frequency spectrum for mb = 0.1 kg (numerical)

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Fig. 14

Velocity frequency spectrum for mb = 0.62 kg (numerical)

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Fig. 15

Snapshot of “line tracking” used to get experimental wave amplitudes

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Fig. 16

Wave form: experimental (dashed) versus numerical (solid)

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Fig. 17

Snapshot of image processing software used to track buoy, including vertical position output graph

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Fig. 23

Frequency response comparison for linear versus nonlinear cases: mb = 0.88 kg

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Fig. 19

Natural frequency comparison: analytical (solid line), numerical (circle), and experimental (square)

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Fig. 20

Horizontal and vertical position comparison: experimental (dashed) versus numerical (solid) for mb = 0.1 kg

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Fig. 21

Horizontal and vertical velocity comparison: experimental (dashed) versus numerical (solid) for mb = 0.1 kg

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Fig. 22

Velocity frequency response comparison over a short time period for mb = 0.62 kg: experimental (dashed) versus numerical (solid)




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