responses of multi point mooring systems under the influence of different sea conditions

8/10/2019 Responses of Multi Point Mooring Systems Under the Influence of Different Sea Conditions

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Responses of Multi Point Mooring Systems Under the Influence of

Different Sea Conditions

Deniz Bayraktar, Abdi Kkner

Istanbul Technical University, Faculty of Naval Architecture and Ocean Engineering, Department of Ocean Engineering

Maslak, Istanbul Turkey

ABSTRACT

In this paper, two multi point mooring systems which are under the

action of periodic wave excitation are considered in order to find out

the approximate responses by using Runge Kutta Method. One of the

mooring systems is anchored to the seabed with four mooring lines

while the other is anchored with six mooring lines. On both systems, a

strong nonlinearity is described by increasing restoring force. For

possible mooring line densities, change in excursion downstream and

tension at the buoys are taken into account respectively. By introducing

different sea conditions into these multi point mooring systems, wave

force acting on each system is observed. For certain excitation

frequencies a number of chaotic phase diagrams are obtained which

yields into instability of these systems. At the end, comparison of the

two systems is figured out to understand the effect of number ofmooring lines on the responses of each system.

KEY WORDS: Multi point mooring systems; Static catenary linemooring; Wave drift force; Runge Kutta Method.

INTRODUCTION

Nonlinear dynamics of ocean structures has been studied out widely.

Abkowitz (1972) presented a hydrodynamic model in order to analyze

the motion of a vessel. Bernitsas and Chung (1990) studied the

horizontal plane motions of a ship moored to two terminals and also

used chains as mooring lines. Moreover, effect of mooring line

arrangement on the dynamics of spread mooring systems was carried

out by Bernitsas and Garza-Rios (1996). On the other hand, Bernitsasand Kim (1998) analyzed the effect of slow drift loads. Effect of second

order slowly varying wave drift loads is studied by Bernitsas, Matsuura

and Andersen (2004). Kim and Bernitsas (1999) gave a general idea

about spread mooring systems by studying the nonlinear dynamics and

stability of these systems. The effect of size and position of supporting

buoys on the dynamics of spread mooring systems was carried out by

Garza-Rios and Bernitsas (2001). Stability analysis and bifurcation

theory are used to determine the changes in spread mooring system

dynamics in deep water. Using harmonic balance method Umar,

Ahmad and Data (2004) studied the stability analysis of a mooring

system. In this multi point mooring system, an additional geometric non

linearity is generated with the association of mooring line angles.

Whether the traditional way of mooring in shallow water is to use steel

chain or wire rope in the form of a catenary, as water depth increases,

the angle of the catenary at a buoy becomes steeper. The steep catenary

initially produces very little horizontal restoring force, and excessive

platform motions can result. For a better understanding about the

nonlinearity under different excitation frequencies, different sea

conditions are introduced into these mooring systems which have the

exact same geometric properties have been compared. Effect of number

of mooring lines on the responses of each mooring systems is analyzed.

MATHEMATICAL MODEL OF THE SYSTEMS

In a multi point mooring system considering a single degree offreedom, the equation of motion for surge under the influence of

harmonic excitation can be given by

FSinXRXCXm =++ )(&&& (1)

Where

[ ][ ]

++

++=

22

22

)sgn(1)sgn()(

XbXd

bdXbXkXR (2)

According to the Eq. 1, mis the sum of msand mawhich are system and

added masses respectively, C is the damping coefficient including

structural and hydrodynamic damping where hydrodynamic mass is

considered to be the 5-8 % of the ships mass which leads the damping

to be small (Journe and Massie, 2001). On the other hand, structuraldamping is idealized as proportional to the mass and stiffness (Umar,

Ahmad, and Data, 2004). It is possible to ignore nonlinear effects due

to hydrodynamic damping. band dare the distances between mooring

lines which are illustrated in Figs. 1-2.

Proceedings of the Eighteenth (2008) International Offshore and Polar Engineering Conference

Vancouver, BC, Canada, July 6-11, 2008Copyright 2008 by The International Society of Offshore and Polar Engineers (ISOPE)

ISBN 978-1-880653-70-8 (Set); ISBN 1-880653-68-0 (Set)

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Fig. 1: Mooring system anchored to the seabed with four mooring lines

Fig. 2: Mooring system anchored to the seabed with six mooring lines

Mooring angles play an important role depending on the geometricnonlinearity of the stiffness which is restoring force R(X)although the

cables in Figs. 1-2 have linear elastic properties.

If Eq. 1 is arranged in order to get first order autonomous equations,

following equalities are obtained.

XY

YX

&&&

&

=

= and =& (3)

(2)

When Eq. 3 is substituted into Eq. 1 and both sides are divided into

total mass m,the following expression will be obtained.

mXRY

mC

mFY )(sin = & (4)

Where

m

Ff = ,

m

XRXr

)()( = ,

m

C= and += t

Finally the above equalities are substituted into Eq. 4. One gets,

)()sin( XrYtfY += & (5)

Restoring force R(X) varies with displacement and the equation ofmotion is derived based on equilibrium of geometric restoring forces

and small body motion under wave and current excitation (Gottlieb andYim, 1992). Restoring force is introduced as a fifth order anti

symmetric polynomial since it describes the actual force displacement

behavior of the system as seen in Eq. 6.

5

5

3

31)( XcXcXcXr ++= (6)

Since the system is under wave excitation, examining the approximate

response of the system for various exciting frequencies is an important

phenomenon in order to have a prevision about the particular situation.

Fourth order Runge Kutta Method is widely used for the solution of

nonlinear systems. It is known to be very accurate and well-behaved for

most of the engineering problems. In this manner, Fourth Order Runge

Kutta Method is introduced as a solution method for the equation of

motion.

The tension forces in the cables are dependent on the cable weight, its

elastic properties and the mooring system. It is assumed that the system

is moored with a static catenary line. Catenary line mooring is the

oldest and most common mooring system that is still in use. In case of

static catenary line, restoring force is derived by lifting and reducing

the weight of the mooring line.

If it is assumed that the buoys diameter is very small compared to the

incident wave length, the wave exciting force can be found by

integrating the pressure field of the incident wave only. Considering

that the system is under the influence of low frequencies which means

the wave is long, diffraction part may be neglected since it is very small

and this may lead one to think that the wave force tends to the Froude-

Krylov force. However, in the behavior of a moored system, the effects

of second order wave drift forces are most apparent. These are

horizontally restrained by some form of mooring system. Analyses of

these horizontal motions are important in order to understand the

response of the system.

CATENARY LINE MOORING

In catenary systems the chain or wire rope must move laterally through

the water in response to applied tension. Long catenary lengths can not

respond in this mode to rapid platform motions. The only effective

elasticity in such a catenary at wave excited frequencies is that of the

very high modulus chain or wire rope, and high mooring loads can

result.

The static forces on a cable element of length dsare showed in Fig. 3where T, tension; dT, change of tension over cable element length, ds;

Dds, normal pressure drag on cable element; Fds, tangential friction

drag on cable element; Pds, gravity force per length, ds; , angle with

flow direction,Vr

; d, change in angle over cable element length; and

ds, length of cable element (Berteaux, 1976).

Fig. 3: Forces on cable element

Under static equilibrium, the vector forces must be zero so the normal

and tangential components of these forces become respectively as in

(Eqs. 7-8) respectively.

0sin)( =+ dsPCosDdsddTT (7)

0)( =+++ FdsdsPSinCosddTTT (8)

Considering a mooring line which is made of steel wire rope with a

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drag force applied at the system, the gravity force becomes more

important and the resistance terms (P>>D and F) of the cableequilibrium is neglected so Eqs. 7-8 may be written as follows

assuming dSind and 0dTd .

dsPCosTd = (9)

dsPSindT = (10)

Integrating from =0 to = yields Eq. 11 as below.

0TTCosT

H == (11)

where T0 is the tension at the origin =0 which shows that horizontal

tension, THis constant.

Integrating over the same domain gives;

PsTSinTV

== (12)

This result shows that the vertical component of the tension TVequals

to the immersed weight of cable length between origin and the point

P().

The tension at this point is given by Eq. 13.

( )( ) 21

22

0PsTT += (13)

Integrating

==

=

0

1

0

0

T

PsSinh

P

Tdxx

ss

yields into Eq. 14.

=

0

0

T

PxSinh

P

Ts (14)

Finally, using Eq. 12 and Eq. 14, the following equation is obtained.

== 1

0

0

T

PxCosh

P

Tdyy (15)

In summary, (x,y)is the coordinate of any point along the mooring line.

The arc length from the origin to this coordinate is given by Eqn. 14

while Eq. 13 shows the tension at this coordinate.

WAVE DRIFT FORCE

It is assumed that the presence of the structure does not affect the wave

field; hence waves propagating past the structure remain unmodified.

For slender body motion such as surge motion, it is accepted as the

wavelength is much larger than the beam of the object moored

(Newman, 1980).

Wave drift forces are second order and consist of a steady-time

independent component and a low frequency component. Due to the

low natural frequency of surge motion of the system and the low

damping at this frequency, large surge motions can result. Assuming

that the wave drift forces are proportional to the square of the wave

height, following empirical formula may be used.

da

RSF

h

h

s

=

2

0

)()(2 (16)

Rh()is the drift force operator which is a function of water depth andwave frequency. Sh()is the wave spectrum according to the specified

sea condition and water depth. The water depth effect on both driftforce operators and wave spectra is introduced in Eq. 16. Also, the

Pierson-Moskowitz wave spectrum has been modified to includeshallow water effects. However, the shallow water correctioncoefficients are ignored in the calculation of surge wave drift force. The

final formula for surge drift forces is given below by Eq. 17.

( )wwwave

CosufXXX += )(21

(17)

NUMERICAL APPLICATION

Two cylindrical buoys which have the exact same geometricalproperties are considered in this study. However, one of them is

anchored to the seabed with four mooring lines and the other isanchored to the seabed with six mooring lines. These cylindrical buoysare hollow with a diameter of 10 m, thickness of 0.7 m, and a height of

6 m with a drag of 1000 N on both of them. Figs. 4-5 shows the relationbetween depth and excursion downstream for possible mooring line

densities 20 kg/m, 30 kg/m, 40 kg/m, 70 kg/m and 100 kg/mconsidering the four point mooring systems. The same relations for sixpoint mooring system are shown in Figs. 6-7. The elasticity modulus ofthis mooring line made of steel chain is taken as 30x106N/m2.

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0 20 40 60 80 100

Depth (m)

Exc

ursionDownstream(m)

20 kg/m

30 kg/m

40 kg/m

70 kg/m

100 kg/m

Fig. 4: Four point mooring system, relation between excursion

downstream and depth for different mooring line densities

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

110000

0 20 40 60 80 100

Depth (m)

Tension

attheBuoy(N)

20 kg/m

30 kg/m

40 kg/m

70 kg/m100 kg/m

Fig. 5: Four point mooring system, relation between tension at the buoyand depth for different mooring line densities

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0

2

4

6

8

10

12

14

16

18

20

0 20 40 60 80 100

Depth (m)

ExcursionDownstream(m)

20 kg/m

30 kg/m

40 kg/m

70 kg/m

100 kg/m

Fig. 6: Six point mooring system, relation between excursion

downstream and depth for different mooring line densities

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

110000

0 20 40 60 80 100

Depth (m)

TensionattheBuoy(N)

20 kg/m

30 kg/m

40 kg/m

70 kg/m100 kg/m

Fig. 7: Sixpoint mooring system, relation between tension at the buoyand depth for different mooring line densities

In Fig. 8, variation of restoring force with excursion is shown for thefour point mooring system.

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

-1 -0.5 0 0.5

X

R(X)

Fig. 8: Variation of restoring force with excursion for the system withfour mooring lines

The coefficients of the polynomial given in Eq. 6 are found by leastsquares approximation which are c1=1.03312, c3=0.178316 and c5=-

0.00677438. Damping coefficient is calculated as 0.05. For thesignificant wave heights of; 6.7 m, 4.6 m, 3.65 m, 3.05 m, 2.1 m, 1.85m and 0.67 m; the surge wave drift forces for both systems are 2609503

N, 1029436 N, 581543.9 N, 372510.8 N, 144689.8 N, 104308.9 N and8635.205 N respectively. Finally, the wave force coefficients f ,corresponding each surge wave drift forces are found as 6.68, 2.64,1.49, 0.95, 0.37, 0.27 and 0.02 respectively.

Equation of motion is solved by Fourth order Runge Kutta method andfor different excitation frequencies of different sea conditions, the

responses in the time domain and phase diagrams of these responses are

obtained.

Fig. 9: Response in the time domain for the excitation frequency=0.40 and wave force coefficient f1=6.68

Fig. 10: Phase diagram of the response for the excitation frequency=0.40 and wave force coefficient f1=6.68



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In Fig. 17 variation of restoring force with excursion is shown for the

six point mooring system.

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

X

R(X)

Fig. 17: Variation of restoring force with excursion for the system with

six mooring lines

The coefficients of the restoring force polynomial given in Eq. 6 arefound by least squares approximation which are c1= 1.03197, c3=0.161673 and c5= -0.00604543.

Equation of motion for this second mooring system is again solved byFourth order Runge Kutta method and for different excitation

frequencies and wave force coefficients, the responses in the timedomain and phase diagrams of these responses are obtained.



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DISCUSSION AND RESULTS

Two symmetric multi point mooring systems which are assumed to beanchored to the seabed with linear inelastic cables are taken into

account and four and six point mooring systems are compared witheach other. In the first mooring system where four mooring lines areused, the change of depth with excursion downstream shows anexponential relation as seen in Fig. 4. The same attribute is current for

the system which has six mooring lines as shown in Fig. 6. In bothsystems as mooring line density increases excursion downstream

decreases with depth. However, for a six point mooring system this isless rather than the system with four point mooring lines. Whether theresult of decrease in excursion seems an advantage as mooring linedensity increases, the tensions at each buoy play an important role too.

Fig. 5 and Fig. 7 show the relation between tension at the buoy anddepth. According to this, as mooring line density increases, tensions at

each buoy increases too. It seems that the change in number of mooringline has no obvious effect on tension; however for a six point mooringsystem tension is slightly more than the four point mooring system.

Whether it is better to obtain a less excursion by a higher mooring linedensity, the increase in tension may lead the mooring lines to a breakoff in a heavy sea condition. According to different sea modes, relatedwave excitation frequencies and wave force coefficients are introduced

and equation of motion is solved by fourth order Runge Kutta method.Generally, increase in the sea modes, the both systems shows an

instable situation. For =0.40 rad/s, the response and phase plots areshown in Figs 9-10 respectively. The response and phase diagramsshow subharmonic response. During the motion of the mooring system,while one mooring line becomes taut, other becomes slack where

subharmonic responses are formed as a result of these nonlinearmooring lines. As excitation frequency reaches 0.50 rad/s, symmetry in

the center is observed as can be seen in Figs. 11-12 which show thatone can say that the system is stable. However, in Figs. 13-14, when the

wave excitation frequency is 0.75 rad/s, symmetry in the centerbecomes apparent which points out a more stable situation for the firstmooring system. Limit cycle interval decreased from 3.9 to 1.6 andfinally to 0.52 as shown in Fig. 10, Fig. 12 and Fig. 14. In Figs. 15-16,

symmetry in the center can be seen clearly and the amplitude of themotion is between 0.35. Considering the six point mooring system,

wave excitation frequency is taken as 0.45 rad/s as f is 2.64. RelatedFigs. 18-19 show similar subharmonic responses as in the four pointmooring system. However the amplitude of the motion is 3.3 which islower than the four point mooring system which verifies that the six

point mooring system is more stable in the same heavy sea condition.

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Similarly, as excitation frequency reaches 0.50 rad/s, symmetry in thecenter with a limit cycle interval of 1.6 is observed as can be seen in

Figs. 20-21. Since the amplitude of the motion is less than the fourpoint mooring system, one can say that this system is more stable.Whether, in Figs. 22-23, for the wave excitation frequency of 0.75rad/s, symmetry in the center becomes apparent, amplitude of the

motion shows a similar stable behavior as in four point mooringsystem. In Figs. 24-25, for the wave excitation frequency of 1.0 rad/s

symmetry in the center can be seen clearly and the amplitude of themotion is very similar with the four point mooring system.It is possible to say that under the influence of heavy sea modes, acylindrical buoy which is anchored to the seabed by means of six

mooring lines is more stable rather than the one anchored to the seabedwith four mooring lines. However as wave excitation frequencyincreases, the amplitudes of both systems motions become similar. In

general, when both of the systems are under the influence of a sea modeof 3-4 they do not show any chaotic behavior.

CONCLUSIONS

Two mooring systems consist of four and six mooring lines areconsidered. Nonlinearity in these systems is assumed to be a result of

nonlinear restoring forces of the mooring lines. Variation of restoringforce with displacement is taken as a quintic anti symmetric polynomial

and using fourth order Runge Kutta method the approximate responsesof these systems are obtained. Different sea conditions and wave

excitation frequencies introduced different responses resulting in perioddoubling and symmetry breaking. In general, it is possible to say thatthe system with six mooring lines is more stable compared to thesystem with four mooring lines considering the diagrams of excursion

downstream versus depth. On the other hand, effect of mooring linedensity should also be considered as another important parameter. It is

possible to obtain a small excursion by using a mooring line with ahigher density but increase in tension constitutes a risky situationespecially at heavy sea modes. The comparison of both systems for the

same heavier sea mode with an excitation frequency of 0.40 and 0.45rad/s, the amplitude of the system with six mooring lines is smallerwhich verify that the system with six mooring lines is more stable.However, the existence of a weak current can generate a chaotic

response even in a moderate sea state since the period doublings createsymmetry loss whether the system is anchored to the seabed by means

of six mooring lines or more.

REFERENCES

Abkowitz, M.A. (1972). Stability and Motion Control of OceanVehicles, The MIT Press, Cambridge, Massachusetts.

Bernitsas, M.M., and Chung, J-S (1990). Nonlinear Stability andSimulation of Two-Line Ship Towing and Mooring, Applied

Ocean Research, Vol 12, No 2, pp 77-92.

Bernitsas M.M. and Garza-Rios, L.O. (1996). Effect of Mooring LineArrangement on the Dynamics of Spread Mooring Systems, JOffshore Mech and Arct Eng, Vol 118, pp 7-20.

Bernitsas M.M. and Kim, B-K. (1998). Effect of Slow-Drift Loads onNonlinear Dynamics of Spread Mooring Systems, J OffshoreMech and Arct Eng, Vol 120, No 4, pp 201-211.

Bernitsas M.M., Matsuura, J.P.J., and Andersen, T. (2004). MooringDynamics Phenomena Due to Slowly-Varying Wave Drift, Trans

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Berteaux, H.O. (1976). Buoy Engineering, John Wiley & Sons, Inc.,New York.

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Position of Supporting Buoys on the Dynamics of Spread MooringSystems,J Offshore Mech and Arct Eng, Vol 123, No 2, pp 49-56.

Gottlieb, O. and Yim S. C. S. (1992). Nonlinear Oscillations,Bifurcations and Chaos in a Multi-Point Mooring System with aGeometric Nonlinearity,Appl Ocean Res, Vol 14, pp 241-257.

Jordan, D.D. and Smith, P. (1999). Nonlinear Ordinary DifferentialEquations An Introduction to Dynamical Systems, Third

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Journe, J.M.J., and Massie, W.W. (2001). OffshoreHydromechanics,Delft Univ of Technology.

Kim, B-K., and Bernitsas, M.M. (1999). Nonlinear Dynamics andStability of Spread Mooring Systems, J Appl Ocean Res, Vol 23,

No 2, pp 111-123.

Newman, J. N. (1980). Marine Hydrodynamics, MIT Press,Cambridge, Massachusetts.

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responses of multi point mooring systems under the influence of different sea conditions

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