Comparative assessment of hydrodynamic models in slow motion mooring dynamics

VIPARATIVE ASSESSMENT OF HYDRODYNAML

MOORING DYNAMICS

Joâo PauIoJ.MatSuura.

Kazuo Nushimoto

Department of Nava! Architecture and Ocean Engineenng

University of Säo;.Paulo,Säò:Pàulò,Braziî

ABSTRACT

The slow motion dynamics of a turret mooring system is analyzed and compared for four of the most commonly used ship hydrodynamic maneuvering models Each of those

utilizes a different approach to model and then to calculate or

measure the hydrodynamic forces and moment acting on the

vessel The four hydrodynamic maneuvering models are studied first by a physics based analysis of each model and then by numerically comparing their prediction of equilibria nonlinear stability analysis bifurcation sequences and morphogeneses of turrèt moonng systems Catastrophe sets are constructed in two dimensional parametric design spaces to determine the qualitative behavior of the system and nonlinear time simulations are used to assess its quantitative properties Static bifuTrcations of the principal equilibhum

are compared to determine the nature of alternate equilibria A turret moored tanker is modeled with anchored catenanes including nonlinear drag External excitation is time

indepeIdent and for the numerical applications it is limited to steady current. Of the four models used the Abkowitz and Takashina models show similar qualitative dynamics The

Obokata and Short-Wing models are also qualitatively

similar but very different from the first group Limited

sensitivity analysis pinpoints the source of discrepancy between the tro schools of thought

A-M CG CG Dc ÑØMENCLATÚRE'. alternate àquilibriurn Abkowitz.Model

cencer of gravity öf' system center of gravity of turret distance between CG and CG

h depth r

L:

length cfyessei

OM.,:

Obokata MOdel

FE ..

principal equilibrium ¿

TECHNISCHE UNIVESrir

Laboratorium voor $cheepshydrornechanjca rchief Mekelweg2, 262á CD DÒIft:

VeL 016 788873 _{Faa. GIS 781830}

ODELS IN SLOW MOTION

-Michael M; Bérnitsa

:LuIS. Garza-Rios.

Department of Nava! Archrtecture

and Manne Engineering

University Of' Michigan, Ann Arbor, Michign

SPM Single:Point Moong

SW M Sort Wing Model T M Takashina Model

TMS r Turret Mooring System

-U current speed

(x y) inertial reference frame with origin at moonng terminal I

(X Y Z) body fixed reference frame with origin located at

CG

.it'apgIe rneasurdd-withrespect co the (i,y)

.' _frame ,.. ' _{headingangle'} , - .

= a - y' relative angle of flow incidence

:

i. INTRODÙC1iON';'

:

¿ The slow thouon dynamics of moored sessels is modeled

by ship hydrodynamic maneuvenng models [1 3 9 10 17

20-22 25 27] Several models are available In each of

these models the

physics of the

hydrodynamics is

approximated in

a unique way resulting

in different

measurements and calculations of the hydrodynamic forces and moment that act on the vessel as it moves relative to

the water The sensitivity of each model to small variations

of its hydrodynamic coefficients and especially to hydrodynamic damping further limit model robustness To

assess qualitatively a maneuvering model in mooring it is important to understand the approximation of its physics

and global dynamic properties of its corresponding mooring system The physics of each model is discerned by studying

the way in which

it approximates the

basic laws of

hydrodynamics and its

range of applicability

The

effectiveness of each model to predict

the nonlinear

dynamics of mooring systems can be assessed by employing

a systematic approach in locating its equilibria identifying

bifurcations of equilibria, and describing morphogeneses as

the behavior of a specific mooring system qualitatively without resorting to trial and error and extensive nonlinear

time simulations.

_{These are used}

to determine the quantitative behavior of the system. i.e. amplitudes of motions, tension in a mooring line, etc.

In this paper, four basic hydrodynamic maneuvering models have been selected as representative of a variety of physics-based models and numerical adaptation methods available today. These are the Abkowitz (A-M) [l-31, Takashina (T-M) [26. 27], Obokata (O-M) (20-22], and Short-Wing (SW-M) [17. 25] models. All four model the slow motion hydrodynamic forces and moment exerted on the hull in the horizontal plane (surge, sway and yaw). A four line Turret Mooring System (TMS) is used to compare

the models based on their

predictions of equilibria, bifurcation sequences and morphogeneses. Other

hydrodynamic maneuvering models (9] can be assessed as

well using the methodology developed in this paper.

The main goal of this paper is to establish a methodology

for assessing maneuvering models and their ability to predict with satisfactory accuracy the nonlinear dynamics of

mooring systems. Comparisons of modeI can be achieved when the numerical applications for comparison use the

same vessel. Unfortunately, data for a single vessel are not

available in the literature for all four maneuvering models. Thus, all numerical comparisons in this work are based on tankers of approximately the same dimensions and with

similar maneuvering characteristics. lt is important to point

out, however, that even if coefficients for the same vessel were available for ali four maneuvering models, scaling problems during model tests would yield different

simulations and predictions by the models [16]. Thus, different predictions are expected to be produced by the four

models not only because each model approximates physics differently, but also because numerical applications do not

use data for the same vessel.

The steps in the approach developed in this paper for

analysis and comparison of the four maneuvering models are: The physics of each of the four models is assessed, and

its parametric range of applicability is surmised.

The numerical representation of each model is assessed

to identify additional limitations.

A design methodology based on nonlinear dynamics and bifurcation theory [4-6. 11. 23) is used to evaluate each

model based on

its ability to predict global system

particulars such as equilibria and bifurc.ton sequences.

Limited sensitivity analysis is used to identify discrepancies between models, and to trace changes in global

system particulars to differences in the physics and the

numerical implementation of the models.

Section lI provides a description of the four hydrodynamic models considered along with a physics-based assessment of relative advantages and limitations. The mathematical model of the mooring system used to compare the four maneuvering models is presented in Section 111. Section IV introduces the

concepts of equilibrium, stability analysis, and bifurcations of equilibria of mooring systems. Numerical applications

producing catastrophe sets and simulations are used in

Section V to compare the

four hydrodynamic models.

Finally. conclusions are drawn regarding the relative merits of each model and suggestions for further comparisons are

provided.

I I . DESCRIPTION OF MYDRODYNAMIC MODELS Each of the four hydrodynamic models compared in this

paper has its own advantages and limitations. The first step in assessing a particular model is to understand how it approximates the physics of hydrodynamic force modeling.

The reader is referred to original papers for a full rendition of each hydrodynamic model. Following, a brief description of

each model is provided presenting some of its capabilities and limitations:

The Abkowitz Model (A-M) assumes

that the

maneuvering characteristics of ships can be described

properly by multi-variable Taylor expansions about the steady state condition of constant forward speed. Tests of deviation from that condition provide the slow motion derivatives [l-3]. In model tests, the hvdrodynamic forces and moment are measured while in

sea trials system identification techniques are used to calculate slow motion derivatives. Usually, terms to

third-order are retained. In addition, A-M assumes that

coupling between velocity and acceleration terms s negligible: thus, only first order accelerations are considered [I]. This model provides a very good

estimation of the hydrodynamic forces and moment on a

vessel in small deviations from a steady forward slow

motion. In mooring applications, A-M is not so accurate for large incident angles of relative flow

9 = a -

These may be caused by large amplitude

yaw or large 9.

A-M for mooring systems can be

improved by model tests and/or sea trials performed for

larger drift angles.

The Takashina

Model (T-M) approximates the

hydrodynamic forces and moment using a Fourier series

with respect to 9 [26, 27]. Kinematic conditions are

used to relate the vessel translational velocities to 9,

thus expressing the hydrodynamic forces and moment in

terms of the vessel velocity vector relative to current. Consequently, this approach makes the velocity components of the vessel dependent variables. Computation of the series coefficients relies on model tests. These consist of fixed O (captive) and free running tests. The captive tests consist of static drift and yaw rotating tests, thus adding the capability of describing the forces and moments on the vessel for larger e [26]. Due to the nature of the experiments, however, T-M does not cake into account the coupled

added mass components in sway and yaw, thus resulting

in a small loss

in the inertial forces and moment components.

(C) The Obokata Model (O-M) is based on the local cross

flow principle [20] which is valid for large e. O-M

incorporates the inertia terms, static damping, and yaw

damping of the hydrodynamic forces and moment. The static damping is modeled after the traditional velocity squared drag forces and moment acting on the vessel,

with measured current force and moment coefficients for

different 9. The yaw damping incorporates the viscous damping forces due to the yaw velocity of the vessel. Both damping components are calculated by a strip theory approach using the sectional speed and 9 [21.

22]. The approximation involved in the O-M approach

the theoretically based inertia term and the seMi

empirical velocity squared drag term are added. The

hydrodynamic coefficients which are measured

experimentally in general do not account properly for

all nonlinearities related to damping and viscosity.

(d) The Short-Wing (SW-M) utilizes a similar approach as O-M. but instead of integrating the static and yaw damping drag components locally in each strip, SW-M

separates them at the global ship level. The static component makes use of a heuristic formulation that

blends the ship hydrodynamics in a cross flow and the

ship hydrodynamics for small drift angles (short-wing

theory), thus resulting in a relatively simple model that

describes the generalized current forces acting on a

vessel as a function of 9, vessel dimensions, and other vessel properties [17]. In addition, SW-M incorporates a linear correction for the presence of the rudder based

on the work by Clarke et al. (7]. Model tests are performed with the ship fixed at a given O to obtain

the coefficients of forces and moment needed.

Additional model tests consist of towing a model which is restricted to one degree of freedom (rotation), which

serve to validate the static component of forces and

moment. A heuristically determined transition function makes sure that the model fits the correct behavior in a

continuous range of 9, and thus has the advantage of being able to preserve the dependence of forces and moment on vessel dimensions. _{The yaw damping} component consists of forces and moment due to the rotation of the ship, and is modeled starting from the cross flow principle [25]. The two components are superposed. The original purpose of this model was to

predict accurately static bifurcation phenomena [17] which it achieves satisfactorily, and it is still under

development. _{The physical limitations of SW-M ¡re:} First, the hydrodynamic forces and moment are measured

for fixed 9, as is the case with O-M. Second, all damping components are computed using methods with different physics bases which are superposed expecting that experimentally measured coefficients account for coupling nonlinearities.

All four models depend on curve fitting of the measured

forces. _{Both A-M and T-M rely on extensive tests to ensure}

that the curve fitting coefficients accurately represent the

hull hydrodynamics. Both methods present experimental

challenges, and are limited by model test precision.

Specifically, higher order hydrodynamic derivatives _may have a strong influence on the prediction of the system

dynamics. _{Yet, different measurements may result even in} sign discrepancies [161.

In SW-M and O-M, testing consists of a limited number of experiments that measure force coefficients with respect to

9. SW-M requires only a small number ofmeasurements to

construct the required expressions heuristically. O-M

requires a larger range of measurements, and is limited in the sense that it employs only a number of force measurements

for the whole range of incident angles, requiring

interpolation to obtain a complete range of force

coefficients. The fact that the force and moment coefficients

are easily measured experimentally in O-M and SW-M, makes

both models simple to validate [17, 20]. Both models.

however, need corrections to account for the rotational

forcesand mòment on the system since the coefficients are measured with the ship fixed.

The four aforementioned models are frequently used in

design of mooring systems (4-6, 8, I l-13, 19, 23, 25). To

make comparisons simple and effective we need co select a simple mooring system. Single Point Mooring (SPM) and Turret Mooring Systems ('l'MS) can be used for that purpose. In this paper, a TMS selected. Its mathematical model is

outlined in the next chapter. Most importantly, the four models cannot possibly be compared using simulations. Any such comparison would be of limited scope. Our

approach for comparison uses a design methodology for mooring systems developed at the University of Michigan since 1983 [4-6, Il, 23]. Design decisions are based on

global properties of a mooring system such as equilibria.

bifurcation sequences of equilibria, and morphogeneses as bifurcations are crossed. Such an approach eliminates the need for trial and error in parameter selection and virtually eliminates the need for nonlinear simulations. Accordingly.

the method can be used to assess the four maneuvering

models described above without resorting to simulations or

exhaustive searches in the design space.

Ill. MATHEMATICAL MODEL

The slow motion dynamics of TMS is formulated by the

three equations of slow motion of the system of vessel and

turret in the horizontal plane (surge, sway and yaw), a

rotational equation for the turret, four kinematic equations, the mooring line model, and the external excitation model.

The geometry of a TMS is shown in Figure 1 with two principal reference frames: (x.y) = inertial reference frame

with its origin located on the sea bed at mooring terminal I:

(X,Y,Z) = body fixed reference frame with its origin

located at the center of gravity of the system CG, i.e. of the

vessel and turret combined. A third reference frame, (X',Y',Z') has its origin at the center of gravity of the turret (CGt), with its Z'-axis parallel to the Z-axis on the body. In addition, n is the number of mooring lines; (x,y)are

the horizontal plane coordinates of the ith mooring terminal

with respect to the (x,y) frame;

!'

is the horizontal projection of the iM mooring line; i(s) is the horizontal angle between the _{x-axis and the iM mooring line,}

measured counterclockwise;

'

is the heading angle; and V1T

is the absolute yaw angle of the turret. The direction of

current excitation (a) is measured with respect to the (x.y) frame as shown in Figure 1: and is the distance between

CG and CGT. A detailed derivation of the TMS

mathematical model is provided by Garza-Rios and Bernitsas [12].

111.1.

Equations of Motion

The horizontal plane, nonlinear, slow motion equations of TMS are modeled by three equations in surge, sway and yaw for the system and one turret rotational equation as follows [12]:

+ {[T,')

-

_{+ cN'}

sinß}

, (1) 1=1 (m+m)v+mr+mur= Yff(u,v,r) +

_{Í{[T'_,]sinßffl_i')cOsß('J}}

(!

+I)r+rnv= NH(u,v,r)

n +

F]sinß°

_cosß'}+NT

. (3) ¿=1 n

ITrT= ía {[r,,')

_-

y(i))}

1=I 'I

-

.2Lf (i){F'(OCOS(ß(i) _i))}

+ N (4)

2

i= I

In equations of motion (l)-(4). rn is the mass of the vessel; ¡ is the moment of inertia of the system about the Z-axis;

rn11, m rn26, m and I are added mass and added

moment; u and y are the relative accelerations of the system in surge and sway with respect to water; r is the relative angular acceleration in yaw; u, y and r are the relative velocity components of the system in surge, sway

and yaw, respectively. In addition, XH, H and NH are the

hydrodynamic forces and moment. These are functions of

the relative velocity vector with respect to water in A-M and

T-M. while O-M and SW-M model these as functions of 8;

NT is the moment transmitted from the turret to the vessel.

All mooring lines are chains with no supporting buoys.

Omitting the index i=1...n to simplify the notation,

for every mooring line, TN is the horizontal tension

component of the catenary; F and FN are the mooring line drag components in the directions perpendicular and normal to the mooring line motion, respectively [14];

ß is the

horizontal angle between the X -axis and the corresponding mooring line,

measured counterclockwise (ß=yti).

Notice that the acceleration terms in sway and yaw in

equations of motion (2) and (3) are coupled, except for T-M. where terms rn,6 and rn6, are not present.

In the turret rotational equation (4), 'T is the turret moment of inertia about, the Z'-axis; r- is the relative angular acceleration of the turret with respect to water; D is the

turret diameter. The terms inside the summation in equation (4) are defined as follows: fa is the fraction of the turret diameter at which the mooring line is attached; F and F, are the drag forces on the ith cacenary in the directions parallel and perpendicular' to the mooring line motion, respectively [II, 14]; ß' is the horizontal angle between

the X'-a.xis and the mooring line measured counterclockwise

(ß'=yy'): y0 is the angle at which the ith catenary is

attached to the turret with respect to the X'-axis (fixed to the turret). In addition, N, is the moment transmitted to the turret by the vessel. The terms N and N are of equal magnitude [12].

(2)

Figure 1. General geometry of TMS

In equations (1 )-(4), it is assumed that the turret is internal to the system and therefore no external hydrodynamic forces

and moments act on it.

111.2.

Mooring Line Mode(

The mooring lines in the TMS are chains modeled by

catenaries anchored to the sea floor and subjected to

nonlinear drag [14]. The horizontal (TN) and vertical (Tp)

tension components are given by

1sinh(

=..Jh(h +

2i.)

P (5)

P h#ih+2!k') Tv=THsinh(j_J=

P)

(6)

where P is the weight of the catenary per unit length (1510 kg/s2); L is the horizontal projection of the suspended part of the catenary; and h is the water depth. The total tension

T in each catenary is given by the vector sum of its

horizontal and vertical components as follows [14]:

T=,ITH2+TI,2 =THcosh(-) (7)

with ari Average Breaking Strength (ABS) for each line of

5159 kN [18].

1113. External Excitation

In the design methodology developed in [4-6, 11, 23],

external excitation includes time independent current, wind,

and mean drift: and memory effect. Without loss of generality for the purposes of this paper, external excitation is limited to current. lt is taken into account in all hydrodynamic maneuvering models by introducing the relative velocities of the system with respect to water in equations (l)-(3). The current speed U and the current direction a are introduced in the kinematics of the system

JI

(8)

wheref ar

che mass/moment of inertia terms including

yusInf+vcosyf+Usina

(9) added mass/moment of inercia H are the hydrodynamic (IO) forces and moment T are the horizontal tension terms and (1I) F,» arc the mooring line drag components Noce that

equations (15) and '(16) result from the combination of where rT is the turret rotationai velocity in equation (II) equations (2) and (3) where the aceleration terms are

coupled due to the added mass components m26 asid m62 for

r all hydrodynami, odels

xceotTM

-

Eauaion(l7).

corresponds to the turret rotational equation (4) The above

four equations are1 combined then with kinematic relations (8)-(1 1) to form a suitable eight dimensional state space

representation of the system

Equilibna of the nonlinear TMS model are found by setting the time denvativ'es of the state vector (12) equal to

zéo,.; i.e.

i.=ucosy-vsini+Ucosa

Vr = TT

.1V. EQUILIBRIA, MÓÁPÑO'GENESESAND., r

BIFURCATIONS OF EQUILIBRIA», - . ',

According to the design methodology for mooring systems developed in (4-6 11 23] global dynamic properties of TMS mïist be obtained to assess a moonng system These global properties include equilibria bifurcation sequences and morphogeneses These are analyzed and compared for all four models by obtaining a suitable state space representation for the system finding its equilibrium properties

and thn applying eigenvalue

analysis at each equilibrium to determine qualitatively its dynamic behavior The eigenvalues reveal whether a particular equilibrium is stable nstable oscillatory

chaotic etc [15 28] Qualitative local information abput all equilibria of a particular system determines qualitatively. the global dynamical behavior of the system Nonlinear

time simulations can be used to determine quantitative properties of the dynamics of the_system such as amplitudes

of motions

mooring line tensions etc Bifurcation

sequences with respect to one or several TMS parameters can

be found analytically _Elli or numerically [13] Further

sensitivity of bifurcations with respect to each design

parameter such as number and pretension of mooring lines fairlead position direction of external excitation on the system dynamics can be performed The required steps re

dèscribed below.

-IV.. State:Ípace ÁepresentatióhandTMS

Equilibria

-:., ,,, L'

The TMS equatlóns of mótion (l)-(4) ani its kinematic

relations (8) (II) are first recast as a set of first order

flonlineâr. cóúpl'ed differential equations of thefoñn '

-

..ï=f. fEC1.

- ':-- (12)

r:

-' where; 9 .i 'the. eightdimensiònai Euclidean space arid çi is the class of continuously differentiable functions The

state space of the system given in terms of equation (12) is

composed of the'follow ng eight variäbles:- '.

V, T',' T, X. Y _{V. P'T]T .} ' (13).:.

By- ,seiecbng state space (13):"equations ol

tió(1)(4j'

càn be wriùèn in the

iJ 6'''l 2)

rai-_-

.:

---;.

' each and everyYeqiiilibriuhi pbsition of,a particular system. --'

-

cnfitation tp-drmin- i

-g1ob,al.behior. ' .

Aiui

f(i)

where the overbar on state space x denotes the equilibnum

solution Depending on the configuration of the system -- TMS may possess:,'either :one, or -three equilibria: '-'The,

-principal equilibrium of the system corresponds to its

weathervaning -posi'cion and .is dendted', by' PE. Two' r,': --additional é4uili,briá, may -exist- as : result -pf a stätic

bifurcation of the system (1 lJ These alternate equilibria are

;denotedbyAE..

-: ..

-H,''

'--: ;-. .,

-lV-.2.

Stability' Analysis

-The stability of a specific equilibrium position can be assessed by performing eigenvalue analysis at that equilibrium Local analysis near an equilibrium solution reveals the behavior of trajectories in its vicinity and can be performed by stúdying the 'linear sysièm -

-=

'1AjExs:.'

where 7p:)=x(:)i is the deviation from equilibrium and [A] is the Jacobian matrix of f evaluatid at that equilibrium

- condition. If älÏ.ei'glü ei ñ'àluei-of [A] have iíegative real

parts equilibrium position i is asymptotically stable and all trajecLoris iñitiated'ñear:i. wiií:àonveige.' toward ita- in-forward time If at least one eigenvalue of [A] has a positive real part i is instable and all trajeccones ihitiated

near such an equilibrium will déviate from it in forward time [28] Depending on thè nature of che positive real parts of the eigenvalues an equilibrium position may be an unstable node an unstable focus a stable limit cycle or chaotic in nature [28] Eigenvalue analysis at each equilibrium position yields the characteristic global behavior of the

system Local stability analysis must be performed near

been established, biftírcatio'n.LsequenàesL of jhe system 'as a -,functiòñ of one or' séveral' design: parameters. are calculated to.

find qualitative than'es i

its 'd'namic' behaviór'. -: To'

-

--

. " '

- , (14)

- ' '(15)

1zh1z'+'TiHFoz +NT . '

-'

- J6)

perform bifurcation analysis, evolution equations (12) are

rect

in the form

i=f(x,p),

XE918, , (20)

where p is the design parameter vector, and N is the

number of parameters in the system. The behavior of the

solutions to the dynamical system described by equation (20)

in the Ne-dimensional space is illustrated graphically by stability or design charts. The set of lines shown in a

stability chart is called a catastrophe set, which constitutes

the boundaries between regions of qualitatively different dynamics [5, 23). These boundaries, also known as bifurcation or stability boundaries, are the result of loss of stability in the system.

V. NUMERICAL COMPARISONS OF THE HYDRODYNAMIC MODELS

To quantify the advantages and limitations of the four

maneuvering models, their ability to predict the dynamics of

TMS is studied. Prediction of equilibria, stability, and

bifurcation sequences is used. For comparison, the design parameters of the TMS are the same for all four maneuvering

models and are summarized in Table 1. As pointed out

previously, however, the hydrodynamic models are compared

based on existing models of tankers of approximately the

same dimensions and with similar maneuvering

characteristics; so, they are not identical.

The qualitative behavior of the TMS as a function of one

or several design parameters is described by a catastrophe

set, as pointed in the previous section. In this section,

bifurcation sequences are derived in the design space defined

by: the distance from the center of gravity of the system to the center of gravity of the turret DCG (nondimensionalized

by the length of the ship L), and the direction of the

current a. Thus, two design variables are used in the

numerical applications(N = 2), and the design parameter vector is

p=[D, a]T.

Figure 2 shows a series of

catastrophe sets in the

(D/L,

a) plane about the

principal equilibrium PE of the system for each of the four

maneuvering models in the following ranges:

O.0

Dc6/L

0.5,

0' a 180'

In Figure 2, four regions of qualitatively similar dynamics exisL These are described as follows:

Region I (R-I): PE is stable. All trajectories starting near

PE equilibrium converge toward it in forward time. Since

there are no other equilibria that may attract or repel the trajectories, the system eventually converges to the PE

irrespective of the initial conditions of the system, resulting in a stable principal equilibrium.

Region II (R-II): PE is unstable with a one-dimensional

unstable manifold. Static loss of stability occurs when crossing from R-I to R-II. Two additional statically stable

equilibria appear in R-II. The trajectories in this region

diverge from PE and are attracted to one of such alternate

equilibria (AE), depending on the initial conditions. Such equilibria, while statically stable, may be dynamically stable

or unstable, resulting in trajectories converging to either

equilibrium or to a limit cycle around it, respectively.

Region Ill (R-III): PE is unstable with a two-dimensional unstable manifold. Dynamic loss of stability, also called

Hopf bifurcation - occurs when crossing from R-1 to R-III. A

limit cycle around PE develops. In this region, no other

equilibrium that may attract the trajectories exists, and thus

the system oscillates around PE. The frequency and

amplitude of oscillations vary with system parameters and

can be found by numerical simulations or approximated

using the Center Manifold Theorem [15].

Region IV (R-1V): PE is unstable with a three-dimensional unstable manifold. In this region, PE is statically and dynamically unstable: i.e. the system has undergone both

static and dynamic loss of stability by crossing both

bifurcation boundaries. The behavior of the system around

PE is chaotic in nature [Il]. It is shown numerically,

however, that the system does not fall into chaos due to the existence of the two alternate statically stable equilibria that result from the static bifurcation which occurs when crossing

from R-III to R-IV. Notice that the boundary between these

regions is approximately the same for O-M and SW-M,

which is not clear in Figure 2.

Figure 2. Catastrophe set about the principal equilibrium; all hydrodynamic models

In the catastrophe sets of Figure 2, A-M and T-M have only regions R-I. R-Il and R-III. O-M and SW-M exhibit regions R-II, R-III and R-IV. To facilitate discussion and

demonstrate the methodology, only the bifurcations of the system around PE are considered in Figure 2. From the

Im . A.M . i

- TM

\

040 sw41 \ '. *40 III T.N IO A.MI T.MI - I : O.M III

/

*4011 SW.M UI

O.M II / _{. T.M II}

/

A.M III

::

SW.M II t 040 IV 5W.M iv \.., 1 i \. TI( III 040W SW.MIII I ¡ A.M III I i

/

... I I O0L 0 0.05 0 I OIt 02 0.23 0.3 0'! 14 045 0.!

Table I. Design parameters of the system

Mooring Line Properties Environmental Conditions

number, n =4

length, wT='95° m

prension = 1327.8 kN/line water depth, h =700 m mooring line spacing =90' current ve). U= 1.8017 mIs

orientation of line 1, y=0'

catastrophe sets in Figure 2, the following observations can

be drawn:

Even though the selected vessel characteristics are as

similar as possible for each model, it is not possible to obtain all hydrodynamic coefficients for all models due to

the nature and limitations of each of them. Thus, the results in Figure 2 are expected to be different for each model. The

bifurcation boundary resulting from static loss of stability

(straight line in all cases) should be approximately the same

for all models. This is indeed the case for T-M, O-M and SW-M. The difference for A-M is due to incomplete data of

the slow motion nonlinear derivatives.

'The catastrophe sets of TMS around PE in Figure 2, show that two groups of models predict dynamic loss of stability differently. The first group is A-M and T-M, and the second group consists of O-M and SW-M. This is expected since the physics behind A-M and T-M is similar and very different from the O-M and SW-M which have similar physics as well. Note that according to Figure 2, neither O-M nor SW-M has a stable PE, and therefore the

system either results in a limit cycle around PE (R-III), or its trajectories are attracted by an alternate equilibrium AE (R-II.

R-IV). Notice also that the relative angle between the

current and the vessel heading in Figure 2 is the same for PE independent of the current angle. The difference in nature of

the catastrophe sets with variations of the current angle a is a consequence of changes in the incident angle of the water with respect to the catenaries. These variations

change the drag in the mooring lines. As the drag in the mooring lines increases, A-M and T-M models become

dynamically stable.

To determine the behavior of the system quantitatively about PE. nonlinear time simulations are performed. In

Figures 3-6, selected simulations are shown. These are arranged in two groups: Figures 3 and 5 for A-M and TM; and Figures 4 and 6 for O-M and SW-M. Figures 3 and 4 correspond to nonlinear time simulations for geometry

Dc/LO.45, a=180;

while Figures 5 and 6 show

nonlinear time simulations for geometry Dc/L=0.45,

a

=l35.

Note that the simulations use different scales.

Figures 3-6 are in accord with the results obtained in Figure

2, i.e. stable PE for A-M and T-M for a =l35 and

oscillatory about PE otherwise.

Figures 3 and 4 depict nonlinear time simulations of the TMS for pair

(D/L

=0.45, a =180). The limit cycles

corresponding to A-M and T-M are shown in Figure 3, while

those of O-M and SW-M are shown in Figure 4. The

amplitudes of the limit cycles for A-M and T-M in Figure 3 are much larger than those for O-M and SW-M shown in Figure 4 . These simulations show that, although the

dynamic behavior of all systems is oscillatory in nature.

major differences exist between the two groups in amplitudes

of oscillations. The frequencies of oscillations, however.

are practically equal except for O-M.

In Figures 5 and 6, nonlinear time simulations of the ThiS

for pair

_(D/L

=0.45,

a

=135') are shown. These

represent a change in current direction from 180' to l35 with respect to the TMS simulated in Figures 3 and 4. As

shown in Figure 5. both A-M and T-M converge to a

principal stable equilibrium 7 = 45' with amplitudes on

the same order of magnitude and frequencies within a I 0Çc

difference. In Figure 6. O-M and SW-M converge to their

corresponding stable limit cycles around PE with significant differences in both amplitude and frequency of oscillations.

In Figure 6. the amplitude of oscillation for SW-M is

substantially higher than that in Figure 4, while O-M shows

approximately the same amplitude of oscillation. These

results show that a larger projected area of the catenaries to

the inflow results in higher drag and higher oscillations.

One would expect that the increased damping would reduce the oscillations. This is not the case.

Figure 3. Drift angle simulation: A-M and T-M;

Dcc/L =0.45, a =180

Figure 4. Drift angle simulation: O-M and SW-M;

DCG/L=0.45, a = 180'

The simulations in Figures 3-6 show that the quantitative

behavior of the TMS is predicted differently by the two

groups of models: The first group consisting of A-M and

T-M, shows relatively large amplitude oscillations (Figure 3) with respect to the other two models (Figure 4) which

comprise the second group. The large amplitude oscillations for A-M and T-M in Figure 3 show that, while the TMS falls

in a stable limit cycle, the system configuration may be

unsafe. Additional dynamic analyses are required to verify

:

:-r

c4-4-t E w.

::

1'T!

i I I Ir

ILl

CI:. w

'theviability of the system. In Figures 4 and 6. the

qualitative behaviór 'of O-M and SW-M is oscillatory, about

PE and thus the system is unstabIe even though the motions

o the vessel are relatively small: Table 2 shows'a summiry ofthe amplitudes (Arhp) and non-dimensional -perio pfthe

limit cycles (Tm) of the simulations in Figures 3 4 and 6

iFigure 6.. Driftâñglesirnúlatioñ: O-M andSW-M;.

=

Dcc/L =O.45 Û

Another basis forj comparison between models is the

nature of the equilibria that result due to static bifurcations

equilibria sequences for each of the models as the system

loses its static stability These can be analyzed by constructing che path of' thé ':static bifúrcatioii for each hydrodynamic model.' ',,:. .

To demonstrate thern pattern of the szatic. bifurcation of PE, a single value of the cúrieñt direction is. selected.' Figure 7

shows che nâture of the scìtic.bifurcaion equilibrium for each of the four hydrodynamic models in terthsof the drift ángle

'-for

â =180',' and for,., ,the range

O

l75Dc/L 0425

In chis range the TMS modeled by

'Figuri5."Dnft àgle.sithúlati'on: k-M and T-M;. Dc/L=O.45,

al35

any of the .fourmodels 'does notassume a geoni'etric:

coñfiguration that rnây rndá mooring line slack..,

Table 2. Oscillation' amplitudes..(A'mp) and noñdimensional

periods (Tx): s bic limit cycles;'

D/L

0.45:

Model":

.,'a=i80

",

a=l35

The' bifurcations shown iñ Figure '7 'about IE 'brañch into two paths with the development of two statically stable AE which are milTor images of each other due io the symmetric nature of the system [4 11] The da_shed line in Figure 7

about the value =0 denotes that PE is statically unstable

to the left of the bifürcatiön. pôint. 'while the. soÌid'.lÏñe to -' the right detiotes static stabiliti' of PE. As' stówn in Figure

7, the. AE curves (secondary: paths. of the bifu'rca,tión) are

similar for AM and T M in

the sense that near the bifurcation point they branch off proportionally Z the

square root of the drift angle ¡ji This is a gneric

supercritical pitch'fork bifurcation The secondary paths for models O M and SW M are very similar and iñcrease linearly

with ¡ji This can still be labeled as supercritical pitchfork

but,it is'due to ('aJiji..4-'b),rather'than (0 ¡j72+b) which is

the generic form. Stich results shoîi again differencesin the nture of static,bifurcations bew,ee,i,the two groups.

Figùre 7. Static bifurcatidns and alierate q'Ùilibtiä;

a = 180

For larger drift angles the curves in figure 7 are almost

identica] for O M and SWM and diverge slightly with

decreaing values of The AE values for T,,M differ for large values of D but approach the equilibnum values of

O M and SW M as ¡ji increases As expected the A M

secondary bifurcation path differs considerably froth che rest...

Amp (deg) T Amp (deg)

AM 3631 331

-TM 2652 337

O-M 023 243 022

3l6

of the models. Such large differences of bifurcations of equilibria can be attributed to the fact that A-M is valid for small relative drift angles.

For the alternate equilibria (AE) in Figure 7, stability

analysis was performed as well.

For the Dcc/L range

selected for the analysis of the system, both AEs are stable

forT-M, O-M and SW-M. The bifurcation path of A-M in Figure 7, however, shows dynamic instability of the system

about AE for the range O.297 Dcc/L O.3l4. with the

development of a two-dimensional unstable manifold about AE. This range is highlighted in the figure with crosses, and indicates that the TMS asymptotically oscillates about either AE, depending on the initial conditions of the system.

The other hydrodynamic models exhibit dynamic bifurcations of AE as well for different current directions.

RELATIVE ASSESSMENT OF MODELS

'The equations of the horizontal plane slow motions of a floating vessel constitute the most important part of the modeling of a mooring system. Numerous models have been

developed in the past thirty years to predict the hull

hydrodynamic forces in slow maneuvering [1-3, 9, 17, 20-22. 26. 27]. Two different major approaches to

approximating the physics

of the problem have been

developed. The first approach is represented here by the Abkowitz and the Takashina models; the second by the Obokaca and the Short-Wing models. In both schools of thought, the goal is to develop a model valid for a wide

range of forward velocities so that model tests can be

avoided for each combination of prescribed maneuvers and forward velocity. In both approaches, a model

approximating the physics is developed which includes several hydrodynamic coefficients which are intended to account for the physics shortcomings of each model. Other models, such as the shallow water maneuvering model of Fujino and Ishiguro [101, are a combination of the two schools of thought.

The first school of thought has a more sophisticated

modeling approach and requires many more model tests. A-M requires measurement of many slow motion derivatives but

does not make any compromises on modeling. As a result, it requires a lot

of experimental or

full scale trial

measurements and data processing. One would expect that coefficients for large O and of high order would be required

for accurate modeling. In practice, however, data are limited

to third order slow motion derivatives and for relatively moderate relative angles of incident flow, O. Such process

requires, however, about one month of model tests.

Extending it

further to capture large

O hydrodynamic behavior has noi been done in towing tests. So. even though A-M appears to have a superior physical basis, its applicability is limited to moderate O for lack of data at the present time. Fortunately, it has been established that the slow motion derivatives are constant for speeds up to IO knots and can be used with satisfactoty accuracy for up to 16 knots [24]. The second model within the first school of thought. the Takashina model, has two advantages: first, it captures the large incident angle behavior; and second, it requires fewer model tests. However, it compromises the nature of dependence of forces and moment on terms with

products of velocity components and products of acceleration components. Specifically, the dependence is modeled via

the kinematic relations which include the dependence on 8. Thus, for a test at a given O ¿he kinematic relations provide

a constraint on the independent variables u and y

The second school of thought is represented by the

Obokata and the Short-Wing models. The intent here is to model as much of the physics as possible theoretically so that fewer tests can be used. The goal of the tests then is to

use experimen;aJ data to

calibrate coupling between

empirical formulas and theoretical equations. Indeed, in that

respect, these models are successful. They are based. however, on the empirical velocity squared terms and modeling approximately damping terms depending on

products of velocity components.

Based on the above observations on the physics of both

schools of though and the four models, we can provide some

guidelines in assessing the validity of predictions made by

those models. At this point we need to discuss our ability to assess a mooring system. The traditional approach of trial and error in selecting values of design parameters to define a configuration along with numerous nonlinear simulations to assess a given configuration cannot provide anything more than a small picture of the richness of the nonlinear dynamics of a given configuration. lt is practically impossible to produce something of global value

for a single model let alone for comparison of all four models. To assess a mooring system we need the ability to

produce global properties of it, that is. quantities independent of time. That is

the basis of the design

methodology for mooring systems developed at the University of Michigan. It produces equilibria, bifurcations

of equilibria, and defines morphogeneses occurring as

bifurcation boundaries are crossed. On the basis of those we

can assess a mooring system. Then, we can compare the four models in the two schools of though in modeling slow

motion hydrodynaznic forces and moment.

The ensuing discussion is based on Figures 2 and 7. In general, they show that models within each school of

thought predict similar qualitative behavior. Also, the onset

of static bifurcation predictions is nearly the same between

the two schools. There are two major differences between

the two schools. The O-WSW-M school models well the inertia terms and, approximates quadratically the damping

terms in the hydrodynamic forces and moment [17, 20-22. 25]. Accordingly, the prediction of the onset of the static bifurcation in Figure 2 is satisfactory, while that of the

dynamic loss of stability (Hopf bifurcation) is not. The yaw

damping terms in these models are nonlinear functions of the rotational velocity which do not affect the catastrophe

set of Figure 2. The lack of rotational measurements in the

development of O-M and SW-M thus omits rotational

damping of first order. The difference observed in the Hopf

bifurcation in Figure 2 is due to these first order rotational terms in the hydrodynamic lateral force and moment. To prove such statement, the linear rotational slow motion

derivatives in A-M (Yr and Nr) and T-M (Ne) are set to zero

and catastrophe sets are developed and plotted in Figure 8. The discrepancy in the Hopf bifurcation in Figure 2 is

virtually eliminated in Figure 8.

The modeling of the static damping terms in the second school of thought also causes the discrepancies in Figure 7

in the nature of the initial slope of the secondary equilibrium path following the onset of the static bifurcation. The static

bifurcation results

for O-M and SW-M show that the

secondary equilibrium paths increase linearly with the bifurcation parameter Dc/L in the vicinity of the bifurcation point. This is due to the fact that the dominant term in the restoring moment of the system is proportional to i,7 For A-M and T-M. the dominant term for the

hydrodynamic restoring moment at equilibrium is proportional in which case the alternate equilibrium angles increase with the square root of the bifurcation

parameter in the vicinity of the bifurcation point.

Figure 8. Effect of rotational derivatives on catastrophe sets

Within the first school of thought, there are some quantitative - not qualitative - differences in Figures 2 and 7 between A-M and T-M. First, the static bifurcation for A-M is located at D0/L =0.4015 instead of 0.3604. This can be attributed to the fact that the slow motion derivatives for the tanker used in A-M have been obtained by system identification techniques of full scale trials. This method concatenates terms and uses a simpler model rather than the complete model [3). The second difference is shown in Figure 2 where the dynamic loss of stability occurs earlier for the T-M. Some of it can also be attributed to the use of a simplified model in obtaining the data for the A-M. Part of it though is due to the more accurate modeling of the A-M

for the terms with velocity component products. The third

difference appears at large angles of relative incident flow in Figure 7 where T-M is superior.

On the basis of the analysis of this paper - which is

indeed limited considering the difficulty of the task at hand

-we can draw, with some skepticism and need for further

analysis. the following conclusions:

Maneuvering models can be classified in two schools of thought. The first which models the physics based on a single model and the second which separates the static damping from the yaw damping. The first school of thought is represented by A-M and T-M. The second school of

thought is represented by O-M and SW-M.

The mathematical modeling of the

first school of

thought does not separate damping into static and yaw terms. Thus, all forces are included implicitly in the

formulation.

The second school of thought requires far fewer model

Both schools of thought - all four models - can provide

a good estimate of static loss of stability and the onset of

static bifurcations.

(y) The second school of thought can provide a quick and

accurate prediction of static loss of stability.

The first school of thought predicts better the

secondary equilibrium path following the onset of static bifurcations because force and moment approximation is

higher than quadratic in velocities.

For small incident angles of relative flow, 8. A-M predicts the system behavior more accurately as it captures better

the damping terms

of products of velocity

components.

For large angles, T-M is superior to A-M.

A-M can be improved by conducting experiments at larger angles of incident relative flow, 8. This is expected

to produce nontrivial values for higher order terms, probably

up to fifth order. Until such data becomes available its

applicability is limited to small angles.

T-M modeling of the velocity product terms via the

kinematic relations introduces dependence between velocity components which are actually independent.

As a final observation it is worth noting that in the future we will investigate the robustness of each model by performing sensitivity analysis of bifurcation sequences on the slow motion derivatives and various hydrodynamic

coefficients. This will shed light on the issue of to what

extent are the discrepancies between the two schools and the

models within each school of thought due to physical modeling inadequacies or to inaccuracies of calculations of

slow motion derivatives and other hydrodynamic

coefficients.

ACKNOWLEDGMENTS

The work at the University of SAo Paulo is sponsored by FAPESP (Research Support Foundation of the State of São

Paulo). Brazil. The work at the University of Michigan is supported by the University of Michigan/Industry

Consortium in Offshore Engineering. Industry participants

include Amoco, Inc.; Conoco, Inc.; Chevron, Inc.; Exxon Production Research: Mobil Research and Development; Shell Companies Foundation: and Petrobrás Research and

Development.

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