Dependence of Roll and Roll Rate in Nonlinear Ship Motions in Following and Stern Quartering Seas

  • Vadim L. BelenkyEmail author
  • Kenneth M. Weems
Part of the Fluid Mechanics and Its Applications book series (FMIA, volume 119)


The changing stability of a ship in waves may have a significant influence on the probabilistic properties of roll in irregular following and quartering seas. In particular, nonlinear effects may lead to dependence between roll angles and rates, which will have significant repercussions on the application of the theory of upcrossings for evaluating the probability of a stability failure related to roll motion such as capsizing. The roll response of a ship in a stationary seaway is a stationary stochastic process. For such a process, the roll angle and its first derivative are, by definition, not correlated and are often assumed to be independent. However, this independence can only be assumed a priori for normal processes, and the nonlinearity of large-amplitude roll motions can lead to a deviation from normal distribution. In the present work, the independence of roll angles and rates is studied from the results of numerical simulations from the Large Amplitude Motion Program (LAMP), which includes a general body-nonlinear calculation of the Froude-Krylov and hydrostatic restoring forces. These simulations show that, for the considered case, roll and roll rate are independent in beam seas, even though the distribution of the roll response is not normal. However, roll angles and roll rates for stern quartering seas are not independent.


Independence Correlation Roll motions Stern quartering seas 



The work described in this paper has been funded by the Office of Naval Research (ONR) under Dr. Patrick Purtell in 2011. The authors are grateful and happy to recognize contributions from their colleagues: Prof. Matthew Collette (formerly of SAIC, currently University of Michigan) discovered problems predicting the upcrossing rate with independence assumption; Prof. Ross Leadbetter (University of North Carolina) pointed out the possibility of dependence without correlation and hinted towards use of squares; and Prof. Pol Spanos (Rice University) motivated the authors to pursue this issue and has contributed to many fruitful discussions. Re-analysis of the results of simulation was carried out in 2017 and funded by NSWCCD Independent Applied Research (IAR) program under Dr. Jack Price. The authors are grateful to Prof. Vladas Pipiras (University of North Carolina) for fruitful discussions and, in particular, for help with confidence interval of the estimate of correlation coefficient. Finally, the authors are grateful Dr. Joel Park (NSWCCD) who made many useful editorial comments.


  1. Belenky, V., Weems, K., and Lin, W.-M. 2016 “Split-time method for estimation of probability of capsizing caused by pure loss of stability”, Ocean Engineering, Vol. 122, pp. 333–343.CrossRefGoogle Scholar
  2. Bishop, B, Belknap, W., Turner, C., Simon, B., Kim, J., 2005, “Parametric Investigation on the Influence of GM, Roll Damping, and Above-Water Form on the Roll Response of Model 5613,” Technical Report NSWCCD-50-TR-2005/027, Naval Surface Warfare Center Carderock Division, West Bethesda, Maryland USA.Google Scholar
  3. Bickel, P. J. and Doksum, K. A., 2001 Mathematical Statistics. Basic Ideas and Selected Topics Volume 1, Prentice Hall, Upper Saddle River, NJ, 2nd ed., 556 p.Google Scholar
  4. Cramer, H., and Leadbetter, M.R., 1967, Stationary and Related Stochastic Processes, John Wiley, New York.Google Scholar
  5. Levine, M. D., Belenky, V., Weems, K.M. and Pipiras. V. 2017, “Statistical Uncertainty Techniques for Analysis of Simulation and Model Test Data”, Proceedings of 30th American Towing Tank Conference, Carderock, Maryland, USA.Google Scholar
  6. Lin, W.-M., Yue, D. K. P., 1990. “Numerical Solutions for Large Amplitude Ship Motions in the Time-Domain” Proceedings of the 18th Symposium on Naval Hydrodynamics, Ann Arbor, Michigan, USA, pp. 41–66.Google Scholar
  7. Shin, Y.S, Belenky, V.L., Lin, W.M., Weems, K.M. and A. H. Engle, 2003 “Nonlinear time domain simulation technology for seakeeping and wave-load analysis for modern ship design” SNAME Trans. Vol. 111, pp. 557–578.Google Scholar
  8. Priestley, M. B., 1981, Spectral Analysis and Time Series, Vol. 1, Academic Press, New York.Google Scholar
  9. Scott, D.W., 1979, “On Optimal and Data-based Histograms,” Biometrika, 66(3):605–610.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.David Taylor Model Basin (NSWCCD)West BethesdaUSA

Personalised recommendations