Response, stability, and bifurcation of roll oscillations of a biased ship under regular sea waves are investigated. The primary and subharmonic response branches are traced in the frequency domain employing the Incremental Harmonic Balance (IHB) method with a pseudo-arc-length continuation approach. The stability of periodic responses and bifurcation points are determined by monitoring the eigenvalues of the Floquet transition matrix. The primary and higher-order subharmonic responses experience a cascade of period-doubling bifurcations, eventually culminating in chaotic responses detected by numerical integration (NI) of the equation of motion. Bifurcation diagrams are obtained through the period-doubling route to chaos. Solutions are aided with phase portrait, Poincaré map, time history and Fourier spectrum for better clarity as and when required. Finally, the same ship model is investigated under variable excitation moments that may result from different wave heights in regular seas. The biased ship roll model exhibits primary and subharmonic responses, jump phenomena, coexistence of multiple responses, and chaotically modulated motion. The stable, periodic, and steady-state roll responses obtained by the IHB method are validated by the NI method. Results obtained by both methods are found to agree very well.
Dynamics, control, and stability of roll oscillations of a biased ship in regular sea waves are investigated. The ship under roll oscillation is modeled as the classical Helmholtz-Duffing oscillator with strongly nonlinear asymmetric restoring moment characteristics. The incremental harmonic balance (IHB) method, amended with a pseudo-arc-length continuation approach, is employed to obtain the uncontrolled and controlled frequency responses. The primary and subharmonic responses of the uncontrolled system are examined through the period-doubling route to chaos path. The same ship roll model is then investigated under state feedback control with time delay. In the control scheme, a moving weight is actuated by the delay controller to generate the anti-rolling moment. The stability of periodic responses is studied using the semi-discretization method with extended Floquet theory. Bifurcation points on periodic response branches are identified from the eigenvalue study. The effect of gains and time delay in the feedback loop in controlling the primary and subharmonic responses is investigated. Several chaotic responses obtained through period-doubling route to chaos paths are successfully controlled. Stable, periodic, and steady-state solutions obtained from the IHB method are verified by numerical integration (NI) of the equation of motion as and when applicable. Solutions are aided with phase portrait, Poincaré map, time history, and Fourier spectrum for better clarity. It is shown that appropriate selections of control parameters can effectively reduce the roll amplitude to a great extent with the improved measure of stability.
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