Homoclinic Phenomena in Ship Motions

Abstract

In the article will be reviewed the changes of ship motion patterns originating from a special class of phase space phenomena identified in the nonlinear dynamics literature as homoclinic (When the term appears in italic, it will imply truly homoclinic as well as heteroclinic events). The key incident is the contact of manifolds of saddle-type (i.e., unstable) objects, which triggers a global change in the phase space flow of the dynamical system. Homoclinic phenomena are more likely to appear in extreme dynamic behavior where, usually, the effect of nonlinearity is large. Their presence can be felt through the sudden termination of oscillatory motions; or, through the emergence of motion irregularity accompanied by loss of system robustness to external excitations. Homoclinic phenomena have been identified to lurk behind some well-known manifestations of ship dynamic instability. All known ship-related cases will be reviewed and explained in the current article. Analytical and numerical tools that can be used for predicting such phenomena effectively will be discussed, as well as, practical assessment methods that are suitable for evaluating ship vulnerability. Three types of ship motion will be reviewedrolling in beam seas;surf-riding and broaching-to in following/quartering seas; anddrifting and yawing in wind. For the roll motions, the investigation will be escalated from free rolling with bias, to resonant (periodically forced) rolling and the mechanism of capsize in beam seas through safe basin erosion; to stochastic rolling and the consideration of wave grouping excitations. In astern seas, the most profound homoclinic event is the transition from ordinary surging to surf-riding. It will be explained, initially, in a deterministic context through simple dynamical analogues. Differences arising from the level of detail in the mathematical model will be discussed (many degrees of freedom, memory effect, and nonlinear waves). Then, a multifrequency wave environment will be introduced, rendering the phase space flow time-varying and incurring significant qualitative effects. The monitoring of Lagrangian structures appearing in such unsteady phase space flows can offer unique insights about nonlinear ship motions and their interactions. Wave celerity is known to correlate with the occurrence of surf-riding in regular seas, although for irregular seas, its role has been uncertain. A phenomenological approach will be reviewed for evaluating this connection. Two possible methods for celerity's calculation in irregular seas will be presented. The third area where homoclinic phenomena have been identified is planar ship motions under strong wind loads. Two particular cases will be addressed: the transition from a course-keeping pattern to a turning motion pattern as the rudder angle is varied; and phenomena of "head wind" ship operation, under sluggish rudder control. The article will conclude with some thoughts as to the future direction in the field.