Atsuo Maki, Lawrence Virgin, Naoya Umeda, Tetsushi Ueta, Yuu Miino, Hiroshi Kawakami, On the loss of stability of periodic oscillations and its relevance to ship capsize, Springer, Journal of Marine Science and Technology, Sep. 2018.
Abstract: This research aims to analytically investigate beam seas roll motion. So far, in the analytical investigation for this topic, the softening Duffing equation has been often utilized because of its similarity to the ship equation of motion, and considerable number of researches have been reported. In this research, we focus on the loss of stability of periodic oscillation and its relevance to ship capsize. In the past researches, the threshold of saddle-node bifurcation, flip bifurcation and heteroclinic bifurcation has been obtained. Among them, the condition with respect to flip was derived from the negative stiffness condition in the variational equation having the form of Mathieus equation. On the other hand, in this paper, the authors reconsider the phenomenon and then that threshold is identical with that of pitchfork bifurcation. Furthermore, the practical capsizing threshold was introduced by using the solution after pitchfork bifurcation. The result shows the fairly good agreement with the numerically obtained capsizing boundary. It means that applying the unsymmetric solution after pitchfork bifurcation into capsizing assessment is reasonable.