Effect of slow-drift loads an nonlinear dynamics of spread mooring systems

Proceedings of the 17th Offshore Mechanics and Arctic Engineering, July 1998 and Journal of Offshore Mechanics and Arctic Engineering, September 1998

EFFECT OF SLOW-DRIFT LOADS ON NONLINEAR

DYNAMICS OF SPREAD MOORING SYSTEMS

by

Michael M. Bernitsas, Ph.D.

Professor, ASME Member

and

Boo-Ki Kim

Research Assistant, Ph.D. Candidate

Submitted for presentation at the

17th International OMAE Conference

Lisbon, Portugal

July 5-9, 1998

and

for publication in

Journal of OMAE, ASME Transactions

Department of Naval Architecture and Marine Engineering

The University of Michigan

Ann Arbor, MI 48109-2145, USA

TECHNISCHE WW1ERSITEII Laboratorium 'icor Scheepshydromechanlca

Mekatweg 2,2628 CD Deift

i& 0t-Nd5?-Fs 015 181S33

B

ABSTRACT

Spread Mooring Systems (SMS) may experience large am-plitude oscillations in the horizontal plane due to slow-drift loads. In the literature, this phenomenon is attributed to

resonance. In this paper, it is shown that this conclusion is only partially correct. This phenomenon is investigated using nonlinear stability and bifurcation analyses which re-vea.! an enhanced picture of the nonlinear dynamics of SMS. Catastrophe sets are developed in a parametric design space to define regions of qualitatively different system dynam-ics for autonomous SMS, including mean drift forces. Lim-ited time simulations are performed to verify the qualitative conclusions drawn on the nonlinear dynamics of SMS using catastrophe sets. Slowly varying drift forces are studied as an additional excitation on the autonomous SMS and simu-lations reveal that slow drift may cause resonance or bifur-cations with stabilizing or destabilizing morphogeneses. The mathematical model of SMS is based on the slow-motion maneuvering equations in the horizontal plane (surge, sway, yaw) including hydrodynaniic forces with terms up to third-order, nonlinear restoring forces from mooring lines, and

en-vironmental loads due to current, wind, and wave-drift.

i INTRODUCTION

During the past two decades, considerable research effort has been devoted to the development of ocean mooring sys-t.ems for station-keeping of tanker and ifoating production, storage and offloading (FPSO) facilities and for oil/gas off-shore operations. Several types of mooring configurations are available such as Single Point Mooring (SPM), Two Point Mooring (TPM), and Spread Mooring Systems (SMS) (API, 1987). SPM and TPM systems aie widely used for relatively short periods of operation during which there is little change in environmental conditions. For longer periods of

opera-EFFECT OF SLOW-DRIFT LOADS ON

NONLINEAR DYNAMICS OF SPREAD MOORING SYSTEMS

Michael M. Bernitsas and Boo-Ki Kim

Department of Naval Architecture and Marine Engineering

The University of Michigan

Ann Arbor, Michigan

tion, the direction of environmental excitation changes sig-nificantly, thus making a SMS more appropriate for

station-keeping.

Time-domain simulation has been used widely to deter-mine the nonlinear dynamical behavior of any specific moor-ing configuration. Results of this approach can be found, for example, in van Oortmerssen et al. (1986), Chakrabarti and Cotter (1989), Kat and Wichers (1991), and Brook (1992). Simulation alone, however, is not adequate for the analy-sis and design of mooring systems because it cannot reveal the complete picture of the nonlinear phenomena associated with system dynamics. A comprehensive design methodol-ogy for mooring systems based on catastrophe theory has been developed by Papoulias and Bernitsas (1988), Chung and Bernitsas (1992), and Garza-Bios and Bernitsas (1996). They developed nonlinear stability analysis and bifurcation theory for mooring systems to produce catastrophe sets in the design space defining regions of qualitatively different system dynamics. This design methodology eliminates both trial and error and extensive nonlinear simulations typically used in the design process. Catastrophe sets have been de-rived numerically by systematic search for bifurcations (Pa-poulias and Bernitsas, 1988; Chungand Bernitsas, 1992) or through analytical expressions of loo of static and dynamic loss of stability (Garza-Bios and Bernitsas, 1996).

Current, wind, and waves are the main sources of vessel excitation in ocean. Papoulias and Bernitsas (1988), Chung and Bernitsa.s (1992), Nishimotoet aI. (1995), Beraitsas and Garza-Rios (1996), and Fernades and Aratanha (1996) have studied extensively the nonlinear dynamics of SPM, TPM, or SMS, but did not include waves in their numerical appli-cations. In this paper, slow-drift loads are considered, along with current, in SMS dynamics. Slow-drift loads are of im-portance in the design of mooring systems because it has been observed that these second-order low-frequency excita-tions may result in large amplitude horizontal plane moexcita-tions

of the moored vessel.

The SMS model is introduced in Section 2. It consists of the maneuvering equations in the horizontal plane includ-ing hydrodynamic terms up to third-order, nonlinear restor-ing forces from moorrestor-ing lines, environmental loads due to current, wind, and wave-drift. In Section 3, solutions by simulation are used to reveal the richness of SMS dynam-ics. In Section 4, the SMS is modeled in a six-dimensional state space and nonlinear stability analysis is introduced. The SMS is autonomous and catastrophe sets are developed in a parametric design space to define regions of qualita-tively different dynamics. The effect of slow-drift loads on SMS dynamics is studied in Section 5. It is shown that even mean drift forces due to head seas may push SMS into al-ternate equilibria or limit cycles. On the other hand, mean drift forces may have stabilizing effect by reducing the un-stable manifold dimension. A slowly varying drift spectrum is imposed as an additional excitation on the corresponding SMS. Resonance - to which large amplitude SMS motions are attributed in the literature - is only one of the possible qualitatively different responses of SMS. Stabilization, ap-pearance of alternate limit cycles or alternate equilibria are possible as well.

2 PROBLEM FORMULATION

The SMS model consists of the mooring line system, the horizontal plane slow motions, and the mooring line model. Their models are presented in the three subsections below.

2.1

_{Spread Mooring System}

Coordinate System

Consider a vessel moored to a spread mooring system,

as shown in Figure 1.

Let O-XYZ be the body-fixed coordinate system with its origin located at the center of

gravity (G) of the moored vessel. The Z-axis is positive upwards and the X-axis points toward the bow. The o-zyz system denotes the inertial coordinate system fixed to the

earth with the origin located at the mooring terminal 1.

The z-y plane coincides with calm water level and the z-axis is positive upwards.

Geometric Relations

The following geometric properties are defined to describe the arrangement of a spread mooring system, as shown in

Figure 1: (z, y,)) are the coordinates of the ith mooring

line fairlead in the body-fixed coordinate system; (zw, y)

and (x, y) are the positions of the ith mooring

termi-nal and attachment coordinates of the ith mooring line with respect to the earth-fixed coordinate system, respectively; 7(') is defined as the angle between the z-axis and the ith mooring line counter-clockwise; ¡(s) is the length of the ith mooring line; is the yaw angle.

The geometric relations of the mooring lines are (i) (i)

COS7' , (1)

DIRECTION OF WiNO ANO WAVES

Figure 1: Geometry of Spread Mooring System

(t) (s) (s) Ihn YT

smy =

=

i,,/(z2

-

z)2 +

4)

= z +zcos'y»sin*,

()

()

.

()

YT =Y+Z1,

sm&+y

cos1i.

For convenience, is defined as

= 2r

+ '.

Kinematic relations

The slow motions of the vessel in the horizontal plane -surge, sway, yaw - are considered. The horizontal compo-nents of vessel velocity (u, y, r) relative to the surrounding water in surge, sway and yaw, respectively, are related to the inertial coordinates (z, y) by the kinematic relations

i = u cos ' - y sin - U cos o, (7)

y =usin,,b+vcos,,b+Usina,

(8)

(9)

where U is the magnitude of the current velocity. The direc-tion of the current is measured with respect to the positive z-axis counter-clockwise, as shown in Figure 1.

2.2

Vessel Motions and Loads

Slow-Motion Maneuvering Equations

The mathematical model is based on the application of

Newton's laws of motion for linear and angular momentum in the earth-fixed coordinate system o-z yr. Forces and moment acting on the vessel are defined in the body-fixed coordinate system O-XYZ. Thus, transformation from the o-z yr sys-tem to the O-XYZ syssys-tem is required to write the equations

of motion in the body-fixed coordinate system. This trans-formation results in the Euler equations of motion for a rigid body

du

- rv) =

dv

m(

+ ru) = F,

where m is the mass of the vessel; Izz is the moment of

in-ertia about the Z-axis; Fx, Fy and M are the total forces and moment acting on the vessel in surge, sway, yaw,

re-spectively. In the body-fixed coordinate system O-XYZ, equations (10)-(12) can be written as

(m - X4) - mrv = XH + + Xw + Xwv,

i=1 (13)

(m - Y - Y + mru = YH + + Yy + Ywv,

=1 (14)

n

-Noi' + (Izz - N

= NH + E(zT2ay -

3,;T2rge)

+Nw+Nwv,

(15)

where X, Y,., and N,. are added masses and moment of

inertia in surge, sway arid yaw directions, respectively; Y,.

is the added mass in sway direction due to yaw motion;

N,, is the added moment of inertia in yaw direction due to

sway motion; and 2)O denote the tension in the th

mooring line in surge and sway directions, respectively; n is the total number of mooring lines. Subscripts H, W and WV indicate hydrodynamic forces due to vessel motions, wind loads, and second-order slow-drift loads, respectively.

Hydrodynamic Forces and Moment

The hydrodynamic forces and moment due to vessel mo-tions can be expressed in terms of the slow motion deriva-tives. The following third-order Taylor expansion is used (Abkowitz, 1969):

XH = X0 + Xu +!Xuuu2+ + X,,,,v2

+ X,.rr2 + !X5852 + Xvuv2u +

+ X55,.62u X,,,.vr + X,.,6v5 + X,.6r6

+ Xvruv'ru + X,,óv5u + XrórSu,

= Y0 + Yo,.0

+You2 + ')

+!YvvV3

2

+ Y,,,.vr2 + Y,55v5

Y,.,vu + Y,,vu

+ Yrr + Yrrrf3 + Yrvvrv2 + Yr66'52 + Y,.0ru

+ Yruuru2 + 1'5 +!y5663+ Y6,,,5v2 + Yorr8r2 + Y5u + Y5óu2 + Y,,,.,çvrô, s?

2 1

Nif = N0 + N,,,u + N0u + N,,v + N,,,,,,v

+ lN,.,.tìf2 + N,,6vS2 + Nvuvu +

!N,,vu2

+ N,.r + !Nrrrr3+!N,.,,,,Tv2 ₊ !Nr65r52

+ N,.Tu+!Nruuru2 + N55 + N6555 + 1N6,.,,,,5v2

+ Nórr6r2 +No6u + No,.6u2 + N,,,.5vró, (18)

where subscripts o and 5 denote propeller and rudder angle effect, respectively. The resistance of the vessel R at velocity u is related to the first four terms in equation (16) by

-R

Xe + X,u + Xu2 +

(19)

The hydrodynaniic forces and moment acting on the vessel

depend not only on the instantaneous vessel motions but

also the history of the surrounding fluid motion. The effect of the history of the surrounding fluid motion is referred to as the hydrodynamic memory effect. This will be studied in a separate paper.

Mooring Line Tension

The tension in the ith mooring line for the surge and sway directions can be represented as

ge =

+Tcosw = Rcos&+ Rsin,

= -T

sin w' = RS»cos

- R»sin '.

The components of the mooring line tension R», R» in the r-y plane are given by

= cosy , (22)

) .

()

= smi'

Current Load

Current loads are implicitly included in the expressions

of equations (16),

(17), and (18) by using the relative

velocities of the vessel with respect to the surrounding

water. Therefore, current loads are accurate to third-order.

This approach is more accurate than the usnaI method based on projected area and drag coeflicients. rEWith no

loss of generality, the current is assumed to be directed to

the negative x-axis in the earth-fixed coordinae system

(a = 00).

Wind Load

1g

Steady wind forces and moment exerted on the vessel are expressed as

(23)

Xw

=

paU,Cxw(ar)Ar,

Yw= -POU,CYW(OT)AL,

where p,, is the density of air; U,,, is the mean wind velocity relative to the air at a standard height of ten meters above the calm water level; AT and AL are the maximum trans-verse and lateral projected areas of the above-water part of the vessel, respectively; L is the length of the vessel. The relative angle a,. of wind direction is defined as

0,. = Oo

-where Oo is the absolute angle of wind direction in the (z, y) coordinate system. The coefficients Cxw(a,.), Cyw(ar), Czw(ar) depend on the vessel type, superstructure loca-tion, loading condiloca-tion, and the relative angle a,. of the wind direction. These coefficients can be obtained from experimental and full-scale measurement data.

Slow-Drift Load

The common approach to formulation of non-linear wave-body interaction problems in ship and offshore hydrodynam-ics is based on potential flow theory using the wave ampli-tude to length-ratio perturbation parameter. The solution of the second-order problem results in three types of wave excitation forces: mean, difference-frequency (slowly vary-ing), and sum-frequency (springing) forces. Mean and slowly varying drift forces are of importance in mooring problems. Newman's approximation (1974) for slowly varying drift forces has been used extensively for many applications in ship hydrodynamics. The important practical consequence of this approach is that slowly varying drift forces depend merely on evaluation of second-order mean drift forces, and this can reduce the computational time significantly. The formula for slowly varying drift forces is

sv__ T

F1 = (Jtwv, Ywv, Nwv)

=

A3AkTkcos[(tk w,)t+( e,)]

j=1 k=1 (24)

where the coefficient 7k =

0.5(D + D)

and i = 1,2, 6

indicate surge, sway and yaw directions, respectively. A,

w, and e are the wave amplitude, circular frequency, and random phase angle of component number j (or k) in the

total number of wave components N. The random phase

angle .e are uniformly distributed between O and 2v and

remaiçonstant in time. The wave amplitude A (or A)

can bomputed from the wave spectrum S(w) as

p,

A, = 2/S(t,)iw,

(25)

where ¿w is the difference between adjacent frequencies. The iwo-parameter Bretschneider wave energy spectrum for-mulatz used to express S(w). The two parameters are the significant wave height H,,,i3 and the mean wave period T2.

The time mean value over the diagonal terms (when k = j) in equation (24) gives the mean drift forces and the mean of the off-diagonal terms (when k j) vanishes. Mean drift

forces can be written as

The transfer functions D' for mean wave drift force are ap-proximated as D'

=

= PWgLCXD cos3(Oo -Ywv = PWgLCYD sin'(Oo

-

---D6 sin 2(90

-C 2

where ,, is the wave amplitude, Pv is the density of water, o is the absolute angle of wave direction, and g is gravi-tational acceleration. The coefficients CXD,CYD,CZD can be obtained from the complete linear (first-order) solution of the seakeeping problem. Further details about the methods of predicting mean drift forces can be found, for example, in the study of Ogilvie (1983). In this paper, it is assumed that these can be given simply in terms of the ship particulars (Hirano et al., 1980).

2.3

Mooring Line Model

In our line of work, we use three types of quasi-static moor-ing line models: nylon or polyester rope, chain, or steel cable. The stability of mooring systems is affected significantly by the line model (Bernitsas and Papoulias, 1990). Chains are usually fully submerged in water and only slightly

exten-sible. They experience large drag force and undergo two-dimensional deformation in the vertical plane modeled by catenary equations. Steel cables are modeled by a three-dimensional, large-strain, finite element model which takes into account drag force, weight, and extensibility. They are lighter than chains and have less drag but are more

extensi-ble. Elastic ropes are light, considerably extensible, nearly buoyant in water. Nonlinear behavior is due to their extensi-bility, and they are modeled as follows (McKenna and Wong,

1979):

ii

q

(30)

where T is the actual tension in the rope, Sb is the average breaking strength, p and q are empirically determined on-stants, and 1 is the working length of the unstrained rse. A wet nylon rope of 120 mm diameter (Sb = 680, 625 ¡bf, p = 9.78, q = 1.93) is selected for the numerical applications ipre-sented in this paper.

Initial pretension is applied to one or more mooring lines. The pretension T1 can be defined in terms of the initial sLiain Ei of the line as T1 = Sbp(EJ)", (31) il - ¡w (32) 'w D2 =

V _{(Xwv,Ywv,Nwv)T = (26) where ¡ denotes the initial length of the pretensioned} moor-ing line.

Table 1: Principal Particulars of Vessels and Mooring Line Arrangements used in Simulations

3 RICHNESS OF SMS DYNAMICS

A barge with stabilizing skegs and a tanker without

propeller axe selected for numerical applications. Their principal particulars and mooring line arrangements used

in simulations are summarized in Table 1. Configuration FinAs denotes a (m+s)-Line SMS with m lines attached on the moored vessel forward of the center of gravity and with s lines aft of the center of gravity. Pretension is imposed on the forward mooring lines and is calculated as a function of initial strain E1. Slow motion derivatives of the two vessels

are provided in Bernitsas and Kekridis (1986).

Nonlinear dynamic behavior of mooring systems such as stable or unstable focus near an equilibrium, limit cycle,

and chaotic motion have been demonstrated by systematic and extensive simulations (Papoulias and Bernitsas, 1988;

Chung and Bernitsas, 1992; Bernitsas and Garza-Rios, 1996). In this section, the richness of SMS dynamics

and the effect of slow-drift loads are demonstrated by

simulation in Figures 2 to 6. The fourth-order Runge-Kutta time-stepping method is implemented because it is relatively simple and sufficiently accurate. The geometric points GI to G4 used to define SMS configuration in Figures 2 to 6 do not cover the entire range of design practice. They were selected to demonstrate SMS dynamics. In Section 5, these

simulations are studied further in relation to catastrophe

sets in Figures 7 to 11.

Barge with Skegs

Figure 2 shows a periodic oscillation (limit cycle) of a

2-Line barge with skegs SMS about equilibrium A which has zero yaw angle. The amplitude of the yaw angle increases when mean drift forces are applied in the current direction

(head seas 80 = 0°). The system still undergoes a limit

cycle and the increased amplitude of yaw angle due to

mean drift forces results in higher tension in the mooring

lines. A different angle of wave direction (8 = 30°) leads to a non-symmetric equilibrium position with non-zero yaw angle. Figure 3 shows that the dynamical behavior of mooring systems can be different qualitatively depending on

003 002

¡

0_01 t003 0.02 o 25 50 75 Im, Nm,,m,s4on.i Tm,. (Ut&) 125 150

Figure 2: Effect of Mean Drift Forces on 2-Line SMS (FiAi, 103/L

0.3,x/L = 0.3,E1

0.013), Barge with Skegs

the mean drift force. As can be seen in Figure 3, the 2-Line barge with skegs SMS experiences stable focus behavior and converges to equilibrium A when only current load is applied. A limit cycle near equilibrium A appears when mean drift forces are applied. This simulation shows that an SMS may exhibit large amplitude slow periodic motions due to mean drift forces. As will be explained in Section

4, this is the result of dynamic loss of stability and not

resonance. In Figure 4, slowly varying drift forces have a stabilizing effect on the SMS; actually higher slowly varying drift forces result in a stable equilibrium A. This shows that there is no resonance in yaw. In surge, however, Figure 5 shows resonant motion on top of the periodic motion due to loss of stability.

Tanker without PropeUer

Figure 6 shows that a 3-Line tanker without propeller SMS converges to equilibrium B in head current without waves; equilibrium A becomes unstable and the system converges to equilibrium B which has a non-zero yaw angle. When mean drift forces are added to the excitation, equilibrium A becomes stable. In this case, mean drift forces have a

favorable effect on the system stability. This is the opposite effect to that observed in the barge with skegs SMS in Figure 3. The system is unstable and the yaw angle fluctuates about equilibrium B when slowly varying drift forces are applied in head seas.

4 NONLINEAR STABILITY ANALYSIS

4.1

State Space Representation

The SMS model presented in Section 2 consists of the equations of motion (13)-(15) and kinematic relations (7)-(9). It can be modeled by six first-order nonlinear coupled ordinary differential equations using the following six state

variables (z1 = u,52 = v,X3 = f,X4 = x,x = y,zs = i'). In

System 2-Line SMS 3-Line SMS

Configuration FIAI F1A2

Vessel Barge with Skegs Tanker without Propeller

L 191.56 ft 1066.3 ft B 35.00ft 173.gOft T 10.42 ft 71.30 ft CB 0.855 0.831 E1 0.013 0.04 U 2 knots 2 knots H113 4.5 ft 8.0 ft

0.03

0.02

10.06

r26

0260

t_/L.O4 x1L=O.3 (G2A E,. aol3

V

'II

160

I

i

t

II

660 200 260 P1di1=0à0r Tm,. (Uk) t - ,,1 * 6th66 L 400

Figure 3: Effect of Mean Drift Forces on 2-Line SMS

(FiAi, 1/L = 0.6,x/L = 0.3,Ej = 0.013),

Barge with Skegs

Figure 4: Effect of Slow-Drift Loads on 2-Line SMS

(FiAi, 1/L = 0.4,z/L = 0.3,E1 = 0.013),

Yaw, Barge with Skegs

Figure 5: Effect of Slow-Drift Loads on 2-Line SMS

(FiAi, 100/L = 0.4,z/L = 0.3,E1 = 0.013),

Surge, Barge with Skegs

Figure 6: Effect of Slow-Drift Loads on 3-Line SMS

(F1A2, 100/L = 0.15, x/L = 0.52, Ej = 0.04), Tanker without Propeller

Canchy standard form, they become = i +_{on - X,} {mx2r3 + Xw(x6) + Xwv(x6)}, +

(IZZ_S){

_D mxlr3+Yw(r6)+Ywv(z6)} +

_yPR»)cosz6}

i=1 n

(mYo)

+ D

{N(zi,z,x3) +

(r,'14 - i4,'R)cosxs}

= i n (m - Y,,)

{>(4'R +

i4

D +

(m Yb)

{N(x6) + Nwv(z6)}, 14 _{z1c06x6X2SiflX6Uco6a,} 15 _{z1 sinz6+z2cosz6+Usina,} SIflX6 } (33)

where D = (IzzN)(mY,)NY,.. Evolution equations

(33}.(38) can be written in the form of one six-dimensional

n

_(tzzNr)

-D (34)

-

+ +

_{{Nw(x6) + N(z6)},}

i=1

13 = {YH(x1,x2,x3)+ >2(Rf'coszs

R»sinx6)}

N,,

first-order vector differential equation as

x = f(x),

f E C',

f: R6 R6 (39)

where * denotes the derivative of state vector x with respect to time t, lR is the six-dimensional Eucidean space, and C' is the class of continuously differentiable functions.

Since the vector field f in equation (39) does not depend explicitly on time t, the system is said to be autonomous.

The mean drift forces as expressed by equation (26) are au-tonomous. The slowly varying drift forces given by equation (24), however, depend explicitly on time and would render the SMS model non-autonomous. In general, stability anal-ysis of a non-autonomous nonlinear system is not possible. Time simulation of a non-autonomous system which includes slowly varying drift forces, however, can be performed with-out any difficulty and is used in Sections 3 and 5 to assess the effects of non-autonomous terms on the stability of the corresponding autonomous system.

4.2

Equilibria of SMS

Equilibria of the vector field (39) are the points x E IR6

such that

O=f(5c),

i=(î,,±2,

16). (40)

These are stationary solutions. Equilibria can be interpreted

as intersections of the null clines. The equilibria of SMS, which are the solutions to a nonlinear equation (40), depend on the mooring line and system configuration, as well as on

the environmental conditions.

4.3

Stability in the sense of Lyapunov

The behavior of a nonlinear system near an equilibrium

point can be determined via linearization with respect to that point. Since linearization is an approximation in the

neighborhood of an equilibrium point, it can predict only

the local behavior of the nonlinear system in the vicinity of that point (Guckenheimer and Holmes, 1983; Seydel, 1988).

Let x be an equilibrium point of the nonlinear system (39). Expanding the right-hand side of equation (39) into

its Taylor series about point x, we obtain

df

x=f()+>6.__ x=t

+

Ix=a +H.O.T.

i,., = I

where deviation ¿ from equilibrium is defined as

¿(t) = x(t) - x.

(42)

Since X is an equilibrium point, we have f() = O. Then, the state equation can be written as

(43)

6

If attention is paid to a sufficiently small neighborhood of the equilibrium point so that the higher-order terms are negligi-ble, the nonlinear state equation (39) can be approximated

by the linear state equation

= [A),

¿ E R6, [A] E R6X6 (44)

where [A] = Df(i) is called the Jacobian matrix evaluated

at the equilibrium point X. If the eigenvalues of [A], ,, have zero real parts (usually called hyperbolic or non-degenerate), then it can be expected that the trajectories of the nonlinear system in a small neighborhood of an

equilib-rium point are close to the trajectories of its linearization about that point.

4.4

Bifurcation and Catastrophe Sets of SMS

A bifurcation is a qualitative change in a limit set of the system as a parameter is infinitesimally perturbed. Bifurca-tions can be interpreted as the appearance or disappearance of a limit set and the change in the stability type of a limit

set. A bifurcation also occurs when a non-stable limit set

of the system remains non-stable but undergoes a change in the unstable manifold dimension. Saddle-node, pitch-fork, and Hopf bifurcation have been found in the nonlin-ear dynamics of mooring systems (Papoulias and Berthtsas, 1988; Chung and Bernitsas, 1992; Bernitsas and Garza-Rios, 1996). A catastrophe set is the set of all loci of bifurcations in a parametric design space. It defines the boundaries be-tween regions of qualitatively different system dynamics and the morphogeneses occurring as boundaries are crossed.

In the numerical applications below, catastrophe sets are constructed for several SMS configurations of a barge with

skegs and a tanker without propeller. The effect of mean

drift forces on bifurcation of SMS dynamics is demonstrated in the catastrophe sets of Figures 7 to 11. Five different SMS with symmetric mooring configuration are studied and their catastrophe sets are constructed for equilibrium A. The five SMS configurations and their geometries are summarized in Table 2. The pretension is set in the forward mooring lines in terms of the initial strain E1. The angle of wave direction

00 is set equal to O degree, i.e., head seas and wind loads exerted on the vessel are not considered.

Four regions of qualitatively different SMS dynamics are identified in Figures 7 to 11.

Region I: All six eigenvalues of the system have negative

real parts. Equilibrium A is stable, and trajectories

(41) _{converge asymptotically to equilibrium A.}

Region II: There exists one eigenvalue with positive real part (one-dimensional unstable manifold). Equilibrium A is unstable, therefore, trajectories deviate from equi-librium A.

Region III: There exists a complex conjugate pair of eigenvalues with positive real parts (two-dimensional unstable manifold). Equilibrium A is unstable, and

trajectories asymptotically reach periodic oscillations (limit cycle).

Table 2: SMS Configurations and Geometries used in Catastrophe Sets

Region IV: There exists a real eigenvalue and a com-plex conjugate pair of eigenvalues with positive real parts. Therefore, equilibrium A is unstable with a

three-dimensional unstable manifold which is the mini-mum dimension for onset of chaos (Seydel, 1988).

Pitchfork bifurcations appear at the boundary between

Regions I and II. Static loss of stability occurs when

crossing from Region I to Region II. Periodic solutions

appear in Region III, and dynamic loss of stability occurs when crossing from Region I to Region III. Region IV is characterIzed by the merging of unstable Regions II and III.

Barge with Skegs

The effect of mean drift forces on the FiAi and F2AO configuration is shown in Figures 7 and 8, respectively.

Figures 7 and 8 show three regions of qualitatively different SMS dynamics about equilibrium A. A stationary pitchfork

bifurcation occurs between Regions I and II. A dynamic

(Hopf) bifurcation occurs between Regions I and III. This boundary has the form of a fold singularity which encloses Region III where periodic solutions exist. As can be seen in Figures 7 and 8, mean drift forces expand the unstable Region III as the significant wave height increases. Mean drift forces have a significant effect on increasing the Hopf

bifurcation domain for the barge SMS, while they have

little effect on the stationary bifurcation locus. Geometric

point G2 (see Figure 7) is in Region I in the presence of very small or no mean drift forces. For higher realistic

mean drift forces, G2 falls in the dynamically unstable Region III. Geometric point Gi remains in Region III.

These observations are verified by simulation in Figures 2

and 3. Figure 9 shows the effect of mean drift forces on

the dynamics of the F2A1 configuration Only Regions I

and II appear in the catastrophe set of Figure 9; therefore,

Figure 7: Catastrophe Set

of 2-Line SMS (FiAi,

(yr/B

= 0.0, E1

=

0.0 13), Barge with Skegs

Figure 8: Catastrophe Set

of 2-Line SMS (F2AO,

yr/B

=

0.0, =5°), Barge with Skegs

this system undergoes only a stationary bifurcation in

the design parameter range. This stationary bifurcation

depends weakly on the mean drift forces. This conclusion is similar to the one obtained from Figures 7 and 8.

Tanker without Propeller

Figures 10 and 11 show the catastrophe sets for a

tanker without propeller SMS in the F1A2 and the F2A2

configuration, respectively. As the significant wave height

increases, the set of bifurcation boundaries moves to the

left. As a result, stable Region I and chaotic Region IV move to the left, unstable Region II shrinks, and Hopf

bifurcation domains expands significantly. Geometric point

G4 (1/L

= 0.15, z,/L

=

0.52) in Figure 10 is in unstable Region H for low mean drift forces and falls in the stable Region I as mean drift forces increase. This is verified by simulation in Figure 6. 0. 0.4-02: -II ... / +G2 /

u'

u'

TT.

-

2.01! . 05 SMS Corifig. Line 4Y

2-Line FiAi i , >0

rr, 0

Vp1 =0

y1O

2 _Vp 0 (2)....,,.

F2AO i w>0

2p 0

Vp O

-(')=2c

2 _{w = w} Z, _{p Vp Yp}

3-Line F1A2 i

,, >0

r, 0

y, =0

"y(')=O

2

Vp 0

2)=ir_f 3 ..y(3)=,r+ 2 F2A1 i w >0

p 0

Vp <O

()=2rÇ2

2 .(2) 3

y, =0

()=r

4-Line F2A2 i

,. >0

xp' 0

Vp' 0 (12)

2 .(2). ç 3

Vp 0

rf

4 w w Xp _{Vp =Yp}

Figure 9: Catastrophe Set of

3-Line SMS (F2A1,

yr/B

= 0.5,

= 15°, Ej = 0.013), Barge

with Skegs

Figure 10: Catastrophe Set of 3-Line SMS (F1A2,

yr/B = 0.5,

= 5°, Ej = 0.04), Tanker

without Propeller

Figure 11: Catastrophe Set of 4-Line SMS (F2A2,

y/Bja = 0.0, y,/B0j

0.5, 2 5°,

E1 = 0.03), Tanker without Propeller

5

EFFECT OF SLOW-DRIFT LOADS ON

SMS DYNAMICS

5.1

Effect of Mean Drift Forces

Nonlinear stability and bifurcation analyses are used to study the effect of mean drift forces on SMS dynamics. This is possible because the system is autonomous. Based on the previous observations, we can summarize the effects of mean drift forces on SMS dynamics as follows:

Mean drift forces have a small detrimental effect on static loss of stability for barge SMS's. Unstable Re-gion II expands at the expense of ReRe-gion I, as shown in Figures 7, 8, and 9.

Mean drift forces may have a detrimental effect on static loss of stability for tanker SMS's. Specifically, Figures

10 and 11 show that Region III may expand at the

expense of Region Il. This implies that a tanker SMS whose trajectories converge to equilibrium B may ex-hibit large amplitude oscillations in limit cycles around equilibrium A. Obviously, this is not resonance. On the other hand, mean drift forces may significantly stabilize tanker SMS's statically. As shown in Figures

10 and 11, Region I expands into Region II, and

Re-gion Ill expands into ReRe-gion IV. That is, the pitchfork bifurcation is deferred significantly.

Mean drift forces have dramatic detrimental effect on dynamic loss of stability for barge SMS's. As shown in Figures 7 and 8, Region III expands at the expense of Region I. Thus, trajectories that converge to equilib-rium A reach a limit cycle around equilibequilib-rium A. Again, these large-amplitude oscillatory slow motions are not instigated by resonance.

Similar is the effect of mean drift forces on dynamic loss of stability of tanker SMS's. Figures 10 and 11 show that Region III dramatically expands into Regions II and IV. Thus, trajectories converge to equilibrium B, and chaotic motion around equilibrium A may become periodic around equilibrium A.

5.2

Effect of Slowly Varying Forces

When the spectrum of slowly varying drift forces is added

to the external excitation of a SMS, the system becomes

non-autonomous. There is no general stability theory to analyze the non-autonomous system subject to spectral

excitation. A preliminary investigation of the effect of slowly

varying drift forces on SMS dynamics is presented here

using simulations of selected SMS systems. The following observations can be made:

-(a) Slowly varying drift forces may stabilize oscillatory mo-tions instigated by mean drift forces. Figure 4 shows significant reduction in yaw angle oscillations. Actu-ally, slowly varying drift forces of higher significant wave

height totally eliminate yaw angle oscillations.

50 4.0 II M--4_5" -z. /L 1.5 1.0 -0.5 . IV . i ::---' ni 444.Øft 5,.l.Oft -Il

-of Moored Vessels in Irregular Seas," Proceedings -of the 6th International Conference of Behavior of Offshore Structures, Vol. 1 , Imperial College of Science, Technology and Medicine, London, pp. 251-264.

Chakrabarti, S.K., and Cotter, D.C., 1989, "Motions of Articulated Towers and Moored Floating Structures,"

Journal of Offshore Mechanics and Arctic Engineering, Vol. 111, pp. 233-241.

Chung, J.S., and Bernitsas, M.M., 1992, "Dynamics

of Tow-Line Ship Towing/Mooring Systems: Bifurcation,

Singularities of Stability Boundaries, Chaos," Journal of

Ship Research, Vol. 36, No. 2, pp. 93-105.

Fernades, A.C., and Aratanha, M., 1996, "Classical As-sessment to the Single Point Mooring and Turret Dynamic Stability Problems," Proceedings of the 15th International Conference on Offshore Mechanics and Arctic Engineering, Vol. l.A., Florence, Italy, pp. 423-430.

Garza-Rios, L.O., and Bernitsas, M.M., 1996, "Analytical

Expressìons of the Stability and Bifurcation Boundaries for General Spread Mooring Systems," Journal of Ship

Research, Vol. 40, No. 4, pp. 337-350.

Guckenheimer, J., and Holmes, P., 1983, Nonlinear Os-cillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, Inc..

urano, M., Takashina, J., Takaishi, Y., and Saruta, T., 1980, "Ship Turning Trajectory in Regular Waves," Transactions of the West-Japan Society of Noval Architects, Vol. 60.

Kat, J.O., and Wichers, J.E.W., 1991, "Behavior of a Moored Ship in Unsteady Current, Wind, and Waves,"

Marine Technology, pp. 251-264.

McKenna, H.A., and Wong, R.K., 1979, "Synthetic Fiber

Rope, Properties and Calculations Relating to Mooring

Systems," Deep Water Mooring and Drilling, Ocean Engi-neering, Vol. 7, pp. 189-203.

Newman, J.N., 1974, "Second-Order Slowly Varying Forces on Vessels in Irregular Waves, Proceedings of Inter-national Symposium on Dynamics of Marine Vehicles and Structures in Waves, London, pp. 182-186.

Nishjmoto, K., Brinati, H.L., and Fucatu, C.H., 1995, "Analysis of Single Point Moored Tanker using

Maneu-vering Hydrodynamic Model," Proceedings of the 14th

International Conference on Offshore Mechanics and Arctic Engineering, Vol. I.B, Copenhagen, Denmark, pp. 253-261. Ogilvie, T.F., 1983, "Second Order Hydrodynamic Effects on Ocean Platforms," Proceedings of International Workshop on Ship and Platform Motions, pp. 205-265.

Papoulias, F.A, and Bernitsas, M.M., 1988, "Autonomous Oscillations, Bifurcations, and Chaotic Response of Moored Vessels," Journal of Ship Research, Vol. _{32, No. 3, pp.}

220-228.

Seydel, R., 1988, From Equilibrium to Chaos, Elsevier Science Publishing Co., Inc., New York.

van Oortmerssen G., Piukster, J.A., and van den Boom

H.J.J., 1986, "Computer Simulation of Moored Ship Be-havior," Journal of Waterway, Port, Coastal and Ocean

Engineering, Vol. 112, No. 2., pp. 296-308. On the other hand, slowly varying drift forces produce

the resonant beating behavior observed in Figure 5 for the surge motion. This behavior is indicative of narrow band excitation near a natural frequency.

Finally, Figure 6 shows that slowly varying drift forces

may render a stable equilibrium A unstable, insti-gate a pitchfork bifurcation, and take a tanker SMS

through transient oscillations to non-resonant oscilla-tions around a stable equilibrium B.

CLOSING REMARKS

A methodology based on stability and bifurcation analy-ses has been developed to perform a comprehensive analysis of the effect of mean drift forces on the nonlinear horizontal plane motions of SMS. Following that, slowly varying drift forces were considered as an additional excitation on the cor-responding autonomous system and their effect on SMS was studied. It was shown that the notion that large amplitude slow motions of SMS are due to resonance of a mooring sys-tem natural frequency with slowly varying drift represents only one of the mechanisms that can instigate such motion-s. Mean drift forces may also cause large amplitude oscillations. Further, slowly varying drift forces may dramatically reduce and even eliminate such motions. Actually, a wide variety of bifurcations and morphogeneses may be caused by mean or slowly varying drift forces.

ACKNOWLEDGMENTS

This work was sponsored by the University of Michi-gan/Industry Consortium in Offshore Engineering.

Indus-try participants include Amoco, Inc.; Conoco, Inc.; Exxon Production Research; Mobil Research and Development; Shell Companies Foundation; and Petrobras, Rio de Janeiro, Brazil.

REFEREN CES

Abkowitz, M.A., 1969, Stability and Motion Control of Ocean Vehicles, MIT Press, Cambridge, Massachusetts.

API, 1987, "Analysis of Spread Moorinç Systems for

Floating Drilling Units," API Recommended Practice 2P (RP2P), Second Edition.

Bernitsas M.M., and Garza-Rios. -L.O, 19.9, "Effect of

Mooring Line Arrangement on the Dynamics of Spread

Mooring Systems," Journal of Offshore Mechanics and

Arctic Engineering, Vol. 118, No. 1, pp. 7-20

Bernitsa.s M.M., and Kekridis, N.S., 1986, "Simulation and Stability of Ship Towing," International Shiçbuilding Progre.ss, Vol. 32, No. 369, pp. 112-123.

Bernitsas M.M., and Papoulias, F.A, 1990, "Nonlinear

Stability and Maneuvering Simulation of Single Point Mooring Systems," Proceedings of Offshore Station Keeping Symposium, SNAME, Houston, Texas,pp. 1-19.

TECHNISCHE UNVERSU hratOriUm voor

M2MSrnitsas

"

'"' hProtessor,

Mem ASME B. -K. Kim Research Assistant, Graduate Student.

Department of Naval Architecture and Marine Engirreerin. The University of Michigan, 2600 Draper Road, Ann Arbor. MI 48109-2145

I Introduction

During the past t o decades, considerable research effort has been devoted to the development of ocean mooring systems for

station-keeping of tanker and floating production. storage and offloading FPSO facilities and for oil/gas offshore operations. Several types of mooring configurations are available such as single-point mooring (SPM). two-point mooring (TPM). and spread mooring systems (SMS) (API, 1987). SPM and TPM

systems are widely used for relatively short periods of operation during which there is little change in environmental conditions.

For longer periods of operation. the direction of environmental excitation changes significantly. thus making an SMS more appropriate for station-keeping.

Time-domain simulation has been used widely to determine the nonlinear dynamical behavior of any specific mooring con-figuration. Results of this approach can be found, for example. in van Oortmerssen et al. (1986). Chakrabarti and Cotter (1989), Kat and Wichers (1991). and Brook (1992).

Simula-tion alone, however, is not adequate for the analysis and design

of mooring systems because it cannot reveal the complete

pic-ture of the nonlinear phenomena associated with system

dynam-ics. A comprehensive design me1hodolog for mooring systems based on catastrophe theory has been developed by Papoulias and Bernitsas (1988). Chung and Bernitsas (1992). and Garza-Rios and Bernitsas (1996). They developed nonlinear stability

analysis and bifurcation theory for mooring systems to produce catastrophe sets in the design space defining regions of

qualita-tively different system dynamics. This design methodology eliminates both trial and error and extensive nonlinear

simula-tions typically used in the design process. Catastrophe sets have

been derived numerically by systematic search for eigenvalue

bifurcations (Papoulias and Bernitsas. 1988: Chung and

Bernit-sas. 1992) or through analytical expressions of loci of static

and dynamic loss of stability (Garza-Rios and Bernitsas. 1996).

Current, wind, and waves are the main sources of vessel

excitation in ocean. Papoulias and Bernitsas (1988). Chung and Contributed b the OMAE Di'.ision for publication in the JOURNAL OF Orr:

SHOR.E MECHANICS AND ARCrIC ENGINEERING. Manuscnpt received by the OMAE

Disision, March 24. 1998: re'.ised manuscript receised May 8. 1998. Technical

Editor S. Liu

Effect of Slow-Drift Loads on

Nonlinear Dynamics of Spread

Mooring Systems

Spread mooring s-cslems (SMS) may experience large-amplitude oscillations in the horizontal plane due to slow-drift loads. In the literature, this phenomenon is

artrib-wed io resonance. In this paper, it is short n thai this conclusion is on/v partial/v correct. This phenomenon is investigated using nonlinear stability and bifurcation

analyses which reveal an enhanced pictu re of the nonlinear dvna,nics of SMS. Catas-trophe sets are developed in a parametric design space to define regions of

qualita-tively different system dynamics for autonomous SMS. including mean drift forces. Limited time simulations are performed to verift the qualitative conclusions drawn

on the nonlinear dynamics of SMS using catastrophe sets. Slowly varying drift forces are studied as an additional excitation on ¡he autonomous SMS and simulations re-teal

that slow drift may cause resonance or bifurcations trith stabilizing or destabilizing

morpho geneses. The mathematical model of SMS is based on the s/ott-motion man eu-vering equations in the horizontal plane (surge. s-tray. vaw,i, including hydrodnamic

forces with terms up to third-order, nonlinear restoring forces from mooring lines,

and environmental loads due ro current, wind, and wave-drift.

Bernitsas (1992). Nishimoto et al. (1995). Bernita and

Garza-Rios i 1996). and Femandes and Aratanha t 19% I have

studied extensively the nonlinear dynamics of SPM. TP\1. or

SMS. but did not include waves in their numerical applications. In this paper. slow-drift loads are considered, along v ith current.

in SMS dynamics. Slow-drift loads are of importance in the design of mooring systems because it has been observed that these second-order low-frequency excitations may result in

large amplitude horizontal plane motions of the moored vessel.

The SMS model is introduced in Section 2. It consists of the maneuvering equations in the horizontal plane including hydrodynarnic terms up to third-order, nonlinear restoring

forces from mooring lines, environmental loads due to current.

wind, and wave-drift. In Section 3. solutions by simulation are used to reveal the richness of SMS dynamics. In Section 4. the

SMS is modeled in a six-dimensional state space and its

nonlin-ear stability analysis is introduced. The SMS is autonomous

and catastrophe sets are developed in a parametric design space to define regions of qualitatively different d namics. The effect

of slo-drif1 loads on SNIS dynamics is studied in Section 5. It is shown that even mean drift forces due to head seas may

push SMS into alternate equilibria or limit cycles. On the other hand, mean drift forces may have stabilizing effect by reducing the unstable manifold dimension. A slowly varying drift spec-trum is imposed as an additional excitation on the corresponding

SMS. Resonanceto which large-amplitude SMS motions are attributed in the literatureis only one of the possible qualita-tively different responses of SMS. Stabilization, appearance of

alternate limit cycles or alternate equilibria are possible as well.

2 Problem Formulation

The SMS model consists of the mooring line system. the horizontal plane slow motions, and the mooring line model. Their models are presented in the following three subsections.

2.1 Spread Mooring System

Coordinate S stem. Consider a sessel moored to a spread mooring system. as shown in Fig. 1. Let O-XYZ be the body-fixed coordinate system with its origin located at the center of

gravity (G) of the moored vessel. The Z-axis is positive up ards

Journal of Offshore Mechanics and Arctic Engineering _{NOVEMBER 1998, Vol. 120 / 201}

DIRECTiON OF WINO AND WAVES

(x,y)

Fig. i Geometry of spread mooring system

and the X-axis points toward the bow. The o-xv: system denotes

the inertial coordinate system fixed to the earth with the origin

located at the mooring terminal 1. The x- plane coincides with calm water level and the z-axis is positive upwards.

Geometric Relations. The following geometric properties are defined to describe the arrangement of a spread mooring

system. as shown in Fig. I: (x, v) are the coordinates of

the ith mooring line fairlead in the body-fixed coordinate

sys-tem; (x». y) and (x. v) are the positions of the ith

mooring terminal and attachment coordinates of the ith mooring

line with respect to the earth-fixed coordinate system, respec-lively: 'y is defined as the angle between the x-axis and the ith mooring line counterclockwise; 1' is the length of the ith mooring line; i/i is the yaw angle.

The geometric relations of the mooring lines are

Kinematic Relations. The slow morions of the vessel in the horizontal planesurge. sway. yaware considered. The horizontal components of vessel velocity tu. y, r) relative to the surrounding water in sur2e. sway and yaw, respectively, are related to the inertial coordinates (x. 'i by the kinematic relations

r = u cos ¿b - L sin i' - U cos a (7)

= u sin ¿li + reos¿r + Usin a ₈₎

(9) where U is the magnitude of the current velocity. The direction of the current is measured with respect to the positive x-axis counterclockwise, as shovn in Fig. I.

2.2 _{esseI Motions and lAds}

Slow-i1orion tfwi4i rj,l I,1nwít ':' The mathematical model is based on the applicarion of It)n' lass of motion

for linear and angular momentum in the earth-fixed coordinate

system o-xvz. Forces and morneni acting on the sessel are de-fined in the body-fixed coordinate stem ()-X}Z. Thus, trans-formation from the o-xv: system to the O-X7 system is

re-quired to write the equations of motion in the bodv-fned coordi-nate system. This transformation results in the Euler equations

of motion for a rigid body f du

m(

-\ dt

(dt

-\ dt

-

= F + ru) = dr - = M

where m is the mass of the vessel; I is the moment of inertia about the Z-axis; F, F, and M are the total forces and moment acting on the vessel in surge. sway, yaw, respectively. In the body-fixed coordinate system O-XYZ. Eqs. ( l0)( 12) can be

written as

(m - X ) i - mr = XH 1 _{+ Xw +} (13)

(mYjU Y,,'+mru=

YH+TV+YW+YWV

(14)

N,r) + (J

- N,),'

= NH +

(x'

- y

Tjgc) + N + Nwv (15)

where X, Y. and N, are added masses and moment of inertia

in surge. sway. and yaw directions, respectively; Y, is the added

mass in sway direction due to yaw motion; N is the added moment of inertia in yaw direction due to sway motion; _TJ5,

and denote the tension in the ith mooring line in surge

and sway directions, respectively: n is the total number of

moor-ing lines. Subscripts H. W, and W V indicate hydrodvnamic forces due to vessel motions, wind loads, and second-order

slow-drift loads, respectively.

Hvdrodvnamic Forces and Moment. The hydrodynamic forces and moment due to vessel motions can be expressed in terms of the slow motion derivaties. The following third-order Taylor expansion is used (Abkowitz, 1972):

XH = X, + Xu + Xu2 ± ±

+ X,52 + Xyu

Xru + X,,62u + X,vr

- X,r6 -

X,r6 -i- X,,uru ± X.,,uóu + X,r6u 16)

= Y, + Y,u + Y,u

Yc + }Ç.u1+ Y,,vr2 +

'4- Yvu + Y,tu2 + Y,r +

+ Y,rv2

+ Y,r62 + Y,,ru - Yru ± Y6 +

*

Yr,6r - Yòu +

Y,.,»5u + Yr,,vri5 (¡7)

N= Pi, + N,,u + N,u2 + Nv +

V,u3 + N,,vr2

-

-'- N LU -

_{Vuu + N,r}

-UI

-

₍₁₎ = (2) VIi)

-¡U) ₌

_{- x) + (y}

_-

yI)2

₍₃₎

x =x + x _{cos ¿Ii - y»}sin ¿b (4)

= y + x sin ¿b + y,'' cos çl' (5)

For conenience. P is defined as

= 2r -

+ ii, (6)

+ + + N,,ru + + Nb

+

+ N)òv +

+ Nóu

+ N6u + N,6uró

(18)

where subscripts o and b denote propeller and rudder angle effect, respectively. The resistance of the vessel R at velocity u is related to the first four terms in Eq. (16) by

R X + Xu +

+ (19)

The hydrodynamic forces and moment acting on the vessel

depend not only on the instantaneous vessel motions, but also

the history of the surrounding fluid motion. The effect of the history of the surrounding fluid motion is referred to as the hvdrodvna,nic memory effect. This willbe studied in a separate paper.

Mooring Line Tension. The tension in the ith mooring line for the surge and sway directions can be represented as

Tg. = +T" cos »'

R' cos i/i + R'' sin ¿1' (20)

= T' sin

.°' _{= RY' cos ii'}

_R'

_{sin ib (21)}

The components of the mooring line tension Rk". RY' in the x-y plane are given by

R'» = T" cos y'

(22)

R» = T' sin y' (23)

Current Load. Current loads are implicitly included in the expressions of Eqs. (16). (17), and (18) by using the relative velocities of the vessel with respect to the surrounding water. Therefore, current loads are accurate to third order. This ap-proach is more accurate than the usual method based on pro-jected area and drag coefficients. With no loss of generality. the current is assumed to be directed to the negative x-axis in the earth-fixed coordinate system (a = O deg).

Wind Load. Steady wind forces and moment exerted on the vessel are expressed as

Xw =

= PUC)W(a,)AL = PUUCZ(a,)L4L

where p is the density of air; U is the mean wind velocity

relative to the air at a standard height of 10 m above the calm

water level; Ar and AL are the maximum transverse and lateral

projected areas of the above-water part of the vessel, respec-tively: L is the length of the vessel. The relative angle a, of wind direction is defined as

a, = a, çt'

where a i th absolute angle of wind direction in the (x. y) coordinate system. The coefficients C(a,). C}W(a,). Cz4(a,)

depend on the vessel tpe. superstructure location, loading

con-dition. and the relative angle a, of the wind direction. These coefficients can be obtained from experimental and full-scale

measurement data.

Slow-Drift Load. The common approach to formulation of nonlinear wave-body interaction problems in ship and offshore hydrodynamics is based on potential flow theory using the wave

amplitude to length-ratio perturbation parameter. The solution of the second-order problem results in three types of wave

excitation forces: mean, difference-frequency (slowly varying).

and sum-frequenc ) springing) forces. Mean and sims h

vary-ing drift forces are of importance in moorvary-ing problems.

Nemans approximation (1974) for slowly varying drift

forces has been used extensively for many applications in ship

hydrodnamics. The important practical consequence of this approach is that slowly varying drift forces depend merely on

evaluation of second-order mean drift forces, and this can

re-duce the computational time significantly. The formula for slowly varying drift forces is

FSV

iv

_{y si r}

k v.

= _{AAT cas ((Wk w)t + (}

_-

_{)J (24)}

j-1 k=I

where the coefficient TL

0.5(D + D)

and i = 1. 2. 6

indicate surge. sway, and yaw directions, respectively. A.

and are the wave amplitude, circular frequency, and random phase angle of component numberj(ork) in the total number of wave components N. The random phase angles are uniformly

distributed between O and 2 and remain constant in time. The

wave amplitude .4 (or A can be computed from the wave spectrum S(.') as

A,=2vS(w)&

(25)

where is the difference between adjacent frequencies. The two-parameter Bretschneider wave energy spectrum formula is

used to express S( .). The two parameters are the significant wave height H, and the mean wave period T.

The time mean ' alue over the diagonal term' (when_{k = j}

in Eq. 24) gives the mean drift forces and the mean of the

off-diagonal terms (when k j) vanishes. Mean drift forces can be written as

F'

(Xwy. Ywv,Nwv)T= A'D (26)

The transfer functions D' for mean wave drift force are approxi-mated as

D' = - = p,gLC,0 cos3 (9 - i/i) (27)

(28)

D6 N _PgLCZDsin 2(8e ib) (29)

where is the s ave amplitude. p, is the density of water. 8

is the absolute angle of wave direction, and g is gravitational acceleration. The coefficients C0. Cr0. C can be obtained from the complete linear (first-order) solution of the

sea-keep-ing problem. Further details about the methods of predictsea-keep-ing mean drift forces can be found, for example. in the study of Ogilvie (1983). In this paper. it is assumed that these can be given simply in terms of the ship particulars (Hirano et al..

1980).

2.3 Mooring Line Model. In our line of work, we use

three t pes of quasi-static mooring line models: nylon or polyes-ter rope. chain, or steel cable. The stability of mooring systems is affected significantly by the line model (Berriitsas and

Papou-lias. 1990). Chains are usually fully submerged in water and

only slightly extensible. They experience large drag force and

undergo two-dimensional deformation in the vertical plane

modeled by catenarv equations. Steel cables are modeled by a

three-dimensional, large-strain, finite element model which takes into account drag force, weight. and extensibility. They

are lighter than chains and have less drag. but are more extensi-ble. Elastic ropes are light, considerably extensible, nearly

Table i Principal particulars of vessels and mooring line arrangements used in simulations

ant in water. Nonlinear behavior is due to their extensibility. and they are modeled as follows (McKenna and Wong. 1979):

(/ _1)

where T is the actual tension in the rope. Sb is the average

breaking strength. p and q are empirically determined constants. and i, is the working length of the unstrained rope. A wet nylon

rope of 120 mm diameter(Sb = 680. 625 lbf. p = 9.78. q = 1.93) is selected for the numerical applications presented in this paper.

Initial pretension ïs applied to one or more mooring lines. The pretension T, can be defined in terms of the initial strain E, of the line as

T, S5p ₍₃₁₎

E,

-

IS (32)

where i, denotes the initial length of the pretensioned mooring

line.

3 Richness of SMS Dynamics

A barge with stabilizing skegs and a tanker without propeller are selected for numerical applications. Their principal

particu-lars and mooring line arrangements used in simulations are summarized in Table I. Configuration FmAs denotes a (m +

s)-line SMS with m lines attached on the moored vessel forward

of the center of gravity and with s lines aft of the center of gravity. Pretension is imposed on the forward mooring lines

and is calculated as a function of initial strain E,. Slow motion derivatives of the two vessels are provided in Bernitsas and

Kekridis (1986).

Nonlinear dynamic behavior of mooring systems such as

sta-ble or unstasta-ble focus near an equilibrium, limit cycle. and cha-otic motion hase been demonstrated by systematic and exten-sive simulations (Papoulias and Bernitsas. 1988: Chung and Bernitsas. 1992: Bernitsas and Garza-Rios. l996}. In this sec-tion. the richness of SMS dynamics and the effect of slow-drift

loads are demonstrated by simulation in Figs. 2 to 6. The

fourth-order Runge-Kutta time-stepping method is implemented be-cause it is relatively simple and sufficiently accurate. The geo-metric points GI to G4 used to define SMS configuration in Figs. 2 to 6 do not cover the entire range of design practice. They v.ere selected to demonstrate SMS d>namics. In Section

5. these simulations are studied further in relation to catastrophe

sets in Figs. 7 to Il.

3.1 Barge With Skegs. Figure 2 shows a periodic

oscilla-tion (limit cycle) of a two-line barge with skegs SMS about equilibrium A. which has zero 'aw angle. The amplitude of the aw angle increases shen mean drift forces are applied in the

T = Shp (30)

current direction (head seas 9,, = O deg). The system still

under-goes a limit cycle and the increased amplitude of yaw angle

due to mean drift forces results in higher tension in the mooring

lines. A different angle of wave direction (&,, = 30 deg) leads

to a nonsymmetric equilibrium position with nonzero yaw angle. Figure 3 shows that the dynamical behavior of mooring systems can be different qualitatively, depending on the mean drift force.

As can be seen in Fig. 3. the two-line barge with skegs SMS

experiences stable focus behavior and converges to equilibrium A when only current load is applied. A limit cycle near equilib-rium A appears when mean drift forces are applied. This simula-tion shows that an SMS may exhibit large-amplitude slow

peri-odic motions due to mean drift forces. As will be e'splained in Section 4. this is the result of dynamic loss of stability and not

resonance. In Fig. 4. slowly varying drift forces have a

stabiliz-ing effect on the SMS: actually, higher slowly varystabiliz-ing drift

forces result in a stable equilibrium A. This shows that there is no resonance in yaw. In surge. however. Fig. 5 shows resonant

motion on top of the periodic motion due to loss of stability.

3.2 Tanker Without Propeller. Figure 6 shows that a three-line tanker without propeller SMS converges to equilib-rium B in head current without waves: equilibequilib-rium A becomes unstable and the system conterges to equilibnum B. which has a nonzero yaw angle. When mean drift forces are added to the excitation. equilibrium A becomes stable. In this case, mean

drift forces have a favorable effect on the system stability. This

is the opposite effect to that obseed in the barge with skegs

SMS in Fig. 3. The system is unstable and the aw angle

fluctu-ates about equilibrium B when slowly varying drift forces are applied in head seas.

4 Nonlinear Stability Analysis

4.1 State Space Representation. The SMS model pre-sented in Section 2 consists of the equations of motion (13)-(15) and kinematic relations (7)(9). It can be modeled by

six first-order nonlinear coupled ordinary differential equations using the following six state variables: (X1 = u. .r i. x, = r.

r4 = t. .r5 = y. r6 = ib). In Cauchy standard form, they become

=

_mX,

{XH(XI.X..X) tRY Cos.t - R sin.v5))

m.x..r, -

+ X(x6)

J m - X,

(1N)

= D f YH(xr r-. x,) +

(Rcosx - R' sin.r5)J

+ (

- V

Ott1.v I.r) D

-f

(x,R

+ v;'Ry') sin .r J +

\'

(x6) + N141 (.)J

{YH(.VI. X:..r) + (R' cos x - sin r5)

+ f

nrr ± },ix)

Y,( .v)f

v, R, ) cos r5)

204 / Vol. 120, NOVEMBER 1998 Transactions of the ASME

System 2.Line SMS 3-Line SMS Configuration FIAi F1A2

\eseI Barge with Skegs Tanker without Propeller

L 191.56 ft 1066.3 ft B 35.00 ft 173.90 ft T 10.42 ft 71.30 ft CB 0.855 0.831 E1 0.013 0.04 U 2 knots 2 knots H413 4.5 ft 8.0 ft.

+ -

t:. .t) (.r D

Fig. 2 Effect of mean drift forces on two-line SMS (FIAI, I_IL = 0.3, x,IL = 0.3, E, 0.013), barge with skegs

0.03

0.02

.-0.01

1/L=0.6, x/L=0.3 (G2), E1 = 0.0)3

w/o mean drift

w mean drift

lt

I $

I

I I t i , , I i , ,

I Ihlilili, t

, , , n 150 200 250 300 350 400

Nondimensional Time (Ut&)

Fig. 3 Effect of mean drift forces on two-line SMS (FIAi, I_IL = 0.6, x,,/L = 0.3, E, = 0.013), barge with skegs

X4 = X1 COS X,, - X2 Sifl X - U cos a 36

= X1 Sifl X X: COS X, + U sin a ( 37

= 3S)

where D = (J - N,)(m - Y) - NY,. Evolution Eqs.

(33-(38) can be written in the form of one six-dimensional first-order vector differential equation as

= f(x).

fC'. f:6-

(39)

where x denotes the derivative of stale vector x with respect to

Journal of Offshore Mechanics and Arctic Engineering NOVEMBER 1998, Vol. 120 / 205

C D . 0.00 Q) C e-0.01 -0.02 -003 t I I I I i 0 50 100 D {,VLV. X', x) .v

R:' - v R) cos

_J

(m - Y)

_(,rR

+ vRY) sin x J L) (m Y)

{N(.) + N5())

(35)

0.03

Fig. 4 _{Effect of slow-drift loads on two-line SMS}

(FIAi, I,/L = 0.4, x,/L = 0.3, E,_{= 0.013), yaw, barge with skegs}

/L=0.4, x/L=0.3 (G3), E, = 0.013

Fig. 5

Effect of slow-drift loads on two-line SMS (F1A1,Ç/L = 0.4, x,,/L = 0.3, E, _{0.013), surge, barge with skegs}

time t. L6 is the six-dimensional _{Euclidean space. and C' is the} class of continuousl _{differentiable functions. Since the vector} field tin Eq. 139 does not depend_{explicitly on time i. the system} is said to be auro,i,_{mous. The mean drift forces as expressed by} Eq. (26) are autonomous. The slov,_{lv varying drift forces given} b Eq. i24). howe _{er. depend explicitly on time and would render} the SMS model nonautonomous._{In general. stability analysis of}

a nonautonomous nonlinear system is not possible. Time

simula-fon of a nonautonomous s _{stem hich includes slowly varing}

drift forces. hoeer. _{can be pertbrmed ithout any difficulty}

and is used in Sections. _{and 5 to assess the effects of}

nonautono-mous terms on the stability of the _{corresponding autonomous} system.

4.2 Equilibria of SMS. _{Equilibria of the vector field (39)} are the points X E R such that

O = f(X). _{X = (., .t.. ). (40)}

These are stationary solutions. _{Equilibria can be interpretedas} intersections of the null clines. The equilibria of SMS. which

0.02 0.01 (e V cc 0.00 (e >-0.01 -0.02 -0.03 /L=0.4, x/LO.3 t (G3), , E, = I 0.013

i)i

- I $ J:tJ $ w meandnft W slowty-vasying $

i.

50 100 150 200 250

Nondimensional Time (ULt)

206 / Vol. 120, NOVEMBER ₁₉₉₈

0.04 0.03 (e 0.02 0.01 0.00 -0.01 0

1,/L=0.i5, x,/L=O.S2 (G4), E, = 0.04 _{w/o mean drift}

w mean drift w slowly-varying

50 100

Nondimensional Tui,e (Utt)

150 200

Fig. 6 Effect of slow-drift loads on three-line SMS (FIA2, I,.,/L = 0.15, x0/L = 0.52, E, = 0.04), tanker without propefler

+G2

wlo mean drift

H,0 = 2.0 ft H,0 = 4.5 ft

IL

Fig. 7 Catastrophe set of two-line SMS (FIAI, (y,/B = 0.0, E, = 0.013), barge with skegs

are the solutions to a nonlinear Eq. (40 i. depend on the mooring Let be an equilihnum point of the nonlinear system (39) line and system configuration. as well as on the environmental Expar.iing the nght-hand side of Eq. (39) into its Taylor series

conditions, about point . we obtain

4.3 Stability in the Sense of Lyapunov. The behaviorof

a nonlinear system near an equilibrium point can be determined via linearization with respect to that point. Since linearization is an approximation in the neighborhoodofan equilibnum point, it

can predict only the local behavior ofthe nonlinear system in

the vicinity of that point (Guckenheimer and Holmes. 1983: SedeI. 1988). df X

= f

_{) +}

-dx,

'-:';

±

± HOT. -.

where deviation from equilibrium is defined as

(41)

Journal of Offshore Mechanics and Arctic Engineering NOVEMBER 1998, VoI. 120 / 207

1.0 0.8 0.6

tì

0.4 0.2 0.000 I H I I :' 0.1 0.2 0.3 0.4 05

Fig. 8 Catastrophe set of two-line SMS (F2AO, yIB = 0.0 fl = 5 deg), barge with skegs

Fig. 9 Catastrophe set of three-line SMS (F2AI, y,/B 0.5. 11 = 15 deg, E = 0.013), barge with skegs

the nonlinear state Eq.(39)can be approximated by the linear

state equation

=[AJ.

[A) (44)

where [Al Df() ts called the Jacobian matrix evaluated at the equilibrium point . If the eigenvalues of [A]. X,. have nonzero real parts (usually called hperbolic or

nondegener-ate),then it can be expected that the trajectories of the nonlinear

system in a small neighborhood of an equilibrium point are close to the trajectories of its linearization about that point.

5.0 4.0 3.0 2.0 1.0 An '00 I

w/o mean drift

I'1,3=2.Oft - H,=4.5ft

-0.1 0.2 IL 0.3 0.4 05

208 I Vol. 120, NOVEMBER 1998 _{Transactions of the ASME}

E(I) = x(t) -

(42)

Since is an equilibrium point, we have f() = O. Then. the state equation can be written as

+ HOT. (43)

If attention is paid to a sufficiently small neighborhood of the equilibrium point so that the higher-order terms are negligible.

1.5

1.0

0.5

w/o mean drift

H,,3 = 4.0 ft I-Ç3 = 8.0 ft II Iv -St_'S--. I III

Fig. 10 Catastrophe set of three-line SMS (FIA2,y0/B = 0.5, lì = 5 deg, E, = 0.04), tanker without propeller

Fig. 11 Catastrophe set of four-line SMS (F2A2, y,,/B,,,,, = 0.0, y0/B.ft = 0.5, 1? = 5 deg, E, 0.03), tanker without propeller

4.4 Bifurcation and Catastrophe Sets of SMS. A bifur-cation is a qualitative change in a limit set of the s stem as a parameter is infinitesimally perturbed. Bifurcations can be interpreted as the appearance or disappearance of a limit set and the change in the stability type of a limit set. A bifurcation also occurs hen a nonstable limit set of the system remains nonstable. but undergoes a change in the unstable manifold dimension. Saddle-node, pitchfork. and Hopf bifurcation have been found in the nonlinear dynamics of mooring systems

(Pa-poulias and Bernitsas. 1988: Chung and Bernitsas. 1992:

Berntt-sas and Garza-Rios. 1996). A catastrophe set is the set of all loci of bifurcations in a parametric design space. It defines the boundaries beteen regions of qualitatiel dIfferent system

dynamics and the morphogeneses occurring as boundaries are

crossed.

In the numerical applications below, catastrophe sets are

con-structed for several SMS configurations of a barge with skegs and a tanker without propeller. The effect of mean drift forces

on bifurcation of SMS dynamics is demonstrated in the

catastro-phe sets of Figs. 7 to 11. Five different SMS with symmetric

Journal of Offshore Mechanics and Arctic Engineering NOVEMBER 1998. Vol. 120 / 209

0.35 0.40 0.45 0.50 IL