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Paper to be presented at the 10th Chesapeake Sailing Yacht Symposium, Annapolis, February1991. Long Version

The Measurement of Weight Distribution of Olympic

Class Dinghies and Keelboats

P. F. Hinrichsen

Abstract

Racing sailors have become increasingly aware of the possible effects of the distributionof weight, as well as the total weight, on boatspeed. Modern construction techniques allowdinghy hulls to be built well under the minimum weight specified by the class rules. This haslead to a trend, notably in the Olympic dinghy classes, towards hulls with light ends, especiallylight bows. Not only does this increases construction costs but, if taken to extremes, could leadto unseaworthy boats. A number of classes, of which the Finn was the first, have thereforeintroduced means of measuring the pitch gyradius of the hull and hence controlling the fore andaft weight distribution. Measurements of the pitch and Yaw gyradii of Flying Dutchman hullsmade at the 1976, 1984 and 1988 Olympic regattas, at the 1990 FD World championships, aswell as data for a number of other classes are presented. The various methods used for gyradiusmeasurement, which include variants of the compound pendulum, the torsional spring-massoscillator and the bifilar suspension, are compared, with special emphasis on their precision,accuracy, worldwide reproducibility and the systematic corrections required. The effects of amplitude, of air damping and of friction at the support on the precision will be discussed.Normally only the hull gyradius is measured, however, it is the moment of Inertia of the totalboat which determines the response in waves. Calculations of the contribution of each of the

components, including the crew, to the total moment of Inertia are presented for FlyingDutchmen.



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1. List of Symbols

a Distance from the horizontal pivot axis to the CG.ac Distance from the trailer axle to the CG of the boat plus trailer, in the bounce test.

at Distance from the trailer axle to the trailer CG, in the bounce test.

b Distance between the axes in a Lamboley test.B(t) Bow displacement during the in the water test. d Half the spacing between the wires of a bifilar suspension.

D Horizontal distance from the axis of rotation to the spring attachment point in thebounce and Snipe tests.

E Total energy of oscillation

g The gravitational acceleration, 9.81 m/s2.H(t) Heave motion during the in the water test. Ib Moment of Inertia of the boat about the trailer axle in the bounce test.

Ic Moment of Inertia of the boat plus trailer in the bounce test.

It Moment of Inertia of the empty trailer in the bounce test.

IT Moment of Inertia of the total boat including the crew. Ip Moment of Inertia about the pitch axis through the CG.

Iy Moment of Inertia about the yaw axis through the CG.

kkeel Gyradius about the pitch axis through a point on the keel rubbing strake vertically

below the CG.kp Gyradius about the pitch axis through the CG.

ky Gyradius about the yaw axis through the CG.

l Length of bifilar suspension wires.

L Horizontal position of the CG forward of the transom.Lb Length from the bow sensor to the applied force W, for the in the water test.

Lbs Length between the bow and stern sensors for the in the water test.Lp Length from the center of pitch to the applied force W, for the in the water test.

Lo Horizontal distance from the standard mass m to the axis for the incline-swing test.

M Mass of the hull.Mc Mass of boat plus trailer in the bounce test.

Mt Mass of the empty trailer in the bounce test.

S Spring constant of the spring used for the Bounce and Snipe tests.

S(t) Stern displacement during the in the water test.

Tc Period of oscillation of boat plus trailer in the bounce test.

Tt Period of oscillation of the empty trailer in the bounce test.

Ts Period of sway oscillation for the bifilar suspension.Ty Period of yaw oscillation for the bifilar suspension.

W Force applied at the bow for the in the water test.

α Angular acceleration.

α(0) Pitch angular acceleration at t = 0 for the in the water test.

θ Angular displacement

θ(t) Pitch motion during the in the water test.



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Γ Torque.

Γg Gravitational torque.

Γs Spring torque.

Note: All equations and data are in SI units as these are standard for all measurements at the

Olympics.

2. Weight Distribution Measurement

With the advance of technology ever more sophisticated sailing dinghies are being built,and this is especially true of the Olympic classes. Although such advances may eventually leadto better production boats, it is the responsibility of the Class Associations to monitor suchdevelopments, and to ensure that expensive and perhaps detrimental developments do notconfer an unfair advantage.

Modern construction methods allow the hulls to be made significantly under theminimum weight specified in the class rules and the question then is where to put the extra

weight. It would be beneficial for the average sailor if it was used to make a stronger andlonger lasting boat, however it seems to be true for nearly all competitive classes, that buildersare concentrating the weight and moving it aft, because many top sailors are convinced thatsuch hulls give them a speed advantage. Past experience has shown that clever builders canusually circumvent scantling rules and that such rules eventually lead to obsolescentconstruction methods and possibly the demise of the class. In contrast determining the weightdistribution with a swing test controls those characteristics of the boat which affect it's speed,while leaving the construction free. However, swing tests requires the accurate timing of sometype of angular oscillation and this is both time consuming and requires carefully controlledmeasuring conditions if the required precision and reproducibility are to be achieved. Thus theintroduction of such a rule is not to be considered lightly, and the wording, the method of measurement and the limiting gyradius specified must be carefully chosen if it is to achieve it'sgoal. This paper will present some of the methods currently used for the measurement of weight distribution, and some of the results obtained.

3. The effect of weight distribution on performance

It is not the intent of this paper to discus the ways in which the distribution of weightcan affect boatspeed. For the present purposes it suffices to say that many sailors are, rightly orwrongly, convinced that light ends are beneficial and this belief is causing significant efforts tobe made to concentrate the weight. However, to put the subject into perspective a shortdiscussion is in order.

The distribution of the weight affects both the average attitude of the boat, and its

dynamic response. The heel and the fore and aft trim both depend on the average position of the weight, i.e. the position of the CG, but not on whether it is spread out or concentrated. For aboat sailing on ideally flat water in a constant wind so that it does not roll, pitch or yaw, theconcentration of the weight would have no effect on the boat's motion, or on its speed.However, except in very light winds we do not sail in this way, and the boat oscillates about anaverage attitude as it progresses. Only the two rotational oscillations, pitch and yaw, aredirectly affected by the fore and aft distribution of the weight. However the hydrodynamic



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forces on the hull depend on these rotational motions and they are thus coupled to the linearmotions.

When a boat starts to pitch the buoyancy tends to counteract the motion so as to bringthe boat back towards an even keel, however, the boat overshoots and then oscillates back andforth with a natural pitching frequency until the energy carried away by wave generation and

drag of the water on the hull plus the air on the rig damp out the pitching motion. When sailingup wind in waves the effect of an individual wave may not be serious, however, if waves areencountered at the natural pitching frequency of the boat then each wave adds to the pitchingmotion until it builds up and reduces the boat's forward speed. The weight distribution affectsthe period of "natural pitching" and hence the synchronism with the wave encounter frequencywhich is necessary for the pitching to build up. A large pitching motion not only increases theresistance to forward motion, but reduces the driving force of the sails (the airflow at the top of the mast can even reverse when the bow goes up!).

This at least is the accepted theory, and has been shown to apply to supertankers,destroyers, and to keelboats such as 12 meters with hulls which are relatively symmetrical foreand aft. After many discussions with top dinghy sailors I am convinced that they believe thatthe weight distribution makes a significant difference to both the "feel" of the boat and to its

speed. However, I am not so convinced that the forgoing resonance theory can be directlyapplied to planing dinghies. Relative to its size, a centerboard dinghy is very light, so the smallamount of energy stored in the pitching motion is rapidly dissipated. A fully rigged FlyingDutchman suspended on knife edges only makes a few oscillations before coming to rest, eventhough this idealized pitching motion does not include the very large damping due to the water,when sailing [Hinrichsen, 1977 #6]. The large damping, together with the pronouncedasymmetry of the hull shape suggests that the resonance will be very broad so that smallchanges in the resonant frequency will have little effect on the pitching amplitude. Differencesin technique, i.e. sailing free or pinching, can change the encounter frequency by as much as theweight distribution changes the resonant frequency. Finally, the influence of the added massdue to the oscillations of the water in contact with the hull, of different masts and crews could

mask the effects of the small differences in the hull weight distribution on the pitching motion.The effect of the fore and aft weigh distribution on the feel of a sailing dinghy may also

be due, at least in part, due to it's influence on the steering response. When sailing in waves it iswell known that the helmsman does not steer a straight course, but is continually altering coursein response to the waves. Sailing to weather he will head up into a wave, and then rapidly bearsoff over the crest, repeating this maneuver for each wave. Similar actions are taken downwind,so that the course of the boat resembles a slalom. This motion can be thought of as a constantaverage motion in the direction of the course, plus a yaw oscillation.

This yawing oscillation is not like the pitching oscillation, for which the changingbuoyancy provides a natural restoring torque, but is purposely introduced by the helmsman's useof the rudder and of sail trim. When the boat is steered off course, there is no natural tendency

for it to be restored to the original course, the helmsman again has to use the rudder. The foreand aft weight distribution of the boat directly affects the boat's response to these rudderactions. A boat with heavy ends will respond more slowly, or will require more violent rudderaction for it to respond as rapidly as a boat with light ends. The sluggish response to the rudderof a boat with heavy ends will mean a less optimum course or alternatively, the more violentrudder action required will induce more drag.



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4. Center of Gravity, CG

The average position of the weight determines the Center of Gravity, i.e. the CG of theboat, which in turn affects the average attitude of the boat when sailing. For stability the CG isgenerally as low as possible and its fore and aft position has a major effect on the trim of the

hull in the water. However for many sailing dinghies the crew together weigh more than theboat and therefore have a very significant effect on the CG of the total boat. Even within theconstraints imposed by boat handling they can significantly alter the position of the CG. Thefore and aft position of the hull CG might therefore seem to be unimportant. However, the crewand the hull can be thought of as the ends of a dumbbell and the closer together their CGs arethe smaller the total moment of inertia. Thus in order to concentrate the total weight the hullCG should be close to the average position of the crew, i.e. well aft. This may be part of theexplanation why many dinghy sailors believe that a light bow is more important than a lowgyradius, and that any suggestion that corrector weights be put on the bow are met with howlsof protest. It therefore makes little sense to regulate the gyradius without also specifying thefore and aft position of the CG.

Fortunately the fore and aft position of the CG can be easily measured as part of any of

the swing tests to be discussed. The vertical position of the CG is measured as part of theLamboley and Incline-swing tests and can also be determined by balancing the hull on agunwale.

5. Principles of weight distribution measurement

To determine the longitudinal distribution of the weight in a non-destructive manner,one must measure the moment of inertia about either the horizontal pitch or the vertical yawaxis, and this requires a dynamical measurement. The second moment of the mass distribution,

or moment of inertia about a given axis can be expressed as I = Mk2, where M is the mass of the hull and k is the gyradius I and k depend on both the orientation and position of the axis of

rotation. The roll moment of inertia is very different to that for pitch or yaw and although inprinciple the yaw and pitch gyradii are also different, in practice they are approximately thesame for typical dinghy hull shapes. The instantaneous axis of pitching rotation is not fixedrelative to the hull, especially for a planing dinghy in waves, However such a motion can betreated as a heave oscillation of the CG plus a pitch rotation about the CG and, except for thecoupling terms, the heave does not depend on the moment of inertia. Thus it is customary toquote moments of inertia and gyradii about an axis through the CG, furthermore these are theminimum values for rotation about a given direction and the gyradius about any parallel axiscan be easily computed from them.

The moment of inertia cannot be measured by any static method as it only enters the

rotational statement of Newtons second law, namely Γ = Iα where Γ is the applied torque and α

is the resulting angular acceleration. This relation is the basis of all measurements of momentsof inertia and hence of gyradii. The method of measurement proposed by Watt Webb [Webb,

1974 #46] uses this equation directly. All the other methods employ a restoring torque Γ(θ)

which is ideally proportional to the angular displacement θ from the equilibrium position andproduce oscillatory motion. A measurement of the frequency or period of the oscillationtogether with the functional dependence of the torque on angular displacement give the momentof inertia. For the "Lamboley test", the "Incline-swing test" and the "Bifilar suspension" theweight and the geometry of the suspension determine the torque while for the methods used by



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the Snipe, Comet and Lightning classes a standard spring at a fixed lever arm provides therestoring torque.

If, in order to enforce a rule, measurements are to be made at regattas then there aresome other requirements on the method to be used. The apparatus must be cheap and reliable,and relatively untrained people must be able to get precise and reproducible results in a short

time. Typically, at a major championship, 50 or more boats will have to be measured in aperiod of two days. No complex adjustments, calculations, or corrections, should be required,as on line results are essential. Measurements made in different parts of the world must give thesame results, without recalibration of the equipment.

It is of great importance that the chosen method has the highest precision possible. Theweight distribution is characterized by the gyradius k which is a length, and the sailor's naturalexpectation is that one can measure it with the same ±1 mm precision with which otherdimensions are measured! Such precision cannot presently be achieved and it is the object of this paper to examine the reproducibility and precision of the methods currently used.

6 The Lamboley Test

Ideally the period of small amplitude oscillation T1 of a rigid body of gyradius k and

with its CG a distance a below a horizontal axis from which it is freely suspended is

T1 = 2π a2 + k2

ag (1)

Unfortunately this period depends on the unknown distance a as well as the gyradius k which one wants to measure. This difficulty can be overcome if two periods of oscillation,about two axes which are a known distance b apart, are measured. The period about the loweraxis is

T2 = 2π (a-b)2 + k2

(a-b) g (2)

The unknown distance a from the upper axis to the center of mass is then

a = b(gT2

2 + 4π 2b)

g(T22 - T 1

2) + 8π2b (3)

and the gyradius is given by

k2 = agT1

2

4π2 - a

(4)

For a hull suspended from a horizontal athwartships axis, see fig. 1, the pitch gyradiuskp about the center of mass can therefore be determined from measurements of the spacing b,

and the two periods T1, and T2. In 1970 Gilbert Lamboley [Lamboley, 1971 #25] introduced



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this method, with b = 200 mm, and the Finn class has successfully used the "Lamboley Test"for the past 20 years. The Europa class has also adopted this test and many other classesincluding the Flying Dutchman and 470 have considered the adoption of the Lamboley Test,and made detailed studies of the weight distribution of hulls using this technique [Hinrichsen,1977 #6].

The practical usefulness of the Lamboley Test has been well established, however it hasa number of disadvantages. The most obvious is that two periods have to be measured, and thisis very time consuming when many boats have to be checked and rechecked. Photocell timingand portable computers speed up the procedure but have the effect of adding an air of incomprehensibility to the results. Finally it can be seen from equations (3) and (4) that thegyradius depends on the difference between the squares of the two periods of oscillation, andthis limits the precision with which the gyradius can be determined.

7. The effects of Damping on Precision

The simple theory above does not take into account pivot friction or air damping, andfor modern light dinghy hulls even small drafts can affect the period of oscillation. Water in the

hull, loose fittings which can flop about, knife edges and/or photocells which are not veryrigidly supported can all limit the precision achievable. The presence of even a few hundredgrams of free running water in a 125 kg hull cause the amplitude to decreases rapidly andnonuniformly. The bifilar test is probably much less sensitive to the presence of water, but nodetailed observations have yet been made. For precision measurements the hulls must beabsolutely dry, but this is not always easily achieved at a regatta!

Pivot friction, Linear and quadratic air damping, can be studied by observing thedecrease of the amplitude with time, see for example fig. 2. Pivot friction would cause this datato curve downwards and there is no evidence for this, thus suggesting that friction wasnegligible. Furthermore, theoretically pivot friction does not affect the period of oscillation[Squire, 1986 #163]. The effect of linear air damping on the period is much smaller than the

scatter of the data, and is in any case constant for all hulls of a given class. The quadratic airdamping (the presence of which is clearly shown by the fact that the graph in fig. 2 is not astraight line) however causes the period of oscillation to depend somewhat on the amplitude[Nelson, 1986 #151]. The observed variation of the period with amplitude, see fig. 3, cannothowever be accounted for by this effect. If the periods measured at a given amplitude are usedto calculate the pitch gyradius kp and CG height "a" then they vary quite significantly with

amplitude as shown in fig. 4. The use of small amplitudes reduces this systematic error but atthe expense of making the measurement more sensitive to drafts and thus increasing the randomerror. Thus the choice of an optimum amplitude is a compromise which may be different foreach class.

Another source of systematic error is the hooks required to suspend the hulls which

produce a measurable change in the observed gyradius. Typical Lamboley hangers, 2.75 kg,produce a decrease of 15 mm, i.e. one percent of the FD gyradius, and for smaller lighter hullsthe effect will be larger. Thus the exact weight and dimensions of the hangers must specifiedand any rule must clearly state that it is "the gyradius of the hull plus hangers" or a correctionfor the hangers must be calculated. All techniques suffer from this problem and the attachmentswhich support the hull and rotate with it should always be as light as possible.

8. The Incline-Swing test



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The vertical position of the CG a can be determined by an inclining test, see fig. 5,

instead of a second period measurement. This has the advantage that the hull only needs to besuspended once, a significant advantage especially for measurements on keelboats. If astandard weight of mass m is placed a horizontal distance l1 forward of the axis the vertical

position a of the CG can be calculated from a measurement of the vertical displacement d1 of apoint which is a horizontal distance Lo from the axis. Then for a hull of mass M

a = mM

. l 1 L o

d1 (5)

The precision and reliability can be considerably improved by measuring a second displacement

d2 with the standard mass a nominally equal distance l2 towards the stern, then

a = m

M

Lo

2

l1

d1

+ l2

d2

- a0 ≈ m

M

Lo

2

l1 + l 2

d1 + d2

- a0

(6)

where for completeness a small correction for the average distance ao of the standard mass

below the pivot is included (the system should be designed to make ao zero). The value of a

together with a single period of oscillation T, about the same axis, when substituted intoequation (1) give the pitch gyradius kp. This method is used by the Dragon class, and was used

for tests on some Tornado hulls at the Olympic regatta in Pusan. The Royal New ZealandYacht Squadron tested the Stewart 34s previously used for the Squadron Cup match races withthis technique. Further advantages are that a static deflection can be easily averaged, and thepresence of water in the hull causes the deflection d to change continuously. The only minordrawback is that the mass of the hull M must be measured accurately, but this is in any case part

of the measurement procedure.For the measurements on Tornado hulls the major sources of uncertainty in the CG

position a (±1 mm) were the mass of the hull M (0.25%) and the deflection d (0.5% with m =200 gm) leading to an uncertainty in the gyradius of ± 4.2 mm. The period of oscillation couldbe measured to ± 5 msec which leads to a further uncertainty in the gyradius of ± 1 mm. Forsimilar timing uncertainties using the Lamboley test the results give similar precision, howeverthe Lamboley test results are much more sensitive to timing uncertainties. The use of anelectronic level and photocell coupled to a computer, could materially improve the precisionand turn around time of this method.

9. The Bifilar Suspension

The bifilar suspension [Newman, 1951 #185; Hinrichsen, 1985 #12] shown in Fig. 6 isan alternative way to generate a torque which is derived from gravity. The hull is suspended bytwo parallel wires of length l, and spacing 2d. When hanging freely with the hull level the CG

is in the plane of the wires, and half way between them. For small angular displacements θ,

from equilibrium, the two wires become inclined to the vertical, the CG rises by z = d2θ2 /2l,and the total energy of oscillation is



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E =

Mky2

2 θ

2 +

Mgd2

2l θ 2

(7)

which is the equation of simple harmonic motion with a period of

Ty = 2πky

d l

g (8)

Then the yaw gyradius ky is given by

ky = d2π

g

lTy

(9)

Thus the gyradius is directly proportional to the measured period Ty, and the constant of

proportionality depends only on the geometry of the suspension which can be chosen so as to

make the constant a round number. Another significant advantage is that the apparatus requiredcan easily be made at home. This method is commonly used on tank test models, has been usedon sailboards, lasers, 470s, an International 14 and to measure Flying Dutchman at the '84 and'88 Olympics, and at the 1990 World Championships in Newport (where 75 hulls weremeasured in three days).

One drawback is that the boat is free to oscillate in a number of ways other than the yawoscillation which is to be measured, i.e. in sway as a simple pendulum, and in coupled pitch andheave. For timing with a vertical photogate at the bow the the heave and pitch are eliminated.Releasing the hull while keeping its center under a plumb bob, which is on the center line of thesuspension, reduces the sway to less than 1% of the yaw amplitude the bow, however, thismodulation is still the major cause of timing uncertainties.

The bifilar suspension prevents any roll rotation thus the sway period is

Ts = 2π lg (10)

and Ty /Ts = ky /d i.e. the sway oscillation can be made a harmonic of the yaw oscillation by

choosing the spacing d to be an integral fraction of ky. The modulation by a harmonic should

then not influence the yaw period, as measured from the zero crossing times at the bow. Thusfor an athwartships suspension, which facilitates measuring the fore and aft position of the CG,a spacing 2d & ky should be chosen (d & ky is much larger than the beam, but is the best

choice if a fore and aft suspension is used). By averaging measurements of the period over one

beat cycle any residual effects of sway can be made negligible.The crossbar or other support which swings with the hull should be light and a

suspension which uses hooks under the gunwales, the separation of which is controlled by ataught cross wire has proved practical. For a fore and aft suspension such as was used forsailboards a spacing d = ky eliminates any correction for the mass of the hooks. The mass of

the suspension wires leads to a negligible correction, however some care must be taken with theend fittings which must not allow the motion of the hull to twist the wires, otherwise thetorsional rigidity of the wires adds an unknown torque. The length l and spacing 2d can be



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measured to ±1 mm. The residual sway modulation of the period Ty, which for the

measurements on FDs was ±7 msec (but much less when averaged) is the limiting factor on theprecision of the method. An uncertainty of ±7 msec in the period corresponds to an uncertaintyof ±2 mm in the gyradius. This is significantly better than can be achieved with a Lamboleytest. The insensitivity of this test to offcenter positioning of the CG has been confirmed by

measurements on an International 14 and its sensitivity confirmed to be better than ±2 mm byplacing up to 5 kg at various positions in the 94 kg hull. Measurements of the decay of theamplitude, similar to those shown in fig. 2, have been made and show that, as expected, the airdamping is less in yaw than in pitch.

The beating of the Sway and yaw oscillations can be used to measure the gyradiusdirectly. If the hull is released from a position which is displaced in sway but with the hullrotated so that the bow is at its equilibrium position it will oscillate in rotation about the bow,which remains stationary, if d = ky. This can be achieved by adjusting the spacing d. After the

system is tuned one just measures d to determine the gyradius ky, without the use of a stop

watch! However despite the fact that only one end of one suspension wire need be adjusted (thewires need only be approximately vertical) the tuning takes too much time for this to be a

practical technique at regattas.

10. Spring Oscillator Methods

In 1965 Robert Smithers [Smither, 1969 #38] developed a method of measuring themoment of inertia of a fully rigged Lightning (without the sails) while it was on its trailer. Hisaim was to investigate the difference in weight distribution between wooden and fiberglassboats as cheaply as possible. The tongue of the trailer was attached to a calibrated spring, of spring constant S, and by rhythmically pushing the trailer tongue the boat was made to oscillatein pitch about the trailer axle. The measured period of oscillation then gives the moment of inertia Ic of the boat plus trailer. A separate measurement on the empty trailer gives its moment

of inertia It which can then be subtracted to obtain Ib i.e. that of the boat alone about the trailer

axle. Separate measurements of the distance a of the CG from the axle are required in order toconvert the result to the moment of inertia about the CG (or other parallel axes such as thatthrough the center of buoyancy). Various methods such as resting the hull on its gunwale andmeasuring the balancing force required at the tip of the mast, or alternatively measuring thetrailer tongue weight and its variation with angular displacement, were used to obtain the CGposition [Smither, 1969 #38].

The principle of this method is similar to those previously described, however therestoring torque is now supplied by the calibrated spring at a lever arm of D i.e. the horizontaldistance of the hitch from the axle. For this setup in which the axis of rotation is below the CGthe weight no longer supplies a restoring torque, thus if it is to be only a small perturbation the

variation with angle of the gravitational torque i.e. Γg = Mga Sinθ , must be much smaller than

that of the torque due to the spring Γs = SD2θ.

For the CG of the boat plus trailer a distance ac vertically above the axle, i.e. zero

tongue weight, the total energy of oscillation is

E ≈ Ic

2 θ

2 +

SD2 - Mcgac

2 θ 2

(11)



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Thus the period of oscillation is

Tc = 2π I c

SD2 - Mcgac (12)

Which gives the Moment of inertia of the boat plus trailer as

Ic = Tc2

4π2SD2 - Mcgac

(13)

A similarly expression gives the moment of inertia It of the trailer alone and then the moment of

inertia of the boat about the trailer axle is

Ib = SD2

4π2T c

2 - T t2 -

g

4π2T c

2Mcac - Tt2Mtat

(14)

By appropriate choice of the spring constant S the last term (which in any case gets somewhatsmaller if the CG is not vertically above the axle) can be made a small correction. For highprecision measurements the constants ac and at can be determined as mentioned above.

The precision achieved was about ±1% and clearly differentiated between the wood andthe glass boats, see table 2. Estimating the uncertainty in the weight measurements as about±0.2% the resulting uncertainty in the gyradius is ±0.7% or ±13 mm which compares favorablywith the Lamboley test. For light hulls on heavy trailers the correction for the trailer will add tothe uncertainty due to the difference between the squares of the two periods in equation (14).The trailer is however unnecessary and in fact can be the cause of added corrections unless boththe wheels and the suspension springs are firmly blocked.

In 1971 Ted Wells [Wells, 1971 #48] eliminated the trailer and simplified this test foruse by the Snipe class (I believe the Comet class also use it now) and Dan Williams refined andcompared it with the Lamboley test, see table 2. For the Snipe test the keel rubbing strakes restson a small steel plate pivoted on a 3/8" steel bar. The hull is adjusted fore and aft until itbalances and then two tangential springs are attached to a jig on the bow, see fig. 7. The periodof oscillation is measured and the moment of inertia about the point on the keel rubbing strakedirectly below the CG is calculated using equation (13) without the last term. The class rulethen specifies a minimum moment of inertia determined in this way, thus obviating the need forthe other measurements required to obtain the gyradius about the CG. One might argue that byadjusting the keel rocker etc it would be possible to concentrate the weight slightly more

without contravening this rule. However as kkeel2 = (Kp

2 + a2), where a is the height of the

CG above the keel, this can only be done at the expense of raising the CG and this may offsetany advantage due to the decrease in gyradius (note that the opposite would be true for a oneperiod Pendulum test).

If results are to be reproducible worldwide then one source of accurately calibratedspring sets is to be prefered and the jig on the bow should be as light as possible and of exactlyspecified dimensions. Springs can age and be abused, however a set in use for 8 years changedby only 0.2%. The fact that the hull rests a pivot on the ground eliminates the need for a very



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solidly supported pivot bar and solid gunwales, as are required for a Lamboley test, and alsospeeds up the procedure.

11. Keelboat tests

Performing swing tests on keelboats is clearly a much greater engineering problem aswell as being more time consuming. The incline-swing test and a yaw inertia test have beenused for out of the water measurements. An in the water test proposed by Watt Webb [Webb,1974 #46] is being implemented by Rick McCurdy for possible inclusion in the IMS rule[McCurdy, 1990 #31].

The Dragon class perform an incline-swing test by hoisting the hulls using a jig,attached to the lifting eyes, which allows it to be levelled and to swing in pitch about a swingcenter which is about 570 mm below the deck. They use a 10 kg weight 4700 mm forward of the swing center and some effort is required to measure the 4700 mm with the requiredaccuracy. An error of 25 mm would lead to an error of 7 mm in the gyradius and timinguncertainties of ±20 msec would produce uncertainties of ±6.6 mm. Tests with known addedweights gave results which reproduced to within ± 0.3% or ±4 mm. The class however have

chosen to specify their rule directly in terms of the period of oscillation and the bow deflectionin order to avoid calculating the gyradius directly.

In 1987 the Royal New Zealand Yacht Squadron wished to ensure that the wooden andfiberglass Stewart 34s, which they used for the Squadron Challenge Cup match race series,were as equal in performance as possible. They therefore performed incline-swing tests on theboats, see fig. 8. Tests on one hull which was loaded with extra weights in specified locationsindicated that both the CG position a and the gyradius kp could be determined to ±1%. Once

organized, the tests required only one hour from haulout to relaunch, however the equipment isnot easily transportable.

Bill Parks of the Star class pioneered a different approach in 1975. This techniquemeasures the yaw moment of inertia by suspending the hull from a crane with a swivelling hook

and attaching a pair of horizontal springs at the bow. The principle is the same as that of theSnipe test except that the rotation is about the yaw axis. The CG is directly below the point of suspension and therefore on the axis of rotation, thus ac in equation (13) is zero. There are

however a number of problems with this elegantly simple method. When the hull swings inyaw the pivot must exert a force which is equal and opposite to that due to the springs at thebow and it is difficult to prevent any lateral motion of the suspension hook. The hook is alsonot at the same level as the bow springs and thus sway and roll motions will develop. Theseproblems together with friction at the hook and varying torsional rigidity of the suspension ledto irreproducible results from site to site. In 1989 the class approached me and I suggested thatthe hull be supported on a turntable using a vertical truck axle bearing and a light frame whichclamp onto the keel to support the hull. The calibrated springs could be incorporated into the

turntable thus making the system self contained. At the present time the Star class has deferedany further action until there is clear evidence that a rule is required.



13

12. In the water tests

For ocean racing yachts there are obvious advantages to an in the water test which canbe performed at the same time as inclining measurements etc. and a technique for suchmeasurements was proposed by Watt Webb [Webb, 1974 #46] in 1974. In 1989 the USYRU

established a Pitching Moment Project to develop a practical instrument based on this proposaland to collect data on yacht performance in waves, on which a handicapping system forinclusion in the IMS rule could be based. The development of the instrument has beendescribed in detail by Rick McCurdy [McCurdy, 1990 #31].

A upward force W applied near the bow which displaces the boat in both pitch andheave such that the buoyancy force is reduced by W and the buoyancy torque balances that dueto W at a lever arm Lp from the center of pitch, see fig. 9. If the applied force W is suddenly

removed the boat will oscillate with a complex damped pitch-heave motion, however at theinstant of release the buoyancy torque responsible for the pitching is equal to the applied torque

Γ = WLp. Measurements of the bow and stern displacements as a function of time, see fig. 10,

then allow both the initial pitch angular acceleration α(0) and Lp to be determined. Provided

that the center of pitch remains fixed the bow and stern displacements B(t) and S(t) are given by

B(t) = H(t) + (Lp + Lb) θ(t) (15)

S(t) = H(t) + (Lp + Lb - Lbs) θ(t) (16)

where Lb is the distance between the bow sensor and the pull and Lbs is the distance between

the bow and stern sensors. then

θ(t) =

B(t) - S (t)

Lbs (17)and

Lp = Lbs B (t) - H (t)

B(t) - S (t) - L b

(18)

By differentiating θ(t) twice and extrapolating to time zero the initial angular acceleration

α (0) can be determined. Unfortunately H(t) is unknown and to overcome this Watt Webbproposed a second measurement with the pull at the stern, then only the measured spacingbetween the two pulls, not Lp, is required. However this complicates the procedure and as the

heave and pitch oscillations differ in frequency by about a factor of 2.2 they can be separated,thus allowing Lp to be determined. In practice the heave has always been found to be

negligible. Then

Ip = W Lp

α(0) =

WLpLbs

B (0) - S(0) (19)



14

In order to allow precise extrapolation to time zero, damped oscillator functions together withoffset and drift parameters were used to model the pitch and heave motions as shown in fig. 10.A nonlinear least squares fitting routine is used to derive the parameters from a simultaneouslyfit to B(t) and S(t).

An analysis of the results for five different boats suggests that the moment of inertia in

the water can be measured with a standard deviation of ±6%, while Lp can be determined to±1.2%. Currently the main source of uncertainty is due to heave of the boat against the liftingtackle which causes W to vary. An alternative approach is therefore being developed, namelythe pitch stiffness will be precisely measured with an electronic inclinometer and then themoment of inertia determined from the period of oscillation as recorded by the sameinclinometer, in a manner similar to the pendulum tests. This technique has the advantages thatneither W or Lp have to be determined and that the system does not require any external fixed

reference point and will thus be self contained.It should be pointed out that the in the water tests do not measure the same quantity as

those performed out of the water. The energy of oscillation of the water in the vicinity of thehull has to be included in the equations of motion and can be represented as an added mass forheave and an added moment of inertia for pitch. The values depend on the detailed hull shapeand precise theoretical calculations cannot yet be made for sailing yachts. The axis about whichthe in the water moment of inertia is computed is also not through the CG. Typically the resultsof mass moments of inertia computed from design weights etc. differ from those measured inthe water by a factor of about two. However it can be argued that it is the in the water valuewhich is more relevant to performance and provided it is measured in a consistent mannershould be used for handicapping purposes. It will now be possible to measure some hulls bothways and compare the data with current theoretical models and the actual pitch-heave motion of the boat in waves. This is an interesting project for the future.

13. Flying Dutchman Results

Fig. 11 shows the close correlation between the yaw and pitch gyradii of the FDs atPusan in '88, thus demonstrating that either can be used for the control of weight distribution.The data for FD hulls at the '76, '84 and '88 Olympics are compared in Figs. 12, 13 and 14,which show the gyradii, the transom to CG and the pivot to CG distances respectively. Thearrows indicate the average values for each year. The conclusion from this data is that theweight is being concentrated, i.e. smaller gyradii, and moved aft. Clearly a major effort is beingput into making the bows lighter, but by how much?

Assuming that the hulls are uniformly lightened, i.e. so the gyradius and CG remain thesame, and then that the saved weight is all added at one point, allows one to to estimate theamount and location of the saved weight. Comparing the "Average '76" and "Average '88" FDssuggests that about 23 kg has been saved and moved back to within 1.83 M of the transom. In

practice the change is more likely due to the empty bows now in favour, but the estimate of 15-20 kg is in agreement with "boatpark wisdom".

14. Total boat Calculations

It is the total boat, i.e. the hull plus equipment and the crew which interacts with thewind and waves, not just the hull. It would be nice if one could measure the moment of inertiaof the total boat, and Robert Smithers has done this for a lightning with everything but the sails.



15

I have performed a Lamboley test on an FD with sails etc. but no crew. However suchmeasurements are not practical as they are very sensitive to drafts. Thus one has to resort tocalculations and these can be assuming a perfectly rigid boat provided the mass, CG positionand gyradius of each component is known. The position of the CG of the boat LCG forward of

the transom is

L CG =

mi L iΣ

i

miΣ (20)

where the sum is over all the components. Similar formulae give the vertical location HCG

below the deckline and the athwartships position BCG of the CG. The total pitch and yaw

moments of inertia about the CG are then

Ip = m i ( L i - LCG)2 + (Hi - HCG)2 + kpi2Σ (21)and

Iy = m i ( L i - LCG)2 + (Bi - BCG)2 + kyi

2Σ

(22)

The results of such a calculation for an FD are given in Table 1. Note that when the crew areadded the CG changes position thus the crew moments of inertia cannot simply be added to getthe total. Provided the rig is tight, and this is realistic for FDs going upwind in waves, thecalculation is probably reliable for the complete boat and this was confirmed by measurement.however, despite what some helmsmen may think of their crews, it is not realistic to assume

they are rigid bodies! Thus the values for the total boat are only for guidance. Robert Smitherhas made measurements with the crew sitting in a Lighting when pitching on it's trailer, bothwith the rig tight and with it loose and has found significant differences [Smither, 1970 #39].One could probably model the system of boat, helmsman and crew as three coupled dampedspring mass oscillators, but that is beyond the scope of this paper.

Although the hull is about 80 percent of the weight of the boat it only contributes 50percent of the pitching moment, the mast (31%) mainsail (6.3%) and rudder (6%) also makesignificant contributions in pitch. For yaw the hull (85%) is the dominant component and onlythe rudder (9.3%) makes a significant contribution. The crew and helmsman, although morethan half of the total weight, add little to the pitching moment but as the crew on the trapeze isperpendicular to the boat he adds substantially to the yaw moment. It is purely fortuitous that

the mast and crew add to the moments in such a way that the total pitch and yaw moments aresimilar.

One might expect that changes in crew position and weight would be much moresignificant than small changes in the hull, however, because the crew is at about the position of the CG of the boat plus helmsman changes in his position have only a small effect, see fig. 15.The helmsman should however be as light as possible and as far forward as is commensuratewith boat handling. It is also advantageous for the CG of the boat to be as far aft as possible,the basic idea is that the CGs of the Helmsman, crew and boat should be as close together as



16

possible as this reduces the first term in equations (21) and (22). Thus the total moment of inertia can be decreased by reducing the hull gyradius or by moving the CG aft. This effect isillustrated in fig. 16 in which the pitch gyradius and fore and aft position of the hull CG areplotted together with contours of the total moment of inertia.

Variations in crew, mast, and equipment can produce about 9 percent changes in total

moment of inertia, while changes in crew position are calculated to produce only 5 percentdifferences compared to about 10 percent variations between hulls. Thus although the hull isthe most important contribution the combined effect of the other components can be as large asthat from the hull. These numbers are for Flying Dutchmen and similar calculations can bemade for other classes given similar data to that in table 1.

15. Data for Other Classes

Many of the measurements summarized in table 2 were made by class measurers on anexploratory basis and was thus buried in class correspondence.

The speed with which a sailboard can be made to respond in both pitch and yaw dependson the weight distribution, and as in this case the rig is decoupled from the hull thus it does not

add to the pitch moment of inertia. Dr Schoop measured a number of sailboards for the IYRUusing a bifilar suspension for both the yaw and the pitch gyradius, and one of the Lechnerboards used in Pusan was also measured. It would be a simple matter to mould a screw socketinto the board so that the eyes could be simply attached for a yaw test if this became arequirement.

The Finn class was the pioneer in the field and an extensive compilation of Finn dataexists [Lamboley, 1975 #27]. The data in table 2 is an early sample which is uninfluenced bythe introduction of the rule which now limits the gyradius to 1140 mm < kp < 1300 mm and the

CG position to 2000 mm < L < 2250 mm. The 470 class has also collected extensive data atboth their 1985 World and 1986 European Championships as well as at the Pusan Olympics.For the latter the IYRU required the 470 gyradius kp >1180 mm, thus the data for the '86

Europeans is given in table 2. Bob Shiels has made measurements of the yaw gyradius using abifilar suspension, but these were for a hull without fittings.

Prior to 1988 the Tornado class had a limitation on the total weight but no restriction onthe weight of the hull alone. Although not used for the Olympic regatta, the Australians built apair of extremely light hulls together with excessively heavy centerboards in order toconcentrate the weight, and these were reputed to be fast, especially downwind. Leif Smitt andI therefore took the opportunity to make incline-swing tests on these hulls together with hullsfrom other tune up Tornadoes in Pusan and the average data is shown in table 2.

The Dragon class is to my knowledge the only keelboat class to have a weightdistribution rule and data from their exploratory tests is given in table 2. As is to be expected heratio kp /LOA = 0.16 is much lower than that for the dinghy classes and compares with ky /LOA

= 0.144 for the Star class. However it is interesting to note that for the Stewart 34s kp /LOA =0.246 which is similar to that for the dinghies. The data from the in the water tests has not beenincluded in table 2 as it is not comparable with the rest of the data.



17

16. Conclusion

A number of techniques which have been used to measure the gyradii of boats have beendescribed and the results summarized in the hope that this will allow future work to benefitfrom this experience.

Opinion on the effect of weight distribution on sailing dinghy characteristics and speedare divided. many sailors believe it has a significant effect while on the other hand there arereports that under some conditions light ends can be a disadvantage. The statistical evidencefrom regatta results is inconclusive and subject to interpretation as light ended boats are likelyto be sailed by top sailors who leave nothing to chance.

To my knowledge no double blind tests, with matched boats sailed in conditions inwhich light ends are likely to be significant, have been made. The data being collected for theIMS project will provide information on the effect of pitching on keelboat performance andsimilar data for dinghies would be of great interest, but much harder to obtain. In the mean timeimprovements in the gyradius measurement techniques will at least improve that aspect of thedata.



18

REFERENCES

1. R. Compton, B. Johnson and C. Van Duyne. "Seakeeping and the Sailing Yachtsman".Second Chesapeake Sailing Yacht Symposium. 1975.

2. W. W. Webb, "The IOR Rule: Moment of Inertia measurement to control empty ends".1974.

3. G. Lamboley, "Influence of the Weight Distribution in a boat, proposed means of control". IYRU Report 1971.

4. P. F. Hinrichsen, "Weight Distribution in Sailing Dinghies" IFDCO Report 1977.

5. P. T. Squire, "Pendulum Damping". Am. J. Phys. 54(November): 984-991, 1986.

6. R. A. Nelson, and M. G. Olsson. "The Pendulum-Rich physics from a simple system".Am. J. Phys. 54(February): 112-121, 1986.

7. P. F. Hinrichsen, "A New Method of measuring the Weight Distribution of SailingDinghies". Bulletin of the IFDCO. 29, 1985.

8. F. H. Newman, and V. H. Searle. The General Properties of Matter. Edward Arnold &Co. London 1951.

9. R. K. Smither, "Measuring Moment of Inertia". One Design and Offshore Yachsman.30-49, 1969.

10. T. Wells, "Moment of Inertia". Snipe Bulletin. 13, 1971.

11. R. S. McCurdy, "Feasability Study of the Measurement of Mass Moment of Inertia inPitch for Cruiser/Racer Yachts". New England Sailing Yacht Symposium, 23 March1990. 1-17, 1990.

12. R. K. Smither, "The Floppiness Factor". One Design and Offshore Yachsman. 35-37,1970.

13. G. Lamboley, "Weight Distribution Experiments made up to July 1975". IYRU Report1975.



19

FIGURES

a

L

CG

b

1

2

Figure 1For a Lamboley test the hull is suspended from a horizontal knife edge and the two

periods of oscillation T1 and T2 about two axes a distance b = 200 mm apart are measured. The

pitch gyradius kp and the vertical position of the CG, a can then be calculated. The horizontaldistance L from the transom to the CG can also be measured.

10

100

0 100 200 300 400 500 600

A m p l i t u d e

θ o m r a d

5

Time sec

FD 88 Axis 1: "Amplitude θo

vs Time"

20

40

θo

= 149.92 mrad

α = 0.001323 s-1

β = 0.12368 rad-1

s-1

Figure 2

The decay of the amplitude of pitch oscillation of an FD in a Lamboley test. The initialcurvature of this "log plot" indicates the presence of nonlinear damping and the absence of anydownward curvature shows that pivot friction is negligible. The data has been fit with afunction representing linear and quadratic damping.



20

0.0

-0.2

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Amplitude mrad.

ΔT/To

vs Amplitude θo

DT1/T1 FD76

DT2/T2

DT1/T1

DT2w/T2w

Δ T / T

o

p e r c e n t

0 50 100 150 200 250

Figure 3The variation of the periods of oscillation in a Lamboley test on FDs as a function of the

amplitude of the swing, where ΔT/T0 is the fractional difference in the period. Data for ΔT1 /T1

were taken in 1976 and repeated in 1988 and ΔT/T2 are from 1988. ΔT2w /T2w is data with 500

gm of water in coke bottles on the bow and stern, to simulate a wet boat.



21

0 100 200 300

550

560

570

580

1460

1470

1480

1490

1500

1510

1520

1530

CALCULATED FD GYRADIUS "Kp"

and CG HEIGHT "a" vs AMPLITUDE

AMPLITUDE mrad

C G

H E I G H T " a " m m

G Y R A D I U S " K

p " m m

Figure 4

If the data of fig.3, at a given amplitude, are used to calculate the pitch gyradius kp and

CG height a, then the results vary with the amplitude as shown.



22

a

L

d

l

C Gm

o

1

1

Figure 5

For the incline-swing test the hull is suspended from a horizontal knife edge and the CGposition a is found from the displacement d1 when a standard mass m is placed at a distance l1

from the axis. The gyradius kp is then found from the period of oscillation.

Figure 6

For the bifilar suspension test the hull is symmetrically suspended by two vertical wiresof length l and spacing 2d. The period of yaw oscillation then gives the yaw gyradius ky

directly.



23

Figure 7 For the Snipes test the hull is balanced on a 3/8" dia. rod and the period of oscillation

with a set of standard springs at the bow is measured.

Figure 8A Stewart 34 undergoing a swing test in New Zealand. Photo by Tom Yates.



24

Lp

Lbs

w

B(t)S(t)

Lb

CENTER OF PITCH

LOAD CELL

SNAP SHACKLE

Figure 9The in the water method uses a string potentiometer B(t) at the bow and S(t) at the stern

to measure the motion following the release from an upward force W near the bow. Lp and the

initial angular acceleration α(0) can be deduced from the bow and stern motion. The torque andhence the effective moment of inertia can then be calculated.



25

20

30

40

50

60

70

0 2 4 6Time Sec

Bow position

Stern position

8 10

P o s i t i o n

Figure 10The bow and stern positions of "Seguin" after release of a 400 lb pull at the bow, see

fig. 9. A nonlinear least squares fit to this data, in terms of damped oscillator heave and pitch

functions plus offset and drift, allows both the initial pitch acceleration α(0) and Lp to be

determined.



26

1350

1400

1450

1500

1550

1600

1350 1400 1450 1500 1550 1600

P i t c h G y r a

d i i

m m

US

Yaw Gyradii mm

F

KC

I

D

Figure 11

The pitch gyradii kp are plotted against the yaw gyradii ky for the FDs at the '88

(triangles) and '84 (diamonds) Olympic Regattas . The close correlation indicates that either kp

or ky could be used to control weight distribution.



27

GYRADIUS meters

1

2

3

4

5

1.35 1.40 1.45 1.50 1.55 1.60

1988

1.35 1.40 1.45 1.50 1.55 1.60

1984

1

2

3

1.35 1.40 1.45 1.50 1.55 1.60

1976

1

2

3

N U M B E R o f H U

L L S

FD GYRADII 1976-88

Figure 12

The distributions of the FD gyradii measured in 1976, '84 and '88 are shown. The 1976and 1988 data are pitch gyradii, while the 1984 data are yaw gyradii. The arrows indicate the

average values. The trend towards increased concentration of the weight is clear.



28

N U M B E R o f H U L L S

1984

1

2

2.5 2.6 2.7 2.8 2.9 3.0

1988

1

2

3

2.5 2.6 2.7 2.8 2.9 3.0

1976

2

3

1

4

5

6

7

8

2.5 2.6 2.7 2.8 2.9 3.0

TRANSOM to CG meters

FD CG POSITION 1976-88

Figure 13

The distributions of the fore and aft positions L of the FD Centers of Gravity asmeasured in 1976, '84 and '88 are shown. The arrows indicate the average values. The decrease

of bow weight of modern FDs is demonstrated.



29

0

1

2

3

4

1988

0.56 0.57 0.58 0.59 0.60 0.61 0.62 0.63 0.64 0.65

0

1

2

1984

0.56 0.57 0.58 0.59 0.60 0.61 0.62 0.63 0.64 0.65

0

1

2

3

4

5

6

1976

0.58 0.59 0.60 0.61 0.62 0.63 0.64 0.650.56 0.57

FD CG HEIGHT 1976-88

PIVOT to CG meters

N U M B E R

o f

H U L L S

Figure 14The distributions of the vertical positions "a" of the FD CGs in 1976, '84 and '88 are

shown. The arrows again indicate the average values. The difference between 1976 and 1988is probably due to the change from wood to fibreglass construction.



30

Figure 15The calculated total moment of inertia of an FD is plotted for different fore and aft

positions of the helmsman and crew. The crew curve assumes the helmsman at L = 1.8 m,while that for the helmsman is for the crew at L = 2.5 m. The solid parts of the curve are thepractical regions.

1350

1400

1450

1500

1550

1600

1650

2500 2600 2700 2800 2900 3000

Transom to CG mm

P i t

c h G y r a d i i

m m

NZ

USKC

F

580

600

620

640

660

680

Figure 16

The FD pitch gyradius kp

is plotted versus the fore and aft position L of the hull CG.

Triangles, diamonds and circles are '88, '84 and '76 data respectively. Contours of constant totalmoment of inertia IT, which assume '88 average values for the other parameters, are also shown.



31



32

Measurers: [1]. P.F.Hinrichsen, [2]. H.Schoop, [3]. M.Oresic, [4]. A.Waine, [5]. J.Clarke, [6]. I.Morton,[7]. S.Forbes, [8]. D. Williams, [9]. R.K.Smithers, [1]0. L.W.Smitt [11]. W.Parks, [12]. A.Watts, [13]. A.Yates

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