New Formula for Gyroscopic Stability of Open Tipped Match Style Rifle Bullets

[Michael Courtney]

New Formula for Gyroscopic Stability of Open Tipped Match Style Rifle Bullets

A paper describing our new formula for the gyroscopic stability of open tipped match bullets has just been published by Cornell University Library. Don Miller knew that the assumption of constant density used to develop his original twist formula would tend to underestimate the stability of open tipped match bullets, and he shared some very good ideas which proved useful in formulating a more accurate equation. I have attached a spreadsheet implementation of the new formula, along with the previous formulas for constant density bullets and plastic tipped bullets.

http://arxiv.org/ftp/arxiv/papers/1401/1401.4187.pdf

Abstract: Earlier work has produced formulas for predicting stability of rifle bullets of near uniform density and also for plastic-tipped rifle bullets. These formulas have been shown to be accurate to within 5%. However, the original Miller twist formula for metal bullets of near uniform density underestimates the stability of match style open tipped rifle bullets having a significant empty volume in the tip. This paper presents a new formula for accurately estimating the stability of these open tipped match style rifle bullets from parameters easily obtained such as the bullet mass, length, and depth of the empty space in the tip. The formula is tested by measuring the aerodynamic drag vs. predicted stability of several bullets over a range of stabilities.

 

[jonbearman]

Could you post a link in pdf format as I cant open it.

 

[sailhertoo]

So to boil it down, Berger's suck?

Proof's in the pudding, honey.

 

[Jay Christopherson]

So to boil it down, Berger's suck?

Proof's in the pudding, honey.

Uh, pretty sure that's not what the paper is stating. He is stating that the original formula *underestimated* the gyroscopic stability of open tipped bullets - such as the Berger's. Which I take to mean that the Miller stability formula is err'ing on the side of a faster-than-required twist to stabilize the longer, heavier bullets. Which I believe since I've successfully shot bullets at longer ranges, out of barrels that the Miller formula stated would be on the edge of stable (or outside altogether).

Or maybe it's not a reference on the twist rate suggestion itself, but more on the fact that the OTM bullets tended to stay stabilized longer than suggested by the older Miller formula. At any rate, anything that calculates stability with more accuracy is a good thing in my book. It means I can make a more informed decision about the bullets vs. barrels that I choose for new builds.

If I'm reading this thing correctly.

 

[Michael Courtney]

So to boil it down, Berger's suck?

Proof's in the pudding, honey.

Uh, pretty sure that's not what the paper is stating. He is stating that the original formula *underestimated* the gyroscopic stability of open tipped bullets - such as the Berger's. Which I take to mean that the Miller stability formula is err'ing on the side of a faster-than-required twist to stabilize the longer, heavier bullets. Which I believe since I've successfully shot bullets at longer ranges, out of barrels that the Miller formula stated would be on the edge of stable (or outside altogether).

Right. There was no intent to denigrate Berger bullets. The data does not denigrate or disparage Berger bullets. Berger bullets were chosen for 2 of the tests because we had them on hand because they shoot so accurately. These two models (62 and 52 grain Varmint) are also favorites for BC testing because the nose shapes are extremely consistent leading to great shot to shot consistency on BC measurements.

The 52 grain Varmint is my daughter's bullet of choice in her .222 Rem for benchrest matches. The 62 grain Varmint is my daughter's bullet of choice in her .223 Rem (1 in 12" twist) for F-Class. I also won the light rifle category in a long range precision rifle contest (steel prairie dogs) with this bullet a few years back in the same rifle.

Berger makes good bullets.

 

[FeMan]

Interesting. Maybe I missed it, but I didn't see anywhere on the spreadsheet or in the paper where the Sg for stable plastic tipped or open tipped bullets was defined. If I missed it, what are we looking for for those two bullet types?

 

[Michael Courtney]

Interesting. Maybe I missed it, but I didn't see anywhere on the spreadsheet or in the paper where the Sg for stable plastic tipped or open tipped bullets was defined. If I missed it, what are we looking for for those two bullet types?

The stability formulas are implemented in cells D11, D12, and D13 for the case of constant density bullets, plastic tipped bullets, and open tipped match bullets, respectively, using inputs from cells D2 through D9. The OTM formula for Sg is equation 2 in the paper, which refers to Sg(CD) and Sg(PT) whose explicit formulas were given in earlier papers as cited. The spreadsheet implements all three of the formulas: Sg(OTM), Sg(CD), and Sg(PT).

 

[Laurie]

It was the 30-cal 155gn Lapua Scenar that made me realise some years back that long OTM types with a substantial amount of airspace above the lead core are given a lower computed Sg value than really applies and will therefore work in slower twists / at lower spin rates than expected. This is the longest bullet in its class at ~1.29 which is still greater than the ultra-long nose Berger Hybrid which followed years after the Scenar's introduction.

Many 'Target Rifle' competitors use the 155 Lapua here in the UK all year round in pretty well 'standard' TR / Palma spec rifles optimised for 155s - 1 in 13, even 1 in 14-inch twist 'tight' 30-inch, short throat chambered barrels usually above Viht N140 loads that will produce anything from 2,850 to 3,000 fps depending on the shooter and what he finds works. In theory, these combinations shouldn't work in the 13 inch twist barrels, never mind 14 (these are disappearing now - a hangover from the days of using 146gn British RG 'Green Spot' milspec ball in TR), but they obviously do. That includes people shooting on low lying Bisley in mid winter with typical southern England temperatures in the mid 30s to high 40s F conditions.

My first 'proper' F/TR rifle had a 1 in 13.5-inch Bartlein barrel and used these bullets despite a computed Sg value of 1.07 in my range's winter conditions and they shot OK to 1,000 yards. This twist rate would stabilise the then new 185gn Berger LR BT too @ ~2,800 fps MV at short range on anything but the coldest days, in fact did so well that I got some stunning 100 yard groups off the bench. A subsequent try-out at 800 yards in temperatures around 4-deg C (40-deg F) saw them just stay in the F-Class target's 'black', but elevation and grouping were poor by F-Class standards. The Miller spreadsheet produced values in the 1.05 to 1.1 range depending on temperature.

The other interesting issue in all this is rpm / Sg value v precision over distance. If I've read Bryan Litz's book ('Applied Ballistics for Long Range Shooting') correctly, if you spin a bullet fast enough to stabilise it for good short range performance, it will perform well at all distances at least so far as that aspect of external ballistics is concerned. The rationale is that spin rates decline much more slowly than bullet velocity over distance as a percentage of their starting speeds at the muzzle. As velocity drops, air pressure on the bullet tip reduces, likewise its propensity and ability to cause the bullet tip to deviate from the line of flight. That seems logical, but there is both anecdotal and scientific evidence that longer distances require higher spin / faster twist rates and a higher Sg value. I did see extracts from a book or papers years back by two Australian ballisticians or scientists who did a lot of work on this using TR / Palma and 'Match Rifle' .308 Win loads and bullets as a primary subject. This showed that faster barrel twists than the norm should be used in TR rifles for 155s if the competitor wanted to compete at 1,000 yards, and considerably faster twist rates if the target is at 1,100 and 1,200 yards. (On the other hand, if you never competed beyond 300 yards, you'd specify 1 in 14, or if memory isn't playing tricks something even slower.) On the MR side which is an any bullet weight discipline, the work showed that the 190gn Sierra MK which was a popular bullet back then would work with a surprisingly slow twist / spin rate at 300 yards, would be optimised by something around the normal MR 1 in 10" rate for mid ranges to 1,000, but would perform better at 1,100 and beyond with a single figure twist rate value.

Unfortunately, I've never managed to rediscover the source of this work or its summarised findings since despite more than a few Internet trawls.

 

[Michael Courtney]

It was the 30-cal 155gn Lapua Scenar that made me realise some years back that long OTM types with a substantial amount of airspace above the lead core are given a lower computed Sg value than really applies and will therefore work in slower twists / at lower spin rates than expected. This is the longest bullet in its class at ~1.29 which is still greater than the ultra-long nose Berger Hybrid which followed years after the Scenar's introduction.

Many 'Target Rifle' competitors use the 155 Lapua here in the UK all year round in pretty well 'standard' TR / Palma spec rifles optimised for 155s - 1 in 13, even 1 in 14-inch twist 'tight' 30-inch, short throat chambered barrels usually above Viht N140 loads that will produce anything from 2,850 to 3,000 fps depending on the shooter and what he finds works. In theory, these combinations shouldn't work in the 13 inch twist barrels, never mind 14 (these are disappearing now - a hangover from the days of using 146gn British RG 'Green Spot' milspec ball in TR), but they obviously do. That includes people shooting on low lying Bisley in mid winter with typical southern England temperatures in the mid 30s to high 40s F conditions.

My first 'proper' F/TR rifle had a 1 in 13.5-inch Bartlein barrel and used these bullets despite a computed Sg value of 1.07 in my range's winter conditions and they shot OK to 1,000 yards. This twist rate would stabilise the then new 185gn Berger LR BT too @ ~2,800 fps MV at short range on anything but the coldest days, in fact did so well that I got some stunning 100 yard groups off the bench. A subsequent try-out at 800 yards in temperatures around 4-deg C (40-deg F) saw them just stay in the F-Class target's 'black', but elevation and grouping were poor by F-Class standards. The Miller spreadsheet produced values in the 1.05 to 1.1 range depending on temperature.

Thanks for posting these observations.

 

[mikecr]

Michael, looking at it from Cp -vs- Cg standpoint, how is the hollow space or a plastic filled space from core to meplat, affecting these factors?

 

[Michael Courtney]

Michael, looking at it from Cp -vs- Cg standpoint, how is the hollow space or a plastic filled space from core to meplat, affecting these factors?

It depends on what you hold constant. The original Miller twist rule applied to a given total length bullet and assumed constant density. If the front part of the bullet has a plastic tip or hollow space, then the center of gravity is further back compared with the constant density bullet of the same length, but the center of pressure has not changed. But this is not the only change. The moments of inertia also change. The tumbling moment of intertia of the plastic tipped or hollow point bullet is smaller than that of the constant density bullet of the same mass and total length. The moment of intertia about the spin axis is larger for the plastic tipped or hollow point bullet compared to the constant density bullet of the same length and mass, because more of the mass is located further from the spin axis.

The formula for Sg of plastic tipped bullets attempted to properly adjust for these various factors, and its accuracy was ultimately confirmed by testing with real bullets as Sg was lowered. Likewise, the accuracy of the open tipped match formula was confirmed by experimental testing.

 

[Michael Courtney]

The other interesting issue in all this is rpm / Sg value v precision over distance. If I've read Bryan Litz's book ('Applied Ballistics for Long Range Shooting') correctly, if you spin a bullet fast enough to stabilise it for good short range performance, it will perform well at all distances at least so far as that aspect of external ballistics is concerned. The rationale is that spin rates decline much more slowly than bullet velocity over distance as a percentage of their starting speeds at the muzzle. As velocity drops, air pressure on the bullet tip reduces, likewise its propensity and ability to cause the bullet tip to deviate from the line of flight. That seems logical, but there is both anecdotal and scientific evidence that longer distances require higher spin / faster twist rates and a higher Sg value. I did see extracts from a book or papers years back by two Australian ballisticians or scientists who did a lot of work on this using TR / Palma and 'Match Rifle' .308 Win loads and bullets as a primary subject. This showed that faster barrel twists than the norm should be used in TR rifles for 155s if the competitor wanted to compete at 1,000 yards, and considerably faster twist rates if the target is at 1,100 and 1,200 yards. (On the other hand, if you never competed beyond 300 yards, you'd specify 1 in 14, or if memory isn't playing tricks something even slower.) On the MR side which is an any bullet weight discipline, the work showed that the 190gn Sierra MK which was a popular bullet back then would work with a surprisingly slow twist / spin rate at 300 yards, would be optimised by something around the normal MR 1 in 10" rate for mid ranges to 1,000, but would perform better at 1,100 and beyond with a single figure twist rate value.

Unfortunately, I've never managed to rediscover the source of this work or its summarised findings since despite more than a few Internet trawls.

There are important differences between Sg (gyroscopic stability) and dynamic stability. If a bullet has gyroscopic stability when it leaves the muzzle, it will maintain gyroscopic stability at all ranges, as Litz describes.

Dynamic stability is different, as shown in the figure attached for the M855 bullet.

Inferences regarding dynamic stability should not be made from Sg values. Some references on dynamic stability:

http://www.dtic.mil/ndia/2011ballistics/11894.pdf

http://www.arl.army.mil/arlreports/2010/ARL-TR-5182.pdf

http://www.fulton-armory.com/faqs/Genl-FAQs/Fly/longr.htm

 

[mikecr]

Michael, looking at it from Cp -vs- Cg standpoint, how is the hollow space or a plastic filled space from core to meplat, affecting these factors?

If the front part of the bullet has a plastic tip or hollow space, then the center of gravity is further back compared with the constant density bullet of the same length, but the center of pressure has not changed.

This would act to decrease stability, as the overturning arm is extended with it.
Are you sure the increased stability from tipped and hollow point bullets isn't a result of Cp moving backwards -moreso than Cg?

In other context, let's consider meplats, and their affect to stability.

 

[Michael Courtney]

Michael, looking at it from Cp -vs- Cg standpoint, how is the hollow space or a plastic filled space from core to meplat, affecting these factors?

If the front part of the bullet has a plastic tip or hollow space, then the center of gravity is further back compared with the constant density bullet of the same length, but the center of pressure has not changed.

This would act to decrease stability, as the overturning arm is extended with it.
Are you sure the increased stability from tipped and hollow point bullets isn't a result of Cp moving backwards -moreso than Cg?

In other context, let's consider meplats, and their affect to stability.

It depends on what you hold constant. Lengthening a bullet by adding a plastic tip to the end and increasing total length is different from cutting off a bit of metal from the front and replacing that metal with plastic without increasing the length of the original bullet.

Consider the 220 grain Nosler Partition in .308. In a standard atmosphere and 1 in 12" twist barrel, Sg(CD) = 1.550 at 2500 fps.

If we add a 0.2" plastic tip, the length grows from 1.370 to 1.570", and the stability decreases to Sg(PT) = 1.352.

If we instead, grind off 0.2" from the original bullet and then add a 0.2" plastic tip for a total length of 1.37", then Sg(PT) increases to 2.088.

If everything else (mass, shape, length) is truly constant, a plastic tip does not change the center of pressure. It only changes the quantities that depend on how the mass is distributed: center of gravity, moment of intertia about the spin axis, and moment of inertia about the tumbling axis.