A simple approach to the mathematical ship form

Ottawa, Canada REPORT

Division of MechanIcàl Engineering Ship Laboratory

Pages - Prefaôb 2 Report: MB-225

Text 11 Date: September 1960

App. - 7 -File:- M2-3-7

FIgures-Text - 6.

App. - 1

For: Internal

Subject: A SIMPLE APPROACH TO THE MATHEMATICAL SHIP FORM

Prepared by: P.A. Hamill Submitted by:

Approved by: D.C. MacPhail Director

SUMMARY

Current work in the Ship Laboratory on mathemtic]. ship forms Is described. Available solutions of the problem of fitting a surface equation to an arbitrary ship are complicated and of limited use. It Is desirable that anâlytical.formsbe introduced in the design stage, to provide a concise description of the hull and also to assist in the design. A very simple process, whereby certain initial design conditions are mathe-matically faired, Is developed and tested on the forebody of a

fine form. The resulting surface would be acceptable as a ship

hUll. It Is suggested that this approach to ship design may

offer considerable advantage - and should be pursued further. NATIONAL RESEARCH LABORATORIES

S.T. Mathews Head Ship Laboratory E.S. TUrner Head Hydrodynamics Section

List of Illustrations

Figure

Co-Ordinate Axes I

Forebody of a Pine Ship Form 2

Analytical Fit of Midship Section 3

Initial Input Data L.a

Modified Input Li.b

The Function Q(z) at n =

0.905

5

Analytical Ship Forebody 6

Page - (ii) MB-225 TABLE OF CONThNTS Page Smrmary (i) 1. Introduction j

2. The Problem: To Fit an Equation to a Given Ship Form I

3.

A Simple Analytical Form 3

Li.. Analytical Solution of a Simple Design Problem 5

5.

Application to a Fine Form 8

6. Conclusion

io

7.

References Ii

Appendix A: A Note on the Waterline Equation and the Selection of Input Data

'Page - I

MB-225.. A SIMPLE APPROACH TO THE MATHEMATICAL SHIP FORM

INTRODUCTION

There are many good reasons for attempting to find analytical expressions for ship hull forms. In a number of current ship research problems a surface equation is a require-ment, e.g. the calculation of' wave making resistance,, damping and wave excitation of arbitrary forms.

Also

the solution of'

many other problems in the fields of hydrodynamics, structures and design could be made easier if suitable expressions for the hull shape were available.

Ideally, a mathematical equation for a ship Surfóe. would be expressed in terms of hull characteristics. In

general, this is not attempted. The ship form being, 'usually, a complicated surface, a large number of parameters are neces-sary to describe It and the difficulty in formulating sueh a set of' parameters in a manner that would result in the required ship form is obvious.

Generally the problem is attempted with the usual table of offsets as a starting point. ,In Section 2, this"

problem, and the available solutions, are discussed. It is

concluded that such solutions are inevitably complicated and offer littlereal advantage'over the original offset

representa-tion.

The question is then asked whether or not a mathe-matically simple form can be acceptable for a ship. A very simple format is proposed in Section

4

and numerica]. experi-mente carried out on the forebody of a fine form ship. It is emphasized that the problem is not to fit the' given offsets but to derive a hull of similar characteristics within the

simple analytical format. The resulting surface is seen to be acceptable as a ship hull.

It iè concluded that an approach of this nature may be of value in the design problem as well as providing the obvious advantages of a conclse'and meaningful ànalytical description. Further applications of'the simple system of

Setion

₄

'will be mad.ein the future and, as required, the method may be extended for more géneral application.

THE PROBLEM:

TO FIT AN EQUATION TO A GIVEN

SHIP FORM

It is quite clear that an attempt to write a surface equation for an arbitrary ship form starting with a set of hull

Page

2 MB-225

characteristics would be à very, complex problem.. In general,

the hull is a.compiicated shape andthe equation which would

accurately describe it would require a large number of

para-meters.

It would be extremely difficult to formulate such a

set of parameters in amariner that would result in the rêquiréd

shIp form.

. .

.

Usuálly the surface çqtion is base on a given

table of offsets.

In References I and 2, two highly successful

'sólutions to the problem of representing a table of offset

are. presented.-

The accuracy of fit is limited only by the

complexity allowed in the finàl,answer and by thé precision

and capacity of the digital computing process in use

.When the 'problem of mathematical ship forms waa

first investigated in the Ship Laboratory, the original thought

was to look for 'a general method whereby an arbitrary hull'

could be fRted, within a given error, by an analytical

expression.

At 'this time the solutions in References I and 2

were nòt available and the initial work consisted in a series

of numerical experiments which were carried out on the; lines

of a fine ship. form.

With no preconceived ideas the suit-.

ability of various functions, polynomials

exponentiala,

trigonometric and combl.nations of each type0 was examined

with the ob.ject.of attaining.a flexible scheme with'a minimum

of complication.

In this regard it should be mentioned' that

simplicity was sought not only in deriving the equation but

also in the final ánswer; for having once found à mathematical

expression for the ship it ±s then desirable that it be suitáble

for further investigations,, e.g

tor hydrostatic calculations,

etc.

It soon 'became clear,' as 'concluded in Reference 2, that

the simplicity inherent in operations with polynomials is a

strong argument for their use.

However, further work showed that all attempts

t'o

cover the whole surface by a single equation were inevitably

complicated and the end product would offer little advantage,

in general usage and manipulation over the original tableof

offsets.

Atlthis time the solutions of References :1 and2.

came to hand and the conclusion that the general ship

form

cannot be rep'esented at all simply was further reinforced.

For example, a-96 term polynomial was required (in Ref. 2)

for a satisfactory ft of the series 60, .0B = 0.6 form.

'

seemsd clear that the solutions now available Would in eneral

principle, be difficult to improve upon and indeed thatthe

possible solutions wei'e not much better, for our purposes.,,

Page - .3

MB-225 It is not proposed to repeat the development of the solutions givén in References I and 2.. For present purposes

lt is sufficient to make thé following comments.

Satisfactory methods (Ref. I and 2) are available for fitting an analytical expression to a given table of

offsets. Both solutions introduce devices which offer

con-siderable advantage in simplicity, computing time and the required precision in the digital process over the more obvious direct methods0

The end product is still very complicated - it

Would appear, necessarily so. In the possile applications mentioned in Reference i, i.e. reduction of the mOuld lofting work in Shipyards, or in obtaining hydrostatic and dynamic characteristics of a given hull, it would offer, once obtained, some advantage over the offset representation. In terms of hull design characteristics it is practically meaulngless.

Some improvement could probably be made on the

solutions of References I and 2. For instance, the treatment of the stem profile In the present note might be irtroduced into the method of Reference I with some advantage. However, such improvement in detail would not have any significant

effect on the final result. It appeared that there Was little to be gained by further concentration on the general problem. Havingthen concluded that the general ship form was of

neces-sity a mathematically complicated surface, It seemed reasonable to enquire Into the possibilities or limitations of simple

forms.

3. A SIMPLE..ANALYTICAL FORM

The aim was to achieve an analytical form in which the hull characteristics would.be recognizable or, at least, easily calculable. This would require that the equation have a small number of terms and suggested, in the simplest possible form, the use of low order polynomials.

A co-ordinate system is shown in Figure 1. The

origin is chosen at the lowest point .of the midship section; XZ Is then the plane of symmetry; YZ the midship section plane and XY the horizontal plané of zero waterline. The forebody and afterbody are treated separately and X is positive toward both ends; Z is positive upwards. _{Defining.the dimensiotless} co-ordinates

x = X/L,

y = Y/iB, z = Z/D where L, B, and D

Page -Li,

MB-225

are the length, the beam and the depth of the ship; the surface equation is then written:

y = f (x,z) (1)

with y = f (XZm) the mth waterline (2)

y = f (x,z) the nth station ₍₃₎

where x is the constant x value at the nth station Zm is the constant z value at the mth waterline.

To fix ideas it was proposed initially to formulate the simplest possible analytical expression for the forebody of the fine ship

form shown In Figure 2.

On inspection it appeared that the individual water-lines of the form in Figure 2 could be very closely represented by fifth order polynomials. It may be noted that this Is true for waterlines In general. However, In this case It appeared possible to represent the general character of the waterlines by a family of cubics. Since the slope dy/dx is zero at the

¡nid-section (x = O) the form of a waterline equation (z = constant) reduces to

y = F + x2 + 4ix3

where F is the midship section ordinate for the particular value of z and and Is are constants to be determined.

The surface equation is then written

y = p(z) + (z) x + z)

(5)

where F(z) is now the equation of the ¡nid-section and and i

are functions of z which mast be determined. Now it is possible at this stage, given a table of offsets, to determine F as a polynomial in z. Suitable values of and hr may then be

evaluated for Individual waterlines and once known numerically these In turn may be fitted by a polynomial in z. In practice, however, this may be difficult. The mid-section itself is usually

Page.- 5 MB-225

a full curve and requires a very high orderpolynomlal for an accurate fit. The individual values of and ir may be found

tO lack precision Since the cubic is descriptive of the

character of the waterlines rather than an accurate fit of the given offséts. Perhaps more important is the fact that such a procedure, by relying on a full table of offsets Ignores the considerable possibilities that a basically simple

analytical form could offer in the earLy design phase. It

was mainly this consideration, i.e. application to design, that governed the following development.

Ideally, a number of hull óharacteristics would be introduced into a suitable analytical process and from the resultant equation the required hull form obtained. A mathe-matical fairing process Is, of cOurse, a very attractive goal

and a ship form obtained by such a process would also have the advantages of a simple analytical description. From such considerations it appeared necessary to attempt a

representa-tion of the secrepresenta-tion curves which would be as simple as the cubic equations of the water lines. This Is possible

only If

the section curve is treated piecewise, that is the curve is broken up in its depth in such a way that each part may be represented by an equation of the form

y = a + bz + cz2 + dz3 (6)

This is best illustrated by an example.

In Figure 3 the broken line shows the mid-section of the sample form. This Is a typical mid-section curve, being rather full, and would require a very high order polynomial for an adequate description. If., however, it is considered in two layers, the first for O < z

0.2667

and the second for z 0.2667, a pair of equations of the form 6 give an acceptable description as shown by the full curve .in Figure 3. Continuity at z

0.2667

Is assured by having the same ordinate y and slope y/dz for both curves. There is no particular difficulty in determining these two-dimensional curves to represent section layers.

11,.

AILYTICAL SOLUTION

OF A SIMPLE DESIGN PROBLEM

The design application will be illustrated if the development is viewed as a solution to the following design

Page MB225

-p rob lé m:

Fr a given layer of the ship.. z0 . 'z z1 the. input

dta (in nunìerical or graphical form) are

The midship section ordinates (x

= o)

'

An intermediate section

(x =m)

(nomal'1y,

aswill

be seen in Appendix A, this section is located abOut halfway between the stern and midshIp section corresponding to Section

5

or6).

The stem profile. In the general case the stem ordinate is brought to zero by a sharp radius. Obviously, a loworder polynomial cannçtzepresent this abrupt ending in the waterlixB. The input data required here consist of the. x, y and z ordinates where the Small fairing arc begins.

The. problem is then tO fit asurfacèequation of te form

y=F(z) +(

to these data.

The first Step is to fit two-dimensional curves of the type 6 to the input data.. Thus the rndship sectiOn x = O is written

y=P(z)

(7)

The intermediate Section X = m becomes

y=p(z)

(8)

The stem may be left. in numerical 'form but for clai'ity is

referred to as . .

y=N( );

X = L(z)

(9)

These Input data are shown diagraniatically In Figure Ija.

In order to avoid fitting equations to the stem and to facilitate the evaluation of (z) and i(z) the following

)

Then

m-n

p-F) - i

_(-)

,

(lii.)

in, - n Page -.7. MB-2?5 device maybe used. Consider an Individuai waterline (z =

constant)

2

This curve is determined uniquely b:y the input data

y.=F for

x = O

y=P for x=m

y=N for x=L

Thus for any constant value of x, say x = n, there is à function y = Q(z), related to the functions F, P,. N and L of z, which satisfies the surface equation. In practice the

relationship, which will be derived later, is fairly complicated and the simplest method of treating it is to first specify it numerically for a number of z vàlues and then fit an equation

of the form

6.

The, two equations 9 for the stem profile may now be replaced by the single input equation

y = Q(z)

x=r1

(-lo)

The input functIons are now as shown in Figure Li.b

and and 4' may be quickly determined as follows:

Since F(z) + (z) in2 +

4(z) in3 =

P(z) (Il)

F(z) + (z) n2 + 4r(z) n3 = Q(z). (12)

in n

- - (p-F)

m

Page - 8 MB-225

Since F, P and Q are all of the form

y = a + bz + Cz2 + dz3

the functions (z) and r(z) are obtained by equating co-efficients in (13) and (14)

The relationship between the "ghost section" Q(z) at x = n and the original input F, P, N and L is found from the equation

N = F + + 1'L3 (15)

in conjunction with (11) and (12).

When and '1' are eliminated we obtain the following equation

for Q:

Q = F + (N-F)

[n2

(m...n)1 _{+ (P-p) Í»}

(Ln)]

₍₁₆₎

L2 (m_L)] [m (m_L)]

In snmmry then the solution is obtained by calcu-lating Q as a numerical function of z, fitting equations of the form (6) to obtain F(z), P(z) and Q(z) and evaluating

(z) and

Ir(z)

by equating coefficients in (13) and (14). The process is then repeated for the remaining layers, z1 to z2, etc., keeping the input data continuous at the coimnon points until the complete hull is specified.

5. APPLICATION TO A PINE FORM

The system will now be applied to the form shown in Figure 2. The input information is:

the mid-section ordinates z = O

an intermediate section ordinate z = 0.6190

the stem profile ordinates which were taken from the lines plan, as explained in Section

4,

where the bow fairing

Page.

-9

MB-225

radii join, the mai waterline curves. It was: convenient

tè

represent the mid-seòtion In two layers, I.e. for z = O to

0.2667

y = F(z) = 0.0237 + 10.472z - 39.

O8Li.z2 +

_L8.855z3

for z >

0.2667

y P(z) =

0.9502 + 0.0l.98z

The Intexdiate section at' X

0.6190

was written

for z =0 to

0.2667

y = P(z) =

-0.0332 + 2.8921z -

3.8014.2z2'

for z > 0.2667

y

P(z) = 0.117L + 1.81i.30z - 2.2878z2 + 1.1273z3

The function Q(z) was evaluated at z = n =

0.905

for

a

number of z values and is shown in Figure

5.

The value öf n was chosen so that the resultant Q(z) could be compared directly with one of the original sections. This comparison

is shown by the broken curve which représenta the original form as specified by the table of offsets. Equations for Q(z) are

z= O to

_Ó.2667

y =.Q(z) = -0.ö067 + 0.0770z +

1.9707z2

-

2.6344z3

z > 0.2667

y =Q(z) =. -0.1069 + 1.16hz -

1.Li.3144z2 +

_Q.7869z3

Finally, the functions and fr as given by equations

(13)

and

(14) are z = O

to 0.2667

IO

.

(z)

-2.9126 - 262.22z + '1365.1z2 - '1996.7z3

io6

_{o (z) =}

_{1.7338 + 1i0.13z - 651.80z2 + 1004.5z3}

z>0.2667

IO o

(z)

-30.23L1j. + 88.545z - 112 79z2 + 511..356z3

ir(z) =

13.471

- 51.551z +

65.597z2

-

31.1142z3

Page - IO

MB-225 _I

Figure 6 shows the sections computed from the surface equation

y = F(z) +

(z)

2

_{+ c(z)}

It is not proposed to discuss the relative merits of the original design and the analytically derived form in Figure

6.

The object was to demonstrate that acceptable ship

forms may be determined by a process of fairing mathematically a given set of basic design data. Figure 6 shows that the simple procedure of Section Li. can be used to derive a satis-factory ship.

6.

CONCLUSION

At the present stage of work in the Ship Laboratory on mathematical ship lines the following conclusions and

suggest ions may be made.

I. The problem of fitting an analytical expression to

an arbitrary ship form (specified by the usual table of offsets) has been briefly examined. There appears to be little prospect

of improving, except in detail, on the available methods (e.g. Ref. I and

2).

The solution is, in general, extremely complicated and offers little advantage over the original offset representation.

It is accepted that a simple mathematical description of a ship surface would be of considerable value. It therefore seems reasonable, all else being equal, to design ships in

this way. A suitable mathematical process whereby certain

basic design requirements are faired mathematically to produce an acceptable ship form would also solve the design problem. In Section 4, a preliminary attempt, of extreme simplicity, to develop such a process is given. This is applied to a fine ship form in Section

5.

It is believed that the method of Section Li. Is capable of wide application and further work along these lines will be carried out. Extension of the method will be made, as required, for more general application.

REFERENCES Pien, C.P. Kerwin, T.E. ¡LES Page -. 11 MB-225

Mathematical Ship Surface.

International Shipbuilding Progress, Vol. 7, No. 68, April 1960, pp. 161-171.

Polynomial Surface Representation of Arbitrary Ship Fórms.

Journal of Ship Research, Vol. Li., No. i,

CO-ORDINATE AXES

FIG. I

LO

0.8

0.6

0.4

02

ANALYTICAL FIT OF MIDSHIP SECTION

FIG. 3

MB-225

02 0.4 Q6 1.0

I

FROM TABLE OF OFFSETS

ANALYTICAL CURVE DEFINED BY THE TWO EQUATIONS:

Z.: O to 0.2667

y 0.0237 i IÖ.472z -39.O84Z2i. 48.855

Z >0.2667

y 0.9502 + O.Ö498 Z

z 1.0 0.8

06

04

0.2

ORIGINAL FORM n:O.905

THE FUNCTION a(z) WHICH REPLACES THE STEM PRÖFILE AS INPUT DATA

FIG. 5

MB-225

THE FUNCTION Q(z) AT

fl:O.905

COMPARED WITH THE

cORRESPONDING SECTION FROM THE ORIGINAL OFFSETS

1.0 0.8 0.6 z 0.4 0.2 I') o d Q2 0.4 0.6 0.8 1.0 y o o (Do ej o ej O c$dO FIG. 6

MB-225.

F-n

Page -:A-.i 'MB-225::

.PENDIX. A

":

.

kNÓTE ON

E WATERLLNE EQUATION AND. THE SLECTLOÑ. OF INPUT DATA With, a surface equation

of the form

=F(z).(z)

x2+(z)

x

the.

equatIon of _a

particular waterliné

(z

='const) 'Is

(A-1)

where

F,

and

r are now constants. F' is

immediately

deter-mined from the midship section (x

= o)

and

and fr depend on.

the Input data, ie. on the stem. profile and on an Inter-mediate section x m.

Considering only

a single waterline

le may write the input data, in the notatiOn of Section L1., as

follows:

x = O

y

=

F

(midship section)

x =

m

y

p

(intermediate section)

(A-2)

X =

L

y

= N (stem profile)

Now It is evident that, for most case6 of interest

one of the requirements on a waterline equation is that y

should decrease over the range of x = O to L. That is

=

2x

+ _34rx2 O. for x =

O

to L (A-3)

By substituting (A-2) in (A-1), and r

are

obtained

In terms of the input conditions. These in tU.rn

_may

be sub-stituted in equation (A-3) to determine what combination of Inputs would give (unacceptable). positive values of dy/dx. TO

illustrate the results we first define the

parameter

r--Page - A-2 MB-225

and refer to Figure A-1. A closed loop is shown in the plane

a. For any combination of and a within this loop dy/dx

will be negative in the range x = O to L. For values outside the loop dy/dx will become positive for some value of x between x = O and X = L.

Thus, in selecting the intermediate section y = p(z) at x = m for the basic data, it is necessary to ensure, at each z value, that the (a, ) combination falls within the

loop of Figure A-1. It is also obvious that a choice of m/L of 0.5 to 0.6 covers the widest range of the parameter a. In general then, as noted in Section I, the intermediate section z = m is chosen about halfway between the stem and midship

NRC MB-225 National Research Council, Canada. Division of Mechanical Engineering. A SIMPLE APPROACH TO THE MATHEMATICAL SHIP FORM. P.A. Hamill. September 1000. 15 p. + 7 figs. Current work in the Ship Laboratory on mathematical ship forms is described. Available solutions of the problem of fitting a surface equation to an arbitrary ship are complicated and of limited use. It is desirable that analytical forms be Introduced in the design stage, to provide a concise description of the hull and aleo to assist in the design. A very simple process, whereby certain initial design conditions are mathematically faired, is developed and tested on the forebody of a fine form, The result- ing surface would be acceptable as a ship hull. It is suggested that this approach to ship design may offer considerable advantage and should be pursued further, UNCLASSIFIED NRC MB-225 National Research Council, Canada. Division of Mechanical A SIMPLE APPROACH TO THE MATHEMATICAL SHIP FORM. P.A. HamlU. September 1960. 15p. + 7 figs. Current work In the Ship Laboratory on mathematical ship forma Is described. Available solutions of the problem of fitting a surface equation to an arbitrary ship are complicated and of limited use. It Is desirable that analytical forms be introduced in the design stage, to provide a concise description of the hull and also to assist In the design. A very simple process, whereby certain initial design conditions are mathematically falred, is developed and tested on the forebody of a fine form. The result- ing surface would be acceptable as a ship bull. It Is suggested that this approach to ship design may offer considerable advantage and should be pursued further. UNCLASSIFIED 1. ShIpbuilding 2. Naval architecture 3. Hulls (Naval architecture) I. HanilU, P. A. II. NRC MB-226 1. ShIpbuilding 2. Naval architecture 3. Hulla (Naval architecture) I. HamiU, P. A. U. NRC MB-225 - NRC MB-225 National Research Council, Canada. Division of Mechanical Engineering. A SIMPLE APPROACH TO THE MATHEMATICAL SHIP FORM. P.A. HamIll. September 1960. 15 p. + 7 figs. Current work in the Ship Laboratory on mathematical ship forma Is described, Available solutions of the problem of fitting a surface equation to an arbitrary ship are complicated and of limited use. It is desirable that analytical forms be Introduced in the design stage, to provide a concise description of the hull and also to assist in the design. A very simple process, whereby certain initial design conditions are mathematically falred, is developed and tested on the forebody of a fine form, The result- ing surface would be acceptable as a ship hull. It Is suggested that this approach to ship design may offer considerable advantage and should be pursued further, UNCLASSIFIED NRC MB-225 National Research Council, Canada. Division of Mechanical Engineering. A SIMPLE APPROACH TO THE MATHEMATICAL SHIP FORM. P.A. HamIll. September 1960. 15 p. + 7 fIgs. Current work in theShlp Laboratory on mathematical ship forms is described. Available solutions of the problem of fitting a surface equation to an arbitrary ship are complicated and of limited use It is desirable that analytical forms be introduced In the design stage, to provide a concise description of the hull and also to assist in the design. A very simple process, whereby certain initial design conditions are mathematically faired, is developed and tested on the forebody of a fine form. The result- Ing surface would be acceptable as a ship hull. It is suggested that this approach to ship design may offer considerable advantage and should be pursued further. UNCLASSIFIED 1, Shipbuilding 2. Naval architecture 3. Huila (Naval architecture) I, Hamill, P. A. II. NRC MB-225 1. Shipbuilding 2. Naval architecture 3. Hulls (Naval architecture) I. Ham.tU, P. A. II. NRC MB-226