Efficient Estimation of Extreme Non-linear Roll Motions using the First-order Reliability Method (FORM)

In on-board decision support systems efficient procedures are needed for real-time estimation of the maximum ship responses to be expected within the next few hours, given on-line information on the sea state and user defined ranges of possible headings and speeds. For linear responses standard frequency domain methods can be applied. To non-linear responses like the roll motion, standard methods like direct time domain simulations are not feasible due to the required computational time. However, the statistical distribution of non-linear ship responses can be estimated very accurately using the first-order reliability method (FORM), well-known from structural reliability problems. To illustrate the proposed procedure, the roll motion is modelled by a simplified non-linear procedure taking into account non-linear hydrodynamic damping, time-varying restoring and wave excitation moments and the heave acceleration. Resonance excitation, parametric roll and forced roll are all included in the model, albeit with some simplifications. The result is the mean out-crossing rate of the roll angle together with the corresponding most probable wave scenarios (critical wave episodes), leading to user-specified specific maximum roll angles. The procedure is computationally very effective and can thus be applied to real-time determination of ship specific combinations of heading and speed to be avoided in the actual sea state.