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

Abstract

This two-part paper deals with impact interaction of ships with one-sided ice barrier during roll dynamics. The first part presents analytical and numerical studies for the case of inelastic impact. An analytical model of a ship roll motion interacting with ice is developed based on Zhuravlev and Ivanov non-smooth coordinate transformations. These transformations have the advantage of converting the vibro-impact oscillator into an oscillator without barriers such that the corresponding equation of motion does not contain any impact term. Such approaches, however, account for the energy loss at impact times in different ways. The present work, in particular, demonstrates that the impact dynamics may have qualitatively different response characteristics to different dissipation models. The difference between localized and distributed equivalent damping approaches is discussed. Extensive numerical simulations are carried out for all initial conditions covered by the ship grazing orbit for different values of excitation amplitude and frequency 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, I.M. Grace · R.A. Ibrahim ( ) · V.N. Pilipchuk

Stochastic response of a vibro-impact Duffing system under external Poisson impulses

“…It is an important issue on how delta function participates in these nonlinear differential equations. Grace et al [11] presented a good interpretation of such a manipulation with delta function. In the transformation manipulation, δ(z) is treated as a specific distribution applied to some testing function rather than the conventional Dirac delta function.…”

“…For example, the stability and bifurcation of the vibrations of vibroimpact systems under deterministic excitation were studied by many researchers, e.g., [4][5][6][7]. Several nonsmooth coordinate transformation techniques were also developed and studied for vibro-impact vibration [8][9][10][11][12]. In particular, the stochastic response of vibroimpact systems has received more and more attention in the past decades [13][14][15][16][17][18][19][20][21][22][23][24].…”

Modeling of Dynamic Systems With Vibro-Impact Interaction and Discontinuity Mapping

“…Equation (9) describes the ship roll dynamics with one-sided inelastic barrier on the entire time interval, where conditions of reflection from the barrier and impact energy loss are already included through transformation (8). The results generated using the two transformations are compared based on the study conducted by Grace et al [7]. Figs.…”

“…Estimates for stability boundaries due ship elastic impact, ( 1) e = , and inelastic impact ( 0.8) e = as predicted by Zhuravlev Model 1 and Ivanov Model 2. Unbounded motion is shown by grey area [7]. …”

Unilateral vibro-impact systems — Experimental observations against theoretical predictions based on the coefficient of restitution

“…al. [46] uses non-smooth coordinate transformations as a preprocessing step to numerical simulations of vibro-impacts systems to weaken the kinematic impact rule. The development of numerical time-integration algorithms for non-smooth models is an active field of research [47][48][49] whose discussion is out of the scope of this work.…”