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Aaron/mods, please don't feature this entry. it's just me thinking out loud to myself to ward off insanity.

Before you start reading, this is not a post full of answers; it's a post full of unanswered questions and conundrums.

We as a sport know surprisingly little about the physics of contracting rubber and there is not much in the scientific literature about it either. I am desperately in need of help from either someone who is a lot smarter than I or somebody who at least remembers their high school physics or math better than I. Reed Richards of the Fantastic Four, please step forward.

I'm in the middle of a logical crisis regarding tapers. We all know that tapers shoot significantly than straight cut bands. I've spent a lot of time effort and money developing ways to counteract side effects (shorter band life) so that I can carry on shooting tapers. It makes sense that I should look into tapers. If I understand why they work, I will be able to theorise the relationship between taper degree and velocity and that will allow me to fully optimise my bands, to find the 'sweet spot'.

I think there are likely three ways that a taper affects velocity.
  • Shifts the centre of inertia forward.
  • Makes the rear of the band elongate more.
  • Some kind of whipping effect
  • I cannot explain the velocity increase we get from tapering in terms of the change in centre of inertia. Every way I calculate this, it has a very small effect on the inertia. Even at 100% taper (taper to zero width at the pouch) the maximum reduction in inertia is about 10% with a regular weight pouch and projectile, when you keep the total bandset weight the same. I know the relationship between mass and velocity really well, and this small change wouldn't create a great change in velocity.
  • I don't think it's an elongation effect; this averages out over the band, plus elongation caps out at the elastic limit and at high elongation, the hysteresis is greater.
  • I don't think it's a whipping effect. I have reviewed lots of high speed video and I don't see any whipping effect.
It just doesn't make sense unless when we test over a chrony and find that taper has very little effect.

I'm grasping at straws here, but maybe it doesn't really have much effect at all. Maybe by removing some rubber, we make a bandset that is weaker and gets drawn out longer so the amount of potential energy stored in the band is greater and combined a lighter inertia it suits lighter ammo better.

I am lost right now and I'm just going to have to shoot lots over a chrony to make any headway. Or maybe someone with a scientific method will publish usable results. Or maybe someone with a better brain will hand me the answer.

Whatever, I'm convinced that there must be an optimum ratio that will give the best energy efficiency.

Why? Because there's two opposing processes going on.
  • Tapers shoot faster than straights. More taper gives more energy efficiency.
  • As width approaches zero, tension per cross sectional area rises so the elongation reaches inelasticity so as taper increases, band efficiency decreases.
That means that eventually it'll get to the point where (2) more than cancels out (1). Hence there must be an optimum ratio. That optimum ratio may depend on other factors like band thickness, length, draw force, temperature, dead mass, etc. I don't know what they are, but for every bandset when the other factors are constant, there should be an optimal taper ratio. Experience seems to show that that ratio will be more aggressive than 2:1 and it can't be as much as 1:0, so it lies somewhere in that range.

Please help!



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ZDP-189
Nov 08 2011 12:39 AM

Trying to get to the bottom of it all with my limited physics and maths skills and a spreadsheet is making no progress and I will have to verify it with data anyway.

It's time to think how I can make some progress with experimental data.

My goal is I have to be able to find the optimum taper and to predict how taper will affect velocity.

Even if I can't understand the underlying physics, I can apply a set of curves to inputs and generate an output because thankfully the inputs don't interact much.

I have a strong feeling that the following (among others) will affect taper's effect on velocity:
  • taper ratio
  • band length
  • projectile mass
  • band thickness and composition
I have a hunch that heavy projectiles will affect tapers' advantages because a taper should boost the maximum unladened velocity (dry firing) of a bandset more than one that's weighed down with lead.

I also have a hunch that tapers have more effect on short bands than long bands. Maybe it's the other way around. Either way, it's probably one or the other.

Same goes for things like thickness. I think a thin band will reach inelasticity slower (faster?) than a thick band and so will have a different optimum taper.

Experience tells me that these are third order curves, constrained by one or two points, so I will need 4 points per curve to interpolate a line.

Just one data point may require an average of 3-5 good reading and maybe 5-8 assuming some bad outliers or error codes.

That means just one factor at a time may take upwards of 12 shots.

Mathematically, combining multiple factors require powers, so two factors is 4x4 points (times 3 good readings). Going up to 4 variables gives 3x(4^4) and we are getting into over a thousand test shots. It's just not practical. The only way to make some progress is independent tests for 16 data points or about 48 good shots.

I then use that data to generalise some relationships and make a predictive model. I will then use the model to make predictions within the data that we have and also in unexplored parts of the envelope. I will then adjust the model to fit the new data.

At least this is easier than modelling markets, economics or human behaviour, because rubber is not capricious or given to bouts of irrationality. I have spent many a year in such a folly.


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ZDP-189
Nov 08 2011 02:42 AM

To illustrate how small a difference I see in the inertial model:

Rectangle Slope Font Parallel Circle


Remember all these figures are modelled based on a several assumptions. They are not real world experimental data!

Above I standardised the average width at 15mm wide multiplied by 4 strands, two per side. I worked out taper dimensions ranging from 1:1("100%) through 3:2("67%"), 2:1("50%"), 3:1("33%") and a theoretical 1:0("0%"). The band set mass has not changed, but the blue lines show how the effective inertia changes by shifting forward. I applied a weighting of the distance travelled against the mass of the latex at that distance. I then applies the dead mass of a lightweight pouch and ties and a lightweight 5g projectile.

I then modelled the effect of the reduction in effective inertia against a standard bandset delivering 21 Joules of kinetic energy, which is about right for a 3:2 taper 190mm long in TBG. In order to compare the effect of I assumed that all bands draw to the same energy and there are no inelastic effects. You can see that the model says it's a teeny bit faster with more aggressive tapers.

Rectangle Slope Font Parallel Plot


I then plotted the model for velocity against projectile mass for various tapers. The model says that the lighter the projectile, the more the effect, but at higher masses, the effect all but disappears.

I believe this pattern broadly reflects what we will observe in practice. However, I think there will be a twist in the model as inelasticity at high elongations starts to affect the efficiency.

I also believe we will see more effect. If the magnitude of effect was this small, I don't think it would be so universally observed and wouldn't have become accepted wisdom.

I am working on the model and supporting data, so please bear with me. Don't take this in any way as my having made any real statement on how things work, because right now, all I have is a bunch of hunches and noway to hang them together into a theory.


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ZDP-189
Nov 09 2011 06:32 AM

Update: I'm making progress. At least I have a clear plan. Until yesterday, I was trying to make a predictive model of a tapered band based on the energy stored in a straight band and then shifting the centre of inertia forwards accordingly. That doesn't explain the magnitude of the difference in performance observed.

I also tried modelling the energy stored in the band released in real time (0.5ms) time slices according to Young's modulus of elasticity. That doesn't work either. It's not even close. I no longer believe that there is such thing as a modulus for tapered bands because different parts of the band are in different parts of their elongation or contraction stress-strain curves at different times in each phase. I could predict velocity by dividing he band into short segments and apply Hook's laws for springs in series, but even that would fail miserably because I would also have to model not only hysteresis, but the Mullins effect which modifies the hysteresis curve at different degrees of stress. It's a solvable by reducing this incredibly complex system by measuring plotting and interpolating the contraction force curve in one dimension and the Mullins effect in another dimension, then use this surface to compute an aggregate contraction-force curve or model each section of the band length individually. I just couldn't attempt such a thing without the power of computers. All that remains is to obtain the hysteresis and Mullins effect curves for a representative sample of latex and run the data through the model, praying that I don't discover that known extant inefficiencies aren't materially significant. I will only know when I test the predictive model against actual bandsets over a chrony.

The reason why I must build a model is I've realised that some of the variables that I had planned to test are not after all independent and that means I would have to test an impractically high number of shots keeping all other variables totally constant. Simply, without a system model to hang it all off, we may be able to understand the nature of how variables tend to affect bands, but we could not actually engineer a set of bands to deliver a certain performance curve. It would remain a matter of educated guesses, trial and error.

I realise this blog topic will make no sense to anyone but myself and 99.999% of shooters are content to just cut bands and maybe tweak the design a little bit, but it's just this reason that leaves me so obsessed.


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JustDavid
Nov 09 2011 03:42 PM

Talk to someone who knows stats reasonably well. You don't have to do every combination to derive the relationships. You really need someone who is working with it on a regular basis. You are looking at doing multi variable a regression analysis. Anyone currently studying stats, or working in stats should be able to help. (I'm a computer guy who can follow the maths/stats, you really need a stats guy for figuring out the tests you need to run).
Regression is primarily linear, but you can deal with non-linear effects too.
or read up a lot on the design of multi variable linear regression experiments.
D


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ZDP-189
Nov 10 2011 01:49 PM

Talk to someone who knows stats reasonably well. You don't have to do every combination to derive the relationships. You really need someone who is working with it on a regular basis. You are looking at doing multi variable a regression analysis. Anyone currently studying stats, or working in stats should be able to help. (I'm a computer guy who can follow the maths/stats, you really need a stats guy for figuring out the tests you need to run).Regression is primarily linear, but you can deal with non-linear effects too.or read up a lot on the design of multi variable linear regression experiments.D
David, thanks for your kind words in my moment of deep stress and confusion. Just to know someone is listening and willing to help makes a big difference.

Ironically, I pretty much used to be the stats guru for a major investment bank. It's not a linear relationship at all you'd need to fit a fourth order curve with origin constraints. It's not that simple either... you couldn't even do a multivariate regression. First, you'd need a vast number of datapoints to do a multivariate high order fit and second, the relationships are interdependent and serially autocorrelated.

Another person who helped was Hrawk, who sent me a PM suggesting several books on elastomers. My original reaction was that it's a kinetics problem, not an elastomer problem, but then I realised that not only can you not use a constant Young's modulus, you can't even depend on a standard contraction force curve. The more I read the more I realise how little I knew. Now I know about fillers and how elastomer crosslink breaking and reformation affects subsequent shots and how temperature and humidity plays a role depending on the elastomer and even the filler particle size. I've progressed to the point of being a conscious incompetent.

I've learned lots of things, like how contraction is an adiabatic process, that fillers may actually improve resilience, why ambient temperature is important, how to warm up a band and why you may not want to draw to the maximum elongation.

The more I learn, the more I learn that that there is that remains to be learned.


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JustDavid
Nov 11 2011 04:01 AM

I think you may know more stats than I do ;-)

Perhaps then a suggestion from an engineering point of view then.

Take various tapers, and a straight band.
Mark them every 2cm with a straight line across the band.
Stretch them out side by side and compare the positions of the lines.
The straight band should show uniform expansion. (lines at equal distances)
The tapered bands won't. . (lines not at equal distances)
A photo and a few measurements may show some interesting relationships.

Also, if you repeat this exercise while stretching the bands more and more, you will quickly see if
you reach the limit of bands expansion in any given section.

If you have a REALLY high speed camera, you could even figure out how they sections contract when you release them, and if the narrow part of the band is contracting much faster than the rest.

Dave

PS, a very interesting problem in general.


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JustDavid
Nov 14 2011 05:04 AM

I tried making up straight and tapered bands and marking them every 2cm.
The straight bands expanded and contracted uniformly. (no big surprise)
The tapered bands behaved as follows.

The narrow end expanded first and expanded more than the wide end.
When contracting them slowly, the wide end contracted first. The narrow end contracted last.

D


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ZDP-189
Nov 16 2011 12:54 AM

David, you seem to be having as much fun at this as I am.

I tried making up straight and tapered bands and marking them every 2cm.The straight bands expanded and contracted uniformly. (no big surprise)The tapered bands behaved as follows.The narrow end expanded first and expanded more than the wide end.When contracting them slowly, the wide end contracted first. The narrow end contracted last.D
Yes, I have done exactly that experiment the first time I had a crack at answering this.

http://slingshotforu...ing-the-answer/

I think I am at a point where I can have a crack at this. Taking a known straight flatband stress-strain curve and assuming no hysteresis and a standardised resilience ratio, I can calculate either a finite element analysis model or a composite contraction-force curve based on the above inputs and weighting according to the relative thickness of the band along the length. This composite curve becomes the band's curve and can be tested against experimental data for bands of various tapers. I can then take the existing mass-valocity model and curve fit this predictive model against velocity data to see what the implied unladened bandset inertia is minus the mass of pouch, ties and projectile and any difference can be accounted for in another curve for internal effects of taper. Repeating this for thickness, I can add another (presumably) independent variable. In short, I have the skeleton of a model which should be pretty well predictive and can be used to model theories about bandset design.

It's funny you should talk about statistics. Statistics are a big part of materials science and not even in the way you describe. When system modellers have little idea about how a system works, we employ a device called the random walk. Economists applied this to free market analysis when they realised they couldn't explain the complexities of human behaviour and elastomer. Random walk is science speak for I'll be boogered if I know, let's call it random and footnote it as an assumption. They employ something called a Gaussian distribution which is a fancy name for a random walk. It means chains and crosslinks will get less the further away from the measured point in a random sort of way.

Surprisingly, this approach generates a crude model that does surprisingly well at predicting the stress-strain of an elastomer up to the point where elongation starts to max out.

We also use the laws of thermodynamics to see how this model will generate a force based on the physical dimensions of a piece of rubber, the temperature and the number of linkages per unit volume of the rubber:

Font Art Number Circle Parallel


Where:
  • f is Force
  • A0 is cross sectional area of unstretched elastic
  • N is the number of connections per volume
  • k is the Boltzmann constant
  • T is absolute temperature
  • Lambda is unstretched length
This can be used in finite element analysis or you can take the area variable and assumption of constant force over the entire band length (and assuming fixed resilience and other huge and unsupported assumptions) to model the relative contraction of different parts of the band based on the average thickness at that part.

It's like what you're doing, but the relationship tells you what shaped curve to fit the model to (L-1/L^2) vs F/A0NkT to get a least squares curve fit. I'd test that into the model above as the way of computing the composite curve for the taper.


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ZDP-189
Nov 16 2011 01:07 AM

One thing I find amusing about the pursuit of a proper engineering model for elastic bands is that I suspect we've already gotten to the very precipice of current scientific knowledge. To use another metaphor, we are standing on the edge of the map. We see the road behind us and have a good idea the direction we're heading, but we don't know what's over the ocean where we want to go.

Rubber is surprisingly badly understood in the context of thin slingshot bands. Static and vibratory states are better studied, but not exactly what we do. If we were studying something with economic merit, such as physical properties of human organs (brain tissue, aorta, ligaments) or tyres or o-rings or vibration dampers, then we could probably whip up a couple of million dollars of funding to pursue what we're doing.

As it is, a lot of the google hits for the scientific terms I need to look into come back to this blog and its embedded images.

Oh, how I'd live to speak to a materials scientist who wants to explain the dynamic properties of thin rubber bands to a layman. Coooeeee! Scientists, drop me a PM please!
 
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