| The ability to resist capsizing is a fundamental requirement in ship design. Traditional ship stability analysis is based upon either hydrostatics or linear dynamics. However, vessel capsizing is a large amplitude dynamic phenomenon requiring the consideration of nonlinear dynamics. Nonlinear dynamics exhibit varied and rich phenomena not found in the familiar linear dynamics. These phenomena may include multiple steady-states and even aperiodic response due to periodic excitation.; In the past, weakly nonlinear systems could only be studied using approximate perturbation techniques (local analysis), while highly nonlinear systems required numerical simulation. With simulation, however, little can be said about the eventual state of the motion without precise knowledge of initial conditions and external forces. In lieu of numerical simulation or approximate local analysis (near a single equilibria), we use modern geometric methods. These modern methods are often not limited to size of the nonlinearity and are capable of analyzing the global (trajectories near one or more singular points) system behavior.; In this thesis, we apply these geometric methods to analyze the large amplitude (near capsizing) ship rolling motion due to a regular wave excitation. The time dependence of the waves is periodic. Because the system is periodically forced, we study the dynamics sampled at each period. This sampling is called a Poincare map or stroboscopic sampling. In the Poincare map, we analyze the boundaries, which are called invariant manifolds, between initial conditions which eventually lead to qualitatively different types of behavior (e.g., unsafe capsizing or safe non-capsizing). For moderate forcing, these boundaries are simple and distinct; however, as the forcing exceeds a certain threshold value, these boundaries can intersect. We predict this value using Melnikov's method. Following intersections of these boundaries, it is no longer easy to determine whether a given initial condition inside one of these intersecting regions, called lobes, is safe (the motion eventually achieves a bounded steady-state and will not capsize) or is unsafe (the motion will eventually become unbounded leading to capsizing). Along with these manifold intersections, aperiodic response (motion) due to a periodic excitation (force) may occur. Finally, we use the methods) of lobe dynamics to predict the eventual safety (non-capsize) and danger (capsizing) of initial conditions that are inside the lobes. |
