Mathematical definition of ship surfaces

DANSK SKIBSTEKNISK FORSKNINGSINSTITUT

DANISH SHIP RESEARCH INSTITUTE

Lyngby

Danmark

REPORT No. DSP-14

s s

.1...;

_.

_{. ...}

0

_Ò

s O s l

.

¶ 0

.

s

..

'

.

s

.0

s Ss _s

Is

.

s s s

... s.

_s

\

s.

555

5 s

S

sS

_s

?

₅

. .

.

s

.5

s.

.

5

.

₅₅₅₅

S

5s5

.

.

S

050

S S s

.

0

.0

o.. .

.0

s S o

.

o o

Mathematical Definition

of Ship Surfaces

IN COMMISSION:

DANISH TECHNICAL PRESS COPENHAGEN V - DENMARK

b

by

DANSK SKIBSTEKNISK FORSKNINGSINSTITUT

Is a self-supporting Institution, established fo carry eut research in the fields of shipbuilding, marine engineering end ship-ping. According of Its by-laws1 confirmed by his Majesty the King of Denmark, it is governed by e council of members elec-ted by shlpbuilders, shipowners and the Academy of Technical Sciences.

The report s are on sale through the Danish Technical Press at the prices stated below.

The views expressed in the reports are those of individual authors.

DSP No. Author Title Price D. Kr.

J. STRØM-TEJSEN Damage Stability of Ships 8,00

2 J. STRØM-TEJSEN Damage Stability during the Period of

Flooding 8.00

3 ULRICH JOHANNSEN Recording Ship Stocks and Spare Parts 5,00

4 GERT ADRIANSEN ALGOL-program tu brug ved

oplgning af propelleraksler i skibe 5,00

5 ULRICH JOHANNSEN Rationalized marine diesel engine

operation and maintenance 5,00

6 Ship Trial Trip Code 1964 10,00

7 GERT ADRIANSEN An ALGOL-programme to Facilitate the

Allingment of Ship Propeller Shafts 5,00

8 C. W. PROHASKA Factors Influencing the Choice of the

Principal Dimensions and Form

Coefficients of Ships 10,00

9 ULRICH JOHANNSEN Skibsberegningspregrammer for

elektronregnemaskine 5,00

10 HARALD WESTHAGEN

and

Launching Calculation and Analysis 8,00

E. KANTOROWITZ

11 ANDERS BØGELUND ALGOL-program Iii brug ved beregning

at rammekonstruktioner

-

5,00

12 RICHARD NIELSEN Jr. Analysis of Plane and Space Grillages

under Arbitrary Loading by Use of the

Laplace Transformation 50,00

13 G. R. DICOVI Damage Stability Calculations by

Electronic Computer 5,00

Den polytekniske Lreanstalt, Danmarks

tekniske Højskole, har antaget denne

afhandling tu

forsvar for den tekniske

doktorgrad.

E. Knuth- Wint erfeldt

rektor

Paul Carpentier

inspectør

Lyngby den 4 april 1967.

Forsvaret finder sted

torsdag den 6 juli 1967 kl. 14.00

i auditorium 12 bygning 308

MATHEMATICAL DEFINITION

OF

SHIP SURFACES

by

E. Kantorowitz.

t25D

_Z&I/

1967

Lyngby Denmark.

Eliezer Kantorowitz 1967,

This report has been typed by Mrs. B. Cribb

on punch tape controlled Friden

Flexowriter,

and has been printed by Polyteknisk Trykkeri,

Acmovledgement.

The present method was

developed

at the

Danish Ship Research Institute

in the period

1961

1 966.

The first experiments vere done

with

a primitive

assembly

laxiguae on the heavily booked DASK computer,

and vere thus very.

cumbersome

_{The installation of a GIER computer at the premises of the}

In-stitute was, therefore,

of decisive importance.

The GIER computer, which is

produced by the Danish firm Regnecentralen, baa one of the most efficient

ALCOL compilers

hitherto written, which enabled many different methods to be

tried with relatively little menual effort.

Late in 1962, it was decided to try to fit polynomials

directly to the

ship surface

A computing procedure

for this purpose was

developed In

the

spring of

1963

with the advice of Mr. P. lvbndrup, Regnecentralen.

Thiring the

same period preliminary

curve fitting and data test prograes vere prepared

with the assistance of Mr. Z. Zarhy. These programs were used for the

ex-periments

required

for the further development

of this method. There were

many problems

to be solved and a multitude

of parameters which could be

varied. The techniques developed at that time solved the problem in principle,

but there vas still much to be done before

the method

could be used practi

cally. During that period, reports arrived concerning poor results obtained

elsewhere with the same subject, and doubts arose in some circles whether the

problems could be solved at all. Deputy director H. Fogli of Nakskov Shipyard,

therefore,

examined the results which bave been obtained with a tanker arid a

cargo liner, and recomnended the continuation and Intenaivation of the work.

Mr. Anders Bøgelund and Mr. Gert Adiansen thus joined the project in the sum..

mer and aitumn of i 96i. These highly qualified co-workers enabled a thorough

discussion of the different

aspects

of the work. Furthermore

Mr. Bøgelund

prepared

_{the fina], curve fitting program, and some auxiliary}

_progras,

while a progrune for preparation of mould loft tables

was prepared by Mr. G.

Adriansen and Mr. A. Bøgelurid.

_{A very fast machine code version of the}

sur-face fitting prograimne

_{was prepared by Mr. P. Fleron. Finally}

_{in November}

1965

the programme was released for the use of the customers of the Institute,

and has since been used for

33

different ships.

Further to the above mentioned contributions

the success of the present

work is due to the information and encouragement obtained from the shipyards

of Nakskov,

Helsirigør

and

Odense,

and

to the assistance

of the computer

center staff,Mr. S. Velschou M.Sc., Mrs. I. Nielsen, Mrs. B. Michaelsen,

and

have checked the &glish wording of this report,and Mi.a I. Larsen bas drawn

the figures.

The author is very grateful that he has been given the chance to do this

work. The confidence and support shown by the director of the Institute,

professor C.W.Prohaska, and later by director Otto Petersen and deputy direc-tor H. Fogli of Nakskov Shipyard have been of decisive importance to the accomplishment of this project.

Lyngby E. Kantorowitz.

Sumnry.

1.

Introduction. 11,

2.1.

!.thematical versus graphical definition of ship forms.

1.

2.2.

Aim of the work.

5.

Review of earlier works,

7.

3.1.

Sectional curves methods.

7.

3.2.

Draught functions methods, lo.

3.3.

Special draught functions methods..

12.

1.

Description of' the method, 15.

14.1. Principal characteristics of the present work.

15.

Ii,2.

The inathemtical representation of a ship.

20.

14.2.1.

Definition of the hull surface.

20.

1,2.2,

Definition of' curves,

22.

14.2.3.

General considerations on the calculation methods employed.

27.

14.2,14. Calculation of

the

curve coefficients.

28.

14.2.14,1.

Calculation of a chain composed of' a single polynomial.

29.

14,2,14,2,

Calculation of a chain of polynomials.

30.

14.2.14.3.

The numerical power of the method.

32.

14.2.14.14, Calculation of the conic sections,

55.

14,2.14.5.

Calculation of circular arcs with prescribed radii. 314,

li,2,5,

.Calculation of the surface coefficients.

35.

14.2,5,1.

The conditions imposed on the surface polynomials.

35.

Ii.2.5.2.

The method used for obtaining smooth transition between

56.

the polynomial surfaces.

14,2,5.3.

The economy of two alternative methods for the calculation 57. of a polynomial surface.

14.2.5.14.

The method used for the claculation of' a surface polynomial. 38.

14.2.5.5.

The method used for imposing the boundary conditions on a

39,

surface polynomial.

5.

Implementation of the method on the HyA-GIER computer. 143.

5.1.

Programming techniques employed.

5,2.

Check of the input data.

5.3.

Calculation of the ship definition, 145.

5.14.

Check of the results.

5.5.

Preparation of the data required for the claculation of' 118.

5.6.

preparation of data for the surface calculations.

51.

Discussion of the results.

53.

6,1.

The polynomials suitable for the surface representation.

5i.

62.

An accurate representation of any ship surface is 511. obtainable.

6.3.

The connection betveen the different parts of the ship

55.

surface.

6.Ii.

The value of the ability to represent any ship shape

55.

accurately.

65.

' The manual effort needed to prepare a ship definition.

56,

6.6.

The computer time and computer storage required.

57.

6.7.

The ship definition is suitable for future applications, 58.

6.8.

Comparison the sectional curves and the surface expression

59.

method.

Conclusions.

60.

References.

62.

ApPENDXCF.

The method used for fitting a polynomial to a surface.

67,

The data forms used for the GIER computer.

72.

Real roots of a polynomial having real coefficIents. 78.

i. Calculation of a chain of polynomials.

81.

1. Sumn.ry.

The coming of electronic _{computers and numerically controlled tools has} enabled the automation of a considerable part of the shipbuilding process. The implementation of such techniques necessitates, however, a mathematical definition of the shape of the ship surface instead of the graphical

repre-sentation hitherto used. The object of the present work is to provide a

method for the preparation of a mathematical definition of a ship surface given by a small scale design lines drawing. The design lines drawing is still considered as the most useful], tool for designing ship lines as it is

a clear way of elaborating a compromise between the different technical

requirements.

Earlier work in this field can be divided into draught functions methods and sectional curves methods. In the former, the parameters of the waterline expression are defined as functions of the draught. The sectional curves methods are a mathematical analogy to the traditional graphical methods. _The thin elastic battens used by draughtsmen are thus substituted by .,Spline

functions,,. _{These functions are fitted to a number of waterlines and} trans-verse sectional curves in an iterative process, which is continued until the two sets of sectional curves represent the same surface. In some of the

earlier works mathematical transformations are employed, whereby the ship shape is modified into a surface which is more easily fitted. Only the sec-tional curves method has hitherto enabled an accurate definition of a wide range of ship shapes. There are, however, some difficulties in obtaining

sufficient smooth surfaces with this method. The solution of the surface definition problem by means of draught functions, sectional curves methods and the employment of transformations introduces the problem of how to handle these devices.

In the present method, mathematical _{expressions are fitted directly to} the ship surface and to the different contour and boundary curves. The ship

surface and the different curves are divided Into the parts of which they are composed, to which polynomials and arcs of conic sections are then fitted. Polynomials are employed In most cases, but arcs of conic sections are mostly used for the parts which are perpendicular or nearly perpendicular to the abscissa plane/axis.

The division of the surface and the different curves into the parts of which _{they are composed is done manually, and is based on a visual analysis} of the lines drawing. The limits between the different parts of' the surface are usually the lines of _{discontinuity In the second and higher order} deriv-atives found on the ship surface (Fig. _).

-1-

-2-The coeffioientB of the expressions defining the ship are calculated

from fixed data, such as the maximum breadth of the ship, and from coordin-ates lifted from the design lines draving. As the lifted coordincoordin-ates are in-accurate, the coefficients are calculated such that the fixed data are main-tained. and such that the curves and surfaces give the best fit to the lifted coordinates using the least squares criterion.

When calculating the curves, any group of connected polynomial curve elements is calculated separately. AU the coefficients of the polynomials constituting such a group are calculated simultaneously by solving the normal equatiOns. The system of normal equations is extended such that the different conditions imposed on the curve are satisfied. e.g. that the curve passes through fixed points or that a&jacent polynomial elements have common

ordin-ate and tangent at the point of connection. The extension of the normal

equations is obtained by applying Lagranges multipliers.

A surface polynomial P(x.z) defines the halfbreadth y of the ship as a

function of the distance x from amidships and the draught z, i. e.

y = P(x.z).

The polynomial P(x,z) is calculated as a linear combination of a set of orthogonal polynomials. These polynomials are generated so that they are orthogonal to each other with respect to the configuration of the abscissas (x,z) of a number of points where the halfbreadths have been lifted from the lines drawings. Boundary conditions are imposed on the surface polynomial by introducing a number of points that satisfy the conditions, and by assigning high weights to these points during the calculations.

Since ships have quite complicated shapes, a considerable amount of data is needed to specify all the details of a ship surface. The manual lifting

and the checking of these data has proved to constitute

a major part of the

whole process. Special prograes have therefore been prepared for check of'

the input data. Furthermore, a programme was prepared to test whether a

cal-culated mathematical surface is acceptable. This programme compares, the

halfbreadths lifted from the drawings, with the corresponding mathematical ones, and checks a number of properties such as the concave and convex

character of a number of selected sectional curves. Furthermore the body plan

of the faired ship is drawn by the computer for visual analysis of the char-acter of the lines. All the programmes, except the procedure for fitting polynomials to surfaces, were prepared in ALGOL. These programmes have been employed successfully for the preparation of accurate definitions for ships

-5-of all normal types and for a number -5-of specilized vessels.

The straight _{forward approach of the present work resulted in a method}

which _{is relatively easy to learn and use. The computer time and computer} storage required for the preparation of _{a mathematics], ship definition, and}

for the use of the definition _{in future programmes are moderate. The present} method has thus proved suitable as a routine _{procedure in the shipbuilding} industry.

2. Introduction.

2.1. Mathematical versus graphical definition of ship forms.

In most shipyards today, shipbuilding technique is based on the use of a graphical definition of the ship surface. The shape of the ship surface is thus defined by a lines drawing, which shows a number of sections through the ship. The dimensions _{needed when building a ship are lifted or derived from} this lines drawing (Fig. 2). An alternative to this graphical technique is to define the ship surface by means of a number of mathematical functions which can then be used for the calculation of any dimension needed. Such a mathem-atical-numerical ship definition was, however, not practically possible for

shipyards before the coming of

high

speed automatic computers. Mathematics]. def initions of somewhat idealized. shipforms have nevertheless been used for

many years for hydrodynamic calculations on the research level.

The coming of' numerically controlled tools enables a considerable part

of the ship production process to be automated (Fig.

3).

The data which are fed into these machines and control their functions can either be prepared manually or by means of digital computers. As the amount of work involved in the preparation of these data is considerable, the use of computers can be

expected to be the rule. In order to be able to use a digital computer for this purpose the shape of the ship must be mathematically defined.

One of the most important concepts In the field of computer applications is that of Integrated data processing. i.e. letting a principal part of the Information-flow and information-processing within a concern occur within one common electronic data processing system. In such a system all Information

used more than once can be stored In a common store, where it Is accessible for later processing in a fast and reliable way. The ship form Is an example of information which is used many times during the shipbuilding process. 1'he

ship form Is thus used for detérmination of the shapes of plates and frames, for the positioning of equipment and machinery inside the hull, for hydrosta-tic calculations, for calculations of volume of cargo spaces and so on.

A ship surface Is composed of a number of parts having different shape characteristics such as the stem, stern, parallel-body and flats of bottom. An accurate mathematical definition of such a complicated surface involves a considerable number of parameters, and is therefore not suited for manual calculations. 1hen computers are used, the evaluation of a complicated math-ematical expression does not, however, constitute a practical difficulty. It

- ___;-__

-

/

';

:

-J

/

T:Y/7.

:-..-.: :.Z

-'T

-

A -.

--

...,

I

. -.

-i--.

. ...-.. .

L

- J.. _-

-,.__----/ ¿7

T---.,

Ii,. i

t

('--I..

11

i

--.---.

.,-._.1 .-.-. --

.,. »

t---I .. - s - .

..'!7

:r,i

. I

C

1Ir,.t

.

rii--

?

;'ik

Fig. 2. Mould loft in a

traditional yard.

The

figure shows the

pro-duction

of

wooden

templates

for

some

ship

plates using the

sectional curves drawn

on the floor of the loft

in

full

scale.

The

templates

are

later

used for marking on

the plates the edges to

be cut (the figure stems

from L.A. Kisby,

Ski-.

bet, Copenhagen 1949).

i

Fig. 3. Numerically controlled plate cutting machine (made by Messer).

The computer produces a punched tape giving the coordinates of a number

of points on the plate along the edge

_{to be cut. This tape is put into the}

director which then automatically controls

_{the cutting machine. The ESSI}

director which is seen to the right is

_{produced by Kongsberg}

Vaapen-fabrikk, Norway (Werkfoto Messer Grisheim

G.M. B. H.).

'I,

_J T'

-5-replace lines drawings i the building processes, such that electronic data processing systems and numerically controlled tools can be utilized.

The complexity of mathematical ship definitions causes some difficulties when new ships are to be designed. During the process of designing new lines. a ship form that satisfies a number of different requirements is developed. The naval architect operates with a conception of the shape which is easily

expressed graphically Mathernatial expressions are not suited to this Job as the creative power of the designer is burdened with the irrelevant problem of how to describe a given shape mathematically. It is therefore likely that

lines drawings will continue to be a tool for designing new ships, and that the mathematical definition of the surface will be produced from these draw-ings.

It is possible that computers some day will be used for the preparation of the design lines drawings, such that the design process results directly

in a mathematical ship definition. Such a technique seems possible as devices for in/output of graphical data are fast developing. One may imagine a design. process where the lines drawings appear ori a ,,screen,. connected to the com-puter. and where the designer Improves the lines on the screen by means of a

special pen. The process may for instance start by the designer specifying a modification to be employed ori a parent ship, the data of which are stored In the computer. The lines of the resulting ship then appear on the screen, such that the designer can either accept, or try to Improve the shape. The search for a satisfactory solution is facilitated by the ability of the computer to calculate the different characteristics of the designed hull In a short time. The techniques of computer-aided design have already been tried experimental-ly [21]. There are, however, many problems to be solved before these tech-niques can be fully utilized. One such problem is that of representing a ship surface in a digital computer. i.e. of preparing a mathematical definition of a

ship

surface.

2.2. Aim of the

work.

The purposes of the present work are:

To devise

a

system of mathematical functions that can be used for the

representation of most ship surfaces with the accuracy necessary when building a ship.

To develop a method for the calculation of the numerical values of the coefficients of a mathematical representation of a ship surface given by a design lines drawing.

-6-As the

design

drawings are made to a very small scale, the lifted data do not represent a surface which is smooth

enough

for

building

a ship. The

method for the calculation of the mathematical surface representation must

therefore include a fairing of the lifted data. The deviations between the faired mathematical surface and the lifted data should, however, be of the same order of magnitude as the errors in the lifted data. This requirement for , .practical identity,, between the surfaces represented by the design

lines drawing and the mathematical expressions, is introduced though consider-able deviations may be accepted in some cases. Discussions regarding the ac-ceptance of such big deviations are thus avoided, helping to reduce the time needed to arrive at a mathematical ship definition, and eliminating a possi-ble source of friction. It is furthermore undesirable that designers of new

ships should be restricted by a mathematical method which limits the shapes that can be used.

The method developed should be suitable for routine procedures in the shipbuilding industry. It should therefore permit the production of the

mathematical

ship

definition within a short time, and with a minimum of

3. Review of earlier works.

The subject of mathematically defined ship hulls was treated by Chapman

[10] as long ago as the middle of the 18th century. and has since attracted many researchers. A review of works prior to 1955 is found in [3h.), while later works are reviewed in [28] [148] [32]. Furthermore, _{a comparison between} hi. specific methods is found in [13].

The subject has been considered from two different points of _{view, that}

of the practical shipbuilder and that of the hydrodynamist. _{The interest of} the hydrodynamist stems from the need for a mathematically _{defined ship hull} when calculating the hydrodynamic forces on ships at sea. For this purpose the definition should preferably be relatively _{simple, in order to}

facili-tate the manipulations _{called for by the equations involved. As hydrodynamic} calculations made hitherto, are based on a number of rough approximations, the mathematical definition has not needed to describe _{the ship form very}

accurately. Mathematical hull definitions, _{which are primarily intended for} hydrodynamic calculations, are _{treated in [20]. [25]. [26], [38] and} [14i].

The interest of the practical shipbuilder stems from the need for a mathe-matical definition of the ship in order to utilize digital computers and

numerically controlled tools in ship production. The ship definition neces-sary for this purpose must be very accurate. Complicated mathematical expres-sions may, however, _{be accepted for this task as computers will be involved} in such production techniques. As the approach of the present work is that of the practical shipbuilder, only works related to this approach will be reviewed.

The different methods published in the field rep?esent all degrees of completion, from proposed methods which have never been tried, to methods which have been tried on a number of ships. The different _{methods will,} how-ever, be reviewed without regard to the degree to which they have been

developed.

The methods published may be divided into two groups, sectional-curve methods and draught-function methods.

.1. Sectional curve methods

_{11 1171 1181}

₁₂₁₄₁

_F211.

The numerical methods reviewed In this chapter are mathematical ana-logies of the traditional graphical methods for the fairing and definition of

:3hip

surfaces. These methods are based on the use of up to four different

systems of sectional curves, namely transverse sections, waterlines,

but-tocks and diagonals. These curves are obtained as the intersection of the

-7-ship surface with the four systems of planes given by equations of the types x=a, z#b, y=c and ridz+e. where x, y, z are the variables (see Fig. i) and

a, b, c, d, e are constants.

The ship

surface may be considered as being composed of an infinite number of transverse sectional curves distributed over the length of the ship. The surface may similarily be defined by an

infinite set of any of the three other types of sectional curves. The ship

surfaces defined by the different sets of sectional curves should of course coincide. The reason for employing different types of sectional curves dur-ing the fairdur-ing process is to ensure that the ship surface obtained, is

smooth in ali directions.

On the lines

drawing it is of course only possible to draw a finite

number of the sectional curves, and any additional curve which might be

needed is obtained by graphical interpolation between the drawn curves. The drawing of the sectional curves and the graphical interpolation between them

is done by means of either a thin elastic batten or French curves. French

curves are templates, _{the edges of which have shapes cormnonly met with when} drawing ship lines.

Lidbro [273 divides the ship surface into about 200 parts. In each part the transverse sectional curves are defined by polynomials or by arcs of conic sections. This mathematical method thus corresponds roughly to the drawing of ship lines by the means of French curves. Unfortunately the pa-per gives no information about the methods used for calculating the coef-ficients of the functions used. The other workers in the field use the

spline function which is the mathematical analor of the elastic batten.

The spline function concept vas developed by Theliheimer [1i] _from

elementary theory for elastic beams. According to this theory an elastic

batten supported at a number of points, as is the case when drawing ship

lines, is represented by a third order polynomial _{(a cubic) for each} inter-val between two adjacent support points. At a support point the two acLjac-ent cubics will have common first and second derivatives, while the values of the third derivatives will normally be different. Such a serie of cubics is therefore defined, as a sTline function.

The assumption made in the elementary theory of elastic beams, i.e. that the deflections are small and that the beam material Is linearly elas-tic, do not apply very well to actual deflections _{and to the wooden or} plas-tic battens used when drawing ship lines. The usefuilness of' the spline

function _{stems, however, from the fact that its third derivative Is} discon-tinuous at a number of points. This enables the spline function to change its character significantly at these points of discontinuity. The

flexibil-

-9-ity of the spi inc function may be illustrated by the fact

that a apUne

curve can be composed of both curved and straight parts.

It seems at a first glance that the apune function has the advantage of being composed of very simple elements, namely cubics. Cubica are, how-ever, not very suited for defining parts of the curves where the angle

between the abscissa axis and the tangents is bigger than 3o-4o degrees.

Such curve parts are therefore avoided in the apUne function methods, either

by dividing the curve into a number of sections to each of which a suitable rotated coordinate system is applied ([i] and [24]) or by transforming the ship lines mathematically to a new set of lines which are less steep ([4]). These measures unfortunately outweigh the advantages of the simplicity of

the cubic representation.

RSsingh [7] avoids the need for a proper orientation of the coordinate system to the spline curve by using arcs of circles instead of cubica for the

definition of the spline function. He arrived at this method by studying the

changes of curvature in different sectional curves.

Gospodnetil

[iO]

avoids the shortcomings of the cubics by applying

eliptic integrals for the definition of the apUne function. Gospodnetics representation is obtained by a carefull

use of the

theory of the thin elastic beam.

The spline function is calculated from coordinates which are lifted

from drawings and are therefore not very accurate. Some smoothing of the errors must conseq.uently be included in the calculation method. The coeffi -cients of the spline function are therefore calculated such that the quantity

f

-rj

- 'n +

t' 2

n n

has the smallest possible value. In this expression f is the spline ordinate which replaces the rith lifted ordinate y, f is the second derivative of the spline function and r the corresponding second divided difference calculated from the lifted coordinates, and S is a weighting factor, the value of which is found by triai and error such that undesirable fluctuations in the apune

curve are avoided. The two quantities of the above expression may be consi-dered as an ordinate and a curvature conservation condition. Bakker {1] uses a smoothness condition instead of the curvature condition namely

(i)

(2)

-10-the nth lifted ordinate. Bakker f inds empirically that

k5

is the most suit-able value.

The ship surface is faired in the same way as in the graphical methods, i.e. by successive fairing of different types of sectional curves1 which is repeated until ali the changes in the ordinates _{in a fairing cycle are below} a prescribed tolerance.

It is of course. necessary that the ship surfaces defined by the

dif-ferent systems of sectional curves, e.g. waterlines and transverse sections. are practically identical. The spline function used for defining the sec-tional curves

_{must therefore be so flexible,,}

_{that both the waterlines} and transverse sectional curves can be fitted the same ship surface with

sufficient _{accuracy. If this is not the case, the iterative fairing}

process-described above will not be convergent. Such

_convergence

_{difficulties are}

likely _{o occur in highly curved parts of the surface (see [is] page 25). If} on the other hand, the function used for defining the sèctional _{curves is too}

flexible,,,

it vil], not be suitablé for smoothing out the errors in the

lifted coordinates. The .,flexibility,, _{of the spline function must thus be} balanced between two inconsistent requirements. _{Adjustments of the}

.

flexib-ility, of the functions _{are made either by changing the numerical value of}

the coefficient S in equation (i) or by changing the number of lifted

coordinates,

The difficulty of obtaining coincidence between the _{surfaces defined by}

the different types of sectional, curves, stems from the fact that the _method only operates with tvo dimensional curves, which of course was a physical

necessity in the original _{graphical methods. On a computer it is, however1} possible to fit a mathematical _{surface directly to the ship surface, so that} the problem is avoided.

Such methods

are reviewed in the two following chapters.

5.2. Draught function methods ref. [3][5][7]{8][_{11][22][23][ 28]} [53][37][4i].

A group of methods for fitting a mathematical expression to a ship surface are the draught function methods _{which are based on the fact that} waterlines can normally be defined by polynomials, _{and that the difficulties} in defining the ship form stems from the changes _{of the form when moving in} the vertical direction. _{In all these methods the waterline}

polynomials are

therefore represented by a number of parameters _{which are given as functions}

of the draught. i.e. the draught functions. _{The task of fairiri.g and defining}

draught functions.

The calculation of the draught functions is performed in two stages.

First, polynomials are fitted to a number of waterlines. These polynomials

are calculated such that they give the best fit, in the least squares _sense,

to the ordinates lifted at the respective waterlines. _{From these polynomials} the corresponding parameter values are then computed. In the second stage, each parameter is faired against draught. This fairing is performed by

fitting a function to the values calculated _{in the first stage. The} expres-sions obtained are the draught functions.

Two criteria have been applied in the choice of waterline parameters to be defined by draught functions, namely the numerical suitability, and the applicability to mathematical ship design. While the first criterion is a necessary one, the application of the second criterion is dependent on the approach to the problem, and has therefore _{only been employed In some of the} works.

The problems raised by the numerical suitability _{of the parameters}

cho-sen may be illustrated by the fact that the coefficients of the waterline polynomials, _{calculated by some of the commonly used least squares methods,} are not suitable as parameters, because the coefficients _{of two almost} iden-tical waterlines might be very different. By applying other methods, coeffi-cients are obtained which are very suitable as parameters. This Is for Instance _{the case in Hayes work [ii] [23], where the waterline polynomial}

is

expressed as a linear combination of Chebychef polynomials. Geometrical

quantities, _{which are known to vary with the draught In a simple manner, are} suitable as parameters. _{Examples of such geometrical quantities are} water-line area and halfbreadths at certain stations. Such parameters are very

suitable, as it Is possible _{to express each of the polynomial coefficients}

as a linear combination _{of these geometrical quantities, and vice versa (for} a verification see [28]).

Some workers claim that they have chosen parameters _{which are suitable} for mathematical _{ship design, i.e. designing a new ship merely by specifying} the draught functions, _{without the need of making a lines drawing. Some of} the parameters used in these works, such as for instance high order water-line area moments, have, however, not hitherto been used for ship design. Furthermore,

_{if mathematical sMp design}

_{Is aimed at, it will be necessary} to study the dependence of the different hull properties on the numerical values of the coefficients _{which define the draught functions. This is not} very practical because of the high number of coefficints _{Involved and} be-cause different _{types of draught functions might be used for different types}

-12-of ships. The number of draught functions necessary to define a ship is

roughly the same as the number of sectional curves found on a design lines drawing, which reflects the fact that the tvo sets of curves contain the same amount of information, namely that necessary for defining a ship surface. It seems much more convenient for the naval architect to design ships by means of lines drawings, which directly show the form of the ship, than by the use of a set of draught function curves which define the ship form in a compli-cated way.

Difficulties in obtaining a close fit to actual ship surfaces by draught function methods are reported in [13] and [28]. In

[28]

maximum deviations of about 6 inches between the design lines drawings and calculated surface are

considered to be the rule. This poor fit is due to the fact that the

assump-tion that all waterlines can be defined by polynomial of an order of about 5 is not always valid. It is thus often impossible to fit the waterlines through the lower part of a cruiser stern with such low order polynomials. Furthermore, all the published draught function methods, with the exception

of the not completely developed methods of [7), neglect the lines of discon-tinuity found on ship surfaces (discussed in chapter Li.i. point 4).

3.3.

Special draught function methods.

This chapter treats four draught function methods, which deviate

con-siderably from the methods discussed in the previous chapter. In the four

methods described below the centre plane of the ship is used as the abscissa

plane and the ship surface is defined by giving the ha]fbreadths as a f

unc-tion of two variables, most often the draught and the distance from amidships.

Berghuis and R6sing&s methods

_[7][8][33J

not only use polynomials but also other functions for defining the waterlines. Furthermore, the waterline expression is calculated by fitting a function to the second derivative of the waterline and by Integrating this function twice. The numerical values of the second derivative are calculated at a number of points using

dif-ferences between the ordinates lifted at these points. This approach is based on a study of the role of change In curvature on the ship form. From a later paper

18]

it seems, however, that the method has been modified so that the waterline function is fitted directly to the lifted ordinates. Unfortunately two of the parameters used in the first two paper; were numerically unsuit-able, such that it was necessary to fair and define these two draught

func-ticns graphically. It was therefore proposed in the last paper [7] to use ₃ transverse sections as draught functions.

Atkens

and Tapia

[2){

6)

define the ship surface by a thematical

expression which is defined such that the transverse seàtlons and waterlines

are apune curves

(described in chapter 3.i.). The coefficients of the

expressions are calculated such that the maximum deviation between the

math-ematica]. surface and the lifted offsets is the smallest possible. i.e. the

so called minimax criterion. Furthermore, unwanted Inflection points in the transverse sections and waterline curves are to be avoided. This Is achieved by requiring that the second derivative of the surface along the transverse sections and waterlines have the correct sign. The coefficients of the

sur-face expression are calculated using linear prograsing techniques (see e.g.

[12]). For each of the lifted ordinates, four constraints are thus set up to

express the requirements, that the second derivative of the mathematical, surface in the vertical and longitudinal directions will have the prescribed sign, that the ordinate of the mathematical surface Is smaller than or equal to the lifted ordinate plus the maximum deviation, and bigger than or equal to the lifted ordinate minus the maximum deviation. As every one of the lif-ted points gives rise to 4 constraints, the linear prograzmning problem

becomes so bulky, that only parts of the ship surface could be fitted at a

time, even though a big computer (i1

₇₀₉₀₎

was used. In the later work [6]. it is reported that considerable savings were obtained by replacing the ciinirnax criterion with the requirement that the sum of the deviations should be the smallest possible. The computer capacity needed is still somewhat

large. The problems of connecting different surfaces and the treatment of

surfaces nearly perpendicular to the centre plane are yet only solved in a sketchy manner.

Pien [31] uses a polynomial for defining the ship hull. As polynomials are not suited for describing the parts of the surface w'iich are almost per-pendicular to the abscissa plane, the ship hull is first transformed

mathe-matically into a new and much smoother shape. As a result of such a modif Ic-atiori, an afterbody, _{shown in the paper, could be defined by a polynomial of}

only eighty coefficients. Unfortunately little mention is made in the paper of the difficult problems of defining the stern/stem regions and of devising the interpolation formula to be used for the transformation given by equation 9 of the paper. The last point may reflect the problem introduced when

employing transformations, which Is the choice of a suitable transformation formula. The solution of this problem Is not obvious, as the ship Is modified by the transformation into an unfamiliar shape.

Hayes and Clenshaw' s method

_111)123]

_{has the same basic ideas as Piena} method. The stem/stern region are however in this method treated In a satis-factory manner, and the transformation used is more simple and suitable. As

..j14_

a result of these transformations, the waterlines vil], automatically have the correct tangent at their entrance to the parallel middle body. As, however, the

transformed surface is defined by a single polynomial, it is doubted whether the surface representatin obtained is flexible enough to cover all normal

ship types.

In the draught function methods reviewed above, the surface expressions

are derived by first setting up a waterline expression. whereafter the con-stants in the waterline equation are substituted by functions of draught. It is, however, also possible to derive a surface expression directly. This approach is found in the present work which is described in the following chapters.

-15-11.. Description of the method.

4.1. Principe.], characteristics of the present work:

j,) The faired ship surface is defined by mathematical expressions.

The results of the computer process are the numerical values of the

coefficients in the eqiaation

y = f(x,z)

giving _{the halfbreadth y of the ship as function of the draught}

z and the distance x from amidships _{(see Fig. i). The ship surface may thus be} repre-sented in future computer programmes in a convenient manner by the

function

f(x,z).

By using appropriate _{functions the ship surface may furthermore be}

represented by a relatively small number of coefficients.

2) The surfaces represented by the design drawing and the mathematical expressions are practically identical.

With the present method, the calculation of an accurate numerical repre-sentation of the surface can probably be made for most ship types. The

observed deviations between the calculated surfaces _{and the design lines}

drawings _{are of the same order of magnitude as the}

accuracy of the drawing. The Intentions _{of the designer are thus fully preserved In the mathematical} representation of the surface.

5) The mathematical expression Is fitted directly to the ship surface.

The techniques of transforming _{the surface into a more suitable surface} before fitting the mathematical _{expression are not applied.}

The problem of choosing proper transformations for different ship types Is not simple and Is therefore avoided. Instead, the mathematical functions are fitted directly to the ship surface. _{The experience gained}

when using the method is therefore directly _{related to shapes that can be studied on the lines drawing. This is} both a practical and efficient technique _{as one becomes familiar with the}

shapes vhlch may be represented by the different functions.

Li) _Division

of the ship surface Into parts having shapes of different

charac ter.

Different corJslderations are employed when designing the different parts of the ship hull. For the submerged _{part of the hull, a number}

of hydrodyn-ami.c and hydrostatic requirements _{must be satisfied. Above the water,}

regard is given to space for cargo and equipment. _{The difference}

between these two parts is, for most ships, apparent at the after part of the aftbody. _The

transverse sectional curves at the after part of such ships are norm ly composed of three parts (Fig. 14), The submerged part vhich is nearly straight

and the part above the water which is roughly elliptical and convex. These

two parts are connected by a third part which is rougly circular and concave. The second and higher order derivatives of such a curve are obviously

discon-tinuous at the two points where the three parts are connected. One may thus

trace on the ship surface two lines of discontinuity in the second and higher

order derivatives. Lines of discontinuity may also be introduced in order to

facilitate building the ship or for other reasons. A special case is the knuckle line, which represents a discontinuity in the first derivative. A

study of the lines of discontinuity is found in [7].

The lines of discontinuity represent boundary curves between parts of the ship surface which have shapes of different character. In the present work a separate mathematical function is fitted to each of these parts of the surface (Fig. 5. 6, 7 and

8).

These functions are calculated such that atija-cent parts of the ship surface are connected in the manner intended by the

designer, i.e. either only having common ordinate (a knuckle line) or also having common tangents (normal case).

The division of the ship surface along the lines of discontinuity is

made manually, and is based on a visual study of the shapes of' the surface

as shown on the lines drawing.

Lines of discontinuity

N

-i.

6-parallel middle body

z

y

Fig. 14. Lines _{of' discontinuity in}

the

_{second and higher order derivatives}

of the surface expression y=F(x.y).

flat of bottom

-17-Application of powerful functions.

Most parts of the ship surface are represented by polynomials in the two variables x and z (Fig. i). Polynomials have been chosen for the representa-tion of the ship surface because of the ease vith which they and their deriv-atives may be manipulated, and because of the role of polynomials in the field of function approximation. In practice, polynomials proved suitable for the representation of most parts of the ship surface. Polynomials are not suitable for the parts of the surface that are almost perpendicular to the

abscissa plane y=O, i.e. the bottom, the stem and stern. These regions,

how-ever,, only constitute a small part of the ship surface, and in the present work they are defined by surfaces, the sections of which are conic sections. This method is in agreement with the traditional techniques, where the stem and stern are designed by conic sections, and where the bilge region amid-ships is usually defined by a circular arc.

Check of the input data by computer.

The amount of data necessary for a detailed specification of a compli-cated ship surface is quite considerable. As these many data are manually prepared, errors can hardly be avoided. If the calculations start vith erroneous data, a lot of computer and operator time is wasted, and the total time needed for preparing the mathematical representation of the ship surface is prolonged. The input data are therefore thoroughly checked by the computer before the calculations are begun.

Check of the results by the computer.

As the intention is to use the method as a routine process in

shipbuild-ing, is it necessary to check in an easy and fast way whether or not the

mathematical representation obtained for the ship surface is acceptable. Such a fast and easy check can of course only be achieved by means of a computer. A special program for this purpose is described in chapter 5)i.

Another reason for having a fast computer test for the applicability of a calculated ship surface is that a number of trials might be necessary when fitting a set of polynomials to the surface of a new ship type. By utilizing the error indications given by the test prograe, a suitable set of

polyno-mials may be found after a few attempts. As the calculation of the surface expression and the tests are performed by computer, the whole process can be

performed in a short time even in cases where a number of trials are neces-sary.

lore rvtip limit

flat of bottom

Fig.

_5.

Mathentica1 representation of a fore body. up limit of side

bottomlimit

polynomial t y= P1(x.y)

bitgetimit

Transverse sect ions

defined by Conic z bilge section s X -18 foreend

/stem

'uiiîL!Ze WI entrance waterlines defined by conic Sections limit of side

ore midship timil

flat of bottom

fore midship limit

02 bit ql imit z bilge polynomial 1 y: P1 (x,z) yhitae bOttOmlimit

fore WI entrance! foreend

Stem lore wi entrance fore end Conic

i N

sections z keel ol nomiol

Fig. 6. _{Mathematical representation of a forebody with bulibous boy.}

Conic section

X conic section alt end conic sections stern oft end aft wt entrance . X conic Sect ion keel X polynomial aft haltbreadth alt wl entrance polynomial 3 y: l(x,z) polynomial 4 y bilge aft wI entrance

oft half breadth

u limit of side polynomial 2 polynomial t y :P1 (z. z) c aft wI entrance

aft midship limit

polynomial 4 y:R,(x.z)

-19-flot of bottom bottom imi z mid

oft midship limit

bil.e li

z bilge

y bilge bottom limit

o ô

v-t--t

flat of bottom

aft midship limit ê Y

K

z

Fig. 7. Matheniatica]. representation of an after body.

Fig.

8.

Mathematical representation of an after body. The polynomial

sur-faces are divided at the station xanid. up limit of Side

z

aft end aft wi entrance

aft midsfljp limit conic sect Ons polynomial 3 y r P3 (x,z) Stern oft end polynomial 2 y: P (x,z) polynomial i

,.inflS

y P1 (x__pilaelimjt c't wi entrance ! bliqe keel ca

-20-14.2 The mathematical representation of a ship.

14.2.1. Definition of the hull surface.

The coordinate system used is shown on Fig. 1. As ships are symmetrical about their longitudinal centre plane (ro). it is sufficient to define half

of the ship. The surface of the ship is hence defined by giving halfbreadth y (ao) as function of x and z, i.e.

rf(x.z)

f(x, z) stands for a number of functions, each defining the ship surface with-in a specific region. Fig. 5 shows the different regions used when defining the fore body of a ship. The after body is treated in the same way (Fig. ₇

and

8).

The different regions shown in Fig. 5 are:

i) Parallel

middle body

- Here the surface of the ship is defined by the transverse sectional curve amidships giving halfbreadth as function of draught. i.e.

y=f(z)

Flats of bottom - Here the ship surface is defined by the keel plane, and by the plane of the rise of floor.

Curved surface betveen the parallel middle body and the stem region. This surface is divided in the manner illustrated in Figs. 5. 7 and

8.

In each part the ship surface is defined by the polynomial equation

y=P(x.z)=

1=0. J0

where I and J are the orders of the polynomial

P(x,z).

_{Fig. 22 shows an} example of the polynomial coefficients ajj.

Normally the different pOlynomial surfaces together constitute one smooth surface. Iii some ships, however, the hull is composed of two

dif-ferent smooth surfaces that meet each other along a longitudinal knuckle line. In such a case the knuckle curve is defined mathematically and is

used as the boundary between the polynomial surfaces above and below lt. 14) BIlge region between the flats of bottom and the lowest polynomial surface

Here the ship surface Is defined, such that any transverse section (x=con-stant) constitutes an arc of a conic section, Each of these conic sections is calculated such that it has common tangents with the polynomial surface and with the flat of bottom at the upper and lower limits respectively of

the section (Fig. 9a). Furthermore the conic section Intersects a longi-tudinal construction curve. The projections of this curve on the two long-itudinal coordinate planes are denoted (ybilge, zbilge) and are shown

in

Figs. 5 to

_9.

In

some ships the bilge sections are not tangent to the flat

ii

of bottom at the after part of the after body (Fig. 9b). The angle y shown in Fig. ₉ is zero in the interval between ai1dships and a point which is

denoted , . xtana.ft,, in the data form shown in appendix 2. When moving aft from this point. y increases steadily.

bottom limit ttots a -21-bottom limi buI. b

Fig.

9.

Transverse sections through the bilge region:

case a: O<xxtan aft. case b: xtan aft.

5) The stem region. Here the ship surface is defined, such that any horizon-tal section, i.e. a waterline, constitutes an arc of a conic section. Such a conic section is calculated, such that lt is tangent to the polynomial surface at the aft boundary of the stem region, and such that its axis of symmetrl is parallel to the x-axis (Figs. 5 and io).

casel : b<o stem tore wi entrance 'I) -22-case 2: b>a

Fig. 2.0. Ends of waterlines.

4.2.2. Definition of curves.

Further to the mathematical representation of the hull surface, a number of curves must be defined. _{These curves are listed in appendIx}

2 (page 0.1 and 0.2) and Include:

Contours of the hull. e.g. stem, stern, midship section. sheer and camber. Boundaries between the different parts of the hull _{surface, e.g. the limit} of the stern region, and the limits of the parallel middle body.

The three coordinate axes shown In Fig. i define three coordinate planes

x=O, y0 and z=0. Each of the above curves is defined by its projection on

one of these three planes. This corresponds _{to the common ship yard practice} of defining a curve by its projection on one of the principal design planes.

The proj ected curve is defined by a mathematical function

corresponding

to the abscissa and ordinate axes specified _{for the curve in appendix 2. The} function used to describe the projected curve is composed of a number of' func-tions, each

defining

the curve in its specific interval, _{such that the curve} is built up of a chain of curve elements. _{Such a curve element may be} repre-sented by:

i) A polynomial,

An arc of a conic section,

An arc of circle having a prescribed _radius.

An arc of a circle can of course be defined as an arc of a conic section, _but is treated as a special case because ship designers often define parts of cur-ves by means of circular _{arcs of prescribed}

radii, e.g. the bilge radius in the transverse, sectional curve amidships. _{Fig. il illustrates how a typical} stern is divided into curve elements.

stem

tore WI entrance

X

straight line ellipse straight line

circle

straight line circle _{straight line}

10000

curve; eters .hlp nos 12811 Edition i/3-66j lIt number of elementas 7

Fig. i.i.

Fig.

12.

Division of a stern into

a nmiber of curve elements.

The coefficients

of the functions

which

define the stern shown on

Fig. 11..

)

The stern is eXpressed ai

f(z) in the

coordinate

system shown in fig. I First curve element order .1, i.e. o straight

line

given by the equation: z

31.74 -0.2531(z -0.6000) 05z 1.4O68ß limits 0.00000 ordertp: 1, 0.6000 coeff: .317381250 2 -.253125019 limit: 1.140686

ordenO rmdiue;-1.0000 centret 1.65225

32.503)1

limiti

1.79713 ordertp: 1. 2.2000 coeff: .315728571 ,2, .11e6428601 lisait: 2.514866 order; O 5.280

radivas-0.9500

_{centre: 2.lIIl}

32.56389 limit: 3.31727 order.tp: l 3.71100 4.180 coeft: .336231999 2. .Ji80000

i.

limIt: 4.18000 $4 orders-i z2 2 z 3317 coeff:-.132108551 1 -.168801666 1 .153326775

i

.8529119319 2 _.%914371e117 2 -.100000000

i

-.1140737541 Ie z sign constant limit; 5.28000 2.549 order.tp: 1. 7.6400 coeff: .369988401 2 .331000018 limit of curve; 10.0000 dMa end

*

arc of o conic section

4.185 z

5.28

The coefficients of the

different

terms in the equation ore shown.

When jpar ranging

the equation in order to

eXpress z = I (z), o squoPWerm op

-pears. This term is

in this cose given the sign

- in order to

optain the correct branch of the

conic section.

DSRI

Ljort.kr.v.

99 ?AZRE4 A )4AT)AT!CAL R3iTATXQI Lyngby -Un.rk SHIP LIPES .3

cf' code r,t.mber crder of con-s

limit

tcr type of nectton with

L

elemer.t subsequent e lenar.t

/

o

1'

/

end element CIKA'1

--polynomial

oukar of points should exceed the order et the poinoaal.

Circle - state the reittu.. in both coluna.s.

COnic section

stat. at }Aest

oint. Point. near the limits of the conic

section alld not be etated here, since the sha,e at these regIons I

s

adequately d.i.n.d by the end point tangents of the s4jacent

ynomials.

n the coni: section be.. s vertical tangent at one ut its ilz..ta

a new

curve .lan. oust be int.ruticed. ThIs .lsm.nt is a vertical ,.Ine which

represent.. the tangent

end

Is specifIed by two .othts. The fIrst I

the end

point

of the couic ere. vhi..e the second is positIoned above or below

the

first one dapenttng on whether the conIc fts.s a positive

or negatIve derIvatIve

adjacent to 'I.

vertical tangent.

F - if it ta rsç'..Ir.d to force the curve thrcugJ a ,otoZ

state

befOre the

-abscissa (? for fIxed). In the case of a conIc section on.l,y

Is permitted.

Limita - point.. havt.zlg abscissa outside the interval s,.ecified

for i.'ts curve

can a.so be stated. ¿so overlaptng between tAs points speeifi.d for

tve sdje,cen; eements mey result in s ours natural transItIon between

tnem.

llame of the curve ttabi.e om

pes

i..2 aM

F'Abeciseacrd-At' F AbscIssa ordinate 'F bscIa ordInate

o

jír.,,

f

1.6

i/.-10

'/6/ ..3.ÇPÇ

e

i

3/. 'íF5 31. 69

F sbsctesa'ord.Inat.e 'P ..bsclssa ordinate' 7 bsc tesa ordinate

7

,bsCi;se. ordinate

co

j

it titis tore Is l&et pese of a....l curve nate

',a'.e

.uAtb. end

Fig. 13. The data supplied to the

computer for the calcula:ion of the

stern shown on Fig. 11.

e

Ejort..J 99

FAIPIPO AND N&4ATICAL RrtTIr2f C?

-5 U

L

-. 3

/

I

Put order of connection after laut

I. .3.3 o Il sienent 5

4/f

/

/

6

-/

/

i

/o

I, o

ta form, Curve Description (see poges 0.1, 0.2 end 0.3).

A curve i. d.tin.4 by a eMin of curve l.nte, which er. to b. speeift.d

b.3'. The curve sl.nt. p.rmittsd and tb. cod. ,u.r. to be used for

the

epec ificetione urs:

[eodJ

type J J I

tLßt_.!.tte

5 k I_poLynomials

h:'-+

:;;:s::;;e. htLA

boÏcti

or circle). The computer vili choos, the type of conte

that fits best to the given condition...

0L_!ote

circle specified by Lt. radius.

Thu computer vili position the ere, of conte sections and circle.

so that .aeh

of them is tangential to its two adjacent curve elemente, which cust be

spec it led

a. polynomial..

Abscisse., o? limite. The coordinate ext. to be used are stated in u table

on

pages 0.2 aM 0.3. The curve elemente are specified below such that the colu

,,abucieeas of the limits,, vili have increasing valuas downwards.

1h. order of connection te defined as:

[orúer

The two adjacent .lementi havait the point of conr.ectton7

-t

dttf.ret ordinates i... jump in the curve.

con ordinate at a point, which fits bet for the curve.

O

con ordinate at the atetad limit.

L_'

I

con ordinate and tangent.

in ca,. of a drei. specified by its radius (eods O) or conOectIon order

-the accuret. value. of -the abscissae of -the limite vili be calculated by -the

eca-poter. State neverthelese approxinate values for these abseissas.

To the cas, of a circle or coni

section, etat. order of connectton I with the

two adjacent pol.ynomie.is.

!4ame of curve (see table on pages 0.2 and 0.5

A text to identIfy the curve. State at least the date.

ThIs tenttfication te useful! if different proposals

are trled.

The curve t, composed of the eener.te:

Sraeet abeetesa of fIrst, element

o

t' 2

2

I .sb3C.sse. .rdInate

______p=7

41. 15

p.1

-25-p ceder the polynomial used.

Fig. i. Definition of a

bottom limit1, curve by a chain of three

polyno-mials. bottom 1376 itT-66 j 29 number of element,: 3 limit: -107.79000 order.tp: 7. -80.3950 coeff: .7T1166873 1 .355819866 -.584(22358 -.3693063(4 3. limit of CU1Ve: 94.5000 data end

Fig. 15. Coefficients of the .bottom _{limit,1 curve shovn on Fig. 114.}

.725783937 limIt: -53.00000 order.tp: 1. -5.9250 coeff: .115200001 2 .2654181645 -.728986059 _6 e-9. -.429739361 ,-8 -..5100ie512-i0. 1imt: i.1500Q order.tp: 7. 67.8250 coeff: .851585(48 1. -.290628750 -.626876741 -2 .1817149052 3. .574Q9114814 _{,-5. -.1140530755 .,_6 -.778(90699 ..9} .5?SS4957S..iQ -53.00 -107. 79 X 94 50

tjort.rsvuj 99

Lnby Den.r4'.

,Ai::.

.-:' uT

uilAL R .sfrATzczi IXP L4E5

Data tore: Curve Description

sea

.l. C.2 end 3.31.

A curv, is defined by a chain of cur,. .l.isent.

vhtch sr. to be specIfied

bijou. The curve eleeinta permitted and the cods .uwar.

to b. ua.d for the

ap. ficatio..e 52.I i straight lin. pulynomls.l. l p p..order jo tal

-arc of s conic .etion (elltpee, itpeitol. peao.a

cc,ntc sectton

or circle). The coput.r utLi chocei thi

ty'e of contO

that fits beat to the given eondtttor.a.

L._

src of

circle puctfted by Its redlu..

The coeput.r vt.l position th. arcs of conic

section. ami etre'es si

that each

of thee Is tangential to it. two adjacent curvi

.mente, wnlch cuat b. .pectfied

55

b.eI..e.s of Atmits. The coordInate axt. to b. used are

staid In a table on

paces

.2 arid

.'. The curve e.ents are epectflsd beLa, auch

that the colui

,abectesaa of the lstt, vtl

have increa.tng values downwards.

The order of conriectior. Is defIned ae

order

' The two adjacent .lenents have at the point of conr.ectton:'

¡

-2

different ordInate. i.e. ,jwap In the curve.

cor ordOnate at a point. which ?Ite best

for the curve.

O

eon ordInate at the eteted

tait.

coin ori r.a'.e and tangent.

_j

in case of a circle specified by ite radius

(code C) or connectthn order

-the accurate values of -the abscissas of -the iteits

wIll be calculat.d by the

com-puter. state nevertheless approxtte velu..

tor these abscieeae.

Tn the caes of a circle or coni

setiOn

atete order of connection i with the

two adjacent polynomiale.

curve

!iams of curve (see table on pe.ea 0.2 and

A text to identIty the curve. State at least

the dati.

Thu

identification le usetull uf dIfferent proposals

,,,

are tried.

Th

curve i. composed of the elemer.ta:

[scfii

f code numbeiT orr o? ¿t

limit

for type of neetton with

elemer.t

subsequent

ScelIst abscIssa of fIrst elenent

i

-4ja7?Q

element

Pt ord.r of conn.ction after last

eleeent - O

L

Fig. 16. The data

supplied to the computer for th

r namy lijorteiue&Ye 99 Ijngby - Der.r

F;G

RRIPTLJi ..F

LiliES

CDl1

-.- cRvE.

t

order of the lnoma..

-PoInts near the .lmita

the conic

the ahae

t tr.ese reglns Is

of the eijacent

lyn,oaa.

tangent at or.. .if Its lImIta. a new

alant to a YertIca. lin, which

by two

s.ilnts. The fIrst Is the end

is .osltIuned a&'we or below the

has a 'o6Itve or nega'Ive derIvatIve

through a poInt

stat.

before the

f e Conic sectlOr. only F la jera.ittcd.

interval :.ctf led for iia curve

between L-se ,ozsts

eeitte4 ror

natural tressa tion between tn,a.

olyTilal - number

nt

nad a..at cI redits sectIon tangent verllcs.. fI.ed). (tania

.olnta should eaceed

In both cos .

at iaa.st

e

otht.

be st&.ed here obOe

by the and ,.olnt tangents

&s a vertis.l

se Introduced. This

sod Is seclf.ed

arc

vh.Lla the second

on whether th, cOnIc

tergent.

to tozo. the curve

n the case

abscIssa csststda the

be sW.ad. n .sveria.tng mey result In a on egeo &. i ami bsclssa ordInate

Ciro:. - atete ta

- sta4

section s..l.t

edaquately da f

When the co.:

curvs .lant

rereesnts the ?oLnt of the 0.45.10 fIrst one d ending a&je.cent to tse

- lt lt Is reç...red

abadesa (P for

Limita - j,'oInts having

Name

1Absctssa'crd.r.at.'

element can s..so

two adjacent elemsnts

o1 the curve .

bot om

'F Absctsa ordtnatr / Abscsia ordInate

-i7.9

a . -/00.4 t 4"f

-90.0 q i,

Iao

7.9-1 -70.0 105$'

-- ¿5.04/OS

II-2$

_-64.0

/. -40

1/33 F '10 vg. 32 -o P AbscIssa ortttnat.' -. Y5 11.3/5

lo

if. Ç

Sj

10.

fl

¿o

,o.v

;

SO 4'2 90

lii

c,

9$'5 0.01'?

) 5'1SfS7.11,

-. -.

j5g if.3/f

!

Abscissa ' Abpcissaord.inat 7 ..bscissa orijisiate' .,rdinat.

J

Lt this form i. last jage of a.Lj curve data.

'tate

-27-14.2.3. General considerations on the calculation methods employed.

Fig. 17. The GIER computer which has been used for the present work. The

unit closest to the door is the drum storage. On the top of this unit is seen the Benson - Frace plotter coupled to the computer. The computer which is produced by Regnecentralen of Copenhagen has an outstanding ALGOL .O

compiler.

The choise of calculation method is a function of the capabilities of the computer to be used. i.e. the storage capacity, the computing speed and the accuracy of the numerical representation. The methods described in this work were prepared for the FIyAGIER computer, and can therefore be used on

installations equivalent or superior to this computer.

The HyA-GIEB Is a parallel binary computer with a 42-bit word length.

The storage comprises a 1O214word core store, and a 38L400_word drum store.

Originally the capacity of the drum storage was only 12800 words, and the first edition of the fairing programme was therefore adapted to this limited capacity. All the calculations described in and 14.25 are made with built -in floating-point arithmetic. The accuracy of a floating point number cor-responds to 29 significant binary digits (bits), which is closely equivalent to 9 d'irrial digits. The time needed to perform an addition and a multiplica-tion between two r;-ting point numbers Is 0.12 and

o.i8

millisecond. Most

other GiER instruct.Loii. are performed In 0.05 millIsecond. Early in 19(7 a