You know it's a juicy thread when it splits off into other threads. Right my friends?
Anyways...
In the last couple days I have been trying to think of a way to explain why tiller shape should correlate with the width profile of a bow. In this post I will explain why this is focusing primarily on mass movement in a bow limb. I will admit that I make some assumptions here that I don't have the knowledge of physics, engineering, or mathematics to prove on my own. The biggest one being that my initial assumptions about mass movement and efficiency are correct. What I have come up with feels intuitive and straight forward enough that hopefully it is close to what actual tests/calculations would come up with.
Of course I could be way off and have missed some obvious errors. Please let me know what I messed up or got wrong! This is for straight limb designs and I have not explored other designs such as recurves, r/d, etc.
In the direction from fade to tip, the distance the limb has to travel at any point along the limb increases (tip moves more than mid limb, moves more than fade)
Because the distance of limb travel increases in this direction, the effect of mass on the speed of limb return increases in the same direction. (more mass at the tips has the greatest effect).
For pure efficiency of mass movement based solely on tiller shape, the width taper approximates the tiller shape and vice versa. The corresponding limb with taper based on tiller shape can be calculated as follows.
Taper from starting width = (limb movement/tip movement) x fade width (does not account for tip width. May assume a taper to a point at the tip and leave the last couple inches wide enough for a string nock.)
Taper from starting width = fade with - limb with at that point
Limb movement and tip movement are measured as the length of the arc that any point along the limb has to travel.
Example: Limb movement at mid limb is 5.9" and tip movement is 22.5". Fade width is 2". 5.9/22.5 x 2 = 0.52". Therefore the width at mid limb should be 1.48" wide
So now that we can get the width from the tip movement we can do the reverse, finding the optimal tiller shape for mass movement as follows.
Limb movement = (width taper/fade width) x tip movement.
Example: You have pulled your bow to 15" draw. You measure tip movement at 7.5", your width at the fades is 1.5" and the width at mid limb is 1.25" giving a 0.25" taper at mid limb. 0.25"/1.5" x 7.5" = 1.25" of limb movement at mid limb.
You would calculate this for several points along the limb, measure the distance of limb travel at those points, and then tiller to get to the desired shape.
You might figure out that these equations would not work for parallel limb tapers because the taper from starting width would be 0. Because I am basing this solely on mass movement it becomes apparent that parallel width tapers in limbs are inherently inefficient as the constant width does not correlate with changes in movement along the limb.
Stiff outer limbs also do not work with the equations as the abrupt width change that more quickly approaches tip width would assume more movement in that spot. In this case we need to use other tillering logic to account for that shape.
If we take a typical mollegabet tiller profile of half working inner limb and half stiff outer limb and use the first equation to get limb width from the tiller shape, the stiff outer limb will result in a straight taper to the tip. We know that because this is a stiff and nonbending section, this portion of the limb can be as narrow as possible given that it does not have to bend to store energy for the bow. The inner limb could follow the previous taper equation but width should be enough for desired energy storage and low set.
Another finding was that perfectly circular tiller results in great than linear (exponential?) increases in limb movement from fade to tip. The resultant width taper for most efficient mass movement would be a bulging convex curve and not a straight taper from fade to tip. Interestingly this somewhat correlates with the width profile needed for uniform stress and perfectly circular tiller as explained by David Dewey. I cannot assume that these curves would be to the same degree.
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.
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.
Picture included belowDecreasing the width of the bow at any given point will make the bow more efficient for mass movement but could result in set so the width always needs to be wide enough to do the work being asked of it without taking set or taking minimal set.
Radius of the bend limits the thickness that the limb can be without taking set. For example, tiller in the shape of a circular arc can have the same thickness through out it’s length. An elliptical tiller shape with gradually decreasing (tighter) radius of bend will require a taper in thickness along it’s length. Rigid outer limbs should be as narrow as laterally stable and thick enough to remain stiff as thickness is not limited if the limb doesn’t bend.
Speaking of thickness, this dimension takes up mass too. For the sake of consistency I understand that my initial assertions do not account for a thickness taper, focusing solely on width.
I have more thoughts regarding things like paddle bow shapes and strain on the portions of the limbs but it is not quite as fleshed out. I also believe that these with and tiller tapers would result in the necessary mass for low set tillering but I haven't been able to connect this yet.
Thoughts anyone?
