We note that for a fixed value of *σ*_{2}, *a*_{1} and *a*_{2} decrease as *µ*_{3} increases. The curves of unstable fixed points around the region of perfect tuning, *σ*_{2}≈0, converge in both modes. The introduction of quadratic roll damping *µ*_{3}, by say attaching antirolling devices like bilge keels, causes the region between the two Hopf bifurcation frequencies (close to perfect resonance) to shrink. However, it does not eliminate complicated motions completely in this region.

Figures 6 show the frequency-response curves for the case in which the values of all the parameters are the same as those in Figures 5 except that *σ*_{1}=0.2. In this case, the curves are shifted slightly to the right and the peak amplitudes of the right branches of the roll mode are smaller than those of the left branches. The opposite occurs in the response of the pitch mode. If *σ*_{1} is chosen to be negative, the frequency-response curves would be shifted to the left. When *σ*_{1}=0, the case of perfect tuning, the curves would be symmetric with respect to *σ*_{2}=0. The qualitative behavior of the solutions in the three cases is the same.

To design more comfortable and safe vessels, one must understand the complicated dynamics of a vessel moving in a general environment. Included among the important dynamic parameters are the ratios of natural frequencies and the nonlinear interactions among the hydrostatic and hydrodynamic forces and moments. One of the objectives of the present work is to investigate the undesirable and potentially dangerous characteristics of the dynamics of a vessel.

It has been believed for a long time that the linear-plus-quadratic model could adequately describe the hydrodynamic damping of the roll motion. However, many investigators avoided using this model because of the difficulties in the analyses. In the present paper, a quadratic nonlinear damping model is introduced into the equation of the roll mode. We investigated the nonlinearly coupled pitch and roll response of a vessel in regular waves when the natural frequency in pitch is twice that of roll (a condition of a two-to-one internal or autoparametric resonance). The method of multiple scales was used to derive four first-order autonomous ordinary-differential equations for the modulation of the amplitudes and phases of the pitch and roll modes when either mode is excited. The modulation equations were used to determine the influence of the quadratic nonlinear damping on the periodic responses and their stability.

When the encounter frequency is near the pitch natural frequency, the jump phenomenon exists for both zero and nonzero quadratic roll damping (*µ*_{3} is the coefficient of the quadratic roll damping.). The saturation phenomenon is broken if a quadratic roll damping term is introduced. This implies that there is no critical value of the excitation amplitude beyond which all of the extra energy input to the pitch mode is spilled over into the roll mode. The amplitude of the roll mode decreases while that of the pitch mode increases as the magnitude of the quadratic roll damping *µ*_{3} increases. In the subcritical case of force-response and frequency-response curves, the overhang regions narrow down as *µ*_{3} increases. The fixed points of the modulation equations undergo a Hopf bifurcation as one of the control parameters is varied. Between the Hopf bifurcation points, the response is an amplitude- and phase-modulated motion consisting of both the pitch and roll modes.

When the encounter frequency is near the roll natural frequency, the amplitudes of both the roll mode and the pitch mode decrease as the magnitude of the quadratic roll damping coefficient *µ*_{3} increases. Again, Hopf bifurcations occur as either the encounter frequency or excitation amplitude is varied.

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