Nonlinear Coupling of Pitch and Roll Modes in Ship Motions

TLDR

In regular seas, pitch mode is excited near its natural frequency at low wave amplitudes, whereas roll mode shows no saturation near its natural frequency and may lack a steady‑state response at near‑perfect resonance. The study analyzes the nonlinear coupling between pitch and roll modes of ship motions when their natural frequencies are in a 2:1 ratio in regular seas. The authors conduct a theoretical analysis of the nonlinear interaction between pitch and roll modes under a 2:1 frequency ratio in regular sea conditions. The analysis shows that pitch amplitude saturates at a small critical value while roll amplitude grows large as excitation increases, and that near roll resonance the roll mode can reach very large amplitudes with no steady‑state response.

Abstract

An analysis is presented for the nonlinear coupling of the pitch (heave) and roll modes of ship motions in regular seas when their frequencies are in the ratio of two to one. When the frequency of encounter (excitation frequency) is near the pitch frequency, the pitch mode is excited if the encountered wave amplitude (excitation amplitude) is small. As the excitation amplitude increases, the amplitude of the pitch mode increases until it reaches a critical small value. As the excitation amplitude increases further, the pitch amplitude does not change from the critical value (i.e., the pitch mode is saturated), and all of the extra energy is transferred to the roll mode. Thus, for large excitation amplitudes, the amplitude of the roll mode is very much larger than that of the pitch mode. When the excitation frequency is near the roll frequency, there is no saturation phenomenon and at close to perfect resonance, there is no steady state response in some cases. The present results indicate that large roll amplitudes are likely in this case also.