On the parametric rolling of ships in a following sea under simultaneous nonlinear periodic surging

A new approach to the study of the parametric rolling of ships in a followi ng sea is presented. The new aspect is the consideration of the interferenc e of surge with the roll dynamics. When the waves are long and steep, the o scillatory component of the surge velocity can become large when compared w ith the mean value. As the oscillatory surge grows in amplitude, it tends a lso to become asymmetric due to the existence of nonlinearity The nature of asymmetric surging is such that a ship spends more time on the crests than on the troughs of the waves. This means, however, that the probability of capsize is increased because a ship's roll-restoring capability around the crest is at a minimum. We propose a new second-order differential equation of roll which incorporates automatically the surge effect through appropria te position-dependent coefficients. We explore numerically how this asymmet ry in surge influences the build up of parametric rolling. The layout of th e stability transition lines of the coupled system was found to be notably different from that of a Mathieu-type system. We pay attention also to the vicinity of surf-riding, where the capsize is more of a 'pure-loss' type.