Stability criteria for the slow motion nonlinear dynamics of towing and mooring systems

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STABILITY CRrFERIA FOR T}IESLÔW MOTION NONLiNEAR DYNAMICS OF TOWING AND MOORING SYSTEMS

by

L 0. Garza-Rios

M. M. Bernitsas

Prepaid for the University of Michigan/Sea Grant/Jndustzy Consortium in Offshore Engineering

No. 332

November 1996

Department of Naval Architeiure and Marine Engjneedrg College of Engineering The University of Michigan Ann.Aibor, Michigan 48109-2145

Analytical expressions for the necessary and sufficient conditions for Stability of towing and mooring systems with several lines are derived based on the Routh stability criteria. These expressions are combined to provide physics-based relations for the bifurcation boundaries occurring in towing and mooring systems as a parameter is varied, and thus can be used for design of mooring and towing systems. Further, these expressions prove that several rules of thumb generally are not applicable The special case of symmetric towing/mooring systems is covered as well. in this particular case, the analytical expressions can be simplified to derive a number of rules that can be implemented readily in design The mathematical model consists of the horizontal plane maneuvering equations (surge, sway, yaw) of the towed/moored vessel

without memory under steady current excitation, and a quasistatic model fOr the

mooring/towing lines. The line static behavior is modeled by (I) nonlinear elastic strings

(nylon, polyester); or (ii) heavy, inextensible catenaries. Othet mooring line models (i.e. three-dimensional extensible finite element model) and external excitations (i.e. wind and second order mean drift forces) can be incorporated in the model

ACKNOWLEDGMENTS

This report is a result of research sponsored by the University of Michigan/Sea

Grant/Industry Consortium in Offshore Engineering under Michigan Sea Grant College

Program, projects number RIF-29 and RJT-35 under grant numbers NA89AA-DSGO83C

ad

DOC-NA36RG0506 from the Office of Sea Grant NatiOnal Oceanic and Atmospheric Administration (NOAA), U.S. Departthent of Commerce, and flUids from the State of Michigan. Industry participants include Amoco, Inc.; Conoco, Inc.; Exxon Production

Research; Mobil Research and Development; Shell CompaniesFoundation; and ElfAquitaine, Pau, France.

p1

ABSTRACT

ii

ACKNOWLEDGMENTS

iii

LISTOFFIGURES

vi

LIST OF TABLES

vii

LIST OF APPENDICES

viii

NOMENCLATURE

ix

INTRODUCTION

1

CHAPTER

1. THEOREI1CAL BACKGROUND 2

1.1. Routh's Stability Criterion 2

MATKEMATICAL MODEL

4

2.1. Equations of Motion 5

2.2. Kinematic and Geometric Relations 7

2.3. Towing/Mooring Line Models 8

STATE SPACE REPRESENTATION AND

EQUILIBRIA OF TOWING/MOORING

SYSTEMS

10

3.1. State Space Representation : 10

3.2. Equilibria of Towing and Mooring Systems 11

ANALYTIcAL STABILITY

CRITERIA AND

LOSS OF STABILITY

14

4.1. ExpansionoftheEquatioflsOfMotiOfl 14

4.1.1. Expansion of the Surge Equation of Motion 16 4.1.2. Expansion of the Sway Equation of Motion 19 4.1.3. Expansion of the Yaw Equation ofMOtion 21

4.2. Stability Coefficients and Characteristic Equation 24 4.3. Stability Criteria for Towing/MooringSystems 28

4.4. Loss of Stability in Towing/Mooring Systems 29

4.5. Numerical Examples

31

V. SYMMETRIC TOWING/MOORING SYSTEMS

34

5.1. Equilibrium an4 Expansion of Variables 35

5.2. Equations of Motion in Symmetric Towing/Mooring Systems 38

5.3. Stability Coefficients and Characteristic Equation 39 5.4. Stability Criteria for Symnietdc Towing/Mooring Systems 40 5.5. Active Stability Criteria for Symmetric Systems 41

5.6. Loss of Stability in Symmetric Systems 42

5.7. Numerical Examples

45

APPENDICES

47

BIBLIOGRAPHY

68

figure

2.1. Geomey of general towing/mooring system

4

4.1. Values of Cp and CRC, 3-ic barge with skegs: £1IJL 0.5, yb/B

03,

£2=15°,E,=O.012

33

5.1. Góometiy of sythmetiic towing/mOoring system 34

5.2. Geometric relations of parameters xi,, y, and y iii syrninetric systems 36 5.3. Values of R1 and R2, 3-line barge with skegs: £/L = 0.5, y1/B = 0.5,

LIST OF TABLES

Table

P.21

4.1. Eigenvalues for 3-line barge with skegs 32

4.2. Sixth-orderstabilityciiteriafor3-linebargewithskegs 32 5.1. Fourth-Order stability criteria for. 3-line barge with skegs 45

App efldi*

First Order Expansion of Geometric Angles - - 48

First Qrder Variation of Towing/Mooring Line Length 51

First Otder Variation of Towing/Mooring Line Tension 53

First Otder Variation of Vessel Velocities and Accelerations in Terms of the Absolute Position Vector and its Time Derivatives 55

First Order Variation of Hydrodynamic Forces and Moment in, Terms of

the Absolute.Position Vector and its Time Derivatives = 57

6 First Order Expansion of the Equations of Motion in Terms of the Position

NOMENCLATURE

a,b,c

dummyindependent variables representing u,v,r and ö infunctions X(u,v,r),

Y(U,v,r) and N(u,v,r) B beam of the vessel

CB block coefficient of vessel

CG center of gravity of vessel D depth of vessel

DIS

Dynamic Loss of Stability

DUM DimensiOn of the Unstable Manifold initial strain inthefth line

h depthofimmersionOfthelifleS

H draft of vessel

mass moment of inertia of vessel about the Z.axis horizontal length of the it/i line

L, LBP length of the vessel

initial length of the ith pretensionul line working length of the undefOrmed ith line m mass of vessel

A nUmber of towing/mooring lines

NSC Number of Sign Changes

N(u,v,r) velocity dependenthydrodyflaifliC moment about the Z-axis

Na derivative of yaw hydrodynamic nnnt with respect to a

N01, derivative of yaw hydrodynamiC moment with respect to a and b

N derivative of yaw hydrodynamic momentwith respect to a b and c

N0 yaw hydrodynamic moment dUe to propeller

N,. hydrodynarnic added moment of inertia in yaw due to a unit sway acceleraton

N,

hydrodynamic added moment of inertia in yaw due to a Unit yaw acceleration

P

vertical force per unit catenary length

r

yaw angular velocityof vessel with respect to wter R vessel resistance

SLS t (i) i)

I

U Uc U V (x,y) (X,Y,Z) X(u1 v, r)

xc

Xab Xabc xo (i) (i) Xm 'Ym (i) (1) vXp Yp t (i) (i) XT 'Y xu Y(u, v, r) 1'ab 'abc yo Yr Yv

Static Loss of Stability

Tension in the ith line

Initial pretension in the ith line

relative vek)city between vessel and surrounding water velocity of current

forward velocity of vessel with respect to water lateral velocity of vessel with respect to water earth-fixed coordinates

body-fixed coordinates

velocity dependent hydrodynamic force in the X-direction derivative of surge hydrodynamic force with respect to a derivative of surge hydrodynamic force with respect to a and b derivative of surge hydrodynamic force with respect to a, b and c surge hydrodynamic force due to propeller

mooring coordinates of the ith line relative to theearth fixed coordinates body thed coordinates of the ith fairlead

attachment coordinates of the ith line relative to the earth fixed coordinates hydrodynamic added mass in surge due to a unit surge acceleration

velocity dependent hydrodynamic force in the Y-direçtion derivative of sway hydrodynamic force with respect to a derivative of sway hydrodynamic force with respect to a and b derivative of sway hydrodynamic force with respect to a, b and c sway hydrodynainic force due to propeller

hydmdynainic added mass in sway due to a unit yaw acceleration hydmdynamic added mass in sway due to a unit sway acceleration

Greek Symbols

Ai) _{angle between the x-axis and the ith line, measured counter-clockwise}

8 rudder angle

A displacement of vessel

ith eigenvalue Of the linearized system in the stability criteria yaw (or drift) angle

angle between the ith line and the X axis, meaired counter-clockwise angle of aperture between the x-axis and the ith line

Special Symbols

D time derivative operator = eqüililriurn of (.)

The objectives in preliminary towing and mooring analysis axe to study the motions and stability properties of the vessel. Systematk parametric analysis can be used to determine the effects of various parameters such as length, orientation, position of the towing/mooring lines,

hull hydrodynamics, etc., on the system response. The problems involving maneuvering,

towing and mooring have been studied widely theoretically, numerically and experimentally [1,

2, 3, 10, 12, 13, 16, 17, 18, 24, 25]. Extensive analyses based on nonlinear dynamics and

bifurcation theories have been conducted for Single Point Mooring (SPM) [7], Two-Point Mooring (TPM) (4, 11] and Spread Mooring Systems (SMS) [5, 14), including their related towing problems [6,8]. The methodology for design of towing/mooring systems developed in our previous work consists of defining an. appropriate mathematical model for a particular

system, evaluating numerically its eigenvalues to determine its stability, and performing

bifurcation analysis to determine the qualitative changes of the system dynamics as a parameter, or a limited numbet of parameters, are varied. This approach drastically reduces trial and error in design, and nearly eliminates extensive nonlinear time simulations.

Further, the dynamics of the system can be. discerned without numerically obtnining the system eigenvalues and directly applying bifurcation theory by the methodology developed in this report. Specifically, analytical expressions for the necessary and sufficient conditions for the system stability are derived. These expressions, which are directly related to the system

eigenvalues (see Chapters I, IV and V), provide physics-based relations that reveal the morphogeneses ocurring in towing and mooring systems as bifurcation boundaries are crossed. Such expressions can be combined to provide analytical (closed form solutions)

means for determiningpoints in the parametric design space where static and dynamic loss of stability occur, as well as elementary singularities and roots to chaos. The expressions derived in this report also serve to determine analytically the dependence of the system on any design parameter.

I. THEORETICAL BACKGROUND

The stability of towing and mooring systems can be evaluated by establishing an appropriate mathematical model, and numerically obtaining the eigenvalues of the system around an equilibrium position.. If all eigenvalues have negative real parts, the system is stable [15]. If at least one eigenvalue has a positive real part, the system. is unstable, and all

trajectories initiating near that particular equilibrium position will deviate from it Alternatively, the stability characteristics of a linear, time-invariant system can also be determined from the

characteristic equation of the system, as described in Section 1.1. The stability criterion

developed by E.J. Routh, provides a means for determining the stability of the system without evaluating explicitly the roots (or eigenvalues) of the characteristic equation.

1.1. Routh's Stability Criterion

The Routh stability criterion establishes a Simple method to determine the number of

eigenvalues in the system with positive real part [22]. Consider the characteristic equalion

a_1a'

+a,_2a"2

a,_3a'

+ ... + a1a 00=0

The coefficients aj of(1.1) are arranged in the patterns Shown in the Routhian array below, andthese coefficients are used to evaluate the remaining constants to complete the array.

Routhian Array 2 an an_2

0n3

bi b2 b3 C _C2 C3 d1 d2 d3 (71 gi

a0

The coefficients b1, b1, b3, etc. in the Routhian array are defined as follows:

_an_1

aa_

an_i

The pattern is continued until the rest of the b's are zero. The roW corresponding to the

coefficients c1, c2, c3, etc, is formed by using the

a'1 and o"

rows. The constants are:

C₁ I

_b1a5a..1b3

_b1a7a1b4

b1

,

b1

,

c3

This is continued until no more c terms are present. The rest of the rows are formed by

following the same procedure to the

a°

row. The complete array is triangular,and the last twQ rows contain only one term eaciL

Necessaiy and sufficient conditions for the system to be stable axe

All the coefficients of the characteristic equation (1.1) must exist.

All the coefficieflts of the first column in the Routh array must have the same sign

If the second condition above is not met, the number of sign changes will indicate the number of roots of the characteristic equation (1.1) (or eigenvalues) which have positive real parts.

In this report, characteristic equations for general and symmetric towing/mooring systems of the form (1.1) are derived. The Routh stability criterion is then applied to specific systems to evaluate their stab ility. The expressions derived for stability in this report are analytical. Analytical expressions for the bifurcations occurring ifl the system as a parameter varies are derived in Chapters lv and V. and constitute the major cOntribution of this report

II. MATHEMATICAL MODEL

Figure 2.1 depicts the geometry of a generaltowing/mooring system, and has two different coordinate systems: (1) A body fixed reference frame (X,Y,Z) with its origin located at the center of gravity (CG) of the Vessel. In this frame, (X, Y) is the horizontal plane and (X, 2) the center plane Of symmetry. (ii) An inertial frame (x,y) fixed to the earth and with the origin located at Terminal 1. The coordinate x is defined perpendicular to the initial direction of the current; yis defined normal to x. The towing/mooring lines are numbered counter-clockwise starting from the line at Terminal 1.

Figure 2.1: Geomeu7 of general towing/moofing system

In Figure 2.1, the geometric design variables required in the study of towing and mooring systems are the following:

of the Wi line with respect to the (x,y) frame; y is the angle ftom the x-axis to the iih line,

measured counter-clockwise; is the tension of the ith line; L' is the length of the ith line; ' is the yaw (or drift) ang]e. It mustbe mentioned that, in general towing systems use only a limited nuniber of towing lines 2), directed toward the towing vessel or vessels.

2.1. Equations of Motion

The nonlinear equations of motion in surge, sway and yaw areexpressed in the (x,y) system of coordinates as follows:

(mXu)ü_m,v=X(u,v,T)+ZTge+Fwge,

(rn_}_Yt+mru=Y(uV,r)+7ay+Fsw

N

+(1 - Nfr)? =N(u,v,r)

+

-+

where m is the mass of the vessel; I is the moment of inertia about the Z-axis; u, v, and r are the relathre velocities of the vessel with respect to water in surge, sway and yaw, respectively. lh the above equatiofls, the linear acceleration terms are included in the left band side and

represent added mass and added moment

of inertia terms. The expressions X(u,v,r),

Y(u,v,r) and N(u,v,r) are the velocity dependent hydrodynanñc fozces and moment in the

surge, sway and yaw directions, respectively. These expressions have been expanded in Taylor series following [1]. In nonlinearanalysis terms up to third order are used

X(u,v,r) X0 + Xu+ +

!u

+ Xypv2+ X,rr2 +

+Xv2u+

Xrrur2u+

.X&&ô2u+ Xvr+ X6v8+ X,.orô

Y(u,v,r)= 1',, +

}u+

+ yy.

!y:y3.,. !y:yr2

.

!yMvo2

1 ₂ 1

31

21

2

+ Yvu + Yvu + }.r+ Yr +

Y,..rv + YrMrô

yo!y53+!y,fr2+!yfr2

+y&s+

!yt2

+ }'vrö,

N(u,v,r)= N0+Nu+

N0u2

+ Nv+ Nv+

NyMv82

+Nw+

!NvuuVU2

+ Nrr+

!Nr,.T3 +

!

N,rv2

+ !Nr&SröZ

+Nru+

NTUUTUZ + N66+

!Nô3 +Na,öv2 +

N&.rôr2

+N&i+

N&S2

+ Nvr6.

The terms in the above equations represent minIy: damping t&ms associated with viscous effects and energy dissipation due to waves. Subscripts "0" and "3' represent propeller and rudder angle effects.

in equations (2.1X2.3), n is the nunitr of

towin/knooring lines in the system; Fe

F, and N

are the external forces and moment exerted on the vessel, and include steady wind and second order wave mean drift forces; Tsue and. are the tension in each of the

towing/mooring lines for the surge and sway directions respectively

Tswge =

Tcosø= R1cosVf+ Rsiny'

(i= 1 ... (2.7)

,=Tsinw= Rcosv Rsiny,

(i= 1,...

n),

where R and R represent the reaction forces in the x and y-directions as f011ows:

(2.8)

= Tcosy,

(2.9)

I4i)

_{= rj}

_sin

)(0,

(2.10)

(2.5)

2.2. J(inematic andGeometjic Relations

The kinematics of the system relate the velocity components (u, v, r) of the vessel with respect to water to the absolute velocities (i, y, v) with respect to the (x, y) frame as

In. equation (2.11), U is the absolute value of the current speed in the negative x-directiOn.

For thç towing probleth, this accounts for the relative speed between the current and the

tugboat(s). The directional changes between the current and the vessel (i.e. headings) can be integrated in the mathematical model either bymodifying the kinematic relations (2.1 1)-(2.13),

or by rotating the system appropriately.

The geometry of the system, as well as the position of the vessel With respect to its center of gravity and orientation of each of thetowingFmooring lines, are described by the following relations:

w=2ic-y+!1,

(i=1,....n)

(2.14) Xm - XT

_(i=1,...n)

_(2.15)

cosr=

-YrnYT _n) (2.16)

siny=

2

2.

£J(xrnxr)

+(ymYT) ,

(i1,...n)

(2.17)

(i1,...n)

(2.18)

y=y+xsinly+ypcOSIV.

(i=1,...n)

(2.19)

i=ucosjfvsinJFU,

y=vcosy+usin,

ji=r.

(2.11) (2.12) (2.13)

2.3. Towing/MooringLine Models

In towing and mooring, three different types of mooring lines may be used during operations: chains, cables, and/or synthetic ropes.

Chains are heavier and have high

hydtodynamic resistance, are usually fully submerged, only slightly extensible, and have

deformation in the vertical plane. Chains are modeled by the catenary equation (21]. Cables are lighter than chains, have less resistance but are more extensible. Cables may be partially

submerged, and their deformation is in general three-dimensional [21]. Steel cables are

modeled by a nonlinear cable model that takes into consideration partial submergence, three-dimensional deformation, effective tension, extensibility, and nonlinear drag. Elastic ropes (nylon, polyester) are even lighter, have lower resistance, are considerably extensible, function primarily out of the water, and the major nonlinear effect comes from their extensibility. For the analysis presented in this report, nonlinear synthetic ropes and inextensible catenaries are used to model the lines, since they produce analytical expressions for the stability criteria. For the nonlinear elastic string, the line tension is given by the semi-empirical eqiation [20]:

T=SbP(e3).

(2.20)

where T is the actual tension in the synthetic rope, Sb is the average breaking strength; p and q are empirically determined constants, and ,is the working length of the unstrained rope. For

the numerical exainpios presented in this work, Sb = 3025 KN (680,625 lbf), p =9.78, and q = 1.93 [21]. These properties correspond to a wet nylon rope of 120 mm diameter.

For the inextensible catenary model, the horizontal tension is obtained by solving the

equation

i'e

p

Swu)

8

(2.21)

where P is the net vertical force per unit catenary length, and h is the depth of immersion of the line. in this particular case. £ is the horizontal component of the line length [21]. Equation

(2.21) assumes that the catenary line(s) is not in contact with the sea bed. In deep-sea mooring, where a portion of the catenary line touches the sea bed, the expression for the

mooring line tension is different since the horizontal projected length in this case is not equal to the horizontal component of the mooring line length. This case will betreated in a later repolt

retensio: For systems with multiple lines, a vessel maybe placed insuch a position relative to the towing/mooring terminals, that one or several lines may be set to an initial tension (pretension), denoted by Tj. For the nonlinear elastic string, such pretension exists if £ > £.,

andi

fined interms of the initial straihEjas:

_1I-tw

E1

,-,

where £j denotes the initial length of the pretensioned line. The expression for the initial pretension is therefore given by:

T1=S(E,).

(2.23)

For the catenary line, 0< £

<4,

and thus the pretension Tj must be defined accordingly. In this report, the catenary pretension is defined relative to the resistance of the vessel R as follows:

Thus, the horizontal projection of the catenarylength,!, can be defined as

=

2TJRSi_1[Pf4;h2]

(2.25)

Mathematically, the concept of pretension is

used to define the position of the

moored/towed vessel with respect to the (x, y) reference frame. It should be noted, however,

that the mean tension of a line at equilibrium will be different generally from its initial

pretension.

(2.22)

III. STATE SPACE

REPRESENTATION AND EQUILIBRIA

OF TOWING

AND MOORING SYSTEMS

3.1. State Space Representatiofl

The nonlinear model presented inChapter Ills autonomous. Ti-regardlessof the number of

towing/mooring lines, the system can be reduced to a system of six first order nonlinear

coupled differential equations by ellininaling all but six variables [11). For systems where the number of towing/mooring lines is unspecified it is convenient torepresent the system in terms of the positionand velocity vectors of thevessel. Selecting the following state variables vector

x=(x1=u,

x2=v,x3=r,X4X,X5=Y,X6MT,

(3.1)

the system nonlinearmodel can be reWritten in Cauchy standardform as

x1 =

_{(m_Xü){X(123)}

(i0cos.c6+40sinx6)+n1x2x3+ Fsurge(X6)I ,

(3.2)

(I

Nr) _{Y(x1, x2, x3) + .:(}

_{i4' c}

_{_M0sinx)nixix3+}

Fsway(x6)}

N(x,x,x3)+

f(x)R0

y)R0)cosx6}

= }'(x1 , x2 ,x3) + cosx6 smx6)- mx1x3 + Fsway(x6)} +

_(rn;A){

_N(x11x2,x3)+

(x40_ yI)cosx}

+(m_($R)+

yPR0)sinx6+Nyaw(x6)} 10 (3.3)

(34)

where D= (' - N)(m - Yr) NV}.

The:reactiofl forces R

and R, defined by auxiliary

equations (2.9) and (2.10), have been used instead of T, and are indirect functions of the

state variables X5 and xfj.

Hereafter, evolution equations (3.2X3.7) will be denoted as:

x=f(x), feC', f:96_)96,

(3.8)

where R6 is the six-dimensional Eudidean space and C1 IS the class of continuously differentiable functions.

For the state space representation selected above, all variables, vector field f. and the corresponding Jacobian Df(x) are continuousfunctions of i The Jacobian off ex sts even if variables corresponding to slack lines are notdefined (Se, y(O,

ø), because they do not

appear explicitly in the vectorfield. Instead, functions R and - which are continuously differentiable functions in terms of x when one or several towing or mooring lines are slack -appear.

32.

quilibria of Towing and Mooring Systems

Equilibria of the nonlinear system are stationary flows of the vector field (3.1), and canbe computed as intersections of null dines (23]. Accordingly, all equilibria (singular flows or staiionaiy solutions) can be foundby setting the tiflie derivative of the state variable vector to zero, i.e.

isanequilil)riUmof x=f(x) C

f(JEO.

(3.9)

The overbar on the state variable vector denotes an equilibrium state. The solution to the nonlinear component equations in (3.9) produces all possible equilibria in the system.

x4=x1cosx6x2sinx6U,

x5 = x1 snx6 + x2C0Sx6

(3.5)

(3.6)

1z

Relations (3.10X3.12) above can be substituted into equations (3.2)-(3.4), to yield three

nonlinear coupled algebzic equations in terms of

i and

as follows:

0= X0+

+Xö2+

Xô2coSi_ UX66sini6

u2xocos;sin;+

(0cos+ °sin)+

Fsurge() (3.13)

o₌

(1 - N)(Y0 +

Y8+

Yö3)+ }.(N0 + N5ö+ N:3)

-

N)(Yg + Y6)+ Y,.(N0 +

No)] cos;

- N)(Y

+

ly2)

}(N +

!No2)]

siii;

+U2[(Iz_NrXYouu+ Yj) Y(NNo)Jcos2;

4U21(iz N)Y,y+ YrNjy6Sifl26U2[(1z N)Y +

- N)Yywg+

yN,lW]cos2;s1n6_13[(1z

- N)1jy+

- N)+

Y,.x(](t)cos;

-

°sin;)_ y(0cos; +

0sin;)}

- N)Fr,,()+

}N)l(i),

(3.14)

0 N(Y0+ Y68+ Yä)+(m Y)(N0+

N6o+N83)

+UIN,(Yc,,+

Y&8)+(m }XN0+Nö)}cosi

N(Y +

ly2)

_{(m Y)(N +}

x6

The equilibrium velocity vector can be recast in terms of state variable from equations

(3.5X3.7) as:

11=Ucosi6,

x2=mx6,

=0.

(3.10) (3.11) (3.12)

Y6)+(m Y)(N01+ N6)]cos2

+jU2ENvYw+(m_ )No]&sin16U2[NJw+(m

Yv)NvulCOS16Sifl16

yY)N%lU1cos2is 6_U[Ny+(m },)N]sin316

+{[Nv(m

)xJ(°cos; 0sin14}

_±{m_ )y(°COS16+

0sin;)}+NFy(16)+(m Yy)Nyw(16)i (3.15)

where

$)

and are functions of x4, 1 and 16. ThC equilibrium values.for the position vector are obtained by solving the three simultaneous nonlinear algebraic equations (3.13)-(3.15). Due to the complexity of the system,these equations must be solved numerically. For single point towing/mooring systems, these equations may be combined to yield a single algebraic equation that can be solved for the position variable x6= iji [7].

The number of equilibria that exist in the system strongly depends on the number and tension of the lines, the system configuration and the environmental conditions. This is an important design issue since appropriate line and environmental parameters may allow the system to reach different plausible eqUilibria. Towing andmooring systems With relatively small pretension in the lines, for example, may inherently have several equilibria, since the vessel is not restricted from moving. In such cases, one or several lines maybecome slack, and, thus, an initial configuration may be transformed into a different oneafter equilibrium is

achieved. Highly pretensioned systems, on the other band, are configured to restrain the

vessel from moving about its center of gravity, thus limiting the numberof equilibria that can be pttained.

IV. ANALYTICAL STABILITY CRiTERIA AND LOSS OF STABILITY

The stability criteria of the system dynamics introduced in Chapter I can be applied readily in towing and mooring applications based on the mathematicalmodl described in Chapter U,

by converting the system into the form (1.1). The procedure to obtain the characteristic

equation of the system consists of fmding an appropriate equilibrium position for the vessel, perturbhig, expanding and liAiearizirg the equations of motion (2.1X2.3), near anequilibrium position. Then, the resulting equations are combined to yield an equation of the form (1.1).

4.1. gxpansion of the Equations of Motion

Once the equilibrium position and vólocities of the vessel have been identified, the

equilibrium values of the variables appearing in the equationsof motion (2.1X2.3) can be found, and these variables perturbed in the form

( )

= ()

+ 4( ) , (441)

where ()is the variable in question; the overbar indicates the

value of the variable at equilibrium, and the term 4() represents the first order variation (or perturbation) from

equilibrium.

The expansions for the variables involved in (2.1)(2.3) are shown in

Appendixes 15. After expansion of each variablein the form of(4.l), the equations of motion can be written as:

$urge

(in xXcos+ sini4j+ 4') mvz,

(2_xg)cos+(y)sinix

i=

)

(()

+

4t(°)

Sway

(m })(_s

4+cosii74ii4y') YI+miZ4=?

(

ãP.'t.(P.-P

.PPi

+

(7()3T°

,p)'f(x_xm)srnVt(YYm)C0SVYp

_((i)

+ +

4t0 j((i) 4())}4x

-+

AtO)

((0M(0)}4

fit. +

NSinVi4+cOSij74

4v)+(I Nr)4i

+

J7o ( -

x)(xj) sin i +

y

cosi)

-

(

-cos

- y

sini)

(7(i) +M(0)

46i)

(7(i)+ (0) (i) 461)]

[y

c sin 14

(7(i)+460)f

y (4.2) (4.3)

)

w} _±{[o+

)

1 J

(7(0+4(i))

i)

+

o)[4'cosr_Yi'sm]4Y

_

i=1

____46'

-

cos

y)

sin =i

((0 si))

+

4()

(

-sin ₊ cosii) ₁

(e(0+

z=

K,

;;_

_F(j)

b=-cos7

a=-siniji,

cos7+

+_y9CO5!J7] (4.4)

In the following subsections, each of the equations of motion (4.2)-(4.4) is further

expanded, separately.

4.1.1. expansion of the Surge Equation of Motion

The Surge equation of motion (4.2) is of the form:

1=

2+(1+ZM)

(aibøx+:1Y+$iAV)

(4.5)

where:

The expansion of an equation of the form (4.5) is shown in Appendix &

After substituting the relations involved in (4.2) from Appendices 1-5, the surge equation

of motiOn becomes:

.i.

1.

(m-X)cosx-

cosv.s1n

+(mX,jsin'i'4y .s1nV,+cosV( Y

1+ {T(0coso9

'{

±{cosV}±{T00±()}4x

{

7}4y_

.{

}+ {

oo(__

) }

+{çsoo}4Y ±{

+x{

s

o0}_

{

(

-xsin)+Ycosã]}4

_cos(xpsino0 +

cos

(4'sin+

YcOS?o(0)}4Vf.

(4.6)

Further, noticing that 2

? cos

} =0 (equilibrium), the surge equation of motion can be Simplified to the form (A6.9) as

+(m Xü)sini4j

where '8

-

X)cosc74+

(ax

Ei

(ax

ax..

Ay (4.7) fl 17&)

E1 =

Ei(s_sn?(cos0)}

I siny(Ocos(00 (4,9)

)+

{isina0}

+{[cos

±{

cosäi((x

sihö + (i)cosöi(0)}

=

(

_{)_±{7(0sin(}

-V)+ySin(

_i)]}

cos(?

i)[4 sin(f

- 7)

y

cos(?°

-4)

sin Zö ₊ cos (i)

- yjP

(4.10)

Ei =

(s_coso

o)}+{

} =

?(') sin(?'

-

+ cos?

cos(?°

(4.8)

4.1.2. expansion of the Sway Equation of Motion

The sway equation of motion (4.3) is of the form (4.5), where.

f=(rn-

)s

cos7_sin)4i+(m_

})cosjiAj

z=Y,

and A are defined as previously,

b1=sini,

a=cosi,

$

Following the procedure of Appendix 6, and substituting the relations of Appendixes 1-5, the sway equation of motion (4.3) can be fuither expanded as:

dY(dYdY

(rn

_(co$7- ?sin,)M+(m

Y)cosi4

_(sini+

Y,4+

(_ J4fr

-

= V

{(i)

sin 9

{c0(

} +

_g{T(i)

_si

[°7]}&

n{(i)

_cos?(i)}Ar

j(i) cosY(0}4x.

Y±{5m}4y

{o

_Sin0)X(

k))}

(in

E4x

+(m

sini)L+ E4y

4.12) whete (4.14) n =

{

(i)

-(cosy0sin0

_sin)}

(4.13)

cos?(0sin0}

g(i)

cosy(0sin(y0

-

j?) =

_{sincos0

-+Sin0)}_E{)Sm?(05m0}

{(cos0

+v±{4

siny cos?o

}4

j7(0[cosã(0

sin

<i) kI[x)

sin

k)(k)

_cosJ

+±{ç

sin sin +

Cos0)}4vF

sin ₊

$

cos))}4Y1.

(4.11)

Noticing that the term V

±{7(0

sin 0}=0 (equilibrium condition), the sway equation of motion can be simplified to the fonti (A6.9):

4.1.3. xpansionof the Yaw Equation of Motioji

The yaw equation of motion (4.4) is of the form (4.5), where

._..(av _dN.2.

f=Nsinr4x -cos,s1nv' 1xN1,cosy'4y

_(sini+

cosJi)4+(Iz

- N,)Aiji+

(Nii_)

Z=N,

and RaredefinedasprevicMis1y,

aj

=[(-cos

)(i)]

(xr

+

Following the expansion in Appendix 6, and substituting the relations of Appendixes 1-5, the yaw equation of motion becomes:

- sin sin ,I:i) + $:

sin öj +

cos

-fl[xjP cos(Ø0 ?) + y sin(?

22

-

,'

_{r(i)(x(i)

sino

cos0)}

-

{cos7(0

}

sin i+

cos?)}4x

+{r(1)(x

sin + )cosö

))(D;(k)

J}

+

(4)

_siñ +

y cos )cos?(0}Ax

-

±{°

}

cosi -

y,i)sinJi)

}4Y+{T(0

(xP sin?i +y cosö<°)

(sin

_)}4Y

_±{c;;.($)

j11(i)

cos°)sin

yto},iy

+{:

(4'

+

Ycos))sin?(0}AY

+N

±{4)sin

+(0cos0}

cosö0 -

_y sin o<

[(i)21)2

J]}

I ,

( (k)

(k) + (k)_{cos ?ii} \ 1

_1T(i)(x(i)si+y(i)cosoi(i))

5111 P i=1 k=1

5j<i)

+_y cosa5<0)2}Avf (x sinãi< ₊ (4.16)

Noticing that

-

E{T)(x

sin

io°

+

yPcos0)}

0 (equilibriun.), the yaw equation of motion can be further ftduced to the form (A6.9):

Nsm+s1nvf_cosY(r+E3x

&

_(aN _aN._L

_NcosYcosV+SmV(pY+E3Y

Ay (4.17) with

}

{.!

[cosy(x(° sin ?ii + y cos?J(°) (x

sini +

(i)

_±{o

_co

0(xsinw +ycos°)

=

_sy(0[(xsin

_{+ y°}

cosi)siny(0 +($) cosi - y

sin

i)cos0]

,y(i)[(4)sinr+

_Y)cos?(0]}

+{

cos0[(xcos_

ysin47)sin?(0JI

(4.18)

4, =

[sin(°(xPsino°

n{1)

_{jy(i)(x(i)sin(i)}

+

Ycosw(0)}

=

f

{

os[(xt sin i

+ y cos4r)sin? +

(x cosT

-

sin )cos)

{(i)

siny°[(x sin i +ycos

i)cosY(0i}

24 E3, =

-

+

{?(0(40cos0

-

sin +

-(4

sin + (j) _.(0)2]}

(4

)+ ) i))2 } =

-+

_±{794

_cos(?

-

i)

+y

sin('

cos(

- i) + y

_sin(Ø' )2 } (x sin(?

-

i) - y

cos(f _))2 } (4.20)

4.2. StabilityCoefficients and Charteristic Equation

The expanded linearized equations of motion (4.7), (4.12) and(4.17)can be combined into

: form of(1.I). By introducing the differential operator

fD=dI4,

these equations can

further be written as

[(m_

Xü)cos2

+(sin _cosii7}z Eix}ix

[(m_X)sinc7D2 _(sini+cosv)D+Eiy]AY

{X4)D+El!V}1=O.

[_(m_Yv)sin2 (sE2]A

+[(m)cosiJi

(4.2 1) (4.22)

[Nsin2

+-+]

+{N1cos2

+cos+E]AY

+[(IzN+(Nç)+E3]41=0.

Thecharacterislic equation for Ax, Ay, or deiived from (4.21)-(4.23) is:

Acr6Bc75Ca4+DcT3+EO2+FO+G=O,

where

A=(m},XIzNr)NyYr,

B=(mX4N(mzi4)_(Iz_Nt)f(m_Y4_Y*]

(4.26)

C=(m_X4u_NJ+*+(1z_Nr)(E2yCOSV(_E2xShu11)j

-

)(1 - N,) -

N1,Y](Ei cos7 +

sini)

axl

aN

ay (a

a

- N,)+

N(Xi7

+

-

[(m

-

_YdXUV + + }. (4.27)

D=(mXâ)ftNiii-+(m -

x4

ai'

1ax1(a1'(ai'aN

]+-[(NU

(423)

(4.24)

26

_N)[(E2xsini??_E2Ycos?)+(E2xcos7E2Ysi147)]

(1

_N4(E1cosi

-

Ei

sin7) -

sini +

Ei _cosi)1

-(m-

Yv)[E3

-(m-

(Eicosji+

E1sJ_N[1]

+N4(mii_)(Eixcosi+

Elxsin1)1 (E1Ysin7+

Eicos1)]

(

-

+)

dXYdYdN aYari'' (

--zJ+YiU )NU

-

dYaxav (

d1qay

dN

E= (m _xu)[(siyE3r -

_{- E2E3v)sini]}

os)+(Yii_

)(E3,cosiii_

E3sini)]

E

E3pr]₊

-

-

E21E1y}+Yr[EixE3y

-

E3xEiyJ

i-(m- },) (Eicosi+ E1sin7)E -(Ecos7+ E3ysini)Ei,}

(Ei+ysinflEivJ

(àY'I(dN._aw i

(dN._dN

-It.

Y -

v

+

cos)+

EIy(.sm Vt

-

COSVt

(Yvi4IFE3xcosvi+

ysm]+(Nvr-ç Ixcosv+ sm]

- .IEl4is1n

Vt+

-

Eiy(ZcosVr

-

.sin)]

(

,x'f

(aN.

-

aN

(aN.

-

dN

-Xv +) EsmV +.COST)+ E3(smVt --COSV

(axi(ay.dy i

(ai

ai..

Xuv+JLE3xS1flY(

+COSY(J_E3ycOSVSlhV

(àNay dYdN,.

dXdN

axa

F =

- EE,)sini

+(E2),E, - E3E,)cosV]

-

EE,)cosi+(E2yE3V, -

Ey,)sin7]

- E3Ei,)

COSiT +(E3xEiw - EixEy,)sin

₁

-

EE3r)cos7 + (E3E1, - EiyE3v)sifl

IJ

+[(E2yElpr EiyEy,)cosi+(EixE2cr -

E2xEiyr)siniYJ

- ExElyr)Cos7+(ElyE2vr

E2Eir)sin]

Equation (424) is in the form of(1.l), and expressions (4.25X4.31) are the Sixth-order Routh-Hurwitz stability coefficients that appaer in

(Li). Constant A in (4.25) depends

exclusively on inertial vessel properties, and corresponds to the first Routh-Hurwitz coefficient in the maneuvering stability criteria [1]. Coefficient B in (426) representS coupling between inertia and daniping, and depends on vessel properties, and the value of the equilibrium forward velocity ii. The rem2ining coefficients depend on vessel properties, line arrangement

values of the state variables evaluated at

equilibrium

and on equilibrium velocities i and iY.

Coefficients C, D, E, and F

represent coupling between inertia, damping and mooring line stiffness terms, while coefficient

G cQrresponds to coupling terms betweendamping and mooring line stiffness exclusively.

(4.29)

E2E3] +

(Y

-

E3Ei -

EiE3y]

+(x+

E3XE2Y

-E2xEiy], @.30)

+(Nii_?)ElE2_

B _(4.39)

4.3. Stability Criteria for Towing/Mooring Systems

The charaçteristiç equation (4.24) in can be written in the form:

A(a_aIXOa2Xa-03X0a4XaSXc7--06)=O,

(4.32)

where a1, 02, 03, a4, 05 and

6 are the eigenvalues of the system, or the roots of equation

(4.24). As explained in Chapter I, stability follows if all real parts of the six roots are negative. Comparison between equations (4.24) and (4.32) prodtices the following relations between the Routh-Hurwitz coefficients and the roots of the sixth-order system

B/A=-(ai+a2+c73+a4+a5+06),

(4.33) C/A₌

(0102+ o1a3 +0104 + 0105+0106+0203 + 0204+0205

+ 0206

+0304+

0305 +

0306+0405+ a4a6

(4.34) VIA=

(010203+010204 + 0(705 + l26 010304+010305+

a103a6

+010405+010406+010506+0203(74+020305+020306+ a20405

20406 + a2a5a6 + 030405

+ 00406 + 030506+ 456)'

(4.35)

EJA (01020304+ oaao + 010203a6 + aioa4as + 01020406

+ a1a2a5a6 +01030405+01030406+01030506 + 01040506 02030405 + 02030406

+02030506+02040506+ c3a4ap6),

(4.36)

F/A =-(0102030405+ (7102030406+ 010203050 + 01020405(76

+010304006 +0203040506),

(4.37)

GIA = a1a2a3a4a5a. (4.38)

Expressions (4.33)-(4.38) give the necessary conditions for the stability of the sixth-order system without having to solve explicitly for the eigenvalues: B/A >0, C/A >0, DIA

>0,

EIA

>0,

F/A >0, and GIA >0.

Following the theory presented in Chapter I, the

necessary and sufficient conditions for stability are compiled into the following six criteria

CD

CR2=

D B(AFBE)

CR3=

_{A A(BCAD)}

+

>0,

E (BGCFXBC.-Ap)AF(AF BE)

_>0

R4-A

A[D(BCAD)+B(AFBE)]

CRS

g[(sc_

_{Bfl+D(AF_BE)BG1 >0,}

A AL (BC_AD)(CFDEBG)+(AFBE)2

J G CR6=

>0.

(440):

(4.41) (4.42) (4.43) (4.44)

For a system to be stable (i.e. thc real parts of all six eigenvalues be negative), relations (4.39X4.44) must hold. The number of eigenvalues with positive real parts in the system is equal to the number of sign changes between the values of the stability criteria in the ordà presented as mentioned in Chapter L

Coefficients A and B are always greater than zero [14], and therefore the first stabilitS'

criterion (4.39) is always satisfied.

Expression (4.33) shows that the sum of the si

eigenvalues therefore must be real and negative. This implies that the number of eigenvalus with positive real parts in towing and mooring system dynamics can be at most five, and thus, the maximum dimension of the unstable manifold cannot exceed five.

4.4. L.oss of Stability in Towing/Moori'gSystems,

As mentioned previously, the stabilityof the system thpends on the signs of the real parts of the System eigenvalues, or conversely, on the number of sign changes in the stabilitycriteria

(4.39)-(4.44). An initially stable system may become unstable, and viceversa, if a single

system parameter is changed. The types of loss of stability that towing and mooring systems exbibit are classified as static and dynamic [7].

o..

$tafic Loss of Stability

Static Loss of Stability (51$) occurs when one real eigenvalue crosses zero from the negative to the positive real axis as a system parameter is varied, and occurs in towing and mooring systems as a result of the action of external excitation (i.e. current, wind, second order drift forces) which maintain a constant magnitude and heading. Pitchfork, saddle node and imple stationary bifurcations are of static nature [11]. SLS canbe represented analytically by setting the term GIA (4.38) to zero. This term is the product of the six eigenvalues present in the system. As one real eigenvalue crosses from the negative to the positive real axis or vice

versa, one root becomes zero and expression (4.38) becomes active, thus establishing the

bifurcation point where static loss of stability occurs, and therefore

St=C

=

(4.45)

The bifurcation corresponding to (4.45) provides the necessary and sufficient condition for static loss of stability, i.e. Cpj = 0 results in static loss of stability and viceversa.

Pynamic Loss of Stability

Dynrniic Loss of Stability (DLv) occurs when a complex pair of eigenvaiues crosses the real axis fromnegative to positive with nonzero speed as one parameter in the system is varied, and is a result of following forces such as the hydrodynamic pressure applied to the hull. Dynamic loss of stability can be determined analytically by multiplying the first five stability criteria and setting this new expression to zero. The expression obtained is

(BC- AD)(DEF+ 2BFG -D2G

-

cF2)

CRC= = 5

-F(BE-4F)2 +BG(ADF+B2G-BDE)

-A5

and dynamic loss of stability occurs if

DLS=CRC=0.

(4.47)

After extensive algebra, expression (4.46) can be recast in terms of the system eigenvalues

as:

CRC = [(0t + O2)(I + O3)(01 + 04 )(°i + 05)(0i + 06)(02 + 03)(02+ 04 )(°2 + 05) (2 +06)(03 +04)(O'3 +05)(O3+06)(04 +O5Xa4 +06XC75

+06)].

(4.48)

The procedure used to derive the expression fOr CRC aboveis described in [14]. Equation (4.48) consists of the product of fifteen terms, eachcontining the sum of two roots of the sixth-order system. When a complex pair of eigenvalues crosses the real axis from positive to

negative or vice versa (irrespective of the eigenvalue pair combination), the sum of twa

eigenvalues becomes zero and expression (4.46) vanishes, thus establishing a bifurcation point for dynamic loss of stability. Such expression is necessaxy for determining dynamic loss ol' stability in the system. Setting CRCto zero, however, does not necessarily imply that the

system has undergone dynamic loss of stability. A pair of real eigenvalues with oppositó signs, or two or more zero real eigenvalues, for example, would also satisfy (4.47). The bifurcation point generated by setting CRCequal to zero, provides a necessary and sufficient

condition for dynamic loss of stability, provided the exceptions noted above do not occur.

43. Numerical Examples

Consider a 3-line barge with skegs (1 line placedforward of the CG, 2 lines aft), which is

moored by nonlinear elastic strings under a 2 knot current. The barge particulars and hydrodynamic properties are shown in [19]. The barge system possesses a symmetric

mooring line configuration (see Chapter V) and is originally aligned to the direction of the current The mooring line parametersfor this system are as follows:

Linel: L/L=0.5, x/L=0.45

y/B=0.0, Q=0.O°, E1=0.012

Line2: t/L=0.5, x/L=-0.45

yIB0.5, £7 = 15.0°, E10,O

Line 3: Lw IL = 0.5, x, / L = 0.45, Ypi

B 0.5, £2 = 15.00, E1

0.0,

where £2 is the angle of aperture between the x-axis and the towing/mooring line. This mooring configuration possesses three different equilibria, which have been found by solving equations (3.13)-(3.15) and are denoted as A (symmetric equilibrium), B (nonsymmetric equilibrium) and B' (mirror image of B).

32L.

The eigenvalues for this system about equilibria A, B and B' have

been obtained

numerically and are depicted in Table 4.1. The Dimension of the Unstable Manifold(DUM) is

equal to the number of eigenvalues with positive real pail. For equilibrium A, DUM =1; for equilibria B and B', DUM=0, thus showing that equilibrium Ais unstable while equilibria B

and B' are stable.

Table 4.1: Eigenvalues for 3-line barge with skegs

Table 4.2 shows the corresponding values for the stability criteria for the same mooring system about equilibria A, B, and B'. The Number of Sign Changes between the stability criteria are denoted as NSC in Table 4.2, and correspond to the dimension of the unstable mnifo1d.

Table 4.2: Sixth-order stability ciiieria for 3-line barge with skegs

In Table 4.2, there is one change in sign between the stability criteria around equilibrium A. This corresponds to a positive real eigenvaluewith a one-dimensional unstable manifold. The number of sign changes between the stability criteria about equilibria B and B' is zero, and therefore all eigenvalues have negative real pails, and the system is stable around equilibria B and B'. These results are in accordancewith Table 4.1.

Criterion Equilibrium A Equilibria B and B

CR, 2.48753 2.48787 CR2 8.21111 8.98351 cRi 111.28097 123.7772 CR4 106.14232 115.79580 CR5 157.64610 150.10839 CR6 19.11216 16.96155 NSC 1 0

Eigenvalue Equilibrium A Equilibria and B

a'

0.1189+17.995 0.1230+18.342 a2 0.1189-17.995 0.1230-18.342 a3 1.4933 1.4018 a4 0.1233 0.1109 as -.0.4398+11.196 0.3645+il.197 -.0.4398il.196

0.3645il.

197

The regions where static and dynamic loss of stability occur in this particular system around equilibrium A, can be discerned by vaiying a parameter. Figure 4.1 shows the values of CRS and CRC as a function of the parameter x/L, which is varied simultaneously for all mooring lines from 0.0 to 0.5. The maximum values for Cp and CRC in Figure 4.1 for the range of values have been scaled to 1.0. The value of x1,/L for which CRS = 0, where static loss of stability occurs, is 0.3475. Dynamic loss of stability occUrs for x1/L =0.1292,

where the value of CRC is zero.

15°,E,O.012 0.75- 0.5- 0.25--0.25 0.5 0.75 -- CRS CRC -1 0.00 0.05 I 0.10

-I

I 0.15 0.20 -I 0.25 0.30 0.35 0.40 0.45 0.50

V. SYMMETRIC TOWING1MOORING

SYSTEMS

The analytical expressions derived in the previoUs chapter show the complexity associated in the design of general towing andmooring systems. The complex form of those expressions demonstrate that rules of thumb that can be used in design, cannot be presently derived. Such expressions however, can be simplified greatly for symmetric systems, where a number of rules can be discerned and can be applied subsequently to the design of symmetric as well as general systems. A symmetric towing/mooringconfiguration is such that the arrangement and length of the towing/mooring lines are selected as to maintain geometric symmetiy about the X-axis. A symmetric configuration which is aligned to the direction of the predorninint external excitation is labeled as a symmetric towing/mooring system, as shown in Figure 5.1. In this figure, the X and x-axes are aligned to the direction of the current (relative direction between the current and the tugboat(s) in towing).

As mentioned in (lapter 1V, the angle Q appearing in Figure 5.1 is a measure of the aperture between the x-axis and the towinglmooring line (0 n/2). The relationship between the angles Q and

y

in each of the four quadrants is as follows: First quadrant 17= y; second

quadrant £?=,cy;third quadrant £2=.ir;fourthquadrant

£2=2ir-y.

5.1. quiibrium and Expansion of Variables

Symmetric towing and mooring systems have the following characteristics: propeller and rudder effects are zero.

equilibrium drift angle iji is zero with tespect to the relative direction of the flow and the X.axis.

Th55eieiatjveveIocitiesateqUillbriumaregivenbY ii=U, i=O, and the

following relations apply:

-

-

(xmXp'

cosw=cosy=

7

),

-

.

-

(YmYYp

sm(o=sin7

TI hydrodynamic forces at equilibrium and their derivatives are reduced to:

(5.3)

(5.6) (5.1)

36

In expression (5.3), R is the resistance of the vessel evaluated atequilibrium.

Figure 5.2 depicts the relationship that exists between the values of the parameters Xp. Yp' and yin each of the four quadrants.

xpo

Yp =0

co'=1

siny= 0

II

..xpo

xPo

YpO

YpO

cosyO

y

cosyO

smyO

sinyO

xpo

yp =0 cosY= I siny= 0

Figure 5.2: Geometric relations of parametCrs xp1 Yp and 7 in symmetncsystems

dY _(5.8)

dYy

_(5.9) (5.10) (5.11) dN

dr

Yp Yp cosy

cos.r 0

sinyO,

Siny 0

III

Iv

From the relations shown in Figure 5.2above, the sunmation terms that include coupling

between hydrodynamic damping and towing/mooring line stiffness (see equations (4.8)-(4.10), (4.13)-(4.15), and (4.18)-(4.20)) have been reduced to the following:

Ely =

_{7(Osjfl?(O]...

i=1' - i=l

(0

Xp

_f)cos?))}

(5.17)

=

cosy0 +

sin (i)

cos(xsiny(O

_(i)

=0 (symmetry), (5.18)

Ei =

{sin2 ?(0}

₊ cos (5 12) (5 13)

sIn?cosY0}

=0 (symmetry), E1

=_±{:

SIyi

cos(0}+

=0 (symmetry),

(5 15)

n7$0

m?0cosø0}4{1j60

Ei

=_{

sin?cOS?°J

= + (5 16) Sm2

{-cos2 ?()}

+±{;

cos(4sin7

) =0 (symmetry),

(5.14) E2 =

YU{7)cosY

E3v, = N U +

±{r[xcos7(° +

sin7(

4)

cos7(1)+ sm

7()2

]}

52. Equations of Motion in SymmetricTowing/Mooring Systems

For symmetric towing/mooring systems,equations of motion (4.7), (4.12) and (4.17) redüceto:

(m-})4j--}4j

I

(5.fl)

Equation (5.21) is only a function of x and itstime derivatives, while equations (522) and

(5.23) are fuictions of y, and their time derivatives exclusively. The surge equation of

motion is therefore decoupled from the sway and yaw equations, and thus its stability can be studied separately. Futiher, this equation canbe written as:

',i+

I

_E4x=0,

(mXu)du

(mXu)

with solution of the form:

(5.24)

(5.25)

E3

{j)COS?O(4Os?ü

+ysin7(0)}

sin sin7(i)

T

(5.19)

n{(i)

(x sin7(i) i))2} (5.20)

_Xu)A+M+Eix4X=0,

(5.21)

The equation is stable if a, a2 have negative real parts, Thc eigenvalues can be obtained reidily and are:

1

al.2=2(mX)

The tenDs (mXu),

,

and E

are positive, and therefore Re{a1,c72}<0, and the

du surge equation of motion is stable.

Further,

-

-

i2

(i)+

2d:

0

,the eigenvaiues can

bewrittenas: aI,2₌

_2(m

x)[

X)Ei

()2]

(5.27)

Since the surge equation of motion is always stable forsymmetric systems, the stabilit' depends exclusively on the coupled sway and yaw equationsof motion(5.22) and (523), by introducing the differential operator 1)=did:, such equations can be written as:

(5.28

(5.29)

5.3. Stabilit Coefficientsand CharacteristiC Eqation

The characteristic equation for either iiy or Aip' derivedfrom equations (5.28)and (5.29) (5.26)

is:

Expression (5.30)is a fourth-other characteristic equation of the form (1.1). The constants

A-E are the Routh-Hurwitz coefficients forthe fi th-order system, and are given by:

A=(mY)(1zNt)NYr,

(5.31)

B = N(mU_-i) l';(1 Ne) N(m I,) Y. N, (5.32)

+YE3y+(mYy)E,.

5.4. Stability Criteria for SymmetricTowingfMoor'ng Systems

In terms of the rempining eigenvalues, the quartic characteristic equation

in a can be

written as:

A(o_03)(4)(a_05)(0-0'6)=0.

(5.36)

A symmetric towinglmooring is stable if all real parts of the four roots 03, 04, 05, and

06 are negative. Comparison between (5.30) and (5.36) produces the following relations

between the RouthHurwitz coefficientsand the roots of the fourth-order systems:

BIA=(a3+cT4+cYs+C6)

(5.31)

CIA=a3a4+a3as+a3ao+0405+0406+0506

(5.38)

DIA=d3O405+030406+030506+O4O5C6

(5.39)

E/A=a3a4a5G6.

(5.40) (5.33) (5.34) (535)

B

CRI=rO

CD

Cj>0t

BCDAD2 B2E

Co,=

>0,

A(BCAD)

E

CR4 =>

For stability, the necessary conditions are therefore B/A >0, C/A'> 0, DIA > 0 and EIA >0. Following the theory presented in Chapter I, the set of necessary and sufficient

conditions are compiled in the following four stability ciiteria

(5.41)

(5.42)

(5.43)

(5.44)

Constants A and B are always greater than zero [14], and therefore the first stability criterion for symmetric systems (5.41) is always satisfied. This also implies, fromequation (5.37), that the sum of the fOur roots of the characteristic equation (5.30) must be negative an4 real, and thus there must be at least one eigenvalue with negative real part. The maximum number of eigenvalues with positive real parts, therefore, cannot exceed three in symmetric

towing and mooring systems.

5.5. Active Stability Criteria for Symmetric S'stems

In symmetric systems, the four stability criteria (5.41X5.44) canbe reduced into twc, dominant stability criteria [14] which are denoted by R1 and R2 These active criteria are recast in terms of the Routh-Hurwitz coefficients and the system eigenvalues as:

R==a3a4a5a6,

(5.45)

R2

BCDAD2 B2E

=[(a3+a4)(a3+a5)(a3+a6)(a4+a5)

The stability of the system depends exclusively on the signs of criteria Ri and R2. The

necessary and sufficient conditions for stability in symmetric mooring become:

R1>O,

(5.47)

R2>O.

(5.48)

5.6. J...oss of Stability in Symmetric 5ystems

As mentioned in Chapter LV, Static Loss of Stability (SLS') occurswhen a teal eigenvalue crosses from the negative to the positive real axis. This phenomenon can be represented

analytically by setting criterion R1 to zero. This expression is the product, of the four

eigenvalues of the system as shown by (5.45), and is the equivalent of (4.45) for general systems. setting expression (5.45) to zero, gives the necessary and sufficient condition for static loss of stability, i.e.

SLS=R1=O.

1(5.49)

AlsO, as mentioned in Chapter LV, Dynamic Loss of Stability (DLS) occurs when a complex pair of eigenvalues crosses from the negative to the positive real axis. The boundary where DLS occurs is defined by making R2 active (i.e. by setting R2 to zero). This is shown in equation (5.46), in which R2 consists of the product of six terms, each conthining the sum of two roots, or eigenvalues, of the fourth-order system. This expression is equivalent to (4.48) fOr general systems, and provides a necessary conditionfor DLS, .e.

DLS=R2=O.

(5.50)

Notes Regarding Static Loss of Stability (SL5

As mentioned earlier in the chapter, static loss of stability corresponds to the fourth stability criterion (Le. R1 = EIA = 0). Since constant A is alwayspositive, R1 depends exclusively on

the sign of E. LettingRi* = E, the expression for the first active criterion can be expanded

to:

}

-

u{cos0(x

cos + sin

+:

s?(0(xs'?(0_Ycos?()}

+ cos

?()(x() cos

+y

in0))}

i=1 j=1

cos cosy

+Ysin?(0)}

i=lj=1 {

[cosy0(xcos

+ysifl?0))]2}

i=1 j=1

-

_{&=1 j=1}

+yPsin?°)

(4'cos?° +

ysin.?')}

+

j

sn2(xcosY°

1=1

-i=1 j=1 n

nI7(OJi)r

+

it:l(i)

,jJ)

_sn(4cos?°

(i)sin(i))]2}

+ [s _{')(x(J)sin?(J)} YP(J)cos?Q))]2}

-

{:

-y

cosy())

(xiny)_y17cos?th1)}.

U)i,i)

-

y)cosf(J))

(5.51)

From above,Ri*is composed of 11 terms. Terms3,5,7,9 and 10 are always positive or zero. Furthermore,the sum of terms 3-ills always greater or

equal to zero [14]. The firt

two terms mav be positive, negative or zero. For an unstable system in static loss, the sum of

n

[(i)

(i

+9lcU[)

_1TjYP

44

the first two terms must be negative, inespective of the value of terms 3-11. These two terms can further be recast as:

R11 = - y.,

xP)u[cos2

7(0 sin2

(5.52)

The term Y, is always negative; N may be positive, negative or zero but small compared to

in magnitude. In addition, the term ysiñycosy is greater or equal to zero for lines

attached forward of the. CG, and less or equal to zero for lines attached aft the CG. Expression (5.52) shows that, an originally statically unstable system may become stable by applying the following rules:

o For lines attached forward of the CG, increase xj, and/or decrease Yp.

ForlinesattachedafttheCG,increaseyandfordeCreaSeXp.

Increasing or decreasing the angle y (or £2) for either forward or aft lines may help an original!y unstable system to Stabilize depending on the relative position of the line(s) in terms of and Yp

Increasing the distance between the towing/mooring terminals with respect to the vessel results in a.reductjon of the angle £2 and an increase in line tension. This appipachwill

render, in general, an originally unstable equilibrium stable, and is widely used in practice. Cases for which this does not hold true are documented in [5], and generally occur for small values of x, for lines attachedforward and large values of .xp for lines attached aft the CG.

It must be noted, however, that a strongly statically stable system may undergodynamic

IQSS of stability, as shown by (5.46), where R2 is proportional to the negative of Ri*. Increasing the value of R1 (and therefore producing a large positive value for R11) may

5.7. Numerical Examples

The fourth-order stability criteria for symmetric systems of Sections 5.4 and 5.5 provide the same infonnation as their sixth-order counterparts derived in Section 4.3. Table 5.1 shows the values of the fourth-orderstability criteria, as well as the values for the dominant criteria R1 and R2 for the 3-line barge example considered in Section 4.5 about its symmetric equilibrium position (A). The Number of Sign Changes between the stability criteria are also shown on the table and are denoted as NSC.

As shown the table, the number of sign changes between the fourth-order stability

criteria is one. This corresponds to a statically unstable system, with one dimensional unstable manifold ('i.e. one real positive eigenvalue), and is characterized by the negative value of R1

This result agrees with those presented in Tables 4.1 and 4.2, which were obtained with

eigenvalue analysis and the sixth-order stability criteria derived in Chapter IV, respectively.

Figure 5.3 shows the values of R1 and R2 as a function of the parameter x1,/L, which 5 varied simultaneously for all mooring lines from 0.0 to 0.5. These are scaled by setting their maximum values in the xfL range to 1.0.

As shown in Figure 5.3, the system is stable for 0.1292 <xpfL < 0.3475. In this rnge,

both criteria are satisfied. Criterion R2 is negative to the left of the lower stability boundai (x/L <0.1292), indicating that the system oscillates about equilibrium A in this range AL

Hopf bifurcation occurs at = 0.1292. To the right of the upper stabilityboundary; (x/L> 0.3475), R1 is negative, and thus the system diverges from equilibrium A. Atthe point x1,/L = 0.3475, R1 is zero and a pitchfork bifurcation occuis.

Table 5.1: Fourth-order stability aiteiia for 3-line barge with skegs

criteiion gquilibiiwn A CR1 2,24967 CR2 1.72794 CR3 2.45215 CR4 -0.29891 -I.73247x105 R2 +1.856O4x1O2 NSC I

0.75- 0.5- 0.25- 0.250.5 0.75 - a-1 1 1 I I I I I 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 x,t. 0.50 Fige5.3:Va1uesofR1andR3IinebargeWithSkegS:t/L=03,Y/B0.5,17 15°,E=0.012

The points in Figure 5.3 where Static and dynamic loss of stability occur, agree with those

obtaihed with the expressions for the bifurcation boundaries derived with the sixth-order

stability criteria, as showfl in Figure 4.1.

It is noteworthy that Figures 4.1 and 5.3 are practically the same in terms of the values of the bifurcation boundaries. This is due to the scaling of each figure. For the symmetric system

of Figure 5.3, however, the signs of R1 and R2 determine whether or not the system is

statically and/or dynamically unstable. The signs of the sixth-order expressions for Cp,ç and

CRC in Figure 4,1 do not necessarily imply stabilitylmstability. Two complex pairs of

eigenvalues with pOsitive real parts, for example, would yield CRC positive eventhough the System is dynamically Unstable.

R1

APPENDIX 1. FIRST ORDER EXPANSION OF GEOMETRIC ANGLES

The angles involved in the geometry of a towing or mooring system (p,y,w) can be expanded about their equilibrium values (4i,7,) by perturbing to linear terms. Let

(Al.1)

cosy=cos(7+4y)

Xm

-6xx(cosicos4VrsmVsm4!V)

y(siniicos4 + cosisin

_iW)

7+4e

siiiy=sin(y+47)=

_4xpsin(+V)YpC0SQiJ+11V)

₁₊₄₁

Ym

_y_x(scosiW+ossinL1Vf)

-y(c?cos4'sifli7sin4V1')

(Al.2)

(D+L1W.

(Al.3)

By applying standard trigonometric relations [9], the sine and cosine expansions for the angle aregivenby:

cosip = cos(i+4)= sTcos4

sinisin4Vi,

(Al.4)

sin (Al.5)

Following relations (2.15X2A9), and applying the above equations (A1.4) and (Al.5), the sine and cosine expansions for the angle y are written as:

(Al .6)

Letting sináQ(), and cos4Ql,expressions(A14)-(A1.7)reduceto:

cosV=cossin74i,

sin,sini+cos74y,

cosy

-£+4e

sin

The expansion for the angle w can be obtained readily frOm relation (2.14), i.e.

which gives the following relations:

cosw=cosycosV+sinysinVf,

sinø=sinycosf+cosySinVf.

Expanding the terms of each of the two expressions above and linearizing dter expansion we arrive at:

cos(7+ 4y)cos(T+Af)=

(xm cos

(AL14)

(ymDSyDl0SP)SihhiV

smUT

sin(?+4y)sin(i+iiV')'

- 2xpSin

_{2ypcos.,)cosV+yp]4V}

(A1.12) (A1.13) (A1.15)

50

n(7+4y)cos(+4)=

_Y_xsin7ypcos?)cosi

cosv

1+41

J(Y-"

+2xsT+2ycosqi)si47_x]4,

1+41

(A1.16)

cos(7+ 4y)sin(7+4,j

(xm

-

xpcosii+ypsinv)sinig

sini

t+4e

-I - 2x cos7 + 2Yp sin i)cosi

₊x,,]

(A1.17)

Combining equatiOnS (A1.14)-(A1.17) with (A1.12) and (A1.13), the sine and cOsine expressions forthe ang1 w are:

J(_xm)cOW+(Y_ym)SiflP+xp]

x L COSW

+41

_j

i+se'

s[(xrn)sin(Y_ym)cosii11,,

-F4e (A 1.18)

sini

1.

ji+4e.

+c

4y [(I_xm)COSP+(Yym)S1fl4u14

+4t

-

7+t

-v'.

(A 1.19)

The horizontal component of the towing/mooring line length is perturbed about it

equilibrium pOsitiOn as follows:

(A2.1)

where, from expression (2.17),

£2(xmxT)2+(ymyI)2

=[xm_i_AX_xpcos(7+4v!)+yph1(ivf)r

+[Ym

=

Xm2 2m

- 2Xm Ax - 2Xm Xp COS!IICOSAV( + 2Xm X Sill 7sin 4'

+2Xm Yp SiflWcOSAVT+2Xm Y SIfl4V(+X2 +24x+ Ax2

+2x

cosicos44f+2x

sin4,r+2xcosiAxcOS4yf

2x sin74xsin4V-2y

fltCOSAVF2Yp ICOS1VSIIIAV(

2ysinAxcQsAyf-2y cOSIJTAXSIJ14Yf+Yrn22Ym Y2Ym 4y

2Ym Xp sinjicoS4V2ym Xp C

Sin4JJ(2ym yp cos7cos4y'

2Ym Yp

sii47sin4yf+2 +24y+4y2 +2x ysin7cos4V

+2x ycossiliAp'+2x sin74ycos4+2x cos74ysin4V(

+2Yp y sicOsAVf-2yp

ysin7sin4t+2y cos74ycos4V

2YpSifllVAySiflhlVt+Xp2 Yp2

=(?+2ZM+4e2).

(A2.2)

Letting cos4y

1 and sinAV( 4' , and ignoring nonlinear terms

in A, the expression

above can be expanded as:

£ +2ë4Xm2 2xm_2xm4X_2xmxp+2xmtpsmVVt

+2xmypSifli + 2Xm Yp COW 'I + 2

+2iAx +2x

cosi

52

2YpICOStW+Ym2

2Yn* Y2YrnAY2YmXpi7

_2ymxpcos?42yrnypcOS7+2yrnyp5in744!V+Y2

+24y + 2x ysin iT+ 2x ycos74yf

+2x sin ç74

+Yp2'

where

=Xm2- 2XmX 2xm Xp COSVI +2Xm Yp SIflV(+21p XCOSV

2ypsiniji+xp2 +y2 Ym2

2YYm ZYmYp0!VY2

_2xpymSifl+2Ypyc0Si+2XpYthhui7

and

2tLtt _2xra4x+2xmXpSifl74!V+2XpSY271V4W

2YpYmflW'Y

+2Xm YcosVFA1V+ 2x cos ,4x 2Yp CO5V4Y2Ym 4y

_2yxpcos4V+2Y4Y+2XpYC+2YpF04Y

Relation (A2.5) above gives the expression for. 41 in terms

of the position vector

(4x,4y,44#) by:

41 [(Y._Xm ((Y_Ym

+sin1_YPcosI)F

JxP(Y_Ym)_YP(2_xm)]c0sExP(_xrn)4YP(_Ym)]h1ci}iivF

(A2.6)

From relations (2.18) and (2.19), the equation above can be reduced further to:

(A2.3) (A2.4) (A2.5)

41=

£ £ £ =

_cos?iix_Sin?4y+(x SiflW+Yp

sw)4

(A2.7)

The first order peiturbation for the towing/mooring line tension isObtained as follows:

T=7+4T.

(A3.l)H

The term 4T can be obtained as a function of the towing/mooring line longth perturbation 4t,

which, as Shown by (A2.7), it is a function of the position vector

(iix4y,si). Following the

Taylor Series Expansion:

f(x)=f(;)+j

(xx)

(xx,)

x=xo

xxo

the line tension can be expanded about the equilibrium length ë as follows:

+

=7+4M or

74L%e.

For the nonlinear elastic string

with

43=SbP(-/)

(_)q1,

and therefore 53 (A3.2) (A3.3) (A3.4)

and, thus,

54

For the inextensible catenary, the expression for the equilibrium tension can beobtained by solving the equation:

P7 _(A3.7)

27)

2?[,

U

Once the equilibrium tensiofl 7 is found, the first derivative of the tension with respect to the towing/mooring line length evaluated at equilibrium is given by

(A3.8)