Journal of Engineering Mathematics

, Volume 89, Issue 1, pp 137–146 | Cite as

Roll dynamics of a ship sailing in large amplitude head waves

  • E. F. G. van Daalen
  • M. Gunsing
  • J. Grasman
  • J. Remmert


Some ship types may show significant rolling when sailing in large-amplitude (near) head waves. The dynamics of the ship are such that the roll motion is affected by the elevation of the encountering waves. If the natural roll period (without forcing) is about half the period of the forcing by the waves, then a stationary solution will have an amplitude that is much larger than for other forcing frequencies. This phenomenon is called parametric resonance. For certain hull shape types the transverse stability may vary considerably due to the waves passing the ship. Moreover, near head waves will also have a direct effect on the roll dynamics. For these processes a differential equation model—a Mathieu type of equation—is formulated. Furthermore, the waves considered are of a type that is encountered in open seas. As a parameterization of these waves the Pierson–Moskowitz spectrum is used. The risk that the ship will reach a critical state is characterized by the time of arrival at this state, starting from an arbitrary pattern of the waves and the dynamic state of the vessel in the stationary situation. Large-scale Monte Carlo simulations of this process are carried out. The percentiles of the arrival time distribution indicate the risk of significant rolling to which the vessel is exposed. Furthermore, a method is proposed to estimate the maximum roll angle in a stationary state by taking into consideration only the part of the wave spectrum that relates to the state of parametric resonance. The result is compared with the outcome of the large-scale Monte Carlo simulations.


Critical roll amplitude Parametric resonance Stochastic waves 



The authors would like to thank the organizers and participants of the Study Group Mathematics with Industry for initiating a first exploration of the subject; see [11].


  1. 1.
    France WN, Levadou M, Treakle TW, Paulling JR, Michel RK, Moore C (2003) An investigation of head-sea parametric rolling and its influence on container lashing systems. Mar Technol SNAME News 40(1):1–19Google Scholar
  2. 2.
    Santos-Neves MA, Rodriguez CA (2007) Influence of non-linearities on the limit of stability of ships rolling in head seas. Ocean Eng 34:1618–1630CrossRefGoogle Scholar
  3. 3.
    Dunwoody AB (1989) Roll of a ship in Astern Seas – Metacentric height spectra. J Ship Res 33:221–228Google Scholar
  4. 4.
    Dunwoody AB (1989) Roll of a ship in Astern Seas – Response to GM fluctuations. J Ship Res 33:284–290Google Scholar
  5. 5.
    Levadou ML, Van’t Veer R (2006) Parametric roll and ship design. In: Proceedings of the 9th international conference on stability of ships and ocean vehicles, vol 1, pp 191–206Google Scholar
  6. 6.
    Shin YS, Belenky VL, Pauling JR, Weems KM, Lin WM (2004) Criteria for parametric roll of large container ships in longitudinal seas. Trans Soc Naval Archit Mar Eng 112:14–47Google Scholar
  7. 7.
    Pierson WJ, Moskowitz L (1964) A proposed spectral form for fully developed wind seas based on the similarity theory of S.A. Kitaigorodskij. J Geophys Res 69:5181–5190ADSCrossRefGoogle Scholar
  8. 8.
    David HA, Nagaraja HN (2003) Order statistics, Wiley Interscience, HobokenGoogle Scholar
  9. 9.
    Gardiner CW (1990) Handbook of stochastic methods for physics, chemistry and the natural sciences. Springer, BerlinMATHGoogle Scholar
  10. 10.
    Tondl A, Ruijgrok T, Verhulst F, Nabergoj R (2000) Autoparametric resonance in mechanical systems. Cambridge University Press, CambridgeGoogle Scholar
  11. 11.
    Archer C, Van Daalen EFG, Dobberschütz S, Godeau M-F, Grasman J, Gunsing M, Muskulus M, Pischanskyy A, Wakker M (2009) Dynamical models of extreme rolling of vessels in head waves. In: Molenaar J, Keesman K, Van Opheusden J, Doeswijk T (eds) Proceedings of the 67th European Study Group Mathematics with Industry, pp 1–27. ISBN 978-90-8585-600-9Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • E. F. G. van Daalen
    • 1
  • M. Gunsing
    • 1
  • J. Grasman
    • 2
  • J. Remmert
    • 3
    • 4
  1. 1.Maritime Research Institute Netherlands (MARIN)WageningenThe Netherlands
  2. 2.Mathematical and Statistical Methods GroupWageningen University and Research CentreWageningenThe Netherlands
  3. 3.Department of Maritime TechnologyDelft University of TechnologyDelftThe Netherlands
  4. 4.Maersk Maritime TechnologyCopenhagenDenmark

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