Nonlinear stability and simulation of two-line ship towing and mooring

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Nonlinear stability and simulation of two-line ship

towing and mooring

Michael M. Bernilsas and Jin-Sug Chung

Department o f Naval Architecture and Marine Engineering, The University of Michigan, Ann Arbor, M I 48109-2145, U.S.A.

The horizontal plane motions - surge, sway, yaw - o f a ship towed by two tug boats, or moored to two terminals, are modelled by the nonlinear third order maneuvering equations without memory. Nylon lines, chains, or steel cables are used as towing[mooring lines. Excitation consists o f time independent current, wind and mean wave drift forces. The resulting Nonlinear Time Varying (NTV) model is studied by simulation, local linear, and global nonlinear stability analyses. Two-Line Towing/Mooring (TLT/M) systems exhibit, in general, three equilibria. Linear analysis reveals that equilibria may be attractor or repellor nodal or focus points. Near first equilibrium, it is shown that o f the five R o u t h - H u r w i t z criteria only two can be active in T L T J M systems o f conventional ship hulls. Subsequent parametric study reveals the complexity o f stability boundaries which may exhibit fold or cusp singularities. Global analysis is used to find attractor limit cycles and chaotic behavior. A barge with or without stabilizing skegs, a mariner type cargo ship, and a tanker with and without propeller which have different stability characteristics are used in nonlinear numerical simulations to verify all theoretical conclusions.

Key Words: Two-line towing/mooring, maneuvering, equilibria, stability boundaries, cusp and fold singularities, attractors, repeltors, local and global stability.

I N T R O D U C T I O N

The slow motion large amplitude dynamics o f a ship towed by two tug boats is studied in this paper. The related problem o f two-line mooring, where ship station keeping is achieved through two mooring terminals or two anchors, is also analyzed. The dynamics o f two-line towing and mooring systems can be modelled using the same equations but different values o f design parameters. The primary objective o f this work is to derive qualitative conclusions regarding global dynamic behavior o f those systems by local linear and global nonlinear stability analyses, which can be used to produce safe and economic designs. Time simulation o f system dynamics is not adequate for design purposes. Suffice to mention at this point the following four limitations o f designing nonlinear systems based only on simulation:

(i) The designer can run his nonlinear analysis code only for a limited number o f values o f the system parameters. For those runs he can derive quantitative conclusions which are helpful but not conclusive regarding the global qualitative system behavior. Some attractors and repellors may be identified using that approach,~ while others may be missed. Boundaries o f stability domains 2 and their singularities, 3 bifurcations o f equilibria, static or dynamic loss o f stability, and

Paper accepted September 1987. Discussion closes October 1990.

regions o f chaotic dynamics will probably be missed altogether. 4 Such information can be derived only by nonlinear global stability analysis. 5

(ii) T i m e s~-muiations o f nonlinear systems may be misleading since a repellor may appear as an attractor, should the selected set o f initial conditions be in the stable submanifold o f the repellor. That is, a natural set o f initial conditions may result in a trajectory converging to an unstable equilibrium as shown in Ref. 4. Only local stability analysis can be conclusive regarding this issue.

(iii) Unanswered still remains the question o f how long the time simulation interval should be in order to reveal the qualitative behavior o f the simulated system. Bifurcation analysis can provide information on the various regions in parametric stability domain which could provide qualitative answers to that question. Such information may be missed by short simulation intervals.

(iv) Last, but probably most important from the point o f view o f fast system dynamics (seakeeping), is that the derivation o f added mass and damping coefficients requires knowledge o f the relative slow velocity. It is usually assumed that the latter is zero or steady. 6 A time integration technique which is under development 7 will soon be capable o f solving this problem - without making that assumption - provided that

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Nonlinear stability and simulation of two-line ship towing: M. M. Bernitsas and Jin-Sug Chung the slowly varying relative velocity is known. It

is understood of course, that memory effects can further complicate this matter by introducing interaction between slow and fast dynamics. An extended dynamics method used in Ref. 8 may still be used to convert the resulting non- autonomous system to an autonomous one and proceed with the approach introduced in Refs. 9 and 10.

Several different approaches have been developed to contribute to the understanding of ship towing, Single Point Mooring (SPM), and Multi-Leg Mooring (MLM) as early as 1950. '~ A simple linear stability method has been developed using ship maneuvering equations and inextensible towline, t'''2 It was then concluded that towing stability can be improved by shortening the towline. Observations on towing of Canadian barges'3 and model tests'4 resulted in contrary conclusion. This contradiction was resolved by deriving boundaries of stability domains in a space of system design parameters. 2 Deeper understanding of the global nonlinear behavior of one-line towing and SPM systems was gained by global nonlinear stability analysis performed in Refs. 3,4, 9 and 10. Memoiy effect on global stability of a SPM tanker system was analyzed in Refs. 15 and 16 correcting results presented in Ref 8 employ- ing an extended dynamics technique to derive an equivalent autonomous system. MLM systems have been studied first by static analysis where optimization techniques were used to derive mooring line forces. ,7-20 Such techniques were necessary to derive a solution because the system elasticity was neglected and the resulting problem was indeterminate. Nonlinear simulations were performed producing useful quantitative results for specific MLM systems, zl-23 Material properties of mooring line models and mooring cable dynamics have also been studied to improve our understanding of SPM and MLM behavior. 24-27 Conclusive qualitative results that can be used in design, however, can be derived only by local and global stability and bifurcation analyses of MLMs. In this work, such effort is initiated by studying the dynamics of two-line mooring and towing systems. Those are the simplest of multi-line systems yet the complexity of their stability boundaries and the richness of their dynamics makes analysis of such systems a necessary step to understanding multi- line system dynamics.

The nonlinear time varying model of the slow motion system dynamics is derived in Section 1. The horizontal plane maneuvering equations of third order are used to describe the ship slow motions. Three different line models are used, namely a nonlinear elastic string for synthetic fiber ropes, a catenary for chains, and a three- dimensional large deformation extensible finite element cable model with deformation and response dependent hydrodynamic load. In Section 2, simulations of systems with different stability characteristics are used to show asymptotically convergent, divergent, periodic, or chaotic behavior. Guidance for identifying systems with such different kind of dynamic behavior was provided by results of local and global analyses performed in Section 3 and 4. Qualitative conclusions regarding global system behavior are derived and used for recom- mending practices for safe and economic designs.

1., PROBI,EM FORMULATION

The configuration of the system described in the intro- duction is depicted in Fig. 1. In this figure some basic symbols are defined. The towed/moored vessel is shown in equilibrium (BI). In general, two more equilibria exist, (A) and (B2). In symmetric Two-Line Towing/Mooring (TLT/M), (A) represents symmetric equilibrium configuration while (B2) is the mirror image of (BI). In nearly symmetric systems where asymmetry may be caused by propeller or unequal lines, (A) and (B2) will be close but not identical to symmetric towing and the mirror image of (BI), respectively. The dynamics of TLT/M systems is studied assuming con- stant velocity of tugboat (Ur) or fixed mooring terminal for the following reason. In simulation of MLM systems, mooring terminal motion can be modeled by introducing two additional state variables. 3' Those result in additional oscillatory motion components of small magnitude which hardly change the richness of TLT/M system dynamics. The set of equations used in this paper to model TLT/M systems consists Of the horizontal plane slow motions including environmental loading, a quasistatic approximation to the slow motion dynamics of the two tow/mooring lines, and appropriate initial conditions. Those are presented in the following three subsections, respectively.

1.1. Horizontal plane slow ship dynamics

In Fig. 1, three different coordinate systems are shown:

(i) Body fixed reference frame (X, Y , Z ) with its origin at the center of gravity of the moored vessel, where (X, Y) is the horizontal plane and (X, Z) is the center-plane of symmetry.

y" f-2

y . t} . . /

H r ,

....x

4 , ) ~ - ~ 01" Y U,, " r l -

Fig. 1. Geometry o f two-line ship towing or mooring syslem

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Nonlinear stability a n d s h m d a t i o n o f two-lhte ship towing: M . M . Bernitsas a n d Jin-Sug C h u n g (ii) Inertial frame ( x ' , y ' ) with the origin located at

Mooring Terminal 1 or Tugboat 1, y ' defined by the two mooring or towing points, and x ' normal to y ' . In towing, the direction o f the velocity U r or the two tugboats is x ' .

(iii) The inertial frame ( x , y ) which has the same origin as frame ( x ' , y ' ) and relative rotation o~. In the mooring system o~ =/3, where/3 is the angle between x ' and the direction o f the current velocity. In the towing system, ~ is the angle between x ' and velocity U = U r - U~, that is the relative velocity o f the tug boat with respect to current.

The nonlinear equations o f motion in surge, sway and yaw are given by equations (1), (2), and (3) respectively,

m { duk- ~ - rv) = Ix, (1)

(':t )

m + ru = Fs, (2)

dr

I~ - ~ = M z , (3)

where the right hand sides represent excitation forces in the X and Y directions, and moment about the Z axis, respectively. The rest of the symbols are explained in the Nomenclature. Those equations can be written as

2

( m -

X,O,'l = X + ~ 7",. c o s o~i + Fx~,~o + Fx~,,~,

i=1 (4) 2 (m - Y ~ ) f i - Y i / = Y - ~_j 73 sin wi + Fr,,,~D+ Fr,,,,E, i = I (5) - N ~ 0 + ( h - N : ) / = 2

N - ~ (Xv, sin wi + Yv, COSCoi)Ti + Aiza,xo + /~//za,,,t,

i = I

(6)

where the hydrodynamic forces and moment have been expanded in Taylor series following Abkowitz' approach. 12 In equations (4), (5), and (6), all linear ac- celeration terms are included in the left hand side, while the remaining terms up to third order are included in X,

Y, and N, as follows

X = X o + X,, ~ u + J X,,,, A u 2 + ~ X , ... A u 3 + J X~.~.v 2

+

89

X r r r 2 +

89

X a a 6 2 +

89

X v r u V 2 A r t +

89

X r n , r 2 A l l

+ 89 X ~ , , a z A u + (X,.~ + m ) v r + X , . w a + X ~ r 6

+ XvruVr A It + Xt.&,v5 Art + g r & , r 6 A l l , (7) y = Yo + Yo,, A u + Yo .... A u z + Y,,v + ~ Y,,,.~.v ~

+ 89 Y~.,,vr 2 + 89 YraaV62 + Yru Art + J Yt.~,,v Art z + ( Y r m u ) r + ~ Yr.r 3 +

}

Yn.rrv 2 + ~ Yraar6 2 + Y,,,r Au + J Y,,,, + r Au 2 + Y~5 + ~, Y~aa8 3 + } Y 6 , 6 v 2 + J Y ~ r 2 + Y~,,a A u + ~, Y6,,,,a A u z

+ YrraVr6, (8)

and

N = N o + No,, A u + Nouu A u 2 + Nrv + ~ NrrrV 3 + ~ N r - v r z + 12 N,.aav62 + Nruv AU + J Nr,,uv Art 2 + N , r + ~ N~,,r 3 + ~Nn.,.rv 2 + J N, aarb 2 + N~,,r A u

i 3

+ J Nn.,r AU 2 + NaB + g N~666 + J N6vrSv 2 + J N 6 , A r 2 + N~,,6 A u + ~ N~,,,,a A u 2 + N ~ n v r S .

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The first four terms in X arc related to the towed vessel resistance R, by

- R = Xo + X,, A u + J X,,,, A u z + ~ X, ... Au 3, (10) where subscripts "0" and "6" indicate propeller and rudder angle effects. All these effects are included in the simulation process in codes MLMSIM (Multi-Leg Mooring S I M u l a t i o n ) and M L M S T A B ( M L M S T A B - ility). Further, 7"/and 7i are the tension in t h e / - t h line and the angle between the latter and the x-axis,

W i : ' ~ i q - ~ , i = 1 , 2 (11)

= drift angle, that is the angle between X and x. The forces and moment exerted on the towed vessel by the wind in the horizontal plane are given by

Fx~ ,so = 89 P, C x~ ,~oA TV 2., (12)

I 2

Fr,,,~t, = ~p~Cy,,,~DA L V,,., (13)

Mzw,.~o = 89 LL V2,., (14)

where the aerodynamic drag coefficients are given in Ref. 28 as functions o f ship type and the relative angle o f direction o f the wind.

The mean drift forces and moment in the horizontal plane are given in Ref. 29 as

= , r F , a ( w o ) ] . , .

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where 0 d = l o e a l angle of wave direction, a = w a v e amplitude oo F~.d(~o) , . (16) ,

F

M a ( w o ) ]dw AIz .... T = P ' g L 2 Io S(~176 o 9 sin 2(Od-- r (17) and the expressions in square brackets in the above integrands are the drift force operators given in terms o f nondimensional wavelength. Codes M L M S I M and M L M S T A B convert each function to the frequency domain by the deep water dispersion relationship. The two-parameter Bretschneider wave spectrum is used to express S(wo) in terms o f H~/3 and T~/3 the significant wave height and period in feet and seconds, respectively.

In addition to the horizontal plane equations o f motion, we have the following kinematic relations

A-= u cos r - v sin r - U, (18)

.~ = u sin r + v cos r (19)

r = r (20)

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Nonlinear stability and simulation o f t wo-line ship to wing: M. M. Bernitsas and Jin-Sug Chung where U = [ U [ . The geometric relations for the two

towing/mooring lines are

sin "rl = y + (yp' cos r + .x~, sin r It d c o s o t

Y+(ymcos

r r sin 7 2 - - - -~ 12 12 (21) (22) I ~ = ( x + x w cos r sin r + ( Y + y v , cos r sin r (23) /~= ( - d sin ~ + x + x m cos r sin r

+ ( - d c o s a + y + y m c o s r r (24) where (Xp,, yp,) are the coordinates o f the fairlead o f the i-th line.

1.2. Towhlglnlooring line nlodel

Synthetic fiber rope, chain, or steel cable are used in towing/mooring operations and are analyzed quasi- statically since only slow dynamics are considered in this paper. The former is modeled by semiempirical equations (25) below 24. Chain tension and configuration are calculated by the classical catenary equations. Steel c a b l e response is computed by a three dimensional, nonlinear, large strain model which takes into account the Poisson effect; the hydrodynamic load is deformation and response dependent. That model was developed in Ref. 25 and is not repeated here because it is very extensive and in the following simulation and stability analyses the synthetic fiber rope is used. For the latter the following relation holds

Ti = Sbp ~/---~,.~, i = 1,2, (25)

where T = a c t u a l tension, Sb= average breaking strength, / w = w o r k i n g length o f unstrained rope, 1 = length o f strained rope, (p, q) = empirically deter- mined constants. For a wet nylon rope used in numerical applications below p = 9.78 and q = 1.93. 25 AII three models are implemented in M L M S I M and M L M S T A B . From the computational point o f view the steel cable finite element model is about sixty times more expensive than the synthetic fiber rope and the chain models. Stability boundaries o f mooring systems and their fold and cusp singularities may be significantly affected by the line model used. 3o

1.3. bdtial conditions

Initial conditions are needed in order to complete the above model and start computer simulations. Initial values must be assigned to state variables x, y, ) , tt, r r = ~. The values o f the line tensions T~ and T2 are t h e n computed by geometric relations (21) to (24) and the cable constitutive equation (25) using the initial values o f x, y and r Attention must be paid to generate reasonable initial values o f T~ and 7"2 otherwise a line may break and simulation will stop.

2. S O L U T I O N BY S I M U L A T I O N

Two simple time integration techniques are implemented in M L M S I M , Euler or rectangular integration and K u t t a - M e r s o n integration. Time intervals o f different integration step may be selected. Thus, small steps are used in transient o f relatively high frequency response and large steps beyond those transients. Atten- tion must be paid to avoid numerical resonance o f the towing]mooring line extensibility oscillatory mode. Such resonance may produce incorrect large amplitude oscillations and error accumulations which may even result in simulations which violate theoretical stability predictions, as in Ref. 8.

Several numerical applications are studied in which four different vessels are towed or moored. Those vessels are:

(i) A barge without skegs which is highly unstable in forward self propelled motion. In symmetric single line towing]mooring equilibrium (A) is unstable for all practical values o f xp and 1,.. 2'x~ (ii) A barge with skegs which is marginally unstable

in forward self propelled motion. In single line towing equilibrium (A) may be stable for practical values o f xp and/,,.. 2 In symmetric single line mooring it may be stable only for nonpracticai

1o values o f xp.

(iii) A mariner class cargo ship which is stable in forward self propelled motion and in single line symmetric towing and mooring. 2

(iv) A tanker which is unstable in forward self p r o - pelled motion. In single line towing and mooring equilibrium (A) is also unstable and has no other finite equilibrium. 2

The particulars o f the above four vessels and their slow motion derivatives are provided in Ref. 2. Some basic properties o f the corresponding T L T / M systems are summarized in Table 1. It is obvious that their stability

Table I. Basic p r o p e r t i e s o f T L T [ M s y s t e m s

Variable

T L T ] M System

1 2 3 4

Barge, Barge,

w/o skegs with skegs C a r g o T a n k e r

no propeller no propeller no propeller no propeller

(.vp, = xp~)l L 0.5 (Yr, = Ypz )] L 0.05 (I.., = I..,)l L 1.0 all 1.0 L = L B P , ft 191.56 Uc 2 knots ,6' 0.0 0.5 0.55 0.5 0.05 0.05 0.1 1.0 1.5 0.2 1.0 1.0 0.3 191.56 528.0 1066.3

2 knots 2 knots 2 knots

0.0 0.0 0.0

* Propeller is used only in Fig. 5 where simulation of complicated dynamics is sho~vn

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N o n l h w a r stability attd s i m u l a t i o n o f t we-line s h i p to whtg: 3/1. M . Bernitsas a n d J i n - S u g C h u n g

~OORING OF BARGE w / o SiCEGS

Unstable equilibrium (A)

S t a b l e equilibrium (B1) i 1 t ~ ~ ~ ~ i ~ ,~ ,', t l U [ ( U t l t | ( ( t @ l ) • Iddc=, (.~c:~ (3::1 tJ3c= C : t ' i - W..t ..-II

@,=

i.-ic r) c::ri- I (}. Stable equilibrium (B1)

Unstable equilibrium (A)

20. 40. 60. 80. IC

O I N E N S I O N L E S S T I N E

Fig. 2. Barge w i t h o u t s k e g s s y s t e m : C o n v e r g e n c e to stable e q u i f i b r i u m (B), or unstable e q u i l i b r i u m (,4); 1~ = lz = 1.0, xp, = xm = 0.5, ym = y m = O.05, Uc = 2 k n o t s , 13=0.0, d = l . 0 * 2~ "A *$ 5 - 1 0 CARGO .UOOR I NO A . A _.

(UT/L) ~ ([+ol1 ~ ~ ,'o ,',

CARGO MOOR ING

TlU~ (UTItL) ~ C[toO ~ ~ ,', ,,,

Fig. 3. Cargo s y s t e m : C o n v e r g e n c e to stable equilibriunl (A); !1 = lz = 1.5, Xp, = x m = 0.55, yp, = Ym = 0.05, Uc = 2 k n o t s , [3 = 0.0, d = 1.0

F i g .

4 . d = 0 . 3

L~OORING OF TANI~ER w / o PROPELLER

lU (UIIL)

~OORING Of TANKER w/O PROPELLER

,L ,'o ,'~ Io 2'~ lo A

tt~E (u~/c) ([.oI) 4',

T a n k e r s y s t e m : Convergence to lintit cycle; lz = 12 = 0.20, xp, = xp~ = 0.50, yp, = Ym = 0.1 Uc = 2 knots, 13 = 0.0,

* All quantities s h o w n in Figs 2 - 9 are dimensionless.

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N o n l i n e a r stability attd s i m t d a t i o n o f t wo-line ship to wing: M. M . Bernitsas a n d J i n - S u g C h u n g

MOORING o r TANKER W I I H PROPELLER

A

TIV( (UI/L i'o r

M O O R I N G OF T A N ~ E R . W I T N P R O P E L L E R

\

/

i

r ,,

ill

Fig. 5. T a n k e r systenl (with propeller): Chaotic dynanlics; 1~ = 12 = 3.5, Xv, = xm = 0.57, yp, = Ym = 0.0, Ue = 2 knots, B=O.O, d = O . O

characteristics are markedly different and cover a broad spectrum of dynamic behavior of towing/mooring systems. By studying the dynamics o f TLT]M systems involving the above four vessels, we can investigate the richness o f the dynamics of those systems. Four sample runs of TLT/M systems with qualitatively different behavior are shown in Figs 2 to 5.

Figure 2 shows a simulation o f a barge without skegs system. Time histories o f the norm of the basic six state variables defined by equation (68) and distance diverge from equilibrium (A) and converge to (B1). Simulations o f other system properties like u, v, r, fi,

~q, ~t, Y, if, T~, Tz, are computed but not shown due to space limitations. In this figure a second trajectory is shown converging to unstable equilibrium (A). Initial conditions are far away from (A) but in the stable submanifold o f (A). Actually, for TLT]M systems that set o f initial conditions is a reasonable one to select. Com- putational accuracy is so high that no component of the state variable vector appears in the unstable manifold. Thus, the simulation result alone - not accompanied by stability analysis - may mislead the designer to believe that equilibrium (A) is stable. This issue will be discuss- ed further in Subsection 4.3, where the stable and unstable submanifolds of equilibria can be identified from the eigenvectors. In Fig. 3, simulation of a cargo system with stable equilibrium (A) is shown. Con- vergence to a limit cycle is shown in Fig. 4 for a tanker system. Finally in Fig. 5, simulation o f a tanker system shows complicated dynamics (chaos) as proven in Ref. 3. This results in breaking of Line I. Analysis of T L T / M systems based on linear and nonlinear stability is performed in the next two sections in order to identify parametric regions of qualitatively different behavior. 3. STABILITY NEAR EQUILIBRIUM (A)

It is desirable, safe, and economically efficient for T L T / M systems to operate near a stable equilibrium (A). A methodology is developed in this section to define regions o f stable equilibrium (A) in the domain o f design parameters. In order to simplify the analysis without limiting validity o f the conclusions derived below, symmetric systems are considered where

1,,, = 1,.~, /3 = 3,~ = 0e = 0, Xp2 = xp,, yp, = - Yo,. In this case equilibrium (A) is symmetric, that is ~TA = 0 and

YA = d]2, where overbar indicates equilibrium state and index A indicates the particular equilibrium. Using the above notation and linearizing equations (4), (5), (6) we derive the following equations of motion which are valid in the vicinity o f (A),

[(nl - X ; , ) D - X u ] A u = (ATI + A T2)cos ~qA (26)

2_ ,q y

(111- Yc,)D 2 - Y v D + lla

_J

+ [ - Y I D z + ( Y ~ h - Y r ) D + Yvu +2TI~ ]

+2r

/-/~. x., ,k

= - (A T, - A T2)sin ~'IA + 2TI..~A, (27)

+ ~2Tl~ ( X v , - Yv, tan ~.,~)] y

- Ni, D z - N v D

+ [(Iz - N i ) D 2 + (Nfft - N r ) D + Nvh

+ (xm cos "YtA- Ym sin ~l,)

/l~ \cos ~%})j

= - A T t ( x m sin 5qA + Ym cos -],,,)

- AT2(xp2 sin 5'2~ + Ym cos 5'2A)

+ 2TIA

(Xp, -- yp, tan 5%))5A, (28)

llA

where prefix A indicates variation from equilibrium states, and D = d[dt = time derivative operator. The linearized equation o f surge (26), is decoupled from equations (27), (28). So we can analyze the system dynamics in the vicinity of (A) by studying the coupled sway and yaw linearized equations (27) and (28). The resulting characteristic equation for either y or r is

A o 4 + Ba 3 + Co 2 + D o + E = 0, (29)

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Nonlinear stability and shnulation o f two-line ship towing: M. M. Bernitsas and Jin-Sug Chung where A = Ao = (Yc. - m ) ( N i - lz) - N~Yi, (30) B = B 0 = ( Y + - m ) N , + Y , . ( N i - 1:) + N , ; ( m - Y r ) - YiN,., (31) C = Co + 27"t,Ci + ~ C2, (32) Co = ( m - ]r)N~. + Y~N~, (32a) C, = [ (m - Yi.)Xp + N~'.]cos ~l,, (32b) C, = (m -- Yi.)Xpxp, + YiXp + Nf.xp, + (Iz - Ni),

(32c)

DI = (N~,- Y~.Xp)cos ~1~, (33a)

D , = ( N ~ . -

Y~.Xp)xp, +(Y~- Y~.)Xp+(N~.- N~),

(33b)

E = ~ E2, (34)

E2 = N v - Y~.X;, (34a)

Xp = xm - Ym tan 5'1~. (35)

Further, since the T L T / M system considered in this section is symmetric, the following equations hold at equilibrium (A):

7"~, = T2,, (36)

5'1, = 27r - 5'2., (37)

/~, = / ~ . (38)

T L T I M is stable if all real parts o f the roots o f the characteristic equation (29) are negative. This occurs if B]A > 0 , C]A > O, D]A > O, E]A > O and ( B C D - A D 2 - B Z E ) ] A > O.

If RI is satisfied, then inequality (40) will also be satisfied for the following reasons:

2TtA )

D = 2 T l ~ c o s ~ ' l . , + / - ~ . . X ~ [ N v - Y r X p ]

+ \ I~, ] [ ( Y r - Y i . ) X p + N ~ - N , - ] ; (44) D consists of two positive terms. The first one is the pro- duct o f a positive expression o f Tr,, 5'r,, ~.,, xp, and E which will be positive if criterion RI is satisfied. The second term in D is the product of a positive expression o f 7"c, and ft., and the term in the braces. In the latter, the large positive terms - Y,~ and - N,- dominate the terms Y,. and Ni. respectively which may be positive, zero or negative but are small in magnitude.

Inequality (42) produces the second active stability criterion R2 R2 : 27"1,oq + ~2 > 0, (45) where

1

oq = ~ (BoC2D2 - AoD~) 1

+ ~ (BoCIDz + BoCzDI - 2AoDIDz)

+ (BoCIDI - AoD~), (46)

1

a2 = ~r, (BoCoDz - B~E2) + BoCoDi. (47) Thus, the five R o u t h - H u r w i t z criteria for T L T ] M systems have been reduced to R1 and Rz. The former has a very simple form. The latter appears complicated but it is in fact only a quadratic and fourth order polynomial expression in 11, and xv,, respectively. Therefore, it is a simple task computationally to find the stability boundary corresponding to R2 = 0, in the space o f design parameters hA, xp,, ;rr,, h, yp,.

3.1. Active R o u t h - t h t r w i t z criteria

For single line towing]mooring systems o f conventional ship hull forms, it has been shown that only two o f the above five R o u t h - H u r w i t z criteria may be active. 2 Thus, stability boundaries in the parametric design domain were defined and rational guide lines for safe operations were derived. In this subsection, a similar analysis is performed for T L T ] M Systems. It has been shown in Ref. 12 that A > 0 for ships. Therefore, for LTI (Linear Time lnvariant) T L T ] M the R o u t h - Hurwitz criteria are reduced to

B > 0, (39)

D > 0, (40)

E > 0, (41)

and

B C D - A D 2 - B2E > 0. (42) Abkowitz tz has proved that inequality (39) always holds for conventional ship hull forms. Inequality (41) can be reduced to RI which is the first possible active criterion for T L T ] M systems

RI : N o - Y~.Xv > 0

3.2. Stability boundaries and their singularities The stability domain o f a T L T ] M system near equilibrium (A) is defined by criteria R~ and R2, where

Rl = R l ( x m , %A, Ym), (48)

Rz = R2 (/~A, xm, ;rl~, h, Yv,), (49) The boundaries o f stability domain are defined by making R~ and Rz active. For the four T L T / M systems considered in this work, those boundaries are plotted in the space o f design parameters Xp,, ~ , ~!~. Designers and operators have little or no control on Ym- The steady forward velocity h is an important parameter both from the design and operation points o f view. In our analysis though we could identify all qualitative behaviors o f T L T ] M Systems without changing h. Thus in our applications below h = 2 knots.

Figure 6 shows the stability boundaries for the barge without skegs system. The two three-dimensional graphs show only surface Rz = 0. The two-dimensional graph shows both boundaries R2 = 0 in the (xm,;rl~) plane with ~ as parameter; RI = 0 does not appear because R~ < 0 in the entire domain shown in Figure 6. S and Uindicate stable and unstable regions respectively with respect to R2 only, not R I . This convention is used

(8)

Nonlinear stability and simulation o f two-line ship towing: M. M. Bernitsas and Jin-Sug Chung CRiTERIOI'| R2 ~, ~ . ~ ~ ' . ~ U I ~ ~ , ~ , , -

I

I ~ ' + .-'JY++~+~-

.r~, I ~'~-~..~+~.:+-~++~ ~ " X l l l O . S I x . ? - x . 4 CRITERION R2 '~.~. ,- ~X.L~ ~ I

i

/

I /

C R I T E R I O N R2 : U = U n s t a b l e , S = 5 t a b l e ~ ~+~ U -~" 1~ . . . 1 0 0 s X p I 0 . 7 1

Xe~ CRITERIA R I & R2 ( R1 < 0 )

O. 6050 ;.- . . . q . . . 9 - - - ; I I I i i i I i I i i 0.5343 r . . . U . . . r . . . -, L I . $ , I I o <8+6~ . . . ~ - ~ , ~ ' : ~ ' , I I i I 0 . ~ 9 3 9 ~ . . . ~ . . . - ~ . . . L . . . k I ! 1 1 I I , i I I i I i , ! + I : 0.2514 ~ ' - ~ ~ r n I I i i I i i I i I i I i I I I i I 0.1801 ~ . . . ~ . . . ~ . . . ~ . . . I I I ~ I I I I I i O l I i v Huv'.''"+ ' ' ~ ~ s L . . . J . . . L . . . J . . . L . . . J 0.0000 0 . 1 1 3 6 0.2272 0.3408 0.4644 0.5680

BARGE M ~ I N G ISOSURFA~ W ~ I L.GAggA.xp Y~x

Fig. 6. Barge without skegs system: Stability criteria R I and R2 f o r equilibrium (A)

in Figs 7 - 9 as well. That T L T / M system is unstable for all practical values of (Xp,, %.). Higher values of u may alter boundary Rz = 0, but will not contribute to our understanding of T L T / M system stability. Boun- dary R z = 0 appears to have a cusp singularity for negative values of 5q~which are o f no practical interest. For a given value of !~, the (xp. ~'t.) plane is divided in

xpl C R I T E R I A R I & R 2 ( R I ) O ) 0.6050 ;- . . . ~- . . . ' S ' I I o.5143" . . . ,'- . . . i i I I - 0 . 4 I .. .. . .. . .. . .. .. . .. . .. . .. . ' . . . o.++++ ," . . . . . o',- ~ ~ - ~ I I I i I i 0 . 3 9 2 9 ~- . . . - ' . . . ~ . . . + . . . ~- - - - ~ _ + ~ - ~ - + ~ ' ~ I i I 0 9 4 3 , 2 I 0 . 3 2 2 1 L . . . L . . . L .. . . ' i i 1 0 5 0.2514 ~ , i i t 1.1+ (~ IIu 0.1807 ~ . . . ~'-~'-'---'~----'=~--- 9 - . . . -~ i i I t - ~ I i i i i ~ I i i 0 . I I 0 0 L . . . J . . . [ - . . . J . . . L . . . d 0.0000 0 . 1 1 3 6 0.~272 0.3408 0.4544 0.5680

BARGE WITH SKEG ISOSURFACF WRT L,GA~./A,xp T~A

Fig. 7. Barge with skegs system: Stability criteria RI and R2 f o r equilibrium (A)

two parts, the unstable above the boundary and the stable below. For this particular system increasing xp,, ~ , and increasing 5'1~ destabilizes TLT/M. The former conclusion is contrary to common practice.

Figure 7 shows the stability boundaries for the barge with skegs system. Obviously, the addition o f stabilizing skegs improves the system stability at the expense o f increased resistance. The system has a finite stability domain in the practical range, and therefore the designer and/or operator may select practical and appropriate values of .x),, ~ , 5"]4 for safe operations. For a given ~,~, the barge with skegs system exhibits fold singularities for practical values of xp, and ~ . For high values of :r~, near +r/2, which are of no practical interest, fold singularities do not appear within the domain o f practical values o f xp,, ~A. In the two- dimensional graph in Fig. 7 only Rz = 0 is shown, while R~ > 0 in the entire domain shown.

Due to the form of fold singularities for a given 5']4, the following interesting conclusions can be derived. The stability of the T L T / M system which is unstable, that is operates inside the fold, can be improved by either increasing or decreasing either xp, or ~ . O f course there will ahvays be a practical optimal combina- tion of xp,, ~ values that will take the system out of the fold and into the stable domain. This conclusion is

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Nonlinear stability and simzdation o f t wo-line s/zip to wing: M. 31. Bernitsas and Jin-Szzg Chzzng C R I T E R I O N R 2 . . . , 2 2 " ~ ' - " 15 O l .~c-. -1,4 Xpl C R I T E R I O N R 2 ~ : . : 2 . , d U c. A _- " .-~--~-A.~: .: ~ " . ~ . i " - - ~ : " ~:~_.'~,'2-'~;~7r I

,,.I

- ~ 4 o o l xPl o.s6 9 ~ C R I T E R I A R 1 & R 2 0 . 6 0 5 0 . . . 0 A 8 1 2 r ... - I ... r " ... r " ... , S

(re>o)

',

' 0 . ~ 5 7 5 , ' - - . . . ~ . . . J . . . ,L . . . t 1,-1 ,.3.5.7.9.11 l ( R I < O ) ~ ~ ~ _{| | I}' - - ' - ' - - - ' - - ' = ~

,--

,_.__

- , u i 0 . 2 3 ~ 7 . . . 7 - r - ~ . . . . , , , ~ , 0 . 1 1 0 0 t.. . . . J . . . L . . . ~" . . . J ~ . . . . J 0 . 0 0 0 0 0.113G 0 . 2 2 7 2 0 . 3 4 0 8 0 . 4 5 4 4 0 . 5 5 8 , ~

CA)IC, O I4~RIN3 ISOSURFACE IA'RT t,OAl/!Jk,xp 7t~

Fig. 8. Cargo system: Stability criteria RI and R2 f o r equilibrium (,4) x=, I C R I T E R I A i l l & P.2 0 0,0 . . .

i )i l

0 . 2 6 1 6 ' i" . . . , . . . T - ~ . . . ! i i i 0 . 2 1 2 1 ~ . . . . "1 . . . "r . . . . ' ' ' I R 1 < 0 ' 0 . 1 7 9 7 ~ . . . . - ~ . . . * . . . t - . - - i l i i i i i i i i i i 0 . 1 5 7 5 - - ... .... ..... .... 0 . 9 4 8 G E - 0 1 . . . - i . . . T . . . r . . . . " 1 / -

,,

',

ti~_"~

,

...

,,,

,,

0 . 1 0 0 0 [ - 0 1 . . . . ..J . . . .t . . . t.. . . . . 0 . 5 7 0 0 0 . 7 1 2 9 0 . 8 5 5 7 0 . 9 9 8 G 1.141 1 . 2 8 4 1 . 4 2 7 1 . 5 7 0 IAN[ER UOORINC WRT L , C A I ~ A , x p

in agreement with that derived in Ref. 2 which explains the contradiction between theoretical analysis and model tests and observations regarding the appropriate tow line length in towing system s. In general, a simple rule of thumb for selection o f the appropriate value of ~ , cannot be derived. In addition, the common prac- tical belief that higher values of xp, will improve towing stability is, in general, incorrect.

Figure 8 showns that for all practical values of .~,, /~,, 5'1~ the cargo TLT/M_system is stable. Only unrealistically high values of lt~ or Z(l., could destabilize that system.

Finally, Fig. 9 shows the stability boundaries for the tanker system. The tanker propeller was taken out to ensure symmetry of the TLT]M system and its equilibrium (A). Within the domain of practical values o f x;,, h,, 5't~, no cusp singularity was identified. A cusp singularity, shown in Fig. 9, was found for 1 = 0 . 0 1 . The tanker towed]moored via a single line exhibits another cusp singularity for yp = 0 and prac- tical values of dimensionless (1,., Xp) = (3.5, 0.53). Near that cusp the system behaves chaotically. 3 Studying bifurcations and chaos of TLT/M systems is beyond the goal of this paper. It is the authors' intention to study this matter in the future.

C R I T E R I O N R 2

o.ol x N 0 3

Fig. 9. Tanker system: Stability criteria R1 and R2 f o r equilibriunz (,4)

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N o n l i n e a r stability a n d sinndation o f two-line ship towing: M . M . Bernitsas a n d Jin-Sug Cluing

4. N O N L I N E A R S T A B I L I T Y A N A L Y S I S

In the previous section, the T L T / M system equations were linearized and the system behavior was studied near equilibrium (A). T h o s e conclusions, however, are valid only in the vicinity o f (A) a n d therefore characterize the system locally and not globally. Qualitative conclusions regarding the global b e h a v i o r o f T L T / M systems are derived in this section by finding all equilibria points a n d studying the system stability near each equilibrium.

4.1. State space representation

T h e T L T / M n o n l i n e a r model presented in section 1 consists o f fourteen equations, n a m e l y (4), (5), (6), (11, i = 1,2), (18) to (24), a n d (25, i = 1,2). Selecting the following state variables

(XI =

it,

X2 = U, ).'3 = r , 24 : / l , X5 : "/l,

X6 = Wl, X7 = /2, X8 = "Y2, X9 = 092), ( 5 0 )

the system nonlinear model can be written in the f o r m o f nine n o n l i n e a r coupled differential e q u a t i o n s as:

.(. X ( X I , X2, x 3 )

pSb

k (.X3i+l

)q

--

m -- X:~ + m---

X:~

i= l

\ ~ . -

1

/ c o s x3ti+x)-~ , (51) m - X , ; 22 = Y ( x l , X2, X3) + N ( X l , X2, x3) - ~

pSb k (X3i+t

)q

A i=~ ~ T , , - 1 I ( l : - N / + x p , Yi)sinx~(i+O + Y I y p ' c ~ I=-Ni-A + ( M z . . . . . + M z , , , , , ) - ~ , Yi ( 5 2 ) .x'~ = Y(xt, x2, x3) N~ + N ( x t , x2, x3) m - Y: A A

A ,=, t--7-,., 1)

x [ [ X p , ( n / - Y~)+ N~]sin x3(i+l) + ( m - Yr')Ys., c o s x 3 . + ,)1 + (Fr,,,~t, + F r , , , , t ) ' - ~ 1 1 1 - Yg. - - , (53) where A is defined by e q u a t i o n (30), a n d X ( x t , xz, x j ) , Y ( X l , x2, x3), N ( x t , xz, x3), are defined by e q u a t i o n s (7), (8) a n d (9) respectively;

24 = x2 sin X6 -- Xl COS X6 -'1- U COS X5

+ X3Xp, sin x6 + X3yp, cos x6, (54)

1

-('s = - - [x2 cos x6 + x~ sin x6 - U sin x5 A'4

+ X3Xp, COS X6--x3yp, sin x6], (55)

I

9 (6 = x3 + - - [ x2 cos x6 + x~ sin x6 - U sin x~ x4

+ X3Sp, cos x 6 - x 3 ) b , sin x6], (56) 27 = Xz sin x9 - xt cos x9 + U cos xs

+ xj.x% sinx9 + X3yp: cos xg, (54)

1

.('8 = - - [ x2 cos x9 + xt sin x9 - U sin xs

x7

+ X3Xp: cos x 9 - x3)'m sin Xg], (58)

1

29 = .x'3 + - - [ x2 cos x9 + x~ sin x9 - U sin x8 x7

+ XaXv: cos x 9 - x3ym sin x9], (59) T h e T L T / M System depicted in Fig. 1 c a n be uniquely defined in terms o f the first six state variables. T h e remaining three can be eliminated using geometric relations (60) to (64)

~5 = ~ol - yt = w2 - 72, (60)

sin yl = y + (yp' cos ~, + xp, sin ~,) (61) 11

d cos ee y + (yR., cos ~ + xp: sin ~,) sin y2 = - - +

6 l,

(62) 1~ = (x + xm cos ~ - Yv, sin ~,)2

+ (Y + Yv, cos ~, + .vv, sin .r (63) /22 = ( - d sin ~ + x + x m cos ~, - yp: sin t)) 2

+ ( - d c o s e e + y + y m c o s ~ , + . V p 2 s i n ~,)2 (64) which p r o d u c e the following expressions:

x72 = d 2 + (Xp: - Xv,) 2 + (yp. - yp,)Z + x42

+ 2d sin eel x4 cos .v5 + ( Y m - y p , ) s i n ( x 6 - .x3)

-- (Xpz-

Xp,)COS(X6 -- 2 5 ) 1

- 2d cos eelx4 sin x5 + (yp: - )%)cos(x6 - Xs) + (xm - Xp,)sin(x6 - Xs) }

+ 2x4 cos x51 (Ym - yp,)sin(x6 - x s )

- (x, o 2 - Xp,)COS(X6 - x s ) l

+ 2x4 sin .x'51 (Ym - yp,)cos(x6 - Xs)

+ ( x m - .x~,)sin(x6 - xs)}, (65) x T s i n x s = - d c o s e e + x 4 s i n x5

+ (Ym - yp,)COS(X6 - xs) + (.x%, - Xp,)sin(x6 - x~), (66)

X9 = X6 -- X5 "1- X8. (67)

T h u s , we can define the following state variable vector

X=(XI=ll, XZ=O, x3=r,

3 . ' 4 = / 1 , X5 = - y i , x 6 = ~ l ) .

(68) Hereafter, evolution e q u a t i o n s (51) to (56) will be d e n o t e d as

9

X = f ( x ) , l E G t, f :

R6--+R 6,

(69) where R 6 is the six-dimensional Euclidian space and C ' is the class o f the c o n t i n u o u s l y differentiable functions.

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Nonlinear stability and simtdation o f two-line ship towing: M. M. Bernitsas and Jin-Sug Chung

4.2. Derivation o f system equilibria

All equilibria o f the nonlinear TLT]M System can be found by setting the time derivative of the state variable vector equal to zero. Then, due to equation (69), we have

f(x) = 0. (70)

Solution of the coupled nonlinear component equations in (70) produce the system equilibria. Using auxiliary equations (11), (25), (61) to (64), equation (70) can be recast in the following form o f simultaneous equations

) ( + S b p ( ~ - - I ) q C o s ( Z y t + ~ ) + S b p ( ~ - - l ) q c o s ( : y 2 + ~ ) + P x , , , , o ( ~ ) + P x , , , , ~ ( ~ ) = o,

]: + S o p ( ~ - l)qsin(~yl + fJ)

- S b p ( ~ - l)qsin(;y2 + ~ ,)

+ P~-,,...,~(~) + Fr,,-.,,~(~) = O, N - S b p ( ~ - l ) q x p t s i n ( : Y l + ~ )

- - S b p ( ~ - - l ) q x p ,

sin(5.2 + if)

- S b p ( ~ - - l ) q y p ,

C O S ( ~ ' I + ~ ' ) + S b p ( ~ - - l ) q y p , C O S ( 3 ' 2 + i f ) "l- /~ZWIND(~) "{- J~ZwA'~E(:) = 0,

) + (yp,

cos f +

Xp,

sin ~) sin 5'1 =

(71)

(72)

(73) (74)

d

y, + ( - yp,

cos ~ + Xp, sin ~) sin zt2 = - / z cos a +

(75) (76) .~" + Xp, cos ] , - yp, sin

C O S ~ 1 = - - ~ ,

cos "]'2 = d sin o~ _ .~ +

Xp,

cos : +

yp,

sin ~ (77)

t~ '

where the overbar indicates equilibrium, and kinematic relations (18) to (20) were used in equations (7), (8) and (9) to eliminate state variables xt, x2, x3 at equilibrium, and produce X, Y, N. In general, the system of simultaneous nonlinear equations (71) to (77) has three solutions, equilibria

Ej

Ej(Yc,),

if,

~tl,/'1,

~Z, 12),

j = 1,2, 3. (78) The numerical method used to compute

Ej, j = 1, 2, 3,

is based on subroutine NLSYS residing in MTS (Michigan Terminal System) which implements an algorithm developed in Ref. 32. This algorithm is approximately quadratically convergent and requires fewer function evaluations per iteration step than the N e w t o n - R a p h s o n method.

It should be noted that in equilibria (B2) and (B2), lines 11 and

12

bear the entire towing]mooring force, respectively, while the other line remains slack; that is 7"2 and Tl are zeros, respectively. In such a case the system of simultaneous nonlinear equations (70) can be reduced t o

(No - ypR - xpYo) + ( - UNv + xpUYv)sin ~/

+ (UNou- xpUYo.)COS ~, + (~ypXvvUZ)sin2~

+ (U2Nouu - xpU 2

Yo,u)COS2~

+ (-- UZNv,, + xpUZYvu)Sin f c o s

+ ( - ~ U3Nvuu + .x'p~ U3yvuu)sin ~

cos2~

+ (89 U3Xvruyp)sin2~/cos

+ ( - ~ U3Nm, + ~ U 3 YrcvXp)sin3]/.

(79)

For the four TLT[M systems simulated in Section 2 and Figs 2 to 5, computed equilibria are shown in Table 2. System 1, Barge without skegs, has three finite real

Table 2. Critical points of TLTIM s)'stems

T L T / M system E q u i l i b r i u m

I Barge without skegs

(A) ( B I ) (B2)

T L T / M system

2 3 4

Barge T a n k e r without propeller

with skegs C a r g o ( A ) (A) (A) ( B I ) (B2) Variable .~/L - 1.3931 - 1.0729 )IL 0.4999 0.7435 ~, 0.0 0.2793 :2~ = D/U** 1.0 0.9613 Ycz = ~l U 0.0 - 0 . 2 7 5 7 YO = iLIU 0.0 0.0 Yc, = [t[L 1.0 + 6 X 10 -7 1.0144 9 ~ = "tl 0.4667 0.9639 -~6 = w~ 0.4667 1.2432 xT=12[L 1 . 0 X 6 X 10 -~ * -v~ = "~z 5.8165 * ~9 = s 5.8165 * - 1.0729 - 1.3931 - 1.9809 - 0 . 6 9 3 6 - 0 . 6 4 8 6 - 0 . 6 4 8 6 0.2565 0.4999 0.5000 0.1500 0.2359 0.064 I - 0 . 2 7 9 3 0.0 0.0 0.0 0.0181 - 0 . 0 1 8 1 0.9613 1.0 1.0 1.0 0.9998" 0.9998 0.2757 0.0 0.0 0.0 - 0 . 0 1 8 1 0.0181 0.0 0.0 0.0 0.0 0.0 0.0 . 1 . 0 + 6 • 10 - 7 1.50001 0 . 2 + I • 10 -7 0.2064 * 9 0.4667 0.30469 0.25268 0.7789 * 9 0.4667 0.30469 0.25268 0.7970 * 1.0144 1 . 0 + 6 X 10 -7 1.50001 0 . 2 + 1 • 10 -~ * 0.2064 5.3193 5.8165 5.97850 6.03051 * 5.5043 5.0400 5.8165 5.97850 6.03051 * 5.4862

* indicates slack t o w J m o o r i n g line ** U = 2 knots

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Nonlinear stability and shnulation o f two-line ship towhtg: M. M. Bernitsas and Jin-Sug Chttng equilibria (A), (B1), (B2). System 2, Barge with skegs,

has only one equilibrium (A). The Cargo ship system also has one equilibrium. System 4, Tanker without propeller, has three equilibria (A), (BI), (B2).

We can draw the following important design conclusions regarding the number of TLT/M system equilibria. TLTIM systems may have up to seven equilibria. Specifically, in equilibrium (A) both towing/mooring lines are taut. In all other cases one line is slack. Then the corresponding (B) equilibria could be any of the possible three equilibria of the corresponding SPM system. Thus we define the following equilibria:

(i) Equilibrium (A): both lines are taut.

(ii) Equilibrium (B1A): line 1 is taut and the corresponding SPM IS m equilibrium (A).

(iii) Equilibrium (B1B1): line 1 is taut and the corresponding SPM is m equilibrium (BI). (iv) Equilibrium (B1B2)" line 1 is taut and the cor-

responding SPM is in equilibrium (B2). (v) Equilibrium (B2A): line 2 is taut and the cor-

responding SPM is m equilibrium (A).

(vi) Equilibrium (B2BI)" line 2 is taut and the corresponding SPM IS in equilibrium (BI).

(vii) Equilibrium (B2B2): line 2 is taut and the corresponding SPM is in equilibrium (B2).

According to this classification, System 1 exhibits equilibria (A), (B1A), (B2A); System 4 exhibits equilibria (A), (BIA), (B2A). All these possibilities make the global behavior analysis and the bifurcation problem of TLT]M's challenging.

4.3. Stability near equilibria (A), (B1), and (B2) Equilibria of nonlinear systems, also called stationary solutions, singular points, critical points, rest points, or fixed points I are in fact special degenerate trajectories. They possess significant properties, and local analysis near those points reveals the behavior of trajectories in the vicinity. Local analysis is performed by studying the linear system

4 = [ Z ] ~ , ~ E R 6, [ A ] E R 6 x 6 , ( 8 0 ) where ~ ( t ) = x ( t ) - K represents deviation from equilibrium .~,

[A] = [025] i, j, = 1, 2 ... 6, (81)

[OxA'

[A ] = Jacobian matrix of first partial derivatives of f evaluated at i , which exist since f E C 1. Expressions for [ A ] are provided in Appendix A. For the four TLT/M

Systems - simulated in Section 2 and Figs 2 to 5 - for each one of the equilibria shown in Table 2 the six system eigenvalues are computed and presented in Table 3.

4.4. Global behavior o f nonlhlear TLT/M system For a specific TLT/M system we can derive several conclusions regarding its global dynamic behavior by studying its local stability properties near equilibria. For the systems simulated in Figs 2-5, local stability properties near equilibria were considered in Subsection 4.3 and summarized in Tables 2 and 3. On the basis of that information we infer the following.

For TLT/M System 1 (Barge without skegs) trajectories initiated near equilibrium (A) diverge and converge to either (BI) or (B2). Equilibria (B1) and (B2) represent mirror image configurations and are attractors since all real parts of their eigenvalues are negative. Convergence to those attractors is oscillatory since they have three pairs of complex conjugate eigenvalues which indicated focal point behavior. Equilibrium (A) is a repellor with focal point behavior. Thus, we expect, in general, trajectories to oscillate away from equilibrium (A) and oscillate towards equilibrium (B1) or (B2). This behavior is shown in Fig. 2.

TLT/M System 2 (Barge with skegs) has one unstable equilibrium (A). All trajectories initiated near (A) diverge and all trajectories initiated far away from (A) result in very high induced line tensions which attract the towed-moored vessel towards (A). Therefore, an attractor limit cycle must exist. System 2 is located close to the stability surface R2 and can be slightly modified to make equilibrium (A) stable (see Fig. 7). That would result in a safe system. Alternatively, the system may be modified to allow setting in equilibrium (B1A) or (B2A) which are stable. (The system particulars cause the slack line to become taut before (B1A) or (B2A) are reached.) Then the system would operate with only one line in tension but it would be in stable equilibrium. In its present form, System 2 is unsafe and the above discussion explains the periodic large amplitude oscillations that MLM systems exhibit and are often attributed to time dependent excitation. 10

System 3 (Cargo) has only one real equilibrium (A). The eigenvalues corresponding to equilibrium (A) show that is a stable (attractor) focal point. This qualitative conclusion is verified quantitatively in Fig. 3.

System 4 (Tanker without propeller) has three unstable equilibria (A), (B1A), (B2A). By the same token used above for System 2, a limit cycle must exist.

Table3. Eigenvahtes o f TL T[Al systems

Eigenvalues T L T / M System Equilibrium M X2 X3 X4 X5 X6 (A) - 0 . 1 6 9 - 1.892 0.837 - 0 . 8 4 5 • 10 -3 0.179 x 10 -4 _+ ~,0.004 I (BI) - 0 . 1 9 3 + /,9.261 - 0 . 1 1 2 + /,0.529 - 0 . 6 0 1 _+ L0.070 (B2) - 0 . 1 9 3 _+/,9.261 - 0 . 1 1 2 + /,0.529 -0.601 + /,0.070 2 (A) - 0 . 1 7 7 - 1.978 - 0 . 2 7 2 - 0 . 9 9 8 • 10 -3 0.263 • l0 -3 +/.0.014 3 (A) - 0 . 2 3 7 - 3 . 0 0 6 - 0 . 3 6 2 - 0 . 2 4 0 x 10 -a - 0 . 6 5 0 x 10 -5 _+ /,0.002 (A) - 3 . 0 1 4 - 0 . 0 4 8 0.113 - 0 . 8 0 7 x 10 -3 - 0 . 5 3 1 x l0 -4 + /,0.004 4 (B1) - 0 . 3 1 0 + /,3.743 - 2 . 2 7 2 0.047 + /,0.473 - 0 . 1 5 3 (B2) - 0.310 +/,3.743 - 2.272 0.047 + /,0.473 - 0. ! 53

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Nonlhtear stability and shntdation o f two-line ship towing: M. M. Benlitsas attd Jin-Sug Chung The global system behavior can be improved by modi-

fying its configuration to make any of those equilibria stable.

The above conclusions were inferred for specific parameter values of the four TLT/M systems studied in this paper. Similar observations can be made on their dynamics for any other set of parameter values. Fur- ther, qualitative and design conclusions can be drawn regarding all equilibria by performing bifurcation analysis and identifying both static and dynamic loss of stability. 9

5. CLOSING REMARKS

A mathematical model for slow motion dynamics of two-line towing/mooring systems was developed using third-order nonlinear horizontal plane maneuvering equations and quasi-static tow/mooring line analysis for synthetic nylon ropes, chains or steel cables. Environ- mental excitation from current, wind, and mean wave drift force was considered. Simulations of four systems with qualitatively different dynamic behavior were used to illustrate conclusions derived by local linear analysis near equilibria and global nonlinear analysis of the corresponding autonomous system. It was shown that TET/M systems may be asymptotically stable, unstable, periodically stable or unstable, or exhibit complicated dynamics (chaos). Those conclusions explain large amplitude oscillatory motions observed in practice and often attributed to time dependent excitation from wind or waves. Extensive analysis near equilibrium (A) has shown that system boundaries may exhibit fold or cusp singularities. Further, it was shown, that of the five Routh-Hurwitz stability criteria only two may be active for conventional ship hull forms. Those two criteria provide relatively simple expressions for rational design of TLT/M systems. Simple rules of thumb regarding appropriate values of towing/mooring line length and fairlead location are incorrect, in general.

To gain deeper understanding of the slow motion dynamics of TLT/M systems, bifurcation analysis should be performed to define regions of qualitatively different behavior. Thus, a designer will be capable of defining parametric domains of safe and economic operation of TLT/M systems. The methodology developed in this work can be extended to provide similar qualitative understanding of the dynamics of Multi-Leg Mooring systems.

ACKNOWLEDGEMENTS

This work has been sponsored by The University of Michigan/Sea Grant/Industry Consortium in Offshore Engineering under Michigan Sea Grant College Pro- gram, projects number E/GLE-14 and R/T-23, under grant number NA85AA-D-SG045C from the Office of Sea Grant, National Oceanic and Atmospheric Ad- ministration (NOAA), U.S. Department of Commerce, and funds from the State of Michigan. Industry par- ticipants include the American Bureau of Shipping; ARCO Gas and Oil Company (1988-); Conoco, Inc.; Exxon Production Research; Friede and Goldman, Ltd.; Noble, Denton & Associates, Inc. (1985-1986); Shell companies Foundation (1985-1986); and the U.S. Coast Guard (1986-). The U.S. Government is author- ized to produce and distribute reprints for governmental

purposes notwithstanding any copyright notation appearing hereon. NOMENCLATURE a, b, c d

F ~ , ~

/z

h

lwi

L, LBP n t MLM Mz N No

N~

N.o

l%bc P , q r R

Sb

t TLT/M

T~

It U

U~

UT 0 (x, y)

(Xp,. yp,)

X

x0

xa

Xab

X~bc

Y

Yo

Y~

Y~b

Y.bc

dummy independent variables representing u, v, r and r in functions X, Y and N distance between tugboats or mooring terminals*

excitation forces in the X and Y directions, respectively

mass moment of inertia of moored vessel about the Z-axis

length of ith towing/mooring line

working length of ith towing/mooring line length of moored vessel

mass of moored vessel Multi-Leg Mooring

excitation moment about the Z axis yaw moment

yaw moment due to propeller

derivative of yaw moment with respect to a derivative of yaw moment with respect to a and b

derivative of yaw moment with respect to a, b and c

constants in expression for synthetic fiber rope tension

yaw angular velocity

moored vessel resistance, or set of real numbers

breaking strength of synthetic fiber rope time

Two Line Towing]Mooring

tension of the ith towing/mooring line vessel forward velocity with respect to water resultant relative velocity of vessel with respect to water

velocity of current velocity of tugboat vessel's lateral velocity

coordinates of vessel's centre of gravity body fixed coordinates of ith fairlead surge force

surge force due to propeller

derivative of surge force with respect to a derivative of surge force with respect to a and b

derivative of surge force with respect to a, b and c

sway force

sway force due to propeller

derivative of sway force with respect to a derivative of sway force with respect to a and b

derivative of sway force with respect to a, b and c

Greek symbols

fl, [3,., Od direction of respectively

6 rudder angle

current, wind, waves,

* quantities appearing in Figs 2 - 9 are non-dimensionalized using standard factors, zz

(14)

N o n f i n e a r stability a n d s h m t l a t i o n o f two-line ship t o w i n g : M . M . Bernitsas attd J i n - S u g Chttng

X eigenvalue

~, drift angle between current direction and vessel longitudinal axis in radians

t}' y a w a n g l e

Special s y m b o l s

D = d ( )]dt t i m e d e r i v a t i v e o p e r a t o r

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A P P E N D I X A : J A C O B I A N O F L I N E A R I Z E D F L O W A T E Q U I L I B R I A T h e t e r m s o f t h e J a c o b i a n M a t r i x , d e f i n e d b y e q u a t i o n ( 8 1 ) , e v a l u a t e d a t e q u i l i b r i u m ,~ a r e g i v e n b e l o w . 1 O X A ( I , 1 ) = - - tn - Nil OX~ ' 1 O X A ( I , 2 ) = - -m - X , ~ 3 . v ~ ' 1 O X A ( 1 , 3 ) = - -I t / - X l i 0.X'~ ' A (1,4) - - -A ( 1 , 5 ) - - -S b p q X 4 _ l c o s x6 m - X,i l,,., I cos .,.9 X7 0X9 n , - Xs ~ \0-~'5,] ~ - ] COS X9