.50 Caliber Conical Bullet Stability

[ETipp]

For altitudes above 3000 feet, Idaholewis' 50-415I bullet stabilizes well in a 1-48 if used at typical Montana hunting temps. Idahoron's 500 S&W bullet doesn't stabilize well in a 1-48 twist barrel at typical temperatures unless used at altitudes of about 6000 feet. Ron tells me that all of his rifles have 1-28 twists, and I'm sure that the 500 S&W bullets he casts for his rifles work very well for him.

Indeed. As I have said, Lewis and Ron has it figured out. Custom barrels with custom fit conical.

 

[WP79Vet]

According to Wikipedia, the Greenhill formula is still used today. I assume it's for Muzzleloader bullets, which tend to be more cylindrical. Yes, there are hollow based designs and hollow points, but evidently, that doesn't change the outcome of the formulas very much. The 1861 Springfield used a 1:72 twist to stabilize the Minnie ball. Springfield target rifles use a 1:66 for the same projectile.

Greenhill used two sets of constants for his equations. For velocities below 2,800 fps he used 150. Above 2,000 fps he used 180. When Greenhill came up with these constants, black powder was the propellant used and velocities for most rifles were low. He obviously, knew this. Cordite wasn't invented until 10 years later.

I hadn't read Miller's statement on the Greenhill formula. The book I read said the results are relatively similar; However, Miller is designed for modern bullet designs, which Greenhill ignores. Greenhill used a cylinder for the bullet type. He uses the bullet length, and the caliber, which gives the circumference and the radius. With those 3 numbers, you can calculate the volume, which is a cylinder.

That said, there is a point at which increased stability does not yield more accuracy. Both, Greenhill and Miller came to this conclusion. Accuracy will eventually level off to its limits with increasing stability. This is the reason both formulas state the best round ball twist is a slow one.

Using Greenhill, because it's quick and easy, a .50 rifle, using a .490 patched round ball, the optimum twist rate is 1:72. The numbers I used are .49 squared (.2401) x 150 = 36.015. Divide that by the bullet length, I'll use .5 = 72.03. I chose those number to gice the quickest twist given the bullet design. Bullet obturation will cause it to almost be .50. I used .49. If I had used .50, the twist would have been slower.

Let's say Greenhill's constant of 150 for lead bullets is too high. We can use 130 instead and the result is 1:63. Since the 150 constant is still used today, it must be the most accurate.

My purpose in writing this is to explain Stability and Accuracy are not synonymous and unfortunately, people tend to think they are. Accuracy requires stability, but only to a point. Then they diverge. Accuracy levels off to its limits, as stability increases.

No, accuracy and stability are not the same. Over stabilization, in fact, can actually reduce accuracy.

 

[DavidAR10T]

BAZINGA!

I truly love this bar. And it doesn't take the integral f(x) by separation of parts to figure out why.

 

[drpatton]

I truly love this bar.

Don't drink too much.

 

[DavidAR10T]

Don't drink too much.

Roger That, sir. The conductive conversations at the MML are a strong libation.
This back slindin' Lutheran will practice Habits #2 & #3 as possible while here.

 

[drpatton]

practice Habits #2 & #3 as possible while here.

Stephen Covey?

 

[DavidAR10T]

A friend of my brother, Louis, had a plaque that listed the 7 HABITS of a Mason to strive for.
I could not find a picture of the plaque but found this on the net that states the 7:

BTW, I was in '76-'86. Taught at Stone Bay '85-'86. I would have loved to been able to purchase my M40A1.
Would have rather had an ARMALITE AR-10 in some circumstances. God Bless.

 

[drpatton]

7 HABITS of a Mason to strive for.

I thought those were based on Stephen Covey's "The 7 Habits of Highly Effective People." They are not. Maybe influenced by him.
Obviously, you know how to shoot. I was issued the standard M16A1. No need for a sniper rifle.
1977-1981

 

[WP79Vet]

According to Wikipedia, the Greenhill formula is still used today. I assume it's for Muzzleloader bullets, which tend to be more cylindrical. Yes, there are hollow based designs and hollow points, but evidently, that doesn't change the outcome of the formulas very much. The 1861 Springfield used a 1:72 twist to stabilize the Minnie ball. Springfield target rifles use a 1:66 for the same projectile.

Per my original post, Greenhill's work was done in the 1870s, and it was based on football-shaped projectiles fired from rifled cannons at sub-sonic speeds. Cannon projectiles are easily seen with the naked eye (In clear weather I could still see the 8 inch 'joes fired at velocities up to about 2100 fps by the howitzer battery I commanded when they reached the peaks of their trajectories.... and yep, that's a whole lot of fun!), so it would have been easy to see projectiles that were unstable. The fact that Greenhill's formula works at all for shoulder-fired rifle bullets traveling at 2800 fps is a lucky accident, as there were no projectiles that traveled at anywhere close to those speeds in the 1870s, and absolutely nothing quantitative was known about aerodynamics then.

The basic idea of "simple rules" for estimating bullet stability (or, equivalently, for estimating the twist required to stabilize a given bullet when fired at a given velocity) is that both the angular momentum of a bullet and the aerodynamic forces on a bullet depend on its weight, length, diameter, and shape. Even though angular momentum depends on the rate of spin while the aerodynamic forces do not, and aerodynamic forces depend on air density while angular momentum does not, it should be possible to find a simple formula (ie "Rule") that incorporates bullet weight, length, diameter, rate of twist, and air density which fits the results of actual stability tests pretty well - without even getting into the details of bullet shape. Miller's Rule is more accurate than Greenhill's Rule because Miller had a much bigger and more accurate library of test results to work with, because Miller developed correction factors for air density (air temperature, pressure, and altitude) and velocity, and because modern science and engineering experiments frequently involve vast amounts of data, so techniques for finding equations which fit experimental results are more advanced today than they were in the 1870s.

In 2009 Miller published a paper which compares the accuracies of different simple rules for estimating the stability of a wide range of bullets fired at various velocities (Miller, Precision Shooting, June 2009 - https://www.jbmballistics.com/ballistics/bibliography/articles/miller_stability_2.pdf). The Miller Rule works much better than the others, including the Greenhill Rule and the Modified Greenhill Rule, which is borne out by the fact that Miller's Rule is used for all of the online stability calculators that I found.

Minnie bullets are inherently stable because their full-diameter, hollow bases mean that their centers of mass are FORWARD of their aerodynamic centers of pressure. Very slow twists are therefore adequate to stabilize a Minnie bullet.

Round balls require very little stabilization because the aerodynamic forces on them are almost completely independent of bullet orientation - and spin axis orientation - with respect to the direction of flight: A very slow spin is all that's needed to overcome the random "knuckle-ball" drifts associated with the flight of a spinless round ball.

It's been recognized that Miller's Rule is less accurate for plastic-tipped and hollow-point bullets, because the Miller Rule assumes that all bullets have uniform density (although it doesn't assume that all bullets are made of lead - the Miller Rule works equally well for bronze, brass, and steel bullets). Modified Miller Rules have therefore been developed for plastic tipped bullets (https://arxiv.org/ftp/arxiv/papers/1410/1410.5340.pdf) and hollow-point bullets (https://arxiv.org/ftp/arxiv/papers/1401/1401.4187.pdf).

 

[drpatton]

it was based on football-shaped projectiles fired from rifled cannons at sub-sonic speeds

​

Greenhill's equations assume a lead cylinder for a bullet. He uses Bullet Length and Diameter, which will give you the volume of a cylinder. His equations do NOT suppose a football shaped object.

Volume of a cylinder is Volume = Pi x Radius squared x Length.
Greenhill's constant of "150" already has Pi and the density of lead calculated in it.

Greenhill may have used Football shaped objects to imagine his formula, but in reality he is using the volume of a lead cylinder to do the calculations.
For some reason, he goes beyond that. Instead of the radius squared, he used the diameter squared.
Somehow, he must have found using the diameter will give a better twist ratio.

 

[drpatton]

Miller's Rule is more accurate than Greenhill's Rule because Miller had a much bigger and more accurate library of test results to work with

This is correct. Greenhill only had Black Powder & muzleloading projectiles to perform tests. The modified Greenhill you speak of, I believe is the change in the Constant from 150 to 180 for bullets whose velocities are over 2,800 fps. I am not aware of any muzzleloader projectiles capable of that velocity.

The Greenhill formula was initially designed for shoulder fired weapons. He did this for the Royal Military Academy for Lead Core Bullets. During his day, Cannon did not shoot lead. They shoot iron or steel shot (ball) and Greenhill's formula doesn't work for them. It was designed for shoulder fired weapons, shooting a lead cylinder of a given length in a given caliber. I assume the cylinder was close enough to conical bullets of this time.

 

[WP79Vet]

​

Greenhill's equations assume a lead cylinder for a bullet. He uses Bullet Length and Diameter, which will give you the volume of a cylinder. His equations do NOT suppose a football shaped object.

Volume of a cylinder is Volume = Pi x Radius squared x Length.
Greenhill's constant of "150" already has Pi and the density of lead calculated in it.

Greenhill may have used Football shaped objects to imagine his formula, but in reality he is using the volume of a lead cylinder to do the calculations.
For some reason, he goes beyond that. Instead of the radius squared, he used the diameter squared.
Somehow, he must have found using the diameter will give a better twist ratio.

Greenhill's equation is T = 150*D*D/L, where T is the twist rate expressed a inches per turn, D is bullet diameter in inches, and L is bullet length in inches. I'm not sure where you're seeing the volume of a cylinder, which is V=pi*D*D*L/4, in Greenhill's equation: Note that in Greenhill's equation, L is in the denominator, while L is in the numerator in the equation for the volume of a cylinder.

Greenhill's rule is NOT "imagined," it is empirical, meaning that it is a fit to observations of the stability of projectiles, rather than a theoretical equation which is derived from the laws of physics: Since the projectiles Greenhill observed were football shaped and made of lead, that shape and material are built into the number 150. The number 150 does NOT come from some calculation involving the volume of a cylinder, the density of lead, and so forth. It is just the number which happens to give the best fit to the observed stability data.

The same is true for Miller's rule. It is a fit to observations of the stability of projectiles, rather than a theoretical equation which is derived from the laws of physics. The shapes (mostly spitzer boattails) of the bullets used for the observations on which Miller's rule is based are therefore built into the constant 30 which appears in Miller's rule. Bullet length, bullet diameter, and bullet weight are explicitly part of Miller's rule, but bullet shape is implicit in the empirical number 30 because that number is a fit to data which came from observations of spitzer boattail bullets.

 

[drpatton]

Greenhill's rule is NOT "imagined," it is empirical, meaning that it is a fit to observations of the stability of projectiles, rather than a theoretical equation which is derived from the laws of physics:

I disagree with that. Greenhill was a mathematician and used math and physics to come up with his formula. He may have used his knowledge from watching projectiles, but his formula was created using Math & Classical Physics. Newton's laws of physics were over 200 years old by then and could not have been ignored. In fact, it shows up within the relationship of spin and forward velocity.
We will have to agree to disagree. I really don't believe Greenhill stopped being a mathematician and came up with his formula empirically.

 

[drpatton]

A.G. Greenhill had a PhD in Mathematics and wrote papers on Physics. I would definitely expect him to use his knowledge of math & Physics when he came up with his rule of thumb for the Royal Military Academy at Woolwich. His rule of thumb became the standard for the British Military and the world until others decided to use bullet shape in their formulas.
"[Greenhill's] shortcut uses the bullet's length, needing no allowances for weight or nose shape"
Meaning it is calculated as a cylinder. (only length and bore size is used, which gives the volume of a cylinder.)

From WikiPedia.
"His 1892 textbook on applications of elliptic functions is of acknowledged excellence. He was one of the world's leading experts on applications of elliptic integrals in electromagnetic theory."

When he uses the term "elliptic" he is referring to magnetic waves.

Textbooks​

• A. G. Greenhill Differential and integral calculus, with applications ( London, MacMillan, 1886) archive.org
• A. G. Greenhill, The applications of elliptic functions (MacMillan & Co, New York, 1892)[8] University of Michigan Historical Mathematical Collection
• A. G. Greenhill, A treatise on hydrostatics (MacMillan, London, 1894) archive.org
• A. G. Greenhill, The dynamics of mechanical flight (Constable, London, 1912) archive.org
• A. G. Greenhill, Report on gyroscopic theory

https://en.wikipedia.org/wiki/Alfred_George_Greenhill

 

[WP79Vet]

I disagree with that. Greenhill was a mathematician and used math and physics to come up with his formula. He may have used his knowledge from watching projectiles, but his formula was created using Math & Classical Physics. Newton's laws of physics were over 200 years old by then and could not have been ignored. In fact, it shows up within the relationship of spin and forward velocity.
We will have to agree to disagree. I really don't believe Greenhill stopped being a mathematician and came up with his formula empirically.

It's now been 336 years since Newton's laws of physics were first published.... and much of experimental physics and the majority of practical engineering are STILL empirical.

That's particularly true in aerodynamics, where EVERYTHNG is empirical: Practical ballistics are therefore mostly empirical too. That's why we STILL can't compute artillery firing tables, even with computers which can do billions of calculations per second, that's why ballistic coefficients and all other aerodynamic forces on projectiles still have to be measured rather than computed, and that's why Donald Miller, PhD, still used the Army projectile stability database to formulate his empirical stability rule in 2005.

This is something I understand pretty well: In addition to my West Point weapons systems engineering courses and extensive service as a cannon battery officer, I are a Stanford physics PhD my own self: I spent 25 years leading DOE and DoD science research projects, and for four years I was one of the senior scientists who, as part of the Army Science Board, review Army research - including ballistics research - every year. There is way too much Army research going on for the board to review everything, even over a period of 4 years, so during my time I never got introduced to Army projectile stability research - and that's why bullet stability is something new and exciting for me.

Empirical science and engineering often require far MORE math - and more sophisticated math - than theoretical calculations: The experimental laser-based remote (non-contact) chemical detection research I led at Los Alamos National Laboratory for six years culminated in an airborne system aboard an Air Force RC 135 flying at speeds of up to 500 mph and altitudes of up to 30,000 feet. At aircraft to target distances of 30 km, we generated 10,000 laser pulses per second, each at a different wavelength, held the laser beam on a 1 meter target for many minutes at a time, and then collected, kept track of, and did the initial data processing on the returning laser pulses. Steering the laser beam, controlling laser frequency and pulse width, collecting the data, keeping track of returning laser pulses, and doing the coarse data analysis in real time meant processing more than 3 million pieces of information and generating more than 1 million instrument contol commands per second. You can imagine how sophisticated the math which goes into something like that has to be.

Greenhill didn't STOP being a mathematician while he was formulating his empirical rule. He used a different kind of math - developing a curve which best fits the experimental data - which was much less developed in his day than it is now. In some ways this kind of math is more difficult than the conventional calculus and differential equations of Newtonian physics, and it involves a much higher volume of calculations to generate a good curve fit. The success of Greenhill's rule is a tribute to his ability in applied mathematics.

 

[user 31408]

With my 1-48 twist Renegade and Invest Arm barrels shooting 460 grain conicals. I get my best accuracy with 65 to 80 grains of Swiss 2F and a wool wad. Most others I have talked to and read of get about the same with their 1-48 twist barrels. With the Hornady 385 grain Great Plains conicals, best loads seem to be 80 to 100 grains of powder. (no wads needed)
Best thing to do is stop reading, analyzing and over thinking things. Just start shooting.

Did you ever try Buffalo Bullets? I got a rifle used and has some 425 grain Buffalo Bullets with it.

 

[edmehlig]

If one is searching for a proven heavy conical for their ML, I would recommend searching threads from Idaho Lewis and Idaho Ron. Those two has it figured out. I know that if I ever move back out west and ML hunt for elk again, I would use the same approach as they do.

I just received my 50-415I mold that I will be casting soon for my TC Renegade. I have to cast some as I just received it to see what diameter they come out. I requested Accurate Molds that I wanted them to cast a bullet 504.5. This way in case some DWB's wanted them for their 504 Cannons.
I'm going to order a Lee 501 sizing die so I can shoot them in my Renegade.