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Del the cat:

--- Quote from: willie on March 08, 2021, 11:37:28 pm ---if you draw your ellipse such that the vertices, or the P to Q distance is 120% of the NTN.   (86.5" for a 72" bow), the curve will give a bend that is slightly stiffer in the handle. :OK

this can be a flat very flat ellipse, say one that resembles the bend half way to brace or more. I draw many ellipses on my backboard to compare with occasionally as I tiller out. but getting the right bend at about brace height makes the rest of the job go much easier.

The 120% makes a very nice shape, but you will have to ignore the last 6" or so on each tip. let them be straight and not bend around so much as the ellipse.

--- End quote ---
Yes, nice... but that's the problem with ellipses... just how much ellipse? Personally I'd prefer it a bit less flat (e.g more circular).
It would be nice to get some sort of consensus on what is considered optimal.
Of course the other problem is how far round the ellipse do you expect the bow to follow... obviously not right round to the horizontal line PQ, but the shorter the portion of the ellipse that is chosen, the closer it becomes arc of a circle.
Like I said very hard to define what we actually mean by "elliptical"
I've drawn a circle matched reasonably to the ellipse, up to the point where they intersect... it shows how subtle the difference is over a realistic arc.
Del

Digital Caveman:
I think the important part is having the right radius of curvature at any given point given it's width and distance from the limb tip.  On a pyramid bow this should result in a constant radius which means it will have a circular arc shape.

RyanY:

--- Quote from: tradcraftsman on March 09, 2021, 09:16:32 am ---I think the important part is having the right radius of curvature at any given point given it's width and distance from the limb tip.  On a pyramid bow this should result in a constant radius which means it will have a circular arc shape.

--- End quote ---

This isn’t entirely true but a close approximation. Here’s a post by Woodbear on Paleo Planet from years ago. Without getting technical it’s much easier to just monitor for set and adjust tiller accordingly.

I suppose it depends on what you call a pyramid bow. If the sides absolutely must be straight line taper, then uniform thickness is not going to give you circle of arc tiller, or uniform stress either. In order to get uniform stress and that perfect circular tiller the sides of the pyramid must bulge out a bit. The bow can and should taper as though aiming to get zero width at the nock, but then deviate from this shape and stay at a reasonable width for the last few inches prior to the nock.

A straight line taper to a point with uniform thickness is indeed the shape that gives uniform stress, and a circular bend in a cantilevered beam with a load at the end. But this assumes that the total deflection is small, such as in an architectural building application. The strength of the beam must be proportional to the distance to the load, and the thickness is the same for the length of the beam, so the width of the beam must be proportional to the distance to the load.



However in a bow the bend is large, and the bending force in not proportional to the distance along the bow arm. The bending force is proportional to the distance from the drawn bow string to the bow arm. Since the bow is bending in an arc, the force is not exactly proportional to the distance along the arm. (Note the difference between the straight dotted green line and the bow arm in the diagram.) In order to get circle of arc tiller, and uniform thickness, the width of the bow must be proportional to the distance from the location on the bow arm to the string at full draw. (This assumes a rectangular cross section bow arm pyramid bow.) This is called Hickmann corrected in archery the technical side. If you know the length of bow, and stiff handle you want, you can draw this out to scale on paper, and measure the distances fairly easily. Then use these widths when you lay out the bow. The maximum width of the bow should not make any difference to the need for a convex bulge to the sides, except that the width effects the draw weight.

If you want a pyramid bow with an absolutely straight taper, and also a circular tiller, you will have to adjust the thickness so that the bending strength of the bow arm is proportional to the distance to the string by adjusting the thickness instead of the width. The stress will not be uniform, being greater by the amount that the wood is thicker along the bow arm. If the bow tapers to a non zero tip width, then the tip area will be stronger and deviate from the circular tiller, but this seems to be advantageous most of the time.

I hope this satisfies the desire for a scientific explanation of pyramid tiller, and uniform thickness. The Hickmann corrected pyramid bow is quite elegant in its simplicity of concept. I hope I have conveyed it adequately.

Dave.

Digital Caveman:
I wish I had that book.   

William M.:
Wow interesting things. I gonna read more into that.

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