ON THE ABILITY OF DIFFERENTIAL TRANSFORM METHOD AND ITS IMPROVEMENTS TO SOLVE PROPERLY THE SHIP’S ROLL EQUATION

A major concern regarding the ship safety is related to severe rolling oscillations which, in certain circumstances, can even lead to its capsizing. Assuming the rolling motion decoupled from its other five possible motions, one results a mathematical model associated to a second order differential equation where the main parameters (inertia, damping, stiffness and excitation) reveal a significant nonlinearity. In the absence of exact analytical solutions, the amplitude and period characteristics of the ship rolling can be evaluated by approximate numerical or analytical methods. In this work, we checked if the performant differential transform method (DTM) and its improvement with Pade approximants is able to provide approximate analytical solutions for nonlinear roll equation, valid for both the transitory and stationary states. The obtained results were verified against those produced by MatLab numerical generated simulations. We noticed that for the linearized roll equation, which describes quite correctly a significant part of the situations of practical interest, the DTM doubled by the Pade approximation [4/4] offers the exact solution. If the nonlinear terms from damping and restoring moments are included in the study but the sea is considered waveless, the investigated technique proves a good accuracy in describing the attenuation over time of the initial excitation. DTM and Pade [4/4] cease to offer reasonable solutions as soon as the exciting moment of the waves is included in the procedure. We gave clear explanations for this impasse and showed that the use of higher-order Pade approximations can solve (even if with additional efforts) totally or at least partially this problem.