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Surf-riding and oscillations of a ship in quartering
waves
KJ. SPYROUX
National Research Institute of Fisheries Engineering. Ebidai, Hasaki, Kashima, [baraki 314-04, Japan
Abstract: The behavior of a ship encountering large regular p waves from astern at low frequency is the object of investiga- r tion, with a parallel study of surf-riding and periodic motion R(, x)
paterns. First, the theoretical analysis of surf-riding is extended Res(u)
from purely following to quartering seas. Steady-state con- t tinuation is used to identify all possible surf-riding states for u
one wavelength. Examination of stability indicates the exis- U
tence of stable and unstable states and predicts a new type of y
oscillatory surf-riding. Global analysis is also applied to de- W
termine the areas of state space which lead to surf-riding for a x given ship and wave conditions. In the case of overtaking
waves, the large rudderyawsurge oscillations of the vessel rG, ZG are examined, showing the mechanism and conditions re- xS
sponsible for loss of controllability at certain vessel headings. Key words: surf-riding, nonlinear, wave, ship motion,
stabi-lity, chaos List of symbols
C wave celerity (m/s)
C(p) roll damping moment (Ntm) g acceleration of gravity (mis2) GM metacentric height (m) H wave height (m)
lx,jz roll and yaw ship moments of inertia (kgm2)
k wave number (m)
KH, K, KR, hull reaction, wave, rudder, and propeller
K9 forces in the roll direction (Ntm)
m ship mass (kg)
n propeller rate of rotation (rpm)
NH, N, NR, hull reaction, wave, rudder, and propeller moments in the yaw direction (Ntm)
* European Union-nominated Fellow of the Science and
Technology Agency of Japan, Visiting Researcher, National Research Institute of Fisheries Engineering of Japan Received for publication on Dec. 12, 1994; accepted on June
5, 1995
X,1, X, X,,, xP
y
Y,,, Yw' YR, Yp Zy ZW Greek symbols ß o I, p We Hydrodynamic coefficients K roll added mass
N , N; yaw acceleration coefficients N, j'j, 1"rr'
Nrrv, yaw velocity coefficients
Marine Science
and TechnoIoy
©SNAJ 1995
rol! angular velocity (rad/s)
rate-of-turn (rad/s) restoring moment (Ntm) ship resistance (Nt)
time (s)
surge velocity (mis) vessel speed (mis) sway velocity (mis) ship weight (Nt)
longitudinal position of the ship measured from the wave system (m)
longitudinal and vertical center of gravity (m) longitudinal position of a ship section (S), in the ship-fixed system (m)
hull reaction, wave, rudder, and propeller forces in the surge direction (Nt)
transverse position of the ship, measured from
the wave system (m)
hull reaction, wave, rudder, and propeller
forces in the sway direction (Nt)
vertical position of the point of action of the
lateral reaction force during turn (m)
vertical position of the point of action of the lateral wave force (m)
angle of drift (rad) rudder angle (rad)
wavelength (m)
position of the ship in the earth-fixed system (m)
water density (kg/rn3)
angle of heel (rad) heading angle (rad)
K. Spyrou: Ship behavior in quartering waves 25
Introduction
In following/quartering seas, ships normally perform
periodic motions engaging all their degrees of freedom.
At high Froude numbers however, increase of wave
steepness can lead to a stationary and distinctively
nonlinear type of behavior, the so-called surf-riding condition.' This brings about a split of the state space of the vessel into two regions: one leading to periodic
and the other to stationary behavior. In even higher
waves, the initial conditions domain leading to surf-riding tends to occupy the whole state space, resulting in a global attraction toward surf-riding. The dynamic stability properties in relation to both surf-riding and periodic-type behavior are of paramount importance because of their direct relation to safety. Loss of
dyna-mic stability during surf-riding can incur loss of
steera-bility and potentially lead to vessel capsize, the fearful condition termed broaching. A similar outcome is
pos-sible when the vessel is in low-frequency-of-encounter
periodic motions if the restoring force provided by the
rudder is inadequate to impede the increasing oscillatory
yawing motion. Because stationary and periodic
mo-tions represent alternative patterns of behavior for
the same dynamical system, it is suggested that it is more effective if the two are examined in parallel, thus obtaining an overall view of the dynamics of
the system.
Until recently, surf-riding was investigated in purely
following waves, essentially on the basis of the surge equation of motion.12 This has led to the notion that
for one wavelength, two states of surf-riding can exist:
one nearer to the trough and the other nearer to the
crest. However, if an arbitrary heading of the vessel in
relation to the wave is included in the analysis, it is found that the surf-riding states in fact belong to a
closed curve extending from trough to crest.3 In this paper the investigation is furthered, and a specific
framework is presented for the systematic analysis of the stability and dynamic behavior of a multidimen-sional nonlinear system such as a steered or unsteered
vessel free to perform surgeswayyaw and roll
mo-tions in waves, including steady-state analysis on the basis of continuation methods and transient analysis employing global techniques. A unique advantage of this approach is its effectiveness in guaranteeing the identification of all qualitatively different types of
surge acceleration coefficient
surge velocity coefficients sway acceleration coefficients sway velocity coefficients
behavior of the examined dynamical system. Application
of this approach has revealed surf-riding states with
different stability
properties and has allowed the
theoretical prediction of an oscillatory type of
surf-riding in quartering waves. This approach has also
shown that, in the surge veJocityheading subspace of
the system. the initial conditions resulting in surf-riding
belong to the interior of a parabola-like curve that
serves as the separatrix with the periodic motions
do-main. For the periodic motions themselves, it has
indi-cated the large oscillations performed by a vessel in quartering waves and has illustrated the mechanism of losing controllability as the rudder oscillations reach
their physical limits.
Mathematical model
A small fishing vessel is taken as the basis of this
investigation, with length between perpendiculars
7.14m, beam 187m, average draft 0.455m and block coefficient 0.447 (see Fig. 1). This vessel has been the object of extensive research in Japan by Fuwa et al.4 and Motora et al.5
The motion of the vessel is analyzed from a system of
axes fixed on a wave trough and thus moving with the
wave celerity c. Four degrees of freedom are considered
adequate for this analysis, namely, surge, sway, yaw, and roll, while the variation of the vessel's position in
the heave direction in relation to the wave is considered
as a geometric effect on the roll-restoring curve. The equations of motion, written for a horizontal system
placed at the middle of the ship, are as follows (Fig. 2)
surge: m(i - rv - xGr2 + zGpr)
= XH + X, + X + XR sway: m( + ru + XG? - ZGP)
= R + Y, + Y + YR
Fig. I. Body plan of the vessel (From Fuwa et al., 1982, with
wave crest
wave trough
A
Fig. 2. Systems of coordinates
yaw: 1i + mxG(v + ru)
= NH + N + N + NR roll:
1j -
ThZG(V ± ru)= KH + Kw + K + KR (1)
XH =
- Y,,vr - Y;.r2 + Xvr - Res(u)
= Y + Y;i + Yv(J + YrU + 'vv1)IvI + YrvJrI
+ YrIrI +
NH = N;1 + N + NrU + N,,i'U + Nrrrlrj + NrrvrrvlU + NrV vr! U + NHÇS
KR = [Kp + C(p) + R(q5, x)] - zyYff (2) The derivation of the hydrodynamic coefficients for this
model was discussed previously.3 For the selected
horizontal system, q = p and ç' = r. The polynomial approximation of the hull reaction forces is based on the work of Mikelis.6 The terms YHÇb' NHQ, represent
the effect of heel on the sway force and yaw moment
and are calculated according to Hirano and Takashina7
YH = (Y
+ YvkìI + Yrq!)U
= (N,,Uq5 + NwVI4 + Nr,rIq5j)U (3) The coefficients appearing in Eq. 3 are essentially heel corrections on the linear sway and yaw coefficients.
The restoring term R(, x), is approximated by the
expression
R(q5, x) = W(GM)[1 + ecos(kx)]h(f) (4)
where e indicates, albeit in a rather qualitative way, the
degree of variation of restoring force from the wave
through to the wave crest, and h() is an odd power polynomial of the heel angle. In principle, detailed
position-dependent calculation o restoring forces for specific waves and with due account of the distinctive
hull characteristics is essential for assesing roll stability, especially at frequencies of encounter which can lead to low-cycle resonance. Here, however, the focus is placed on the stability of horizontal motions, and the restoring lever is maintained large enough even at the wave crest.
Thus reference to this effect serves mainly the purpose of model completeness.
The rudder characteristics such as normal force,
effective rudder inflow velocity and angle, equivalent drift angle at the rudder position, and rudder-to-hull interaction coefficient are calculated according to the MMG recommendation (see for example Ogawa and Kasai8).
The wave excitation is expressed as
X. = pg(H!2)k cos wj a(
)e kd(x)A(x)sin k(x + x cos yi)dx
Yw = pg(H12)k sin
sin k(x + x cos = pg(H12)k siny
'La( W)e_kA (x5) x sink(x + xcos i)dx
K = pg(H!2)k sinyi L W)e_kdA (X)Zw
sin k(x + x cosw)dx (5)
In (5), A(xs) represents a local sectional area of the sin(kB(x) sin w!2) (normally very
ship and a() =
(kB(x5) sin w!2) sinx
near to 1.0 since hm =
i).
with B(x) the localxo X
sectional beam.The above formulation, corresponding to a Froude-Krylov calculation, is more accurate at near-zero fre-quency of encounter between the ship and the wave, We, where:
= (2irIA)[c - Ucos(
-
ß)] (6)A certain departure from near-zero encounter frequency can arise when considering also the periodic motions of the vessel. In this case, the results should be interpreted
as only showing the behavior of a ship under Froude-Krylov excitation. The suitability of Froude-Froude-Krylov, the importance of inclusion of diffraction, or the need
of also taking into account the ship's own wave system have been matters of extensive debate (see for example
Yoshino et al.9). This debate, however, is not entered
into here.
In Eq.5, use was made of the kinematic relationship x = - Cr,which removed the direct time dependence, allowing transformation of the system described by Eq. ito reach an autonomous dynamical system formulation
i =J(z;a)
(7)where z is the state variables vector, z = (u, y, r, p, 4,
K. Spyrou: Ship behavior in quartering waves 27 of the main ship control parameters, i.e.. propeller
rotation rate, n, rudder angle, ô, and metacentric height
GM, and also the principal wave parameters. i.e.,
wavelength), and height H, thus, a = [(n, 6. GM), (i.,
H)].
To consider the behavior of an automatically steered
vessel, an extra differential equation describing the
rudder response to yaw angle and rate is added to the
system in Eq. 7
= r4ô - a41( - Wr) - air]
(8) in which case the rudder angle ô is transferred to the state variables of the vector equation (Eq. 7). In Eq. 8,Wris the requested heading, a, ar are yaw angle and
rate gains, with a,. = abr, where b, is a constant. Also, t, is the rudder's time constant.
Surf-riding in quartering waves: steady-state analysis Steady states are the organizing centers of the behavior of a dynamical system. and in this sense their analysis is
normally the source of valuable information. For surf-riding in particular, the steady solutions would appear
associated not only with the long-term but also with the
actual motion pattern of the vessel, and thus a steady-state investigation would offer direct insight into be-havior in this case. To obtain the whole range of states of surf-riding for an arbitrary heading. the equilibrium points of the autonomous vector equation = f (z; a)
must be identified by varying the key control parameter,
i.e., the rudder angle, between ±35°. This corresponds to a fundamental investigation of a dynamical system, aiming to trace automatically the dependence of the state vector, z. on a critical control parameter, say ¡, i.e..f[z(,u), i] = O, to find out the solutions' branching and possible bifurcating behavior. Steady-state con-tinuation algorithms are the standard tools for carrying out such an investigation.10 In principle, the points of equilibrium of a system of first-order, nonlinear
dif-ferential equations can he found with repetitive solution
of the corresponding algebraic system of equations by using standard numerical tools (e.g., those in the NAG or IMSL Mathematical Libraries). However, this
me-thod is inefficient in practice, and it may conceal
impor-tant information related to bifurcation points, due to the difficulty for a nonlinear system in predicting how many solutions might correspond to a certain value of
a control parameter, and where these solutions are
located. Thus the search is arbitrary. Continuation, on the other hand, generates the whole sequence of solu-tions, normally passing without difficulty over limit points and over crossings of the solutions' branches.
The specific method applied here was originally
developed by Kubicek and Marek» built upon certain
existing ideas, like those of Keller.'2 Our problem is the solution of a system of algebraic equations written in vector form asf(zCu), i) = O. First, we differentiate
with respect to a parameter I accounting for the
length of the solutions curve, thus we obtain df/dl =
[(&f!ôz1) (dz1)/dl)J, for i = 1, n, while incorporating n+1
iI
the simple Pythogoras relationship, dl = dz to
nj-I
our system of algebraic equations (the summations go
up to n + i because it is convenient to use the
par-ameter as the z, unknown). Solving for dz1!dl leads
to an initial value integration problem with I as the
independent variable, which can be processed in any standard way. This integration works essentially as a
predictor of the solution, while a corrector must also be used to control accumulation errors which arise during
consecutive integrations, leading to divergence from
the correct solution path. The method of
Newton-Raphson offers a straightforward corrector of this kind.
An introductory account of continuation can be found
in the book of Seydel. '
To analyze the stability of the obtained equilibria, local linearization is performed around each solution
point by calculating the Jacobian matrix of f, C = {dfIdz} and by examining the signs of the real parts of the eigenvalues of C. 4The partial derivatives appearing in C are calculated analytically, wherever feasible, and verified numerically.
Application of these ideas to surf-riding has led to
the graph and associated stability map shown in Fig. 3 (position on the wave represents longitudinal position). For one wavelength, the stationary states of surf-riding
to\et
\ìF
V
region '4- of crest / ,-.I4.
/4v-t, "o tj 110 S?ó1ei ocus/
St6Ie ¿o.., region of trough .?JL2.O H/2=1Ì2O. GM=O.535 m Fn=O.56 -7 6 -5 - -3 -2 -1 Oposition on the wave (m)
Fig. 3. States of surf-riding for one wavelength with their
stability properties 0.75 0.5 0.25 'ti I-, o 0 -0.25 -0.5 -0.75
constitute a closed curve, which in the (x, w) projection of the system's state space approximates a circle. This is a direct, multidimensional extension of the single-surge case of Kan, as the two points of surfriding found with
the latter method would be obtained by intersecting this curve with the horizontal line corresponding to zero heading. Plots of heading w. or position i against
the rudder angle (5were presented previously.3 Figure 4
gives three-dimensional views of the relation between the three parameters.
To understand the dynamical features associated
with the stability map of Fig. 3, the attracting or
repel-ling characteristics of the steady solutions for a simple two-dimensional dynamical system are summarized in
Fig. 5. In principle, surf-riding states were found to be
unstable in the absence of active control. Two very
small stable regions arose at large headings nearer to the wave trough. although these would be difficult to realize in practice. Llnstahle states of saddle, nodal, and oscillatory type exist at different positions of the
wave down-slope. The transitions from one type of
stability to another are realized through limit points
(from stable nodes to saddles and from saddles to
unstable nodes) and also Hopf bifurcation points (stable
-
û.,
0.1
- o
-0.1
-
32
position on the wave(m)
- -3 -2
0.5 0.2
0.1
to unstable focus) as discussed in the following section.
Addition of the autopilot stabilizes the saddles nearer to the wave trough, and in this sense, stable states for
the steered vessel are to be found near to this region of the wave, if they exist at all. Such a case of stable
surf-riding for the steered vessel in quartering waves is
shown in Fig. 6. In the presented example, initiating
the run from the wave trough leads to stationarity,
while repeating the run from the crest gives rise to a
periodic pattern.
From the control parameters, the wave steepness is
critical for surf-riding. Figure 7 illustrates the generation and growth of surf-riding by increasing the wave height, for AIL = 2.0. It is also possible to observe the variation
of steady heel during surf-riding. The value of GM affects the range of rudder angles corresponding to equilibrium points. Lowering GM results in larger rudder angles.
Self-sustained oscillatory-type surf-riding
As indicated in Fig. 3, a Hopf bifurcation occurs at
relatively large headings signifying a transition from
Fig. 4ac. Surf-riding in three dimensions from different view angles. Surf-riding states exist at the outline of the surface -0.5 -0.25 0 0.25 0.5 C heading (rad) 0.5 -0.5 -0.25 0 0.25 -0.2 heading (rad) .5 a -0.25 0 0.25 heading (rad)
K. Spyrou: Ship behavior in quartering waves 29
stable stationary to stable oscillatory behavior, while stationary surf-riding is rendered unstable. The range of rudder settings and initial conditions giving rise to
this type of behavior is generally very limited, implying that it may easily pass unnoticed. However, it should be
mentioned that oscillatory-type surf-riding has been
observed experimentally by Kan.1 From an engineering
point of view, and unless otherwise proven, this type
of behavior may be of limited importance. Yet, its
dynamics present considerable interest, because this region was found to also include a domain of chaotic
behavior. To explain this, first, Fig. 8a presents a
typical self-excited oscillation after the Hopf bifurcation
corresponding to rudder angle ô = 0.1566 rad, for )./L = 2.0, HI). = 1/20 at En = 0.56. The associated time history of the vessel's position x, is shown in Fig. 8b.
Taking the power spectrum (i.e., taking the Fourier
transform of this data set) shows clearly one frequency
with its harmonics, Fig. 8e. The power spectrum is a
powerful technique to reveal the number of frequencies hidden in a certain set of numerically or experimentally
produced data, as explained by Berge et a1.'
The phase plane plot and the time history corres-ponding to the lower rudder angle ô = 0.1560 rad,
Figs. 9a and 9b, appear considerably different to their
0.34
0.32
0.28
0.26
Fig. 5. Basic attractors and repellers
u0=2.0 ni/s. W0.3 rail r-0.3 rad, Fn =0.35
Fig. 6. An example of stable surf-riding in quartering waves (with autopilot)
previous counterparts, suggesting either a quasiperiodic ora chaotic pattern. Observation of the power spectrum
of the motion leads to the conclusion that the vessel
behaves chaotically, since the power spectrum presents
a Continuous appearance as if there was broad-band
noise in the system, Fig. 9c. This is a classical fingerprint
of chaotic behavior. A visually clearer picture is given
in Fig. 10, for ô = 0.1564 rad, resembling the so-called Rössler attractor. The rudder angle difference between the occurrence of the first bifurcation and the emergence
of chaotic behavior is very small (it was measured as
= 0.00173 rad). Details of the transition and characteristics of this chaotic domain are currently
under investigation. Global behavior
The question following the identification of steady
surf-riding states is, which initial conditions of the system
state space lead to surf-riding and which lead to "other"
motions. This question is raised only for the steered vessel, since, as mentioned in the section on steady-state behavior, this is mainly when stable surf-riding
can occur. As "other" motions, four different conditions
are specified: normal periodic motions around the
re-*
Stable node Stable focusS)
'I Unstahl b -d Cno e (Oscillatory Unstable focus instability) Eige,waluec Ewatue. imginv p Part rcl p\ \
s ddl e porn Typical system and a saddle in
with two attractors the middle
iryt
C
Unstable limit-cycle Stable
j/
limit-cycle .1 . 3 4 surge velocity (m/s) 5 6 6a o 0.6 0.4 0.2 -0.4 -0.6 0.3 0.2 0.1 o -0.2 -0.3 -0.2 -0.1 0 0.1 0.2 -0.2 -0.1 0 0.1 0.2
c rudder angle (rad)
Fig. 7a-d. Effect of wave height
quested heading, turning, entrapment in the beam
wave condition, or even vessel capsize. To provide
an answer, we should attempt to unravel the global
morphology of the system state space. However, the difficulty we are immediately faced with is the multi-dimensional character of the problem. To overcome this, it is suggested we restrict our attention to the part
of the phase flow which crosses the plane of the system's most critical variables, namely, the initial surge velocity and heading. The position of this plane in state space is
rudder angle (rad)
2
-3
-6
-7
-0.2 -0.1 0 0.1 0.2
rudder angle (rad)
-0.2 -0.1 0 0.1 0.2
sway velocity (rn/see)
defined by a preselected set of values of the remaining variables. It is also assumed that the initial heading coincides with the requested heading of the vessel. In this case the initial conditions vector (ufl, v, r, Po. çb0, v'o, X0, d) of the system takes the form (u0, Vf, Pf. j, Wo Xf, öf) with the controls vector being (nf, w. (Wo)) All state components carrying the subscript fare meant as fixed initial conditions during global analysis. Their effect can be assessed with a parametric study at a later
K. Spyrou: Ship behavior in quartcring waves 0.86 0.85 0.84 -o '; 0.83 0 rj - 0.82 a 0.8]. 08 i 3.5 3 0.5 i5=-O.1566 rtd 100 200
position on the wave (m)
C Period (4096/value (sec))
To demonstrate the effectiveness of this approach,
Fig. lia shows the domain occupied by the initial
conditions resulting in surf-riding, in the initial surge velocityinitial and requested heading plane, with all other components set initially to zero (but of course free to obtain nonzero values during the evolution of
the system in time). These data were obtained by
applying direct integration from a suitably dense grid of
initial conditions in this plane. The vessel's nominal
Froude number is taken as Fn = 0.33, which is just below the threshold for attraction to surf-riding
-1.8 -2 E -2.2 -2.4 n o o -2.8 31 300 400 500 600
Fig. 8. Oscillatory surf-riding, limit-cycle behavior, a in the
xw plane,b time history,c power spectrum
from any initial surge when the heading relatively to the wave is zero (Fn = 0.34). The boundary
be-tween surf-riding and other motions (mostly normal periodic) appears with a well-defined parabolic shape.
As explained earlier, the vessel was initially assumed at
X() = 0, i.e., at the wave trough. Repeating the inves-tigation for x0 2/4, i.e.. the middle of the wave
up-slope, produces the picture of Fig. lib, suggestinga shift of the initial surge velocities toward higher values,
with a parallel shrinkage in the range of initial and
requested headings.
-1.8 -2.8 -2.6 -2.4 -2.2 -2
O 100 200 300 400 500
-1.6 -1.8 J 0.86 0.84 0.82 0.8 0.78 3.5 1.5 0.5
The periodic motions of the steered vessel
Having examined steady and transient behavior in
rela-tion to surf-riding, attenrela-tion
is now turned to the
overtaking waves' periodic motion pattern, which has
its own interesting dynamical characteristics. In the low
range of wave heights. under normal conditions, the
periodic motion accounts for the only possible mode of
operation, since surf-riding cannot exist. In larger
waves, it is regarded as the safe alternative to surf-riding. However, in following/quartering waves in
general. it is known that appreciable, occasionally
-2.2
critical, rudderheading oscillations can be induced in the vessel as reported by Eda.16 Such a condition is illustrated in Fig. 12 with the vessel striving to keep a 40° heading in large waves. lt is noted that the rudder
oscillations have grown to such an extent that they
almost reach the maximum allowed rudder deflection.
The vessel cannot advance with a heading beyond 40°,
and if this is attempted broaching is a likely outcome. The question that naturally arises therefore is, what is
the range of headings that can be safely maintained by
the vessel, as a function of its nominal Froude number and wave parameters. For this we need to understand
-3
-2.8 -2.6 -2.4 -2.2
-2 -1.8 -1.61 00 200 300 400 500
a position on the wave (m)
time (sec) b
100 200 300 400 500 600
Fig. 9. Chaotic behavior a in the .t-çupiane. b time history, e
K. Spyrou: Ship behavior in quartering waves
-2.8 -2.6 -2.4 -2.2 -2
position on the wave (m)
Fig. 10. Variation with the rudder angle
a 0.86 0.55 0.84 0.83 - 0.82 0.81 0.8 0.6 0.4 (.1.2 2/L=2.O HR=1/20 (Fri =0 330) 2.5 3 3.5
surge velocity (mis)
-1 .8
Fig. 11. The domain of initial conbitions for surf-riding, a X1) 0.0. b x0 =
)IJL=2.O, H/)i.=rl/15, Fn=0.20 V1 =0.7 rad
0.5 0.4 0. 2 o -0.2 -0.4 130 135 time (s)
Fig. 12. Largo oscillations in quartering waves
surge (mis) heading (rad) rudder (rad) 140 145 33 150 -0.6 2.25 2.5 2.75 3 3.25 3.5 3.75 4
= E o Q o . = E C Q o 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8-4
how the variation of the requested heading affects
oscillation amplitudes.
Identification of steady periodic states
Two different numerical routes exist to identify the main features of this oscillatory motion as a selected control parameter varies. One is with repetitive inte-gration of Eq. 7, which provides information on both
the transient and steady motion of the vessel. However, it is probably more systematic and efficient, in general, to start by studying the steady-state amplitudes and fre-quency of motion with direct steady-state identification,
Fig. 13a-c. Steady oscillation amplitudes as a function of requested heading. The autopilot constants are given with
the order, t,i, aw, br (Eq. 8)
and subsequently to focus on specific cases for applying
integration in time. This is similar to the thinking
applied for the stationary solutions case.
A steady periodic solution with period T would
satisfy the relation
z(t + T) = z(t) (9)
Considering only one period interval and normalizing
time in the [0, 1J range, by applying the transformation t Tr, where r E [0, 1j, Eq. 7 would read
Tf(z;a) (10) ..rauropilor: H/A.i/l5 5,5,1 Fn=O.331
A,
surf-riding1j"rudder
limit 0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8
0.9 00.1 02 0.3 04 05 06 07 08 0.9
a specified heading (rad) specified heading (rad) b
0.2 0.3
0.4 0 5 0.6
0.7 0 8c specified heading (rad)
0.9 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.] o
K. Spyrou: Ship behavior in quartering waves 35 Eq. 9 would then provide n mixed boundary conditions
on Eq. lO
z(1) - z(0) = 0 (11)
Considering the general problem where the period Tis
an unknown, we can incorporate it in the variables
vector, thus, z,,.1 = T, with dz,,1/dr = O. To solve then for the unknowns z1, with i = 1, n + 1, we have the
n + i equations of the enlarged Eq. 10 and n boundary conditions from Eq. 11. The necessary extra boundary condition comes from a phase relation which essen-tially gives us the opportunity to choose a new time origin for the periodic solutions. One possibility is to
select a value for one of the variables, say,Zk = i,with the condition of course that this value must fall within the range of values taken byZ. over one period. This phase condition is enough to find the period and
amp-litude of motion, but is inadequate for locating the
maxima/minima of the motion variables which are of
particular interest in this study. To circumvent this, we
apply instead the phase condition, dzk/dr = fk(z; a) = 0, which is equivalent to requesting the shifted time
origin to coincide with a singular point of the considered
variable. Choice of. say, a maximum, is then possible with addition of the constraint, d(fk(z; a))/dr < 0.
Oscillations aWL=2.0,H/X= 1/15
The steady oscillation amplitudes for the surge, yaw, and rudder motions, taking as control parameter the
requested heading. u,., are shown in Figs. 13a, 13b, and 13e, at Fn = 0.20 and Fn = 0.33. At the lower Froude number, rudder oscillations of very large amplitude are needed for effective course keeping, especially for
headings beyond vi, 10°. The limiting situation is
defined by the heading at which the rudder reaches 35°.
This occurs with starboard rudder at a heading of
approximately 210. Tn Fig. 13, the rudder has been
allowed to go beyond the physical limit of 350 for better
illustration purposes. A slightly different picture arises at the higher Froudc number. In this case, the vessel can exhibit the periodic motion pattern with certainty
only for headings in excess of 22° since below this value
surf-riding is as an equally possible condition. The oscillations of the rudder are of reduced amplitude,
never reaching their limits, and in this case the peak of rudder oscillation occurs to the port side.
Concluding remarks
A dynamical systems approach for investigating the behavior of a ship in large quartering waves was
pre-sented. Although not all questions have been answered, a
number of notable findings have emerged. In relation
to surf-riding, it was shown that these states belong to a
closed curve with the following stability properties: In the vicinity of the trough. all states are saddle points
that with the addition of suitable autopilot can be
rendered stable. For the unsteered vessel, a very limited
range of stable states may exist at large headings. and these would be difficult to identify in a real situation.
The wave crest is dominated by instability in the
longi-tudinal direction, and is in general not liable to stabi-lization with conventional means. The largely ignored
range of surf-riding states near the middle of the wave's
down-slope is probably the most interesting in terms
of dynamics. In this range, a Hopf bifurcation sets
the scene for an oscillatory type of surf-riding in the unsteered vessel. Slight variations of a control
para-meter such as the rudder angle give rise to further
transitions leading to chaotic behavior. However, the
physical significance of this behavior has yet to be
demonstrated. For stationary-type surf-riding, global analysis was applied to identify the associated initial
conditions domain. In the uií plane, this domain
coincides with the interior of a parabola-like curve,
separating surf-riding from other motions: mostly
normal oscillations with the waves overtaking the vessel. The dynamical features of these periodic motions were
also investigated. A method was first put forward for the identification of periodic states as a function of a certain control parameter. For large waves, significant
yawsurge--rudder oscillations are associated with
effective course keeping, where the physical limitations
of rudder oscillation define the threshold values for
requested headings.
Acknowledgments. I would like to thank Dr. Umeda,
my host researcher in Japan, for helpful discussions on
the subject. I also thank Dr. Kan and Dr. Fuwa of the Ship Research Institute for their valuable comments. This research was carried out on the basis of a
fellow-ship offered by the Science and Technology Agency of Japan in collaboration with the Directorate General for
Research and Development of the European Union.
References
Kan M (1990) Surging of large amplitude and surf-riding of ships in following seas. In: The Society of Naval Architects of Japan (eds), Naval Architecture and Ocean Engineering, vol 28. Ship and Ocean Foundation, Tokyo, pp 1-14
Umeda N, Kohyama T (1990) Surf-riding of a ship in regular seas (in Japanese). J Kansal Soc Nay Archit Jpn 213:63-74 Spvrou Ki(1995)Surf-riding, yaw instability and large heeling of ships in following/quartering waves. Schiffstechnik 42(2): 103-112 Fuwa T, Sugai K, Yoshino T. et al. (1982) An experimental study on broaching-to of a small high-speed boat. Papers of Ship Research Institute, No 66. Tokyo
Motora S, Fujino M, Koyonagi M, et al. (1981) A consideration on the mechanism of occurrence of broaching-to phenomena. In:
The Society of Naval Architects of Japan (eds) Naval Architecture and Ocean Engineering, vol 19. Ship and Ocean Foundation, Tokyo, pp 84-97
Mikelis N (1985) A procedure for the prediction of ship manoe-uvring response for initial design. [n: Proceedings of the Interna-tionai Conference on Computer Applications itt the Automation ofShipyard Operation and Ship Design (ICAAS 85). Elsevier/Nrth Holland. pp 437-446
Hirano M, Takashina J (1980) A calculation of ship turning motion taking coupling effect due to heel into consideration. Trans West-Japan Soc Nay Archit 59:71-81
Ogawa A. Kasai H (1978) On the mathematical model of nianoe-uvring motion. Im Shipbuíld Prog 25(292):306-319
Yoshino I, Fujino M, Fukasawa T (1988) Wave exciting forces on a ship travelling in following waves at high speed (in Japanese). J Soc Nay Archit Jpn 163:160 172
Spyrou KJ (1991) A new approach for assessing ship manoeuvra-bility based on dynamical systems' theory. PhD thesis. University of Strathciyde. Glasgow
Kubicek M, Marek M (1983) Computational methods in bifurca-tion theory and dissipative structures. Springer, Berlin Keller HB (1977) Numerical solution of bifurcation and nonlinear eigenvalue problems. In: Rabinowitz PH (ed) Applications of Bifurcation Theory. Academic, New York
Seydel R (1988) From equilibrium to chaos: practical bifurcation and stability analysis. Elsevier, New York
Spyrou KJ (1995) Yaw stability of ships in stationary wind. Schiftstechnik 42(1):21-30
iS. Berge P, Pomeau Y, Vidai C (1984) Order within chaos: towards a deterministic approach to turbulence. Wiley. New York 16. Eda H (1972) Directional stability and control of ships in waves.
