Numerical definition of ships hulls by means of elastic interpolation

NATIONAL RESEARCH LABORATORIES Ottawa, Canada

RE PORT

Division of Mechanical Engineering Ship Section

Pages - Preface -

6 Report: MB-258

Text

-

32 Date: January 1965

Tables

- 13

Lab. Order: 13435A

Figures.

-

22 File: M2-29-2

For: Internal

Subject: NUMERICAL DEFINITION OF SHIPS' HULLS BY MEANS OF ELASTIC INTERPOLATION.

Submitted by: S.T. Mathews Author : D. Gospodneti Section Head

Approved by: D.C. MacPhail Director

SUMMARY

By the interpolation method described in this report a computation of the offset of any point on the ship's hull is made possible, . once the usual table of offsets is given. A thin elastic beam, forced to pass through the given points,, is used as the interpolator. Examples and descriptions of computer programs illustrate the practical use of the method.

Page (ii) MB-258 TABLE.OF CONTENTS :1.0 2.0 3 0

4.0

SUMMARY LIST OF TABLES LIST OF ILUSTRATIONS LIST OF SYMBOLS INTRODUCTION ELASTIC INTERPOLATION

2.1 Introductory Defi.nitions and Relations 2.2 Interpolation for the Given Points 2.3 Relation Between g*, .y* an4 .G, r

2 4 Practical Method of Obtammg G, r from Given Data 2.5 Practical Method of Obtaining

1from G,

P

2.6

Description of Computer Proams

2.7

Examples .

2 7 1

Simply-Supported Beam

2.7.2

NACA Profile 63, 3-6.18 2. 7.3 Curve With Loops

NUMERICAL DEFINITION OF SHIP'S HULL 3. 1 Classification of Parts of the Hull

3 2 Information Needed for One Part of the Hull 3.3 Generation of New Information.

3.4 Example APPLICATIONS', - - '. Page (i) (iii) (iv) (v) 1 1 1 6 13 16 20 23 25 25 25 26 26 26 26 27 28 31

LIST OF TABLES

Table Simple Beam: Reduction Output

Simple Beam: Evaluation Output

Wing Profile: Reduction Output 3

Curve With Loops: Reduction Output 4

Input Data: Sections

Input Data: Waterlines 6

Input Data Sections: Reduction Output 7

Evaluation Output

Transposition Output 9

Comparison 10

New Input Data: Sections 11

New Input Data: Waterlines 12

New Comparison 13

Page (iii) MB-258:

Page- (iv)

MB-258 LIST OF ILLUSTRATIONS Figure Elastic Beam Conditions at Supports 2 Forces at Supports Single Section

g*, y* versus G, r

o'

versus G, r

6 Function V 7 Simple Beam 8 Wing PrOfile 9 Leading Edge V

curve With Loops

10

Curved Surface 12

Boundary Curves 13

Boundary Curves 1

Contour and Boundary Conditions

Body:Plan V V : V, 15 VV16 Waterlines V 1.7 Comparison Discrepancies. . 18

Body Plan DetailV

V V

V

V

19

New. ComparisOn Discrepancies 20

New Body Plan Detail . 21

LIST OF SYMBOLS

Symbol Meaning Page Equation

a,b,c,d

coefficients 16 (49) g function 9 (33) g* _{function 10 (37)} j index of section k constant 4 (10) I length of section

r

chord length 6 (22) s arc length

x,y

co-ordinates C1 constant 4 (9), (10) E elliptic Integral II 6 (17) EJ rigidity F elliptic integral I 5 (16) G constant 10 (36) L index, end-point M bending moment 3 (3)

0

index, initial point

P force

V constant 21 (57)

Z constant 21 (57)

Page - (v)

Page - (vi) MB-258

LIST OF SYMBOLS (Cont'd)

Symbol Meaning Page

...

Equation

a

chord slope angle 6 . (21)

coefficient (5) y function 9 (34) fUnction 10. . (37)

60

angle 2 . (.1) angle slope angle slope angle (19) p radius of curvature (2) TO angle (29) variable (10) coefficient 3 (5) constant 10 (36) function 8 (27)

1.0 INTRODUCTION

The description of a ship's hull is usually given as a set of mutually orthogonal cross-sections, known as waterlines, stations, and buttocks, from which a table of discrete offsets is prepared. It often occurs in practice that the designer needs a point, or a set of points on the hull, that are not coincident with those given in the table. He is then compelled to refer to the drawing and draw in some supplementary àurVes. Therefore it seems desirable to develop a method of computing the new offsets from the given ones, avoiding as far as possible any reference to the original drawing. The way in which the cross-sections are drawn points to an appropriate method of interpolation. The draughtsman uses a thin, elastic beam, made of wood or plastic, and forces it, by means of suitable weights, to pass through the given points. This is obviously a natural way of doing it, for it corresponds to the wa in which the steel plates are actUally fixed on the frames, erected on the berth in the shipyard. This repOrl provides the desired information by calculating the form of. a thin, elastic beam passing through the given points. It also demonstrates that, by proper sequence in application of such elastic inter-polation, the number of ship' s hull offsets increases, self-correcting, until the surface definition, viz, offsets density, reaches the prescribed accuracy. This might be the accuracy of the numerically-controlled milling machines for model making in research institutes, or, as well, the mould-loft floor accuracy for the purpose of plate cutting in shipyards.

2.0 ELASTIC INTERPOLATION

2. 1 Iiltroductory Definitions and Relations1)

Consider a thin, homogeneous, symmetrical elastic beam of a constant cross-section, loaded at its ends by forces and moments acting in the symmetry plane (Fig. 1).

Page - 1

MB-25,8 NUMERICAL DEFINTTIOW OFSHIPS' HULLS

BY MEANS OF ELASTIC INTERPOLATION

Page - 2

MB-25 8

0 denotes the initial and L the end-point of the beam that is straight when unstressed. The eqüilibriuthcoñditions yield

It is convenient to choose a special co-ordinate system with the origin at .0 and pOsitive direction of the abscissa coinciding with

denotes the slope angle of the tangent in (x', y') system U denotes the slope angle of the tangent in (x, ,y) system. Positive direction of the tangent. comcides with the positive direction of the curve at the contact point

x0,

₀

are the co-ordinates of the point 0 in (x, y) system.

60 is the angle between x' and x directions.

s isthe length of the curve between Oand some point on the curve. Its positive directio

is from. 0 towards L.

is the total length of the curve between 0 and. L. 1/p is the curvatUre at some point on the curve. EJ is the rigidity of the beam.

M is the bending moment at some point on the beam. Transformation formulae are

1/p

= d/ds = M/EJ

(2)

+

x'

coso

+ y'

sinO0

(1)

y = y0

cosä0 - x'

sinó0 It is

Page - 3 MB-258 where

M

= (3)

From (2) and (3) it follows that

d/ds

₌

- y')/EJ

_{+ ML/EJ}

Observing that dy'

= ds

sin- and

dx' =

cost, and

differ-entiating with respect to s, we obtain

2/dS2

= -(P

sin)/EJ

(4)

Let us introduce the coefficients

Ia

1IP/EJ,

M0/v,.

WL

= ML/

(5)

that satisfy the relations

=

_{f (i/p)0}

₌

I (d/ds),

_{/3WL = I (1/p)L =}

_{I (d/dS)}

₍₆₎ Using (5), we give to (4) a non-dimensional form

i2 (d2/ds2)

=

_-

2 (7)

that can be transformed into

i2 d(d/ds) = _2/32 sin(/2)

cos(/2)

ds By multiplying with (d /ds) and integrating, we obtain

Page MB-258

Further integration depends upon the value of the constant C1 There are three possible cases

(sin2 ('/2)) C1 < I

1<c1+

1

We shall concenate on the case (9a) only, for the reason given later.

Put

Cl

sin(/2) = k

sin,

< ir

1 () (s)

d/ds will, be

OsltiVe and .- < + °o. if we choose the initial value according to Equ.ations (11).

Sign Of the curvature . . Sign' of the slope, angle

< 7r iT > > (1/p)0 >

0 =

> -

iT Range of - ir/2 '.(9a)' -(9b) 0 < ir/2 ir/2 < iT

-

r < i,b,-. - IT/2

-

_{7T <} 0 (1/p)0 O (9c) (10)

Page - 5 MB-258 Using (10) and (11) we transform (8) into

I (d/ds)

= 2 (3 k COS (12)

or _{i 2 2}

I (d/ds)

= /3

I 1 - k

sin çb (13)

that, integrated from

s = 0,

=

to s,

yields

/3 (s/I) = F (k, ₎

- F (k,

) (14)

where F (k, ) denotes the elliptic integral of the first kind. Equation (14) gives

for the point L

/3 = F (k, F (k, ₎ (15)

Combining (14) with (15), we obtain F (k, ?,b) [F (k,

.PL) - F

o]

5/I + F (k, b)

(16)

that States that the function = (s), and, so the cu.rve itself is completely deter-mined by the values of k, and For that reason we call k, the elliptical parameters of the elastica.

Using the relations

dx' = ds

cost =

_{[2 cos2U/2) -, 1] ds} dy' ds

_sine

=

2 sin(/2)

cos(/2)

we obtain from (13)

(3

dx'/I

= 2

'Ii

-

k2

sin2i

d

- d

/I

-, k2 sin2çli

Page - 6 MB-25 8 Integrating, we obtain = [F (Ic, )

-

F(k

) 2E (k, +. 2 (k, (17)

2k (cos0

cos)/j3

where E (k, ) denotes the elliptic integral of the second kind. 2. 2 Iñterpolatidn for the Given Points

A thin elastic bèam Of a onstànt cross-section and flexural rigidity

EJ, is forced to deflect by a set of krife-edged supports that, .athe absence of

friction, exert forces normal to the deflected beam; The co-ordinates of the supports are

xl,

_{xn+1, Yn+1}

Let the index Oj denot the initial and U the end point of the th section Conditions to be met at the supports are

(1/P)Ljl

(1/p)0. (18)

' Lj-1

a0.

± E (19)

P;j_1 COSLi1

P. cosç.

(20)

The conditions (18) and (19) state that the sections should fit smoothly, having equal slope angles and curvatures at the supports (Fig. 2.

We define (Y ₊₁

- Y) /

(x ₊₁

- Xj)

oj

r

J I 2 = ((x

- x.)

J-+-1 J

(If the curve is bound to make loops, care should be exercised to place a in the.. proper quadrant.)

The condition (20) states that the reaction forces at the supports are normal to the deflected beam (Fig. 3). Equation (20) combined with equation (5) yields

cosL_l/_l =

/3

cos0./

which, applying (6) and (18), becomes

Lj-

/

Oj

/

Equations (2), (5) and (8) combined together yield

M2/2PEJ

= 2C1

- 2

sin2(/2)

which, applying (5) and putting. 2C1 = 1 + q, becomes 2

w/2

= q

+ cost

or

w /2 cost

= q/cos + 1 (25)

Finally, (23) and (25) lead to the relation

q_1/ COSLJl

=

qj/ cos

(26)

Suppose

M01 = 0

and

I = ir/2 at the first support.

These particular values, substituted into (24), give t1e value of the constant C1 for the first section:

Page - 7. MB-258

Page- 8

MB-258

The recurrence formula (26) gives, then

C1 = = 1/2 = OOnstant for all sections.

Therefore, one of three elliptical parameters, viz. k, is constant and equal v/2

for all sections of the beam.

The use of the conditions (18) and (19) involves the expressions for 1/p and (Fig. 4).

Let us introduce the function defined by

(k, a,

)

= F (k, a)

-

F (k, j3) -

2E (k, a) + 2E (k, f3)

(27)

which enables us to give the equations (17) a more concise form. FOr the point L

xj/

W/2,

= (COS

- cosL)

/

(28)

There is also

tanr

= _(cos

_{- cos?pL) /}

('/2,

0'

(29)

-T0,

From (29), (30) and (10), it follows that

=

2 arc sin

[V/2) sin0]

= 2arc sin

[V/2)

sin?PL]

At. the same time, there is.:

or

=

[(v/2

L]

+ 2 (coscbo - cosL)./$

which, taking into account that

= w2/(1/p)2 (2

cos2)

/

(1/p)2 yields the expressions for (1/p)0 and _(1/p)L

(1/p)0 = coS,0

(J/2,

L/ (r

cosT0)

(32) (1/p

L = \I COSIL

('/2,

L

) / (r

cos r0)

EquatiOns (31) and (32), combined with the ôonditions (18) and (19) respectively, form a system of functional equations that determines the elliptical parameters

and Lj

Let us introduce another tWo special fUnctions:

g(0,

cos0

_L)/c0sTO

(33)

0'

2 arc sin

[(r5/2) sinp0]-

r0

(34)

+

Page - 9. MB-258

Page - 10 MB-258

With their aid, the abovè-mentioied system can be written in the form

g(0,

Lj =

(oj_i

-)' cosi)Lj_1/(rj_1

cos0_1) =

=

+ 2 arc sin

W/2)

j number of the section = 1,

G0 (or I'o) and G (r r) are determined by the imposed boundary conditions

at the ends of the beam, because from (31), (32), (33), (34) and (35) it follows that (V'/2) r. (1/p)0.

(36) G._j-:L

r.

_j-i

Consider atthê same time the pair of equations

g b1)

= G

Let us now introduce functions g* and y*, that derive from functions g and y by the interchange of positions between variables

tT* I7I !

-

o á21) I b UVo _{'1'L' b ''FL' ''O}

+ (35)

Lj-i

Oj1

-Lj1] -

2 ar sin

(Vi/2) sin0.1] =

..., n

To any pair G, r coEreSponds a pair _L by (38) and hence a pair .g*,

y' by

(37). This functiOnal. relationship between G, I' and g*, y* is shown graphically on Figure 5. The system (35) transforms through the use of (37) into the system

- ç.

r/r1

= G1

Oj-1

a) = r1

that is less cumbersome to solve (ia respect to G, I') once the functional relation-ship between G, I' and g*, .y* has been numerically estãblishéd.

The diagram on Figure 6 displays the system (38). For ease of graphical representation, variables are transformed according to the formulae

Through this diagram elliptical parameters for given G, I' can be found easily. Once they have been found, each section. can be treated separately by means ofequation (1) and the relations (40):

x'/l

('I/2

_{çl, b)/f3}

Y' / = '1 (cos . cos

/3 =

r

(I/?,

_L'COSTO

tanr0 =

'-.v/ (cos _{- COS/)L}

)/() (1/2,

₍₄₀₎ =

ft

)

-

F (/2,

o)] /

/3

o = 2 arc sin

[(v'/2) sin]

-Page -11.

MB-2 58

Page = 12 MB-258 We add that

(''L'2, a,ir/2)

where a0 2

arc sin (cos

(fi/2)

0

p

r

1 r-1

3+4r-4

J".

4 2

p1=O l)=o ii:=O

r

+ = ir/2 ..1

rr+1

2r + 1)! J'

* p1 = > = 0

or, numerically, for up to

r = 4

a1 = - 1/28

a2 1/5280 - 1/120960 - 67/183859200

2.3 Relation Between. g*,

y* and G, I'

For a given pair G, r we obtain the corresponding pair

_L by solving the pair of equations

coso

/ COSTO

T0.=

where, as before

(5/2,

_0'

?1L

tanr0 = V

(cos40

_{- cosL)/(I)}

sin (/2)

= (V/2) sinp0

For smafl G, r, the corresponding 0

L

are either near ir/2 or near - ir/2.

Suppose the first case. Put

= ir/2

_b'

. = ir/2

-Hence, and

L

Jing small, we can write.

cOs0 = COS)T =

The relation connecting and can be transformed into

2

=

(5moui sm

) arc cos (cos (42)

Page- ;3.

MB-258

that has the roots

The system (41) reduces to

-

= G

2 2 2

-

+

O +

0 L

+ L

)'3

=r

(44) Page - 14 MB-258

or approximately, for around ir/2

= ir/2

_-

2 Further approximations are

//2,

_O'

_'L

= (V/3) (ç

- cc)

tauT0

3/(2

+

O'L

+ TO = 7r/2

-

(ç ₊ Po + ç )/3

7* =

TO = (VI G/6)

/J

VI G/2

+ r

=

_-

_{WI G/6}

_{+ .J}

r)/1/v'I

_{G/2 + I'}

(43)

The signs of the roots satisfy the condition that and G should have the same sign. The values for g* and y*, which correspond to the given pair of values for

G and r, follow from (41) and from

We obtain

or approximately

g* =

(46)

7* 2 2

Inserting (43) into (46), we obtain

g*2G+3r

(47)

Gv/2-2r

(It is easy to show that (47) remains valid also for and

L being in the neighbourhood of - 7r/2.)

By a simi]ar, but somewhat lengthy procedure, which takes account of higher

order terms, we obtain the expansions

-=

_-

_{Gcos(IL/costtO} =

-

+ r

a Gm_n

mn n

Page 15 MB258 (45) (48) =

ml

O

amn

_n

Gmn

r

Page - 16 MB258

Values of the coefficients for m = 1, 3, 5 (for even m they are zero) are

This 5th degree approximation for

all practical cases.

r.

= cG.

3 3

2.4 Pra.ctical Method of Obtaining G, r from Given Data

To compute the set of values G, r, one pair for each section of the curve, co-ordinates of the given pOints and conditions at the first and last point must be known. At the ends of the curve either siope angle or curvature might be given. Assume that slope angle is given at the first point.

Let by GJ"1, r3"1 denote exact roots for the 3th section In the neighbourhood of exact roots there is, neglecting higher order terms

G._j-1

=a.G.+b..

j a10

= z

a01

= 3

a30 = .- 1/7 a21 =

-

13

f/14

a12 = 27/7 a03 =

-

23 a50 =

-

43/6468 a41 =

-. 169

ff/4312 a32

= -

599/4312 a23 = 2951 V/25872. a14

-

.801/539 a05 = 3958 /2695 - a10

--2

a30 ₌

a

= - 1/14 a1 ._ a03 = 2/7 a5,0 = 109 ff/129360 a41 =

-

109/12936, a32

- .265

ff/17248 a23 = 103/12936 = 215 /3234 a05

= 40/539

G. J

Therefore, (dg.*/dG.

is a

J 3- _I

We can solve (50) for G.1 and

G. 3 a. I b. = 3 c. j-1 function of G)_-j-1.

, r

_j-1

and c..

j have, by comparison with (49a)

i-i

r./r.1

(dg.*/dG.1) *1 g (dg.*/dG. ₎ i

.311

Page - 17 MB-258. Let us assume that we know the coefficients a1, b1, C1, d. and that we also know

a set of approximate roots

G.'1 for all sections, that is for

_j =

1, 2,.... k.

We compute from (49b)

= c. G.' + d.

j-1 j-1 j-1

3-Because of (39) we can write

r.1 /r.

and expand around

G'1

r+i

[(d:)

(G_1 -

G'1)

(50)

g*

_{is a function of G.}

_{and r.}

, that is in its turn also a function of G. . Hence,

3 3-1 j-1 j-1

dG. = 8G. +

3-1 J-1

_i-i

(51)

P Page 18 MB-258

a, b. are fimctions of

G.T 3 J 3-1' BecaUse of (39)wehäVe

Expanding around G _-1 we obtain

= (dy*/dG)

_{(G1 -'}

Inserting (49a) and comparing with (49b) we obtaIn

c

= a

(d */dG)

= (dy.7dG.1)

3 3

r.I

and. hence funOtions of G.'.

3-1 .. j-1 *

-

Oj+1 *1 + Yj ₊

_-

Oe +1

-C.

j1

and.d..: 3-1

Again, by the same reasoning as before, coefficients c., d. are fi.thctions of

Gi: _{, c. and. d. . Hence, .th coefficients of. one section can be obtained from} j=i. j-1 j-1 . - ..

the coefficients Of the previous one. This fèatue Of the system enables us to devise an iterative scheme.

According to our starting supposition, there is always c

= oaxal

d =

I's, because

-1. 6)).

We set G.'1 = 0 for j

1,

...,

k. .

Using First Order Expansion for g.*and y.", we. compUte from(52) and (53) the

coefficients a1, b1, c1, d1, and subsequently. Ti from (49b).

Repeatmg that scheme all over we obtain the whole set of values a

c, d, J = 1, ...

k.

roots stabilizes

Page 19 MB-258 Two cases might now occur :

slope angle at. the end _{point, 0Ok+1'} is given

curvature at the end point, (1/p)ok1, is given.

In either case we are free to. assign any particular value to aOkl an1 rk+l,.

since they are not prescribed by the given points.

We choose in the case a, the values

aOkl

=

90k1

rkl

=

rk.

This choice makes. = o.; From (49b) we obtain, as a better approximation to

the exact root G, the value

Working backwards all the way down through equations (49a) and (49b) we obtain II II

anew set G.

,

r.

j-1 _j-1

In the case b, our choice is

ok+1 =

0., .rkl

= V'. This choice makes G

_{= (1/p)ok1,}

that has to be inserted into (49a) to start the succession of opera-tions that yield the new set G.111, r.111.

A. repetition of the scheme with the newly acquired set G.111, instea4

I.

. I

.of G1 would not yield any improvements because we used a linear, First Order Expansion in conjunötionw.th the linear system (49a, b). .

But, using a Third Order Expansion for g* and y., and using the set of

U. II I I . .

Page- 20

MB-258.

Needless to say, further improvement could be obtained by. successive application of Fifth, Seventh, etc. Order Expansion to the same computational

scheme. :

Much the same ideas apply to the case whe'e the curvature is given at the first point. Here we start from the given set

r'1.

(= 0) and recurrent

formulae . . . . =

a.F.

). :1 a. =

l/(d.y.*/dr.1)

= .o.r.

+ d. 3 We find that b =

(d*/dl)

c =

-

(dg */dr

)..

d =

-

r+

[(dgj*/drj1

..yj*I +

(a0j

(54) (55)

The reasoning regarding the conditions imposed at the end of the curve remains the Same as exposed befOre.

2.5 Practical Method of Obtaining

from G, I'

Once we know the set of values G1, r1 for j = 1, ..., k that is

values.

g*

and Put

= _{G/g and Z} =

_-

₍₅₆₎

There is, by virtue of (45)

V COS O/COSL

=

-From these two relations we find that

4

_-

1

(V - cosZ

sinZ

.COS0 V _cOs4!?L

The problem is to allocate proper quadrants to and

Before attempting it, we should decide about the number of character-istic points that might occur on any of the sections. (Under characteristic points on elastica we understand iiflexion points, where the bending moment is zero, and compression or dilatation points, where transverse forces are absent.) We restrict oui discussion, and hence its length too, to one characteristic point per section at most, because appearance of more than one of such points on a particular section seems to be extremely improbable in practice.

Conditions to be satisfied by and

are:

> byvirtue

) of

Io,J

) (1O)and(11)

Page - 21 MB-258

Page = ₂₂

MB-258

signum (cos0)

= sinum (G), by Virtue of (33), (35) and (27) V.

signum (cos).

= signum (V) x.signum (G), by Virtue of (56) and (57).

With the help of these conditions we can sort out the proper quadrant for and

L for all, possible cases: .

V

IfG>O, V<0,

V

then

iS in],

_L in II quadrant. (There is an inflexion point on thesection.)

IfG<O, V<O,,

then IS in

_-

_{1'L in} -I quadrant (There is an inflexion, point on the section.)

V

if G > 0,

v' -. V'Ib'I' Z, then

"

in I quadrant 4.

if G > 0, V .

1.,

-

<)Z, L then is in . in I quadrant.

(There is a compression point on the section.)

5

If'G. >0, 0 <V < 1,

_-

₌ then is, in -I, in -I quadrant.

6;

if' G >0, 0.

V.<i,

_ILL

_o '

I'I

then. is in

_-

in I quadrant. .. V .

(There.is a compression point on the section. ) ..

7.

If G <0, 0 <V<'l,

_ILI

' LI

then in II quadrant.

8

ffG<0, 0<V<1, lLI.

O >

lLI

then

'is in

in III quadrant.

(There is a compression point on the section. ) 9.

II G <.0, V >1,

_L'

_Iol

< Z

_L

then is in -II, in -II quadrant.

if G < 0, V >

I O!

Z

L +

1 o1

> z

then is in II, _L

' m quadrant.

(There is a compresSiOn point on the section.)

ifG=0,

_ILI

-

ir/2 = Z,

then

= ir/2,

zp1 is in II quadrant.

IIG=0,

- ILJ'2 =

Z,

then = 7r/2,

L S in UI quadrant.

(There is one compression point on the section.)

If G = 0,

_{ILJ +}

ir/2 =

then.

- 7r/2,

L is in I quadrant. (There is one infleion point on the Section.)

if G = 0,

_.-

_ItLI

+ ir/2 = Z,

then

= -

7r/2,

L s in -I quadrant.

ifG>0, g*

0,

then

= arc cos

(Pv'5jfl 2.

ifG<0, g* =0

then

= arc cos ('Isin ZI)

-=

-

7r/2.

17.. IIG>0, Z =0,

then we compute Z' using. G' = G/2, r' = r, from which

arc cos (v'iin

_{I Z} )

(There is an inflexion point on the section.)

if G < 0, Z

0,

then we compute Z' using G' = G/2, r' = r, from which

= ai cs (''I

I )

-

'

= -

arc cos Z' ).

(There is an inflexion point on the section.)

IfG=r=0,

then we set

= = ,r/2. (The section is a straight line.)

Page - 23 MB258

Page . 24 MB-258

2. 6 Description of Computer Programs

Obviously, even an attempt to cOmpute elliptic parameters by hand for an elastica passing through only a few given points would take a prohibitively long time. This is the reason why some earlier applications of the elastic beam did not

go. beyond polynomials of the third degree for each seOtion,, dragging. all the

inadequacies of polynomial interpolation along. To perform the numerical work in Elastic Interpolation, several computer programs were written in Fortran .11

language fOr the SDS 920 8K machine.

A program, called Reduction, accepth?ondfflons at both ends and takes up to 65 given points. It computes the necessary constahts and elliptic parameters for each section and punches the results on a tape. Computing time varies with

the number of given points and with the complexity of thefl curve

For a "reasonable"

cürvè of about 20 sections, one minute is a typical time.

Three evaluation programs aeready to accept the tape. The first, called Abscissa Evaluation, returns ordinate, slope angle and curvature of the curve for a given abscissa. The second, called Arä -Evaluation, returns ordinate of the curve for a given length of the curve. The third, called Psi Evaluation, returns co-ordinates of the curve for a given on a given section, together with corresponding arc length, slope angle and curvature.

All three. programs, optionally, either tabulate or punch, or plot the

results on a Mqseley X-.Y Plotter (15 in. x 10 in.) via Digital to Analogue Converter

of the

macmeW.

Computing speed ranges from 5O to 5 points per second, depend-ing on the program used and the complexity of the curve.

Because the double-precision floating-point arithmetic used in Fortran works with an accuracy too high for the job at the expense of computing speed, a translation of the programs into the single-precision. fixed-point machine language would shorten the time by a factor of about 20.

(1)

Figures 5, 6, 7, 8, 9, 10, 11, 15, 16, 17, 18, 19, 20 and 21, except for

2.7 Examples

The following examples illustrate the methods and procedures in Elastic Interpolation.

2.7.1 Simply-Supported Beam

A very special case of a beam supported in three equidistant points was selected. The given points were (0, O)

(1, 1.66926), (2, 0) with 1/p = 0 at

both ends. For suôh an arrangement slope angles at both ends are ir/2 and - ir/2 respectively. Table 1 shows the output of the Reduction program. Table 2 shows the outputs of the Abscissa EvaluatiOn program. Figure 8 shows the plotted results

of the Psi Evaluation program. 2. 7.2 NACA Profile 63,3-618 Initial curvattire 0

End slope angle = 6. 1 radians

Given.points, as chosen from NACA tabulations:

(1) _{Compare, D. Gospodneti6: Deflection of a Simply-Supported Beam.} JOurn. of Appl. Mech.., Dec. 1959.

Page - 25 M.B-258 xi _yj 1 100.000 0.000 2 80.178 5.073 3 . 60.125 9.667 4 . 34734 12.086 5 . 14.404 9.219 6

6.868

6.542 7 _{1.965 3.616} 8 0.156 1.511 9 0.000 0.000 10 0.844 -1.211 11 _{3.035 -2.500} 12 8.132 -3.998 13 15.596 -5.181 14 35.266 -5.906 15 59.875 -3.241 16 _{79.822 -0.297} 17 100.000 0.000

Page - 26

IvLB-25 8

Table 3 shows the output of the Reduction program. Figure 9 shows the whole profile plptted (1 inch = 10).

Figure 10 shows the leading edge (1 inch = 1), with omitted points from NACA tables cross-marked.

2.7.3 Curve With Loops

Table 4 shows the output of.the. Reduction program, that also contains given data and end conditios for bothparts of the curve.

Figure 11 shows plotted results.

3.0 NUMERICAL DEFINITION OF SHIP' S HULL 3. 1 Classification of Parts of the Hull

The surface of the ship ' s bOdy is a Oontinuous surface composed of several contiguous parts. The boundaries of the parts are curves along which the designer wishes to have either the discontinuity of the tangent plane,' s slope angle or discontinuity of rathi of curvature, or both. The boundary of a part might as well be any other curve inside one of the parts just mentioned, if the designer

chooses so for some convenient reasons.

The first type of boundaries includes. the profile, kni.ickles, parallel middlebody boundary, flat, bottom boundary, stem-to-body joining lines, keel-to-body joining lines, and similar curves.

The second tylie might occur when the designer wants to consider only a specific portion of the hull's surface.

3.2 Information Needed for One Part' of the Hull

The designer usually 'provides .a set of offsets for the surface, that is, ordinates of points of intersections between lines of intersections of the surface

with two sets of parallel, not necessarily equidistant planes, preferably mutually orthogonal (Fig. 12).

Conveniently, we might refer to the lines of intersections with the planes z = const. as the waterlines (WL), and to the lines of intersections with the planes x = const. as the sectiOns (S), following the naval .rchitects' practice.

It must now be noted that, by knowing offsets only, neither waterlines nor sections can be drawn by an elastic batten or computed by Elastic Interpolation. In fact, the designer, when using the batten, always puts some extra weights outside' the range of the curve in order to impose on the boundary point that slope angle or curvature that he wants. Hence, for a surface to be uniquely defined by the Elastic I.nterpo]ation, the designer must also provide this additional information that is usually withheld. The choice is his to give either slope angle or curvature distribu-tion of waterlines, and either slope angle or curVature distributioti of secdistribu-tions along the boundary curves.

3.3 Generation of New. Information

Elastic Interpolation is able to generate new information on the surface of the hull much in the same way as it is generated on the mould-loft floor, being the digital counterpart of that analogue and graphical method of computation.

Suppose we need to compute the ordinate Y for a point P(X, Y, Z), X and Z being given (Fig. 1.2).

There are two ways of reaching that same goal.

First, we can compute section ordinates, boundary ordinates and water-line boundary conditions for given Z, and use them to compute Y on the waterwater-line passing through P.

Second, we can compute waterline ordinates, boundary ordinates and section boundary conditions for given X, and use them tO compute Y on the

section passing through P.

Page - 2.7 MB-258

Page - 28 MB-258

It is obvious that there are two possibilities for computing Y because we use. a one-dimensional interpolation in a two-dimensional case.

This should not be disturbing, because it only emphasizes the fact that a curved surface cannot, in principle, be represented by a two-dimensional drawing, but only by a three-dimensional model.

-On the áontráry, a discrepancy in the value of y obtained in two different ways serves, if it is Out of tolerances, as an indication that the information we fed into the computer or the information we obtained from the designer is. either erroneous or not äonclusive enough to define the ship's hull as accurately as the tolerances prescribe.

While the.first case is-a trivial one, the second case calls .for.a detailed examination and insertion of new sections and waterlines. Because, almost

invariably, such discrepancies are grouped around a large one, dying out rapidly with the distance from it, itwill very often be possible to confine our examination and recomputatiôn to that small region containing the unwanted "bump" or

"hollow". How such acquisition of new information can be carried out will be best explained bi a numerical example.

3.4 Example

The forebody of a ship, contoured by parallel middlebody boundary, deck, stem boundary and keel, has been defined by Tables 5 and 6 that have been read off from the lines plan in the proportion 1 inch on the drawing = 1 unit in the

Table.

The Tables comprise all necessary data for each of the tabulated lines to be accepted and processed by the program Reduction.

The code fOr indices is as follows:

LINE is set equal to 0 LI Y are to be interpreted as lengths, equal to .1 if they are slope angles (in radians) and equal to 2 if they represent curvatures (in radian/unit length)

MO, ML are set equal to 1 or 2 depending upon whether boundary conditions. are given as slope angles or given as curvatures.

The tag function is best explained on the example itself. In Table 5, the lines tagged with -1. 0 and -2. 0 represent the boundary conditions of waterlines at midships and bow respectively; lines -3. 0 and -4. 0 inform us of the shape of

Page - 29 MB-258 The organization of the Tables follows the scheme

NL

Z, I, LINE

K, BEGIN, END, MO, ML

x1, y1

XK+l, _K+1 etc.

where

NL is the total number of lines to be processed

Z is the tag of the line

I is the number of parts in the line LINE shQws the type of the line

K is the number of sections in that part of the line that follows

BEGIN, END are boundary conditiOns at the respective extiemities of the line

MO, .ML show the type of respective boundary conditions

Page -. 30 MB-258

middleship bouindary curve in contour and section planes respectively. This also applies to lines -5. 0. and -6. 0, referring to the stem boundary curve (Fig. 13).

The rest. of the lines are sections tagged by their respective distances from the midship section.

Similarly, in Table 6, lines -1. 0 and -2. 0 represent the boundary condi-tions of seccondi-tions at keel and at deck, whereas lines-3. 0, -4. 0, -5. 0 and -6. 0 show the boundary curves in contour an4 waterline planes (Fig. 14). The rest of the lines are waterlines tagged by their respective distances from the keel.

Both sets of data are sufficient for complete definition of the same

Surface, which offers the possibility of mutual checking and generation of additional information on the surface. . .

Some of the data in Table 6 were derived by computation from the data Table 5, to ensure the uni4ueness in definition Of contour lines.

After both sets. of data were processed by the program Reduction, the corresponding: lines were plotted by the plotting routines (Fig. 15, 16 and 17).

Once we have, the Reduction output tape, containing the same information as the typed output, one part of which is shown in Table 7, we can, through the

program Abscissa Evaluation, obtain "y's" of all lines for any number of "x's"

we want In Table 8, for instance, we computed the offsets at WL 0. 00, 0. 05, 17. 95 and 18. 00. Anothét program, called Transposition, accepts thiS Abscissa Evaluation tape and edits it in subh a way (Table 9) that it is acceptable by program Reduction, which then prOduces basic data for all waterlines chosen. In such a way the first set of input data can generate basic data fOr any number (not necessarily

equidistant) of waterlines that can then serve for plotting, computing or model cutting purposes. ..

A strictly dual. procedure can be applied to the second set of input data, making Possible plotting, computing or cutting of any number of sections at what-ever lOcation we might speOffy. . . . .

Page - 31 MB-258'

The type-out of the Abscissa or Psi Evaluation programs can provide us with slope angles and curvatures at desired points, whereas the Arc Evaluation program can furnish the length of' any of the waterlines or sections between. specified points.

One of the possibilities of mutual checking and correcting;is "mid-panel point" comparison. Starting from both sets of input data we compute, by the

methods described, offsets of points lying half-way between given waterlines and half-way between given sections, and compare them by the program Comparison. A type-out of this program, applied to our example, is shown in Table 10. Notable

differences are graphically displayed on Figure 18. The most affected region of the body plan is plotted on Figure 19.

Proper interpretation of these differences led to correction and 'iniplernen-tation of input data (Tables 11 and 12). Repeated processing turned out a new

Comparison (Table 13 and Fig. 20). A new body plan detail was plotted (Fig. 21). The greatest difference was now less than 0.01, which would be less than 0. 010 inch 'for an 18-foot model.. Another iteration seemed tobe hardly justifiable. A bilge

domain detail was plotted '(Fig. 22) to illustrate this.

A valuable help in judging and smoothing of curves proved to be the curvature versus arc length that is independent of the co-ordinate system, being an intrinsic characteristic of the curve itself. A program, producing a curve to the given curvature versus arc length, was found to be useful to the same end.

'4.0 APPLICATIONS

Programs for definition of ships' hulls, based on Elastic Interpolation, were devised with a specific purpose in mind. The model-milling machine in the Ship Section of the Mechanical Engineering Division, National Research Council, Ottawa, which is now steered by an' operator who manually follows waterlines drawing, would give improved over-all efficiency if modified into a numerically-controlled' machine. The reason lies in a disproportion of the accuracy and efficiency

Page - 32 MB-25 8

of the machine, itself and. the accuracy and efficiency with which it is

used. The

tolerances l;o which the machine was built and the precision with which it can perform are like those of a first class tool machine, that an operator, following a drawn line manually, cannot matôh by far. Moreover, outs are never less than

a half -inch apart, as otherwise the lines planwould become too dense, especially at the ends The cutting speed is therefore low, which also impairs the accuracy because the operator needs movement of they feeler to judge and control its directiOn. The accuracy of the finished model is influenced by all these factors, because they leave too muOh excess material that has to be rernoed by hand

A numerically-controlled machine could cut as many waterlines as we want quickly, accurately and automatically, reducing finishing time considerably The machine, used as a plotter, could perform a variety of tasks for th drawing office, such as preparing diagrams fOr hydrostatic, stability, resistance and similar computations.

HOwever, the possibility of anapplication. of the methods described could be extended to mould-loft floor jobs and plate cutting in

shipyards, or, generally,

to döfinition of all empirically given, lines and surfaces obeyiig the law of Elastic Interpolation.

Table 1

MB -258

TABLE 1. SIIIPLE L3EAIi; REDUCTION OUTPUT

1 1.00000 1 0 2 .1.57079 -1.57079 1 1 0.00000 0.00000 1.94587 1.03106 0.00.000 0.53974 1.57079 3,14159 1.00000 1.66926 1.94587 -1.03106 -1.64857 1.03106 -3.14159. -1.57079. 2.00000 0.00000 1.41421 -1.57079 0.00000 . 0.00000 1.57079 1.57079

x R ALPHA (3 . GAMMA PSI PSI

TARLE 2. SIf1PLE BEAu; EVALUATION OUTPUT 1 Y SLOPE CURVATURE AfJG LE Table 2 MB -258 1 0.00000 20 0.10000 .Ø000Ø 0 0.00000 0.00000 1.57080 0.00000 0.10001 0.93516 1.25154

-0.67123

0.20000 1.16491 1.06213 -0.83614 0.30000 1.31507 _{0.90121 -0.94392} 0.40000 1.42426 0.75539 -1.02229 0.50000 1.50643 0.61905

-1.08127

0.60000 1.56835 0.43907

-1.12573

0.70000 1.61381 0.36346

-1.15835

0.80000 1.64500 0.24079

-1.18073

0.90000 1.66324 0.11996

-1.10383

1. 00000 1 66926 - 0. 00000 1 19 3 14 1.10000 1.66325 -0.11996

-1.19383

1.20000 1.64500 -0.24079

-1.18073

1.30000 1.61382 -0.36346

-1.15835

1.0000

1.56837

-0.48907

-1.12573

1.50000 1.50644

-0.61904

-1.08128

1 60000 1 42427

-0.75539

-1.022 2.9 1.70000 1.31:508

-0.00121

-0.94392 1.80000 1.16492

-1.06213

-0.83614 1.90000 0.93516

-1.25155

-0.67122 2.00000 0.00000

-1.57079

0.00000

1 0 - ISd ISd VIJIiV) VHd1V A X

SIZ6°0-

6L0881- 000000

6c6c'Ø-

0000I'9 6IO8tOZ 000000

00000'OOI 6IO8IO O6L6°9 ØlitiliØ°Ø-

68itI'Ø

øc°i

O69IZ

ØOL6°O-

OOZZ8°61

00I'i'-

80c91'Oz

ZL6i°9

£69tO°Ø LtIZØO'Ø

SIS°I

9øZL°I

0OSL865 00906°c-

88L11

90t6c9

£LcI°0

t,SL9O°Ø-

SØcI

SS8TSI

0099Z°Sc

989°6I

t,c9I°9

9StP0 tIS6LO°Ø_ 9LL9S0 SZG9°Ø

OOIBt'S-

0096S°SI OØZIB øi866°c- LILSSL OO9P9

8I'iØ'O

BIØSO°O-

9SO°I

8Z8c°t

£øsI1t,°r

9LGøI

56'iII°O-

£1I'Ø

'iL66S

LSZIcS OØOOYZ-

øocco'

OOIT'I-

SO1iSZ OtiISL°S

LI9ti°O.

tOG2t°O- 8cc1i6'O

UL6I

OØIit,8'O 6Sc°O- Z8189Ø 'scØc°t 86O6SØ

9OI'S

01911i1 ØØ000'O ØØØØO°O

666i°Ø-

18L6O°t-

990O-

OS'itcO IS6O9ti O6tS°I 0011S1

0095P0 øØgt9°

zccu°

L1iOO1i 08OI°O L0161O-

ti98°I-

ti9S6L°Ø-

00S96t OØ8989 ØOZ'1S9 ?L6ØL'S tJ96L9'c

tIit°O

iccØi°ø- cS68Z'I-

9I86Ø-

S6O°I-

9tic°t-

6cLO°O- 8cSL0O Z618'i°c ccLG6'L OO6I'6

OOOt1I

0øcLt,

OO98O'I 9IIS°OZ 69I8Z° £L9St°O ZIOZE°O- 8tØ82°Ø- 5O999°O-

cøici-

1iLt6ØO- SS9600 I99$iO' 16c0c°cz 00L99'6

OOS1°Ø9 OOLOS OSZLSO 6c916Z ZIJLØØØ 98IO°O- 8ZL1S°t

SIz°I-

ØØSLt08 000øro LsOgli'Oz 'OI68'Z OØØOO°Ø £LI00°Ø- 08oLS'I- 6$l86tT-

ø000rooi

000000 0000E°9 Z

I

91 0 1 0ø000 lfldIflO UUI!3fl(J3I 31I.IOHd NIM ]1HV.L 8c - c

eqej

TABLE 4. CURVE WITH LOOPS; REDUCTION OUTPUT 1 3.00000 2 0 30 -2.37000 _13.27000 1 1

Table 4

MB-258 3.80000 8.60000 4.92367 -2.17570 0.31240 -0.19430 0.75615 1.11112 1,00000 4.55000 2.66663 -1.86764 0.10318 -O..1428O -1.40988 -0.91581 0.22000 2.00000 1.03325 -1.31643 0.15200 -0.33953 -1.43755 -0.47201 0.48000 1.00000 0.45695 -0.40489 O.4506'i -0.35256 -0.55261 -0.02551 0.90000 0.82000 0.56921 0.32175 0.65931 -0.35475 0.71350 1.41084 1.44000 1.00000 1.45726 0.75628 0.35559 -0.20614 0.85613 1.28440 2.50000 2.00000 1.43210 1.01960 0.15063 -0.10541 0.20085 0.35189 3.25000 3.2200.0 2.26645 1.29371 0.22838 -0.17020 -0.56767 -0.30832 3.87000 5,40000 1.60112 1,60828 0.18232 -0.13730 -0.73001 -0.50629 3.81000 7.00000 1.21248 2.00493 0.16204 -0.25185 -1.38830 -0.66426 3.30000 8.10000 0.59857 2.72918 0.40503 -0.34036 -0.81425 -0.25498 2.66000 8.38000 0.80056 3.49890 0.65435 -0.38852 0.5532.6 1.20836 1.91000 8.10000 1.34272 4.10166 0.45735 -0.3C791 0.28197 0.73387 1.14000 7.00000 1.00603 4.60283 0.26493 -0.21846 -0.92839 -0.53611 1.03000 _{6.00000 0.79202 5.14215 0.29928 -0.29321 -1.12300 -0.52349} 1.36000 5.28000 0.70682. 5.84485 0.53426 -0.33628 0.49506 1.03117 2.00000 .4.98000 1.00404 6.37294 0.44315 -0.24595 0.83732 1,35662 3.00000 5.07000 0.93048 6e64573 0.13039 -0.09920 -0.91046 -0.72822 3.87000 5.40000 2.22551 6.96354 0.37946 -0.21.192 0.88676 1.35340 5.60000 6.80000. 1.453'il 7.25453 .0.084.58 -0.1386.6 -1.44688 -0.92506 6.42000 8.00000 0,62642 7,56252 0.17750 -0.09136 1.16945 1.50075 6.60000 8.60000 0.50160 7.93381 0.02546 -0.31433 -1.55114 -0.47208 6.56000 9.10000 0.39294 9.16745 0.90386 -0.63230 -0.01034 0.88651 6.18000 9.20000 0.37736 9.98338 0.54873 -0.27385 0.86.963 1.51987 5.86000 9.00000 0.60108 10.19841 0.06896 -0.06225 -1.32592 -1.1.1285 5.43000 8,58000 1.80854 10.48740 0.37839 -0.21333 0.87508 1.33600 4,55000 7.00000 1.02611 10.76951 0.07792 -0.12626 -1.45074 -0.95780 4.32000 6.00000 0.84758 11.3225 0.30921 -0.35599 -1.22510 -0.42571 4.60000 5.20000 0.72801 12.28807 0.71385 -0.4.5874. 0.31295 1.00403 5.30000 5.00000 1.27016 . 13.04058 0,70.283 -0.37178 0.70612 1.44302 6.43000 5.58000 1.27016 13.27000 0.11771 0.00000 1.40343 1.91011 2 0.00000 0.00000 ₂ 2 6,43000 5,58000 2.70496 -1.98151 0.00000 -0.01789 -1.57080 -1.33704 5.35000 4.52000 3.100000.77000 2.473421.41421 -1.913000.00000 -0.000000.06939 -0.03272-1,89664 1.34744 -1.57080 1,57080 -0.24765

x R ALPHA G GAMMA PSI PS I

13 -1.'0000O 1 1 1

0.00000

0.00000

0.00000

0.00.000

18.00000

0.00000

-2.00000

1 2 1

_0.00000

_0.00000

0.00000

0.00000

18.00000

0.00000

-3.00000.

2 0 1

_0.00000

0..00000

0.00000

.17.45000

3.00000

17.45000

3

1.57079

_0.00000

1 2

3.00000

17.45000

4.2ØØØØ

_26.17500

10.00000

34.90000

18.00000

43.62500

-4.00000

2 0 1

0.00000

_0.000O 12 .

1.57079

0.00000

0.00000

4.46.000 Ø.5ØØØØ

7,55000

8

1.57079

0.00000

0.00000

9.39000

0.05000

9.93544

0.20000

10.46703

0.50000

11.04831

.1.00000

11.62607

1.5ØØØØ

11.08808

2.00000

12.21843.

2.50000

1.2.34804

3.00000

1.2.39000

3.00000

12.39000

.18.00000

12.30000

.26.17500

1 0 5

1.57079

_0.00000

0.00000

9.00000

0.50000

10.84000

1.00000

11.45000

2.00000

12.07000

3.00000

_12.32000

4.20000

12.39000

34.90000

1 0 8

1.57010

0.00000

0.00000

7.38500

0.50000

9.72000

.10O00

_10.54000

2.00000

11.37500

3.00000

11.81500

4.00000

12.07500

6.00000

12.31000

8.00000

12.38000

10.00000

_12.39000

43.62.500

1 :0

Table 5

MB -258 TABLE 5. INPUT DATA; SECTIONS

12

_1.54400

_0.16370

0.00000

1.66000

0.5.0000

4.81000

1.00000

5.89000

2.00000

7.21000

3.00000

3.06000

4.00000

8.68000

6.00000

9.58000

8.00000

10.21000

10.00000

10.68000

1.2.00000 1.1.07000

14.00000

11.42000

16.00000

11.75000

18.00000

12.07000

61.07500

1 0 12

1.33720

_0.42210.

0.00000

1.02000

0.50000

2.56000

1.00000

3.37000

2.00.000

4.46000

3.00000

5.27000

4.00000

5.93000

6.00000

6.94000

8.00000

7.60000

10.00000

8.28000

1.2.00000 _.8.31000

14.00000

0.33000

16.00000

9.06000

.18.00000

10.76000

69.30000

1 0

Table 5

(Cant' d)

MB -258

1.00000

8.58000

2.00000

9.73000

3.00000

10.43000

4.00000

10.93500

6.00000

11.59000

8.00000

11.014000

10.00000

12.12.500

12.00000

12.21i500

14.00000

12.33000

16.00000

12.37500

18.00000

12.39000

52.35000

1 0 12-

0.68720

_0.57300

0.00000

0.78500

0.50000

1.18500

1.ØØØØØ

_1.55000

2.00000

2.21000

3.00000

2.77000

4.00000

3.26000

6.00000

4.05000

8.00000

4.71000

1,0.00000

5.21000

12.00000

5.69000

14.00000

.6.28000

16.00000

7.11000

1:8.00000

8.24000

78.52500

1 0 10

0.22940

O.51&IOØ

0.64710

0.47000

.00000

0.78600

3.00000

1.01200

4.00000,

1.24400

6.00000

1.67.000

8.00000

2.01000

10.00000

.2.27000

12.00000

2.59000

14.00000

3.04000

16.00000

3.78000

18.00000

4.85000

-5.00000

3 0 1

1.47343

1.47343

0.00000

71.90000

0.92500

81..370OO 3

1.47343

0.18132

0.92500

81.37000

2.00000

8'i.14000

4.00000

.

85.71500

6.00000

86.28500

1

0,18132

0.18132

6.00000

86.28500

18.00000

88.48500

-6.00000

2 0

I. -0.25892

-0.25892.

0.00000

0.64000

0.92500

0.39500

11

_-0.25892

_0.00000

1 2

0.92500

0.39500

1.50000

0.29700

2.00000

0.24500

3.00000

0.17100

4.00000

0.13000

6.00000

0.08200

8.00000

0.07500

10.00000

0.08600

12.00000

0.11000

14.00000

0.16200

16.00000

0.23700

18.00000

0.36600

Table 5

(Cont'dJ

MB-258

0.00000

0.00000

17.4SØØØ

0.00000

71.90000

0.00000

1

0.09736

_0.09736.

71.00000

0.00000

31.37000

.

0.92500

Table 6

MB- 258 18

_0.00000

_0.00000

.1 2

17.45000

-0.33.333

20.00000

-0.32500

24.00000

-0. I8OO

26.17500

-0.06141

28.00OOO

-0.02400

30.53750

_-0.00500

3i..9O00O

_-0.00036

39.26250

-0.00468

43.62500

-0.00808

4.7.98750

-0.00270

52.35000

0.01000

56.71250

0.02582

61.07500

0.04006

65.11.3750

0.04078

69.80000.

0.04900

74.16250

0.04247

78.52500

0.03209

82.88750

0.01798

88.48500

0.00000

-3.00000

4 0 17

-1.00000

3 1

0.00000

17.45000

43.62500

0

0.00000

43.62500

47.08750

5.2.3500.0 56.7125.0

61.07500

63.25625

6r.4375Ø

(57.61375

69.80000

7.1.90000 9

0.00000

71.OØØØØ

74.16250

76.34375

78.52500

30.00000

8.2.00000

83..5O00

85500O0

87.50000

38.48500

-2.00000

1 TA'JLE 6. 1

0.00000

1.57070

1.57079

0.00000

1.57070

1.56400

1.54400

1.48450

1.33700

1.21000

1.06400

0.87650

0.68700

0.53500

0.00000

2

0.53500

0.43800

0.33500

0.22940

0.14000

0.03300

0.00000

0.02600

0.10000

0.16200

2

INPUT PATA; WATERLINES

7

0.09736

1.38047

81.37000

0.02500

83.35034

1.50000

84.14000

2.00000

84.70072

2.50000

85.12680

3.00000

85.71500

4.00000

36.06507

5.00000

86.28500

6.00000

1

1.38947

1.38947

86.28500

6.00000

88.43500

18.ØØØØØ

-4.00000

3 0 7

0.00000

_0.00000

17.45ØØØ

9.39000

26.17500

9.00000

34.90000

_7.38500

43.62500

4.46000

52.35000

_1.66000

61.07500

1.02000

69.8ØØØØ

_.0.78500

71.90000

0.64000

1

0.00000

_0.00000

71.90000

0.64000

81.37000

0.39500

16

_0.00000

_0.00000

31.37000

0.39500

81.89107

0.37643

33.35034

0.29700

84.14ØØØ

_0.24500

34.70072

0.2032.3

85.12630

0.17100

85.71500

0.13000

26.06507

0.10118

86.23500

0.08200

36.65167

_0.07500

37.01833

0.08600

87.38500

0.11000

87.75167

0.16200

88.11833

_0.23700

88.30167

0.20650

88.43917

0.34823

88.485ØØ

0.36600

0.50000

1 0 7

_0.00000

_0.00000

17.45ØØØ

11.04831

26.17500

10.84000

34.90000

9.72000

43.62500

7.55000

52.35000

4.81000

61.07500

2.56000

69.3ØØØØ

1.18500

77.01892

0.50757

1.00000

1 .0 8

0.00000

_0.00000

/

Table 6

(Cont'd)

MB -258 1 2 2 2 2 2

çjØ

0.Or1cc

3

0.00000

7

0.00000

0.00000

1 2 3.2.53945

12.30000

3 1 2

17.(50Ø0

12.2.181i3 2G.17500

12.07000

34.90000

1l.37i

1t3.62500

9.73000

52.35000

7.21000

61.07500

4.46000

69.30000

2.21000

78.52500

0.73G

34.14000

i.2Ii500

3.000. 1

0 3

0.00000

0.00000. .1

2

17.45000

12.39000

26.17500

1.2.32000

34.00000

11 .31500

43.52500

I: 43000 52.3500.0

61.07500

5.27000

60.3i00

2.77000

73.52500

1.01200

85.12630

0.17100

1!.000Ø0 1 0

0.00000

1 2

25.58023

_12.39000

26.17500

12.33365

34.9ç0ØØ

12..;7500

43.62500

10.03500

52.35000

3.63000

61.07500

5.93000

69.80000

.3,26000

73.52500

1.244Oc 8.5.71500

0.13000

5.00000

I

0 7

0.00000.

_C.00ØØØ 1 2

20.77231

12.30001

34.00000

12.31000

43.62500

11.5.9000

52.35000

9.52000

G1.075ØC 6.94ØØØ

59.80000

4.05000

78.52500

1.57000

86.22500

0.03200

8.00000

1 0

Table 6

(Cant' d)

MB -258 17.li.5000 11.1,2.607

26.17500

1i.'50

l0.510C'.0

3.62500

52.35Ø.Ø

5.8(0c1

6.1.07500 3.371Ø0

60.80000

1.55000

78.52500

.55277

31.89107

0.376l3

2.00000

1 0

34.00000

12.38000

43.62500

11.94000

52.35000

10.21000

61.07500

7.69000

6980000

4.71000

78.52500

2.01.000

36.65167

O.O75ØØ

10.00000

1 0 7

0.00000

0.00000

.1 2

34.90000

12.39000

34.90000

_12.39000

43.62500

12.12500

52.35000

10.68000

61.07500

8.28000

69.30.000

5.21000

78.52500

2.27000

87.01833

0.08600

12.00000

1 0 6

0.00000

_0.00000

_{1 2}

37.14393

12.39000

43.62500

1.2.24500 52.35Ø0Ø

11.07000

61.07500

8.81000

69.SOØØØ

5.69000

78.52500

2.59000

87.38500

Ø.11ØØØ

14.00000

2. 0 .6

0.00000

0,00000

_{1 2}

39.33335

_12.30000

43.62500

12.33000.,..

52.35000

11.42000

6i.07500

9.33000

.60.80000

6.28000

78.52500

3.04000

87.75167

0.16200

16.00000

1. ₀ 6.

0.00000

0.00000

_{1 2}

41.43774

.

12.39000

43.62500

1.2.37500

'52.35000

11.75000

61.07500

9.96000

698ØØØØ

_7.11000

78.52.500

3.78000

88.11833

0.23700

-5.00000

2 0 15'

_0.00000

.

0.75289

17.45ØØØ

3.00000

25.58923

4.0.0000

28.01670

5.00000

29.77231

_6.00000

31.23698

_7.00000

32.53945

8.OØØØØ

33.74610

9.00000

34.90000

10.00000

36.03009

11.0,0000

Table 6

(Corit'd)

MB -258

37.14393 12.00000 38.2ti417 13.00000 39.33335

14.00ø0

40.41378 15.00000 41.48774 16.00000 42.55744 17.00000 43.62500 18.00000 1

0.00000

0.00000. 1

43.62500 18.00000 83.48500 18.00000 -6.00000 2 0 1

_0.00000

0.00000

17.45000 12.39000 43.62500 12.39000 5

0.00000

0.00000

1 2 43.62500 12.39000 52.35000 12.07000 61.07500 10.76000 69.80000 8.24000 78.52500 4.35000 88.48500 0.36600

Table 6

(Cont'd)

MB -258

Table 7

MB -258

lADLE 7. INPUT DATA SECTIONS; REDUCTIOfI OUTPUT

13 -1.00000 1 1 1 0.ii0000 0.00000 1 1 0.00000 0.00000 .18.00000 0.00000 0.00000 0.00000 1.57080 1.57080. 18.00000 0.00000 18.00000 0.00000 0.00000 0.00000 1.57080 1.57080 12 -2.00000 1 2 1 0.00000 0.00000 1 1 0.00000 0.00000 18.00000 0.00.000 0.00000 0.00000 1.57080 1.57080 18.00000 0.00000 18.00000 0.00000 0.00000 0.00000 1.57080 1.57080 11 -3.00000 2 0 1 0.00000. 0.00000 1 1 O.00O00 17.45000 3.0,0000 0.00000 0.00000 0.00000 1.57080 1.57080 3.00000 17.45000 3.00000 0.00000 0.00000 0.00000 1.57080 1.57080 3 1.57079 0.00000 1 2 3.00000 17.45000 8.8.0713 1.43412 -0.08568 0.13667 1.69857 2.21096 4.20000 26.17500 10.47691 0.98412 -0.47778 .0.23807 -2.22032 -1.62223 10.00000 34.90000 11.83747 0.82872 -0.04588 0.02163 -1.75100 -1.57080 18.00000 43.62500 1.41421 0.00000 0.00000 0.81790 1.57080 364744 10 -4.00000 2 0 8 1.57079 0.00000 1 1 0.00000 9.39000 0.54773 1.47938 -0.12890 0.09140 3.074'i7 3.20380 0.05000 9.93544 0.55235 1.29577 -0.13002 0.09219 3.0791 3.20954 0.20000 10.46703 0.65413 1.09435 -0.15393 0.10924 3.07052 3.22507 0.50000 11.04831 0.76407 0.85742 -0.17963 0.12772 3.05027 3.23102 1.00000 11.62607 0.61729 0.62667 -0.14515 0.10303 3.06197 3.20777 1.50000 11.98808 0.55051 0.43172 -0.12957 0.09188 3.07548 3.20548 2.00000 12.21843 0.51653 0.25364 -0.12159 0.08619 3.08027 3.20220 2.5OOOO 12.34804 0.50176 0.08372 -0.11812 0.08372 3.081'i.7 3.19990 3.00000 12.39000 0.50176 0.00000 -0.11813 0.00000 -1.73847 -1.23084 0.00000 0.00000 1 1 3.00000 12.39000 15.00000 0.00000 0.00000 0.00000 1.57080 1.57080 18.00000 12.39000 15.00000 0.00000 0.00000 0.00000 1.57080 1.57080 9 26.17500 1 0 5 1.57079 0.00000 1 1 0.00000 9.00000 1.90672 1.30546 -0.33128 0.26533 2.37580 2.80684 0.50000 10.84000 0.78873 0.88417 -0.17956 0.13111 2.67112 2 86619 1.00000 11.45000 1.17661 0.55500 -0.28918 0..19468 -2.70836 -2.41343 2.00000 12.07000 1.03078 0.24498 -0.20834 0.12758 -2.20042 -1.92309 3.00000 12.32000 1.20204 0.05826 -0.14237 0.0795,5 -1.98021 -1.71730 4.20000 12.39000 1.20204. 0.00000 -0.05220 . 0.00000 -1.68196 -1.34711 8 34.90000 1 0 8 1.57079 0.00000 1 1 000000 7.38500 2.38793 1.35985 -0.27760 0.21094 2.52631 2.85144 .0.50000 9.72000 0.96042 .1.02323 -0.13101 0.11415 1.94357 2.22790 1.00000 10.54000 1.30278 0.69572 -0.29804 0.19234 -2.'s9208 -2.16937 2.00000. 11.37500 1.09252 0.41451 -0.17684 0.10965 -2.17457 -1.93130 '3.00000 11.81500 . 1.03325 0.25437 -0.10391 0.06633 -2.08837 -1.92697 4.00000. 12.07500 2.01376 0.11696 -0.14273 0.07831 -1.96702 -1.69623 6.00000 12.31000 2.00122 0.03498 -0.04597

f1,t1'3

-1.81384 -1.67663 8.00000 12.38000 2.00002 0.00500 -0.02016 0.00963 -1.69209 -1,57393

10.00000 12.39000 7 43.62500 1 0 12 1.57079 0.00000 2.00002 1 1 0.00000 -0.00052 0.00000

Table 7

(Cont' d) MB -258 -1.58138 -1.514862 0.00000 'e.46000 3.13019 1.41037 -0.17563 0. 16042 '1.95758 2 337i 26 0.50000 7.55000 1.14495 1.11837 -0.11775 0. 09624 2.00387 2. 2 2775 1.00000 8.58000 1.52398 0.85505 -0.22809 0. 15489 -2.68.900 -7.1* 5393 2.00000 9.73000 1.22066 0.61073 -0.15697 0. 09 705 -2.13267 -1.00520 3.00000 10.43000 1.12028 0.4676fi -0.08875 0. 06027 -2 324214 -2. 2 1512 4.00000 10.93500 2.10452 0.31649 -0.14638 0 093143 -2.. 18852 - 1 988 8 74 6.00000 11.59000 2.03039 0.17325 -0.09898 0.06016 -1. 98787 -1. 80621 8.00000 11.94000 2.00854 0.09223 -0.05638 0. 0 30 75 -1 81410 -1. 6461*9 10.00000 12.12500 2.00360 0.05992 -0.01764 0.01067 -1 74116 -1. 666 88 12.00000 12.24500 2.00181 0.04247 -0.00997 0.00858 1 67517* 1.71*357* 14.00000 12.33000 2.00051 0.02249 -0.01645 0.00086 -1., 73090 -1. 6 571*7 16.00000 12.37500 2.00006 0.00749 -0.00892 0.00690 1.72104 1. 7638 * 18.00000 12.39000 2.00006 0.00000 -0.0111i4 0.00000 -1. 622 76 -1. 46673 6 52.35000 1 0 12 _{1.54400 0.16370} 1 1 0.00000 1.66000 3.18944 1.41338 -0.10476 0.13062 1.. 75240 2.19606 0.50000 4.8.1000 1.19013 1.13721 -0.1.2669 0. 089 73 _3.10049 3.22739 1.00000 5.89000 1.65602 0.92246 -0.17578 .0. 12510 3.03268 3. 2 0981* 2.00000 7.21000 1.31244 0.70449 -0.13981 0. 002 72 -2.32633 -2.. 15639 3.00000 8O6ØOO 1.17661 0.55500 -0.10103 0.06318 -2. 03721 -1. 86751 4.00000 8.68000 2.10317 0.42285 -0.12245 0.. 0 7729 -2. 1.06 52 -1. ¶32 17f* 6.00000 9.58000 2.09688 0.30516 -0.07884 0.04083 -1. 9901* 7 -1.85726 8.00000 10.21000 2.05448 0.23081 -0.05251 0 .0 3046 -.1.83623 -1. 60198 10.00000 10.68000 2.0,3767 0.19258 0.02400 0.01443 -1.76666 -1. 6 7843 .12.00000 11.07000 2.03039 0.17325 -0.01320 0. 00743 -1. 60473 -1. 61892 14.00000 11.42000 2.02704 0.16353 -0.00512 0,0.0418 1 .659 '*9 1.70068 16.00000 11.75000 2.02544 0.15866 -0.00749 0. 00012 -1.6 1335 -1.148868 18.00000 12.07000. 2.02544 0.16370 0.01444 0. 0 00 00 1.5121*1 1.68777 5 61,07500 1 0 12 1.33720 0.42210 1 1 .0.00000 1.02000 1.61911* 1.25686 -0.03197 0.08034 1.62525 2 . 0 5 8 2 0.50000 2.56000 0.95189 1.01776 -0.1617.9 0.10118 -2.16614,9 -1. 94128 1.00000 3.37000 1.47922 0.82843 -0.16224 0.10172 -.2.17330 -1.04914 2.00000 4.46000 1.28690 0.68081 -0.09199 0.05026 -2.07941 -1 93429 .3.00000 5.27000 1,19817 0.58337 -0.06253 0.044O3 -2.03954 -2. 8 7655 4.00000 5.93000 2.21*056 0.46764 -0.11521 0.07102 -2.06672 -1. 88256 6.00000 6.94000 2.13600 0.35877 -0.07079 0.OL661 -2.07747 -1,961*91 8.00000 7.69000 2.08521 0.28686 -0.05469 0.02878 -1.79714 -1. 6229 5 10.00000 8.28000 2.069O3 0.25905 -0.01260 0.00893 2,67076 2 .6 8418 12.00000 8.81000. 2.06649 0.25437 -0.0126.7 0.00427 -1.61579 -1.ti 1587 14.00000 9.33000 2.09688 0.30516 0.04413 -0.03327 -1.16637 -1.07994 16.00000 9.96000 2.15'O7 0.38051 0.05430 -0.04000 -0.99004 -0. 9 1715 18.00000 10.76000 2.15407 0.42210 0.06103 0.00000 1.45057 1 81298 4 69.80000 1 0 12 0.68720 _0.57300 1 1 0.00000 0.78500 0.64031 0.674.71* -0.00539 0.01245 1.594t7 1.75657 0.50000 1.18500 0.61905 0.63058 -0.04067 0.02305 -1. 7921*2 -1.65904 1.00000 1.55000 1.1.0817 0.58337 -0.03188 0.02085 1.72365 1.87560

Table 7

(Cont'd)

MB -258 2.00000 2 .2 1000 1. 1461.2 0. 51049 -0.06010 0.03573 -1.87316 -1.72667

3.00000-2.77000

1.11360 0.45562 -0.03044 0.02592 1.75945 1.87772 4.00000 3. 26000 2.15037 0.37619 -0. 004 72 0.04011 -1 86402 -1.62827 6.00000 4. 05000 2.10609 0.31875 -0.018 4 3 0. 02622 1. 6364 1 1.83829 8.00000 4.71000 2.06155 0.24498 -0.07275 0.03437 - 1.8 0002 -1. 57165 10.0000d 5. 2 1000 2.05679 0 23554 -0. 00027 -0.00784 -1.57203 -1,41611 12 00000 5.69000 2.08521 0.28686 0,03430 -0.03542 -1.43785 -1.25001 14.00000 6.23000 2 16539 0. 39 337 _0.03474 -0.05992 .0.00010 0.03485 16 00000 7.11000 2 29715 0.51429 0.08957 -0.06106 0.77805 0. 885 32 18. 00000 8. 2 4000 _{2.29715 0.57300} O . 7961 1 143337 1 84 820 3 78.52500 1 0 10 0.22940 0.54100 1 1 0.64710

.0.47000

1.38931 0.22946 0.00714 -0. 0 0115 1.52952 1.65198 2.00000 0.78600 1.02522 0.22227 -0.01035 _0.00226 -1.63025 -1.50606 3.00000 1.01200 1.02656 0.22797 0.01113 -0.00290 1.50629 1.62847 4.00000 1.24400 2.04487 0.20086 -0. 01982 0.01603 1.75200 1. 8 3166 6.00000 1.67000 2.02869 0.16830 -0. 02314 .0.02342 1.76560 1. 8 7156 8.00000 2.01000 2.01683 0.12923 -0.04280 0.01217 -1. 70109 -1. 46755 10.00000 2.270110 2.02544 0.15366 0 034 10 -0.02346 1.00743 1.0 5817 12.00000 2.59000 2.05000 0. 22131 0.03170 -0.03985 1 . 4 726 11 - 1 2 38 31 14.00000 3.04000 2.13251 0. 35'I38 0. 109 78 -0.07579 11 5343 4 11. 6 5212 16.00000 3.781100 2.26824 0.49125 0.10787 -0. 06 305 1.17959 1.38782 18.00000 4.85000 2.26824 0.54100 0.05147 0. 00 0011 1.46042 1.70288 2 -5.00000 .3 0 1 1.47343 1.47343 .1 1 0.00000 71.90000 9.51507 1.4 7343 -0.00000 0. 00000 1.57080 1. 5 7080 0.92500 81.37000 9.51507 1,47343 0.00000 0.00000 1.570811 1.57080 3 1.47343 0.18132 1 1

0.92500 -81.37000

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