斜め追波中の船の転覆につて : 第2報転覆現象におけるカオスとフラクタル

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In the previous paper, since a new mode of capsizing accompanied with the period doubling bifurcation, which was regarded as a precursor of the chaos and crisis in the dynamical system theory, was observed in the model tests, the authors pointed out that an investigation into the relation between capsize and chaos should be carried out. In the present paper, a simple capsize equation with nonlinear cubic term in the restoring force is solved by the numerical time simulation. A safe basin, which is defined as a non-capsizing region in the phase plane of rolling angle and rolling velocity, is obtained for the fixed control parameters such as forcing frequency, forcing amplitude and damping coefficient. In order to draw a complete safe basin in some square area of the phase plane, 301×301=90,601 grid points are used as initial conditions for the time simulation. By the results of simulation until the prescribed number of forcing cycles, e. g. 50 cycles, each grid point is distinguished in accordance with capsize or non-capsize. The boundary of safe basin obtained by such a method is metamorphosed complicatedly and in a fractal way as the control parameter is varied. This means the sensitivity and complexity of dependence of capsize on initial conditions. Melnikov analysis is also carried out in order to find the critical condition of homoclinic tangency, which means the beginning of the fractal metamorphoses in the basin boundary. The area of the safe basin is also decreased in a fractal way like a devil' staircase as the forcing amplitude is increased. This signifies that the basin boundary has a feature of the Cantor set. In the control parameter plane of the forcing amplitude and forcing frequency, the capsizing or non-capsizing region can be separated for some fixed initial condition as well as fixed damping, by using the similar numerical method. In this case, more than 1,350.000 grid points are used. The boundary between both regions is again fractal and complicated. Analytical expressions for the boundary are farely good approximations to the numerical results, which suggests that the simple analytical expressions may be used as criteria for capsizing. An investigation should be continued to cover the wide range of control parameters and extended to asymmetrical capsize equation of a biased ship and a Mathieu type capsize equation.
社団法人日本船舶海洋工学会の論文

斜め追波中の船の転覆につて : 第2報転覆現象におけるカオスとフラクタル

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斜め追波中の船の転覆につて : 第2報 転覆現象におけるカオスとフラクタル

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斜め追波中の船の転覆につて : 第2報転覆現象におけるカオスとフラクタル