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I've recently gotten back into shooting slingshots again and have followed with interest the discussion on Chinese tubes. As a retired physicist, I have time to try to make sense of the unexpected result that tapered bands (e.g., 4 strand mixed with 2 strand on each side) can outperform single bands (e.g. 2 strands per side). One can algebraically model a slingshot in which each side is composed of 2 strands in series, with each strand having different stiffness, mass, length, etc.. In the simplified case where the draw force is proportional to the amount of draw beyond the unpulled length (="draw length"), I did not find any advantage to pseudo tapered bands over untapered at CONSTANT draw length AND pull force; (by untapered, I mean constant band configuration all the way between pouch and slingshot "Y") . This calculation includes the mass of the rubber bands, the variation in velocity along the bands, and the mass of the pouch and projectile.

To get constant draw length and pull force for different types of bands, one needs to change the unstretched length of band(s). It seems that many on this forum are not aware of some general rules that can help understand observations. In particular, the pull force at a given draw can be doubled by doubling the number of identical strands side-by-side (i.e., in parallel) or by HALVING the UNstretched length of all bands. Alternatively, the pull force at a given draw length can be halved by halving the number of bands in parallel or by doubling the UNstretched length of all bands. This means that it is very important to carefully monitor the UNstretched length of bands as this can make more difference in stiffness than the band type (e.g., 2040, 1745, etc.).

I'm not sure why I didn't find it favorable to have pseudo-tapered bands at constant draw length & pull force. It is hard to know what the pull force is, so the various speed measurements that are being reported in this forum are hard to do at constant pull force. Also, if the pull force is reduced by tapering, it becomes easier to pull the pouch back and draw lengths are likely to increase as a result. Projectile speed is pretty sensitive to draw length, so an unplanned extra draw length could be contributing to a perception of higher speed for pseudo-tapered.

For the numerically oriented, the stiffness constants k (which have units of force / "draw length" ) combine in parallel like

k.effective = k1 + k2 (k1 & k2 in parallel),

while in series combine like

1/k.effective = 1/k1 + 1/k2 (k2 added to end of k1 (="series" connection)).

So be sure to report unstretched band lengths, and when possible control draw lengths and measure the force needed to achieve such draw lengths.
 

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I want to add to the analysis I described yesterday. There I looked at pseudo-tapered vs standard performance at constant pull force AND draw length BEYOND the unstretched length. A more appropriate comparison is at constant pull force and distance between anchor point (e.g. ear) and slingshot frame. This changes the conclusions but with a twist: pseudo-tapered configurations are beneficial over untapered configurations ONLY if the pseudo-taper has LESS band mass than the untapered-- a pseudo-tapered design with more mass than the untapered performs worse than the untapered design at constant pull force and anchor distance. For example, a standard chinese tube configuration for an untapered design has 2 bands on each side. Relative to that, a pseudo-tapered configuration with 1 plus 2 bands on each side gives better performance, while a pseudo-taper with 2 plus 4 bands on each side gives worse performance. This is not a surprise if you measure the various masses because the rubber bands (e.g. for 2040 bands) dominate all masses. E.g., for a typical untapered design, the bands weigh 8g, the pouch 3g, and the 3/8" steel ball weighs 3.5g.

I've included a pdf document that gives some calculation results plus commentary, and the function that calculated them. I am now comfortable that a rather standard physical model gives reasonable results-- and is much more convenient for understanding trends than building & testing different rubber band configurations. You may want to look only at the calculated results, but to understand the symbols used you may want to read the documentation comments at the beginning of the shown function. An example result:

For an anchor length of 30" & with 13.8 lb pull, a 3/8" steel ball has:
pseudo-tapered: 238 ft/s speed & 6.9 ft-lb energy
untapered: 205 ft/s speed & 5.1 ft-lb energy.
 

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Hrawk,
I am aware that F=k* DeltaX is an approximattion for rubber bands, and that there is hysteresis etc., but one would hope that it would be sufficient to understand the nature of pseudo-taper benefits or their lack, and I think that now this has been illuminated. By calibrating the rubber elastic (i.e., k) properties by using someone's measured velocity data (as I have done), one is closer to getting the velocity right at the expense of pull force. In any case, pull force is usually not measured carefully, and would depend on how long one holds the draw before releasing (because of the hysteresis effects). If there were more observations of time-dependent pull force and energy lost inelastically in the rubber, one might be able to make better fudge factors to account for these extra complications (perhaps an effective mass for rubber bands greater than the actual value?). From what I have seen in brief google searches, F=k*DeltaX is not off by more than ~25% when hysteresis & nonproportionality are included (except at extreme pulls near the rubber breaking point), so I am not motivated to get heroic about improving the calculations I have used, but if anyone has some practical suggestions, I'm all ears.
 

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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.
 

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Henry--
The concern is that your pseudo-taper worked better because you were either exceeding a maximum stretch factor on your pseudo-taper or not being at maximum stretch factor for your untapered design. To get at this, you need draw lengths, rubber type (2040 or ?), and lengths of each section of your pseudo-tapered or untapered designs, and the number of bands in each section. I totally agree that one should not be trustful of untested theory, on the other hand, theories can be a lot easier to design with than experiments once they are established. To establish a theory, one makes predictions and then everyone tries to shoot it down. Happy hunting!
 

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To tie a loop for a taper I use what looks like a leather washer: a 0.75 inch diameter piece of leather (similar to pouch leather) with a 0.25 inch diameter hole punched into it. This leather washer is located at the junction of the looped and single tube sections. One end of the hole in the washer is connected to the single tube that goes to the pouch. The opposite end of the hole in the washer is connected to a folded-in-half single tube that becomes the loop when the 2 ends of the folded tube are fed together into the hole in the washer. I've been using soft cotton thread to tie Body jewelry Twig Jewellery Wire Metal
the tube ends. I find this gives rather precise control over the length of the looped section and is easy to reproduce (e.g., so that both sides of the slingshot have very nearly the same length of their looped sections). I had trouble getting this control with the method described by Tobse, which otherwise worked well and is easy. Another advantage of the leather washer method is that it allows use of different types of tubing in the single and looped sections. For example, it can be beneficial to use 1745 in the single section and 2040 in the loop section. I've attached a picture.

The method shown by Henry recently also looks attractive, but I had a devil of a time trying to get the sleeve tube in place-- maybe wider forceps or forceps with a more wedge-shaped tip would have helped.
 

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My newest configuration is 1745 tubing cut to 17¼". It pitches a .38 cal. lead ball w/34" draw 182 fps, but the clincher is the life. I retired the first set after 1467 shots. It was still working but power was down about 1 fpe. I am on the second set with over 1500 shots and see no real noticeable power lose. I still use double bands pseudo tapered and a cocktail, but for general shooting that single loop of 1745 working great for me and my pocket book is thanking me.
Those are great lifetimes. Can you describe your pseudo-taper tie and, if relevant, your fork tie that lets you get so many shots? (What is a "cocktail"?). Also, can you tell me the lengths of the single and doubled sections of your pseudo-taper that had 182 fps. Also, when you say "single loop" of 1745, do you mean a single strand (e.g., tied at fork and pouch), or a single strand folded back and tied only at the pouch, which really acts like two strands? Thanks.
 

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I use a partial butterfly draw when shooting that comes out to 62in or 157cm. I also shoot 3/8" steel (and 5/8" marbles that weigh close to the same) and I'm looking for the right setup.

I've been using 1745 tubes but they are a little too slow for my liking. I want a FAST setup, something that will turn that 3/8" steel into a penetrating projectile. But I'm a little lost because it looks like Henry's tests were done with a draw that is almost half the length of mine. Should I just scale it up?
You should get Henry's take on this, but to me "scaling it up" would be to use the same stretch factor (ratio of stretched total length to unstretched total length, with same ratio of 1-tube to 2-tube section unstretched lengths). This will give you the same draw force as Henry, but the increased draw length from butterfly will increase the projectile energy by about the ratio of draw lengths, and the projectile velocity by about the square root of that number.
 

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Just sanding and sanding while the tubing arrives... [email protected] is a long time from dankung.
But the result will stays for... im sure
Here's what I do for attaching tubes to a wood slingshot:
. Wouldn't do it any other way.

By the way, over time I have become disenamored with pseudo-tapered tubes- they break too easily, tend to slip creating uneven sides, and take a lot more time to tie. I prefer a single tube on each side of the fork (1745, 2550, or 2050) and use a stretch factor of 5 (e.g., 30 inch draw for unstretched length of 6").
 
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