Research Papers: Offshore Technology

Modeling of Response Amplitude Operator for Coupled Sway, Roll and Yaw Motions of a Floating Body in Sinusoidal Waves Using Frequency Based Analysis

[+] Author and Article Information
Samir K. Das

Department of Applied Mathematics,
Defence Institute of Advanced Technology,
Girinagar, Pune 411025, India
e-mails: samirkdas@diat.ac.in;

Masoud Baghfalaki

Department of Mathematics,
Kermanshah Branch,
Islamic Azad University,
Kermanshah 6718997551, Iran
e-mail: masoudbaghfalaki@yahoo.com

1Corresponding author.

2Formerly, Senior Research Officer, Mathematical Modelling and Coastal Engineering Centre, Central Water and Power Research Station, Khadakwasla, Pune-411024, India.

Contributed by the Ocean, Offshore, and Arctic Engineering Division of ASME for publication in the JOURNAL OF OFFSHORE MECHANICS AND ARCTIC ENGINEERING. Manuscript received November 25, 2013; final manuscript received March 3, 2015; published online April 8, 2015. Assoc. Editor: M. H. (Moo-Hyun) Kim.

J. Offshore Mech. Arct. Eng 137(3), 031303 (Jun 01, 2015) (18 pages) Paper No: OMAE-13-1111; doi: 10.1115/1.4030019 History: Received November 25, 2013; Revised March 03, 2015; Online April 08, 2015

The paper investigates the characteristics of response amplitude operators (RAO) or transfer function of a floating body in frequency domain for coupled sway, roll and yaw motions in sinusoidal waves. The floating body is considered to be initially at rest and waves act as beam to the floating body with varying frequency (ω) between 0.3 rad/s and 1.2 rad/s. The hydrodynamic coefficients (HC) are computed using strip theory formulation and the general expression of RAO is derived. The behavior of RAO under coupled conditions is examined by considering two asymptotic cases, corresponding to ω0 and ω. For the intermediate frequency range, analytical expression for system frequency is derived. The effects of viscous damping for uncoupled and coupled transfers have been compared with the result of nonviscous case. A mathematical analogy with respect to Mathieu and Hill equations has been established using frequency based classifications of governing equations. This modeling approach can provide useful guidelines to determine RAO for coupled motions and computing of wave loads and sensitivity analysis with respect to initial conditions of a floating body for the wide range of frequencies.

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Weinblum, G. P., and St. Denis, M., 1950, “On the Motions of Ships at Sea,” Soc. Nav. Archit. Mar. Eng., Trans., 58, pp. 184–231.
Cartwright, D. E., and Rydill, L. J., 1957, “The Rolling and Pitching of a Ship at Sea. A Direct Comparison Between Calculated and Recorded Motions of a Ship in Sea Waves,” Trans. R. Inst. Nav. Archit., 99(1), pp. 100–135.
Cummins, W. E., 1962, “The Impulse Response Function and Ship Motions,” Schiffstechnik, 9, pp. 101–109.
St. Denis, M., and Pierson, W. J., 1953, “On the Motion of Ships in Confused Seas,” Soc. Nav. Archit. Mar. Eng., Trans., 61, pp. 280–354.
Frank, W., and Salvesen, N., 1970, “The Frank Close-Fit Ship-Motion Computer Program,” NSRDC, Washington, DC, Report No. 3289.
Faltinsen, O. M., 1990, Sea Loads on Ship and Offshore Structures, Cambridge University Press, Cambridge, London.
Faltinsen, O. M., Newman, J. N., and Vinje, T., 1995, “Nonlinear Wave Loads on a Slender Vertical Cylinder,” J. Fluid Mech., 289, pp. 179–198. [CrossRef]
Holappa, K. W., and Falzarano, J. M., 1998, “Application of Extended State Space to Nonlinear Ship Rolling,” Ocean Eng., 26(3), pp. 227–240. [CrossRef]
Jacobsen, K., and Clauss, G. F., 2006, “Time-Domain Simulations of Multi-Body Systems in Deterministic Wave Trains,” ASME Paper No. OMAE2006-92348. [CrossRef]
Korvin-Kroukovsky, B. V., 1955, “Investigation of Ship Motions in Regular Waves,” Soc. Nav. Archit. Mar. Eng., Trans., 63, pp. 386–435.
Korvin-Kroukovsky, B. V., and Lewis, E. V., 1955, “Ship Motions in Regular and Irregular Seas,” Int. Shipbuild. Prog., 2, pp. 81–95.
Kukkanen, T., 2010, “Wave Load Predictions for Marine Structures,” J. Struct. Mech. (Rakenteiden Mekaniikka), 43(3), pp. 150–166.
Kukkanen, T., and Mikkola, T. P. J., 2004, “Fatigue Assessment by Spectral Approach for the ISSC Comparative Study of the Hatch Cover Bearing Pad,” Mar. Struct., 17(1), pp. 75–90. [CrossRef]
Lewis, E. V., and Numata, E., 1957, “Ship Model Test in Regular and Irregular Seas,” Stevens Institute of Technology, Hoboken, NJ, Experimental Towing Tank Report No. 656.
Taylan, M., 2000, “The Effect of Nonlinear Damping and Restoring in Ship Rolling,” Ocean Eng., 27(9), pp. 921–932. [CrossRef]
Price, W. C., and Bishop, R. E. D., 1974, Probabilistic Theory of Ship Dynamics, Chapman and Hall, London.
Taylan, M., 1999, “Solution of the Nonlinear Roll Model by a Generalized Asymptotic Method,” Ocean Eng., 26(11), pp. 1169–1181. [CrossRef]
Surendran, S., and Reddy, J. R., 2002, “Roll Dynamics of a Ro–Ro Ship,” Int. Shipbuild. Prog., 49(4), pp. 301–320.
Surendran, S., and Reddy, J. V. R., 2003, “Numerical Simulation of Ship Stability for Dynamic Environment,” Ocean Eng., 30(10), pp. 1305–1317. [CrossRef]
Newman, J. N., 1962, “The Exciting Forces on Fixed Bodies in Waves,” J. Ship Res., 6(3), pp. 10–18
Newman, J. N., 1977, Marine Hydrodynamics, MIT Press, Cambridge, MA.
Newman, J. N., 1990, “The Quest for a Three-Dimensional Theory of Ship-Wave Interactions,” Philos. Trans. R. Soc. London, Ser. A, 334(1634), pp. 213–227. [CrossRef]
Clauss, G., Lehmann, E., and Östergaard, C., 1992, Offshore Structures, Volume I: Conceptual Design and Hydrodynamics, Springer-Verlag, London.
Clauss, G. F., Klein, M., Sprenger, F., and Testa, D., 2010, “Evaluation of Critical Conditions in Offshore Vessel Operation by Response Based Optimization Procedures,” ASME Paper No. OMAE2010-21071. [CrossRef]
Cotton, B., and Spyrou, K. J., 2001, “An Experimental Study of Nonlinear Behaviour in Roll and Capsize,” Int. Shipbuild. Prog., 48(1), pp. 5–18.
Low, M., and Grime, A. J., 2011, “Extreme Response Analysis of Floating Structures Using Coupled Frequency Domain Analysis,” ASME J. Offshore Mech. Arct. Eng., 133(4), p. 031601. [CrossRef]
Das, S. K., Sahoo, P. K., and Das, S. N., 2006, “Determination of Roll Motion for a Floating Body in Regular Waves,” Proc. Inst. Mech. Eng., Part M, 220(1), pp. 41–48. [CrossRef]
Salvesen, N., Tuck, E. O., and Faltinsen, O. M., 1970, “Ship Motions and Sea Loads,” Soc. Nav. Archit. Mar. Eng., Trans., 78, pp. 250–287.
Tick, L. J., 1959, “Differential Equations With Frequency-Dependent Coefficients,” J. Ship Res., 3(2), pp. 45–46.
Vugts, J. H., 1968, “The Hydrodynamic Coefficients for Swaying, Heaving and Rolling Cylinders in a Free Surface,” Shipbuilding Laboratory, Delft University of Technology, Delft, The Netherlands, Report No. 194.
Das, S. K., Das, S. N., and Sahoo, P. K., 2008, “Investigation of Coupled Sway, Roll and Yaw Motions of a Floating Body: Numerical Modelling for Non-Linear Roll Restoring,” Ships Offshore Struct., 3(1), pp. 49–56. [CrossRef]
Das, S. N., Shiraishi, S., and Das, S. K., 2010, “Mathematical Modeling of Sway, Roll and Yaw Motions: Order-Wise Analysis to Determine Coupled Characteristics and Numerical Simulation for Restoring Moment's Sensitivity Analysis,” Acta Mech., 213(3–4), pp. 305–322. [CrossRef]
Baghfalaki, M., Das, S. K., and Das, S. N., 2012, “Analytical Model to Determine Response Amplitude Operator of a Floating Body for Coupled Roll and Yaw Motions and Frequency Based Analysis,” World Scientific4(4), pp. 1–20. [CrossRef]
Das, S. K., and Baghfalaki, M., 2014, “Mathematical Modelling of Response Amplitude Operator for Roll Motion of a Floating Body: Analysis in Frequency Domain With Numerical Validation,” J. Mar. Sci. Appl., 13, pp. 143–157. [CrossRef]
Das, S. K., and Das, S. N., 2006, “Modelling and Analysis of Coupled Nonlinear Oscillations of Floating Body in Two Degrees of Freedom,” Acta Mech., 181(1–2), pp. 31–42. [CrossRef]
Nayfeh, A. H., and Mook, D. T., 1979, Nonlinear Oscillations, Wiley, New York.
Das, S. N., and Das, S. K., 2005, “Mathematical Model for Coupled Roll and Yaw Motions of a Floating Body in Regular Waves Under Resonant and Non-Resonant Conditions,” Appl. Math. Modell., 29(1), pp. 19–34. [CrossRef]


Grahic Jump Location
Fig. 1

Schematic diagram of a floating body with sign convention

Grahic Jump Location
Fig. 2

Schematic diagram of strips of a floating body

Grahic Jump Location
Fig. 3

Comparison of sway, roll and yaw exciting force/moments at time t = 7.85 (s) and at frequency, w = 0.8 (rad/s)

Grahic Jump Location
Fig. 4

Real part, imaginary part, and norm of transfer functions for uncoupled (1DOF) and coupled (3DOF) sway, roll and yaw motions

Grahic Jump Location
Fig. 5

Roll amplitude for case C with and without damping: (a) w = 0.3 (IC-1), (b) w = 0.56 (IC-1), (c) w = 0.74 (IC-1), (d) w = 1.2 (IC-1) (linear damping), (e) w = 0.74 (IC-1), and (f) w = 1.2 (IC-1) (unbounded damping)

Grahic Jump Location
Fig. 6

Comparison of roll amplitude of case C with the full form and approximation of Hill and Mathieu equations: (a) w = 0.3 (IC-1), (b) w = 0.56 (IC-1), (c) w = 0.74 (IC-1), (d) w = 1.2 (IC-1), (e) w = 0.56 (IC-2) Eq. (73), and (f) w = 0.74 (IC-2) Eq. (76)

Grahic Jump Location
Fig. 7

Comparison plots of ɛ in δ–t plane indicating parametric excitation

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Fig. 8

Plots of X·4 with time for the frequencies ω = 0.56 (where b44 = 0) and 0.74 Eq. (80)

Grahic Jump Location
Fig. 9

Sway and yaw amplitude for case C new: (a) w = 0.3 (IC-1), (b) w = 0.30 (IC-1), (c) w = 0.56 (IC-1), (d) w = 56 (IC-1), (e) w = 0.74 (IC-1), and (f) w = 0.74 (IC-1)




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