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My kids are one of my main driving forces. They ask pertinent questions that I might otherwise not think to ask myself and they won't stop till you run out of answers.

Why is the Earth round? Because things stick together and the most efficient shape for a clump is round. Why? Because of gravity. Why? because mass curves space.
Why is the sky blue? Because blue is refracted more by the atmosphere and oranges and reds less. Why? Blue light is at the higher frequency end of the visible spectrum.
Why are you cutting the bands narrower at that end, Daddy? Because it makes them faster. Why are they faster?

I don't know.

Maybe it's because there's less inertia on a taper at the end that must accelerate most? But that doesn't explain why I get the impression that heavy shot seems to still move quite a bit faster.

Maybe there's more elastic potential energy stored in the band.

Let's get a tape measure, clamp, a plastic bag and a whole heap of lead sinkers and see.

So I measured and noted the elongation of a 25mm (1") wide strip of 0.65mm (.0256") latex exercise band that was 200mm (8") between the clamps as I added weight. I did the same for a band that tapered 25mm (1") to 12.5mm (.50").

Here is the chart:

Rectangle Slope Font Plot Parallel


That's a pair of classic stress-strain curves for an isotropic elastomer. You can see it takes a little force to get the polymer out of the unstressed state and then a flatter section as the molecules align and then it takes a lot of force to continue stretching as the polymer begins to crystallise at the elastic limit. From here on it gets harder and harder to stretch till the band breaks.

Next I chopped the data about.

Here's an estimation of elastic potential energy plotted against elongation.

Slope Rectangle Font Plot Line


What I did was find the area under the first chart. As the stress-strain curve is not a straight line, I had to do it step by step, so for every increase in distance, I multiplied the increase to the average of the force and the force for the length before. This gave the incremental potential energy for that little increase in distance, I added all these together to get a running cumulative total Potential energy as shown above.

This shows that the straight band held more energy than the tapered band, but that's not surprising because the wider straight band took more force to achieve the same elongation than the tapered band.

Therefore...

Key Message 1: A wider straight band holds more energy at a given elongation that a narrower tapered band.

What's the big deal about potential energy? If the band is nearly 100% efficient, i.e. no energy lost to heat (actually bands get cooler, BTW), sound and moving the air about, then the bands will deliver the same amount of kinetic energy as the elastic potential energy and if the mass (kind of the weight) of the shot is the same each time, then kinetic energy is a proxy for velocity.
Eye Pink Gas Font Magenta


So, assuming a person can pull a given draw force, which style of band will contain the most elastic potential energy?

It depends. The chart below shows that until the bands approach the elastic limit, tapered bands hold more potential energy for a given amount of draw force. Thereafter, it's much the same.

Plot Slope Rectangle Font Parallel


Key Message 2: Regardless of inertial effects, tapered bands give more speed for a given draw force.

What's this all happening? What's the nature of the beast?

Let's brush up on Hooke's Law. Hooke said "Ut tensio, sic vis" or in modern lingo, "Dude, as you pull harder, it gets longer". i.e. F = -kx or Force is a number multiplied by the elongation. What's that 'number'. We'll call it the Spring Constant, which defines the relationship between force and elongation.

Slope Rectangle Plot Font Line


You can see above that as I said a long way up this page, it's hard to pull, then easier and easier until the elastic gets almost to its limit. You can also see that the tapered band is easier to stretch than the straight band. A low spring constant means it's stretchier (more elastic).

Another thing to note: as you get to the elastic limit, it doesn't work so well. You're wasting force. You can pack more energy into your band that way, but it isn't drawn that far for long in the shot, so it doesn't act on the shot for very long and it takes a lot of force to do it. You're better off with a thicker or wide band and almost but not quite reaching the elastic limit. Actually, it's not that simple, because a wider band is heavier, but the effects of the elastic limit are so pronounced that it still holds true. Therefore...

Key Message 3: Don't draw to the elastic limit. Draw close to it, but not so much that it starts to tighten up.

Test to see what force you can draw without tremor and how long you want your draw to be and tune your bands till you can get the thickness, width, taper and length to be just so the force applied at full draw is right and the bands don't quite reach their elastic limit. You can only do this with hands on experimentation. So....

Key Message 4: Tailor your bands to your body's requirements.

Another thing; elastic creeps, especially when it's never been stretched befoer. In order to get consistent results, I had to pre-stretch the elastic and take the force off the bands immediately after each measurement and even then I was left with some creep that had to be adjusted in the elongation figures. This creep has three effects. Firstly, it makes the bands easier to draw which to an extent is good because it stores more potential energy for a given draw force. Secondly, it makes it hard to tune the bands to your ideal length. Thirdly, it's bad in the long run because your bands lose their snappiness.

Key Message 5: Pre-stretch your bands before tuning, do not hold your draw position too long and replace your bands.

Later I'll post more about why tapered bands work and I'll also write about thin vs thick bands if i have the time.


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ZDP-189

May 25 2010 09:35 AM

A bit more about how tapers work...

If you took a regular straight band and marked out lines across it every 25mm (inch) dividing it into segments and then stretched it, each segment would stretch by the same amount.

I have noticed tapers don't work like this. The narrow end elongates much more at first than the wide end, but as you reach the elastic limit, the narrow end can no longer stretch as much and the wide end contributes more of the elongation. The charts below illustrate this.

Colorfulness Rectangle Slope Font Parallel


Rectangle Slope Plot Font Parallel


This is all because you have a range of elasticities along the band and so the line along the elastic portion of the stress-strain curve and is smoother and straighter and the transition into the elastic limit is more progressive. Maybe this means it gives a smoother draw and shot and stays efficient past the point where the straight band becomes more abruptly inelastic?

Sometimes I think of the way a sausage party balloon inflates and deflates. It's hard to pump air into until a little blip of an inflated pocket appears somewhere and this goes to its elastic limit, stretching the part next to it which balloons and so it grows to fill the whole length. It works almost like a tear, except it's not tearing but just kicking the next bit 'over the hump'. I'm all out of scientific words, but I mean the early part of the curve below where it's hard to stretch:

Slope Rectangle Plot Font Line



Carnivore Fawn Dog breed Grass Terrestrial animal

ZDP-189

May 25 2010 09:46 AM

The obvious next question is "How radical a taper should be used?"

I would imagine that as the taper increases (a narrower narrow end compared to the wide end) the effects above would become more pronounced. There is a limit though. The on most of my slingshots, the bands tend to wear out or snap at the pouch tie. The more I taper the bands, the quicker they go. It would be best then not to over do it; to find a taper that has the narrow end reaching the elastic limit, but comfortably within breaking strain. This will depend then on many factors, including the band thickness and width at the fork, how far it is drawn and how it is tied. Certainly, I am looking at how my pouches are tied with a view to improving reliability.

You might also ask "What is the ideal shape for a taper?"

Who's to say that it has to be straight taper? This could do with some experimentation, but for now I am happy with straight tapers. One thing's for certain: although the experiment above doesn't consider whether the thin end is at the pouch of the fork, having the thin end at the pouch end reduces the energy wasted overcoming the inertia of the mass band by moving the centre of mass forward.


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whipcrackdeadbunny

May 25 2010 02:37 PM

The obvious next question is "How radical a taper should be used?"

I would imagine that as the taper increases (a narrower narrow end compared to the wide end) the effects above would become more pronounced. There is a limit though. The on most of my slingshots, the bands tend to wear out or snap at the pouch tie. The more I taper the bands, the quicker they go. It would be best then not to over do it; to find a taper that has the narrow end reaching the elastic limit, but comfortably within breaking strain. This will depend then on many factors, including the band thickness and width at the fork, how far it is drawn and how it is tied. Certainly, I am looking at how my pouches are tied with a view to improving reliability.


You might also ask "What is the ideal shape for a taper?"


Who's to say that it has to be straight taper? This could do with some experimentation, but for now I am happy with straight tapers. One thing's for certain: although the experiment above doesn't consider whether the thin end is at the pouch of the fork, having the thin end at the pouch end reduces the energy wasted overcoming the inertia of the mass band by moving the centre of mass forward.
That really is brilliant, I wonder if you teach physics, or really love catapults? I think perhaps a small brace could be used to protect the tying area (say a bit of tape) it might protect the rubber enough? here though I'm presuming that the cut/tear/snap is caused by the tie itself eventually penetrating the rubber; and that the protector would not conflict with the pull and release of the projectile or the tightness of the tie.
How about a piece of tighter elastic tying the pouch? or maybe a small clamp?


Drinkware Tableware Liquid Highball glass Barware

Darb

Nov 02 2010 06:16 PM

(3) I learned from hammocking about hyperbolic catenary curves. Essentially, the main draw force goes up the middle and the part in the middle of the side is under less tension. On a tarp, it'd be flapping in the wind, on a slingshot band, it's just a part that's doing less work. I also learned that catenary curves don't need to be precise hyperbolas. An exponential curve or similar will approximate.
This stuff is a bit outside my technical wheelhouse, but the above may indirectly hold part of the answer to Tex's comment ...

Next ZDP its time to address why thinner bands shoot faster than thicker ones with the same pull weight (especially in cold weather) -- Tex
... if you apply it to the angular elastic forces at work across the band's cross-section (rather than the lengthwise ones being discussed earlier under band tapering).

It's just an educated guess.


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Darb

Nov 02 2010 06:28 PM

Bands tend to fail at either the front or the back, especially where the pouch tie is, or where the pouch cuts into the band. ...

... Maybe there is a better tie than passing a band through a hole in the pouch. This attachment is good and sturdy, but on very thin bands and extreme tapers, the leather can cut through the band. If I replaced the hole with a small leather tab with a bulb shaped end, then the bands would tie onto that.

I could also improve the pouch. I could skive the leather in the centre of the pouch where it needs to be wide to make it lighter. Top leather is stronger and more durable than split. I could make the pouch from two layers of very thin top but apply only one layer of top in certain locations: the outside edges of the band especially. The layers would be adhered and/or stitched together. Stitches alone might do the trick. Having the outsides of the pouch stronger than the centre would make the pouch more ready to stretch and so conform to the shot's shape this may work better than wet forming a bulge, which always loses its shape quickly in use.
Any experimental progress to report on this Dan ?

The failure point came up again over in this thread, and I experienced a similar failure on my pair of Tex's express bands, after around 400 shots.


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ZDP-189

Nov 03 2010 12:15 AM

Not much new to report; I have been busy trying to get the Scallops and Fastbands commercially ready for Hogancastings.


Drinkware Tableware Liquid Highball glass Barware

Darb

Nov 03 2010 06:29 AM

Attachment idea:

What about a sturdy metal attachment arch, atop a free-spinning "pancake sandwich" pivot base that mounts atop the fork head ? That could provide an efficient and low-wear attachment point for both flatbands (if cross-folded to fit inside) and tubes alike. Actually, it'd be slightly more efficient to use the arch just for tubes, and a horizontal "[" bar (think miniature towel rack) for flatbands. Tubes or double-cross-folded flatbands would be mounted through their respective curved or squared arch bar.

I may have seen something similar atop one of the recent Chinese SS pics, but this is the idea that's been kicking around my head just now.

I'd draw it, but my drawing skills are permanently stuck at the 6-yr old finger painting level.
Cuisine Dish Font Gas Tableware



Hairstyle Eyebrow Eye Neck Jaw

ursu5

Mar 13 2011 06:09 AM

Like you described, different regions of the tapered band function most strongly to store and release energy at different points in the draw / release. It approximates a progressive spring where the rate changes with the amount of compression or extension.On release, while one part of the band is most rapidly changing length and accelerating the load, another part of the band is just getting started. The effects of hysteresis might be included by filming a release using a high speed camera with the bands marked into segments for measurements.

A bit more about how tapers work...If you took a regular straight band and marked out lines across it every 25mm (inch) dividing it into segments and then stretched it, each segment would stretch by the same amount.

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Danny0663

Oct 23 2011 10:40 PM

Thanks for the fixed charts, now i can understand abit beter


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ZDP-189

Oct 25 2011 10:05 PM

Thanks for letting me know there was a problem!


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ZDP-189

Nov 05 2011 12:01 AM

Tapers have become the flavour of the month again. This is just super. Tapers are one of the lest well understood aspects of bands, both in the community and by me.

Here is a great thread started by pop shot.

http://slingshotforu...-vs-cut-tapers/

He starts with holes punched in straight bands and quickly advances to third degree polynomials. Talk about a learning curve! {apologies for the math pun}

I reproduce below our exchange.

yes, i meant a curve, a broad concave curve on the top and bottom of the bands (facing outwards) that leaves the pouch tips parallel at say, 15mm and the fork end (uncut) at 25 or 30mm. i drew it up in paint (hacked it up in paint)

Rectangle Slope Font Parallel Tints and shades


And yes, Z, i acknowledge this is extremely hard to do (and replicate) without some serious machinery.
Mine are curved, but it's hard to notice on sight. It's not hard to do, but yes, you need tooling.

maybe puka tapers
Punched bands aren't linear. There's a lot of 'waste' rubber and it pulls in different directions. Can you put a pattern on the rubber? Something like a stamp pad with letters on it. Then stretch. You'll see they distort ... and not linearly.

Tubes? I have no idea. Well I do have an idea. I think they'd work, but they'd fail much faster.

Punching bands and tubes creates lots of local stress risers, like the saw-back on a 'Rambo' knife. Also known as failure points. If you have no alternative and need the speed, then that's when they make sense. But as a matter of general best practice, tapering is best.

Make it fit, PLEEEEAAAASSSEEE!
The best way is to show you a picture and then illustrate with data so you can see the difference. You need some decimal place millimeters (0.1mm is three thou') to show it, i.e. it's a rounding error.

Rectangle Font Parallel Pattern Number


x-position Straight Curve (all in mm)
0 27.00 27.00
5 26.82 26.80
10 26.65 26.60
15 26.47 26.41
20 26.29 26.21
25 26.12 26.02
30 25.94 25.83
35 25.76 25.64
40 25.59 25.45
45 25.41 25.26
50 25.24 25.08
55 25.06 24.89
60 24.88 24.71
65 24.71 24.53
70 24.53 24.35
75 24.35 24.17
80 24.18 23.99
85 24.00 23.81
90 23.82 23.64
95 23.65 23.46
100 23.47 23.29
105 23.29 23.12
110 23.12 22.95
115 22.94 22.78
120 22.76 22.61
125 22.59 22.44
130 22.41 22.28
135 22.24 22.12
140 22.06 21.95
145 21.88 21.79
150 21.71 21.63
155 21.53 21.47
160 21.35 21.31
165 21.18 21.16
170 21.00 21.00

Mean width: 24.0 23.9

Exponent: 0.007364343

It starts to make a difference if the bands are short and aggressively tapered:

Rectangle Slope Font Parallel Pattern


But not so much. Therefore, exponential curved tapers are close to straights and straights are close to exponential curves.

Now, you can hand draw a more aggressive curve, but each section of band no longer draws the same amount of band in front/behind it. You can also use different mathetical models; a catenary curve for example, is not valid in this setting.

In the case of a convex section, like the one you illustrated above, that section, particularly the convex edges, are not drawing efficiently. Yet they might be faster than straights, but if so that's because you are drawing the tail end to extreme elongation. For the same draw force, I suspect an exponential taper will be more efficient. I say I suspect, because I haven't fully explored tapers and curves. But that doesn't mean that I haven't done a fair amount of testing and modelling. I just need more before I can be definitive.

That's why I'm so interested in this thread. People like you are advancing our understanding of our sport.

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ZDP-189

Nov 05 2011 12:06 AM

Here's MJ's analysis of the physics:

http://slingshotforu...nds-are-faster/

I was looking at Pop Shot's post about punch-tapered bands just now http://slingshotforu...-vs-cut-tapers/ and it got me thuinking again about why tapered bands are faster. I've seen it said several times on here that it has to do with the lighter weight of the pouch side but I'm convinced that this is only a very small part of the equation. It's about stretch and acceleration.

I made up a quick illustration using chained bands. I used these because 1. I have a bunch 2. they're easy to make and 3. because they illustrate my point very well.

Here's two sets of chained bands made from the same #32 rubberbands both in a two link configuration. The top set is made up of two links of two bands for an effective straight taper, just like a set of flats cut straight. The bottom set is in a 2-1 taper, or 50%, like if you tapered a set of flats from 1" to 1/2". Both are the same static length of 6" long. I have colored the mid-point of both sets red for illustrative purposes.

Atmosphere Sky Rectangle Font Display device

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static.JPG

Stretching the "straight taper" set to 300% or 18" shows just what you would expect, the mid point is at right about 9".

Atmosphere Sky Rectangle Font Display device

Grey Wood Font Twig Electric blue
straight.JPG

Stretching the "50% tapered" set to the same 300% shows the mid point to be only at about 6", which means that the single-band end is stretching 50% more over the same draw length.

Atmosphere Sky Rectangle Font Display device

Grey Wood Font Twig Electric blue
taper.JPG

This super-stretched band has way more stored energy than the straight-cut set due to this extra stretch. Assuming that your ammo is properly matched with your bands this will deliver more speed with the same draw length but with increased wear since the tapered end is working 50% harder than an otherwise identical straight set would be.

Clear? Was this already obvious to everybody or does this help your understanding of band dynamics? Let's hear your thoughts.
Except it's not quite that simple.

What he's done is to store more energy into the system. It's actually less efficient because of hysteresis in the tail band and incomplete elongation in the in front band. The million dollar question is would a non-tapered band of the same draw weight shoot as fast if it is drawn to a high elongation at the same draw weight?


Hairstyle Eyebrow Eye Neck Jaw

learnin'

Sep 22 2014 12:14 PM

I'm curious why you compared a 1" straight to a 1"x0.5" taper, rather than keeping the areas the same and comparing it to a 1.25" x 0.75" taper?
 
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