012 Analysis of Parametric Roll Resonance using Poincaré Maps

A ship sailing in head or following sea may encounter an unexpected roll motion. This roll motion can build up to 40◦ or even more and can cause serious damage to cargo and ship. The unexpected motion is induced by a phenomenon called parametric resonance. Especially modern container carriers are vulnerable, due to their particular hull shape used to achieve an optimal trade-off between high service speeds and maximum container payload above deck. To reduce the chance of accidents in the future, detection and stabilization of the roll oscillations caused by parametric resonance is important. Since the Mathieu equation can be used as a simplified 1-DOF ship model, it is used to investigate the onset of parametric resonance. One of the conditions for parametric resonance to occur is that the encounter frequency is approximately equal to twice the natural frequency. The wave encounter frequency is peak frequency of the wave spectrum corrected for the ship’s forward velocity and heading angle. The analysis of the Mathieu equation is done using Poincaré map theory. The determination of a Poincaré map is often difficult and requires to solve the system. However, numerical methods can be used to calculate the Poincaré map of a system. For maps with a simple structure a fitting procedure can be used to find an expression for the map. Investigation of the linear (damped) Mathieu equation showed the presence of a transient effect in the period time. Neglecting this transient behavior the Poincaré map of the Mathieu equation is, depending on the amount of damping, an increasing or decreasing exponential function. The Poincaré map with the transient effect taken into account is also calculated but gives rise to a more elaborate expression. After the investigation of the Mathieu equation, a 2-DOF ship model is considered and the Poincaré map is calculated. The nonlinearities in the ship model give rise to a complex map, therefore it is not possible to use a plain fit and numerically calculate the Poincaré map. However, the graphical representation of the Poincaré points is still useful to analyze the behavior of the system. The analysis indicated the presence of the transient behavior in this ship model as well. To stabilize the roll motion of the ship, different control strategies are investigated. The control strategies are based on a change of the encounter frequency which is enabled by changing the surge velocity and/or the heading angle. The surge dynamics are feedback linearized and different velocity profiles are implemented. Changing the speed to a desired end velocity using a constant acceleration or deceleration is able to stabilize the roll motion if the change of speed is large enough, i.e. the encounter frequency is changed so much that the resonance condition is no longer fulfilled. In all the simulated situations a change of speed initially leads to a decrease of the roll motion. This effect is used to make triangular and sinusoidal wave velocity profiles that stabilizes the roll motion, although the (average) velocity is close or equal to the initial condition, and thus has an encounter frequency close to the resonance condition. The use of a change of heading as the only control strategy turned out to be not beneficial. But heading control can be used in addition to velocity control. This is investigated with a controller based on an extremum seeking control strategy. This controller is able to reduce the roll motion to zero, but uses relatively large accelerations to do so. These accelerations are not feasible for a container ship, like the one considered here.