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Rolling of Biased Ships in Quartering Seas N. Sanchez (University of Texas at San Antonio, USA) A. Nayfeh (Virginia Polytechnic Institute and State University, USA) ABSTRACT The equation governing the nonlinear rolling of a biased ship in quartering seas can be reduced to a nonlinear orclinary-differential equation with parametric and external 'excitations. The solutions of this equation are analyzed by performing analog-computer simulations to locate bifurcation points in a two-climensional parameter space consisting of the waveslope and encounter frequency. The predicted instabilities for various levels of the parametric excitation are summarized in bifurcation diagrams that display the Stable regions and the instabilities that take the ship into "dangeroust'responses. INTRODUCTION I n the study of the roll motion of a vessel it is common to have an equation of motto', with time-varying coefficients, which arise from time-dependent restoring moments dire to the ship's position on the wave (7,2) or to changes in the displacement volume resulting from coupling with other modes (3). These time-varying coefficients constitute what is known as parametric excitations (4). A considerable number of roll studies (e.y., 1-3, S-10) analyzed the i'~fluenc.e of these excitations on the stability of a ship in the presence of resonances, which can lead to capsizing under rather mild sea conditions (9,1 a). In pre\/iot~s work (9,10), we applied an analytical-nt~merical procedure to characterize the stability of the steacly-state response of a ship when a parameter is slowly varied. Using this procedure, we analyzed the stability of an approximate analytical solution by using Floquet theory and elementary concepts of bifurcation theory. In (10), we have shown that a ship model under a purely parametric excitation, which is the case of rolling in longitudinal \vaves in the absence of heeling moments and pitch-roll coupling, displays self-similar behavior near the resonances. Instabilities can lead to capsizing through two scenario`,: one evolving from a large oscilla1 ion through the disappearance of a chaotic attractor (crises) and a second' potentially more <dangerous, developing from a small oscillation through a sudden tangent instability. Similar behaviors occur for excitation frequencies near the linear natural frequency, twice the natural frequency, and the 133 superharmonic frequencies (at higher excitation levels). In (9), we treated the behavior of the same ship model under an external excitation; that is, the case of rolling in beam seas. In this case we have also found a qualitatively similar behavior in the neighborhood of the primary resonance. These results are in agreement with the conjecture of Parlitz and Lauterborn (11) that this behavior represents a manifestation of some fundamental physical structure and should be observed in the neighborhood of every resonance of the system if the excitation amplitude is sufficiently large, Wellicome (7) and Feat and Jones (2), among others, considered a second-order equation with parametric and external excitations. It models the rolling motion of a ship in (a) beam seas when the coupling with heave or the Smith effect is included or (b) longitudinal waves when bias is incl'~decl. One of the important features of the nonlinear motions observed in these studies (2) is the difference in stability characteristics between ships with positive and negative bias angles, which confirms the experimental observations of Wrlyht and Marshfield (12). In this work we expand the above studies by analyzing the nonlinear rolling response of a biased ship to both parametric and external excitations. Our aim is to gain general understanding of the dynamics by identifying the instabilities that appear wl~en a parameter is slowly varied and characterize th.F! locus of these instabilities in a parameter space of pl~ysical significance. Results are presented for a broad portion of the amplitude and Frequency of the external excitation, keeping the level of the parametric excitation fixed. Following Wright and Marshfield (12) and Feat and Jones (2), we derive the equation of rnotio'' as a function of the relative motion of the ship and the wave. We let q5 be the absolute roll angle and ~ be the relative roll angle with respecl to the local wa`'eslope a. Applying Newton's second law, we find that the equation of motion can be written as .. . (1 + [/~§ -F D(~) -I- K(~), r) = B-/oc (1a) B=+-~. (1b) where the o,/erdots represent derivatives with respect to the time I-, I iS the roll moment of inertia HI is the added moment of inertia, which is assumed to be constant (12), and B is a constant bias moment, which might be due to a steady wind, or a shift in cargo, or water or ice on deck. The righting moment K(d,~) has

an explicit time dependence, which might come from two sources (2): the position of the ship on the wave or variations in the displacement volume due to heave coupling. Win are mostly interested in the latter effect; however, consideration of the first effect only changes the nu merical values of the coefficients. The righting-moment function is approximates! as (5,12) K(0, t) = too td + a393 + a565 + he cost (2) where coo is the linear undamped natural frequency of the ship, the odd polynomial fits the ship's righting moment curve, and the parametric term represents heave - roll coupling expressed by the coefficient K`'zaz h= 2(oo Here, Koz is the magnitude of the coupling coefficient and az is the amplitude of the heaving motion, which is assumed to be harmonic with frequency S2. The damping moment D(~) is expressed as D(~)-id a- q3B3 Assuming that the wavelength of the wave is large compared with the ship's beam, we can write the waveslope elf a regu lar beam sea as a harmonic ^ function or = Am COS fat, where Em is the maximum waveslope. Using (1b)-(4), we rewrite (1a) as + 2,u~ ~ p3~73 ~ o'2 t~ + a303 -a ~505 + kit) cos(S2~] = 070 its + (X305. + WAS] + ~ ~ 5' COS(~QT + ]~) where y represents a phase angle between the wave and heave motions and As is the bias angle produced by the moment B. l-o simplify the governing equation, we introduce the time scaling t= (idols which transforms (5) into B+2~-~3t}3- 0+a364-~505+h~cos(fit' = As -a ~30S + ASKS + f, cos(S2t) ~ f2 sin(52t) where tier, overdot re~,i-esents the derivative with respect to t, ^ °`mli2 f1 = t1 ~ ~-1) IS y and amlQ 2 (/+,SI) n y Using the transformation = Bs + u we rewrite (6) as u+2pu+q3`i31- u+ bju+b2u2+b3u3 (7) + b4u4 + b5u5 -t hu cos(S2t) = f, cos(S2t)-f2 sil,(<at) b, = 3~30s + 5~50S, b2 - 3~30S + 10~50s ~ be = ~3 + 1Oa502 b4 = 5~50S , b5 = ~5 and f1 = f1 has (11) Equation (to) is general and can model the roll motion of biased and unbiased ships in a variety of regular sea conditions, including beam seas, head or following seas, an`] quartering seas. The cases treated in the two previous publications (9,10) are particular forms of this equation. When y =0 and h--0, (1()) models the rolling of a biased or an unbiased ship in beam seas, depending on whether fls is zero or nonhero. Similarly, when am = 0 and As = 0, (10) models the rolling motion of a biased or an unbiased ship in heal or following seas. The natural frequency has been scaled lo unity to make the results applicable to a wide number of cases. Because of the generality of the cases covered, we have a large number of excitation parameters to consider, namely, Q. A, h, ~s, and am. Because our goal is to study the effect of the parametric amplitude h, we study primarily the stability boundaries in the orm-Q parameter space at a set level of h. The phase angle y is set equal to zero because this angle produces large amplitudes near Q- 1. The bias angle is taken to be either +6 or -6 degrees. The combined effect of the external and parametric terms complements the results previously sho\A/n (9,10) for the cases of a purely parametric or external excitation. B/FURCAT/ON DIAGRAM The results obtained by using an analog~computer simulation of (5) for Y - 0, (9s=0°, h - 0.3, <and the other coefficients as given in Table 1 are shown in the bifurcation diagram in Figure 1. In this diagram, we present the bifurcations of three basic attractors which are stable near Q =- 1 and Q = 2. Figure 2 shows the phase portraits of the two attractors at 52 = 0.~356, and Cam= 0.048. The small attractor loses its stability/ wheel the pararnet~`rs are varied across S., which is the locus of saddle-node bifurcations that produce a jump to another attractor. It the crossing occurs for values of Am > /2, the ship capsizes. For crossings with values of Ccm</2, the jump is to the large attractor in Figure 2. The curve S2 is the locus of saddle-node bifurcations that produce a jump from the large to the small attractor. The large attractor also u nciergoes a period-doubling sequence to chaos in the region between P. and J.. Figure 3 shows phase diagrams and power spectra of various attractors at selected locations in this region. In the dotted region all initial conditions lead to capsize. When J. is crossed the chaotic attractor loses stability and the ships capsizes or its response jumps to the small attractor, depending on whether the crossing occurs above or below /. The large attractor in Figure 2 also loses stability near Q = 2. Between P3 and P4 this limit cycle is unstable. When P3 iS crossed from left to right the attractor undergoes a period-doubling bifurcation and a jump to the subharmonic response is observed. When P4 iS crossed from right to left another periocl-doubling bifurcation is observed. Therefore, the period-one response is unstable in the region between P3 and P4. 134

The third attractor represented in the bifurcation diagram in Figure 1 is the subharmonic response near Q = 2, which is stable in the region between P4 and J2. Figure 4 shows selected attractors in this region: (a) shows the period-one response, (b) shows the subharmonic resonant response obtained after crossing P4 from right to left, (c) and (d) show quantitative changes in the subharmonic response, (e) shows the subharmonic response after a period-doubling bifurcation across P2, and (f) shows a chaotic attractor to the right of J2 Figure 5 shows the bifurcation diagram obtained from an analog-computer simulation of (5) when the parameters are set at V = 0, 49s= 1- 6°, and h = 0.3. In this case the dotted region is much larger than that for the case of an unbiased ship. The region of stability of the subharmonic response shrank and the period-one and subharmonic responses only coexist in a narrow region between Q = 1.5 and 2.0. The curves S. and S2 represent saddle-node bifurcations of the large and small attractors of the period-one resonant response, which produce jumps in the corresponding attractors. Figure 6 shows the two attractors for Q - 0.768 and 0Cm= 0.217. In Figure 5, S. represents a jump from the large to the small attractor; S2 represents a jump in the small attractor, which takes the system to the large attractor if am is below tire black dot or to capsize if it is above this point; and P.' represents period-doubling bifurcations, leading to either chaos in the portion of the curve enclosed by E' or directly to an instability in the rest of the curve. The instability results in either capsize when on, is above the black dot on P. or a jump to the subharmonic attractor when it is below P.. Figure 7 shouts the coexisting attractors of the period-one and subharmonic responses at Q -- 1.758 and 0Cm = 0.046 . The curve E, in Figure 5 represents the point of capsize. For values of Q below the black dot on E', a period-doubling sequence to chaos such as the one shown in Figure 8 is observed. For crossings of E, between the black dot and the point of merge with P., a saddle-node bifurcation is observed, making the period-2T solution unstable and capsizing takes place. Figure 5 also shows that the stability region of the subharmonic response is narrower than that in Figure 1. Crossing P3 from right to left causes the period-one response to lose stability and the subharmonic becomes stable. When P2 is crossed, the system undergoes a sequence of period-doubling bifurcations leading tie chaos, which disappears when E' is rear bed, causing the ship to capsize. Figure 9 shows selected phase portraits of the above changes in the solution: (a) shows the period-one response, (b) and (c) show the subharmonic response after crossing P3, (d) shows a period-doubling in the subharmonic after crossing P2, and (e) shows a second period doubling in the subharmonic response. Figure 10 shows the bifurcation diagram obtained from an analog-computer simulation of (5) when the parameters are set equal to: y -- 0, As--- 6°, and h = 0.3. Figure 11 shows the attractors of the primary resonance. Across curve S in Figure 10, the small attractor undergoes a saddle-node bifurcation that produces capsizing. Across P.' the large attractor undergoes period-doubling bifurcations. In the portion of P. enclosed by E' a sequence of period-doubling bifurcations culminating in chaos is observed, in the rest of the curve capsizing occurs after the first period-doubling bifurcation. The large attractor also undergoes period-doubling bifurcations across P4. The subharmonic response undergoes period-doubling bifurcations across P2 and P3. Capsizing is observed after crossing P.; and E2. Figure 12 shows the coexisting attractors that are found between P2 and P4. CONCLUSIONS The present resmelts show that the rlyna'~irs of a positively biased ship is different from that of a negatively biased sl~ip, which confirms the experimental observations of Wright and l'/larshfield (12). However, it is difficult to conclude which of the two cases is more stable because the stability depends on the region of the parameter space tinder consideration. Nevertheless, it is clear that the motion is very sensitive to bias and the region where capsizing is observed grows considerably and appear to be bigger for positively biased ships. VVe have also found the bifurcation diagram to be sensitive to changes in the phase angle y. The most interesting feature is the major qualitative changes that the bifurcation diagrams undergo as the bias angle changes. This clearly suggests the need for a complete analysis before a particular design can be considered safe to operate. ACKNOWLEDGE/\fENT This work was supported by the Office of Naval Research under Contract Nos. N00014-83-K-0184/NR 43227 53 a n d N 0()0 1 4 -90-J - 1 149 ~ REFERENCES Kerwin, J. E., "Notes on Rolling in Longitudinal Waves," International Shipbuilding Progress, Vol 2,1955,pp.597-614. Feat, G. and Jones, D., "Parametric Excitation and the Stability of a Ship Subjected to a Steady Heeling Moment," International Shipbuilding Progress, Vol. 28, 1984, pi:). 263-267. Paulling, J. R. and Rosenberg, R. M., "On Unsl-able Ship Motions Resulting from Nonlinear Coupling," Journal of Ship Research, Vol. 3, 1959, pp. ~,6-46. 4. Nayfoh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley-l nterscience, New York, 1979. . Blocki, W., "Ship Safety in Connection with Paramet'ic Resonance of the Roll," International Shipb`'ildincl Pro'3ressq Vol. 27' 1980, pp 36-53 6. Abicht, V]., "On Capsizing of Ships in Regular and Irregular Seas," Proceedings of the International Conference on Stability of Ships and Ocean Vehicles, Glasgow, 1975. 7. 135 Wellicorne, J. F., ",Nn Analytical Study of the Mechanism of Capsizirig," Proceedings of the International Conference on Stability of Ships and Ocean Vehicles, Glasgow. 1975.

8. Skomedal, N. G., "Parametric Excitation of Roll Motion and its Influence on Stability," Second International Conference on Stability of Ships and Ocean Vehicles, S111-3a, October 1982. 9. Nayfoh, A. H; and Sanchez, N. E., "Stability and Complicatecl Responses of Ships in llegular Beam Seas," to appear, International Shipbuilding Progress., 1990. 10. Sanchez, N. E, and Nayfoh, A. H., "Nonlinear Rolling Motions of Ships in Longitudinal Waves," to appear, International Shipbuilding Progress, 1990. 11. Parlitz, U. and Lauterborn, W., "Superstructure in the Bifurcation Set of the Buffing Equation," Physics l esters, Vol. 107A, 1985, pp. 351-355. 12. Wright, J. H. G. and Marshfield, W. B., "Ship Roll Response and Capsize Behavior in Beam Seas," Transactions Royal Institution of Naval Architects, Vol. 122, 1980, pp. 129-147. 13. Nayfoh, A. H., Introduction to Perturbation Techniques, Wiley-1 nterscience, New York, 1981. 14. Berge, P., Pomeau, Y. and Vidal, C., Order Within Chaos, Wiley-lnterscience, New York, 1984. 0.20 Of 0.1 5 0.05 0.00 - Tact 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Q Figure 1. Bifurcation diagram from analog-computer simulations of equation (5) for ~' ~ O. Figure 3. ds-0°,andh = 0.3. utt) u(t) Figure 2. Phase portraits of coexisting attractors at Q = 0.856 and Am = 0.048. Both attractors correspond to primary resonance. O. u u U~-Q_ ~ ~ U W U ~ U ·~"Q; U US U t. . .1. . . .1. . .1. . . 18 UO W 2~1.,. W Phase portraits and power spectra of the attractor and the excitation~at selected locations near the primary resonance: (a) T-periodic solution for [2 -= 0.933 arid An = 0.258, (b) 2T-periodic solution for Q - 0.966 and 0tn7 = 0.268, (c ) 4T~periodic solutions for Q = 0.942 and °{m - 0.282, (d) 8T-periodic solution for Q = 0.939 and am = 0.284, and (e) chaotic attractor for Q - 0.937 and An,- 0.285. 136

U O 1 u up ~Ill ° ~ .,1.. w 3 U U U U U 1 U ~ U W ~ ~ = U(t) 1 ... w ,,, t..1. .1. .1. · -! .1.... (O ) w w I. I I l 'w U(t) LU Figure 6. Coexisting attractors near the primary resonance for 52 = 0.768 and ~7 = 0 217. ^3 ^1~1~l U. W ° ~ .1 W Figure 4. Phase portraits and power spectra of the attractor and the excitation at selected locations near the subharn~onic resonance: (a) primary response fo`- Q - 2.158 and Am = 0.107, (b) subharmonic response for (2 = 2.102 and Em = 0.1 13, (c) subharmonic response for 52 - 1.920 and am = 0.136, (d) subharmonic response for Q -- 1.514 and Am - 0.218, (e) 2T-periodic subharmonic response for ha = 1.505 and Am = 0.221, and (f) chaotic attractor in the subharmonic response for Q- 1.480 and An, = 0.228, 0.30 0.25 0.20 o( 0.1 5- m 'A E, i::: 2 1l p p ~ ~ 2 1 ~I U(t) ( o ·~.2'..'2:jl j lS2 \ '"I ~ 0.101 ~Al 1 1 u(t) 0.05 l Ill l l Figure 7. oexisting attractors near the /'l ~subharmonic resonance for 52 = 1.758 0.00- ~ ' ~ ~and Am - 0.046. O.O 0.5 1.0 1.5 2.0 2.5 3.0 Q P3 Figure 5. Bifurcation diagram from analog-computer sirr~ulations of equation (5) for y = 0, B. = + 6 , and h = 0.3 137

up ~ Cu u is flow u ut it u u Ou u 3 . . . 1. W If.. 1, . . . 1. U. Figure 8. Phase portraits and power spectra of the response and the excitation near the primary resonance: (a) T-periodic response fcr Q = 1.098 and an, - ().228' (b) 2T-periodic response for Q =- 1.098 and con, = Q.231' (c) 2T-periodic response for Q - 1.098 and Am - 0.245, (d) appearance of broad-band frequency content in the 2T-periodic response for Q = 1.093 and Am = 0.248, and (e) chaotic attractor for Q = 1 067 and Em = 0.249. 0.30 0.25 0.20 at 0.15 m 0.10 0.05 0.00 _ . 0.0 0.5 1.0 1.5 2.0 2.5 3.0 n Figure 10. Bifurcation diagram from analog-computer simulations of equation (5) for ~ -- 0, Bs = - 6°, and h = 0.3. 138 ~ ~1 O . w O _ w 3 u33 f 1 1 1 ...... 1 . UD cage m~ u Ha ~l.-l.l I ,.ll ~ tD . W o 1 ..... .... . w Figure 9. Phase portrait and power spectra of the response and the excitation near the subharmonic resonance: (a) primary response for 52 = 2.319 and am = 0.058, (b) subharmonic response for Q -- 1.997 and am ~~ 0.078, (c) subharmonic response for Q = 1.794 and ten' - 0.097, (d) 2T-periodic attractor of the subharmonic response for 52 = 1.780 and a,n = 0.099, (e) 4T-periodic attractor of the subharmonic response for Q = 1.763 and Em = 0 100.

u u c) u to u _L 1 3 - c~ co ° ll ........ .1.., . .1 ..1.,..1,,,l,, 0 ~ · · I.. l. · · · J I ' Figure 11. Phase portraits of the attractors near the primary resonance and the power spectra of the response and excitation for Q = 0 702 and Am = 0.063 and Q - 0.827 and Em 0 057. U(t) OF U(t) Figure 12. Coexisting attractors near subharmonic resonance for S2 = and Am = 0.061. Table 1. Coefficients of ship model considered. I K ·~>0 ~ P3 2 2 [ tO0 (~0 mK2 110.mm 5.278 0.086 0.108 - 1.402 0.271 0.25/ the 1 .698 139

DISCUSSION Alberto Francescutto University of Trieste, Italy First, I would congratulate the authors for this very interesting paper indicating the possibility of new terrible possibilities in the already complicated behaviour of a ship in longitudinal or quartering seas. The author, in his presentation, mentions the possibility of jumps, involving the oscillation amplitude, taking place sharply in time. In my experience, this sharpness indicates always a few cycles of the oscillation. Would the author comment on this point? Finally, it seems that the consideration of the angle between the wave train and ship's heading has been neglected in the construction of the equation of motion (5). This, in fact, would include an explicit external excitation also in the limiting case of pure longitudinal sea. AUTHORS' REPLY Dr. Francescutto's first question has to do with the time it takes an unstable solution to grow from a given initial position to the point of capsize. The process may involve a fraction of a cycle to many cycles of oscillation, depending on the initial conditions and the scenario through which the vessel capsizes. In the case of capsize through saddle-node bifurcations, the capsize time may be very short. On the other hand, in the case of capsize through period-doubling bifurcations and chaos, the capsize time may be very long. In reference to the consideration of the angle between the wavetrain and the ship's longitudinal axis, we agree that it is not explicitly shown in equation (5). However, the governing equation is general and can treat arbitrary orientations by properly adjusting the excitation parameters in equation (10). DISCUSSION Hongbo Xu Massachusetts Institute of Technology, USA (China) You have shown that the nonlinear analysis is a very useful tool of studying the ship motion. When this one-degree freedom system (rolling of a ship in quartering seas) bifurcates, the magnitude of the ship motion undergoes a jump. There must be energy exchanges between the ship and waves or energy transfer between the potential component and kinetic component. I think it is important to identify the energy-sharing mechanism in order to understand the dynamics of the system. Perhaps you also have this kind of results to show us? AUTHORS' REPLY Dr. Xu's question addresses the energy transfer mechanism between the wave and boa, as well as the exchange between the potential and kinetic energies. Understanding of these fundamental mechanisms is of primary importance as Dr. Xu points out. The basic mechanism is the adjustment of the phasing between the vessel motion and the wave. DISCUSSION Hang S. Choi Seoul National University, Korea The equation you have used seems very complicated; cuffing oscillator for righting moment and Van der Pot oscillator for damping. In addition to these, the biased initial position is introduced as a control parameter. However, the resulting motion of this complex system depends strongly on the initial condition of the motion. Would you comment how the initial condition can be determined in your mathematical model? AUTHORS' REPI~Y Dr. Choi's question addresses an important aspect of the mathematical model. The solution to the differential equation governing the oscillation of the vessel depends on the initial conditions in the form of position and velocity. Furthermore, there might be many possible solutions to the differential equation, depending on the initial conditions. This is exactly the essence of the basin of attraction shown in the paper. Each set of initial conditions, as an independent variable, is located on the phase plane and the solution reached from that point is determined. Therefore, the motion is strongly dependent on the initial conditions. In summary, the initial conditions are independent variables which must be specified for the particular situation. To assess the stability of the vessel, we have to determine the solution for any physically meaningful set of initial conditions in phase space. This generates the domain of attraction of all possible solutions. From this information, we need to determine which solutions pose risk and use the analysis to evaluate the seaworthiness of the vessel under consideration. 140