Future design of ship lines by use of analogue and digital computers

FRAN

STATEN S SKEPPSPROVNINGSANSTALT

(PUBLICATIONS OF THE SWEDISH STATE SHIPBUILDING EXPERIMENTAL TANK)

Nr 59 GÖTEBORG 1966

FUTURE DESIGN OF SHIP LINES

BY USE OF ANALOGUE

AND DIGITAL COMPUTERS

BY

AKE WILLIAMS

Paper presented for the

6t1 Shipbuiiders' Scientific Session Gdansk 1966

'Ships after 1970"

4s

SCANDINAVIAN UNIVERSITY BOOKS

SCANDINAVIAN UNIVERSITY BOOKS

Denmark: MmeXSGAARD, Gopenha gen

Norway: UNIVERSITETSFORLAGET, 081o, Bergen SWedeTI, AXADEMIFÖRLAGET-GUMPERTS, Göteborg SVENSKA BOEKFÖRLAGET/NOrStedtS - Bonniers, Stockiwim

PRINTED IN SWEDEN BY

In this report some results are given of current work at the Swedish State Shipbuilding Experimental Tank (SSPA) concerning mathema-tical ship lines. In an earlier report an account was given of the method

of calculation and the practical procedure for delineating ships

mathematically.

The present part of the investigation deals with special problems in connection with tracing ship lines by use of digital and analogue

drafting equipment. Some of the methods mentioned in this report are outlined only and will be the subject of further research work.

1. Summary

Alternative methods for design of ship lines by use of computers

are discussed. Special interest is devoted to the use of drafting equip-ment in combination with digital and analogue computers.

As an introduction some general points of view are given regarding

mathematical definition of ship lines and its application to

ship-building in theory and practice.

The construction of ship lines can be performed graphically and mathematically and will be based on single form coefficients as well as form-defining parameters of hydrostatic and hydrodynamic meaning.

For an analysis of the hull form in order to find connections with

results from model tests, the single form coefficients are unreliable.

For a strict definition of the hull shape it is necessary to introduce

form parameters affecting the entire surface of the ship.

The hull form is mathematically defined by waterline polynomials. The number of polynomial terms is equal to the number of prescribed form parameters, which are faired in vertical direction to draft func-tions valid for the whole ship.

The numerically controlled drafting equipment provides good ac-curacy for full scale use and permits drawing of all kinds of sections.

4

The analogue system is easy to handle but is restricted regarding

tracing the lines. In order to take advantage of both systems special "hybrid" computers can be built up.

Form parameters of two alternative hulls for a side trawler are

given in Appendix 1. In Appendix 2 the practical procedure for design of ship lines is outlined.

2. Introduction

A requirement of the shipbuilding industry is that the drawings first made up by the ship lines' designer must be enlarged up to loo

times in the case of a big ship. Moreover the accuracy must be high also in full scale.

When the preliminary ship lines are treated mathematically it is

essential that important criteria for stability, propulsion, manoeuvring

are not affected. The ship hydrodynamicist's point of view is that those parameters must be included in the mathematical form which

allude to the hydrostatic and hydrodynamic properties.

However, the mathematical method must also be adoptable to the

fabrication of the structural members of the ship otherwise it is

possible, that an important argument for mathematical representation is lost, namely from the economical point of view. Thus, the

mathe-matical method must take care of discontinuous sectional lines (in

first or second derivative) in the boundaries between different parts of the ship, such as constant radius in the bilge, simplified construction of keel, stem and stern post and also plane surfaces in bottom and sides. The main problem can also be formulated inversely in this way: The mathematical treatment must not only meet the demands of

plate and profile fabrication but also make it easier for the ship design department to project new ships. Thereby it would be a great

advan-tage if the method could permit a controllable modification to new

ship lines stipulated by altered claims of size, speed, stability, etc.

It is also essential for the whole organization in shipbuilding that the form-defining method is suitable also for the ship hydrodynamicist. Now and in the future, large systematic work will be performed for investigating the influence of the hull form upon resistance,

propul-sion, cavitation, manoeuvring, seakindliness, etc. Much would be

achieved if the results from these investigations could be linked with the entrance data of the mathematical form definition.

') The number within brackets refer to the list of references on page 25.

3. Methods for construction of ship lines

With regard to basic conditions and working procedures, the con-struction of ship lines can be performed in different ways.

A common feature for the methods mentioned below is that the

requirements on cargo capacity, stability and other hydrostatic data can be satisfied in the same degree on basis of fixed main dimensions.

Also the demands on the hydrodynamic properties such as

resis-tance, propulsion and manoeuvring can be met in general. In that case however, there will be differences between the methods with respect

to the control of these qualities. The desirable connection between hull lines and hydrodynamic properties, as mentioned at the end of the previous chapter, is not realized to the same extent for all methods. Regarding the working procedure to full scale hull coordinates the methods differ in many respects. This is an important consideration for the future treatment of mathematical ship lines.

3. 1. Graphically defined ship lines

During the early design stage a preliminary lines drawing must be

available for the different departments of a shipyard. Information concerning hold and tank capacities, hatch and rig plans and also

engine room arrangements are preliminarily made up on basis of this drawing. Later modifications of the hull form due to model test results, etc. must not be so extensive that the main points of the ship's speci-fication are affected.

It is therefore of great importance that a suitable hull form can be

selected at once, and that the designer is in a position to use all

experience which has been collected up to date. In making use of this experience two different methods can be followed.

3.1.1. Ship lines based on single form coefficients

The conventional design of ship lines begins with selection of proper coefficients for main dimensions, fullness, angles of entrance and run,

bilge radius, characters of some specified waterlines and transverse sections, etc. Examples of Swedish investigations giving recommenda-tions about single form coefficients and general description of suitable

hull forms (see Chapter 4.1) are publications of the Swedish State Shipbuilding Experimental Tank [1], [2]1) and Chalmers University of Technology [3].

6

3.1.2. Ship lines defined by hull shape parameters from standard forms The method of designing ship lines by use of single form coefficients (block coefficient, prismatic coefficient, etc.) in combination with

recommended character of sections will result in several alternatives,

i.e. when the procedure is repeated by another draftsman using the same basic data, the final lines drawings will not be in conformity

with each other.

An attempt to introduce more complete form parameters for graph-ical delineation of hull forms has been made by GIJLDHAMMER [4], [5]. Direct use of these form parameters for waterline area, statical moment, moment of inertia, etc. involves a dependence on the given standard

forms, which can not always be recommended. However, the form

factors can be altered in accordance with latest practice, and keeping this modification in mind, the method is most valuable.

3. 2. Mathematically defined ship lines

Mathematical representation of ship forms means that the entire

hull surface is defined by a number of analytical functions, which are able to represent the ship satisfactorily and thus replace all graphical

description. The functions must be in such a form that they can be easily evaluated for every set of independent variables in the same way as when the hull is defined by a lines plan or a table of offsets. As in the case of graphically defined ship lines, mathematical delinea-tion work can start either from an empirical basis or from a set of form-defining geometric parameters.

3.2.1. Ship lines based on fairing of preliminary hull coordinates In this method the calculations start from the coordinates read off from the preliminary lines drawing. The procedure can involve simple interpolation as well as complicated smoothing procedures. An example of a fairing method where interpolation as well as smoothing are used

is the work of THEILHEIMER and STARKWEATHER [6].

Interpolation in the meaning that the mathematical hull surface must realize a number of coordinates from the preliminary lines

drawing must be avoided. After reading and enlarging, every coordi-nate from the designer's first drawing is in general affected with large errors. A great deal of the aim of the mathematical fairing can be lost

if such points are accepted to be included in the mathematical

hull surface.

SCANTIJNGS ANO /NJLL FORM PARAMETERS O/O/TAL COMPUTER ANALOGUE COMPUTER COMPUTER ALL SECTIONS DIRECTOR UNIT

WATERLINE SECTIONS ONLY

WATERLINE ANO TRANSVERSE SECTIONS OJO/TAL ORA WINO MACNINE ANALOGUE O RA WINO MACHINE

Fig. 1. Methods for construction of mathematical ship lines, block diagram.

3.2.2. Ship lines based on hull form parameters

The method of fairing preliminary hull coordinates by interpolation and smoothing gives, in general, good coincidence with original lines. Care can easily be taken to different details in the hull forni. On the

contrary, the ship surface can not be related to important hydro-dynamic form parameters (see Chapter 4.2). Certainly these can be

calculated afterwards, but there is no possibility to carry out desirable modifications by use of these parameters.

A study regarding mathematical ship lines based on essential hull

form parameters has been carried out at the Swedish State

Ship-building Experimental Tank [7].

The further account will deal with construction of ship lines on basis of hull form parameters with hydrostatic and hydrodynamic meaning according to [7].

A block diagram showing the main aids which can be used in the calculation and drawing of mathematical ship lines is given as Fig. 1. Details of the different calculation and drawing methods are reported in the following chapters. First, however, some further comments will be given about hull form parameters and the mathematical form.

4. Hull form parameters

4. 1. Single form coefficients

In discussions about hull form, it is common to make use of a num-ber of coefficients for main dimensions, fullness, slenderness, etc. Also

8

special terms are used regarding the character of the hull lines at critical spots.

The discussion will often concern length to beam ratio, beam to

draft ratio, block coefficient, length to displacement ratio and water-line angles fore and aft. For a general description of the water-lines regarding larger areas of the hull it is common practice to use expressions like U- and V-formed sections, convex and concave waterlines, etc.

The above is reasonable for an approximate description of the hull

form. On the other hand, if there is demand for an analysis of the hull lines, for example in order to investigate the connection with

results from model tests, the single form coefficients are unreliable. It is easy to find two hull forms with different hydrodynamic properties although they have same main dimensions, fullness, slenderness, centre of bouyancy, CWL angles fore and aft, etc.

4. 2. Form parameters as functions of draft

In order to keep a strict definition of the hull shape it is necessary to introduce form parameters affecting the entire surface of the ship. Thereby it is convenient from many points of view to characterize the form by use of a small number of draft functions.

The draft functions consist of the form parameters of the waterlines smoothed in vertical direction. The selection of waterline parameters is here in accordance with a special investigation reported in [7]. The conclusion of that investigation was that the waterlines may be satis-factorily defined by first and second derivatives at ends, which effec-tively direct the curves at stem and stern, and also by the area, which secures the over-all coincidence. This combination of parameters is used in the following account.

It follows that the waterline end coordinates are included in the

draft functions. In the normal case the end coordinates are represented

by the midship section, stem and stern contours and the boundary lines between the cylindrical part of the ship and the entrance and

run respectively. At special hull forms also other draft functions must be included, such as the contour of a transom stern or a bulbous bow. It is now important that the draft functions can give the hull form directly. Thereby a certain mathematical form will be prescribed for the waterlines. The equations must also realize the values of the draft functions at the actual waterline. A block diagram showing the calcula-tion of draft funccalcula-tions is given as Fig. 2.

O

MOOIPLCATION OP PRELIMINARY LINES

Fig. 2. Draft functions from main dimensions and hull form parameters, block diagram.

The introduced draft functions (see also Appendix 1) have primarily a purely geometric import. Moreover, it will also be possible to find connections between the draft functions and hydrodynamic properties of the hull, such as required propulsive power, course stability, sea-kindliness, etc. Considering this, there are further reasons to base the mathematical lines on form parameters of this kind.

5. Mathematical definition of ship lines

The hull form is defined by waterline polynomials. The coefficients

of each polynomial depend of the values and combination of the

selected form parameters. A complete account regarding the choice of mathematical form, alternative analytical functions and formulation of waterlines by polynomials is given in [7]. A short summary is to be found below.

A waterline section is expressed non-dimensionally, see Fig. 3, as

= 1--a

The number of terms is equal to the number of geometric conditions (prescribed form parameters) including boundary conditions. Before

the calculation of the waterline equations, the waterline form para-meters have been faired to draft functions valid for the entire ship.

The polynomial coefficients are to be calculated from the linear

system of equations formed by the geometric conditions (Vi). As an

example the condition for waterline area is given below, in

non-dimensional presentation.

i

nl

a.

o

ii+l

Thus the conditional equations form a set of linear equations, which can be written symbolically in matrix form.

SCANTLINGS CALCULATION PRELIMINARY

AND OP PAIRING Of FIXATION

PRELiMINARY HULL FORM DRAFT FLiNG, YES OF

y

'o

/

n-' L w

Fig. 3. Presentation of ship lines in coordinate systems.

2 X

/

/

/

/

D L.. f

L/2

..:llhII

B/2

- 8/2

/

X0 X, xn_, Xfl L WI-IF LWE Bw 2 2

FV11 V12

1

F'1]

FKvi

L21

]x ial]

= LV2

where V1, V2, ... is the matrix of coefficients belonging to the equa-tions for condiequa-tions Vi, V2,

a represents the coefficients of the waterline polynomials

K1, K2, ... represents the known terms of the conditional

equations, which enclose the given form parameters.

6. Application of digital computer to construction

of mathematical ship lines

The main calculations for mathematical lines start from the draft

functions, which have been fixed before by aid of a separate proce-dure, see Fig. 2.

The known terms of the conditional equations are taken from the

draft functions, and after that, the system of equations can be solved, waterline polynomials determined and hull coordinates calculated. The

computer application in this connection appears from Fig. 4. EVEN TUAL

LINES DRAWING RY HAND

COMMAND SIGNAL GENERATOR AUXILIARE FUNCTION COMMAND SIGNAL CONTROL UNIT AUXILIARY FUNCTION CONTROL SIGNAL POSITION FEED RAC/( SIGNAL

DIGITAL

DRAWING

MACMINE

Fig. 4. Application of digital computer to construction of mathematical ship lines,

block diagram. CALCULATION TRANSFORMATION OF WATERLINE INPUT ORAFE TO CURVE UNIT OF

FUNCTIONS POL l'NON/ALI ELEMENTS FOR _DIRECTOR

ANO HULL _{DIRECTOR UNIT} COORDINATES

CONTOURING CONTOURING

12

Fig. 5. ARISTOMAT numerically controlled drawing machine.

On basis of the calculated coordinates a lines drawing can be made by hand This is however a hard work in case of numerous coordinates. It is therefore desirable to replace that work by use of a digital drawing machine connected to the computer.

This stage of the investigation work has been carried out in coopera-tion with the Swedish Shipbuilders' Computing Center, where an

ARISTOMAT numerically controlled drafting equipment is installed,

see Fig. 5. A separate service program [8] has been worked out for

transformation of the numerical data into curve elements suitable for the director unit connected to the drawing machine.

The director unit, see Fig. 6, is of the ESSI/KONGSBERG type {9], which permits information for continuous path control in the form of parabolas, circle arcs and straight lines. For automatic control of all

machine operations, the director must perform the following three functions:

Input unit with tape reader

Control panel with

selectors for scale

factors and other

operations Fig. 6. ESSI/KONGSBERG director unit.

Generation of command signals for the plotting mechanism. Control of machine operations.

These functions are performed by separate units within the director, see Fig. 4.

A number of waterlines traced by the digital drawing machine are shown in Fig. 7.

7. Application of analogue computer to construction

of mathematical lines

In the preliminary design stage it will sometimes be necessary to

dimen-Fig. 7. Waterlines traced by numerically controlled drawing machine. (Notice,

sions and principal form coefficients, a number of preliminary lines

drawings can be worked out according to the actual method, taking

into consideration proper form-defining parameters from favourable hull lines for similar ships. For rapid work an analogue computer with a connected tracing machine is most valuable, especially for studying the influence of varied form parameters.

The geometric conditions V for a waterline section, taken from the draft functions, can be included among the coefficients of the corre-sponding polynomial. Thus

= 1+a

where

a = 1cV

Then the general polynomial coefficients consist of one constant part

k and one variable V, which represents the parameter values. The

coefficients k1» which are determined separately, depend only on the

selected set of parameters and the polynomial forni, which are the

same from ship to ship.

According to the above, the input data of the analogue computer

consists of the draft functions and the fixed coefficients k13 connected

to the parameter set. Generation of powers of

, calculation of

resulting coefficients a and summation of polynomial terms are carried out by the integrators and summators of the analogue computer. Setting of coefficients and parameters are performed by use of the potentiometers of the control unit of the computer, which also provides easy digital control of entrance data.

The output signal transmitted to the drawing machine represents

the polynomial, which can then be traced directly. In this case there

are no complications in form of required transformation to curve

elements, as when the ship lines are to be traced numerically.

A block diagram showing the application of an analogue computer to the construction of mathematical ship lines is to be found as Fig. 8. A corresponding principal scheme is presented as Fig. 9. Photographs

of control panel and drawing machine are given in Fig. 10 and 11. The lines drawing according to Fig. 11 is small and could have greater accuracy. If, however, the method can be further developed and a larger drawing machine can be used, it might be possible to

16

produce the first proposals to ship lines in this manner. As the total

time for drawing the lines is very short, a number of alternative

forms can be studied, which all can meet same special requirements as defined by some of the draft functions.

DRAFT FUNCTIONS ç FIXED COEPF CONN WITH PORN PARAN DIGITAl. CONTROl. OF FAÑAN VALUES CONTROl. UNIT GENERATION OF POWERS REFERENCE SIGNAL

Fig. 8. Application of analogue computer to construction of mathematical ship lines,

block diagram.

basic form (VO)

p

o

d7

o

o

.summator integra/or

o

potentiometer R reference s/gnat H SU/INATION OF PARAN ANALOG/JE DRAWING tRACH/Nr

Fig. 9. Application of analogue computer to construction of mathematical ship lines,

Digital reading of coefficients and parameters Connections for generation of Dowers of

and

summation

Setting of coefficients and parameters

Setting of breadth of waterline at; et em Setting of maximum breadth of waterline Setting of longitudinal position of stem Setting of longitudinal. position of max. breadth

Fig. i L EAI-VARIPLOTTER tracing a number of preliminary waterline sections for a container ship. (Notice, waterlines are compressed in the longitudinal direction and

are not equally spaced.)

8. Application of "hybrid" computer to construction

of mathematical lines

As pointed out in the previous chapters, the digital and analogue

DRAFT FUNCTIONS

with tracing ship lines. Summing up, following points of view can be given, see also Fig. 1.

The digital system provides good accuracy and permits the drawing of ali kinds of sections. However, when used for tracing small preliminary drawings at the projecting stage, the system will be too expensive.

The analogue system is easy to handle and will be capable to trace lines with sufficient accuracy for ship project purposes. By use of

the present method, it is however impossible to draw sections

other than waterlines.

The rapid development of computer elements during recent years

now makes it possible to build up "hybrid" computers consisting of special "intermediate" units and ordinary components from digital as well as analogue computers.

A block diagram showing the application of such a "hybrid" com-puter is given in Fig. 12. The basic elements are the analogue comcom-puter and analogue drawing machine, as before, but now completed with an analogue to digital converter and a digital memory unit.

Starting from the draft functions the analogue computer will produce an output signal varying with the actual polynomial. During the trans-mission of the signal to the drawing machine its tension at longitudinal stations (waterline ordinates) are registered and brought to a digital memory unit via an analogue to digital converter.

Fig. i 2. Application of "hybrid" computer to construction of mathematical ship lines,

block diagram. ANALOGUE COMPUTER WATERLINE SEC TIONS GENERATION OP POWERS AND SUM/RATION OF ANALOGUE STORING OF WC ORDINA TE, GROUPING OF MEMORIZED ANALOGUE DRAWING PA RA ME TERS TO DIGITAL CONVERTER IN DIGITAL COORDINATES

FOR TRANSVERSE MACHINE MEMORY

SECTIONS

INTERPOLATION TRANSVERSE

TO CURVE SECTIONS

When all waterlines are traced the digital memory will contain coordinates for drawing also transverse sections. Reading of the

memory is then so arranged that consecutive ordinates, valid for the transverse sections, are supplied to the computer passing the converter. Separately, an analogue interpolation to curve elements is carried out, and after that, all required transverse sections can be traced, see Fig. 12.

9. Acknowledgement

The author wishes to thank Dr. HANS EDSTRAND, Director of the

Swedish State Shipbuilding Experimental Tank,

for having been given the opportunity to perform this investigation,

as well as for the interest he has shown in it. Thanks are also due to

the staff of the Tank, specially to Mr. Sigurd Göransson, who worked out the analogue computer operating procedures.

Appendix 1. Form parameters of two alternative

hulls for a side trawler project

As mentioned in Chapter 4, it is necessary for a strict definition of

the hull shape to introduce proper form-defining parameters. In the first instance, the parameters are introduced with regard to the fol-lowing mathematical treatment as these form the draft functions,

from which the different procedures can be started.

Also in connection with other problems, draft functions of this kind are useful e.g. for analysis of model test results [10]. An example will be given here.

Fig. 13 and 14 show transverse sections and stem and stern contours

of two alternative hulis of a side trawler project. Suppose that pro-pulsion model tests have been performed and the results have been

analysed according to [10], i.e. required shaft horse power for each of the hull versions have been compared with the results for a standard hull form in combination with a comparable standard propeller. Sup-pose also that version B has shown better propulsive qualities than A.

However, the above result is not complete, one question has not

been answered: Why is version B "better" than A?

It is evident, that the result of the analysis, given as "propulsive

quality" coefficients, will have greater value if the corresponding hull

WLI

BL

Fig. 13. Side trawler project, transverse sections. Alternative A.

- - - Alternative B.

Fig. 14. Side trawler project, stem and stern contours.

WL WL 4 WL 7 WL 5 WL 2 WL 7

II&4

___MMMßE/áì

L'

rDWM/IMIIIWL

LrßisIIMIIiIWL4

ßf7/MuII/.

1W7/ÄW.

k

i

i

f

lIz 2 8/z 9 9Va 10 1/2

22

work, this is extremely important, as the aim is to find connections

between hull form and attained hydrodynamic qualities.

Neither the hull versions A and B nor the differences between them

can be satisfactorily defined by an optical survey of the body plan given in Fig. 13. For example, the differences in hull form can only be expressed by phrases like "somewhat slender sections in forebody", "more hollow waterlines near DWL", "more U-form in afterbody", etc. It appears from Fig. 15 and 16, that the hull forms as well as the dif-ferences between them can be better expressed by the draft functions.

Appendix 2. Outline of the practical procedure when designing

ship lines by use of analogue and digital computers

The procedure in practice, from the main data stage of the project

to the final mathematical lines drawing, may follow two possible

WI. 7 W1 6 WL 5 WI. 4 WI. j WI. j WL ßL. 0 0.03 A wp 75 -0.3 -04 0.01 0 -02 -0.1 -002 XO

Fig. 15. Side trawler project, forebody draft functions. Alternative A.

- - -

Alternative B.

AwF = waterline area, forebody.

(dyfdx) = ist derivative at stem. (dy/dx2) = 2nd derivative at stem.

= ist derivative at midship section.

Wi 7 Wi 6 Wi 5 wt_ 4 Wi 3 WL 2 WL /

/

I

/

I AW4

//

/

/

/

_/

/

/

/

/

/

/

7

7

7

7

alternatives. These are represented by the two branches in the block diagram in Fig. 17.

The working scheme indicated by the left hand branch in Fig. 17 might possibly be followed in the first instance, as this alternative does not differ so much from present practice. A preliminary lines drawing is traced by hand in the normal way and then model tests

are undertaken. If the results are not satisfying, the lines are redrawn and tested again.

Now at last the work begins to define the lines mathematically, see Fig. 17. The form parameters of the model tested lines are

cal-culated, the draft functions are faired and a preliminary

mathe-matical lines drawing is traced for survey. If the differences are greater than what is owing to fairing, or are not satisfying from other respects, the procedure must be repeated.

The above outlined method may have one disadvantage. In special

0 25 50

02 84 O6 08

Q; 005 0 -005 -a;

fdy/dL -002 Fig. 16. Side trawler project, afterbody draft functions.

Alternative A.

- -

Alternative B.

AWA = waterline area, afterbody.

(dyJdx) = ist derivative at stern.

(d2y/dx') = 2nd derivative at stern.

(dy/dx)0 = ist derivative at midship section.

24 Calculation of form-parameters. Fairing oC draft functions Pro liiuinary linos drawing yes no Prel. mathematical lines drawing Satisf. coincidence? no Project main data no Model tests Analysis Patisfying result? -y es Final mathematical lines drawing Choice of proper draft functions Mathematically def.

Prel. lines, digital or analogue mach. Satisfying result? no Yes Eventual complementary draft functions

Fig. 17. Outline of the practical procedure when designing ship lines by use of analogue and digital computers, block diagram.

tested lines, that the hydrodynamic properties are affected. It is

there-fore desirable to introduce mathematically defined lines bethere-fore the

model tests.

All systematic work regarding ship forms as well as analysis of model test results for ships to be built, will result in experiences regarding favourable form parameters, which can suitably be presented as draft functions. It will then be possible to start the design of hull

lines for an actual ship project with choice of a set of proper draft

functions, see right branch of Fig. 17.

If the first combination of draft functions does not give acceptable lines, some of the functions must be modified and a new tracing must be performed. When the analogue method is used, a number of pre-liminary lines drawings can be traced within a short time.

An eventual modification after the model tests can be directly applied to the draft functions, which can be then adjusted in accor-dance with recommendations based on the analysis of the model

test results.

Before the final mathematical lines are calculated, some

comple-mentary draft functions (regarding stem and stern roundings, etc.)

may be incorporated.

References

EDSTRAND, H., LINDGREN, H.: "Systematic Tests with Models of Ships with

SSPA Publication 38, 1956.

FREIMANIS, E., LINDOREN, H.: "Systematic Tests with Ship Models with

0.600-0.750", SSPA Publication 44, 1959.

LINDBrAD, A.: "On the Design of Lines for Merchant Ships", Transactions of

Chalmers University of Technology No. 240, Oothenburg, 1961.

GULDHAMMER, H. E.: "Formdata, Some Systematically Varied Ship Forms and

Their Hydrostatic Data", Danish Technical Press, 1962.

GULDUAMMER, H. E.: "Formdata II, Hydrostatic Data for Ship Forms of Full and Finer Type", Danish Technical Press, 1963.

THEILHEIMER, F., STARKWEATRER, W.: "The Fairing of Ship Lines on a High Speed Computer", DTMB Report 1474, 1961.

WILLLMS, A.: "Mathematical Representation of Ordinary Ship Forms", SSPA

Publication 5.5, 1964.

HÖOLUND, N.: "Algol Procedure for Tracing Functions on an Aristo Drawing Machine

with EssiJKongsberg Director Unit", VBC Subprogram 1, 1966, (in Swedish).

"The ESSI Continuous Path Control System for Machine Tools", Kongsberg

Vàpenfabrikk, Norway.

WILLIAMS, A., JonssoN, L. G.: "Analysis of Results from Resistance and Propul.

Contents

l'ago

Pref ace ₃

Summary 3

Introduction 4

Methods for construction of ship lines 5

Hull form parameters 7

Mathematical definition of ship lines 9

Application of digital computer to construction of

mathe-matical ship lines 11

Application of analogue computer to construction of

mathe-matical ship lines 13

Application of "hybrid" computer to construction of

mathe-matical ship lines 18

Acknowledgement 20

Appendix 1. Form parameters of two alternative hulls for a side

trawler project 20

Appendix 2. Outline of the practical procedure when designing ship lines by use of analogue and digital computers 22