Simplified model of the nonlinear oscillations of two ships in tandem

Transcription

1 4th International Offshore and Polar Engineering Conferene, Toulon, Frane, 004 Simplified model of the nonlinear osillations of two ships in tandem Jessé Rebello de Souza Junior Claudio Gomes Fernandes Department of Naval Arhiteture and Oean Engineering, University of São Paulo São Paulo, São Paulo, Brazil Abstrat This work presents a simplified mathematial model of the osillations of two ships (FPSO plus shuttle vessel) onneted through a hawser. It is shown that the model an be restrited to just one degree of freedom per vessel: the heading angle. The derivation of the model is based on an extended potential funtion that takes into aount wind and urrent fores, as well as elasti restoring terms. The model also inludes nonlinear (an der Pol-like) damping terms. The model is first validated against redued-sale experimental results. A detailed study of nonlinear phenomena exhibited by the model, their analytial and numerial predition, and onfirmation of results from omparison with an industrial-strength numerial ode are show here. Important undesirable behavior, suh as oexisting equilibrium solutions and fishtailing instabilities are investigated. Another relevant aspet of the problem the athment regions of ompeting solutions, i.e. their basins of attration is also studied. Keywords: FPSO; nonlinear dynamis; nonlinear osillations; tandem. Introdution Reliable mathematial models of the dynamis of FPSOs under the ation of wind and urrent have been available for some time now, see for instane Aranha et al. (00), Tannuri et al. (00), Simos et al. (00), eite et al. (998) and referenes therein. They allow for detailed quantitative preditions of motions for a wide range of environmental and operational onditions. The prie paid for suh lose agreement has, however, been high in terms of the ensuing omplexity of the models involved, whih, in turn, hinders in-depth investigations of the fundamental dynamial phenomena at play. It is the purpose of this paper to present a simplified mathematial model of the osillations of two ships (FPSO plus shuttle vessel) onneted to eah other through a hawser. Arguments are developed to show that the model an be restrited to just one degree of freedom per vessel: its heading angle. As shown below, the derivation of the model is based on an extended potential funtion that takes into aount wind and urrent fores, as well as elasti restoring terms. The model also inludes nonlinear (an der Pol-like) damping terms. The model is first validated against redued-sale experimental results. The main body of the work is dediated to a detailed study of nonlinear phenomena exhibited by the model, their analytial and numerial predition, and onfirmation of results from omparison with an industrial-strength numerial ode. Important undesirable behavior, suh as oexisting equilibrium solutions and fish-tailing instabilities are suessfully investigated. Another relevant aspet of the problem the athment regions of ompeting solutions, i.e. their basins of attration is also studied. This latter problem is of partiular interest to the engineer beause it quantifies the relative importane of desirable and undesirable solutions for any given environmental ondition, Morishita et al. (00). It is thus proposed that a simplified model suh as the one desribed in this study an be an effetive tool in the analysis and design of floating prodution systems, providing a better understanding of the omplex dynamis involved, and also delimiting ritial onditions where more refined models an be employed to produe detailed quantitative preditions. Mathematial Model Complete Model An FPSO-DICAS and an FPSO-Turret system with a shuttle tanker in tandem onfiguration an be seen in Figure a and b, respetively, along with the oordinate systems adopted. OXYZ is a right-handed referene frame fixed to the earth, while G i x i y i z i, i=, are right-handed referene frames with their origins at the ship s enter of mass and fixed to the ship (i= for the FPSO and i= for the shuttle vessel). The equations of motion for eah of the vessels, when expressed in a referene frame attahed to its body and under the ation of urrent and wind an be written as follows (Morishita et al, 00): Paper No. 004-JSC- Souza Junior

2 4th International Offshore and Polar Engineering Conferene, Toulon, Frane, 004 The fores and moment due to urrent an then be expressed as (eite et al, 998): (a) (b) ( m m ). u = ( m m ( m m ). v. r + X & ( m m ). u. r + Y Figure : Coordinate systems ). vr ( m. x m ). g 6 ( m. x m ). ( m m ). v& = ( m m). ur g 6 r& ( m m ). u. v ( m. x m )(. v + r u) N I z m ). r& = & 66 g. + ( 6 where - m is the ship mass; - m ii, i =,, 6 are the surge, sway and yaw additional masses, respetively; - u and v are the surge and sway veloities in a referene frame attahed to the ship; - u and v are the urrent veloities; - r is the angular (yaw) veloity; - I z is the moment of inertia in the Z-diretion; - X, Y and N are the total external fores and moments (urrent, wind and elasti restoring fores). In the present work no hydro- or aerodynami oupling is onsidered between the vessels: they are oupled only through elasti fores oming from the hawser. The latter are modeled in the omplete model onsidering onventional atenary equations; see for instane Garza- Rios et al (997), and as linear spring fores in the simplified model. r X ψ = 4 (, r ) ρt. C ( ψ ). r Y ( ψ, r ) = ρt. C ( ψ ). 5 r N ψ = 6 (, r ) ρt. C6 ( ψ ). r where: ψ is the heading of the vessel w.r.t the earth-fixed referene frame OXY, and = u u ) + ( v v ). T is the vessel s draft, r ( its waterline length, and ρ is the water density. The drag oeffiients above are given by: S C ( β ) = os( ) β (log(re) ) T 7 πt + (os(β ) os( β )) 8 πt πt C ( β ) = CY sin( β ) sin( β ) sin ( β ) + 8 πt CBB sin( β ) os( β ) + T l g πt πt C6 ( β ) = C sin( β ) sin( β ) sin( β )os( β ) Y 9 + os( β ) πt T.4 sin( β ) os( β ) where B is the breadth of the ship; C B is its blok oeffiient; C Y is the lateral fore oeffiient in transversal steady urrent; Re is the Reynold s number; l g measures the longitudinal distane between the hull s enter of mass and the midship setion; β is the angle of attak defined as: β = artan( v v, u u ) 0 The wind omponents of external fores and moment are given by (Morishita et al, 00):. C w( rw). v. ψ ρ w X w =. A T. C w( wrv ). v. ψ ρ w Y w =. A. C w( rw). v. 6 ψ ρ w N w =. A. BP where ψ rw = ψ w ψ. C are oeffiients determined w, Cw, andc6w experimentally; w is the wind speed; A T and A are the transversal and lateral projeted areas of the vessel, respetively, and BP is the vessel s length between perpendiulars. Simplified Model Paper No. 004-JSC- Souza Junior

3 4th International Offshore and Polar Engineering Conferene, Toulon, Frane, 004 Obtaining a simplified model that retains all the relevant dynamis of the original model and is at the same time simple enough to allow analytial studies is not a trivial task. There exist tehniques to redue a ertain model, suh as the enter manifold theory, whih simplifies a model to a ertain domain of the phase spae of displaements and veloities, Carr(98). However, the limitation in domain is an undesirable feature that limits the appliability of the model. One wants to obtain a simplified model that an be valid for the entire domain of study. One way to define suh a model is to investigate the omplete model from an energy point of view. This an be done by desribing the omplete model in terms of its potential wells. In the present ase, however, it is not possible to obtain a unique energy funtion ( x, x,..., ) onsidering the system as a whole. To see that let us reall that for gradient systems one an write: 6 ψ = M. dψ M. dψ and the potential funtion related to the sway and surge displaements as: xy = X. dx X. dx Y. dy Y. dy 7 Taking the potential wells of the omplete system desribed above one an see that the degree of freedom orresponding to angular motions aptures most of the relevant dynamis of the entire system, see Figures and. In those figures one an observe that the potential well related to motions in the X and Y diretions is haraterized by just one equilibrium point (a minimum). This is due to the elasti nature of the restoring fores in those diretions. ( x, x,..., ) = f( x, x,..., ) dx + ( x, x,..., ) x, x,..., x ) = f ( x, x,..., x ) dx + ( x, x,..., x ) 4 ( n n n.. ( x, x,..., ) = f n ( x, x,..., xx ) dxx + n ( x, x,..., ) where the f i are the restoring (displaement-dependent) fores appearing in the equation of motion for the system desribed in Equations to one an write: x i. Applying equation 4 for ( x, x,..., ) = { f ( x, x,..., x ) dx n f + ( x, x,..., )} = ( x, x,..., ) =,, { f ( x, x,..., x ) dx ( x, x,..., ) = { f ( x, x,..., ) dx n f + ( x, x,..., )} = f + ( x, x,..., )} =, Figure : Potential well related to angular motions (angles in radians) If the potential funtion is unique then: f f = and f f = 5 However, for the FPSO system desribed previously, an inspetion of the equations of motion reveals that f, f and f are funtions just of the f f heading x (=ψ). Then and must be zero and therefore different from f and f funtion for the omplete system.. So one annot obtain a unique potential One way to irumvent this problem is to define a separate potential funtion for the angular displaement ψ keeping onstant the other degrees of freedom at the values they assume at a speifi equilibrium point. et us then define the heading potential funtion as: Figure : Potential well related to linear displaement (in meters) With this feature in mind it is possible to propose a qualitative study of the stability of the system based only on its angular motions. It is useful Paper No. 004-JSC- Souza Junior

4 to observe additionally that the equilibrium points of the omplete model are largely determined by the partiular form taken by the oeffiients in Equations 7 to 9. To determine the exat form of the potential funtion for the simplified model a areful analysis of the zeroes and relative magnitudes of the terms involved in the equations of motion was performed, taking into aount the number and relative position of equilibrium points as well as the overall shape of the resulting potential funtion. The resulting expression has the following form: 4th International Offshore and Polar Engineering Conferene, Toulon, Frane, 004 f hawser = Khllsin( ψ ψ) where K h is the hawser elasti oeffiient and l and l are the hawser moment arms for the main vessel and shuttle vessel lengths (the longitudinal distane between the vessel s enter of gravity and the hawser winh), respetively. So the final equation of a simplified model of an FPSO-Turret system an be written as follows: simp = A.os( ψ).(. sin + C. sin ( ψ) + ) ( ψ) + σ.os( ψ ψ) B.os( ψ).os( ψ) v 8.[logRe ] && ψ + D. E. & ψ + f ( ψ) + fhawser( ψ, ψ) = 0 Parameters A, B, and C an be adjusted to yield quantitative agreement with experimental and/or detailed numerial models. σ is the ratio between wind and urrent speeds. Another point to be observed is the relationship between the degrees of freedom. It an be seen from the omplete model that the wind and urrent fores have the mathematial form of a an der Pol osillator, with a displaement variable multiplied by a veloity. This is an important feature of the system, sine it will give rise to osillatory behavior, and as suh, must be present in any simplified model to be reated. The equation of motion for the simplified model therefore inorporates a term of the form:.[log Re ]. D E. ψ& 9 where: - D and E are parameters to be determined, and σ is the same = as above ( w σ ). The ensuing equation of motion is then given by:.[log Re ] & ψ + D. E. ψ& + f ( ψ ) = 0 0 with simp f ( ψ ) =, where simp is given by Equation 8. ψ.[logre ] && ψ + D. E. ψ& + f ( ψ ) + f hawser( ψ, ψ ) = 0 Experimental alidation The first step in the verifiation of the simplified model proposed here was a omparison with experimental data. Suh a omparison also provided a quantitative basis for the adjustment of parameter values. Experimental data was taken from Simos, et al (00). That experiment onsisted of towing a redued sale model simulating the ation of different urrent veloities. Wind was not inluded in the experimental setup. Typially, after a short transient, the model would undergo periodi osillations of various amplitudes. Here we have ompared the steadystate amplitudes obtained from the simplified model with those reorded in the experiments. Parameters were alibrated for the lower speed and used throughout the tests to examine the ability of the model to adapt to hanges in urrent speeds. Figures 4a and 4b show experimental data along with numerial results obtained with the simplified model. The simplified model shows good agreement with the experimental data. When the urrent veloity was hanged, the model was able to predit well the new amplitude of the response, whih has an important role in terms of engineering analysis. The higher is the amplitude of the motion, the higher is the likelihood of a ollision between the tankers. In the tandem onfiguration Equation 0 an represent the FPSO- DICAS system. The mooring system prevents large displaements of the main ship in the XY plane, and therefore one an onsider an inertial referene frame attahed to the main ship. However, in order to represent the interation between the main ship and the shuttle vessel for the FPSO-Turret system, the elasti fores oming from the hawser must also be inluded. To do so one has to add one more mathematial funtion to the potential well, whih represents the interation between tankers through the ation of the hawser. For that purpose, the omplete expression of the elasti moment between vessel (onsidering linear spring fores) was further redued to just the term dependent on both heading angles (main vessel and shuttle vessel), whih results in the following: Paper No. 004-JSC- Souza Junior 4

5 (a) 4th International Offshore and Polar Engineering Conferene, Toulon, Frane, 004 (b) (b) wind at 0deg; urrent at 80 deg Figure 4: alidation of simplified model against experimental results. Figure 5: Bifuration diagrams of omplete model Comparisons with the omplete model The simplified model must display behavior similar to the omplete model. An important point to be observed onsidering a non-linear model is that one annot just ompare time series resulting from a few numerial simulations. Instead one an use bifuration diagrams, whih enapsulate a greater wealth of information about the system. Bifuration diagrams for the omplete model were available from our previous studies (see, for instane, Morishita et al, 00 and Morishita and Souza Junior, 004). In this way the strategy here is to ompare bifuration diagrams obtained from the two models, simplified and omplete. The heading of the shuttle vessel was used to represent the fixed points in the bifuration diagrams. The ontrol parameter seleted was the wind to urrent speed ratio σ. One an see in Figure 5 the bifuration diagrams for the omplete models for angles on inidene of the wind of 0 deg and 0 deg, both with urrent inidene at 80 deg. (a) wind at 0 deg; urrent at 80 deg (a) wind at 0deg; urrent at 80 deg (b) wind at 0 deg; urrent at 80 deg Paper No. 004-JSC- Souza Junior 5

6 Figure 6: Bifuration diagrams of simplified model 4th International Offshore and Polar Engineering Conferene, Toulon, Frane, 004 It is possible to identify in both ases (omplete and simplified results) three different regimes in the bifuration diagram. The first region as well as the third one with well defined equilibrium points and a seond region with several equilibrium points and displaying fold bifurations. The simplified model aptured very losely the qualitative behavior of the omplete model, espeially when onsidering the substantial hange in shape and nature of the equilibrium points between these two onditions. Quantitative agreement is also very good. Basins of attration From a pratial point of view, one of the most important aspets to be evaluated is the robustness of attrating equilibrium points and limit yles. Suh robustness results from the presene of at least two features. Firstly, the solution must be observed for a wide range of parameter values suh as speeds and angles of inidene of urrent and wind. Seondly, in order to be of pratial relevane, the solution must attrat to itself a sizable region of initial onditions (positions and veloities). In other words, it should have a basin of attration (or athment region) whih is large with respet to other attrating solutions. Failure to exhibit these features will result in a solution that is not likely to be seen in pratie, either beause environmental onditions will differ slightly from those neessary for its appearane or beause other attrating solutions will have aptured the behavior. (a) Basin of attration For the present study we have developed an algorithm to automatially identify and display the basin of attration of all attrators present for a given environmental ondition. The maps reated by this algorithm show the kind of attrator identified by a symbol. A limit yle is identified by a small irle (o) and a point equilibrium by a star (*). The basins of attration of different attrators are distinguished by different olors and identified by a retangle. The maps seleted for display in this text are all typial in the sense desribed above, i.e., their general features are ommon to a large range of environmental situations. They also orrespond to onditions examined in the preeding figures. Two suh onditions are exemplified here. The first senario an be inspeted in Figures 7a and 7b, whih is typial of low wind situations (σ=0.). Here the dominane of urrent fores destabilizes the stati equilibrium, giving rise to steady-state osillations. The basin of attration shown in Figure 7a effetively overs the whole phase plane, indiating that the periodi attrator is the only attrating solution. The phase diagram shown in Figure 7b helps to illustrate the behavior of the system. It also shows quite learly a prevalent feature of this system: yaw motions are heavily damped ausing trajetories to onverge rapidly onto the stable manifolds of the yli attrator. Note that the presene of a single stable periodi solution is onsistent with the results shown in Figure 6b above and, also importantly, with the no-wind ondition of the experimental setup (see Figures 4a and 4b). The seond senario is depited in Figures 8a and 8b where due to stronger wind fores (σ=.) two stati equilibria are now stable (it should be realled that wind fores tend to stabilize stati equilibria aligned with them, see for a instane Morishita et al, 00). As expeted from the symmetry of this system, the two basins of attration are largely symmetri, see Figure 8a. Figure 8b depits the two attrating equilibria, again displaying very learly the onvergene of trajetories onto their stable manifolds aused by the heavy yaw damping. The passage from one senario to the other is made through a superritial Hopf bifuration, whih is easily visible in Figure 6b roughly at (σ=0.5). (b) Phase Diagram Figure 7: σ = 0., ψ v =0 0 and ψ =80 0 It should perhaps be noted here that from an engineering point of view diagrams suh as those shown in Figures 6 and 7 an be of onsiderable interest. isualizing the athment regions of undesirable attrating solutions helps identifying potentially dangerous situations, be it in terms of environmental onditions or in terms of hawser length, vessel draft or whatever other parameter under study. Simply knowing that unaeptable solutions exist does not give engineers any real measure of how dangerous the situation may be, beause many or all of the undesirable solutions might have insignifiant basins of attration, in whih ase the real situation is atually safe. Conversely, the existene of just one unaeptable solution among many safe solutions might represent a real problem if it happens to apture a large portion of the phase spae. Paper No. 004-JSC- Souza Junior 6

7 4th International Offshore and Polar Engineering Conferene, Toulon, Frane, 004 simplified model to apture hanges in behavior with inreasing urrent speed. Additional investigation of the model was arried out with the prodution of maps of basins of attration for a two of the main senarios enountered. The engineering relevane of the results was also assessed and it was argued that basins of attration an be a valuable tool in assessing the real safety of operations from the point of view of the dynami behavior of these systems. Referenes Aranha, JAP, Silva, S, Martins, MR, eite, AJP (00). A weathervane ship under wave and urrent ation: an experimental verifiation of the wave drift damping formula, Appl Oean Researh, ol., No, pp 0-0. (a) Basin of attration Carr, J (98). Appliations of Centre Manifold Theory, Springer- erlag, New York. Garza-Rios, O, Bernitsas, MM and Nishimoto, K (997). Catenary Mooring ines with Nonlinear Drag and Touhdown, Report to the University of Mihigan/Sea Grant/Industry Consortium in Offshore Engineering, and Department of Naval Arhiteture and Marine Engineering, University of Mihigan, Ann Arbor, Report No., January 997. eite, AJP, Aranha, JAP, Conti, MB, Umeda, C (998). Stati equilibrium of turret systems based on heuristi expressions for the urrent fores, Pro 7 th Int Conf on Offshore Meh and Arti Eng, isbon. (b) Phase Diagram Conlusions Figure 8: σ =., ψ v =0 0 and ψ =80 0 The present work introdued a simplified mathematial model for the study of the dynami behavior of two ships in tandem, with partiular referene to the offloading of FPSOs by a shuttle vessel. It was shown that good results an be obtained restriting the model to just one dynami variable per vessel: the angular displaement. The struture of the model was derived from energy arguments, in whih the reprodution of a (generalized) potential funtion allowed the seletion of mathematial terms to be inluded. These also inluded an ad ho an der Pol nonlinear damping term. The model was first validated against experimental data obtained from redued sale tests. Its preditions were then ompared with those of a detailed model used at the University of São Paulo to simulate ship motions. Results were ompared using bifuration diagrams overing a range of relevant environmental parameters, and showed very good qualitative and quantitative agreement, inluding the ability of the Morishita, HM, and Souza Junior, JAR (004). Dynamial features of an autonomous two-body floating system, in: Dynamial Systems and Control, FE Udwadia, and HI Weber (Eds.), Taylor and Franis, ondon. Morishita, HM, Souza Junior, JAR, Fernandes, CG (00). Nonlinear dynamis of a FPSO and shuttle vessel in tandem onfiguration, th Int Offshore and Polar Eng Conf, Stavanger, ol, pp 6-4. Simos, AN, Tannuri, EA, Aranha, JAP, eite, AJP (00). Theoretial analysis and experimental evaluation of the fishtailing phenomenon in a single-point moored tanker, Pro th Int Offshore and Polar Eng Conf, Stavanger, ol, pp Souza Junior, JAR, Fernandes, CG (00). Nonlinear dynamis of an arhetypal model of ships motions in tandem, th Int Workshop on Dynamis and Control, U Southern California, os Angeles. Tannuri, EA, Simos, AN, eite, AJP, Aranha, JAP (00). Experimental validation of a quasi-expliit hydrodynami model: Fishtailing instability of a single-point moored tanker in rigid-hawser onfiguration, J Ship Researh, ol 45, No 4, pp 0-4. Paper No. 004-JSC- Souza Junior 7