The Parametrically Excited Nonlinear Rolling Of Ships And Analysis Of Its Dynamic Stability

The ship motions in waves are complex.Even in longitudinal waves, there may be large amplitude rolling which contains complex nonlinear phenomena. The serious nonlinear rolling will lead ships to capsizing. In this paper,nonlinear equations of ship motions in longitudinal waves describing the couplings between heave,pitch and roll are deduced based on the theories of rigid body dynamics and Newtonian mechanics and use nonlinear dynamics method analyse the solution's stability.Firstly,considering the mathematical expressions of nonlinear damping and nonlinear righting moment and a third order mathematical model is introduced associated with the ship righting moment,the nonlinear equations of ship motions about heave,roll and pitch are established. Secondly,the safe basins are used to analyse the stability of ship roll motion.The results show that when parametric resonance occurs in longitudinal waves,the area of safe basin decrease rapidly.,it may occur safe basin broken when there are in high seas.The ship is in danger especially in this case.A simple equation governing the ship motion is deduced from the linear approximation of the heave equation and the pitch equation .The method of multiple scales is used to determine a second-order approximate for the nonlinear harmonic responses. The perturbation solutions are compared with solutions obtained by numerical intergration of the nonlinear governing roll equation and the results show that the approximate analysis solutions are valid. A Floquet analysis is used to predict the stability of steady-state harmonic responses, and study the solution's jump, bifurcation and chaos phenomena. Finally,the stability of trivial and period solutions are analyzed with dynamic stability and bifurcation theory and chaos of ship motion are discussed..the double period bifurcation saddle-node bifurcation are found in ship nonlinear rolling through the analysis of bifurcation.The calculated results shows that double period bifurcation leads to chaos consequentially,saddle-node bifurcation can cause losing stability of trival and period solution and results in chaos motion of ship.