lysanderxiii
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Oh, and if the acceleration were constant, you will find the pressure-time curve would have to look like this:
Dwell Time in barrel
- Thread starter Archie
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ExtremeSquared
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Pressure going to zero (and by extension force, acceleration) means the SLOPE of velocity goes to zero. Velocity would flatten.
Edit: I think I misread your graph. The vertical red bit is an axis, not a plot.
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ExtremeSquared
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The last element to all this calculus is barrel length.
x = position
x' = velocity
x'' = acceleration
going backwards...
integrating acceleration(time) over time=0 to time=t gives you velocity(t)
integrating velocity(time) over time=0 to time=t gives you position(t)
position(t) is useful, because you can set position(t) = barrel-length, and then solve for t if you know the equation.
The hardest part is going to be the unknown function acceleration(time) because in the underlying equation ( acceleration(time) = F(time) / mass ) -- force or pressure is dependent on time AND x position, and x position is already a function of time, not to mention needing a bunch of thermodynamic and chemistry chaos thrown in, so this should turn into an absolutely horrible differential equation pretty quickly if you're trying to model it from scratch without looking up pressure curves in a book.
Props to quickload.
x = position
x' = velocity
x'' = acceleration
going backwards...
integrating acceleration(time) over time=0 to time=t gives you velocity(t)
integrating velocity(time) over time=0 to time=t gives you position(t)
position(t) is useful, because you can set position(t) = barrel-length, and then solve for t if you know the equation.
The hardest part is going to be the unknown function acceleration(time) because in the underlying equation ( acceleration(time) = F(time) / mass ) -- force or pressure is dependent on time AND x position, and x position is already a function of time, not to mention needing a bunch of thermodynamic and chemistry chaos thrown in, so this should turn into an absolutely horrible differential equation pretty quickly if you're trying to model it from scratch without looking up pressure curves in a book.
Props to quickload.
denton
Member
As you probably realize, that is the numerical method approximation of integral calculus.
You can integrate with distance on the X axis to get work, or you can integrate with time on the X axis to get kinetic energy. If you know bullet mass, you can work out MV either way.
not that this idea will add anything, but popped into my head as you guys are solutioning this. it is overly simplistic, but I think of it like a slingshot. once pressure starts to spike in a cartrdige, the behavior is then similar to letting go of the slingshot being like when the bullet starts to move. the tension on the cord would be similar to chamber pressure, you're going to have an early spike and then a decline. my limited understanding of powders is this is why they add things in there to slow ignition, is to delay the pressure spike, to be nicer to the chamber, but also to give the brass a chance to seal.
Remington1911
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I remember reading somewhere years ago that barrel life is measured in minutes, with 10 minutes being about the max.
Seemed like a really short amount of time, but when you think of just how long it is in there, that is a great deal of shooting.
I want to say this was with average.....whatever that is....shooting, and not something with 10 minutes of ammo and the trigger wired shut.
I also seem to remember one of those "torture" tests where a guy had something like 100 30 round magazines, and just shot as fast as he could, actually it was a couple people, and if I remember it took hours to finish it. I want to say they lit cigs off the barrel, and cooked bacon on it.
I could be getting those numbers wrong, but it was just a crazy test. As I remember the barrel was not as bad as everyone thought.
Seemed like a really short amount of time, but when you think of just how long it is in there, that is a great deal of shooting.
I want to say this was with average.....whatever that is....shooting, and not something with 10 minutes of ammo and the trigger wired shut.
I also seem to remember one of those "torture" tests where a guy had something like 100 30 round magazines, and just shot as fast as he could, actually it was a couple people, and if I remember it took hours to finish it. I want to say they lit cigs off the barrel, and cooked bacon on it.
I could be getting those numbers wrong, but it was just a crazy test. As I remember the barrel was not as bad as everyone thought.
lysanderxiii
Member
Yes, I do.
But, solving the problem by numerical method does not require knowledge of anything more advanced than algebra.
And, BTW, when you ask a computer to integrate something, it does it that way - numerically.
F-111 John
Member
The statement you originally commented on was "You'll find that acceleration is almost all accomplished in the first couple inches," to which you said "The "almost" doesn't quite qualify your statement. If that were true, barrel length would make no difference to muzzle velocity, and as we all know, longer barrel = greater MV (to a practical maximum) for a given cartridge."
My response was that the "almost all" was a valid qualification, because most of the cartridge's ultimate muzzle velocity is obtained within the first couple of inches of barrel travel, with the remainder of the travel down the bullet adding much less to the muzzle velocity.
Depending on bullet, powder, and barrel length, the first couple of inches of barrel travel can provide upwards of 80% of the bullet's ultimate muzzle velocity, whereas the final 14 inches of a 16 inch barrel provides an additional 20%.
To me, 80% qualifies as "almost all."
lysanderxiii
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Generally speaking, you are correct, but the 80% is a bit high for rifles. The first 2 inches of a 16 inch barrel accounts for just under 50% of the velocity.
I would say that it is typical that the first 20% of travel accounts for just under 50% of the velocity, approximately.
Even for pistol cartridges in pistol length barrels your 80% is a a little high, 5/8 being a good approximation
GHinNH
Member
If I understand correctly, in the graph I posted the mistake made was thinking that x=time, instead it does in fact indicate time in milliseconds, and was being acknowledged as such. That's all there was to that.
I will note, that is from a preliminary model I have for 6MM ARC. Since I just recently finished the rifle, I do not have any real chrono results yet to input into the model. The time, pressure, and FPS are not entirely accurate yet. When I do, I will be able to calibrate the model to reflect what is actually happening with ONLY my rifle AND components.
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F-111 John
Member
My apologies, Archie. It was not my intent to insult you.
One variable you've all missed is pressure curve from the burning powder. If it peaks and falls before the bullet leaves the barrel, then velocity will start to fall. For any calculations to be valid, that peak must occur just as the bullet is at the muzzle.
F-111 John
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Thank you very much for the very thorough research and the inclusion of the various graphs. I concede that 2" = 80% is inaccurate, but in fairness it wasn't meant to be an exact metric.
GHinNH
Member
Not at all good sir... If you care to, see my original post. I don't have a problem understanding the how's, why's, and many other variables involved with measuring barrel time, and why there is no simple answer. Fortunately enough the egghead slide rule math involved (that I cannot comprehend) is built into GRT.
lysanderxiii
Member
The velocity will not fall, the rate at which it increases (acceleration) with decrease, but the velocity will always increase provided the force from the gas pressure behind the bullet is more than the friction.
Look at the posted graphs.
And, the peak velocity does not always have to be at the muzzle. Given a long enough barrel, the pressure decay will eventually cause the force due to pressure drop below the friction force and the bullet will start to slow down. Granted, it would be a long, long barrel for a typical rifle cartridge, but for a pistol it is possible to see with reasonable barrel lengths. For a 9mm Parabellum a barrel of about 30 inches would probably start to see a velocity drop in the barrel. In a .22 LR it starts to occur after about 20 inches, that's why .22LR rifle rarely have barrels longer than 20 inches.
lysanderxiii
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Shoot indoors?
doubleh
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SSN Vet
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It may not sound like a long time, but there are several distinct events that happen in that millisecond.
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