Mathematical representation of bodies of revolution by use of a digital computer

FRAN

STATENS SKEPPSPROVNINGSANSTALT

(PUBLICATIONS OF TRE SWEDISH STATE SHIPBUILDING EXPERIMENTAL TANK)

Nr 51 GÖTEBORG 1962

MATHEMATICAL REPRESENTATION

OF BODIES OF REVOLUTION BY USE

OF A DIGITAL COMPUTER

B Y

AKE WILLIAMS

SCANDINAVIAN UNIVERSITY BOOKS

SCANDINAVIAN UNIVERSITY BOOKS

Akademiförlaget-Gumperis. Göteborg

Munksgaard. Copenhagen

Norwegian Universities Press, Oslo, Bergen

.Svenska Bokförlaget / P. A. Norstedi & Söner - Albert Bonnier, Stockholm

PRINTED IN SWEDEN BY

In the following is given an account of a calculation method con-cerning mathematical representation of bodies of revolution.

As an introduction, the practical applications of bodies of revolu-tion, investigations carried out, actual parameters used, etc. are mentioned.

The mathematical treatment is based on the sectional-area curve of the body, which, on the basis of given geometrical quantities, is expressed as a polynomial. By using additional parameters the

shape of the body can be varied, while the given geometrical

quan-tities are kept constant.

The method of calculation permits a simple classification of bodies

of revolution. The bodies are classed according to the number of geometrical parameters that define them. A consistent symbolical

characterisation for bodies of revolution is also introduced.

As an illustrative example a series of bodies of revolution with certain geometrical parameters systematically varied is computed,

see Appendix 1.

In Appendix 2 it is shown how an empirically given body of revolu-tion can be approximately described by a mathematically expressed body.

1. Introduction

Axially symmetrical forms are mainly used for such bodies which

are working in one single medium. Provided that the body is far

enough from any adjacent media the meeting flow will be axisymmetri-cal. From the resistance point of view the body of revolution is then the most advantageous. Consequently in the case where a body is intended

to be pushed forward in one single medium, and when frictional resistance is the most important component, the body is usually given an axially symmetrical form. Well-known examples of this

are airships and torpedoes.

At present there is a growing interest in bodies of revolution and their hydro- and aerodynamical properties. The reason being the

4

essentially increased range of application of this type of hull form. Earlier use of the axially symmetrical form was mainly confined to

airships and torpedoes, but nowadays this form lends itself to many different commercial and military uses e. g. submarines, rockets and

missiles.

In the course of time a great number of more or less systematic

investigations have been performed with bodies of revo]ution. Many

of the investigation results are not published, as they have direct

connection with actual military projects. In Fig. i is shown a general survey of most known experimental investigations, for which results are available. In the figure the investigated forms have been indicated on the basis of the two most important geometrical parameters,

namely the prismatic coefficient and the length to breadth ratio.

The figure also shows the kind of investigation performed.

When carrying out systematic investigations it is essential that the forms considered are defined in a suitable manner. The most

common method of describing a hull-form is to express it empirically

by means of either a lines plan or a table of offsets. With either of

these presentations an alteration of a parameter value is often a troublesome procedure. Moreover, this can generally be worked out

only in an approximate manner. Many advantages can be gained if the hull form is exactly described by a mathematical expression, such that actual parameters can be varied in a simple way. Among the more important advantages of mathematical definition of forms

the following may be mentioned: Exact formulation of the body form

No fairing problems when drawing the body

Direct and accurate variation of the geometrical parameters Simple and exact calculation of displacement, center of buoyancy, moment of inertia, etc.

Simplified treatment of special hydrodynamical problems.

2. A short survey of literature and comparison with the present method of calculation

In the available literature there are only a few methods given for

defining mathematically bodies of revolution. Mention wifi be made below of the most important investigations which have been published.

In a report from DTMB [1]1) LANDWEBER and GERTLER give an

0,9 0,8 0,7

u)

0,6 0,5

Bodies adth ainoidal runs, Weitbrecht l9'il _{Bodies wth pointed ends, Petersohn 9j}

----f----rBodos with rounded nose Lock,Johansen 1129

¶ Body dimensions s3sternatically varied Body dimensions riot systematically varied

Akronl Subos tanker Todd

958

-f

Plow separation, Palkemo

953

V

Mar

--I

8 7 S--JAIbacorej & E

e

Bodies with pointed ends, Amtsberq 1931, 1950,1955

Subie, tanker Mitsubiohi

IQSâj

Body with pointed E!_nds, Astrup

957

¿

I I 8 9, lO LEbGTk/BR[4DTII

Bodies with sinoidal runs, Knopf

921

Subrn. tankers Craqo 1958

I?

53 ceo torpedo Albrinq 1942

Fig. 1. General survey of the moat known experimental investigations, for

which results are published.

13 14 Test medium Resistance Propulsion Stability Pressure distr Water 1? Air

G

account of a method of calculation, by which a body of revolution

can be expressed mathematically from prescribed geometrical quanti-ties. For usual forms the method is relevant for bodies of revolution

having prismatic coefficients up to about 0.75. The results of the calculation are given as linear combinations of tabulated functions;

on the basis of a number of selected valuesof the longitudinal variable

and the location of the maximum section, the ordinates of the

sec-tional-area curve are read off in the tables provided.

Starting from a treatment developed in [2], WEINSLUM extends his theory of analytical waterlines and sectional-area curves to include

also bodies of revolution. The "basic forms" introduced by him are

used to express the symmetrical parts of bodies of revolution. These basic forms are applicable to full or fine forms, see e. g. [3].

The following method given for the mathematical definition of axially symmetrical forms is partially based on the three references mentioned, but on certain points it implies some extensions. It was intended that the method of calculation should be applicable to full and fine forms, and further, that the procedure should be so con-structed that it could also be used for an approximation of bodies not too extremely shaped.

Below a short comparison is carried out between the present work

and other earlier methods, particularly the method of LANDWEBER

and GERTLER, with respect to prismatic coefficient, system of axes,

geometrical parameters and the solution of conditional equations.

2.1. Prismatic coefficient

Bodies of revolution with high values of the prismatic coefficient

often have a cylindrical middle part. This does not exclude the fact

that in certain hydrodyiiamical investigations such a body is approxi-mately expressed by one single equation. Using the present method, forms with a prismatic coefficient up to 0.85 and even more, can be expressed mathematically. When applying the method of calculation according to [1] to bodies with a prismatic coefficient of about 0.75

it should be noted that it is possible to obtain unrealisticforms, even

if the remaining geometrical parameters are given normal values.

2.2. System of axes

In the present analysis the mathematical expressions are referred

body. According to WEINBLIJM, this approach has definite advantages

when investigating the wave resistance; the equation of the body can be divided in a simple manner into one symmetrical and one asymmetrical part, the expressions obtained are then used for cal-culating the wave resistance of these two parts independently.

2.3. Geometrical parameters

When approximating a given body by a mathematical expression, the question arises as to how many geometrical parameters are required to define the actual form. In their report, LANDWEBR and GERTLER remark that the primary geometrical parameters introduced

by them are not alone sufficient to fix an arbitrary body. For that reason they enlarge the mathematical expression with an additional

term, which does not affect the prescribed geometrical quantities.

In the present method of calculation the centre of buoyancy and

moment of inertia can also, if desired, be introduced as geometrical parameters. Moreover, after the example of WEnqBLTJM, two addi-tional parameters have been introduced, one even and one odd,

these affect the symmetrical and asymmetrical parts of the body, respectively. The additional parameters have no influence upon the

given values of the geometrical parameters. By use of a mathematical

expression which is based on six geometrical parameters and two

additional parameters, it has been found that there exists considerably increased possibilities of defining a normally shaped body of revolu-tion.

2.4. Solution of conditional equations

In the two mentioned methods, as well as in the present analysis, the developed conditional equations form a set of linear equations. In [1] this is solved by an indirect method where the result is given as a linear combination of the prismatic coefficient and the radius

of curvature at the nose and tail. The coefficients of this combination consist of polynomials, which are evaluated in formulas and tables. The tables are arranged on a basis of selected values of the longìtudì-nal variable and the location of the maximum section.

In the present method of calculation the developed set of

equa-tions is directly solved by use of a digital computer, which then

determines the equation of the sectional-area curve for every value required from those geometrical parameters which are intended to

8

y

t

X

Fig. 2. Systems of axes for bodies of revolution with and without cylindrical middle parts.

define the body. On the basis of the obtained expression, the

com-puter works out an arbitrary number of ordinates for the symmetrical

and asymmetrical parts of the sectional-area curve, and the contour of the body.

3. Symbols and definitions

In the following, the discussion is based on bodies of revolution without any middle part. When defining a body of revolution fitted with a middle part, calculations are performed only for its entrance and run.

A body of revolution with a cylindrical middle part is drawn up

in an xy-system. Examples of a body of revolution without cy-lindrical middle part, and the entrance and run of a body with a cylindrical middle part, are presented in ì1-coordinates, see Fig. 2.

Thus the following list of symbols and definitions is applicable

to a body formulated in a --system. When defining the body in

non-dimensional coordinates its length is = 2 and breadth at the half length ( = O) also = 2. For this reason, the geometrical

quanti-ties are made non-dimensional by dividing by the half length or the half breadth at = O.

The symbols used essentially agree with the nomenclature which is to be found in SNAME, Technical and Research Bulletin No 1-5, [4J.

R un Middle Erirance

L/2 L/2

Symbol Definition Non-dimensional form a a. B

B0

E k,, LER R RE BR 8 Ta TE TR T VER ER As index, asymmetry Polynomial coefficient

Polynomial coefficient used as additional parameter

Maximum breadth of body

Breadth of body at =O

As index, entrance of body Radius of inertia of body about the

,j-axis

Radius of inertia of the ,2-curve about the fl-axis

Length of body As index, run of body

End radius of entrance of body End radius of run of body As index, symmetry d??2a (LERI2) value = -B

b'-Bi_0/2 b' =2 k' k,7 L_ER/2 -, k L_ER/2 1ER = 2 RE

r'=

B2oI2LER BR r'R - B2o/2LER dfl20( I) Tangent dp72(LER/2) value ₌ ta d2(l) Tangent d2(LER/2) value = 1E= Tangent d,72s(LER/2) value = -d d25(l) Tangent Volume of body System of axes for body Longitudinal coordinate Coordinate of maximum breadth Coordinate of conter of buoyancy Ordinate of meridian contour

ER VER LER B20/8 çn, ER ç ER LER/2

lo b'0 -2 4-1 o ER 1ER/2l

Fig. 3. Non-dimensional representation of a body of revolution.

The non-dimensional representation of the contour and the squared contour curve (of-curve) of the body is shown in Fig. 3. The squared

contour curve consists of one symmetrical and one asymmetrical part. In the figure the most important geometrical quantities are

stated.

In the list of symbols and definitions the occurring geometrical

quantities have been made non-dimensional in a consistent manner according to the preceding treatment. However, some of the coef f

i-cients do not agree with those used in practice. For this reason the coefficient for volume of body, VR, is replaced by the more

com-monly used prismatic coefficient, which is defined as V

- LB2/4

A similar argument applies to the end radii of entrance and run

of body. In the preceding list of symbols the dimensionless representa-tion of the end radii is based on the half length and the half breadth at

= O, while in practice the total length and the maximum breadth

is used for the same purpose. Therefore the coefficients r and r

are replaced by r'+1 and r'1 which are defined

RE , RR

r±1

= B2ILER r_1 = B2ILER

-e)

r'i

V LER /4 RE BR B2ILER = prismatic coefficient = end radius of entrance

= end radius of run

4. Characteristic form parameters

4.1. Primary form parameters

In order to characterise the shape of a body of revolution, a number

of geometrical parameters can be used. The greater the number of parameters introduced, the more complete is the description of the body. On the other hand care must be taken not to introduce more parameters than can be controlled in the actual case. Thus, when

presenting mathematically a body of revolution a suitable selection

may be made from the geometrical parameters mentioned below.

The dimensions of these parameters are length and (length)3. length of body

B maximum breadth of body = volume of body

RE = end radius of entrance of body BR = end radius of run of body

= coordinate of maximum breadth

eRR = coordinate of center of buoyancy

= radius of inertia of the ì72-curve about the q-axis

For reasons which are discussed later (see Chapter 5.3.2) we

intro-duce the radius of inertia of the squared contour curve (2-curve)

instead of that of the body itself. The geometrical parameters stated above are here called primary form parameters because, when used as shape characteristics, they directly determine the mathematical expres-sion o/the body.

When presenting a body shape analytically, it is convenient to

express the geometrical parameters in a non-dimensional form.

Length and breadth (and of course also the length to breadth ratio) are therefore eliminated as parameters and finally we get the

12

The symbols occurring in the above are also to be found in Chapter 3 and Appendix i and 2.

4.2. Additional parameters

To make it possible to easily and systematically change the con-tour of a body of revolution, mathematically defined by a number

of primary form parameters, we introduce additional parameters (see

also Chapter 2.3.). These are two in number, one even and one odd as stated previously, and they influence the symmetrical and the

asymmetrical parts of the body respectively. The selected values of

the additional parameters are inserted in the conditional equations

for the primary form parameters (see Chapter 5.4.) and are then

to be found as coefficients in the final analytical expression.

4.3. Classification of bodies of revolution according to the number of

primary and additional parameters

When engaged in design work concerning bodies of different kinds it is common to meet one of the two problems mentioned below:

We are required to determine the shape of a body on the basis of a number of geometrical quantities. Thereby there are often special requirements concerning only the most important primary

form parameters. The case is general when the object is to make

non-postulated investigations of different kinds.

We require to describe a given body mathematically. When

de-fining the body by means of form parameters according to Chapter

4,1., we introduce a suitable number of these (see also Chapter

2.3).

Considering the above we select the following two sets of primary form parameters to be used in the present method.

d'i _'m'

- T

= coordinate of maximum breadth ER

ER

e) _ER

₌

= coordinate of center of buoyancy

LERI2

f'/ -i-''f11

-'I

= radius of inertia of the 2-durve about the

e) ø

= prismatic coefficient

r1

= end radius of entrance

r = end radius of run

= coordinate of maximum breadth Parameters as above and, moreover,

= coordinate of center of buoyancy

k radius of inertia of the 2-curve about the 1-axis

A simple system for classification of bodies of revolution can now

be introduced. Division into classes can be made according to the

number of geometrical parameters that define the body:

Four-parameter bodies, - of which the shape is defined by four primary form parameters according to c)

Five-parameter bodies, - of which the shape is defined by four

primary form parameters according to e) plus an additional

parameter (odd or even)

Six-parameter bodies, - of which the shape is defined by either oc) Four primary form parameters according to c) plus two

addi-tional parameters

or fi) Six primary form parameters according to d)

Seven- and eight-parameter bodies, where the shape is described

by six primary form parameters according to d) plus one and

two additional parameters respectively

Consequently, when working with problems of type a) we

com-monly use body-forms defined by e), f) or g oc) and with problems of

type b) body-forms defined by g fi) or h). This is exemplified in Appendix i and 2.

5. Equations for the geometrical conditions

5.1. Choice of mathematical form

As has been already pointed out in [1], it is convenient to base the

mathematical representation upon the sectional-area curve of the body. The fact is, that there exists a linear connection between the

end slopes of the sectional-area curve and the radius at tail and nose

of the body. This simplifies the mathematical treatment to a high

degree. Furthermore, the sectional-area curve is a good approximation for the distribution of doublets which generate the shape of the body

14

in uniform flow, a fact which simplifies the solution of certain hydro-dynamical problems.

In order to express the sectional-area curve analytically, there

are available different kinds of functions such as polynomiaLs,

ex-ponential functions, trigonometric functions or a combination of

these. In this treatment we choose the polynomial, though it is

possible that other functions may be more efficient, in the sense

that they give better coincidence by fewer terms. However, the

polynomial is easy to handle and admits a simple analytical

deter-mination of desired geometrical quantities. Some theoretical methods

e. g. those for computing the potential-flow field about a body of

revolution start simply from the sectional-area curve expressed as a polynomial.

5.2. Conditional equations for louT-parameter bodies

In this chapter we derive the equations which satisfy the

geome-trical conditions for four primary form parameters according to

4.3.c. For such a body we first introduce a general polynomial for

the sectional-area curve. When representing the body in

non-dimen-sional coordinates the sectional-area curve may be replaced by a squared contour curve (2-curve), which is expressed as

= i +a1Ea22+ .

.. (1)

or

Zn

= 1- -

(-1)1a1E'

Starting from four geometrical conditions (corresponding to the four primary parameters) and two boundary conditions we shall

determine six of the coefficients in equation (1). In principle the set

of coefficients to be evaluated (ai) may be choosen arbitrarily. In

practice, however, one or possibly several sets of coefficient-indices

and exponents can be attached to equation (1), in order to obtain a particular combination of primary quantities to give a body-shape

of realistic appearance.

Thus it may be shown that, in general, a fine four-parameter body (say ø = 0.6) is well defined by a polynomial with

- ø

a-- (

-) a 0

a3 + ( ¿-) a4...

- 0 ,a2_1+ (

'2i') a2= ø

- i

Whereas, the above indices and exponents could not be used for

the case of a full four-parameter body (say ø = 0.8). In this case

the body is best defined by a polynomial with e. g.

i = 1, 6, 7, 8, 9, 10,

these presuppose normal body-shapes. We shall now begin to esta-blish the equations by which the six coefficients a- in equation (1) can be determined. At first we divide the squared contour curve into symmetrical and asymmetrical parts

= 1a22a44 .

. .

a22

₍₂₎

= a1--a33-j- . . .

+a2_2''

(3)

We bring in the geometrical conditions according to Chapter 4.3.c. and the boundary conditions.

5.2.1. Condition for prismatic coefficient

Considering non-dimensional coordinates we get +1

0 =

For the numerator is obtained

±J +1

f q2d

_{== J (1+a1a22+}

+a_i2l_a2nE21)d=

-1 -1

2 2 2

= 2 --a2 --a4 -. .

_{- 2n+1}

a2 (4)

When determining b'/4 we put

=

,, into equation (1)

b'2/4

= 1+a1a2,Ç+ .

.

+a2_i1l_a2?z

₍₅₎

If equations (4) and (5) are brought into the expression for ø we

obtain

16

5.2.2. Condition8 for end radii

The end radii of the body are parameters which are entirely

asso-ciated with the contour curve (generatrice). However, the mathe-matical treatment is based on the squared contour (2) curve and therefore we introduce new parameters which are equivalent to the

end radii and which refer to the last mentioned curve. Thus the

expressions for the end radii are replaced by expressions for the

slope at the ends of the 2-curve i. e. the tangent value (see Chapter 3).

The question now is how to find a connection between these two

geometrical quantities.

Let the fore end of the body have the e-coordinate (dimensional

representation, see Fig. 2) and set the nose radius = RE. At the fore end the body coincides with a sphere of radius = RE and therefore

at =

the generatrice coincides with the circle

= R

or

2 D2_

1 -.Ll

Differentiating the above with respect to gives d(q2)

= -2+2(Ej-RE)

According to definition in Chapter 3 d(2)

TE =

_{- dE}

is valid for

=

and therefore when = we obtain

TE = 2RE (7)

Considering the manner of making the geometrical quantities non-dimensional the following is also valid

t = 2r

(8)

Moreover, with respect to definitions concerning system of axes and

tangent values we obtain

2

- (r+r' )a+ [(r+r')+

1]a

-' _'2i

- m (r+i+ri)ai+[m (r+1+r1)+n]a2,,

= r1+r'1

For the asymmetrical part of the 2-curve we have

d-q(1)

a

dE

Now we shall express the tangent values t and tÇ as functions

of the coefficients r1 and r1. According to definitions we have

= (t+t)

I f4 4t

- !

If equations (8) and (9) are brought in, the above may be rewritten as

t; = r+r

t,= rr

In conformity with Chapter 3 the coefficients r and r are replaced by r'1 and r1, hence

= +

b' (r+r')

(10)

t = + b' (r'1r'1)

For the symmetrical part of the 2-curve we have

d(1)

d

Thus, differentiating equation (2) and substituting = i we obtain

t = 2a2--4aH- . . +2na,, (12)

b'2 in equation (10) is substituted from equation (5). Then equations (10) and (12) give

18

Thus, differentiating equation (3) and substituting = 1 we obtain

t = a1 - 3a3 .

. . (2m-1)a,.,_1 (14)

As before b'2 in equation (11) is substituted from equation (5). Then

equations (il) and (14) give

t _'2

- [cm(

1r_1)+ --

ai+m(r+iri)ao-1j3 31 t4

-

rn

(r+ir_i)+ -i-j

a3+C,(r+ir_1)a4-2n-1

-

[1_1(r'

r' )+

₂

] a2fl1+(r±1r 1)a2fl =

= r1r'1

(15)

5.2.3. (Jondition for location of maximum breadth

The contour curve of the body may have an extreme value for

=

and consequently the following is valid for the squared

contour curve = b'2/4 for

₌

We differentiate equation (1) d2 de

= a1-2a2+ .

. . (16) d2

Setting = O for

=

we obtain the conditional equation

'2 -2

",1

ai-2ma2+

+(21)m

a21 1-2n;

a2 = O (17)

J.2.4. Boundary conditions

Because of the choosen system of axes, we have the following

boundary conditions when the body is expressed non-dimensionally:

= + 1 when = O and = O respectively.

5.3. Conditional equations for sir-parameter bodies

In this chapter we discuss the conditional equations for six-para-meter bodies, where the shape is described by six primary form

parameters according to 4. 3. d. Thus the object is to determine eight of the coefficients which are included in the polynomial (1). To

con-form with the case of four-parameter bodies, a set of

coefficient-indices and exponents (i) for equation (1) is selected. When we have a fine six-parameter body these indices may be

i= 1, 2, 3,4. 5, 6, 7,8

and in the case of a full six-parameter body e.g. i = 1, 6, 7, 8, 9, 10, 11, 12

The conditional equatious expressed in Chapter 5.2. are also valid

for six-parameter bodies. According to Chapter 4.3. there are two additional conditions, one concerning the center of buoyancy the

other concerning the radius of inertia.

5.3.1. Conditional equations for center of buoyancy

In non-dimensional coordinates the center of buoyancy of the

body is expressed as

f2d

ER +1 (20)

Inserting in equation (2) gives

+a2 = i

(18)

Inserting in equation (3) gives)

20

Solving the numerator, we obtain

p2" -2i+1

f

2d

= f (+a12a23+

+a21 a2

)d =

-1 -1

=a1+a3+...+

_2n+1 (21)

Equations (4), (20) and (21) give

¡ 2 2 2

ER t2_ --a2 -- a4 ... -

_2n+1 a2,,

2 2 2

=--a1+--a3+.

_{+2fl+l2n_l}

'LR 1 _¿ER

5.3.2. Conditional equation tor radius 0/inertia

In dimensionless coordinates, the radius of inertia about the 1-axis may be expressed by the following formula

a2,, = ¿ER

(±)

(23)

Substituting for 2 from equation (1) we obtain for the radius

of inertia of the body a conditional equation which is not linear in a.

However, our intention is to make the geometrical conditions form a linear system of equations. For this reason the radius of inertia

as a primary form parameter cannot be referred to the body itself in

this case. On the other hand, if we put 2/4_{= O in the numerator of}

equation (23) i. e. neglect the moment of inertia of every disc element

i _¿ER

Fig. 4. Disc element of a body of revolution according to formula (23)

about its own axis (see Fig. 4) we obtain for the radius of inertia a

conditional equation which is linear in a. Thus, the radius of inertia

is not referred to the body itself but to the squared contour curve of the body. So the radius of inertia of the squared contour curve about the n-axis can be written

7r

f22

(24)

f2d

It is clear that the radius of inertia about the n-axis can be exactly calculated for a body of revolution whose shape is already given as a polynomial, according to equation (1). If, however, the polynomial contains a great number of terms, such a calculation can be

trouble-some, while a similar calculation by equation (24) can be easily performed. Therefore it will be of some interest to investigate to

what extent the radius of inertia of the 2-curve may be approximated

to the radius of inertia of the body itself.

Primarily, the question concerning the approximation of k by

Ïc of course depends on the length to breadth ratio of the actual body, and to some extent on the shape of the body. As an example we here

determine the relation between X and k for a prolate spheroid of

varius length to breadth ratios according to Fig. 5.

Equation of the generatrice

b2

= - (a2 2'

22

Fig. 5. Scheme of a prolate spheroid.

Assuming the volume of the prolate spheroid to be = V we obtain

the following expression for the radius of inertia about the 1-axis.

b2 +0 _1b2

Fk = r

_-

_{f (a2_2)}

(a2_2)+2

We integrate the above

4

Vk2 - iab2(a2+b2) 15

In a similar way we obtain

4

V=

Then the result will be

a

- Va+b

(25)

In Fig. 6 the ratio /kq is drawn on the basis of the length to

breadth ratio.

We return to equation (24), the numerator of which can be written

f22d =

1

-

f(2+ai3_a24+

... +a2iE2

l_a2+2)d =

2 2 2

as

C

Fig. 6. The ratio ,,/k5 as a function of a/b for a prolate spheroid.

Inserting equations (26) and (4) into (24) we obtain

F2(9

--a2 -a4 ...

- 2n±1

a2n)

=

2 2 2

= - - a2 ...

_{- 2n+}

3a2 a

/1

1

2n+3 - 2n+

(27)

5.4. Introduction of additional parameters

The conditional equations determined in the preceding chapters

are intended for bodies of revolution which are defined only by primary form parameters. To make it possible to alter the contour of

such a body while its primary form parameters are kept constant, two addtional parameters have been introduced, see also Chapter

49

In the present account two coefficients, a, in the general polyno-mial for the squared contour curve, have been used as additional parameters. One of these belongs to the asymmetrical part and is

denoted aia and the other to the symmetrical part and denoted

24

As indices j we choose the two integers next larger than the sequence of integers i, which, according to the foregoing, represents the coeffi-cient-indices and exponents of equation (1). Thus, for a body which

is also defined by two additional parameters, the squared contour curve may be expressed as

= 1+a1a22+ .

.

+a2fl_i2 'a2

(28)

where indices and exponents 1,2, . . . , 2n belong to the sequence of integers i.

When determining the conditional equations for a body as above, we therefore substitute equation (28) instead of (1) as the expression for 2 and as a0 and are known coefficients in equation (28), the

known terms of the conditional equations will be increased by ex-pressions containing the additional parameters.

6. Computation of the polynomial of the squared contour curve The conditional equations for the primary form parameters

deter-mined in Chapter 5 form a set of linear equations from which the coefficients a in equations (1) or (28) can be found. As a summary

the system of equations is written down in Fig. 7 in matrix form, in

which the additional parameters are also taken into consideration.

The whole system can be written in the shortened form

Aa=c

(29)

where A denotes the matrix of coefficients and a and c denote the column vectors. It must be emphasized that indices and exponents

1,2, . . . , 2n consist of the sequence of integers i, which belong to the

squared contour curve of the body, according to equations (1) or(28).

The numerical solution of the linear equations and calculation of

the ordinates of the sectional-area curve and generatrice is performed

by a medium-size digital computer of the type W E G E M A T I C

1000. Thecodingworkhasbeencarriedoutbythe

ADB-insti-tutet at Chalmers University of Technology,

G ö t e b o r g, where two machines of this type are in service.

On account of the relatively small size of the matrix it has been

solved by using the elimination method according to GAUSS. However, the matrix is "ill-conditioned" i. e. when solving the system by normal

i

(2n-1)22

2nE21

i F

i

ER "ER .3 3 2n+1 2n+1 i i i i o

___i'2

o 5 3 '

2n+32n+1"

o i o i i o i o

øi+øEaa

(øE_

aj.

(r+r')(1 + Eaja)

[(ri

+ r) +

[E(ri_ri)+

ja]

(i-_jaa_laj

_{+isE 'a15}

i ¿ER

_{ja+2 - js+1}

i / i i I a 3 - \js-1-3 js+1 ' i ' 2g-1

-E1(r1+r1)

2n i

' _{' t_L.} m r+1 r_1, o

Fig. 7. System of equations in matrix form for determining the coefficients in the polynomial (28).

(øE:: 2n±1)

F97 F F Em (r+1+r1)+n F F Em (r+1r_,) i

-0

øE2

--., F F F F

,(r+1+r_i)

Em (r+1+ r_i) +1 a1 X a2

0.8/1.9/0/0.2 (i = 1,6,7,8,9,10) 26

methods, significant characters may be lost, partly due to rounding-off errors and partly due to exceeding the limited spaces in the arith-metic registers of the computer. For this reason the elimination pro-cess has been partially modified in order to obtain an accurate result.

One modification was to incorporate automatic shift of the binary point into the program to avoid overflow in the registers.

After fixing the coefficients of the polynomial, the computer will determine an arbitrary number of ordinates for both the symmetrical

and asymmetrical parts of the sectional-area curve and finally the

generatrice of the body. The well-known method of HORNER'S scheme is used to carry out these calculations.

7. Symbolic representation of mathematical bodies of

revolution

By means of a mathematical representation it is possible to identify different kinds of bodies by short and distinct expressions. According

to previous chapters, the equation of a non-dimensional body of

revolution is determined on the basis of some or all of the following parameters.

0 prismatic coefficient

= end radius of entrance

r_1 = end radius of run

= coordinate of maximum breadth ¿ER = coordinate of center of buoyancy

radius of inertia of the 2-curve about the n-axis

a0 = additional parameter, asymmetrical part of body = additional parameter, symmetrical part of body

Moreover, the mathematical expression of the body may contain a

selected set of terms a1E' according to equation (1) and (28). Thus, a four-parameter body defined by the first four parameters above,

may be identified by the symbolical expression 0

(i=

E.g.

E.g.

0.8/1.9/0/0.2/0.0838/0.2353

(i = 1,6,7,8,9,10,11,12; a13 = 0.1, a14 = 0.5)

S. Acknowledgement

The author wishes to express his thanks to Dr. HANS EDSTRAND,

Directorofthe Swedish State Shipbuilding

Experi-m e n t a 1 T a n k, for having been given the opportunity to work out this calculation method as well as for the interest he has shown in it. Thanks are also due to the staff of the Tank for all their

assi-stance.

9. References

[i] LANDWEBER, L., GERTLER, M.: "Mathematical Formulation of Bodies of Revolu. tion", DTMB Report 719, September 1950.

WETNBLIJM, G. P.: "Exakte Wasserlinien und Spantflächenkurven", Schiffbau Vol. 35 No. 6, March 1934.

WEINELUM, G. P., BLUM, J.: "The Wave Resistance of Bodies of Revolution", DTMB Report 758, May 1951.

"Nomenclature for Treating the Motion of a Submerged Body through a Fluid", Technical and Research Bulletin No. 1-5, SNAME, 1950.

In a similar manner the symbolical expression of an

eight-para-meter body will be

0 /r±l/rl//R/kl

Appendix i

Calculation of a number of bodies of revolution with their qeometrical

parameters systematically varied.

Let us assume that systematical investigations concerning

re-sistance, propulsion, stability and other aspects of a body, are to be performed for a project, and in addition, let us presuppose that the

body has an axially symmetrical shape. Thereby the main dimensions,

and some fundamental demands of the project are available at an

early stage. The final optimum shape of the body will afterwards be determined by other considerations. (See also Chapter 4.3.a).

In investigations of this kind it is often convenient to study a

family of body, shapes, in which the given geometrical quantities are kept constant. Thus, the parent form may correspond to a body

shape, which is supposed to be the optimum one. Other forms in the

family are given systematically varied values of those geometrical

parameters, the influence of which, we require to investigate.

As a numerical example we here choose a project which is supposed

to be a submarine tanker or something similar. Length, breadth,

displacement and position of maximum breadth of the hull form are

fixed geometrical quantities. These are given below as

non-dimen-sional coefficients.

Tail radius is made = O and a suitable value of nose radius is supposed to be r1 = 1.3. Thus, when planning for a family of bodies of revolution, the parent form may coincide with the following body

(symbolically indicated) 0.7/1.3/0/0.2 (i = 1,2,3,4,5,6)

L/B=6

Length/breadth, LIB = 6 Prismatic coefficient,

0 = 0.7

Now we require to make the following systematic variations of the parent form

a) Variation of nose radius of body:

= 1.0 1.3 1.6

(Underlined value indicates the parent form) b) Variation of asymmetrical part of body:

= a7 = 0.5 0

0.5

When increasing the odd additional parameter the asymmetry

of the body will increase so that, in this case, the bulk of the volume is slightly shifted towards the entrance of the body. e) Variation of symmetrical part of body:

a=a8==1

1

When increasing the even additional parameter, the symmetrical

part of the body is altered so that, in this case, the bulk of the

volume is shifted more towards the middle of the body.

The effect of variations of the parameters according to a)c)

are shown in Figs. 8, 9 and 10, where also the polynomials of the computed mathematical forms are given. Since all the bodies are drawn up at constant length/breadth ratios the ordinates here

cor-respond to i/0.5b' instead of .

Appendix 2

Mathematical formulation of a given axially symmetrical form

In order to give another illustration of the present method of

mathematical formulation we shall express mathematically an

empirically given body of revolution. (See also Chapter 4.3.b.)

As an example, we choose that form, which corresponds to the well-known airship "Akron". The aerodynamical properties of this

body have been the subject of extensive experimental and theoretical

0,51,

Fig. 8. Example of systematical variation of nose radius with constant values of other

primary form parameters

- 0.7/1.0/0/0.2 ,j = (i = 1,2,3,4,5,6) 0.8786135H-0.5303O95 0.7/1.3/0/0.2 , = 1 + 0.170951_0.670I962 + 0.976667-0.341040 (i = 1,2,3,4,5,6) 1 .147618+0.01 12366 - 0.7/1.6/0/0.2 = 1 1.2O9414+0.393631 (i = 1,2,3,4,5,6) 1.417583-0.509692

Fig. 9. Example of systematical variation of asymmetrical part with constant values - 0.7/1.3/0/0.2 (i = 1,2,3,4,5,6; 0.1 1229O + a7

0.5)

0.7/1.3/0/0.2 (i = 1,2,3,4,5,6) 1. 147618e -f- 0.01 12366 . . - 0.7/1.3/0/0.2 (i = 1,2,3,4,5,6; 2. 182945-0.032584° + = 0.5)

b'

0,2

0,4

-Fig. 10 Example of systematical variation of symmetrical part with constant value8

of primary form parameters

- 0.7/1.3/0/0.2 = i +O.254758_0.9479992+ 0.8l7843+ 1.223357e (i = 1,2,3,4,5,6; a8 = - 1) 0.7/1.3/0/0.2 ,j = i + 0.17095 1-0.6701 962 + 0.976667-0.341040 (i = 1,2,3,4,5,6) 1.I47618+ 0.01 12366 --. - 0--.7/1--.3/0/0--.2 ,j = 1)0.087144_0.3923942+ 1.l35489l.9054344 (i = 1,2,3,4,5,6; _1.222634,5+2.2978286_8 a8 = 1)

'o 0,5 o _{- 0,5} 'o

Fig. II. Contotii of the airship "Akron" and its approximate mathematical form.

Akron

- O.66/O.7375/O/O.1910.1093/O.1966

(i = 1,2,3,4,5,6,7,8)

i + O.218O34_O.74O7262+

34

all six primary form parameters of the original body (according to

Chapter 4.3.d). Thus

= 0.1900

R = 0.1093

= 0.1996

In Fig. 11 the calculated mathematical form is to be found together with the original body. The agreement seems to be good, this can aLso be seen from the following table

e-values i 'Akron" -values Approximate mathematical form -1 0 0 -0.9 0.207 0.204

-0.8

0.389 0.385

-0.7

0.543 0.542

-0.6

0.672 0.673

-0.5

0.775 0.778

-0.4

0.854 0.859

-0.3

0.914 0.918

-0.2

0.956 0.959

-0.1

0.984 0.985 0 1.000 1.000 0.1 1.006 1.008 0.2 1.008 1.010 0.3 1.006 1.006 0.4 0.996 0.996 0.5 0.974 0.973 0.6 0.932 0.930 0.7 0.857 0.857 0.8 0.733 0.737 0.9 0.535 0.541 1.0 0 0 0 = 0.6600 r = 0.7375 r'1

=0

Page

Summary 3

Introduction 3

A short survey of literature and comparison with the present

method of calculation 4

Symbols and definitions 8

Characteristic form parameters 11

Equations for the geometrical conditions 13 Computation of the polynomial of the squared contour curve 24

Symbolic representation of mathematical bodies of revolution 26

Acknowledgement 27

References 27

Appendix i Calculation of a number of bodies of revolution with their geometrical parameters systematically varied 28

Appendix 2 Mathematical formulation of a given axially