Examining Nonlinear Stability of a Pitch-Roll Ship Motion: Innovative Approach

Abstract Purpose It is essential for maritime safety, particularly in severe sea conditions, to investigate the nonlinear stability of a ship's pitch-roll motion. It reveals how nonlinear interactions can lead to unexpected instabilities, providing a more accurate design framework. Analysing two-degrees-of-freedom (2DOF) of roll-pitch motion of a vessel explicates complicated coupled dynamics, essential in comprehending nonlinear resonance events and parametric instabilities. Beyond linear approximations, nonlinear stability analysis of coupled pitch-roll ship motion helps capture genuine vessel behavior, particularly in strong sea conditions and large-amplitude waves. Consequently, the current study examines the 2DOF of an excited harmonically pendulum scheme, recognized in the works as an effective model of coupling between pitch and roll motions of a ship. We focus primarily on the dangerous condition of a vessel in which the excitation period is close to the pitch period, and the pitch frequency becomes twice as high as the roll frequency. Method The existing methodology is based mainly on a non-perturbative approach (NPA), which facilitates a unique analysis that is independent of Taylor expansion. He’s frequency formula (HFF) represents the principal tool employed in constructing NPA. The principal purpose of NPA is to transform weakly oscillating nonlinear ordinary differential equations (ODEs) into linear ones. The quick evaluation of frequency-amplitude correlation is essential in acquiring successive approximations of responses to parametric nonlinear variations. The inspiration for some criticisms on the stability of steady states is examined. A chaotic analysis of specified models is conducted using bifurcation diagrams, phase portraits, Poincaré maps, and Lyapunov spectrum. This strategy enables us to identify and distinguish distinct forms of motion exhibited by every system.