Graphically Designing a Bow Version 2.0

[Wooden Spring]

Because my better judgement is usually no better than my worst judgement, and because people keep asking me to, here is the two pager summarizing some important parts of Archery: The Technical Side. This one will stay up, but I probably won't be commenting so as to not create controversy. As I said before, I do not wish to create offense. 

Taken from: Archery: The Technical Side
by C.N. Hickman, Forrest Nagler, and Paul E. Klopsteg

This document describes a graphical method for laying out a bow in order to mathematically produce a width and a thickness for any wood of any draw weight and of any draw length. Since wood is a natural material and subject to variation, these numbers should serve as guidelines only rather than an exacting reference. Please note these formulas are for rectangular cross section self bows that lay flat when unstrung – any backing, reflex, or string follow will alter the finished bow to some degree.

Referring to the attached picture:
BT = Bow limb from fade to nock
R = Center of path of T, of radius t = (3/4)(BT)
G = Length of arrow
H = Brace height of string
CA = ET¹, Upper half of string
Arc BT¹ = The drawn limb; it is the arc of a circle of radius r, with the center on the perpendicular BF to BT¹, and on the perpendicular bisector of BT¹ 
D = Distance from fade to string measured perpendicularly to string at full draw along BD

To Draw the Diagram:
1)   Establish your draw length G, and draw a vertical line from P to E.
2)   The length of the bow from nock to nock can be established based on your draw length (see below). When you decide on handle length and fade length, the actual bow limb length BT can be drawn. PB is the length of fade and ½ of handle length.
3)   Once PB and BT has been drawn, calculate R. This is BT times .75
4)   Draw radius t at R
5)   Decide on your brace height, H. This can be established by your fistmele – in giving a “thumbs up,” the distance between the tip of your thumb to the bottom of your hand, say, 6”. Once you have decided on your brace height, offset PT by this distance to obtain CA
6)   Since CA = ET¹, draw an arc from E of length CA to intersect the arc t
7)   Draw a construction line from B to F (F will be established next, for the moment, pick an arbitrary distance straight down)
8   Draw a construction line from B to T¹
9)   Draw a construction line as a perpendicular bisector of BT¹, and extend it to meet BF at F
10)   The bending radius of the limb can now be drawn. Draw an arc r with center at F from B to T¹. This will be R used to determine the Thickness in the formula
11)   Draw BD. This line is drawn from B and perpendicular to ET¹. This will be D used to determine Width in the formula

General Notes for any given draw:
The Modulus of Elasticity of a wood fixes the relation between curvature, thickness, and stress.
1) Bow length limits limb curvature
2) Limb curvature limits thickness (for any given wood)
3) Thickness determines stress (for any given wood)
4) Thickness then practically determines the cast of the bow (for any given wood)
5) Width fixes power of bow

The back of the bow is laid out such that a given width at the fades tapers in straight lines to a point of zero width at the nocks. The nocks are then given a width of 1/2", and parallel lines are then drawn from here, back to the tapers.

The thickness remains constant for the length of the tapers, then it tapers where the limb is parallel in order to maintain a perfect arc of a circle tiller.

The handle is 4" long by 1 1/4" deep by 7/8" wide with 2" long fades. Handle cross section is shown to the left of the bow - the narrow portion faces the archer.

Draw Length
Your height divided by 2.5

Draw Weight
This is personal preference

Bow Length
Generally, 68" NTN for a 28" draw.  Adjust bow length 2" for every 1" difference in draw length

Bow Thickness
R/T = E/(2S)      Solving for T yields:   T = [(2S)R]/E
                          
Where:
R = Radius of curvature in inches (r in drawing)
T = Thickness
E = Modulus of Elasticity
S = Modulus of Rupture

Bow Width
PD = (SWT²)/6   Solving for W yields:    W = (6PD)/(ST²)

Where:
P = Tension in string (12% more than draw weight)
D = Distance in inches from fade to string measured perpendicularly to string at full draw (BD in drawing)
S = Modulus of Rupture
W = Width
T = Thickness

[blackhawk]

Welp...since no one else wants to tango on this anymore i will..and all i have to say is this....

     ..... 

[willie]

Thanks for putting this up, I have come across various references to this Book, and always wondered about where wood tech was going before the invention of f****g****.

[Wooden Spring]

Thanks for putting this up, I have come across various references to this Book, and always wondered about where wood tech was going before the invention of f****g****.

Nathanael Herreshoff referred to fiberglass as "frozen snot."

[Marc St Louis]

I can bring the deleted thread back unless you prefer to leave it where it is

[AndrewS]

@wooden spring

thumbs up

[PlanB]

Thanks for putting this up, I have come across various references to this Book, and always wondered about where wood tech was going before the invention of f****g****.

Nathanael Herreshoff referred to fiberglass as "frozen snot."

L. Francis.

[PatM]

  Comparing a fiberglass boat made on a mould with the inside left rough has little relation to the glassing technique used on bows.

[scp]

I have no way of figuring out whether the formulas are optimal or not. For that matter, I don't even know how to use them to design better bows. I would really appreciate if someone provide a more detailed description of how to apply them properly.

[willie]

It looks like the formulas are for any bow limb of radius "r", whether you should consider them optimal, probably depends on your preferred amount of set and/or any safety  factors you may wish to employ.

to find radius r, it seems to me that you must graphically locate T', shown at the end of arc t. The drawing might be easier to understand if the draftsman had shown an arc of radius E T', swung from the arrow nock at full draw and intersecting arc R T, at T', rather than a straight line. Once T' is graphically located, the chord T' B is drawn and bisected. Then a perpendicular is erected to find the intersection F with a vertical dropped from the fade. The distance F T' is the radius of curvature for the limb, used in the calculations. maybe the book has more

and if you have a slide rule, you can compute limb width and thickness without a tedious arithmetic or an electronic calculator.        

willie

[Wooden Spring]

Thanks for putting this up, I have come across various references to this Book, and always wondered about where wood tech was going before the invention of f****g****.

Nathanael Herreshoff referred to fiberglass as "frozen snot."

L. Francis.

DOH!  My bad...   Well hey, at least I got his last name right, and that's hard enough to spell! 

[Wooden Spring]

Willie, thanks for the elaboration... I pretty much just copied the drawing out of the book, but I suppose some drawing instructions would not go amiss. I modified the original post to include this:

To Draw the Diagram:
1)   Establish your draw length G, and draw a vertical line from P to E.
2)   The length of the bow from nock to nock can be established based on your draw length (see above). When you decide on handle length and fade length, the actual bow limb length BT can be drawn. PB is the length of fade and ½ of handle length.
3)   Once PB and BT has been drawn, calculate R. This is BT times .75
4)   Draw radius t at R
5)   Decide on your brace height, H. This can be established by your fistmele – in giving a “thumbs up,” the distance between the tip of your thumb to the bottom of your hand, say, 6”. Once you have decided on your brace height, offset PT by this distance to obtain CA
6)   Since CA = ET¹, draw an arc from E of length CA to intersect the arc t
7)   Draw a construction line from B to F (F will be established next, for the moment, pick an arbitrary distance straight down)
8   Draw a construction line from B to T¹
9)   Draw a construction line as a perpendicular bisector of BT¹, and extend it to meet BF at F
10)   The bending radius of the limb can now be drawn. Draw an arc r with center at F from B to T¹. This will be R used to determine the Thickness in the formula
11)   Draw BD. This line is drawn from B and perpendicular to ET¹. This will be D used to determine Width in the formula

[willie]

Wooden Spring

Does the book go into much detail about working practice for the builder?  I think I read somewhere that the limb thickness, as calulated with these formulas, is at the max strain.

I did note the caveat.
 

these numbers should serve as guidelines only

willie

[Wooden Spring]

Does the book go into much detail about working practice for the builder?  I think I read somewhere that the limb thickness, as calculated with these formulas, is at the max strain.
willie

On page 190, under Forrest Nagler's article on Bow Design, he does not go into much more detail other than to say: "With any wood there is a maximum permissible combination of thickness and curvature that will avoid failure."

The relationship of thickness to curvature is defined as /T) = (MOE/2MOR). Any one wood or any other homogeneous material will have its own combination figure which one cannot go below without incurring excessive stress. Each side of the equation must yield the same result, ergo, when an exacting number for the ratio of MOE to MOR can be obtained through experimenting on the very piece of wood you intend to make a bow with, and the radius is also known since that number is ultimately determined by the archer's draw length, then the thickness can be readily determined.

For example, if the ratio (MOE/2MOR) = 55.21, such as with Argentine Osage Orange, then the ratio /T) must be the same, /T) = 55.21. If the ratio of R/T goes less than the ratio of MOE/2MOR, then the bow will incur excessive stress.

[willie]

the book is called the "technical side", so they seem to have the failure point worked out for any particular stiffness.
 I  thought that there might have been some discussion on the proportional limit of wood, as the practical working limit for a bow is usually  "how much set did it take"? Perhaps it is assumed to be some percentage of the MOR?  Aren't the differences in how far the wood can bend before taking set where we so often find our preferences for woods like osage or yew?