Preliminary Note on the Measurement of Length

PRELIMINARY NOTE ON THE MEASUREMENT OF LENGTH

by

J. E. Kerwin and J. N. Newman April 1976

H. Irving Pratt

Ocean Race Handicapping Project

Report No. 76-2

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Report No. 76-2

PRELIMINARY NOTE ON THE MEASUREMENT OF LENGTH

by

J. E. Kerwin and J. N. Newman April 1976

(Prepared for the 6 May 1976 Meeting of

the

SYRTJ Handicap

Rule Contittee)

This resé arch was carried out under the. H. Irving Pratt Ocean Race

Handicapping

Project, MIT OSP

Project No.

81535.

The generoüS support of the

div-idual

donors to

this program

is gratefully acknowledged.

Introduction

To handicap yachts one must consider both the effect of size and of shape For geometrically similar yachts hull.

size can be characterized equally well by the waterline length, beam, the square-root of the wetted surface, the cube root

of the displacement, etc However, size is much more difficult to define if yachts are nt geometrically similar. In this case, size has meaning only if we define precisely the charac-teristic which we are intending to measure. In this regard, a rating rule may be thought of as a weighted average of Several suitably chosen meaSureS of size. r

Length is intended as a measure of the effeçtive hull length for wave-making purposes, predominantly at high speeds Consider two hulls of

denti.cal

displacement, with resistance curves as Sketched below

Speed

Clearly Hull 2 has a longer effective length han Hull 1. We define Hull i as the "base boat", and plot its resistance curve on the basis of speed-length ratio

V/ff

set equal to its length of Hull 2 speed portion of as possible with

waterline length. 'We now define the effective as that length which will result in the high-itS resistance curve coinciding as closely

that f Hull 1.

The resulting effective length for Hull 2 is not necessarily equal to its actual waterline length One would expect, for example, that if Hull 2 is relatively fine-ended compared

with Hull 1, then its effective length will be correspondingly less than its LWL. In any event, the effective length defined in this way would permit a fair evaluation of the performance of Hull 2 relative to Hull i in the regions of high speeds.

MoSt rating rules contain measurements intended to estimate this effective length Inevitably, hull shapes have evolved in which the ratio of "true effective length" to "rule effective length" has been maximized. This has resulted in characteristic hull ends associated with the particular measurement rule in force at the tie of design.

The objective of this note is to explore means ò.f deter-mining effective length in as fundarnental a way as possible.

The wavemaking characteristics of a hull are an integrated phenomenon, and the same should be true o-f the ienqth. In

particular, it should not depend significantly on local point measurements such as the LWL. (-Fastening a long needle

length of te needle, but will not change the wave resistance.) he ogt important characteristic öf the hull, from the standpoint of wave resistance, is the sectional-area curve

It seems logica], to base the length on this curve and the properties thereòf.

The effective length changes with boat speed and heel

angle. Generally boats with long overhangs, arid to a lesser

extent beamy overhangs, will gain effective length under

these

conditions

_Thus

one must try to estimate the sectional

area curve not only for the static case, but also for assumed sailing conditions. Conversely, portions of the hull, which are above the water under normal

conditions

should not affect thé length.

With these guiding principles, it sééms reasonable tò divide the overall problem in two pieces One is the optimum way of determining length in terms of a given sectional-area

curve. The second is the estimation of the relevant

sectional-area curve.

Detenination. of frOm S

It may be well to begifl with some definitions. .he

coor-dinate x measures fore-and-aft distance along the hull, and

S(x) is the sectional area at the position x, below some

prescribed datum waterline Typically S(x) will be defined by a curvé of the general form:

'C

The total submerged volumé V iS the area undér thiS curve, or the integral

v=fS(x)dx

The fistmoment of the sectioPal-area curve is dSf-ïed by the intégral.

3--I

=

f X

S(x) dx

and the second-moment or moment-of-inertia by

12=fx2S(x)

dx

The:

longitudinal

position of the center of buoyancy, LCB,

is determined by the ratio

The (longitudinal) prismatic the; ratià of the area under scribed rectangle. Thus, if

secional aréa,

V

Sma X (LWL)

Typically, fOr modern, ocean racers, c will be between .O..5,5

and 0.6 , based on the canoe body..

The simplest definition of the length of the sectiönal-area curve is its waximum longitudinal length, i. e the

waterline length LI'L. There are some exceptions where the LWL is less than the overall length of S(x) , i.e., a submerged

bow: bulb or extended submerged rudder. In all such cases' it

seems preferable to use the overall submerged length, rather

than the LWL. To emphasize this we replace LWL by LWS

hereafter, to denote "length on waterline or submerged"

LWS is a point measurement, with the well known disadvantages

thereof. In particular, adding the above-mentioned needle

to the bow will extend the LWS by the length of the needle, but will have no significant effect on the effective hull lenth.

There appear to be two basically different types of length which warrant our attention here. One is a replacement for

the LWS , 'which measures the

effct-ve

overall length of the

submerged portion of the hull We shall call this "Lg"

A second type of length is one which makes allowance for the overall distribution of submerged volume, ascribing to a high-prismatic hull a longer length and vice

versa

We shall call

this "Lv" At this stage it is not possible to

say which

of these is more important, although one suspects the importance

of L will increase as the speed-length ratio increases, for

a given hull. For the time, being we assume that both L9 'and

L are relevant parameters., to be averaged in. sorné optimum

f

4

coefficient c is defined as

S to the areaof the circum S is the maximum croSs

manner depending on the Deift tests with MQdels 8 and 9. (If desired, this averaging can be varied on the basis of

anticipated Speed-length ratios for a given raöe. For example, one might prefer to give more weight to L in the Trans-Pac,

by comparison to the Bermuda Race.)

Before proceeding to discuss L and Separately, it may be noted that some preliminary studies of their relative importance can be made on the basis of Taylor's Standard Series, as well, as wave-resistance theory as this has applied to ships

The

conclusions

suggest that at a moderate speed-length ratio

on the order of i O to 1.2 , L is of

dominant

importance.

There is a simple explanation for this, the prismatic coef fi-oient can be optimized for any speed-length ratio, and at the speed-length ratios of 1.0 to 1.2 , a prismatic coefficient on the order of 0.55 to 0.6 iS generally optimum. Thus,

in this range, the curve of resistance vs. prismatic is stationary, and one should not penalize high prismatics We look forward

to the Deift tests as an opportunity to confirm this for ocean-racing yachts, and to provide data for higher speeds.

heEffectiveWaterline Length (Le)

The problem here s to measure thé effective òverall length of a sectional-area curve, so that this will not be

unduly sensitive to local irregularities at the ends, nor to measurement errors associated with measuring LWL in the usual manner For example, one would like to ascribe to the three following sectional-area curves equal (or only slightly

different) values of L5 ; (conventional) (needle-nose, extended skeg òr rudder) (snub nose, transom stern)

Moreover, one wishes to do this in such a way that the

result is not sensitive to measurement inaccuracies or small variations in the static waterplane

Our present ideas for achieving this goal are to fit, to the sectional-area curve, a suitable

faired

curve which is substantially identical to S(x) in an integrated sense, but insensitive to local irregtiarities at the ends. Having done this, the maximum overall length of the faired curve can be ascertained. As an initial effort we have done this with a f ifth-order polynomial which is fit to the sectional-area curve in a least-squares sense.

Our first attempt in this direction is shown in Figure 1, for the base boat of the Deift Series (Model 1). Here the curve with crosses C _{I j} ) is the actual sectional-area

curve, and the plain cúrve is the fifth-Order polynomial, fit to minimize the RMS error. The. proposed length L can be determined from the zero-crossings of the polynomial, and ïs slightly less than the actual LWS due to the fairing of

the ends. However,, this polynomial fit is affected at the.

ends by the local inflection of the sectional-area curve, to

a degree which may be úxidesirable To avoid this we have

therefore adopted a weighting of the least-squares fit, pro-portional to the magnitude of the product S(dS/dx) . This product is zero at the ends as well as at and a

maximum over the steep porti.ons of the sectiònal-area curve. The result of this procedure is shown in Figure 2 for Model 1, and the modified treatment of the

ends

is apparent. Figures 3 and 4 show the corresponding results with the weighted polynomial for beift Model 8 (wide ends, hiqh prismatic) and Model 9

(narrOw eñds, low prismatic). The values of LS determined

f rout the weighted polynomials are as follows:

A minor disappOintment here is that the full hull (8) has a smaller value of Lg , and vice versa Moreover, our initial

attempts to use the same procedure with less conventional sectional-area curves has led to disaster For both reasons, further refinement of the curve-fitting process appears warranted

LWL

Model i 33.08 30.29

Model 8 33.08 30.21

The Meásurernent of _LV

Hee the

_{bjetive is to còrect the length for the}

distri-bution of sectional area along the hull length, i e to

discrjjn-mate f mne vs full-ended hull forms. _{The simplest way to} do this is by multiplying the length times the prismatic coefficient, and dividing by a suitable base. Prismatic, thus

(c) (LWL)

p--.

-v _0.57

O57Smax

This _{egth is effectively the ratio of the vo1e to the} maximum sectional area, normalized by a suitable constant so as to approximate the LWL for

conventional

boats. Since

V and _{are integrated quantities, they cannot be altered} by local deformations of the hull surface. _{However this}

formula might encourage large maximum _{cross-sectional area} of the hull, in other _{words the following could prove to} be arule-beater: .. - . .. -. ...-. ... .

As a possible alternative, which is less sensitive to

_Smax

the following length can be defined, based on the second

moment and analogous to thé radius of gyration:

142)

₌

_{5(Iz/V_xcB2)V2}

The Second-moment rather heavy weights sectional area

near the ends, and thus

Ljj( 2)

_{also may encourage design}

trends toward low prismatic coefficients, but one cannot

decrease this length simply by increasing the cross-sectional

area locally as with L(').

A deficiency of

L(2) is that

it is dependent weakly on the asymmetry of the sectional-area

curve, in particular,

Lv(2)

will be less for a symmetrical

sectional area curve than for one with more fullness aft

(This may be regarded by sOme as an ädvàntage.)

To test these two proposed length measurements, they can

be applied to simple mathematical sectional-area curves, as

well as to actual hull f ons.

The following table gives some

simple mathematical results, with widely differing prismatic

coefficients,:

I

sectional-area

prismatic

r4/Lwt4

z42)/LwI

rectangle

1.0

l..5

1.44

parabola

2/3

1.17

1.12

quartic (symmetric)

0.6

1.05

1.04

0.57

1.00

1.00

0.54

0.95

0.96

diamond (synimnetric)

0.5

0.88

1.02

diamond (x.05 LWL) 0.5

0.88.

1.04

To illustrate these length measurements, for actua±

hulls, the following table is for the same three Deift Models:

prismatic _Lv(1) L IOR "L"

The distinction between these two lengths is minimal, and both ascribe to the high prismatic hull a longer 1ength, and vice versa However, the penalty for a high prismatic coef

fi-cient is more for L(') by comparison t

Determination of' Waterplane Datum

The need is clear to penalize long overhangs, as well as extra length which accrues in the heeled condition The CCA 4% waterline is a simple example of the former, and the quarter-beam length is an example of early attempts to account

for heel Our approach here is to define several sailing

conditions, such as:

Static waterplane (beneaththeconventionaJ.

LWL);

A waterplane correspoflding to high-speed upright sailing, with a bow wave and

stern wave defined pragmatically, and the total submerged volume, maintained constant equal to the static

Y;

and

A similar but inclined datum, say with 20 degree heel, and reduced bòw and stern waves.

A total of three waterplanes, three sectionâl area curves, and three corresponding lengths will result, to be averaged into a single value of "L" . One can debate the weighting

of these, as well as the need for more than three typical conditions.

Model I 0.573 32.54 32.27 35.54

Model 8 o. 595 33.69 32.99 36.71

To illustrate the anticipated results from these definitions of the waterpla.ne datum, Figures 5-13 show the sectiona1-ara curves of Deift Models 1, 8, and 9 for

heél angles of 0, 10, 20, and. 30 degrees These results

include a static waterplane (Figures 5-7), a small wave system (Figures 8-10) and a large wave system (Figures

11-13) The wave which has been chosen for these

cai.cu-lations has a crest at station 1 (10% aft of the forward end of the static waterp lane) and a second crest at station 12 (20% aft of the after end) This wave has been defined as a sine wave of linearly increasing amplitude. Far the small wave the bow and stern crests are 1% and 2% respec-tively of the LWS, and for the large wave these are doubled. For the static waterplane case each hull is balanced

hydro-statically at all heel angles, with the displaced volume and LCB maintained constant. In waves this pro.ved unrealistic, however, since a resulting severe trim by the stern would have resulted1 which is approximately offet by the bow-down moment exerted by the sails. Thus for cases of waves, shown in Figures 8-13, the procedure adopted was to maintain constant submerged volume, but t shift the LCB forward and preserve the static trim angle.

For the conditions illustrated in the 3bove figures, the corresponding values of

L(')

and Lvt2) _{are tabulated}

on the next page, along with the LWS value for each condi-tion Generally speaking, for these three hull shapes, heel angle decreases the LWS by one to two feet, with a smaller reduction in the values of the integrated lengths L. For the narrow-ended hull this change is larger, while for the wide-ended hull the values of L are insensitive to heel

angle Regarding the lengthening of the hull due to waves,

the LWS increases uniformly for all three hulls, typically by five to six feet over this range of wave heights. The integrated length LV(a) réf lects this change. but

_L(')

suffers, apparently because the prismatic coefficient is

Small in the largest. wave. To the extent that L should

increase in. waves by the same amount as LWS , the parameter

Lv(2) appears to be relatively satisfactory.

Static Waterplane LWS

(i)

2)

MCDEL i

SE) HEL= O

31. C0

2. 539

.32. 2'

MODEL t

(BASE) HEL

10

2.6 10

3:.

3L1 33

MODEL i

(BAS!) HEEL

20

.2. 250

3.2.306

7

MODEL 1 (BASE) TEEL 30

31.8 '40

31, g4'4 3 1.

MODEL 8

(WIDE) FEEt0

33. oea

ii . 6

CL

'2.

? i

tODEL ä (WZD!) HZX. TO

32.33 C

33.760

32. 9 i CDEL 8 (WIDE) EEL 20 32. Ó70 33. R61

12.

MODEL 8

(WTDE) HEEL = 30

31

31,

5 32! 20

MODEL 9(NA2ROi)

33.080

31,352

31.468

MOCEL 9

(NARROJ) iL

IC

32. 530

31.221

31.310

MODEL 9 (N?RPO4) HEEL 20

32.050

30.796

30.939

MODEL 9 NkRO.i)1!L= 30

30.

IsO

3.136

30.118

Small Wave

MODEL 1(A3E) HE!L0

16. 3)

34. '5!

_3. '.)17

MODEL 1

(BSE) HEEL= 10

35 .2

?..L

.785

3'4 1

MODEL 1

(B.SE)

REEL 20 3 5 03

l_. 55

34L, 735

MODEL I

(BASE) HSL= 3D

33 I0

34.21)

'.t, 23C

NODEL 8(WIDE) EE3L=0

35.0CC

097

35.

¶

MODEL 8(WIDE) HEt

10

1. 6Ö

36. 101

35,

i

MODEL 8 (WDE)

EL= 20

L0O

1 'J )

:351 16L

MODEL .8(W.ID!) HEEL= 30 3t, ¿iCO 15 , J'4') 3

MODEL 9(NARR0)

EL0

35.300

33.524

34. 105

MODEL 9 (NEOW)

!EEL= 10

3e.L&50 3 3 .3 83.

33. 913

MODEL 9 (NA0W) HZL= 20

.J.

32.959

33.. 400 ODEL 9 (N.ROW.)

¡{EL= 30

33.100

32 .3 06 3.2 32

Large Wave

OCEL 1(BPL5E) iEEt=0 37,

'4.0

72.611

. e,-'ç1

MODEL 1

(BASE) KEL= 10

7. 2..('

3..50

21. Q

MODgL i

(BASE) UEE.

20

7. 000

32

7.

MODEL i

(BAS!) HEEL= 30

'. 0c

2.

.7

MODEL 8(WtDE) HEELO

3 - .700

33., 3-32

MODEL S (WIDE) REEL 10

35. 300

LV 3 .

MODEL 8 (WIDE)

I!EL

20

17.910

3 .3 . 2

.OÒEL R (VIDE) HEE. 0

37.750

3 5L2.)

Li..)

MODEL 9(NAPROW) HEL=O 1

31, 7J

MODEL 9 (NAROW) HEEL 10 3's.. 233

31 71

3c.

MODEL 9 (NRRO)

20

?7 9.3

31 3:.,.') 23

Summary. and Discussion

Our purpose here has been to

determine

and compare some

possible

measurements of effective hull length, on the

assumption that the hull offsets are available.. _Three possible approaches have been set forth, each of which can be applied with a variety of datum waterplanes to represent realistic sailing conditions. The scientific merit of

these proposed lengths is in our opinion intermediate between the strictly empirical approaches of past and present rating rules, and the more scientific but impractical approaches of wave-resistance theory r tank testing. With further refinement these or similar integrated length parameters appear to öffer substantial improvements in the measurement of hüll length. Future efforts should be devoted not only to refinement of these parameters, but also to testing thém against a larger nùiber of hull Shapes.

9 ti: U?

Lu ir

Cr-Ci '-J -J. (J -i i c-2O.ÜO -16.00 -12.00 -l).00 -'1.00 0.00 4.00 6.00 12.00 16.00 X-F T

[1E5 tOOEL

i

U3HE) ÑEEL-O NO i4flVE PFUIFILE UNIFORM

WGI -'J c-J Q -('J Ci Q Ci- (S.J Figure 1 - Actual fit (, C

0

U-) and polynomial curves. ) sectional-area

I__J cc

il

Li I* i CJ. (J;). -. t) L) 2ci.00 -ilL 00 --l.(It) -iJ.Ù0 Figure 2

- Actual and. welghtéd

polynomial results -(J.00 X-1 i U.. (ICI

)[Ii II:

i MUllE t i

uHE)

l-IE.E t U ni-. NO

1AVE IflhiF i LIE

C) n, ej

&j9 Ui Ñ - (n.-.'

I?igure 3

- Actual and weighted polynomial

results 16.00 20.00 12.00 C)

L(LÌI _Ú_

X -f Ï

EiEFi 11:5 MODEL D

(HI Df [NOS)

tiEEi. :.fl NO l'lAVE PIIEE I LE:

C) ('J 'J Ci 4' c_. -, .0. ÛU I 1. 110 -I ' (iii

Figure 4 - Acttial and weighted

polynomial

resulte

-q

1ii I I li.00 12.Uû -q lit) U. (lu Li.ÙQ

X-i T

il il I, E ' fulFil I

'i

t I1-I1iIflH UN[)5i

lIEU_L . U Ñ(l iHvF. PFi(1F I L i. 18.00 2(1.00

C) C)

¿

('J

o

OEuJ-I-'

uJ''

-J a: (1 Symbol Heel AngIe °-d(I.UOE -1.6 [JO

-'l2.ÛO-.fjO-í[oo

o'.uo:

q'oo

X -F T SEf1JE5 MODEL. I 'LBRS[.) I:If'L,L

O, iú. ¿.0,30 LEG NO

I'RVE FIIOF ILE

Figure 5

- Effect of heel in

calm water

113.00 20.00

n

o

o

10 20

+

30

C)

OEU- ILl OE

Figure 6

- Effect of heel in calm Water

C) 16.00

-2.û0

-.00

-.00

û'.ÚO q'.oü O.00

¡2.00

X-FI

EíiI[-.5 I10ÍWL O

u4IOEJ hEEL-OP lU, 20.30 DEG

NO WAVE 1'ÙF ILl.

-(Jo

Figure 7 - Effect of

heel in calm water

C) -±OE.ÛO -16.(iO

-12.00

-.00

-1Mû

0.('

'L'diO O0 12.00 16.(iO

li.00

x-il

EfiIf3 MODEL 9NAtThOWJ MEEL-Q, 1û.20.30

a

a

('J 1%.) Lu.-:

o

cr -i.

(i

'-J c;l)OE. 00

Figure 8 - Effect of heel in small

wave

-T-

--

I-i

r

i i i I i

-16.00

-I.U0

-D.00 -'1.00

0.00

'LOO B.00 12.00

X-il

311111.5 Ñ'OOEL 1(13113E)

IIEEL-O, 102.0,30 DEG 0.5

WAVE Píuif[LE

16.01) 2Ù.1i0

Q Q Crut u_j.-. a: a: CJ Q

-r---

---20.fi0 -16.00 -12.00 -0.00 T

r

i. r. -.

--00 12.00 16.00 20.,Ú0.

X-f T

3EF1.1L.5 MoorL ii cHI0EJ

ME[L.-0. 10.20,30 DEG

0.5 NAVE P10F

ILE C) ci (r;- "J

Figuré 9

Effect o

heel in 8n1a11

wave

Q C.' c-J Q C). "J

Ci - j I;. (Io I I

-12.00

-0.00

Figure lO

- Effect of heel in small

wave

-L00

0úo

oo

a.ao

X-f T

12.00 16.00 FI1 I F MUON ¶1 tNÑRt1W) MEE- L -O

I 0,20. 30 DECI 0.5 WMVE

PuDE I LF L

C)

LO

LLJ a: ci -LJ (n.1 C-) C) a) C) C.) Figure 11

- Effect of heel in large

wave -20.00 -(JO -'I2.0O -h.00 -LCiO O'.00 'LOO 8'.00 1!.0O 16.00 b.oû X -f T

tflU;5 t10EWL

I WHSfJ, HELL=D. 10, 20. :30

I1EG

1 .0 WAvE Fft0F1LE:

a

OEu' '-J.-. cr OE I-C) (-JC U.ej (n

C) -j. t.)

-20.00

-16.00

-'12.00

-J.00

-I,.û0

D0fl 8.00 12.00

X-FI

SEHILS MODEL 8t1410.EJ

ÑEEL-0,I0.,20,30 DEG

1.0 HAVE PHOFILL:

20.00

C) a (D

J

Figure 12

- Effect of heel in large

wave

D D D

tD

t'.)

cr L)9

Figure. Ï3

Effect of heel in large

wave

-20.00 '16.00 hI2.00

-Thø

-1L00 01.00

X-FT

12.00 16.00 20.00

3Efi1E,S tiflOEL 9(NARI3ÔW)

ME[LL.:O. tÚ', 20,30 DEG 1.0 WAVE