Table of Contents Author Guidelines Submit a Manuscript
Table of Contents Alerts

To receive news and publication updates for Mathematical Problems in Engineering, enter your email address in the box below.

Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 934714, 32 pages
http://dx.doi.org/10.1155/2010/934714
Review Article

Modeling of Ship Roll Dynamics and Its Coupling with Heave and Pitch

R. A. Ibrahim and I. M. Grace

Department of Mechanical Engineering, Wayne State University, Detroit, MI 48202, USA

Received 14 May 2009; Accepted 11 June 2009

Academic Editor: Jose Balthazar

Copyright © 2010 R. A. Ibrahim and I. M. Grace. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. H. Nayfeh, D. T. Mook, and L. R. Marshall, “Nonlinear coupling of pitch and roll modes in ship motions,” Journal of Hydronautics, vol. 7, no. 4, pp. 145–152, 1973. View at Google Scholar
  2. A. H. Nayfeh, D. T. Mook, and L. R. Marshall, “Perturbation-energy approach for the development of the nonlinear equations of ship motion,” Journal of Hydronautics, vol. 8, no. 4, pp. 130–136, 1974. View at Google Scholar
  3. A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations, Pure and Applied Mathematics, John Wiley & Sons, New York, NY, USA, 1979. View at MathSciNet
  4. B. M. Suleiman, Identification of finite-degree-of-freedom models for ship motions, Ph.D. thesis, Virginia Polytechnic Institute and State University, Blacksburg, Va, USA, 2000.
  5. T. I. Fossen, Guidance and Control of Ocean Vehicles, John Wiley & Sons, New York, NY, USA, 1994.
  6. E. M. Lewandowski, The Dynamics of Marine Craft Maneuvering and Seakeeping, vol. 2 of Advanced Series on Ocean Engineering, World Scientific, Singapore, 2004.
  7. T. Perez, Ship Motion Control; Course Keeping and Roll Stabilization Using Rudder and Fins, Advances in Industrial Control, Springer, Berlin, Germany, 2005.
  8. J. R. Paulling and R. M. Rosenberg, “On unstable ship motions resulting from nonlinear coupling,” Journal of Ship Research, vol. 3, no. 1, pp. 36–46, 1959. View at Google Scholar
  9. W. D. Kinney, “On the unstable rolling motions of ships resulting from nonlinear coupling with pitch including the effect of damping in roll,” Institute of Engineering Research, University of California, Berkeley, Calif, USA, 1961.
  10. M. R. Haddara, “On the stability of ship motion in regular oblique waves,” International Shipbuilding Progress, vol. 18, no. 207, pp. 416–434, 1971. View at Google Scholar
  11. M. A. S. Neves and C. A. Rodríguez, “On unstable ship motions resulting from strong non-linear coupling,” Ocean Engineering, vol. 33, no. 14-15, pp. 1853–1883, 2006. View at Publisher · View at Google Scholar
  12. M. A. S. Neves and C. A. Rodríguez, “Influence of non-linearities on the limits of stability of ships rolling in head seas,” Ocean Engineering, vol. 34, no. 11-12, pp. 1618–1630, 2007. View at Publisher · View at Google Scholar
  13. J. H. Vugts, “The hydrodynamic forces and ship motions on oblique waves,” Tech. Rep. 150 S, Netherlands Ship Research Center, 1971. View at Google Scholar
  14. J. N. Newman, Marine Hydrodynamics, The MIT Press, Cambridge, Mass, USA, 1977.
  15. L. V. Edward, Principles of Naval Architecture, The Society of Naval Architects and Marine Engineers, Jersey City, NJ, USA, 1989.
  16. H. Chu, Numerical modeling of ship motions in shallow water waves, Ph.D. thesis, Florida Institute of Technology, Melbourne, Fla, USA, 1998.
  17. Y. Lee, Non linear ship motion models to predict capsize in regular beam seas, Ph.D. thesis, University of Michigan, Ann Arbor, Mich, USS, 2001.
  18. M. S. Khalid, Simulation of Euler equations of motion and blended—nonlinear hydrodynamics for multi-hulled vessels, Ph.D. thesis, University of Michigan, Ann Arbor, Mich, USA, 2007.
  19. M. A. Abkowitz, “Lectures on ship hydrodynamics—steering and maneuverability,” Tech. Rep. Hy-5, Hydro- and Aerodynamics Laboratory, Lyngby, Denmark, 1964. View at Google Scholar
  20. M. A. Abkowitz, “System identification techniques for ship maneuvering trials,” in Proceedings of the Symposium on Control Theory and Navy Applications, pp. 337–393, Monterey, Calif, USA, 1975.
  21. M. A. Abkowitz, “Measurements of hydrodynamic characteristics from ship maneuvering trials by system identification,” Transactions of the Society of Architects and Marine Engineering, vol. 88, pp. 283–318, 1981. View at Google Scholar
  22. W. Y. Hwang, Application of system identification to ship maneuvering, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, Mass, USA, 1980.
  23. C. G. Källström and K. J. Åström, “Experiences of system identification applied to ship steering,” Automatica, vol. 17, no. 1, pp. 187–198, 1981. View at Publisher · View at Google Scholar
  24. K.-H. Son and K. Nomoto, “On the coupled motion of steering and rolling of a high-speed container ship,” Naval Architecture and Ocean Engineering, vol. 20, pp. 73–83, 1982. View at Google Scholar
  25. A. Ross, Nonlinear Manoeuvring Models for Ships: a Lagrangian Approach, Ph.D. thesis, Norwegian University of Science and Technology, Trondheim, Norway, 2008.
  26. G. Kirchhoff, Advanced Engineering Mathematics, John Wiley & Sons, New York, NY, USA, 1869.
  27. L. M. Milne-Thomson, Theoretical Hydrodynamics, MacMillan, London, UK, 1968.
  28. T. I. Fossen, Marine Control Systems: Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles, Marine Cyberentics AS, Trondheim, Norway, 2002.
  29. Z. Rong, Studies of weak and strong nonlinear sea loads on floating marine structures, M.S. thesis, Norges Tekniske Hoegskol, Trodheim, Norway, April 1994.
  30. L. Alford, A. Banik, V. Belenky et al., “Discussion on analytical approaches to the study of vessel dynamics-outcomes of the two-part mini symposium at the 2007 SIAM conference on applications of dynamical systems,” Marine Technology, vol. 45, no. 4, pp. 211–220, 2008. View at Google Scholar
  31. J. O. de Kat and J. R. Paulling, “The simulation of ship motions and capsizing in severe seas,” Transactions of the Society of Architects and Marine Engineering, vol. 97, pp. 139–168, 1989. View at Google Scholar
  32. J. A. Witz, C. B. Ablett, and J. H. Harrison, “Roll response of semisubmersibles with nonlinear restoring moment characteristics,” Applied Ocean Research, vol. 11, pp. 153–166, 1989. View at Google Scholar
  33. J.-P. F. Denise, “On the roll motion of barges,” Transactions of the Royal Institution of Naval Architects, vol. 125, pp. 255–268, 1983. View at Google Scholar
  34. D. W. Bass and M. R. Haddara, “On the modeling of the nonlinear damping moment for the rolling motion of a ship,” in Proceedings of the International Association of Scientific Technology (IASTED) Symposium: Identification, Modeling and Simulation, pp. 346–349, Paris, France, 1987.
  35. D. W. Bass and M. R. Haddara, “Nonlinear models of ship roll damping,” International Shipbuilding Progress, vol. 35, no. 401, pp. 5–24, 1988. View at Google Scholar
  36. M. Taylan, “The effect of nonlinear damping and restoring in ship rolling,” Ocean Engineering, vol. 27, no. 9, pp. 921–932, 2000. View at Publisher · View at Google Scholar
  37. J. F. Dalzell, “A note on the form of ship roll damping,” Journal of Ship Research, vol. 22, no. 3, pp. 178–185, 1978. View at Google Scholar
  38. A. Cardo, A. Francescutto, and R. Nabergoj, “On damping models in free and forced rolling motion,” Ocean Engineering, vol. 9, no. 2, pp. 171–179, 1982. View at Publisher · View at Google Scholar
  39. J. B. Mathisen and W. G. Price, “Estimation of ship roll damping coefficients,” Transactions Royal Institution of Naval Architects, vol. 127, pp. 295–307, 1984. View at Google Scholar
  40. A. F. El-Bassiouny, “Nonlinear analysis for a ship with a general roll-damping model,” Physica Scripta, vol. 75, no. 5, pp. 691–701, 2007. View at Publisher · View at Google Scholar
  41. M. Taylan, “Effect of forward speed on ship rolling and stability,” Mathematical and Computational Applications, vol. 9, no. 2, pp. 133–145, 2004. View at Google Scholar
  42. A. H. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics, Wiley Series in Nonlinear Science, John Wiley & Sons, New York, NY, USA, 1995. View at MathSciNet
  43. L. Arnold, I. Chueshov, and G. Ochs, “Stability and capsizing of ships in random sea-a survey,” Nonlinear Dynamics, vol. 36, no. 2–4, pp. 135–179, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  44. S. Surendran, S. K. Lee, J. V. R. Reddy, and G. Lee, “Non-linear roll dynamics of a Ro-Ro ship in waves,” Ocean Engineering, vol. 32, no. 14-15, pp. 1818–1828, 2005. View at Publisher · View at Google Scholar
  45. G. Bulian, “Nonlinear parametric rolling in regular waves–a general procedure for the analytical approximation of the GZ curve and its use in time domain simulations,” Ocean Engineering, vol. 32, no. 3-4, pp. 309–330, 2005. View at Publisher · View at Google Scholar
  46. J. B. Roberts, “A stochastic theory for nonlinear ship rolling in irregular seas,” Journal of Ship Research, vol. 26, no. 4, pp. 229–245, 1982. View at Google Scholar
  47. J. B. Roberts, “Effect of parametric excitation on ship rolling motion in random waves,” Journal of Ship Research, vol. 26, no. 4, pp. 246–253, 1982. View at Google Scholar
  48. J. Falzarano and F. Zhang, “Multiple degree of freedom global analysis of transient ship rolling motion,” in Proceedings of the ASME Winter Annual Meeting on Nonlinear Dynamics of Marine Vehicles, J. M. Falzarano and F. Papoulias, Eds., vol. 51, pp. 57–72, New York, NY, USA, 1993.
  49. X. Huang, X. Gu, and W. Bao, “The probability distribution of rolling amplitude of a ship in high waves,” in Proceedings of the 5th International Conference on Stability of Ships and Ocean Vehicles, Melbourne, Fla, USA, 1994.
  50. I. Senjanović, G. Ciprić, and J. Parunov, “Survival analysis of fishing vessels rolling in rough seas,” Philosophical Transactions of the Royal Society A, vol. 358, no. 1771, pp. 1943–1965, 2000. View at Google Scholar
  51. N. K. Moshchuk, R. A. Ibrahim, R. Z. Khasminskii, and P. L. Chow, “Asymptotic expansion of ship capsizing in random sea waves-I. First-order approximation,” International Journal of Non-Linear Mechanics, vol. 30, no. 5, pp. 727–740, 1995. View at Google Scholar · View at MathSciNet
  52. G. H. Bryan, “The action of bilge keels,” Transactions of the Institute of Naval Architects, vol. 42, pp. 198–239, 1900. View at Google Scholar
  53. T. Hishida, “Study on the wave making resistance for rolling ships—parts 1 and 2,” Journal of Society of Naval Architecture of Japan, vol. 85, pp. 45–61, 1952. View at Google Scholar
  54. T. Hishida, “Study on the wave making resistance for rolling ships—parts 3 and 4,” Journal of Society of Naval Architecture of Japan, vol. 86, pp. 277–283, 1954. View at Google Scholar
  55. T. Hishida, “Study on the wave making resistance for rolling ships—parts 5 and 6,” Journal of Society of Naval Architecture of Japan, vol. 87, pp. 67–68, 1955. View at Google Scholar
  56. H. Kato, “On the frictional resistance of the roll of ships,” The Science of Naval Architects of Japan, vol. 102, pp. 115–123, 1958. View at Google Scholar
  57. M. Martin, “Roll damping due to bilge keels,” Tech. Rep., Iowa University, Institute of Hydraulics Research, Iowa City, Iowa, USA, 1958. View at Google Scholar
  58. M. Tanaka, “A study on the bilge keel—part 1,” Journal of Society of Naval Architecture of Japan, vol. 101, pp. 99–105, 1957. View at Google Scholar
  59. M. Tanaka, “A study on the bilge keel—part 2,” Journal of Society of Naval Architecture of Japan, vol. 103, p. 343, 1958. View at Google Scholar
  60. M. Tanaka, “A study on the bilge keel—part 3,” Journal of Society of Naval Architecture of Japan, vol. 105, pp. 27–32, 1959. View at Google Scholar
  61. M. Tanaka, “A study on the bilge keel—part 4,” Journal of Society of Naval Architecture of Japan, vol. 104, pp. 205–212, 1961. View at Google Scholar
  62. H. Kato, “Effect of bilge keels on the rolling of ships,” Journal of Society of Naval Architecture of Japan, vol. 117, pp. 93–114, 1965. View at Google Scholar
  63. C. G. Moody, “The effect of bilge keels on the roll damping characteristics of the large bulk-oil carrier SS World Glory,” Tech. Rep., David Taylor Model Basin, West Bethesda, Md, USA, 1961. View at Google Scholar
  64. L. E. Motter, “Ship motion and bilge keel studies for the CVA,” Tech. Rep. OPNAVINST S5513.3A-17, David Taylor Model Basin, West Bethesda, Md, USA, 1967. View at Google Scholar
  65. H. Jones, “Bilge keel size investigation for an attack aircraft carrier (CVA-41),” Tech. Rep. DTNSRDC/SPD-0861-01, David Taylor Model Basin, West Bethesda, Md, USA, 1978. View at Google Scholar
  66. H. Jones, “Bilge keel size investigation for an attack aircraft carrier and vstol (vertical short takeoff and landing,” Tech. Rep. DTNSRDC/SPD-0861-02, David Taylor Model Basin, West Bethesda, Md, USA, 1979. View at Google Scholar
  67. R. E. D. Bishop and W. G. Price, “A note on structural damping of ship hulls,” Journal of Sound and Vibration, vol. 56, no. 4, pp. 495–499, 1978. View at Publisher · View at Google Scholar
  68. Y. Himeno, “Prediction of ship roll damping: state-of-the-art,” Tech. Rep. 239, Department of Naval Architecture and Marine Engineering, The University of Michigan, Ann Arbor, Mich, USA, September 1981. View at Google Scholar
  69. R.-Q. Lin, W. Kuang, and A. M. Reed, “Numerical modeling of nonlinear interactions between ships and surface gravity waves—part I: ship waves in calm water,” Journal of Ship Research, vol. 49, no. 1, pp. 1–11, 2005. View at Google Scholar
  70. R.-Q. Lin and W. Kuang, “Nonlinear ship-wave interaction model—part 2: ship boundary condition,” Journal of Ship Research, vol. 50, no. 2, pp. 181–186, 2006. View at Google Scholar
  71. R.-Q. Lin and W. Kuang, “Modeling nonlinear roll damping with a self-consistent, strongly nonlinear ship motion model,” Journal of Marine Science and Technology, vol. 13, no. 2, pp. 127–137, 2008. View at Publisher · View at Google Scholar
  72. N. Salvesen, “Five years of numerical naval ship hydrodynamics at DTNSRDC,” Tech. Rep. DTNSRDC-79/082, David W. Taylor Naval Ship Research and Development Center, Bethesda, Md, USA, 1979. View at Google Scholar
  73. W. M. Lin, M. J. Meinhold, and N. Salvesen, “Nonlinear time-domain simulation of fishing vessel capsizing in quartering seas,” Tech. Rep. SAIC-CG-D-17-98, Science Applications International, Annapolis, Md, USA, April 1998. View at Google Scholar
  74. S. Chakrabarti, “Empirical calculation of roll damping for ships and barges,” Ocean Engineering, vol. 28, no. 7, pp. 915–932, 2001. View at Publisher · View at Google Scholar
  75. Y. Ikeda, “Roll damping of ships,” in Proceedings of the Ship Motions, Wave Loads and Propulsive Performance in a Seaway, 1st Marine Dynamics Symposium, pp. 241–250, The Society of Naval Architecture in Japan, 1984.
  76. Y. Ikeda, Y. Himeno, and N. Tanaka, “Components of roll damping of ships at forward speed,” Journal of Society of Naval Architecture of Japan, vol. 143, pp. 121–133, 1978. View at Google Scholar
  77. Y. Ikeda, Y. Himeno, and N. Tanaka, “A prediction method for ships roll damping,” Tech. Rep. 00405, Department of Naval Architecture, University of Osaka Prefecture, Osaka, Japan, 1978. View at Google Scholar
  78. Y. Ikeda, T. Fujiwara, and T. Katayama, “Roll damping of a sharp-cornered barge and roll control by a new-type stabilizer,” in Proceedings of the 3rd International Offshore and Polar Engineering Conference, pp. 634–639, Singapore, June 1993.
  79. J. P. Morison, M. P. O'Brien, J. W. Johnson, and S. A. Schaaf, “The force exerted by surface waves on a pile,” Petroleum Transactions, vol. 189, pp. 149–154, 1950. View at Google Scholar
  80. T. Sarpkaya and M. Isaacson, Mechanics of Wave Forces on Offshore Structures, Van Nostrand Reinhold, New York, NY, USA, 1981.
  81. M. R. Haddara, “On the random decrement for nonlinear rolling motion,” in Proceedings of the 11th International Offshore Mechanics and Arctic Engineering Symposium, vol. 2 of Safety and Reliability, pp. 321–324, Calgary, Canada, 1992.
  82. X. Wu, L. Tao, and Y. Li, “Nonlinear roll damping of ship motions in waves,” Journal of Offshore Mechanics and Arctic Engineering, vol. 127, no. 3, pp. 205–211, 2005. View at Publisher · View at Google Scholar
  83. N. Salvesen, O. E. Tuck, and O. Faltinsen, “Ship motions and sea loads,” Transactions Society of Architects and Marine Engineering, vol. 78, pp. 250–287, 1970. View at Google Scholar
  84. W. G. Meyers, D. J. Sheridan, and N. Salvesen, “Manual: NSRDC ship-motion and sea-load computer program,” Tech. Rep. 3376, Naval Ship Research and Development Center, Carderock, Md, USA, 1975. View at Google Scholar
  85. A. H. Nayfeh, S. A. Ragab, and A. A. Al-Maaitah, “Effect of bulges on the stability of boundary layers,” Physics of Fluids, vol. 31, no. 4, pp. 796–806, 1988. View at Google Scholar
  86. J. H. G. Wright and W. B. Marshfield, “Ship roll response and capsize beviour in beam seas,” Transactions of the Institute of Naval Architects, no. 3, pp. 129–148, 1979. View at Google Scholar
  87. C. Kuo and Y. Odabasi, “Theoretical studies on intact stability of ships (Phase 2),” Department of Shipbuilding and Naval Architecture, University of Strathclyde, October 1974.
  88. W. M. Lin and N. Salvesen, “Nonlinear time-domain simulation of ship capsizing in beam seas,” Final Report, Science Applications International, Annapolis, Md, USA, November 1997. View at Google Scholar
  89. S. Surendran and J. V. R. Reddy, “Roll dynamics of a Ro-Ro ship,” International Shipbuilding Progress, vol. 49, no. 4, pp. 301–320, 2002. View at Google Scholar
  90. S. Surendran and J. V. R. Reddy, “Numerical simulation of ship stability for dynamic environment,” Ocean Engineering, vol. 30, no. 10, pp. 1305–1317, 2003. View at Publisher · View at Google Scholar
  91. G. Contento, A. Francescutto, and M. Piciullo, “On the effectiveness of constant coefficients roll motion equation,” Ocean Engineering, vol. 23, no. 7, pp. 597–618, 1996. View at Publisher · View at Google Scholar
  92. A. Francescutto, G. Contento, M. Biot, L. Schiffrer, and G. Caprino, “Effect of excitation modeling in the parameter estimation of nonlinear rolling,” in Proceedings of the 8th International Offshore and Polar Engineering Conference, vol. 3, pp. 490–498, Monteral, Canada, May 1998.
  93. A. Francescutto and G. Contento, “Bifurcations in ship rolling: experimental results and parameter identification technique,” Ocean Engineering, vol. 26, no. 11, pp. 1095–1123, 1999. View at Publisher · View at Google Scholar
  94. A. B. Mahfouz, “Identification of the nonlinear ship rolling motion equation using the measured response at sea,” Ocean Engineering, vol. 31, no. 17-18, pp. 2139–2156, 2004. View at Publisher · View at Google Scholar
  95. W. E. Cummins, “The impulse response function and ship motions,” Schiffstechnik, vol. 9, pp. 101–109, 1962. View at Google Scholar
  96. J.-S. Chung and M. M. Bernitsas, “Hydrodynamic memory effect on stability, bifurcation, and chaos of two-point mooring systems,” Journal of Ship Research, vol. 41, no. 1, pp. 26–44, 1997. View at Google Scholar
  97. L. J. Tick, “Differential equations with frequency-dependent coefficients,” Journal of Ship Research, vol. 3, no. 2, pp. 45–46, 1959. View at Google Scholar
  98. W. R. McCreight, “Ship maneuvering in waves,” in Proceedings of the 16th Symposium on Naval Hydrodynamics: Ship Wakes; Large Amplitude Waves; Real Fluid Effects in Ship Hydrodynamics; Fluid-Structure Interaction; Frontier Problems in Hydrodynamics, pp. 456–469, National Research Council, US Naval Studies Board, University of California, 1986, US Office of Naval Research, USA.
  99. T. Jiang, T. E. Schellin, and S. D. Sharma, “Maneuvering simulation of a tanker moored in a steady current including hydrodynamic memory effect and stability analysis,” in Proceedings of the Royal Institution of Naval Architects (RINA) International Conference on Ship Maneuvering, London, UK, 1987.
  100. S. D. Sharma, T. Jiang, and T. E. Schellin, “Dynamic instability and chaotic motions of a single-point-moored tanker,” in Proceedings of the 17th Office of Naval Research (ONR) Symposium on Naval Hydrodynamics, The Hague, The Netherlands, 1988.
  101. M. Schmiechen, “Zur kollisiondynamik von Schiffen,” Jahrbuch der Schiffbautechnischen Gesellschaft, vol. 68, 1974. View at Google Scholar
  102. M. Schmiechen, “Equation for non-quasi-steady ship motion,” in Proceedings of the 14th International Towing Tank Conference, Ottawa, Canada, 1975.
  103. A. Francescutto, A. Serra, and S. Scarpa, “A critical analysis of weather criterion for intact stability of large passenger vessels,” in Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering (OMAE '01), vol. 1, pp. 829–836, 2001.
  104. K. Spyrou, “A basis for developing a rational alternative to the weather criterion: problems and capabilities,” in Proceedings of the 6th International Ship Stability Workshop, October 2002.
  105. J. M. T. Thompson, “Designing against capsize in beam seas: recent advances and new insights,” Applied Mechanics Reviews, vol. 50, no. 5, pp. 307–324, 1997. View at Google Scholar
  106. A. N. Lansbury, J. M. T. Thompson, and H. B. Stewart, “Basin erosion in the twin-well Duffing oscillator: two distinct bifurcations scenarios,” International Journal of Bifurcation and Chaos, vol. 2, pp. 505–532, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  107. J. M. T. Thompson, “Loss of engineering integrity due to the erosion of absolute and transient basin boundaries,” in Proceedings of IUTAM Symposium ‘Nonlinear Dynamics in Engineering Systems’, W. Schiehlen, Ed., Springer, Berlin, Germany, August 1989.
  108. J. M. T. Thompson, “Chaotic phenomena triggering the escape from a potential well,” Proceedings Royal Society London A, vol. 421, pp. 195–225, 1989. View at Google Scholar · View at MathSciNet
  109. J. M. T. Thompson and F. A. McRobie, “Intermediate bifurcations and global dynamics of driven oscillators,” in Proceedings of the 1st European Nonlinear Oscillations Conference, E. Kreuzer and G Schmidt, Eds., vol. 72 of Mathematical Research, pp. 107–128, Academie, Hamburg, Germany, 1993. View at MathSciNet
  110. W. Froude, “Remarks on Mr. Scott Russell's paper on rolling,” Transactions of the Institute of Naval Research, vol. 4, pp. 232–275, 1863. View at Google Scholar
  111. W. G. Price and R. E. D. Bishop, Probabilistic Theory of Ship Dynamics, Chapman & Hall, London, UK, 1974.
  112. A. R. J. M. Lloyd, Seakeeping—Ship Behavior in Rough Weather, Ellis Howood, Chichester, UK, 1989.
  113. W. G. Price, “A stability analysis of the roll motion of a ship in an irregular seaway,” International Shipbuilding Progress, vol. 22, no. 247, pp. 103–112, 1975. View at Google Scholar
  114. M. R. Haddara, “A study of stability of the mean and variance of rolling motion in random waves,” in Proceedings of the International Conference on Stability of Ships and Ocean Vehicles, University of Strathclyde, Glasgow, UK, 1975.
  115. P. K. Muhuri, “A study of the stability of the rolling motion of a ship in an irregular seaway,” International Shipbuilding Progress, vol. 27, no. 310, pp. 139–142, 1980. View at Google Scholar
  116. J. B. Roberts and N. M. C. Dacunha, “Roll motion of a ship in random beam waves: comparison between theory and experiment,” Journal of Ship Research, vol. 29, no. 2, pp. 112–126, 1985. View at Google Scholar
  117. M. R. Haddara, “Note on the power spectrum of nonlinear rolling motion,” International Shipbuilding Progress, vol. 30, no. 342, pp. 41–44, 1983. View at Google Scholar
  118. S. Kwon, D. Kim, and J. Chung, “Application of path integral solution to ship rolling motion,” in Proceedings of the 3rd International Offshore and Polar Engineering Conference (ISOPE '93), pp. 657–660, Singapore, 1993.
  119. H. Lin and S. C. S. Yim, “Chaotic roll motion and capsize of ships under periodic excitation with random noise,” Applied Ocean Research, vol. 17, no. 3, pp. 185–204, 1995. View at Publisher · View at Google Scholar
  120. J.-Y. Gu, “Calculation of ship rolling probability using a new path integration method,” Journal of Ship Mechanics, vol. 10, no. 6, pp. 43–52, 2006. View at Google Scholar
  121. J.-Y. Gu, “Probability analysis of ship nonlinear roll-motion excited by white noise,” Journal of Ship Mechanics, vol. 10, no. 3, pp. 17–25, 2006. View at Google Scholar
  122. S. C. Yim, T. Nakhata, W. A. Bartel, and E. T. Huang, “Coupled nonlinear barge motions—part I: deterministic models development, identification and calibration,” in Proceedings of the 23rd ASME International Conference on Offshore Mechanics and Arctic Engineering (OMAE '04), vol. 1, pp. 281–291, Vancouver, Canada, 2004.
  123. S. C. Yim, T. Nakhata, and E. T. Huang, “Coupled nonlinear barge motions—part II: deterministic models stochastic models and stability analysis,” in Proceedings of the 23rd ASME International Conference on Offshore Mechanics and Arctic Engineering (OMAE '04), vol. 1A, pp. 293–302, Vancouver, Canada, 2004.
  124. L. Liu and T. Yougang, “Stability of ships with water on deck in random beam waves,” Journal of Vibration and Control, vol. 13, no. 3, pp. 269–280, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  125. E. Mamontov and A. Naess, “An analytical-numerical method for fast evaluation of probability densities for transient solutions of nonlinear Ito's stochastic differential equations,” International Journal of Engineering Science, vol. 47, no. 1, pp. 116–130, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  126. M.R. Haddara and Y. Zhang, “On the joint probability density function of non-linear rolling motion,” Journal of Sound and Vibration, vol. 169, no. 4, pp. 562–569, 1994. View at Publisher · View at Google Scholar
  127. C. Jiang, A. W. Troesch, and S. W. Shaw, “Highly nonlinear rolling motion of biased ships in random beam seas,” Journal of Ship Research, vol. 40, no. 2, pp. 125–135, 1996. View at Google Scholar
  128. J.-Y. Gu, “Nonlinear rolling motion of ship in random beam seas,” Journal of Marine Science and Technology, vol. 12, no. 4, pp. 273–279, 2004. View at Google Scholar
  129. Y.-G. Tang, J.-Y. Gu, H.-Y. Zheng, and H.-X. Li, “Study on the ship capsize in random beam seas using Melnikov method,” Journal of Ship Mechanics, vol. 8, no. 5, pp. 27–34, 2004 (Chinese). View at Google Scholar
  130. L.-Q. Liu, Y.-G. Tang, H.-Y. Zheng, and J.-Y. Gu, “Analysis method of capsizing probability in time domain for ships in random beam waves,” Journal of Tianjin University Science and Technology, vol. 39, no. 2, pp. 165–169, 2006 (Chinese). View at Google Scholar
  131. L.-Q. Liu, Y.-G. Tang, and H.-X. Li, “Stochastic chaotic motion of ships in beam seas,” Journal of Marine Science and Technology, vol. 15, no. 2, pp. 123–128, 2007. View at Google Scholar
  132. L. S. Pontryagin, A. A. Andonov, and A. A. Vitt, “On statistical consideration of dynamical systems,” Journal of Experimental and Theoretical Physics, vol. 3, no. 3, pp. 165–172, 1933. View at Google Scholar
  133. G. Q. Cai and Y. K. Lin, “On statistics of first-passage failure,” Journal of Applied Mechanics, vol. 61, pp. 93–99, 1994. View at Google Scholar · View at Zentralblatt MATH
  134. G. Q. Cai, J. S. Yu, and Y. K. Lin, “Ship rolling in random sea,” in Stochastic Dynamics and Reliability of Nonlinear Ocean Systems, R. A. Ibrahim and Y. K. Lin, Eds., vol. 77, pp. 81–88, ASME, New York, NY, USA, 1994. View at Google Scholar
  135. N. K. Moshchuk, R. Z. Khasminskii, R. A. Ibrahim, and P. L. Chow, “Asymptotic expansion of ship capsizing in random sea waves-II. Second-order approximation,” International Journal of Non-Linear Mechanics, vol. 30, no. 5, pp. 741–757, 1995. View at Google Scholar · View at MathSciNet
  136. R. A. Ibrahim, N. G. Chalhoub, and J. Falzarano, “Interaction of ships and ocean structures with ice loads and stochastic ocean waves,” Applied Mechanics Reviews, vol. 60, no. 5, pp. 246–290, 2007. View at Publisher · View at Google Scholar