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A Stochastic Analysis of Nonlinear Rolling in a Narrow Band Sea A. Francescutto and R. Nabergoj (University of Trieste, Italy) ABSTRACT The extensive analysis carried out i n last years on the nonlinear rolling of a ship, through the use of different versions of perturbation methods, allowed many researchers to discover some very peculiar features of this motion. In a deterministic beam sea, it appeared the possibility of resonances different from synchronism, in particular subharmonic oscillations in both the upright and heeled conditions. Moreover, the resonance peaks as a function of tuning ratio are bent, mainly as a consequence of righting arm nonli nearity, so that i n a suitable range of frequencies the oscillation can have three steady states of very different amplitude. Later on, the possibility of successive bifurcations to chaotic behaviour was detected and it is actually widely and deeply studied. This "deterministic chaos" is in fashion and is very interesting in principle; on the other hand it is still related to the appearance of a strictly mono- chromatic sea, which is a possibility that can be hardly realized in a real seaway. At the other extremum, stochastic nonlinear rolling was treated mainly as a process with sufficiently broad band to have no possibility of jump phenomena and no research was carried out taking the bandwidth as a parameter. To clarify this puzzling aspect, the nonlinear model introduced in previous work with the excitation represented by white noise filtered through a linear filter was studied in detail. The filter was allowed to vary centerpeak and bandwidth of the excitation. A representation as a sum of two sinusoidal processes with Gaussian slowly varying amplitudes was introduced. This description is particularly suitable for description of narrow band processes. The system of differential equations rolling motion plus filter, was solved by means of the perturbation method of multiple scales. Several curves roll variance/tuning ratio were obtained varying the bandwidth for an excitation variance corresponding to a moderate sea. The results i ndicate that, provided the bandwidth of the 141 excitation is not greater than a sharp JONSWAP spectrum, at least in the considered cases, the main resonance exhibits the same characteristic features shown in the deterministic, i.e. multivaluedness and jump possibility. This similarity is related to the fact that narrow bandwidth means high autocorrelation. The perturbation method allows to obtain simplified equations giving the maximum roll variance as a function of the excitation variance. These results are confirmed through the envelope analysis of a preliminary stochastic time- domain numerical simulation. The curve giving the probability density as a function of the variance of the motion, appears, in fact bimodal. Finally, the region of the first subharmonic has been analysed giving the response curves for the roll variance of the subharmonic component and the excitation variance threshold for its onset. INTRODUCTION The connection between large amplitude rolling and capsizing is not very clear. In principle, the large amplitude rolling motion of a ship can be a stable motion, provided it be included in some stability boundary. In practice, the ship is a very complicate system, so that many dramatic scenarios can appear once large amplitude rolling is in some way originated. The reasons of thiese practical differencies can be summarized as depending on: - wrong operation of the ship. This is for example the case when the rudder is released in the attempt to recover from an excessive heeling in manoeuvering, non properly adjusted antirolling tanks are employed to reduce excessive rolling, a route change is adopted to avoid the beam or quartering action of wave trains; - structural failure with consequent opening of holes in the hull and flooding of some compartment; - effect of additional heeling moments due to water loading on deck and through the deck openings caused by the large amplitude rolling or

shifting of cargo due to the large rolling accelerations. This last phenomenon can in turn give raise to a positive feedback with final . . capsizing or originate a structural failure - and of course, the concurrency of more than one of this phenomena. The analysis of casualties at sea reveals that this is often what happens when a ship is lost. Actually, it is not very simple to quantify the relative i mportance of these consequencies of large amplitude rolling. On the other hand, neither the probability of large amplitude rolling has been stated in a satisfactory way. This paper is thus devoted to a first analysis of this problem. In previous papers [1-4], it has been shown that large amplitude rolling can be a consequence of a jump between the antiresonant state and the resonant one, or between non resonant and subharmonic states, due to the strong deviation from linearity of the rolling motion. This originates the possibily of bifurcations that can be effective as a consequence of some change in the parameters in the case of a deterministic excitation, and due to the very nature of the oscillation when a narrow band stochastic excitation is considered. The phenomenon is not very fast, but quite difficult to forecast, inasmuch its occurrence is not related to some dramatic change in the excitation statistics. Moreover, the amplitude differences can be dramatic also in presence of non very intense excitation. Therefore, it appears of interest a parametric research to find the probability of its occurrence. To this end, the probability density function of the response levels should be computed by means of approximate analytical methods in the following three cases of interest: - bifurcation between the anti- and the resonant oscillation in the main resonance region in stochastic beam sea; - bifurcation between non resonant and subharmonic in the first subharmonic region in stochastic beam sea; - bifurcation between zero amplitude (or negligible one) and parametric suharmonic rolling in stochastic following or quartering sea. As a consequence of the possibility of bifurcations, the probability density functions (pdfl can be bimodal functions for some values of the parameters. This allows an evaluation of the probabilities of the considered different oscillation states, and in particular of the probability of the onset of large amplitude rolling. In this paper, we will devote our attention to ship rolling in a beam sea, with particular regard to the possibilites offered by the bifurcations in the regions of synchronism and of the main subharmonic. The analysis is limited to the obtaining of the frequency response curves in terms of roll variance in synchronism and to the frequency response and excitation threshold in the case of subharmonic. NARROW BAND EXCITATION The inclining moment due to the action of the sea waves, is usually represented by means of the introduction of the energy spectrum of the sea. Extensive campaigns of measurements allowed to obtain a statistical description of the different regions of the world. From these tables, some standard descriptions have been obtained, as the ITTC and ISSC standard spectra, and successive corrections and integrations. These spectra are usually considered to be sufficiently narrow band to allow a simplified statistical treatment, and at the same time sufficiently broad band with respect to the system transfer function to allow a simplified treatment of the response [5,6~. In addition, hypoteses such as Gaussianity, stationarity and ergodicity are often assumed to further simplify the analysis. For a detailed discussion of these problems see Ref. [7] .Actually, this procedure is a Ratifying one when quasi-linear and non extreme phenomena are analysed. If we devote our attention to ship rolling, this is unfortunately not true, since it is a highly nonlinear, large amplitude motion, that can lead to extreme phenomena such as capsizing. It is not the scope of this paper to investigate in detail the connection between large amplitude rolling and capsizing. We will assume that large amplitude rolling is in itself an extremely interesting topic and we devote our attention to some conditions, non sufficiently investigated until now, that can lead to large amplitude rolling. As shown in previous papers [1-4] in the deterministic domain, the rolling motion, due to its inherently high degree of non linearity, can exhibit peculiar phenomena leading to non expected large amplitude oscillations, also in weak to moderate seas. This is due to the appearance of different steady states of oscillation corresponding to the same excitation intensity and the consequent possibility of bifurcation with jumps among these different states. In the deterministic approach, the problem is an initial value problem, so that the phase plane, or the Van der Pol plane are divided in different regions, called domains of attraction, each leading to one of the steady state solutions. Neglecting the appearance of chaotic behaviour, that seems not playing a dramatic role, the jump is thus only possible when there is a change in the parameters defining the system or the excitation, such as cargo shifting, water on deck, wind gusts, change of heading. The build-up of a jump is not an immediate process, requiring a few cycles of the excitation to be completed. In a very broad band stochastic excitation case, it is thus very unlikely to occurr due to the very poor correlation among successive cycles. Actually this only means that its probability is very low, i.e. it is a 'rare event'. On the other hand, ITTC spectrum is only an average one, so that its parameters are subject to statistical uncertainty. Moreover, a peak magni- fication factor has been suggested for the description of North Atlantic. With this correction, 142

one gets the quite sharp JONSWAP spectrum. The description of the sea action through the spectrum, is a short term description, valid for the well developed sea. It does not exclude the possibility of appearance of narrow and very narrow spectra in particular conditions, as for example crossing channels or in port regions or in the developing zone, or simply as a statistical fluctuation of broader spectra. In this case, there could be sufficient correlation among cycles of the excitation to allow for the jump between different amplitude states, without the need of any change in the system parameters. The spectrum of the sea excitation f will be described through a simple linear filter of the type: Sf = ~co02SO/~(cof2-~2~2+c,~2~,2] (1 ) shaping Gaussian white noise W of level SO The excitation is zero mean and has variance of2=~So. It is suited to describe narrow band excitation, as in the limit for y - 0 it reduces to the sinusoidal excitation f = ewcoscoft , (2) with eW2=2of2, whose spectrum is represented by 6(o) In the following, the perturbation method of the multiple scales will be adopted. For this, it is convenient to represent the narrow band process f in the Stratonovich form [8~: fits = Fc~t~cos~ft + Fs~t~sinc3t where Fc and Fs are slowly varying independent Gaussian processes represented by the equations: dt = (2) WC- (2 - ) Fc dt = (2) Ws~ (27) Fs where We and Ws are independent Gaussian white noise processes with the same spectrum level of W. For further details see Ref. [9,103. Actual values of the spectrum level SO, bandwidth parameter ~ and centerpeak frequency of can be obtained through a nonlinear fitting of the spectra to be approximated. This procedure applied to normalized standard wave height spectra Sly using IMSL library routine RNLIN, provided the following values [1 13: - ITTC So=.35 cof-1.16 ~=.50 is= 0 143 - JONSWAP with sharpness magnification factor equal to 7 (maximum value) So=.13 c~=1.01 y=.14 The transformation of wave heights into inclining moments actually broadens a little the spectrum. In any case, it appears that, also in the frame of standard spectra, there is the possibility of effectively narrow band spectra, in the sense of . . . . . giving rise to Jumps in ro Ing. In the following, a value for If Will be used so as to give an excitation intensity corresponding to ew =.2, a value used in previous calculations in the deterministic domain. This value represents a sinusoidal wave with effective wave slope well below the limit for breaking waves. The bandwidth factor ~ will be varied as a parameter. THE EQUATION OF MOTION In the light of the preceding discussion, the rolling motion in beam sea will be represented by the following nondimensional nonlinear differential equation: ddt2 + (2~+81 x2) dt + 62 (d-t) + coO2x + a3x3 = fits (3) We do not discuss here the meaning of the different terms, that can be found in Ref. [13. The excitation f is represented by the expression (2) in the deterministic case, whereas in the stochastic it is represented by the solution of the following differential equation: d2f +,~,df + c'~f2f = ~/2mfW (4) As mentioned, an approximate solution i n terms of the variance O2 of the motion will be obtained in the synchronism region Cof _ Coo and in that of the main subharmonic Of ~ 3c,)O. To get some insight into the nature of the solutions, attention was preliminarly focussed on a form of Eq. 3 containing the minimum of nonlinearities, i.e. only the righting arm one. In these conditions, the rolling motion is described by an equation of the Duffing type. This is tied to the fact that, in the deterministic approach, it is this nonlinearity that is responsible of the presence of bifurcations, whereas the damping moment nonlinearities play an important role in determining the maximum values that can be reached by the oscillation amplitude. Once obtained a solution valid in the stochastic narrow band case, the deterministic sinusoidal solution can be recovered by means of the following positions:

c2 ,s2= 2 e 2 ~ W tuft = 2 2 Q2 6Q = 2 (5) being C the amplitude of the nonlinear response in resonance and Q the amplitude of the linear response out of resonance, both in the deterministic case Only the results relative to the application of the method of multiple scales [12] will be reported. SYNCHRONISM REGION The rolling motion in the region of synchronism is statistically governed by the following equation: ja32(S6 + 3a3(tI)o2-cof2~4 + [(t,)o2 t~f2~2 + (2~fJ2~1 +/2 = (1 +~/2~) of 2 (6) that, using the positions represented by Eq's 5, reduces to: a 0.5 ~0.0 ~ ~5 o.o~ , 1 0)f/Ct)o2.0 1.0 1.5 Fig. 1 Roll variance ~ as a function of tuning ratio of/mot The following values have been used for the parameters: oc3=4.0, lo=. 05, rs~=.1414 (eW=.2~. The number on the curves indicates the value of y. ~OC32C6 + ~X3(COo2-COf2)C4 + 0-5 [(COo2-Wf2~2 + (2~2]C2 = eW2 (7) in the sinusoidal case. Eq. 7 agrees with previous results [1 ]. Eq. 6 represents a very important result inasmuch as it allows a relatively simple calculation of the frequency response curve in presence of stochastic excitation. Due to the particular way through which it was obtained, its validity is limited to the narrow band case. In Fig. 1 and Fig. 2, the response curves given by Eq. 6 are reported as a function of tuning ratio centerpeak frequency/natural frequency of small amplitude oscillations too for different values of the bandwidth parameter including the limiting case of sinusoidal excitation. Fig. 1 represents a tipical case of modern containership, whereas Fig. 2 refer to a more traditional case [13~. As one can see, the possibility of bifurcation is preserved also in the stochastic case, provided that the bandwidth is sufficiently small. In particular, in the considered cases, the most sharp JONSWAP is not far from the higher limit of values of That gives multiple regime of oscillations. In the sinusoidal case, in the frequency region where three states are possible, only the two extreme represent stable oscillations and the bifurcation involves them. In the stochastic case, ~ 0 Of/mo Fig. 2 Roll variance ~ as a function of tuning ratio Cof/Coo. The following values have been used for the parameters: a3=-0.5., ,u=.05, Of=. 1414 (eW=.2~. The number on the curves indicates the value of y. the concept of stability of an oscillation state looses part of its meaning. Actually, in the frequency region where three levels of variance are possible, the roll variance oscillates, having local maxima of probability in correspondence to the two extreme values. This is intrinsically connected with the nature of the stochastic excitation [143. Neglecting the contribution of the nonlinear term, the solution far from resonances is obtained as:

aQ2 =(1 +~y/2~) c~f2 / [(COo2 c,3f2)2 + (2pc'Jf)2(1 +~/2t,)2] _ tI~2/(mo2-mf2)2 (8) A perturbative analysis on Eq. 6, considering r,)O+of_ Woo gives a second degree algebraic equation on the variable Ace = Cof-mo. The reality condition of the solution gives a simple formula relating the maximum value of the roll variace am2 to the excitation variance: am2 =CIf2 / 4,u2cl)o2~1+~/2~) that reduces to the sinusoidal expression: or Cm2 =eW2 / 4~2coo2 These expressions correspond to a linear approach and thus generally they are not sufficiently reliable. A corresponding analysis conducted on the complete nonlinear rolling Eq. 3, allowed us to obtain the general equation for the roll variance in the implicit form: of2 = ~/2 [(Oo2 cl)f2~2 - ~a362 method of multiple scales permitted to obtain an equation describing the variance case of the resonant part of the solution, i.e. the amplitude of the subharmonic component that has to be added, in some way, to the non resonant component of variance aQ2. Here, only the results relative to the Duffing case will be presented. The analysis of the full Eq. 3 is actually in progress. As a consequence of the stability of the non resonant component, we have the following possibilities: 6S2 = 0 ;~32~}s4 + [3a3Qn-~a32~1 +~y/6~1)CJQ2 ]aS2 +[Qn2+~2~2~1+~/6l,l,)21 _ O ~10) with Qn = C)02-mf2/9 + 3a3~,Q2 Eq. (10) is a second degree algebraic + (2cl)o~eq)2~1~l2peq)2]cs2 equation in the variable css2. The reality condition with Req - ~ + 4 O2 ~eq = (~, +3C)o262) 0.4 A perturbation on this equation gives the implicit equation for the maximum roll variances: 0 (2moJ2~2~1 +~y/2Req)om2+~L(1 +~/2~eq) 0.2 ~eq6m4+~eq2Csm6] = (Sf2 `9' the corresponding sinusoidal expression proved good in the forecasting of the maximum roll amplitude in synchronism [153. Here, the predictions are, of course, of a statistical nature, but sufficient for many practical scopes. When a bifurcation is possible, Eq. 9 refers to the highest variance level, irrespectively of its actual probability of occurrence.. SUBHARMONIC SOLUTION In the frequency region COf_ 3mo the non resonant solution given by Eq. 8 is generally stable. Nevertheless, the analysis conducted in the sinusoidal case, indicated that there is the possibility of excitation of a resonant oscillation with frequency of/3 _ mo The application of the o., ao , {th . 3.0 35 Of/Oo 40 Fig. 3 Variance os of the subharmonic component, excitation threshold CIfth for subharmonic oscillation onset and variance aQ of the out of resonance component as a function of tuning ratio cl)f/cOo. The following values have been used for the parameters: a3=4.0., p=.005, of=.1414, ~.01. 145

for the solution gives the threshold value ofth2 Of the excitation variance for the onset of a subharmonic component of the oscillation: with = - 3a (1+~/21l) [~mo2 c,~f2~2 + (2~2~1+~/2~2]{Wo2-~2l9 + ~ 3 t~ 2 2/9~2 4 2 2~1 /6 (7-~16~] ~ /2 } /~7-~/6~) This expression also reduces to the previous one [1 ~ i n the sinusoidal li mitt In Fig. 3 and Fig. 4, the values Of `ss, 6Q, and fifth are reported as a function of centerpeak 0.0 Fig. 4 Variance as of the subharmonic component, excitation threshold Fifth for subharmonic oscillation onset and variance aQ of the out of resonance component as a function of tuning ratio Of/Coo. The following values have been used for the parameters: a3=-1.75., p=.005, 6f=.1414, y=.01. frequency of the excitation. As one can see, the excitation of a subharmonic component is a phenomenon possible also in the stochastic case. The required bandwidth is in general quite narrow and the roll damping quite low, so reducing the probability of occurrence. NUMERICAL SIMULATION AND CONCLUSIONS Numerical time domain simulation in the stochastic excitation case is not, in general, an easy task. This is due to a lot of complications arising in the realization of a true stochastic time series for the excitation as explained in [73. In addition, is not clear what happens of the domains of attraction that define the set of initial conditions that lead to a particular steady state solution. On the other hand, in the limit of narrow band the system has to recover the sinusoidal features. Looking forward to the possibility of presenting a more complete numerical picture of the behaviour of this nonlinear system in stochastic domain in next future [14], we report here only some conclusions obtained from preliminary calculations. Time domain simulation of Eq. 3 with the forcing term given by the following expression [163: fits = ( N ) TCOS(okt+¢k) This gives a pseudorandom signal when the COk are chosen independently from a random population with a pdf of the same form of the spectrum of f and the phases Ok are indipendent and uniformly distributed in (0,2~. A time domain solution was obtained in the synchronism region where multiple regime was indicated. A statistical analysis on the time series indicated a bimodal pdf. These results confirm that jump possibility is likely to occurr in the domain of parameters indicated by approximate perturbative solutions. The probability of the highest state depends on the tuning ratio, decreasing as this increasingly deviates from the synchronism condition. ACKNOWLEDGEMENTS The authors would expresse their gratitude to Prof. H. G. Davies for the interesting discussion on narrow band processes during his visit in Trieste. REFERENCES [1] Cardo, A., Francescutto, A., Nabergoj, R., "Ultraharmonics and Subharmonics in the Rolling Motion of a Ship: Steady-State Solution", International Shipbilding Progress Vol. 28, 1981, pp. 234-251. [2] Cardo, A., Francescutto, A., Nabergoj, R., "Deterministic Nonlinear Rolling: A Critical Review", Bulletin de ['Association Tec h n i q u e Maritime ed Aeronautique, Vol. 85, 1985, pp. 119- 141. [3] Cardo, A., Francescutto, A., Nabergoj, R., "Transient Nonlinear Rolling: The Domains of Attraction", Proceedings 14th Scientific and Methodological Seminar on Ship Hydrodynamics, Varna, Vol. 2, 1985, pp. 23-29. [4] Francescutto, A., '~Jump phenomena in Nonlinear Rolling in a Stochastic Beam Sea", Bulletin de ['Association Technique Maritime ed Aeronautique, Vol. 88, 1988, pp. 504-524. [5] Roberts, J. B., "A Stochastic Theory for Nonlinear Ship Rolling in Irregular Seas", Journal of Ship Research, Vol. 26, 1982, pp. 229-245. [6] Cardo, A., Francescutto, A., Nabergoj, R., 146

"Stochastic Nonlinear Rolling: Which Approach?", Bulletin de ['Association Technique Maritime ed Aeronautique, Vol. 87, 1987, pp. 491-505. [7] Francescutto, A., "On the Nonlinear Motions of Ships and Structure in Narrow Band Sea", Proceedings of the IUTAM Symposium o n Dynamics of Marine Vehicles and Structures in Waves, London 1990 [8] Stratonovich, R. L., Topics in the Theory of Random Noise' Vol. 1, Gordon and Breach, New York, 1963. [9] Rajan, S., Davies, H. G., "Multiple Time Scaling of the Response of a Duffing Oscillator to Narrw- Band Random Excitation", Journal of Sound and Vibration, Vol. 123, 1988, pp. 497-506. [10] Davies, H., G., Rajan, S., "Random Superharmonic and Subharmonic Response: Multiple Time Scaling of a Duffing Oscillator", Journal of Sound and Vibration, Vol. 126, 1988, pp. 195-208. [11] Francescutto, A., Cardo, A., Contento, G., "On the Representation of Sea Spectra Through Linear Filters" (in Italians, University of Trieste, Institute of Naval Architecture, Technical Report N. 67, 1988. [12] Nayfeh, A. H., Mock, D. T., Nonlinear Oscillations, Wiley-lnterscience, New York, 1 979. t13] Cardo, A., Francescutto, A., Nabergoj, R., "The Excitation Threshold and the Onset of Subharmonic Oscillations in Nonlinear Rolling", International Shipbilding Progress,Vol. 32, 1985, pp. 210-214. [14] Francescutto, A., "On the Probability of Large Amplitude Rolling and Capsizing as a Consequence of Bifurcations", Accepted for presentation at the 10th International Conference on Offshore Mechanics and Arctic Engineering 'OMAE', Stavanger, June 1991. [15] Cardo, A., Francescutto, A., Nabergoj, R., "Nonlinear Rolling in Regular Sea", International Shipbilding Progress,Vol. 31, 1984, pp. 3-7. t16] Shinozuka, M., "Simulation of Multivariate and Multidimensional Random Processes", Journal of Acoustical Society of America, Vol. 49, 1971, pp. 357-367. DISCUSSION All H. Nayfeh Virginia Polytechnic Institute and State University, USA How realistic are the results presented in Fieures 1 and 3 for the care of a hardening-type righting arm? What are the assumptions underlying the derivation of egs. (6), (7), (8), (9), and (10)? Are they the same as those used by Davis and Raj an? How different are your results from their results? What are the limitations of the method? AUTHORS' REPLY The validity of the procedure adopted to get the perturbative approximate solutions is implicitly proved by the goodness of the comparison with the results of a numerical simulation (see [14] for more details). As regards the meaning of considering the case with hardening stiffness (`x370), we can deserve the following. A research on a consistent specimen of ships in a variety of loading conditions [ 17], indicated that the validity of a model of righting arm with ~3 > 0 can extend to quite large amplitude inclinations (30° - 40°) especially for modern container ship forms. This is sufficient to assess the importance of the inclusion of this case in a true nonlinear seakeeping context, as far as rolling motion is considered. Capsizing, instead, is generally too a complicated phenomenon to be explained by means of relatively simplified models like those considered. On the other hand, when large amplitude rolling is generated, it is likely to think that additional mechanisms can deteriorate the stability qualities of the ship, leading to capsize as indicated in the introduction (see also ref. [14]). Additional references [17] Cardo, A., Francescuto, A., Nabergoj, R., "The Excitation Threshold and the Onset of Subharmonic Oscillations in Abulinear Rolling, International Shipbuilding Progress, Vol. 32, 1985, pp. 210-214. DISCUSSION Odd Faltinsen Norwegian Institute of Technology, Norway Is the frequency dependence of hydrodynamic coefficients included in the model? AllTHORS' REPLY Actually, the frequency dependence of the hydrodynamic coefficients is not included in the model. This is not due only to the obvious overcomplications involved, but to the necessity of a better clarification of the exact meaning of frequency in presence of a stochastic excitation with the possibility of multiple regime of oscillation and of the techniques for transfering this frequency der~endence when doing time domain simulation with nonlinear - -r v models. On the other hand, in a full nonlinear approach, it is not very easy to separate added mass and damping contributions. 147