assessment of the effect of mooring systems on the horizontal motions with an equivalent force to...

ARTICLE IN PRESS

Ocean Engineering 33 (2006) 1644–1668

0029-8018/$ -

doi:10.1016/j

�CorrespoE-mail ad

www.elsevier.com/locate/oceaneng

Assessment of the effect of mooring systemson the horizontal motions with an equivalent

force to model

R. Pascoala, S. Huangb, N. Barltropb, C. Guedes Soaresa,�

aUnit of Marine Technology and Engineering, Technical University of Lisbon, Instituto Superior Tecnico,

Av. Rovisco Pais, 1049-001 Lisboa, PortugalbDepartment of Naval Architecture and Marine Engineering, Universities of Glasgow and Strathclyde, UK

Received 12 February 2005; accepted 21 September 2005

Available online 18 January 2006

Abstract

The present work was carried out to assess the performance of an equivalent force model for

the rapid analysis of mooring lines subject to horizontal motions. The verification is performed

for cases of practical interest. Due to the very distinct behaviour of slack, moderately slack

and taught moorings, these are analysed separately and the variation in the model coefficients

is justified. It is shown that the methodology provides reasonable estimates of the mooring line

behaviour. The final equation is very simple to introduce into a vessel motion program and is

also thought to be useful in controller design or stability analysis.

r 2005 Elsevier Ltd. All rights reserved.

Keywords: Nonlinear mooring model; Equivalent force; Slow drift motion

1. Introduction

The inclusion of fully dynamic cable analysis into an existing code for vesselhydrodynamics may turn out to be a difficult task due to mismatch of the relevant

see front matter r 2005 Elsevier Ltd. All rights reserved.

.oceaneng.2005.09.005

nding author.

dress: [email protected] (C. Guedes Soares).

www.elsevier.com/locate/oceaneng

ARTICLE IN PRESS

Nomenclature

CDn, CDt normal and tangential drag coefficientsE Young’s modulusg gravitational acceleration (9.81m s�2)H water depthHD horizontal distance from anchor to fairleadm cable mass per unit arc lengthT0 initial tension at the fairlead of each cable when at static mean position

R. Pascoal et al. / Ocean Engineering 33 (2006) 1644–1668 1645

time scales (e.g. Leonard et al., 2000). Furthermore, the code may become quiteinefficient in terms of calculation time.

If model based controllers or the stability of a system are being analysed, then it isgood to have low order approximations, as the inclusion of a fully dynamic cabledynamics formulation will usually be too complex for real time operation or analyticstudies.

Several simplified solutions to estimate the forces due to mooring lines have beenpublished, with special attention given to the mooring line damping. Most of themethods rely on use of a quasi-static solution and possibly perturbation analysis.Banduin and Naciri (2000), for instance, give some discussion on mooring linedamping, comparing their results with those of other authors.

Pascoal et al. (2005) have proposed a somewhat different methodology because itis designed to provide an equivalent force model containing all the relevant terms. Itis based on the assumption that the force is differentiable with respect to position,velocity and acceleration. The physical reason for writing force as a function of thesevariables is that then it is naturally expressed as stiffness, damping and mass termsthat are easier to interpret as the changes in the variables tend to very smallamplitudes about a mean position, in which case they will reduce to those of a linearmass-spring-damper system.

A Taylor expansion, containing terms with the relevant variables, is used inconjunction with a simple system identification procedure. The solution is linear inthe coefficients and the identification scheme is based on a least squares criterion andclosely resembles one of the procedures used to determine hydrodynamic derivativesfrom manoeuvrability tests.

In this paper, the methodology proposed by Pascoal et al. (2005), and the resultingmodel for the effect of a mooring system on the horizontal forces, are assessed byusing it to provide the coefficients of the Taylor expansion for significantly differentmooring systems.

The difference in mooring behaviour is most noticed as load-carrying shifts froman almost purely geometric balance to a strain driven equilibrium. There are threevery distinct behaviours that correspond to a slack, moderately slack and a taughtcable geometry. The slack moor has the possibility to geometrically change and thatis its response to different loads. The moderately slack moor crosses from a purely

ARTICLE IN PRESS

R. Pascoal et al. / Ocean Engineering 33 (2006) 1644–16681646

geometric to a partially elastic load-carrying mode. The taught moor has an almostpure elastic load-carrying mode.

The moderately slack moorings were expected to be the most troublesome whensimplified modelling is considered and indeed presented itself as such due to the shiftfrom a geometric to an elastic response. The low order model provides onlyacceptable results for this case. Anyhow, in real application, this is probably theleast-favoured load-carrying mode because it leads to lower fatigue life, snap loadingof the cables, etc.

The ease of including the resulting model into a vessel dynamics program isachieved by its algebraic equation form. Once implemented, it allows long runs withmultiple dynamic mooring lines to be used when calculating the statistics of mooringline dynamic tensions and vessel dynamic response.

2. Theoretical background

According to Newtonian mechanics, any external force applied at a given point ina system, with given physical properties, has a reaction that is function of relativemotions of the system constituents and their time derivatives up to second order. Fora one-dimensional motion, this is symbolically expressed as

F ¼ F ðx; _x; €xÞ. (1)

The idea is to find F to sufficiently high order only for the force at the fairlead of amooring system such that they may be determined without having to solve the fulldifferential equations for each run of a vessel dynamics code or if someapproximation is to be found for analysis of stability or controller design forinstance. To make such an approximation, the usual Taylor expansion of (1) isconsidered about some equilibrium static position for the fairlead, without loss ofgenerality considered to be the origin.

F ¼XN

n¼0

1

n!xqqxþ _x

qq _xþ €x

qq €x

� �n

~0

F þ RN . (2)

Only configurations with symmetric response will be considered and thus only oddpowers are of interest. Eq. (2) can be written in the form of a polynomial withunknown coefficients and with some of them collected to better establish a relationwith the cable behaviour. For N ¼ 3 this is represented as:

F ffi ða1 þ a6 _x2 þ a8 €x

2Þxþ ða3 þ a5x

2 þ a11 €x2Þ _x

þ ða4 þ a7x2 þ a10 _x2Þ €xþ a2x3 þ a9 €x

3þ a12 _x

3 þ a13x _x €x. ð3Þ

The coefficients will be used throughout the paper and have to be identified usingsome system identification procedure. One of the problems with cable systems is thatthey can become completely slack and kink, but this behaviour is just as unwanted inthe identification process of a mooring system as for instance the saturation ofmechanical actuators in a chemical power plant (see for instance Braun et al., 2000)

ARTICLE IN PRESS


or of electronic components (e.g. Godfrey, 1993). It was in electronics that specialphase relations for the sum of sine waves were discovered to produce signals with thesame energy as any random phase signal, but having smaller maximum excursions.For a flat spectrum, i.e. one with the same spectral energy density in all of itsbandwidth, the phases satisfying the minimum peak-to-peak excursion are know asSchroeder phases (Schroeder, 1970).

There are immediately two possible ways to identify the derivatives; by exciting thetop with a known force and to record the motions, or vice-versa. The application ofknown motions is easier and more intuitive to perform. The fairlead point has to bemoved according to some law so that a relationship between the kinematics and theforce may be found. Here it was chosen to have a slow drift motion that encompassesthe maximum estimated value plus a wave frequency motion spectrum synthesizedusing Schroeder-phased sum of sine waves. Schroeder phases can thus be used toexcite the fairlead point at wave frequency, and the excursions due wave frequencymotion will have limited maximum amplitude while all the energy will be stillpresent. This is a desired situation because the taught slack behaviour and kinkingare reduced to a minimum and the identification is carried out using a high-energysignal.

The multi-sine is written as (Godfrey 1993, p. 134):

xðtÞ ¼XNf

i¼1

ffiffiffiffiffiffiffi2Pi

pcosð2pf i þ jiÞ (4)

while Schroeder phases are defined as ji ¼ 2pPi

j¼1jPj, or for flat energy density:

ji ¼pN

i2. (5)

The highest power of the expansion in (2), the number of harmonics in (4) and theidentification procedure have to be chosen. Here, terms up to the fifth order have beenconsidered, and the number of harmonics was chosen to be 161, because it gives afundamental frequency of 0.0125Hz when the wave frequencies range 0.05–0.25Hz(typical bandwidth of wave frequency motion), thus allowing for 800 s of simulationwithout repetition, which was determined to be sufficient for this application. Theidentifications have been carried out using linear least square fit (e.g., Kalaba andSpingarn 1982, pp. 165–169). The procedure is fully documented by Pascoal et al. (2005).

3. Numerical examples

Arrangements with very distinct behaviour are presented as examples to assess thepredictive capacity of the model. The model is subject to different types ofexcitations in order to reduce the probabilities of the results being a coincidence.

The properties common to all the mooring arrangements are: H ¼ 400m, cablelength ¼ 1200m, line diameter ¼ 0.103m, m ¼ 44:0 kg=m, CDn ¼ 1:2, CDt ¼ 0:01,E ¼ 4:0� 1010 N=m2.

The initial properties that differ in the arrangements are given in Table 1.

ARTICLE IN PRESS

Table 1

Properties of the mooring arrangement

Slack Moderately slack Taught

HD [m] 1080 1110 1140

T0=mgH 2.44 3.94 15.3

Table 2

Matrix of performed runs, the code used for cross-reference, and their main parameters

Slow drift parameters Configuration

Period [s] Amplitude [m] Slack Moderately slack Taught

100 20 S10020 MS10020 T10020

40 S10040 MS10040 T10040

200 40 S20040 MS20040 T20040


The test matrix, with the corresponding code used to refer to the set of results, isrepresented in Table 2.

The 20 and 40m amplitudes correspond to 5% and 10% of the water depth,respectively. The 100 and 200 s periods are considered to be representative of theperiods of slow motion encountered in practice. The taut line configuration givesloads that cannot be sustained by single line but is given for comparison ofqualitative behaviour.

3.1. Slack mooring with 20 m drift and 100 s period

The symmetric mooring system used to simulate the slack configuration is shownin Fig. 1.

In Pascoal et al. (2005), the selection of a signal that is useful in the identificationof the unknown coefficients was found to be of the utmost importance.

In the figures to follow, Model Force pertains to the one given by the simplifiedmodel as described in Pascoal et al. (2005) while Mooring Force is the one obtainedfrom a lumped mass formulation.

The coefficients resulting from a multi-tone composed of two sine waves, a multi-sine and a static analysis are presented in Table 3. In this case, the multi-tonecorresponds to a low frequency signal with amplitude modulation and 20mmaximum amplitude plus a single 2m amplitude wave frequency. The relative erroron the restoring coefficients is small; the change of the linear is from 4% when usinga single wave frequency (1WF) to 0.26% with the flat spectra multi-sine, and thesehave been compared with values from a quasi-static formulation approximated tothe fifth order (Static 5). Using a multi-sine, as may be observed by looking at thepairs such as linear and cubic restoring (corresponding to a1 and a2), linear and cubic

ARTICLE IN PRESS

0

-200

-400

z [m

]

-1500 -1000 -500 0 500 1000 1500x [m]

Fig. 1. Symmetric mooring system for simulation.

Table 3

Full estimation of coefficient from the dynamic code with multi-sine

a1 a2 a3 a4 a5 a6 a7 a8 a9

1WF 10530 1.327 96135.8 32658.1 63.8 �36.7 �68.0 828.7 �81526.1

M-Sine 10939 1.342 131643.2 �63818.5 56.5 36.1 �40.9 169.5 337.0

Static 5 10968 1.291

a10 a11 a12 a13

1WF �39288.2 95206 7301.9 86.2

M-Sine 1221.8 �1473.1 �49.5 33.7

Static 5 �39288.2 95206 7301.9 86.2


damping (a3 and a12), a4 and a9, the higher order terms of the proposed power serieshave decreasing modulus, which is a sign of possible convergence within a certainsubset, while this is not the case when using a multi-tone.

In Fig. 2 the power-spectral density of the force response to the Schroeder-phasedinput is shown. In all figures, the waviness of the spectral density is due mostly to thediscrete nature of the excitation spectrum and the absence of smoothing. Althoughthe system may turn out to be strongly nonlinear, spectra are probably the simplestway to compare the response from the model with the parameters obtained from theidentification procedure and the one from the cable dynamics program.

Next, the predictions to the response from signals of single sinusoids arepresented. The signals are given in Table 4 and the results are plotted in Figs. 2–6.Although there are some fairly large errors in trying to predict the behaviour underlarger periods, higher frequencies give good prediction, e.g. Signal 2, and this ismostly due to the damping forces.

The regression for the coefficients of the fifth order approximation was notproblematic. In Table 5, the coefficients that exist in both the third- and fifth-orderapproximations are presented, also added is the fifth-order regression of coefficientsto the results from a quasi-static formulation. The velocity cubed term, (c10), hasbeen estimated in such a way that it enables predictions that extend to relatively largeamplitude wave frequency (which in practice is not important for the case in study).The cubic restoring is different and indicating a slight softening but also notimportant for this case, the linear restoring relative error is 1.9% and the relativechange in linear damping is 5% of the smallest value. There is still indication of apossibly converging power series.

ARTICLE IN PRESS

Power-Spectral Density of Responsex 106

2.5

2

1.5

1

0.5

0

S (f

) [k

N2 s]

0.05 0.1 0.15 0.2 0.25 0.3f [Hz]

Mooring ForceModel Force

standard deviation from dynamic code = 389.15 kNstandard deviation from model = 380.94 kN

Fig. 2. Resulting power-spectral densities from the Schroeder-phased input (not smoothed and zoomed).

Table 4

Matrix of sinusoidal signals to assess the quality of the multi-sine input coefficients

Identification Period [s] Amplitude [m]

Signal 1 100 15

Signal 2 90 20

Signal 3 130 30

Signal 4 100+10 20+2


Good results are obtained if, after estimation, only a3, a4, a7, a8, a11 and linearstiffness are considered. There was no benefit from considering fifth-order terms; theestimation is actually not as good as for the third order.


The system simulated here is exactly the same as the one in S10020, simply thelargest amplitude of the slow drift has been changed. The new kinematics are plottedin Fig. 7. The system outputs are plotted in Figs. 8 and 9.

It has been shown that the predictive capacity is fairly good under a multi-sineexcitation. In order to reduce the number of simulations, which is actually part of thepoint behind the simplified model, the predictions of this S10040 case to the signalsused in S10020 are presented. This is basically a simulation of a situation that wedesire will encompass the other. The signal properties are given in Table 6 and theresults are plotted in Figs. 10–14.

ARTICLE IN PRESS

Model Response300

200

100

0

-100

-200

-300100 150 200 250 300 350 400

time [s]

Model Force [kN]Mooring Force [kN]

Fig. 3. Model with multi-sine coefficients; prediction under Signal 1.

Model Response300

200

100

0

-100

-200

-3000 50 100 150 200

time [s]


Fig. 4. Model with multi-sine coefficients; predictions of Signal 2.


Only third-order predictions are shown because fifth order was seen to produce nobetter result. Once more, as may be seen from looking at the response under signal 2,in Fig. 11, the model gives better prediction at higher frequencies. Other examplesare given in Figs. 13 and 14.

ARTICLE IN PRESS

Model Response600

400

200

0

-200

-400

-6000 50 100 150 200

time [s]


Fig. 5. Model with multi-sine coefficients; prediction under Signal 3.

Model Response600

400

200

0

-200

-400

-600200 250 300 350 400 450 500

time [s]


Fig. 6. Model with multi-sine coefficients; prediction of multi-tone signal, Signal 4.


When small amplitude monochromatic motion, e.g. Signal 1, is imposed, somevery large errors occur in estimating the peak value, 40% in this case as may be seenin Fig. 10. This is trying to predict the response of a 15m motion, having no wave

ARTICLE IN PRESS

Table 5

Comparison between third and fifth-order Taylor coefficients

a1 a2 a3 a4 a5 a6 a7 a8 a9

M-Sine 3 10939 1.342 131643.2 �63818.5 56.5 36.1 �40.9 169.5 337.0

M-Sine 5 10756 �3.094 124851.4 �50994.4 �11.3 402.7 �181.9 353.0 �974.8

Static 5 10968 1.291

a10 a11 a12 a13

M-Sine 3 1221.8 �1473.1 �49.5 33.7

M-Sine 5 �398.1 1553.1 2262.7 27.0

Static 5

Prescribed Top-End Kinematics50

0

-50200 250 300 350 400 450 500

time [s]

Offset [m]Acceleration [ms-2]Velocity [ms-1]

Fig. 7. Imposed kinematics with Schroeder phasing.


frequency components, with a model of 40m motions possessing wave frequencycomponents. The error is, as anticipated, due mostly to the damping term (Figs. 10and 12).


Now, instead of making new identification for the S20040 cases, which may beconsidered an established procedure, it will be seen how the S10040 adjusts to thelarger slow drift period while maintaining the wave frequency components. Theseresults are presented in Figs. 15 and 16. Only third-order predictions are shown

ARTICLE IN PRESS

Model Response1500

1000

500

0

-500

-1000

-1500200 250 300 350 400 450 500

time [s]


Fig. 8. Adjustment of the model to the top-end force.


6

5

4

3

2

1

0

S (f

) [k

N2 s]

0 0.05 0.1 0.15 0.2 0.25 0.3f [Hz]



Fig. 9. Power spectral density of the top-end force zoomed to wave frequency component.


because it was seen that fifth order gives no significant enhancement. The resultmight be classified as being fairly reasonable, with 15.3% relative error in standarddeviation and the model giving overestimation. A new identification would be better.

ARTICLE IN PRESS

Model Response300

200

100

0

-100

-200

-300100 150 200 250 300 350 400

time [s]


Fig. 10. Prediction of system response under Signal 1.

Table 6

Signal used for establishing the predictive capacity


Signal 1 100 15

Signal 2 90 20

Signal 3 130 30

Signal 5 100+Wave 20m+Schroeder superposition


3.4. Moderately slack mooring with 20 m drift and 100 s period

The performance with taughter geometry presented in Fig. 17 is now assessed. Theprocedure used for identification is considered the same. Thus, only the results arepresented and analysed.

Fig. 18 shows the power spectral density for the force signal that is obtained fromthe cable dynamics program and from the simplified model when subject to theSchroeder-phased input used in identification.

In Table 7, it is seen that the cubic order restoring, a2, coefficient was changeddramatically by this top 30m shift in the horizontal distance from fairlead to anchorpoint (from 1080m in the previous case to 1110m). This may be justified by anincrease in elastic stiffness and transition of stiffness mechanism, because with anoffset of 5m (1115m from anchor to fairlead) there is no cable on the bottom andthus less capability for geometric accommodation.

ARTICLE IN PRESS

Model Response800

600

400

200

0

-200

-400

-600

-8000 50 100 150 200

time [s]


Fig. 12. Prediction of system response under Signal 3.

Model Response400

300

200

100

0

-100

-200

-300

-4000 50 100 150 200

time [s]


Fig. 11. Prediction of system behaviour under Signal 2.


The prediction under this taughter configuration is not so good. It seems that inrelatively slack case there was still some truth to the superposition of wave frequencycomponents and linearity of response. Webster (1995) mentioned that, for a slackconfiguration, damping coefficients were constant with amplitude; apparently for aslack configuration it also does not vary much within the range of wave frequencies.

ARTICLE IN PRESS


4.5

4

3.5

3

2.5

2

1.5

1

0.5

0

S (f

) [k

N2 s]

0 0.05 0.1 0.15 0.2 0.25 0.3f [Hz]



Fig. 14. Power spectral densities of the outputs with Signal 5 excitation.

Model Response800

600

400

200

0

-200

-400

-600

-800200 250 300 350 400 450 500

time [s]


Fig. 13. Prediction of Schroeder-phased 20m amplitude, Signal 5.


For the taughter configuration, not much movement is allowed without changing themeans of load accommodation from geometric to elastic and a dramatic change indrag as well; this may be the reason for a less capable estimation.

In Figs. 19 and 20, the predictions, using the coefficients determined with thespectrum from multi-sine with Schroeder phases, are worst than for a slack case. The

ARTICLE IN PRESS


4.5

4

3.5

3

2.5

2

1.5

1

0.5

0

S (f

) [k

N2 s]

0 0.05 0.1 0.15 0.2 0.25 0.3f [Hz]



Fig. 16. Resulting power spectral densities from the S10040 model and S20040 simulation.

Model Response800

600

400

200

0

-200

-400

-600

-800200 250 300 350 400 450 500

time [s]


Fig. 15. Prediction from S10040 model to the signal of S20040.


signal is a 2m amplitude 10 s harmonic motion superimposed on a 20m-amplitude100 s slow drift. The relative error in standard deviation is 26.5% and the relativeerror in the peak value (not considering the phase shift) is 33%.

For any of the examples shown, the relevant contribution to the final predictioncomes from a3, a4, a8, a9, a10, a11 linear K and cubic K.

ARTICLE IN PRESS


9

8

7

6

5

4

3

2

1

0

S (f

) [k

N2 s]

0 0.05 0.1 0.15 0.2 0.25 0.3f [Hz]



Fig. 18. Power spectral density of response to Schroeder-phased input.

Table 7

Coefficients for a taughter configuration

a1 a2 a3 a4 a5 a6 a7 a8 a9

M-Sine 3 25315 43.7 233959.6 �182022.5 �98.1 �234.4 �52.1 101.4 3274.3

M-Sine 5 25498 25.9 228166.6 �179529.4 �430.9 430.2 �715.6 �205.1 6372.4

Static 5 24133 21.0

a10 a11 a12 a13

M-Sine 3 4414.4 �5795.8 �344.0 �72.5

M-Sine 5 3771.6 �10268.8 9237.8 201.7

Static 5

0

-200

-400

Z [

m]

-1500 -1000 -500 0 500 1000 1500x [m]

Fig. 17. Simulated moderately slack symmetric mooring.


As in previous cases, estimation in SLS should be performed with all terms. Afterthe coefficients are estimated, some may be insignificant and disregarded.

A fifth-order Taylor series estimation does not render a noticeable improvementupon the third order.

ARTICLE IN PRESS


S (f

) [k

N2 s]

0 0.05 0.1 0.15 0.2 0.25 0.3f [Hz]


15

10

5

0


Fig. 19. Estimated by the fifth-order Taylor series.

Model Response1500

1000

500

0

-500

-1000

-15000 50 100 150 200 250 300 350

time [s]


Fig. 20. Time trace of Fifth-order prediction.


3.5. Moderately slack with 40 m drift and 100 s period

The configuration of the mooring arrangement is the same as for the MS10020,but the slow drift amplitude has been increased to 40m. New coefficients weredetermined and the adjustments may be visualized in Figs. 21–28.

ARTICLE IN PRESS

Model Response


6000

4000

2000

0

-2000

-4000

-6000200 250 300 350 400 450 500

time [s]

Fig. 21. Model as during the identification procedure.


2

1.5

1

0.5

0

S (f

) [k

N2 s]

0 0.05 0.1 0.15 0.2 0.25 0.3f [Hz]



Fig. 22. Power spectral densities of the signal and the model as identified.


The signals in Table 8 are used in order to verify the adequacy of the newlydetermined coefficients.

The simulation for MS20040 is also used, but, as before, the new identification isnot performed and instead an attempt to predict the behaviour with the MS10040model.

ARTICLE IN PRESS

Model Response2000

1500

1000

500

0

-500

-1000

-1500

-20000 50 100 150 200 250 300 350 400

time [s]


Fig. 23. Prediction of system response to Signal 1.


7

6

5

4

3

2

1

0

S (f

) [k

N2 s]

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14f [Hz]



Fig. 24. Power spectral densities of prediction and response to Signal 1.


The results are plotted in Figs. 23–28 and comprise all the coefficients from thethird-order approximation. Naturally, if the coefficients were to be identified again,the adjustments would be better, but even with the previous values they are stillquite good.

ARTICLE IN PRESS

Model Response2000

1500

1000

500

0

-500

-1000

-1500

-2000200 250 300 350 400 450 500

time [s]


Fig. 25. Time trace of prediction and response to Signal 2.

2

1.5

1

0.5

0

S(f)

[kN

2 s]

x107

0 0.05 0.1 0.15 0.2 0.25 0.3

f [Hz]


Power-Spectral Density of Response

Standard deviation from dynamic code = 887.86 kN

Standard deviation from model = 1036.3 kN

Fig. 26. Power spectral densities of response and prediction under Signal 2.


It is interesting to see that this 40m model actually predicts the 20m caseof Signal 1 more adequately than with the identification previously performed for‘‘up to’’ 20m situations.

ARTICLE IN PRESS

Model Response

200 250 300 350 400 450 500time [s]


6000

4000

2000

0

-2000

-4000

-6000

Fig. 27. Prediction of MS20040 with the MS10040 model, Signal 3.


S (f

) [k

N2 s]

0 0.05 0.1 0.15 0.2 0.25 0.3f [Hz]

Mooring Force

Model Force2

1.5

1

0.5

0


Fig. 28. Power spectral densities of MS20040 with the MS10040 model, Signal 3.


3.6. Taught mooring with 20 m drift and 100 s period

A taught configuration is analysed next. It will be shown that the behaviour is quitedifferent from that of previously simulated configurations and most likely is a case thatmay be subject to a reasonable analytic solution even with currents (Fig. 29).

ARTICLE IN PRESS

Table 8

Signals used to assess validity of the coefficients


Signal 1 100 & 10 20m+2m

Signal 2 100+wave 20m+Schroeder superposition

Signal 3 200+wave 40m+Schroeder superposition

0

-200

-400

z [m

]

-1500 -1000 -500 0 500 1000 1500x [m]

Fig. 29. Symmetric mooring system for simulation.

8000

7000

6000

5000

4000

3000

2000

1000

0

-10000 100 200 300 400 500 600 700 800

time [s]

Horizontal Force from Mooring Lines

Offset [m]Hor. Force Moor 1 [kN]Hor. Force Moor 2 [kN]

Fig. 30. Time trace of top end tension on each moor.


In Fig. 30, a typical time trace of a taught mooring configuration is shown. Evenwith slow motion of the top end, horizontal tension in the mooring lines reachesstates of null values. This very asymmetric signal, departing considerably from asinusoid, has been found in previous studies, amongst others by Huang and Vassalos(1995), Webster (1995) and Aranha and Pinto (2001).

In Fig. 31 the response spectrum is almost flat, just as is the excitation. Thisindicates frequency-independent proportional response only attainable when thesystem is governed by constant restoring coefficient.

ARTICLE IN PRESS

18

16

14

12

10

8

6

4

2

00 0.05 0.1 0.15 0.2 0.25 0.3

f [Hz]

S (f

) [k

N2 s]

x 106 Power-Spectral Density of Response


standard deviation from dynamic code = 5040 kN

standard deviation from model = 5027.5 kN

Fig. 31. Power spectral density of response to Schroeder-phased input.

Table 9

Coefficients for a taught configuration

a1 a2 a3 a4 a5 a6 a7 a8 a9

M-Sine 3 380635 �139.6 57664.4 �63623.1 203.4 �212.5 �229.2 �30.1 2454

Static 3 368011 �123.3

a10 a11 a12 a13 a10

M-Sine 3 2971.8 �3310.1 1501.5 280.5 2971.8

Static 3


Taught situations render the possibility of further simplification. As wasmentioned and may be observed in Table 9, or confirmed by inspection ofFig. 31, this problem is governed by restoring coefficients with relativelysmall contributions from a3, a4, a5 and a7. Ultimately, apart from restoringforces, all other coefficients may be disregarded, and if non-linear effects of snaploading or the mooring itself are not important for the analysis, quasi-staticcalculations suffice.

When the dynamic code is used, top-end horizontal tension may reach zero in oneof the mooring lines, normally this won’t happen in the static code, but this is alsorelatively unimportant to the resultant force owing to the very large tension the othermooring line has attained at that instant.

ARTICLE IN PRESS

1

0.5

0

-0.5

-10 100 200 300 400 500 600 700 800

time [s]

x 104 Model Response


Fig. 32. Time trace of T10020 model and T20040 simulation.


3.7. Taught mooring with 40 m drift and 200 s period

This case is not very realistic because tensions in the individual mooring lines arealready too large for a real case, but anyhow, the behaviour is seen not to besignificantly different from the one experienced with the T10020. Here, in Fig. 32,only a sample prediction using the coefficients as determined in T10020 is presented.T10040 is not presented because it is redundant in this context. The relative error instandard deviation is less than 9%.

4. Conclusions

A simplified model, for estimating mooring forces arising from horizontalmotions, has been assessed. The model starts from physical reasoning, but itscoefficients are estimated from a least square identification procedure. Thecoefficients are estimated from a single quick run of a cable dynamics code usingspecial Schroeder-phased signal.

Three examples shown were: a slack mooring, a moderately slack mooring and ataught mooring configuration. The identification procedure was shown to work wellfor single catenary slack and taught moorings, but has some problems with theintermediate configuration. The estimation problem is due to the strong, butcontinuous, behaviour change as the geometric capability to accommodate loadsends and the elastic load bearing is triggered.

ARTICLE IN PRESS


Complex multileg arrangements with localised buoyancy, etc, could be analysedusing the same technique.

The final equation is very simple to include into a time domain moored vesseldynamics program and eventually may be used to monitor the forces at the fairleadduring long time domain simulations.

Acknowledgment

The work of the first author has been financed by Fundac- ao para a Ciencia e a

Tecnologia under grant SFRH/BM/4244/2001.

References

Aranha, J.A.P., Pinto, M.O., 2001. Dynamic tension in risers and mooring lines: an algebraic

approximation for harmonic excitation. Applied Ocean Research 23, 63–81.

Banduin, C., Naciri, M., 2000. A contribution on quasi-static mooring line damping. Journal of Offshore

Mechanics Arctic Engineering 122, 125–133.

Braun, M.W., Ortiz-Mojica, R., Rivera, D.E., 2000. Design of Minimum Crest Factor Multisinusoidal

Signals for ‘‘Plant-Friendly’’ Identification of Nonlinear Process Systems. In: Proceedings of the 12th

IFAC Symposium on System Identification, June 21–23, Santa Barbara, CA.

Godfrey, K. (Ed.), 1993. Perturbation Signals for System Identification. Prentice-Hall International, Great

Britain.

Huang, S., Vassalos, D., 1995. Chaotic Heave Motion of Marine Cable-body Systems. In: Proceedings of

the 1st ISOPE Offshore and Mining Symposium, Tsukuba, Japan.

Kalaba, R., Spingarn, K., 1982. Control, Identification and Input Optimization. Plenum Press, New York.

Leonard, J.W., Idris, K., Yim, S.C.S., 2000. Large angular motions of tethered surface buoys. Ocean

Engineering 27, 1345–1371.

Pascoal, R., Huang, S., Barltrop, N., Guedes Soares, C., 2005. Equivalent Force Model for the Effect of

Mooring Systems. Applied Ocean Research, in press.

Schroeder, M.R., 1970. Synthesis of low-peak-factor signals and binary sequences with low autocorrela-

tion. IEEE Trans. Inf. Theory IT-16, 85–89.

Webster, W.C., 1995. Mooring induced damping. Ocean Engineering 22 (6), 571–591.

assessment of the effect of mooring systems on the horizontal motions with an equivalent force to...

Documents

mooring behaviour

equivalent force model

model coefcients

mooring line behaviour

mooring line damping

effect of mooring systems

moderately slack moorings

cable mass