Comparison of global technique and direct evaluation of capsizing probability on French frigates

Transcription

1 International Shi Stability Worksho 03 Proceedings of the 3 th International Shi Stability Worksho, Brest 3-6 Setember Comarison of global technique and direct evaluation of casizing robability on French frigates ABSTRACT Angelo Brodu, Etienne Mauger, Jean-Yves Billard Ecole navale - IRENAV Yves-Marie Scolan ENSTA Bretagne Pierre Vonier, Jean-François Leguen DGA Techniques Hydrodynamiques The dynamic stability of warshis is analyzed by evaluating the roension of the shis to overcome a rescribed rolling angle. One technique is the evaluation of the robability to reach that angle. If the dynamical behavior can reduced to a time differential equation, Melnikov theory and the analysis of the integrity of attraction basin is also useful. In the resent study, it is shown that all three techniques allows to class French warshis in terms of their casizability. KEYWORDS Casize, casizing robability, dynamical stability, global analysis. INTRODUCTION Casizing robability evaluation is a tedious task involving numerous arameters and a rather long comutational effort. Moreover the result of the comutation is, but for comarison uroses, biased by an arbitrary choice of some of those arameters. Among those arameters the casizing angle above which the shi is considered to casize has not received a clear definition. Chosen constant in a revious work (Beauuy et al., 0) and equal to 45 as measured on one of the shis involved in the comarison, this arameter decreased drastically the oerability of the other shis and thus biased the comuted values of oerability. In the resent work we roose a resentation of two methods the goal of which is the classification of shis by risk level. The first section resents the method of oerability reviously introduced (Beauuy et al., 0) and, in the second section, a global technique. Then we focus on the estimation of the casizing angle based on the global technique and then used to define more recisely the oerability of each shi. OPERABILITY DEFINITION Oerability, o, can be defined as the robability for the shi to be oerated safely from her launching to the end of her oerative life. If ca is the robability of casizing determined for the same eriod, we can write: o ca To define the oerability the following hyothesis are followed: The shi is suosed to be oerated in the Atlantic Ocean. She has an equal robability to be everywhere in the ocean.

2 Probability [-] International Shi Stability Worksho 03 Proceedings of the 3 th International Shi Stability Worksho, Brest 3-6 Setember The sea state is mono directional. Her headings are sread from 0 to 360 with an equal robability. Her seed is sread between 5 and her maximum seed with an equal robability. Each simulation is realized for duration of 3600 s. The limit roll angle for casize is arbitrarily fixed at 45. Yaw DDL has been blocked to avoid broaching. For each run the successive maximum of roll angle are ket and ost treated using a Weibull method roosed by (Derbanne, Leguen, Duau, & Hamel, 008) in order to define the casize robability for articular conditions. In this method robabilities are alied to the roll angles and fitted by a Weibull law of robability. Extraolation to the defined value of roll angle (45 in our case) allow the evaluation of casize robability (green line). This rocess is illustrated on figure Roll Angle [ ] Data Weibull Série3 Fig : Rough data and Weibull law of robability fitted on the data. For each run the shi seed is evaluated in waves and ket constant in mean by a slight variation of the rotational velocity of the shafts. This long and tedious rocedure avoid bias effects that may imact the evaluation and the reresentation of casizing risk as reorted on olar lots of Figure. Fig : Polar lot of casize robability (Hs = 4.5 m, T = 8 s, T = 3600 s). For the evaluation of long term robability, an atlas of sea state describing the relation Hs-T for North Atlantic has been used (US Coast Guards Table) and a Pierson-Moskovitz sea sectrum is assumed. The long term robability if then evaluated by: Pca X,T i i X i T SS TZ i Where i is the robability of occurrence of conditions i (issued from the atlas), T Zi is eriod of zero u-crossing of the roll movement on the i st condition, T SS is the characteristic duration of one sea state, 4 hours in our case, and T is the total duration of the robability estimation, 30 years in the resent study. From these comutations a general oerability has been deduced both function of the oerability on a secific sea state and of the robability of aearance of the sea state in the chosen atlas of waves. This value, resented in figure 3 for North Atlantic, is thus a function of the chosen atlas. T T SS

3 Aire (m.rad) International Shi Stability Worksho 03 Proceedings of the 3 th International Shi Stability Worksho, Brest 3-6 Setember Aire sous la courbe des GZ (jusqu'à 70 ou chavirement) en fonction de l'oérabilité Equation de droite y = 7.03x Coefficient de détermination R = FLF CMT Aconit Jeanne d'arc Colbert Galissonnière V. Schœlcher Limite Branlebas 89.90% Corse réglementaire % 83.00% 88.00% 93.00% 98.00% Oérabilité (%) Fig 3: Correlation between GZ area curva and oerability. GLOBAL TECHNIQUE The dynamic stability of several warshis is analyzed and comared according to their resective rolling dynamics due to beam seas. Mathematically the dynamical system reduces to a second order differential equation. The excitation is due to a regular wave. Nonlinearities are introduced in the restoring moment and a quadratic daming moment. Standard hydrostatic softwares rovide the restoring moment. As a result this moment (Virgin, 987), reresented with a discrete function, is fitted with a high order olynomials. It aears that an eleventh order olynomials is high enough to cover the wide range of shis. The other coefficients of the differential equation follow from standard seakeeing codes: "in air" and added inertia, linear and quadratic daming coefficients and excitation moment. In the resent analysis, all coefficients of the differential equation are constant. Among the methods that allow a global analysis of nonlinear dynamical system, Melnikov's method (Guckenheimer & Holmes, 983) is roosed to quantify the loss of stability by measuring the distance between stable and unstable orbits. This distance is known as Melnikov's function and its zeroes define the Melnikov's criterion. The latter rovides an easy way to link all the coefficients which define the dynamical system, rovided those coefficients are constant in time. The sace of these coefficients (or arameters) can be hence searated in "safe" or "unsafe" areas and that searation follows from an analytical formula. Indeed the comutation of that formula does not require a great amount of comutational resources. For engineering urose, as shown in Syrou (Syrou, 0), such formula is of crucial interest since it allows to easily class shis with resect to redefined criteria. However it is well known that Melnikov's criterion might be too conservative. For examle, given the eriod of excitation, the redicted critical wave height above which the shi may casize, is too small. Then it is an issue to correlate Melnikov's criterion with other aroaches. One of those aroaches is rather simle but might require tremendous comutational resources if the imlemented algorithms are not otimized. Here the technique which analyses the erosion of attraction basin (Thomson, 990) is combined with an interolated Cell-to-Cell Maing(Tongue & Gu, 988). The illustrative results of such a method are the tyical "Dover cliffs" lots, showing the area of the eroded basin lotted in terms of two arbitrary arameters among: the wave amlitude, the wave frequency, the daming coefficients, the olynomials coefficients of the restoring moment,... It is then observed that, in many situations, Melnikov's criterion follows the edge of the "Dover Cliff". However that deends on several conditions such that: The tye of selected items listed above, The thresholds which define whether the comuted orbits in the hase sace escaes or not from the attraction basin. The tyical differential equation which simulates the dynamics of the rolling motion reads: I B B C Q Af sin( ) v v () where single over dot and double over dots denote first and second derivative with resect to time resectively. Polynomial Q(y) is non

4 International Shi Stability Worksho 03 Proceedings of the 3 th International Shi Stability Worksho, Brest 3-6 Setember 4 dimensional. It never vanishes whatever the value of and it reads N Q(y) a y i i i so that the restoring force is reresented with a olynomials of order N + 3. In the resent study N = 4 and the highest olynomials is of order eleven. The solutions of time differential equation are searated in two categories: either bounded or unbounded. Indeed the resent choice of olynomials lead to asymtotic infinite value of roll angle after the angle vanishing stability is reached. That is not hysical but that essentially means that casizing is a dramatic and not reversible event. That is why the word unbounded means that the rolling angle is greater than a threshold. In ractice it suffices to set this threshold at (or close to) the angle of vanishing stability. Equation () is made non dimensional by introducing the following arameters v x,, a, b, b, v v CI Hence yielding x bx bx x x x Q(x) Aasin( ) () where from now on the over dots denote the derivatives with resect to It aears that equation () covers all the ossible shis. Hence that makes it ossible to comare them. Figure (4) shows the (x) x x Q(x) for 8 of the olynomials studied warshis. Fig 4: Variation of the restoring moment (x) with the non dimensional rolling angle x. It should be noted that such variations could not be fitted with simle third or quintic order olynomials. The Hamiltonian of the unerturbed dynamical system follows from the sum of the kinetic energy Ec x and x otential energy E (x) Q( ) d. The heteroclinic orbits which links the two symmetric equilibrium oints ( x, x 0 ) follow from the integration of the conservation law 4 a n Ec E E () 4a n 4 n n (3) Equation (3) is sace integrated yielding the time variation of x and x, hence the boundaries of the unerturbed attraction basin, denoted S I f B B I 0. In ractise the Melnikov function deends on C C I C the following two integrals k k D x(t) dt, k, (4) F( ) x(t)cos t dt We end u with Melnikov criterion which reads D ub D u B A,(5) F f F f with FF I Ca 4 0 Ca 4 and u v I where we isolate the critical amlitude A in terms of the dimensional arameters. This simle equation is of crucial interest for designers since it searates the sace of arameters into "safe" and "unsafe" areas. It can be concluded that the smaller the function F, the safer the corresonding shi. In that direction, it is worth reminding that F() may vanish for an infinite set of frequencies as soon as a 0 (Scolan, 997). However in the resent context that character is of little interest. In the three dimensional sace of

5 International Shi Stability Worksho 03 Proceedings of the 3 th International Shi Stability Worksho, Brest 3-6 Setember 5 arameters B, B, A, the higher the coefficients of (B, B ) the higher the critical wave height and the safer the corresonding shi. Hence, we can comare or classify shis deending on the criterion (5). Figure 5 shows the classification of 9 warshis by lotting their associated coefficients (D, D ). It is worth noting that D and D can be easily calculated analytically. The former is exactly half the area of the undisturbed attraction basin 0 D x(t) dt x dx 8 E () E (x) dx The latter is the area below the otential energy curve, 3 0 D x(t) dt x dx 4E () 4 E (x)dx Fig 5: Coefficients (D, D ) (see equation 4) for 9 warshis of the French Navy. As it is done in (Bikdash, Balachandran, & Nayfeh, 994), a arametrical analysis of D i in terms of a i would bring new insights into dynamical stability criteria. We erform a first arametric study in terms of the threshold of the Cell-to-Cell maing. It is decided that the rolling motion is suosed to be critical above a threshold between 70% u 0% of the vanishing stability angle v. Figure 6 shows how fast the erosion occurs deending on that modified threshold. It is observed that above 00% of v, the erosion varies similarly with A. Below, the erosion occurs more and more abrutly. That can be correlated with the direct assessment of casizing robability. Fig 6: Variation of the area of the attraction basin with the amlitude of the excitation wave for warshi F70. The arrow corresonds to an increasing threshold from 70% u 0% of the angle of vanishing stability. The next arametric study confirms that Melnikov's criterion may be too conservative. Figures (7 a) shows the variation of the attraction basin area S(A,) with the wave amlitude A and the nondimensioned circular frequency for three vessels: jda (Jeanne d'arc), F70 and CMT. Indeed the suerimosed curve which corresonds to the equation (5) linking A and follows the edge of the "Dover cliff" at least for low frequencies. The rate at which erosion occurs is a feature which has been barely exlored so far. In figure (7 b), shows the variations of the quantity S ; that is the modulus of the gradient of the function S(A, ). A criterion which can range the shi follows from the location of the maximum of S in the arameter sace (A, ), i.e. the sea state. The revious classification of the shis regarding the location of the greatest rate of erosion relatively to the highest wave amlitude A, is hence confirmed.

6 International Shi Stability Worksho 03 Proceedings of the 3 th International Shi Stability Worksho, Brest 3-6 Setember 6 Fig 7 a: variation of the area of the attraction basin S(A, ) made nondimensioned with S 0. Melnikov's criterion is suerimosed with a green line. From to to bottom: CMT, F70, Jeanne d'arc. Fig 7 b: isocontour of the gradient S in the arameter sace (A, ), horizontal and vertical axes esectively. From to to bottom: CMT, F70, Jeanne d'arc. CONCLUSIONS Different conclusions can be drawn from this research: The two methods based on comletely different hysical hyothesis lead to the same classification of the shis. The analytical

7 International Shi Stability Worksho 03 Proceedings of the 3 th International Shi Stability Worksho, Brest 3-6 Setember 7 method revails by its raidity and the second one allows an evaluation of the casize risk; As far as the analytical method takes into account the shi characteristics and a eriodic excitation we can infer that the two henomena that control the robability of casizing for frigates are the amlitude of excitation and the nonlinear behavior of the oscillator. The random character of the swell acts secondary; The rate of erosion of the attraction basin gives an interesting insight in the imortance of the choice of the angle of casizing. From this observation a ersective work is the determination of the angle of casizing that can be different for each shi in the database. This association of a angle of casizing different for each shi is consistent with the well-known variation of this arameter from one shi to another. REFERENCES Beauuy, B., Stachelhausen, N., Billard, J., Mogicato, E., Vonier, P., & Leguen, J. (0). Oerability of French frigates over 50 years. th International Conference on the Stability of Shis and Ocean Vehicles. Athens, Setember 3-8. Bikdash, M., Balachandran, B., & Nayfeh, A. H. (994). Melnikov analysis for a shi with a general roll-daming model. Nonlinear Dynamics, 6, 0 4. Derbanne, Q., Leguen, J., Duau, T., & Hamel, E. (008). Long term non-linear bending moment rediction. 7th International Conference on Offshore Mechanics and Arctic Engineering (. 0). Estoril, Portugal, June 5-0. Guckenheimer, & Holmes. (983). Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (Sringer V.). Scolan, Y.-M. (997). Technical note on shi rolling associated to high degree olynomial restoring moment using the Melnikov method. Alied Ocean Research, 9(3-4), doi:0.06/s04-87(97)000-9 Syrou, K. J. (0). A Basis for Develoing a Rational Alternative to the Weather Criterion: Problems and Caabilities. Series: Fluid Mechanics and Its Alications, 97. Thomson, J. M. T. (990). Loss of Engineering Integrity due to the Erosion of Absolute and Transient Basin Boundaries. Nonlinear Dynamics in Engineering Systems International Union of Theoretical and Alied Mechanics 990, Tongue, B. H., & Gu, K. (988). A higher order method of interolated cell maing. Journal of Sound and Vibration of Sound and Vibration, 5(), Virgin, L. N. (987). The nonlinear rolling resonse of a vessel including chaotic motions leading to casize in regular seas. Alied Ocean Research, 9(),