View attachment slingshot3pdf.pdf
Recently I've reported on a physical model of a 2-section (pseudo-tapered) slingshot and its predictions. In the most recent discussion I gave results showing that velocity at constant draw (anchor) distance and pull force was better for a 2-band per side slingshot than for a 2+4 band (per side) pseudo-tapered slingshot, but worse than for a 1+2 band per side pseudo-tapered slingshot. (In my terminology, a long strand shaped into a loop with both ends connected at the pouch is considered to be 2 bands or strands.) Further analysis of this situation exposed new issues, and, I think, one approach for optimum slingshot design.
First of all, while the 1+2 band per side pseudo-tapered design seemed an improvement over the 2 band per side design, the next question was what would be the optimum length of the bands in the 2-band section of the 1+2 band design. Well, the result was that the optimum length would be zero-- which is to say a single 1 band per side (= UNtapered) slingshot is best. So, the overall drive seemed to go in the direction of minimizing the mass of the rubber bands, all the way down to one band per side. But here's the problem: the designs with 1 band per side, or 1+2, have excessive stretch factors (total length of a stretched strand divided by its unstretched length). These stretch factors at rather modest pull forces exceed what seems to be considered the maximum safe value of perhaps 5.0 or 5.5.
It turns out that one can show that the optimum slingshot design for velocity, when constrained by a maximum stretch factor, occurs at a pull force that causes the stretch factor to be the maximum allowed stretch factor. For example, if you have a 30 inch draw distance with unstretched band length of 7.5", untapered, you are operating at a Stretch Factor of 30/7.5=4.0, which is below the maximum stretch factor of (say) 5.5. A better design would use reduced unstretched band length of 30/5.5 = 5.45 inches. Such a design benefits both from allowing a larger pull force, AND lighter bands (because of their shorter length), both of which increase projectile speed.
When considering pseudo-tapered designs, calculations always show that the optimum length of the stiffer section approaches zero for highest velocity; i.e., it is best not to have the stiffer section. So optimum designs constrained by Stretch Factor will be UNtapered, and the choice of 1 or 2 or more strands of "rubber" per side will be set by how much pull force the user is comfortable with.
In the table in the attached pdf file, I show various cases illustrating these points for a 30 inch draw (from slingshot frame) shooting a 3/8 inch diameter stainless steel ball. The rows of the table highlighted in yellow give optimum designs having pull forces of 10.8, 13.8, 17.3, and 21.6 lbs, with corresponding velocities of 208, 230, 254 and 278 ft/s, all at a maximum stretch factor of 5.5. The corresponding configurations (per side) are 1 band of 2040, 1 band of 1745, 1 band of xxxx (=unknown) rubber having 1.6 times the 2040 rubber density (in g/inch), and 2 bands of 2040.
The attached pdf file also goes into my experimental determinations of the stiffness constants k1 (previously referred to as ko), and rubber density (g/inch) for 2040 and 1745 bands. I found k1=1.20 lb/inch for 2040 & 1.52 g/inch for 1745, while densities were 0.247 g/inch for 2040 and 0.317 g/inch for 1745. A priori
, one expects the stiffness ratio of 1745 to 2040 should match the density ratio. In fact they do match: 1.27 vs 1.28. (For example 2 bands of 2040 in parallel would have twice the stiffness and twice the density if they were considered as a single effective band.). This means that one can probably skip the pull force vs draw measurements for types of bands other than 2040 and 1745, and just do mass & length measurements to get the g/inch of the other rubber types, from which you could infer the stiffness relative to 2040 or 1745. (But you need to weigh long lengths (e.g., 30 feet) to get accuracy.) If anyone has such density data for other rubbers, please share.
Using my measured force vs draw (i.e. k1 extraction) and density data for 1745 rubber, predicts a velocity of 199 ft/s for the conditions and setup that "Rayshot" used (earlier in this forum) when he measured 196-199 ft/s at "30 - 31 inch draw" (I used 30 inches in the calculation). So our physical model gives good predictions.
See the attached pdf file for more details.