Skip to main content
SpringerLink
Log in

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

  • Chapter
  • First Online:
Contemporary Ideas on Ship Stability

Part of the book series: Fluid Mechanics and Its Applications ((FMIA,volume 119))

Abstract

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 .

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 219.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 279.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    All symbols are explained in the Nomenclature at the end of the paper.

Abbreviations

A :

Wave amplitude

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

Added mass coefficient

A R :

Rudder area

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

Proportional, differential gain

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

Damping coefficient

c :

Wave celerity

\(F_{N}\) :

Rudder normal force

\(Fn\) :

Froude number

H :

Wave height

\(H/\lambda\) :

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

\(K_{T}\) :

Propeller thrust coefficient

k :

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

L :

Ship length

m :

Ship mass

\(q,p,r\) :

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

t :

Time

\(t_{p}\) :

Thrust deduction coefficient

t r :

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

\(U_{R}\) :

Inflow velocity at rudder

\(X,\,Y,\,Z\) :

Forces in surge, sway and heave respectively

x :

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

\(x_{O}\) :

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

\(x_{S}\) :

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

δ :

Rudder angle

Λ :

Rudder aspect ratio

\(\theta\) :

Pitch angle

λ :

Wave length

\(\rho\) :

Water density

\(\varphi\) :

Roll angle

\(\psi\) :

Heading angle

\(\psi_{r}\) :

Desired heading angle

\(\omega_{e}\) :

Encounter frequency

\(\omega\) :

Wave frequency

References

  • 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 

  • 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.

    Article  Google Scholar 

  • Cummins, W.E., 1962, “The Impulse-Response Function and Ship Motions”, Schiffstechnik, Vol. 9, No. 47, pp. 101–109.

    Google Scholar 

  • 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 

  • 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 

  • 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 

  • 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 

  • 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 

  • Grim O., 1951, “Das Schiff in von Achtern Auflaufender See”, Jahrbuch S.T.G., Bd. 45.

    Chapter  Google Scholar 

  • Grim O., 1963, “Surging Motion and Broaching Tendencies in a Severe Irregular Sea”, Deutsche Hydrographische Zeitschrift, Bd. 16.

    Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • ITTC, 2005, “Report of Specialist Committee on Ship Stability in Waves”, Proceedings of ITTC Conference, Edinburgh.

    Google Scholar 

  • Jefferys, E.R., 1984, “Simulation of Wave Power Devices”, Journal of Applied Ocean Research, No. 6, Vol. 1, pp. 31–39.

    Article  MathSciNet  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • Krauskopf B., Osinga H. G. and Galan-Vioque J. (editors), 2007, “Numerical Continuation Methods for Dynamical Systems”, Springer Publications.

    Google Scholar 

  • 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 

  • 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 

  • 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 

  • 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 

  • Rydill L.J., 1959, “A Linear Theory for the Steered Motion of Ships in Waves”, Transactions of RINA, pp. 81–112.

    Google Scholar 

  • Schmiechen M., 1975, “Equations for Non-Quasi-Steady Ship Motions”, Seakeeping Committee Report, Proceedings of 14th ITTC Conference, Ottawa, Canada.

    Google Scholar 

  • SNAME, 1952, “Nomenclature for Treating the Motion of a Submerged Body Through Fluid”, Technical and Research Bulletin, No. 1–5.

    Google Scholar 

  • 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 

  • 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 

  • Spyrou K.J., 1996b, “Homoclinic Connections and Period Doublings of a Ship Advancing in Quartering Waves”, Journal of Chaos, Vol. 6, pp. 209–218.

    Article  Google Scholar 

  • 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 

  • Spyrou, K.J., 2010, “Historical Trails of Ship Broaching-to”, Transactions of RINA, Vol. 152, Part A4, pp. 163–173.

    Google Scholar 

  • 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 

  • 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 

  • 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 

  • 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.

    Article  Google Scholar 

  • Tick L.J., 1959, “Differential Equations with Frequency-Dependent Coefficients”, Journal of Ship Research, Vol. 3, No. 2, pp. 45–46.

    Google Scholar 

  • 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 

  • Trident F-D Waveload, 2006, “User’s Guide and Theory Manual”, Martec Ltd, Halifax, Nova Scotia, Canada.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Weinblum G. and St. Denis M., 1950, “On the Motions of Ships at Sea”, Transactions of SNAME, Vol. 58, pp. 184–248.

    Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. Streamlined Naval Architects (Previously at, National Technical University of Athens), Piraeus, Greece

    Ioannis Tigkas

  2. School of Naval Architecture and Marine Engineering, National Technical University of Athens, Athens, Greece

    Kostas J. Spyrou

Authors
  1. Ioannis Tigkas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kostas J. Spyrou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kostas J. Spyrou .

Editor information

Editors and Affiliations

  1. David Taylor Model Basin/Naval Surface Warfare Center Carderock Division, West Bethesda, MD, USA

    Vadim L. Belenky

  2. School of Naval Architecture and Marine Engineering, National Technical University of Athens, Athens, Greece

    Kostas J. Spyrou

  3. MARIN-Maritime Research Institute Netherlands, Wageningen, The Netherlands

    Frans van Walree

  4. Department of Naval Architecture and Ocean Engineering, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

    Marcelo Almeida Santos Neves

  5. Department of Naval Architecture and Ocean Engineering, Osaka University, Suita, Osaka, Japan

    Naoya Umeda

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Tigkas, I., Spyrou, K.J. (2019). Bifurcation Analysis of Ship Motions in Steep Quartering Seas, Including Hydrodynamic “Memory”. In: Belenky, V., Spyrou, K., van Walree, F., Almeida Santos Neves, M., Umeda, N. (eds) Contemporary Ideas on Ship Stability. Fluid Mechanics and Its Applications, vol 119. Springer, Cham. https://doi.org/10.1007/978-3-030-00516-0_19

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-00516-0_19

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-00514-6

  • Online ISBN: 978-3-030-00516-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation