Accuracy of computer programmes for the calculation of hydrostatic curves and static stability

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L

ß

Report no. 212

a

LABORATORIUM VOOR

SCH EEPS BOU WKUNDE

TECHNISCHE HOGESCHOOL DELFT

r

ACCURACY OF COMPUTER PROGRAMMES FOR THE CALCULATION

OF HYDROSTATIC CURVES AND STATIC STABILITY.

by

A. Versluis.

L

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Summar.

J,. Introduction

H

Flow diagram of the stability programme.

-2-2. Description of the computer programme for hydrostatic curves.

3. Description of th computer programme for static stability. 9

4. Mathematical model. ₁₂

5. Determination of the accuracy of the hydrostatic calculations. ₁₃

6. Determination of the accuracy of the stability calculation. '4

7. Conclusion, ₁₈

Appendix 1: ₁₉

Flow-diagram of the programme of the hydrostatic calculations.

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Summary.

The reliability of computer calculations is an important question for the user. Generally he has only a small influence on the accuracy which can be obtained and only a rough idea of the method underlying the calculations.So in most cases the user cannot judge the reliability of the computed results unless distinct programme specifications are given. Possible checks on the results are the variation of the number of input data, comparison with differently obtained results or comparison with exactly known results. Accuracy and the principle under which the computation or approximation is carried out must not

be mixed up. It will e.g. be clear that the results of static stability

calculations where the ends of the ship outside the perpendiculars, camber, sheer or superstructures are or are not taken into account may differ out of principle. Therefore it may be misleading to check the accuracy of the computer

output with a number of differently obtained results.

The best way appears to be to check the programme results for a mathematical ship form, for which the hydrostatic data and the static stability can be determined analytically. To that end the form y = (1-x2)(1-z4) has been

chosen, where x, y and z are non-dimensional coordinates given by x = X/L,

y = Y/iB, z = Z/D. The origin of the coordinate system lies in the deck at half length. The form is symmetrical forward and aft, with vertical lines as stem and stern. The waterlines are parabolae. There is no sheer, camber or

super-structures, so that -1 x +1; -1 y + 1; 0 z +1. The draught has been

varied between z 0.075 and z 0.875, the inclinations between 10 and 90 degrees in intervals of 10 degrees. Details of the computations will be found

in the report.

This procedure has been applied to the computer programmes developed by the Shipbuilding Laboratory of the Technological University Delft. Details these

programmes are also found in the report. The results show no material difference between the numerically computed values and the analytic values; see table I to IV. For this hypothetical ship form there is neither a distinct difference between a longitudinal division in 40, 20 or 12 sections. The latter conclusion is not

necessarily true for ordinary ship forms as well. But the programmes have the possibility to adapt the longitudinal division to the form under consideration

by placing the sections at unequal intervals and by changing the number of

sections. Since the user can also decide for himselfWhetherhe wants to include

the ship ends, sheer, camber or superstructures he has a fair understanding of

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-3-the value of -3-the computed results.

A point which has riot been discussed is the accuracy and suitability of the input data, supplied by the user. This point is equally relevant to

different types of computations. Computer progranmies have the possibility to prínt the coordinates of the ship form or to plot the form with which they carry out the computations. These data must be compared with the original lines to avoid erroneous results.

H

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-4-I .-4-Introduction.

4

The purpose of this investigation is the determination of the accuracy of computer programmes for the hydrostatic and stability calculations, which have been developed in the Shipbuilding Laboratory of the Technological University of Delft. This accuracy is found by comparing the numerical calculations with the results of an analytical approach.

The analytical computations are carried out for a mathematical shipform:

2 4

y = (1-x )(1-z ).

H

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-5-2.Description of the computer programme for hydrostatic curves,

The integrations for the determination of areas, volumes, statical moments and moments of inertia are carried out by Simpson's second rule with

unequal intervals. The position and the numbe of the sections (with an even number of intervals) are adjustable for each specified case.

The ship form is determined by the cross-sections which are represented by an arbitrary number of measured points.

The programme is also suitable for ships with a non horizontal keel and for ships with a trimmed draught.

The depth is divided into two groups of waterlines. The mutual distance of the lower group of waterlines is 0.10 m. while the distance of the higher group depends on the depth of the ship. It may vary between 0.10 and 0.50 meter for small, respectively for large ships.

The width of the sections on these waterlines is found by a third degree interpolation from the measured points. (see fig.1).

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-6-Fig. I

Plotted body-plan of the mathematical form

from, the. measured points.

x measured points

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Cc

Now the following quantities can be found:

-

+

Xto

Fig .2. Sect

n

Waterplane area = 2 . E p(r). y(r,q1 = Wp areatql.

r=0

Moment of the waterplane about the midship section: n

momtqi = 2. E p(r). xtr). yr,q

r=0

Transverse moment of inertia: n

= 2/3

. E p(r). yr,q33 r =0

Longitudinal moment of inertia about the midship section:

n ₂

2. E p(r). xtr] . y(r,q).

r=0

Waterp lane coefficient: n

C (q) = 2 . E p(r). ytr,q3 /(L .B)

r=0

where:

r refers to the rth section along the length. rg refers to the number of the midship section.

q refers to the qth waterplane of the body. n = total number of sections

p(r) = associated integration factor.

With the areas, statical moments, and moments of inertia of the waterplanes

known, the following characteristics are found:

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z 8 X 5tc. qq n qq V 2. E p(z). _{p(r). yCr,q]=} E p(z).wparea(q1.

q0

r=O q=O

The moment of the volume about the basic line (momb) is:

qq n Momb 2 . E p(z). z E

p(r). yr,q

q0

r0

qq = E p(z).wparearq],. z[q)

q0

and the moment about the midship section (mom ms):

qq n

mom ms 2 . E p(z) E p(r) . y[r,q . xtr

q0

r=O

Now the following characteristics are found directly:

displaced weight t = hulifactor * y * displaced volume.

(y = specific weight of water)

= momb/V B1 momms/\7 = It/v Bm1

_=KB+BM

Km1

KB+BM1

= V/(L

*B)

pp. q, =

= t/

_pp Sect

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-9-For the determination of the projected wetted surface the sections are assumed to consist of a number of straight lines (dl). The projected wetted surface is then given by:

n qq

S = E p(r) E dlCq].

r0

q=1

Finally is: Section area until waterline qq: qq

2. Z p(z). ytr,q]

q0

Moment of section area about waterline qq: qq

2. E

p(z). y(r,q].

(ztqql-z(q]).

q0

Midship section coefficient ß:

qq

CM = 2. E

p(z).

yrg,q/(B*qq));

q=O

3. Description of the computer programme for static stability.

The method used has been developed in principle by F. Taylor and is published in Transactions of the R.I.N.A. of May 1962.

The data requirements of this programme are similar to the usual data requirements for the hydrostatic calculations.

Except for normal ships the programme is also suitable for: ships with non horizontal keel.

ships which have a trimmed draught.

the stability of a ship ma regular sine wave with a wave length equal to

the ship's length and a variable wave height and an arbitrary position of the crest of the wave along the ship.

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-10-'s, QJ L u

yyCo

¿[LU

Io

It is possible to take account of the influence of camber and/or superstructures.

Notations:

q refers to the qth waterline of the hull ii refers to the th inclined waterplane

tt refers to the angle of inclination

-- -- ________________________________________

i

Pig.5.

The distance from the centre plane to an inclined waterplane for waterline (q] is (see fig.6):

yycq] = yyO]+ qdy

dy = waterline distance tr]/tanqrtt)

zt°

e

wtq) = ytr,ql-yytq]. m(q] (ytr,q3+yy1q])

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-II-The following restrictions must be made:

if wq = 2.yUr,q1

take wLq) = 2.yr,q) and mq) = 0,

if wtq3 O take wtq = O and mtq] = 0.

If the above mensioned criteria are satisfied for all the waterlines of a section, the re-immersion problem as illustrated in figure 7 is automatically solved.

f ig.7,

The areas and the moments of area are determined as follows:

The required area of the section is qq

E p(z). wtq].

q0

The required moment of area about the base line:

qq

E p(z). wtq). z(ql

q =0

where:

qq = number of waterlines

(z) = associated integration factor.

Longitudinal integration of areas and moments of each section up to individual waterplanes and inclinations gives the volume and centroids of volutne.

The volume for inclination 4 tt to the waterplane is:

n qq

Vttt,ii3 = E

p(x). E

p().

_wtq.

r=0

q0

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-12-12

The transverse centre of buoyancy is:

n qq E

p(x). E

p(z). wqL

mtq]. r=O q=O BYttt,iil = n qq E p(x). E

p(z).

wtq]. r=O q=O

The vertical centre of buoyancy is:

n qq E p (x) E p(z). w q3. z (q] r=O

q0

BZttt,ii) -n qq E

p(x)

E

p(z).

wtq). r=O

q0

1< f ig.8.

KNsintt,ii) = BZ(tt,ii .sinq + BY[ttii]

_.cos4.

A. The mathematical model.

For the determination of the accuracy of both programmes they have been applied to a mathematical model.

For this model the hydrostatic calculations can be determined analytically.

The mathematical model is established by:

2 4

y = (1-x )(I-z )

The fore- and afterbody are symmetrical. The waterlines are parabolae.

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As can be seen from the mathematical model the calculations are carried out in a non dimensional form1

5. Determination of the accuracy of the hydrostatic calculations.

The waterline area, moment of inertia and the volume to any waterline can be calculated in a simple way.

Waterline area: ₊₁

A 2 (l-x2)(l-z)dx 8/3

(1-z)

Transverse moment of inertia:

+1 64

43

= (l-x ) (1 z,) dx It 2/3

I

2 = /105(1-z,) x=- I Volume: +1 32 8 = 2

J j

(l-x2)(1-z4)dzdx = 'lS -+ x=-1 z=z

Comparison of these data with the results obtained by the computer programme for the hydrostatic calculations are found in the following tables.

The parabolic waterlines have been integrated by Simpson's second rule; therefore there is no influence by varying the number of the sections. draught

z,

Table I: Waterplane Area.

12 sections analytically 40 sections numerically 20 sections 0.075 2.66658 2.66659 2.66659 2.66659 0.175 2.66416 2.66418 2.66417 2.66417 0.275 2.65141 2.65142 2.65143 2.65142 0.375 2.61393 2.61394 2.61394 2.61394 0.475 2.53092 2.53093 2.53093 2.53093 0.575 2.37517 2.37517 2.37517 2.37517 0.675 2.11308 2.11309 2.11309 2.11309 0.775 1.70467 1.70468 1.70468 1.70467 0.875 1.10352 1.10352 1.10352 1.10351

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14

Table II: Transverse moment of inertia.

Table III: Volume.

No significant differences were found in the numerical and analytical

computations as is shown in the above tables.

6. Determination of the accuracy of the stability calculation.

The mathematical model y = (l-x2)(1-z4) has no sheer and no camber.

The deck is situated in the X-Y plane.

When the deck edge is not immersed by inclination, the modellength is divided

draught z' Analytically 40 sections num?rically 20 sections 12 sections 0.075 6.09466 6.09468 6.09436 6.08976 0.175 6.07810 6.07817 6.07781 6.07320 0.275 5.99126 5. 99 129 5.99099 5.98645 0.375 5.74074 5.74075 5.74045 5.736 12 0.475 5.21096 5.21105 5.21073 5.20679 0.575 4.30691 4,30693 4.30671 4.30344 0.675 3.03273 3.03275 3.03260 3.03029 0.775 1.59222 1 .59224 1.59216 1.59096 0.875 0.43194 0.43 194 0.43191 0.43157 draught z' Analytically 40 sections numerically 20 sections 12 sections 0.075 1.933335 1.933337 1.933336 1.933333 0.175 1.666754 1.666756 1.666756 1.666753 0.275 1.400839 1.400840 1.400840 1.400837 0.375 1.137288 1.137289 1.137289 1.137287 0.475 0.879563 0.879564 0.879564 0.879561 0.575 0.633523 0.633522 0.633522 0633521 0.675 0.408067 0.408067 0.408067 0.408065 0.775 0.215777 0.215776 0.215776 0.215775 0.875 0.073551 0.073551 0.073551 0.073551

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into 50 intervals. When the deck edge is immersed by a larger inclination, the intersections S1 and S2 are determined (see fig.10).

A

0 y

When (1-z,/tg4) O there is no intersection; the deck does not immerse. If the deck immerses, the model parts are divided into 50 ordinate intervals each (fig.10).

These three parts are separately integrated.

For each of the sections the intersections of the inclined waterplane with the section can be determined with a very high degree of accuracy.

ZA and ZB are determined with an accuracy smaller than 0.5 * ,o_6.

The area and the moments of the section can be calculated analytically.

-16-5o

5ø LtrvoI

5e vt.rve

Fige 10. - z,/tg4 i s, S2 = = X2 = /i-z,/tg = -v'1-z,/tg4

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16 z Area

JB(

(1-x2)(1-z4) Z Z

J

2 4 } dz + 2 (1-x )(1-z )dz

ZA

tg _Z=ZB z 2 1 Mom

= I

z(1-x2)(1-z4) -

jdz

+

J

z(I-x2)(1-z4)dz y ./

ZZA

tg4

_ZZB

z

z Mom

=1/

22

B 1-z4)2dz

-Z,Z

)2 dz. z 2 _J (1-x ) (

ZZA

z=zA

The integration over the length of the ship has been carried out numerically by means of the Simpson rule.

In connection with the great number of sections, the error made by using Simpson's integration rule is negligible.

A comparison of KN sin with the numerical results obtained by the computer programme is given in the following table.

Table IV: KN sinc

angle of

mcl that ion

A comparison of the cross-curves of stability between the analytical and numerical calculations is shown in figure 12.

-17-analytical va lue s degrees 40 sections Numerical values 20 sections 12 sections 10 0.15208 0. 152Q8 0.15211 0.15210 20 0.29893 0. 29894 0. 29894 0. 29896 30 0.41646 0.41647 0.4 1644 0.41666 40 0.50118 0.50 I 19 0.50118 0.50 129 50 0.56069 0. 56069 0.56061 0. 56073 60 0.59730 0.59731 0. 597 22 0. 59730 70 0.61217 0.61214 0.6 1224 0.61223 80 0.60665 0. 60665 0. 606 79 0.60681 90 0.58333 0. 58332 0.58332 0.58332

Hence it follows that the difference

negligible for this form.

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o

I1cc

#OLy4S

Vo1u&

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Conclus ion.

Comparison of the results of both programmes computed numerically and analytically shows no significant differences.

One can expect that the results will hold for normal ship forms when a suitable subdivision is made.

18

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-19-APPENDIX 1. FLOW-DIAGRAM OF THE PROGRAMME OF THE HYDROSTATIC CALCULATIONS.

read parameters

read measured points of the ordinates

calculate required waterlines

calculate half breadth of the ordinates on the waterlines by 3rd degree interpolation

no,-ordinates finished

no

calculate the developed frame length for all the ordinates until the subsequent waterlines

e

waterplane integration routine calculate

waterplane area

moment of waterplane about midship section

transverse moment of inertia longitudinal moment of inertia

wetted surface

oWt.p

-20-waterplanes finished es

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flO/.

.

çf1n1shed until the subsequent waterlines

rio

no,

ordinates finished

20

ordinate integration routine calculate:

ordinate area

moment of ordinate about waterlines

es

volume integration routine calculate;

displaced volume V

moment of displaced volume about basic-line moment of displaced volume about midship section

calculate:

displaced weight i

centre of buoyancy above moulded base KB

longitudinal centre of buoyancy from amidship section metacenter above moulded base 1ZÏ

longitudinal metacenter above moulded base

idi,

finished until all subsequent waterplanes'es

calculate:

waterplane coefficient

midship section coefficient CM block coefficient CB

longitudinal prismatic coefficient C

vertical prismatic coefficient Ci,.,,

nut out results

21

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APPENDIX 2 FLOW-DIAGRAM STABILITY PROGRAMME.

no no

read parameters

read measured points of the ordinates

calculate required

calculate half breadth of the ordinate on the waterlines by 3rd degree interpolation y[r,q]

ordinates finished

calculate: yy(q], mtql

waterlines finished es

ordinate integration routine calculate:

ordinate area until waterplane(ii moment of ordinate about center line moment of ordinate about basic line

\yes deck edge Lmmersedj

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(w&s 6970)

22

noi.

is there any

camberes

calculate camber contribution

no,

Çordinates

finishedyes

volume integration routine calculate:

displaced volume

moment of displaced volume about basic-line

moment of displaced volume about center line

Calculate: BY [ tt,ii) BZ [tt,ii] KN (tt,ii] KN sin(tt,iij no i Ç'terplanes finished) no i -Ç angles finished)?Tes