THE NONLINEAR DYNAMICS OF SHIPS IN BROACHING

Kostas J. Spyrou1, spyrou@deslab.ntua.gr

1 Department of Naval Architecture and Marine Engineering, Ship Design Laboratory, National Technical University of Athens, 9 Iroon Polytechniou, Zographou, Athens 157 73, Greece.

Abstract

Recent research on ship motion dynamics has led to the discovery of the underlying causes of broaching. This is a type of ship motion instability which is manifested on the horizontal plane with a sudden divergence from the initial course and it may end with a rapid ship capsize. We shall summarise the key dynamical phenomena which are responsible for broaching.

1. INTRODUCTION

Ship survivability against capsize in heavy seas has become one of the areas of primary concern among ship researchers, designers and regulators in recent years. When a ship is subjected to the effect of large waves it may capsize according to a number of different scenarios, depending on the magnitude and direction of the wave excitation and the ship’s own capability to resist such excitations. Resonant or breaking waves approaching a ship from the side (“beam-seas”) have a potential to excite large rolling which could result in capsize, especially if the intensive oscillation of the ship causes shift of cargo or, if a considerable quantity of green water is shipped on the deck.

More dangerous still can be a group of steep and relatively long waves approaching a ship from the stern (“following-sea”). Waves of this kind are known to incur significant reductions in roll restoring capability (i.e. the tendency to return to the upright position) for many types of vessels and they may also instigate dangerous coupled motions. According to a popular classification, in following-seas a ship may capsize in at least three different ways: Pure-loss of stability is a sudden, non-oscillatory type capsize taking place around a wave crest due to slow passage from a region of the wave where roll restoring has become negative. Parametric instability is the gradual build-up of excessively large rolling created by a mechanism of internal forcing, the result of a fluctuating restoring that depends on where the ship lies in relation to the wave.

The third distinctively identified, yet until recently not fully understood, type of capsize is broaching (Fig. 1). Broaching is an unintentional change in the horizontal-plane kinematics of a ship. Broadly, it may be described as the “loss of heading” by an actively steered ship, that is accompanied by an uncontrollable build-up of large deviation from the desired course. Broaching is more

commonly arising in waves which come from behind and propagate in a direction forming a small angle, say 10-30 deg, with the longitudinal axis of the ship. Although the inception of broaching represents a problem of instability on the horizontal plane, capsize may be incurred at the post-critical stage due to development of large heel as energy is transferred into the roll direction.

Broaching could happen to small as well as to larger ships. It is notable however that the dynamics involved do not seem to fit always into a single pattern. Frequently, broaching is manifested as a sudden divergent yaw, which peaks within a single wave length. Control is lost when the middle of the ship lies somewhere on the down-slope and nearer to the trough. In other cases there is a gradual, oscillatory-type build-up of yaw as successive waves impinge on the ship from behind. In moderate sea states a ship is more likely to broach-to if it runs with a high speed and is slowly overtaken by the waves. Broaching may also occur at lower speeds if the waves are very steep.

In this paper we shall present an overview of our research on broaching which spans a period of five years and has taken place in Japan, the UK and Greece, on the basis of fellowships provided by, or linked with, the EU [1-13]. Key contribution of this research has been the association of specific nonlinear phenomena with broaching behaviour. Nonlinearity due to the effect of the wave force is a key factor when the surge dynamics are involved in the creation of broaching. Consideration of stiffness nonlinearity in the roll direction is also essential when studying the possibility of a capsize occurring in the wake of controllability loss. Simple analogues of the surge and roll dynamics are shown in Fig. 2, while some basic concepts about nonlinear systems are presented in the Appendix.

Fig. 1: A typical example of broaching behaviour

2. BACKGROUND 2.1 The international regulatory regime

The stability regulations which non-military vessels need to satisfy are usually agreed at the International Maritime Organisation (a United Nations Agency usually referred-to as IMO) and then enforced by the Governments of the Member States. Currently there exist two sets of criteria: The first rely on a statistical analysis of accidents that was carried out in the thirties and involved only the static roll characteristics. They are supplemented by a second set, providing some simplistic account of the roll dynamics due to side wind and waves (“weather criterion”) [14]. Broaching remains basically unaccounted although two requirements are of some relevance: The limit of a maximum 100 heel, due to a prescribed heeling moment associated with a ship turn on the horizontal plane (applicable to passenger ships); and the requirement of a minimun acceptable turning rate of the rudder (the prescribed minimum is rather too low for preventing broaching however) that appears in SOLAS, the IMO Convention for ship safety [15]. Notable is also a recently developed at IMO guidance for the ship Master which is intended to help him to avoid dangerous situations. It is based on a combination of theoretical considerations with extensive experimental results and is assumed to cover, albeit in an empirical way, wave environments where broaching is likely to occur. 2.2 Summary of earlier research

In the USA in 1948 Davidson [16] proved that a ship which could keep a straight-line course in calm water might be unable to achieve this if it encountered following waves. In 1951 in Germany Grim [17] pointed out the possibility of occurrence of surf-riding if long and steep waves approach a ship for the stern. Surf-riding is a peculiar type of behaviour where the ship is suddenly captured and then carried forward by a single wave. This has been believed by many to work as a precursor of broaching. It has been studied in Germany, Russia Japan, Australia and elsewhere, with most notable a relatively recent study in Japan by Kan where were identified the fundamental aspects of this phenomenon [18]. All these studies concentrated however on the single-degree surge dynamics and, because of this, an explanation on how surf-riding is linked with broaching could not be developed.

Well known theoretical studies on broaching were carried out in the UK by Rydill [19], Du Cane & Goodirch [20] and Renilson & Driscoll [21]; in Holland by Wahab & Swaan [22]; in the USA by Eda [23]; in Japan by Motora et al [24] and Umeda [25]; and in Russia by Ananiev & Loseva [26]. Broaching has been studied also with experimental methods: Tests with scaled radio-controlled physical models have taken place in large square ship model basins of the UK, Japan and the USA, in the hope that the pattern of a ship’s behaviour in the waves could be

understood by observing the motions of a scaled model and by analysing their time-series. Very notable were the experiments of Nicholson [27], Fuwa [28], Marshfield [29] and more recently of Umeda [30] and DeKat & Thomas [31]. Perhaps the most complete investigation of broaching took place at the Ship Research Institute of Tokyo in the early eighties. It combined tests on a radio-controlled model of a fishing boat, independent measurements of hydrodynamic forces, comparisons with data obtained from the real, full-size, boat and theoretical analysis [24, 28].

Several important issues came out of all these studies; like for example that the instability should occur when a ship lies on the down-slope of the wave and the stern is “resting” on a crest. Also, that in a broaching situation the ship is overpowered by an excessive wave yaw moment which cannot be counterbalanced by the maximum moment produced by the rudder. Some suggested also that the reduced effectiveness of the rudder on the wave’s down-slope might be an important factor [21] while others considered this to be a secondary effect [24].

2.3 Possibilities for improvement

Earlier analyses were mostly content with an ordinary stability examination based on the sway, yaw and possibly surge motion equations, with different positions of the ship considered on a selected reference wave. A number of crucial simplifications are innate to these approaches however. Firstly, they have relied on linear motion equations. However, nonlinearity can give rise to responses that are not deducible, even at a qualitative level, from a linear low-amplitude analysis. Therefore critical effects may remain unaccounted and terms which could probably explain broaching were dropped from the equations. Another weakness is their exclusive concentration on static and steady-state behaviour. The ship is usually assumed to be at quasi-static equilibrium and to travel at exactly zero frequency of encounter (or momentarily 'frozen' at certain positions of the wave). However, broaching is a phenomenon of dynamic nature and therefore, the study of transient behaviour is essential. Steady-state analysis usually results in overestimation of a system’s safety margin.

Fig.2 : The concept of a ball rolling in a potential well offers some simple analogues for the roll (left) and surge (right) ship dynamics.

3. ELEMENTS OF THE NEW APPROACH

Our research on broaching was comprised of the following stages: (1) Firstly, a mathematical model of coupled ship motions

in long and steep regular waves was developed, suitable for frequencies-of-encounter2, between the ship and the waves, near to zero.

(2) The analysis of surf-riding was extended to cover a quartering sea (i.e. waves meeting the ship from behind but with a non-zero angle). This is the wave environment where broaching is most likely to occur. Bifurcation analysis (see Appendix) was carried out.

(3) The physics behind the connection of surf-riding with broaching and capsize was explored. Furthermore, the range of ship control parameter values (for given wave characteristics) where broaching takes place was identified.

(4) The occurrence of broaching at lower speeds where surf-riding is not involved in the generation of the horizontal-plane instability was studied.

4. BASICS OF THE MATHEMATICAL MODEL We applied Newton’s second law for linear and angular momentum in order to describe a ship’s motion in reference to an inertial system of axes fixed at a trough of the incoming regular wave system. These axes move with a constant velocity which equals the wave celerity. The motion equations were transformed subsequently in order to fit a non-inertial axes system, having its origin at the middle of the ship.

Ship motion is controlled through the magnitudes of the hydrodynamic loads produced by the propeller and the rudder. But the type of motion depends crucially on the hydrostatic and hydrodynamic loads acting on the ship hull. To facilitate the calculation of the hull forces these were separated into the so-called manoeuvring forces due to hydrodynamic reaction caused by the ship motion, and forces produced by the existence of the waves. The hydrodynamic reaction part is influenced by viscosity especially in the stern area and for this reason it cannot be predicted accurately. For simplicity we assumed these forces to be suitable analytic functions whose coefficients were identified from experiments with a scaled model. With similar experiments we could measure also the wave excitations, though for these we could reliably follow also a theoretical approach.

The derived set of motion equations was brought into the standard dynamical system’s representation

( )azfz ,=dtd through suitable transformations. t is time

and z is the state variables’ vector comprised of the displacements and velocities for the considered degrees of

2 This is the modified wave frequency which the ship actually experiences due to forward speed, a Doppler-like phenomenon.

freedom (surge, sway, yaw and roll) plus the angle of the rudder when automatic control through an autopilot is present. The controls vector a is comprised of two sets of parameters: the ship based and the exogenous. In the first group belong the desired heading and the constants (gains) of the autopilot. These are substituted by the rudder angle when there is no continuous rudder control (then ship behaviour is simply examined for specific rudder angles). Also in the ship based controls belong the nominal Froude number and the ship’s metacentric height, the latter dictating initial roll stability. The Froude number is a nondimensional version of the ship’s forward speed which is a state variable. However in the nominal form it is based on the steady speed obtained in still water, which is a representative of the propeller’s rate of rotation. Hence, the nominal Froude number is a control variable. The exogenous control parameters considered were, the length and the maximum steepness of the wave. Since the wave excitations could be expressed as functions of the ship position on the wave, the direct time-dependence could be removed from the excitation terms.

5. NONLINEARITY IN SURGE DYNAMICS [5, 18] In a following or quartering sea the ship motion pattern may depart from the ordinary periodic response. In Fig. 2 we have considered the changes in the geometry of surge motion under the gradual increase of the nominal Froude number, Fn . We have assumed an environment of steep waves coming from behind, say with 0.2=Lλ ,

201=λH where L,λ are respectively wave, ship

length; and H is the wave height. The Froude number corresponding to a speed equal with the celerity of this wave is 0.564. For a low Fn there is only a harmonic periodic response [plane (a)] at the encounter frequency. However, as the speed approaches the wave celerity the response becomes asymmetric [plane (c)]. The ship stays

Fig. 3: Transformations of a ship’s forward motion. On each plane the horizontal axis is the speed while the vertical is the ship’s relative position on the wave. The horizontal axis across the planes is a representative of the propeller’s rate of rotation.

longer in the crest region and passes quickly from the trough. In the literature this referred as asymmetric large-amplitude surging, or, surfing on a crest.

In parallel an alternative, stationary behaviour starts to coexist, owed to the fact that the resistance force which opposes the forward motion of the ship in the water, can be balanced by the sum of the thrust produced by the propeller and the wave force along the ship’s longitudinal axis. This is known as surf-riding and the main feature is that the ship is forced to advance with a constant speed that equals to the wave celerity. On a plane having as axes the position and velocity of the ship along the wave direction, the surf-riding states appear in pairs: one nearer to the wave crest and the other nearer to the trough. Nearer to the crest they are unstable. Stable surf-riding can be realised only in the vicinity of the trough.

Surf-riding is characterised by two speed thresholds: the first is where the balance of forces becomes possible [plane (b)]. For a fishing vessel that we examined in detail this arose at a Fn about 32.0 (for a wave with

0.2=Lλ , 201=λH ) which was about 57% of the wave

celerity value. The second threshold (occurring at a Fn slightly higher than 4.0 ) flags the complete disappearance of the periodic motion. This critical for safety event is manifestation of a global bifurcation phenomenon, known as homoclinic saddle connection5. It occurs when a periodic orbit collides in state-space with an unstable equilibrium (the surf-riding state near to the wave crest). A dangerous transition towards some alternative, nearby or distantly located, state is the practical consequence.

It has been shown that the underlying dynamics of surge resemble those of a pendulum with constant torque. A generic form of the surge motion equation is as follows:

dkxfdt

dxb

dt

dxb

dt

dxb

dt

xd=+

+

++ sin

3

3

2

212

2

where x is the relative position of the ship on the wave measured from a trough, t is time, 3,2,1, =ibi are

coefficients that depend on the nominal Froude number and the wave celerity; f is the amplitude of the wave force for

surge; and d is the difference between thrust and resistance for a speed equal to that of the wave.

The periodic motion of a ship overtaken by waves is represented by full rotations of the pendulum whereas surf-riding corresponds to the equilibrium states. The key nonlinearity is the sinusoidal nature of the stiffness term.

6. THE QUARTERING SEA

6.1 Continuation studies [1]

From a continuity consideration one should expect surf-riding to be realisable also for a range of non-zero angles of heading. This range however should not extend too far; because as a beam sea situation is approached, the wave force in the surge direction will diminish and equilibrium

should become impossible to reach. We have found that when the heading of the ship does not coincide with the wave direction, the points of surf-riding equilibrium belong to a closed curve. 2-D projections of this curve, identified by coupling our mathematical model with a continuation algorithm, are shown in Fig. 4. Stability is very dependent upon the method of rudder control. We investigated stability with the rudder: (a) fixed at certain angles, and, (b) controlled by an autopilot. For the latter feedbacks based on the yaw angle (the heading error from the desired course) and the yaw rate were used.

Very interesting is the occurrence of “Hopf” bifurcations (points B in Fig. 4) creating stable oscillatory surf-riding (Fig. 5). It is notable that such oscillatory motion on a single wave had been observed in experiments performed in Japan [18] and it is believed that an explanation has been provided by our analysis. Below will be summarised the key phenomena, as identified from an investigation based on fixed-controls (no autopilot). This case was found to be the most interesting in a dynamical sense. 6.2 Chaos and further homoclinic events [4] We have used the angle of the rudder as the bifurcation parameter. The self-sustained oscillations created at a point near to the middle of the wave’s down slope (point B in figures 4 and 5) went through a period doubling cascade as the rudder angle δ was increased slightly. A projection of a chaotic response is shown in Fig. 6 together with the corresponding power spectrum showing the frequency content of the response. The existence of chaos was confirmed also through calculation of Lyapounov

Fig. 4: 2-D projections of the curve of states of surf-riding in a quartering sea.

Fig. 5: Self-sustained oscillations in the surf-riding mode

exponents. These are basically time averages of the logarithms of the moduli of eigenvalues. The important point is that they can indicate whether initially close by orbits tends to diverge with an exponential rate.

Further increase of the angle δ brought the response back to periodicity; but due to the increasing δ the oscillatory response came nearer to the (unstable) points of surf-riding (see Fig. 5). This had as a result a new homoclinic connection (unrelated with the one discussed earlier). This event ended the oscillations performed by the ship while it was carried forward by the wave. The ship was then forced to return to a more usual motion pattern where it is overtaken by the waves. 6.3 The link of surf-riding and broaching [5-8]

The key for explaining why surf-riding is conducive to broaching lies in the role of the nonlinear surge motion dynamics. As already was explained, in large and relatively long waves (particularly for a ratio of wave-length to ship-length between 1 and 2 where there is evidence that the tendency for broaching is greater) a ship does not approach smoothly the condition of zero frequency of encounter but it jumps to it as soon as a certain speed threshold is exceeded. If the ship’s longitudinal axis lies at some small angle relatively to the direction of the waves, this dynamic effect is ‘imported’ into yaw. At the ensuing stage, the ship passes from a trough, where it should experience a tendency to turn because around

the trough the wave yaw moment produces a destabilising effect. The homoclinic connection shown in Fig. 3 is therefore the key event behind broaching. Once the periodic response is vanished, the likely types of motions are, either surf-riding or broaching. Which of the two should prevail depends on whether the steering system is capable to check the turn before the yaw angle becomes too large.

A broaching case is shown in the upper graph of Fig. 7. The closed orbit is the initial periodic motion as the waves overtake the ship. The periodic motion passes from the vicinity of the unstable surf-riding point (near the crest) and for this reason its shape is distorted. As the ship is set on the verge of the homoclinic connection any slight increase of the propeller rate by an unaware Master would be enough to instigate broaching. Broaching could be avoided simply by selecting a higher proportional gain for the autopilot Such a case can be seen in Fig. 7 (lower). After the homoclinic connection the ship reached a state of surf-riding lying near to the trough.

7. GLOBAL DYNAMICS [7]

Generally, it is sensible to regard broaching as a change of state caused by variation of some ship-based or exogenous control parameter. This defines a problem of transient, and in our case also multi-degree, dynamics which we have tackled by assuming that the change of state was effected upon a steady initial motion pattern. It is not unreasonable to imagine a ship to be in a nearly steady periodic motion, running ahead or before the waves, when suddenly a group of steeper waves approach it from behind. Or perhaps that the Master of a ship caught in surf-riding is attempting to alter the propeller revolutions, or the rudder angle, in order to escape and return to the ordinary periodic motion pattern.

The variation of a control parameter should produce a transition towards some other state. What is essential here is that, in certain instances, an erratic type of behaviour, namely broaching and/or capsize, is possible to emerge. In Fig. 8 is shown the result of a systematic exploration of the outcomes when a ship escapes from surf-riding. The ship is assumed to be lying initially at a fixed point with its nominal Froude number, Fn , ‘tuned’ on the wave celerity.

The desired heading, rψ is taken in the range where stable

surf-riding can exist. These control settings determine uniquely an initial (surf-riding) condition. Then, Fn is suddenly reduced to lowFn and the emerging type of

behaviour is noted. The procedure is repeated until the control plane [ ]lowr Fn,ψ is covered with sufficient

resolution. Surf-riding is seen to occupy the high Fn region. Immediately below is located the domain of periodic motion which appears to be ‘in competition’ with broaching. At higher headings broaching and surf-riding tend to interface directly. Most notable is how the broaching domain becomes ‘eroded’ by capsize [4].

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

-1 0 1456789

1011

head

ing

(rad

)

relative position on the wave, cos(2πx/λ)

surg

e ve

loci

ty (

m/s

)

-1 0 1456789

101112

head

ing

(rad

)

relative position on the wave, cos(2πx/λ)

surg

e ve

loci

ty (

m/s

)

saddle near crest

broaching

Fig. 7: Simulation of broaching.

Fig. 6: Chaotic attractor and power spectrum.

During broaching a ship is subjected to combined excitations which include: a direct roll moment from the waves, an ‘internal’ forcing caused by a time-varying roll stiffness, and a moment due to hydrodynamic reaction arising from the sway and yaw motions. We proposed the following simplified roll model for broaching [10]:

( )( ) ( )rvffd

d

d

dcp ,coscos12 3

2

2

+Ω=−Ω+++ τϕϕτετϕ

ζτϕ

ϕ is a scaled roll angle; ζ is the damping ratio; τ , Ω are,

scaled time and encounter frequency; ε , pf are the

amplitudes of ‘internal’ and direct moment due to the wave (both depend on heading); and cf is the roll moment due to

the sway, v , and yaw, r , velocities.

8. BROACHING AT LOW SPEED [8]

In a review by Conolly [32] of broaching incidents, those that had happened at speeds much lower than the wave celerity were distinguished from the more classical high-speed events. At low to moderate speeds, surf-riding is rather unlikely to happen, unless the water is very shallow. Therefore, if broaching has occurred in that speed range, there should be a different mechanism in action which does not involve surf-riding.

We considered whether in that case the instability could be an intrinsic feature of the yaw motion of a ship overtaken by very large waves. Assuming a nearly following sea and an autopilot capable to limit large oscillatory course deviations, the fundamentals of yaw motion is captured by the following equation [8]:

( ) ahd

d

d

d=Ω−++ ψτ

τψ

ζτψ

cos122

2

ζ is the damping ratio of the steered ship in yaw which receives relatively large values unless there is no differential gain in the autopilot; τ , Ω are scaled with respect to the natural frequency in yaw. The amplitude h of the parametric forcing is the ratio of the wave yaw excitation to the ship’s own static gain3. The bias parameter a plays the role of a constant external excitation that depends on the angle between the desired course and the wave propagation direction. It is increased with the proportional gain of the autopilot, with the desired heading and with the ship’s static gain (a characteristic of a hull). On the other hand, it is reduced if a hull is not very responsive to the rudder. As the yaw equation is Mathieu type, parametric instability should be expected. Due to the large damping however, the internal forcing required is much higher (i.e. steeper waves) than that at zero encounter frequency.

We considered this instability further on the basis of the mathematical model outlined in Section 4. We found that there is a critical desired heading (given the ship and the wave) where a bifurcation occurs creating a stable subharmonic response (Fig. 9). Further increase of the desired heading caused a rapid increase of the amplitude of yaw oscillation, leading shortly to a turn backwards of the steady yaw response curve. This precipitated a sudden and dangerous jump to resonant yaw. The transient behaviour was a growing oscillatory yaw which corresponds closely to what has been described as cumulative-type broaching.

9. CONCLUDING REMARKS In this review article were summarised the most important of our findings regarding the basic nature of the broaching instability of ships. Recently, we have widened the scope of our research by considering also other, single or combined, types of dynamic instability. Our long term objective is to develop a rigorous technical basis for the assessment of ship stability in a seaway with full account of the large-amplitude ship dynamics. This could lead to more rational stability regulations, improved design and a better guidance for Masters on how to cope with adverse weather.

ACKNOWLEDGEMENTS

The EU grants with contract no: ERBFMBICT982963 (TMR) and ERBCHBGCT930427 (HCM) are gratefully acknowledged. Similarly is acknowledged the award of a

3 Proportionality constant between steady yaw and rudder angle.

Fig. 8: Possible outcomes of an escape from surf-riding

Fig. 9: Jump to resonant yaw.

STA Fellowship (ID no 193065) offered by the Japanese Government for which the author was proposed to the Japanese authorities by the EU.

REFERENCES

1. K.J. Spyrou (1995) Surf-riding, yaw instability and large heeling of ships in following/quartering waves, Ship Technol. Res., 42/2.

2. K.J. Spyrou (1995) Surf-riding and oscillations of a ship in quartering waves, J. Mar. Sci. Technol., 1/1.

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4. K.J. Spyrou (1996) Homoclinic connections and period doublings of a ship advancing in quartering waves. CHAOS, 6/2.

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6. K.J. Spyrou (1996) Geometrical aspects of broaching-to instability. 2nd Int. Workshop Stab. Oper. Saf. Ships, Osaka.

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11. K.J. Spyrou (1997) On the nonlinear dynamics of broaching-to. Int. Conf. Des. Abnormal Waves, Glasgow.

12. K.J. Spyrou (1999) Similarities in the yaw and roll behaviour of ships in extreme astern seas. IUTAM NOMES ’99, Hanoi.

13. K.J. Spyrou ,(1999) On course stability and control delay. Accepted in Int. Shipbuilding Prog.

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15. IMO (1997) SOLAS, Consolidated Edition, London. 16. Davidson, K.S.M. (1948) A note on the steering of ships in

following seas. 7th Int. Congr. Appl. Mech., London. 17. O. Grim 1951 Das Schiff in von Achtern Anlaufender See.

JSTG, 45. 18. M. Kan (1990) Surging of large amplitude and surf-riding of

ships in following seas. Sel. Pap. Naval Archit. Ocean Engin., Soc. Naval Archit. Japan, 28.

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21. M.R. Renilson & A. Driscoll (1982) Broaching - An investigation into the loss of directional control in severe following seas. Trans. RINA, 124.

22. R. Wahab R. & W.A. Swaan (1964) Course-keeping and broaching of ships in following seas. J. Ship Res., 7/4.

23. H. Eda (1972) Directional stability and control of ships in waves. J. Ship Res., 16/3.

24. S. Motora, M. Fujino, M. Koyonagi, S. Ishida, K. Shimada & T. Maki (1981) A consideration on the mechanism of occurrence of broaching-to phenomena. JSNA, 150.

25. N. Umeda (1998) New remarks and methodologies for intact stability assessment, 4th Int. Ship Stab. Workshop, St. Johns.

26. D.M. Ananiev & L. Loseva (1994) Vessel’s heeling and stability in the regime of manoeuvring and broaching in following seas, STAB’94, Melbourne, Florida.

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28. Fuwa, T., Sugai, K., Yoshino, T. & Yamamoto, T. (1982): An experimental study on broaching-to of a small high-speed boat. Pap. Ship Res. Inst. 66, Tokyo.

29. W.B. Marshfield (1987) HASLAR model experiments 1977 to 1986. Rep. RE TR87315, Ministry of Defence.

30. N. Umeda (1998) Broaching experiments (still unpublished) 31. J.O. deKat & W.L. Thomas III (1998) Broaching and capsize

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32. J. E. Conolly (1972) Stability and control in waves: A survey of the problem. J. Mech. Eng. Sci., 14/7.

APPENDIX: Elementary concepts of nonlinear dynamics Nonlinearity: Non-proportional relationship between the force and the variables describing the system’s response. Steady and transient motion: The long resp. short term response that follows the application of some steady excitation on a dynamical system. State vector: The set of variables that can describe fully the motion. State space (or phase space) is the enlarged physical space spanned by the state vector. Control vector: The set of parameters which we can vary during the investigation. They define a control space. Continuation (or path- following): The tracing of steady responses under variation of a control parameter. Bifurcation: A change in the qualitative characteristics of the response; such as a change of stability and a smooth or abrupt creation/disappearance of a pattern of behaviour. Bifurcations are distinguished into local and global. Typical cases of local bifurcation are, the creation of an oscillatory response despite the absence of external periodic excitation (Hopf bifurcation); the doubling of the period of an oscillation (flip bifurcation); and the folding backwards of the steady response curve as a certain influential parameter is varied (fold bifurcation). This bifurcation usually creates a sudden jump phenomenon. Chaos: Random output from a regularly forced system. Also, sensitivity to initial conditions which leads to unpredictability.