Key Takeaways & Executive Findings
- •• Incorporation of a linear-plus-quadratic damping term in the ship rolling equation preserves key dynamics of the physical system, as confirmed by model tests. • A new Melnikov-based method provides an upper bound on the domain of chaotic rolling motion (erratic rocking) under random excitation and response. • The proposed Melnikov criterion is verified using phase plane diagrams and Poincare maps. • This study is the first to systematically modify system parameters in the rolling equation for ship stability analysis.
Abstract
To study the rolling motion of a ship in the presence of water on its deck, a linear-plus-quadratic damping term was incorporated into its equation of motion. Ship model tests indicate that the key dynamics of the physical system are preserved in the ship rolling equation with the linear-plus-quadratic type damping term. To take into account the presence of randomness in the excitation and the response, a new method was developed and a Melnikov criterion was obtained to provide an upper bound on the domain of the potential chaotic rolling motion (erratic rocking). Additionally, the Melnikov criterion proposed in this study was verified by the utilization of phase plane diagrams and Poincare maps. Furthermore, this research has made the initial endeavor to systematically modify the system parameters in the rolling equation of motion for ship stability analysis.
1. Introduction
Presently, there is still a lack of comprehensive understanding regarding the large magnitude or strongly non-linear motions of ships, particularly in relation to non-linear roll motions. The significance of comprehending large roll motions cannot be emphasized; nonetheless, its intricate nature can be vexing. The challenges essentially arise from two domains: the determination of hydrodynamic forces and the derivation of solutions to the consequent nonlinear equations of motion for rigid bodies.
Given the challenges associated with precisely estimating the comprehensive hydrodynamic forces, it is a common practice to employ a roll model that effectively separates the six degrees of freedom (Wang [1−2]). Regrettably, a closed form solution for a 1-DOF nonlinear ordinary differential equation cannot be found in general, even when the excitation is sinusoidal. The equation under consideration demonstrates a wide range of qualitative characteristics in its dynamics, including harmonic, sub-harmonic, super-harmonic, periodic, and aperiodic behaviors. Additionally, the system may undergo state transitions, exhibit bifurcations, or display chaotic behaviors.
Current research endeavors in the field of nonlinear rolling motion have primarily focused on the derivation of approximate solutions for the one-degree-of-freedom (1-DOF) roll equation, as shown by Eq. (2). Numerous methodologies have been devised, each grounded in distinct assumptions and mathematical ideas.
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WANG Ying-guang (2025). A New Approach for Melnikov Analysis of the Stability of a Ship with Water on Deck. SinoTechIntel Verified Research. https://doi.org/10.3969/j.issn.1007-7294.2025.06.006
Research & Educational Purpose Only:The translations, structured abstracts, analytical annotations, and data reports provided by SinoTechIntel are intended exclusively for academic research, internal corporate R&D, and educational benchmarking. They do not constitute formal engineering, chemical safety, legal, or professional advice.
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Frequently Asked Questions
What is the main contribution of this paper?
The paper introduces a new Melnikov-based method to analyze the stability of a ship with water on deck, providing an upper bound on the domain of chaotic rolling motion. It also verifies the criterion using phase plane diagrams and Poincare maps.
What damping model is used in the ship rolling equation?
A linear-plus-quadratic damping term is incorporated into the equation of motion, which better preserves the key dynamics of the physical system compared to linear-plus-cubic damping.
How is the Melnikov criterion verified?
The Melnikov criterion is verified by utilizing phase plane diagrams and Poincare maps, which confirm the presence of chaotic rolling motion within the predicted domain.
What is the significance of this research for ship stability analysis?
This research is the first to systematically modify system parameters in the rolling equation of motion, providing a new approach to assess ship stability under random excitation and response.
What are the key dynamics preserved by the linear-plus-quadratic damping model?
Ship model tests indicate that the linear-plus-quadratic damping model preserves the key dynamics of the physical system, including the onset of chaotic rolling motion, which is crucial for stability analysis.
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