Modelling and analysis of ship roll oscillations interacting with stationary icebergs

Abstract

Impact dynamic interaction of ships with solid ice or stationary rigid structures is a serious problem that affects the safe operation and navigation in arctic regions. The purpose of this study is to present two analytical models of impact interaction between ship roll dynamics and one-side rigid barrier. These models are the phenomenological modelling represented by a power law in stiffness and damping forces, and Zhuravlev non-smooth coordinate transformation. Extensive numerical simulations are carried out for all initial conditions covered by the ship grazing orbit for different values of excitation amplitude and frequencies of external wave roll moment. The basins of attraction of safe operation are obtained and reveal the coexistence of different response regimes such as non-impact periodic oscillations, modulation impact motion, period-added impact oscillations, chaotic impact motion, and unbounded rotational motion. The results are summarized in the bifurcation diagram in terms of response-excitation amplitudes plane. The stability fraction index is obtained for different values of excitation frequency based on the ratio of the area of bounded roll oscillations to the total area of the grazing orbit.

Dynamics and bifurcations of nonsmooth systems: A survey

“…Systems of oscillators with piecewise smooth springs and the related grazing bifurcations find applications in many other engineering systems, e.g. gear pairs , Parker-Vijayakar-Imajo [289], Luo-O'Connor [246]) vibrating screens and crushers (Krukov [214], Wen [376]), vibro-impact absorbers and impact dampers (Ibrahim [178]), ships interacting with icebergs (see Ibrahim [151,178]), offshore structures (see ), suspension bridges , de Freitas-Viana-Grebogi [129]), and pressure relief valves (see the paper by Hos-Champneys [171] in this special issue 5 ). Similar differential equations with nonsmooth continuous right-hand sides describe the so-called Chua circuit, for which border-collision and grazing bifurcations are discussed in Yu.…”

Inelastic impact dynamics of ships with one-sided barriers. Part I: analytical and numerical investigations

“…By dropping the non-conservative terms and excitation term from (4), the resulting equation was earlier solved for different initial conditions. It was shown by Grace and Ibrahim [6] that, at a given critical initial energy, the ship experiences grazing orbits, which separates manifolds of smooth and impact motions. The next two subsections will introduce two models for the impact roll dynamics described by differential equations of motion on the entire time interval despite the presence of a barrier.…”

“…Regarding the second class, where ships experience impact with stationary barriers, Grace and Ibrahim [6,7] and Grace et al [8] considered the cases of purely elastic and inelastic impacts with one-sided barriers using Zhuravlev [9] and Ivanov [10] non-smooth coordinate transformations. The unperturbed dynamics of the ship roll oscillations was examined in the absence of damping and external moment.…”

Constrained Control of Coexisting Attractors in Impact Oscillator with Delay

“…To the same purpose, impacting and non-impacting multistability can be exploited to avoid impact and impact-caused damages in elastic beams [12,13]. The impact of ships with icebergs due to the roll motion can also be controlled in grazing point [14][15][16]. Jumping from impacting as one stability status to a coexisting non-impacting one with different actuation behaviour [17,18] is another example of such an attractor exchange.…”