A Stochastic Analysis of Nonlinear Rolling in a Narrow Band Sea

The extensive analysis carried out in last years on the nonlinear rolling of a ship, through the use of different versions of perturbation methods, allowed many researchers to discover some very peculiar features of this motion. In a deterministic beam sea, it appeared the possibility of resonances different from synchronism, in particular suharmonic oscillations in both the upright and heeled conditions. Moreover, the resonance peaks as a function of tuning ratio are bent, mainly as a consequence of righting arm nonlinearity, so that in a suitable range of frequencies, the oscillation can have two different amplitude steady states. Later on, the possibility of successive bifurcations to chaotic behaviour was detected and is presently under study. This “deterministic chaos” is in fashion and is very interesting in principle, although it is still related to the action of a strictly monochromatic sea, a possibility hardly realized at sea. At the other extremum, stochastic nonlinear rolling was mainly treated as a sufficiently broadband process to have no possibility of jump phenomena. To clarify this puzzling aspect, the nonlinear model introduced in previous works with the excitation represented by filtered white noise is studied in detail. The filter was allowed to vary centerpeak and bandwidth. A representation as a sum of two sinusoidal processes with Gaussian slowly varying amplitudes was introduced. This description is particularly suitable for the description of narrow band processes. The system of differential equations rolling/filter was solved by means of the perturbation method of the multiple scales. Several curves roll variance/tuning ratio were obtained varying the bandwidth for an excitation variance corresponding to a moderate sea. The results indicate that, provided the bandwidth of the excitation doesn’t exceed that of a sharp JONSWAP spectrum, at least in the considered cases, the main resonance exhibits the same characteristic features shown in the deterministic, iee. the multivaluedness and jump possibility. This similarity is related to the fact that narrow band means high autocorrelation. The perturbation method allows to obtain simplified equations giving the maximum roll variance as a function of the excitation variance. These results are confirmed by means of the envelope analysis of a preliminary stochastic time-domain numerical simulation. The curve giving the pdf as a function of the variance of the motion appears, in fact, bimodal. Finally, the region of the first subharmonic has been analysed giving the response curves for the roll variance of the subharmonic component and the excitation threshold variance for its onset.