Nonlinear Effects of Stabilizing Ship Motion with P-Control.

For a 3 DOF model of ship motion with P-controls, we present a detailed analysis of the possibilities to stabilize the straight motion, and the resulting nonlinear effects. We identify existence, location and geometry of the stability boundary in terms of yaw damping and yaw restoring control strengths, including the dependence on the propeller diameter and the thruster position. To facilitate the analysis, we consider a combination of rudder and propeller forces into an effective thruster force. Using a recently developed analytical technique, we determine the nature of bifurcations associated with the stability change. This is required by the non-smooth character of the nonlinearities, and we find that supercritical Andronov-Hopf bifurcations are predominant. By means of numerical continuation we provide a global bifurcation analysis in the P-control strengths, which identifies the arrangement and relative location of stable and unstable equilibria and periodic orbits. We illustrate the resulting stable quasi-periodic ship motions in Earth-fixed coordinates and present some direct numerical simulations.