bird's mouth spars revisited

8/11/2019 Bird's Mouth Spars Revisited

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7/11/13 Bird's Mouth Spars revisited

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Design ReviewDatabase

Bird's Mouth Spars revisited

By Gatan Je tt

Contents

What is a bird's mouth spa r?Some limitsand possibilities exploredSimple s ize calculations, the manual methodEasiest size calculations, let your PC do the job!Dealing with a tapered mastWeight saving and s trength lossAppendix

Introduction

A few months ago, a n interesting articleon bird's mouth spa rs was p ublished right here, atDuckworks. The author, David Farless, p resented a series of equa tions to accurately calculate thesize o f such a construction. This ha s inspired me to study this subject in greater depth. In thisarticle, we will explore som e o f the limits and potentials o f bird's m outh spa rs. You will also beprovided with tools to let your computer do the size calculations for you. But first, a quickreminder...

What is a bird's mouth spar?

A bird's mouth spa r is a way to ma ke a hollow mast or spar from wood for use on a sailboat.Figure 1 below shows what the cross se ction look s like. It typically uses 8 identical pieces of woodor staves glued together. The outside is then trimm ed to produce a round spar. The m ain goal isto save weight.
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Figure 1- Bird's Mouth Spar : Definitions

O.D.is the outside diameterI.D.is the largest inside diameterR1is the shortest inner radius

R2is the largest inner radius o r half I.D.

Nis the numbe r of sides (eight in this case)(alpha) is the angle between a djacent staves or 360 / N

Lis the width of materialHis the thickne ss of m aterial

Important Ratios

Kis the thickness to width ratio of staves or (H / L)Mis the "conversion factor" or (O.D. / L)

A is the inside to outside diam eter ratio or (I.D. / O.D.)

The d efinitions us ed he re are pretty much the same that were used in the previous article. Ihave on ly added a few names for the ratios that will be of interest here. The m odel sh own inFigure 1 is of "classic" proportions, that is, a s tave's thicknes s ratio of 0.5 in an 8-sidedconfiguration. Although I have never se en other configurations, it is po ssible to b uild a bird'smouth s par with a different number of side s. Figure 2 shows so me of the pos sibilities.


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Figure 2- The world of Bird's Mouth Spars

Oka y, you might think that the ma th for size calculations would be far too complex for suchconfigurations, right? Well, no! The equa tions presented by David Farless in the previouslymentioned articlehad a hidden potential. With just a m inor modification, these e quations canhandle any pos sible configuration! From now on, we will refer to these as the Farless e quations.(Always give credit when credit is due!) The revised equations can be found in the Appendixsection.

The trick for building a spa r with more or less than 8 sides is in the V-notch angle s use d on thestaves. Figure 3 shows them in a little bit more detail than Figure 2. Table 1 gives you theangles u sed a nd a few other points to take in consideration.

Figure 3- Stav e shape

Table 1- Basic rules for bird's mouth spars

Numberof sides

Angles for theV-notch (degrees)

Theoretical limitsfor ratio(K) or (H /L) Practical range

for ratio (K)N

Angle1

Angle2

Minim um Maxim um

5 72 18 0.1407 1.0515 not recommended

6 60 30 0.1200 1.1547 0.195 - 0.85

8 45 45 0.0929 1.4142 0.15 - 1

9 40 50 0.0834 1.5557 0.135 - 1.15

10 36 54 0.0757 1.7013 0.125 - 1.312 30 60 0.0637 2.0000 0.105 - 1.5

15 24 66 0.0515 2.4586 0.085 - 1.875

16 22.5 67.5 0.0483 2.6131 0.08 - 2

18 20 70 0.0431 2.9238 0.07 - 2.25

20 18 72 0.0389 3.2361 not recommended

Angle 1, shown in Figure 4 below, is located o n the inside of the s par. Its value is always thesame as " " (alpha), shown in Figure 1. Angle 2is on the outside and is equal to 90 degreesminus the first angle.

Figure 4- Stave Angles

Table 1 also lists minimum and maximum limits for the thickness ratio of the staves (K). Theminimum theoretical limit is reached when I .D. is equal to O .D. Figure 5 below shows an exa mpleof K going too far in that direction. In practice, you obviously need to use a h igher ratio to havesome thickness left! Otherwise your spar will fall apart when you try to round the outside. That'swhere the p ractical range comes into play. The minimum practical limits shown in Table 1corresponds to the m axim um weight saving possible for a given s trength. (More on that later.)
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Figure 5- Going too far: K is too low, I.D. is larger than O.D.!

The m axim um theo retical limit is reached when I.D. equa ls zero (See Figure 6B). You canactually exceed that limit: the inside will open up again (Figure 6C ). However, the formulas forcalculating size can produce som e weird results when you go in that zone . Better stay below thatmaximum limit. Here aga in, a practical upper limit for K has bee n shown. This max imumpractical limit was chosen at the point where the weight saving falls to 5 percent compared to asolid spar of e qual strength. (More on that later too.)

Figure 6- Going too far, high v alues of K

A- "Practical Maximum"

B- "Theoretical Max imum", I.D. = 0

C- Gone too far, formula results unpredictable

You may have noticed that both the 5 and 20-sided configuration have the comment "notrecomm ended ". Here is why. For these, on e of the angles use d is very sharp. This me ans thatthe corresponding side of the V-notch becomes very thin (See Figure 3). Th is ma y be too fragile,at least during asse mbly. But this ma y very well be a weak spot unde r stress, even when gluehas cured. This potential problem would only get worse for any larger number of sides and s etsthe limit of what you can do. T his is one reason why 8-sided s pars are far mo re common. For an8-sided spar, the angles are equal and each side is of equal strength. And the glue area of theV-notch is also the largest at 45 degrees. So it is probably not a good idea to go very far fromthat 45-degree value. It may be wise to use a number of sides between 6 and 12. The smallestangle ne ver falls be low 30 degrees with these .

Configurations using 7, 11, 13, 14, 17 and 19 sides a re not discussed here: the angles requiredwith these are just too odd.

Now that we ha ve looke d at som e of the limits of bird's m outh spars, let's look a t ways tosimplify size calculations.

Size Calculations

For those of you who have read the previous articlewritten by David Farless, you already knowthat accurate size calculations are not really simple. That a rticle also mentions a "rule-of-thumb"solution originating from a W oodenBoat article (I must admit that I have never read that article).The o utside diameter was de fined as 2.5 times the stave width. However that solution is notentirely accurate. In fact, both the width and thickness of the s taves have an effect on theresulting diame ter.

Wouldn't it be nice to have a solution that offers the simplicity of the Wo odenBoa t rule-of-thumband the accuracy of the Farless equations at the s ame time? Better still, wouldn't it be great to

'
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,

First, our new rule-of-thumb. If you look back a t Figure 1, among the d efinitions, there is a ratiocalled "M". It is defined a s O.D. divided by L. That is exactly the sam e thing as the "2.5x " fromthe Wo odenBoa t article. W e "on ly" need to figure o ut what is the real value for "M". Tha t ratio isdefined in the Appendixif you are really "math curious". Table 2 below, however, will save youtime. It gives you pre-calculated "M" values fo r 6 to 12-sided spa rs. The cells are highlighted inpinkwhen K falls outside the "Practical Range " defined in Table 1. You should avoid these. Novalues a re shown when K falls o utside the theoretical limits of Table 1.

Table 2- Improved "Rule-of-Thumb" : Conversion Factor

(M)

StaveThicknessRatio (K)

(M)6 sides

(M)8 sides

(M)9 sides

(M)10 sides

(M)12 sides

0.100 - 2.444 2.771 3.097 3.745

0.125 1.795 2.451 2.777 3.102 3.749

0.150 1.807 2.458 2.783 3.106 3.752

0.175 1.820 2.465 2.788 3.111 3.755

0.200 1.832 2.473 2.794 3.116 3.759

0.225 1.845 2.480 2.800 3.121 3.762

0.250 1.857 2.487 2.806 3.125 3.766

0.275 1.870 2.495 2.812 3.130 3.769

0.300 1.882 2.502 2.818 3.135 3.772

0.325 1.895 2.509 2.824 3.140 3.776

0.350 1.907 2.517 2.829 3.145 3.7790.375 1.920 2.524 2.835 3.149 3.782

0.400 1.932 2.531 2.841 3.154 3.786

0.425 1.945 2.539 2.847 3.159 3.789

0.450 1.957 2.546 2.853 3.164 3.792

0.475 1.970 2.553 2.859 3.168 3.796

0.500 1.982 2.561 2.864 3.173 3.799

0.525 1.995 2.568 2.870 3.178 3.802

0.550 2.007 2.575 2.876 3.183 3.806

0.575 2.020 2.583 2.882 3.187 3.809

0.600 2.032 2.590 2.888 3.192 3.812

0.625 2.045 2.597 2.894 3.197 3.816

0.650 2.057 2.605 2.900 3.202 3.819

0.675 2.070 2.612 2.905 3.207 3.822

0.700 2.082 2.619 2.911 3.211 3.826

0.725 2.095 2.627 2.917 3.216 3.829

0.750 2.107 2.634 2.923 3.221 3.833

0.775 2.120 2.641 2.929 3.226 3.836

0.800 2.132 2.649 2.935 3.230 3.839

0.825 2.145 2.656 2.940 3.235 3.843

0.850 2.157 2.663 2.946 3.240 3.846

0.875 2.170 2.670 2.952 3.245 3.849

0.900 2.182 2.678 2.958 3.250 3.853

0.925 2.195 2.685 2.964 3.254 3.856

0.950 2.207 2.692 2.970 3.259 3.859

0.975 2.220 2.700 2.976 3.264 3.863

1.000 2.232 2.707 2.981 3.269 3.866

1.025 2.245 2.714 2.987 3.273 3.869

1.050 2.257 2.722 2.993 3.278 3.873

1.075 2.270 2.729 2.999 3.283 3.876

1.100 2.282 2.736 3.005 3.288 3.879

1.125 2.295 2.744 3.011 3.293 3.883

1.150 2.307 2.751 3.017 3.297 3.886

1.175 - 2.758 3.022 3.302 3.889

1.200 - 2.766 3.028 3.307 3.893

1.225 - 2.773 3.034 3.312 3.896

1.250 - 2.780 3.040 3.316 3.900

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1.300 - 2.795 3.052 3.326 3.906

1.325 - 2.802 3.057 3.331 3.910

1.350 - 2.810 3.063 3.336 3.913

1.375 - 2.817 3.069 3.340 3.916

1.400 - 2.824 3.075 3.345 3.920

1.425 - - 3.081 3.350 3.923

1.450 - - 3.087 3.355 3.926

1.475 - - 3.093 3.359 3.930

1.500 - - 3.098 3.364 3.933

Choo se the stave thickness ratio and the num ber of sides that you want to use, look up thecorresponding value of "M" in the table, and you have e verything you need to do the sizecalculations. Let's do an example.

Example 1Let's say we want to build a 3-inch diameter, 8-sided spar. If we choose a stavethickne ss ratio (K) of 0.5, the resulting value for M is 2.561 (highlighted in yellowin Table 2). The value for the stave width (L) is then:L = O.D. / ML = 3 inches / 2.561 = 1.171 inches

Since we chose a s tave thicknes s ratio of 0.5, the thickness of the stave (H) is(0.5 X 1.171) or about 0.586 inches.

You can do wnload a text version of Table 2and print it on a single page. (You m ay have toadjust the page m argins and the font size to achieve that). With this table at your disposal, youcan do the calculations by ha nd if you have to.

Let's now look a t the eas iest me thod: your PC doing the calculations for you!

Bird's Mouth Spar Size Calculators

Instructions

Choo se the Calculator that suits your needs. Enter the values of your choice on the left, thenclick on the "Ca lculate!" button. Re sults will appea r on the right. An error messag e will appea r inthe Status box if your inputs do n't work out. You can clear all entries by clicking on the Re load o rRefresh button of your Web browser. Or you can just enter new inputs and click on "Calculate!"again. C licking on the pictogram in the m iddle will lead you back to Figure 1, in case you ne ed areminder for all those definitions. Dimensions must all be in the s ame units. Ex.: Inputs ininches give results in inches.

Hints to avoid error messages (in the Status box)

Input values must be num bers higher than ze ro (no negatives, no letters).The numbe r of sides m ust be 5 or more.

I.D. must be sma ller than O.D. (obviously).H or K mus t not be too sma ll or too large (less obvious).H must be sma ller than the radius or half the diameter.

Calculator 1: Given N, calculate the two angles

used for the V-notch on the staves.

Input

N:

(Number of sides)

Calculate! Results

Angle 1:

degrees

Angle 2:

degrees

Status :

Ready

Calculator 2: Given N, O.D. and I.D., calculate L, H and K

Inputs

N:

Calculate!

Results

L :

H :
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O.D.:

I.D.:

A :

K :

M :

Status :

Ready

Calculator 3: Given N, L and H, calculate O.D., I.D. and K

Inputs

N

:

L

:

H

:

Calculate!

Results

O.D.:

I.D.:

A :

K :

M :

Status :

Ready

Calculator 4: Given N, O.D. and H, calculate L, I.D. and K

Inputs

N :

O.D.:

H :

Calculate!

Results

L :

I.D.:

A :

K :

M :

Status :

Ready

Calculator 5: Given N, O.D. and K, calculate L, H and I.D.

Inputs

N :

O.D.:

K :

Calculate!

Results

L :

H :

I.D.:

A :

M :

Status :
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Calculator 6: Given N, L and K, calculate O.D., H and I.D.

Inputs

N

:

L

:

K

:

Calculate!

Results

O.D.:

H :

I.D.:

A :

M :

Status :

Ready

Dealing with a Tapered Mast

There is one case where the Ca lculators above will prove handy: building a mast with a tape r. Ataper means that the mast diame ter will gradually be reduced in size as you go up. I have neverbuilt a bird's mo uth mast so far; I ha ve only drafted a few in TurboCAD. So, I thinkthe easiestway to build a tapered bird's mouth m ast is to cut a taper in the width of the s taves, leaving the

thicknes s constant all the way. Figure 7 below illustrates this. As you can gues s, the le ngth hasbeen condensed to better show what is going o n.

Figure 7- Creating a taper : j ust cut away the yellow section

The yellow section would be removed to produce a tap ered ma st. This is the e asies t cut to do:your piece o f wood would be lying flat on its largest side . It sure be ats cutting a taper on the V-

notch side. It is also the be st compromise for strength. By cutting this way, the thicknes s ratiowould go up, m inimizing the strength loss caused by a sm aller end diame ter. It also k eeps theglue joint surface at its maximum.

Let's do an e xam ple. You can try it for yourself as we go a long.

Example 2

Step 1

Let's say we want to build a 3-inch diameter mas t with a tapered top diam eter of 2 inches. Wewill do a 12-sided ma st this time . We can start with the bo ttom section of the m ast using e itherCalculator 2, 4 or 5. Let's sa y we want to use O.D. and I .D. as inputs. This m eans we m ustchoose Calculator 2 to do the job. Let's choose an I.D. of 1.75 inches. The results obtained forL and H are:

L = 0.780 inch at the bottomH = 0.655 inch

Step 2

Now, for the final dimens ions at the top, we use C alculator 4 and enter our final O.D. of 2inches and the value for H we have just calculated. The final stave width (L) we obtain for thetop is:

L = 0.512 inch at the top
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So we have to remove (0.780 - 0. 512) or 0.268 inch at the top, gradually reducing to zero atthe point where we go back to the full 3-inch diam eter.

This approach could be ex tended to multiple levels of taper.

Weight Saving and Strength Loss

We now arrive at the core o f the subject: the weight saving that a bird's mouth spa r can a chieve.The information presented h ere is base d in part on the "Farless equa tions", with one refinement.The formulas for relative m ass and s trength presen ted in that article were (and still are) firstapproxima tions. In the case of the strength formula, that approxima tion is also the worst-casefigure, so it is a goo d idea to k eep it as is.

The relative mas s equa tion, though, can't be used to compare relative ma ss (or weight saving)

for various configurations. It doe s not take into account the num ber of sides. Figure 8 belowshows you why we m ust take that in account. It shows, using the sam e I.D. and O.D., a 6 - and a12-sided spa r, superposed. The 12-sided spa r area is s hown in yellow; the 6-sided spa r hasadditional area (in grey). Although these two ex amples ha ve the same theoretical strength, the6-sided spa r is heavier.

The accurate formula for weight saving is s ubstantially longer than the first approxima tion (seethe Appendix Section).

Figure 8 - Cross section area comparison, 6- a nd 12-sided configuration

So how does a bird's mouth spa r compa re to a solid spa r in terms of weight saving and strength?To a void comparing apples and oranges, we will use the ratio A or (I.D. / O.D.). First, Figure 9below compares bird's mouth spars to a so lid round spa r of the sa me size or diame ter. We cansee that the weight saving increas es with the numbe r of sides. The difference is m ore importantfor high values o f ratio A (thinner spar). The weight saving can loo k fantastic there, if you forgetto look at the s trength loss curve (in grey), which is the same for all configurations. Yo uobviously don't get som ething for nothing. As ratio A come s closer to unity, you s tart to loosestrength faster than you loose weight. Not an improvement. The range use d for ratio A in these

graphs stops at 0.2: a t that point, the weight saving falls to about 5%, our "practical limit" for

maximum K (See Table 1).
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Figure 9 - Weight saving compared to a solid round spar of same s ize ...Round Exterior

There are two solutions to get a big weight saving without losing too much strength. First, use alarger diameter than the original solid spar. You can a chieve the s ame theoretical strength andstill save weight. Sounds too good to be true? There is o ne d rawback: you e nd up with a largerspar. The la rger the weight saving, the larger that spar becomes. You might end up with a sparthat is just too large to be practical. Figure 10 below shows you what you can expect to achieve.The grey curve s hows you ho w much bigger the s par mus t be to a chieve the corresponding weightsaving. We can see, for example, that a 12-sided spar can achieve a maximum weight saving of70%! But to do so, its diame ter must be abou t 90% larger than a solid spar! A less e xtremeexa mple, the "classic" configuration (8-sided, K=0.5), requires a size o nly about 5% larger forequivalent strength, while saving about one-third in weight. Not bad!

(A note, here, you can use the Size Calculator #2 to figure what is the value o f K for a given ratioA. Enter the num ber of sides you want to use, enter "1" for O.D. and enter the value A in the"I.D." input box. Th ere is also a conversion table in the Appendix s ection.)

Figure 10- Weight sav ing compared to a solid round spar of same strength ...Round Exterior

This graph reveals so me thing interesting. As you go for a thinner spar, the weight saving goe sup, until it reaches a p eak . That peak, b y the way, is the practical minimum ratio K proposed in

Table 1. If you try a thinner value pa st that point, the weight saving starts to fall down again.


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This happens more or less in the graph area located on the left side of the "Size Increase"curve. You should a void using values in that area: no t only you won't save more weight, but alsothe size increase becomes enormous! There are two explanations for this phenomenon. Thisproblem is due to the fa ct that the inside and outside s hape are different. If both the inside andoutside were the sam e shape , the weight saving would continue to climb on that graph. Butbecause of the "corners" or vertex of the polygon shape, you e nd up with a big differencebetween minimum and maximum thickness. Those vertex are weak spots that put a limit onoptimum weight saving. Another way to look a t it is in the amo unt of ma terial that you mus tremove to obtain a round exterior (See Figure 11). As a 12-s ided spa r is much "rounder" than a6-sided, there is less waste. But even with a large numbe r of sides, fo r very thin staveconstruction, you waste more ma terial, weakening your spa r.

Figure 11 - Outside waste that must be remov ed to obtain a round exterior

The graph of Figure 10 tells you how to a chieve the bes t weight saving, for a given strength. Thehighest results are a chieved with a pretty large increas e in s ize, perhaps too large. In real life,you would probably want a good compromise between weight, strength and size. Figure 12 belowis an attempt at finding such a compromise. This Arbitrary Score is not a "scientific" calculation.The curves s hown were obtained by subtracting the Size Increase curve from the Weight Savingcurves of Figure 10. This is a "two-thumbs up" kind of rating. It rewards good weight saving butpenalizes large size increases. It doe s shows where the overall good compromises are. You haveto take the results with a grain of s alt. You can get equal scores for very different weight savingvalues. O n the le ft portion of the graph, high score values reflects great weight saving. O n theright portion o f the graph, high score value s reflects m inimal size increase. If weight saving ismore important to you, go "left". If you want to replace an e xisting solid mast and you a relimited in space, go "right".


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Figure 12- Arbitrary Score compared to a solid round spar of same strength ...Round Exterior

The se cond method to max imize strength is to keep the exterior in the sa me polygon shape asthe inside (See Figure 13). T his is not as pretty, but you don 't hit the sam e weight saving limitas a rounded bird's mouth spa r. That's because the ratio A (I.D./O.D) is constant all around. Andfor a beginner like m e, it might be e asier to build. A 12-sided spa r done this way wouldn't looktoo bad. For weight saving and strength comparison, we will use the minimum diame ter for thatpolygon. This will allow us to use the sam e overall size calculations a s a rounded spa r.


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Figure 13- Polygon exterior shape

Figure 14 compares a po lygon-shaped b ird's mo uth spar to a solid round spa r of equivalentstrength. You can see , if you compa re with Figure 10, that the weight saving can reach highervalues. And the increase in size required to m aintain strength is sm aller. That's for high valuesof A. Although the graph stops at a m axim um of o ne, ratio A can actually exceed one with such aconfiguration. That's because I.D. has been de fined in diagona l, while O.D. is mea sured straightacross.


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Figure 14- Weight sav ing compared to a solid round spar of same s trength ...Polygon Exterior

Table 3 below gives you the maximum limit for (A). At that point, the stave thickne ss ha sreached zero.

Table 3- Maximum theoretical valuesfor ratio (A) - Polygon Exterior

Numberof sides (N)

Maximum Ratio (A)(I.D. / O.D.)

5 1.2361

6 1.1547

8 1.0824

9 1.0642

10 1.051512 1.0353

15 1.0223

16 1.0196

18 1.0154

20 1.0124

When ratio (A) and the hollow area be comes s ma ll, you can reach a point where you no longersave weight! You end up with a higher weight than a solid round spar. So, for a thick stave, this

configuration is no t a go od idea ; the rounded version is a be tter choice for that.

Figure 15 below shows the Arbitrary Score for a po lygon shape . The peak s core is a bit higherthan for a rounded version. But you hit negative values a t the right end of the graph, just likeFigure 14.


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Figure 15- Arbitrary Score compared to a solid round spar of same strength ...Polygon Exterior

The o utside waste, for a po lygon exterior, is obviously much smaller than for a round exterior, ascan be se en in Figure 16 below. The waste increases a s the hollow area be comes sma ller.


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Figure 16- Outside w aste for a polygon exterior shape

You may have noticed, throughout these graphs, a large gap between the 6-sided and 8-sidedconfiguration. That's because no 7-sided curves were done . This gap a lso tells you that an 8-sided spa r would be a much better choice than a 6-sided one.

All the strength calculations are theoretical strength values . The formulas u se what is called themom ent of inertia. This is a fa ir estimate a s long as the stress is uniformly distributed. In reallife, there are always some localized stress points, like at the m ast partner, for exam ple. So youmay have to add some e xtra reinforcements in critical areas to avoid ba d surprises.

This abo ut covers what I had to sa y about bird's mo uth spars. I hope you ha ve found this"exploration" interesting. Feel free to ha ve a look at the Appendixse ction where the formulasused a nd the graph results in Table format are shown.

- The End -
http://www.duckworksmagazine.com/04/s/articles/birdsmouth/appendix.htm

bird's mouth spars revisited

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