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Parametric roll vulnerability of ships using Markov and Melnikov approaches

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Abstract

The designs of modern container ships, roll-on–roll-off vessels and cruise vessels have evolved over the years, and in recent times, some of them have been observed to experience dynamic instabilities during operation in the open ocean. These catastrophic events demonstrate that satisfying prescriptive stability rules set forth by International Maritime Organization (IMO), national authorities (e.g., Coast Guard) and other classification societies are not sufficient to ensure dynamic stability of ships at sea. In light of these events, IMO is organizing efforts to make way toward a second generation of intact stability criteria that are better equipped to deal with these dynamic instabilities. This paper discusses the development of such a tool for parametric rolling in a realistic random seaway, which is one of the critical phenomena identified by IMO. In this study, a previously developed analytical model for roll restoring moment, which was found to be effective in modeling the problem of parametric roll, is analyzed using the Melnikov approach. The stability of the system is quantified in terms of rate of phase space flux of the system. This approach is further compared with another technique known as the Markov approach that is based on stochastic averaging and quantifies stability in terms of mean first passage time. The sensitivity of both of these metrics to environmental parameters is investigated. Finally, the nature of random response is analyzed using Lyapunov exponents to determine whether the vessel exhibits any chaotic dynamics.

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References

  1. Bulian, G.: Development of analytical nonlinear models for parametric roll and hydrostatic restoring variations in regular and irregular waves. Ph.D. Dissertation, Università degli studi di Trieste, Trieste (2006). https://www.openstarts.units.it/dspace/handle/10077/2518

  2. Carmel, S.: Study of parametric rolling event on a Panamax container vessel. Transp. Res. Rec. J. Transp. Res. Board 1963, 56–63 (2006). https://doi.org/10.3141/1963-08

    Article  Google Scholar 

  3. Chai, W., Naess, A., Leira, B.J., Carlo, M.: Stochastic dynamic analysis and reliability of a vessel rolling in random beam seas. J. Ship Res. 59(2), 113–131 (2015). https://doi.org/10.5957/JOSR.59.2.140059

    Article  Google Scholar 

  4. Devaney, R.: An Introduction to Chaotic Dynamical Systems, 1st edn. Addison Wesley Publishing Company, Redwood City (1989)

    MATH  Google Scholar 

  5. Esparza, I., Falzarano, J.: Nonlinear rolling motion of a statically biased ship under the effect of external and parametric excitation. In: S.C. Sinha, R.M. Evan-Iwanowski (eds.) Proceedings of the 14th Biennial Conference on Mechanical Vibration and Noise: Dynamics and Vibration of Time-Varying Systems and Structures, vol. 56, pp. 111–122. ASME, Albuquerque (1993)

  6. Falzarano, J., Somayajula, A., Seah, R.: An overview of the prediction methods for roll damping of ships. Ocean Syst. Eng. 5(2), 55–76 (2015). https://doi.org/10.12989/ose.2015.5.2.055

    Article  Google Scholar 

  7. Falzarano, J.M.: Predicting complicated dynamics leading to vessel capsizing. Ph.D. Dissertation, University of Michigan, Ann Arbor (1990). http://deepblue.lib.umich.edu/handle/2027.42/91674

  8. Falzarano, J.M., Holappa, K., Taz-Ul-Mulk, M.: A generalized analysis of saturation induced ship rolling motion. In: Falzarano, J.M., Papoulias, F. (eds.) Proceedings of the 1993 ASME Winter Annual Meeting: Nonlinear Dynamics of Marine Vehicles, vol. 51, pp. 73–92. ASME, New Orleans (1993)

  9. Falzarano, J.M., Shaw, S.W., Troesch, A.W.: Application of global methods for analyzing dynamical systems to ship rolling motion and capsizing. Int. J. Bifurc. Chaos 2(1), 101–115 (1992). https://doi.org/10.1142/S0218127492000100

    Article  MathSciNet  MATH  Google Scholar 

  10. Falzarano, J.M., Su, Z., Jamnongpipatkul, A., Somayajula, A.: Solving the Problem of Nonlinear Ship Roll Motion Using Stochastic Dynamics, pp. 423–435. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-00516-0_25

    Book  Google Scholar 

  11. France, W., Levadou, M., Treakle, T.W., Paulling, J.R., Michel, R.K., Moore, C.: An investigation of head-sea parametric rolling and its influence on container lashing systems. Mar. Technol. 40(1), 1–19 (2003)

    Google Scholar 

  12. Frey, M., Simiu, E.: Noise-induced chaos and phase space flux. Phys. D Nonlinear Phenom. 63, 321–340 (1993). https://doi.org/10.1016/0167-2789(93)90114-G

    Article  MathSciNet  MATH  Google Scholar 

  13. Froude, W.: On the rolling of ships. Trans. INA 2, 180–227 (1861)

    Google Scholar 

  14. Greenspan, B., Holmes, P.: Repeated resonance and homoclinic bifurcation in a periodically forced family of oscillators. SIAM J. Math. Anal. 15(1), 69–97 (1984). https://doi.org/10.1137/0515003

    Article  MathSciNet  MATH  Google Scholar 

  15. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Applied Mathematical Sciences, vol. 42, 1st edn. Springer, New York (1983). https://doi.org/10.1007/978-1-4612-1140-2

    Book  MATH  Google Scholar 

  16. Hsieh, S.R., Shaw, S.W., Troesch, A.W.: A predictive method for vessel capsize in random seas. In: Falzarano, J.M., Papoulias, F. (eds.) Proceedings of the 1993 ASME Winter Annual Meeting: Nonlinear Dynamics of Marine Vehicles, vol. 51, pp. 103–123. ASME, New Orleans (1993)

  17. Hsieh, S.R., Troesch, A.W., Shaw, S.W.: A nonlinear probabilistic method for predicting vessel capsizing in random beam seas. Proc. R. Soc. A Math. Phys. Eng. Sci. 446(1926), 195–211 (1994). https://doi.org/10.1098/rspa.1994.0099

    Article  Google Scholar 

  18. Huang, X., Gu, X., Bao, W.: The probability distribution of rolling amplitude of a ship in high waves. In: Proceedings of 5th International Conference on Stability of Ships and Ocean Vehicles (STAB94), vol. 4, pp. 77–100. STAB International Standing Committee, Melbourne (1994)

  19. Ikeda, Y., Himeno, Y., Tanka, N.: A prediction method for ship roll damping. Technical Report, University of Osaka Prefecture, Osaka (1978). Report No. 00405. http://www.marine.osakafu-u.ac.jp/lab15/papersearch/ papers/PDF/A23.pdf

  20. Jiang, C.: Highly nonlinear rolling motion leading to capsize. Ph.D. Dissertation, University of Michigan, Ann Arbor (1995)

  21. Jiang, C., Troesch, A.W., Shaw, S.W.: Capsize criteria for ship models with memory-dependent hydrodynamics and random excitation. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 358(1771), 1761–1791 (2000). https://doi.org/10.1098/rsta.2000.0614

    Article  MathSciNet  MATH  Google Scholar 

  22. Jordan, D.W., Smith, P.: Nonlinear Ordinary Differential Equations: An Introduction to Dynamical Systems, 4th edn. Oxford University Press, New York (1999)

    MATH  Google Scholar 

  23. Lou, J., Xie, G.: Solution methods of complex nonlinear dynamic systems in offshore engineering. Technical Report, Offshore Technology Research Center, Texas A&M University, College Station (1993)

  24. Melnikov, V.: On the stability of the center for time-periodic perturbations. Trans. Mosc. Math. Soc. 12, 1–57 (1963)

    MathSciNet  Google Scholar 

  25. Moideen, H., Falzarano, J.M., Sharma, S.: Parametric roll of container ships in head waves. Ocean Syst. Eng. 2(4), 239–255 (2012). https://doi.org/10.12989/ose.2012.2.4.239

    Article  Google Scholar 

  26. Moideen, H., Somayajula, A., Falzarano, J.M.: Parametric roll of high speed ships in regular waves. In: Volume 5: Ocean Engineering, p. V005T06A095. ASME, Nantes (2013). https://doi.org/10.1115/OMAE2013-11602

  27. Moideen, H., Somayajula, A., Falzarano, J.M.: Application of Volterra series analysis for parametric rolling in irregular seas. J. Ship Res. 58(2), 97–105 (2014). https://doi.org/10.5957/JOSR.58.2.130047

    Article  Google Scholar 

  28. Moseley, H.: On the dynamical stability and on the oscillations of floating bodies. Philos. Trans. R. Soc. Lond. 140, 609–643 (1850)

    Article  Google Scholar 

  29. Nayfeh, A.H., Khdeir, A.A.: Nonlinear rolling of ships in regular beam seas. Int. Shipbuild. Prog. 33(379), 40–49 (1986)

    Article  Google Scholar 

  30. Parker, T.S., Chua, L.O.: Practical Numerical Algorithms for Chaotic Systems, 1st edn. Springer, New York (1989). https://doi.org/10.1007/978-1-4612-3486-9

    Book  MATH  Google Scholar 

  31. Roberts, J.: Nonlinear analysis of slow drift oscillations of moored vessels in random seas. J. Ship Res. 25(2), 130–140 (1981)

    Google Scholar 

  32. Roberts, J.B., Vasta, M.: Energy-based stochastic estimation for nonlinear oscillators with random excitation. J. Appl. Mech. 67(4), 763–771 (2000). https://doi.org/10.1115/1.1330546

    Article  MathSciNet  MATH  Google Scholar 

  33. Scolan, Y.M.: Application de la méthode de Melnikov pour l’étude du roulis non linéaire des navires dans la houle. C. R. Acad. Sci. Ser. IIB Mech. Phys. Astron. 327(1), 1–6 (1999). https://doi.org/10.1016/S1287-4620(99)80002-7

    Article  MATH  Google Scholar 

  34. Simiu, E.: Chaotic Transitions in Deterministic and Stochastic Dynamical Systems: Applications of Melnikov Processes in Engineering, Physics, and Neuroscience, 1st edn. Princeton University Press, Princeton (2002)

    Book  MATH  Google Scholar 

  35. Somayajula, A., Falzarano, J.: Large-amplitude time-domain simulation tool for marine and offshore motion prediction. Mar. Syst. Ocean Technol. 10(1), 1–17 (2015). https://doi.org/10.1007/s40868-015-0002-7

    Article  Google Scholar 

  36. Somayajula, A., Falzarano, J.: Estimation of roll motion parameters using R-MISO system identification technique. In: Chung, J.S., Muskulus, M., Kokkinis, T., Wang, A.M. (eds.) 26th International Offshore and Polar Engineering (ISOPE 2016) Conference, vol. 3, pp. 568–574. International Society of Offshore and Polar Engineers (ISOPE), Rhodes (Rodos) (2016)

  37. Somayajula, A., Falzarano, J.: Application of advanced system identification technique to extract roll damping from model tests in order to accurately predict roll motions. Appl. Ocean Res. 67, 125–135 (2017). https://doi.org/10.1016/j.apor.2017.07.007

    Article  Google Scholar 

  38. Somayajula, A., Falzarano, J.: Critical assessment of reverse-MISO techniques for system identification of coupled roll motion of ships. J. Mar. Sci. Technol. 22(2), 231–244 (2017). https://doi.org/10.1007/s00773-016-0406-x

    Article  Google Scholar 

  39. Somayajula, A., Falzarano, J.: Volterra GZ approach—a new method to accurately calculate the non-linear and time-varying roll restoring arm of ships in irregular longitudinal seas. Ships Offshore Struct. 13(4), 423–431 (2018). https://doi.org/10.1080/17445302.2017.1409457

    Article  Google Scholar 

  40. Somayajula, A., Falzarano, J., Lutes, L.: An efficient assessment of vulnerability of a ship to parametric roll in irregular seas using first passage statistics. Probab. Eng. Mech. (2019, under review)

  41. Somayajula, A., Guha, A., Falzarano, J., Chun, H.H., Jung, K.H.: Added resistance and parametric roll prediction as a design criteria for energy efficient ships. Ocean Syst. Eng. 4(2), 117–136 (2014). https://doi.org/10.12989/ose.2014.4.2.117

    Article  Google Scholar 

  42. Somayajula, A.S.: Reliability assessment of hull forms susceptible to parametric roll in irregular seas. Ph.D. Dissertation, Texas A&M University, College Station (2017)

  43. Somayajula, A.S., Falzarano, J.: A comparative assessment of simplified models for simulating parametric roll. J. Offshore Mech. Arct. Eng. 139(2), 021103 (2017). https://doi.org/10.1115/1.4034921

    Article  Google Scholar 

  44. Somayajula, A.S., Falzarano, J.M.: Non-linear dynamics of parametric roll of container ship in irregular seas. In: Proceedings of 33rd International Conference on Ocean, Offshore and Arctic Engineering—Volume 7: Ocean Space Utilization; Professor Emeritus J. Randolph Paulling Honoring Symposium on Ocean Technology, p. V007T12A018. ASME, San Francisco (2014). https://doi.org/10.1115/OMAE2014-24186

  45. Somayajula, A.S., Falzarano, J.M.: Validation of Volterra series approach for modelling parametric rolling of ships. In: Proceedings of 34th International Conference on Ocean, Offshore and Arctic Engineering—Volume 11: Prof. Robert F. Beck Honoring Symposium on Marine Hydrodynamics, p. V011T12A043. ASME, St. John’s (2015). https://doi.org/10.1115/OMAE2015-41467

  46. Spyrou, K.J.: Dynamic instability in quartering seas: the behavior of a ship during broaching. J. Ship Res. 40(1), 46–59 (1996)

    Google Scholar 

  47. Spyrou, K.J., Thompson, J.M.T.: Damping coefficients for extreme rolling and capsize: an analytical approach. J. Ship Res. 44(1), 1–13 (2000)

    Google Scholar 

  48. Spyrou, K.J., Thompson, J.M.T.: The nonlinear dynamics of ship motions: a field overview and some recent developments. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 358(1771), 1735–1760 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  49. Su, Z.: Nonlinear response and stability analysis of vessel rolling motion in random waves using stochastic dynamical systems. Ph.D. Dissertation, Texas A&M University, College Station (2012). http://repositories.tdl.org/tdl-ir/handle/1969.1/ETD-TAMU-2012-08-11759

  50. Su, Z., Falzarano, J.M.: Gaussian and non-Gaussian cumulant neglect application to large amplitude rolling in random waves. Int. Shipbuild. Prog. 58, 97–113 (2011). https://doi.org/10.3233/ISP-2011-0071

    Article  Google Scholar 

  51. Su, Z., Falzarano, J.M.: Markov and Melnikov based methods for vessel capsizing criteria. Ocean Eng. 64, 146–152 (2013). https://doi.org/10.1016/j.oceaneng.2013.02.002

    Article  Google Scholar 

  52. Thompson, J.M.T., Rainey, R.C.T., Soliman, M.S.: Mechanics of ship capsize under direct and parametric wave excitation. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 338, 471–490 (1992). https://doi.org/10.1098/rsta.1992.0015

    Article  MATH  Google Scholar 

  53. Umeda, N., Francescutto, A.: Current state of the second generation intact stability criteria—achievements and remaining issues. In: Proceedings of the 15th International Ship Stability Workshop, June, pp. 13–15. Stockholm, Sweden (2016)

  54. Virgin, L.N.: The nonlinear rolling response of a vessel including chaotic motions leading to capsize in regular seas. Appl. Ocean Res. 9(2), 89–95 (1987). https://doi.org/10.1016/0141-1187(87)90011-3

    Article  Google Scholar 

  55. Webster, W.: Motion in regular waves—transverse motions. In: Lewis, E. (ed.) Principles of Naval Architecture, vol. III. SNAME, Jersey City (1989)

    Google Scholar 

  56. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, 1st edn. Springer, New York (1990)

    Book  MATH  Google Scholar 

  57. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Texts in Applied Mathematics, vol. 2, 2nd edn. Springer, New York (2003). https://doi.org/10.1007/b97481

    Book  MATH  Google Scholar 

  58. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Physica D Nonlinear Phenom. 16(3), 285–317 (1985). https://doi.org/10.1016/0167-2789(85)90011-9

    Article  MathSciNet  MATH  Google Scholar 

  59. Zhu, W.Q.: Lyapunov exponent and stochastic stability of quasi-non-integrable Hamiltonian systems. Int. J. Non-Linear Mech. 39(4), 569–579 (2004). https://doi.org/10.1016/S0020-7462(02)00223-8

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank Dr. Frans van Walree of MARIN for making available to us the modified C11 hull form description for our analysis. The authors would also like to thank Dr. Loren Lutes, emeritus professor with the Department of Civil Engineering at Texas A&M University, for his guidance in developing the Markov approach. Finally, the authors would like to thank Dr. Paul Hess for facilitating the funding from Office of Naval Research (ONR).

Funding

The research described in this paper was supported by the Office of Naval Research (ONR)—ONR Grant N00014-16-1-2281.

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  1. Department of Ocean Engineering, Indian Institute of Technology, Chennai, 600036, India

    Abhilash Somayajula

  2. Department of Ocean Engineering, Texas A&M University, College Station, 77843, USA

    Jeffrey Falzarano

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Somayajula, A., Falzarano, J. Parametric roll vulnerability of ships using Markov and Melnikov approaches. Nonlinear Dyn 97, 1977–2001 (2019). https://doi.org/10.1007/s11071-019-05090-7

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