Investigation on Parametrically Excited Motions of Spar Platforms in Waves

  • Claudio A. Rodríguez
  • Marcelo A. S. NevesEmail author
Part of the Fluid Mechanics and Its Applications book series (FMIA, volume 119)


As offshore oil exploration goes into deeper waters, spar platforms appear as a good alternative for oil field developments due to their inherent hydrodynamic behaviour regarding its vertical motions response in waves. Spar’s long natural periods in heave and pitch guarantee the good linear responses of the associated motions; however, nonlinear unstable motions can be triggered. Many numerical and experimental investigations have put forward the reasoning that this kind of floating structure is prone to parametrically induced motions due to changes in their pure hydrostatic restoring. Based on an analytical model, the present paper demonstrates that the main contribution to parametric excitation comes from variations in the pressure field associated to the incident wave and not from purely nonlinear hydrostatic actions. A nonlinear mathematical model based on Taylor series expansions, Neves and Rodríguez (Ocean Eng 33(14):1853–1883, 2006) , is employed to explain the underlying mechanism that leads to the phenomenon of Mathieu instability in vertical deep drafted cylinders. Analytical expressions are derived for the nonlinear hydrostatic and Froude-Krilov actions. A set of coupled time-dependent equations is obtained. Based on that, general conditions for the appearance of principal resonances are derived. These analytical results are verified by means of numerical simulations. Numerical analyses are carried out for regular wave conditions and a domain of parametric amplifications is obtained.


Spar platforms Stability Parametric roll Nonlinear dynamics 



The present investigation is supported by CNPq within the STAB project (Nonlinear Stability of Ships). The Authors also acknowledge financial support from CAPES, FAPERJ and LabOceano.


  1. Bin–Bin Li, Jin-Ping Ou and Bin Teng, 2011. “Numerical Investigation of Damping Effects on Coupled Heave and Pitch Motion of an Innovative Deep Draft Multi-SPAR.” Journal of Marine Science and Technology, Vol. 19, No 2, pp-231–244.Google Scholar
  2. Haslum, H.A., and Faltinsen, O.M., 1999. “Alternative Shape of Spar Platform for Use in Hostile Areas”, Proceedings of Offshore Technology Conference, Paper No. OTC10953, Houston.Google Scholar
  3. Hong, Y., Lee, D., Choi, Y., Hong, S. and Kim, S., 2005. “An Experimental Study on the Extreme Motion Responses of a SPAR Platform in the Heave Resonant Waves.” In: Proceedings of the 15th International Offshore and Polar Engineering Conference (ISOPE’2005), Seoul, Korea.Google Scholar
  4. Kleiman, A. and Gotlieb, O., 2008. “Nonlinear Dynamics and Internal Resonances of a Ship with a Rectangular Cross-Section in Head Seas.” In: Proceedings of the 27th ASME International Conference on Offshore Mechanics and Arctic Engineering, OMAE 2008, Paper OMAE2008-57691, Lisbon, June, 15–20.Google Scholar
  5. Koo, B.J., Kim, M.H. and Randall R.E., 2004. “Mathieu Instability of a Spar Platform with Mooring and Risers”. Ocean Engineering, (31), pp. 2175–2208.CrossRefGoogle Scholar
  6. Liao, S.-W. and Yeung, R. W., 2001. “Investigation of the Mathieu Instability of Roll Motion by a Time-Domain Viscous-Fluid Method”. The 16th International Workshop on Water Waves and Floating Bodies. Hiroshima, Japan, April.Google Scholar
  7. Liaw, C. Y., Bishop, S. R. and Thompson, J. M. T., 1993, “Heave-Excited Rolling Motion of a Rectangular Vessel in Head Seas”. International Journal of Offsore and Polar Engineering, vol. 3, no. 1 (Mar), pp. 26–31.Google Scholar
  8. Liaw, C. Y., 1994. “Dynamic Instability of a Parametrically Excited Ship Rolling Model”, International Journal of Offsore and Polar Engineering, vol. 4, no. 2 (Jun), pp. 106–111.Google Scholar
  9. Neves, M.A.S. and Rodríguez, C.A., 2006. “On Unstable Ship Motions Resulting from Strong Non-Linear Coupling”, Ocean Engineering, Vol. 33, no. 14 (Oct.), pp. 1853–1883.CrossRefGoogle Scholar
  10. Neves, M.A.S., Sphaier S.H., Mattoso, B.M., Rodríguez, C., Santos, A., Vileti, V. and Torres, F.G.S., 2008. “Parametric Resonance of Mono-column Structures”. In: Proceedings of the 6th Osaka Colloquium on Seakeeping and Stability of Ships, 26–28th March, Osaka, Japan.Google Scholar
  11. Paulling, J.R., 1961. “The Transverse Stability of a Ship in a Longitudinal Seaway”. Journal of Ship Research, vol. 4, no. 4 (Mar.), pp. 37–49.Google Scholar
  12. Rho, J.B., Choi, H.S., Lee, W.C.Shin, H.S. and Park, I.K., 2002. “Heave and Pitch Motions of a SPAR Platform with Damping Plate.” In: Proceedings of the 12th International Offshore and Polar Engineering Conference (ISOPE’2002), Kitakyushu, Japan.Google Scholar
  13. Rodríguez, C.A., 2010. “On the Nonlinear Dynamics of Parametric Rolling”. D.Sc. Thesis, COPPE - Universidade Federal do Rio de Janeiro, Brazil. (in Portuguese).Google Scholar
  14. Rodríguez, C.A. and Neves, M.A.S., 2012a. “Nonlinear Instabilities of SPAR Platforms in Waves”. Paper OMAE2012-83577, Rio de Janeiro, July.Google Scholar
  15. Rodríguez, C.A. and Neves, M.A.S., 2012b. “Domains of Parametric Roll Amplification for Different Hull Forms”. In: Fossen, T.I., Nijmeijer, H. (eds.), “Parametric Resonance in Dynamical Systems”,, Chapter 6, pp 107–127.Google Scholar
  16. Yumin Liu, Hongmei Yan and Tin-Woo Yung, 2010. “Nonlinear Resonant Response of Deep Draft Platforms in Surface Waves”, Paper OMAE2010-20823, Shanghai, China.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.LabOceano, COPPE/UFRJRio de JaneiroBrazil

Personalised recommendations