Mathematical representation of ordinary ship forms

FRAN

STATEN S SKEPPSPROVNINGSANSTALT

(PUBLICATIONS OF THE SWEDISH STATE SHIPBUILDING EXPERIMENTAL TANK)

Nr 55 GÖTEBORG 196E

MATHEMATICAL REPRESENTATION

OF ORDINARY SHIP FORMS

BY

AKE WILLIAMS

Also published in "Schiff und Hafen" Itamburg 1964

kÄ

SCANDINAVIAN UNIVERSITY BOOKS

SCANDINAVIAN UNIVERSITYBOOKS

Denmark: 1UKSGAATD, Copenhagen Norway: UIVERSITETSFORLAGET, Oslo, Bergen

Sweden: AKADEMIFÖRLAGET-GUMPERTS, Göieborg

SVENSA BOKFÖRLAc4ETJN0r5tedtS -Bonmers, Stockholm

PRINTED IN SWEDEN BY

In this report the results are given of an introductory study carried out at the Swedish State Shipbuilding Experimental Tank (SSPA) concerning mathematical ship lines. The method is, as yet, not en-tirely applicable for full scale use. Primarily, the programming work, choice of computer and to some extent also the numerical methods, have been planned with the intention of defining an arbitrary ship form mathematically.

The investigation which will be further developed at SSPA will include enlargement and improvement of the mathematical methods in order to enable complete mathematical descriptions of ordinary as well as special ship forms to be obtained.

Research work is also in progress to systematize ship forms using the introduced geometrical form parameters.

1. Summary

An account is given of a method of calculation concerning mathe-matical formulation of ship forms. The hull surface is defined by a

small number of draft functions, which indicate the size of the selected

form parameters of the waterline sections. These draft functions are extremely useful when carrying out systematic variations of ship forms and also for single ship projects, as the functions may include curves for displacement, statical moment, moment of inertia, etc.

As an introduction, a short review is given regarding mathematical ship lines and their use when carrying out hydrodynamical calcula-tions, model testing, projecting ships and building ships. A short survey of literature is also given with the intention of classifying the different delineation methods.

Ordinary polynomials are used for expressing the waterline sections and sorne of the draft functions. Using these the calculations are carried out by a "main program". The numerical determination of entrance data is performed by an "auxiliary program". Investigations are made to show the influence of varying sets of form parameters

4

A practical procedure for delineating ships mathematically is in-dicated and is then applied to the hull of a tanker. Some correcting procedures are proposed to be used if the first result is not satis-fying, whereby a repetition of the whole work can be avoided.

2. Introduction

2.1. The purpose of mathematical representation of ship forms

The expression "mathematical representation of ship forms" involves

a procedure which is often very complex. It may consist of relatively simple mathematical methods of fairing and interpolation in a 2-dimensional way, as well as extensive smoothing procedures for hull surfaces by means of complicated analytical functions. The various

parts of a ship's hull may also be treated in quite different ways

using equations, whose characteristics are very divergent from each other.

In general, mathematical representation in this case 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 of 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

common table of offsets.

2.2. A short review regarding mathematical ship forms

The background to the attempts to define ship forms mathematically

can be given in outline as the following stages in chronological order. As early as the 18th century there were shipbuilders who tried to express certain hull lines using simple analytical functions. It is pos-sible that some of their ideas could be applied to the ship hulls of that

time and also to some extent could be entered in the hydrostatical calculations. However, the possibilities at that time of developing the analytical methods and performing the necessary calculations were very limited, and therefore the efforts did not give useful results.

The work of defining ship forms analytically seems to have been taken up again in the decades before and after the turn of the last century. It was among others the advancing model testing technique which could take advantage of ship forms so defined, that the shape

parameters could be easily modified. In conjunction with the planning of D. W. TAYLOR'S standard ship forms, a mathematical procedure

was developed resulting in shapes which were not fully practical but which were fully acceptable for model testing purposes.

During the last decennium many of the calculation procedures re-garding the hydrodynamics of single bodies have been extended to more complicated 3-dimensional forms such as ship forms. In most cases a necessary condition has been that the actual body must be

defined by a suitable mathematical expression. Those

hydrodyna-micists, who began to treat these problems, have given considerable

contribution to the development of mathematical ship lines, notwith standing the fact that they introduced only simplified forms.

2.3. The situation now and in the future

After touching upon three stages of the development of the

mathe-matical ship form, the actual situation in present day shipbuilding

may be considered with its highdemand for optimizing every proce-dure and construction from the technical and economical point of view. Also the access to digital and analogue computers,

coordinato-graphs and other electronical equipment for calculation, control and governing of procedures must be taken into consideration.

An attempt to describe the present and future situation regarding

the use of mathematical ship lines is to be found below, starting from the most theoretical aspects and proceeding to the structuralelements of the ship itself.

2.3.1. Hydrodynamics

At many institutions research work is in progress regarding

theore-tical determination of potential flow, wave resistance,

damping

coefficients, etc. Starting from simplified forms, the hydrodynamicists

now concentrate their energies upon the calculation of these quantities for actual vessels in service and under construction. The

approximat-ing equations thus derived must enclose the entire ship,

and at the

same time, have forms that are suitable for the actual calculation method. In general, the claim of fully smoothed sections and

coinci-dence with original lines is however not so rigorous as in the case of

6

2.3.2. Model testing

As mentioned before, it is essential to have a system of ship forms available when planning model test series. Mathematically defined hull lines will make it possible to vary those shape parameters, which are the most important in conjunction with resistance and propulsion tests, wave tests, manoeuvring performance experiments, etc. 2.3.3. Projecting ships

For an actual newbuilding the first preliminary lines are designed

in accordance with the special demands for the actual ship with

regard to economical speed, propeller size, seaworthness, space

limi-tations, hold cubics, hull strength considerations, etc.

In order to optimize the lines with regard to propulsion, steering, etc. in still water and in waves, the primarily corrected hull will be model tested, whereby a number of alternative forms may be inves-tigated.

Wheii the final shape of the hull has thus been established, the lines are to be mathematically formulated and an extensive table of hull coordinates valid for the full scale ship can be determined with

minimum delay. Any readjustments of some of the main dimensions, cargo volume or displacement can be rapidly performed by modifying some of the form parameters which are included in the equations for the hull surface.

2.3.4. Building ships

When the final hull is mathematically formulated, the hull surface is represented in the main memory of the digital computer either as a set of equations or as a number of associated coordinates. To this mathematical model are added the boundary lines of all shell plates and also the location or ordinary frames, web frames, bulkheads, gir-ders, etc. The shape and weight of the building elements can thus be determined with great accuracy. Precise information can also be given to the workshops regarding e.g. bending of frames and shell plates and

also flame cutting of bulkheads, girders, brackets and floors.

Information for flame cutting is given as coded orders on punched or

magnetic tape. Regarding that part of the information which applies to flame cutting along an intersection line with the ships surface, data are taken directly from the digital computer. Via the tape, this data is forwarded to the digital differential analyser, which is included in the controlling and governing equipment of the flame cutting machine.

1) _{The numbers within brackets refer to the list of references on page} 67»

3. A short survey of literature

A survey of the available literature regarding mathematical ship

lines seems here to be rather unnecessary as, in the firstplace,

SAUN-DERS [1]') has given an excellent summary of the most important mathematical methocts published up to 1955. However, an attempt will be made to divide the different methods into some groups

depen-dent on the main intentions of the authors and also to give some

comments on the delineating procedures. A number of papers pub-lished since 1956 will also be discussed.

3.1. Some classical works

Although the problem of defining very simple ship-like forms

mathematically must have been treated during several centuries, the

first one to write about it was probably the Swedish

shipbuilder

CHAPMAN in the middle of the 18th century. He describedthe hull

lines by use of a system of parabolic curves. About one hundred

years later another Swedish navalarchitect, JOHN NYSTRÖM, followed

CHAPMAN'S work and improved it by using parabolas of different

orders, including fractional. He was thereby also able to express

hollow waterlines.

A standard work regarding mathematical ship lines is the

well-known method by D. W. TAYLOR [2]. His mathematicaltreatment of

the problem, which is described shortly in the next chapter, is the starting point of several modern fairing procedures.

Fundamental aspects of graphical and mathematical presentation

of ship lines have been given by WEINBLUM [3]. An important

state-ment by WEINBLUM is that the ship's hull may be delineated on the basis of an origin amidskips if theoretical hydrodynamic calculations

are to be carried out. He develops mathematical expressions for an

entire ship from given shape characteristics, which, when varied, will

produce a series of forms suitable for systematic investigation of

wave resistance.

3.2. General treatment of the problem with regard to building of

ships and hydrostatical culculation

D. W. TAYLOR'S method makes use of 4th-degree parabolas for fine

transverse sections and hyperbolas for full sections. The waterlines and the sectional-area curve are expressed as 5th-degree polynomials

8

based on certain geometrical parameters. A number of warships in USA have been delineated according of this method. A flat bottom based on a constant rise of floor can not be reproduced, nor can a fixed bilge radius or flat side of large extent. It is therefore presumed

that the equations can not be used for the full form of e.g. a

super-tanker of to-day.

The TAYLOR standard series of ship forms as well as the connected mathematical method have been presented again in recent years by GERTLER [4] in coni unction with a reanalysis of the original test

results.

F. W. BENSON [5] has adopted TAYLOR'S expressions for waterline

sections namely y = ax+bx2+cx3+dx4+ex5 where x = longitudinal coordinate y = athwartships coordinate z = vertical coordinate

The coefficients are determined on basis of given form parameters and random conditions. Thus the waterline ordinates are calculated

from

y = A+Bt+Cw+D

where

t tangent value at stem (stern) w = area coefficient

= curvature amidships

The coefficients A, B, C and D are 5th degree polynomials in x and are tabulated for selected values of x. B, C and D are influence-func-tions belonging to the form parameters t, w, and ¿.

BENSON introduced a f airing process in vertical direction which

in-volved an expression of the form parameters as functions of the ver-tical coordinate z, see Fig. 1. The main difficulties appear to be the establishment of proper analytical functions for the waterline para-meters in z. Several later authors, including the present, have more or less followed the main ideas of BENSON even if there are great differences in the choice of form parameters, mathematical expres-sions for the waterlines, parameters for fairing in vertical direction,

Fig. 1. Principal scheme for mathematical fairing of ship lines by use of draft functions.

WATANABE [6] has proposed a method similar toTAYLORand

BEN-SON, but improvements are said to be made on two points:

Application is possible to any special type of bow and stern

includ-ing Maier forms, bulbous bow and cruiser sterns.

The length of the parallel waterlines at each draft is taken into

consideration, resulting in a fairly good reproduction of bilge radii.

WATANABE uses waterline polynomials of the same type asBENSON

but with terms of only even orders up to the 10th. The coefficients

are determined on the basis of the

form parameters area, statical moment and angle of entrance (run). The vertical fairing is performed by expressing each waterline parameter as a function of the vertical

variable.

The manner of defining a waterline mathematically by means of

useful geometric parameters has also been adopted by THIEME [7],

[8]. He lias added a square root term to the general polynomial and

also two additional parameters, by which it is possible to modify

the derived polynomial without changing the primary form para-meters, which are the area and ist and 2nd derivatives at the ends of the waterline.

Another set of geometrical form parameters for proper definition

of waterline sections and transverse sectional area curves has been

introduced by SPARKS [9]namely

Area coefficient

Transverse moment of inertia Centre of flotation

Longitudinal moment of inertia

Two equally spaced breadths in fore and after body respectively

CALCULATION OF MATHEMATICAL

PRfl tM/NARY CHA RACTERISTIC

FAIR/NO OF DEFt NI TION

MATHEMATICAL

L INES

FORM PARAMETERS OF WATER _SHIP

PARA METERS DRAFT LINES FROM

DRA WINO LINES

FOR FUNCTIONS DRAFT

lo

The methods described above, [2] and [5][9], involve mathematical

procedures which are not too complicated. The hull surface is expressed

as a set of two-dimensional equations linked together graphically _or by another set of relatively simple analytical functions.

ID. W MARTIN [10] has tried different approaches to express the sectional area curve in conjunction withwave resistance calculations.

His conclusion is that the least

squares method using Chebyshev polynomials has definite advantages compared with the Lagrange interpolation polynomials and a Fourier sine series.

Also F. TAYLOR [il] prefers Chebyshev polynomials

as it can be

shown that a sum of these polynomials wifi approximate a given

function so that the maximum deviation is less than for every other

polynomial of same degree. The main property of the Chebyshev

formula is that, generally, the total deviation is more evenly

distri-buted along the range.

The authors of [10] and [il] present their methods as considerably better alternatives to the method of representing hull waterlines by ordinary polynomials in powers of x fitted to the ordinates of the

given curve. The determination of the coefficients of these polynomials involves the solution of simultaneous equations. According to F. TAY-LOR, the polynomial coefficients become increasingly uncertain as the

degrees of the polynomials are increased; the system of equations will be ill-conditioned. There is also a general tendency for the resulting

polynomials to oscillate to a greater extent at the ends of the range

than in the centre.

It must be observed, that the matters mentioned above are valid

in the case when the waterline polynomial is based entirely on a set of half-breadths of the given curve. If the polynomial is prescribed to agree with properly selected geometrical parameters, the risk for wiggles will be considerably reduced. Moreover, it is possible with modern computer routines to solve also a "troublesome" set of simul-taneous equations with good accuracy. Thus, it is not a matter of course that the methods [10] and [11] give more reliable results than the formulas given in [2] and [5] [9], though the former procedures contain more advanced numerical analysis.

The approach of P. C. PIEN [12] is to make some modifications to

the original hull surface before a polynomial approximation is applied.

An "end condition factor" is introduced which eliminates the com-plexity due to the end profile. Stations 1/2 and 19 1/2 will thereby serve as vertical boundary lines of the modified hull, y = f1(x,z).

By interpolation, a new hull form y = /2(x, z) is established, which nearly follows the modified hull and coincides with y = f (x, z) at

three vertical sections. / is a relatively simple surface equation;

only x2- and x4-terms are used for waterlines.

e) The surface, which will be the object for the main calculations, is t3(x,z) = f1(x,z)f9(x,z). / is considered to be a surface which is much more suitable for mathematical fairing in three dimensions

than the original surface.

The surface f3(x,z) is determined as a polynomial f3(x,z) =

and PIEN prefers to solve the corresponding system of equations in two steps, determining first the longitudinal sections and then the

vertical sections by use of the method of least squares. It shouldbe

noted that / may also be a complicated surface, e.g. containing nearly flat areas, and if this is so, difficulties may arise when approximating it by the advised method of least squares with ordinary polynomials

having exponents up to the 10th.

RösINGH and BERGHTJIS [13] treat the problem in similarity with

PIEN, using 3-dimensional expressions. The smoothing of the lines is achieved by double integration of the second derivatives. Thus, from the coordinates of the original waterlines a generating function (x)

is obtained after two differentiations. The condition is now

that

must be a continuous function of x in the actual interval. (x) may

properly be expressed as a function with two terms, and the two

coefficients a and b are determined by the least squares method. The smoothing in vertical direction is then carried out by expressing a and b as functions of z using the same method as above for approxi-mating waterlines.

The special form of the generating function will be applicable

only for limited parts of the hull, e.g. the after body below CWL.

It is supposed that difficulties regarding continuity may occur atthe "joints" where (x) shifts form.

THEILHEIMER [14] starts from the elementary beam theory valid

for a batten which is supported by weights at a numberof stations.

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first and second derivatives at the support points. The third deriva-tive is discontinuous and proportional to the reaction force at the support. The waterlines are expressed as polynomials which must

pass through the given half -breadths. That curve is primarily accepted,

for which the sum of the squares of the reaction forces is minimum. Nevertheless, the received curve may have wiggles and this may be tested by studying the second differences. If the test fails, the given ordinates are modified and the procedure is repeated. The continuity in vertical direction is secured by expressing the polynomial coeffi-cients as functions of z.

Together with STARKWEATHER [15], THEILHEIMER has improved the

method by introducing a smoothing procedure as _{an addition to the} interpolation formulae. Smoothing is applied when the test for wiggles

mentioned above fails. If y, and /(x,,) are original and derived ordinates

and r and f "(x) are original and derived second differences, a linear combination of N

[/(x) _y]2

and N [f "(x,) _r]2 is minimized.

An interpolation practice is also used by LIDBRO [16], who divides

the hull surface into regions and then delineates the parts one by one.

A transverse sectional curve is expressed as a polynomial, whose coefficients are determined from the condition that the curve must pass through some given points. In the longitudinal direction these points form parametric curves, which are also expressed mathematic-ally by interpolation.

3.3. Mathemgtical formulation of ship forms for hydrodynan4icai

cal-cul ations

As mentioned in Chapter 3.1, mathematically defined ship forms are

utilized also for theoretical investigation of wave resistance, potential flow, damping coefficients, etc. The hull surfaces are in these cases sometimes very simplified and are represented by uncomplicated formulas. It can also be dealt with forms, which very nearly coincide with entirely practical hulls, which however not can meet the require-ments necessary for actual shipyard production.

The work of KERWIN [16] may belong to the latter group above.

Every transverse sectional curve is approximated, undependent of

each other, by a polynomial

(principally a Legendre-polynomial) which in jumps will reach very high degrees (up to 200). The coeffi-cients of the polynomials are considered as functions of the longitu-dinal coordinate and are approximated by Legendre-polynomials (see also [13]). Bilge radii and a flat bottom can not be realized. The waterlines are smoothed to vertical stem and stern as the representa-tion formulae are primarily intended for wave resistance calcularepresenta-tions

(Michell ships).

In cooperation with GERRITSMA and NEWMAN [17], KERWIN has

used his method for polynomial representation and calculation of damping coefficients for Series 60 hull forms including the fullness

= 0.60, 0.70 and 0.80.

Also the fairing method worked out by HERSHEY [18] is intended for the calculation of wave resistance. The principal program involves an assumption that the geometry of the ship can be expressed piece-wise by continuous quadrilateral surfaces which are bounded by con-tinuous curves. Each quadrilateral surface is subdivided by a

coordi-nate net. The surface is specified by a matrix of coordinates for the intersections of the coordinate net. The matrices are calculated by an

iteration procedure, which uses the Lagrange interpolation method.

Examples of those methods which deal with simplified ship forms, are the extensive works of WEINBLIJM, part of which is published in [3] and [19]. WEINBLIJM has elaborated 3-dimensional equations which are valid for the entire surfaces of different simple ship forms.

He presents also very important ideas regarding the choice of

geo-metrical parameters, especially in the case when systematical

resist-ance and stability investigations are to be carried out.

An approach to a simplified mathematicalform has also been made

by P. A. HAMILL [20]. A process is achieved, whereby certain initial

design conditions are mathematically faired. The waterlines are ex-pressed as

y =

F+Ax2+Bx3,

where F is the ordinate at largest section and A and B are derived from the ordinates at two other sections, one in the middle of fore (after) body and one near the stem (stern). A and B are parameters also depending on the vertical coordinate, see also [13] and [16].

14

3.4. Systematica' variation of ship forms

Evidently, most of the delineation methods _{referred to above are}

suitable for systematical variations of the_{hull geometry, especially}

those which make use of form parameters of geometrical import. Some

authors will be mentioned here, who have studied the problem of formulating mathematical ship forms for systematical investigations.

D. W. TAYLOR and WEINBLTJM have _{already been mentioned, and a}

later authority in this sphere is LACKENBY [21], who has given

funda-mental views of the problem.

In order to form appropriate _{groups of ship forms, VÖLKER [22]}

applies sectional area curves defined by such geometrical parameters,

which are used in several of the referred_{works. In his "Formdata",}

GIJLDHAMMER [23] does not treat mathematical_{lines but he has}

per-formed a great deal of work in establishing families of practical ship forms based on a limited number of form factors. These form factors

seem to be extremely suitable for application in _{conjunction with}

mathematical delineating by many of the methods referred in Chap-ter 3.2 and 3.3.

3.5. The shipyard production aspect

The works previously referred to have only slightly covered the

concluding problem of the mathematical f_{airing procedure, namely}

how to apply the analytical expressions for the hull surface to the production of the actual components of the ship, such as shell plating, frames, floors, bulkheads, etc.

TAGGART [24] has showed how the layout _{work on the mold loft}

can be facilitated by expressing certain important_{curves (waterlines,} bilge diagonals) mathematically, using ordinary polynomials of sixth

degree based on four shape parameters, i.e. _{area, statical moment,}

moment of inertia and cubical moment.

Among those who have treated this _{application problem, MILLER}

and Kuo [25] may be mentioned. Their mathematical method is in main the saine as TAGGART'S and the vertical f airing process is the same as in [13], [16] and [20]. Many valuable _{aspects are given} re-garding the accuracy at different stations of the production.

AP

4. Coordinate systems, symbols and definitions

4.1. Coordinate systems

When describing the whole ship or its fore and after body, the

dimensional xyz-system is commonly used here (see Fig. 2)

whereby:

x is the longitudinal coordinate with

x = O at L/2

y is the transverse coordinate with

y = O at center line

z is the vertical coordinate with z = O at base line

All primary data and final results for the actual ship are expressed dimensionally in the xy - z-system.

.51n X z WL z

/2

L/ 2

J

WL z CWL D X 8/2 8/2 PP

Fig. 2. Presentation of the ship lines in a dimensional xyz-system.

Sin FP

8/2

Run Midd'e pari Enirance

L, / 2 Bw 2 X0 X, 16 7 LWF LWE

/

N X,,, X,

nI

n CWL I ßK/2 8/2 A 2 X

Sometimes it is suitable to describe the ship in a non-dimensional way. This is the case, expecially when the form is to be systematically varied without any connection with actual ship dimensions. Here an

x'j'z'-system can be used with the same origin as in the

dimen-sional xyz-system, and also:

= i at forward perpendicular

= 1 at after perpendicular

= i at maximum breadth of ship = i at constructed waterline

The three-dimensional systems of axes mentioned above are appli-cable to the whole ship or its fore and after body. In the following, however, the mathematical treatment is mainly based on entrance and run of the ship's waterlines and these parts are referred to a non-dimensional system, see Fig. 3.

4.2. Symbols and definitions

In Table 1, all symbols and definitions are based on the

xyz-and --systems, which are the most common coordinate systems

in the following discussion.

As is seen in Table i and Fig. 3, the non-dimensional a-coordinates have common indices with the dimensional x-coordinates as below:

Boundary between middle part and entrance (run). = O as defined.

Coordinate for first station of entrance (run) next to the middle part.

Coordinate for first station of entrance (run) next to fore stem (stern).

Coordinate for fore stem (stern). = i as defined.

4.3. The relation between geometrical properties in the ay-system and - .sqstem

Regarding the mathematical treatment of the lines of a ship pro-ject, all primary values are available in meters or feet. The following introductory calculations are generally also carried out in a dimen-sional way, i.e. using the xy-system. This is done in order to get as

18

Table 1. Symbols and definitions

many calculation results as possible to be valid for the actual pro-ject, e.g. volumes, angles of entrance and run and length of different parts of the ship.

However, when the real mathematical, treatment starts, all geo-metrical properties of the ship, which are to be used in the further

calculations, must be transformed to the --q-system. Below are

mentioned some transformation formulae which will be used in the further discussion: Symbol Definition a. a A A A», B BK B», D D», E

F

F L L», M M M», R R V W x, y, z X', y', z' x0, x,

;-x,,_1

, - ¿ Polynomial coefficient

Polynomial coefficient used as additional parameter Area

As index, after body of ship

Area of waterline Maximum breadth of ship

Breadth of (flat plate) keel

Maximum breadth of waterline Draft at constructed waterline (OWL)

Draft at waterline (WL)

As index, entrance of ship Rise of floors

As index, fore body of ship Length of ship

Length of ship between perpendiculars Length of waterline

Moment (statical)

As index, middle (cylindrical) part of ship

Moment of waterline area with respect to

Bilge radius

As index, run of ship

Volume of ship As index, waterline

Dimensional coordinate system for ship Non-dimensional coordinate system for ship

Boundary x-coordinates for entrance or run of ship x-coordinates for stations in entrance or run of ship

Non-dimensional coordinate system for entrance or run of waterline

Boundary E-coordinates for entrance or run of ship e-coordinates for stations in entrance or run of ship

Longitudinal coordinate:

Transverse coordinate:

yiyn

yo_yn

Area of entrance of waterline:

AWF B»

Jo

XX0

(valid for LWFLWE and LwMLwFLwE)

.1 0

- B(xx0)

Area of run of waterline:

AWA

Ixo!

J o x--x0 (4.1) X - X0

J'

o

Bwxx0I

ist derivative at boundaries of entrance (run):

d-q dy

xx0

= dx

_YoY

2nd derivative at boundaries of entrance (run):

d2-q d2bl (XX0)2 dC2 - dx2

yy

(4.2) (4.3) (4.4) (4.5) (4.6) (4.7) (4.8)

5. Characteristic form parameters

As pointed out previously, the mathematical fairing in the present method is primarily carried out in a longitudinal direction and the analytical expressions finally obtained are valid for the ship's water-lines. The vertical smoothing is secured by draft functions consisting

of a selected set o form parameters belonging to the waterlines.

Essentially, each geometrical property of a waterline can be used as a form parameter and can thus be included among the draft func-tions. With regard to the way of calculating and using the form para-meters, the following division has been made.

5.1. Form parameters of ist order

For a normal case, the form parameters of ist order are the coordi-nates of the end points for the different waterlines. Thus, the corres-ponding draft functions of ist order agree with the boundary lines for the entrance, middle and run of the ship's hull. Consequently, for a normal case, these draft functions are:

LWF = length of fore body LWA = length of after body

LWMF = length of cylindrical part in fore body LWMA = length of cylindrical part in after body B = maximum breadth of waterline

As an example, a set of draft functions of ist order is shown in Fig. 4. Regarding special ship forms, other characteristic boundary lines must be included among the draft functions of ist order, i.e. knuckle lines and the vertical contour of a transom stern or a bulbous bow.

Other geometrical characteristics, which also for a normal ship can be introduced as form parameters of ist order, are the coordinates for the boundary lines near the stem and stern, which indicate the be-ginning of local roundings. As is shown in Fig. 2 a great part of the stern beyond the after perpendicular is here considered as a "local" rounding, see also Chapter 9.i.2. If the boundary lines are used as draft functions, it is evident that these substitute LWF and LWA as these are defined in Fig. 3.

However, in the following account and LWA are entirely used

Fig. 4. Examples of draft functions of ist order.

the above mentioned boundary lines for the local roundings are more suitable expressions for the terminal points of the waterline functions. When preparing the preliminary lines drawing, from which the actual work may start, care must be taken when designing the diffe-rent boundary lines. It will facilitate the following mathematical treatment if these lines are given an uncomplicated appearance or, preferably, can be formulated in advance by simple analytical

ex-pressions.

5.2. Form parameters of 2nd order

The parameters discussed in the previous chapter express the limits of the waterlines and can be established in an early stage of the work. Form parameters of 2nd order are such geometrical quantities, which express the shape of a waterline.

The most common way of characterizing a particular sectional line of the ship's hull is to give a number of ordinates corresponding to a set of an independent variable. According to the above, each of these ordinates can be said to make a form parameter of 2nd order. In some methods of fairing such a set of coordinates is just included in the main mathematical operations.

22

With the present method of f airing, the aim is to make use of such shape parameters which can lead to a limited number of draft func-tions. Consequently, it is essential to select a set of parameters, each of which is able to characterize the shape of the curve, or a part of it, as well as possible. Further, it is desirable, that these draft func-tions should have appearances which are not too complicated. Such geometrical quantities are area, statical moment, moment of inertia,

/irst and second derivatives at certain points,etc.

An example: In order to approximately characterize the end slope of the different waterlines, which must be controlled during the cal-culations, two or three coordinates will be entered in the main calcula-tions, if these start from a set of coordinates as shape parameters. Similarly, to characterize the end radius of curvature of a waterline by the same method, three or four coordinates are needed. For the same purpose, the end slope is here expressed by the first derivative, the radius of curvature by the first and second derivatives. Thus, in this case, only one or two figures are entered in the main calculations instead of three or four. The first and second derivatives are deter-mined by the auxiliary program, see Chapter 7.2.

Form parameters of 2nd order can be selected among the following quantities valid for a waterline either in the fore or the after body:

A

= area

M statical moment = moment of inertia M3 = moment of 3rd power M4 = moment of 4th power dy(x0) dx dy(x) dx d2y(x) dx2

- ist derivative at boundary to middle part (or entrance and run respectively)

- 1st derivative at fore stem (stern) = 2nd derivative at fore stem (stern)

As an exemple, a set of draft functions of 2nd order is showed in Fig. &

When defining mathematically a waterline on basis of some of the

shape parameters just mentioned, the result is dependent on the

d /d2

(xx)

dg/dA / AWF

MWF (AS X,,)

Fig. 5. Examples of draft functions of 2nd order.

) The combination of form parameters which has been selected (see

Chapter 8).

b) The form of the mathematical expression (see Chapter 6).

e) The accuracy with which the form parameters have been calculated

(see Chapter 7.2.).

Summing up; the question regarding form parameters of 2nd order for waterlines is how to find a combination of parameterswhich:

) Are few in number (seeChapter 6.2.).

b) Are easily calculated from the preliminary lines drawing (see

Chapter 9.2.1.).

e) Can be used practically also for other purposes, e.g. hydrostatical

calculations.

d) Are able to characterize the waterlines in vital regions, e.g. at stem and stern.

-e) Give draft functions which are suitable for mathematical formula-tion by simple funcformula-tions.

5.3. Form parameters oJ 3rd order

As stated in the previous chapters, form parameters

of ist order

can be established when preparing the preliminary lines drawing.

Form parameters of 2nd order are to be treated in connection with

the main mathematical operations.

C WL

24

Form parameters of 3rd order can be entered in the _{calculations,}

when the final result has been obtained and this result

_{must be}

corrected for some reason. This parameter may consist of a coefficient,

which afterwards is included in the mathematical expression for a waterline and whose influence on the curve is known. The final result

may also be modified by introducing a new shape parameter according

to Chapter 5.2, which is then considered as a form parameter of 3rd

order.

An account is given in Appendix II regarding the use of form

parameters and draft functions of 3rd order.

6. Choice of mathematical form

6.1. _{Alternative analytical functions}

As pointed out in the previous chapters, the main mathematical calculations are applied to the ship's waterlines. Thereby entranceand

run are treated separately, i.e. an entire waterline running from fore to aft is in main defined by two separate equations. Locally near fore

stem and stern there will also be special expressions for stem rounding,

soft nose, bulbous bow and stern, transom stern, cruiser stern, etc. Regarding the choice of mathematical form, there are several alter-native functions which may be considered; polynomials, exponential functions and trigonometrical functions or a combination of these. In this treatment, the general polynomial is choosen, though it is possible that other functions may be more efficient in the sense that they give better coincidence with fewer terms. Moreover, the _common polynomial cannot describe a curve whose first derivative is equal to infinity at some point e.g. an afterbody waterline including a cruiser stern.

However, the polynomial is easy to handle and admits a simple

analytical determination of desired geometrical quantities.

Integra-tion and differentiaIntegra-tion can be carried out in an easy way and some

of the properties of the curve are to be found as coefficients in the polynomial, e.g. the different derivatives when the independent vari-able is equal to zero.

As is evident from the literature survey in Chapter 3, thereare some

theoretical methods concerning wave resistance, potential flow and damping coefficients which just use polynomials when describing the shape of the actual body.

11

-valid for a waterline as it is shown in Fig. 6. The expression includes the boundary condition'7 = i when = O.

The coefficients of the polynomial are determined on basis of a combination of form parameters mentioned in the previous chapter. The number of terms in the polynomial is equal to the number of geometrical conditions (mcl. boundary conditions) which are

intro-duced. Before calculation of polynomials, the selected parameters have been f aired in the vertical direction as pointed out previously.

Or

6.2. Formulation of waterlines by polynomials

The general expression of a polynomial in the e'7-system will be

'-j = 1+a1e+a2e2+a3+...

+ar

(6.1)

y,

o

Fig. 6. A waterline section in the xy-system and ta-system respectively.

R0 X, X X1 Rk Rn-, X,,

y

26

There are two essential points of view regarding polynomials in

this connection:

The number 0/terms must be kept as low as possible. The fewer the

terms the fewer extreme points, and thus the risk of producing uncontrollable wiggles will be less. This means that only a limited number of form parameters can be introduced despite the fact

that the curve wifi be more accurately defined by use of a larger number of parameters.

The exponents o/the polynomials must be the same for all waterline8

in the tore or after body. This condition is, in general, not hard to satisfy, but will sometimes cause problems when there are great differences in the fullness of the waterline sections. A fine section is best defined by a polynomial with terms of lower powers, a full

section will require terms of higher powers for good coincidence.

7. Calculation of polynomials on basis of a number

of geometrical parameters

In this chapter an account will be given regarding determination of waterline polynomials and also form parameters of 2nd order. Both

calculation methods are programmed for a digital computer and are here called "mainZ program" and "auxiliary program" respectively.

It must be emphasized, that the programs are not intended pri-marily for full scale use. The programming work, the choice of

com-puter and to some extent also the numerical methods, have been planned with the intention of investigating the possibilities of defining an arbitrary ship form mathematically by the present method. For that reason, the main program is constructed in a flexible manner so that investigations can be made regarding influence of varying sets of geometrical conditions, varying degrees of polynomial, etc. '7.1. Geometrical conditions

The general expression for a waterline polynomial in the

non-dimensional

--system has been given in Chapter 6.2. In this

account, polynomials up to the 16th degree will be treated:

b: Area C: Statical moment d: Moment of inertia e. Moment3 Moment4 ist derivative at e = i 2nd derivative at

= i

ist derivative at = O 2nd derivative at = O

= O when = i

(Boundarj condition) $ CdC

$q2d

S C3dC

$q4d

dr1( 1) dC d2'( 1) d'q (0) d2q (0) dC2 (l) = O

The polynomial must realize one or more ofthe following geometrical

conditions (bk);

7.2. NumericaL calculation of form parameters of 2nd order

When the 1st order draft functions have been fixed, the calculation of form parameters of 2nd order will start. In order to make use of the advantages of the digital computer, the determination of area,

different moments and derivatives is carried out by numerical methods.

The calculation starts from a table of offsets based on the preliminary lines drawing.

Numerical methods and precision of calculations should be choosen bearing in mind the non-accurate initial values.

7.2.1. Calculation of area and moments

According to Chapter 4.2. (see also Fig. 3) the sets x1, x2, . . . and C1, C2, . . C,1 represent the equally spaced stations in the dimen-sional and non-dimendimen-sional system respectively. The area between 1

and C,_1 can be determined by the well-known Simpson's formula when n is an even number. The remaining areas at ends may be approximated by the trapezoidal rule.

2S

The total area can then be expressed by the following modified

Simpson's formula:

where

SM is Simpsons multiple (modified)

n is an even number not less than 6

x2x1

X,, X0

3 xx0

SM0 = -.

4

x2x1

xx

S.ZIn

= -

n n1 4

x2x1

(1 2 Jo

d=

--n-2 n-3

+ 2

¡=2,4,... - spacing of stations SM1 = SM0+O.5

SM1 = SM+O.5

In the same way the formulas for statical moment and moment of inertia can be fixed

Statical moment with respect to C0:

C' 2 ¡

JCdC =

--.J

(SMOCOO+SM1C11+ o n-2 n-3

+ 2

C1m+ _{i+5MniCn.in_i+8MnCnn)} i=2, 4,... i'=3,5, . (7.2) (7.3) e1 according to formula (4.1) according to formula (4.2)

Moment of inertia with respect to dy 2

j

= -

(sM00+sM1e1+

n-2 n-3

± 2

Th+SMn_i_in_i+SMnn

7.2.2. Calculation of derivatives

ist and 2nd derivatives at ends of the waterline are determined by curve interpolation. A polynomical curve of 2nd degree passing through the first three coordinates (from table of offsets) is formed by Newtons general interpolation formula with divided differences. The demanded derivatives can then be obtained in a simple way. The accuracy will evidently not be the highest, especially not concerning

the 2nd derivative, but it is considered to be sufficient in view of the relatively rough base material.

First a general polynominal passing through three arbitrary points located on the given curve will be determined by Newton's formula. The coordinates are (x1, y), (xi, y.) and (xk, Yk) according to Fig. 6.

y = y1+(x-x)[x xJ]+(x-x)(x-xJ)[xxJxk]

dx = [x1x] + 2[xlxJxk]x-x[xxJxk] XJ[XXJXk]

d2y

dx2 = 2[xxJxk]

When x = x0, x = x1, Xk = x2 and further x = x0, the derivatives can be written as

dy(x0)

-

[x0x1]+(x0x1)[x0x1x2]

dx

When x = x,_2, x1 = x_1, Xk = x, and further x = x,, the deri-vatives can be written as

The corresponding derivatives in the --system will be evaluated

by use of formulas (4.7) and (4.8).

7.3. Equations Jor the geometricai conditions

7.3.1. Condition /or area (condition b) r1 I 'qd

= I

(1+a1e+a2+a3+ ... +a)d =

Jo Jo i i i i

= l+ai+a2+a3+...+fl+lan

dy(x) (7.7)

(7.)

dx

-

_1)[x_2x_1xJ dy (x) y,, -1 YnYn-i

Y11-iY-y,, -2

X,,-X,, -1 X,,_1 -X,,_2

-

+(2x,,x,,_2x,,)

uX X,,_-X,,_2

YY-

Y-1Y-2

d2y(x,,)

x,,x1

-dx2

x,,x,,2

30 or (7.5) (7.6) dy(x0) Y2Y1 YiYo YiYo X2X1 X1X0

-

+(xxj)

ax

x1x0

x2x0

Y2Y1 YiYo d2y(x0)

x2x1

dx2

-i i

7a1+ --a2+

--a3+ .

i. .

+

i

= Jd_1

r1 (7i))

7.3.2. Condition for statical moment (condition c)

r'

r'

I vede

= I

(1+a1e+a22+a33+ .

. .

+a,)d =

Jo Jo

i i

i

i i

a,+

a2+ a3+ . .

+

n2

a, =

Jrd

-7.3.3. Condition ¡or moment of inertia (condition d)

(1 ('.1

j2de =j

o O

(1+a,+a22+a+... +a,2d =

i i

i

i

=+_a1+a2+ja3+...+fl3afl

i

i

i i "l

+n+3a,=j2de_

7.3.4. Condition for (moment)3 (condition e)

r'

rl

j3d =j

o O

(i+a1+a2+a3+... +a,3d =

i i i

i

i

=+--a1+--a2+-a3+ ..

.

+fl4

i

i

i

i

r'

a1-j- --a2+ a, ... +

a, =

f

'd

-6 .

7.3.5. Condition for (moment)4 (condition f)

r'

ji4d=

J

(1+a,+a22+a3+ ... +a,)4d =

0 .10 i i i i

i

i

i i i '1

a+ a2+a3+

+n+5 a,

=j

(7.10) (7.11) (7.12) (7.13)

32

7.3.6. ist derivative at = i (condition g) d-11

= a1+2a2C+3a2± ... +na,C1

Substituting C = 1: a1+2a2+3a3+

+na, -

_d 7.3.7. 2nd derivative at = i (condition h) dC2 = 2a2+6a3C+ . . .

+n(n-1)aC"2

Substituting C = 1: d2(1)

2a2+6a.3+ ... +n(n-1)a

7.3.8. ist derivative at C = O (condition i)

It is evident from 7.3.6. that:

d,1(0)

a1=

7.3.9. 2nd derivative at C = O (condition j)

It is evident from 7.3.7. that

d2-q(0)

"a -

2

dC2

1.3.10. Boundary condition, = O u'hen C = i (condition k) 77 = 1+a1C+a2C2+a3C3+ ... +aC? Substituting C i: a1+a2+a3-]-

. +a = O

d77 (1) (7.14) (7.15) (7.16) (7.17) (7.18)

where

b, C, d1, . . .k. is the matrix of coefficients belonging to the equations for conditions b, c, d, . - k

the column vector a represents the coefficients of the waterline

poly-nomial

the column vector Kb, K, Kd, . . . K, represents the known terms of

the equations for conditions b, c, d, . . . kand which enclose the given

form parameters of 2nd order.

Suppose that a polynomial is to be determined on basis of area, moment of inertia, ist derivative at = i and ist derivative at = O

(and also the boundary condition at = O), i.e. conditions b, d, g, i

and k. The selected set of indices is i = 1, 2, 4, 6 and 8. Further, the polynomial will realize a given parameter value of 3rd order, which is included in the polynomial as a known term (see Chapter 5.3, Table 3 and Appendix II). The power of this additional parameter is selected as j = 10 in this example. The matrix will then have the

following appearance: 3 Kb b10a10 - d10a10 K5 - g10a10 - i10a10 Kk - k10a10

7.4. Solution of the system of conditional equations

The conditional equations for the form parameters of 2nd order form a set of linear equations from which the coefficients a in the waterline polynomial can be found. As a summary, the system of equations can be symbolically written in matrix form:

b1 d1 g1 '1 k1 h2 d2 g2 2 k2 b4 d4 g4 '14 k4 b6 d6 u '16 k6 b5 d8 ti ?8 k8

+

a1 a9 a4 a6 a8 b1 Cl d1 k1 b2 d2 k2 b3 C3 d3 k3 bu Cu d >( a1 a2 a3 au Kb K K K (7.19)

34

The solution of the system of equations has been carried out by use of Gauss' elimination method. However, the system can be ill-conditioned, i.e. when solving the system, significant characters can be lost, partly due to rounding-off errors and partly due to exceeding the limited space in the arithmetic registers of the computer. For this reason, the elimination process has been modified in order to obtain a more accurate result. One modification was to incorporate automatic shift of the binary point into the program to avoid overflow in the

registers.

When the coefficients a of the polynomial have been determined,

these wifi form the basis of another calculation of all parameter

values. This is performed by the main program and the result is pre-sented in the result table, see table 3. By comparing the last received parameter values with those which are originally given, the accuracy of the program can be checked.

Finally, an arbitrary number of coordinates for the waterlines are to be determined The well-known method of Homer's scheme is used for the computation.

All calculations in connection with the main and the auxiliary

pro-grams have been performed using the W E G E M A T I C i O O O

computer. The coding work has been carried out by the A D B

-institutet at Chalmers University of

Techno-i o g y, G o t e b o r g, where two machTechno-ines of thTechno-is type are Techno-in servTechno-ice.

8. Selection of a suitable set of form parameters

to describe the waterlines

8.1. Investigation regarding influence from number of parameters and degree of polynomial

By use of the main program, a waterline polynomial can be

deter-mined on basis of up to nine geometrical conditions. Thereby the

expo-nents of the polynomial can be selected among integers from i to 16. As pointed out before, the main program is constructed in this manner just in order to make it possible to calculate polynomials of different degrees based on widely varied sets of form parameters.

When the vertical fairing is based on draft functions containing form parameters, the sets of exponents and parameters must be the

same for all waterlines in the fore or after body of the ship. The

question is now, how to determine the most suitable combination of exponents and form parameters by means of which all waterlines can

Combination H:

A, '(0), "(0), '(1), "(l);

i = 1,2,3,4,5,6

The calculations are carried out for three different waterlines of an ordinary cargo liner, namely WL 1/4 Entrance, WL 5 Entrance (CWL) and WL 8 Run, see Fig. 7-io.

be defined, from the finest in the bottom to the fullest in the upper regions. To make an effort to obtain an answer to the question, a short experimental investigation is here carried out as it seems to

be very difficult to get any satisfactory results by use of strictly

mathematical methods.

Bearing in mind the general points of view mentioned in Chapter 5.2 and 6.2, the present investigation will be performed for the fol-lowing combinations of parameters and exponents.

Combination A: Area Statical moment ist derivative at e = o ist derivative at

= i

Exponents of polynomial: 1, 2, 4, 6, 8 or shorter: A, M, '(0), 77'(l): = 1, 2,4, 6, 8 Combination B: A, M, M2, M3, '(0). '(1): i = i, 2, 3, 4, 6, 8, 10 Combination C: A, M, '(0), '(i); i = 1, 2, 3, 4, 5 Combination D: A, M, M2, M3, '(0). '(1): i = 1, 2, 3, 4, 5, 6, 7 Combination E: A, M, '(0). "(0), '(i), "(1); i = 1, 2, 3, 4, 6, 8, 10 Combination F: A, '(0), "(0), '(1), "(1); = i, 2, 3, 4, 6, 8 Combination C. A, M, '(0), i "(0), '(1), "(1); i = 1, 2, 3, 4, 5, 6. 7

1.0 0.8 0.6 0.4 0.2 0 1.0 0.8 0.6 0.4 02 o y

Fig. 7. Mathematical formulation of three waterlines for an ordinary cargo liner.

Influence from number of form parameters and degree of polynomial. Original waterlines from preliminary lines drawing

- - CombinationA:AM'(0) ,j'(l); i= 1, 2,4, 6,8 Combination B: A MM2 M3 '(0) '(1); i = 1, 2, 3, 4, 6, 8, 10

rAg.

5 ENTI? 2IIENTR

-___

_

'u

rA

8 RUN 5 TR

u

-'u

-11 -1 0 -0 9 -0 8 -0 7 -0 6 -0 5 -0 4 -03 -02

0/

02 03

04

05

06

07 08 09 10 -11 -1,0 -09 -08 -07 -06 -05 -04 -03 -02 01 02 03 04 05 06 07 08 09 10

Fig. 8. Mathematical formulation of three waterlines for an ordinary cargo hner.

Influence from number of form parameters and degree of polynomial. Original waterlines from preliminary lines drawing

- - - Combination C: A M '(0) '(1); i = 1,2,3,4,5

!.0 0.8 0.6 0.4 0.2

0/

Fig. 9. Mathematcal formulation of three waterlines for an ordinary cargo liner.

Influence from number of form parameters and degree of polynomial. Original waterlines from preliminary lines drawing

- Combination E: A M '(0) (0)

i(l)

,(1); i = 1, 2. 3. 4, 6, 8, 10 - - Combination F: A '(0) '(0) '(l) (i); i = 1,2, 3,4,6, 8

-___

-EN TR -

A

A

8RUN "N

WL¼Uì

-11 -10 -09 -08 -07 -06 -05 -04 -03 -02 01 0.2 03 04 05 06 07 08 09 1.0 -LI -10 -09 -08 -07 -06 -05 -04 -03 -02 01 02 03 04 05 06 07 08 09 10

Fig. 10. Mathematical formulation of three waterlines for an ordinary cargo liner.

Influence from number of form parameters and degree of polynomial. Original waterlines froni preliminary lines drawing

- - - Combination G: A M 17'(0) ,'(0) '(l) (1); i = 1,2,3,4,5,6,7 - - Combination H: A i'(0) (0) '(1)

i(i);

i = 1, 2, 3, 4, 5, 6

38

8.2. Discussion of the investigation results

The accuracy by which the form parameters of combination A H

have been calculated is in accordance with the auxiliary pro gra7n,

whereby the number of ordinates included has been about 10. This must be remembered when critizing the result, as the influence of smaller errors in the parameter values (due to approximate computa-tion methods, non-accurate preliminary drawing or table of off-sets, etc.) will also be evident as a result of the investigation.

Regarding the coincidence between the given and the mathematical

line, the attention may be drawn to the end parts rather than the

middle of entrance and run. Also relatively small differences at the

ends often involve a distortion of the waterline's character. The

remaining differences at the middle parts of entrance and run, when the end parts have been brought into coincidence, often are of no greater disadvantage.

The following comments may be made regarding the investigation

results.

Combination A and B, Fig. 7

Combination A gives good coincidence for WL 5 Entrance and WL 8 Run even at the end regions. WL 1/4 Entrance is badly defined both by combination A and B, especially as the curvature at fore end can not be realized at all. When using two additional form para-meters, M2 and M, the coincidence for the medium waterline seems to be the same, while there has been a distinct change for the worse at the fullest waterline, especially as the curve has been so distorted at the fore end that the maximum breadth has been exceeded.

Combination C and D, Fig. 8

By introducing terms of lower powers in the polynomial (compared with combination A and B) the general effect will be a better

coin-cidence for finer waterlines and an inferior for fuller. As is clear from Fig. 7 and 8, somewhat smaller differences for WL 1/4 Entrance have been obtained by combination C compared with A, while the result has been the opposite for WL 8 Run. In this case however, when

using terms of lower powers, the influence of M2 and M seems to have

Combination E and F, Fig. 9

In order to make the agreement better at the ends, M2 and M in combination B have here been replaced by the second derivates at

O and

= 1. For all

waterlines considered, the intentions have

also been realized as is seen in Fig. 9. By excluding the statical mo-ment as a geometrical condition the result seems to be equivalent, with the exception of the finest curve.

Combination G and H, Fig. 10

By introducing terms of lower powers in similarity with combina-tion C and D, the result for WL 1/4 Entrancehas here become

satis-factory even when the statical moment is excluded as aform

para-meter. Also the other waterlines seems to be well defined by

com-bination G as well as comcom-bination H.

Summing up, the coincidence between given and mathematical

lines can not radically be improved by introducing moreform

para-meters such as moments of higher powers. Thereby an increased number of terms in the polynomial is required, which involves a greater risk for wiggles. It seems also that even small deviations from the exact parameter values (for A, M, M2 and M3) will cause

distor-tions especially at the ends of the curve.

The investigation result indicates that it is necessary to also include

parameters which are able to effectively direct the curve at ends. Considering also the points of view in Chapter 5.2 and 6.2, the

con-clusion is that the waterlines may be satisfactorily defined by the derivatives '(0), j "(0), '(1) and "(1), which secure the coincidence at

the end parts, and also the area A which secures the "over-all"

coinci-dence. This combination of parameters is used in the following

cal-culations.

9. The practical procedure when defining ship lines

mathematically

In the following is given an outline of the calculation procedure.

With regard to the mathematical work, the routine can be divided

into four parts, corresponding to the following headings 9.1-9.4. The subheadings below refer to the block scheme in Fig. 11.

PREL IM/NA NY L INES DRA WING MODIFICA lION OF PRELIMINARY LINES DRAWING

PRELIMINARY FA/RING OF _{DRAFT FUNCT.}

OF I ST ORDER

\ SA I/SF. RESULT2/

NO

FIXATION OF

DRAFT FUNd OF 3 RO ORDER

/

CONTROL OF SMOOTHNESS OF SECTIONAL

LINES

SAT/SF RESULT 7

NO

YES

CALCULATION OF POLYNOMIAL S LARGE NUMBER

OF HUL L

COORDINATES

Fig. 11. The practical procedure for mathematical representation of ship

lines according to the present method.

NO STOP/NO OF HULL COORDINATES ON MAGNET/C TAPE YES

FIXATION OF _{DRAFT FUNd.}

OF / ST ORDER CALCULATION OF FORM PARAMETERS OF 2 ND ORDER PREL IM/NARY ¡

FA/RING OF _{DRAFT FUNd.} OF 2 ND ORDER SAT/SF RESULT

CALCULATION OF FIXATION OF POLYNOMIALS DRAWING MATHEMATICAL DRAFT FUNCI s LIMITED NUMBER OF 2ND ORDER OF HULL SHIP LINES COORDINATES p

CALCULATION OF DIFFERENCES FOR DRAFT

FUNCTIONS

9.1. Preliminary lines drawing and draft functions of ist order 9.1.1. Preliminarj lines drawing

When working out the preliminary lines, the successive mathema-tical treatment must be taken into consideration. Attention must be given to those parameters, which wifi later form the draft functions. It will, for example, facilitate the introductory work if the stem and stern contours are designed in conformity with the character of the vertical sections near the ends of the ship, thus enabling the draft functions for ist and 2nd derivatives to be less complicated.

Regarding the required size of the preliminary drawing, this should be of such a size so that all the intentions of the designer are entirely evident. A natural starting point is the body plan and other lines drawings made up in conjunction with the model tests.

9.1.2. Modification of preliminary lines drawing

The modifications carried out in this connection include smoothing of local irregularities and extending the ordinary waterline to sharp contours. In the after body, this extensions will be considerable as the stern part of the hull beyond the after perpendicular can not be defined by the same set of functions, which is valid for the after body in the main.

9.1.3. Preliminary /airing of draft functions

The smoothing process in vertical direction begins with drawing up the draft functions of ist order. This is carried out preliminarily on a millimetre-paper in size AO. In the normal case, the contours fore

and aft and boundary lines between entrance (run) and cylindrical part of the ship are drawn up, whereas the maximum breadth of the waterlines is generally defined analytically by the breadth of the flat plate keel, rise of floors, bilge radius and the vertical side. Also other geometrical characteristics, which in other cases are incorporated

among the draft funct.ions of ist order (see Chapter 5.1), are

preli-minarily drawn up here.

Often the contours and especially the boundary lines mentioned above are subject to modification in order to facilitate a succeeding mathematical formulation. If the corrections are considerable, these must be brought back to the preliminary drawing after which the waterlines will be justified accordingly.

42

9.1.4. Fixation o! draft functions

When the preliminary fairing of the draft functions of ist order has given a satisfactory result, these functions are to be expressed

definitely. This can be done by:

Tracing the functions in full scale in the mold loft.

Drawing the functions to a scale 1/5 or 1/10 with high precision in the drawing office.

Approximating the functions as closely as possible by one or more

analytical expression.

Evidently, the last alternative is the most desirable and consequent, as thereby also the vertical fairing will be carried out mathematically. Moreover, it implies exact information regarding actual parameter values for any arbitrary waterline. An example of mathematical for-mulation of these draft functions is given in Appendix III.

9.2. Draft functions of 2nd order

9.2.1. Calculation of form parameters

The numerical calculation of the form parameters of 2nd order starts when the draft functions of ist order have been fixed. In the present case, area and ist and 2nd derivatives at the boundaries of entrance and run are determined by the auxiliary program on basis of a table of offsets from the corrected preliminary lines drawing.

The calculations, which will take some minutes when using a

medium-speed digital computer, are carried out for iois waterlines in

entrance and run respectively.

9.2.2. Preliminary /airing of draft functions

As with the draft functions of ist order, a preliminary f airing is carried out on a mm-paper in size AO. The actual parameter values determined by the auxiliary program are thereby presented in the xy-system. This will in most cases enable draft functions of more

uncomplex appearance compared with a presentation in the

--system. If the functions for ist and 2nd derivatives at stem and

stern are complicated, this may lead to a correction of the preliminary lines drawing as suggested in Chapter 9.1.1.

!1.2.3. Fixation of draft functions

The actual draft functions will be expressed definitely in the same manner as these of ist order. When approximating the functions by

analytical expressions, it is important that the deviations are as

analogous as possible to each other. If this is not achieved, there will be a greater risk for getting an unsatisfactory final result in the form of distorted transverse sections. In this point of view, it is ad-vantageous to work with a small number of draft functions.

9.3. Calculation of poiynonials. Mathematical lines drawing

9.3.1. Calculation of polynomials, limited number of coordinates

The ship s hull will be completely defined by the fixed draft func-tions, by which also the vertical continuity is secured. Thus, the draft functions are the only values to be entered into the main

pro-gram. whereby an arbitrary number of waterline polynomials now

can be determined.

Next stage is to make a preliminary mathematical lines drawing. Therefore, i000 2000 coordinates are calculated in the entrance and

run respectively, corresponding to 100 200

stations and 20-25

waterlines for the whole hull. When using a medium-speed computer, polynomials and coordinates will be determined within a quarter of an hour.

9.3.2. Drawing, mathematical ship lines

The first set of coordinates computed from the main program according to the previous chapter will now be used to set up a preli-minary body plan. The coordinates are set off on millimetre paper in AO, one for the fore body and another for the after body. This work is easily performed by aid of a high-precision coordinatograph.

Drawing the waterlines may also be considered, especially when these have an unconventional appearance. Also when the waterlines in some ranges are nearly straight, a checking is necessary in order to ensure that no wiggles have arisen.

9.3.3. Checking the smoothness of sectional lines

The purpose of niaking up a preliminary mathematical lines plan

44

The constructor will get an idea of the agreement between the approximating mathematical lines and his intentions presented in the original preliminary drawing.

An optical survey will be performed regarding the smoothness of the transverse sectional lines. As for the waterlines, this control is advisable, especially when there are nearly flat areas in the hull surface. It is assumed, that drawings to a scale corresponding to size AO are sufficient for the detection of wiggles and other irregu-larities of the mathematical lines.

If the result is not satisfactory, a correction should be undertaken regarding the form parameters already introduced. New parameters may also be included in order to overcome the difficulties. The

mea-sures may be some different depending on the character of the

"failure"; point a) or point b) above, or both of them. An account concerning these matters is given in Appendix II.

9.4. FinaZ calculation and storing of hull coordinates

9.4.1. Calculation of hull coordinates; large number of coordinates

Final calculation of hull coordinates starts when the mathematical lines plan is approved, i.e. when the draft functions for all the intro-duced paranieters give a mathematical hull surface which is free from irregularities and is in proper agreement with the constructor's ori-ginal intentions.

A ship of 150 m length and 12 m depth will require say 45 000 coordinates for a suitable description of the hull surface. This

corres-ponds to all intersectional points between 60 waterlines and 750

trans-verse sections. The distribution of known points may be so arranged,

that the half breadth at any arbitrary point of the hull can be

deter-mined by simple interpolation in any direction. Thereby the result must be the same within say some tenths of a millimetre.

9.4.2. Storing of hull coordinates

The calculation and storing on magnetic tape of 45 000 coordinates as mentioned above will take about two hours when using a digital

computer of the type SAAB D-21.

When using numerically controlled flame cutting, the information regarding actual curves must pass through a digital differential ana-lysator, which with very short time intervals adjusts the flame cutters

to due position. Suppose that a certain process requires 200

coordi-nates to be brought out from the computer's memoryand supplied

to the analysator. The total computer time for access, interpolation, feeding out of punched tape and control tables will thereby

be 10-30

seconds.

io. Acknowledgement

The author is indebted to Dr. HASs EDSTRAND, Director of the

Swedish

State

Shipbuilding Experimental

T a n k, for having been given the opportunity to carry out this investigation work, as well as for the encouragement he has given it. Thanks are also due to the staff of the Tank for all their assistance.