Comparative Approximate Studies on the Ship’s Rolling Motion

The new geometrical and analytical techniques of nonlinear dynamics offer to the naval engineer ideal tools for studying the ship dynamics and stability. Between them, the bifurcation diagram, the Poincare map, the phase plane, and the time histories are considered in the paper in an attempt to demonstrate the low period solutions of the asymmetric roll equation derived by Kan and Taguchi. The model assume a linear damping, a restoring moment represented by a third-order polynomial and a single frequency harmonic excitation. For moderate forcing amplitudes, the system exhibits a period 1 motion, characterized by a period doubling bifurcation to a period 2 motion that warns about the beginning of a period doubling cascade to chaos. Two different approaches, the Fast Fourier Transform and the Harmonic Balance Technique are used to obtain approximate solutions for the period 1 and 2 motions and to predict the period doubling bifurcation by a stability analysis. The two sets of solutions match reasonably well with the numerical solution, especially for the first mentioned approach.