a numerical feasibility study of a parametric roll advance

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Leigh Shaw McCueAerospace and Ocean Engineering,

Virginia Polytechnic Institute and StateUniversity,

Blacksburg, VA 24061e-mail: [email protected]

Gabriele BulianDepartment of Naval Architecture, Ocean and

Environmental Engineering (DINMA),University of Trieste,

Trieste, Italye-mail: [email protected]

A Numerical Feasibility Study of aParametric Roll Advance WarningSystemThis work studies the practicality of using finite-time Lyapunov exponents (FTLEs) todetect the inception of parametric resonance for vessels operating in irregular longitu-dinal seas. Parametrically excited roll motion is modeled as a single-degree-of-freedomsystem, with nonlinear damping and restoring terms. FTLEs are numerically calculatedat every integration time step. Using this numerical model of parametric roll and throughtracking trends in the FTLE time series behavior, warnings of parametric roll are iden-tified. This work serves as a proof of concept of the FTLE technique’s viability in detect-ing parametric resonance. The ultimate aim of the research contained in this paper, alongwith future work, is the development of a real-time, onboard aid to warn of impendingdanger allowing for avoidance of severe, even catastrophic, vessel instabilities.�DOI: 10.1115/1.2746399�

Keywords: capsize, chaos, Lyapunov exponents, parametric roll

IntroductionParametric roll is a phenomenon arising, most typically, in

aves of length approximately equal to the ship length at an en-ounter frequency �e twice that of the roll natural frequency �0.owever, because of both nonlinear effects on the restoring term

nd Doppler effects, the most dangerous situations in terms of rollr roll accelerations very seldom occur at the exact 2:1 resonance,ith respect to any characteristic frequency of the excitation spec-

rum at the encounter frequencies. Such uncertainty, however, re-uces when the spectrum of the parametric excitation is suffi-iently narrow-banded. In ships subject to these conditions,articularly vessels with large bow flare and a wide transom stern,he roll restoring arm changes substantially due to changes in thenderwater hull geometry from when the wave peak is amidshipsersus peaks at the bow and stern. In addition, pitch affects thehape of the underwater hull geometry and heave motion changeshe instantaneous displacement, inducing additional parametricxcitation with respect to the “pure geometrical” effect of waves.his periodic oscillation in the roll restoring arm can result in

apidly, and unexpectedly, increasing roll motions. Although itsheoretical existence has been known of for decades, parametricoll has only recently garnered attention from a regulatory andrevention standpoint �1,2�. Such incidents as the APL China �3�,hich involved litigation on the order of $100 million, haveroven that parametric roll exists beyond theory and carries sub-tantial financial risk and grave potential for loss of life. In addi-ion, several other cases of large amplitude roll motions, likelyue to parametric excitation, seem to occur worldwide to otherypes of vessels �4�, and this is raising concerns regarding safetynd even the actual economic risk involved in transportation �5�.n the other hand, new future giant container designs are pres-

ntly being tested specifically for parametric roll �6�, and classifi-ation societies are trying to provide means for prevention anditigation of this problem �2�.Today, parametric roll represents potentially as great a stability

ssue as the beam sea condition in the past. This is not to say that

Contributed by the Ocean Offshore and Arctic Engineering Division of ASME forublication in the JOURNAL OF OFFSHORE MECHANICS AND ARCTIC ENGINEERING. Manu-cript received July 2, 2006; final manuscript received January 23, 2007. Reviewonducted by Eddie H. H. Shih. Paper presented at the 25th International Conferencen Offshore Mechanics and Arctic Engineering �OMAE2006�, June 4–9, 2006 Ham-
urg, Germany.
ournal of Offshore Mechanics and Arctic EngineeringCopyright © 20

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the beam sea condition is no longer a problem, but simply that,due to the change in the ship typologies and the almost constantenvironmental conditions �at least with respect to the “mean hu-man time scale”�, there are some ships more likely to encounterenvironmental conditions able to trigger the parametric roll phe-nomenon. Often these types of ships are less endangered in abeam sea, especially when the ships are particularly large. Ofcourse, other types of ships, especially small ships, are still proneto suffer very dangerous motions, and even capsize, in a beam sea.A correct definition of the most dangerous phenomena for everyship is thus of utmost importance: the designer should indeedprovide information to the master of the vessel regarding the bestway to respond to harsh weather conditions. The usual way ofkeeping bow/head sea at reduced speed in rough weather has beenshown by facts to be sometimes not only ineffective, but evendangerous �3�. Experiments, in addition, have demonstrated thatthe increasing of speed, in certain conditions, can be an effectivemeans for the reduction of the parametrically excited rolling mo-tion �7,8�. On the other hand, again, experiments have shown thatchange of heading could not be an efficient way of escaping fromparametric resonance �7�, especially in short-crested irregularwaves, where the pure parametric �autoparametric� excitationcombines with a “beam-sea-type” excitation �excitation of theideal source of energy type according to �9��. Of course, an in-crease in speed is likely to increase undesired effects, such asslamming or water on deck. For this reason, a correct ship han-dling could be thought of as a sort of multiobjective optimizationproblem, where the parametric roll phenomenon is to be takeninto account due to its capability of inducing large amplitude mo-tions and consequent accelerations. Accelerations are, indeed, tobe properly considered because of the fact that cargo shift, a typi-cal initiating event �10� able to lead to capsize, is strongly relatednot only to the instantaneous rolling angle �that is, by definition, astatic quantity�, but also to the actual accelerations �11�. It is thenimportant to get insight into the parametric roll phenomenon inrealistic environmental conditions, i.e., mainly irregular sea, andfor this reason analytical models are very effective in a prelimi-nary phase of the research. Although they are not fully reliable foraccurate predictions, they can give important information on theunderlying physics of the phenomenon under analysis, tracing themain way to be followed by means of more sophisticated calcu-lation tools. In addition, analytical models can more easily be
tackled by means of theoretical tools, which can hardly be applied
AUGUST 2007, Vol. 129 / 16507 by ASME

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o fully numerical computations. In particular, in this paper,yapunov exponents are considered with respect to an analytical-DOF model forparametrically excited roll motion in irregularongitudinal long-crested waves.

Lyapunov exponents measure the rate of convergence or diver-ence of nearby trajectories. A positive Lyapunov exponent indi-ates exponential divergence or a strong sensitivity to initial con-itions. This sensitivity is a standard sign of chaotic behavior. Theim of this research is to detect this sort of rapid divergence of theoll/roll-velocity trajectory. Therefore, measuring rates ofivergence/convergence of neighboring trajectories while trackingrends of instabilities can be used as an indicator of foreseeableanger.

The use of Lyapunov exponents to study capsize has beenouched on in the literature for both naval architecture and non-inear dynamics. In recent years, the asymptotic Lyapunov expo-ent has been calculated from equations of motion for the moor-ng problem �12�, single-degree-of-freedom capsize models13,14�, single-degree-of-freedom flooded ship models �15–17�,orks studying the effects of rudder angle while surf riding as it

eads to capsize �18�, and investigation into the use of the tool foralidating numerical simulation to experimental data �19,20�. Inrder to anticipate chaotic vessel motions in real time, one cannstead use the incremental variant, i.e., finite-time Lyapunov ex-onents �FTLEs� to detect instantaneous changes in ship behavior21,22�.

Modeling

2.1 Analytical Model. Analytical nonlinear modeling ofarametric roll in regular seas has received, in the past, a consid-rable attention, both with respect to 1-DOF modeling �e.g.,23–25�� and multi-DOF approaches �26�. Much less effort seemso have been devoted to the development of nonlinear models forhe more realistic irregular sea conditions. Nonlinearities in thenalytical model are of fundamental importance both in dampingnd restoring, in order to allow the model for providing boundedtatistics inside the stochastic instability region. In this paper, annalytical 1.5-DOF model is used. The model was developed in8� and has been exploited in the past for the analysis of para-etrically excited roll motion in longitudinal long-crested irregu-

ar sea �25,27�. The analytical model is based on the followingundamental simplifying assumptions:

1. The ship speed of advance is constant.2. The actual sea surface is substituted by the Grim’s effective

wave �28�.3. The pressure under the Grim’s effective wave is hydrostatic.4. Heave and pitch motions are considered in a quasi-static

way, i.e., the ship is in instantaneous equilibrium on theGrim’s effective wave with respect to heave and pitch.

5. Sway and yaw are neglected.

Under the aforementioned assumptions, it is possible to con-ider roll motion as a 1.5-DOF system, where the .5 recalls theact that heave and pitch are instantaneously determined, in prin-iple, by the quasi-static assumption. Although the used assump-ions are quite crude, they allow for the expression of the rollynamics by means of a simple second-order differential equation,here the parametric excitation is given by the time evolution of

he Grim’s effective wave amplitude �eff.

� + d�� + �02GZ„�,�eff�t�…

GM= 0

GZ��,�eff� = �NG

Kn��eff��n

n=0

66 / Vol. 129, AUGUST 2007


Kn��eff� = �j=0

Nk

Qjn�effj

d�� = 2�� + �� + ��3 �1�

Damping parameters in Eq. �1� are to be considered as speeddependent, in order to account for the increase of linear dampingwith speed due to lift effects, and the usual reduction of the non-linear damping due reduced efficiency in vortex shedding.

The effective wave is, according to �28�, a fictitious wave hav-ing length equal to the ship length, and its wave crest is assumedto be always amidship. A positive effective wave amplitude meanswave crest amidships, whereas negative effective wave amplitudemean wave trough amidships. If the wave amplitude and the meanvalue of the so-defined substitute wave are determined, at eachinstant, by a least-squares fitting of the actual sea surface’s profile,it is possible to determine two transfer functions relating the spec-tra of the effective wave amplitude and of the mean value with thesea elevation spectrum. Because of the quasi-static assumption forheave motion, only the amplitude �eff is of interest, and its spec-trum can be determined, in longitudinal long-crested irregular sea,starting from the sea elevation spectrum at the real frequencies.

�2�

According to the actual ship speed, the Doppler effect is to beapplied to S�� in order to obtain the spectrum of the effectivewave amplitude at the encounter frequencies.

The results of the analytical model in Eq. �1� have been com-pared, in �8�, with experimental results on the same ship used inthis paper, obtaining a general overestimation of roll motion fromthe analytical model in Eq. �1�. However, by introducing an em-pirical tuning coefficient kc on the parametric excitation, a goodagreement has been found in terms of predicted statistical, non-Gaussian, characteristics of roll motion, this meaning that themodeling in Eq. �1� could be considered a good archetypal modelfor the analyzed phenomenon, although the accuracy of the modelis strongly dependent on the tuning factor kC. In particular, thetuning process consists of the substitution of the original Grim’seffective wave process �eff with a corrected process �eff,C=kC�eff, where, due to the necessity of reducing the predicted rollamplitude, it is kC�1. Thanks to the linearity of the involvedoperations, this correction actually corresponds to a linear stretch-ing of the scale of the significant wave heights when the used seaspectrum is proportional to the square of the significant waveheight.

2.2 Ship and Environmental Conditions. The sample shipused in this paper has been tested in the past for parametric roll inregular and irregular long-crested longitudinal waves �see�8,27,29–31��, showing the very easy inception of large amplitudemotions even for very moderate wave amplitudes �both in regularand irregular sea�. The bodyplan of the ship is reported in Fig. 1,and main particulars are reported in Table 1. Damping coefficientshave been determined from the direct analysis of experimental rolldecays with advance speed, in order to allow for the speed depen-dence of linear and nonlinear damping. In particular, a linear +cubic model has been assumed as appropriate for this ship, due tothe absence of bilge keels that usually make a linear + quadraticmodel more suitable.

The sea spectrum has been assumed of the Bretschneider type,
i.e., Eq. �3�:
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As a “nominal worst case” condition, the modal wave fre-uency of the spectrum �m is assumed to be 0.683 rad/s, corre-ponding to a wave length equal to the ship length, leading to theegular wave 2:1 resonance occurring at a speed of advance ofbout 4.5 knots �2.315 m/s, Fn=0.064� in head seas. For this typef spectrum, a series of experimental results are available for dif-erent significant wave heights �30�.

2.3 Lyapunov Exponent Calculation. If one considers a ballf points in N-dimensional phase space, in which each point fol-ows its own trajectory based on the system’s equations of motion,ver time, the ball of points will collapse to a single point, willtay a ball, or will become ellipsoid in shape �32�. A commoneasure of the rate at which this infinitesimal ball collapses or

xpands is the Lyapunov exponent. For a system of equationsritten in state-space form x=u�x�, small deviations from the tra-

ectory can be expressed by the equation �xi= ��ui /�xj��xj �33�.x is a vector representing the deviation from the trajectory withomponents for each state variable of the system. Lyapunov ex-onents are asymptotic parameters taken in the limit as time ap-roaches infinity. For the case of real-time detection of ship mo-ions and/or observation of discrete events, such as capsize, theTLE can be used to investigate time-dependent behaviors with

he aim of lending insight as to the nature of chaos in vesselotions. Initial studies indicated potential for advance detection

f instabilities leading to capsize through tracking of variations inhasing and/or magnitude of the FTLE time series �19,21,22�. Forhe sake of reference, standard equations for the Lyapunov expo-ent and FTLE are given in Eqs. �4� and �5� �33�, where t repre-ents time, T is a finite time step, and � and �T are Lyapunovxponents and FTLEs, respectively.

Fig. 1 TR2 body plan

Table 1 Main particulars of the ship TR2

ull RoRo TR2ength �m� 132.22readth �m� 19.00raught �m� 5.875olume �m3� 7714

etacentric height GM �m� 0.865atural roll frequency �rad/s� 0.397

ournal of Offshore Mechanics and Arctic Engineering


� = limt→

1

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�x�t��x�0�

�4�

�T�x�t�,�x�0�� =1

Tlog

�x�t + T��x�t�

�5�

3 Results

3.1 Non-capsize. The equation of motion given by Eq. �1�was simulated over 5000 s for a number of runs featuring identi-cal roll/roll velocity initial conditions �2.0 deg, 0.0 deg/s�. Theship is set to travel with speed equal to 1.414 m/s �Fn=0.04� inhead seas emulated by a Bretschneider spectrum with 2.644 msignificant wave height 1 �H1/3 /Lpp=1/50� and modal period of9.2 s ��=Lpp�. Note, the selection of speed and modal periodallows the ship to be excited with a spectrum of the parametricexcitation process having mean encounter frequency equal totwice the roll frequency, i.e., traditional conditions for excitationof parametric roll. Each simulation was conducted with a differentrandom seed resulting in varied random wave trains. The effectivewave elevation spectrum Doppler adjusted for forward speed isgiven in Fig. 2.

Ten sample roll time histories are presented in Fig. 3. For allcases, as a result of parametric roll, initially small motions rapidlygrow to dangerously large oscillations. These motions appear toarise unpredictably.

Typically, in the study of capsize, one finds distinct peaks in theFTLE time series prior to a capsize event �21,22�. Applying thesame methodology to parametric roll cases, however, yields littleinsight. Figure 4 illustrates that for a sampling of 500 numericalsimulations, those runs that experienced moderate to large mo-tions, as defined by exceeding 30 deg of roll, within the first 500 sof simulation had a nearly identical distribution of maximumFTLEs as those which did not roll to angles in excess of 30 deg.

Consider instead the finite-time Lyapunov exponent time seriespresented in Figs. 5 and 6 for the first sample history of Fig. 3. If,rather than focusing on the magnitude of the FTLE time history,one examines the behavior of the FTLEs and the sum of theFTLEs, qualitative and potentially quantifiable information is

1As described in Sec. 2.1, according to experimental data, the analytical modeldescribed leads to an overestimation of roll; therefore, the results presented for asimulated significant wave height of 2.644 m correspond to experimental cases

Fig. 2 Effective wave amplitude spectrum from Bretschneiderspectrum with modal frequency �m=0.683 and H1/3 /Lpp=1/50

nearer a significant wave height of 3.7 m.

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leaned. In Fig. 6, at 610 s, the shape of the FTLE time seriesegins to noticeably alter. This is accentuated upon examinationf the sum of the two Lyapunov exponents, an entropy-like term34� corresponding to the rate of growth of the area defined by thewo principal axes of the system �35�, whose oscillations increasen magnitude at 590 s, reflecting the change in the FTLE timeeries. In review of Fig. 5, these events just precurse the onset ofotions exceeding 30 deg of roll. As the amplitude of roll motion

iminishes, the FTLE time series returns to its original behavior. Ithould be noted that this phasing and/or idiosyncratic FTLE be-avior has been identified in capsize simulations as well.�21,22�hile the idiosyncratic behavior in the FTLE time series is likely

ig. 3 Sample parametric roll time histories for TR2 subject1/3 /Lpp=1/50 for Fn=0.04

ig. 4 Histogram of maximum FTLEs separated by runs expe-iencing, and not experiencing roll in excess of 30 deg as simu-
ated over 500 s for Fn=0.04
68 / Vol. 129, AUGUST 2007


tied to the increase in motions, the sum of the finite-timeLyapunov exponents �SFTLEs� time series is demonstrating be-havior related to the nonlinear roll damping model, and through it,the vessel motions, as described in Sec. 3.2.

The same process was conducted for runs with zero forwardspeed. Five sample time histories are given in Fig. 7 illustratingmuch the same behavior as in Fig. 3 for Fn=0.04. Again exam-ining in detail the behavior of the first time series from Fig. 7,Figs. 8 and 9 illustrate again that the FTLE time series changescharacter in the vicinity of increased motions. Looking closely atthe parametric roll event occurring shortly around 1650 s, it isclear that the behavior of the FTLE time series alter behaviorbeginning at 1600 s. Additionally, during the event, from 1650 sto 1675 s, the phase and shape of FTLE oscillations change dra-matically before returning to the steady-state behavior after1700 s.

3.2 Relationship to Roll Damping. From the analyzed ex-amples, it can be seen that a strong correlation exists between thebehavior of the SFTLEs and the amplitude of motion. In fact, it isquite likely that what the SFTLE time series is detecting is theonset of large motions via changes in damping because, for non-linear damping, large motions imply large velocities and, thus,more than linear damping. Recall, as previously stated, the sum ofthe FTLEs is measuring the rate of contraction or expansion of aunit ball of initial conditions around the fiducial trajectory. Whenthe sum is negative the volume, or for this two state-space vari-able scenario, the area shrinks �dissipative condition�, converselythe area expands when the sum is positive. Large negativeSFTLEs corresponds to a large dissipation rate, this correlating,approximately speaking, to large damping. A general form of theequations of motion �1� can be written as

� + G��,�,t� = 0 �6�

which, as stated previously, can be written in state-space form as˙

aves defined by Bretschneider spectrum with �m=0.683 and
to w
x=u�x�. First-order perturbations of this trajectory are then writ-



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ig. 5 Roll time history „top…, finite-time Lyapunov exponent time history „middle…, sum of FTLE time history „bottom… for firstase in Fig. 3 for Fn=0.04

Fig. 6 Magnification near first parametric roll event in Fig. 5 for Fn=0.04

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ig. 7 Sample parametric roll time histories for TR2 subject to waves defined by Bretschneider spectrum with �m=0.683 rad/snd H /L =1/50 for Fn=0.0
1/3 pp
ig. 8 Roll time history „top…, finite-time Lyapunov exponent time history „middle…, sum of FTLEs time history „bottom… for first
ase in Fig. 7 for Fn=0.0
70 / Vol. 129, AUGUST 2007 Transactions of the ASME

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en as �xi= ��ui /�xj��xj, where the terms �ui /�xj comprise theacobian matrix for the system. Returning to the general form ofur oscillator, namely, Eq. �6�, we can write the Jacobian of theystem at any time t+T �referring to the language of Eq. �5�� asollows, where ��o , �o� represents the solution of Eq. �6� for theducial trajectory at time =t+T.

J�t + T� = � 0 1

− � �G

�x�

��o,�o˙ ,t+T�

− � �G

� x�

��o,�o˙ ,t+T�

�7�

Recalling that unlike Lyapunov exponents, FTLEs are definedn terms of finite time and thus are local with respect to state spacend time. If we assume T to be small, we can consider the matrix�t+T� to be almost constant during the numerical determinationf the FTLEs over the time interval �t , t+T�. Starting from theseonsiderations, one can derive the relation between the SFTLEsnd the function G. Assuming the smallness of T, the equation ofhe perturbation can be written as:

�x � � 0 1

− � �G

�x�

��o,�o˙ ,t�

− � �G

� x�

��o,�o˙ ,t� · �x = Jt · �x �8�

here the subscript t recalls that the matrix is assumed constanthen dealing with the determination of FTLEs, although it is

ctually a time dependent matrix. We can say that the character-stic time scale of Jt is much longer than the calculation time T,.e., Jt is a slowly varying matrix.

Equation �8� is now a linear differential equation with constantoefficients and the solution is known to be of the form:

�x�t + T� � Av1 exp��1�t� · T� + Bv2 exp��2�t� · T� �9�

here � j are the eigenvalues of Jt �and they are thus slowly vary-ng functions of the time t� and v j are the corresponding eigen-

Fig. 9 Magnification near first para

ectors. A and B are constants, depending on the initial conditions.



Therefore, Eq. �9� is saying that, considering a unit ball in statespace in the vicinity of ��o�t� , �o�t��, it deforms into an ellipsoidand the principal axes of the ellipsoid stretch in time with ratesexp��1T� and exp��2T�. Because the area of the ellipse is propor-tional to the product of its principal axes, it can be said that locallythe area V�t� changes according to �36�.

V�T� exp��1 + �2�T� �10�The SFTLEs is actually measuring the exponential rate of

change of the area, i.e.,

SFTLEs�t� �log V�T�

T= �T1

�x�t�� + �T2�x�t�� 11�

Equation �11� states that the SFTLEs at each time instant can bedetermined by summing the eigenvalues of the matrix Jt. Becausethe sum of the eigenvalues of Jt is equal to the trace of the matrix,we obtain that

SFTLEs�t� � − � �G

��

��o�t�,�o�t�,t�

�12�

In the particular case of the modeling used herein, we have

� �G

��

��o�t�,�o�t�,t�

= 2 · � + 2� · �o�t� sign��o�t�� + 3 · � · �o2�t�

�13�It is important to note that, in the field of 1-DOF models for roll

in naval architecture, the function G contains damping, restoring,and forcing. Usually, forcing and restoring are not dependent onthe roll velocity, and the only term of G having an influence on thepartial derivative with respect to � is the damping term. Becausethe linear plus quadratic plus cubic damping in velocity is mostlyused, the result in Eq. �13� should apply to several models �beam

tric excitation in Fig. 8 for Fn=0.0
me
sea, longitudinal sea, etc.�. Although the derivation above is not

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igorous from a mathematical perspective, its results are quiteccurate. As an example of the strong relation between the nu-erically determined relation SFTLEs��t�� and the curve deter-ined using Eq. �13� is reported in Fig. 10. It is important to

mphasize that, although the reported idea allows one to under-tand part of the behavior of the FTLEs, other aspects, such as thedd behavior in time/frequency domains, are still undescribedathematically.This approach will result in a maximum limit for the SFTLEs

ime series of −2�, yet it can be seen that the SFTLEs at times,ear large amplitude motions, exceed this −2� value. This is dueo the numerical orientation of the orthonormal principal axes ofhe ellipsoid surrounding the fiducial trajectory. In the analytical

ig. 10 Numerical and theoretical dependence of SFTLEs onoll velocity

ig. 11 Roll time history „top…, finite-time Lyapunov exponent
eading to capsize for Fn=0.04
72 / Vol. 129, AUGUST 2007


approach described above, the basis vectors remain in their origi-nal orientation over the interval �t , t+T�, while, numerically, thefirst Lyapunov exponent will direct itself over the interval definedby T toward the direction of largest expansion with the followingexponents arranged sequentially along the next most rapidlygrowing directions normal to the first exponent. Therefore, in thismanner, at particularly rapid growth areas, the −2� limit is ex-ceeded.

3.3 Capsize. A study of FTLE behavior when the vessel issubjected to very large amplitude parametrically induced roll, andcapsize, was conducted using the analytical model in Eq. �1� atforward speed of 1.414 m �Fn=0.04� in waves defined by theBretschneider spectrum, Eq. �3� with modal period of 9.2 s andH1/3 /Lpp=1/25. As with the non-capsize runs studied previously,phase and other qualitative idiosyncratic behavior was noted priorto increases in roll response and/or capsize. Examine Fig. 11, inwhich a sample roll time history leading to capsize is presentedalong with the FTLE time series and the sum of the FTLE timeseries. In Fig. 12, areas of particular interest are magnified. Notethat “bumps” in the FTLE time series are apparent prior to sub-stantial increases in roll angle and the largely divergent behaviornear capsize. These phenomena are simply exaggerations of theeffect detected for the more moderate roll motions discussed inSec. 3.2.

Under parametric excitation, the righting arm of the vessel istime dependent; therefore, it becomes difficult to define a fixed“angle of vanishing stability.” However, clearly, it is important toensure that any oddities in FTLE behavior are noted before themotions leave the initial stable attractor. Reflecting on a phaseportrait given in Fig. 13 for the data of Figs. 11 and 12, illustratesthat for this case the practical angle of vanishing stability is at61.3 deg.

According to the analytical model in �1�, it can be seen that therestoring term is a function of the effective wave amplitude. Be-cause the effective wave amplitude �eff is a function of time, the

e history „middle…, sum of FTLE time history „bottom… for case
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ngle of static vanishing stability is also a function of time. Fromhe analytical modeling of GZ�� ,�eff�, it is possible to define aeterministic relation �v��eff�, where �v is the angle of vanishingtatic stability. Such relation is reported in Fig. 14, where it can beeen the detrimental effect of the wave crest amidships. Althoughot reported in the Fig. 14, it should be noted that the static re-toring curve shows a negative derivative at the origin when theffective wave amplitude exceeds 2 m, this leading to a lollngle of the order of 15–20 deg in the range �eff� �2m ,5m�.

It is then possible, in principle, to trace the time evolution ofv��eff�t��=�v�t�, and to check whether the instantaneous abso-

ute value of the roll angle ��t� �the system is symmetric� crosseshe critical level given by �v�t�. Equivalently, relying on the defi-

Fig. 12 Magnified portions of FTLE time series f

ig. 13 Roll-roll velocity phase space of time series from Fig.
1 for case leading to capsize for Fn=0.04


nition of the angle of vanishing stability being a crossing frompositive to negative righting arm, a time-dependent angle of van-ishing stability can be calculated. E.g., capsize was defined as achange in sign of the righting arm value from positive to negativefor positive roll angles beyond the angle of loll �or negative topositive for negative roll angles�. Thus, the GZ curve at eachinstant in time is used to update the angle of vanishing stability forthe vessel in the conditions it encounters. Although such crossingdo not necessitate that the ship capsizes, it is assumed that theprobability that the ship is pushed back into the region of stabilityby the parametric excitation is sufficiently small to be neglected.For this reason, the capsize instant is defined as the aforemen-

Fig. 11 for case leading to capsize for Fn=0.04

Fig. 14 Angle of static vanishing stability �v as a function of

rom

the effective wave amplitude

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F1

1

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ioned crossing.Five-hundred separate simulations, each 1000 s in length, with

dentical roll/roll velocity initial conditions and different randomave trains were conducted to investigate the relationship be-

ween peak FTLE and capsize. The time-dependent angle of van-shing stability was used to define capsize. For the 500 runs stud-ed, the angle of vanishing stability ranged from 54.9 deg to3.6 deg. Note that this is related to the probability density func-ion of �eff�t�. If the standard deviation of �eff is small, you willave only small deviations in this quantity. As shown in Fig. 15,hose runs resulting in capsize encountered larger peak FTLE val-es than non-capsize cases. Additionally, Fig. 16 displays the timerom the occurrence of the peak FTLE to the capsize event. Al-hough peaks in the FTLE time series occur, oftentimes they occuroo close to the capsize event for initiation of corrective measuresr too far from the capsize event to be certain that it is, in fact, theapsize instability detected. However, for some cases, peaks in theTLE time series are discernable prior to capsize with sufficient

ime in which to issue an operator warning. A system based onetection then of peaks along with qualitative changes in theTLE time series �see, e.g., Fig. 12� could provide the basis of, orupplement to, a real-time motion monitoring system.

ig. 15 Histogram of maximum FTLEs separated by runs cap-izing or not capsizing as simulated over 1000 s for Fn=0.04

ig. 16 Histogram of time from maximum ftle to capsize over
000 s for Fn=0.04
74 / Vol. 129, AUGUST 2007


4 ConclusionsThis work demonstrates that FTLE time series give indicators

of the onset of parametric resonance based on simulation of a 1.5DOF analytical model. Although the warning signs of increasingmotions are largely qualitative, ongoing research is devoted toimproving on and quantifying phase and/or shape irregularities inthe FTLE time series along with the clearly quantifiable measuresassociated with capsize, i.e., peaks in the FTLE time series. Thequestion could be posed, then, of why bother with FTLEs; couldnot the same information be gleaned from solely a roll time se-ries? There are a number of benefits to incorporating FTLE cal-culations into real-time stability detection software. Specifically,

1. Although a roll time series indicates that large roll motionshave occurred, FTLEs are of use for indicating the existenceof dangerous conditions, e.g., nearing the instantaneousseparatrix from stable to unstable motions, which may notbe obvious from a roll time history prior to a catastrophicresult. The roll time history gives only an indication of whathas occurred, with no insight as to if that behavior is abnor-mal or a threat. As the ultimate goal is not simply to detectlarge amplitude motions, but to warn of hazardous condi-tions, FTLEs yield greater information into the physical be-havior of the system.

2. FTLEs capture rates of growth in all degrees of freedom.Rather than attempting to track deviations in multiple pa-rameters, monitoring the FTLE time series provides a repre-sentation of divergent behavior in multiple dimensions.

3. Tracking trends in FTLE peaks, troughs, phase, and shapeirregularities have been demonstrated in concept to indicatethe onset of capsize �21,22�, quiescence �37� and parametricroll as described herein. They can thus serve as a generaltool that can be versatile for use in an on-board real-timemotion monitoring system.

4. FTLEs provide another data point that can be used in anyvariety of tools for motion predictions as part of an intelli-gent system design.

Areas of ongoing and future study include quantification of theidiosyncratic FTLE behavior and development of appropriatetools to monitor FTLEs in real time from realistic, e.g., noisy,experimental data. The authors imagine that this technique couldserve as part of a multicomponent extreme roll motion detectionsystem. The ultimate aim of work proceeding along this path is towarn vessel captains that they have entered dangerous systemstates in time for corrective measures, and likely before such dan-ger is obvious from roll motions alone.

AcknowledgmentPart of this work has been carried out while G. Bulian was at

Osaka University as a research fellow, under the gratefully ac-knowledged financial support of the Scholarship No. PE05052from the Japan Society for the Promotion of Science �JSPS�.Leigh McCue’s contribution to this work was funded under anAdvanceVT Research Seed Grant and ONR Grant No. N00014-06-1-0551.

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AUGUST 2007, Vol. 129 / 175


a numerical feasibility study of a parametric roll advance

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