pitch is twice that in roll __(____2____)__ has undesirable seakeeping characteristics. This observation was a manifestation of the two-to-one internal resonance whose significance cannot be determined using linearized equations __(____3____,__ __4____,__ __5____,__ __6____,__ __7____)__.

For a century after Froude, however, no further research on this phenomenon was pursued. In 1959, Paulling and Rosenberg __(____8____)__ studied the coupled heave-roll motion of a vessel using a set of nonlinear ordinary-differential equations. They simplified the equations of motion and obtained a single roll equation having the form of a simple linear Mathieu equation which contains a time-varying coefficient due to a simple harmonic motion of the heave mode. In this model, the heave influences the roll but the roll does not influence the heave. Experimentally, they tested the case of unstable rolling motion excited by the heave mode only. This study has two principal shortcomings. First, due to the lack of consideration for damping and nonlinear coupling terms, the analytical model was not capable of yielding realistic results. Second, in the experimental setup, the heave mode was given a prescribed motion and hence the effect of roll motion and waves generated thereafter on the heave mode are not taken into account.

Kinney __(____9____)__, Kerwin __(____10____)__, Blocki __(____11____)__, and Sanchez and Nayfeh __(____12____)__ also studied the response of the roll to longitudinal waves. Except for Blocki's study, all other studies are theoretical ones. In his experiment, Blocki considered coupling of the heave and roll modes of a ship that possesses fore and aft symmetry, which is placed in beam waves. As in the study of Paulling and Rosenberg, he studied the case of parametrically excited roll motions in which energy is fed to the roll mode by a prescribed heave or pitch motion, or equivalently, wave motion. His equations reduce essentially to a one-degree-of-freedom model governing only the roll mode.

To explain Froude's observation, Nayfeh, Mook, and Marshall __(____13____)__ and Mook, Marshall, and Nayfeh __(____14____)__ modeled the ship motion by two nonlinearly coupled equations involving the pitch and roll modes; they included the dependence of the pitching moment on the roll orientation. Thus, the pitch (heave) motion is not prescribed but is coupled to the roll motion, and consequently, the pitch (heave) and roll orientations are determined simultaneously as functions of a prescribed excitation. They clearly showed the significance of the frequency ratio in causing undesirable roll behaviors, such as the “saturation” phenomenon. They offered an explanation of the observations of Froude.

Nayfeh __(____15____)__ considered the nonlinearly coupled roll and pitch motions of a ship in regular head waves in which the couplings are primarily in the hydrostatic terms when the pitch frequency is approximately twice the roll frequency and the encounter frequency is near either the pitch or roll natural frequency. He demonstrated the saturation phenomenon when the encounter frequency is near the pitch natural frequency.

In the present paper, we use a linear-plus-quadratic damping model for the roll motion and investigate the cases of primary resonances. The linear-plus-quadratic damping model has long been recognized by investigators to describe closely the dissipation of energy in the roll mode. However, it was not used so far because of some analytical difficulties. We obtain various complicated responses, which are common features of the nonlinear dynamics of many mechanical and elastic systems. These responses include supercritical and subcritical instabilities, periodic motions, and coexistence of multiple solutions and associated jumps. The quadratic damping eliminates the saturation phenomenon. Such phenomena can never be addressed by the linear approach because it is incapable of representing not only the strong nonlinear interaction between the two modes but also the effect of the viscous damping in the roll mode.

We consider the response of a ship that is restricted to pitch and roll to a regular wave. We assume that the ship is laterally symmetric. We use the right-handed coordinate systems: a body-fixed coordinate system *oxyz* such that its origin *o* is at the center of mass, the x-axis is positive toward the bow, the y-axis is positive toward starboard, and the z-axis is positive downward. The orientation of the ship with respect to an inertial frame *OXYZ* is defined by the Euler angles and *θ* as follows: *θ* is a pitch rotation about the original y-axis, and is a roll rotation about the new x-axis. The components *p* and *q* of the angular velocity about the x- and y-axes are related to *,θ,* and by

(1)

The equations of motion can be written as

*I*_{xx}*–I*_{xz}*pq=K+K*_{o} cosΩ*t* (2)

*I*_{yy}*+I*_{xz}*p*^{2}*=M+M*_{0} cos(Ω*t*+*τ*) (3)