Nonlinear dynamics of ship capsizing due to broaching in following and quartering seas

Marine Science

and Technology

©SNAJ 1999

Nonlinear dynamics of ship capsizing due to broaching in following

and quartering seas

NAOYA U M E D A

National Research Institute of Fisheries Engineering, Ebidai, Hasalci, Ibaraki 314-0421, Japan

Abstract: To provide a theoretical methodology to predict the critical condition f o r capsizing due to broaching, a nonlinear dynamical system approach was applied to the surge-sway-y a w - r o l l motion of a ship running w i t h an autopilot in following and quartering seas. 'Fixed points of a mathematical model f o r the ship motion and unstable manifolds of the fixed point near the wave crest were systematically investigated. A s a result, the existence of heteroclinic bifurcation was identified. W i t h nu-merical experiments, it was confirmed that this heteroclinic bifurcation reasonably well represents the critical condition f o r capsizing due to broaching. Thus the nonlinear dynamical approach can be substituted f o r tedious numerical experiments.

Key words: capsizing, broaching, surf-riding, heterochnic bifurcation, following seas, quartering seas

List of symbols

A„ rudder area b control vector B ship breadth c wave celerity Cj block coefficient D ship depth d„ aft ship draft

fore ship draft d,„ mean ship draft Dp propeller diameter Fu nominal Froude number g gravitational acceleration GM metacentric height

GZ righting arm » H wave height

I„ roll moment o f inertia yaw moment o f inertia / j , . added roll moment of inertia

Address correspondence to: N . Umeda, Department of Naval Architecture and Ocean Engineering, Osaka University, 2-1 Yamadaoka, Suita 565-0871, Japan

Received f o r publication on N o v . 20,1998; accepted on March 16, 1999

added yaw moment o f inertia

derivative of roll moment with respect to roll rate K derivative o f roll moment with respect to yaw rate

rudder gain

K, derivative o f roll moment with respect to sway velocity Ks derivative o f roll moment with respect to rudder angle

derivative o f roll moment with respect to roll angle K„ wave-induced roll moment

l.c.b. longitudinal center o f buoyancy Lpp ship length bet\veen perpendiculars LQA ship length overall

JU ship mass

/», added mass in surge '"v added mass in sway 11 propeller revolution

N, derivative of yaw moment with respect to yaw rate J\ derivative of yaw moment with respect to sway velocity

derivative o f yaw moment with respect to rudder angle derivative o f yaw moment with respect to roll angle wave-induced yaw moment

P roll rate

P' nondimensional roll rate (pL^J^u^ + v~) r yaw rate

r' nondimensional yaw rate {rL^Jyu^ + v-) R ship resistance

Sf wetted surface area t time

f nondimensional time (Jc/Lpp) T propeller tln'ust

T, time constant for differential control TE time constant for steering gear T, natural roll period

u surge velocity

u' nondimensional surge velocity {ii/c) V sway velocity

X state vector

X„ wave-induced surge force

Yr derivative of sway force with respect to yaw rate 1'v derivative of sway force with respect to sway velocity

derivative of sway force with respect to rudder angle derivative of sway force with respect to roll angle wave-induced sway force

Zf, height of center o f lateral force 8 rudder angle

K^^. gyro radius in pitch fc, gyro radius in yaw X wavelength A rudder aspect ratio

(^G longitudinal position o f center of gravity from a wave trough

^(j' nondimensional longitudinal poshion <p roll angle

% heading angle %^ autopilot course

Introduction

A ship running in waves usually has a periodic motion, the period being equal to the encounter frequency of the ship to waves. W h e n a certain condition is satisfied, the ship may meet a transition f r o m a harmonic motion to a nonharmonic motion. Examples of such nonhar-monic motions involve surf-riding and broaching. Surf-riding is a phenomenon whereby the ship is forced to run with wave celerity. Broaching is a phenomenon whereby the ship cannot maintain her course despite maximum steering effort. Neither phenomenon can be overlooked because either may result i n sea disasters, such as capsizing.

A m o n g seafarers, these phenomena have been known f o r years.^ Recent model experiments demon-strated that an intact ship complying with the Inter-national Maritime Organisation ( I M O ) stabihty criteria can capsize owing to broaching.^ However, i t has been difficult to prevent these phenomena by understanding their dynamics even i n regular waves because a hnear theory as a basic t o o l f o r engineering cannot deal, i n principle, with any behaviors other than harmonic mo-tions. Surf-riding and broaching are strongly nonhnear, so it is essential to use a nonlinear approach. Neverthe-less, methodology based on nonlinear dynamics has not yet been fully estabhshed f o r engineering apphcations. Research on surf-riding started with a mathematical model of an uncoupled surge motion i n pure following seas. A wave-induced surge force is modeled here as a periodic function of surge displacement, so this simplest mathematical model is nonlinear. Grim^ explained, us-ing the analogy of a pendulum, that surf-ridus-ing critically occurs when a ship moves f r o m one unstable equilib-rium to another unstable equilibequilib-rium. Makov* clarified whole initial value dependence of surf-riding by a phase plane analysis. K a n et al.' proposed that surf-riding occurs when the maximum ship velocity reaches wave celerity. Finally, Umeda and Kohyama«>^ concluded that these three proposals are different interpretations f o r the same heterochnic bifurcations as the critical condi-tion f o r surf-riding i n pure following seas.

I n the case of quartering seas, because coupling between surge and lateral motions cannot be avoided, it is essential to take both surf-riding and broaching into account. A pioneering work for broaching was carried out by Davidson,^ who used a hnear sway-yaw coupled model. H e showed that even a ship that is directionally stable i n calm water can be directionally unstable i n fohowing seas. Wahab and Swaan,' Eda,^° and others further pursued directional instability i n waves, but they could not explain a direct relation between directional instability and broaching itself because broaching is not a linear phenomenon but a fully nonlinear one.

Later developments in computer technology realized time domain simulations of nonhnear motions with some sets of initial conditions. W i t h this technique, Motora et a l . " and Renilson'^ concluded that broaching occurs when the wave-induced yaw moment of a ship exceeds the rudder-induced yaw moment w i t h the maximum opposite rudder angle.

These time domain simulations, however, are limited in their ability to identify the critical conditions for broaching because, hke any other nonlinear phenom-ena, broaching depends very much on the initial con-ditions. T o overcome this difficulty, it is desirable to analyze behaviors of solution sets instead of the solution itself. This practice is known as the nonlinear dynamical system approach. The first attempt i n this direction was reported by Umeda and Renilson."-!'' They identified fixed points of a surge-sway-yaw-rudder system and showed that the fixed point with the maximum opposite rudder angle can be a saddle and their unstable mani-f o l d represents a typical trajectory omani-f broaching. These fixed points correspond to unstable surf-riding equilib-ria i n quartering seas. Periodic orbits, another major steady state of the system, were investigated with an averaging method by Umeda and-Vassalos.^' They showed that the periodic orbit can be unstable when the encounter frequency becomes small. These outcomes f r o m the analyses on fixed points and periodic orbits suggest that when a periodic motion becomes less stable a ship can be attracted by an unstable surf-riding point as a saddle and then repelled with a violent yaw motion despite the apphcation of maximum opposite rudder. This phenomenon is known as broaching."

This approach was extended to a surge-sway-yaw-roll-rudder model. Repeating time domain simula-tions, S p y r o u ^ " ' discussed the topology of boundaries between capsizing, surf-riding, and broaching; and he f o u n d chaotic surf-riding and a j u m p - o f periodic orbit to be extreme cases. Umeda et al.^" presented a method f o r predicting the approximate critical condi-tion for broaching with stabihty analyses on fixed points and periodic orbits but without time domain simulations; they vahdated their method w i t h model experiments.

Recent progress in research into broaching is remark-able, as mentioned above. However, a direct method for predicting the critical condition has not yet been estab-lished f o r broaching, whereas i t was estabestab-lished f o r surf-riding of uncoupled system. This is mainly because stable surf-riding is an attractor of the system, and broaching is not. Broaching i n regular waves results in capsizing or periodic orbits. Therefore, by focusing on capsizing due to broaching as a kind of attractor, I generalized a phase plane analysis of uncoupled system to apply to a high-dimensional system and developed a methodology for more directly assessing the critical condition f o r capsizing due to broaching i n regular waves. This paper first provides a mathematical model and then discusses the fixed points and global topology of the system described by the model. Finally, it pre-sents a close relation between a global bifurcation of the system and a critical condition obtained by numerical experiments.

IVIathematical model

Ship motions i n waves are often described w i t h a six-degrees-of-freedom model as three-dimensional mo-tions of a rigid body. On the other hand, it is desirable to use a simple but still reasonable model f o r a nonlinear system because the increase i n the number of degrees-of-freedom makes nonlinear dynamical system analysis difficult. When a ship runs in quartering seas w i t h some-what high speed, it is exposed to the possibility of broaching. I n this situation, the encounter frequency of the ship i n waves becomes much smaher than the natural frequencies in heave and pitch. Surge, sway, roll, and yaw motions, which have zero or small restoring terms, significantly respond to such small encounter frequency. Therefore, heave and pitch motions can be reasonably approximated by simply tracing their stable equilibria. This approximation was well vahdated with a systematic comparison between model experiments and a strip theory in quartering waves with zero and very low encounter frequency.^' Hence this investigation uses a four-degrees-of-freedom model, a surge-sway-y a w - r o l l model. Here it is noteworthsurge-sway-y that all hsurge-sway-ydro- hydro-dynamic terms should be obtained with the heave and pitch i n static equilibrium* at zero encounter frequency. Because of the low encounter frequency, hydro-dynamic forces acting on a ship, including wave-induced forces, mainly consist of hydrodynamic l i f t and buoy-ancy. Wave-making effects that depend on the encoun-ter frequency, which can be dealt with using a strip theory, are neghgibly smah. O n the other hand, vortices generated by hull sections immediately flow down-stream with ship forward velocity and f o r m trailing vortex sheets, which induce hydrodynamic l i f t forces. Therefore, a maneuvering mathematical model

focus-ing on hydrodynamic l i f t components'^ ^an be recom-mended f o r broaching, but a seakeeping model that focuses on wave-making components cannot. Because wave steepness is much smaller than 1, that is, at least up to 1:7, the drift angle and non-dimensional yaw rate due to waves can be assumed to be as small as the wave steepness. Thus, wave effects on maneuvering coeffi-cients with respect to sway and yaw can be ignored as higher-order terms. Thus, the mathematical model used in this paper is based on a maneuvering one with linear wave-induced forces but without nonhnear maneuver-ing coefficients due to sway, yaw, and waves.

As can be seen in Fig. 1, two coordinate systems are used: (1) a wave fixed with its origin at a wave trough, the ^ axis in the direction of wave travel; and (2) an upright body fixed with its origin at the center of ship gravity, with the x axis pointing toward the bow, the y axis to starboard, and the z axis downward. The latter coordinate system is not aUowed to turn about the x axis.'^ The symbols and nondimensionalization are de-fined i n the nomenclature.

The state vector, x, and control vector, b, of this system are defined as follows:

x = {^^/^.,u,v,x,r,(!),p,sY (1)

b = {n,xX (2)

The dynamical system can be represented by the fohow-ing state equation:

X = F ( X ; b) = {/i(x; b),/,(x; b),... ,f,{x; b)}' (3)

where

/i(x; b) = {ucosx-vsmx-c]/X (4) /,(x; b) = {T{U; n)-R{u) + X^X^JX, 4 / ( m , + m , ) (5)

ƒ3 (x; b) = [-{m + in^ )ur + Y^, {u; n)v + Y^ («; n]r + Y^ {u)(p

+ Y,{ir, n)5+Y„.{^jA, u, x\ «)}/(/" + /",)

(6)

/4(x;b) = /- (7)

ƒ5(x; b) = {yV„(t(; n)v + A?,(«- n)r + N + N.{u; n)5

+ y v , . , ( ^ , A , » , 2; " ) } / ( ! . + J . z ) (8)

/6(x;b) = p (9)

f,{x\ h)^\in^ZHur + K^{u\ n)v + K^{ii; n)r + Kp{u)p + K^[u)<l) + Ks[u\ n)ö + K,,{^a/^> X; ")

- m g G Z ( 0 ) } / ( / „ . + / „ . ) (10) /,(x; b) = [-Ö-K,{x-X.)-K^T^>]/T, (11)

Table 1. Principal particulars of purse seiner and wave pa-rameters ^ T] 43.0 m 0.242 Lpp 34.5 m 0.242 B 7.6m GM 1.0m D 3.07 m _T, 7.4s 2.50m Ap 3.49 m ' 2.65 m A 1.84 d.. 2.80m r. 0.63 s Q. 0.597 Kp 1.0 l.c.b. (aft) 1.31m To 0.0 s Sf 324 m^ H/X 1/9.2 Dp 2.60m •r X/Lpp 1.5

Step, these effects are ignored herein but should be included in the near future.

I n this study, based on the above-mentioned math-ematical model, numerical calculations were carried out f o r a Japanese purse seiner, whose principal particu-lars and wave parameters are shown i n Table 1. The loading condition here marginally comphes w i t h the Intact Stabihty Code of the I M O , and hydrodynamic coefficients under this condition can be f o u n d i n the literature.'

Fig. 1. Coordinate systems _{Fixed points}

Because the external forces are functions of the surge displacement but not time, this equation is nonlinear and autonomous.

The wave forces and moments are predicted as the sum of the Froude-Krylov components and hydro-dynamic l i f t due to wave particle velocity by a slender body theory. Umeda et al.'"* well validated this predic-tion method with a series of captive model experiments. Even under the typical broaching conditions (i.e., the runs w i t h zero encounter frequency i n extremely steep quartering waves), reasonably good prediction f o r wave-induced yaw moment in both amplitude and phase was reported. The resistance, propeller thrust, and maneuvering coefficients are determined with cir-cular motion tests. The added inertia terms, which can-not be obtained by circular motion tests, are estimated theoretically in sway and yaw and empirically i n surge. The longitudinal distance between the center of the added mass and the center of the mass is assumed to be negligibly small. The roll damping moment is estimated w i t h roU decay tests at zero forward velocity and Takahashi's empirical formula f o r f o r w a r d velocity ef-fect.'^ The roll restoring moment is calculated hydro-statically. As the rudder angle and r o h angle are not so smah, the wave effect on the rudder force and roU mo-ment may be taken into account. However, as the first

A nonlinear dynamical system, such as the system de-scribed by E q . (3), may have several steady states: fixed points, periodic orbits, quasiperiodic orbits, and chaos. A m o n g these options, this paper focuses on fixed points because fixed points of the present system correspond to riding. Previous experiments" showed that surf-riding is a prerequisite f o r broaching.

First, the fixed points

X = ( | G / A , iT, V, X. r, ^,P, S) (12)

are obtained by solving the following equation:

F ( X ; b) = 0 (13)

I f a solution exists, the ship is in an equilibrium position surfriding on the wave w i t h a drift angle and heading angle. F(x; b) is linearized at x, putting x = x + y to obtained the fohowing equation:

y = OT(x;b)y (14)

where

D F ( x ; b ) = ^ / . ( x ; b ) l < l , j < 8 (15)

I f an eigenvalue of £)F(x; b) has a positive real part, local asymptotic behavior at x is unstable.

Equation (13) can be solved by the Newton method if an appropriate set of initial values is provided. As an example, Figures 2-5 show fixed points obtained f o r various auto-pilot courses, Xc-, with a certain nominal Froude number, Fn. Here the nominal Froude number means the Froude number of a ship when a certain propeher revolution number, n, is commanded i n stiU water. Thus, instead of n and x^, Fn and Xc can be re-garded as control variables. I n the case of Xc = 0, E q . (13) tends to the equation f o r the uncoupled surge model because the lateral motions (i.e., v, x, <P, and 5) are zero. This equation can be easily solved by a regule falsi method.'' I n this case there are usually two fixed points: an unstable one near the wave crest and a stable

one near the wave trough.^ Then the fixed points of Eq. (13) can be traced by increasing Xc f r o m these two fixed points at Xc - 0- H couplings with lateral motions are taken into account, even the fixed point near the wave trough can be unstable. Whereas at the fixed point near the wave crest (shown i n Figs. 2-4) the rudder angle and the heading angle are both positive, at the fixed point near the wave trough the heading angle is negative despite the positive rudder angle. This is because the wave-induced yaw moment near the wave trough acts to increase the heading angle, and the rudder acts to pre-vent its increase.

As shown i n Figs. 6 and 7, f o r Xc = 10 degrees each fixed point i n the eight-dimensional phase space has one

auto pilot course (degrees)

Fig. 4. F k e d points f o r Fn = 0.3248 Fig. 2. Fixed points f o r Fn = 0.3248

1.6

0 10 20 30 40 50

auto pilot course (degrees)

- i

1.2-

SO.8-o

^ 0 . 4

0

0 10 20 30 40 50

auto pilot course (degrees)

Ï

0-

.10 -8 -6 - 4 - 2 0 2

Real part o f eigenvalue

Fig. 6. Eigenvalues of the fixed point near the wave crest f o r Fn = 0.3248 and Xc = 10 degrees bl) 9-o ft Ö

0¬

-2¬

4¬

-6--10 -8 -6 - 4 - 2 0 2

Real part o f eigenvalue

Fig. 7. Eigenvalues of the fixed point near the wave trough f o r Fn = 0.3248 and Xc = 10 degrees

eigenvalue having a positive real part without an imagi-nary part and seven eigenvalues having negative real parts. Thus, each fixed point is a saddle of index 1. Figure 8 shows the eigenvector of the eigenvalue, having a positive real part, f o r the fixed point near the wave crest. Instability in surge is dominant here. O n the other hand, as shown i n Fig. 9, instability i n yaw, roll, and rudder is dominant f o r the fixed point near the wave trough. Thus, the fixed point near the wave trough is responsible f o r course instability, and that near the wave crest is responsible for longitudinal instabihty.

M - 0 -0 -0 6 4¬ 2¬ 0¬ 2¬ 4¬ 6¬ 8¬ 1

-1

1

1

_i

Fig. 8. Eigenvector of the fixed point near the wave crest f o r Fn = 0.3248 and Xc = 10 degrees. The numbers i n the abscissa indicate those of the components of the vector x shown i n Eq. (1) 6 4¬ 2¬ 0-2^ 4¬ 6¬ 8¬ -1 1——1

1

i

1

i

1

i

1

i

Fig. 9. Eigenvector of the fixed point near the wave trough f o r Fn = 0.3248 and Xc = 10 degrees. The numbers i n the abscissa indicate those of the components of the vector x shown i n E q . (1)

Unstable manifolds and numerical experiments

A h h o u g h only local asymptotic behavior at the fixed point has been discussed so far, Hartman's theorem and the stable manifold theorem'^ enable us to investigate the local topological structure of the system by E q . (3). That is, there exist local stable and unstable manifolds, W^^(x; b) and W/^X^; b), tangent to eigenspaces, spanned by eigenvectors of Z3F(x; b) at x. Then the global stable and unstable manifolds and W" are obtained by letting points i n Wi^ how backward in time and those i n Wf^^ flow forward.

I n the dynamical system discussed here, because the fixed point is a saddle of index 1 i n the eight-dimensional phase space, its unstable manifold is one-dimensional and its stable manifold seven-one-dimensional. Thus, although visuahzation of the stable manifold is not easy, the unstable manifold is easily visuahzed as a trajectory. That is, the unstable manifold, or an outset,

nondimensional l o n g i t u d i n a l position -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 nondimensional longitudinal position

Fig. 10. Unstable manifold of the fixed point near the wave

crest f o r Fn = 0.3248 and Xc = 10 degrees Fig. 12. Unstable manifold of the fixed point near the wave crest f o r Fn = 0.3248 and Xc = 10 degrees

>

/

id

0 10 20 30 40 50 60

heading angle (degrees)

Fig. 11. Unstable manifold of the fixed point near the wave

crest f o r Fn = 0.3248 and Xc = 10 degrees

Ö o a • 1—( d o fl

roll angle (degrees)

Fig. 13. Unstable manifold o f the fixed point near the wave

crest f o r Fn = 0.3248 and Xc = 10 degrees

is obtained by numerically integrating the state equa-tion f r o m the fixed point with a small perturbaequa-tion f o r the positive or negative direction of the eigenvector of the eigenvalue having a positive real part. I f a stable manifold were visuahzeÖ, the domain of attraction could be identified. This study, rather, attempts to derive useful knowledge f r o m the unstable manifold.

Figures 10-13 show the unstable manifolds of the fixed point near the wave crest f o r Fn = 0.3248 and

Xc = 10 degrees. The trajectory toward the downslope,

namely, 0.5 < (^g/A < 1.0, shows that the heading angle rapidly increases to starboard despite the maximum opposite rudder angle; the ship then capsizes to port. D u r i n g this behavior the ship center remains on the

wave downslope, and at the final stage the ship center situates on the downslope near the wave trough. This is a typical example of capsizing due to broaching, which was often observed in model experiments.' The trajec-tory toward the upslope, 0 < < 0.5, approaches a saddle next to the original saddle and then realizes capsizing due to broaching.

Figures 14 and 15 show the unstable manifolds f o r

Fn = 0.3247. Here the trajectory toward the downslope

tends to capsizing because of broaching, as i n the previous example. However, the trajectory toward the upslope approaches the saddle next to the original one and then tends to a periodic attractor. Thus, we can presume that between Fn = 0.3247 and Fn = 0.3248 there

O O O O O O O O T - - . . . . . ,

' ' ' ' Initial longitxidinal position

nondimensional longitudinal position

Fig. 16. Results of numerical experiments f o r Xc ~ 10 degrees. Fig. 14. Unstable m a n i f o l d of the fixed point near the wave xhe initial condition is set to be a periodic attractor f o r Fn =

crest f o r Fn = 0.3247 and = 10 degrees 0.1 and Xc = ^ degrees. Circle, periodic motion; cross, capsizing due to broaching

0 . 6 7

0 10 20 30 40 50 60

heading angle (degrees)

Fig. 15. Unstable m a n i f o l d of the fixed point near the wave

crest f o r Fn = 0.3247 and Xc = ^'^ degrees

is a nominal Froude number whose unstable manifold of a saddle tends to a different saddle. This is a "heterochnic connection." Thus, i f the ship situates be-low this heteroclinic trajectory, capsizing due to broach-ing is not likely to occur and a periodic motion is likely. That is, the nominal Froude number for the heteroclinic bifurcation is closely related to the critical condition f o r capsizing due to broaching.

To validate this outcome more directly, numerical experiments were carried out by solving Eq. (3) i n the time domain with sets of initial values. The initial values were provided based on the "sudden change concept" I proposed previously at the Workshop on Numerical and Physical Simulation of Ship Capsize i n Heavy Seas held i n 1995 at the University of Strathclyde. That is, the

initial values are set to be a steady periodic orbit f o r a set of control variables, and the control variables are then suddenly changed to specified values; transient behavior is observed. This sudden change of the control variables corresponds to actual operational practice at sea and can be easily realized in physical model experi-ments. Moreover, this sudden change concept can re-duce the number of initial variables f r o m eight state variables to two control variables and one initial longi-tudinal position, fi^g/Ajo. This means that the concept can exclude unrealistic initial conditions i n advance.

First, the initial steady state was set to be a harmonic periodic motion for Fn - 0.1 and Xc = 0 degrees; the control variable was then suddenly changed to the specified nominal Froude number and Xc = 10 degrees. Figure 16 shows the result of this numerical experiment, with the abscissa the initial longitudinal position. The capsizing boundary obtained here does not depend on the initial position and agrees with the heteroclinic bifurcation hne. For the uncoupled surge model, the heteroclinic bifurcation corresponds to critical condi-tion f o r stable surf-riding. Thus, we can say that capsiz-ing due to both broachcapsiz-ing and stable surf-ridcapsiz-ing can be explained under one umbreUa of nonlinear dynamics.

The transient behavior just above the capsizing boundary is shown in Figs. 17-20. A t the final stage, the ship center situates on the downslope, and the ship vio-lently turns to starboard despite application of maximum opposite rudder; and it then capsizes to port. A s shown in Fig. 2 1 , the trajectory obtained here at the final stage closely coincides with the unstable manifold discussed before. Whereas stable surf-riding is an attractor f o r the uncoupled surge model, capsizing acts a kind of attractor i n this case. I f the metacentric height is large enough,

0.5-,

-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5

nondimensional longitudinal position

Fig. 17. Transient behavior obtained by the numerical

experi-ment f o r Fn = 0.3248 and Xc = 10 degrees. The initial condition is set to be a periodic attractor f o r Fn = 0.1 and 2c = 0 degrees

B

.2

B

roll angle (degrees)

Fig. 19. Transient behavior obtained by the numerical

experi-ment f o r Fn = 0.3248 and Xc = 10 degrees. The initial condition is set to be a periodic attractor f o r Fn = 0 1 and Xc = ^ degrees

0.6

0.4

^ 0.5

I

I

0-2-.§

0.1-S

0--0.1

0 10 20 30 40 50 60 70

heading angle (degrees)

Fig. 18. Transient behavior obtained by the numerical

experi-ment f o r Fn = 0.3248 and Xc = 10 degrees. The initial condition is set to be a periodic attractor f o r Fn = 0.1 and Xc = ^ degrees

repeat of broaching without capsizing could be a limit cycle. I t is noteworthy that both trajectories almost tend to the fixed point near the wave crest. Although previous research"'^'''" for broaching has mainly focused on the fixed point near the wave trough, the fixed point near the wave crest governs the critical situation.

Outcomes f r o m the numerical experiments can de-pend on the initial periodic state. Thus, another experi-ment was carried out with a different initial periodic motion for Fn = 0.3 and Xc - 45 degrees, as shown i n Fig. 22. I n this case, the boundary obtained by the numerical experiments depends slightly on the initial longitudinal poshion, and the heterochnic bifurcation corresponds

90- 60- 30- -30- -60-

-90-heading angle (degrees) roll angle (degrees) rudder angle (degrees)

20 22 24 26 28 non-dimensional time

30

Fig. 20. Transient behavior obtained by the numerical

experi-ment f o r Fn = 0.3248 and Xc = 10 degrees. The initial condition is set to be a periodic attractor f o r Fn = 0.1 and = 0 degrees

to the maximum variation of boundary. The amount of this variation is negligible f r o m a practical viewpoint, although an analysis of stable manifold is expected.

The effect of the autopilot course on the capsizing boundary is shown in Fig. 23. Here the initial variables f o r numerical experiments were a periodic m o t i o n for

^ 1 . 3

0.35

-0.5 -0.4 -0.3 -0.2 -0.1 0

nondimensional longitudinal position

Fig. 21. Comparison between the numerical experiment (squares) based on the sudden change concept and the un-stable manifold (filled circles) of the fixed point near the wave crest f o r Ffi = 0.3248 and Xc = 10 degrees. The initial condition f o r the numerical experiment is set to be a periodic attractor f o r Fn = 0.1 and Xc = 0 degrees

I

^ 0

p

^ 0,

0.34

33¬

32¬

31¬

0 5 10 15 20 25 30

auto pilot course (degrees)

Fig. 23. Critical condition for capsizing due to broaching. The

initial condition for the numerical experiment (squares) is set to be a periodic attractor f o r Fn = 0.1 and Xc = 0 degrees with the initial longitudinal position, (^c/-^)o = 0.5. Fdled circles, heteroclinic bifurcation

^ 0 .

2

^ 0,

13

0.33

).328

).326

.324

,322-0.32

Heteroclinic

bifurcadon

-ë-0 -ë-0.2 -ë-0 4 -ë-0.6 -ë-0.8 1

initial londitudinal position

Fig. 22. Results of numerical experiments f o r Xc = 10 degrees.

The initial condition is set to be a periodic attractor f o r Fn = 0.3 and Xc = 45 degrees. Circles, periodic motion; cross, capsiz-ing due to broachcapsiz-ing

heteroclinic bifurcation is shghtly larger than the boundary obtained by numerical experiments. Thus, we can say that the heteroclinic bifurcation can predict the critical condition f o r capsizing due to broaching with practical accuracy.

Conclusions

This study identified the existence of heteroclinic bifur-cation f o r a four-degrees-of-freedom model of a ship

in following and quartering seas by calculating the unstable manifold of the fixed point near the wave crest. This heteroclinic bifurcation reasonably well represents the critical condition f o r capsizing due to broaching.

Acknowledgments. This research was partly supported by a Grant-in A i d f o r Scientific Research of the Ministry of Education, Science, Sports and Culture of Japan. The author acknowledges the support of its project leader. Professor Hamamoto f r o m Osaka University.

References

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