The mathematical fairing of ship lines

SHIP DESIGN

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TCAL

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Summary

A critical examination is made of present prac-tice in the design of ship lines and in the sub-sequent processes whereby particulars of the hull shape are conveyed to shipyard production

de-partments. It is shown that the present systems suffer 'from fundamental weaknesses which could be overcome if accurate numerical data existed for material, ordering and machine tool control. A discussion is given of the geometrical properties of the hull and tie factors which are vital to the designer. Details of a method used to obtain a mathematical surface which will satisfy these boundary conditions are outlined. This method has been applied to several ships which have been built or are under construction and an indication of the practical difficulties encounterd and the degree of success achieved is given. To illustrate some of the computational procedures involved the mathematical definition of the bow profile is

de'-scribed.

introduction

The mathematical delineation of a ship hull sur-face is a problem which has received considerable

attention in the past, but became a practical pos-sibility only with the advent of electronic com-puters. Since these machines have become widely available, many different approaohes to the pro-blem have been tried without any general method being established. The pressure for this work has come largely from research workers who wish to use mathematical equations to the s'uiiface for other calculations connected with wave resistance, ship motions etc. and only recently from these shipyards wishing to have more accurate constructional data. There is to a certain extent a conflict between these

Dept. of Naval Architecture, University of Clasgow.

N.

S. Miller0) and C. Kuo0)

[.ab.

_{y. Scheepiriw!cide}

kchnsche Hogsco3I

De If i

two requirements. The first problem may be con-sidered solved but the application of mathematical fairing to 'design and production has not been studied in detail. It is therefore necessary to exa-mine present practice in 'both these applications to decide what should ibe t'he aims and require-mènts of a mathematical fairing. system.

Present practice

The initial design of a ship is usually formulated as a 1/48th scale drawing of the hull surface and deck outlines. This drawing is prepared by geome-trical variation of known successful ship forms. either previously 'built by the shipyard undertak-ing the design or taken from the published results of experiment tank tests. The present state of our knowledge of the hydrodynamic bhavio'ur of ship hulls is such that it is only possible to say that cer-tain 'combinations of principal dimensions

(Length-breadth ratio, Breadth-draught ratio etc.) in con-junction with certain underwater volume

charac-teristics, (ratio of immersed volume to volume of circumscribing block, longitudinal position of

centre of volume etc.) give the best performance. This has 'been esablished as a result 'of the ana-lysis of a large number of experiment tank tests on models. The designer will check that the initial design satisfies these empirical criteria and also gives the 'desired cargo carrying weight and volume. The shape of the hull surface is expressed by drawing approximately 25 vertical transverse sec-tions along the length, horizontal transverse secsec-tions (wàterplanês) at intervals of 2 to 4 ft'in. the vertical direction, longitudinal vertical sections (buttock lines) at four o five positions in the half width of the ship and two or three longitudinal diagonal sections (diagonals). A typical lines plan used in the fairing process is shown in Fig. 1, where the

above terms are explained. Sheer and decklines are not included in the figure. Each of these sections is smoothed or «.faired» (in ship terminology) by the use of battens or splines and weights and care-fully cross checked against one another.

When the designer is satisfied that his drawing will fulfil all the design criteria, the shape of the hull is transmitted to the mould loft in the form of offsets. There the same lines drawings are pre-pared as in the design office. Traditionally these were prepared on the full scale and they still are in many shipyards in Britain. This involves a very considerable area of the order of 15,000 sq. ft. or more in a shipyard of medium size.

On the Continent after 1945 there was a marked tendency to do all mould loft work on 1/10th full scale and this is now the practice in a few British yards. The main reason for using full scale lofting was on grounds of accuracy but yards with experi-ence of the two systems generally state that they belive the 1/10th scale drawing provides greater accuracy than the full scale. No published figures are available whereby these accuracies

can be

assessed. Applying the criteria of fairness given

below to the mould loft offsets made available to us by several shipyards, led us to belive that errors of up to 1.0 in. in offsets can occur frequently in present practice. The cause of these errors is diffi-cult to determine. Errors may occur in reading scales and transcribing data rather than errors in obtaining a smooth curve with the use of battens. Indeed, from experience gained during the

de-velopment of a mathematical fairing process, it appeared that by far the commonest errors in data were due to these causes.

The offsets determined from the mould loft are supplied to the drawing offices to enable accurate

European Shipbuilding No. 4 - 1963

drawings to be prepared for use during fabrication and, ideally, for ordering the material for the hull. In practice the mould loft work takes some time and steelwork has to be ordered before the final dimensions are available so that a margin must be allowed on all plates and frames, resulting in a certain quantity of scrap.

From the full scale or 1/10th scale drawings in the mould loft the shell plating is developed and templates prep.ared to enable plates and frames to be formed to the correct shape. The latter operation requires several transferences of data and the final shape is conveyed in a form which is easily distort-ed by mechanical damage, thermal and humidity

changes, etc. Conveying information regarding

Shape o machine tools in the form of templates

greatly limits both the speed of operation of the

machine tool and the accuracy which can be

achieved.

In modern shipyard practice ships are built in large sections which are fabricated under cover and then assembled on the building berth or in a dry dock. Frequently trouble is experienced in getting these sections to. Lit togther smoothly with consequent resort to cutting on the berth, the use of force to deform the section to the required shape and the 'deposit of excess weld metal to fii gaps. This is believed to be a costly process and usually results in a certain amount of residual stress being built into the ship. Unfortunately no figures have been published for such rectification costs

in a

Shipyard, and how much allowance for these re-sidual stresses is made when determining the struc-tural requirements 'of the hull is not known. The lack of fit between sections is due to a number of causes among which the most important are poor definition of shape in mould loft, errors in

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i-I O t .,-,..

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* -t o Fig. 1.

European Shipbuilding No. 4 - 1963

C c»

Fig. 2. Flat of side curve'» on shell expansion of forebody.

ferring data» errors in machine tool operations (cut-ting and bending) welding distortions, mechanical distortion during lifting operations, thermal effects and flexible foundations on building berths. The variety of sources of error makes it difficult to

detect the precise influence of cach but it is thought that the accurate numerical definition of the smooth hufl shape is ali essential preliminary to the eli-mination of the distortion errors.

To sum up, the disadvantages of the mould loft

are:

It requires a fairly large amount of space, some capital equipment and labour.

The smoothness of the

hull depends on the

judgment of the loftsman and not on some ab-solute standard.

The process involves several transferences of data which leads to errors at the assembly stage with considerable rectification costs.

It takes a considerable time and therefore does not permit the accurate ordering of material, hence causing excessive scrap.

It is difficult to convert mould loft data into a form suitable for the automatic control of machine tools.

Any mathematical fairing process must overcome all of these drawbacks without loss of any other information at present available from the mould loft process.

Geometrical properties of the hull

Before going on to consider mathematical me-thods of :fairing, the geometrical properties of the hull and the required accuracy of the hull surftce will be examined. Only a typical sargo vessel or tanker is discussed and special 'features such as bulboús bows, bossings, skegs, etc. will require extra mathematical conditions to be satisfied.

The profile drawing in Fig. i shows that the hull can be 'divided into three regions bounded by what are termed «fiat Of side curves», the forward and after regions being curved surfaces, while the

centre section has a constant cross section shape. The extent of this centre section depends on the purpose of the ship and may be very small in fast vessels, while in tankers, ore carriers, etc., it may occupy 40 0/0 or more of the length.

The transverse section in the centre region is

generally designed as a rectangle with a small

radius at the bottom corners or bilge and some-times a slight slope (rise of 'floor) to the bottom. For the axes given in Fig. i dy/dz = dy/dz = O on the side and dy/dz becomes infinite on the bottom if there is no rise of 'floor or an angIe ap-proaching /2 if there is rise of floor. The mathe-matical equations for midship designs employing circular, 'parabolic and elliptic bilges have been programmed for the DEUCE computer at

Gin:;-gow LTniversity.

The region where these equations hold is bound-ed by what has been termbound-ed in Fig. i the flat of side curves. These are the lines passing through the point where each waterline becomes straight and parallel to the longitudinal centre line. On fine vessels there may be no flat of side but each water-line will 'have a point where the slope is zero. In the ordinary drawing process these curves are poor-ly defined but their position is not critical and a considerable error can be made in drawing them without altering the form to any significant extent. At the bottom the flat of side curve will follow the flat of bottom line so that generally this curve reverses direction in the region of the bilge and its shape on an expansion of the ship's surface is n »shown in Fig. 2. 'hus the whole centre region is

exactly defined by the equations for the midhip section shape and these boundary curvos.

The fonvard curved portion is bounded at its

aft end by the flat of side curve (x = f(z) with

dy/dx = d2y/dx O and y = midship section off-set) and at the forward end by the bow shape. A great many different bow shapes are in use, the above water design

being largely a matter of

aesthetic considerations. In the x- z plane (profile) the shape is generally made up of a combination of straight lines, conic sections and curves drawn using splines and weights or ship curves. When the 'esign employs definite mathematical curves the geometric conditions can readily be used to formulate mathematical equations for the profile (see Appendix III). Where ship curves are used dx/dz may change from zero to infinity in the depth of the profile and it may be difficult to express the curve mathematically with sufficient accuracy.

The waterline offsets at the bow vary from the half width of the stem bar with perhaps dy/dx equal to a specific angle of entrance in the lower

region to y = O dy/dx = co or an angle approach-ing -/2 in the upper region where the bow fiares out and a soft nosed stem is fitted. In many cases the soft nosed stem is a mathematical surface and its line 'of tangency with the waterlines can be considered as the waterline endings (giving a speci-fic offset and slope). These endings must be

satis-fied exactly both in the x - z and y - z planes.

The stern profile (see Fig. Sa.) normally consists of a combination of straight lines and curves, the latter sometimes being conic sections but are more generally drawn with splines or ship curves. Ma-'thematically dx/dz may vary from zero to infinity and back to zero or a negative angle over the depth of the ship. In the x - y plane the 'stern endings vary from y equal to the half width of the stern post at the lower 'regions to y = O, dy/dx = co in

the upper region.

Because of the very sharp curvature in this

region it is sometimes treated as an appendage and the fairing of the main hull carried up to a point immediately forward o'f the stern post. In the yz plane this section is 'of similar shape to the stern profile (see Fig. Sb) Le. dy/.dz varies from zero to an angle approaching infinity and back to nearly zero. In this region the designer's lines must be closely adhered to since the performance of the propeller and the hydrodynamic forces transmitted to the hull are sensitive to small changes in the aperture clearances an'd to the flow 'of water to the propellers.

In the curved surfaces in tile fore and aft body apart 'from the boundary conditions discussed above the only geometrical condition which must be satisfied is that each transverse section should be tangential to the base line in a flat bottomed ship, or to the rise of floor line in other cases. This holds true for all sections except those close to the stern and stem when it may be necessary to introduce

some discontinuity in dy/dz

for construeonal

reasons.

Required accuracy of fairing process

There is a tendency' in shipyards to regard the lines plan drawn by the naval architect on 1/48th scale as sacrosanct and for the mould loft to follow slavishly the lines of the designer. Thus an impres-sion has arisen that on the 1/48± scale the lines can be faired to within approximately i in. of the final offset and the mould loft rarely makes alter-ations in excess of this amount.

Mathematically the best indication of the smooth-ness of a line is the 'behaviour of the successive differences of equally spaced points. The detection.

Euro pean Shipbuilding No. 4 - 1963

of errors in data for curves by the use of diff eren-ces is illustrated in several text-books on numerical methods [1] [2J [S]. This approach has been used by Heigh [4] [5] for fairing curved surfaces en-countered in mechanical engineering work but it has now been extended to lines fairing in a few shipyards. In the latter process offsets are measur-ed from the 1/48th scale drawing on a rectangular gri'd so that first 'and second differences can be obtained in both horizontal and vertical directions (in some 'shipyards the grid lines are at 450 to the horizontal plane). These dif'ferences are smoothed

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n

STCR PROFILE

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of 3 ins, to 4 ins, spread over lengths of 40 it. or by arithmetic process or drawn to a base of length and faired by the use of splines and weights and cross checked so that a smooth surface is obtained. The smoothness of the ist. 2nd and 3rd

dif-ferences of a curve is not an absoIue criterion since smooth differences would be obtained from a periodic curve (such as a sine curve') which might be quite unsuitable for the lines of a ship. The additional qualification must be used that there should be no more than the required number of points of inflexion in each line, whether it be a waterline or a section. The latter information can easily be detected from a drawing but is difficult to determine from tables of offsets obtained from 1/48th scale drawings, due to the presence of er-rors in the offsets. This will be discussed in greater detail later.

Tables of differences were prepared for a large number of sets of mould loft offsets and these indicated that errors of i inch were common. A typical set of mould loft offsets for a water line are given in Table i together with their first, se-cond and third differences. This indicates the lack of smoothness of present methods and the impos-sibility of using such data directly for the control of machine tools. In many cases apart from the local unfairness caused by poor fairing or measure-ment, there was evidence of unfairness of the order more, i,e, bulges or hollows on the lines. lt is con-sidered that these were chiefly due to tI'e mould loft following the 1/48th scale oFfsets too slavishr and were not due to a genuine desire of the de-signer to have bulges or hollows;

The fact that these fairly large errors occur in practice immediately prompts the question: «What is the maximum alteration that can be made to the 1/48th scale lines plan which will not affect ship economics assuming the final surface is fair?» Ship economics here means the initial cost of the hull plus all subsequent running expenses. The chief factors are thought to be the resistance of the hull and its cargo carrying capacity, since any fairing process which tends to remove local unfairnesses is likely to reduce the cost of fabrication.

As mentioned earlier, in the present state of our knowledge of ship resistance no one is able to state that one hull is better than another, provided they have the same overall dimensions, same displace-ment; prismatic coefficient, L.C.B. position, V.C.B. position, angle of entrance, stern arrangement and the rate of change of slope of the lines in the hori-zontal and vertical directions are sensibly the same, i.e. there are no shoulders or hollows intro-duced in the one or the other of the designs con-European Shipbuilding No. 4

Table I Frame Offset Nos. x103

-¿2 1962 62 82,479 -31 61 32,448 O -31 -11 60 32,417

-11

-42

+1 59 32,375 -10 -52 -21 58 32,825 -31

-88

+20 57 32,240 -11 -94 +22 56 32,146 ±11 -83 -33 55 32,063 -22 -105 ± 13 54 31,958

-9

-11.4

-2

53 31,844 -11 -125

-10

52 31,719 -21 -146 ±21 51 31,573 O -146 50 31,427 -21 -167

-10

49 31,260 -31 -198 + 11 48 31,062 -20 -218 ± 29 47 30,844 ±9 -209 50 46 30,635 -41 -250 -r21 45 30.385 -20 -270 ±8 44 30,115 -12 -232

-29

43 29,833 -41 42 29,510

-325.

+11 ç, -312 33 41 29,198

-42

-354 ±31 40 28,844 -11 -365 +32 39 28,479 + 21 -344

-72

38 28,135 -'395 +29 37 27,740 C--417 +22 36 27,823 O -417 35 26,906 -10 -427 -32 .34 26,479

-42

-469 3.3 26,010 +1 -468 32 25,542

-22

-490 31 25,052

sidered. It aupears that much larger differences than are at present accepted between 1/48th cale

drawing and final faired form would have no

significant effect on resistance provided the above criteria were met. It is clear that the loftsman can only be allowed to make small changes or he might upset one of these criteria. However if it is pos-sible in some way to formulate the mathematical equations to the surface so that all these criteria are guaranteed in the mathematical form, then it should be possible to allow larger differences be-tween the 1/48th scale drawing offsets and te mathematical surface. It should be clear that the above conditions will give the same cargo carrying capacity 'for the two forms but that there may be slight changes in the capacity of each hold or tank. This would not appear to 'be of any importance in the great majority of cases.

The above discussion relates purely to the hull form criteria which have to be satisfied but the practical requirement's of mathematical 'airing must

also be considered. The prime aim must be to

provide precise data for ordering the material for the hull and its subsequent fabrication and erection. For the latter operations the ultimate aim must be to use numerical control for all machine tool processes and to eliminate templating, drawing or any other process which involves transfer of data. This means that one must be able to specify the offset, the slope and the rate of change of slope in two mutually perpendicular planes at any arbi-trary point on the 'hull surface. Such data are ne-cessary to control plate and frame bending Oper-ations. Those conditions automatically infer that a mathematical equation for the hull surface must be available since it would be very 'difficult and expensive to interpolate such data from a know-ledge of discreet faired point's either on waterlines

or sections.

Finally, there is the question of accuracy. Pre-sent mould loft o'ffsets are usually quoted to 1/16 inch indicating that they are accurate to ± 1/32 inch or 1/384 ft. Numerical control of machine tools o'f the size and ñature of those used in ship-yards could achieve muh 'higher accuracies and it seems reasonable at this stage to aim at an ac-curacy of 1/1000 'ft. This is much greater than that normally achieved in shipyards but since one of the objects must be the elimination of rectification

costs, it is desirable to eliminate completely the

effects of errors in the machine tool operations (plate cutting and plate and frame bending).

To sum up, the essential features of a successful fairing process are:

European Shipbuilding No. 4 - .2963 the method of fairing a given lines' plan must produce no change in volume or shape which will affect the vessel's earning power in any way, i.e. it must not affect the cargo capacity or the hull resistance.

the criterion of 'fairness shall 'be that the first, second and third difference be smooth in both horizontal and vertical directions and satisfy the geometrical 'boundary conditions with the re quired number of points of inflexion in both horizontal and vertical directions.

8) the shape of th'e hull must be expressible by a mathematical equation, ideally over the whole hull, but if not then each c.uaUon must cover large areas and at the boundary between any two areas there must be continuity.

4) the accuracy required is 1/1000 ft. on the full

scale.

Review of mathematical fairing of ships lines The problem of fairng ship lines mathemtically has a long history sating back to Admiral Taylor [6] and his Standard S;-ies of model's 'of mathe-matical forms and even earlier. Prior to the advent of the electronic digital computer the chief interest in mathematical ship forms was taken by scientists engaged in the problem of calculating the wave resistance of ships. Forms having ship-like charac-teristics were produced by Weinblum [7], [811, Watana'be [9] and Jinnaka [10] amongst others. The mathematical expressions they developed vere too simple to express practical ship fouiis to any-thing like the accuracies expected in the normai fairing process, but some were very close to actual designs and could well have been considered further. An account of this early work is given by

Saunders [11]. Its chief characteristic was an endeavour to achieve accuracy of representation with the minimum number of terms and as this can only the achieved by trial and error process, it is considered 'unsuitable for computer applications

F. W. Benson [12], [13] attempted to obtain 5th degree polynomial water lines which would satisfy certain of the design conditions for a hull, i.e. they would give the correct displacement, centre of buoyancy, angle of entrance, position of parallel middle body, etc. Hi's work was probably limited by the computational facilities available at the time and the present work in some ways may be re-garded as the logical continuation ofBenson's, ideas. With the advent of electronic digital computers and computer. controlled machine tools, a great impetus was gi'ven to the problem of fairing ship lines mathematically and several reports have been published on the subject (Ref. [14] to [19]).

European Shipbuilding No. 4 - 1963

The methods described in the above references,

with perhaps the exception of [15], attempt to

obtain as close a fit as possible to the original offsets without due regard to the accuracy of the data on which the original design was based. Few of them state explicitly how they sátisfy all the geometrical conditions mentioned earlier and only the method described in ref. [15] appears to have been used in shipyard practice. To obtain a close fit one requires high order polynomials, the order depending on the block coefficient of the ship be-fng faired but for a fine ship such as the 0.60 block

model of the series 60, which Pien [16] and Ker-win [17] used, it is undoubtedly necessary to use àpproximately one hundred terms in the equations to the fore body and aft body surfaces. Such an equation is liable to introduce unwanted waviness in areas where the waterlines or sections arenearly straight. It has been found that this would give a maximum difference from the quoted offsets of approximately 3" on a 400 ft. ship of that'design.

However, if only 50 terms are used, this difference has only increased to 6" and en only in the region of the bilge where the offsets are inevitably poorly defined. The latter shape is certainly smoother than the former, in that shoulders and hollows in the original form are removed and the possibility of waviness greatly reduced.

It was decided therefore to abandon the search for extreme closeness of fit and to concentrate on ensuring that the characteristics which the designer required from the hull were satisfied by the surface quations developed. By approaching the problem in this way it was hoped that a design tool would be developed which could finally dispense with

the need for an initial lines pian.

Before going on to describe in detail the methods used, it is perhaps worth noting one important point

about the geometry of the ship. With the axes

selected as in Fig. 1 one of the most difficult points to overcome is 'the requirement that dy". z = on the base in a flat bottomed 'ship and equal to a specific large angle on the rise of floor line in the other case. To overcome this difficulty, Kerwin had to use very high powers and the condition' was omitted in at least two other methods. It can overcome if instead of taking theàfs 6n he centre-line of the ship, it is taken on the periphery of th midship section and the difference between the midship section offset and the thection offset

is fitted with the surface equations. At the base, if the axis is taken where the midship section be. comes tangential to the base line, then the differ-ence between the slope of the midship section and the slope of any other section is zero at this point.

This is equivalent to finding the equation to the surface that would remain if the form was removed from a rectangular block 'having the same breadth,

depth and length overall as the ship. Method used at Clasow University

The ship is treated in longitudinal halves. Sur-face equations are derived for the forebody be-tween the stem profile and forward flat of side curve and for the aft body between the aft flat of side curve and a station immediately forward of the stern post.

The cruiser stem is faired locally and matched to the main hull surface. For a ship with rise of floor, the surface equations terminate at a

vater-line just above rise of floor; conics are used to join the lower region to the main body of the hull.

To illustrate the method 'of setting up the equat-ions, the forebody of a hat bottomed ship is

con-sidered below.

Let origin be amidships and the x, y and z axes lie along the longitudinal, transverse and vertical directions respectively. Let the flat 'of side curve

and stem profile 'in the x - z plane be

x = xi(z) = Flat of s'i'de curve (1)

x = x2(z) y = y1(z) y = y2(z)

Equations (1) to (4) define the boundaries for the 'fairing process and before any curve fitting is done, these four equations must be obtained ac-curately and must be contin'uous functions.

In order to avoid rapid changes in curvature and magnitude of the offsets, the surface is smoothed with respect fo urve y1(z) as the datum line i.e. the difference between the offsets at any point on the waterline and the maximum offset for that

waterline is faired. Thus the transformed o'ffse becomes after normalisation.

-yj(z)_. y2(z)

It has been 'found useful to move the axis of x along the x = xi(z) and make the length of each waterline between x x1 (z) and x = x2(z) unity, i.e. the x variable has been transformed to

x-

_x()xi(z)

xxi(z)

Likewise for the z-axis, a shift of origin to the midpoint would simplify the solution of the equat-ion. Thus

2z - (zh ± Zg)

(zh Zg)

(T)

= Stem profile . (2)

= Curve of maximum offsets at flat

of side (3)

where zh = Maximum Waterline Zg.

=

Minimum Waterline.

The conditions to be observed at the boundaries are:

X0 ZZ Y0

(S)

X=0 Z==ZO

(9)

d2Y = O for certain

X0 ZZ--

_{dx- cases} (10)

X = i.o

z = Z

Y = 1.0 (11)

XX Z-1.0

q- =0

(12)

X = X

Z = 1.0

- O (iS)

If one assumes that the surface can be expressed by an ordinary polynomial of the form

let x = X z = Z Y = f(x, z) (14

then, to fulfil the boundary conditions (a), (b) and (c) the first three coefficients of x become zero and the polynomial has the form

Y=CO3x3±C04x4+CO5x5+

±Coxi

±C13x3z + C14x4z + C15x5z +

+C1xiz

(15)

±

CK3xSzK± CK4x4zK + CxjxzK

Since this polynomial surface must have the same volume and moments of volume in both hori-zontal and vertical directions as the original ship, these conditions together with (d), (e) and (f) are then used to determiiie coefficients in equation

(15), e.g. this volume condition gives

= LO 'x = 1.0 dz Yclx=

J = - io)

= o 2 {v4c03±1/0c04± ± O /4C23 ±'/sC24

+

±2/3 . .

+coj

.

±cli

+c2i of'/4C() +°/5C() +(.) C(Kl)j }

+

( {i/.:±'lOCK4 ±...

±

( 1)CSi}

forK

even, and in the case for '

odd, then the

last row is zero.

Oc03 Oco4

0co

X = 1.0

y = 1.0

European Shipbuilding No. 4 1963

The integral on the left hand side is determined by numerical integration techniques, e.g. Simpson's rule over the region where the data is evenly

spaced and trapezoidal method for The, portions

whfch lie between the last data point and the

boundary.

The number of integrals used depended on the order of polynomial required in the horizontal and vertical directions. As an example, consider a poly-nomial up to x0 and z4 for the horizontal and

verti-cal planes respectively. Then equation (15) be-comes Y CO3x8+C04X4+CO5x5 ± c13x3z+c14X4Z ± c15x5z + c23x3z2 + c24x4z2 ± c25x5z2 (17) +c33x3z3+c34x4z3±C35X'Z3 +c43x3z4 + c44x4z4 ±c45x5z4

The conditions used may be as follows

2e03 ±0c13 +2/3C23 + OC33 ±2/5C43 2c04 +0e14 +2/3C24 ± Oc34 +2/5c44 2co ±0c15 +2/3c25 + Oc35 ±2/5c45 (18) X = 1.0

yl.O

fzdz

0=

+2/3C13 + Oc23 +2/5c33 + Oc43 +2/3c14 + Oc24 +2/5C34 + Oc44 +2/3C15 ±0e25 ±2/5C35 + ₄₅ (19) X = 1.0 y = 1.0 5z2dz = 2/3 2/3CØ3 + Oc13 2/3C04 + Oc14 2/3CO5 + Oc15 X = 1.0 y = 1.0

5'dz = O =

Oc03 ±2/oc13 + Oc04 +2/5C14 + Oc05 +2/5c15 + +2/3C20 + +2/c2 ± ±2/5c25 ±

fzdz

= 2/5

=

and the following integrals

Oc33 ±2/7c43 Oc34 +2/7C44 OC:J5 ±/c45 Oc2:; ±2/7C33 + Oc24 +2/7C34 + Oc25 +2/7c35+ Oc43 Oc44 OC45 (20) (22) 2/oc03 +Oc13 +2/7C23 ± Oc33 ±2/SC.;:1

21oc04 +Oc14 +2/7c24 ± Oc34 +2/oc44

2/oco5 + Oc15 +2/7C25 + Oc35 ±2/c43

(16) ₍₂₁₎

X = 1.0

European Shipbuilding No. 4 - 1963

JJYdzdx: 5fYzdzdx: $5 Yz2dz.dx; ffYz3dzdx;

ffYz4dz dx; ffYxdzdx: f f Yxz dzdx; 5f Yxzdzdx; f $Yxzdzdx; f fYxz4dzdx.

all integrals being between the limits

O to

i

forx

and 1 to +1

for z.

It may also be noted that this method does not need any offsets to generate a surface so long as one has the numerical values of those integrals. For this reason it is possible to design ship lines with integrals obtained by a regression analysis of resistance characteristics to give the optimum com-bination of integrals.

or

Solutions to mathematical equation for ship surface One approach to the solution of the mathematical equation for the ship's surface would be to cal-culate all the double integrals associated with the required designed volume and moment character-istics and solve the coefficients in the three dimen-sional equations, i.e. expressing equation (15) in matrix notation gives

[A] [c] = [b]

(23)

where [A] is a square matrix associated with the boundary conditions.

[c] is a column vector of the coefficients. [b] is a column vector of boundary

con-ditions and double integrals.

However, the elements of [A] are closely related

to the elements of the Hilbert Matrix and the

solution becomes very ill-conditioned. Hence it

vas found impracticable to solve the equation on a DEUCE digital computer in one single operation. Instead, the problem was treated as a two 'dimens-ional problem, the horizontal fairing being carried out first.

Using the same notation as before, the equation for any waterline at a given height above base can be expressed as

Y = a0+ajx±a2x2+

+ax

.... (24)

for any value of z,

and from the boundary conditions given in equat-ions (8) to (ii), coefficients a0 = a1 O and for some ships with large area of flat of sides a2 = 0. The other coefficients are determined from the area and first, second, third, etc. moments of that waterplane. The actual values of these integrals are determined by numerical techniques. In prac-tice the highest order of fit found necessary has been j = 7.0. e.g. 1.0 i 1.0 vdx Jo r1 .0 J yx.dx t 1.0 yx2. dx

Je

f1.0 vx0. dx .Jo (25)

[w] [a] = [M]

(26)

The solution to the above equation can be ob-tained readily by expressing matrix [w] in terms of lower and upper triangular matrices which are evaluated exactly by using vulgar fractions through-out the calculation. (See Appendix I).

The coefficients 'determined from equation (26) for each waterline are then faired vertically in the

form

f(a)co+cjz+ce1z2+caz3 +

+CKjZ

(27)

The solution of the above equation may be ob-tained in the same way as for waterlines or by the use of Forsythe's Method (20) which has been modified and programmed on DEUCE by

CIen-shaw (21). (See Appendix II).

Where there is a rise of floor the same procedure could be applied directly if the offsets vere given with reference to an axis at right angles to the rise of floor line. This is inconvenient and the re-sulting equations would be in a difficult form for further calculations. The effect of the rise of floor is to make every section extend over a different length in the z-direction and this cannot easily be allowed for in the equations. It was found

prefer-able to treat the portion of the 'hull below the

waterline immediately above the rise of floor as an appendage.

The procedure is the same as before but fairing is done between a waterline immediately above the 'height of rise of floor and the highest water-line. The region between this waterline and the tangents of the section on the rise of floor is fitted with conics s'o that continuity will be achieved be-tveen this region and the main 'body of the hull. It will be noted that this method may not always 'give the designed volume characteristics, since the lower appendage is not fitted in the least square

11111

1/4 1/ _1/o 1/7 1/ _a4

1/5 _1/6 1/7 1/8 1/9 _a0

1/6 1/7 1/5 1/9 _{1/lo a0}

sense but the error has been found to benegligible. In the case of the aft body, the region between the cruiser stern and the station immediately for-ward of the stern post is faired locally. The offsets and slopes at the latter station are matched at each waterline and the equations set up so as to pro-vicie the saine volume and centre of volume.

i)oto /UitCflWflLS

It is to be expected that for some time to corne the integrals which form the right hand sides of

the

above equations will be determined from

numerical integration of offsets supplied by the designer. The number of offsets required depends on the type of ships to be faired and on the order of polynomial used. Obviously, if only the same number of offsets were used as there are coeffi-cients in the polynomial expression the mathemati-cal surface would tend to pass through the given points and no fairing would take place, i.e. it would represent an interpolation procedure. Additionally, if only a few offsets per waterline are used the process of numerical integration is open to

con-siderable errors, especially if an integration rule of the Simpson type, where ordinates receive unequal weighting, is used. On the other hand, the object

of this work is to reduce the amount of

offset

measurement etc. required in the shipyard so that the use of a large number of offsets is to be avoid-ed. In practice it has been found that a minimum of ten points per waterline (including the two end points) are adequate for even the finest block

coef-fici ent.

The highest waterlines in the ship will have the fewest data so that if the data is obtained on a rectangular grid over the whole length, this usually involves data being supplied at forty equally spaced sections in the length on the lowest waterline.

The position of the flat of side curve and the correct representation of the bow profile and stern profile are vital in obtaining a smooth surface. It must be emphasised that the same criteria of

smoothness must apply to these lines as

to the

whole surface. A discontinuity in the boundary curve will appear later as a discontinuity in the surface. The fore and aft position of the flat of side curve is not of vital importance but generally speaking it

has been found that draughtsrnen

estimate its position too far away from amidships and any error preferably should be in the other direction. It is hoped eventually to detect this

point from the given offset data, but so far a satis-factory method of doing this has not been evolved.

In the vertical direction the fairing process must he taken up to the highest point in the ship for

European Shipbuilding No. 4 - 1963 which data are required, i.e. generally the top of the forecastle bulwark at the extreme forward end. means that the lines plan must be extended up to this height over the whole length

of the

forebody and the unwanted data removed later. Similarly for, the aft body it-must be extended up to the top of the poop deck bulwark. The normal waterline spacing used in shipyards which varies from 2 to 4 ft. gives sufficient data for the vertical fairing to be carried out. The waterlines have to be more closely spaced in the bottom region. Data inspection

It has been found necessary to subject all the data taken from 1/48th scale drawings to differ-encing programmes which give the first, second and third differences for all waterlines and sections in order to detect errors in reading and recording the offsets. Where visual inspection of the differen-ces indicates data errors in exdifferen-cess of two to three inches the data are returned to the shipyard for checking. This method does not detect errors in the immediate area of the boundaries and most trouble has been experienced due to errors in this region, especially at the stern and stem.

Practical results and experience

A number of ships covering the complete range of block coefficient encountered in merchant ship-building practice have been faired using the method described earlier.

Although generally speaking fine block coeffi-cient ships require higher degree polynomials, some fine vessels have been found to have lines which could be fitted by a 15 term expression for the fore body and aft body surfaces. Thus it is impossible to state which order of polynomial should be used for a given design and it is general practice to pro-duce a number of surface equations employing different orders of polynomials horizontally and vertically and to choose that form which best mat-ches the designer's requirements.

The hydrodynamic requirements have been dis-cussed earlier. In practice, it was found that some of the designers took the view put forward in this report, i.e., that a smooth 'surface with the correct volume characteristics was what they desired and that differences of up to six inches between their 1/46th scale lines plan and the mathematical sur-face were tolerable. Others held the view that they wish'ed their lines reproduced closely - within 3 inches maximum difference with an average differ-ence of rn '/ inch.

In both cases it was found

possible to satisfy the designers but the second

European Shipbuilding No. 4

-view involves the use of high order polynomials nd leads to other diffi: ' ties.

In particular it has been found that in the great majority of lines plans certain transverse sections will be nearly straight over a considerable extent of their length. This is always encountered in the stern half when the frames are changing from the concave up curvature in the midships region to the concave down curvature encountered at the stern. In ships having very U shaped sections in the fore body - (as for instance series 60 forms) large num-bers of frames with straight portions are encounter-ed. These straight portions are regarded as desir-able from the practical point of view since they make the plating easier to. develop and bend and involve less frame bending. If one attempts to it a high order polynomial to a surface containing one of these «fiat» portions, waves will be set up in the fIat surface. These can be removed by fitting the flat area with low order polynomials

determin-ed by a least squares method with the correct

boundary conditions to match the rest of the hull. Programmes were developed to do this in the cases where the designers wished ose fit to their lines. It should be noted that this will not invalidate any of the volume characteristics. Such a process is

time consuming and expensive and could be avoid-ed completely by the use of lower order polynomi-als for the surface equation. When numerically controlled machine tools become available for plate and frame bending, the practical desire to introduce straight parts will largely disappear and the in-centive to use low order polynomials be increased since they will reduce computing costs in the con-trol process.

Elsewhere, undoubtedly the greatest problem lies in obtaining the equations to the boundary curves. A large number of programmes have been developed to cover combinations of various types of mathematical curves used in designing the bow and stem profiles. Despite the range of these pro-grammes and of other curve fitting propro-grammes available on DEUCE, designs have been encounter-ed wh'ih could only be fittencounter-ed by specially develop-ed equations. This is time consuming and where such cases. arise the fairing process can never be automatic. The difficulty can only be overcome the designer allowing greater flexibility in these regions, which is not generally possible especially at the stern, or the boundary design must be de-veloped from basically hematicai curves. With-in the range of the curves programmed on DEUCE an infinite variety of shapes can be produced which should provide no practical restriction on the

de-signer.

Conclusions

To use numerically controlled machine tools efficiently and to reduce the tolerances on the finished prefabricated sections, it is essential to

define the hull shape accurately. The desired de-gree of accuracy can be achieved by expressing the hull surface mathematically but not by the present mould loft, drawing and template methods.

To carry out numerical control of plate and frame bending. surface development etc., it is

necessary to have an equation which represents large parts of the hull surface so that offsets, slopes, curvatures, developed lengths etc. may be evaluated readily for any point on the hull surface.

e) It is possible to obtain equations for the com-plete hull shape from a knowledge of the required volume and longitudinal and vertical moments of volume of the fore and aft bodies, provided the boundaries are defined precisely by smooth curves without discontinuities. A 1/4Sth scale lines plan is therefore not an essential starting point for the generation of the surface equation. The number of moments of volume used in the solution should be the minimum required to satisfy the specified resistance and propulsion characteristics.

d) Before the fairing process can be employed automatically, i.e. a single computer programme written to do the complete operation, the designer must specify a limited ringe of curves 'for use in defining the required bow and stern profiles. Acknowledgements

The work described in the report was financed by a grant from the Department of Scientific and Industrial Research and carried out in the Naval Architecture Department at Glasgow University for the period ist October, 1960, to 31st October, 1962. We are most indebted to the D.S.I.R. 'for making this work possible.

At the same time a parallel approach to the

problem was made by Mr. J. G. Hayes of the Mathematics Division, National Physical Labor-atory and the two groups exchanged data and ideas throughout this period. The success of the work at Glasgow owes much to the advice and assistance of Mr: Hayes.

We would also like to record our indebtedness

to:

-Miss M. A. Blair for 'her work in programming. Several Clyde shipyards who supplied data for the fairing process. Particular mention should be made of the Fairfield Shipbuilding Company for their co-operation in carrying out a critical examin-ation of the results in their mould loft.

Computer and to Dr. D. C. Cilles, Director of

Computing Laboratory for his co-operation. Finally, we are most indeb.tcd to Professor J. F. C. Conn for his direction and encouragement.

REFERENCES

Modern Computing Methods. National Physical Laboratory - Notes on Applied Science. No. 16.

i-I.M SO. 2nd Ed. p. 62.

Buckin'gham, R. A.: «Numerical Methode.» Pitman Press, p. 25.

Kunz, K. S.: «Numerical Analysis.» McGraw Hill Book Co., p. 38.

Hoigh, W.: «Economy by Efficient Calcula.tiorts.» I E.S.S. 1948.

Heigh, W.: «The Geometry of the Job.» I.E.S.S.

1937.

Taylor, D. W.: «Ship Calculations, Resistance anti Propulsion.» International Congress of

Engineer-ing, Sin Francisco, 1915.

Weinbium, G.: «Exakte Wassc'rlinien und Spant-flächenkurven.» (Exact Waterlines and

Trans-verse-Sections) Schiffbau, 1934 April and May. [S] Weinbiuni, G.: «Beiträge zur Theorie der Schiff

s-oberfläche.» (Contribution to the Theory of Ship

Surfaces). W. R. H. 1929-1930.

Watanabe, K.: «Mathematical Expression of Ship

Form.» Journal of Zosen Kiokal, Vol. LXXVII,

July 1945.

Jinnaka: «On Ship Forms and Wave-making Resistance.» Journal of Zosen Kiokai, Vol. XIV

Part 1, Feb. 1961.

Sounders, H. E.: <Hydrodynamics in Ship De-sign». Soc. of Navsi Arch. and Marine Eng. Vol.

II, 1957.

Benson, F. W.: «Mathematical Ships Lines.» Inst.1 Naval Arch. 1940.

Benson, F. W.: «Mathematical Curve Tracimg». B.S.R.A. Technical Memo. 23, Jan. 1951.

Taggart, R.: «Mathematical Fairing of Ships Lines

for Mould Loft Layout.» A.S.N.E. May 1955, p.

337.

Rosingh, W. H. C. C. E. and Berghuis, J.: «Ma-thematical Design of Hull Linee». Inter.

Ship-building Progress. Vol. 6. Jan. 1959.

Pien, P. C.: «Mathematical Ship Surface. DT.

N.B. Report 1398. June 1960.

Kerwin, J. E.: «Polynomial Surface

Represen-tation of Arbritrarzj Ship Forme». Jour, of Ship

Tteseorch, Vol. 4. July 1960.

[is] Theilheimer, F. and Starkweather, W.: «The

Pair-ing of Ship Lines on a High-Speed Electronic

Computer». D.T.M.B. Report 1474. Jan. 1961. Taylor, F.: «Computer Applications to

Shipbuild-ing». R.I.N.A. April, 1962.

Forsythe, G. E.: «Generation and Use of

Ortho-gonal Polynomials for Data.Fivting with a Digital Computer». J. Soc. Indust. Applied Math. Vol. 5, p. 74, 1957.

Clenshaw, C. W.: «Curve Fitting with a Digital

Computer». The Computer Journal, Vol. 4, 1961.

European Shipbuilding No. 4 - 1963

APPENDiX I

Derivation, of equations for fitting waterlines The equation for any waterline at a given height tove base may be expressed as

Y = a0+ajx+a2x2± .

±ax

I. 1.

where y, x have been defined by equations (5) and (6) respectively.

The coefficients in equation I. 1. are determined from the boundary conditions and the area and moments of the w. .rplane about the flat of side cklrve. Thus on applying these conditions we have:

At the flat of side curve:

-atx0,

y=O

dx

and in ships with large flat of side

d2y

at

x0,

,

=0

:.

a2-0

clx-At the forward boundary,

at x = 1.0,

y = 1.0 .. a2±a3- ....±aj = 1.0

Taking integrals we have

Area =JYdx

a2± a3+.

. . ±

ist Moment

Jx dx a2+ a3+

.

±

p.10

Kth Moment = Yxk dx = J0

i ,

i

(k-+3) a2 (k±4) a3Th

The order of moment used depends on the

order of equation and available boundary con-ditions. The integrals on the left hand side must be determined using numerical methods over the correct length of each line. The number of data points per waterline should be at least 4 more than the order of polynomial it is proposed to use.

In matrix notation the equation for the solution of I. 1. may be expressed as

[W] [a] = [M]

I. 2.

The values of column vector [a] can be deter-mined as

follows:-Let

[W] = [L] [U]

1.3.

where [L] is a lower triangular matrix and [U] is an upper triangular matrix whose diagonal elements are .pnity.

The elements of [L] and [U] are obtained by identification of the elements of [W] with the ele-ments of the product [L] [U].

On substituting 1.3 into 1.2

[W] [a] =

[L] [U] [a] = [M]

I. 4

at x = 0,

:. a00

European Shipbuilding No. 4 - 1963 By putting

[U] [a] = [N]

I. 5

Then

[L] [N] = [M]

I. 6

The values of [N] can be determined by a pro-cess of back substitution.

The values of [.a] will be evaluated from 1.5 by a second back substitution process.

In the fairing method developed at Glasgow, the values of upper and lower triangular matrices [U] and [L] have been calculated using vulgar fraction for a number of bOundary conditions and order of polynomials. The following solutions have been

found useful: - See Table IIIII.

APPENDIX II

Derivation of equations for fitting sections The coefficients derived from fitting of water-lines are faired vertically by least squares 'methods. Each set of the coefficients is faired 'separately

with an equation of the form

f(aj) = co±cijz+c2z2+

±ckfzk II. 1 The coefficients of the above equation may be evaluated in a manner similar to that described in Appendix I, with the exception that no con-ditions are applied where rise of floor exists. ifl

matrix notation

[V] [C] = [J] II. 2

where [V] = square matrix of elements associated with the solution of II. 1.

[C] = 'column vector of coefficient's. [J] = column vector of integrals.

This approach was applied for

_{orders up to}

k = G. Since the matrix is symmetrical the lower and upper triangular matrices are identical.

Alternatively, the coefficients in e

'on II. 1.

may be replaced by coefficients of Tchebyshev polynomial, viz.,

f(a) = SoTo+S1j-Ti±S2T2+

+ SkjTk II.

The solution in terms of triangular matrices has been calculated for k = 8.

The modified Forsythe Method programmed on

DEUCE by Clenshaw, [21] can also be used.

/1O5540 i 2 55/7 ₁ 1 10/7 45/j4 /s i [W1] = [L1] [U1] [W2] = [L2] [U2] 1 (1.0 ydx Jo r' .0 yxdx -0 I"oyxcL,c

.0

(1.0 yx5dx Jo [W1] = 1/4 1R 1/ /i 1/5 1/6 1/.. Ìo

11111

Table 1/ 1/7 1J 1/ II 1/ 1fg 1/ 1/ 1/7 1/ 1/ i/io Table III

1111

a3 5/4 5j _/c 1/7 _a4 [W2] = 1/ 1j /7 _{V8 a5} 1/« 5/7 1/8 1/6 _a6 i 1/4

/20

[L2] = 1/5 _1/35 1/030 h/ _1/., _...1)554

i

[U1] = 1 1 1 /s ₁ 1 15/7 s/ [L1] [U2] = _1/ __I/oo _..__1/30 _1/4 1 /s _i /100 _/5?5 ____1/50 _1/3555 /588o 1 1/j 1/4 1/5 1/ 1 1 a2 a3 a4 a5 a6 i rio ydx Jo vx2dx Jo 1

APPENDIX Ill Geometry of bow profiles

Mathematical equations for certain bow shapes The shape of the bow is largely governed by aesthetic considerations and practice varies greatly.

Two distinct types, have been encountered: where the profile

is a continuous curve

(AHEFQ in Fig. 4) which is 'specified in terms of a series of coordinates and which can be fitted using Least Squares technique with provision for any special boundary requirements such as zero slope at base, etc.

a combination of straight lines (AH, EI), an arc of a circle or a parabola (HE), and a curve IQ

whith can only be treated by least

squares

technique with allowance for the continuity of

slope, etc. at the given point I.

The equations for type (b) are given below:

Circular arc _{From z = zH to z}

z-T

Let Forward Perdendicular, FP, coincide with axis OZ.

The straight

lines CEI and AHJ have the

equations: From z = zE to z = OF z = m1x ± e1 III. i and z = m2x + c III. 2 where e1 OF, c2 = OJ

Let the equation of the line bisecting CEI and

AHJ at B passing through the centre of the circle, G, be:

Z 1fl3X ± C III. O

From these equations it is, possible to obtain the points:

r = V (xExG)+(zE_zG)2

III. 6 Any point on the arc of the circle will be given

by:

From z O to z = z H

X =

OJz

1112

European Shipbuilding No. 4 - 2963

X = X +

_Vr2(zzG)2

III. 9

X-Fig. 4.

0Ey

rn2 iii. 10 Parabolic curve

Th'e conditions that equations III. 1 and III. 2 may 'be tangents to the parabola

Vx/a ± \/z/b

1.0

are:

f/a + g/b = 1.0 III. 12

XB, ZB, XE, ZE, (given), XH, zH

a ± k/b = 1.0 III. 13

On substitution of III. i in III. 3 eq'uatiòn of BG _{where a, b, are parameters defining the parabola,}

can be written as: _and

z = m.3x+(mi m3)xB ±c1 III. 4

f = OA

g OJ

The normal at E is: h = OC k = OF

From III. 11 and III. 12 the parameters a, b can

z _{i/rn1 (XE _X)±ZE}

be determined as

The solution of III. 4 and III. 5 will give the C0 _(fk--hg)

_(fk-hg)

ordinates of G and radius of the circle as: a _{(k_a)}

b

-(fh)

and again the distances from F.P. are:

zH g2/b, XH = f2/a; ZE = k2/b, XE = h2/a From z = O

to z = z

x = (gz)/m

III. 14

From zzHto zzE

X = a(l-Vz/b)2

III. 8 III. 15

,Europcan Shipbuilding No.

4 - 1963

I enlighet med avtal mellan de i det nordiska skeppstek-niska samarbetet deltagande organisationerna inbjuder

Fin-lands Skeppstekniska Kommitté till det VIII Nordiska skeppstekniska mötct, NSTM-G!3, som iir öppet fOr alla

intresserade. Mötet hMlcs i Ab den 4 och 5 oktober 1963.

P R O G R A M:

Fredagen den 4 oktober: Lokal: Konserthuset, Aningaisga-tan 9. Mötesledarc: Dipl.ing. Erik A. Heino.

Kl. S.15-9.00: Registrering.

« _{9.15: MOtets öpnandc.}

« 9,25: Civ.ing. Rutger Bonnet, Sverige: Bestimning p projektstadiet av vâgmoment for fartygsnybygge.

« 10.45: Paus.

10.55: Giving. E. J. Nielsen, Danmark: Elekisk

dks-maskineri tu losning, lastning og forhaling.

« 12.15: Lunch i Teaterrestaurangen, Auragatan 10 (buss fr5n mötesplatsen till och frAu luneheri).

« 13.45: Overingenier Harald Aa Walderhaug, Norge:

Un-dersokelse av en hydrofoilbât med mekanisk

kontrollerte foilvinkler.

« 15.05: Paus.

15.15: Dipl.ing. Jan Lindblom, Finland: Orn lisnmade triikonstruktioner i fartygs- och bâtbyggeri.

« 19.35: MOtets av.slutande.

19.00: Micldag pA Abo SIott (buss frAn ho.tellen till och

frAn middagen).

Lördagen den 5 oktober: Exkursioner: Kl. 8.30-11.30: Avfärd framför Societetshuset, Humlegârdsgatan 2, Wiirtsilä-koncernen A/B, Crichton-Vulcan och Ab Abo

Bâtvarf.

Valmet Ab, Pansie Skeppsvarv ooh Oy Laivateollisuus

Ab.

T

MATI-MATCtL FA

(Cont. from previous page).

From z = ZE to z = OF

x = (kz)/mi

III. 16

The curved portion IQ is fitted by an equation

b0+b1z-4-bz2±b3z3 III. 17

' with the condions that it will have the same off-set and slope at I as for straight line CEF and the

v

C::L

AnmOlan och betalning: Alla intresserade, utan hiinsyn till utbuldning, praktik eilen medlemskap i de arrangerande or-ganisationerna, kan deltaga. Kongressavgift, betalning fOr middag och exkursioncr, samt pris für foredrag (fOr icke mötesdeltagane) finns angivct pA anmiilningsblankettcn.

Anrniilningar sAndes till: Arrangcinangskommitén fOr

NSTM-63, do Finlands Metsilundustriförening, Södra Kajen

10, Helsingfors, och avgiftcrna inbetalas pA konto «Finlancls Skeppstekn.iska Kommitté» i Nordiska Förenungshankcn,

Södra Kajen 10, Hels'ingfons. Anmälan bör göras samt ay-gifterna inrbetalas snarast möjiigt och senast den 10. septem-ber 1963.

Hotellreservation: Inkvarteringen pA hotell ombesörjes av Abo stads turistbyrâ, Slottgatah 14, tel. 13 538, som Aven bekröftar rumsbestiullningarna direkt till deltagama.

Önske-mAl angAende hotelirum meddelas pA anrnälningsblanketten,

dAr riktpris fOr olika rumstyper Ar an.givna. DA det i Abo rAder brist pA enkeltrum hoppas arrangörerrsa att sA mAnga mötesdeltagare sam möjligt vili flOja Sig med dubbelrum. I sâdant fall torde ocksâ den önskade rumskamratens namn antecknas pA anmAlningsblanketten.

Diskussion: Efte'r föredragshâllarens inledning (max. 1

minuten) Ar diskussionen fri. Diskussuonsdeltagarc med lAnga

inliig anmodas att utarbeta dessa skriftligt och endast ge en resnmé frAn talarstolen. Inleveras 200 kopior av inlägget i god rid till sekretariatet, utsändes dessa till mötesdeltagar-. na. FOr att underlätta referenternas arbete ombedes cadra diskussionsdeItagare att lAmna kopia av inlägget till sekre-tariatet. Apparat fOr ljusbilder och tavla stAr till disposisjon. Andra uppivsninger: I enlighet med avtal i samarbets-komnuittén komrner det ej att arrangeras nâgot damprograim aller turustmïssigt program i anslutning till mötet.

Arrangemangskommittén nstrn 1963.

offset at highest waterline together with the cor-rect area for determining the unknown coefficients: b0. b1, b2, b3.

Considerable variation in the bow profile can be obtained by varying the slopes m1 and m2 and the curve IQ.

A number of programmes are available to cover a large variety of bow shapes.