Abstract
A nonlinear model for simulating the heave-roll motions of ships in following waves is presented. The parametric excitation is modeled by a Hill's type equation, instead of the conventional Mathieu's equation. The model includes not only the linear but also the quadratic coupling term. Instability conditions for parametrically excited rolling motions are derived using the harmonic balance method. The results are verified by numerical analyses. The effects of including the quadratic coupling term on the instability conditions and nonlinear responses are studied. The complex dynamic behaviour of the coupled system in the various instability regions is also investigated. Bifurcations of the flip, fold and pitchfork types are observed in the Poincaré mapping of the numerically simulated responses. Chaotic motions leading to boundary crises and inevitable capsize are also reported.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blocki, W., ‘Ship safety in connection with parametric resonance of the roll’, International Shipbuilding Progress 27, 1980, 36–53.
Chantral, J. M. and Berhault, C., ‘Parametric resonance of C.A.L.M. buoy motions in regular waves’, Behaviour of Offshore Structures 1, 1985, 269–280.
de Kat, Jan, O. and Paulling, J. R., ‘The simulation of ship motions and capsizing in severe seas’, SNAME Transactions 97, 1989, 139–168.
Grebogi, C., Ott, E., and Yorke, J., ‘Crises, sudden changes in chaotic attractors and transient chaos’, Physica D 7, 1983, 181–200.
Hsu, C. S., ‘On nonlinear parametric excitation problems’, Advanced Applied Mechanics 17, 1977, 245–301.
Hsu, C. S., Cell-to-Cell Mapping: A Method of Global Analysis for Nonlinear Systems, Springer-Verlag, New York, 1987.
Jordan, D. W. and Smith, P., Nonlinear Ordinary Differential Equations, Clarendon Press, Oxford, 1977.
Liaw, C. Y., Bishop, S. R., and Thompson, J. M. T., ‘Heave-excited rolling motion of a rectangular vessel in head seas’, International Journal of Offshore and Polar Engineering 3, 1993, 26–31.
Mei, C. C., The Applied Dynamics of Ocean Surface Waves, World Scientific, Singapore, 1989.
Nayfeh, A. H., Mook, D. T., and Marshall, L. R., ‘Nonlinear coupling of pitch and roll modes in ship motion’, Journal of Hydronautics 7, 1973, 145–152.
Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, J. Wiley, New York, 1979.
Nayfeh, A. H., ‘On the undesirable roll characteristics of ships in regular seas’, Journal of Ship Research 32, (2), 1988, 92–100.
Newman, J. N., Marine Hydrodynamics, The MIT Press, Cambridge, MA, 1977.
Parker, T. S. and Chua, L. O., Practical Numerical Algorithms for Chaotic Systems, Springer-Verlag, New York, 1989.
Paulling, J. R. and Rosenberg, R. M., ‘On unstable ship motions resulting from nonlinear coupling’, Journal of Ship Research 3, 1959, 36–46.
Sanchez, N. E. and Nayfeh, A. H., ‘Nonlinear rolling motions of ships in longitudinal waves’, International Shipbuilding Progress 37, 1990, 247–272.
Thompson, J. M. T. and Stewart, H. B., Nonlinear Dynamics and Chaos, J. Wiley, Chichester, 1989.
Tondl, A. and Nabergoj, R., ‘Model simulation of parametrically excited ship rolling’, Nonlinear Dynamics 1, 1990, 131–141.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Liaw, C.Y., Bishop, S.R. Nonlinear heave-roll coupling and ship rolling. Nonlinear Dyn 8, 197–211 (1995). https://doi.org/10.1007/BF00045774
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00045774