On the Undesirable Roll Characteristics of Ships in Regular Seas

Abstract

In 1863 Froude observed that a ship whose frequency in pitch (heave) is twice its frequency in roll has undesirable roll characteristics. To explain this phenomenon, Paulling and Rosenberg as well as Kinney assumed the pitch (heave) motion to be a simple harmonic independent of the roll motion. Substituting the pitch (heave) expression into the roll equation, they obtained a Mathieu equation. They found that exponentially growing instabilities can occur for certain pitch amplitudes and frequency ratios. The exponential growth is unrealistic, the result of their neglecting the influence of the roll motion on the pitch (heave) motion. To improve their results, the present author offers an analysis for the nonlinear coupling of the pitch and roll modes of ship motions in regular seas. When the encounter frequency is near the pitch frequency, only 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 in accordance with linear theory until it reaches a critical small value. As the excitation amplitude increases further, the pitch amplitude does not change from the critical value (that is, the pitch mode is saturated), and all the extra energy is transferred to the roll mode. Consequently, for large excitation amplitudes, the response is a combined roll and pitch motion, with the amplitude of the roll mode being very much larger than that of the pitch mode. More dangerously, the nonlinear theory predicts instabilities in regions where the linear theory predicts stability. Moreover, the nonlinear theory predicts conditions for the nonexistence of steady-state periodic responses. Instead, the responses can be amplitude- and phase-modulated roll and pitch motions or even chaotic. When the encounter frequency is near the roll frequency, there is no saturation phenomenon and, at close to perfect resonance, there are no steady-state periodic responses in some cases. The present results indicate that large roll amplitudes are likely in this case also. Further, the results predict the possibility of large amplitudes in the roll motion even when the ship is moving through a head or following sea.