Range Report Bullet Stability Factor Formula

[nick338]

So reading a post here yesterday about poor accuracy at close range turning into better accuracy at longer ranges got me thinking about something that I read in one of Bryan Litz's books that's puzzling to me. There must be information that is missing or I'm just not smart enough to comprehend the physics behind it so here goes.

Any one can plug in numbers to a ballistic app and come up with an stability factor for the particular bullet they want to shoot. It's widely known that an sg of 1.4 should be adhered to for proper stability across a broad range of environmental factors. In reading the books, Brian has stated that a bullet that leaves the muzzle stable will always be stable and that sg can actually grow to 4-5 as it travels down range because it sheds velocity at a faster rate than the aerodynamic forces acting on it can force it to tumble.

This is where I'm getting lost. In looking at the actual formulas for calculating twist rates, all the variables of input are constant as the bullet travels down range with the exception of velocity which is always decreasing so the rpms of the bullet should also decrease at an equal rate, should they not?

Launching a bullet at 2800 fps out a 9 twist barrel equates to:

2800x720)9=224,000 rpm's with roughly an sg of 1.4 based on my environmental conditions. If I change the mv to 1120 fps we have this:
1120x720)9=89,600 rpm's with roughly an sg of 1.03 which is much lower.

So what part of this equation that makes sg grow down range am I missing?

 

[KYpatriot]

No the angular momentum is not lost at the same rate as forward velocity decreases.

Aerodynamic drag on the bullet has much more effect on the forward velocity, and decreases that velocity at a much higher rate than it does the rotational “velocity” or spin rate. Since the bullet is conserving its spin much more than the forward velocity, spin stability grows as the projectile travels downrange.

The aerodynamic forces acting on the bullet trying to tumble it, the overturning moment, is proportional to forward velocity. Since that velocity decreases faster than spin, the stability increases.

Bullets can even keep spinning with zero forward velocity:

 

[nick338]

No the angular momentum is not lost at the same rate as forward velocity decreases.

Aerodynamic drag on the bullet has much more effect on the forward velocity, and decreases that velocity at a much higher rate than it does the rotational “velocity” or spin rate. Since the bullet is conserving its spin much more than the forward velocity, spin stability grows as the projectile travels downrange.

The aerodynamic forces acting on the bullet trying to tumble it, the overturning moment, is proportional to forward velocity. Since that velocity decreases faster than spin, the stability increases.

Understood. Is there a formula to calculate the sg and rotational spin rate for a given bullet and velocity? The reason I'm asking is because it seems that some bullets shed a ton of bc from the muzzle to 1500 fps, while others maintain a similar bc or actually improve on it. I just want to understand the reasoning behind this as it may help in understanding what bullets perform better at longer ranges than others.

 

[KYpatriot]

Understood. Is there a formula to calculate the sg and rotational spin rate for a given bullet and velocity? The reason I'm asking is because it seems that some bullets shed a ton of bc from the muzzle to 1500 fps, while others maintain a similar bc or actually improve on it. I just want to understand the reasoning behind this as it may help in understanding what bullets perform better at longer ranges than others.

Yes, there have been several through the years that have been tweaked here and there, one of the more prominant ones is the Miller Stability formula. There has been a ton of stuff written on this and similar topics, and if you are interested in them it would be tough to do better than the books by Bryan Litz.

https://www.amazon.com/Applied-Ballistics-Long-Range-Shooting/dp/0990920615

 

[nick338]

Yes, there have been several through the years that have been tweaked here and there, one of the more prominant ones is the Miller Stability formula. There has been a ton of stuff written on this and similar topics, and if you are interested in them it would be tough to do better than the books by Bryan Litz.

https://www.amazon.com/Applied-Ballistics-Long-Range-Shooting/dp/0990920615

I have read all the books. The Miller formula, Greenhill formula and so on give you only the stability factor at the muzzle, which I understand is the most important but I'm trying to understand the stability factor and how it increases downrange.

If you have the books, check out the different bc numbers as they are stepped from the muzzle down to 1500 fps. For example most of the Hornady AMAX and ELD bullets lose .120-.130 in G1 bc from muzzle down to 1500, yet a 215 Berger may actually gain bc. I'm guessing some bullets even though their initial bc is not as high, have less aerodynamic drag as they travel down range and I'm wondering if it's because they remain more stable then other bullets even though the sg for both bullets is acceptable at the muzzle.

 

[KYpatriot]

I havent considered that question much....but Id say it has to do with modeling. BC is how you tell a computer model the shape of your bullet so it can calculate drag.

There are two common models for drag, and one of them is based on a relatively high drag flat base bullet, the G1 standard. The other is more representative of the kind of bullet we shoot for long range, and its a boat tail bullet, the G7 standard. Still, while that standard is likely closer to the specific bullet you are interested in, it probably wont be an exact match. So ballistic calculators can use either standard to calculatory a trajectory curve, but it wont match your bullets actual trajectory because it isnt the standard bullet shape. You can get around this by adjusting the BC that you feed the calculator in speed bands... this will true the calculation. It should stand to reason that the G1 model will need more adjustment in our case than the G7 model since it was built for a different style of bullet...but with enough adjustment it can give you trajectory data just as good.

Put another way, the different speed bands for G1 BCs are an attempt to true the calculation for a much different bullet shape which is why you will see bigger adjustments than the G7 numbers, because the G7 numbers start out closer to real world drag profiles for long range bullets, though they will still exhibit some variation unless you are shooting a bullet exactly the same shape as the G7 standard.

In the end, the bullet isnt getting more or less aerodynamic as it flies...the BC variations are simply what is needed by the ballistic calculator to true the model to the real world trajectory. The G1 or G7 standard you are using to predict the actual trajectory needs adjustment.

You could get around all this by skipping the G standard/trueing process and skip the middleman by using actual trajectory data for the bullet you are shooting by referencing the actual drag curve of that bullet instead of a correcting a standard. The trajectory data calculated from these custom curves are available from applied ballistics and I think Hornady as well...but gathering that data is expensive and it isnt a single number or a few banded numbers so it isnt just going to be printed on the box of bullets. You will need to buy their calculator to use that research. We in effect do this ourselves when we gather real world dope on our shots...we use that data to build our own custom curves. In my opinion that is most accurate of all because it incorporates the rifle, scope, and shooter into that data.

Bottom line...published BCs change with velocity because we are using them to adapt a standard trajectory curve...they are not an absolute true measure of the actual aerodynamic performance because they arent truth data the way a custom curve is, even though we often use them that way.

 

[nick338]

I believe some bullets perform very differently as they slow down to transonic speeds. One of these days I will load up some rounds and get a good group at 100 yards, but with a light enough charge so that the bullet flies into the the transonic zone as it hits the 600 yard mark at my range and put them on paper. Compare them to a known load at the same distance and determine if there is a large enough difference in the group size to attribute to the design of the bullet and how well it transitions

 

[b2lee]

If I shoot my 300 PRC at 3000fps out of a 9 twist barrel......at 1450 yards it is going to be a TOF...Time Of Flight of a mere 2 seconds. There is no way in hell that the bullet will lose enough rpms to go from a stable rpm to an unstable rpm.

If it leaves my barrel stable...it will be stable when it hits the dirt at 1mile in a mere 2.7 seconds.

If it leaves my barrel unstable...and it reaches a distance where the speed goes transonic to subsonic and that CLAP of shock waves hits that bullet...and it isn't stable...it will go all knuckle ball on you.

If it leaves my barrel super unstable...the ass end will try to pass the front and keyhole the target.

 

[KYpatriot]

There are some bullets than dont transition to subsonic well even though they are adequately spin stabilized. Likely has to do with the fact that the aerodynamic center of pressure on the bullet changes as the shockwave angle changes and then disappears entirely once it is subsonic. When the distance between the center of pressure and the center of gravity changes, so do the moments acting on the bullet.

Seems like the 308W Fed 168SMK has some trouble with that. Very popular and accurate bullet in the federal gold medal load. Also known for not transitioning well even though it is very stable. I bet there is some reading on that you might be interested in but I dont have any links off the top of my head. If you look at it compared to the 175smk that transitions pretty well, you notice the 168 has a pretty short boattail. Not sure if that is causal though.

Also check out papers on spin stabilizing artillery...I have read some stuff somewhere that talked about overstabilization being a problem, causing a shell to fall nose up instead of the long axis of the shell being inline with the velocity vector. This would obviously create a lot of extra drag.

Interesting subject.

 

[nick338]

There are some bullets than dont transition to subsonic well even though they are adequately spin stabilized. Likely has to do with the fact that the aerodynamic center of pressure on the bullet changes as the shockwave angle changes and then disappears entirely once it is subsonic. When the distance between the center of pressure and the center of gravity changes, so do the moments acting on the bullet.

Seems like the 308W Fed 168SMK has some trouble with that. Very popular and accurate bullet in the federal gold medal load. Also known for not transitioning well even though it is very stable. I bet there is some reading on that you might be interested in but I dont have any links off the top of my head. If you look at it compared to the 175smk that transitions pretty well, you notice the 168 has a pretty short boattail. Not sure if that is causal though.

Also check out papers on spin stabilizing artillery...I have read some stuff somewhere that talked about overstabilization being a problem, causing a shell to fall nose up instead of the long axis of the shell being inline with the velocity vector. This would obviously create a lot of extra drag.

Interesting subject.

I believe the issue with the 168 Sierra is not the length of the boattail but the angle. Optimum bt angle is between 7-9 degrees and the 168 is 13 degrees which is very sharp and causes instability as the bullet slows down.

All the new Berger long range target hybrid bullets have a short boattail in comparison with their other offerings and just talking a stab at it, maybe the shorter bt shifts the center of gravity closer to the center of pressure to help keep the bullet balanced better when slowing down.

As far as the artillery that is correct, we also see to some degree when launching bullets at a high angle to shoot extreme distances, once the bullet reaches it's highest point the noise remains pointed at the same angle as when it left the muzzle but is now descending along its trajectory to its intended target so that would effect bc for sure.

 

[KYpatriot]

If I shoot my 300 PRC at 3000fps out of a 9 twist barrel......at 1450 yards it is going to be a TOF...Time Of Flight of a mere 2 seconds. There is no way in hell that the bullet will lose enough rpms to go from a stable rpm to an unstable rpm.

If it leaves my barrel stable...it will be stable when it hits the dirt at 1mile in a mere 2.7 seconds.

If it leaves my barrel unstable...and it reaches a distance where the speed goes transonic to subsonic and that CLAP of shock waves hits that bullet...and it isn't stable...it will go all knuckle ball on you.

If it leaves my barrel super unstable...the ass end will try to pass the front and keyhole the target.

See, this here is why people dont care about subsonic transition like they used to...guys like b2lee will need a forward controller by the time his rounds go subsonic with all these awesome new loadings and bullets?

 

[rball]

If I shoot my 300 PRC at 3000fps out of a 9 twist barrel......at 1450 yards it is going to be a TOF...Time Of Flight of a mere 2 seconds. There is no way in hell that the bullet will lose enough rpms to go from a stable rpm to an unstable rpm.

If it leaves my barrel stable...it will be stable when it hits the dirt at 1mile in a mere 2.7 seconds.

If it leaves my barrel unstable...and it reaches a distance where the speed goes transonic to subsonic and that CLAP of shock waves hits that bullet...and it isn't stable...it will go all knuckle ball on you.

If it leaves my barrel super unstable...the ass end will try to pass the front and keyhole the target.

I like your analogy on this Gunny! Keeps it simple for us simpletons.