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Calculating recoil energies.

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Calculating recoil energies.

Unread postAuthor: Ragnarok » Wed Oct 31, 2007 2:09 pm

Hi there. I know a few of you know who I am.
For those of you that don't, I'm (currently, at least) the UKSGC's 3rd top poster with over 1900 posts, I've built many cannons (mostly pneumatic), I've been reading here for a long time and I've got some plans for some other pretty neat stuff. ... so I am pretty experienced,don't take my post count here at face value.
I fancied a bit of more theoretical discussion today, and although UKSGC is a great community, they don't take to scientific stuff that much.
I've acquired a pretty heavy lump of steel rod that fits my usual barrel better than I could have hoped for, and naturally, I developed some concerns about the safety of firing it. Would the recoil generated be dangerous?

As the student engineer I am, I set up a model of the situation to get an idea of what should be expected.
Some assumptions were made, the main one being that the system is adiabatic. Over a matter of 50 milliseconds or so, the effects of heat transfer are low, so no major errors should arise by assuming the heat transfer to be negligible.

In a pneumatic, on firing, the potential energy in the chamber will be converted to:
    Recoil kinetic energy
    Piston kinetic energy
    Projectile kinetic energy/Sabot kinetic energy
    Heat energy lost in the cooling from the rapid decompression
    Potential energy losses from blow-by, either past the projectile or the pistion
    The kinetic energy of the muzzle blast.
Other effects, such as the sound of the report are later forms of the same energy.

With the aid of GGDT, both the projectile's energy and losses to the decompression cooling are known.
To actually obtain a result, there is a need to cut back heavily on variables. So, Piston KE and blowby losses are taken to be zero. Patently not true, but the numbers are many times smaller than the other energies.
<sup>1</sup>/<sub>2</sub>m<sub>Projectile</sub>*v<sub>Projectile</sub><sup>2</sup> + <sup>1</sup>/<sub>2</sub>m<sub>Gas</sub>*v<sub>Gas</sub><sup>2</sup> + <sup>1</sup>/<sub>2</sub>m<sub>Launcher</sub>*v<sub>Launcher</sub><sup>2</sup> + nRT<sub>final</sub> - nRT<sub>initial</sub> = pV<sub>initial</sub>

The initial chamber energy is known, as is KE<sub>Projectile</sub> and (nRT<sub>final</sub> - nRT<sub>initial</sub>). This can therefore be taken as a single constant, E, for which the value is calculated as:
pV<sub>initial</sub> - nR(T<sub>final</sub> - T<sub>initial</sub>) - <sup>1</sup>/<sub>2</sub>m<sub>Projectile</sub>*v<sub>Projectile</sub><sup>2</sup>
So: E = <sup>1</sup>/<sub>2</sub>m<sub>Gas</sub>*v<sub>Gas</sub><sup>2</sup> + <sup>1</sup>/<sub>2</sub>m<sub>Launcher</sub>*v<sub>Launcher</sub><sup>2</sup>

It is also known that the recoil momentum is equal, but opposite, to the sum of the projectile and gas momentum.
m<sub>Launcher</sub>*v<sub>Launcher</sub> = m<sub>Projectile</sub>*v<sub>Projectile</sub> + m<sub>Gas</sub>*v<sub>Gas</sub>

Bear in mind that the masses of both the gases, and the launcher are known constants, which leaves us with only the velocities of the launcher and gas unknown.

To put some numbers in the situation, I'll use the data I used myself.
m<sub>Launcher</sub> = 4.5 kg (10 lbs)
m<sub>Projectile</sub> = 0.3 kg (300 grams, or 0.66 lbs)
m<sub>Gas</sub> = .018 kg (18 grams or .63 oz )
v<sub>Projectile</sub> = 58.7 ms<sup>-1</sup> (193 fps)
pV<sub>initial</sub> = 1359 J (1002 ft lb)
T<sub>initial</sub> = 293 K (68 <sup>o</sup>F)
T<sub>final</sub> = 206 K (-89 <sup>o</sup>F)
Which makes nR = 4.64 J K<sup>-1</sup>
And the constant E is equal to 488.5 J (331 ft. lb.)

So from: E= <sup>1</sup>/<sub>2</sub>m<sub>Gas</sub>*v<sub>Gas</sub><sup>2</sup> + <sup>1</sup>/<sub>2</sub>m<sub>Launcher</sub>*v<sub>Launcher</sub><sup>2</sup>
we get: <sup>1</sup>/<sub>2</sub>* 0.018 kg*v<sub>Gas</sub><sup>2</sup> + <sup>1</sup>/<sub>2</sub> * 4.5 kg*v<sub>Launcher</sub><sup>2</sup> = 488.5 J

And: m<sub>Launcher</sub>*v<sub>Launcher</sub> = m<sub>Projectile</sub>*v<sub>Projectile</sub> + m<sub>Gas</sub>*v<sub>Gas</sub>
becomes: 4.5 kg *v<sub>Launcher</sub> = .3 kg * 58.7 ms<sup>-1</sup> + 0.018 kg *v<sub>Gas</sub>
rearranged to: 17.61 Ns = 0.018 kg *v<sub>Gas</sub> - 5 kg *v<sub>Launcher</sub>

So, this leaves a pair of simultaneous equations. Solving them to find v<sub>Gas</sub> and v<sub>Launcher</sub>:
v<sub>Launcher</sub> = 4.62 ms<sup>-1</sup>
v<sub>Gas</sub> = 177 ms<sup>-1</sup>

v<sub>Gas</sub> is of course, an average value. But, looking briefly at it, you can see that 177 m/s for 18 grams of gas is not an insignificant figure. With a lighter projectile, and noting the trend that lighter projectiles are accompanied by higher gas velocities, it might in fact turn out that the momentum of the gases is actually far greater than the momentum of the projectile.

The number we needed was v<sub>Launcher</sub>, which we then convert to recoil energy:
<sup>1</sup>/<sub>2</sub>m<sub>Launcher</sub>*v<sub>Launcher</sub><sup>2</sup>
For an energy of 48.1 J (35.5 ft lb.)

To get an idea of what this meant, I also calculated values for a few other projectiles I've used before:
Paintball recoil energy: 4.1 J (As I said before, in this case, most of the recoil energy is generated by the gases, not the projectile.)
25g spud slug recoil energy: 7.9 J
50g steel dart: 12.1 J
80g steel dart: 16.9J

Judging by these numbers, I need to do some more thinking about what 48.1 J actually means.

So, I went and found some typical firearm recoil energies. Seems that 48.1J is in the realm of a mid range game hunting rifle.
Personally, I've only ever fired a couple of different firearms cartridges, the .22 LR, and the 5.56x45 L2A1 Ball NATO, both well known for their low recoils, of about 1.8 and 5 J respectively, in typical weight rifles.
Ignoring the fact that I can get nearly as much recoil from firing a paintball as from the most common military round, I think that 48 J should be a manageable, but very solid kick.

However, I have a strong warning:
We have all heard tales of water filled cannons breaking. Launchers, particularly PVC ones, won't respond well to high recoil energies.
The 300g projectile I am considering is not much under the mass I would be firing with a full barrel of water, but I have a very carefully constructed metal cannon, which was specifically designed to handle recoil as best as possible, and even then, I will be taking extreme care.
Above certain limits, recoil may harm you or your launcher - please consider the potential consequences of heavy projectiles before you fire them.


I know this model only takes into account launcher weight, and not the user's hands, shoulder or anything else, but in that split second, the rigid launcher itself will primarily absorb the recoil energy over the more elastic user, so it's not too inaccurate.
Also, I apologize, I haven't got any information on combustions or hybrids. The heat losses complicate the matter significantly, and the increased dependence on C:B ratio makes it much harder to create a good algorithm, so I can't currently offer much advice to combustion spudgun users other than just being very careful with heavy rounds.

To get an idea of recoil distance, I took a definite integral of the area under a trapezium using a simplified model for the human system, treating it as just a preloaded spring.
This is an edgy model, but at a guess I can expect between 2 to 4 inches of recoil.

And that about wraps it up. I hope I didn't bore you to death, send your brain dribbling from your ears, or confuse you too heavily.
If you got this far, thanks for reading. I hope this gets you thinking a bit about recoil energies, and also helps you stay safe or avoid damaging your launcher.
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Unread postAuthor: Fnord » Wed Oct 31, 2007 2:38 pm

Wow.

Although the math you've done isn't too hard to understand, the huge volume of it kept me from reading most of that part :)

I prefer to just "calculate" recoil by trying different (smaller) weights, and then either guess based on proportions, or do some BOTE math based on the ggdt results.

Though math is good for your brain, in this case you could easily get an idea of what to expect just by building up to the weight you plan on shooting.

But still, kudos for figuring it out.


Where is Joanna at right now, by the way?
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Unread postAuthor: ammosmoke » Wed Oct 31, 2007 3:21 pm

Last I heard joanna was on some world touring thing I think... She is going to be gone for about a year..... Too bad.

But, yeah, if it is a metal gun, I think that you should be just fine. Although, you may want to pm boilingleadbath, he is quite the math genius.
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Unread postAuthor: Ragnarok » Wed Oct 31, 2007 4:02 pm

_Fnord wrote:Where is Joanna at right now, by the way?

About halfway through a stay in New South Wales.

To all you guys who're falling asleep, the maths you see is already somewhat simplified.
In reality, you need to calculate for decompression losses after the projectile has left the muzzle, and my original calculations did cover that nicely, but when I was writing it up, I took it out because I've only recently covered it on my engineering degree, so putting it in would have confused more than a few people - and if I'm trying to get a couple of people to poke some numbers into calculators, making it more complex again won't help much.

@ammosmoke: If you're interested in the launcher, I'll be posting it up in the pneumatics section in a little while.
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Unread postAuthor: sandman » Wed Oct 31, 2007 9:17 pm

ooo, very interesting and informative, as i also plan to go into an engineering major even though that is 2 years off, i can understand most of it, but can you elaborate on the recoil distance, as when i googled trapezium i got geometric astronomical shapes and a bone in your wrist, and i dont think either of those have to do with it.
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Unread postAuthor: Ragnarok » Wed Oct 31, 2007 10:16 pm

Certainly. I'll explain as best as I can.
The recoil energy is already (approximately) known, so to get a rough conversion of that into recoil distance, it has to be converted backwards through:
Work = Force * Distance.

In this case, work is 48.1 Joules. (Note: I'm using metric, as it's easier to work with but the same principles apply in imperial)

There are three simple models that can be used to describe the human system for this purpose.
The simplest (and first diagram below), but least accurate is just assuming the user applies a continous force, regardless of recoil distance. A little inaccurate, as force will increase with recoil distance.

To find the distance, just divide the energy by the force - simple.

The second is assuming the human body to be an initially unloaded spring, which where extension is the x-axis on a graph, and force is the y-axis, (for your convience, this graph is provided below) will create a right angled triangle (assuming the spring does not become plastically deformed of course, but that's beside the point here), relating to Hooke's law.

The area of the triangle is work, and if the spring constant is K (where K = Force on spring/extension), then the area is found by the equation:
Area = Extension<sup>2</sup>*K/2

Note, of course that the area can also be found by the usual Base*height/2, but as the height (or force) can be found as extension * the spring constant, this is simplified to give only one variable.
This is swapped around so that it is in terms of extension:
Extension = SQRT(2*Work)

The trapezium model is essentially the same, except the spring has a preload. The area under the graph is again work.
The maths is a touch more complex, but it can be found as
Area = Intial load*extension + extension<sup>2</sup>*K/2

It's more accurate again, but not still not that suited to the human body. However, if someone were making a preloaded spring recoil damper, then the system should be millimeter perfect.

Hope that makes that a bit clearer.

Image

Technically, all these three shapes should have the same area as they are all absorbing the same energy. Put bluntly, they won't, so no pedantics.
It's past three o' clock in the morning as I write this, and I have lectures at 9 tomorrow, so I'm surprised there's a picture there at all.
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Unread postAuthor: spudgunnerwryyyyy » Wed Oct 31, 2007 11:02 pm

That is amazing, but kind of pointless for standard spudding uses. You could however make a fully auto recoil based system with this math.
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Unread postAuthor: jackssmirkingrevenge » Thu Nov 01, 2007 1:19 am

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quite a gestation period but welcome on board :)

complex equations aside, when I fired a 1.4 kilogram steel ball from my 3" combustion, it recoiled enough to break the ignitor against my finger - on that scale things start to get nasty. However I've fired a 4 foot long 1.31" burst disc pneumatic with a barrel full of water and the recoil, while noticeable, was nothing spectacular.
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Unread postAuthor: Novacastrian » Thu Nov 01, 2007 1:47 am

I fired my 700ml Hybrid at a two times mix today, with a 25grm round.
The recoil is amazing, kicks like a mule.
Btw, the penetration is amazing too.
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Unread postAuthor: jackssmirkingrevenge » Thu Nov 01, 2007 2:00 am

you should post videos :D
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Unread postAuthor: Novacastrian » Thu Nov 01, 2007 2:04 am

jackssmirkingrevenge wrote:you should post videos :D


I've seen the vid before but thanx, i laugh my ass off whenever i see it.
Have you seen the re-coil of the Nitro 700?

BTW *should post some vids* did i write my sig in vanishing ink? :lol:
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Unread postAuthor: jackssmirkingrevenge » Thu Nov 01, 2007 2:21 am

Novacastrian wrote:Have you seen the re-coil of the Nitro 700?


higher muzzle energy than a 50 cal sniper rifle and with no muzzle brake or recoil absorbtion system other than the rifle's sheer weight, rather like the Mauser T-gewehr, the grand daddy of all modern anti-material rifles that was responsible for many a broken collar bone :shock:
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Unread postAuthor: Novacastrian » Thu Nov 01, 2007 2:42 am

jackssmirkingrevenge wrote:
Novacastrian wrote:Have you seen the re-coil of the Nitro 700?


higher muzzle energy than a 50 cal sniper rifle and with no muzzle brake or recoil absorbtion system other than the rifle's sheer weight, rather like the Mauser T-gewehr, the grand daddy of all modern anti-material rifles that was responsible for many a broken collar bone :shock:


now you are just "rubbing in" my certain inadaquate errr.. personal facilities :lol: :lol: :lol:
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Unread postAuthor: jackssmirkingrevenge » Thu Nov 01, 2007 2:55 am

you started!

penetration is amazing too


:D
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Unread postAuthor: Novacastrian » Thu Nov 01, 2007 3:00 am

jackssmirkingrevenge wrote:you started!

penetration is amazing too


:D


Who'da thunk it-but you :D look at the sig
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