A NUMERICAL INVESTIGATION INTO THE CAPSIZE

International Journal of Modern Manufacturing TechnologiesISSN 2067–3604, Vol. VII, No. 1 / 2015A NUMERICAL INVESTIGATION INTO THE CAPSIZE PHENOMENON OFA VESSELDumitru DeleanuConstanta Maritime University, 104, Mircea cel Batran str, Constanta, RomaniaCorresponding author: Dumitru Deleanu, dumitrudeleanu@yahoo.comthe ship’s rolling motion, so the mechanical system canbe reduced to a forced nonlinear oscillator with linear/nonlinear damping and restoring force.In the present study, the following equation, derivedby Thompson et al [5], is taken as a model for avessel capsize in periodic beam seas:Abstract: Each year, almost one hundred vessels are lostdue to the capsizing, particularly in high seas, leading toheavy losses of human lives and ships. This is why thestability against capsizing is a fundamental requirementwhen designing a ship. In our work, the second-ordernonlinear differential equation developed by Thompson etal. is taken as a model for a vessel capsize in periodic beamseas. It describes the behavior of a ship that issimultaneously heaving, swaying and rolling in waves.With the forcing amplitude as bifurcation parameter, theanalyzed system exhibits either periodic or chaoticbehavior, the route to chaos being realized by a perioddoubling sequence of periodic motions. Some acceptedindicators, like bifurcation diagrams, phase planes andPoincare maps have been computed and they confirm thetransition from order to chaos. The paper investigates alsothe fractal erosion of safe basin of attraction and proposesa geometric way to evaluate quickly this process leading tocapsize.Key words: Capsize Direct and Parametric Excitation,Order and Chaos, Safe basin. x x x x 2 1 G cos t F sin tWhere: Ak 2x , f , F , G Ak V n V(1)(2)Here, represents the roll angle, the angle of vanishingstability, the non-dimensional damping coefficient, thewave frequency, the natural frequency of the boat, A thewave height, and k the wave number. A dot denotesdifferentiation with respect to time. The details aboutEq. 1 can be found in [5]. Despite of its simplicity, Eq. 1shows a wide spectrum of qualitatively distinct types ofbehaviors, including steady-state solutions, jumps toresonance, or period doubling cascades leading tochaos. This rich dynamics will be investigated in thenext sections.1. INTRODUCTIONBy capsizing or keeling over of a ship we understandthat situation when the ship is turned on its side or itis upside down. In the language of nonlineardynamics, capsizing is a transition from a stableequilibrium point near the upright position to a stableequilibrium point near the upside-down position.Despite of an extensive research on the capsizingproblem of a vessel, not much practical progress hasbeen made so far and a large number of shipscontinue to sink because of this phenomenon. This isdue of the huge number of factors involved, includingsevere winds and waves, water on deck, liquid cargo,lack of information about environmental conditionsduring the accident or resonance conditions [1].According to the IMO (International MaritimeOrganization) there is a necessity for proposingmathematical models that could simulate the ship’scapsizing taking into account as much as possible of thesefactors. Such examples can be found in Kan and Taguchi[2], Soliman and Thompson [3], Umeda and Hamamoto[4], and others. Most of these approaches consider only2.BIFURCATIONDIAGRAMS,PLANES AND POINCARE SECTIONSPHASEThe response of the system (1) has been investigatednumerically by use of a Runge – Kutta - Gillprocedure with constant step for the followingparameters: 0.1, 1.8, G 5F [5]. With theforcing amplitude F as bifurcation parameter, it wasfound that the analyzed system exhibits eitherperiodic or chaotic behavior, the route to chaos beingrealized by a doubling sequence of periodic motions.The bifurcation diagram F – x showing the differentbifurcation points is given in Fig. 1. To construct thisdiagram, we started from equilibrium conditions ( x(0) x(0) 0 ) and plotted x at every one forcingcycle with the same phase angle. The first 200 cycleswerediscardedtoavoidtransients.12

1.8 0.2790.280.2810.2820.2830.284FF(a)(b)Fig. 1. a) Bifurcation diagram F - x for 0.1, 1.8, Gb) The upper – right part of (a) is zoomed for detailsFor small forcing amplitude F the system executesperiodic oscillations with period corresponding to theforcing period T 2 / , as shown in Fig. 2(a). AsF is gradually increased different types of periodicmotions are obtained. Starting with F 0.235, theperiod 1 motion bifurcates into a period 2 motion. 5F ;techniques including cell-to-cell mapping and coarsegrid-of-start method. The simulation show that anincreasing of the forcing amplitude leads to a processof fractal erosion of the safe basin, finished with asharp decrease of safe area.To illustrate this, Eq. 1 was solved numerically justup to ten cycles of the forcing. Our investigationwas restricted at this time interval becauseexperiments and numerical simulations haveshowed that if escape has not occurred within 8 –10 cycles than it is unlikely to appear in thefollowing cycles. The phase – plane x x indicates a continuously andsignificantly change of the trajectory’s shape (seeFigs. 2 (b)-(d)). For further increase in forcingamplitude period 4, 8 and 16 are obtained, asillustrated in Figs. 2 (e)-(g). When the forcingamplitude outruns the value 0.286 the responsebecomes chaotic, as given in Fig. 2(h). In all thepanels of Fig. 2, the points of the Poincare mapcorresponding to one period of forcing are alsoplotted in the phase planes, and are indicated by ablue big dot. Finally, if F exceeds 0.287 the schemebecomes unstable, and the amplitude goes to infinity(the ship is in capsizing state).The behavior described above remains unchanged forother initial conditions around equilibrium position. The initial conditions ( x(0), x(0)) have beenselected from a vast set having 40,401 201 x 201elements, obtained by dividing the rectangle [- 0.8,1.2] x [- 0.9, 0.9] in equally spaced segments. Eachpoint is tested against the escape criterion, andclassified as safe or not. If it is safe then a smallblack rectangle is drawn around it in the phase plane ( x, x ) , otherwise the rectangle is maintainwhite. In this way, the safe basin is given by theblack area in the phase plane.The fractal erosion experienced by the safe basincould follow different routes [7]. Fig. 3 presentsa slow and permanent reduction of safe basinowing to the increasing of the forcing amplitudeF.Every panel in Fig. 3 requires about 80 min CPUtime on our computer, so the computational costfor having a complete image of the fractalerosion of safe basin for a given , , G pair isremarkably high. To avoid this, we plot insteadthe intersection between safe basins and x, The point ( x(0), x(0)) is considered safe if theamplitude of the displacement x(t) not exceeds unity,that means the roll angle remains smaller than theangle of vanishing stability, V . For a non-safepoint, the associated trajectory goes out in the phaseplane, like a spiral with expanding amplitude [6]. Fora given configuration of parameters , F , G, , theset of all initial conditions that do not lead to capsizeis called safe basin of attraction. We will commentmore on this topic in the next section.3. SAFE BASIN OF ATTRACTION ANDINTEGRITY CURVES respectively x axis, for any value of excitation’samplitude F that leads to emptiness safe basins.Safe basins can be generated numerically by various13

F 0.1F 030.040.050.06xabF 0.24F x-0.3-0.2-0.100.1c0.40.50.6F 0.28050.50.50.40.40.30.30.20.2dx/dtdx/dt0.3dF 30.40.50.60.7xefF 0.281F 60.7xhgFig. 2. Phase planes and Poincare maps for 0.1, 1.8, G 5F .(a) Period 1 orbit (F 0.1); (b) Period 2 orbit (F 0.239); (c) Period 2 orbit (F 0.24); (d) Period 2 orbit (F 0.25); (e)Period 4 orbit (F 0.27); (f) Period 8 orbit (F 0.2805); (g) Period 16 orbit (F 281); (h) Chaotic orbit (F 0.2853)14

The results are given in Fig. 4. Panels have beenconstructed for 200 values of F chosen uniformly inthe mentioned interval. Although less than an hourhave been necessary for the whole computationalprocess (for every panel), we have now a clear imageof both exterior and interior erosion of safe basin.F 0.0, G 5F, 1.8F 0.2, G 5F, g. 3. Erosion of the safe basin for Eq.(1) with 0.1, 1.8, G(a) F 0.0; (b) F 0.2; (c) F 0.4; (d) F 20.4 5F :G 5F, 1.8G 5F, 1.81.2-0.40.4x(0)x(0)F1-0.2-0.4-0.60.8F 1.0, G 5F, 1.8-0.80-0.80.6(b)F 0.4, G 5F, 60.811.2dx/dt(0)x(0) Fig. 4. Intersection between safe basins and: a) x axis; b) x axis, forThe integrity curves show the relative influence ofwave excitation’s amplitude F on capsize relative to 0.1, 1.8, G 5Fvessel safety in the absence of incident waves. Theycould be generated by plotting the safe area,15