Bifurcation Analysis of Ship Motions in Steep Quartering Seas, Including Hydrodynamic “Memory”

  • Ioannis Tigkas
  • Kostas J. SpyrouEmail author
Part of the Fluid Mechanics and Its Applications book series (FMIA, volume 119)


Steady-state ship dynamics in steep harmonic waves encountering the ship from stern quartering direction is under investigation. Bifurcation analysis is performed by applying a numerical continuation method. Stationary as well as periodic states are traced, as selected control parameters are varied. Regions with coexistence of different ship responses are identified. The main novelty of the paper lies in the extension of the continuation analysis to a 6-DOF model, for a quartering sea environment, with inclusion of memory effects within a potential flow framework. Complete, vessel-specific stability diagrams, for horizontal plane motions, are produced in an automated and time-efficient manner. These could provide useful guidance to ship masters for avoiding the occurrence of surf-riding and broaching-to.


Manoeuvring Surf-riding Broaching-to Bifurcation Homoclinic Continuation Nonlinear dynamics 



Wave amplitude

\(A_{ij} \left( \omega \right)\)

Added mass coefficient


Rudder area

\(a_{\psi } , \, a_{r}\)

Proportional, differential gain

\(B_{ij} \left( \omega \right)\)

Damping coefficient


Wave celerity


Rudder normal force


Froude number


Wave height


Wave steepness

\(I_{x} , \, I_{y} , \, I_{z}\)

Roll, pitch and yaw ship mass moment of inertia

\(K, \, M, \, N\)

Moments in roll, pitch and yaw respectively

\(K_{ij} (\tau )\)

Impulse response function


Propeller thrust coefficient


Wave number \((k = {{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right. \kern-0pt} \lambda })\)


Ship length


Ship mass


Pitch, roll and yaw angular velocity in a body-fixed system, respectively

\(S\left( {x_{s} ,T_{s} } \right)\)

Vertical hull sectional area below instantaneous waterline




Thrust deduction coefficient


Rudder’s time constant

\(T_{s} \left( {x_{s} x_{0} ,t,z,\theta } \right)\)

Draught of ship at vertical section S

u, v, w

Surge, sway and heave velocity in a body-fixed system, respectively


Inflow velocity at rudder


Forces in surge, sway and heave respectively


Longitudinal distance travelled by the ship, with respect to a system fixed at a wave trough


Longitudinal distance travelled by the ship in an earth-fixed system


Longitudinal distance of a vertical ship section S in the body-fixed system

\(x_{G} ,\,\,z_{G}\)

Longitudinal distance from amidships and vertical distance from keel of ship’s centre of gravity, respectively

Greek Letters


Rudder angle


Rudder aspect ratio


Pitch angle


Wave length


Water density


Roll angle


Heading angle


Desired heading angle


Encounter frequency


Wave frequency



Ioannis Tigkas would like to thank “Alexander S. Onassis” Public Benefit Foundation for a scholarship that supported his Ph.D. studies at NTUA.


  1. Belenky, V.L. and Sevastianov, N.B., 2007, Stability and Safety of Ships. Risk of Capsizing. Published by The Society of Naval Architects and Marine Engineers, ISBN 10: 0939773619.Google Scholar
  2. Clarke, D., 1972, “A Two-Dimensional Strip Method for Surface Ship Hull Derivatives: Comparison of Theory with Experiments on a Segmented Tanker Model”, Journal of Mechanical Engineering Science, Vol. 14, No. 7, pp. 53–61.CrossRefGoogle Scholar
  3. Cummins, W.E., 1962, “The Impulse-Response Function and Ship Motions”, Schiffstechnik, Vol. 9, No. 47, pp. 101–109.Google Scholar
  4. Davidson, K.S.M., 1948, “A Note on the Steering Of Ships in Following Seas”, Proceedings of 7th International Congress of Applied Mechanics, London.Google Scholar
  5. de Kat J.O. and Paulling J.R., 1989, “The Simulation of Ship Motions and Capsizing in Severe Seas”, SNAME Annual Meeting, Presentation, No. 5.Google Scholar
  6. Dhooge A., Govaerts W., Kuznetsov Yu. A., Mestrom W., Riet A.M. and Sautois B., 2003, “MATCONT and CL_MATCONT: Continuation Toolboxes for MATLAB”. Report of Gent and Utrecht Universities.Google Scholar
  7. Doedel E.J., Champneys A.R., Fairgrieve T.F., Kuznetsov Y.A., Sandstede B. and Wang X.J., 1997, “AUTO97-00: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont)”, User’s Guide, Concordia University, Montreal, Canada.Google Scholar
  8. Fuwa T., Sugai K., Yoshino T. and Yamamoto T., 1981, “An Experimental Study on broaching of Small High Speed Boat”, Journal of the SNA of Japan, Vol. 150.Google Scholar
  9. Grim O., 1951, “Das Schiff in von Achtern Auflaufender See”, Jahrbuch S.T.G., Bd. 45.CrossRefGoogle Scholar
  10. Grim O., 1963, “Surging Motion and Broaching Tendencies in a Severe Irregular Sea”, Deutsche Hydrographische Zeitschrift, Bd. 16.Google Scholar
  11. Hamamoto M., 1988, “Study on Ship Motions and Capsizing in Following Seas: 1st Report - Equations of Motion for Numerical Simulation”, Journal of the SNA of Japan, Vol. 163, pp. 173–180.Google Scholar
  12. Hamamoto M., 1989, “Study on Ship Motions and Capsizing in Following Seas: 2nd Report - Simulation of Capsizing”, Journal of the SNA of Japan, Vol. 163, pp. 123–130.CrossRefGoogle Scholar
  13. Hamamoto M., Fujino M. and Kim Y.S., 1994, “Dynamic Stability of a Ship in Quartering Seas”, Proceedings of STAB 94, Melbourne, Florida.Google Scholar
  14. Holappa K.W. and Falzarano, J.M., 1999, “Application of Extended State-space to Non-linear Ship Rolling”, Journal of Ocean Engineering, Vol. 26, p. 227–240.CrossRefGoogle Scholar
  15. ITTC, 2005, “Report of Specialist Committee on Ship Stability in Waves”, Proceedings of ITTC Conference, Edinburgh.Google Scholar
  16. Jefferys, E.R., 1984, “Simulation of Wave Power Devices”, Journal of Applied Ocean Research, No. 6, Vol. 1, pp. 31–39.MathSciNetCrossRefGoogle Scholar
  17. Kose K., 1982, “On a New Mathematical Model of Manoeuvring Motions of A Ship and its Applications”, Journal of International Shipbuilding Progress, Vol. 29, No. 336.CrossRefGoogle Scholar
  18. Koushan, K., 2006, “Dynamics of Ventilated Propeller Blade Loading on Thrusters due to Forced Sinusoidal Heave Motion”, Proceedings of 26th Symposium on Naval Hydrodynamics, Rome, Italy.Google Scholar
  19. Krauskopf B., Osinga H. G. and Galan-Vioque J. (editors), 2007, “Numerical Continuation Methods for Dynamical Systems”, Springer Publications.Google Scholar
  20. Motora S., Fujino M., Koyanagi M., Ishida S., Shimada K. and Maki T., 1981, “A Consideration on the Mechanism of Occurrence of Broaching-to Phenomena”, Journal of the SNA of Japan, Vol. 150, pp. 84–97.Google Scholar
  21. Ogilvie T.F., 1964, “Recent Progress towards the Understanding and Prediction of Ship Motions”, Proceedings of 5th Symposium on Naval Hydrodynamics, Bergen, Norway, pp. 3–79.Google Scholar
  22. Paik B.G., Lee J.Y. and Lee S.J., 2008, “Effect of Propeller Immersion Depth on the Flow around a Marine Propeller”, Journal of Ship Research, Vol. 52, No. 2, pp. 102–113.Google Scholar
  23. Renilson M., 1982, “An Investigation Into the Factors Affecting the Likelihood of Broaching-to in Following Seas”, Proceedings of STAB 82, Tokyo, pp. 17–28.Google Scholar
  24. Rydill L.J., 1959, “A Linear Theory for the Steered Motion of Ships in Waves”, Transactions of RINA, pp. 81–112.Google Scholar
  25. Schmiechen M., 1975, “Equations for Non-Quasi-Steady Ship Motions”, Seakeeping Committee Report, Proceedings of 14th ITTC Conference, Ottawa, Canada.Google Scholar
  26. SNAME, 1952, “Nomenclature for Treating the Motion of a Submerged Body Through Fluid”, Technical and Research Bulletin, No. 1–5.Google Scholar
  27. Spyrou K.J., 1995, “Surf-Riding Yaw Instability and Large Heeling of Ships in Following/ Quartering Waves”, Journal of Ship Technology Research, Vol. 42, pp. 103–112.Google Scholar
  28. Spyrou K.J., 1996a, “Dynamic Instability in Quartering Seas: The Behaviour of a Ship During Broaching”, Journal of Ship Research, Vol. 40, No. 1, pp. 46–59.Google Scholar
  29. Spyrou K.J., 1996b, “Homoclinic Connections and Period Doublings of a Ship Advancing in Quartering Waves”, Journal of Chaos, Vol. 6, pp. 209–218.CrossRefGoogle Scholar
  30. Spyrou K.J., 1997, “Dynamic Instability in Quartering Seas – Part III: Nonlinear Effects on Periodic Motions”, Journal of Ship Research, Vol. 41, No. 3, pp. 210–223.Google Scholar
  31. Spyrou, K.J., 2010, “Historical Trails of Ship Broaching-to”, Transactions of RINA, Vol. 152, Part A4, pp. 163–173.Google Scholar
  32. Spyrou K.J., Tigkas I. and Chatzis A., 2007, “Dynamics of a Ship Steering in Wind Revisited”, Journal of Ship Research, Vol. 51, No. 2.Google Scholar
  33. Spyrou K. and Tigkas I.G., 2007, “Principles and Application of Continuation Methods for Ship Design and Operability Analysis”, Proceedings of 10th PRADS Symposium, Houston, Texas, USA.Google Scholar
  34. Spyrou K.J. and Tigkas I.G., 2011, “Nonlinear Surge Dynamics of a Ship in Astern Seas: “Continuation Analysis” of Periodic States with Hydrodynamic Memory”, Journal of Ship Research, Vol. 55, No. 1, pp. 19–28.Google Scholar
  35. Taghipour R., Perez T. and Moan T., 2008, “Hybrid Frequency–Time Domain Models for Dynamic Response Analysis of Marine Structures”, Journal of Ocean Engineering, 35, pp. 685–705.CrossRefGoogle Scholar
  36. Tick L.J., 1959, “Differential Equations with Frequency-Dependent Coefficients”, Journal of Ship Research, Vol. 3, No. 2, pp. 45–46.Google Scholar
  37. Tigkas I.G., 2009, “Nonlinear Dynamics Analysis of the Directional Instabilities of Ships in Wind and Waves”, PhD Thesis, National Technical University of Athens, Greece.Google Scholar
  38. Trident F-D Waveload, 2006, “User’s Guide and Theory Manual”, Martec Ltd, Halifax, Nova Scotia, Canada.Google Scholar
  39. Umeda N., Hamamoto M., Takaishi Y., Chiba Y., Matsuda A., Sera W., Suzuki S., Spyrou K.J. and Watanabe K., 1995, “Model Experiments of Ship Capsize in Astern Seas”, Journal of the SNA of Japan, Vol. 177, pp. 207–218.CrossRefGoogle Scholar
  40. Wahab R. and Swaan W.A., 1964, “Course-keeping and Broaching of Ships in Following Seas”, Journal of International Shipbuilding Progress, Vol. 7, No. 4, pp. 293–301.CrossRefGoogle Scholar
  41. Weinblum G. and St. Denis M., 1950, “On the Motions of Ships at Sea”, Transactions of SNAME, Vol. 58, pp. 184–248.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Streamlined Naval Architects (Previously at, National Technical University of Athens)PiraeusGreece
  2. 2.School of Naval Architecture and Marine EngineeringNational Technical University of AthensAthensGreece

Personalised recommendations