Sailboat test technique

24 JLJU 1978

ARCHIEF

DAVIDSON LABORATORY CASTLE POINT STATION

HOSOKEN. NEW JERSEY

Sailboat Test Technique

by

Paul G, Sperm

December 1958 _{Technical Memorandum No. l2Lf}

(Revised April 1959) (Reviséd April 1960) (Revised October 1966)

Lab.

y.

STEVENS INSTJTUTE OF TECHNOLOGY

Technisc

_{e Hogeschool}

INIR0DUCiION

1. The purpose of this note is to describe in some detail the procedure used at the Davidson Laboratory in tank testing sailboats in the heeled

condition,

and to explain the method by which the speed made good to windward, etc. is calculated from test results.

Most of the material in the present note, and xmich additional information, also appears in earlier D.L. publications, notably:

T,M. 10 which describes the full size sailing trials of Gimcrack T.M. 16 which gives test results for the

_{model of Gimcrack, and}

sail coefficients derived from the model and full size tests T.M. 17 which further discusses the sail coefficients

T0M. 108 which gives a complete set of calculations for heeled

and upright tests of a yacht

Note _{which gives Schoenherr friction coefficients, sail} coefficients, solutions or velocity

triangles

etc. in convenient forTs for use in calculation.

Notation

4

The

_{oursc refers to the line along which the vese1 moves.}

X = leeway ang1e, between course and ship s head

anglo between ship s head and apparent wind

_(VA)

(p

X)

angle between conree and apparent wind (VA)

6

angle between shipvs head and truo wind

_(VT)

C -

angle between ship' s head and resultant wind force

on the sails

(all the above angles are measured in the horaontal plane)

¿

- angle between ship's head and resultant wind force on the seils

(measured in plane perpendicu1r to mact, (neglet1ng mast ral4

s heel anglo

Sail Forces

The sail coefficients are defined by the relationships

= KR SA. VA2

o SA. VA2

FR

Driving Force (lbs) parallel to course

FH

Heeling Force (lbs)

FH 0.05

Horizontal ccponant of Heeling 1oro acting at

right angles to couro

sin

Vertical component of Heeling Force

apparent wind (knots)

SA

Sail area (squimre feet)

-

*

Sail AÑa

3

/

of Fore TriAangle

₍

+100

0,

of

.tinsail

_J

+100 /o of

_J

exclUding Roach

(-*

in the sail-area notation used. in Rating Rules

b xp

).

_{The Centre of Effort (CeE.) is taken as the centre of}

_{the foUdng}

areas

1700/O

_{of Foretx-iangle at Centre of Area of Foro Triangle}

100°/o of

ainsai1 at its Centre of Area

Balance predic*ions Lising this C .E. are bnd to be in reasonablz agreeìt with

hein angles observed on fuilsize yachts.

Confirmation is provided by Warners

pressure measurements on sails(Trns. SNANE l92) which showed pressures

on jib about double those on mainsail.

.

_{The total effect of wind on sails and hull is taken}

_{as the reau.Ltant}

of

F

FH cos Q

and

sin Q

acting through the C .E.

peeled Test!

- Applied Force System

,

The rol1owin

syst

oÍ» forces

_{nd couples is applied to the model}

when it ïs towed in the tanks

a.

Resistance

_(rt)

This force, acting f oriard alo

_{the course}

_{is applied at}

deck level near the bow1

_{by the tow-bar, which is adjusted}

to be horizontal and parallel

_{to the course0}

h.

Lateral Porco

_{(FH cos}

_{or F)}

Two horizontal forces, at right angles to the course,

are

applied within the hull,

near the W.L, one near each end of the nadel.

The dynamometers

_{are arranged with a simple mechanical}

linkage, so that

_{a aingle moveable weight balances these}

two forces.

_{Trie value of this wc.ght is ecual to the sum}

of the two force3,

_i.e

,

to the total transverse

_horizontal

force

_('FR

_{Q) .}

_{The position of the weight indicates}

where

cos

_{icts i.e0. the C.L0R. position}

i'.. cos Q

is, of course, oqud

_{to the horizontal}

_{component of the sail}

fcrce0

C.

_{Vertical Fo}

_{e - Added Weiht}

(F

sin Q )

_{(See Figure 1)}

Weights are added within

_{the hull at the mid section to}

repre-sont th

_{vertical component of}

_{the sail force}

The Resultant

of i-. cos Q

and

F

sin Q

_{is the heeling force}

_F..

*

_Tt

should be noted that the C.L.R. position referred to

_{in this}

note has a dffferent

_{meaning from the C.L.R.}

_{discussed in tcxts}

Sliding Weight Shift (S) (See Figure 1)

There is a sliding weight of two lb., which is adjusted

transversely to

get the desired heel angle. The sliding weight is initially Bet

ifl the position which gives zero

heel angle

at

rest and the sliding weight shift (3) is measured from this position to the position

_{which gives}

the desired heel angle in any particular running conditioñ.

Rig Shift (See

Figure 1)

The sliding weight and its mounting (total _{weight 2.LO lb.)} can also be moved fore and aft. It is

_{moved forward from}

the position which gives correct trim at rest at zero heel angle by a distance _{known as the 'Rig Shift' which providea}

a couple of appro]cLmately

_{the value necessary to allow for}

the fact that R _{is aDpl'ied at deck level, not at the Centre}

of Effort of the rig.

On the assumption that slight variationz in fore and aft trim are not 1mportant, _{standard values of rig shift}

are used,, which are selected according to the type of model (Racing _or Cnising) and the heel _angle.

The correct rig shift, of course,

depends also on the model

_{resistance and}

_{height of C.F. The}

¿ justification for using _{a fixed rig shift l'or a given heel angle}

¿

is that the models are all of about the saine size, _{and therefore}

have roughly

_{the same resistance at the}

speed corresponding to a given heel angle.

If it were ever desired _{to have more exactly the correct trim,} it would not be difficult to move a ballast _weight fore-and-aft _{to provide a coutle closely}

approximating the correct value. _{The correct value for this couple is such that}

as applied, combined with this

_{couple, is equivaìent to}

acting at the C .E. _{combined with}

R_R8m applied at the level

where

the excess akin friction _{acts (i.e. somewhere}

below W.L.)

(R

_{Fiiflsize Resistance}

_x

Node_

_{DisPlacement)}

8m

bhip Displacement

However, it must not _{be forgotten that}

there is no exact data as to the height of C.E.,

for real sails.

i

i':-1214

-h-Heeled Tests - Tank Procedure

The sailboat test technique

used at D.L. is based on the fact that

the model can be run down trie tank

every two ninutes or leas; (ìt is considered desirable to maintain a two minute interval between rune to

avoid changes in turbulence).

Therefore, several runs are ide anu the

apparatus is adjusted eetween runs unti! the mode], runs in the desired condition.

Before running, suitable values of Heel Angle (Q , speed

(Vm) and Lateral

Force

_(FN

cosQ)

are

_{selected, and Rig Shift and Added Weight} are adjusted to the appropriate values. _{Then runs are made}

_{varying the}

Sliding weight shift, CLR position, and Leeway angle _{) , until the} model runs at the desired heel angle with the desired lateral force.

For the condition so obtained the following _{are recorded:} Q

,

F cosQ

, CLR position, Sliding weight shift, X

, and

At each speed it is usual to run with three different lateral _forces. At each heel angle, it is usual to run three _speeds.

For a

coiiplete test three heel angles, 100, 200 and 300 _{are usually}

run.

_{This involves 9 points at each}

_{heel angle and 27 points in all.}

Sometimes it is found that extra points may be required; sometimes points can be omitted. _{The object is to cover the}

_{range in}

_{which the}

best speed made good to windward (V)

_{will lie.}

_{Ecperience gives a}

very good

guide to the

_{values of}

Vm

which should be used.

It will be noted that running at chosen _{values of Heel Angie ()} and Lateral Force _{(FN coQ) makes it possible to fix appropriate valuez} of added weight (FR sinQ) .

_{Furthermore, the subsequent calculations are}

very much simplified by use of fixed values _el'

_iieel

_{Angle. The equipnent} is so designed that tests may aleo conveniently be made at heel angles _of

O

150, 2S°, _{and 3°.}

T14-12

-heeled Tests - Calculation Prcedurt

9

In testing,

_{FF coso}

and

sinQ

_{are applied within the bulle.}

whereas for the ship

_{FH co5}

_{and FH sins are applied at the}

_Centre

of Effort of the eail plan.

Furthermore, the C.G. of the model is not

_{in 'eneral}

_{in the sarx}

position as the C.G. of the ship.

However, in the model test a heeling couple is

_{applIed y the shift}

of the slIding weight, which is adjusted until the model

_{rune at the selec}

-ted heel angle

lO

_{Given the heel angle}

_{(Q) ,}

_{the C .G. positions of moael and}

ship, and

the G.E0 posItion for the ship,

_{it is not difficult to calculate,}

_{for any}

value of

F

_coser,

_{what couple nrì$t be combined}

_{with all the transverse}

gravity and dyràmoreter forces

_{acting on the model to give}

_{a resultant in}

the seme line as the resultant of the gravity and sail

_{rorces applied to}

the ship. In the teSt c couple is provided

_{by raovin}

_{the sliding weight}

and we thus arrive at a relationship between sliding

_{weight shift () and}

cosG

_{which must be satìsf led}

_{if the model forces}

re to have the s.

line of actn as the

ship f orces.

_{The plot 01 this relationship}

_{is a}

straight line and is

_{referrod to asthe stability}

_{line (or stability curve)}

We aleo have values of

COSQ

_and

_{obtained in the test}

_runs

and these are plotted

_{on the san* a}

_{as tbe stability lino}

Then

the point of interscction

_{gives the value of}

FtT Co5

_{with which the ship}

would rim at

_{the selected heel}

_angle.

_{This is referred}

aa the

equi-libriun value,

In theory, different curves relating the values of

_{FH cos}

_and

S

_{observed in tests at}

a given heel angle might be expected Lcr

different speeds.

_{In fact, the difference is not usually not}

ieekble for

the snail

_{¡'ange of speed ordinarily investigated, end}

so a single value of

FE cos

is ordinarily obtained

_{for each heel angle}

T '-l?1

.6

-TM-l2I

IL The values of ail

the other quantitiee observed aro plotted against FR cosQ for each speed and heel angle The equilibrium value of FR having boon dterminod as above, the corrosponding values of the other

quantities are piked off from the curves for uso in subsequent calculations. This plotting process tonds to draw attention to any readings which aro

inconsistant with the others, and oy be in error.

12 It dl1 be noted that the equilibrium condition obtained as above, involvo the righting couple provided by

the hufl in the actual running

condition. Any chnngo in stability due to wivaiaking, etc. is thus auto-matically taken into account.

It may also be noted that if it is desired to dotermino the effect on perforrrrìce of a snail change in stability, due to alterationc to the bal last or sail plan for example, it is necessary to redraw the stability line and repeat subsequent calculations but not to do any additional toet-ing, provided that the intersection still lies within the range of FR co tested, which cìunonly is the case.

13. The equilft'iuxa raluee of FR co and rust nr be scaled up

to full sie

FR cos is assumed to be unaffected by rernolds Number, c, and is therefore scaled up in the ratio of the disp].acoraente of ship and model. must be corrected for Peynolds Number and for drag of the sand-strip used to stimulate turbulence.

The sandetrip extends from the f ortmurd end of the ter1ino to the point at which the bottc of the keel bocomo approximately horionta1.

Heeled tests are run with sandstrip 1" wide (i.e. 1/2" on each ido of centreline), Upright tests are run with 1/2" and 1" sandstrips. It is acsumed that the added resistance dde to the sandetrip is proportional to ita width and that the added resisthnco is the carne for heeled and up-right tests. Furtheraoro, it is found that tho incroase of resist&nco coefficient resulting from increasing

sandstrip froro l/2

to 1" is to a

first approximation independent of speed. An average ovar the spood rango

is thoref oro takon.. The ìncretise

of resistance coefficient

due to a

1" sand-strip is do±lo the difference 'cetween 1" and 1/2" and is of the order of ,COO.

After tiis correction has been applied the ndal resistance is exnanded _{full iize using the li,e. Owing to the usually}

i e.

a -

-.i 'case

_{(3 ô4k}

CLÂht.

j

frbJ

c "

'

6

This idea behind the above assumption is roughly that the effect Of trimming the sails is approximately the same as the effect of rotating the whole rig about the mast so that its angle to the apparent wind re-mains the same as it was in

the Gimrack

sailing tests.

On this basis, for any heel angle, a range of values of KR and cosQ maybe calculated corresponding to d1?'ferent values of + X These calculations and results appear in TM 17,

and

Figure 17-3 prèvides the data in a convenient form for use. Figure 17-3 is reproduced as Figure 2 of this note.

It will be noted that in this fi: e an ä.ngle

defining

the trim of the sail ( ô) , appears as the independent variable. However, this variable does not appear elsewhere in the usual calculations and may,: for most purposes, be disregarded.

The

conditions of

the original Gimòrack, sailing trials are represented

by ô-2O<

and the vues of KR , and KR/KS cosQ for ô 200 are the

same as the values obtained in the trials. Th Values of ( + ?..) given

for ô - 200 are 10 larger than the values estimated for Gimcrack, since it was considered that the values for Gimcrack were rather lower than would apply general1y.

TM-121

'f

cut-away profile shape of sailing yachts, a length equal to 0.7 LWL is used in calculating Reynolds Niunber for

model and ship.

llj. Comparison of ests of Gimcrack sailed by a good helmsman with tank

results for

her model gave the values

for KR K , , and X for ¡2' Gimcrack sailing at any value of heel angle () ,

However, it is not necessarily true that the value of at which Gimcrack sailed is optimum for all boats To overcome this difficulty the assumptions are made that, for a given

heel

angle (Q) , if is changed by ¿$3 , (with a suitable adjustment of sail trin)

a) The resultant of KR and K1 is unchanged,

& b) The angle whìch this resultant

makes

with the centre-plane of the yacht changes by

1.

Values of

R arAd F cooG,

for

era3. speeds at taL heel aïg1e

being asiuilabls frcì Uia ìiio&1 testh

Ui closehaL1d p rfor.n

is

calculated as

follows;-Rs K

&

The ratio

,

i

used to ontor

Figure 17-3 and corresponding values of

and

aro read off,

b.

V,

is obtained froe the oquation

R,

o

The velocity triangles now give true wind volootty

_(VT),

Angle of true ind to ehtpe head (')

Speed made good to windward

Vì,g

The above calculations are nude for several speeds at each heel

angle and the reult are plotted as

Vm

(ordirate) agint

(abscissa)

The

nvelope touching those

curvos gives the opthrium

ele eebau.lsd perfonnc e

o

50

+

-Vrv

Sorietios ari extra speed znay bava to be run to obte.in the part of

the fixed

Q

curvo '$iich touches the envelo, but oporionco gives a

ery good guide to the poods

id.ch should be run at each. heel anglo0

16,

L _{icH cosQ}

does not change iuch whon

(j3 + X)

is vur'ied, tho

rition of

for a fid hoel ançle is

ul1 within the rance

con-adored.

Thus it is true that the envelope touches the fxod Q

curves

va'y nearly at the na. value of V, obtained at fixed

G

This fact is oorncti os used as a short cut to save p1ottin without

appreciably affecting the rosults

(This is dono in T

105)

TM-1

Tri- 1214

- 10

-For routine computing certain charts are used to save labour. These give:

KR and ( + A) for all values of KR/KR cos within the

range used, for = 0, 5, 10, 20 and 30 degrees

Vmg/Vs and VT/vs for all values of VA/VS and ( + A)

within the usual range. (These charts are ¡n Note 514.)

An estimate of weather or lee helm is obtained from _{a graphical plot} based on the hydrodynarnic forces determined from the model tests and the sail force acting at the CE defined ¡n section 14 (p.1. above). For this procedure see TM 108. _{The balance predictions so obtained are ¡n} reason-able agreement with helm angles observed on fulls ize yachts.

ADDENDUM TO TM 12k

October 1966

Numerical Calculation of Sailboat Performance

There is an alternative to the graphical procedure for performance calculations which is particularly suitable for use with a digital computer and may be convenient for other purposes.

It follows from section 114, and ¡s implicit in DL TM 17, Appendix A, that for fixed heel angle

JKR3 + KH3 = constant = KT and

+ A - tan1 KR/KH = constant = E , say

The Gimcrack sail coefficients can be concisely stated by the following tabulation of K.1. and E , taken from TM 17:

r

R

H

,V

Values interpolated from smooth curves

In practice it was found that the graphical presentation of the sail coefficients given in TM 17, Fig. 17-3, was too cumbersome for routine use, and the information was re-worked into a nomograrn given in DL Note 54. Values of + À , K., and Rs/FN Gos e were taken from Note 54 and worked backwards to give values of

Kr and E The latter

agree exactly with the above tabulation. Thus the riomogram ¡n Note 5k and the above table of

Kr and E are exactly equivalent and the use

of either the nomogram (which has been in routine use for years) or the above table will result ¡n identical performance prediction.

The model tests give values of _Rs and FN cos e for several speeds at each heel angle, where of course _{FN =} (FN cos 8) sec 8

For the given heel angle, K1. and E are obtained from the above table. The relative wind angle is found from

' À = E + tan1 RS/FN

and the apparent wind speed, _VA , from

Kr SA. VAa = \J R _{+ FH} 1M-124 I I -e , deg Kr X 10 _E , deg 0 6.23 10.16 5*

₅₁₄

9.70 10 4.77 9.29 15* _4.20 9.15 20 3.70 9.30

25*

_{3.24 9.80} 30 2.80 _10.67 35* 2.40 12.00

Knowing + A. _{, VA} and V from model tests, the wind speed, VT , wind angle, 'y , and speed made good, _VMG , may be found as follows

VAcos(P+A)_Vs

VA sin (ø + À)

tan' b/e

b cosec y

V5 cosy

A Fortran computer program based on this procedure has been developed at the Davidson Laboratory. The program will either take model values and expand them to full-size or accept full-size values of R5

, FH cos 9

and V and compute S + X

, VA , VMG , V1 y , etc. This program

also has provision for computirrg the balance predictions mentioned ¡n section 18.

The sanie method has been used, with the aid of a digital computer, to prepare a table, for heel angles of 10°, 20° and 30°, in which

VT/VS , S + X and y are tabulated as funGt ions of two

arguments:

cos O/R and R5/V52 . SA

The table covers the range of the arguments usually encountered, and a fairly simple double interpolation gives _VMG , VT , etc., for the

actual values of the argument obtained from the tests.

TM- 124 - 12 -(see TM 108, P. 44): e ₌ b ₌ y = VT = VMG =

Sling Woi)it

xiol at Rost

Trimsd to O - 0°

Mded voight, applied at midsection,

aecouiit& for

daiwnrcI coiiiponnt of

rig forces.

LIi)I

;rEIGHT 3IiIFT

liing Woiht

r'

Shift, in.

s,'

Added Weight

!odol Underiay

Tinnd to required heel angle by

shift of sUding weight arid by

Lyth'od,yiiainic Lift of koel.

As sail force, increace, entire

sliding uiht ri

iust be

hií'ted foririì to account for

the forrard rotational

omont

of tí sail forcuso The rig is

shifted tho distanccs given in

t.0 tai)lC bilow.

of extrcme

weight

FJ.g Shifts For

O

Racin- Liol

Cruising ?dels

k

Due to the lot: stability

racing models, the slU.ing

-\

0,,,

j

Lu..

o.6l

l.&DT

_l.O2

)

_'['tI

4. '-.?

h-I

i

7"

ri

is zlung

e1ow decks in these

casurencnts arc rac from O - 0° position,

classes,

in cruising models the

cliWLnig weight is above deck8.

--- ---h

. .E I

I

s . P..' -t i . _,

-.

, çi

r

%,._. I Ai (J .j: C. O 'Ñ Q) (::

-"u

M : 1 ;

.1-I ' T t Il. i'

I:J.:

' :'

i::L-;3

,À )

:\

: I I 4]

1 L.'

i;î

:. .. -I -t I-: I t .; t i_ _1_

:::ri

_i.J

'oLj

: . .

cLI 3èinJ

-' , . : ' -1zì:.wL. I I :

:;

-. ;

-LÇ05

V _ ; i j 1-(i :! ---i -

:-Li;s-ji:,

tIjiÌ

I

Prediction of Ship Resistance from Model

The total resistance of Ship (R) and model (r) _{is made up of}

two parts, one due to friction, the other due mainly to wavemaking but including also various smaller effects _{such as eddy-making. The former} is caLled the skin friction resistance, _(rf , Rf) _{or frictional} re-sistance; the latter the residual resistance. (r _{R )}

r r

i.e., or the model _{r = r + r}

For the ship

_{P=R +R}

f r

Ft can be shown that a model tested at speed y _{will have the}

Rp.

same wave pattern as a ship tested at y _{and the value of will}

then he the same for model and ship.

or in other words, if the value of -y--- _{is the sanie for both model and}

Rr

ship, then -s-- _{is the same for both.}

Having calculated the model frictional _{resistance, (rf)}

, as

overleaf, we subtract this from the _{total resistance measured in the} tank (r) _{to get the model residual resistance (r).}

The ship re-sidual resistance is ohtaned by scaling _{up the model resistance in} pro-portion to the ratio of the displacements _{of ship and model. To this is} added the calculated frictional _{resistance (Rf) to arrive at the ship} resistance

The ship resistance so obtained is for _{a ship speed equal to the} model speed multiplied by the _{square root of the scale ratio.}

TM -124 15 -Thus ff we have Ship Speed y = [Scale Ratio Model Speed R r

_Aship

A model r

where

The skin friction resistances may be expressed as:

and 1 2 r

C .p y

A f fm 2 m m 1 2 R

C .p V

A f fs 2 s s

where and _{are skin friction coefficients appropriate to} model and ship

density of water in test tank (fresh)

p _{- density of water around ship (usually salt)}

A and A are wetted areas of model and ship

n s

Re P =

p = density of the fluid = viscosity of the fluid v= 1t/p = kthematic viscosity

for model

for ship

Skin friction resistance is not measured in a tank test; it is computed from tabulated skin friction coefficients, as follows:

lE the flow is turbulent, which may be ensured by _{sand strips on} the model, and is more-or-less inevitable for a hull of the size and speed of the fuilsize yacht, there is a known relationship _between and the Reynolds Number.

X _{is the length of the wetted surface in the direction}

of flow. or normal displacement vessels X _{is usually taken as equal to the LWL.,}

From the tabulated density and viscosity of fresh and salt water, the Reynolds numbers for model and ship at each speed tested are readily

The Reynolds number used in skin friction calculations _{is defined as}

but for sailboats X is taken as 0.7 x LWL, owing to their cut-away form.

calculated. _{Then the values of for model and ship are obtained from} tables of Schoenherr or other friction coefficients. (c is rather sensitive to changes of water temperature.)

A convenient procedure for carrying out the calculations is as follows:

Measure model resistance r at speed y

Evaluate C

tm

where C

=r.x

2

tm 2

Apv

_mm

(If a sandstrip is used to stimulate turbulence _Ct is reduced by a small amount (about .0005) to allow for the extra resistance of the sandstrip.)

Determine Reynolds number for model at y _{and ship at} V (R and R ₎

em es

Look up C and C for values of Reynolds number R and R

fm fs _{em es} since

C =C

-C

f fm fs C C

-C

ts tm f A

SS

Resistance of Ship = R = C x ts 2

This resistance is for ship at speed V v

Repeat for other speeds.

This calculation isidentical in principle to that described earlier,

and therefore A 2 V L s L and

APV2

ship s s L - 2 model A p y

mm

TM-124 -

17

-DA(IT.A14 5'HÖW(, FÖR(ES

CON4T?uTIN& TO SAWBOT SrALJT>1

)N H1iE

_C(jNtflTON

G

\

j//

V

ws

f

(

')t_._)_/ .1

WI-\

UTÂY VTE

MCrEL OR SHIP

Ncte.

r+tly Hì

v;ûceI VCG-

c± G

'v

'Hic uprrqhT c

d+«v, with

1i

s!ict,

u..r

cs shcu.'r.

ur

s

ìY

S

p(,+(fl

r'

±;

_{VC& fo Cr}

kflCI COUSIYK1 c

hccrI

ftc. Yvìt9

df'br

TM-12t

-TM-i 24 A-1

APPENDIX

STABILITY CALCULATIONS

Appendix A has

been added to provide additional information as to the derivatives of the "Stability Linest' outlined on pages 6 and 7, paragraphs 9-12

of this

technical memorandum.

Bu t

FHx2 W(b sìnQ + c cosQ)

2.00 x S

C

STABILITY CALCULATIONS

Refer to the figure on page 16 which shows the geometry of the ship reduced to model size, and that of the model itself. Because the over-turning moment of the model is less than that of the ship by the amount F1xa , the righting moment nst he correspondingly reduced by shifting

G laterally. Thus, to produce the sanie static equilibrium,

where all factors are for model size. Rearranging this expression,

FHcosQ = (b sin9 cos9 + c cos2Q)

where 2.00 = sliding weight

S = sliding weight shift

Wb . 2.00 S cos79

FHCOS

= - smn9 csQ +

_{a a}

This equation establishes the necessary relation between _FHcosQ and S

to make the model forces equivalent to the ship forces. The experimental data establish the actual relation between _FHCOS and S for the model

in question, as governed by its form stability.

The intersection

of

the two straight lines which represent these two relationships gives the result sought for, namely, a value

of

_FHCOS consistent both with the actual stability of the form and with the

dimen-sions

of

the ship.

In practice, the lines representing the two relationships lie approxi mately at right angles. Thus, their intersection can be determined graphi cally with considerable accuracy.

TM-121+

STABILITY CALCULATIONS

Rearranging terms,

APPENDIX B (March 1968)

Refer to the figure on page B2 which shows the geometry of the ship reduced to model size, and that of the model itself. Moments are taken about the intersection of LWL and centerline plane of the model.

Heeling Moment Righting Moment CE x H secp

K +(G-G) Am smp

where all factors are for model size and

G -G _{is distance between model vertical center of gravity and}

s m

ship center of gravity location

K ¡s moment (inch-lb) measured by the test apparatus and represents the righting moment due to the water forces.

(G-G)

K + s m sirip H CE secp CE m seccp (G -G ) A sinp cosp

Kcosp

s m m H CE + CE

This equation establishes the necessary relation between H and K to make the model forces equivalent to the ship forces. The experi-mental data establish the actual relation between H and K for the model ¡n question, as governed by its form stability.

The intersection of the two straight lines which represent those two relationships gives the result sought for, namely, a value of H

consistent both with the actual stability of the form and with the di-mensions of the ship.

In practice, the lines representing the two relationships lie

approximately at right angles. Thus, their intersection can be determined graphically with considerable accuracy.

TM-

DIAGRAM SHOWING FORCES

CONTRIBUTING TO SAILBT STABILITY IN HEELED CONDITION A m WATER FORCE TM-124 32