Nonlinear Container Ship Model for the Study of Parametric Roll Resonance

Transcription

1 Modeling, Identification and Control, Vol. 8, No. 4, 7, pp. 87 Nonlinear Container Ship Model for the Study of Parametric Roll Resonance Christian Holden Roerto Galeazzi Claudio Rodríguez Tristan Perez 4 Thor I. Fossen Mogens Blanke, Marcelo de Almeida Santos Neves Department of Engineering Cyernetics, Norwegian University of Science and Technology, Norway. {c.holden, fossen}@ieee.org Department of Electical Engineering, Technical University of Denmark. {rg, m}@elektro.dtu.dk Department of Naval Architecture and Ocean Engineering, LaOceano/COPPE, Federal University of Rio de Janeiro, Brazil. {claudiorc, masn}@peno.coppe.ufrj.r 4 Centre for Complex Dynamic Systems and Control, University of Newcastle, Australia. tristanp@ieee.org Centre for Ships and Ocean Structures, Norwegian University of Science and Technology, Norway. Astract Parametric roll is a critical phenomenon for ships, whose onset may cause roll oscillations up to ±4, leading to very dangerous situations and possily capsizing. Container ships have een shown to e particularly prone to parametric roll resonance when they are sailing in moderate to heavy head seas. A Matla/Simulink R parametric roll enchmark model for a large container ship has een implemented and validated against a wide set of experimental data. The model is a part of a Matla/Simulink Toolox (MSS, 7). The enchmark implements a rd -order nonlinear model where the dynamics of roll is strongly coupled with the heave and pitch dynamics. The implemented model has shown good accuracy in predicting the container ship motions, oth in the vertical plane and in the transversal one. Parametric roll has een reproduced for all the data sets in which it happened, and the model provides realistic results which are in good agreement with the model tank experiments. Keywords: parametric roll resonance; nonlinear systems; model validation; parameter identification; ships Introduction Parametric roll is an autoparametric resonance phenomenon whose onset causes a sudden rise in roll oscillations. The resulting heavy roll motion, which can reach -4 degrees of roll angle, may ring the vessel into conditions dangerous for the ship, the cargo, and the crew. The origin of this unstale motion is the time-varying geometry of the sumerged hull, which produces periodic variations of the transverse staility properties of the ship. Parametric roll is known to occur when a ship sails in moderate to heavy longitudinal or olique seas; the wave passage along the hull and the wave excited vertical motions result in variations of the intercepted waterplane area, and in turn, in relevant changes in the restoring characteristics. The onset and uild-up of parametric roll is due to the occurrence of concomitant conditions: the wave length is close to the ship length (λ w L PP ), the ship approaches waves with encounter frequency almost twice the roll natural frequency (ω e ω ), and the wave height is greater than a ship-dependent threshold (h w > h s ). ISSN c 7 Norwegian Society of Automatic Control

2 Modeling, Identification and Control The risk of parametric roll has een known to the maritime community since the early fifties, ut only for small vessels with marginal staility e.g. fishing oats sailing in following seas. However, the phenomenon has recently attracted significant interest y the scientific community after accidents occurred with container ships sailing in head seas, incidents that involved significant damage to cargo as well as structural damages for millions of dollars (France et al., ; Carmel, 6). Several different types of vessel have reported to experience parametric roll in head seas, e.g. destroyers (Francescutto, ), ro-ro paxes (Francescutto and Bulian, ) and PCTC (Palmquist and Nygren, 4). Container carriers, however, are the most prone to parametric roll ecause of the current particular hull shape, i.e. large ow flare and stern overhang, and hence arupt variation in the intercepted water-plane area when a wave crest or trough is amidships. This has called for deep investigations into the nature of parametric roll in head/near head seas, and for the development of mathematical models ale to capture and reproduce the physical aspects driving the resonant motion. In the last six years mathematical models of different complexity have een proposed y the scientific community, most of them relying on the Mathieu Equation to descrie the dynamics of the ship suject to parametric resonance. One-DOF models considering the uncoupled roll motion have een widely used to analyze the critical parameters of the phenomenon and derive staility conditions. Examples can e found in the papers y France et al. () and Shin et al. (4) where the authors employed the -DOF roll equation to show that, in regular waves, the Mathieu Equation can explain the onset of heavy roll motion in head seas. Bulian (6) proposed a.-dof model where the dynamic interaction etween the vertical motions and the roll oscillation was relaxed y the assumption of quasi-static heave and pitch. Moreover, that assumption allowed an analytical description of the GZ curve that was approximated as a surface varying with roll angle and wave crest position. This model is considered valid for moderate ship speed in head seas, and has lead to reasonale results in predicting parametric roll. A -DOF nonlinear fully coupled model was first developed y Neves (). A first attempt was done y using Taylor series expansion up to nd -order to descrie the coupled restoring forces and moments in heave, pitch and roll. This model, although it provided a quite thorough description of the nonlinear interactions among the different modes, tended to overestimate the roll oscillation aove the staility threshold. Neves and Rodríguez () proposed a rd -order analytical model where the couplings among the three modes are expressed as a rd -order Taylor series expansion. In this new model the nonlinear coefficients are mathematically derived as a functions of the characteristics of the hull shape. This -DOF model has een applied for the prediction of parametric roll to a transom stern fishing vessel (Neves and Rodríguez, 6a,) providing outcomes which etter match the experimental results than the nd -order model. It is noted that the aove-mentioned literature have attempted to model parametric roll from an analytical points of view. Jensen (7) takes a statistical approach instead, motivated y the difficulties inherent in descriing the interaction etween a -dimensional wave pattern and the motion of a ship hull. He shows how the statistical distriution of nonlinear ship responses can e estimated very accurately using a firstorder reliaility method. A commercial implementation in a system to predict parametric roll (SeaSense R ) was reported in Nielsen et al. (6). The direction of this paper is the analytical one, aiming at providing simulation tools that could e.g. e used in studies of active stailization and control. The model proposed y Neves and Rodríguez () is applied to descrie the dynamics of a container vessel suject to parametric roll resonance conditions. The model parameters are identified ased upon the ship line drawings and the loading conditions. A Matla/ Simulink implementation of the aove model is then presented. The reliaility of the implemented model in simulating parametric resonance ehavior is validated against experimental data. The validation has shown good agreement with the experimental results for roll oth in the experiments where parametric roll resonance occurred, and in the experiments where it did not occur. The main goal of this work is to provide a enchmark for simulating parametric roll of a container ship over a large range of ship speeds and sea states. This enchmark has een designed to e a fully integrated part of Matla/Simulink Toolox for marine systems (MSS, 7). The availaility of such a powerful tool opens up a great wealth of opportunities, notaly the design and testing of novel model-ased roll motion stailizers. Mathematical Model for Parametric Roll The proposal and the adoption of an analytical model for representing a specific phenomenon should e driven y a trade-off etween complexity and agreement with 88

3 Holden et al., Nonlinear Container Ship Model for the Study of Parametric Roll Resonance physical laws governing that phenomenon and/or experimental results. Tondl et al. () define an autoparametric system as follows: e the generalized coordinate vector, where z is the heave displacement, φ is the roll angle, and θ is the pitch angle, as shown in Figure. Definition Autoparametric systems are virating systems that consist of at least two constituting susystems. One is the primary system that will usually e in a virating state. This primary system can e externally forced, self-excited, parametrically excited, or a comination of these. The second constituting susystem is called the secondary system. The secondary system is coupled to the primary system in a nonlinear way, ut such that the secondary system can e at rest while the primary system is virating. An autoparametric system is, hence, characterized y these main aspects:. two nonlinearly coupled susystems;. a normal mode where the primary system is in a virating state and the secondary system is at rest;. the presence of instaility regions where the normal mode ecomes unstale; 4. in the region of instaility of the normal mode the overall system is in autoparametric resonance: the secondary system is parametrically excited y the virations of the primary system and it will not e at rest anymore. Considering Definition, DOF models have too little complexity to descrie an autoparametric system, since the roll motion for a ship sailing in longitudinal seas represents only the secondary system. They are useful to otain insight in the parametric roll resonance phenomenon, ut they will have difficulty predicting the real amplitude of the oscillations aout the transverse plane. The model proposed y Neves and Rodríguez () is complex enough to capture the dynamics of a container vessel ehaving as an autoparametric system; it includes oth the primary system (heave and pitch dynamics) which is externally excited y the wave motion, and the secondary system (roll dynamics) which is parametrically excited y the primary.. Equations of Motion The -DOF nonlinear mathematical model of the container vessel is presented in the following way (using the notation of Neves and Rodríguez ()): Let s(t) = [ z(t) φ(t) θ(t) ] T () Figure : Definition of motions Then the nonlinear equations of motion can e expressed in matrix form as where (M + A) s + B( φ)ṡ + c res (s, ζ) = c ext (ζ, ζ, ζ) () M R is the diagonal rigid-ody generalized mass matrix; A R is the generalized added mass matrix; B R is the hydrodynamic damping (nonlinear in roll); c res R is the nonlinear vector of restoring forces and moments expressed as functions of the relative motion etween ship hull and wave elevation ζ(t); c ext R is the vector of the external wave excitation forces and moments which depends on wave heading, encounter frequency, wave amplitude and time... Generalized Mass, Added Mass and Damping The generalized mass matrix can e written as m M = I x () I y where m is the ship mass, I x is inertia in roll and I y is inertia in pitch. 89

4 Modeling, Identification and Control The hydrodynamic added mass and damping matrices are expressed as Z z Z θ A = K φ (4) B( φ) = M z M θ Zż Z θ K φ( φ) Mż M θ () where all entries except K φ( φ) can e evaluated y means of potential theory (Salvesen et al., 97). The hydrodynamic damping in roll may e expressed as K φ( φ) φ = K φ φ + K φ φ φ φ (6) where the linear term represents the potential and linear skin friction, whereas the nonlinear term takes into account viscous effects. The coefficients K φ and K φ φ can e calculated y the formulae given in Himeno (98). The roll damping characteristics may also e derived from data of roll decaying tests at appropriate forward speeds of the vessel... Waves In regular seas, the incident wave elevation according to the Airy linear theory, see Newman (977), is defined as ζ(x, y, t; χ) = A w cos(kx cos χ ky sin χ ω e t) (7) where A w is the wave amplitude, k is the wave numer, χ is the wave heading, and ω e is the encounter wave frequency. For head seas (χ = 8 ), the wave elevation reads as ζ(x, t) = A w cos(kx + ω e t). (8).. Nonlinear Restoring Forces and Moments The nonlinear restoring actions are given y the comination of the effects of the vessel motion in calm water and the effect of the wave elevation along the hull. Therefore, the vector of restoring forces and moments can e written, up to rd -order terms, as c pos c pos,s + c pos,ζ + c pos,s + c pos,sζ + c pos,ζ + c pos,s + c pos,s ζ + c pos,sζ + c pos,ζ (9) where c pos,s i ζ j = i+j c pos s i ζ s i ζ j. j The st, nd and rd -order components in (9), which are independent of the displacement vector s, must e included in the external forces and moments acting on the vessel. These terms descrie the linear and nonlinear Froude-Krylov forces/moments. The nd and rd -order nonlinear effects due to hullwave interactions must, instead, e included in the restoring vector c res ecause of their affinity, from the mathematical point of view, with the hydrostatic actions. Then the restoring force and moments due to ody motion are given y c res (s, ζ) = c pos (s, ζ) c ext,fk (ζ) () where c ext,fk (ζ) = c pos,ζ + c pos,ζ + c pos,ζ. Therefore the restoring force/moments in each degree of freedom are given y the following terms: st -order ody motions (c pos,s ) Z () = Z z z + Z φ φ + Z θ θ K () = K z z + K φ φ + K θ θ () M () = M z z + M φ φ + M θ θ nd -order ody motions (c pos,s ) Z () = (Z zzz + Z zφ zφ + Z zθ zθ + Z φθ φθ + Z φφ φ + Z θθ θ ) K () = (K zzz + K zφ zφ + K zθ zθ () M () + K φθ φθ + K φφ φ + K θθ θ ) = (M zzz + M zφ zφ + M zθ zθ + M φθ φθ + M φφ φ + M θθ θ ) nd -order hull-wave interactions (c pos,sζ ) Z () h/w = Z ζz(t)z + Z ζφ (t)φ + Z ζθ (t)θ K () h/w = K ζz(t)z + K ζφ (t)φ + K ζθ (t)θ () M () h/w = M ζz(t)z + M ζφ (t)φ + M ζθ (t)θ rd -order ody motions (c pos,s ) Z () = ( Zzzz z + Z φφφ φ + Z θθθ θ 6 + Z zzφ z φ + Z zzθ z θ + Z φφz φ z + Z φφθ φ θ + Z θθz θ z + Z θθφ θ φ + 6Z zφθ zφθ ) K () = ( Kzzz z + K φφφ φ + K θθθ θ 6 + K zzφ z φ + K zzθ z θ + K φφz φ z + K φφθ φ θ + K θθz θ z (4) + K θθφ θ φ + 6K zφθ zφθ ) 9

5 Holden et al., Nonlinear Container Ship Model for the Study of Parametric Roll Resonance M () = ( Mzzz z + M φφφ φ + M θθθ θ 6 + M zzφ z φ + M zzθ z θ + M φφz φ z + M φφθ φ θ + M θθz θ z + M θθφ θ φ + 6M zφθ zφθ ) rd -order hull-wave interactions (c pos,s ζ +c pos,sζ ) Z () h/w = Z ζzz(t)z + Z ζφφ (t)φ + Z ζθθ (t)θ + Z ζzφ (t)zφ + Z ζzθ (t)zθ + Z ζφθ (t)φθ + Z ζζz (t)z + Z ζζφ (t)φ + Z ζζθ (t)θ K () h/w = K ζzz(t)z + K ζφφ (t)φ + K ζθθ (t)θ + K ζzφ (t)zφ + K ζzθ (t)zθ + K ζφθ (t)φθ + K ζζz (t)z () + K ζζφ (t)φ + K ζζθ (t)θ M () h/w = M ζzz(t)z + M ζφφ (t)φ + M ζθθ (t)θ + M ζzφ (t)zφ + M ζzθ (t)zθ + M ζφθ (t)φθ + M ζζz (t)z + M ζζφ (t)φ + M ζζθ (t)θ The time varying terms depend explicitly on the wave elevation ζ(t) and thus implicitly on the time t. Looking at the st, nd and rd -order coefficients, a strong cross-coupling etween all three degrees of freedom ecomes evident...4 External Forcing The interaction etween ship motion and wave passage is modeled as a variation of the geometry of the sumerged hull defined y the instantaneous wave position. The external forcing vector c ext (ζ, ζ, ζ) includes only contriutions independent of ship motions, such that c ext (ζ, ζ, ζ) = τ w + τ w. (6) τ w represents the st -order wave excitation forces generated y the wave motion. These forces are characterized y two contriutions: the first one is due to Froude-Krylov forces, which are caused y incident waves considering the hull restrained from moving and that the presence of the hull does not influence the wave field. The second contriution gives the diffraction forces, which provide the corrections necessary for the variation of the flow field produced y the hull. τ w are the nd -order wave excitation forces which include three important components. The first contriution is given y the mean wave drift forces caused y nonlinear wave potential effects; the second one is due to low-frequency wave drift forces caused y nonlinear elements in the wave loads; and the third component is given y high-frequency wave drift forces. In the present analysis the external force and moments are defined as eing proportional to the first order wave motion, whereas higher order terms are neglected. Therefore the external force/moments vector c ext reads as c ext (ζ, ζ, ζ) τ w = c ext,fk + c ext,dif. (7) The wave excitation forces are defined y the waveforce response amplitude operator (force RAO) for each degree of freedom. The Force RAO is computed (Perez, ) as F i (ω e, χ) = τ wi (ω e, χ) ζ ej arg[ τwi(ωe,χ)] (8) where τ wi is the complex st -order wave excitation forces, and ζ is the complex wave elevation. Since the model only considers head seas, (8) simplifies to F i (ω e ) = τ wi (ω e ) ζ ej arg[ τwi(ωe)]. (9) With these force RAOs, it is possile to otain the wave excitation loads in each degree of freedom as τ wi (t) = F i (ω e ) A w cos(ω e t + α i ) () for i =, 4,, where α i = arg[ F i (ω e )]. For example, the external force acting on heave is given y Z ext (t) = F (ω e ) A w cos(ω e t + α ). () Identification of Model Parameters from Hull Form and Wave Characteristics The identification of model parameters is completely ased upon the hull shape of the container vessel and upon the wave characteristics. In this section the formulas are presented. The numerical values of those parameters, computed for the considered container ship, can e found in Appendix A. In Tale the main characteristics of the containership are reported. 9

6 Modeling, Identification and Control Tale : Main characteristics of the container ship Quantity Sym. Value Length etween perpendiculars L PP 8 m Beam amidships B.6 m Draught amidships T.7 m Displacement m Roll radius of gyration r x. m Transverse metacentric height GM t.84 m. Body Motion Coefficients The st -order ody motion coefficients refer to calm water hydrostatics and are given y Z z = ρga Z θ = ρga x f K φ = GM t () M z = ρga x f M θ = GM l where ρ is the water density, g is the acceleration of gravity, A is the waterplane area, x f is the longitudinal coordinate of the centre of floatation, and GM l is the longitudinal metacentric height. The nd and rd -order ody motion coefficients correspond to the variations in the restoring characteristics of the ship due to the changes in pressure related to the vessel motions. In order to compute them numerically, it is necessary to express the nonlinear hydrostatic actions as function of the three modes heave, pitch, and roll. In particular, it is possile to demonstrate that Z(z, φ, θ) = ρg( ) K(z, φ, θ) = ρg[ z G sin φ + (y B cos φ z B sin φ)] () M(z, φ, θ) = ρg[ z G cos φ sin θ (x B cos θ + y B sin φ cos θ + z B cos φ sin θ)] where is the mean displacement, = (z, φ, θ) is the instantaneous displacement, z G is the vertical coordinate of the centre of gravity, x B, y B, and z B are the coordinates of the instantaneous centre of uoyancy. Tales - show the nd and rd -order coefficients for each degree of freedom.. Hull-Wave Interaction Coefficients Under the assumption of regular waves, the periodic wave passage along the hull produces cyclic variation in the restoring characteristics of the vessel. These Tale : nd -order hydrostatic restoring coefficients Heave Roll Pitch Z zz = Z z K zz = M zz = M z Z zφ = K zφ = K z φ M zφ = Z zθ = Z z θ K zθ = M zθ = M z θ Z φφ = Z φ K φφ = M φφ = M φ Z φθ = K φθ = K φ θ M φθ = Z θθ = Z θ K θθ = M θθ = M θ changes are taken into account y the nd and rd - order coefficients included in the nonlinear interactions c pos,sζ and c pos,s ζ + c pos,sζ. In order to determine the hull-wave interaction coefficients, the Froude-Krylov forces must e defined. The velocity potential for the undistured wave, as defined in (7), is given y ϕ I = A wg ω e e kz sin(kx cos χ ky sin χ ω e t). (4) Therefore, the st and nd -order Froude-Krylov forces are: F F ϕi K j (t) = ρ t n j ds () F F K j (t) = ρ ( ϕ I ϕ I ) n j ds (6) where n is the normal to the hull surface and j addresses the specific mode for which the force is computed. The coefficients are then given y the formulas is Tales 4. Due to the assumption of regular waves, the coefficients can e descried as a sum of a sine and a cosine term. For instance, the nd -order term K ζφ (t), which is proportional to wave amplitude, can e written as K ζφ (t) = A w (K ζφc cos ω e t + K ζφs sin ω e t) (7) where K ζφc and K ζφs are constants. Analogously, the rd -order terms K ζzφ (t) and K ζφθ (t) are given y K ζzφ (t) = A w (K ζzφc cos ω e t + K ζzφs sin ω e t) (8) K ζφθ (t) = A w (K ζφθc cos ω e t + K ζφθs sin ω e t). (9) These functions play an important role since they parametrically excite the coupled system, eing multiplied with, respectively, z(t)φ(t) and φ(t)θ(t). The rd -order term K ζζφ (t), which is proportional to the wave amplitude squared, is given y K ζζφ (t) = A w(k ζζφ + K ζζφc cos ω e t + K ζζφs sin ω e t) () 9

7 Holden et al., Nonlinear Container Ship Model for the Study of Parametric Roll Resonance Tale : rd -order hydrostatic restoring coefficients Heave Z zzz = Z z Z zzφ = Z zzθ = Z z θ Z φφz = Z z φ Z φφφ = Z φφθ = Z φ θ Z θθz = Z z θ Z θθφ = Z θθθ = Z θ K zzz = K φφz = K θθz = Roll K zzφ = K z φ K zzθ = K φφφ = K φ K φφθ = K θθφ = K φ θ K θθθ = Pitch M zzz = M z M zzφ = M zzθ = M z θ M φφz = M z φ M φφφ = M φφθ = M φ θ M θθz = M z θ M θθφ = M θθθ = M θ Z zφθ = Heave-roll-pitch coupling K zφθ = K z φ θ M zφθ = Tale 4: nd -order hydrostatic restoring coefficients due to wave passage Heave Roll Pitch Z ζz (t) = F F K z K ζz (t) = M ζz (t) = F F K z Z ζφ (t) = K ζφ (t) = F F K 4 φ M ζφ (t) = Z ζθ (t) = F F K θ K ζθ (t) = M ζθ (t) = F F K θ where it can e noticed the presence of a constant term plus a super-harmonic term of doule the encounter frequency.. Nonlinear Restoring Forces and Moments Redux Rewriting the restoring forces and moments () (), according to the equations derived in this section gives: st -order ody motions (c pos,s ) Z () = Z z z + Z θ θ K () = K φ φ () M () = M z z + M θ θ nd -order ody motions (c pos,s ) Z () = (Z zzz + Z zθ zθ + Z φφ φ + Z θθ θ ) K () = K zφ zφ + K φθ φθ () M () = (M zzz + M zθ zθ + M φφ φ + M θθ θ ) nd -order hull-wave interactions (c pos,sζ ) Z () h/w = A w(z ζzc z + Z ζθc θ) cos ω e t + A w (Z ζzs z + Z ζθs θ) sin ω e t K () h/w = A w(k ζφc cos ω e t + K ζφs sin ω e t)φ () M () h/w = A w(m ζzc z + M ζθc θ) cos ω e t + A w (M ζzs z + M ζθs θ) sin ω e t rd -order ody motions (c pos,s ) Z () = 6( Zzzz z + Z θθθ θ + Z zzθ z θ + Z φφz φ z + Z φφθ φ θ + Z θθz θ z K () = 6( Kφφφ φ + K zzφ z φ + K θθφ θ φ + 6K zφθ zφθ ) (4) 9

8 Modeling, Identification and Control Tale : rd -order hydrostatic restoring coefficients due to wave passage Heave Z ζζz (t) = F F K z Z ζζφ (t) = Z ζζθ (t) = F F K θ Z ζzz (t) = F F K z Z ζzφ (t) = Z ζzθ (t) = F F K z θ Z ζφφ (t) = F F K φ Z ζθθ (t) = F F K θ Z ζφθ (t) = Roll K ζζz (t) = K ζζφ (t) = F F K 4 φ K ζζθ (t) = K ζzz (t) = K ζzφ (t) = F F K 4 z φ K ζzθ (t) = K ζφφ (t) = K ζθθ (t) = K ζφθ (t) = F F K 4 φ θ Pitch M ζζz (t) = F F K z M ζζφ (t) = M ζζθ (t) = F F K θ M ζzz (t) = F F K z M ζzφ (t) = M ζzθ (t) = F F K z θ M ζφφ (t) = F F K φ M ζθθ (t) = F F K θ M ζφθ (t) = M () = 6( Mzzz z + M θθθ θ + M zzθ z θ + M φφz φ z + M φφθ φ θ + M θθz θ z ) rd -order hull-wave interactions (c pos,s ζ +c pos,sζ ) Z () h/w = Z ζzz(t)z + Z ζφφ (t)φ + Z ζθθ (t)θ + Z ζzθ (t)zθ + Z ζζz (t)z + Z ζζθ (t)θ K () h/w = K ζzφ(t)zφ + K ζφθ (t)φθ + K ζζφ (t)φ () M () h/w = M ζzz(t)z + M ζφφ (t)φ + M ζθθ (t)θ + M ζzθ (t)zθ + M ζζz (t)z + M ζζθ (t)θ 4 Matla Implementation of the Model A Matla/Simulink model for the container ship model was developed for the purposes of simulating parametric roll resonance, ased on the model of Section. For each time instant and system state, a function generates the instantaneous value of [ṡ T s T ] T. Numerically integrating with an explicit Runge-Kutta method of order 4, with the fixed time step h = s, the state [s T ṡ T ] T is calculated for any given time instant. The parameters used in the calculations are listed in Appendix A. While this was not done for the results presented in this paper, for other encounter frequencies than the ones used in the experiments, interpolation can e applied to calculate approximate parameter values. The code is part of the Marine Systems Simulator (MSS, 7). Validation of the Model Against Experimental Data A comparison of the simulation and the experimental results can e seen in Figures 4. The experiments were conducted with a :4 scale model ship in a towing tank. The experiments were done with varying forward speed, and wave frequency and height. This is summarized in Tale 6. All data in the tale and in the figures are in full scale. The simulations were done with the code descried in Section 4. All simulations were made allistically. Initial conditions can e found in Tale 7. Initial conditions not listed in the tale (θ, ż, φ and θ) were all zero. The experiments were all assumed to start at t =. A comparison of the simulation results with the experimental results can e seen in Tale 8. The first column is the experiment numer. The second column is wave amplitude A w. The third column is wave frequency ω. The fourth column is the ratio of encounter frequency to natural roll frequency (ω e /ω ). The fifth 94

9 Holden et al., Nonlinear Container Ship Model for the Study of Parametric Roll Resonance Tale 6: Experimental conditions Exp. U [m/s] ω [rad/s] A w [m] ω e [rad/s] Tale 7: Simulation initial conditions Exp. z [m] φ [rad] e e e e e e e e e e e e e- 8..6e e e e e e e e e- Tale 8: Simulation results Exp. A w ω ω e/ω max φ sim max φ exp Error %

10 Modeling, Identification and Control column is maximum roll amplitude for the simulations (max φ sim ). The sixth is maximum roll amplitude for the experiments (max φ exp ). The seventh and final column is the percentage error given y max φ sim max φ exp, max φ exp rounded to integer value. Note that most of the experiments were stopped efore the final steady-state roll angle could e achieved due to fear of vessel capsizing. Figures show heave, roll and pitch as functions of time, oth experimental and simulated Figure : Exp. 7. Exp. dashed red, sim. solid lue Figure : Exp. 7. Exp. dashed red, sim. solid lue. In Figure 4, we can see the maximum roll angle achieved in the simulations and experiments for certain conditions, plotted against the ratio of encounter frequency to natural roll frequency (ω e /ω ). The data in the figure is all for A w =. m, and ω =.464 rad/s Figure 4: Exp. 74. Exp. dashed red, sim. solid lue Figure : Exp. 7. Exp. dashed red, sim. solid lue Figure 6: Exp. 76. Exp. dashed red, sim. solid lue. 96

11 Holden et al., Nonlinear Container Ship Model for the Study of Parametric Roll Resonance Figure 7: Exp. 77. Exp. dashed red, sim. solid lue Figure 8: Exp. 8. Exp. dashed red, sim. solid lue Figure 9: Exp. 78. Exp. dashed red, sim. solid lue Figure : Exp. 8. Exp. dashed red, sim. solid lue Figure : Exp. 79. Exp. dashed red, sim. solid lue Figure : Exp. 8. Exp. dashed red, sim. solid lue. 97

12 Modeling, Identification and Control Figure : Exp. 8. Exp. dashed red, sim. solid lue Figure 4: Exp. 86. Exp. dashed red, sim. solid lue Figure : Exp. 84. Exp. dashed red, sim. solid lue Figure 6: Exp. 87. Exp. dashed red, sim. solid lue Figure 7: Exp. 8. Exp. dashed red, sim. solid lue Figure 8: Exp. 88. Exp. dashed red, sim. solid lue. 98

13 Holden et al., Nonlinear Container Ship Model for the Study of Parametric Roll Resonance Figure 9: Exp. 89. Exp. dashed red, sim. solid lue Figure : Exp. 9. Exp. dashed red, sim. solid lue Figure : Exp. 9. Exp. dashed red, sim. solid lue Figure : Exp. 9. Exp. dashed red, sim. solid lue max( φ ) [deg].9.. ω e /ω Figure : Exp. 9. Exp. dashed red, sim. solid lue. Figure 4: Max. roll angle vs ω e /ω for A w =. m, ω =.464 rad/s. Exp. dashed red, sim. solid lue. 99

14 Modeling, Identification and Control 6 Analysis of the Model Based Upon the Validation Results The rd -order model developed for the 8m long container ship shows high capailities in reproducing the vertical and transversal dynamics of the vessel under parametric resonance conditions, as shown y the comparison of the experimental results (Figures 4). Considering the experiments where parametric resonance did occur, the implemented model performs well: starting from similar initial conditions and eing sujected to the same excitation forces used during the experiments, the model develops parametric resonance within the same time frame as the :4 scale model ship in most of the cases. The most ovious differences etween the simulation and the experimental results consists of the amplitude of the oscillations. In all the experiments where parametric resonance occurred, the peak value of the roll oscillations is higher than the saturation level at which the model settles. Although the model has a general tendency to underestimate the peak value of the roll motion, the gap is relatively small in most cases. Considering the 9 experiments where parametric roll did not occur, the model produced false positive cases developing resonant motion. In order to understand this disagreement etween model ehavior and experimental results, the tuning factor ω e /ω must e taken into consideration. In fact all the false-positive cases occur with a tuning factor close to the limits of the first instaility region of the Hill-Mathieu Equation (ω e ω ), as shown in Figure 4. Looking at the peak value of the roll oscillations (Figure 4 and Tale 8), it is seen that the largest differences are in the region of high tunings ( ωe ω.) for which the model predicts large roll motion whereas the experiments showed no amplification. It seems ovious that when the experimental conditions are close to the limits of staility the model does not match exactly the frequency at which the arupt variation in roll motion take place. The errors indicated in tests 7, 79 and 8 have no real physical meaning, since the initial condition of degrees was chosen aritrarily high in order to indicate a decaying motion. For all experiments, heave and pitch dynamics have shown relatively good agreement with the experiments. In all the test runs the two modes oscillates at the excitation frequency, matching the experimental records. The amplitude of the oscillations is close to that of the experimental values. 7 Conclusions A Matla/Simulink enchmark for the simulation of parametric roll resonance for a large container ship has een implemented and validated against experimental results. The implementation reflects the coupled rd - order nonlinear model for parametric roll first introduced y Neves and Rodríguez (). The mathematical model for the container ship at hand, already presented in Rodríguez et al. (7), has een reviewed, illustrating in details the ship dynamics that has een taken into account. st, nd and rd -order contriutions have een descried and analytical formulas of all the couplings coefficients due to heave, roll, pitch, and wave motion have een given. Furthermore, numerical values of all coefficients are computed ased upon the hull geometry and wave characteristics. The enchmark has een tested on a set of different conditions, which have een chosen to match the experimental conditions. Each test run differs from the previous for at least the value of one parameter among ship speed, wave frequency, and wave height. The heave and pitch dynamics descried y the model are in good accordance with the experimental results. The model seems to reproduce the pitch motion slightly etter, catching the right amplitude in most of the cases. The results otained in roll have shown good agreement with the records of the experiments. In particular, the model agrees with the experiments in all the cases where parametric roll occurred, although the amplitude of the roll oscillations does not quite reach the experimental peak value in most cases. For the experiments where there was no parametric roll, then the model produces false positives in aout % of the cases. This disagreement etween the simulation and the experimental results is elieved to e related to the specific values of the tuning factor ω e /ω which were too close to the limits of the first instaility region of the Hill-Mathieu Equation. In these cases, it is very difficult to get the correct response with allistic simulations. The availaility of this enchmark offers a wide range of opportunities for development of new model-ased control strategies for counteracting or preventing parametric roll resonance. A Tales of Coefficients The parameters can e found in Tales 9 6. All numers are given in the kg-m-s (SI) system. Only non-zero numers are listed. Tale 9 contains the rigid ody inertia matrix, and Tale the added mass. Tale contains the hydro-

15 Holden et al., Nonlinear Container Ship Model for the Study of Parametric Roll Resonance dynamic damping parameters, while Tale contains the ody motion parameters. Tale contains the wave motion parameters for heave. Tale 4 contains the wave motion parameters for roll. Tale contains the wave motion parameters for pitch. Tale 6 contains the external wave excitation parameters. Note that α and α are given in radians. Tale 9: Rigid ody inertia m I x I y 7.7e7.4e.99e Tale : Restoring force (motions) Heave Roll Pitch Z z = 7.988e7 K φ =.44e9 M z = 7.66e8 Z θ = 7.66e8 K φφφ =.7844e M θ = 4.6e Z zz =-.4e6 K zφ =-8.468e7 M zz =-.498e8 Z zθ =-.4986e8 K φθ =-.49e M zθ =-4.9e Z zφ =-.9468e8 K zzφ = 7.978e7 M zφ =-.64e Z θθ =-4.9e K φθθ =.4e M θθ =-4.87e Z zφφ =.887e8 M zφφ =.7e Z φφθ =.7e M φφθ = 4.64e Z θθθ =.4e9 M θθθ = 8.664e Tale 4: Restoring force roll (wave) ω K ζφc K ζφs e8.e e8.98e e8.948e e8.797e e8 -.44e e8 -.7e e8 -.9e7 Acknowledgements The authors would like to thank MARINTEK for having provided the experimental facilities and the financial support. Many thanks are due to Dr. Ingo Drummen for having collected the experimental data. This work was partially supported in Brazil y CNPq within the STAB project (Nonlinear Staility of Ships), y LaOceano-COPPE/UFRJ and CAPES, and in Norway y CESOS (Centre of Excellence for Ships and Ocean Structures). References Bulian, G. Development of analytical nonlinear models for parametric roll and hydrostatic restoring varia- Tale 6: External wave forces ω e F α F α.9.89e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e tions in regular and irregular waves. Ph.D. thesis, Università degli Studi di Trieste, 6. Carmel, S. M. Study of parametric rolling event on a panamax container vessel. Journal of the Transportation Research Board, 6. 96:6 6. France, W. N., Levadou, M., Treakle, T. W., Paulling, J. R., Michel, R. K., and Moore, C. An investigation of head-sea parametric rolling and its influence on container lashing systems. In SNAME Annual Meeting.. Francescutto, A. An experimental investigation of parametric rolling in head waves. Journal of Offshore Mechanics and Arctic Engineering,. :6 69. Francescutto, A. and Bulian, G. Nonlinear and stochastic aspects of parametric rolling modelling. In Proc. of the 6th International Ship Staility Workshop.. Himeno, Y. Prediction of ship roll damping - state of the art. Technical report, Department of Naval Architecture and Marine Engineering, The University of Michigan, 98. Jensen, J. J. Effcient estimation of extreme non-linear roll motions using the first-order reliaility method (FORM). Marine Science and Technology, 7. (4):9. MSS. Marine Systems Simulator - Matla/Simulink Toolox. < 7. Accessed Novemer. Neves, M. A. S. On the excitation of comination modes associated with parametric resonance in

16 Modeling, Identification and Control Tale : Added mass ω e Z z Z θ K φ M z M θ e7.986e8.7e9.4e9 4.7e e e8.7e9.6e9 4.68e e7.74e8.7e9.8e9 4.9e e7 6.97e8.7e9.4e9 4.69e e7.96e8.7e9.7e9 4.6e e7 6.4e8.7e9.6e9 4.97e e e8.7e9.67e9 4.77e e7.87e8.7e9.74e9 4.79e.9 8.e7 6.8e8.7e9.e9 4.76e e7.66e8.7e9.86e9 4.9e.6 8.9e7.79e8.7e9.8e9 4.68e e e8.7e9.9849e9 4.9e e7.46e8.7e9.87e9 4.4e e7 4.77e8.7e9.e9 4.4e e7 6.8e8.7e9.98e9 4.9e e e8.7e9.e9 4.e e7 4.9e8.7e9.8e9 4.4e Tale : Hydrodynamic damping ω e Zż Z θ K φ K φ φ Mż M θ e7.9e9.9e8.999e8.648e8.74e e7.46e9.467e8.74e8.67e8.7e e7.8e9.9e8.999e8.67e8.74e e7.e9.96e8.4696e8.9e8.74e e7.79e9.9e8.999e8.689e8.766e e7.e9.46e8.e8.984e8.797e e7.77e9.9e8.999e8.699e8.77e e7.96e9.44e8.7877e8.484e8.74e e7.77e9.9e8.999e8.76e8.697e e7.47e9.9e8.667e8.74e8.7e e7.e9.4e8.448e8.74e8.766e e7.67e9.9e8.999e8.79e8.6866e e7.498e9.9e8.779e8.98e8.767e e7.89e9.4898e8.e8.98e8.744e.6 4.4e7.8e9.9e8.999e8.76e8.664e e7.88e9.9e8.886e8.9e8.698e e7.74e9.878e8.774e8.8e8.694e Tale : Restoring force heave (wave) ω Z ζzc Z ζzs Z ζθc Z ζθs Z ζφφc Z ζφφs e e -.446e8.999e8 8.7e7 -.49e e6.979e -.8e8.8e8 9.87e e e6.7e -.9e8.e8 9.6e7 -.9e e6.6e -.6e8.9e e7 -.4e e6 -.7e -.64e8.84e8.684e7 -.77e e6 -.49e -.6e8.64e8.887e7 -.66e e6-6.9e -.674e8 -.96e8.49e7.7e7 Tale : Restoring force pitch (wave) ω M ζzc M ζzs M ζθc M ζθs M ζφφc M ζφφs e8.999e8-4.e.48e9.8e -8.84e e8.8e8-4.86e 9.96e9.78e -6.79e e8.e e 7.67e9.e -.8e e8.9e e 4.948e9.46e -4.84e e8.84e8-4.69e.646e9.47e e e8.64e8-4.78e -.e9.496e -.48e e8 -.96e8-4.7e e9.84e -.8e9

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