Stochastic nonlinear ship rolling in random beam seas by the path integration method

Abstract Nonlinear ship rolling in random seas is a serious threat to ship stability. In this work, the dynamic stability of the vessel in random seas is evaluated by means of probabilistic methods. Specifically, the Markov diffusion theory is applied in order to study the stochastic aspects of the roll motion driven by random wave loads. The roll motion is modeled as a single-degree-of-freedom (SDOF) model in which the nonlinear damping and restoring terms as well as the random wave excitation are all incorporated. The stationary wave excitation moment in the SDOF model is represented as a filtered white noise by employing a second order linear filter. Therefore, a four-dimensional (4D) Markov dynamic system is established by combing the SDOF model with the linear filter model. Because the probabilistic property of the 4D Markov system is governed by the Fokker–Planck (FP) equation, the response statistics of roll motion can be obtained by solving the FP equation via an efficient 4D path integration (PI) method, which is based on the Markov property of the coupled dynamic system. Furthermore, the random wave excitation is approximated by an equivalent Gaussian white noise, and a two-dimensional (2D) PI technique is applied in order to obtain the response statistics of the dynamic system driven by this Gaussian white noise. The rationality and accuracy of applying the equivalent Gaussian white noise to simulate nonlinear ship rolling in random seas is studied. Moreover, the accuracy of the response statistics computed by the 4D and 2D PI techniques is verified by means of the versatile Monte Carlo simulation technique.

[1]  Hans Christian Karlsen,et al.  Statistics of Wave Induced Nonlinear Loads and Responses , 2006 .

[2]  Alberto Francescutto,et al.  LARGE AMPLITUDE ROLLING IN A REALISTIC SEA , 2004 .

[3]  Torgeir Moan,et al.  Stochastic Dynamics of Marine Structures: Index , 2012 .

[4]  Mircea Grigoriu,et al.  Response of stochastic dynamical systems driven by additive Gaussian and Poisson white noise : Solution of a forward generalized Kolmogorov equation by a spectral finite difference method , 1999 .

[5]  Miao Guo-ping.  Calculation of Ship Rolling Capsizing Probability by Path Integral Solution , 2013 .

[6]  Pol D. Spanos,et al.  Maximum roll angle estimation of a ship in confused sea waves via a quasi-deterministic approach , 2014 .

[7]  A. Naess,et al.  Response statistics of nonlinear, compliant offshore structures by the path integral solution method , 1993 .

[8]  Bernt J. Leira,et al.  Stochastic Dynamic Analysis and Reliability of a Vessel Rolling in Random Beam Seas , 2015 .

[9]  Arvid Naess,et al.  Pendulum's rotational motion governed by a stochastic Mathieu equation , 2013 .

[10]  Alberto Francescutto,et al.  A simplified modular approach for the prediction of the roll motion due to the combined action of wind and waves , 2004 .

[11]  J. B. Roberts.  Response of an oscillator with non-linear damping and a softening spring to non-white random excitation , 1986 .

[12]  Zhiyong Su,et al.  Nonlinear Response and Stability Analysis of Vessel Rolling Motion in Random Waves Using Stochastic Dynamical Systems , 2012 .

[13]  A. Naess,et al.  Response probability density functions of strongly non-linear systems by the path integration method , 2006 .

[14]  A. Jazwinski.  Stochastic Processes and Filtering Theory , 1970 .

[15]  Huan Lin,et al.  Chaotic roll motion and capsize of ships under periodic excitation with random noise , 1995 .

[16]  Katrin Ellermann,et al.  Nonlinear Dynamics of Offshore Systems in Random Seas , 2008 .

[17]  Marcello Vasta,et al.  Markov modelling and stochastic identification for nonlinear ship rolling in random waves , 2000, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[18]  B. Spencer,et al.  On the numerical solution of the Fokker-Planck equation for nonlinear stochastic systems , 1993 .

[19]  E. Kreuzer,et al.  Non-standard stochastic averaging of large-amplitude ship rolling in random seas , 2012, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[20]  Edwin Kreuzer,et al.  Probabilistic approach to large amplitude ship rolling in random seas , 2011 .

[21]  Naoya Umeda,et al.  Capsizing Probability Prediction for a Large Passenger Ship in Irregular Beam Wind and Waves: Comparison of Analytical and Numerical Methods , 2006 .

[22]  Armin W. Troesch,et al.  HIGHLY NONLINEAR ROLLING MOTION OF BIASED SHIPS IN RANDOM BEAM SEAS , 1996 .

[23]  A. Naess,et al.  Efficient path integration methods for nonlinear dynamic systems , 2000 .

[24]  Arvid Naess,et al.  Stochastic spur gear dynamics by numerical path integration , 2007 .

[25]  Pol D. Spanos,et al.  Stochastic response analysis of the softening Duffing oscillator and ship capsizing probability determination via a numerical path integral approach , 2014 .

[26]  Marwan Bikdash,et al.  Melnikov analysis for a ship with a general roll-damping model , 1994, Nonlinear Dynamics.

[27]  A. Pirrotta,et al.  Probabilistic response of nonlinear systems under combined normal and Poisson white noise via path integral method , 2011 .

[28]  Bernt J. Leira,et al.  Filter models for prediction of stochastic ship roll response , 2015 .