THE NONLINEAR DYNAMICS OF SHIPS IN BROACHING
Kostas J. Spyrou1, spyrou@deslab.ntua.gr
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 (“beamseas”) 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 postcritical 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 [113]. 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.
1
Department of Naval Architecture and Marine Engineering,
Ship Design Laboratory, National Technical University of Athens,
9 Iroon Polytechniou, Zographou, Athens 157 73, Greece.
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.
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 downslope might be an important factor [21] while others
considered this to be a secondary effect [24].
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
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.
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.
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.
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]
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
dz dt = f (z, a ) 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.
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 λ L = 2.0 ,
H λ = 1 20 where λ , L 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 largeamplitude 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 0.32 (for a wave with
λ L = 2.0 , H λ = 1 20 ) which was about 57% of the wave
celerity value. The second threshold (occurring at a Fn
slightly higher than 0.4 ) 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:
2
3
d 2x
dx
 dx 
 dx 
+ b1
+ b2   + b3   + f sin kx = d
dt
dt
dt 2
 
 dt 
where x is the relative position of the ship on the wave
measured from a trough, t is time, bi , i = 1,2,3 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 surfriding 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
Fig. 4: 2-D projections of the curve of states of
surf-riding in a quartering sea.
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 Fig. 5: Self-sustained oscillations in
dynamical sense.
the surf-riding mode
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
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 Fig. 6: Chaotic attractor and power
surf-riding (see Fig. 5).
spectrum.
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
12
11
tendency for broaching
10
9
8
is greater) a ship does
7
6
not approach smoothly
5
4
1
0
-1
the condition of zero
relative position on the wave, cos(2πx/λ)
frequency of encounter
but it jumps to it as
0.20
0.18
soon as a certain speed
0.16
threshold is exceeded. If
0.14
0.12
the ship’s longitudinal
0.10
axis lies at some small
0.08
0.06
angle relatively to the
0.04
direction of the waves,
0.02
0.00
this dynamic effect is
11
10
9
‘imported’ into yaw. At
8
7
6
the ensuing stage, the
5
4
1
0
-1
ship passes from a
trough, where it should
relative position on the wave, cos(2πx/λ)
experience a tendency
Fig. 7: Simulation of broaching.
to turn because around
surge ve
locity (m
/s)
heading (rad)
surge
veloci
ty (m/s
)
heading (rad)
broaching
saddle near crest
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, ψ r 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 Fnlow and the emerging type of
behaviour is noted. The procedure is repeated until the
control plane [ψ r , Fnlow ] 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].
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]:
d 2ϕ
dϕ
+ 2ζ
+ (1 + ε cos Ωτ )(ϕ − ϕ 3 ) = f p cos Ωτ + f c (v, r )
2
dτ
dτ
ϕ is a scaled roll angle; ζ is the damping ratio; τ , Ω are,
scaled time and encounter frequency; ε , f p are the
amplitudes of ‘internal’ and direct moment due to the wave
(both depend on heading); and f c is the roll moment due to
the sway, v , and yaw, r , velocities.
Fig. 8: Possible outcomes of an escape from surf-riding
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 highspeed 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]:
d 2ψ
dψ
+ 2ζ
+ (1 − h cos Ω τ )ψ = a
2
dτ
dτ
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
Fig. 9: Jump to resonant yaw.
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 largeamplitude 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.
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.
3. K.J. Spyrou & N. Umeda (1995) From surf-riding to loss of
control and capsize: A model of dynamic behaviour of a ship
in following/quartering seas. PRADS’95, Seoul.
4. K.J. Spyrou (1996) Homoclinic connections and period
doublings of a ship advancing in quartering waves. CHAOS,
6/2.
5. K.J. Spyrou (1996) Dynamic instability in quartering seas:
The behaviour of a ship during broaching. J. Ship Res., 40/1.
6. K.J. Spyrou (1996) Geometrical aspects of broaching-to
instability. 2nd Int. Workshop Stab. Oper. Saf. Ships, Osaka.
7. K.J. Spyrou (1997): Dynamic instability in quartering seas:Part II: Analysis of ship roll and capsize for broaching. J. Ship
Res., 40/4.
8. K.J. Spyrou (1997): Dynamic instability in quartering seas:Part III: Nonlinear effects on periodic motions. J. Ship Res.,
41/3.
9. K.J. Spyrou (1997) On the multi-degree, nonlinear dynamics
of ship motions with application to the broaching problem.
IUTAM Appl. Nonl.Chaotic Dyn. Mech., Ithaca.
10. K.J. Spyrou (1997) A new method to analyze escape
phenomena in multi-degree ship dynamics applied to the
broaching problem. STAB'97, Varna.
11. K.J. Spyrou (1997) On the nonlinear dynamics of broachingto. 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.
14. IMO (1995) Code on Intact Stability for All Types of Ships
Covered by IMO Instruments, Res. A.749(18), London.
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.
19. L.J. Rydill, (1959) A linear theory for the steered motion of
ships in waves. Trans. RINA , 101.
20. P. DuCane & J.R. Goodrich (1962) The following sea,
broaching and surging. Trans. RINA, 104.
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.
27. K. Nicholson (1974) Some parametric model experiments to
investigate broaching-to. Int. Symp. Dynam. Mar. Veh. and
Struct. in Wav., London.
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
model tests for validation of numerical ship motion
predictions, 4th Int. Ship Stab. Workshop, St. Johns.
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.