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I think you’ll find that running an imaginary line from the tips to the center of the grip it is going to be a mirror image of the string itself….
You guys are making this way too complicated….. It’s the “String Angle” we are talking about here. And it’s the angle of the string itself at full draw that matters.
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Yes that imaginary line must mirror the string.
But the problem of stacking occurs because of changing the angle of attack at the tips.
But practically you probably very right also (-; just because of course there must be a direct relationship between all those angles in the square.
Dont wanna be dogmatic here - its just a topic i was interested for a while. Give you the original words of my bowyer physician - maybe someone finds it interesting too:
Concerning the string angle - I guess that you mean the angle between the string and the tangent of the limb at the nock, see fig. 1). If so, this angle really does not matter, or at least not as you think it would. You can imagine a small circle at the end of the limb in front of which the string is attached (at the back side). Then the angle is always zero, because the string is always tangent to the limb irrespective of the shape of the limbs (recurve, straight, etc.) and stage of drawing. Obviously this can not affect the shooting characteristics of a bow - a straight limbed bow will perform as a straight limbed bow.
What really matters is the angle between the string and the line segment connecting the middle of the bow and the point where the string is attached to (the nock), see fig 2). The shape of the force-draw curve, fig 3), is mostly determined by this angle, NOT the string angle (other quantities on which it depends include material properties etc.).
You can always approximate the limb of a bow by the line segment connecting the middle and the nock. If you recurve a bow, it acts as adding any kind of (non-contact) reflex into the bow. You pre-stress the material, which results in a force-draw curve, which is more concave, thus you store more energy in the bow (total energy is the area under this curve). This is the reason why it seems to you that low string angle "adds" energy to the bow.
From this consideration it also follows that reflexing the handle and tips has no effect when the limbs are deflexed and the overall reflex is zero. It has no advantage, except it looks fancy.
On the other hand, when you feel "stacking" it can be caused by several factors. 1) The bow is short - the angle phi in my drawing is low, so the cos(phi) gets high and the drawing force rises rapidly. This happens in short composite bows.
2) The bow is deflexed - the force-draw curve is more convex - see my drawing. This is the typical cause of stacking in longbows.
3) You push the material beyond the linear dependence of force on the strain (Hook's law) and it gets stiffer.