Nonlinear Dynamics

, Volume 36, Issue 2–4, pp 135–179 | Cite as

Stability and Capsizing of Ships in Random Sea – a Survey

  • Ludwig Arnold
  • Igor Chueshov
  • Gunter Ochs


This report is a survey of methods of stochastic and nonlinear dynamics in ship stability. After a brief introduction we describe the sea as a stationary random field. We then derive the general equations of motion of a ship from ‘first principles’, specializing to the case of the equations of motion for roll, heave and sway using strip theory from which eventually the ‘archetypal’ nonlinear random differential equation for the roll motion follows. This determines in particular how and where the stochasticity of the sea enters the equation. We then analyze simple nonlinear models of ship motion by means of the theory of random dynamical systems which amounts to studying invariant measures, Lyapunov exponents, random attractors and their (random) domain of attraction and to using stochastic bifurcation theory to describe qualitative changes.

Conley index random attractor random dynamical system random field random invariant set random seaway roll motion ship capsizing ship stability stochastic bifurcation stochastic stability 


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  1. 1.
    Adler, R., The Geometry of Random Fields,Wiley, New York, 1981.Google Scholar
  2. 2.
    Arnold, L., Random Dynamical Systems, Springer-Verlag, Berlin/Heidelberg/New York, 1998.Google Scholar
  3. 3.
    Ashwin, P. and Ochs, G., 'Convergence to local random attractors', Dynamical Systems, 18, 2003, 139–158.Google Scholar
  4. 4.
    Azaïs, J.-M., León, J. R., and Ortega, J., 'Geometrical characteristics of Gaussian sea waves', Preprint, Universidad Central de Venezuela, Caracas, Venezuela, 2001.Google Scholar
  5. 5.
    Baumgarten, R., Kreuzer, E., and Wendt, M., 'Nonlinear dynamics in marine technology', European Journal of Mechanics, A/Solids 16, 1997, 25–44.Google Scholar
  6. 6.
    Blocki, W., 'Ship safety in connection with parametric resonance of the roll', International Shipbuilding Progr. 27, 1980, 36–53.Google Scholar
  7. 7.
    Chakrabarti, S. K., Hydrodynamics of Offshore Structures, Springer-Verlag, Berlin/Heidelberg, New York, 1987.Google Scholar
  8. 8.
    Clauss, G., Lehmann, E., and Östergaard, C., Offshore Structures. Vol. II: Strength and Safety for Structural Design, Springer-Verlag, Berlin/Heidelberg/New York, 1994.Google Scholar
  9. 9.
    Cramér, H. and Leadbetter, M. R., Stationary and Related Stochastic Processes,Wiley, New York, 1967.Google Scholar
  10. 10.
    Crauel, H., 'Random Dynamical Systems', Ph.D. Thesis, Bremen, Germany, 1987.Google Scholar
  11. 11.
    Cummins, W.E., 'The impulse response function and ship motions', Schiffstechnik9, 1962, 101–109.Google Scholar
  12. 12.
    Embrechts, P., Klüppelberg, C., and Mikosch, T., Modelling Extremal Events for Insurance and Finance, Springer-Verlag, Berlin/Heidelberg/New York, 1997.Google Scholar
  13. 13.
    Faltinsen, O.M., Sea Loads on Ships and Offshore Structures, Cambridge University Press, Cambridge, 1998.Google Scholar
  14. 14.
    Froude, W., 'On the rolling of ships', Transactions of the Institute of Naval Architects 11, 1861, 180–229.Google Scholar
  15. 15.
    Gayer, T., 'Bifurcations of the controlled escape equation', in Proceedings of the Fifteenth International Symposium on the Mathematical Theory of Networks and Systems, MTNS, South Bend, Indiana, 2002.Google Scholar
  16. 16.
    Lloyd, Germanischer., Vorschriften und Richtlinien des Germanischen Lloyd. Volume 1: Schiffstechnik, Teil 1, Kapitel1, Selbstverlag des Germanischen Lloyd, Hamburg, 1992.Google Scholar
  17. 17.
    Hooft, J. P., Advanced Dynamics of Marine Structures,Wiley-Interscience, New York, 1982.Google Scholar
  18. 18.
    Hsieh, S.-R., Troesch, A., and Shaw, S., 'A nonlinear probabilistic method for predicting vessel capsizing in random beam seas', Proceedings of the Royal Society of London, Series A 446, 1994, 195–211.Google Scholar
  19. 19.
    Institute of London Underwriters. Hull Casualty Statistics. IUMI Conference Paris, 1997.Google Scholar
  20. 20.
    International Maritime Organization. Code on Intact Stability for All Types of Ships Covered by IMO Instruments. Resolution A.749(18).IMO, London, 1995.Google Scholar
  21. 21.
    Jaglom, A. M., Einführung in die Theorie stationärer Zufallsfunktionen, Akademie-Verlag, Berlin, 1959.Google Scholar
  22. 22.
    Jiang, C., Troesch, A. W., and Shaw, S. W., 'Capsize criteria for ship models with memory-dependent hydrodynamics and random excitation', Philosophical Transactions of the Royal Society of London, Series A 358, 2000, 1761–1791.Google Scholar
  23. 23.
    John, F., 'On the motion of floating bodies. I', Communications on Pure and Applied Mathematics2, 1949, 13–57.Google Scholar
  24. 24.
    Kinsman, B., Wind Waves: Their Generation and Propagation on the Ocean Surface, Prentice-Hall, Englewood Cliffs, New Jersey, 1965.Google Scholar
  25. 25.
    Krée, P. and Soize, C., Mathematics of Random Phenomena, Reidel, Dordrecht, The Netherlands, 1986.Google Scholar
  26. 26.
    Kreuzer, E., Markiewicz, M., and Pick, M.-A., Streifenmethode f ür instationäre Strömung und Pfadverfolgung, Preprint, TUHH, 2001.Google Scholar
  27. 27.
    Kreuzer, E., Markiewicz, M., and Wendt, M., Zwischenbericht zum Teilprojekt 'Systematisierung von Dynamikuntersuchun-gen extremer Schiffsbewegungen', Technical report, Technische Universität Hamburg-Harburg, 2000.Google Scholar
  28. 28.
    Kreuzer, E. and Wendt, M., 'Ship capsizing analysis using advanced hydrodynamic modelling', Philosphical Transactions of the Royal Society of London, Series A 358, 2000, 1835–1851.Google Scholar
  29. 29.
    Kreuzer, E. and Wendt, M., 'Ship capsizing as a nonlinear dynamics problem', in IUTAM/IFToMM Symposium on Synthesis of Nonlinear Dynamical Systems, Kluwer, Dordrecht, The Netherlands, 2000, pp. 37–48.Google Scholar
  30. 30.
    Krylov, A. N., 'A general theory of the oscillations of a ship on waves', Transactions of the Institute of Naval Architects 40, 1898, 135.Google Scholar
  31. 31.
    Kuznetsov, N. G., 'On uniqueness and solvability in the linearized two-dimensional problem of a supercritical stream about surface-piercing body', Proceedings of the Royal Society of London, Series A 450, 1995, 233–253.Google Scholar
  32. 32.
    Kuznetsov, N. G. and Mazya, V. G., 'Unique solvability of the plane Neumann-Kelvin problem', Mathematics USSR Sb. 63, 1989, 425–446.Google Scholar
  33. 33.
    Lewis, E. V. (ed.), Principles of Naval Architecture, Society of Naval Architects and Marine Engineers, New Jersey City, New Jersey, 1989.Google Scholar
  34. 34.
    Lloyd, A. R. J. M., Seakeeping: Ship Behaviour in Rough Weather,Wiley, New York, 1989.Google Scholar
  35. 35.
    Longuet-Higgins, M. S., 'The statistical analysis of a random, moving surface', Philosophical Transactions of the Royal Society of London, Series A 249, 1957, 321–387.Google Scholar
  36. 36.
    Mischaikow, K. and Ochs, G., 'A Conley index for random homeomorphisms', Preprint, 2002.Google Scholar
  37. 37.
    Moshchuk, N. K., Ibrahim, R. A., Khasminskii, R. Z., and Chow, P. L., 'Asymptotic expansion of ship capsizing in random sea waves-I. First-order approximation', Internatinal Journal of Non-Linear Mechanics 30, 1995, 727–740.Google Scholar
  38. 38.
    Moshchuk, N. K., Ibrahim, R. A., Khasminskii, R. Z., and Chow, P. L., 'Asymptotic expansion of ship capsiz-ing in random sea waves-II. Second-order approximation', International Journal of Non-Linear Mechanics 30, 1995, 741–757.Google Scholar
  39. 39.
    MTG Marinetechnik GmbH, 'Programmsystem SIMBEL', Technical Report, MTG, 1995.Google Scholar
  40. 40.
    Neumann, G., ' Zur Charakteristik des Seeganges', Arch. Meteorol. Geophys. Bioklimatol. 7(A), 1954, 352.Google Scholar
  41. 41.
    Neves, M., Pérez, N., and Valerio, L., 'Stability of small fishing vessels in longitudinal waves,' Ocean Engineering 26, 1999, 1389–1419.Google Scholar
  42. 42.
    Ochi, M., Ocean Waves. The Stochastic Approach, Cambridge University Press, Cambridge, 1998.Google Scholar
  43. 43.
    Ochs, G., 'Random attractors: Robustness, numerics and chaotic dynamics,' in Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems,B. Fiedler (ed.), Springer-Verlag, Heidelberg, 2001, pp. 1–30.Google Scholar
  44. 44.
    Ogilvie, T. F., 'Recent progress toward the understanding and prediction of ship motions,' in Proceedings of the ONR Fifth Symposium on Naval Hydrodynamics, Bergen, Norway, 1964, pp. 3–80.Google Scholar
  45. 45.
    Oh, I. G., Nayfeh, A. H., and Mook, D. T., 'A theoretical and experimental investigation of indirectly excited roll motion in ships', Philosophical Transactions of the Royal Society of London, Series A 358, 2000, 1853–1881.Google Scholar
  46. 46.
    Pagani, C. D. and Pierotti, D., 'Exact solution of the wave resistence problem for submerged cylinder, II. The nonlinear problem', Archive for Rational Mechanics and Analysis 149, 1999, 289–327.Google Scholar
  47. 47.
    Pagani, C. D. and Pierotti, D., 'The Neumann-Kelvin problem for beam', Journal of Mathematical Analysis and Applications 240, 1999, 60–79.Google Scholar
  48. 48.
    Pierson, W. J. and Moskowitz, L., A proposed spectral form for fully developed wind seas based on the similarity theory of S. A. Kitaigorskii. Technical report U.S. Naval Oceanographic Office, 1963.Google Scholar
  49. 49.
    Price, W. G. and Bishop, R. E., Probabilistic Theory of Ship Dynamics, Chapman & Hall, London, 1974.Google Scholar
  50. 50.
    Rainey, R. C. T. and Thompson, J. M. T., 'The transient capsize diagram-a new method of quantifying stability of waves', Journal of Ship Research 35, 1991, 58–62.Google Scholar
  51. 51.
    Roberts, J. B. and Vasta, M., 'Markov modelling and stochastic identification for nonlinear ship rolling in random waves,' Philosophical Transactions of Royal Society, London, Series A 358, 2000, 1917–1941.Google Scholar
  52. 52.
    Rozanov, Y. A., Stationary Random Processes, Holden-Day, San Francisco, California, 1967.Google Scholar
  53. 53.
    Senjanović, I., Ciprić, G., and Parunov, J., 'Survival analysis of fishing vessels rolling in rough seas,' Philosophical Trans-actions of the Royal Society London, Series A 358, 2000, 1943–1965.Google Scholar
  54. 54.
    Sobczyk, K., Stochastic Differential Equations. With Applications to Physics and Engineering, Kluwer, Dordrecht, The Netherlands, 1991.Google Scholar
  55. 55.
    Soliman, M.S. and Thompson, J.M.T., 'Stochastic penetration of smooth and fractal basin boundaries under noise excitation', Dynamics and Stability of Systems5, 1990, 281–298.Google Scholar
  56. 56.
    Spyrou, K. J. and Thompson, J. M. T., 'The nonlinear dynamics of ship motions: A field overview and some recent develop-ments', Philosophical Transactions of the Royal Society London, Series A 358, 2000, 1735–1760.Google Scholar
  57. 57.
    Spyrou, K. J. and Thompson, J. M. T. (eds.), 'The nonlinear dynamics of ships', Philosophical Transactions of the Royal Society London, Series A (Theme Issue 1771) 358, 2000, 1733–1981.Google Scholar
  58. 58.
    St. Denis, M. and Pierson, W., 'On the motions of ships in confused seas,' Transactions on SNAME 61, 1953, 280–357.Google Scholar
  59. 59.
    Thompson, J. M. T., 'Designing against capsize in beam seas: Recent advances and new insights', ASME Applied Mechanical Review 50, 1997, 307–325.Google Scholar
  60. 60.
    Thompson, J. M. T., Rainey, R. C. T., and Soliman, M. S., 'Mechanics of ship capsize under direct and parametric wave excitation', Philosophical Transactions of the Royal Society London, Series A 338, 1992, 471–490.Google Scholar
  61. 61.
    Wendt, M., Zur nichtlinearen Dynamik des Kenterns intakter Schiffe im Seegang, VDI Verlag, Düsseldorf, Germany, 2000.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Ludwig Arnold
    • 1
  • Igor Chueshov
    • 2
  • Gunter Ochs
    • 1
  1. 1.Institut für Dynamische SystemeBremenGermany
  2. 2.Department of Mechanics and MathematicsKharkov UniversityKharkovUkraine

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