Solving the Problem of Nonlinear Ship Roll Motion Using Stochastic Dynamics

  • Jeffrey M. Falzarano
  • Zhiyong Su
  • Arada Jamnongpipatkul
  • Abhilash Somayajula
Part of the Fluid Mechanics and Its Applications book series (FMIA, volume 119)


Due to nonlinear viscous damping and the softening characteristic of the stiffness, the roll motion of a ship exhibits complex dynamics. Specifically predicting the probabilistic characteristics of roll response in an irregular seaway is still a challenging problem and continues to be of interest for both practitioners and researchers. In this work two techniques from the theory of stochastic dynamics are applied to study the probabilistic nature of roll motion in irregular seas. The first method is the “Moment Equation method” where the roll response moment equation is formulated from a six dimensional state space rolling model with a fourth order linear filter using the Itô differential rule. The resulting moment equations are solved using a cumulant neglect technique. Alternatively in the second approach, the probability density function of the rolling response is evaluated by solving the corresponding Fokker Planck Equation of the system using “Path Integral method”.



The work has been funded by the Office of Naval Research (ONR) T-Craft Tools development program ONR Grant N00014-07- 1-1067 with program manager Kelly Cooper.


  1. Chai W, Naess A, Leira BJ (2015) Stochastic Dynamic Analysis and Reliability of a Vessel Rolling in Random Beam Seas. Journal of Ship Research 59(2):113–131CrossRefGoogle Scholar
  2. Falzarano J, Vishnubhotla S, Cheng J (2004) Nonlinear Dynamic Analysis of Ship Capsizing in Random Waves. In: 14th International Offshore and Polar Engineering Conference, The International Society of Offshore and Polar Engineers, Toulon, France, vol 1, pp 479–484Google Scholar
  3. Falzarano J, Somayajula A, Seah R (2015) An overview of the prediction methods for roll damping of ships. Ocean Systems Engineering 5(2):55–76CrossRefGoogle Scholar
  4. Falzarano JM, Shaw SW, Troesch AW (1992) Application of Global Methods for Analyzing Dynamical Systems To Ship Rolling Motion and Capsizing. International Journal of Bifurcation and Chaos 02(01):101–115, Scholar
  5. Falzarano JM, Vishnubhotla S, Juckett SE (2010) Combined Steady State and Transient Analysis of a Patrol Vessel as Affected by Varying Amounts of Damping and Periodic and Random Wave Excitation. Journal of Offshore Mechanics and Arctic Engineering 132(1):014,501, Scholar
  6. Francescutto A (1990) On the Non-linear Motions of Ships and Structures in Narrow Band Sea. In: IUTAM Symposium on Dynamics of Marine Vehicles and Structures in Waves, Elsevier, London, UK, pp 291–304Google Scholar
  7. Francescutto A, Naito S (2004) Large amplitude rolling in a realistic sea. International shipbuilding progress 51(2):221–235Google Scholar
  8. Guha A, Somayajula A, Falzarano J (2016) Time domain simulation of large amplitude motions in shallow water. In: 21st SNAME Offshore Symposium, Society of Naval Architects and Marine Engineers, Houston, FebruaryGoogle Scholar
  9. Hsieh SR, Troesch aW, Shaw SW (1994) A Nonlinear Probabilistic Method for Predicting Vessel Capsizing in Random Beam Seas. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 446(1926):195–211, Scholar
  10. Jamnongpipatkul A, Su Z, Falzarano JM (2011) Nonlinear ship rolling motion subjected to noise excitation. Ocean Systems Engineering 1(3):249–261, Scholar
  11. Jiang C, Troesch AWA, Shaw SWS (2000) Capsize criteria for ship models with memory-dependent hydrodynamics and random excitation. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 358(1771):1761–1791, Scholar
  12. Jiang CB, Troesch AW, Shaw SW (1996) Highly nonlinear rolling motion of biased ships in random beam seas. Journal of Ship Research 40(2):125–135Google Scholar
  13. Lin H, Yim SC (1995) Chaotic roll motion and capsize of ships under periodic excitation with random noise. Applied Ocean Research 17(3):185–204, Scholar
  14. Moideen H, Falzarano JM, Sharma S (2012) Parametric roll of container ships in head waves. International Journal of Ocean Systems Engineering 2(4):239–255, Scholar
  15. Moideen H, Somayajula A, Falzarano JM (2013) Parametric Roll of High Speed Ships in Regular Waves. In: Proceedings of ASME 2013 32nd International Conferences on Ocean, Offshore and Arctic Engineering, ASME, vol 5, p V005T06A095,
  16. Moideen H, Somayajula A, Falzarano JMJ (2014) Application of Volterra Series Analysis for Parametric Rolling in Irregular Seas. Journal of Ship Research 58(2):97–105, Scholar
  17. Naess A, Moe V (2000) Efficient path integration methods for nonlinear dynamic systems. Probabilistic Engineering Mechanics 15(2):221–231, Scholar
  18. Roberts JB, Spanos PD (2003) Random Vibration and Statistical Linearization. Dover Publications, Mineola, New YorkGoogle Scholar
  19. Roberts JB, Vasta M (2000) Markov modelling and stochastic identification for nonlinear ship rolling in random waves. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 358(1771):1917–1941, Scholar
  20. Somayajula A, Falzarano J (2015a) Large-amplitude time-domain simulation tool for marine and offshore motion prediction. Marine Systems & Ocean Technology 10(1):1–17, Scholar
  21. Somayajula A, Falzarano JM (2014) Non-linear Dynamics of Parametric Roll of Container Ship in Irregular Seas. In: Proceedings of ASME 2014 33rd International Conferences on Ocean, Offshore and Arctic Engineering, San Francisco, pp 1–10,
  22. Somayajula A, Falzarano JM (2015b) Validation of Volterra Series Approach for Modelling Parametric Rolling of Ships. In: Proceedings of ASME 2015 34th International Conferences on Ocean, Offshore and Arctic Engineering, American Society of Mechanical Engineers, St. John’s, NL, CanadaGoogle Scholar
  23. Somayajula A, Guha A, Falzarano J, Chun HH, Jung KH (2014) Added resistance and parametric roll prediction as a design criteria for energy efficient ships. International Journal of Ocean Systems Engineering 4(2):117–136, Scholar
  24. Somayajula AS, Falzarano JM (2016) A comparative assessment of simplified models for simulating parametric roll. Journal of Offshore Mechanics and Arctic Engineering Scholar
  25. Spanos PTD (1983) ARMA Algorithms for Ocean Wave Modeling. Journal of Energy Resources Technology 105(3):300, Scholar
  26. Spyrou KJ, Thompson JMT (2000) The nonlinear dynamics of ship motions: a field overview and some recent developments. Philosophical Transactions of the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences 358(1771):1735–1760MathSciNetCrossRefGoogle Scholar
  27. Stark H, Woods JW (2002) Probability and random processes with applications to signal processing. Prentice Hall, Upper Saddle River, NJGoogle Scholar
  28. Su Z, Falzarano JM (2011) Gaussian and non-Gaussian cumulant neglect application to large amplitude rolling in random waves. International Shipbuilding Progress 58:97–113, Scholar
  29. Su Z, Falzarano JM (2013) Markov and Melnikov based methods for vessel capsizing criteria. Ocean Engineering 64:146–152, Scholar
  30. Su Z, Falzarano JM, Su Z (2011) Gaussian and Non Gaussian Response of Ship Rolling in Random Beam Waves. In: Proceedings of the12th International Ship Stability Workshop, Washington DC, USA, pp 189–193Google Scholar
  31. Thompson JMT (1997) Designing Against Capsize in Beam Seas: Recent Advances and New Insights. Applied Mechanics Reviews 50(5):307, Scholar
  32. Vishnubhotla S, Falzarano J (2009) Effect of More Accurate Hydrodynamic Modeling on Calculating Critical Nonlinear Ship Rolling Response. In: Lecture Notes in Applied and Computational Mechanics, vol 44, pp 269–274, Scholar
  33. Vishnubhotla S, Falzarano J, Vakakis A (2000) A new method to predict vessel/platform critical dynamics in a realistic seaway. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 358(1771):1967–1981, Scholar
  34. Webster W (1989) Motion in Regular Waves - Transverse Motions. In: Lewis E (ed) Principles of Naval Architecture Vol III, SNAME, Jersey City, New JerseyGoogle Scholar
  35. Yim SCS, Lin H (2001) Unified Analysis of Complex Nonlinear Motions via Densities. Nonlinear Dynamics 24(1):103–127, Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Jeffrey M. Falzarano
    • 1
  • Zhiyong Su
    • 2
  • Arada Jamnongpipatkul
    • 3
  • Abhilash Somayajula
    • 1
  1. 1.Texas A&M UniversityCollege StationUSA
  2. 2.COTEC Offshore Engineering ServicesHoustonUSA
  3. 3.Houston Offshore EngineeringHoustonUSA

Personalised recommendations