Mooring Line Dynamics Volume II, Phase I: Development of a Mathematical Model, Report VM-V-5-II

(1)

MaT

SNetherlands

Marine Technological Research MOORING LINE DYNAMICS

Phase I: Development of a mathematical model MaTS report VM-V-5 II November 1984

L

_J

P1984-1

VOL.2

0

,

Industriele Raad voor de Oceanologie

...

Netherlands Industrial Council for Oceanology

(2)

The Netherlands Industrial Council for Oceanology (IRO) and its Marine Technological Research (MaTS)

From the start in the early sixties Dutch industry was involved in the development of the oil and gas resources of the North Sea. The first platforms on the southern part of the UK Continental Shelf were

constructed and installed by the Dutch. From then on the Dutch industry has been building up its name and reputation in all activities related to design, construction and installation of equipment for exploration and development of oil and gas.

Soon the need was felt for a co-ordinating body to further the interests of the Dutch offshore industry.

To this end the Netherlands Industrial Council for Oceanology (IRO) was founded in 1971. In this context the term oceanology referred to coastal engineering, underwater technology, sea-mining, shipbuilding, energy

production, equipment manufacture, offshore supply, fishery and recreation and related advisory and supervisory activities amongst which pollution control.

The activities of IRO were, however, soon focussing on the production of oil and gas offshore. By now some 250 companies involved in

above-mentioned activities have become member of IRO. Through the years the IRO has grown to the following set of tasks:

it forms a platform for all people involved in offshore activities in the Netherlands;

it provides information on offshore activities in the world. One of the channels of information is formed by the 'IRO-Journal' a weekly which gives a short overview of up-to-date information;

it provides information on its members to interested parties. Amongst others the IRO is present on the main offshore exhibitions in the world representing its 240 members. Furthermore IRO publishes the Netherlands Offshore Catalogue, in which it gives descriptions of its member

companies and their activities;

it takes care of contacts with government authorities, taking a seat in scientific committees and other consultative bodies;

it stimulates and draws attention to new possibilities in the field of oceanology which might become of economical importance;

it co-ordinates combined efforts of groups of several companies to operate on foreign markets;

it initiates, co-ordinates and desseminates results of applied research in the offshore field through its Marine Technological Research (MaTS) efforts.

MaTS projects are jointly financed by government and industry. They are meant to raise the standard of Dutch offshore technology and they are aimed at satisfying the need for knowledge on middle long term. It is the responsibility of the MaTS organisation to sort out strategic research fields within the offshore context and to develop relevant projects in these fields; futhermore to promote and manage these projects and to disseminate the results.

(3)

g

Netherlands Ship Mode/ Basin

The Wageningen Ede Laboratories of Maritime Research Institute NetherIands (MARIN)

2. Haagsteeg P.O. Box 25. 57: AA, Wagen.ngen. The Netherlands

Telepnone + 31 8370 93911. Tee x 45148 nsmb ni

Ede Laboratory 10, Nels Bohrstraat, 6716 AM Ede

Teiephone 31 8380 37177

Report No. 45064-2RD

MOORING LINE DYNAMICS Phase I: Development of a

Mathematical Model November 1984.

(4)

Report No. 45064-2-RD

MOORING LINE DYNAMICS

Phase I: Development of a Mathematical Model

N.S.M.B. Order No. Z 45064

HOM

DD)

Ordered by: Industri6le Raad voor de Oceanologie Marien Technologisch Speurwerk (MaTS) Project VM-V-5

Postbus 215

2600 AE

DELFT

Reported by: Ir H.J.J. van den Boom Approved by: Dr Ir G. van Oortmerssen

Netherlands Ship Model Basin

(5)

-Report No. 45064-2-RD

CONTENTS

-3

Page

INTRODUCTION 3

1.1. Background 3

1.2. Objective and scope 4

LITERATURE REVIEW 7

PROBLEM DEFINITION 12

MATHEMATICAL MODEL 15

FLUID REACTIVE FORCE EXPERIMENTS 19

PARAMETER STUDY 23

CONCLUSIONS 28

REFERENCES 29

NOMENCLATURE 31

APPENDIX I : TRANSFORMATION MATRICES FOR 2-D LUMPED MASS

MODEL 33

APPENDIX II: SOLUTIONS OF TRIDIAGONAL EQUATIONS BY THE THOMAS

ALGORITHM 34

-2-Tables ( 5)

Figures (29) Photo pages ( 2)

(6)

1. INTRODUCTION

1.1. Background

The advent of large moored and guyed offshore structures has put

high demands upon mooring arrangements. Important parameters in

this respect are the large displacement of the structure, deep and

hostile waters and the required round-the-year workability. The wide variety of mooring systems may be elucidated by shallow and

deep water single point moorings with temporarily or permanently moored tankers, clump weight systems used for guyed towers and wire moorings of semi-submersible crane vessels.

Current design procedures for offshore mooring systems comprise the following items:

Catenary calculations to obtain the static load versus displace-ment relationship of the mooring lines.

Motion analysis of the moored structure taking into account the static catenary results. In general these motions are character-ized by mean displacements, low frequency motions and wave fre-quency motions.

Maximum mooring loads found from the extreme upper-end position of the line.

From both theoretical and experimental research

it is

known that the dynamic behaviour of a mooring line induced by high frequency

upper-end oscillations may contribute significantly to line ten-sions and motions. Therefore these dynamic effects may be of im-portance in the design of the mooring arrangement. In some cases mooring line dynamics may also affect the motions of the moored

object.

(7)

HOU

In the past decades various authors have reported on experimental results and numerical techniques. Unfortunately the validity of the presented numerical models is not clearly demonstrated. Therefore an extensive research program has been carried out at the N.S.M.B. Laboratories of MARIN on behalf of IRO, as part of the MaTS

pro-gram. The project was sponsored by the following parties: Ministry of Economic Affairs

Van Rietschoten & Houwens Gusto Engineering

Heerema Engineering Service MARIN

Shell Internationale Petroleum Maatschappij

The project was carried out under supervision of a project group

(ProMaTS), consisting of the following members: A. van der Wel (Chairman) - Gusto Engineering

F. Groen - Heerema Engineering Services

Bouquet (later: G. Moeyes) - Shell Internationale Petroleum

Maatschappij

Twigt (L. Kuik) - Van Rietschoten & Houwens

1.2. Objective and scone

The objective of this study is to investigate the dynamic behaviour of mooring lines by means of model tests and mathematical models. The validity of the mathematical models was assessed by determinis-tic comparison with model test results.

The ultimate goal of this project is to derive a reliable and

effi-cient computer program for dynamic analysis of mooring lines. To

this end the research program contained the following phases:

(8)

-4-Report No. 45064-2-RD

HgaEM

-5-I Development of the mathematical model

A numerical model of the motions of the mooring line and the associated forces is built on the basis of existing literature and experimental data. Information on fluid reactive forces (added inertia and drag) have been obtained from oscillation

tests with rigid model chains and towing tests with

free-hanging chains.

Comparisons between the developed mathematical model, a

so-called lumped mass method, and finite element calculations were

made. In addition a parameter study has been carried out in

order to investigate discretization effects and to gain insight in the governing factors in dynamic tension amplification.

II Harmonic oscillation tests

In this phase various line configurations were exposed to hori-zontal and vertical oscillations at chosen frequencies and am-plitudes. During these tests the oscillation and line-end

forces were measured while line motions were recorded by under water video.

For all situations the model test results were correlated with numerical results.

III Irregular wave tests

In order to check the validity of the computer program for rea-listic excitations model tests were conducted with the mooring lines attached to a semi-submersible vessel and a barge in ir-regular waves.

(9)

1.3. Re2ort outline

This report, which describes Phase I, is followed by five other

parts:

No. 45064-1-RD Research Program Summary No. 45064-3-RD Phase II - Part I

Correlation Study ; Analysis

No. 45064-4-RD Phase II - Part II

Correlation Study ; Data Report

No. 45064-5-RD Phase III

Irregular Wave Tests No. 45064-6-RD Computer Program DYNLINE

Reference Guide and User's Manual

HOMM

(10)

2. LITERATURE REVIEW

HOE

213)

Traditional approaches to solve the dynamic behaviour of cable

systems were based on semi-analytical techniques. The obstacles for

a pure analytical approach caused by geometric non-linearitites

were removed to reduce the equations of motion to ordinary differ-ential equations. Other approaches such as the perturbation tech-niques derived linear equations of motion by evaluating small

variations about an equilibrium configuration.

Application of chains and cables in various underwater systems

required more general approaches to the problem. It was found by

assuming the line to be composed of an interconnected set of

dis-crete elements that the system of partial differential equations

describing the variables along the line could be replaced by equa-tions of motion in an earth-bound system of co-ordinates.

The two most successful discrete element techniques, the "lumped

parameter method", better known as the Lumped Mass Method (LMM) and the Finite Element Method (FEM) will be discussed here.

Lumped Mass Method

This technique involves the lumping of all effects of mass,

exter-nal forces and interexter-nal reactions at a finite number of points

("nodes") along the line. By applying the equations of dynamic equilibrium and continuity (stress/strain) to each mass a set of descrete equations of motion is derived. These equations may be solved in the time domain directly using finite difference

tech-niques, [1]. Material damping, bending and torsional moments are normally neglected. This procedure implies that the behaviour of a

continuous line is modelled as a set of concentrated masses connected by massless springs.

(11)

HOHE3

-8-Walton and Polacheck, [2], were the first authors who suggested

this method to solve mooring problems caused by transient motions

of the moored vessel. Their spacewise discretization neglected

material elasticity. Moreover no data on fluid reactive forces were available and no validation of the algorithm was given. The explic-it central difference method, [1], was proved to provide condexplic-ition- condition-ally stable solutions for the given schematization.

In the following decades LMM's were applied to the dynamics of

oceanographic and naval cable systems. Cable-mass configurations

suspended in the ocean were studied by Goeller and Laura, [3],

McLaughlan et.al., [4], and Palo et.al., [5]. Comparisons with ex-perimental results showed a reasonable agreement even for snap load conditions. Anchor and buoy deployment simulations were carried out by Tresher and Nath, [6], using a second order predictor-corrector scheme.

In recent years the LMM is used to solve offshore mooring problems. Wilhelmy et.al., [7] and [8], investigated the response of clump

weight moorings. They used a Newmark integration method. Using

several discretizations it was concluded that even a small number of nodes represented the global response of the mooring system. No explanation was given for the high frequency secondary vibrations. Special attention was given to the clump weight motions and energy dissipation by hydrodynamic drag and seafloor friction. It was con-cluded that the hydrodynamic drag of the line governed the

damping.

In case of the guyed tower concept, the energy dissipated by the

line motions contributed significantly to the global motions of the structure.

Nakajima, Motora and Fujin°, [9], were the first inv.estigators who presented deterministic comparisons between the results of forced

oscillation tests and LMM simulations. The model of Walton and

Polacheck was extended with elastic deformation and seafloor

(12)

HOEM

-9--[10] an excellent agreement was found for several multi-component

lines. Unfortunately no slack conditions were investigated.

The use of the Houbalt finite difference scheme, [1], provided an

efficient computer program with respect to the required computer

time. The 3-dimensional version of the model is presented in [11].

A similar algorithm extended with bending moments was used by

Ractliffe, [12], for the analysis of flexible catenary risers. Ac-cording to Ractliffe computational efficiency can be increased by applying the simplest possible integration procedure and small line increments. In this way an easy control from the point of view of numerical stability is possible.

Finite Element Method

The Finite Element Method utilizes interpolation functions to des-cribe the behaviour of a given variable internal to the element in

terms of the displacements of the nodes defining the element (or

.

other generalized co-ordinates). The equations of motion for a

single element are obtained by applying the interpolation function

to kinematic relations (strain/displacement), constitutive rela-tions (stress/strain) and the equarela-tions of dynamic equilibrium. The

solution procedure is similar to the LMM.

Various models based on the FEM have been presented either using

linear or higher order shape functions, [13]. Recently FEM-models for mooring line analysis were developed by Fylling and Wold, [15] and Larsen and Fylling, [16]. They found that the normal components of excitation have only small effects on line tension variations. According to Larsen and Fylling the dynamic response of wire

sys-tems at high frequencies of oscillation can be approximated by

using the material elasticity in a quasi-static analysis. Results

of field measurements agreed fairly well with FEM calculations

(Ormberg et.al., [17]). The maximum dynamic amplification of ten-sion during these measurements was 30 percent.

(13)

HOE

-10--Lindahl and Sjoberg, [17], found a reasonable agreement between 2-D and 3-D FEM simulations. A good correlation was found between the results of analytically derived "eigen periods" and 2-D FEM calcu-lations for a free undamped taut cable with small amplitudes of os-cillation.

Model Tests

Several authors have reported on model tests to quantify dynamic

effects in mooring line motions and tensions. Van Sluijs and Blok,

[19], found from systematical series of forced oscillation tests

that the ratios of maximum dynamic tension and maximum quasi-static tension strongly depend on the frequency of oscillation. This dyna-mic ratio increased with increasing oscillation amplitude and

pre-tension and with reduction of line mass.

Suhara et.al., [20], carried out vertical and horizontal harmonic

oscillation tests with model chains. Line tensions were measured

and vertical line motions were recorded by means of cinematography and photo transistors. The authors proposed an approximative calcu-lation method for nearly sinusoidal tension responses. It was also

concluded that the LMM models can be applied for taut conditions

and large amplitudes of oscillation.

The following conclusions were drawn from the literature survey: The dynamic effects in line tension are mainly due to the global,

"fist mode shape" motion of the line.

The frequency regions of dynamic amplification of tension often include the wave frequencies.

Dynamic effects are mainly due to forced oscillations of the

upper-end in the direction of the line. In strong transverse cur-rent 3-D effects may become important.

In the vertical 2-D situation seafloor friction is of minor im-portance for usual line lengths.

(14)

HOMM

Analytical and semi-analyical solutions can be derived for special situations only. Discrete element methods simplify the

equations of motions considerably.

Due to the line geometry it is desirable to express the fluid

forces in normal and tangential components.

Though of great importance for the dynamic behaviour of mooring lines, experimental data on fluid reactive forces (drag and added inertia) are scarce.

Two discrete element methods, the FEM and the LMM have been ap-plied successfully to mooring line dynamics for harmonic oscilla-tion. However, no systematic validation studies have been carried

out.

Various finite difference methods such as the Central Difference-, the Houbolt- and the Newmark-8-method have been used to solve the discrete equations of motion.

Computational efficiency varies strongly with the mathematical

models and numerical techniques chosen.

Based on the results presented and from the point of view of

required computer time the LMM developed by Nakajima et.al., [9], is the most promising model.

No time domain results for irregular upper-end oscillations have been presented.

(15)

3. PROBLEM DEFINITION

HOMM

-12-A mooring line connected to a structure floating in irregular

waves, wind and current is subject to the following loads: Upper- and lower-end forces

Weight and buoyancy

Seafloor reaction forces Inertia

Fluid loading due to wave orbital velocities, current and induced line motions.

The first three categories are taken into account in the

quasi-static analysis. Knowing the line elasticity, the line position and tension can be derived from the well known catenary solutions as, among others, given by Korkut and Hebert [21]. Solutions of 3-D

cable problems as caused by current, are found by shooting methods as suggested by Bendenbender [22] and De Zoysa [23].

Inertia and fluid forces originate from the motions of the line, wave and current action. The fluid loading may be divided in in-phase components proportional to the relative fluid acceleration

("added inertia") and quadrature-phase components proportional to the relative fluid velocity squared.

In order to determine the

importance

of the separate loading compo-nents the following approximations are used for the 2-D forces per unit line length:

Inertia Fl = + ,

D2 3v (1)

(M CI ¶/4 p)

--at

Drag FD = /5) CD D v1v1 (2)

(16)

where:

M = line mass

D = characteristic line diameter v = relative fluid velocity

P = fluid density (1.025 kg/m3)

g = gravitational acceleration (9.81 m/s2)

Let a harmonic upper-end oscillation be given by x = A cos wt. The line velocity and acceleration at this point will be:

= w A sin wt

0 OOOOOOOOO 0

. (4)

= w2 A cos wt

The orbital velocity due to regular waves is given by: cosh k(Z-H)

u = w ;

a sinh kH cos (wt) where:

;a = wave amplitude

k = wave number = 21T/A

For deep water (kH >> 1, w2 = kg) equation (5) may be approximated

by:

2

(Z-H) u = w ; eg

a cos(wt) (6)

Taking into account that the amplitude of upper-end oscillation due

to a floating body will be of the order of magnitude of ;a, for

taut conditions the normal displacement S (Figure 1) will be of the order ;a or larger. With respect to the total relative fluid-line

velocities the orbital velocities may be of importance near the

water surface only. Due to the large line length-wave length ratio, however, these wave effects will be of minor importance for the

lower node shapes.

HORE13

(17)

Report No. 45064-2RD

HOU

123

14

In general current velocities are not beyond 1.5 m/s. Hence, in

deep water the fluid loading is governed by the fluid reactive forces induced by the line motions. For shallow water (A/H > 27) this does not necessarily hold true.

Fluid ineria forces are assumed to be proportional to the relative fluid acceleration while drag forces are proportional to the velo-city squared. For a 76 mm chain the terms (M + CI

n/4

D2p) and (1/2p CD D) have approximately the same magnitude. Assuming that S is of order a or larger, the drag force will be dominant. For small nor-mal displacements which may occur at the ends of the mooring line, in case of low pre-tension or at high frequencies of oscillation, the inertia term will be dominant.

Transverse, horizontal forces caused by hydrodynamic lift and

vortex shedding may be of importance for the 3-D case but will not be discussed here.

Basically the dynamic behaviour of a mooring line will be non-lin-ear due to:

Catenary effect.

Geometry influence on fluid forces (normal and tangential compo-nents).

Sea floor contact.

Non-linear drag forces.

Therefore in general the motions and tensions of a line should be solved in the time domain or by utilizing proper linearization

(18)

4. MATHEMATICAL MODEL

The mathematical model chosen for the simulation of the dynamic

behaviour of mooring lines is a modification of the so-called

lumped-mass method as presented by Nakajima, Motora and Fujino [9]. The model is applied for two dimensions assuming that the mooring line remains in the vertical plane through both line ends. For

three dimensions the same approach can be used.

The spacewise discretization of the mooring line is obtained by

lumping all forces to a finite number of nodes ("lumped-masses"). The finite segments connecting the nodes are considered as massless springs accounting for the tangential elasticity of the line (Fig-ure 2). The line is assumed to be fully flexible in bending direc-tions. The hydrodynamic forces are defined in the local system of co-ordinates (tangential and normal direction) at each mass.

In order to derive the governing equations of motions .for the j-th lumped-mass, Newton's law is written in global co-ordinates (Figure 2).

HM.]-1-[m.(T)])

x.

(T) = F.(T)

-3 where:

[Mi.]

= inertia matrix

[Ini]

= added inertia matrix = time

x. = displacement vector

3

F. = external force vector

3

(x1, _{x2' x3)}

HOREE3

-15-(8)

The added inertia matrix can be derived from the normal and tangen-tial fluid inertia forces by directional transformations:

[n.(T)] =

(19)

where a_n3 and _at represent the normal and tangential added mass:

2

= P CIn

11/4 Dj

tj

ani

2

at _{= P CIt} IT/4 Dj

The matrices [A .] and

[A]

are given in Appendix I.

n3 t3

The nodal force vector F. contains contributions from the segment -3

tension T, the drag force FD, buoyancy and weight FW and the soil forces FS. F.(T)

= T.(T)Ax.(T) - T.

1

(T).6xj-1(T)

+ FD. J(T) + FW. + FS.(T) 3 _J 3-(10) where:

Ax = the segment basis vector (x. x )/Z

--j -3+1 -j j

= original segment length

The drag force may be derived from the normal and tangential force components.

FDj(T) = [0(T)] fDj(T)

(11) fD .(T) n3 = 12P CDn Djlj

unj(T)

lu

n3

.(T)I

. fD t.(T) =3 1/2 CDt D

j

utj(T)

lutj(T)I P u.(T)

= [r.(T)1(c.

where:

fD. = drag force in local co-ordinates -3

u. = relative fluid velocity in local co-ordinates

3

c. = current vector in global co-ordinates

L73,s-lif

[ri

_{= transformation matrices (Appendix I)}

HOM

.

(12)

...

.

(13)

(20)

HOE

Sea floor contact may be simulated by spring-damped systems.

Tan-gential soil friction forces may be of importance when the line

part on bottom is extremely long. Normal soil reactive forces may be of importance for 3-D problems. Both effects are neglected here.

FS(3) = - (cjxj(3) + bjki(3)) x.(3) < 0

FS(3) = 0.0 x.(3) > 0

or:600

The time domain relations between nodal displacements, velocities and accelerations may be approximated by finite difference methods such as the Houbolt scheme [22]:

1

k.(T+AT)

_=6A

Illx.(T+AT) - 18x.

(T)

+ 9x(T-r) - 2x.(T-2AT)1

7 T 1,

...

(16) 1

j(T+AT) =

{2x(T+AT) - 5x

(T) +

4Ij(T-AT) xj(T-2AT)1

3

AT2 (15) 2 AT

-X.(T-FAT) = 5/2 X.(T) 2X.(T-AT) 1/2X.(T-2t) 2

xj(T+AT)

3

(17) The segment tension

Tj(T+AT)

is derived from the node positions by

a Newton-Raphson iteration using the additional constraint equation for the constitutive stress-strain relation.

T. (T)

2 r

*.(T) = k.

AX.(T)

-

(1(18)

--3 EA.

3

Tk+1(T+AT) = Tk(T+AT) [Alpk(1)]-1 Tk(T) . 0 . (19)

where:

V = segment length error vector (V1, _{..., Vj,}

Tk = tentative segment tension vector at the k-th iteration (T1, ..., Tj, ..., TN)

A* =length error derivative matrix

[4]

_{obtained from equations}

(17) and (18).

(21)

HOED°,

For each time step the system of equations (19) should be solved

until acceptable convergence of Tk(T+AT) is obtained. The initial tentative tension can be taken equal to the tension in the previous

step. Each node j is connected to the adjacent nodes j-1 and j+1,

hence equation (19) represents a tridiagonal (Nx3) system. Such equations may be efficiently solved by the so-called Thomas algo-rithm (Appendix II). The computational procedure is illustrated by Figure 6. In order to avoid instability and transient behaviour the simulation is started from the quasi-static condition of the

moor-ing line. This condition may be found from catenary calculations

[21] or numerical methods [22], [23]. The simulation is initiated by applying a starting function to the upper-end boundary condition:

IN(T)

aN(T)

cosh (4.0 IITINF) where:

TINF = starting time

On the basis of the known line angles the fluid inertia matrix is found from eq. (9). Line velocities are obtained from eq. (16) re-sulting in drag forces by use of eq. (11) through (14). Soil reac-tion forces are derived from eq. (15). Knowing the inertia matrices and the right hand side of equation (8) the accelerations for the

new time step are solved. Displacements follow from eq. (17). A

correction of

tension is

predicted by eq. (19) using the segment length error from eq. (18). The whole procedure is repeated until an acceptable accuracy in tension is obtained.

In that case the simulation is proceeded with the next time step by applying the next boundary condition (20).

(22)

5. FLUID REACTIVE FORCE EXPERIMENTS

Test set-up

As discussed in Section 2 fluid reactive forces play an important role in the dynamic behaviour of mooring systems. In order to de-rive these forces for chains, harmonic oscillation tests with a 4 mm rigid chain type DIN 766 (Table 1) were carried out in the Basin

for Unconventional Maritime Structures, measuring 220 x

4 x

4 x m.

To this end the chain links were soldered and pre-tensioned in a U-frame. Two-component transverse strain-gauge transducers connected the chain to the frame while a tension-transducer was incorporated.

Photographs of the test set-up are presented at the end of this report. The upper-end of the chain was at approximately 0.50 m

beneath the water surface.

Scaling

Insight in the hydromechanic characteristics of the chain oscillat-ing in water may be gained from evaluatoscillat-ing the followoscillat-ing numbers:

Reynolds _Rn V.D

Keulegan-Carpenter Kc - D

where:

D = characteristic diameter

'Or = maximum velocity

T = period of oscillation V _{= kinematic viscosity}

The typical range of Rn-values for mooring chains and wires is 104-106. This is the typical "critical flow regime" for smooth isolated cylinders (Figure 7). Mooring wires and in particular chains can be considered as complex rough bodies with rather constant drag forces.

HOMM

(23)

HOMM

-20-For a line oscillating in still water the Keulegan-Carpenter number reduces to 27rS/D. For situations of interest the Kg-values will be in the range of 10 to 100. Hence, the drag forces correspond to the forces at constant velocity.

Though free surface effects were not present, the oscillation tests were carried out according to Froude's law of similitude at scale 1

to 19 thus representing a 76 mm chain. V

Froude Fn - ---IgL

This procedure ensured a proper scaling of the inertia forces

(added mass). The Reynolds number during the model test varied from

102 to 104. The drag forces which require a Rn-scaling, however,

are assumed to be approximately constant in the Rn-range 102-106

due to the complex geometry and roughness of the chain (Figure 7). By scaling on the basis of contant Froude numbers, comparisons with Phase II results could be made.

Results

A review of the test program and the measured force components in

the direction of oscillation are given in Table 2. Typical force

records are shown in Figure 4. From these records it appears that

the lift forces contain a considerable amount of high frequency energy. The magnitude of the harmonic components is an order

smaller

compared to the drag forces.

The oscillation tests were repeated with the force transducers

only. Harmonic analysis was carried out to derive the amplitudes of

the in-phase and quadrature phase forces at the oscillation

fre-quency. The fluid inertia force was derived from the difference be-tween the in-phase force components of the chain oscillation test and the transducers oscillation test by subtracting the chain

iner-tia force. The drag force was obtained from the quadrature phase

force components. Dimensionless hydrodynamic coefficients were derived by taking into account the chain volume and the volumetric chain area (Table 3):

(24)

Report No. 45064-2-RD Inertia Cin -Drag CDn -where: FIn 7/4 p dc2 X R FDn 1/2 p dc lki 8/3

dc = volumetric chain diameter

X = chain length

The factor 8/37 results from the harmonic analysis procedures where

the non-linear drag force record is approximated by a pure sine

function. Assuming an equal energy dissipation during each cycle of oscillation, the amplitude obtained from the harmonic analysis

should therefore be multiplied by 31T/8.

The drag coefficients in still water increase nearly linearly with the frequency of oscillation. The drag values are indepenent of the amplitude of oscillation. This may be explained by the large ampli-tude-diameter ratio (KC-number). CDn values of 1.3-1.4 were found by towing the test set-up with several constant speeds (Table 4).

Rotation of the complete chain about its own center line over 45

degrees provided the same results.

Accurate derivation of fluid induced inertia forces is strongly

hampered by the large mass of the chain. The fluid inertia force has been deducted from the difference of the measured in-phase

force and the inertia force due to the mass of the chain which had the same order of magnitude. For low frequencies of oscillation the inertia coefficients cannot be considered to be accurate due to the small amplitudes of the measured forces and the large corrections for chain mass and force transducers. For high freqUencies of os-cillation, dynamic amplification induced by the test set-up

reson-ance cannot be neglected. The natural frequency of the system

amounted to 3.3 rad/s. On basis of this knowledge and taking into account that added inertia is independent of frequency in the ab-sence of free surface effects, it may be concluded that the inertia

HOE

21 (21)

(25)

FDT = -Fx cos a + Fz sina

CDt - FDt

1/2 p dc Z (V sina)2

HOEM

coefficients amount to approximately 1.3-1.6. Drag coefficients

were also obtained from towing tests with a free hanging chain

section at several constant velocities. By measuring both vertical and horizontal force components at the tow connection, the normal

and tangential drag force coefficients have been derived directly from (Figure 5): Fx a = arctan

Fz-

+B

(22) G FDn = -Fx sina + Fz cosa

A review of all towing test parameters and the results is given in

Table 4. Tangential force coefficients were derived for tests with large inclination only.

From these towing tests the following conclusions can be drawn: - The normal drag coefficients, amounting to 1.2-1.4, correspond

fairly well to the oscillation tests at low frequencies. - The tangential drag coefficient amounts to 0.3-0.4.

- No scale effects were found in the tested ranges of scale factor (9.5-76) and Reynold number (0.22 103 - 16.5 103).

- For steel wire the normal drag coefficient is approximately 1.2 while the tangential drag coefficient is 0.1.

Taking into account that the volumetric chain diameter was used to

derive the hydrodynamic force coefficients, it is expected that

these coefficients are also valid for other types of chain links

with small shape differences.

-22-(23) FDN CDn 1/2 p dc 2. (V cosa)2

...

. (24)

(26)

6. PARAMETER STUDY

HOED,

-23-Computer simulations were carried out with the 2-D Fortran 77 com-puter program DYNLINE which has been developed on the basis of the mathematical model presented in Section 4. The objectives of these simulations were to investigate possible discretization effects in the lumped mass method, to compare the results with finite element analysis and to study the governing factors in the dynamic

behav-iour.

The approximations which form the basis of the simulation algorithm may be summarized as follows:

The mooring line remains in the vertical plane through both line

ends.

The mooring line is completely flexible in the bending

direc-tions.

All external forces are lumped to a finite number of nodes. Elasticity is linear with elongation.

Fluid reactive forces can be described by eq. (12) and (13)

using constant force coefficients.

Sea floor contact is simulated by critically damped springs. Friction is neglected.

The following situation was simulated:

Line : 76 mm, chain DIN 766

Water depth: 75 m Line length: 525 m Pre-tension: 1275 kN

Harmonic horizontal oscillation

This configuration corresponds with situation No. 1-of Phase II. A static load-excursion relation is given in Figure 8 while the ini-tial condition is presented in Table 5. This situation represents a

"pre-tension" equal to 30 percent of the breaking load. This ten-sion is assumed to incorporate the mean and maximum low frequency excursion of the moored structure.

(27)

Input was derived from the chain particulars (Table 1) and the

Discretization

KOMM,

-24-Geometry discretization is an important aspect of the lumped mass modelling. The number of nodes should be sufficient to describe the

line position. Moreover parasitical motions of the lumped masses

may occur. The tangential stiffness of a single lumped mass is lin-early dependent on the relative tangential displacement:

C. = 2 EA 2 (25)

The normal stiffness is non-linearly dependent on the normal dis-placement 6n. For small deflections this stiffness equals:

Cn = EA 6n/22 (26)

Neglecting the damping the resonance frequencies of these parasiti-cal motions may be approximated by:

w =

n (27)

From this information it is clear that for the usual types of moor-ing lines resonant response of separate masses in the lumped para-meter model will not provide significant parasitical motions. This even holds true for clusters of masses. The occurrence of such may

be prevented by increasing the number of nodes thus reducing the

nodal mass and element length.

experimental results from Section 5: CDn = 1.30

CDt = 0.40 CI n = 1.60 CIt = 0.20

(28)

HOU73.3

-25-In order to investigate the effect of the geometry discretization on motions and tensions, computer runs were carried out for w = 1.0 and 1.5 rad/s, with 6, 12 and 24 nodes. The initial condition of these models is given in Table 5. By oscillating at 0.02 rad/s it

was concluded that all models represented the same quasi-static

load excursion characteristics. The integration time step was 0.025 s while 12 seconds starting time was taken into account.

The results are presented in Figures 9 through 14.

From these results it appears that parasitical tension peaks at the

upper line end occur at the end of each period of slackness. The

number of peaks fairly correspond with the number of nodes. Hence it may be concluded that these peaks are due to the successive ten-sioning of slack segments. The corresponding nodal motions are neg-ligible. The global (primary) tension is hardly affected by the sec-ondary peaks except for low frequencies and large segments (Figure 9).

The effect of time discretization was investigated by reducing the integration time step from 0.0250 s to 0.0125 and 0.0050 s. By com-paring Figures 11, 15 and 16 it appears that secondary tension

ef-fects in global tension peaks increase with time step reduction.

These effects are due to normal motions and occur along the entire line. The same result was found with the 24 node model (Figure 13).

On the basis of these results it may be concluded that the 12 node model can be used to derive the primary (global) motions and ten-sions for this configuration.

Validation

The results of the 12-node lumped-mass model have been correlated with finite element calculations. To this end use has been made of the ANSYS package [24]. Fluid reactive forces were taken into ac-count similarly to the model derived in Section 4. Slack conditions could not be taken into account. A good agreement with the DYNLINE results was found, see Figures 17 and 18.

(29)

-26-Drag

The sensitivity of the results to the value of the normal drag co-efficient in the wave frequency region is elucidated by Figure 19. From this figure it appears that the influence of the normal drag

coefficients is large for frequencies in the range of 0.5 to 1.5 rad/s. Taking into account the drag coefficients obtained from

forced oscillation tests (Table 3) it may be concluded that the

frequency dependency of the coefficients plays no important role in the dynamic behaviour.

In Figure 20 dynamic tension-excursion relations are compared with the quasi-static values for various frequencies of oscillation. The area enclosed by the tension loops represents the energy dissipated by drag during one oscillation.

Am2litude of oscillation and 2re-tension

The dynamic tension amplification, here defined as the maximum dy-namic tension divided by the maximum quasi-static tension, strongly depends on the amplitude and frequency of motion (Figure 21). For large amplitudes the normal drag force will cause dynamic effects at low frequencies. With increasing frequency the drag and inertia

will equal gravity forces resulting in an "elevated equilibrium"

and small normal motions. For high frequencies only the upper part

of the line will follow the upper-end oscillation yielding lower

dynamic amplifications.

Figure 21 clearly shows the effect of pre-tension. In this study

the pre-tension represents the tension due to the maximum low fre-quency excursion of the moored structure. At a pre-tension of 850 kN the maximum dynamic amplification is approximately constant.

-This maximum, however, shifts towards lower frequencies with in-creasing amplitude of oscillation. In case of the high pre-tension

the effect of the amplitude of oscillation seems to have a large effect on the dynamic amplification of tension. It should be born in mind that for this situation the quasi-static tension also

(30)

HOMM

-27-creases rapidly with increasing amplitudes. Figures 23, 24 and 25

show the tension and motion records for a horizontal oscillation

amplitude of 4.0 m at 1.0, 1.5 and 2.0 rad/s. The vertical

dis-placement at node 10 clearly shows the change of equilibrium. The primary tension characteristics are hardly affected by the

frequen-cy of oscillation.

For small amplitudes of motion the dynamic tension amplification in

the wave frequency region is small (Figures 26 trough 29).

In-teresting information can be derived from the phase lags between line motions and tensions and the forced oscillation. At extreme

low frequencies (the quasi-static situation) both line motion and

tension will be in-phase with the forced oscillation. Figure 26

shows a vertical line motion which is completely in-phase while the tension is approximately 70 degrees heading of the forced oscilla-tion. With increasing frequency the phase difference in tension de-creases while the motion time lag inde-creases. From all tests it

ap-pears that all primary tensions as well as the vertical motions

have the same phase shift along the mooring line.

In Figure 22 the maximum dynamic tensions are plotted together with the quasi-static values versus the amplitude of oscillation. From this figure it appears that the maximum dynamic tension may be

ap-proximated by applying the material elasticity to the high frequen-cy oscillations directly. For oscillations with amplitudes for which the elastic region is reached the dynamic amplification is

(31)

7. CONCLUSIONS

Mooring line dynamics have been investigated by several authors. Experimental data on fluid reactive forces are scarce. No sys-tematic validation of existing mathematical models is given. Dynamic coefficients derived from towing tests with free hanging

chains fairly agree with those obtained from forced oscillation tests up to 1.0 rad/s. The damping can be presented dimension-less using the volumetric line diameter. Typical normal drag co-effcients were approximately 1.30. No scale effects were found in the towing tests for scale ratios between 9.5 and 76.

Lumped mass models can be used efficiently to simulate the pri-mary motions and tensions due to forced upper-end oscillations. A good agreement with finite element results was found.

Lumped mass simulations may contain parasitical secondary ten-sions for small line segments and integration time steps.

These tensions do not affect the global (primary) motions and

tensions.

Wageningen,

August 1984. NETHERLANDS SHIP MODEL BASIN

Dr Ir M.W.C. Oosterveld

Head Research and Development Division

AGvD/mt

HOEM

(32)

REFERENCES

HORE

-29-Bathe, K. and Wilson E.L.: "Numerical methods in finite ele-ment analysis", Prentice Hall Englewood Cliffs, 1976.

Walton, T.S. and Polachek, H.: "Calculation of non-linear transient motion of cables", D.T.M.B. Report 1279, 1959.

Goeller, J.E. and Laura, P.A.: "Analytical and experimental

study of the dynamic response of cable systems", OTC paper

1156, 1970.

Maclaughlan, R.A. et.al.: "A dynamic analysis of moored and free floating cable systems", OTC paper 1742, 1973.

Palo, P.A., Meggitt, D.J. and Nordell, W.J.: "Dynamic Cable Analysis Models", OTC paper 4500, 1983.

Thresher, R.W. and Nath, J.H.: "Anchor-last deployment simu-lation by lumped masses", ASCE Journal of the Waterways, Har-bours and Coastal Engineering Div., November 1975.

Wilhelmy, V., Fjeld, S. and Schneider, S.: "Non-linear re-sponse analysis of anchorage systems for compliant deep water platforms", OTC paper 4051, 1981

Wilhelmy, V. and Fjeld, S.: "Assesment of deep-water anchor-ings based on their dynamic behaviour", OTC paper 4174, 1982. Nakajima, T., Motora, S. and Fujino, M.: "On the dynamic

analysis of multi-component mooring lines", OTC paper 4309,

1982.

Ando, S.: _{"On the hydrodynamic forces of mooring wire ropes} and chains", (Part 1, _{Partial models). Transactions of the}

West-Japan Society of Naval Architects, No. 50, August 1975. Nakajima, T., Motora, S. and Fujino, M: "A three-dimensional

lumped mass method for dynamic analysis of mooring lines",

Journal of the Society of Naval Architects of Japan, Vol.

154, December 1983.

Ractliffe, A.T.: "Dynamic response of flexible catenary risers", Int. Symposium on Developments in Floating Produc-tion Systems, London 1984.

(33)

non-Report No. 45064-2-RD

HOEM

-30-Leonard, J.W.: "Curved finite element approximation to non-linear cables", OTC paper 1533, 1972.

Webster, R.L.: "Non-linear static and dynamic response of

underwater cable structures using the finite element method", OTC paper 2322, 1975.

Fylling I.J. and Wold, P.T.: "Cable dynamics - comparison of experimental and analytical results", Report-8979, The Ship Research Institute of Norway, Trondheim, 1979.

Larsen, C.M. and Fylling, I.J.: "Dynamic behaviour of anchor lines", BOSS 1982, Boston.

Ormberg, H., Fylling, I.J. and Morch, M.: "Dynamic response

in anchor lines: Numerical simulations compared with field

measurements", OTC Paper 4493, 1983.

Lindahl, J. and Sjcberg, A.: "Dynamic analysis of mooring

cables", Second International Symposium on Ocean Engineering and Ship Handling, Gothenburg, 1983.

Van Sluijs, M.F. and Blok, J.J.: "The dynamic behaviour of mooring lines", OTC paper 2881, 1977.

Suhara, T. et. al.: "Behaviour and tension of oscillating

chain in water", Journal of the Society of Naval Architects of Japan, Vol. 148, December, 1980;

Vol. 152, January, 1983; Vol. 154, December, 1983.

Korkut, M.D. and Hebert, E.J.: "Some notes on static anchor chain curve", OTC paper 1160, 1970.

Bendenbender, J.W.: _{"Three-dimensional boundary value} pro-blems for flexible cables", OTC paper 1281, 1970.

De Zoysa, A.P.K.: "Steady-state analysis of undersea cables", Ocean Engineering, Vol. 5, 1978.

(34)

NOMENCLATURE

HME

A = amplitude, material area cross-section

a = added inertia

= buoyancy per unit length

CD = hydrodynamic drag coefficient

CI = hydrodynamic inertia coefficient

= current vector = line diameter dc = volumetric diameter = force FD = drag force Fl = inertia force

FW = weight minus buoyancy

fl) = drag force in local co-ordinates

= gravitational acceleration = weight per unit length = water depth

= subscript: node number = wave number

= length

1 = line segment length

= line mass (matrix) = added inertia (matrix) = node

= subscript: normal direction = stroke

= tension, period of oscillation

0 = pre-tension

Tdy = maximum dynamic tension

Tqs = maximum quasi-static tension

= subscript: tangential direction = fluid velocity

= fluid velocity = displacement

(35)

-31-Report No. 45064-2-RD x1,x21x3 = 3-D system of co-ordinates x,z = 2-D system of co-ordinates a _{= angle} d _{= displacement} a = wave amplitude

A _{= transformation matrix (Appendix I)}

A = wave length

= kinematic viscosity = fluid density

= transformation matrix (Appendix I) = time

= vertical line angle

* _{= segment length vector}

= angular frequency of oscillation = transformation matrix (Appendix I)

HOU

(36)

APPENDIX I

TRANSFORMATION MATRICES FOR 2-D LUMPED MASS MODEL

[czi] =

[ri]

= -sinTicosTi

2

cos 4). cosTj

HOU

-33-[At.]

=

2

cos cl). sin -j

cosT.

sin n. cosT3. sin2 -[Ani] = .2 -sin (1). -sinT .3

cos

T3.

(37)

HOhlE3

APPENDIX II

SOLUTIONS OF TRIDIAGONAL EQUATIONS BY THE THOMAS ALGORITHM

Let a Nx3 system of equations be given by:

-Aj Fi...1 + BjFj - CjFj+1 = Dj j = 2, 3, N

The solution Wj of this system is found from:

Ej = Cj/(Bj - AjEj_i)

j= 1, 2, ... .. (N-1)

ej = (Dj + Ajej_i)/(Bi - AjEj....1)

Wj = Ej Wj+1 + ej

Taking into account that:

Al = 0 WN = eN

(38)

-34-Report 45064-2-RD

Mooring Line Dynamics

PARTICULARS OF ANCHOR CHAINS Chain type: DIN 766 (D = 0.076 m)

PARTICULARS OF bitEL WIRES (D = 0.076 m)

HOE

Table 1

Model chains Prototype chain

6 BS = 4.38*10 N - EA = 0.694*109 N D B L M W d EA A M W d EA 9 mrn mm mm kg/m N/m mc N*105 kg/m N/m mc N*10 _ 1.0 4.2 7.9 0.021 0.177 0.0019 0.03 76 124 1048 0.144 1.19 2.0 6.8 12.0 0.080 0.690 0.0036 0.11 38 119 1021 0.137 0.60 4.0 14.2 22.8 0.338 2.874 0.0076 7.00 19 125 1063 0.144 4.90 8.0 26.9 40.1 1.383 11801 0.0151 22.00 9.5 128 1092 0.144 1.93

-Model wires Prototype wire

8 8 EA = 3.2*10 N - 4.4*10 N _ D M W d EA A M W d EA mm kg/m N/m mc N*105 kg/m N/m mc N*1010 1.0 0.00401 0.034 0.001 0.5 76 23.2 204 0.076 2.3 2.0 0.0164 0.144 0.002 2.3 38 24.3 213 0.076 1.3 4.0 _0.0608 _0.540 _0.004 _5.1 19 22.5 199 0.076 0.36

(39)

Report No. 45064-2-RD

HARMONIC ANALYSIS OF OSCILLATION TESTS

HOREM

Table 2

w in Speed in Stroke in F xl Fx2 -Fx total rad/s m/s m Fxa FX E Fxa FX E FXa

i

E in kN in kN in ° in kN in kN in ° in kN in kN in 0.456 0 1.9 1.53

-

312 1.24

-

312 2.77

-

312 0.921 0 1.9 6.50

-

311 5.20

-

311 11.70

-

311 1.157 0 1.9 10.50

-

309 8.65

-

308 19.15

-

309 1.381 0 1.9 15.75

-

309 13.32

-

308 29.07

-

309 1.609 0 1.9 23.31

-

308 19.45

-

310 42.76

-

309 1.838 0 1.9 31.82

-

306 28.53

-

305 60.35

-

306 0.456 0.436 1.9 1.69

-0.49

301 1.49

-0.54

304 3.18

-1.03

303 0.920 0.436 1.9 6.74

-0.96

307 5.52

-1.00

308 12.26

-1.96

308 1.378 0.436 1.9 15.49

-1.18

307 12.94

-1.34

307 28.43

-2.52

307 1.841 0.436 1.9 31.00

-1.62

305 28.71

-1.42

306 59.71

-3.04

306 0.459 0.872 1.9 2.51

-1.56

295 2.08

-1.46

294 4.59

-3.02

295 0.917 0.872 1.9 7.33

-2.63

301 5.82

-2.08

307 13.15

-4.71

304 1.378 0.872 1.9 15.92

-3.47

304 12.54

-2.91

308 28.46

-6.38

306 1.839 0.872 1.9 30.64

-4.95

302 26.56

-3.72

306 57.20

-8.67

304 0.457 0 0.95 0.56

-

325 0.46

-

323 1.02

-

324 0.921 0 0.95 2.46

-

326 2.06

-

326 4.52

-

326 1.148 0 0.95 3.79

-

324 3.11

-

324

6.90 -

324 1.380 0 0.95 5.85

-

323 5.00

-

323 10.85

-

323 1.614 0 0.95 8.42

-

323 6.82

-

324 15.24

-

324 1.838 0 0.95 11.29

-

322 9.14

-

322 20.43

-

323 1.150 0 1.425 6.56

-

315 5.44

-

313 12.00

-

314 1.613 0 1.425 14.86

-

314 11.88

-

316 26.74

-

315

(40)

Report No. 45604-2-RD

OSCILLATION TESTS (PHASE I)

Chain type: DIN 766

= 0.076 m

dc

= 0.144m

Scale =

1: 19

HOREM

_{Table 3}

w

in

rad/s Fx total Stroke S in m C In cDn Current in m/s F xa e in kN

in0

0.456 2.77 312 1.90 1.8Q 1.27 0 0.921 11.70 311 1.90 1.58 1.40 0 1.157 19.15 309 1.90 1.53 1.51 0 1.381 29.07 309 1.90 2.16 1.63 0 1.609 42.76 309 1.90 3.09 1.79 0 1.838 60.35 306 1.90 3.12 2.03 0 0.456 3.18 303 1.90 1.07 1.78 0.436 0.920 12.26 308 1.90 1.42 1.58 0.436 1.378 28.43 307 1.90 1.43 1.65 0.436 1.841 59.71 306 1.90 2.94 2.00 0.436 0.459 4.59 295 1.90 2.17 2.99 0.872 0.917 13.15 304 1.90 1.20 1.85 0.872 1.378 28.46 306 1.90 1.18 1.68 0.872 1.839 57.20 304 1.90 1.82 1.96 0.872 0.457 1.02 324 0.95 _0.54 _1.55 0 0.921 4.52 326 _0.95 _1.31 _1.67 0 1.148

6.90

324 0.95 0.86 1.72 0 . 1.380 10.85 323 0.95 _1.54 _1.96 0 1.614 15.24 324 0.95 1.93 1:95 0 1.823 20.43 322 0.95 1.91 2.11 0 1.150 12.00 314 1.425 0.78 1.60 0 1.613 26.74 315 1.425 2.21 1.81 0

(41)

Report No. 45064-2-RD

TOWING TESTS - _{Chain type: DIN} ₇₆₆ _{(D =} _{0.076 m,} _dc

_{= 0.144}

m)

TOWING TESTS - Steel wire (D = _0.076 _{m, dc =} _{0.076 m)}

Table 4

rn II

0

,..., Z 4, Under F

--a ^

L V FX FZ water

a

FDn Dt CDn CDt ,-1 A weight in in in m/s in kN in kN W in deg. in kN in kN o E

z >

_-

in kN --1.42 19 25.25 0.877

-1.89

-

0 _1.89

_-

_1.31

-3.46 19 25.25 2.167

-12.43

-

_-

₀ _12.43

_-

_1.42

-6.91 19 25.25 4.358

-50.80

-

0 _50.80

-

_1.45

-5.24

19 25.25 3.303

-28.80

-

0 _28.80

-

_1.42

-4.13

9.5

28.72 0.930

-2.28

0.04 35.66 3.66 2.28

-

1.26

-8.33

9.5

28.72 1.864

-8.38

1.46 35.66 13.77 8.49

-

1.23 -16.50

9.5

28.72 3.692

-19.48

10.57 35.66 37.83 21.87 3.60 1.22 0.33 1.40 19 57.10 0.866

-4.03

-0.06

59.28 3.89 4.02

-

1.28

-2.80

19 57.10 1.753

-14.73

1.89 59.28 14.37 14.74

-

1.22

-5.50

19 57.10 3.468

-36.23

18.34 59.28 41.51 39.29 10.28 1.38 0.46

5.50

19 57.10 3.474

-36.04

19.10 59.28 41.89 39.58 9.84 1.41 0.43

0.49

38 114.0 0.916

-8.36

0.43 113.60 4.22 8.37

-

1.25

-0.99 38 114.0 1.864

-31.62

3.97 113.60 16.09 31.48

-

1.23

-1.96 38 114.0 3.693

_-71.36

37.46 113.60 43.14 77.68 21.46 1.33 0.42 0.22 76 228.0 1.121

-22.52

6.66 232.50 5.69 23.07

-

1.10 -0.45 76 228.0 2.275

-88.66 22.85

232.50 22.92 90.56

-

1.23

-0.89 76 228.0

4.502 -177.20 90.36

232.50 51.27 181.36 81.70 1.36 0.39

'

m II

0

-4 Under

az

4,0 w ., 71 0 ro 0 E

E >

A L in in V in m/s FX in kN FZ in kN water weight W

in

kN

a

in deg. FDn in kN-FDt in kN cDn cDt 0.64 19

_57.0

_0.762

_-1.45

_0.04 _11.07 _7.49 _1.44

_-

_1.14

-1.27 19

57.0

1.520

-4.65

1.78 11.07 26.59 4.95

-

1.21

-2.53 19

57.0

3.030

-6.51

6.03 11.07 52.25 8.75 1.46 1.15 0.11

(42)

RepOrt No. 45064-2-RD

LINE DISCRETIZATIONS

HOE):3

Table 5

N x1 in m x3 in m cl) in rad T in kN 1 1 1 0.0 0.0 0.0 1198 2 25.0 0.0 0.0 1198 2 3 50.0 0.0 0.0 1198 4 75.0 0.0 0.0 1198 2 3 5 100.0 0.0 0.0 1198 6 121.0 0.1 0.012 1198 4 7 142.3 0.5 0.031 1198 8 163.6 1.4 0.050 1199 3 5 9 184.8 2.6 0.069 1201 10 206.0 4.3 0.088 . 1202 6 11 227.1 6.4 0.116 1205 12 248.3 8.8 0.125 1207 4 7 13 269.3 11.7 0.143 1210 14 290.4 14.9 0.162 1214 8 15 311.3 18.5 0.180 1218 16 332.2 22.5 0.198 1222 5 9 17 353.3 26.9 0.217 1226 18 373.8 31.6 0.234 1232 10 19 394.4 36.8 0.252 1237 20 414.9 42.3 0.270 1243 6 11 21 435.4 48.1 0.287 1249 22 455.7 54.3 0.305 1256 12 23 476.0 60.9 0.322 1263 24 496.1 67.8 0.339 1270 7 13 25 516.1 75.0 0.355 1278 _

(43)

-Mooring Line Dynamics

MOORING LINE CONFIGURATION

tAlilik.

Allelb

4111111111.v

1171 W frdiZilir

Aggillihk

1P11.1r

0 ir

doidill01111

Extreme catenary position

NM--10111101111Fr

10r

(44)

CO-ORDINATE SYSTEM FOR LUMPED MASS MODEL

(45)

Report No. 45064-2-RD

Tj 1

NODAL FORCE DEFINITION

G.

HOM

_{Fig. 3}

(46)

S 2 (N X 1

X2

7Y 2 <N 3ECOND.3

Report No. 45064-2-RD

0 50. 00

TEST NO. 660203 -

Oscillation test zero speed

Lil = 1.381 rad/s S = 1.9 m 0 V A

HOMM

if-\\

Fig. 4

/0/A0,1

(47)

Report No. 45604-2-RD

FREE HANGING DRAG TEST ARRANGEMENT

HORM

(48)

Report No. 45064-2-RD

Next time step

Mooring Line Dynamics PROGRAM FLOW DIAGRAM

Line data

Quasi-static solution Boundary condition

Correction of tension

HOM

Output Fig. 6 Upper-end position (20) Inertia matrices (9) Soil forces (15) Velocities (16) Drag forces (11)-(14) Accelerations (8) Displacements (17)

(49)

0

102

NORMAL DRAG FORCE AS FUNCTION OF REYNOLDS NUMBER

result for smooth cylinder of infinite length result for rough cylinder

drag test test Typical Typical 0 Chain 0 Wire drag

Model tests Reality

0 qy. 0 8 0 _

..

_.

-\

- -

_

Report No. 45064-2-RD

HOWL3

Fig. 7

105 106 107 Vd R -V 4.0 3.0 2.0 A (NI 1 . 0

(50)

Report No. 45064-2-RD

20,000 16,000 12,000 4,000 o _

STATIC LOAD DISPLACEMENT

2

Displacement in m

HOM33

_{Fig. 8}

(51)

Report No. 45064-2-RD

Number of masses : 6 Frequency : 1.0 rad/s Tensions in N 4.00L0 T-NODE2 0 -4.00x 2.00 _ Z-NODE2 0 -2.00 4.00 Z-NODE5 0 -4.00 4.00 _ X-NODE5 0 -4.00 _ 4.00x T-NODE7 0 -4.00x 2.00 _ X-NOL77 0 -2.00 SECONDS

HOH

_Fig.

9

Li

(52)

Report No. 45064-2-RD

4.00xle

T-NODE2 0

-4.00xle

2.00 Z-NOCE2 0 -2.00 4.00 Z-NOCE5 0 -4.00 .._ 4.00 X-NODE5 0 4.00* T-NOCE7 0 -4:00x 2.00 .., X-NOCE7 0 -2.00 SECONDS

Number of masses : 6

Frequency : 1.5 rad/s

-MD°,

_{Fig. 10}

(53)

Report No. 45064-2-RD

4.00x1406 T-NO0E2 0 -4.00xL0 2.00 Z-N00E2 0 -2.00 4.00 _ Z-N00E10 0 -4.00 _ 4.00 _ X-NODE10 0

-4.00 _

4.0o.(1135 T-NODE13 0 -4.00x1.0 2.00 X-NOSE13 0 -2.00 _ SECONDS

5

HOUM

- AL

Fig. 11

(54)

Report No. 45064-2RD

T-NODE2 0

Number of masses : 12 Frequency : 1.5 rad/s 2.00 Z-NODE2 0 -2.00 4.10 Z-NODE10 0 -4.00 4.00 X-NODE10 0 4.00 _ 4.00xLC

ii/n\

T-NCOE13 0 k 4. COx 2.00 X-N00:13 0 -2.00 _ SECONDS 0 10

HOUM

_{Fig. 12}

20

If\

(55)

Report No. 45064-2RD

Number of masses : 24 Frequency : 1.0 rad/s Z-NODE2 0 -2.00 _ 4.00*Le T-NO0E2 0 5 -4.00x 4.00 _ Z-NOCr20 0 -4.00 _ 4.00 X-NOCE20 0 -4.00 _ 4.00xle T-N0025 0 -4.00412 X-N0025 0 -2.00 _ SECONDS

HOED?)

Fig. )3

(56)

Report No. 45064-2-RD

Number of masses : 24 Frequency : 1.5 rad/s 2.00

-Z-NOCE2 0 -2.00 _ 4.00xLe T-N002 0

-4.00xL:

4.00 _ Z-N00020 0 -4.00 _ 4.00 X-NCOE20 0 -4.00 _ 4.00x1O5 T-NODE25 0 -4. 00L0 X-NOCE25 0 _1 on _{_} EECONCS 0

(I\

ka 10

HOE=

1\

20

(57)

Report No. 45064-2-RD

4.00x1.05 T-N0002 0 -4.001.05 2.00 _ Z-NODE2 0 2.00 _ 4.00 Z-NCOE10 0 4.00 _ 4.00 _ X-NOCE10 0 -4.00 _ 4. 00 LOs T-NODE13 0

Number of masses : 12 Frequency : 1.0 rad/s AT : 0.0125 s -4.00x1L X-NCIDE13 0 2.00 SECONDS

,tkrt

HORil

Fig. 15

(58)

Report No. 45064-2-RD 4.00)(10' T-NCOE2 0 -4.00L2 2.00 Z-N00E2 0 -2.00 4.00 _ Z-NODE10 0 -4.00 _ 4.00 _ X-N0010 0 -4.00 _ 4 rr Lc T-NODE13 0 -4. 00x 2.00 X-N0.713 0 -2.00 SECONDS

AT : 0.0050 s

.3

HOE=

Fig. 16

(59)

Report No

45064-2-RD

X -NODE 13 T-NODE 13 (FEM) T-NODE 13 (DYNLINE) T-NODE 13 (FEM) T-NODE 13 (DYNLINE)

Mooring Line Dynamics FEM-DYNLINE CORRELATION Amplitude : 2.0 m Frequency : 1.0 rad/s 2.00- -2.00-4.00*106_

-4.00*106_

4.00*106--4.00*106 Seconds 0 Amplitude : 2.0 m Frequency : 0.5 rad/s 2.00-X -NODE 13 2.00 -4.00*106 0 4.00*106 400* 106 0

4.00*106-Seconds 0 10

HORE

_{Fig. 17}

10 20

(60)

Report No. 45064-2-RD

10.0 7.5

5.0

2.5

0

Mooring Line Dynamics FEM-DYNLINE CORRELATION

Tdy = Dynamic tension Tqs = Quasi-static tension

Dynline 0 Ansys.

W in rad/s

HOREE3

Fig. 18

(61)

Report No. 45604-2-RD

10. 7.5

5-2.5

0

INFLUENCE OF DRAG o

C= 1.3

Dn

----41

CDn = 0.8

W in rad/s

HOHE3

Fig.

19

,

/

./

_//

/

,

/

,

/

,

/

,

//

/r

/7

//

/

..----/ ,

/

.../../.. _

---0

_0.5

_1.0

15

(62)

Report No. 45064-2--RD 20,00 16,000 12,000 8,000 = c 4,000 0 -4

TENSION/DISPLACEMENT RELATION U) = 0.5 rad/s w = 1.0 rad/s w = 1.5 rad/s Displacement in m

HOREE3

_{Fig. 20}

\

/ / / //

/

,....-1

\

\ \ I .

/

-,

.-'1

/ - - - . ..

_\

\ \ Static '

/i

il

/

//

e

.

/

N \ \ _./// .

7

/ /

II

-2 0 2 4

(63)

Report No. 45064-2-RD

10 8 6 4 2

EFFECT OF PRE-TENSION AND OSCILLATION AMPLITUDE

T0 = 850 kN T0 = 1275 kN

HOREE3

_{Fig. 21}

MI= 4 m = 1 m

.

IF

,v

r

vas

pr

id.

A = 24

_____,r

. ./

_ -.. ..-... ..- ,--..-._.

_-.

--- .... ... -.. A = 1 m A = 2 m -.._-..

,.

A

A = 4 m _ 0.8 _1.6

_2.4

_3.2 W

in rad/s

(64)

E.

Report No. 45604-2-RD

20.00 16.00 12.000 8.000 4.000

MAXIMUM TENSION VERSUS OSCILLATION AMPLITUDE

Quasi-static tension --- Max. dynamic tension

HOW Dg

_{Fig. 22}

)

I

/

1

I

/I

/

T + 0 AL.EA

//

/

L i 1

/

II

I

L

dymax. I

i /

qs

I/

-8

-4 0 4 8 Amplitude in m

(65)

Report No. 45064-2-RD

4.0 xL05 T-NODE2 0 4.00xL, Z-NODE2 2.00

-4.00 _ Z-NODE10 0 -4.00 _ 4.00 _ X-NODE10 0 -4.00 _ 4.00xl.D5 T-N0013 0 4.001U X-N00:13 0 2.00 SECONDS

HO M3

I1 k,4 Fig. 23 Amplitude Frequency : : 4.0 m 1.0 rad/s \\-\\N\N-vi,/,/////,/ 0 10 ₂₀

(66)

Mooning Line Dynamics

Amplitude : 4.0 m Frequency : 1.5 rad/s 4.00x1.05 T-NODE2 0 Z-N002 0 4.00 Z-NODE10 0 4.00 _ 4.00 _ X-NOCE10 0 00 _ -NODE13 4.00152,5 T-NODE13 0 2.00 _ SECONDS

f\1\

HOEM

1

10 20

(67)

Report No. 45064-2-RD

Amplitude : 4.0 m Frequency : 2.0 rad/s 4.00xLC' T-NODE2 0 -4.00x 2.00 Z-NOCE2 0

rr

4.00

-Z-NODE10 0 -4.00 -4.00 -X-NODE10 0 -4.00 _ 4.00xLC' 1 -NODE13 0 4.00xLC' 2.00 X-NODE13 0 -2.00 _ HCONDS

KOHM

Fig. 25

(68)

Report No. 45064-2-RD

Amplitude : 1.0 m Frequency : 1.0 rad/s 4.00x1.05 T-NODE2 0 Z-NODE2 0 -2.00 4.00 _ Z-NODE10 0 -4.00 _ 4.00 _ X-NODE10 0 -4.00 _ T-NODE13 0 -4.00xL05 2.00 X-N00713 0

HOE

Fig. 26

-2.00 _ SECONDS

(69)

Report No. 45064-2-RD

T-NCDE2 0

(-\\

4. 00K 2.00 Z-N0:E2 0

it

-2.00 4.00 _ Z-NODE10 0 4.00 4.00 _ X-NODE10 0

-4.00

_ 4.00xLe T-NODE13 0

-4.00*LL'

2.00 _ -NOGE13 0 2.00 SECONDS

Amplitude : 1.0 m

HOMM

Fig. 27

(70)

Report No. 45064-2-RD

Amplitude : 1.0 m Frequency : 2.0 rad/s 4.00x05 T -N00E2 0 -4.00xL05 Z-NO0E2 0 -2.00 4.00 _ Z-N0010 0 -4.00 4.00 _ X-N00:10 0 -4.00 _ 4.00,f1.0' T-N00E13 0 2.00. X-NODE13 0