The differential equations of motion are developed for a "steered ship" that is free to heave, roll, and pitch in regular ocean waves. The hydrodynamic forces and moments acting on the ship are developed as analytic expressions of the heave distance, and roll and pitch angles, using the Froude-Krylov model, "strip theory," and experimental data. The non-linearities of the force and moment expressions are approximated with polynomial non-linearities in the vicinity of the static equilibrium values of heave, roll, and pitch. The method of multiple time scales is used to develop the "solvability conditions," which predict multiple steady-state values of roll, heave, and pitch for some frequencies of ocean waves when the heave, pitch and roll natural frequencies exhibit either a 3:2:1 or 2:2:1 ratio, respectively. Examples are included which include the computation of the coefficients of the differential equations of motion for a typical large modern cargo vessel loaded in such manners so that it has these ratios between its natural frequencies. It is found that such vessels may experience dangerously large amplitude rolls in longitudinal waves when the excitation frequency is approximately twice the roll frequency. Distributions of mass within the ship such that one of the principal axes of inertia is not exactly parallel with the long axis of the ship are shown to have a major effect on the amplitude of the roll motion in these cases. The results are compared to those obtained from numerical integration of the differential equations of motion with good agreement.
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