A coupled roll sway heave model for analysing ship capsize in beam seas on the basis of a nonlinear dynamics approach

Transcription

1 16 th International Conference on Hydrodynamics in Ship Design 3 rd International Symposium on Ship Manoeuvring Gdansk Ostrόda, Poland 7 1 September 5 N. Themelis and K. J. Spyrou A coupled roll-say-heave model for analysing ship A coupled roll say heave model for analysing ship capsize in beam seas on the basis of a nonlinear dynamics approach N. Themelis & K.J. Spyrou School of Naval Architecture and Marine Engineering National Technical University of Athens 9 Iroon Polytechneiou, Zographou, Athens 15773, Greece. Abstract We study the effect of heave and say on roll dynamics for a ship in beam aves. In the first part of the paper e discuss quantitatively the limitations of single-roll models that neglect these couplings, in terms of ave steepness and the long ave assumption. Thereafter e present a roll-say-heave mathematical model that is currently developed and is suitable for systematic investigations of large amplitude rolling and capsize. Preliminary predictions of response ill be shon for a fishing vessel exposed to harmonic beam aves, including her non-zero mean roll and drift as function of ave steepness. Introduction Capsizing of a ship in aves is one of the most basic fears for mariners, passengers as ell as for anyone ith a vested interest on maritime transportation. The keen interest of the international community is reflected by the current efforts at IMO to improve the main ship stability criterion, the so-called eather criterion [Francescutto, ] as ell as by abundance of research papers on the topic [Perez-Rojas, 3]. Approaches based on mathematical modelling are noadays established as the main route toards improving understanding and developing effective design criteria. Contemporary investigations range beteen rigorous analyses of nonlinear dynamics based on some simplified model of roll motion, to simulation studies that rely on a fuller set of motion equations, incorporating interactions ith the other degrees of freedom. So far, these to orlds are quite separate although they target the same problem. Our longer term research goal is to ork toards bridging the to: i.e. to set up an environment enabling rigorous nonlinear dynamics analysis of ship roll and capsize on the basis of a detailed coupled mathematical model. At this stage e are focusing on the development of a detailed roll-say-heave model for regular and 314

2 irregular beam seas. Some preliminary predictions from this model are reported in this paper. There have been already a fe attempts to investigate nonlinear roll dynamics ith couple models [Thompson et al., 199], [Chen et al., 1999], [Mc Cue & Troesch, 3]. These hoever did not consider the lateral drift motion. This aspect as tackled by [Kuroda & Ikeda, ] ho found that the drift (a combined outcome of the ave excitation and the lateral resistance of the ship) could incur sudden jumps of roll amplitude due to the change of the encounter frequency. The behaviour in the shorter ave range here there could be more complex interactions beteen roll, say and heave has not been investigated yet. Limitations of the one degree model and the effective gravitational field After Froude it has been quite common to investigate ship rolling ith a single-degree of-freedom model, assuming that the ship tends to follo the rotational motion of ater particles. This allos use of the concept of effective gravitational field here the effective gravity force can be assumed to ork alays perpendicular to the instantaneous ater surface [Thompson et al., 199]. The concept is valid for long incident aves relatively to the beam of the ship and it implicitly assumes a slave variation of heave in order to maintain a constant immersed volume. Hoever, these assumptions have limitations as the ave steepness is increased. Furthermore, in the shorter ave range (length comparable to the ship s beam) the method deviates and may not be fully representative of the physical system. When a body follos the circular motion of ater particles on a sinusoidal ave in deep ater, a time varied gravitational acceleration g e ( t) act on it, hich is often assumed as nearly perpendicular to the ave slope. Fig. 1 shos the circular motion of a particle on the ave surface in deep ater. We assume that a surface particle stays on the surface. Initially the particle is at position 1 (see Fig. 1) and after time t, it ill be transferred to position having run an angle ω t on a circular trajectory. The accelerations that act on the ave particle at position are: the acceleration of gravity g and the centrifugal acceleration ω A. The vector sum of these to is the effective gravitational acceleration g e ( t), hich forms an angle β ith g (Fig. 1). The effective gravitational acceleration is calculated then as follos: g e = g Aω cos( ωt ) (1) here A, ω are, respectively, ave amplitude and frequency. The slope n y of a sinusoidal ave is: ω ηy = A sin( ωt) () g According to Fig. 1, the condition of perpendicularity beteen the effective gravitational acceleration and the ave slope is expressed as follos: γ = β = (3) n y 315

3 Figure 1: The rotational motion of a ater particle for a ave in deep ater. here γ is the angle that characterizes the error from perpendicularity and the angle β is calculated as follos: β = arcsin ω Asin( ωt) 4 g + A ω gaω cos( ωt) (4) In Figs. and 3 are shon the variation of the angle γ (scaled over the ave slope Ak for more enlightening presentation of the results) ith the ave steepness H λ and the ratio λ/β (as usual k, λ are respectively ave number and length; B is the beam of the ship). It becomes clear that the assumption of perpendicularity is not valid (the error is greater than 1% of Ak ) hen the ave steepness is above 1/; and also, hen the ave length is shorter than about five times the beam (customarily, the long ave assumption refers to a ave that is longer by at least six times to the ship beam). If the body is not folloing the motion of ater particles then it should be drifting, i.e. dynamic say coupling should be taken into account. Furthermore, in shorter aves the ship should also be heaving. As a matter of fact, a roll-say-heave model should be more suitable for investigating ship dynamic response in beam seas, especially for shorter aves. 316

4 γ/ak H/λ Figure : Effect of ave steepness γ/ak λ/b Figure 3: Effect of avelength. The mathematical model of the coupled roll motion From kinematics and in accordance to Fig. 4, the equations of motion in heave, say and roll are ritten as follos: Figure 4: The inertial (OYZ) and body fixed (Gyz) coordinate systems. 317

5 mv ( φ) Fy (5) = + φ = m ( v) Fz (6) I Gφ = MG (7) here v, are the say and heave velocity of the ship s center of gravity and φ is the roll angular velocity, m and I G are, mass and mass moment of inertia around x. The transformation beteen the earth and body fixed coordinate systems is ell knon: Y G cosφ sinφ v Z G = sinφ cosφ φ 1 φ (8) The to forces and the moment that appear at the right-hand-side of (5)-(7) can be analyzed as follos: FK D F = F Hs + F W + F W + F R + F V (9) F Hs is hydrostatic, F is the viscous force. V FK F W is Froude Krylov, D F W is diffraction, F R is radiation and In linear ave theory, the total ave velocity potential is the sum of the potentials of incident ave, diffraction and radiation. The hydrostatic and Froude Krylov (hydrodynamic) forces are estimated by the integration of the incident ave pressure (static and dynamic respectively) over the etted surface of the ship. For regular aves, the incident ave potential is calculated from: Ag kz Φ I = e sin( ky ω t ) (1) ω Z = Z Acos( ky ωt) (11) From Bernoulli s equation the pressure is: Φ I 1 P= ρ gz + + ΦI ΦI t (1) The hydrostatic and Froude - Krylov forces are repetitively: FHSi () t = ρg Z nds i, for i=, 3, 4 (13) S() t FK I FW () t ρ Φ = nds i, for i=, 3, 4 (14) t S() t here i =, 3, 4 correspond to say, heave and roll motion, ρ is seaater density and St ( ) is the instantaneous etted surface. We should mention that the integration is performed over the instantaneous etted surface and pressures are calculated from the exact ave elevation. As a matter of fact, the nonlinear part of the forces is taken 318

6 into account, hich is important for the accurate simulation of the large motions of the ship. The nonlinear Froude-Krylov force has a nonzero mean, so a steady drift force is present, hich may introduce a bias to the rolling motion. The diffraction force should also bring about steady drift. It is ell knon that this steady drift force is proportional to the square of the ave amplitude and it increases in the short ave range here the reflection of aves and relative heave motion are more intense (e.g. [Maruo, 196], [Neman, 1967]). Kuroda & Ikeda () have focused on this force in their investigation of ship roll ith drift. This component is currently implemented in our model. The radiation forces are frequency dependent. In order to study transient behaviour it is necessary to transform these from the frequency domain to the time domain. Using the impulse response function, obtained as the Fourier transform of the frequency dependent radiation transfer function, the radiation forces ill be [Cummins, 196]: + F () t = a ( ) s K ( τ ) s ( t τ) dτ, for j, k=, 3, 4 (15) Rj jk k jk k K jk ( τ ) = bjk ( ωe)cos( ωτ e ) dω π (16) The convolution integral is the ell-knon memory effect. ajk, b jk are the added mass and damping coefficients. s k, sk are velocity and acceleration of the ship in the k direction of motion and ωe is the encounter frequency. In our model e use a statespace approximation of the radiation force in order to maintain the mathematical model in the form of a system of o.d.e.s hich enables easier consideration of nonlinear dynamics. Our model calculates also the say drag force, roll damping, and cross coupling forces beteen say, heave and roll. For example, the drag force due to bilge keels is calculated as follos (see Fig. 5): Figure 5: Bilge keel damping forces 319

7 1 FBY = ρ ( Y G r Aφ cos( θ ) u ) Y G ra φ cos( θ ) u CDABK (17) 1 FBZ = ρ ( Z G r Aφ sin( θ ) u3 ) Z G ra φ sin( θ ) u3cdabk (18) MB = ra[ FSZcos( θ 1) + FSYsin( θ 1)] (19) C D is the drag coefficient and A BK the total bilge keel area. Other symbols are explained in Fig. 5. The method takes into account the local relative velocities along the hull, using the say, heave and roll velocities ( Y G, Z G, φ ), the ave particle velocities u and u 3 (eqn. and 1) as ell as the detailed geometry of the hull. Φ I u = () y ΦI u3 = (1) z To calculate the forces and solve numerically the system of differential equations e have used the program Mathematica. As said, the system contains only ordinary differential equations (the convolution integrals are approximated by sets of o.d.e.s). Input data are as follos: concerning the ship, the hull geometry, her mass and the distribution of mass. For the incident ave, its height and frequency. The code creates panels over the hull hereon the static and dynamic pressures are calculated at successive time steps, as ell as the angle beteen the horizontal plane and the normal vector of the panel. Application of the mathematical model As application e have used a Japanese fishing vessel hose body plan and basic particulars are shon in Fig. 6 [Umeda et al., 1995]. Her panelization is shon in Fig.7. L bp (length) 34.5 m GM (metacentric height).75 m B (beam) 7.6 m T (natural roll period) 9.7 s D (depth) 3.7 m b BK (breadth of bilge keels).35 m T (mean draft).65 m KG (vertical position of centre 3.36 m C b (block coef.).597 of gravity above keel) Figure 6: Body plan and main particulars of investigated ship. 3

8 Figure 7: Half-hull panelization. Characteristic output of the code is summarised belo: the GZ curve in calm ater as calculated numerical from the code, is shon in Fig. 8. Simulation of a roll decay test from extreme angle of release, approximately 85% of the angle of vanishing stability, is illustrated in Fig. 9. In Fig. 1 is shon a numerical simulation of roll response near resonance, for moderate ave steepness. ω o, ω are respectively the natural roll frequency and the ave frequency. H is the ave height. In Fig. 11 is shon the corresponding say response, and it is apparent that there is significant drifting motion that cannot be neglected. We have examined also the effect of ave steepness on the amplitude of steady roll and also on the mean roll angle (Figs. 1 & 13). There seems to be a near linear relationship beteen the amplitude of roll and ave steepness. Furthermore, e notice that there is an increase of a similar nature concerning the mean roll angle (toards the eather side), hich, for H λ = 11, can become about 16% of the roll amplitude. We believe that the mean roll angle comes from the lateral ave drift force combined ith the lateral resistance force and the pair produces an extra roll moment that tends to rotate the ship to the eather side. This bias in rolling motion should be seriously taken into account as it may reduce disproportionally the dynamic stability of the ship [Thompson, 1997].. GZ [m ] deg. Figure 6: The GZ curve. 31

9 4 3 roll [ deg.] sec Figure 9: Roll decay test 3 ω/ωο=1.18, H/λ=1/3 roll [ deg.] sec Figure 1: Roll response for ω / ω = 1.18 and H λ = 13. o 7 ω/ωο=1.18, Η/λ=1/3 6 5 say [m] number of ave periods Figure 11: Say response. 3

10 Roll [deg.] H/λ Figure 1: Effect of ave steepness on roll amplitude ( ω / ω = 1.31 ) o Mean Roll/ Amplitude H\λ Figure 13: Effect of ave steepness on the mean roll angle ( ω / ω = 1.31 ). o The drift motion tends to loer the encounter frequency and, as a result, larger amplitude motion appears later in terms of ave frequency. This effect is shon in Fig. 14 here e present points of the roll response curve taking into account the mean drift velocity in the calculation of encounter frequency. It is ell knon that different initial conditions can affect the amplitude of transient (and some times also of steady-state) motion. In order to investigate this effect, e examined various scenarios for the initial roll angle and the lateral initial position of the ship in aves. When there is an initial heel to the lee side the obtained max roll is significantly higher than that of heel to the eather side and therefore the propensity to capsize is affected (Fig. 15). Concerning the initial lateral position of the ship on the ave, the max transient roll angle is reached hen e place the ship on a trough (Fig. 16). We should note that in steady state the roll amplitude as the same for all scenarios. 33

11 4 35 ω/ω ο ωe/ω ο 3 5 Roll [deg.] Figure 14: Roll response diagram for constant ave steepness H λ = max abs min 5 Roll [deg.] eather side initial roll angle [deg.] lee side Figure 15: Maximum and minimum roll angles (transient response) for different initial heel ( ω / ω = 1.18, H λ = ). Concluding remarks o The limitations of the one degree of freedom model as ell as the need for a more realistic and insightful investigation of roll dynamics have led us to initiate development of detailed mathematical model of ship rolling in beam aves that takes into account the coupling ith heave and say motions. Preliminary results are shon for a fishing vessel. The implementation of a rigorous nonlinear dynamics investigation based on this model and the integration of these ithin a risk assessment context are the next formidable tasks of our research. 34

12 3 5 Max Abs min Roll [deg.] YG/λ Figure 16: Effect on transient response of the initial lateral position ( ω / ω = 1.18, H λ = ). o References Chen, S., Sha, S.W., Troesch, A.W. (1999): A systematic approach to modelling nonlinear multi DOF ship motions in regular seas. Journal of Ship Research, 5, Cummins W. E. (196): The impulse response function and ship motions. Schiffstechnik, 9, 47, Francescutto A. (): Intact stability the ay ahead. Proceedings, 6 th International Ship Stability Workshop, Webb Institute, Long Island. Kuroda, T., Ikeda, Y. (): Extreme roll motion in ide frequency range due to rapid drift motion. Proceedings, 6 th International Ship Stability Workshop, Webb Institute, Long Island. Lloyd, Α.R.J.M (1989): Seakeeping: Ship Behavior in Rough Weather. Ellis Horood Series in Marine Technology, ISBN Maruo, H. (196): The drift of a body floating on aves. Journal of Ship Research, 4, 3, 1 1. McCue, L.S., Troesch, A.W. (3): The effect of coupled heave/heave velocity or say/say velocity initial conditions on capsize modelling. Proceedings, 8th International Conference on the Stability of Ships and Ocean Vehicles, Madrid, Neman, J.N. (1967): The drift force and moment on ships in aves. Journal of Ship Research, 11, No. 1, Perez-Rojas, L. (3) Proceedings 8 th International Conference, Stability of Ships and Ocean Vehicles, Madrid, ISBN Thompson, J.M.T., (1997): Designing against capsize in beam seas: Recent advantages and ne insights. Applied Mechanics Revie, 5, Thompson, J.M.T, Rainey, R.C.T., Soliman M.S. (199): Mechanics of ship capsize under direct and parametric ave excitation. Philosophical Transactions of the Royal Society, 338, Umeda N., Hamamoto M., Takaishi Y., Chiba Y., Matsuda A., Sera W., Suzuki S., Spyrou K., Watanabe K. (1995): Model Experiments of ship capsize in astern seas. Journal of the Society of Naval Architects of Japan, 177,