THE MOTIONS OF A MOORED SHIP IN WAVES

DR. IR. G. VAN OORTMERSSEN

PUBLICATION No. 510 NETHERLANDS SHIP MODEL BASIN WAGENINGEN, THE NETHERLANDS

THE MOTIONS OF A MOORED SHIP IN WAVES

DR. IR. G. VAN OORTMERSSEN

PUBLICATION No. 510 NETHERLANDS SHIP MODEL BASIN WAGENINGEN, THE NETHERLANDS

CONTENTS

1. INTRODUCTION 2. POTENTIAL THEORY DESCRIPTION IN THE FREQUENCY DOMAIN

2.1. A general hydrodynamic approach to harmonic ship

motions

2.2. The equations of motion in the frequency domain

2.3. The determination of the velocity potential 2.4. The potential for a ship along a quay

3. WAVE EXCITED FORCES AND RMRODYNAMIC COEFFICIENTS

3.1. Numerical calculations with the 3-dimensional

source technique

3.2. Experimental verification

3.3. Discussion of the results

3.4. The influence of the water depth on added mass

and damping

3.5. The influence of a quay parallel to the ship

on added mass and damping 4. EQUATIONS OF MOTION IN THE TIME DOMAIN

4.1. Potential theory description for flow due to

arbitrary ship motions

4.2. ~quations of motion in the time domain 4.3. Relation between equations in the time and

frequency domain 4.4. The behaviour of the damping for high frequency

motions 4.5. Numerical computations of retardation functions

and constant inertia coefficients 5. THE APPLICATION OF THE EQUATIONS OF MOTION IN THE

TIME DOMAIN

5.1. General

5.2. Numerical calculations

5.3. Examples of computed moored ship motions and

experimental verification

5.4. Analysis of the results

5.5. Extension to other systems

6. CONCLUSIONS

APPENDIX I Spectral analysis of irregular signals 118

APPENDIX I1 Particular solutions of an equation of motion

with non-linear, asymmetric restoring force 121

REFERENCES

NOMENCLATURE

SUMMARY

CHAPTER 1

INTRODUCTION

Up till a few decades ago, the mooring of ships has been

mainly a matter of practical experience. Ships were moored in

harbours or sheltered areas only, where the external forces are

in general limited to the rather steady current and wind forces.

In a few harbours, as for instance Long Beach and Cape Town,

with an open connection to the sea, difficulties were encountered

with moored ships. Sometimes moored ships showed erratic motions

and even mooring line failures occurred in apparently smooth

weather conditions. Such troubles may be caused by harbour reso-

nance phenomena or seiches, exerting forces on the ship which,

although they are small, result in large motion amplitudes

because of their low frequency, which is close to the natural

frequency of the moored ship. These problems were the challenge

for investigators as Basil W. Wilson 1 to attempt to describe

mathematically the dynamic behaviour of a ship, moored in waves.

With the development of the ocean industry and the advent of

very large ships, which can only be accommodated in a few har-

bours with sufficient water depth, the need arose to moor ships

in exposed areas. To this purpose special mooring facilities were

designed to absorb the loads exerted by the environment on the

moored ship. Nowadays a variety of mooring arrangements is in

operation. The appropriate system for a particular mooring de-

pends on water depth, weather conditons, ship size and the allow-

able motions of the moored ship.

In general it can be stated, that the magnitude of the loads

to be absorbed by the mooring system are the lower, the more

freedom the ship is left to move. How much the ship is allowed to move depends on the nature of the operation which has to take

place during the mooring.

Transfer of fluid cargo, for example, can be done with

floating flexible hoses, so that no stringent requirements have

to be put on the restraint of the ship. Therefore, a frequently

applied type of mooring arrangement for oil terminals is the

single point mooring system: the ship is attached to a buoy or

an articulated or fixed tower by means of a single bowhawser.

The ship has much freedom to move and will take an average equi-

librium position in which the external loads are relatively

small.

Sometimes a spread mooring system is used, in which the ship

is moored by means of a symmetric system of mooring lines, with

or without buoys. A disadvantage of this system is that the ship

is not free to rotate to find a favourable heading with respect

to waves, wind and current.

If the allowable motions of the ship are small, for instance

when cargo has to be loaded or discharged by means of land based

cranes, the ship is usually moored to a jetty by means of mooring

lines and fenders.

Other arraagements of interest are: ships moored to rotating

floating piers, or to other ships or storage vessels. In these

cases as well as for the single or multiple buoy systems the

loads in the mooring lines are determined by the motions of the

moored ship as well as those of the floating body or bodies to

which she is moored.

Because of the short history and fast development of mooring

in exposed areas, the design of terminals can not be based on

empirism. On the other hand the problem is too complicated for

an analytical treatment. Therefore it is common practice to study

the behaviour of a moored ship by means of experiments with small

scale models. Although model testing provides an effective tool

to determine mooring forces and maximum motions of the moored

ship for design purposes, this method inheres a few drawbacks.

First, model tests are expensive and time consuming. The

test set-up is complicated, it is essential that elasticity prop-

erties of mooring lines and fenders are simulated very carefully,

and sophisticated facilities are needed to simulate the relevant

environmental conditions. For this reason test programs are usu-

ally restricted to final design configurations and selected

weather conditions which are assumed to be the most critical.

Further, the fundamental insight gained from model tests on

these complicated systems is limited. Only the resulting output

is measured without learning much of the mechanism which causes

this output. As an example the low frequency motion of a moored

ship observed in tests in irregular waves may be mentioned. Some

investigators believe that it is caused by second order wave for-

ces, others think that non-linearities in the elasticity of moor-

ing lines are the reason for it, while some assume that it is a

free vibration caused by transient phenomena. A definite answer

is hard to give, since for instance first and second order wave

forces can not be separated in a model test. Therefore it would

be helpful to have the disposal of a computer-based simulation

method, which gives more flexibility in this respect.

This thesis will be devoted to the formulation of a mathe-

matical model for the prediction of the behaviour of a moored

ship in irregular waves in a purely theoretical way, which can

be used for practical calculation on a computer. The study will

be restricted to systems with six degrees of freedom, which means

in practical terms that the ship is moored by means of mooring

lines and fenders to a rigid structure (jetty), while the elastic

characteristics of the mooring system may be non-linear and

asymmetric.

It is not the intention to be complete, since each model,

however complex it may be, can not be more than a poor reflection

of nature, and a very complicated mathematical formulation does

not necessarily learn us more than a more approximate one, of

which the solution is feasible. Important is, however, that the

model reflects the typical behaviour of moored ships and can be

extended and adapted whenever this appears desirable from com-

parisons with model experiments or prototype observations.

Although in first instance the excitation of the ship will

be restricted to linear forces due to long crested waves, other

external forces such as wind and current forces and loads in-

duced by passing ships can be incorporated in the model as well.

The basis of the equations of motion is the law of dynamics

of Newton:

or, since the inertia m of the ship may be regarded as constant:

The external force F is composed of

- arbitrarily in time varying forces due to the waves; - hydrodynamic and hydrostatic restoring forces, which are a function of the motions of the ship;

- restoring forces due to the mooring system, which are a func- tion of the instantaneous position of the ship.

In the classical ship motion theory, it is common practice to

formulate the equations as follows:

a, b and c are coefficients which describe the hydrodynamic and

hydrostatic restoring forces.

In fact, (1.3) is not a real equation of motion, in the

sense that it relates the instantaneous motion variables to the

instantaneous value of the exciting forces. It can only be used

as a description in the frequency domain of a steady oscillatory

motion, since the hydrodynamic coefficients a and b depend on the

frequency of motion.

Analytical work on the moored ship problem published so far

has been based on equation (1.3), where three categories can be

discerned with regard to the simplifying assumptions made.

Some investigators, as for instance Kaplan and Putz [l-21,

Leendertse [l-31, Muga [l-41 and Seidl [l-51 linearized the elas-

ticity characteristics of the mooring system. The restoring

forces of the mooring aids can then be incorporated in the hydro-

static term cx and the equations (1.3) of motion in the frequency

domain can be solved easily, with the restriction that only har-

monic excitations can be used.

Others, as Abramson and Wilson [l-1, 1-61 , Yang 11-71 and Kilner [l-81 add non-linear terms to equation (1.3) to account

for the restoring forces of the mooring system, and solve the

equations by means of the method of equivalent linearization,

assuming that the excitation is pure sinusoidal and that, as in

the earlier mentioned method, the response of the ship is simple

harmonic too, with a frequency equal to that of the excitation.

This is not realistic, since observations both in model and full

scale situations have revealed that also other modes of motion

may occur.

The work of Wilson and Awadalla [l-9, 1-10] , Lean [l-111 , Wilson [l-121 and Bomze [l-131 belongs to the third category,

which is characterized by the assumption that the hydrodynamic

coefficients a and b in equation (1.3) are independent of the

frequency, so that this equation is regarded as an actual differ-

ential equation. The solution, which is found either by approxi-

mate analytical methods (ref. [l-91 , [l-14 ) or by finite differ-

ence integration in the time domain (ref. [l-101 , @-l21 and

[l-13]), may contain components with frequencies lower (subhar-

monic) or higher (superharmonic) than that of the forcing func-

tion.

It will be shown in chapter 3 that the assumption of con-

stant hydrodynamic coefficients can not be justified: especially

in shallow water these coefficients appear to be very sensitive

to changes in frequency. Consequently, a time-domain description

of the behaviour of the moored ship is needed which takes into

account the frequency dependency of the fluid reaction forces.

A possible approach would be to use the impulse response

function technique. If for any linear system the response R(t)

to a unit impulse is known, then the response of the system to

an arbitrary force F(t) is:

The moored ship as a whole may, of course, not be thought of as

a linear system, but this difficulty can be overcome by isolating

the free floating ship in still water, for which system the

assumption of linearity holds true, as long as the motions re-

main small. The non-linear mooring forces can be incorporated in

the external forcing function. A drawback of this impulse re-

sponse function is, that it relates the input and output signals

of the system without reflecting the physical processes behind

it, the system of ship-fluid interaction is regarded as a black

box.

In this thesis it is therefore proposed to use the equations

of motion in the time domain as they have first been formulated

by Cummins [l-141 , and which can be considered as true differen-

tial equations; they give the instantaneous relationship between

the motion variables and the external forces. In these equations

the various factors governing the response of the ship are sepa-

rated into clearly identifiable units.

The only assumption involved is linearity of the hydrody-

namic restoring forces. Non-linear and asymmetric mooring char-

acteristics can be dealt with, and the exciting force may be

arbitrary, which means that besides first order wave forces also

slowly varying drift forces and wind- and current forces can be

included in the forcing function, although in this thesis only

the problem of first order wave forces will be discussed.

In the equations of motion in the time domain the hydrody-

namic forces are expressed by constants and functions. It is not

feasible to obtain these directly from the potential theory.

However, there is a theoretical relationship between the equa-

tions in the time domain and those in the frequency domain, and

therefore the second chapter will start with the potential theory

description of harmonic ship motions at zero speed in waves in

shallow water. Since terminals are very often located in shallow

water, it is essential to take into account the effect of the

nearness of the sea bottom on the wave forces and the hydrodynam-

ic coefficients. Also the effect of a quay parallel to the ship

will be included. Since the two-dimensional strip theory, which

is widely used in naval hydrodynamics, is not applicable to the

case of small underkeel clearance, the three-dimensional source

technique will be discussed as a tool to compute wave excited V

forces as well as hydrodynamic restoring forces.

In chapter 3 numerical results will be presented of calcula-

tions of the wave forces and hydrodynamic coefficients of a large

tanker, together with experimentally determined values. From

these data the accuracy and limits of applicability of the cal-

culation method will be discussed.

Chapter 4 deals with the equations of motion in the time domain and their relation with the description in the frequency

regime. Further it will be shown in that chapter how the unknown

coefficients and functions in the equations can be derived nu-

merically from added mass and damping data.

The numerical solution of the set of equations of motion in

the time domain, applied to the moored ship problem, will be dis-

cussed in chapter 5. Example computations will'be given for a

I large tanker moored in oblique, beam or head seas to a jetty, and

the results will be analysed and compared with measurements, ob-

tained from model experiments.

Finally, a review of the main conclusions is given in chap-

ter 6.

CHAPTER 2

POTENTIAL THEORY DESCRIPTION IN THE FREQUENCY DOMAIN

2.1. A general hydrodynamic approach to harmonic ship motions

In this chapter a general formulation of the ship motion

problem will be presented in the frequency domain. This general

formulation, in terms of the linear potential theory, is valid

for deep as well as for shallow water.

A complete theoretical derivation of the formulae will not

be given. This has been done before by other authors, and for a

thorough mathematical treatment reference is made to their work

(see for instance Wehausen and Laitone 12-11, John [2-21, Tuck

[2-31 ). Here, only a summary of the most important equations will

be given, which are needed for the derivation of the equation of

motion and for the numerical computation of the wave exciting

forces and the fluid reactive forces.

Since the aim is to formulate a mathematical model for the

motions of a moored ship, the analysis can be restricted to the

case of zero forward speed, which means a considerable simplifi-

cation of the problem.

In this-section will be dealt with pure harmonic motions

and therefore the mooring system will be left out of considera-

tion.

The ship is considered as a rigid body, oscillating sinus-

oidally about a state of rest, in response to excitation by a

long crested regular wave. The amplitudes of the motions of the

ship as well as of the wave are supposed to be small while the

fluid is assumed to be ideal and irrotational.

In the theory of hydrodynamics it is common practice to

define a system of axes with the origin in the free surface.

For the description of ship motions, however, it is more conven-

ient to use the centre of gravity of the ship as a reference

point. Therefore, to prevent transformations from one system into

the other, the description will be given here in a right handed,

space fixed coordinate system as shown in Figure 2.1.

The oscillating motion of the ship in the jth mode is given by:

WAVE DIRECTION

Figure 3. l. The coordinate system.

in which 5 is the amplitude of motion in the jth mode and td j

the circular frequency.

The motion variables xl, x2 and x3 stand for the transla-

tions, surge, sway and heave, while x4, x5 and x6 denote rota-

tions around the GX1, GX2 and GX3 axis respectively.

In naval hydrodynamics, it is usual to introduce a set of

three independent angular displaceme'nts, the so-called Eulerian

angles: yawing, being about the absolutely vertical axis GX3,

pitching, around the rotated position of the GX2 axis, which

remains in the horizontal plane, and rolling, about the positon

of the GX1 axis after the previous two rotations. Since only

small motion amplitudes are considered, these Eulerian angles

coincide with the angular displacements about the space fixed

axes (see Vugts [2-41).

The free surface at great distance from the ship is defined

by:

~ K ( x ~ COS a + x2 sin a)-iwt x , e c = To

where CO = amplitude of the wave

K = wave number = 2n/h, where A is the

wave length

cr = angle of incidence

The flow field can be characterized by a velocity potential

The potential function Q can be separated into contribu-

tions from all modes of motion and from the incident and diffrac-

ted wave fields.

Following Tuck [2-3) a convenient formulation is obtained

when writing

with the convention S7 = 50. The case j = o corresponds to the incident wave potential:

S@cosh 1 ' ( ~ 3 + C' ir (xl cos a. + x2 sin a) 'P0 = -

V cosh K d e

2 in which v = w /g

c = the distance from the origin to the sea bed

d = water depth

The relation between the wave length and the wave frequency

is given by the dispersion equation, which follows from the free

surface condition: 'I

v = K tanh K d

The cases j = 1, 2, ..., 6 correspond to the potentials due to the motion of the ship in the jth mode, while 'P7 is the

potential of the diffracted waves.

The individual potentials are all solutions of the Laplace

equation

2 a2 a 2 a2 v 'P. = (- + - 3 2 + - ) Q . = 0 axl ax, axi 3

while the following conditions must be satisfied:

- the linearized free surface condition

- the boundary condition on the sea floor

- and the boundary condition on the ship's surface. Due to the linearization, this boundary condition may be applied to the

surface S in its equilibrium position.

in which nl through n6 are the generalized direction cosines on

S, defined by:

n = cos (n, 1 n2 = COS (n, X2) n3 = COS (n,

n4 = x2n3 - X3n2 n 5 = x n 3 1 - X1n3 n6 = x1n2 - X2"1

For the incident and the diffracted waves one finds:

When defining:

then (2.11) may simply be written as:

The potentials v1 through q7 should satisfy moreover the

boundary condition at infinity, the radiation condition, which

states

Supposing that the unknown potentials q l through v7 can be determined, the pressure on the surface S can be found from

Bernoulli's theorem. The linearized hydrodynamic pressure is

given by:

The oscillating hydrodynamic forces (k = 1, 2, 3) and

moments (k = 4, 5, 6) in the kth direction are:

Now the following matrix is defined:

This matrix can be considered as a transfer function which trans-

forms motion variables into force components (see Ogilvie and

Tuck [2-S] ) .

Equations (2.18) can then be written as:

From (2.19) and (2.15) it follows that:

2 aqk k j

'P. dS T = - P w

hence:

The surface S can be closed by adding the free surface, the

sea bottom and a vertical surface at infinity.

For the cases k, j = 1, 2, ....., 7 the integrand vanishes on the extra surfaces, so for these cases we may apply Green's

theorem, which results in:

For k, j = 1, 2, ....., 6 this relation means that a force in the kth mode due to the motion in the jth mode is equal to

the force in the jth mode due to motion in the kth mode.

Similar relations for the ship with forward speed have been de-

rived by Timman and Newman 12-61 and Newman [2-71. For j = 7,

equation (2.23) yields the Haskind relation:

The total wave exciting force amplitude per unit wave

amplitude is given by:

2.2. The equations of motion in the frequency domain

The differential equations which describe the motions of the

free floating ship in response to simple harmonic waves can be

found from Newton's law of dynamics:

Xk, the total external force in the kth mode, consists of hydro-

static and hydrodynamic restoring forces and of wave exciting forces.

%j is an inertia matrix. Since the origin of the system of

axes coincides with the centre of gravity of the ship in its

rest position, it is found that

where m = mass of the ship

Ik = moment of inertia in the kth mode

I = product of inertia k j

The rate at which the mass of the ship changes is very

small, and its influence may be neglected when considering the

motions during a period of time which is small compared to the time required to load or discharge the ship.

Therefore, one may write for the steady oscillating condition

Ckj is a matrix of restoring force coefficients. Besides the

hydrostatic restoring forces, Ckj may also include restoring

forces due to a mooring system, as long as this mooring system

has linear load-excursion characteristics. For the hydrodynamic

force Fk in equation (2.28) use can be made of expression (2.20).

Hence

It is common practice to separate T for k, j = 1, 2, ....., k j

6

into real and imaginary parts

The real parts a are called added mass coefficients, the imag- k j

inary parts damping coefficients. When using these quantities,

the following real representation of the equations of motion is

obtained:

6 2 E {-m + a sin (wt + E.) + b w cos (wt + E.) j=1 k j J k j J

+ C sin (wt + E . ) ~ C = Xk sin (wt + fik) k j I j

in which Xk = wave excited force in the kth mode c j , fik = phase angles.

Thus a set of equations has been obtained which are not

real equations of motion, but merely a set of algebraic equa-

tions, fixing the amplitudes and phases of the six oscillations

of the ship under the action of a train of regular sine waves

at one specific frequency.

2.3. The determination of the velocity potential

In the foregoing sections it has been shown that the flow

around the ship is completely defined by the velocity potential.

Once this potential function is known, the wave exciting forces,

hydrodynamic forces and unrestrained ship motions in harmonic

waves can be obtained easily.

For the computation of deep water ship motions, the so-

called strip theory, which was first formulated by Korvin-

Kroukovsky and Jacobs [2-81, has proven its usefulness. This

theory takes advantage of the fact that for ships the longitudi-

nal dimension is large relative to the lateral and vertical di-

mensions. For such a slender body the three-dimensional problem

can be reduced successfully to a local two-dimensional problem.

After its presentation, the method has been refined by many au- thors and the results are in general reliable. A drawback is,

however, that no information can be obtained about the surge mode.

Unlike the deep water problem, very few studies have been

presented on the motions of a ship in shallow water.

Kim [2-91 has adapted the strip theory for a restricted wa-

ter depth. For the vertical modes of motion this approach yields

useful results, but it can not be used for lateral motions,

especially in the lower frequency range, since the strip theory

is basically two-dimensional, requiring that the flow of water

passes entirely underneath the keel of the ship. In shallow

water, however, three-dimensional effects become important:

the water flows partly underneath the ship and partly around the

ends. In the extreme case, the ship sitting on bottom, water can

move only around the ends of the ship.

A three-dimensional approach has been presented by Newman

12-101 but he neglected the effect of the free water surface.

A very interesting contribution to the shallow water problem

has been given by Tuck (see Tuck [2-31, Tuck and Taylor [2-111

and Beck and Tuck [2-121. He has derived an approximate solution

of the linear velocity potential for the case that the wave

length is much greater than the depth of the water. Application

of this theory is therefore restricted to long waves and low

frequency motions, which poses, especially for mooring problems,

a rather stringent limitation.

A method of solution, of which the validity is unrestricted

as long as linearity is ascertained, is provided by the three-

dimensional source technique. This technique has been applied

successfully during the last few years for the computation of

wave loads on large offshore structures (see for instance Daubert

[z-131 , Garisson and Chow [2-141 , Van Oortmerssen [2-151 , Boreel 12-16] , but it can be used just as well for ship shaped bodies.

According to Lamb [2-171 the potential function rp . can be 3

represented by a continuous distribution of single sources on a

boundary surface S:

where Y . (xl, x2, x3, al, a2, a3) 3 = the Green's function of a source, singular in

alt aZ1 a3 = the vector, describing S

= the complex source strength

For the Green's function y we can choose either a complicat-

ed function, being the solution of the Laplace equation which ) satisfies the boundary conditions at the sea bottom, in the free

surface and at infinity, or the simple fundamental source func-

tion l/r for an unbounded fluid. In the latter case the function

o is such, that rp satisfies all the boundary conditions. The j

boundary surface S consists of the ship's surface, the free wa-

ter surface, the sea bottom and, to close the surface, a cylin-

drical vertical surface at great distance of the ship. By apply-

ing Green's theorem to this boundary surface, an integral equa-

tion is obtained for the unknown function cp in terms of its j

boundary values and its normal derivative on the boundary. By

the method of discretization this integral equation can be re-

duced to a set of linear algebraic equations with the unknowns

being the values of the source strength function U at a discrete

set of control points along the boundary. For more details of

this method reference is made to the work of Yeung [2-181 . A

drawback of this method is that the number of elements, required

to schematize the entire boundary surface, is very large, which

results in an evenly great number of equations to be evaluated

numerically. A favourable feature is the possibility to take the

bottom topology into account, while the approach can be extended

to the case of a fluid domain of finite extent, for example a

canal or a basin. In the present work the other approach will be used, in oth-

er words a Green's function of more complicated form will be ap-

plied on the surface of the ship only. The water is assumed to

have a constant depth.

The Green's function of a source, singular in (al, a2, a3)

which satisfies the boundary conditions in the free surface, on

the sea bottom and at infinity, is given by (see Wehausen and

Laitone [2-l] ) :

0 2 2a (K2 - v cosh "(a3 + c) cosh K(X~ + C) + i 2 2 . JO(~R) (2.33)

~ d - v d - v

in which r = /Ixl - al)2 + (x2 - a2) + (x3 - a3) 2 ' rl = J(xl - al)2 + (x2 - a212 + (x3 + 2c + a3) 2'

R = J(xl - a1)2 + (x2 - a2) 2'

John [2-21 has derived the following series for y, which is the

analogue of (2.33) :

KZ - v 2 y = 2s cosh ~ ( a ~ + C) cosh K(X~ + C )

K2d - v2d + v . {Yo(~R) - i Jo (<R) J

2 2 m 4(ui + V )

+ C . cos pi(x3 + C ) COS ui(a3 + C) % (uiRl i=l dpi + dv2 - v (2.34)

where pi are the positive solutions of:

pi tan (pia) + V = o

Although these two representations are equivalent, one of the

two may have preference for numerical computations, depending on

the values of -the variables. In general, equation (2.34) is the

most convenient representation for calculations, but when R=O

the value of KO becomes infinite, and therefore equation (2.331

must he used when R is small or zero.

The unknown source strength function U must be determined

such, that the boundary condition on the ship's surface S is

fulfilled:

To solve equation (2.36) numerically, the surface S is subdi-

vided into a number of finite, plane elements. The boundary

condition will be applied in one control point on each element,

being the centre of the element. In each of the control points,

the potential is given by:

Where: ipjm is the value of Q in the mth control point. j

U is the source strength on the nth element. jn

Yrn is the contribution of the nth element to the Green's

function in the mth control point.

ASn is the area of the nth element.

Integral equation (2.36) now reduces to a set of N algebraic

equations for the N unknown source strengths:

in which n is the value of n. in the centre of the mth element. jm 3

Equation (2.38) can be written simply as:

For each of the modes j = 1, 2, ...... 7 this formula represents a set of N complex linear equations in the N unknown source

strengths. The unknown source strengths can be found by inver-

sion of the complex valued N X N matrix A:

The complex matrix A can be transferred into a real 2N X 2N ma-

trix, which can be inverted with standard numerical routines.

In general, the Green's function ynmmay be computed with

sufficient accuracy as if the source strength is concentrated in

the centre of the element. There are, however, two notable ex-

ceptions to this rule. First, a difficulty is encountered when

evaluating yrn and - when n = m. In that case, ynm has a an singularity of the type l/r, as can be seen from (2.33). This

singularity can be removed by spreading the source uniformly

over the facet. The total Green's function can be split up in a

regular and a singular part:

Where y ' is the regular part, which can be computed without dif-

ficulties when r = 0. The potential and its normal derivative

due to the singular part can be found as follows.

Consider the potential 9' at P in Figure 2.2 due to a source of

unit strength spread over the plane area AS. The normal from P meets AS at Q, where the distance

PQ = e. The distance from Q to the perimeter of AS is f (Q) , a function of the angular position Q. Then:

1 1 9' = ?;I F 'IS

AS

Figure 2.2.

In the limit as e+O it is found that:

For a circular facet, this results in:

'P' = n

For the rectangular element:

where :

'P' = Ql%@

l + E ) {ln(q+@Z) + q l n Q = - l m

in which q is the aspect ratio of the rectangle.

Figure 2.3 shows the value of Ql on a base of the aspect ratio.

The normal derivative of the potential in P becomes:

Here the limit for e+O is

It is remarkable that, unlike the potential itself, the value

of the normal derivative is independent of the shape of the

surface element.

Q 1

A second case which gives ,,00

some trouble is when a surface

element is situated close to

the sea bottom, for instance

the bottom elements of a

O '9.0 2.0 3 o tion of the reflected element

9 to the potential is given by

(2.411, in which e is now Figure 2 . 3 . Correction factor

a constant, being the distance for rectangular facets as a

from the element to its re- function of the aspect ratio

flection. For a circular g.

element f(8) = a = radius of the element and consequently (2.41)

yields

ship with small-underkeel

clearance. For this case, 0.95

P in Figure 2.2 represents

the point where the potential

is evaluated, while Q is the

centre of the reflected sur-

face element. The contribu-

*

This result means, that when the source strength is taken as con-

centrated in the centre of the element, a correction has to be

applied, which amounts to:

(2.48)

l + - + l

for a circular element. a'' from (2.451, In the same way a correction factor is found for an

being :

a The factors n2 and Q3 are shown in Figure 2.4 on a basis of g. From this Figure it appears, that the influence becomes signifi-

cant when 2 = 0.25, or, when the keel clearance equals the ele-

ment size. In case of non-circular elements the integration of

equations (2.41) and (2.45) has to be done in a numerical way.

When, after solving the set of equations (2.38) the source

strengths are known, the wave exciting forces and added mass and

damping coefficients are found using equations (2.20) and (2.30).

Finally, the motions of the free floating ship can be cal-

culated from the equations of motion in the fie-quency domain

Figure

factor

2.4. Correction

for image facets.

2.4. The potential for a ship along a quay

So far the hydrodynamics of a ship at zero forward speed in

shallow, but otherwise unrestricted water have been considered.

The method of solving the velocity potential can, however, easily

be extended to the case of a ship near a vertical wall. It is ob-

vious that the influence of a quay is present in a lot of mooring

situations. ---_ r-- . IMAGE ,

\ --C-----

7 2-__-__________--/

M'

Figure 2.5. Definition sketch.

Consider a vertical quay near the ship, defined as x2 = h,

as shown in Figure 2.5. For sake of simplicity the wall is chosen

parallel to the ship, which covers most practical cases, although

a wall which makes an angle with the GX1-axis, can be tackled in

the same way.

The velocity potential for this case can be found by the

method of images. The ship is reflected with respect to the wall.

For the numerical calculation this means that the number of sur-

face elements is doubled, while the number of unknown source

strengths remains the same, due to the symmetry of the flow field.

To the Green's function of each facet the Green's function of its

reflection has to be added:

The incoming wave will also be disturbed by the wall. The

problem of progressing waves in an infinite ocean bounded on one

side by a vertical wall when the wave crests at infinity may make

any angle with the shore line has been treated by Stoker [2-191 . It should be emphasized that he was not able to decide whether

the waves are reflected back to infinity from the shore, and if

so, to what extent. From observations, however, it is known that

a wave is reflected by a vertical wall, and by adding the image

in the plane x2 = h to the incoming progressive wave, a wave system is obtained which fulfils the boundary condition at the

quay (see Lamb 12-17] ) :

1 ~(~3'~) ~K(X~COS a + x2sin a) + V,,=-5 V 0 cosh ~d { e

~K(X COS a + 2h sin a - x2sin a)) + e 1

In most practical situations a quay forms part of a harbour

basin and therefore the undisturbed wave will in general be a

much more complicated system of progressive and standing waves.

CHAPTER 3

WAVE EXCITED FORCES AND HYDRODYNAMIC COEFFICIENTS

3.1. Numerical calculations with the three-dimensional source

tech?ique

With the three-dimensional source technique described in

2.3, the wave excited forces, hydrodynamic coefficients and free

floating ship motions were computed numerically for a large

tanker of the 200,000 tdw class. For these computations a FORTRAN

program was used on a Control Data 6600 computer.

Particulars of the ship are listed in Table 3.1 while a

small scale body plan is given in Figure 3.1.

TABLE 3.1.

Main dimensions 200,000 tdw tanker.

Length between perpendiculars 310.- m

Breadth 47.20 m

Draft 18.90 m

Volume of displacement 235,000 m 3

Block coefficient 0.85

Midship section coefficient 0.995

Prismatic coefficient 0.855

Distance of centre of gravity to midship section 6.61 m

Height of centre of gravity 13.32 m

Metacentric height 5.78 m

Longitudinal radius of gyration 77.50 m

Transverse radius of gyration ' 17.00 m

In the calculations, the ship had to be represented by means

of a composition of flat surface elements. For a,ship shaped

body, it is rather obvious to use triangular and quadrangular

elements. For a proper choice of the number of surface elements

the following considerations must be kept in mind.

Figure 3.1. Small scale body plan.

First, as is quite clear, the accuracy of the results will

increase with increasing number of elements, since smaller

elements can describe the curved geometry of the hull better,

and also because a finer distribution of sources will approximate

the pressure gradient along the hull more accurately.

The accuracy of computation will depend also on the frequen-

cy considered: for short waves (or high frequency motions) more elements are required than for long waves (or low frequency mo-

tions). Although it is difficult to predict a minimum acceptable

ratio of wave length to element size, it may be expected that

appreciable errors will occur when the wave length becomes small-

er than five times the length of a surface element. Due to the

size of the ship, the range of frequencies which are of practical

interest, is confined to O<w<0.8 rad sec-'. Therefore, the length

of an element should not be larger than 17 metres.

Further, it is desirable to prevent large variations in the

size of elements, and to keep the aspect ratio as close to 1

(which means a square shape) as possible.

Finally, the computing time increases progressively with

the number of elements. For this reason, the number of elements

should not exceed 200.

Between these conflicting requirements a compromise has

been sought applying a number of l60 elements. The subdivision

in facets of the ship's hull is shown in Figure 3.2. As can be

seen from a comparison with Figure 3.1 the shape of the sections

has been simplified considerably. The bilge radius was neglected

at all, and the rudder was deleted.

Computations of hydrodynamic coefficients and wave forces

and motions were carried out for a water depthjdraft ratio 6 - 1 amounting to 1.2, frequencies ranging from 0.07 to 8.0 rad.sec

and wave directions of 180, 225 and 270 degrees. Also a series of

calculations was performed with a wall parallel to the ship at

a distance of 16.50 m to the ship's side, representing a solid

jetty.

The results are presented and discussed in Section 3.3.

3.2. Experimental verification

In order to verify the numerical results obtained with the

three-dimensional source technique, a series of model experiments

has been carried out in the Shallow Water Laboratory of the

Netherlands Ship Model Basin. This experimental basin measures

210 metres in length and 15-75 metres in width, while the water

depth is variable with a maximum of 1.00 metres. At one end of

the basin a paddle type wave maker is installed, capable to

generate regular as well as irregular waves, while the beach at

the other end can be adjusted in height to match the water depth.

The towing carriage, which accommodates the measuring and

recording equipment, was positioned halfway the basin for these

tests.

The ship model was made of wood to a linear scale ratio of

1:82.5 according to the lines shown in Figure 3.1. Rudder, pro-

peller'and bilge keels were omitted on the model.

The experiments comprised measurements of wave forces and

motions, and forced oscillation tests to determine the hydrody-

namic coefficients.

The tests were carried out in accordance with Froude's law

of similitude.

The wave loads and motions were measured in regular waves.

The undisturbed waves were recorded before the start of the

tests at the position of the centre of gravity of the model by

means of a wave probe of the resistance type.

During the tests the measured signals were recorded simul-

taneously and continuously on ultra violet paper chart and on

analogue magnetic tape. The evaluation of the results was carried

out on a computer. TO this end, the analogue records were con-

verted into digital records by taking samples at time increments

corresponding to 0.28 seconds on prototype scale.

During the wave force measurements the ship model was

mounted to a 6-component dynamometer, which was mounted rigidly

to the towing carriage as shown in Figure 3.3.

Figure 3.3. 6-component measuring rig.

The dynamometer essentially consisted of two large frames, con-

nected by means of 6 force transducers of the strain-gauge type.

Different angles of wave attack were established by rotating

the ship model around its centre of gravity.

The signals of the 6 measured forces were transformed on the

computer into 3 forces and 3 moments in the space-fixed system of

axes as defined in Figure 2.1. From those new signals, the ampli-

tudes of the first harmonic component were obtained. Also a har-

monic analysis was performed on the signal of the incident wave,

and the transfer functions were found by dividing the first har-

monic components of wave forces and moments by that of the wave

elevation.

During the measurements of ship motions, the ship model was

kept in its position by means of two long steel rods, which acted

as soft springs, soft enough not to influence the motions in the

range of wave frequencies applied.

Inertial and stability properties of the model were adjusted

according to the values stated in Table 3.1.

The rotative motions were measured by means of gyroscopes,

translations by means of a pantograph, which instrument trans-

forms a translatory motion into a rotation of a potentiometer. A

sketch of the test set-up is given in Figure 3.4.

Figure 3.4. Test set-up for motion measurements.

Again, a harmonic analysis was performed on the measured

signals, and only first harmonics were used for further analysis.

To determine the hydrodynamic coefficients (added mass and

damping) the ship model was forced to oscillate with a prescribed

amplitude and frequency in one of the six modes, by means of an

excitator, which main component is a Scotch-yoke mechanism. This

excitator is equipped with two legs, spaced 1.00 metre apart,

which can both perform a harmonically oscillating translatory

motion either in phase, or with a certain phase difference. The

motions of both legs are measured by means of potentiometers. An electronic control system is used to keep the number of revolu-

tions of the driving motor constant, which is necessary to a-

chieve a pure harmonic motion.

A six component force transducer was mounted between the

model and the legs of the model excitator. A sketch of this

transducer is shown in Figure 3 . 5 .

Figure 3 . 5 . 6-component transducer for oscillation tests.

Six forces are measured by means of strain gauge transducers.

The test set-ups for the oscillations in the various modes l

are depicted in Figures 3 .6 through 3 . 9 .

The model was always placed in the basin in such a position

that the generated waves traveled as much as possible in the

longitudinal direction, to minimize the influence of reflected

waves from the basin's side walls.

The rotational motions yaw and pitch were achieved by ad-

justing a phase difference of 180 degrees between the motion of

the two excitator legs. The pivots were placed in such a way,

that the model rotated always around an axis through the centre

OSCILLATOR LEG 6- L S COMPOllEH+ FORCE TRANSOUCL.

CARRIAGE

-0SCILLAmR LEG

Figure 3 . 6 . Test Set-Up

for surge.

Figure 3 . 7 . Test set-up for sway and yaw.

-.-----, -- -- -& L 8 COMPONENT FORCE TRANSDUCER

Figure 3 . 8 . T e s t set-up

for heave and pitch.

Figure 3 . 9 . T e s t set-up

for roll.

of gravity. During the tests in roll direction, the model was

attached to the carriage by means of two hinges at the height

of the centre of gravity.

The records of the six measured forces were, after digi-

tation, transferred into three forces and three moments in the

space fixed coordinate system GX1X2X3. Subsequently, a harmonic

analysis was performed on these signals as well as on the motion

signal. Added mass and damping could then be obtained using the

equations of motion in the frequency-domain (2.31):

(Xk cos Ek.) /L - Ck. ak j

- W 2 l - Mlcj

X sin E ~ . - k bkj - W C

where: E is the phase lag between the force Xk and the motion k j

5 - The hydrostatic coefficients C were determined by means of

k j static measurements.

The moments of inertia in air of the model with measuring equip-

ment (Mkj) were measured by means of oscillation tests in air.

3.3. Discussion of the results

The computed transfer curves of the wave excited moments

are presented in Figures 3.13 through 3.12, together with the

experimental results.

In general, the agreement is good. The only notable excep-

tions are the surge force and pitch and yaw moments in beam waves

(see Figure 3-12), which originate from the asymmetry in the hull

shape. The discrepancy between theory and experiments can be

attributed in these cases to the simplified representation of the

hull in the computations.

The experiments were carried out in waves with amplitudes

corresponding to 1.5 metres. At frequencies U ' = 2.25 and 3.38

three wave amplitudes were applied, ranging from 1 to 2.5 me-

tres. In most cases the results obtained for these three wave

THEORETICAL

EXPERIMENTAL

Figure 3 . 1 0 . Transfer funct ions of wave excited forces and

moments ; S = 1 . 2 , a = 180 degrees.

Figure 3 . 1 1 . Transfer functions of wave excited forces and

moments; 6 = 1 . 2 , a = 225 degrees.

THEORETICAL I - EXPERIMENTAL I

Figure 3 . 1 2 . Transfer functions of wave excited forces and

moments; 6 = 1 . 2 , a = 270 degrees.

amplitudes are very close, from which it may be concluded that

the wave loads are linear up to a wave amplitude of at least

2.5 metres.

The added mass and damping coefficients, as obtained from

computations and measurements, are presented in the Figures 3.13

through 3.24 in a non-dimensional way, as defined in Table 3.2.

In general, the agreement found for the main hydrodynamic

coefficients is satisfactory. Relatively small differences as

they occur in the surge, heave and yaw modes may be explained by

the simplification of the ship's geometry. More serious are the

discrepancies found in the added mass moments of inertia in pitch

and roll mode, and in the damping coefficient in roll. A reason

for the higher measured damping coefficient in roll may be sought

in viscous effects, but no reasonable explanation can be given

for the differences between measured and computed values of a'44

and a'55, although it should be remarked that it is rather diffi-

cult to obtain reliable experimental values for these coeffi-

cients in the low frequency range. To find the total inertia

coefficient, the hydrostatic restoring moment has to be subtract-

ed from the total in-phase measured moment, according to equation 2 (3.1). The remainder is then divided by a very small value, w .

Thus, a relatively small error in the measured moment may result

in a large proportional error in the inertia coefficient.

There are a few reasons why more discrepancy may be expect-

ed in the coupling coefficients. First, the computed values will

be less accurate, since coupling effects are influenced largely

by details in the geometry, and the simplification in the shape

is therefore relatively more important. Second, the experimental

error will be larger, since the experimental coefficients are

found from small measured forces.

For the coupling coefficients, there are two sets of experi-

mental data, as found from tests in different modes. Since from

the linear theory it follows that

and bkj = bjk

the difference in the two sets of data gives an indication of the

experimental error.

Fortunately, the hydrodynamic forces due to the coupling

effects are rather small and consequently even a large inaccura-

cy in a computed coupling coefficient will not have much impact

on the computed ship motions. This may be illustrated with the

following example.

The most important discrepancy between a measured and com-

puted coupling coefficient is found in case of the coupling

between sway and yaw. Supposing that the measured values are

the most reliable, it must be concluded from Figure 3.22 that

there is a large error in the computed mass coupling coefficient

of yaw into sway at small frequencies. The resulting error,

however, in the total hydrodynamic restoring force in the sway

mode, will not exceed 7 percent in this extreme case. The motion amplitudes as used in the tests are indicated

in the Figures. In the sway and heave modes three amplitudes

have been applied, to check the linearity. From the results it

appears that the hydrodynamic coefficients are linearly depending

on the motion amplitude in the range of amplitudes tested. In the

sway mode the maximum amplitude applied corresponded to 2.475

metres, in the heave mode to 1.24 metres which is quite large

in comparison with the available underkeel clearance.

The motion transfer curves as found from theory and model

tests are presented in Figure 3.25. The agreement between experi-

ments and theory is good, except for the roll motion at reso-

nance, where viscous damping plays an important role. Since the

roll motion at resonance is overestimated by the potential theo-

ry, small humps occur at this frequency in the computed sway and I yaw transfer curves in beam seas, due to the coupling terms.

The motion measurements were performed in waves with ampli-

tudes ranging from 0.8 to 1.9 metres prototype scale. At three

frequencies, ( W ' = 1.7, 2.25 and 2.8) two wave amplitudes were

applied. Again, the results demonstrate linearity under the

circumstances considered.

Resuming the results obtained, it may be concluded that the

numerical results obtained with the three-dimensional source

technique for hydrodynamic phenomena in shallow water show a

satisfactory agreement with experimental values. Apparently, the

Figure 3.19. Coupling coefficients

of surge into heave; 6 = 1.2.

m . -

0.4

Figure 3.20. Coupling coefficients

of surge into pitch; 6 = 1.2.

0

0.2

YI . - m

- THEORETICAL

EXPERIMENTAL.

FROM HEAVE TEST

0 FROM SURGE TEST

- THEORETICAL

EXPERIMENTAL:

FROM SUROE TEST

o FROM PITCH TEST

'a

m - - n 0

* . . . -

2 " 0

0

I I p THEORETICAL I EXPERIMENTAL: 1 1 I FROM SWAY TEST 1

FROM ROLL TEST

-0.1 00, 2 4 6

Figure 3.21. Coupling coefficients

of sway into roll; 6 = 1.2.

0.2

- THEORETICAL

EXPERIMENTAL:

FROM SWAY TEST

0 FROM YAW TEST

Figure 3.22. Coupling coefficients

of sway into yaw; 6 = 1 . 2 .

EXPERIMENTAL.

FROM HEAVE TEST

0 FRDM PITCH TEST

Figure 3.23. Coupling coefficients

of heave into pitch; 6 = 1.2 . Figure 3.24. Coupling coefficients

of roll into yaw; 6 = 1.2.

rather rough subdivision of the ship's hull into 160 surface

elements is sufficient for a prediction of wave loads, hydrody-

namic coefficients and motions in the frequency range which is

of practical interest (up to a frequency U' = 4.5).

In section 2.3 it has been stated, that the strip theory

will yield wrong results in case of shallow water, in particular

for the lateral modes of motion. How serious the deficiency of

the strip theory is, may be deduced from a paper by Flagg and

Newman [3-l]. In this paper the authors present data on the two-

dimensional sway added mass coefficients for rectangular profiles

in shallow water, computed with a rigid free-surface condition,

which means that their results are solely valid for the case

w = O.For the subject ship, with 6 = 1.2, one arrives at a sway

coefficient ai2 = 11 on a basis of their data, while Figure 3.14

indicates that at zero frequency the real added mass coefficient

will lie in between 3 and 4.

TABLE 3.2.

Definition of non-dimensional added mass

and damping coefficients.

mode added mass

k z j - 1

"l; j PV k = j = 3

k = l , j = 3 - k = j = 4

5. = akj b' = bk j k = j = 5

' pVL2 kj P~L'WL k = j = 6

k = 4 , j = 6

k = l , j = 5 - - 3.i b' = bk

j ~ V L k = 2 , j = 4 kj p v L m k = 2, j = 6

k = 3 , j = 5

damping

Surge 10

a Theory Experiment

180" - m

225' -- 0

Figure 3.25. Motion transfer functions; 6 = 1.2.

3.4. The influence of the water depth on added mass and dampinq

To investigate the qualitative influence of the water depth

on hydrodynamic coefficients the oscillation tests in the sway

and heave mode were carried out for in total four values of the

water depth to draft ratio, 1.05, 1.1, 1.2 and 2.0. The added

mass and damping coefficients for sway and heave are shown in

Figures 3.26 and 3.27.

From Figure 3.26 it appears that the added mass in the

sway mode increases with decreasing keel clearance in the low

frequency range, while the reverse is the case at high frequen-

cies. The slope of the curve increases considerably with de-

creasing water depth, while the hump shifts towards.lower fre-

quencies. The damping in the sway mode is also higher in shallow-

er water, but at high frequencies the curves approach each other

asymptotically.

Although not tested, it may be expected that the coeffi-

cients in the other horizontal modes, surge and yaw, will show

a similar picture.

The shape of the curves of added mass in the heave mode

remain more or less the same, as can be seen from Figure 3.27.

The curve shifts upwards with decreasing water depth. Also the

heave damping increases with decreasing water depth. This result

is in accordance with the data presented by K i m [3-21.

It is expected that the same trends will occur in the other

vertical modes of motion, roll and pitch.

In general, it may be concluded that from the data present-

ed in this section it appears that the influence of the water

depth on the added mass and damping is extremely important. More-

over the data show that the frequency dependency of the coeffi-

cients is obvious especially in very shallow water.

3.5. The influence of a quay parallel to the ship on added mass

and damping

A frequently occurring mooring situation is with the ship

against a quay or solid jetty. Therefore the influence of a

vertical wall parallel to the ship's centreline on the added mass

and damping has been investigated both by means of calculations

Figure 3.26. Added mass and damping coefficient in

sway for different water depths.

Figure 3.27. Added mass and damping coefficient in

heave for different water depths.

and experiments. Only the sway and heave motions have been con-

sidered, under the assumption that they are representative for

the horizontal and vertical modes of motion respectively.

The model tests and calculations were carried out for a

water depth to draft ratio of 1.2. The calculations were per-

formed in accordance with the method described in section 2.4,

while the same schematization of the tanker was used as de-

scribed in section 3.1.

Figures 3.28 and 3.29 show a comparison of calculated and

measured values for the case that the distance between ship and

quay is 16.50 metres (0.35B). The agreement is good. Figures 3.30

through 3.33 show the experimental results for various distances

between ship and quay.

The presence of the quay has quite a remarkable influence

on the hydrodynamic coefficients of the ship. As can be seen

from Figures 3.30 and 3.32 the effect of the quay on the added

mass disappears at very low and very high frequencies. But in

the range of frequencies which is of interest for ship motions

in waves, the added mass is influenced significantly. Most

interesting features are the occurrence of sharp peaks and

negative added mass values. From observation during the tests it

appeared that the peak values may be associated with the occur-

rence of standing waves between quay and ship with nodal lines

perpendicular to the quay.

A physical interpretation of negative "added mass" is diffi-

cult. However, it should be kept in mind that the quantity under

consideration is just the in-phase component of the fluid reac-

tive force, and the denomination "added massn originates only

from the practice to combine this component with the inertia term

in the equations of motion. It could also be combined with the

displacement term, and called then "hydrodynamic spring coeffi-

cient". Anyway it is obvious that in the range of frequencies

where the added mass value is negative, the water between quay

and ship acts like a spring. Consequently, a ship floating freely

in waves near a quay, may experience a resonant motion at two

frequencies in the sway mode. In the heave mode the number of

resonant frequencies may even be greater, due to the presence

of more peaks in the curves of added mass and due to the pres-

ence of a hydrostatic spring constant.

Figure 3 . 2 8 . Added mass and damping coe f f i c i en t i n sway;

distance between ship and quay 1 6 . 5 0 m, 6 = 1 . 2 .

Figure 3.29. Added mass and damping coefficient in heave;

distance between ship and quay 16.50 m, 6 = 1.2.

Figure 3.30. Measured added mass in sway as a function of

distance between ship and quay; 6 = 1.2.

DISTANCE BETWEEN SHIP'S SIDE

AND QUAY:

8.25 m

----m- 16.50 m

15 -.-.- 24.75 m

Figure 3.31. Measured damping coefficient in sway as a

function of distance between ship and quay: 6 = 1.2.

I

DISTANCE BETWEEN SHIP'S SIDE

AND QUAY :

--v-----

-----

--- -..-..-

m - m (0

Figure 3.32. Measured added mass in heave as a function of

distance between ship and quay; 6 = 1.2.

4 0

30

20

DISTANCE BETWEEN SHIP'S SIDE

AND QUAY :

8.25 m

------ 16.50 m

-.-- 25.75 m

--- 33.00 m

-..-..- 41.25 m

-.- -..--.- m m ..... ....

m - m n

1 0

.......... - -.. ---.._ ---......._. ....... -..*.

Oo 2 4 u c

9 Figure 3.33. Measured damping coefficient in heave as a

function of distance between ship and quay; 6 = 1.2.

From Figures 3.31 and 3.33 it appears that the hydrodynamic

damping increases considerably near a quay.

Finally it is remarked that the results presented here for

heave show some resemblance with the results found by Kalkwijk

for a ship, oscillating in a navigation lock [3-31, and the re-

sults found by Lee et. al. [3-4) for heaving catamarans. It is

evident that the problem discussed here of a heaving ship near a quay is theoretically equivalent to the heaving of a twin hull

ship.

CHAPTER 4

EQUATIONS OF MOTION IN THE TIME DOPIAIN

4.1. Potential theory description for flow due to arbitrary ship

motions

As has been outlined in the Introduction, it is desirable

to formulate a set of equations of motions which relate instan-

taneous values of forces and motions. The obvious problem is then

to describe the reactive forces of the fluid due to arbitrarily

in time varying ship motions. To solve this problem, the approach

of Cummins [4-l] is followed.

Cummins describes an arbitrary motion as a succession of

small impulsive displacements. His basic assumption is that at

any time the'total fluid reactive force is the sum of the reac-

tions to the individual impulsive displacements, each reaction

being calculated with an appropriate time lag from the instant

of the corresponding impulsive motion.

Consider a ship, floating at rest in still water. The space

fixed system of axes is defined as in section 2.1. Suppose that

the ship is given an impulsive displacement in the j-th mode,

amounting to A X This displacement is achieved by moving the j'

ship at a constant velocity V for a small period of time At as j

shown in Figure 4.1.

Figure 4.1.

so that

During the impulse the flow can be characterized by a velocity

potential which is proportional to the velocity of the ship, so:

The normalized potential Y. has to satisfy the boundary condi- 3

tions at the ship's surface:

and on the sea bottom:

During the impulse, the free surface will be elevated. The magni-

tude of this elevation amounts to:

a* . Arj = -vj 4 . A.

After the impulse, the surface elevation will dissipate as a

radiating disturbance of the free surface. Since the problem is

assumed to be linear, the potential of this decaying wave will

be proportional to the impulsive displacement:

This potential function must satisfy the initial conditions:

X. (to i- At) = 0 J

and the initial surface elevation equals the elevation due to

the impulse, so that in the free surface:

With 4.5 this yields:

(t = to + At, x3 = d - C)

Moreover, x j satisfies the free surface condition:

and the boundary condition on S:

When the ship performs an arbitrarily in time varying

motion, we can consider this motion as a succession of small

impulsive movements, as shown in Figure 4.2, and the resulting

velocity potential will be:

Taking the limit for At+O yields:

Figure 4.2 .

From (4.3) and (4.11) it is clear, that the total potential

satisfies the boundary condition on the surface of the ship. The

total potential must also satisfy the free surface condition

By substituting (4.13), (4.71, (4.9) and (4.10) it is easily

found that (4.14) is fulfilled only, when:

$j = 0 in the free surface X = d - c 3

From (4.5) it appears,that after an impulsive displacement the

elevation of the free surface amounts to:

Further, Bernoulli's theorem yields in the surface:

In the free surface, 'j

appears to be small to the second

order, so that it may be concluded, that the free surface condi-

tion (4.14) is indeed fulfilled.

The physical meaning of I) is that it describes the instan- j

taneous reaction of the fluid to the impulsive motion. The other

part of the potential, x.(t), describes the motion of the fluid l

after the impulse, which, due to the free surface, will fade a-

way only after a long time. When the ship is moving continually,

it experiences at a certain moment fluid reactive forces which

are a result of the past motions. The past history of the motion

is taken into account by the convolution integral in equation

(4.131, which acts as a memory.

The total potential function as formulated in (4.13) ade-

quately describes the fluid motions due to arbitrary ship mo-

tions. It will not be easy to determine this function, but for

the present purpose the potential is not needed explicitly, as

will be shown in the following sections.

4.2. Equations of motion in the time domain

The linearized hydrodynamic pressure on a ship moving arbi-

trarily in time, is given by Bernoulli's theorem:

The hydrodynamic reaction forces and moments in the k-th direc-

tion are then found from:

The following quantities are now defined:

The equations of motion in the time domain can now be written as

follows, by applying Newton's law of dynamics:

Mkj is the inertia matrix as defined in section 2.2.

C is a matrix of hydrostatic restoring force coefficients. k j

Kkj is a retardation function, and %j is a frequency-independent

added mass coefficient.

Xk(t) is an arbitrarily in time varying exciting force, which

may include all kinds of external influences, as wave forces,

forces due to current, wind, passing ships, and non-linear re-

straining forces.

The equations of motion (2.23) provide an adequate set of

differential equations, relating the instantaneous values of

excitation and resulting motions. Returning to the practical

problem of how to compute the motion response of a moored ship to

excitation by regular or irregular waves it is obvious that at

first the coefficients m and the functions K (t) and Xk(t) k j k j

have to be determined.

The time history of the forcing functions in irregular seas

can be generated easily when the forces in regular waves are

known as a function of the wave frequency (which can be obtained

for instance from diffraction calculations as described in chap-

ter 2).

The general representation of the surface elevation corre-

sponding to a particular energy spectrum is given by:

The amplitude of the harmonic wave components are prescribed by

the spectral density of the waves:

Where S (W,) is the spectral density. S

The phase angle E, is supposed to be a random variable. Now the

time histories of the force components are given by:

How to obtain m and K (t) is less obvious. Ogilvie [4-21 k j k j

has shown how these coefficients and functions are related to

the frequency-dependent added mass and damping coefficients as

they appear in the frequency-domain description. The time-domain

equations (4.23) can describe motions of any kind, also harmonic

motions. Let the ship perform a simple harmonic motion, in re-

sponse to a harmonic excitation:

X = c j e -i(wt + E .) j 3

Substitution in (4.23) of this motion yields:

m

-iwg .e -i(wt + E j ) 1 Kkj(r)e iwr ] l dr l=

0

= Re{% e -i (wt + 6k) l

6 m

i + m - - 1 K ~ ~ ( T ) sin wr dr sin(wt + E.) j=1 kj w

0 l 3

m

+ w 1 Kkj (T) cos wt dr . cos(wt + E .) + C sinfwt + E .))c = 0 3 k j 3 j

This equation is equivalent to the frequency-domain description

(2.31).

Hence : m

- 1 - - I Kkj (t) sin wt dt akj - mkj w c m

bkj = I Kkj (t) cos wt dt 0

The retardation function can now be written in terms of the

frequency-dependent damping coefficient, by taking the inverse

Fourier transform of (4.30) :

The constant added mass coefficient can then be obtained from

(2.29), when the frequency-dependent added mass is known for one

single value of the frequency:

1 mk. = ak. ( w l ) + 7 ;i. Kkj(t) sin w't dt 3 3 0

w' is an arbitrarily chosen value of w, the result for

given by (4.32) is independent of the value of U ' . mkj

4.4. The behaviour of the damping for hiqh frequency motions

The formulae (4.31) and (4.32) give the theoretical rela-

tionship between the coefficients and functions in the frequency

and time domain descriptions, from which m and K can be k j k j

determined. In practice, however, a problem is encountered. As

it appears from (4.31), the damping must be known for all fre-

quencies for the calculation of the retardation function.

In theory, the method for computation of added mass and

damping as described in chapter 2 holds true also for high fre-

quencies, but in practice the numerical process is not suitable

for handling high frequency motions. Moreover, the damping ap-

proaches asymptotically to zero with increasing frequencies, and

therefore one should like to have an analytical description of

the damping as a function of w for the high frequency range.

To investigate the asymptotic behaviour of the damping for

w- a two-dimensional approach may be used. The relation between

the damping coefficient and the amplitude of the radiated waves

at infinity is given by Newman [4-31 to be:

in which Rk(w) represents the ratio of the amplitude of the ra-

diated waves at infinity to the amplitude of the motion, the so-

called wave making coefficient.

The case of high frequency motions in the vertical mode has

been treated by Ursell [4-41, Rhodes-Robinson 14-51 and Hermans

4 6 . From their work, it appears that for U+- to the first or-

der :

so that:

where Ck is a constant.

When the ship is approximated by a vertical barrier, extend-

ing to the sea bottom (thus ignoring the keel-clearance, which

is permissible when the wave length is small compared to the

draft of the ship) the behaviour of the damping for horizontal

motions and the roll motion at high frequencies can be deter-

mined from the work of Ursell, et. al. [4-71 and Biesel [4-81 . For the surge, sway and yaw modes the ship may be regarded

as a piston type wave maker. According to 14-81 the wave making

coefficient then amounts to:

2 Rk = . 2 sinh rd

slnh rd cosh rd + ~d k = l, 2, 6

This coefficient approaches to a constant value, for U+-.

When rolling around the centre of gravity, in other words

around an axis below the water surface, the ship acts as a pad-

dle-type wave maker. Reference [4-81 gives for this case the

following wave making coefficient:

R4 = 2 sinh ~d(l - cosh rd + rd sinh ~ d )

~d(sinh ~d cosh Kd + ~ d )

It appears, that also R4 = constant for U+-, SO that:

Suppose now that the damping is known (from calculations

or measurements) from w = o to w = w It is then possible to 1' choose the value of the constant Ck such, that the high frequen-

cy approximation corresponds to the known part of the damping

curve. Consequently, equation (4.31) can be written as:

where p = 3 for k = 1, 2 , 4 , 6

p = 7 for k = 3 , 5

The first integral can be solved numerically without problems,

while the second one can be treated in an analytical way.

Thus, the retardation functions and constant added mass

coefficients for uncoupled motions can be obtained. The treatment

of the coupling terms is, however, less simple. No information is

available on the asymptotic behaviour of the damping coefficients

which describe the coupling effects. The best thing one can do in

this case is a method of trial and error: fair the curves to zero

(which will be done rather arbitrarily) and compute first K and k j

subsequently % for several frequencies. In case a large scatter j

is found in the values of mk the computation should be repeated j'

using a different extrapolation of the damping curve, and this

iteration process must be repeated until the values of m as k j

found for different frequencies agree within reasonable limits.

From a theoretical point of view this way of determining

Kkj and %j

for the coupling effects is rather unsatisfactory.

But fortunately it appears in practice that the value of + is j not very sensitive for the high frequency approximation of the

damping, so the error in K and m will be small, and since k j k j

the coupling terms in the equations of motion are of minor impor-

tance, the effect of the error on the motions will be even small-

er.

4.5. Numerical computations of retardation functions and constant

inertia coefficients

By means of equations ( 4 . 3 2 ) and ( 4 . 3 8 ) the retardation

functions and constant added mass coefficients have been computed

for a 200,000 tdw tanker, departing from the damping and added

mass data as they have been obtained from the three-dimensional

source technique, which were presented in chapter 3 .

After partial integration, formula ( 4 .38 ) yields for

k = 1, 2, 4 and 6:

2 "'1 Kkk(t) = F bkk(w) COS wt dw + - 0

n rosUilt

sin wlt m

-t w -t2 j O S d..] 1 w t wt

1

and for k = 3, 5:

t cos wlt + ( - - t3 t5 + - - - )sin wlt 15wl 180wl 360wl 1 -7- cos Wtdwt 360 n

The integral in the last term of (4.39) and (4.40) can be expand-

ed in a series (see ref. [4-91 ) :

m m cos wt n w t 2 n

I wt dwt = - ( y + In wlt + z (-1) ( 1 ) ) 2n (2n) ! , (,4.41)

wit n=l

As an example of the extrapolation of the damping curves

Figure 4.3 shows the damping for the sway mode. In this case the

high-frequency approximation

was used with C2 = 2.86 which corresponds exactly to the value

found from (4.33) and (4.35) for the two-dimensional case of a

flat vertical barrier.

The damping curves for the coupling effects were extrapolat-

ed in such a way that the curves reached a value zero between

w = 2.0 and w = 2.5.

For the numerical integrations a Simpson-routine was used. -1 The steps of integration amounted to Aw = 0.01 rad.sec. for

equations (4.311, (4.39) and (4.40) and At = 0.125 seconds for

equation (4.321, while the upper limit of the integral in equa-

tion (4.32) was taken as 25 seconds. Since the retardation func-

tions vanish rather quickly (which means that in the equations

of motion much emphasis is laid on the very near past of the

motion of the ship) this period was assumed to be sufficient.

Figure 4.3. Extrapolation of the sway

damping curve.

The retardation functions thus obtained for the 200,000 tdw

tanker are given in Table 4.1, while the most important ones are

also shown in Figure 4.4 and 4.5.

The constant added mass coefficients were computed for 8

different values of U . Due to the various approximations in-

volved these computations did not yield exactly the same value.

Therefore the outcomes were averaged. Table 4.2 shows these

averaged values of mk together with their root mean square j

TABLE 4 .l. Computed retardation functions;

TIME

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0 19.5 20.0 20.5 21.0 21.5 22.0 22.5 23.0 23.5 24.0 24.5 25.0

K1l

4.4622 2 4.0642 2 3.0052 2 1.6232 2 3.1202 1 -6.3762 1 -1.1289 2 -1.2422 2 -1.1462 2 -9.9512 1 -8.6232 1 -7.4802 1 -6.2225 l -4.7059 1 - 3 1 1 0 l -1.7882 1 -9.9412 0 -7.3465 0 -8.1325 0 -1.0032 1 -1.1872 1 -1.3662 1 -1.5742 1 -1.7862 1 -1.9202 1 -1.8952 1 -1.7132 1 -1.4649 1 -1.2682 1 -1.1962 1 -1.2362 1 -1.3242 1 -1.404 l -1.4632 1 -1.5222 1 -1.5932 1 -1.6492 1 -1.6379 1 -1.5182 1 -1.2952 1 -1.0142 1 -7.3142 0 -4.7952 0 -2.5932 0 -5.7592-1 1.2822 0 2.8132 0 3.8022 0 4.1932 0 4.1692 0 4.0222 0

K2 2

9.9872 3 8.9932 3 6.4502 3 3.3929 3 8.2402 2

-7.6742 2 -1.4749 3 -1.6692 3 -1.6722 3 -1.6202 3 -1.5252 3 -1.3809 3 -1.2372 3 -1.1002 3 -9.9072 2 -9.1122 2 -8.5749 2 -8.1982 2 -7.8475 2 -7.4452 2 -7.0539 2 -6.7792 2 -6.6112 2 -6.3882 2 -6.0359 2 -5.6002 2 -5.2632 2 -5.0819 2 -4.9322 2 -4.6692 2 -4.2955 2 -3.9329 2 -3.662+ 2 -3.4322 2 -3.1482 2 -2.7992 2 -2.4689 2 -2.2142 2 -2.0129 2 -1.8012 2 -1.5659 2 -1.3395 2 -1.1482 2 -9.6972 1 -7.8132 1 -5.9252 1 -4.3102 1 -2.9842 1 -1.6722 1 -2.2372 0 1.1612 1

K33

9.4632 3 8.9122 3 7.4002 3 5.3002 3 3.0802 3 1.1472 3

-2.5982 2 -1.0932 3 -1.4472 3 -1.4682 3 -1.2835 3 -9.7382 2 -5.9162 2 -1.8632 2 1.7422 2 4.1222 2 4.6812 2 3.2672 2 2.5722 1 -3.5952 2 -7.4342 2 -1.0562 3 -1.2552 3 -1.3262 3 -1.2712 3 -1.1092 3 -8.6822 2 -5.8925 2 -3.2072 2 -1.0572 2 1.0072 1 2.7339 1 -4.5592 1 -1.7922 2 -3.3732 2 -4.8642 2 -6.0172 2 -6.6812 2 -6.8099 2 -6.4363 2 -5.6792 2 -4.7042 2 -3.6932 2 -2.7982 2 -2.114f 2 -1.6662 2 -1.4299 2 -1.3542 2 -1.3889 2 -1.4842 2 -1.5982 2

K44

1.1482 5 9.8799 4 6.0432 4 2.0989 4

-3.1082 3 -1.0059 4 -8.9682 3 -8.8282 3 -1.1502 4 -1.3872 4 -1.3712 4 -1.2209 4 -1.1522 4 -1.1939 4 -1.2029 4 -1.100f 4 -9.6102 3 -8.9362 3 -8.9071 3 -8.6432 3 -7.7912 3 -6.9459 3 -6.7032 3 -6.8162 3 -6.5782 3 -5.8042 3 -5.0212 3 -4.7032 3 -4.6701 3 -4.4212 3 -3.8612 3 -3.3962 3 -3.3432 3 -3.5092 3 -3.4825 3 -3.1692 3 -2.8472 3 -2.7212 3 -2.6482 3 -2.3772 3 -1.9122 3 -1.4922 3 -1.2532 3 -1.0712 3 -7.8092 2 -4.2302 2 -1.7109 2 -7.4662 1 2.9379 0 1.7862 2 3.8552 2

. K55

5.2102 7 4.8202 7 3.7712 7 2.3712 7 9.9072 6 -7.8902 5 -7.1459 6 -9.6172 6 -9.6662 6 -8,8412 6 -8.118# 6 -7.7512 6 -7.5442 6 -7.2462 6 -6.8072 6 -6.3712 6 -6.1132 6 -6.0872 6 -6.1912 6 -6.2622 6 -6.1842 6 -5.9492 6 -5.6199 6 -5.2532 6 -4.8579 6 -4.3962 6 -3.8419 6 -3.2092 6 -2.5632 6 -1.9799 6 -1.5022 6 -1.1362 6 -8.5612 5 -6.3672 5 -4.6362 5 -3.3082 5 -2.2892 5 -1.4062 5 -4.8732 4 5.2262 4 1.5192 5 2.3202 5 2.7999 5 2.9702 5 2.9489 5 2.846# 5 2.6919 5 2.4422 5 2.0622 5 1.5932 5 1.1432 5

units: tonf, m, sec.

value in a non-dimensional form.

TABLE 4.2.

Average and root mean square values of constant added mass

coefficients as computed for 8 different frequencies.

In most cases the deviation found between the various m - val- k j

ues is small, particularly when the root mean square value is

related to the total mass term. Most deviation occurs in the

coupling terms.

In theory, there is a possibility to obtain m in a differ- k j

ent way. From equation (4.29) it becomes obvious, that m is the k j

asymptotic value of a ( W ) for W+-: k j

. = a,. ( W ) for W+- 3 3

mode

For this case, the linearized free surface condition degenerates

into :

%j

0.033

0.505

3.741

1.42 1 0 - ~

0.158

0.045

0.024

0.032

-0.005

-0.043

-0.023

0.28 loe3

k

1

2

3

4

5

6

1

1

2

2

3

4

j

1

2

3

4

5

6

3

5

4

6

5

6

U k j

inBofml; j

6

15

9

1

3

17

2 0

14

7 0

32

3 4

51

- a k j

i n % o f (M + m k j ) ' k j

0.2

5

7

0.01

0.5

1

2 0

14

7 0 3 2

3 4

5 1

= o in the free surface

Work has been done on this special problem with regard to

its relevance to ship vibrations. Reference is made to the work

of Lewis [4-101, Taylor 14-111 , Kumai [4-121 , [4-131, Prohaska @-l41 and Todd [4-151, who gives an excellent review of pub-

lished data. Xoch [4-163 determined added mass coefficients for

w = - experimentally by means of an electric analogon, taking into account the influence of a restricted water depth. On

basis of his data the following coefficients are predicted for

horizontal and vertical motion:

m2 2 - = P v 0.5 and - m33 - - 3.5

PV

These figures are in good agreement with the values in Table 4.2,

which increases the confidence in the method of computing the

retardation function and constant added mass coefficient.

CHAPTER 5

THE APPLICATION OF THE EQUATIONS OF MOTION IN TK3 TIME DOMAIN

5.1. General

In the previous sections the tools have been sharpened for

the ultimate job: the prediction of the behaviour of a moored

ship in a purely theoretical way. The equations of motion in the time domain have been discussed, necessary to describe the be-

haviour of a ship with non-linear mooring restraints, while it

has been shown also how the wave forces and fluid reactive for-

ces can be computed by means of the three-dimensional source

technique. In this chapter these tools will be applied to the

analysis of the motion behaviour of a ship, moored to a jetty.

First, the mathematical model to simulate moored ship motions

will be discussed. Then, the results of numerical and experimen-

tal investigations for a certain mooring situation will be com-

pared and discussed. Finally, the general motion behaviour of

the moored ship will be analysed by means of approximate analy-

tical methods.

5.2. Numerical calculations

Consider a ship, moored by means of a number of mooring

lines with non-linear elastic characteristics, to a jetty

equipped with fenders, as sehematized in Figure 5.1.

Figure 5.1.

Initially, the ship is at rest with its centre of gravity

in the origin of the space fixed system of coordinates GX1X2X3.

The coordinates of the chocks of the i-th mooring line are:

on the jetty: pilr pia, pi3.

on the ship : qilr qi2, qi3-

The load-elongation relationship of the mooring line is given by:

where Ti = tension in the line

&li = elongation of the line

li = initial length of the Line.

When t = 0, there is an amount of pretension in the line, Pi.

The length of the mooring line without pretension can then be

found from:

Ali(t=O) where can be found from (5.1).

'i Now the ship is being subjected to wave action. At a certain

moment, the motion of the ship is given by xl (t) , x2 (t) , . . . . . . . ... x6(t). For small rotative motions of the ship,'and neglecting

terms which are small of second order, the chock position on the

ship is given by:

The instantaneous elongation of the line is then:

and the instantaneous line tension is then found from l . In

case Ali(t) < 0 the line is slack and consequently the tension

is zero. The resulting forces and moments of the i-th mooring

line acting about the centre of gravity of the ship are:

Let the coordinates of the point of the i-th fender which

makes contact with the ship be ril, ri2, ri3. The relationship

between the impression of this fender di and the reaction force

in the fender Di is given by:

At a certain instant of time the impression of the i-th fender

amounts to:

The corresponding fender reaction is found from (5.6), and the

resulting forces and moments on the ship are:

for k = 1, 3, 5

Including the reaction forces of mooring lines and fenders, the

complete equations of motion in the time domain become:

where Xk(t) = wave exciting force.

n = total number of mooring lines. 1

nf = total number of fenders.

The wave exciting force is generated as follows:

where n = number of wave components W

E = a random phase angle in case nu > 1 ri The factor (1 - e -O.Olt) is applied in order to prevent a shock load at the start of the simulation. In this way all wave loads

gradually increase from 0 to over 98 percent of their appropriate

value in a period of 400 seconds.

The numerical solution of the 6 coupled second order differ-

ential equations (5.9) is carried through according to the fol-

lowing procedure. Suppose the simulation has arrived to the

moment t, At is the time increment applied, so the equations of

motion have to be solved for the moment t + At. First, the velo- At cities for t + -3 and t + At are predicted by extrapolating the

obtained time histories. To this end, the velocity is expanded

in a Taylor series:

At = *.(t) + At%.(t) + 7 {2.(t) - %.(t - At)} (5.11) 3 3 3 l

At In a similar way b.(t + 3) is found. Subsequently, the new po- 3

sition and orientation are predicted by numerical integration of

the velocities, applying Simpson's rule:

At x.(t + At) =x.(t) + T {Sj(t) + 4*.(t + % ) + 2 (t + At)]. J J I j

(5.12)

The time history of the velocities is now known until the moment

t + At, so the convolution integrals can be computed. The numer- ical integration of these convolution integrals is carried out

by means of Simpson's rule, using a time increment equal to the

time step At, applied for the solution of the equations of mo-

tion. The upper bound of the integrals, in theory infinity, was

fixed at 25 seconds for the case described in this thesis.

Subsequently, the values of the mooring line and fender forces

as well as the hydrostatic restoring forces can be calculated for

the new coordinates. After substitution of these forces in (5.9),

6 linear equations are obtained from which the accelerations

Z.(t + At) can be found. Finally, the predicted velocities are l checked by integration of the accelerations. In case the differ-

ence is acceptable, the computation continues for the next time

step; if not, the time increment has to be decreased. When start-

ing the computation process at t = 0, no reliable prediction of

%.(At) can be made. Therefore, an iteration procedure is used l in that case: the values of the velocities found after solving

the equations of motion are used as a new prediction and the

process is repeated until the predicted and computed velocities

at t = At are in satisfactory agreement. The mathematical simulation process described before has

been programmed in FORTRAN for use on a Control Data 6600 com-

puter.

Besides some steering parameters, the input of the program

consists of:

- inertia matrix of the ship - matrix of hydrostatic restoring coefficients - added mass coefficients and retardation functions of the ship

- coordinates of fenders and bollards - elasticity characteristics of fenders and lines - pretension in mooring lines - frequency, amplitude and phase angle of the wave load components.

The retardation functions and elasticity functions of lines and

fenders are read in as a number of discrete points, sufficient

to fix the curves. Intermediate values are, when needed, found

by means of interpolation sub-routines.

As output of the program, the computed time histories of

motions and forces can be printed, plotted or dumped on digital

tape for further analysis.

The computing time is linearly proportional to the inverse

of the time step. Systematic computations with varying time step

have shown that a step of 0.2 seconds is sufficiently small for

an accurate numerical solution of the equations of motion. With

this time step, the required computing time amounts to 1 second

for 10 seconds real time.

5.3. Examples of computed moored ship motions and experimental

verification

To check the adequacy of the mathematical model for moored

ships, an extensive experimental program has been carried out to

analyse the motion behaviour of a moored ship in regular and

irregular waves. Afterwards, typical test situations were se-

lected for simulation on the computer to see whether the observed

phenomena could be reproduced by means of the mathematical simu-

lation.

The study was conducted for a loaded 200,000 tdw tanker,

moored to an open jetty in water with a depth amounting to 1.2 times the draft of the vessel. The ship is the same as describ-

ed in Chapter 3. The mooring lay-out is depicted in Figure 5.2. The vessel

was moored by means of 4 lines, each representing two or three

wires with nylon tails in reality. The load-elongation curves of

the lines are shown in Figure 5.3. In each line a pretension was

applied of 20 tons.

The pair of stiff fenders had a linear elasticity amounting

to 1575 ton/m.

The model tests were carried out in the Shallow Water Labo-

ratory of the Netherlands Ship Model Basin on a scale of 1:82.5.

The same ship model was used as for the tests mentioned in Chap-

ter 3. Figure 5.4 shows a general view of the test arrangement.

213.25 m 213.25 m 137.20 m ! + ' ~ 7 . ~ r n

t 43.75 m i a . m m i l i

\, zkFEIUYRS,, 2 4 0 m

I l

~--.

2W.W TDW TANKER

no.* m I t3D.45 m 156.aom 1 1SA.40 m

CHOCK FENDER WLPHlN

Figure 5.2. Mooring lay-out of 200,000 TDW tanker.

The non-linear elasticity characteristic of the mooring

lines was simulated by means of a composition of 4 linear springs.

The calibrated load-elongation relationships for the modelled

lines are plotted in Figure 5.3.

The fenders were simulated by means of rocking arms, to

which linear springs were connected, as shown in Figure 5.5. The

friction between the model and the fenders was minimized by using

vertical wheels which were fitted to the fender arms.

The loads in mooring lines and fenders were measured by

means of strain gauge transducers. The motions were measured by

means of potentiometers. All signals were recorded both on mag-

netic tape and paper chart.

The model tests were based on Froude's law of similitude.

Hence, prototype values were obtained by applying a length scale

factor 82.5, time and velocity scale factor m and mass and 3 force scale factors 82.5 .

First an extensive series of tests in regular beam waves

(a = go0) was carried out to investigate the behaviour of the

moored ship over a wide range of periods, varying from 9 to 41

PERCENT ELONGATION

Figure 5.3. Load-elongation characteristic of mooring lines.

seconds. The long period waves are not merely of theoretical

interest, but may also be associated with seiches or harbour

oscillations, which sometimes cause problems in practice.

In all conditions tested, the ship attained a periodic

motion, after a period of transience.

Characteristic of the sway motion was the occurrence of

a mean displacement in addition to the oscillatory motion. In

short waves this motion had the same frequency as the waves, with

bounces against the fenders of equal strength at time intervals

equal to the wave period. In certain long waves, however, a sub-

harmonic motion was observed: the ship motion was composed of a

motion with frequency w equal to that of the exciting waves, on

which a motion was superimposed with frequency either w / 3 or w / 2 .

Figure 5.4. Experimental set-up.

Impacts against the fenders occurred then at intervals of 3 or 2

wave periods respectively. Subharmonic motion was observed only

in waves with such a frequency, that the frequency of the subhar-

monic ( w / 3 or w/2) was close to the "natural" frequency of the

moored ship, which in the sway mode, amounted to approximately - 1 0.07 rad.sec. (due to the non-linear elasticity characteristic

of the mooring system there is no well-defined natural frequency;

in fact the resonance frequency depends on the amplitude of mo-

tion). When the amplitude of motion was decreased, the subharmon-

ic motion disappeared. After this observation, it was attempted

to produce a subharmonic motion of mode w / 4 , by trying waves

with frequencies around 4 times the natural frequency and large

amplitudes. These attempts were, however, without success. In

some cases another mode of regular motion was found, for instance

with alternating light and heavy bounces aqainst the fenders.

Figure 5.9. Fen- s iae~la t ion .

FigUres 5.6 tbrough 5.8 show a review of khe mst representative r e s u l t s of the tests, tlogether with tire r e s u l t s of the coaputa- t ione .

The maputations were carried out with pure harmonic Loscing f a ~ ~ ~ ? t i n n s , which were obtain& from the theore t ica l t ransfer curtr-es, taking into acwunt a wave height equal t o tha t , measured i n the basin without t h e &ip model being there.

Time h i s t o r i e s were computed f o r a period of lOOb seconds, although it appeared that a f t e r 500 seconds r e s u l t s k c m e stat ionary.

It appeared t h a t the modes of motionr found i'n the expexi- raeats, a re aLse predictea by the aathematisal model. Sraall d i f - ferences ate found but of ten tfiese ean be explained by the fact &at the Long Haves i n the basin were not mre s i n e wave@. For i n e a n c e , the wave recard of the test in Figure 5.6 s h w s clear- Ip higher harmonic oompoaents. The second harmonic i n t h i s ease

E SURGE

SWAY

H E M

mu

A F P--A-~~--~A- I - V - - PITCH

Figure 5.6. Computed and measured ship motions and mooring -1 forces. Regular waves from 90 degrees, w = 0.212 rad-sec. ,

wave height 0.9 m.

MEASURED COMPUTED

A - V 1 WAVE

- , \ /-v v-',,'- , SWAY

A A A 0 A n A HEAVE

- - l ,. - A , A A P - - . , - V " V V ROLL

FENDER 2

LINE 1

LlNE 2

LlNE 3

-- A A A A

. l

LlNE 4

-p& YAW

A A A I\ FENMR 1

TIME

Figure 5.7. Computed and measured ship motions and mooring - 1 forces. Regular waves from 90 degrees., w = 0.212 rad-sec. , wave height 0.32 m.

- - - / *vv\

/A" v v v -. A A A A

COMPUTrD

WAVE

SUROE

/-v-v v-v 1. SWAY

A . vnv W W L A A A HEAVE

. vAvAvAvAv

I ROLL

PITCH

YAW

FENOER I

FENDER 2

vvvvv LINE I

LlNE 2

LME 3

W LINE 4

n M E -

Figure 5.8. Computed and measured ship motions and mooring -1 forces. Regular waves from 90 degrees, w = 0.69 rad-sec. ,

, wave height 1.06 m.

is close to the natural roll period. This causes a pronounced

superharmonic roll response, with corresponding secondary peaks

in the mooring line forces, which are not present in the computed

records. In the computed roll motion a superharmonic component

is found too, but its amplitude is much smaller.

Table 5.1 shows for some typical tests a comparison of

numerical values of the most interesting signals, being maximum

peak values of mooring line and fender forces, as well as the

maximum peak to trough value of the sway motion (x2 max

X2 min. 1. From the results it may be concluded that both the qualitative and quantitative agreement between theory and experi-

ments is satisfactory: the differences between measured and

computed values are usually less than 20 percent.

Also long crested irregular waves have been taken into con-

, sideration. One sea condition was used, of which the spectral

density is shown in Figure 5.9, with three angles of wave attack,

cr = 90. 135 and 180 degrees. The measurements lasted a period

corresponding to 2100 seconds prototype scale and began 1000

seconds after starting the wave generator, thus avoiding that

transient phenomena would influence the results.

The computations were performed for a period corresponding

to 2500 seconds. The first 400 seconds represent a period of

transience, thus leaving 2100 seconds for analysis.

Of all measured and computed time histories of motions and

forces a spectral analysis was carried out. Besides spectra, this

analysis yielded the following statistical quantities:

- mean value - root mean square value - significant double amplitude - maximum and minimum value.

For a definition of these quantities and details of the spectral

analysis technique, reference is made to Appendix I.

The wave spectrum was simulated in the computations by means

of 15 sine waves, ranging in frequency from 0.425 to 1.125 rad.

sec.-'. To that end the measured wave spectrum (see Figure 5.9)

was subdivided into 15 bands of constant width. Each band was

represented by a sine component, having the centre frequency of

that band and a height, following from the band area. These wave

Figure 5.9.-Spectral density of the irregular waves as measured

in the basin. Significant wave height 2.6 m, mean period 8.9 sec.

components were summed with arbitrary phase angles, and with the

aid of the computed transfer functions the forcing functions in

the 6 modes were determined.

The first theoretical results for a = 90 degrees thus

obtained, showed a good aqreement with the experimental results

with regard to the shape of the spectra and the maximum values,

but the significant values and the areas under the spectra were

too high for the theoretical results. To some extent this is

caused by the fact that the roll motion is over-estimated in the

computations. After adding some additional damping in the roll

mode, derived from the oscillation tests, the agreement improved.

The results of tests and computations, both with and without

additional roll damping, are given in Table 5.2 and Figure 5.10.

The results for bow quartering waves, ct = 135 degrees,

are presented in Table 5.3 and Figure 5.11. For this case, a

TABLE 5.2.

Comparison of computed and measured results

Wave spectrum Angle of attack 90 degrees

Signal

Mooring line

1

Mooring line

2

Mooring line

3

Mooring line

4

Fender 1

Fender 2

- - - --p - Surge

- - -- - - - Sway

-----A-

Yaw

Quantity

U

F1/3

Fmax

G

F 1/3

Fmax - U

F1/3

Fmax

U

F 1/3

Fmax

a

F 1/3 Fmax

U

F 1/3 Fmax - - - - X1 U

Xlmax X lmin -- - - - X2 U

X2max X 2min -- - x6

U

X6max

X6min

Unit

tonf m

m

" "

m.

W

"

" "

"

- - - m

- --

" m

degr " "

Experiment

18

68

108 ----.---p

2 8

110

188

30

107

166

14

63

89 -- - 241

971

1530 ---_ 177

761

1196 - 0.14

0.10

0.47

-0.05 -- - - --

-0.23

0.47

0.99

-2.10

0

0.11

0.28

-0.43

Computation

With theoretical

roll damping

2 5

92

113 - 4 6

152

211

145

147

214

3 0

100

147

257

1180

1709

255

1150

1580

0.01

0.17

0.39

-0.37 - - p - ---

-0.85

0.79

0.83

-3.20 ---.-----.--------p---

-0.01

0.06

0.12

-0.18

With experi-

mental roll

damping

21

8 4

98 -F-------

39

141

170

4 0

142

184

2 7

9 8

128

229

1060

1208 __U_-_-_____

261

1192

1320 +

0.01

0.08

0.26

-0.22

-0.71

0.66

0.76

-2.46

-0.01

0.05

0.13

-0.14

MOORING LlNE I

E

-- -\

0 0

MOORING LlNE 2

. P

.. l m '\ i \

l '\ ,

0 0

MOORING LlNE 3

z F *

0 0

MOORING LlNE 4

z O 1 E 1 " - \

"P \

S - \ I \

D l I

I \\

l L. .- O 0 0 5 10 O 0 0 5 1 0

w in rad sec-' win rad.se&

Figure 5.10. Computed and measured spectra of ship motions and

mooring forces. Irregular waves (see Figure 5.9) from 90

degrees.

TABLE 5.3.

Comparison of computed and measured results

Wave spectrum Angle of attack 135 degrees

r

Signal

Mooring line

1

---------A-

~ooring line

2

--- Mooring line

3

--A

Mooring line

4

Fender 1

------.---W-- Fender 2

- - - . Surge

- - - -- - - - Sway

- - - W - - - yaw

Computation

8

3 3

'4 0 ----- 11

41

5 1 W - - - -

11

40

52 ---------

8

3 4

43

82

344

418

7 7

300

410 --- 0.03

0.07

0.20

-0.17 - - - - --

-0 -08

0.08

0.15

-0.26 - -0.01

0.06

0.20

-0.25

Quantity

U

F 1/3

Fmax

U

F 1/3

Fmax

U

F 1/3

Fmax -- U

F1/3

Fmax

U

F 1/3

Fmax

U

F 1/3

Fmax - - - U

X lmax

Xlmin - W - - . - X2 U

X2max

X2min . - - - - X6 U

X 6 m x

'6rnin

Unit

tonf I

n

I,

"

p

I

"

m

m

It

m

" "

m

n

n

- - . m m

n

" - - - - "

m

m

n

- - - degr

D

" U

Experiment

8

4 2

57

8

4 1

57 A - - --

12

5 4

8 5

10

4 4

61

7 1

274

476

6 3

254

536 - -

-0.04

0.22

0.58

-0.83 - - - - - -

0

0.07

0.14

-0.29

0

0.13

0.57

-0.40

1 0

- EXPERIMENT 0

X --- COMPUTED

"E

0

FENDER 2 -1

MOORING LlNE 3

Y V)

I

0 0 -

MOOR1NG LlNE 4

P U ' 1 -;Fii,-.: P .

O 0 05 1 D O 0 0 5 1 0 W 10 pad sec-' w m rad S-'

Figure 5 . 1 1 . Computed and measured spectra of ship motions and

mooring forces. Irregular waves (see Figure 5 . 9 ) from 135

degrees.

TABLE 5.4.

Comparison of computed and measured r e s u l t s

Wave spectrum Angle of a t t a c k 180 degrees

S i g n a l

Mooring l i n e

1

--bp-----

Mooring l i n e

2

- - -- -- Mooring l i n e

3

Mooring l i n e

4

Fender 1

-- -- -- Fender 2

- - - -- - Surge

Sway

Yaw

Q u a n t i t y

a

F1/3

F m a ~

U

'1/3

Fmax

U

F 1/3

Fmax

a F

1/3 'max

U

F 1/3

F . max -- -- U

F 113

Fmax - xl

a X lmax X lmin -

X2 a

'2max X 2min p-. -

x6 a

X6max

X6min

Uni t

t o n f " "

" "

- - - "

m

" " " " "

- -- " " "

- - - m " t.

" " " "

degr " " m

Experiment

8

41

57 -- - - 6

31

38 - - - --

8

41

5 4

9

33

50

10

34

132 - - -- -

16

54

139 - - - - -0.32

0.38

0.58

-1.39 ----------.----------A-------.------

-0.01

0.01

0.03

-0.03 --- 0.07

0.08

0.25

-0.13

NO

d r i f t f o r c e

3

24

25 -- - - -

0.5

20

20 - - - --

0.4

2 0

2 o 2.5

24

26

1

33

33 - - - 1

36

36 . - -

0.02

0.07

0.18

-0.15

0.02 -

0.02

0.02 -- - -0.002

0.001

0

-0.003

C o m y t a t i o n

With

c o n s t a n t

d r i f t f o r c e

3

32

34 - - -

0.6

18

19 -- - - -

0.5

22

22

0.3

1 8

20

2

44

46 - - - 2

26

27 - - -

-0.43

0.08

-0 -02

-0.43

0.02

0.02

0.02 -- 0 . O l

0.002

0.01

0.005

With

v a r y i n g

d r i f t f o r c e

11

4 2

55 - - - -- 3

21

25 --W

4

2 6

3 1

10

2 8

4 1

17

6 5 76 - - - - 16

53

68 -- --. -0.23

0.36

0.68

-1.13

0.02 -

0.02

0.02 ---- 0.01

0.01

0.04

-0.03

EXPERIMENT V 3 COMPUTED WITH

*E VARYING DRIFT FORCE

n 0

MOORING LlNE l

i E

0

MOORING LlNE 2

f a *

n "

MOORiNG LINE 3

l

0 0

MOORlW LINE 4 :Ei;Kl P m. a 8

- 1 0 0 0 5 10

O 0 0 5 W in rad sec-' W m rod sac-'

Figure 5.12. Computed and measured spectra of ship motions and

mooring forces. Irregular waves (see Figure 5.9) from 180

degrees.

good agreement was found, without additional roll damping.

For the case of head waves (a = 180 degrees), in first

instance a very bad correlation was found: the theoretical

forcing functions consist only of small oscillatory loads in

surge, heave and pitch mode, resulting in small variations of the

mooring forces around the pretension and a small surge oscilla-

tion around the zero position. In reality, however, the surge

motion consists of a larger amplitude oscillation around a mean

displacement, and the variations in the mooring loads are also

larger. It was assumed that in this case the second order wave

drift force plays an important role. To check this assumption,

two calculations were made: one with an additional constant and

one with a slowly varying drift force, which is much more realis-

tic in irregular waves. As an approximation of the magnitude of

the drift force, measurements by Pinkster 15-11 , performed on a different ship model in a similar sea condition were used. In

Table 5.4 the results of computations without drift force, with

constant drift force and with slowly varying drift force, are

compared with the measured values. Figure 5.12 shows spectra of

motions and forces according to experiments and computations with

varying drift force. Obviously, the influence of the low frequen-

cy drift force is essential in this condition.

5.4. Analysis of the results

The results of the investigations described in section 5.3

have revealed interesting features of the behaviour of a ship,

moored to a jetty.

In regular beam seas, the sway motion consists of an oscil-

lation around a mean displacement. In certain long waves a sub-

harmonic sway motion was found with frequency w / 2 or w/3 .

In irregular seas, the spectra of horizontal motions and

mooring forces show low-frequency peaks, clearly distinct from the

range of wave frequencies. From the computations it followed that

in head waves the second order wave force plays an important role

in exciting this low frequency behaviour, but in the other wave

directions considered, low frequency peaks were found due to

first order wave excitation only. This low frequency behaviour

in irregular waves must also be distinguished from the subharmon-

ic motion in regular waves: the longest wave component in the

spectrum had a frequency of 0.425 rad. sec .-l whereas subharmonic

motions only occurred in waves with frequencies lower than ap- - 1 proximately 0.21 rad.sec. .

As was shown, this typical behaviour of a moored ship can be

predicted by the complicated mathematical model described here,

but to understand why these modes of motion occur, it is helpful

to use a simplified analytical approach.

The above mentioned special features of moored ship motions

occur mainly in the horizontal modes, surge, sway and yaw. When

these motions are considered as being uncoupled, it is observed

that the restoring force and moment in surge and yaw are non-

linear, but symmetric, and hence they can be schematized as:

where f(x) is the restoring force, X is the displacement and

a and f3 are constants. When moreover the frequency-dependency

of the added mass is ignored and the damping is neglected, the

simplified equation of motion becomes

This is the well-known Duffing equation, which has been treated

extensively in the literature of non-linear vibrations (see for

instance Stoker, [5-21 ) . The solutions of the Duffing equation show the following particulars:

- the first order approximation of the motion due to a harmonic excitation F = F cos wt is: X = A cos wt a

- the response curve shows a "backbone" shape as shown in Figure 5.13

- with F = Fa cos wt, subharmonic solutions exist with frequency

w/3 - with excitations consisting of two harmonic components, F =

F1 cos wlt + F2 cos w2t, the motions contain components with

frequencies 2wl 2 w2 and 2w2 2 wl, the so-called combination tones, besides the basic frequencies wl and w2.

The elasticity of the mooring system in the sway mode is

essentially different in that sense, that the restoring force is

Figure 5.13. Response

of non-linear mass- spring system.

asymmetrical: when pushing against the fenders, the force is dif-

ferent from the case that the ship pulls at the lines. The most

simple way to schematize such a restoring force is:

When again ignoring frequency - dependency of added mass and damping, the equation of motion becomes:

Solutions of this equation are discussed in Amendix 11. It ap-

pears, that:

- with an excitation F = Fa cos wt, the first order approximation

of the resulting motion shows an oscillation around a mean

value:

- subharmonic motion may occur with frequencies w/2 as well as 0/3

- with bi-frequency excitation (F = F1 cos wlt + F2 cos w2t) com- bination tones can be found with frequencies 2wl 2 w2, 2w2 2 wl and wl f w2.

Thus this simplified analytical approach shows how a mean

displacement and low frequency motions originate from the non-

linear particulars of the mooring system.

'The presence of damping may have an important influence on the occurrence of subharmonics. Levenson [5-31 has shown for the

Duffing equation, that subharmonics due to harmonic oscillations

exist only if t3e damping coefficient is small compared to the

amplitude of the exciting force, and that the damping coefficient

would have to be taken small of still higher order to obtain

subharmonics of order higher than 1/3 in the presence of damping.

This explains why the subharmonic sway response of the moored

ship of order 1/3 disappeared when the wave height was decreased,

and that it was impossible to obtain subharmonics of order 1/4

in the physical and mathematical models.

It will be clear, that the equations of motion in the fre-

quency domain can only be used to explain certain particulars of

the behaviour of the moored ship, brought about by the peculiari-

ties of the mooring system, the coefficients of the inertia

and damping terms being strongly dependent on frequency. In the

present case, the inertia coefficient at the subharmonic frequen-

cy is around four times larger than at the wave frequency. For

the damping coefficient, the difference is even larger.

The low frequency response in irregular seas from 90 and

135 degrees is obviously a result of the phenomenon of combina-

tion tones, since subharmonic response was not found in the

separate wave components of the spectrum. To check whether the

drift force has an additional effect in these cases, a calcula-

tion was made for the 90 degrees condition with a measured time

history of the drift force added to the first order wave forces.

However, no significant influence was found in beam seas. In

irregular head waves, the computations without second order wave

forces showed also low frequency peaks in motion and mooring

force spectra, but their magnitude was a factor 100 to 1000

smaller than the measured peaks, and the measured low frequency

behaviour could only be reproduced by adding a varying drift

force.

With regard to the low frequency second order wave forces

a difficulty is, that the knowledge of the nature of these forces

is still insufficient. The constant drift force in a regular

wave, which originates from the second order pressure, may be

computed with the linear three-dimensional source technique. But

the variation of the drift force in random seas is still subject

to study, just as the possible influence of second order terms

in the wave potential. Significant work on drift forces has been

done by Maruo [5-41, Gerritsma and Beukelman [5-51, Mei and

Black [5-61, Remery and Hermans [5-71, Hsu and Blenkarn [S-81,

Newman [5-91 and Pinkster [5-l] and [5-101 . The experimental verification of the mathematical model has

been restricted to head waves and such wave directions, in which

the ship is pushed against the fenders by the drift force. To

check the possible influence of the drift force in case the ship

drifts away from the fenders, a computation was made in a wave

spectrum from 270 degrees, with drift force simulation (without

drift force the results would have been the same as for waves

from 90 degrees). Two computations were made, one with a constant

drift force, amounting to the average value of a drift force

record, and one with additional low frequency components. In

Table 5.5 the results are compared with the results for 90 de-

grees. Contrary to 90 degrees waves, the drift force has a sig-

nificant influence on the sway motion and mooring forces in case

of 270 degrees waves, while it is observed that the difference

between results with constant and varying drift force is not very

significant. Obviously, the main effect of the drift force in

this condition is that it gives the ship a mean displacement in

sway, which affects the characteristics of the restoring forces

for the high frequency oscillations substantially. In 90 degrees waves, the mean displacement due to the drift force is negligible

since the ship is pushed against the very stiff fenders, which

explains that no drift force influence was found in that case.

In head waves, it was found that the oscillating character

of the drift force was more important than its average value. For

this angle of wave attack, the first order wave forces are very

small, as is the surge response to these forces, and consequently

the low frequency forces induced by the mooring system through

the phenomenon of combination tones are unimportant compared to

the varying drift force.

The following statement is an attempt to generalize the ob-

servations made on the influence of the second order wave force:

in an analysis of non-linearly moored ship motions the wave drift

force must be included as an essential part of the wave excita-

tion, when the amplitude of the force of second order is rela-

TABLE 5.5.

Com2arison of computed r e s u l t s for wave s p e c t r a w i t h a n g l e s

o f a t t a c k 90 and 270 d e g r e e s .

S i g n a l

Mooring l i n e

1

Mooring l i n e

2

- - - -- - - Mooring l i n e

3

Plooring l i n e

4

Fende r 1.

- --p -- -- Fende r 2

- -- - S u r g e

Sway

- - - - --- Y aw

Q u a n t i t y

a

F 1 / 3

Fmax a

F1/3

Fnax

a

F 1 /3

Fmax

n F

113 Fmax

a

'113

Fmax - - - a

F 1 /3

Frnax -- --- - X1

a X lmax X lmin -

X2 a

X 2max

X2min - - x6 a

Xsmax '6rnin

U n i t

t o n f 11

"

W

n

- - v

" ----------B-----.-------------- "

V

I

- - - " "

m " n

m

------B-.----.-----------.---- A

I

d e g r

"

90 d e g r e e s

NO d r i f t force

25

92

113

4 6

152

211 . --p --- -

4 5

147

214

30

100

147

257

1180

1709 - - - - -- --

255

1150

1580 - -- 0.01

0.17

0.39

-0.37

-0.85

0.79

0 .83

-3.20 . - - - - -- -

-0.01

0.06

0.12

-0.18

270

c o n s t a n t

dr i f t f o r c e

2 6

112

160

5 3

204

331 - - -

4 7

179

258

34

125

174

126

803

1981 -- - --

140

824

2041 --------------p

0.09

0.34

0.97

-0.83

-1.63

0.87

0.79

-4.47 -- - - - -.

-0.07

0.05

0.11

-0.24

d e g r e e s

v a r y i n g

d r i f t f o r c e

2 7

115

187

58

212

386 - -

5 3

187

310 -- ----- -

37

123

216

140

826

2328 -- --p

153

892

2370

0.09

0.34

1.03

-0.90 ---W-----

-1.67

0.95

1.11

-5.22 - - -- - -

-0.07

0.05

0.09

-0.22

tively important (which is the case for waves from 0 or 180 de-

grees), or when the average second order force changes the aver-

age position of the moored ship significantly. In case of ships,

moored by means of a linear system in random seas, it is likely

that the drift force is always important, since it is then the

only low-frequency excitation.

Although the quantitative agreement between results of the

mathematical and physical model is reasonable for cases where the

drift force is of minor importance (90 and 135 degrees), some

discrepancies remain, especially in the 90 degrees waves where,

although the shapes of the computed spectra are very similar to

those of the measured spectra, the peak values of the spectra are

much higher in some cases (in particular the low frequency

peaks). This is reflected by the fact that the computed r.m.s.

values are larger than the measured values. It is not clear

whether these differences are due to experimental errors or due

to limitations of the mathematical model.

Experimental errors are caused by:

- shortcomings in the test set-up, such as flexibility of the mooring structure, friction between ship model and fenders,

damping and dynamic effects in pantograph and mooring line

springs

- measuring errors involved in the limited accuracy of measuring and recording instruments

- evaluation errors due to the process of converting analogue records to digital ones.

The magnitude of the first category of errors is hard to estimate

but the total error due to measuring and evaluation inaccuracies

is estimated at 5 percent. I For an assessment of possible errors in the mathematical

model, a review is given below of approximations and assumptions

involved in the mathematical model.

- The equations of motion are based on the assumption that the fluid reactive forces are linear. In Chapter 3 it has been shown experimentally that this assumption holds true for the

motion amplitudes concerned.

- First order wave forces and fluid reactive forces are computed numerically. The differences with measured values are small in

general, except for the roll damping. The computed roll damping

appeared to be insufficient in case resonant roll motion oc-

curs.

- Certain approximations are involved in computing the constant added mass coefficients and retardation functions, as discussed

in Chapter 4. Errors, involved in these numerical computations

and the effect of cutting-off the retardation functions after

25 seconds have been checked by means of equations (4.29) and

(4.30). It has been found, that the fluid reactive forces are

represented correctly within 2 percent for the range of wave

frequencies as well as for the low frequencies where motion

response was found.

- Wave forces and fluid reactive forces from viscous origin have been neglected. The largest influence can be expected in the

sway mode. An estimate of these forces has been made using the

empirical formula:

where : V = relative sway speed

% = lateral area of the ship

For the drag coefficient CD, measured values as reported in

reference 15-11] were used. Inclusion of this force, in the model

did not change the results significantly.

- The wave spectrum was simulated by means of 15 discrete compo- nents. An increase of this number to 20 yielded differences in

the results of less than 1 percent.

- The equations of motions were solved with a time step amounting to 0.2 seconds. Computations with smaller steps showed that

the possible error does not exceed 1 percent.

- The first 400 seconds of the computed irregular records were regarded as a period of transience. Calculations with longer transient periods did confirm this

assumption.

- In the spectral analysis of motion and force records, inaccu- racies are involved in the choice of sample time and length of

auto-correlation function. This, however, applies both to com-

puted and measured results and will not affect the comparison

of the two.

Finally it is remarked, that it may be advisable to increase

the duration of both the physical and the mathematical simulation

for a more reliable spectral analysis, in particular with regard

to the low frequency parts of the spectra.

5.5. Extension to other systems

In this chapter, the approach of analysing moored ship mo-

tions with the aid of the equations of motion of Cummins in com-

bination with the three-dimensional source technique for obtain-

ing hydrodynamic forces, was applied to a ship, moored in waves

to an open jetty.

The same approach can be used for a ship, moored to a solid

jetty or a quay, since for this case the hydrodynamic forces can

also be computed with the three-dimensional source technique.

Further, the equations of motion enable the inclusion of

other arbitrarily in time varying excitations. The desirability

of adding second order wave forces was already discussed in the

previous section, but also wind and current forces can be includ-

ed, or excitations induced by passing ships.

The approach can also be extended to systems with moving

mooring structures, as for instance the single buoy mooring sys-

tem. In that case equations must be added which describe the

dynamic behaviour of the buoy, while provisions in the model must

be made which enable the adaptation of amplitudes and phases of

the wave loads for changing position and heading of the ship.

CHAPTER 6

CONCLUSIONS

As a result of the investigations the following conclusions can

be drawn:

1. The frequency-dependency of added mass and damping necessi-

tates a time-domain approach for the analysis of the behaviour

of a non-linearly moored ship, in which the fluid reaction

forces are described as a function of the past history of the

flow. The equations of tnotion according to Cummins satisfy

these requirements.

2. First order wave forces and fluid reactive forces in all six

modes of motion in shallow water can be obtained from the

three-dimensional source technique. The influence of a verti-

cal wall can be taken into account. Computed results show a

satisfactory agreement with values, obtained from measurements

on small scale models.

3. A ship, floating freely in waves in the proximity of a quay,

may experience resonant sway motion at one or two frequencies,

and resonant heave motion at more than one frequency.

4. Particular modes of motion, exhibited by a ship, moored by

means of a system with non-linear and asymmetric restoring

force characteristics, can be predicted by a mathematical

model based on the equations of motion in the time-domain

according to Cummins.

5. A ship, moored to a jetty in random seas, can experience three

types of low frequency behaviour:

- subharmonic response to certain harmonic components, with a frequency amounting to 1/2 or 1/3 of that of the exciting

wave component. This subharmonic response is a result of

the non-linearity of the mooring system.

- "combination tones", a low frequency motion induced by the simultaneous action of more than one harmonic wave force

component. This phenomenon is also a result of the non-lin-

earity of the mooring system.

- low frequency motions excited by the low frequency second order drift force.

6. In most mooring situations in random seas, the second order

wave forces will have an important influence. The only prac-

tical exception seems to be the case that a ship is pushed

against stiff fenders by the waves.

APPENDIX I

SPECTRAL ANALYSIS OF IRREGULAR SIGNALS

The record of an irregular phenomenon u(t) is assumed to be

composed of a mean value uo and an infinite number of components

with random phase angles sn:

with: un = amplitude of the.nth component of u(t) with a circular

frequency un.

The spectral density SU(w) of the signal is defined by [

If dw = wn - 'n-1 is chosen small enough, the amplitude un of

the component with frequency wn is determined by:

'Jn = J2SU(wn) dw'

The priniiple of the calculation is as follows:

If u(t) represents the value of the signal as a function of time,

the auto-correlation function RU(?) is determined by:

lim 1 T

%(.C) = T-tm ~lt) u(t + 'C) dt 0

The spectral density function SU(w) is then obtained by Fourier-

transformation of Ru (T) :

This type of calculation is performed in a numerical way.

The analogue records of irregular signals, measured during the

model tests were therefore digitized with a sample time amount-

ing to 1.1 second (prototype scale). For the analysis of the

computed signals, the sample time amounted to 1.0 second. The

auto-correlation function was computed for 90 values of T. The

upper boundary of the integral, T, was equal to the duration of

the record, and the integral was computed with a time increment

i equal to the sample time. The Fourier transformation of the

auto-correlation function to obtain the spectrum was carried out

for 90 values of the frequency, equally spaced. The frequency

range over which the spectrum is determined is bounded by w = 0

and w = ~r/At where At is the sample time. For a given spectrum SU(w) the moments mu of the spectrum

are defined as follows:

If p = 0, then

m

muo = S, dw = area of spectrum 0

If p = 1, then

m

= I o SU dw = first moment of spectrum "ul

etc.

With the aid of the moments of the spectrum the following

quantities can be calculated:

- significant double amplitude: 4 q 0 . For narrow spectra this value corresponds to the average of the

one-third highest double amplitudes [peak to trough value):

- - average period:

For narrow spectra this value corresponds to the average value

of the periods between zero-crossings To.

In addition, the following statistical quantities are de- fined :

- mean value

n=N - 1 U = - ' Un ; N is the number of samples n=l

- root mean square value: U

- maximum value: U m a x . Highest peak value, encountered in the record.

- minimum value: urnin. Lowest trough value, encountered in the record.

APPENDIX I1

PARTICULAR SOLUTIONS OF AN EQUATION OF MOTION WITH

NON-LINEAR, ASYMMETRIC RESTORING FORCE

Consider the following one-degree of freedom equation of

motion with non-linear, asymmetric restoring force:

Duffing's iteration method will be applied here to find certain

particular solutions of this equation.

First, solutions will be searched for the case of a simple harmonic forcing function:

z + ax + 6x2 + yx3 = cos wt

Suppose a solution exists which, in first approximation, has the

form :

xo = A cos wt + B

Substitution yields :

= -aA cos wt - aB - B A ~ cos2 wt - BB 2 1

2 -2ABB COS wt - y~~ cos3 wt - 3yA B cos2 wt 3 -3y?B2 cos wt - yB + Fa cos wt

2 = -aA cos wt - aB - %BA* - %BA cos 2wt

3 3 - B B ~ - 2ABB cos wt - %yA cos wt - kyA cos 3 wt

2 2 -3/2yA B - 3/2yA B cos 2wt - 3yAB2 cos wt

-y~' + Fa cos w t .

Then as a second approximation is found:

3 + - l (%BA' + 3/2y~'~)cos 2wt + $g % cos 3wt 4w 2 W

-(%BB + %BA' + f8B2 + + + constant

From the requirement that the solution is periodic, two equations

are found relating the amplitude A of the motion component with

the same frequency as the excitation and the mean displacement

B to the frequency w and amplitude Fa of the excitation:

2 2 Fa and w2 = (a + 2 BB + ayA + 3yB - Zi-)

As is the case with the Duffing equation, also subharmonic

solutions exist which satisfy the equation with asymmetric

restoring force. A solution with the subharmonic of frequency

2 can be written in the form: X = A wt

4 cos - + AI cos wt + B 2

substitution in the equation of motion yields:

W 2 2 + aA4 + 2BA4B + 3yA B )cos + 4 2

2 2 (-Alw + aA1 + 26A1B + 3yAlB )cos w t +

2 2 2 2 (BAt + 3yA B ) cos f + (BA1 + 3yA1B)cos2 wt +

3 3 wt 3 yA4 COS F + yAl cos3 wt + (2f3A%A1 + 6yA A B)cos wt cos F + ?i l

2 wt 3yA A cos2 wt cos - + 3 y ~ i ~ 1 cos wt cos2 G = Fa cos wt f l 2 2

Making use of the following relations, while neglecting higher

harmonic terms:

3 wt - COS - - wt

2 %COS i- + ..... cos3 wt = %COS wt + . . . . .

wt - wt cos wt cos - - #cos - + ..... 2 2

2 wt wt cos wt cos - = #cos F + ..... 2

2 wt - cos wt cos - - f + *cos wt + ..... 2

leads to the following equations for A#, A1 and B:

Similarly, for the subharmonic with frequency $ the solution can be written as:

Substitution and trigonometric reduction yield the following

equations for this case:

To investigate the effect of multi-frequency excitation, a

forcing function will be considered which is the sum of two har-

monics :

3 + cxx + fix2 + -yx3 = F cos w t + F2 cos w2.' 1 1

Duffing's iteration method can be started with the first approxi-

mation:

xo = A1 cos wlt + A2 COS w2t + B

After substitution it can be observed that xl contains terms

cos wlt cos w2t

cos2 wlt cos w2t

cos wlt cos w2t

which leads for the second approximation xl to terms of frequen-

cies wl + w2, wl - w2, wl + 2w2, w1 - 2u2, 2w1 + w2 and 2wl - W2. So with multi-frequency excitation, the response may show solu-

tions with frequencies higher (originating from sum-frequencies)

or lower (from difference-frequencies) than those of the excita-

tion itself. These solutions are called combination tones.

REFERENCES OF CHAPTER 1

F-l] Wilson, B.W. ; "Ship response to range action in harbor

basins": Trans. A.S.C.E., Vol. 1 1 6 , 1951 , pp. 1129-1157.

[l-21 Kaplan, P. and Putz, R.R.; "The motions of a moored

construction-type barge in irregular waves and their

influence on construction operationw: N By-32206, Marine

Advisers, Inc. La Jolla, 1 9 6 2 .

[l-31 Leendertse, J.J. ; "Analysis of the response of moored

surface and subsurface vessels to ocean waves": Rand

Corporation Memorandum RM-3368 PR. 1963 .

[l-41 Muga, B.J.; "Experimental and theoretical study of motion

of a barge as moored in ocean waves": University of

Illinois, Hydraulic Engineering Series No. 1 3 , 1 9 6 7 .

[l-51 Seidl, L.H.; "Prediction of motions of ships moored in

irregular seas": Proc. N.A.T.O. Advanced Study Institute

on Analytical Treatment of Problems in the Berthing and

Mooring of Ships, Wallingford, 1973 , pp. 221-229.

[l-61 Abramson, H.N. and Wilson, Basil W.; "A further analysis

of the longitudinal response of moored vessels to sea

oscillations": Proc. A.S.C.E. 8 5 , 1 9 5 9 , WW4, p. 1 7 3 .

[l-71 Yang, I-Min; "Motions of moored ships in six degrees of

freedom" : gth Symposium on Naval Hydrodynamics, Paris,

[l-81 Kilner, F.A.; "Model tests on the motion of moored ships

placed on long waves": Proc. 7th Conf. on Coastal

Engineering, The Hague, 1 9 6 0 , Volume 2 pp. 723-745.

1 9 Wilson, J.F. and Awadalla, N .C. I nSubharmonic response in the nonlinear oscillations of moored ships": Offshore

Technology Conference, Houston, 1971 , paper OTC 1 4 2 0 ,

Volume I1 pp. 65-80.

[l-101 Wilson, J.F. and Awadalla, N.G. ; 'Computer simulation

of nonlinear motion of moored ships": Proc. N.A.T.O. Advanced Study Institute on Analytical Treatment of

Problems in the Berthing and Mooring of Ships,

Wallingford, 1973, pp. 277-298.

[l-111 Lean, G.X.: "Subharmonic motions of moored ships

subjected to wave action": Transactions of the Royal

Institution of Naval Architects, 1971, Vol. 113,

pp. 387-399.

p-121 Wilson, B.W.; "Progress in the study of ships moored

in waves": Proc. N.A.T.O. Advanced Study Institute

on Analytical Treatment of Problems in the Berthing

and Mooring of Ships, Wallingford, 1973, pp. 143-213.

[l-131 Bomze, B.: "Analytical determination of ship motion and

mooring forces": Offshore Technology Conference,

Houston, 1974, paper OTC 2072.

[l-141 Cummins, W.E.; "The impulse response function and ship

motionsn: D.T.M.B. Report 1661, Washington D.C., 1962.

REFERENCES OF CHAPTER 2

Wehausen, J.V. and Laitone, E.V.; "Handbuch der Physik":

Vol. 9, Springer Verlag, Berlin 1960.

John, F.; "On the motion of floating bodies": Corn. on

pure and applied mathematics, Part I: 2, 1949, pp. 13-57;

Part 11: 3, 1950, pp. 45-100.

Tuck, E.O.; "Ship motions in shallow water": Journal of

Ship research, Vol. 14, No. 4, 1970, pp. 317-328.

Vugts, J.H.; "The hydrodynamic forces and ship motions

in waves": Thesis, Delft, 1970.

Ogilvie, T.F. and Tuck, E.O.; "A rational strip theory

of ship motions": Part I, University of Michigan,

Department of Naval Architecture and Marine Engineering,

Report No. 013, March 1969.

Timman, R. and Newman, J.N.; "The coupled damping

coefficients of a symmetric ship": Journal of Ship Research, Vol. 5. No. 4, 1962, pp. 1-7.

Newman, J.N.; "The exciting forces on a moving body in

waves": Journal of Ship Research, Vol. 9, No. 3, pp.

190-199.

Korvin-Kroukovsky, B.V. and Jacobs, W.R.; "Pitching and

heaving motions of a ship in regular waves": Trans.

S.N.A.M.E. 65, New York, 1957, pp. 590-632.

Kim, C.H.; "The influence of water depth on the heaving

and pitching motions of a ship moving in longitudinal

regular head waves": Schiffstechnik, Vol. 15, No. 79,

1968, pp. 127-132.

Newman, J.N.; "Lateral motion of a slender body between

two parallel walls": Journal of Fluid Mechanics, Vol.

39, 1969, pp. 97-115.

Tuck, E.O. and Taylor, P.J.; "Shallow water problems

in ship hydrodynamics": ath Symposium on Naval Hydrodynamics, Pasadena, 1970.

Beck, R.F. and Tuck, E.O.; 'Computation of shallow

water ship motionsA : gth Symposium on Naval

Hydrodynamics, Paris, 1972.

Daubert, A.; "Quelques considerations sur les

differentes simulations et methodes de calcul du

comportement des structures immergees ou flottantes

en mer": 'La Houille Blanche' no. 2, 1970.

Garisson, C.J. and Chow, P.Y.; "Wave forces on submerged

bodiesw: A.S.C.E. Waterways and Harbors Div. 1972, 98,

pp. 375-392.

Oortmerssen, G. van; "Some aspects of very large

offshore structures" : gth Symposium on Naval

Hydrodynamics, Paris, 1972.

Boreel, L.J.; "Wave action on large offshore

structures": Conference on off-shore structures, Inst.

of Civil Engineers, London, 1974.

[2-14 Lamb, H. ; "Hydrodynamics": Sixth Edition (1932) . [2-181 Yeung, R.W. ; "A singularity distribution method for

free-surface flow problems with an oscillating bodyn:

Univ. of California, College of Engineering, Report

NA 73-6, Aug. 1973.

[2-14 Stoker, J. J. ; "Water Wavesn : Interseience Publishers,

Inc., New York, 1957.

REFERENCES OF CHAPTER 3

Flagg, C.N. and Newman, J.N.; "Sway added-mass

coefficients for rectangular profiles in shallow watern:

Journal of Ship Research, December 1971, pp. 257-265.

Kin, C.H.: "The influence of water depth on the heaving

and pitching motions of a ship moving in longitudinal

regular head waves": Schiffstechnik, Vol. 15, No. 79,

1968, pp. 127-132.

Kalkwijk, J,P.Th.; "Hydrodynamic forces and ship motions

induced by surges in a navigation lock": Thesis, Delft,

1973.

Lee, C.M., Jones, H.D. and Curphey, R.M.; "Prediction

of motion and hydrodynamic loads of catamarans": Marine

Technology, October 1973. pp. 392-405.

REFERENCES OF CHAPTER 4

Cummins, W.E.; 'The impulse response function and ship

motions": D.T.M.B., Report 1661, Washington D.C., October

1962.

Ogilvie, T.F.; "Recent progress toward the understanding

and prediction of ship motionsn: 5th Symposium on Naval

Hydrodynamics, Bergen, 1964.

Newman, J.N.; "The exciting forces on fixed bodies in

waves": Journal of Ship Research, Vol. 6, No. 3, 1962,

pp. 10-17.

Ursell, F.; "Short surface waves due to an oscillating

immersed body": Proc. of the Cambridge Phil. Soc., A 220

(19531, pp. 90-103.

Rhodes-Robinson: "On the short-wave asymptotic motion due

to a cylinder heaving on water of finite depth, I and

111: Proc. of the Cambridge Phil. Soc., 67(1970) pp. 423-

442 and 443-468.

Hermans, A.J.; "A perturbation method for the radiation

of swface waves": Journal of Engineering Mathematics,

Vol. 6, No. 4, 1972, pp. 323-330.

Ursell, F., Dean, R.G. and Yu, Y.S.; 'Forced small

amplitude water waves: a comparison of theory and

experiment": Journal of Fluid Mechanics, Vol. 7, Part 1,

1960, pp. 33-52.

Biesel, F.; "Etude theorique d'un certain type d'appareil

S houle": ''La Houille Blanchem, Nr. 4, 1951.

Abramowitz, M. and Segun, J.; 'Handbook of mathematical

functions": Dover Publications, New York, 1965.

Lewis, F.M.; "The i n e r t i a of t h e water surrounding a

v ibra t ing ship": Proceedings, S.N.A.M.E., 1929.

Taylor, J.L.; "Vibrations of ships": Proceedings I . N . A . ,

1930.

Kumai, T.; "On the v i r t u a l i n e r t i a coe f f i c i en t s f o r t h e

v e r t i c a l v ibra t ion of ships": European Shipbuilding,

Nos. 2 and 3, 1959.

Kumai, T.; "Added mass moment of i n e r t i a induced by

to r s iona l v ibra t ion of sh ipsn : European Shipbuilding,

No. 6 , 1958.

Prohaska, C.W.; "The vert l ical v ibra t ions of ships":

Proc. A.T.M.A., 1947.

Todd, F .H. ; "Ship h u l l v ibrat ion": Edward b o l d ,

London, 1961.

Koch, J.J.; "Experimental, method f o r determining the

v i r t u a l mass f o r v ibra t ions of ships": D.T.M.B., Report

NO. 225, Washington D.C., 1949.

REFERENCES OF CHAPTER 5

[5-l] Pinkster, J.A.; 'Low frequency second order wave forces

on vessels moored at sea": llth Symposium on Naval

Hydrodynamics, London, March 1976.

[5-21 Stoker, J. 3. ; 'Non-linear vibrations in mechanical and

electrical systems": Interscience Publishers, Inc.,

New York, 1957.

[5-3] Levenson, M.B. ; llHamonic and subharmonic response for

the Duffing equation": Thesis, New York University,

[5-41 Maruo, H.; "The drift of a body floating on waves":

Journal of Ship Research, December 1960, Vol. 4.

[5-5] Gerritsma, J. and Beukelnan, W.; "Analysis of the

resistance increase in waves of a fast cargo ship":

Report No. 334 Laboratorium voor Scheepsbouwkunde,

Technical University, Delft, 1971.

[S-d Mei, C.C. and Black, J.L. ; "Scattering of surface waves":

Journal of Fluid Mechanics Vol. 38, Part 3, 1969, pp.

433-455.

F-71 Remery. G.F.M. and Hemans, A. J. ; "The slow drift

oscillations of a moored object in random seas": Offshore

Technology Conference, Houston, 1971, Paper SPE 3423,

OTC. 1500.

Hsu, F.H. and Blenkarn, K.A.; "Analysis of peak mooring

forces caused by slow vessel drift oscillations in

random seas": Offshore Technology Conference, Houston,

1970, Paper OTC 1159.

E - 9 ] Newman, J.N. ; "Second order, slowly varying forces on

vessels in irregular waves": International symposium

on the dynamics of marine vehicles and structures in

waves, University College, London, April 1974.

[5-14 Pinkster, J .A. ; "Low frequency .phenomena associated

with vessels moored at sea": S.P.E. European Spring

Meeting, Amsterdam, May 1974, Paper 4837.

E-14 Remery, G.F.M. and Van Oortmerssen, G. ; "The mean wave,

wind and current forces on offshore structures and

their role in the design of mooring systems": Offshore

Technology Conference, Houston, 1973 , Paper OTC 1741.

Symbols not included in the list below are only used at a specif-

ic place and are explained where they occur.

lateral area of the ship

breadth of the ship

matrix of restoring force coefficients

reaction force in the i-th fender

general force or moment

hydrodynamic reactive force in the k-th mode

centre of gravity

moment of inertia in the k-th mode

product of inertia

Bessel function of the first kind of zero order

modified Bessel function of the second kind of zero

order

retardation function in the k-th mode due to motion

in the j-th mode

length of the ship

force or moment in the k-th mode due to the i-th

mooring line

inertia matrix

force or moment in the k-th mode due to the i-the

fender

pretension in the i-th mooring line

impulse response function

immersed surface of the ship

spectral density of the waves

tension in the i-th mooring line

velocity in the j-th mode

wave force or moment in the k-th mode

Weber's Bessel function of the second kind of zero

order

added mass coefficient in the k-th mode due to

motion in the j-th mode

non-dimensional added mass coefficient as defined

in Table 3.2.

damping coefficient in the k-th mode due to motion

in the j-th mode

non-dimensional damping coefficient as defined in

Table 3.2

distance from the centre of gravity of the ship to

the sea bottom

depth of water

impression of the i-th fender

acceleration of gravity

h distance between the ship's centre line and the

quay i imaginary unit; as a subscript: integer number

j, k subscripts ranging from 1 to 6 used for a direction or a degree of freedom

li length of the i-th mooring line

m mass of the ship

mkj frequency-independent added mass coefficient in the

k-th mode due to motion in the j-th mode

%j non-dimensional frequency-independent added mass

coefficient as defined in Table 3.2

n normal vector, pointing outside the body

n j

generalized direction cosine

P hydrodynamic pressure

Pill Pi21 Pi3 coordinates of the i-th dolphin

qil, qi2, qi3 coordinates of the i-th chock on the ship

ril, ri2, ri3 coordinates of the i-th fender t time

X j

displacement in the j-th mode

a angle of wave incidence

Y Green's function 6 ratio of water depth to draft

6k phase angle of wave force in the k-th mode

E j

phase angle of j-th mode of motion

5 free surface elevation

50 wave amplitude

'j amplitude of motion in the j-th mode; in section

4.1 applied for a free surface elevation due to

motion in the j-th mode

wave length

= 2n/X wave number

specific mass of water 2

= W /g circular frequency

= (U-

velocity potential

time-independent potential function

source strength: root mean square value

volume of displacement

SUMMARY

It is the intention of this thesis to formulate a mathemat-

ical model which can be used for computer simulations of the

behaviour of a moored ship in waves. In order to achieve this,

two problems must be solved. First, a set of equations of motion

must be drawn up, which can describe adequately the behaviour in

six degrees of freedom of a ship, moored by means of a non-linear

mooring system in regular or irregular waves. Second, a method

must be found for the computation of the wave forces and the

hydrodynamic reaction forces acting on the ship.

For the description of the motions, the six coupled equa-

tions of motion in the time-domain according to Cummins are used.

These equations can bake into account non-linear and asymmetric

restoring force characteristics of the mooring system, while the

exciting forces may be arbitrary. The only restriction of these

equations is that linearity is assumed of the hydrodynamic re-

storing forces.

In general, mooring of ships occurs only in areas with a

restricted water depth. In shallow water the flow around a ship

has a three-dimensional character, and therefore the three-di-

mensional source technique is applied to obtain the wave loads

and hydrodynamic reaction forces on the ship. Based on the linear

potential theory, this technique supposes an ideal fluid and

small amplitudes of waves and motions. The effect of a forward

speed is not included in this method, but this is not of interest

for moored ship problems. The influence of a restricted water

depth and a quay parallel to the ship can be taken into account.

An extensive experimental verification has been carried out

by means of model tests, for a 200,000 tdw tanker in shallow

water (the keel clearance amounted to 20 percent of the draft).

Comparative computations and measurements of wave loads, hydro-

dynamic restoring forces and free floating ship motions have been

carried out. Further, computer simulations have been made for the

case that the ship is mooted in regular and irregular waves to

a jetty, by means of mooring lines and fenders. The results of

these computations are also compared with those of model experi-

ments.

A n analysis of the results obtained shows that the typical

behaviour of the moored ship, with subharmonic motion response,

is represented adequately by the mathematical model. In condi-

tions where the ship is pushed against the fenders by the waves,

the subharmonic motions are caused mainly by the non-linear

characteristics of the mooring system. When the angle between

direction of wave propagation and the longitudinal axis of the

ship is small, or in case the ship is pushed away from the

fenders by the waves, it appears that the second order wave force

has a dominant influence on the low frequency motions of the

moored ship.

EBi, SCf!E€PSBOUWKU#DIG PRBEFSTATIM Uaagsteag 2 - Wageninpn

H O L L A N D