MOSES

NTNU

Norwegian University of Science and Technology

Department of Marine Hydrodynamics

PROJECT THESIS

SPAR

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0

5

10

15

20

25

30

35

40

45

T [sec]

Surge [m/m]

Moored

Free floating

SPAR

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0

10

20

30

40

50

T [sec]

Pitch [deg/m]

Moored

Free floating

Free floating SPAR

-200

-150

-100

-50

0

50

100

150

200

0

10

20

30

40

50

T [sec]

Phase [deg]

Heave

Pitch/Surge

SPAR

0

2

4

6

8

10

12

14

16

0

5

10

15

20

25

30

35

40

45

T [sec]

Heave [m/m]

Moored

Free floating

Pitch motions, H=10m

0

2

4

6

8

10

12

14

0

5

10

15

20

25

30

35

40

T [sec]

Pitch [deg]

Frequency domain

Time domain

SPAR

0

5

10

15

20

88

90

92

94

96

98

100

102

T [sec]

Pitch [deg/m]

Heave motions, H=10m

0

10

20

30

40

50

0

5

10

15

20

25

30

35

T [sec]

Heave [m]

Time domain

Frequency domain

10

15

20

25

30

35

0

500

1000

1500

2000

T [sec]

Air gap [m]

-10

-5

0

5

10

500

700

900

1100

1300

1500

1700

1900

T [sec]

Pitch [deg]

moored, with risers

free floating

-120

-115

-110

-105

-100

-95

-90

-85

-80

500

1000

1500

2000

2500

3000

T [sec]

Heave [m]

moored with risers

free floating

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

0

5

10

15

20

25

30

35

40

T [sec]

Damping coeff. [-]

Hull shape III

Hull shape II

Hull shape I

-98.5

-98

-97.5

-97

-96.5

-96

500

700

900

1100

1300

1500

T [sec]

heave [m]

hull shape I

hull shape II

Acknowledgement

This project thesis has been written under the supervision of two people that I would like to thank, Odd M. Faltinsen, supervisor from the institute and Jon Erik Borgen, supervisor from the company (Inocean as).

In addition I would like to thank the department of Inocean in which I have spent most of these twenty weeks.

This work concludes my education…(

OSLO , ……/…….. - 2002

Truls Jarand Larsen

Introduction

As a termination of my Master of Science degree in Marine Technology at the NTNU (Norwegian University of Science and Technology) in Trondheim, I am writing a thesis at the department of Marine Hydrodynamics in co-operation with Inocean as. This thesis is a conclusion of a 20 weeks work, starting the 28th of January with a hand-in date the 20th of June.

The assignment has the title: “Modelling of wave induced motions of a SPAR buoy in MOSES”, and the exact wording is:

“The candidate has to be familiar with MOSES (Multioperational Structural Engineering Simulator) and find out its limitations in relation to wave frequency motions, slowly varying motions in 6 degree of freedom and dynamic stability (Mathieu instability) in roll and pitch. How MOSES handles currents, wind and viscous damping are details that have to be discussed. Further on it will be clarified how currents in the moonpool and the effect of risers and mooring are handled. The candidate will critically indicate any possible deficiency.

As a part of the thesis a calculation of the linear wave induced motions of a spar buoy has to be carried out. These results are compared with the calculations done by Haslum in his Dr. Ing theses [Haslum 2000]. At the end the candidate will investigate the influence of some changes in the hullshape.

As far as the time allows it the candidate will implement the long wavelength model for linear wave induced motions of a Spar, as in [Haslum 2000], where currents in the moonpool are considered. Further on the model of Haslum for dynamical instability in roll and pitch will be implemented.”

This assignment is based on the use of MOSES, which is an analysis tool for almost anything that can be placed in the water. It is a quite comprehensive programming language that I first got to know during a similar assignment in Stolt Offshore in Paris. I have considerably improved my ‘MOSES-knowledge’ by working on this thesis.

All the linear results will be compared to [Haslum 2000]. In addition a non-linear time domain analysis has been carried out. The results have been used to point out the instability phenomenon as well as other aspects that may be of interest in relation to the time domain, such as the extreme response during a long lasting (2 – 3 hours) hurricane.

One of the more interesting aspects of the assignment is to see how MOSES handles the Mathieu instability. This part has therefore been emphasised in my work.

As said, I will compare most of my results with the one produced in the Dr. Ing. Theses “Simplified methods applied to nonlinear motions of spar platforms”, by Haslum. I will use his results as a reference [Haslum 2000] throughout the report and have therefor chosen the same dimensions of the Spar buoy. This applies to the linear response, RAO, as well as the time dependent motions and the Mathieu instability phenomenon.

In addition to the above mentioned analyses, an evaluation of the coupled effects from mooring line dynamics and riser friction will be carried out.

A lot of effort is made in doing the actual programming and to produce every single result. Behind every figure presented in the assignment it is hidden many days of work. Even though only a few results have been presented, lots of analyses have been carried out without leading to the desirable results. This is explained in the belonging chapters.

Executive summary

After creating the hull the linear motion response were calculated. They are expressed as RAO’s and have been compared to the linear results produced by Haslum. The agreement between the results is good and act as verification for the MOSES model. The effect of mooring system on the linear motion response is investigated. To increase the effect from the mooring system, the mooring lines were modelled quite stiff. Even very stiff lines had small influence on the linear wave frequency response.

Next part of the thesis is based on the time domain. To get the wanted level of confidence from the results a typical three hours hurricane analysis has been carried out, showing the heave and pitch response for a moored buoy. While the deflections in pitch are quite large, the heave response reveals the advantages of a Spar buoy. The hurricane used is a GOM hurricane with Hs=12.2m and T=14sec, which produces a heave response at maximum one and a half meter.

To increase the damping in heave alternative hull shapes have been tried out. When altering the lower part of the buoy, either by adding a circular disc slightly bigger than the rest of the Spar or by increasing the diameter at the bottom section of the Spar, the heave damping increases.

An interesting aspect of the assignment was to see how MOSES handles the Mathieu instability. This instability is a quite newly explained phenomenon that is testified by complicated theory. By showing this Mathieu instability in a time domain analysis, the program reveals its diversity.

The results from the coupled analysis show the importance of including mooring line dynamics and riser friction when predicting the Spar response. The response was significantly reduced and the Mathieu instability phenomenon was suppressed. Existing Spars have deep drafts to reduce the wave loads and consequently the heave motion. Traditionally, the additional damping from mooring lines and risers were ignored in estimating the heave response. Since this effect is important, the draft of the Spar can be reduced while maintaining an acceptable heave response. Reduction in Spar hull draft can reduce fabrication costs substantially and as a result the Spar solution will be more cost effective.

MOSES does not allow you to alter the parameters in the diffraction calculations. Consequently to implement the long wavelength model for linear wave induced motions of Haslum (as proposed in the introduction) is hard to accomplish.

Table of contents

2Acknowledgement

Introduction3

Executive summary5

Table of contents6

1.0 Inocean as – a brief presentation of the company7

2.0 MOSES – calculation procedures and general issues8

3.0 The Spar buoy14

3.1Movement of the SPAR16

4.0 The Processes18

4.1RAO- frequency domain18

4.2Time domain20

4.2.1 Low frequency behaviour20

4.2.2 The damping problem22

4.2.3 Hurricane analysis23

5.0 The Mathieu instability26

6.0 Coupling effects30

7.0 Alternative hull shapes32

8.0 Recommendations for further work35

9.0 References37

List of figures38

Symbols and nomenclature38

Appendix39

Chapter 1

1.0 Inocean as – a brief presentation of the company

Inocean as was established in 1996 in Oslo, and also has offices in Stavanger and Houston. Inocean is a technology company within naval architectural design, engineering and marine operations, serving major offshore companies and ship owners at home and abroad. Inocean is set to take part in the future development of floating structures and marine operations.

Engineering

Inocean deals with every phase of marine engineering, such as global and local structural design and calculations, hydrodynamic and hydrostatic calculations, riser and mooring calculations, technical drawing and documentation and the design of special tools.

Marine operations

Inocean analyses, plans, executes and leads marine operations and mobilises vessels for offshore construction work. The company’s aim is to reduce offshore weather-related delays to a minimum through using special tools and advanced simulations. Inocean also provides the personnel needed to perform the actual offshore operations.

In-house design and products

Inocean offers a range of products, such as:

· Lophius semi-submersible

· Flexistinger

· Anchor handling and stand-by design

· Steel production L-riser design

· 3-CODS subsea drilling derrick

R&D projects

Inocean is currently engaged in joint R&D project on developing next generation propulsion systems with industrial partners and research institutions.

Chapter 2

2.0 MOSES – calculation procedures and general issues

As a part of this assignment it will be explained a few things about the MOSES’ calculation procedure and the handling of some general issues in relation to analysis of the SPAR buoy. This is important for the evaluation of the reliability in the results and in addition being able to make a comparison with the results of the Dr. Ing thesis by Herbjørn A. Haslum. In this context it is important to detect possible limitations regarding the program and critically indicate any possible deficiency.

MOSES (Multi-Operational Structural Engineering Simulator) is a general-purpose simulation program for the analysis of almost anything, which will be placed in the ocean. You can choose the hydrodynamic theory you want to use and what kind of analyses you want to perform. (Motions, stress, forces etc.)

An example of the syntax is shown below:

$Constants and units

&dimen –save –dimen meter m-tons

&model_def –save

&model_def –density 490 –emodulus 2.9E4

$

-6

-4

-2

0

2

4

6

0

2000

4000

6000

8000

10000

T [sec]

Pitch [deg]

$Definition of the macro

&insert macro

$

$Definition of the classes

-97.8

-97.6

-97.4

-97.2

-97

-96.8

-96.6

-96.4

-96.2

0

2000

4000

6000

8000

10000

T [sec]

Heave [m]

~1TUBE 42 0.75-FYIELD 51.204

~10TUBE 34 0.50-FYIELD 34.136

$

$Element definition

BEAM300~1*J100*J101

BEAM301~10*J101*J102

$Defines the co-ordinates for points

*J100

0.00.00.0

*J101

10.0.00.0

*J102

20.0.00.0

*J 10

10.0.05.0

$

$Supplementary loads

&describe load_group

MJNTS

*J1011.75

$

SPAR

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0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0

10

20

30

40

50

T [sec]

Pitch [deg/m]

Moored

Free floating

&instate –loc spar 0 0 -202.5 0 0 0

These are just a few examples from a Moses file. In addition to these commands, the specific commands for performing the analysis and regulating the output (where and how) are many and have to be adapted to each case.

For more details about the Moses files, see Appendix 2 and 3.

Stepwise and briefly explained, this is how MOSES function:

1. Starting out by creating a model of a SPAR with specified degree of accuracy. This is done by defining points at the end face of the cylinder followed by creating panels between these points. The model is now a cylinder consisting of panels, which has to be imparted with physical qualities that corresponds to the buoy, i.e. mass distribution (center of gravity, partial loads etc.) and some material properties. Further on you can model mooring and risers to include their stiffness. The model is then ‘placed’ in equilibrium with specified draft, and all the hydrostatic properties are then calculated.

2. By using the model and the geometry beneath the water surface, the potential distribution of the pressure on the panels is given by the linearized Bernoulli equation

)

(

t

gz

p

d

df

r

+

-

=

(2.1)

where

=

r

fluid density

=

g

acceleration of the gravity

=

z

depth

=

f

velocity potential

By integrating the pressure over the body we obtain the hydrodynamic forces on a portion of the body. The program now calculates the added mass, damping matrices and in addition all the hydrodynamic properties.

3. On basis of the hydrodynamic forces the linear motions in six degree of freedom are estimated as the RAO (Response Amplitude Operator). This is straightforward done by using the equations of motions for a freely floating body:

[

]

)

(

)

(

6

1

t

F

C

B

A

M

j

k

k

jk

k

jk

k

jk

jk

=

+

+

+

å

=

h

h

h

&

&

&

(j=1,…,6)

(2.2)

where

=

jk

M

body mass

=

jk

A

hydrodynamic mass (added mass)

=

jk

B

damping coefficient

=

jk

C

restoring coefficient

=

h

body movement

=

j

F

excitation force

The RAOs are considered as transfer functions and are further found by estimating the coefficients in the equations of motions for each period defined, and finally end up with the corresponding RAO. These are defined as vectors in six degree of freedom and expresses the linearized wave induced motions in the frequency domain.

Further on it is implied that the response is linear such that the RAOs can be multiplied with a chosen wave spectrum, to finally obtain the response spectrum (expressed as RMS values – Root Mean Square):

)

(

)

(

)

(

2

w

w

w

F

x

S

RAO

S

=

(2.3)

where

=

)

(

w

x

S

response spectrum

=

)

(

w

F

S

wave spectrum

The major disadvantage with spectral response is that this response is applicable to a single environment and thus the post processing options are limited.

4. So far, all the static properties have been found. Next one wishes to analyse the movements in relation to a time series. That leads us to the strength of MOSES, the time domain.

As a starting point it utilises the hydrodynamic properties calculated in the frequency domain to satisfy the basic equations of motion. These equations are integro-differential. This means that the unknown is placed in an integral expressed as the derived. In a linear system a known deceleration function is included in the integrand. This deceleration function is estimated from the added mass and damping coefficients, which are dependent on the frequency. This requires great numerical accuracy that in practice can lead to inaccuracy.

The program will persistent calculate and bring up to date the parameters like the centre of buoyancy, waterplane area etc. as the buoy moves and the wet surface changes. The hydrodynamic forces are calculated at the displaced position and the finite amplitude effect of the changing waterplane area is taken into account. The dynamics of the system are in other words taken care of and the non-linear wave induced motions are found, i.e. the presence of one of the Mathieu instability phenomena will be, according to the theory, detected during a time domain analysis. The dynamic stability can be verified in all six degree of freedom. The calculations are based on a current environment (wave spectrum, significant wave height and peak period).

5. A function allows one to scale the wave excitation force in the time domain. This means that you can (in percentage) specify the interaction from the wave excitation force on the model. For example zero, which results in the direct wave force not being applied to the system. By this you can investigate the low frequency behaviour, i.e. slowly varying motions.

6. Currents and wind can easily be modelled by specifying the characteristic area and defining the velocity and direction of the load. As an alternative the wind can be specified by a wind spectrum. Effects from wind and currents are however not applied in this model.

7. The program allows you to model the SPAR with risers and all the geometrical gadgets inside the moonpool. It is difficult to predict how the program manages to simulate the complexity inside the moonpool with respect to the damping and the added mass effects, but it is likely to think that the problem will not be handled satisfactory. Alternatively the buoy can be sealed at the bottom and regarded as a closed cylinder. This is not far from the reality, since the actual opening in the moonpool, between the risers and the buoyancy tanks etc., is quite small. The model used here is therefor considered as a closed cylinder.

8. At last in this brief MOSES ‘introduction’, a few words about the environment.

It is possible to choose wave spectrum (ISSC or JONSWAP) or any sized regular wave, represented by the direction, the wave height and peak period.

The wave is represented by a cosine wave

x

=

x

a cos((t + kx cos( + ky sin()

(2.4)

· = wave heading

R= (RAO( cos((t + ()

(= phase lead

The environment directions are as follows:

Pitch motions, H=10m

0

2

4

6

8

10

12

14

0

5

10

15

20

25

30

35

40

T [sec]

Pitch [deg]

Frequency domain

Time domain

Chapter 3

3.0 The Spar buoy

A spar platform is a large vertical circular cylinder with a large draft that reduces the heave response significantly and permits rigid risers and surface trees, i.e. the motion response are crucial for the Spar buoy. Configured with oil storage and surface completed well, a spar may be able to combine the best characteristics of the TLP (Tension Leg Platform) and FPSO (Floating Production Storage and Offloading) for fields where the reservoir can be reached from one drilling centre [CMPT 1998].

Fig. 3.1. Isometric view with mooring

Fig. 3.2. Front view with mooring

When creating the hull, different levels of accuracy were tried out. First a hull was created in a program called Femgen. The model was made by horizontal panels every 10th meter and 17 panels vertically, before it was converted into a MOSES model. This gave a large number of panels, which was unpractical to work with. To cope with this problem, MOSES has a function that allows you to refine your model at a specified level of accuracy. All the horizontal panels were then deleted and left a model consisting of seventeen vertical panels. This made the work easier and not so time demanding. As explained earlier, the reproduction of the hydrodynamic database required a lot of time. By using a quite rough mesh this was no longer a problem. When the analysis required several different databases (different range of periods), the mesh was chosen quite rough and a fine mesh was chosen when the database was used for the non-linear analyses. Different accuracy in the mesh, showed however good agreement when comparing some of the results.

The main particulars of the spar used in this assignment are the same as used in [Haslum 2000], except some small adjustments in the diameter to obtain the hydrostatic properties as correct as possible. After the impartment of the geometry, MOSES calculated a centre of buoyancy too low, compared to the spar in [Haslum 2000]. This lead to negative GM. By adding more buoyancy in the upper section, the centre of buoyancy ascended and the GM value became closer to the actual one.

A lot of work has been put down to verify the MOSES model and consequently be able to compare it to the model of [Haslum 2000]. Many of the static properties not given in the Dr. Ing thesis, but calculated by MOSES, have been verified by means of hand calculations. MOSES calculates most of these parameters using the input properties, and it is therefore useful to do this verification in order to control the input as well. Here are a few examples:

· By stability reasons the GZ value has to be positive during a time domain process. This is continuously verified.

· Making the hydrodynamic database is time demanding and has to be done for a range of specified frequencies. A need for different periods arises when producing different kinds of linear results. Consequently the hydrodynamic database has to be changed and adapted.

In addition to the things mentioned, lots of extra work is required to get the wanted results. This is mostly time demanding work, which is not explicit shown in this report.

The main particulars of the spar are shown below:

Draft (d) = 202.5m

Diameter (D) = 36.4m (Haslum D=37.5)

Radius of gyration Rxx=80 Ryy=80 Rzz=36.5

Centre of gravity (KG) = 105.25m

Metacentric height (GM) = 3.11 (Haslum GM =4.4m)

Natural period in heave TN,3= 31.3sec

Natural period in pitch TN,5= 95.9sec

The spar is moored with a 16 points taut system, as shown in fig. 3.1 and fig. 3.2. The fairleads are situated vertically at the centre of gravity and on the outside of the spar body. The pretension in each fairlead is set to 400 kN. A discussion of the mooring lines and their influence on the linear wave frequency motions will follow later.

In the effort of modelling the right mooring system, different parameters have been evaluated and tried in the model. The pretension, the weight and the e-modulus of the lines are parameters that are continuously changed and adapted. The whole process culminated in modelling an equivalent mooring line at each fairlead.

However, to detect the Mathieu instability in section 5.0 the mooring lines are deactivated.

3.1Movement of the SPAR

A right hand co-ordinate system is applied as illustrated in fig 3.1.1.

Fig 3.1.1.The six degrees of freedom

Existing spar platforms have deep drafts to reduce the wave loads and consequently the heave motions. Traditionally, the damping from mooring lines and risers was ignored in heave response analyses. By simultaneously predict the dynamic response of the spar, mooring lines and risers one has revealed that mooring lines dynamics and riser friction can have significant effect on the spar heave response. As a consequence the draft of the spar can be reduced and still maintain an acceptable heave response. Reduction in spar hull draft can reduce the fabrication and transportation costs, which will result in making spar solutions more cost effective. [OTC 12082]

Important ‘types’ of motions are the slow drift motions. They are caused by non-linear effects from waves, wind and currents. These motions arise from resonance oscillations and appear in surge, sway and yaw for a moored buoy. Low frequency behaviour is considered in section 4.2.3.

In addition to the slow drift motion (low frequency) a floating structure can experience wave-frequency motion, high-frequency motion and mean drift. Linear excitation forces mainly cause the wave-frequency motion, while the high-frequency motion and mean drift are caused by resonance oscillations [Faltinsen 90].

Damping form the risers is caused by Coulomb friction at the riser guides and the keel as well as from hydrodynamic forces. The risers exert a normal force on the Spar that increases as the Spar pitches or offsets laterally. If the heave response is small enough, the static friction on the guides will prevent the Spar from moving further. If the motions are larger, the friction opposes the heave motions and will consequently produce damping. In addition to the damping, coupling forces between the risers and the Spar occurs in both surge/sway and pitch/roll. When predicting the Spar response, summation of all the risers are important.

When the Spar offsets from the mean position the mooring lines will provide restoring forces. The lines will go slack or taut as the buoy moves. This causes drag load on the lines, which provides damping to the Spar. However, the damping is more pronounced on the heave motion than on surge/sway or roll/pitch motions. The current induced drag on the lines can as well change the restoring force characteristics on of the mooring lines, and are thus useful to consider [OTC 12082]

Chapter 4

4.0 The Processes

Both the time domain and the frequency domain process give adequate solutions in most cases. The time domain process does properly account for all aspects of a problem but is computationally expensive. A solution in the frequency domain is in many cases a good alternative solution, which is much less time demanding.

The theory behind the time and frequency domain is explained in section 2.0.

4.1RAO- frequency domain

The movements of the SPAR are expressed statically by the RAO (Response Amplitude Operator) as a function of the six degrees of freedom (Surge, sway, heave, roll, pitch and yaw).

In the following figures, a presentation of the linearized motions in heave, pitch and surge are given. The calculations are done with and without mooring and the results are shown in the same diagram.

How MOSES calculates the transfer functions, are explained earlier in section 2.0.

SPAR

0

5

10

15

20

88

90

92

94

96

98

100

102

T [sec]

Pitch [deg/m]

Fig. 4.1.1. Pitch RAO Two frequency intervals, to illustrate the natural period in pitch.

Fig. 4.1.2. Heave RAO

Fig. 4.1.3. Surge RAO

The spar is floating with a draft d=202.5m. The main particulars of the spar are given earlier in section 3.0.

However, the transfer functions (RAO – Response amplitude operators) are defined as the frequency dependent steady state motion response amplitude divided by the wave elevation amplitude [Haslum 2000]:

RAOi(T) =

a

i

z

h

[m/m]

(4.1)

i = [1,2,3,4,5,6] = degrees of freedom

In the Dr. Ing thesis by Haslum, these RAO’s are calculated using two different simplified methods (Long wavelength approximation and McCamy and Fuchs theory) and the panel method program WAMIT [Haslum 2000]. The agreement between the three methods shown in [Haslum 2000] is good, as well as the agreement between Haslum and the MOSES-results presented here.

As earlier explained, the mooring system used is very stiff to increase its effect. Despite this the influence from the mooring is quite small and especially the heave motion that is practically identical with the free floating spar. The RAO calculations are performed at a water depth = 700 meters. As production systems extend to water depths beyond 1000 meters, the effects of mooring become increasingly significant when predicting the Spar’s response. For these water depths, the viscous damping, inertial mass, current loading and restoring effects should be included to accurately solve the system’s motion response. By coupling the mooring as well as the riser with the Spar’s motion typically results in a reduction in extreme motion response [OTC 12083].

Fig. 4.1.4 show that pitch and surge motions are in phase with each other, and they are 90 degrees out of phase compared to the wave. This means that they are contributing to displacements of the deck simultaneously. It is shown that the heave between T=11sec and T=31

Fig 4.1.4. Difference of the phases sec is 180 degrees out of phase.

The linear frequency motion response (RAO’s) for all the six degrees of freedom are shown in Appendix 4, presented as “motion response operators”.

4.2Time domain

The calculations done in the time domain are quite complicated, and interpreting time series of the Spar response may be troublesome. When the damping is low, a transient from the start exists for a long time. The simulations done here have consequently at least a 1500 seconds duration, where the response has reached a steady state.

Initiating a time domain simulation in MOSES is not very complicated. The decisive part to get reliable results is to have a stable Spar that reflects the reality as best as possible before starting the time domain calculations. After doing this calculation once, it is possible to retrieve all kinds of results related to the time domain.

A more thorough explanation of the time domain is given in section 2.0 and is also explained in [MOSES manual].

4.2.1 Low frequency behaviour

For a moored spar buoy low frequency motions occurs in surge, sway and yaw. Low frequency motions are resonance oscillations exited by second order, non-linear coupled effects between the wave and the spar [Faltinsen 90].

For moored large structures as the Spar, the natural periods in the horizontal degrees of freedom are much larger than the wave periods with considerable energy. The horizontal low frequency excitation is in general larger than the linear wave frequency motions, despite the fact that second order difference frequency forces (1) are generally an order of magnitude smaller than linear wave frequency forces. This effect is therefore important in relation to the design of the mooring system [Haslum 2000.]

As explained earlier, the design philosophy behind a deep draft Spar, implies that the draft is adequately large to reduce the heave response. The natural periods in heave, pitch and roll is significant larger than wave periods containing important energy. Consequently the second order excitation forces may contribute to the total motion response in vertical degrees of freedom. This motion is a limiting factor for Spar production platforms, with regard to the design of rigid risers and for the drilling operations [Haslum 2000].

In fig. 4.2.1.1 the low frequency surge motion is illustrated as well as the air gap in fig. 4.2.1.2. The environment used is an ISSC spectre with Hs=7m and T=12sec.

Heave motions, H=10m

0

10

20

30

40

50

0

5

10

15

20

25

30

35

T [sec]

Heave [m]

Time domain

Frequency domain

SPAR

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0

5

10

15

20

25

30

35

40

45

T [sec]

Surge [m/m]

Moored

Free floating

Fig. 4.2.1.1. Low frequency surge motion for the 202.5m draft spar.

Fig. 4.2.1.2. Air gap, simultaneously recorded as fig. 4.2.1.1

(1) Second order difference frequency forces occur due to bi-spectral interaction in bicromatic waves. Such second order forces may be represented by quadratic transfer functions, which are dependent on the wave frequencies of the two interacting waves and independent the wave amplitudes [Haslum 2000].

To simulate the low frequency motions in MOSES the direct wave force has not been applied. This simplifies the calculations by allowing investigating this effect without having to use small

time steps to cope with the high frequency behaviour.

The response of the Spar is quite complex especially because of the interaction between wave frequency and low frequency motions in surge pitch and heave.

4.2.2 The damping problem

As a starting point MOSES uses the equations of motion. By supposing that we know the solution at time t1 we can estimate the solution at time t2. After a few steps the equations of motion can be written:

S[q(t2) –q(t1)] =

s

(4.2)

where

S = cI + fC + Kand

s

= s – [aI + dC]

)

(

1

t

q

&

&

-[bI + eC]

)

(

1

t

q

&

a = 1 - 1/2

b = - (1/)

c = 1/2

d = (1 – /2)

e = (1 – )

f =

How the problem is further solved is explained in [MOSES manual]. The customised parameters for the damping problem are the Newmark parameters and To detect the instability phenomenon in section 5.0, the default values .25 and .5 were used. There is almost no numerical damping with these values. In fact, for some problems, the scheme results in small negative damping. This is of no concern here. If these values are changed from the default to .33 and .66, then a small bit of numerical damping is induced. For problems such as decay problems in calm seas, the defaults do not work very well. The following figure illustrates this effect for the heave decay of the Spar.

SPAR

0

2

4

6

8

10

12

14

16

0

5

10

15

20

25

30

35

40

45

T [sec]

Heave [m/m]

Moored

Free floating

Fig. 4.2.2.1. Effect of Newmark parameters

These results are found for the Spar at draft = 202.5 meters and no mooring. A regular wave with H=5m and T=10 seconds was used.

4.2.3 Hurricane analysis

The length of the simulation should be chosen such that it will give a specified level of confidence. To avoid the transient from the start of the simulation and to ensure that the response has reached a steady state the following results are based on a three hours typical GOM (Gulf Of Mexico) hurricane condition with Hs=12.2m and T=14seconds. An ISSC spectre is used.

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

050010001500

T [sec]

Surge [m]

Fig. 4.2.3.1. Heave response for a three hours hurricane

It is quite remarkable that despite the rough environment, the heave response is no more than one and a half meter at most. As explained earlier this is due to the large draft and probably the taut mooring system. As opposed to the linear motions, the second order motions produced during a time domain are clearly affected by the restoring forces from the mooring lines. In addition one should take the damping effect of the risers in account. These would have had an additional damping effect on the heave motions, as will be shown later in section 6.0, where a fully coupled analysis will be carried out.

During the same hurricane the pitch response was recorded, and shows relatively large deflections, with a maximum at almost 9 degrees pitch amplitude.

Free floating SPAR

-200

-150

-100

-50

0

50

100

150

200

0

10

20

30

40

50

T [sec]

Phase [deg]

Heave

Pitch/Surge

Fig. 4.2.3.2. Pitch motions during a three hours hurricane

The evaluation of forces in the four mooring lines is as well investigated. During a 1000 seconds ISSC spectrum environment, with significant wave height at 12.2 meters and a zero up-crossing period at 14 seconds, the force magnitude in the four anchor lines are recorded. They are shown in Appendix 4, labelled “connector force magnitudes”. These results are not discussed, they are attached to illustrate one of the possibilities in retrieving results.

Chapter 5

5.0 The Mathieu instability

Under certain conditions, spar platforms can be exposed to large unexpected motions. This is explained by the Mathieu instability phenomenon, and occurs because of two specified situations. The first and simplest case is trigged due to an abrupt change in waterplane area and therefor a change in the heave restoring force. This is the case when the hull cross section area changes along the height (Fig. 5.1) [Haslum 2000].

The heave response of this hull shape has been calculated in the linear frequency domain,

-97.32

-97.3

-97.28

-97.26

-97.24

-97.22

-97.2

-97.18

-97.16

-97.14

02004006008001000

T [sec]

Heave [m]

shown as the RAO, and by the non-linear time domain method. Calculating the hydrodynamic exciting forces at the mean position produces the RAO, according to the linear theory explained earlier. In time domain, the hydrodynamic forces are calculated at the displaced position and the effect of the changing waterplane area is taken into account.

The unstable wave period is expected in the vicinity of

Fig 5.1

[½TN, TN, 3/2TN,…], where TN is the natural period in heave. This is obviously dependent on the system damping. According to the theory presented in [Haslum 2000] there should be a critical wave period at ½TN = 16.5 sec. By calculating the heave response in frequency and time domain, one should expect a disagreement between the methods at wave periods around 16.5 sec. The model used in MOSES did not show this difference. This kind of instability is quite sensitive when it comes to viscous damping, and the results obtained are probably a consequence of the damping applied to the model.

The other situation that provokes the Mathieu instability is a heave/ pitch amplifying interaction. It may occur even if the hull has a constant cross section. One should expect this instability at a certain wave period that is a function of the natural period in heave and pitch:

3

,

5

,

1

1

1

N

N

Wave

T

T

T

+

=

(5.1)

where

=

3

,

N

T

natural period in heave

=

5

,

N

T

natural period in pitch

When a wave at this frequency occurs, the heave motion will oscillate with both the natural heave frequency and the wave frequency. This produces an envelope process. For a certain wave period, this envelope period coincides with the natural period in pitch, and you get the equation explained above.

For the spar used here (see fig 5.1) with a natural period in pitch TN,5=106,0 sec, and a natural period in heave TN,3=33,0 sec, this critical wave period is:

33

1

106

1

1

+

=

Wave

T

= 25,2 sec

By calculating the heave and pitch response in both frequency and time domain, this critical wave period is found when the two methods disagree. As fig. 5.2 and fig. 5.3 show, the agreement between the two methods is good, except for Twave= 25,5 sec.

-97.8

-97.6

-97.4

-97.2

-97

-96.8

-96.6

-96.4

-96.2

0

2000

4000

6000

8000

10000

T [sec]

Heave [m]

Fig 5.2. Illustration of the Mathieu instability phenomenon in pitch, shown by a disagreement between the frequency and time domain.

-6

-4

-2

0

2

4

6

0

2000

4000

6000

8000

10000

T [sec]

Pitch [deg]

Fig 5.3. Heave motions. The Mathieu instability phenomenon is shown by the disagreement between the frequency and time domain.

To produce the time domain results, a regular wave with H=10m was used and the steady state amplitude was measured. In order to compare the two methods the RAO’s were multiplied by the wave height.

A great effort has been made in producing the time domain results. Since MOSES only allows defining one period at the time, the simulation has been carried out for each period. This is a quite time demanding task. In addition each simulation has been run with different levels of damping, different hull shapes and different types of environment. The alteration of the Newmark parameters (damping) is thoroughly explained in section 4.2.2.

MOSES does not calculate the exact natural periods for a system, but allows you to investigate them by looking at the RAO’s for different degrees of freedom. This means that the natural periods given here are manually found at the peak of the RAO curves. This is probably the explanation why MOSES gives the highest heave amplitude at Twave=25,5 sec. Anyway, it is in the vicinity of the period T=25.2 sec calculated from the formula (5.1).

The envelope process of the heave motion is then illustrated by the first 400 seconds, before the instability accrues at approximately 1100 seconds. The Illustration in fig. 5.4 shows that the envelope for the heave motion has the same period as the natural period in pitch (T=106 sec), i.e. the heave envelope trigs the pitch instability.

Fig 5.4. The envelope process of the heave motions.

T=25.5 sec. H=10m

Generally, this effect is caused by two frequencies in the signal. When these frequencies are close, the envelope period is large and the effect is clearly pronounced. Hence the envelope effect is reduced if the frequencies are moved apart. This effect is caused by the amplifying pitch/ heave interaction.

The non-linear heave excitation causing the instability can also be explained if considering the displaced position instead of the mean position. The vertical component of the horizontal 1st order total force when the Spar has a pitch inclination explains this effect [Haslum 2000]. See figure 5.5.

Fig. 5.5. Second order heave force contribution due to surge and pitch interaction.

Chapter 6

6.0 Coupling effects

When predicting the Spar response the effect from mooring line dynamics and riser friction is important. Their contribution to the total damping can constitute several meters in the Spar response. Results of this coupled analysis reveal that mooring and risers have significant effect on the Spar heave response. A characteristic feature of a Spar platform is the slow oscillatory motion that occurs at resonant frequencies. The damping is low at resonant periods and correct estimation of the damping is therefore important to get reliable results [OTC 12082].

Concerns about excessive heave and pitch response of Spar arising from the Mathieu instability have been raised for long period waves (See section 5.0). This instability occurred for the Spar shown in fig. 5.1 without mooring lines and riser effects included. A new analysis was carried out including these effects showing the Mathieu instability being suppressed. The heave response is shown in fig. 6.1.

-98.5

-98

-97.5

-97

-96.5

-96

500

700

900

1100

1300

1500

T [sec]

heave [m]

hull shape I

hull shape II

Fig. 6.1. Heave response. Two cases: 1) free floating and 2) coupling effects included. Regular wave, H=10m and T=25.5sec

Two cases are presented in the figure. The heave response for a free floating Spar, i.e. no additional damping and coupling effects from mooring and risers included. The mooring lines used are described earlier in section 3.0, and the riser system consists of 16 risers each with a diameter= 346mm and a pretension= 100 kN. As tried with the mooring system, the risers were modelled as one equivalent riser with pretension equal the sum of the 16 risers. The problem arising with the introduction of an equivalent riser, was the adaptation of the stiffness and

weight and attachment to the seabed. With 16 separate risers, the separation of the attachment points at seabed gives a certain effect, which is probably not taken care of by one equivalent riser.

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

0

5

10

15

20

25

30

35

40

T [sec]

Damping coeff. [-]

Hull shape III

Hull shape II

Hull shape I

The results show the importance of including all the damping effects in predicting the response under resonance conditions. This type of analysis is conservatively done by excluding the damping from mooring and risers and does often lead to Spars with hull draft greater than necessary. A reduction in the draft will reduce the costs significantly [OTC 12082].

-120

-115

-110

-105

-100

-95

-90

-85

-80

500

1000

1500

2000

2500

3000

T [sec]

Heave [m]

moored with risers

free floating

As well as the heave motions, the resonant pitch response is substantially reduced by including the effects from mooring and risers. Fig. 6.2 shows the

pitch response from the same analysis as in fig. 6.1. The uncoupled pitch response shown is the Mathieu instability response. According to analysis done [OTC 12082] the coupled effect gets even more important when operating in deep sea.

The response characteristics for a Spar are fairly complex due to the interaction of wave frequency and low frequency motions. When coupling the effects from mooring and risers with the vessel response, large reductions in extremes are obtained. As explained, these reductions are important to take into the design of the mooring lines and risers in an early stage.

Finally, there is a role for coupled analysis in the validation of the design, in particular when designing deep water Spars where lack of experience is a problem. The limitations of model basins to access the full vessel/ riser/ mooring system in very deep water makes the ability to accurately simulate coupled effects practically a requirement for new systems [OTC 12083].

Not many computer programs can handle these effects. Coupled results from MOSES in relation to a Spar buoy, as presented here, are therefore useful and have a certain commercial value.

Chapter 7

7.0 Alternative hull shapes

It is because of its relatively low damping in resonant motions and low natural period in heave the classical spar (hull I, Fig. 7.1) may experience the large heave motion explained. According to [Haslum 2000] some measures are possible to reduce the heave response:

1. Increase the damping in heave

2. Increase the natural heave period out of the wave energy region

3. Reduce the linear heave excitation force

Fig. 7.1 shows alternative hull shapes to cope with these three points. The first point is in theory dealt

10

15

20

25

30

35

0

500

1000

1500

2000

T [sec]

Air gap [m]

with by adding a circular disk at the bottom of the spar (hull II). The heave response for hull shape II from a time domain analysis is illustrated in Fig. 7.2. The time domain shows small deviations from the classical spar (hull I). The RAO’s were also calculated without

Fig. 7.1. Alternative hull shapes

showing any major differences between the two hulls. It is however uncertain whether or not the results are comparable. The physical properties are changed as a consequence of the geometrical differences and the results compared here are the actual heave motion for each spar.

-10

-5

0

5

10

500

700

900

1100

1300

1500

1700

1900

T [sec]

Pitch [deg]

moored, with risers

free floating

Fig. 7.2. Heave response from an ISSC spectrum with Hs=10 m and T=14 sec. Two different hull shapes are considered.

Especially hull shape III has major differences in centre of gravity, metacentric height etc. To deal with this problem in a different manor, the damping coefficients in heave for the three hull shapes have been compared. They are shown in fig. 7.3.

The second point, increasing the natural heave period, is done by adding a pontoon at the keel, i.e. hull shape III has a natural period in heave higher than the classical spar.

An increase in the added mass increases the diffraction term and thus reduces the heave excitation force. For example adding a disk at the keel will in theory increase the added mass, but practical tests shows that the disk has to be very large to have an important effect on the heave excitation force. It is practical troublesome to construct spars with large disks [Haslum 2000]. Hull shape III however, shows large added mass coefficients in heave for a given interval of periods compared to hull I and II.

Fig. 7.3. Heave damping coefficients for three different hull shapes

The damping coefficients are normalized by the mass of the buoy and express the linear heave damping (exclusive of added mass effects).

This figure confirms the theory explained. Hull shape III produces at the most fifteen times the damping of hull I and II (T=20sec), and hull shape II has a few percent more damping than hull I. This difference is slightly expressed in the heave response during a time domain analysis, shown in fig. 7.2. In addition to these structural changes, increasing the draft can reduce the heave response. This is an expensive measure and is seldom the solution used for a spar buoy.

The three hulls shown in fig. 7.1 are the same as used in [Haslum 2000]. Hull shape I is the classical spar shaped as a cylinder. Hull shape II is the same cylinder with a cylindrical disk at bottom. The disk diameter is 1.32D, where D is the cylinder diameter and the thickness=0.2m. The last hull shape consists of two cylinders. The upper is the same as hull I and the bottom cylinder has a diameter=2.596D and height=30m.

The idea behind the increased heave damping, together with the use of this enormous hull is that due to the counteracting diffraction force and the large draft, the motion response of the platform should be adequately low to permit installation of rigid risers with dry wellheads. Therefore, the motion response (in particular heave and pitch) is crucial for the concept.

Chapter 8

8.0 Recommendations for further work

Wave induced motions on a Spar buoy are presented. Motions in frequency and time domain are calculated and illustrated. An analysis including mooring line dynamics and riser friction is also presented. In addition, there are some issues in relation to wave induced motions on a Spar that are not treated in this thesis. These issues are useful to consider in an overall evaluation.

Effects produced by wind forces and currents are not applied to the MOSES model. They will in some cases affect the Spar motions. Especially low frequency motions can be caused by wind gusts with significant energy at periods at the order of magnitude of a minute. This is due to the high natural periods of the Spar [Faltinsen 90].

A well known phenomenon in many fields of engineering is resonance oscillations caused by vortex shedding, typical for cylindrical shaped structures as the Spar. To avoid these vortex-induced oscillations, helical strakes are often used (see the illustration in Appendix 1). To control the instability in pitch that is discussed, more pitch damping is required. Helical strakes contribute to this kind of damping and will consequently play a part in suppressing this instability [Faltinsen 90]. An analysis with helical strake should be carried out.

In section 7.0 alternative hull shapes have been tried out, in the effort to increase the damping in heave. By further investigating the effect of different hull shapes, one should be able to find an optimisation of the Spar hull with respect to the heave motion. By optimising the hull with respect to one degree of freedom, it will probably affect the motion characteristic of the buoy in the other modes. To which extend this geometrical change will affect the motions of the buoy, should be investigated,

The flooded centerwell of the Spar called moonpool, may have some effects on the motion characteristics. For the classical Spar the natural period of the vertical fluid motion is close to the natural period of the platform in heave. This makes it sometimes difficult to simulate. In cases where the moonpool is constructed for large equipment to be lowered through it, the passage between the risers is quite large. The simplified method of considering the Spar as closed at the keel, is in such cases probably too simple. The resonance response of the water column could be important [Haslum 2000].

The Mathieu instability phenomenon could as well be studied more carefully. The effect of altering the wave amplitudes and the wave periods on the instability, could be examined. This problem is treated in [Haslum 2000], where the results are presented as a 3-D chart, to illustrate the influence from the wave amplitude at the range of periods where the instability occurs.

In addition to the mentioned means, there are a lot of possibilities in the use of MOSES. Once the time domain simulation has been turned successfully, several results have been produced and stored in a database. It is then possible to specify the result wanted, everything from evaluation of the forces in the risers to stability verifications of the buoy.

Chapter 9

9.0 References

1. The Centre for Marine and Petroleum Technology (CMPT) (1998). Floating Structures: a guide for design and analysis, Volume One.

2. The Centre for Marine and Petroleum Technology (CMPT) (1998). Floating Structures: a guide for design and analysis, Volume Two.

3. Faltinsen, Odd M. (1990). Sea loads on ships and offshore structures. Cambridge University Press.

4. Haslum, Herbjørn A. (2000). Dr. Ing thesis: Simplified methods applied to nonlinear motions of Spar platforms.

5. OTC 12083 (Offshore Technology Conference - 2000). Coupling effects for a deepwater Spar.

6. OTC 12082 (Offshore Technology Conference – 2000). Effects of Spar coupled analysis.

7. OTC 12085 (Offshore Technology Conference – 2000). Deepwater nonlinear coupled analysis tool.

8. Larsen, T. J. Projet de fin d’etudes (2001). Motion and stability analysis of a pipelay vessel (barge and stinger). Stability verifications of a buoy launch from a barge.

9. MOSES manual. Ultramarine Inc. Offshore Engineering Software (www.ultramarine.com)

List of figures

Figure 2.1

MOSES reference

Figure 3.1

Isometric view with mooring

Figure 3.2

Front view with mooring

Figure 3.1.1

The six degrees of freedom

Figure 4.1.1

Pitch RAO. Two frequency intervals, to illustrate the natural period in pitch

Figure 4.1.2

Heave RAO

Figure 4.1.3

Surge RAO

Figure 4.1.4

Difference of the phases

Figure 4.2.1.1

Low frequency surge motion for the 202.5m draft Spar

Figure 4.2.2.1

Air gap, simultaneously recorded as fig. 4.2.1.1

Figure 4.2.2.1

Effect of Newmark parameters

Figure 4.2.3.1

Heave response for a three hours hurricane

Figure 4.2.3.2

Pitch motions during a three hours hurricane

Figure 5.1

Spar platform

Figure 5.2

Illustration of the Mathieu instability in pitch, shown by the disagreement between the frequency and time domain

Figure 5.3

Heave motions. The Mathieu instability phenomenon is shown by the disagreement between the frequency and the time domain

Figure 5.4

The envelope process of the heave motion, T=25.5sec, H=10m

Figure 5.5

Second order heave force contribution due to surge and pitch interaction

Figure 6.1

Heave response. Two cases: 1) free floating and 2) coupling effects included. Regular wave, H=10m and T=25.5sec

Figure 6.2

Pitch response. Coupled and uncoupled with mooring and risers

Figure 7.1

Alternative hull shapes

Figure 7.2

Heave response from an ISSC spectrum, with Hs=10m and T=14sec. Two different hull shapes are considered

Figure 7.3

Heave damping coefficients for three different hull shapes

Symbols and nomenclature

MOSES

MultiOperational Structural Engineering Simulator

RAO

Response Amplitude Operator

GOM

Gulf of Mexico

RMS

Root Mean Square

ISSC

International Ship and Offshore Structures Congress

TLP

Tension Leg Platform

FPSO

Floating Production Storage and Offloading

d

Draft

D

Diameter

KG

Vertical centre of gravity

GM

Metacentric height

TN

Natural period

Appendix

Appendix 1

Illustration of a moored SPAR with helical strake and risers

Appendix 2

MOSES files

Appendix 2a

Command file:Spar6.cif (*)

Appendix 2b

Geometry file:Spar6.dat(Hull shape I)

Appendix 3

MOSES files, alternative hull shapes

Appendix 3a

Hull shape, fig. 5.1(Spar8.dat)

Appendix 3b

Hull shape II

(Spar7.dat)

Appendix 3c

Hull shape III

(Spar9.dat)

Appendix 4

MOSES file, output (Spar6.out)

(*)This command file applies to all the geometry files.

Calls extern program

Abstract:

This work is based on the use of MOSES (MultiOperational Structural Engineering Simulator), which is an analysis tool for almost anything that can be placed in the water. A quite comprehensive programming language that allows you to do coupled analysis of Spar platforms in which damping effects from mooring lines and risers are included. The diversity of the program is further expressed trough the handling of a newly explained phenomenon, the Mathieu instability.

Alternative hull shapes with improved heave motion characteristics are investigated, showing increased heave damping when differ from the classical hull shape.

The effect of mooring system on the linear motion response is investigated. It is seen that even a very stiff mooring system has small influence on the linear wave frequency response.

The results from the coupled analysis show the importance of including mooring line dynamics and riser friction when predicting the Spar response. The response was significantly reduced and the Mathieu instability phenomenon was suppressed. Existing Spars have deep drafts to reduce the wave loads and consequently the heave motion. Traditionally, the mooring line dynamics and riser friction were ignored in estimating the heave response. Since this effect is important, the draft of the Spar can be reduced while maintaining an acceptable heave response. Reduction in Spar hull draft can reduce fabrication costs substantially and as a result the Spar solution will be more cost effective.

Defines a position for the Spar

Draft = 202.5 m

Trim = 0.4°

Defines the supplementary punctual loads.

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Geometrical properties for the elements

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Heave (z)

Surge (x)

Sway (y)

roll

pitch

yaw

Keywords:

Advisor:

Odd M. Faltinsen

SPAR Buoy

Wave induced motions

� EMBED Excel.Sheet.8 ���

� EMBED Excel.Sheet.8 ���

MOSES

� EMBED Excel.Sheet.8 ���

Availability:

O

y

x

315°

0°

45°

90°

135°

180°

225°

270°

Fig. 2.1. MOSES Reference [Larsen 01]

Number of pages:

94

Delivered:20.06.2002

Student:Truls Jarand Larsen

� EMBED Excel.Sheet.8 ���

Fig. 6.2. Pitch response. Coupled and uncoupled with mooring and risers.

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Title:

Modelling of wave induced motions of a SPAR buoy in MOSES

Address:

NTNU

Department of Marine Hydrodynamics

N-7491 Trondheim

Location

Marinteknisk Senter

O. Nielsens vei 10

Tel.+47 73 595535

Fax+47 73 595528

_1077108881.unknown

_1082286764.unknown

_1083661282.xls

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902902

904904

906906

908908

910910

912912

914914

916916

918918

920920

922922

924924

926926

928928

930930

932932

934934

936936

938938

940940

942942

944944

946946

948948

950950

952952

954954

956956

958958

960960

962962

964964

966966

968968

970970

972972

974974

976976

978978

980980

982982

984984

986986

988988

990990

992992

994994

996996

998998

10001000

10021002

10041004

10061006

10081008

10101010

10121012

10141014

10161016

10181018

10201020

10221022

10241024

10261026

10281028

10301030

10321032

10341034

10361036

10381038

10401040

10421042

10441044

10461046

10481048

10501050

10521052

10541054

10561056

10581058

10601060

10621062

10641064

10661066

10681068

10701070

10721072

10741074

10761076

10781078

10801080

10821082

10841084

10861086

10881088

10901090

10921092

10941094

10961096

10981098

11001100

11021102

11041104

11061106

11081108

11101110

11121112

11141114

11161116

11181118

11201120

11221122

11241124

11261126

11281128

11301130

11321132

11341134

11361136

11381138

11401140

11421142

11441144

11461146

11481148

11501150

11521152

11541154

11561156

11581158

11601160

11621162

11641164

11661166

11681168

11701170

11721172

11741174

11761176

11781178

11801180

11821182

11841184

11861186

11881188

11901190

11921192

11941194

11961196

11981198

12001200

12021202

12041204

12061206

12081208

12101210

12121212

12141214

12161216

12181218

12201220

12221222

12241224

12261226

12281228

12301230

12321232

12341234

12361236

12381238

12401240

12421242

12441244

12461246

12481248

12501250

12521252

12541254

12561256

12581258

12601260

12621262

12641264

12661266

12681268

12701270

12721272

12741274

12761276

12781278

12801280

12821282

12841284

12861286

12881288

12901290

12921292

12941294

12961296

12981298

13001300

13021302

13041304

13061306

13081308

13101310

13121312

13141314

13161316

13181318

13201320

13221322

13241324

13261326

13281328

13301330

13321332

13341334

13361336

13381338

13401340

13421342

13441344

13461346

13481348

13501350

13521352

13541354

13561356

13581358

13601360

13621362

13641364

13661366

13681368

13701370

13721372

13741374

13761376

13781378

13801380

13821382

13841384

13861386

13881388

13901390

13921392

13941394

13961396

13981398

14001400

14021402

14041404

14061406

14081408

14101410

14121412

14141414

14161416

14181418

14201420

14221422

14241424

14261426

14281428

14301430

14321432

14341434

14361436

14381438

14401440

14421442

14441444

14461446

14481448

14501450

14521452

14541454

14561456

14581458

14601460

14621462

14641464

14661466

14681468

14701470

14721472

14741474

14761476

14781478

14801480

14821482

14841484

14861486

14881488

14901490

14921492

14941494

14961496

14981498

15001500

15021502

15041504

15061506

15081508

15101510

15121512

15141514

15161516

15181518

15201520

15221522

15241524

15261526

15281528

15301530

15321532

15341534

15361536

15381538

15401540

15421542

15441544

15461546

15481548

15501550

15521552

15541554

15561556

15581558

15601560

15621562

15641564

15661566

15681568

15701570

15721572

15741574

15761576

15781578

15801580

15821582

15841584

15861586

15881588

15901590

15921592

15941594

15961596

15981598

16001600

16021602

16041604

16061606

16081608

16101610

16121612

16141614

16161616

16181618

16201620

16221622

16241624

16261626

16281628

16301630

16321632

16341634

16361636

16381638

16401640

16421642

16441644

16461646

16481648

16501650

16521652

16541654

16561656

16581658

16601660

16621662

16641664

16661666

16681668

16701670

16721672

16741674

16761676

16781678

16801680

16821682

16841684

16861686

16881688

16901690

16921692

16941694

16961696

16981698

17001700

17021702

17041704

17061706

17081708

17101710

17121712

17141714

17161716

17181718

17201720

17221722

17241724

17261726

17281728

17301730

17321732

17341734

17361736

17381738

17401740

17421742

17441744

17461746

17481748

17501750

17521752

17541754

17561756

17581758

17601760

17621762

17641764

17661766

17681768

17701770

17721772

17741774

17761776

17781778

17801780

17821782

17841784

17861786

17881788

17901790

17921792

17941794

17961796

17981798

18001800

18021802

18041804

18061806

18081808

18101810

18121812

18141814

18161816

18181818

18201820

18221822

18241824

18261826

18281828

18301830

18321832

18341834

18361836

18381838

18401840

18421842

18441844

18461846

18481848

18501850

18521852

18541854

18561856

18581858

18601860

18621862

18641864

18661866

18681868

18701870

18721872

18741874

18761876

18781878

18801880

18821882

18841884

18861886

18881888

18901890

18921892

18941894

18961896

18981898

19001900

19021902

19041904

19061906

19081908

19101910

19121912

19141914

19161916

19181918

19201920

19221922

19241924

19261926

19281928

19301930

19321932

19341934

19361936

19381938

19401940

19421942

19441944

19461946

19481948

19501950

19521952

19541954

19561956

19581958

19601960

19621962

19641964

19661966

19681968

19701970

19721972

19741974

19761976

19781978

19801980

19821982

19841984

19861986

19881988

19901990

19921992

19941994

19961996

19981998

20002000

20022002

20042004

20062006

20082008

20102010

20122012

20142014

20162016

20182018

20202020

20222022

20242024

20262026

20282028

20302030

20322032

20342034

20362036

20382038

20402040

20422042

20442044

20462046

20482048

20502050

20522052

20542054

20562056

20582058

20602060

20622062

20642064

20662066

20682068

20702070

20722072

20742074

20762076

20782078

20802080

20822082

20842084

20862086

20882088

20902090

20922092

20942094

20962096

20982098

21002100

21022102

21042104

21062106

21082108

21102110

21122112

21142114

21162116

21182118

21202120

21222122

21242124

21262126

21282128

21302130

21322132

21342134

21362136

21382138

21402140

21422142

21442144

21462146

21482148

21502150

21522152

21542154

21562156

21582158

21602160

21622162

21642164

21662166

21682168

21702170

21722172

21742174

21762176

21782178

21802180

21822182

21842184

21862186

21882188

21902190

21922192

21942194

21962196

21982198

22002200

22022202

22042204

22062206

22082208

22102210

22122212

22142214

22162216

22182218

22202220

22222222

22242224

22262226

22282228

22302230

22322232

22342234

22362236

22382238

22402240

22422242

22442244

22462246

22482248

22502250

22522252

22542254

22562256

22582258

22602260

22622262

22642264

22662266

22682268

22702270

22722272

22742274

22762276

22782278

22802280

22822282

22842284

22862286

22882288

22902290

22922292

22942294

22962296

22982298

23002300

23022302

23042304

23062306

23082308

23102310

23122312

23142314

23162316

23182318

23202320

23222322

23242324

23262326

23282328

23302330

23322332

23342334

23362336

23382338

23402340

23422342

23442344

23462346

23482348

23502350

23522352

23542354

23562356

23582358

23602360

23622362

23642364

23662366

23682368

23702370

23722372

23742374

23762376

23782378

23802380

23822382

23842384

23862386

23882388

23902390

23922392

23942394

23962396

23982398

24002400

24022402

24042404

24062406

24082408

24102410

24122412

24142414

24162416

24182418

24202420

24222422

24242424

24262426

24282428

24302430

24322432

24342434

24362436

24382438

24402440

24422442

24442444

24462446

24482448

24502450

24522452

24542454

24562456

24582458

24602460

24622462

24642464

24662466

24682468

24702470

24722472

24742474

24762476

24782478

24802480

24822482

24842484

24862486

24882488

24902490

24922492

24942494

24962496

24982498

25002500

25022502

25042504

25062506

25082508

25102510

25122512

25142514

25162516

25182518

25202520

25222522

25242524

25262526

25282528

25302530

25322532

25342534

25362536

25382538

25402540

25422542

25442544

25462546

25482548

25502550

25522552

25542554

25562556

25582558

25602560

25622562

25642564

25662566

25682568

25702570

25722572

25742574

25762576

25782578

25802580

25822582

25842584

25862586

25882588

25902590

25922592

25942594

25962596

25982598

26002600

26022602

26042604

26062606

26082608

26102610

26122612

26142614

26162616

26182618

26202620

26222622

26242624

26262626

26282628

26302630

26322632

26342634

26362636

26382638

26402640

26422642

26442644

26462646

26482648

26502650

26522652

26542654

26562656

26582658

26602660

26622662

26642664

26662666

26682668

26702670

26722672

26742674

26762676

26782678

26802680

26822682

26842684

26862686

26882688

26902690

26922692

26942694

26962696

26982698

27002700

27022702

27042704

27062706

27082708

27102710

27122712

27142714

27162716

27182718

27202720

27222722

27242724

27262726

27282728

27302730

27322732

27342734

27362736

27382738

27402740

27422742

27442744

27462746

27482748

27502750

27522752

27542754

27562756

27582758

27602760

27622762

27642764

27662766

27682768

27702770

27722772

27742774

27762776

27782778

27802780

27822782

27842784

27862786

27882788

27902790

27922792

27942794

27962796

27982798

28002800

28022802

28042804

28062806

28082808

28102810

28122812

28142814

28162816

28182818

28202820

28222822

28242824

28262826

28282828

28302830

28322832

28342834

28362836

28382838

28402840

28422842

28442844

28462846

28482848

28502850

28522852

28542854

28562856

28582858

28602860

28622862

28642864

28662866

28682868

28702870

28722872

28742874

28762876

28782878

28802880

28822882

28842884

28862886

28882888

28902890

28922892

28942894

28962896

28982898

29002900

29022902

29042904

29062906

29082908

29102910

29122912

29142914

29162916

29182918

29202920

29222922

29242924

29262926

29282928

29302930

29322932

29342934

29362936

29382938

29402940

29422942

29442944

29462946

29482948

29502950

29522952

29542954

29562956

29582958

29602960

29622962

29642964

29662966

29682968

29702970

29722972

29742974

29762976

29782978

29802980

29822982

29842984

29862986

29882988

29902990

29922992

29942994

29962996

29982998

30003000

30023002

moored with risers

free floating

T [sec]

Heave [m]

-103.53

-99.7

-103.78

-99.95

-103.94

-100.51

-103.96

-101.27

-103.82

-102.09

-103.51

-102.84

-103.09

-103.4

-102.6

-103.69

-102.16

-103.68

-101.84

-103.4

-101.74

-102.91

-101.89

-102.31

-102.29

-101.7

-102.87

-101.17

-103.53

-100.79

-104.14

-100.58

-104.59

-100.54

-104.78

-100.65

-104.68

-100.87

-104.3

-101.16

-103.7

-101.5

-102.99

-101.86

-102.31

-102.2

-101.77

-102.52

-101.5

-102.78

-101.55

-102.97

-101.9

-103.07

-102.48

-103.03

-103.16

-102.84

-103.8

-102.5

-104.3

-102.02

-104.56

-101.46

-104.55

-100.9

-104.29

-100.43

-103.83

-100.13

-103.26

-100.07

-102.7

-100.28

-102.25

-100.77

-101.98

-101.46

-101.92

-102.28

-102.08

-103.08

-102.38

-103.73

-102.76

-104.11

-103.14

-104.13

-103.45

-103.78

-103.67

-103.1

-103.77

-102.19

-103.77

-101.2

-103.69

-100.29

-103.56

-99.61

-103.43

-99.3

-103.31

-99.39

-103.21

-99.89

-103.12

-100.7

-103.02

-101.69

-102.89

-102.68

-102.75

-103.51

-102.61

-104.05

-102.49

-104.23

-102.42

-104.03

-102.45

-103.49

-102.58

-102.73

-102.83

-101.88

-103.17

-101.08

-103.55

-100.45

-103.9

-100.08

-104.15

-99.98

-104.22

-100.14

-104.09

-100.52

-103.76

-101.03

-103.27

-101.6

-102.7

-102.16

-102.16

-102.64

-101.76

-103.01

-101.59

-103.23

-101.71

-103.31

-102.11

-103.24

-102.73

-103.03

-103.44

-102.7

-104.12

-102.25

-104.63

-101.73

-104.89

-101.18

-104.83

-100.66

-104.47

-100.24

-103.87

-100

-103.14

-99.99

-102.41

-100.24

-101.82

-100.73

-101.49

-101.43

-101.47

-102.25

-101.76

-103.08

-102.29

-103.77

-102.93

-104.22

-103.57

-104.32

-104.08

-104.04

-104.39

-103.4

-104.45

-102.49

-104.28

-101.45

-103.91

-100.42

-103.45

-99.6

-102.96

-99.1

-102.56

-99.04

-102.28

-99.44

-102.18

-100.23

-102.22

-101.3

-102.39

-102.45

-102.63

-103.51

-102.89

-104.3

-103.13

-104.68

-103.32

-104.61

-103.45

-104.1

-103.54

-103.24

-103.59

-102.17

-103.62

-101.09

-103.63

-100.16

-103.63

-99.53

-103.59

-99.27

-103.49

-99.41

-103.32

-99.89

-103.07

-100.61

-102.77

-101.46

-102.46

-102.32

-102.19

-103.05

-102.04

-103.59

-102.04

-103.88

-102.22

-103.9

-102.59

-103.68

-103.08

-103.27

-103.62

-102.73

-104.12

-102.1

-104.48

-101.45

-104.61

-100.84

-104.48

-100.33

-104.09

-99.97

-103.5

-99.81

-102.81

-99.89

-102.14

-100.22

-101.63

-100.78

-101.38

-101.51

-101.47

-102.33

-101.89

-103.15

-102.55

-103.84

-103.33

-104.28

-104.09

-104.4

-104.67

-104.14

-104.98

-103.51

-104.97

-102.59

-104.63

-101.51

-104.03

-100.43

-103.28

-99.52

-102.51

-98.93

-101.87

-98.76

-101.48

-99.08

-101.4

-99.85

-101.63

-100.96

-102.11

-102.25

-102.72

-103.5

-103.35

-104.51

-103.88

-105.12

-104.24

-105.21

-104.38

-104.78

-104.31

-103.88

-104.05

-102.66

-103.67

-101.32

-103.26

-100.08

-102.88

-99.13

-102.59

-98.62

-102.41

-98.62

-102.35

-99.1

-102.39

-99.97

-102.5

-101.08

-102.64

-102.24

-102.8

-103.3

-102.97

-104.09

-103.14

-104.54

-103.3

-104.59

-103.46

-104.27

-103.62

-103.66

-103.76

-102.85

-103.86

-101.96

-103.88

-101.1

-103.79

-100.36

-103.57

-99.81

-103.24

-99.5

-102.83

-99.46

-102.39

-99.71

-102.02

-100.21

-101.78

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