Computer aided design of shipform by affine transformation

TECHNISCHE HOGESCHOOL DELFT

AFDELING DER SCHEEPSBOUW- EN SCHEEPVAARTKUNDE

LABORATORIUM VOOR SCHEEPSHYDROMECHANICA

Rapport No. 438 P

A. Verslu is

COMPUTER AIDED DESIGN OF SHIPFORM BY AFFINE TRANSFORMATION

Delft University of Technology Ship Hydromechanics Laboratory

Mekelweg 2 Delft 2208

Contents Introduction

Transformations of basic shipforms

2.1. Method 1: Separate transformation of the after- and the forebody 2.2. Method 2: Transformation of the whole shipform

2.3. Transformation of the waterline endings Computerprogram

3.1. Input 3.2. Output 3.3. Applications

3.3.1. Example 1 (fine ship) 3.3.2. Example 2 (full ship)

3.3.3. Comparison of the plotted lines drawings from the examples 1 and 2 with their basicshipform

Basicshipforms Appendix 1 Appendix 2

I. Introduction

Two methods to design shipforms by means of a transformation from a basic shipform are being

des-cribed in this article.lt is possible to use different kinds

of shipforms as basicform, and to store these basic shipforms in digital forrn in the computer memory. Within fixed limits of the blockcoefficient and the

longitudinal position of the centre of buoyancy a

basic shipform can be transformed into a new

ship-forrn. It is time consuming to prepare a good faired basic shipform, therefore a "designbooklet" (which now contents 10 basicshipforrns) will be developed gradually. The "designbooklet" will have to contain

enough basic shipforms as to make it possible to

choose a shipform most suited for the desired ship.

The transformed shipforrn is drawn to a given scale by

the x-y plotter coupled to the computer. The

hydro-static calculations are included in the program.

2. Transformations of basic shipforms

For the transformation of a basic shipform to a

desired shipform two methods of computation can be used.

The first method is to work with the two halfs of the ship, namely the afterbody and the forebody and

then to transform these two parts separately to a given

value of the blockcoefficient of the afterbody and of the forebody respectively. In the second method the correct blockcoefficient and the longitudinal position of the centre of buoyancy of the whole ship are used

for the desired transformation. By means of a "switch"

it is possible to choose for one of these two computa-tion methods, the choice to make depending on the desired alternation from the basic shipform. The first method is especially useful, if only the form of the forebody or the form of the afterbody must be

chang-ed. For example: for research into the field of

sea-keeping qualities it is common to vary the forms of the ordinates in the forebody, while the afterbody remains unchanged.

2.1. Method 1: Separate transformation of the

after-and the forebody

The curve of sectional areas shows the distribution of the sectional areas over the shiplength. This curve is the base for the set up of the lines drawing. If the

1 Delft University of Technology, Ship Hydromechanics Laboratory,

The Netherlands.

COMPUTER AIDED DESIGN OF SHIPFORM BY AFFINE TRANSFORMATION

by

A. Versluis

position of the longitudinal centre of buoyancy does

not correspond with the desired one, the general

known con-ection method will be applied, namely

transformation of the curve of sectional areas as shown in Figure 1.

Figure 1.

B1 = longitudinal position of the centre of gravity of

the curve of sectional areas.

B2 = longitudinal position of the centre of buoyancy.

h= vertical position of the centre of gravity of the

curve of sectional areas. The transformation angle a is:

-a = -arct-an

The dotted line is the transformed curve of sectional areas. For small angles a the area of the curve is almost

unchanged, and corresponds with the volume of the

ship below the construction waterline: the area between this curve and the axis is unchanged, and also the prismatic coefficient is not changed. The new

sec-tional area of the design ordinates can be read-off out of this figure.

This procedure means that the design ordinates are transformed with maintenance of their area. To change the longitudinal position of the centre of buoyancy of

the basicship, the curve of sectional areas of that

basicship must be transformed with an angle

a as

shown above. Instead of calculating the new sectional areas of the design ordinates, the distances over which

the original design ordinates are moved (ai) are

com-puted.

Figure 2.

2

a. = F. x tan(a) J

in which

j is the index of the design ordinates,

F. is the area of design ordinate j.

L

1

Figure 6.

By making both half curves dimensionless into x and

y directions by means of dividing by half the length

between the perpendiculars, and the main cross section area respectively, the area of each of these curves will

corresponds with the prismatic coefficient of the ship

part concerned. When these curves are transformed

with different angles a, a relation between

transforma-tion angle aa versus (pa and the relatransforma-tion af versusOf

can be obtained; see Figure 7.

Figure 7.

if: a>

0 > Ø basicshipform

a < < basicshipform.

With given values of the prismatic coefficient 4) (total) and the longitudinal position of the centre of

buoyancy, the values of oa and Of can be read offfrom

the "Combinationdiagram" of Guldhammer (see

Ap-pendix 1). The transformation angles aa and af, for the

after- and the forebody can than be computed from

Figure 7. With these angles the new positions of the designordinates can be computed. In this way a

num-ber of points of the transformed waterline has been

found.

The contour in the plane of symmetrychanges with

the ordinates. Above the constructionwaterline the

contour does not change. Of this transformed

water-line the coefficients of the Theilheimer Spwater-line interpo-lation polynomial (Appendix 2) can be computed, and

after that the half breadths at the original positions of

the design ordinates (i.e. with equalintervals) can be

determined. If this is repeated for a numberof

water-lines, the transfornied form of the

design ordinates

can be determined. Hereafter the ship will be made

dimensionfull by means of the principal dimensions of the ship as desired. The transformation of the curve of sectional areas of a fine basicform without a cilindrical

midship region will produce a

knuckle at half the

length when d) is transformed to a smallervalue than that of the basicform; see Figure8.

4)a f cpa basicship If basicship o 2 6 8 10 12 11 16 18 20 Figure 3.

The new positions of the design ordinates of the ship in Figure 3 of which the longitudinal position of

the centre of buoyancy is shifted backwards with

con-stant displacement, are indicated by the dotted lines.

I f a new shipform has to be designed, starting from a basic shipform, in many cases not only the position of

the centre of buoyancy will change, but also the pris-matic coefficient. To make a ship fuller, the curve of sectional areas of the afterbody is transformed by an

angle a and the curve of the forebody with anangle

a (a > 0); see Figure 4.

0 10 20

A.P.P. F.P.P.

Figure 4.

A combination of the changing of the longitudinal

position of the centre of buoyancy and increasing or reducting the prismatic coefficient will lead to two

angles aa and af by which the curve of sectional areas of the after- and forebody will have to be transformed respectively; see Figure 5.

10 20

A.P.P F P P

Figure 5.

The design ordinates in the afterbody are displaced

over a distance F. tan (aa) and in the forebody over

a distance F. tan (a f).The curve of sectional areas

is now divided into an after- and a forebody with an

overlap of 4 ordinate intervals necessary to make the

fairing of the midship form possible.

The afterbody consists of the part between design

ordinate 0 till 12, while the forebody is the part

10 /2

Figure 8.

curve of sectional areas of

the basocshipform

curve of sectional areas of

the ship desired

This means that this method is not suitable for this case.

If the prismatic coefficient is greater than that of the basicform, a cylindrical midship part will be

cre-ated automatically; see Figure 9.

cylindrical rnidship part

curve of sectional areas ot the

basic ship form

curve of sectional areas

of the ship desired

s _{longer cylindrical midship part}

- curve of secticvvl areas ot the

bas c shmform

curve of sectional areas

Arm of the ship

desired

< basicshipfor shorter cylindrical midship part

Figure 10.

2.2. Method 2: Transformation of the whole shipform

In the second transformation method the curve of sectional areas will not be divided in an after- and a forebody. In the x-direction the curve has been made

dimensionless with the half length between the perpen-diculars of the basicform, while the sectional areas are made dimensionless by the main cross section area.

-1

ord O ord 10 ord 20

Figure 11.

20

In shifting the design ordinates, a goniometric term

will be used: n.l.

a. = A x. sin( ir X- x. I) in which

A = a parameter, tx.

= he distance of ordinate j until '/2L.

a a A X. slrqn,DO I

curve of sectanal areas 0

of the basicshipform

Figure 12.

Figure 12 shows that for A > 0 the basicordinates

are moved to the ends and so the prismatic coefficient will increase. When A < 0 the basicordinates are

shift-ed to the main cross section, which results in a finer

ship; see Figure 13.

a.AviXe sin(x el XI)

curve of sectional areas of the basicshipforrn

Figure 13. 3 -1 ord. 0 loved gn ordinate

A1116

Ifo.

21111

ord. 20 1111111111111 I

11111111/'

I i I ord 20 ord D I I I I I I I i I I , 1

Icurve et sectiona areas

i

af ter Moving the ordinates.

111

ma

ord.20 -1 ord.0 ord.0 .../ I I I I

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...--I I 1 of moving I 1 I ''' I ..- i ....1 I I 1 I -Icurve -7 the section ord ord. 20 1areas after nates. I I I I - 0 N.X 10 12 20 Figure 9.

For basicforms with a cylindrical midship part this first method is very suitable. Increasing of the

pris-matic coefficient results in a longer cylindrical midship part and on the other hand transformation to a smaller prismatic coefficient gives a shorter cylindrical midship part; see Figure 10.

-curve of sectionat areas of the

basic shipform

10 12 20

curve of sectional areas

4

To compute the prismatic coefficients for different

values of the parameter A, the relation between 0 and A can be found.

Figure 14.

In this way a shipform with the correct value of the

prismatic coefficient is found, but this form has an

incorrect place of the longitudinal centre of buoyancy.

To move the longitudinal position of the centre of

buoyancy without changing the prismatic coefficient, the earlier mentioned method is used, namely swinging

the curve of sectional areas.

The shifted ordinates give points for the

transform-ed waterlines. The contour and the ends of the water-lines until the beginning of the rounding are shifted

just as the ordinate forms. By means of the half

water-linebreadths of the shifted design ordinates and the endpoints of the waterlines till the rounding begins, the coefficients of the Theilheimer spline interpola-tion polynomial of that waterline can be computed and after that, the half breadths of the transformed

waterlines at the original places of the designordinates (i.e. with equal intervals). By repeating this for a

num-ber of waterlines the transformed forms of the design ordinates are found. After that the whole transformed ship is made dimensionfull with the half length, the half breadth and the draught till the construction

wa-terline.

As well as for full ships with a long midship cylin-drical part as for fine ships without a cylincylin-drical

mid-ship part, this method can be used very well.

2.3. Transformation of the waterline endings

For the rounding of the waterlines at the

ends, a polynomial is used, which is determined by two points

and the first derivatives in these points. The equation

of the polynomial is:

y2 +by=Ro +R1 x+R2x2

As illustration for a waterline in the afterbody, for

example at the transom, is given in Figure 15.

A is the point where the waterline begins with the

rounding. Point B is the end of the rounding, in this

case the transom at thewaterline. To simlify the

poly-A(xAyA

Ie(xe,yB)

CIOS, Odio

Figure 15.

nomial, the origin of the coordinates system is shifted to point B.

v,

Figure 16.

For further simplification x, and yA are given the

value one.

Figure 17.

MB is the derivative dx/dy in point B, and MA is the

derivative dy/dx in point A. With this conditions the

coefficients of the polynomial can be computed:

= (1 MA )/( MA 2 + 1)

R2 = 1

(1 MB )

and

b = .

3. Computerprogram

Both the described methods are programmed in an

Algol-computerprogram. By means of a "switch" the

desired method can be chosen. The offsets of the basic

shipforms are continuously stored at the discmemory

of the computer. Each basic ship from hasbeen given a codenumber. With this codenumber the computer can

get the data of the desired basic shipform, and

trans-form this basicship to the shipdesired. The computer

time used by the IBM 370/158 system, like the one at

the computer centre of the Delft University of

Tech-nology, amounts about half a minute, and the memory used is 384 K bytes.

In the following parts of this chapter the in- and

output of the computerprogram is given in addition to

3.1. Input

The following data are needed to compute and draw a lines drawing by the computer.

length between the perpendiculars breadth moulded

draught until the construction waterline

switch for chosing one of the two computation

methods

position = 1: method I (separated fore- and

after-body)

position = 2: method 2

if position = I:

- _{(blockcoefficient of the afterbody)}

- _{Cb f (blockcoefficient of the forebody)} if position = 2:

Cb (blockcoefficient of the total ship)

LCB (longitudinal position of the centre of buoyan-cy)

codenumber of the desired basicshipform scale of the lines drawing

switch for plotting the lines

position = 1: only plotting the designordinates position = 2: plotting the whole lines drawing if position = 2 then:

switch for the way of plotting the waterlines position = 1: normal drawn waterlines position = 2: dotted waterlines alternatively

switch for plotting the hydrostaticcurves position = 1: the hydrostatic curves are plotted position = 2: the hydrostatic curves are not plotted.

3.2. Output

The output of the computerprogram contains: - list of ordinates

- hydrostatic calculations - list of sectional areas - the plotted shipform

Dependent on the switch for plotting the lines, only

the designordinates or the complete linesplan is drawn. - the hydrostatic curves

Dependent on the switch, the hydrostatic curves are

plotted or not.

The length of the plotted hydrostatic curves diagram

is always 50 cm, that means independent of the scale

of the lines drawing.

The output of the second method is identical to

the output of the first method.

3.3. Applications

Starting from the same basicform two linesplans

will be computed and drawn. The maindimensions of

both examples are kept equal to another, while the

blockcoefficient will be different.

By example 1 is: C, < Cb basicshipform, and by example 2 is: Cb > Cb basicshipform.

In both examples the second computation method is

chosen.

3.3.1. Example 1 (fine ship)

The input of the computerprogram:

- length between the perpendiculars _{180.00 m}

breadth moulded _{28.00 m}

draught until the construction

water-line _{10.50 m}

switch for choosing the computation

method _{position 2}

Cb _.75

LCB _{+ 2.90 m}

codenumber of the desired

basic-shipform 3

scale of the lines drawing 250 switch for plotting the lines position 2 switch for the way of plotting the

waterlines _{position I}

switch for plotting the hydrostatic

curves _{position I}

The output of the computerprogram is then:

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3.3.2. Example 2 (full ship)

The calculation method is now repeated to design a

shipform with a blockcoefficient larger than that of

the basicship started from.

The input of the computerprogram is now:

- length beteen the perpendiculars 180.00 m

- breadth moulded 28.00 m

- draught until the construction

water-line 10.50 m

- switch for choosing the computation

method position 2 10 Oa

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codenumber of the desired basic-shipform

- scale of the lines drawing

- switch for plotting the lines

- switch for the way of plotting the

waterlines

- switch for plotting the hydrostatic

curves

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13cutpt 013 331030, Basicshipform nr. 7 413,1,00 t.t.36034" 303,34 .1 00,00 tan0.0,30. - 0.1330 c C. WM. 1030.0.601 al luau .36.0.40 o 3 00 2 00 013 LiND5 0.141. SCALE 36 311,311tat 01331161049t 124414 MO. 0.301 30. 00 033006t 3. 760 11 t 1C 100. HYDROSTATIC CURVES tt IC.. 10000

12

IIIEEMIMMMMik

_{_}

)11111.1111111

AhnI11111111,----BEMRe ET 020 22 CT 0 '0 HTOROSTRTIC CURVES

vsFANNPvariagsmoorms

r sin rim m or.

111111=1111

rIl

111111/11MIN

MIL 1111//ir

015:.4 OP"

W..;,-, -

,../....abaarniMI

.0111/115MERVAtt

tr=020.,

411,

iimmommotasmiww-111111%11111111111MINIIIIIIP/

_{Nimwa...Imm....}

01

.1-1/4110151MILISIII

/A Mk' Ì I

I00 .0 uolo 222. Rum HYDROSTATIC CURVES 414,.5444, MI NI 4.44 or 0454414, IN 445. a.255N IL ,[4. 0000 r 7 Basicsinpform nr. 9 HYDROSTATIC CURVES LINES DR. NG SCALE 0 00 0 00 5 00 Basicshipform nr. 8 Basicshipform nr. 10 FAINCIPAL01.251., mina, 20.0[0 1,200 2 022.21. 404.0 11154040,5. .1111 alNOW,IMLENOVN.5. O.12n 22.22/, miimm Tm. x.om .2. on, oisnmemm

[rim/ womm .2.212 Tes. 4, A

cm.... ...ow, 2202.22. 2.7

11111111MIMIESIIIMPW11-MIN=

DIM"

wmomniNLAI/Nm=

_///%11%/MINE

CURVE Or SECTIONAL AREAS % OF TME MAINFRAME AREA PRINCSPAL 044.I045.

LINES DRAWING SCALE I:

LEMON tomo.o.-zol us.. II

4.401. NOULOCO 9.000 N

0.001

LINES DRAWING SCALE I: _{1.1.14 1000.020.} 0.10

0.4414. 44.0.1300 4100 N 0.4104 0 0 2 OP 6 00 5.00 0 CO I 20 0 20

Combination diagram

At a very early stage of the designing process, the

principal dimensions, the blockcoefficient and the LCB are chosen and fixed. If the FORMDATA is to be used

in the further procedure, it is necessary to know the

blockcoefficients 5, and 5, for the afterbody and

forebody separately. This can be done by simple

esti-mating, but a quicker and more reliable way is to use the diagram above. It replaces the

combination-dia-gram given in the FORMDATA (I) and is more logical and easier to use.

EXAMPLE:

Given 5 and LCB: say 5 = 0,682 and CB: 1,2% abaft

ML

(D. By entering the diagram at 5 = 0,682 and =

LV

0,012 aft, we read 5 A = 0,705 and 5F = 0,659.

Appendix 1

NOTE!

The curves are based on the FORMDATA forms but

by concentrating all the variations into one set of

curves, some small errors may sometimes be found,

and in critical cases the results has to be checked and corrected if necessary. The error will be very small in all normal cases, corresponding to an error in LCB of less than 0,1% of L.

Taken from: Formdata IV by H.E. Guldhammer Danish Technical Press, Copenhagen, 1969.

Centre of Buoyancy from 231as Fraction of L = =

14

Appendix 2

Theilheimer spline interpolation polynomial for N

points.

f(x) = C1 + C2 x + C3 x2 + C4 x3 + C5 (x

x3) +

CN-1(x xN-3)+3

3

+ COx xN_2 )+

INTERNATIONAL PERIODICAL PRESS Rotterdam - The Netherlands Matrix N x N for computation the N coefficients

1 x1 x21 X13 o o O O o 2 3 O O o 1 x2

x2'2

o o 2 3 o O o 1 x3 x3 x3 O O 2 3 3 o O o 1 x4 x4 x4 ("4-'3) O

23

3 3 o O o '5 '5 "5 ("5-'3) ("5-'44 x2 3 3 3 _{o O o} x6 6 x6 (x6-x3) (x6-x4)+ 2 3 _{)3( )3 )3 o O} 'N-3 2

3)3t

₎₃ )3( )3 O 'N-2 'N-2 'N -2N -2 -'3'+'"N -2 -'4'+ 2 3 3 3 3 3 3 1 xN-1 'N-1 'N -1('N -1 -'3)+('N -1 ('N -1 -'N -4)+('N -1 -'N -3)+('N -1 -'N -2)+ ("N-N-4)3+ ("N-"N-3)3+ (N-1-2)3+ XN 3 3 x2 (xN -x3) (x,-x4)3,. N

Given : (xi, Yi) 1=1 N

CI C2 C3 C4 C5 C6 cN-3 CN-2 N-1 cN y1 Y2 Y3 Y4 Y5 Y6 YN-3 YN-2 Y N-1 YN mdlere: (x = (x xi) 3

if x> xi

=0

if x < = xi

This curve consists of cubic segments which have continuous first and second derivatives and a discon-tinuous third derivative. Such a batten of curves is