Greenhill's Twist Formula *NOT* Suitable for Round Balls

[ScottRiqui]

Bottom Line Up Front - The Greenhill Twist Rate formula is not valid for round balls, either mathematically or physically, and the real-world reasons why rifling is beneficial for round balls are not predicted or modeled at all in Greenhill's formula.
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I don't shoot muzzleloaders, but I'd always wondered why some rifles that shot round balls had rifled barrels. From a physics standpoint, it shouldn't matter if a round projectile spins or not, as the conventional meanings of "stability" don't apply to spheres (their center of pressure and center of mass are always co-located at the center of the sphere, so they can't really be dynamically "stable" or "unstable" the way a dart or a conical bullet can be.) Also, no matter how a flying sphere rolls, pitches or yaws, the frontal area presented to the air remains perfectly unchanged, so they can't "tumble" the way a pointed bullet can.

It's pretty well known that the traditional "Greenhill formula" you find in books and on the Web is just an approximation/simplification of a more complicated (and complete) formula that Greenhill derived, but I could never find anything that gave the original complete formula and explained how it was derived and how the approximation came to be.

I recently found an article in the International Journal of Impact Engineering where the author did just what I was looking for - went back to Greenhill's original formula and examined his assumptions, models, and approximations. Here's what I found after reading the paper and talking to its author:

1) The Greenhill formula assumes the air is frictionless, so it ignores any boundary effects such as turbulence. The formula also assumes that the density and shape of the projectile are consistent and flawless.

2) The mathematical model Greenhill used for the bullet is a "prolate spheroid" - a round shape where the length is longer than the diameter (think of a rugby ball).

3) Further, the artillery shells Greenhill was modeling in his work were about 2.5 times longer than their caliber, and all of the experimental data used to calculate the value of the constant 'C' in the simplified formula (usually given as 150 or 180) are from projectiles whose length was between 2.5 and 8.0 times longer than their caliber.

Since Greenhill chose to model the projectile as a prolate spheroid, many of his assumptions and simplifications don't work for round balls. When I asked the author about it, he agreed that the full version of the Greenhill formula gets "very strange" when the projectile length is less than about twice as long as the caliber. And when the length/caliber ratio gets down to 1:1 (as in a round ball), the full formula "blows up" entirely (some terms shrink to zero, while other terms go to infinity, so the formula gives no result.)

With all this said, there's no doubt that round balls benefit from a small amount of rifling in the barrel, but this is only because the balls are never perfect spheres, and the spinning motion helps "average out" the imperfections. There are also things like the Magnus effect, but the Magnus Effect would actually be an argument *against* rifling, since it states that a crosswind on a spinning projectile causes it to deviate high or low, depending on the direction of the crosswind. With a non-spinning ball, there would be no Magnus Effect. However, there's nothing in Greenhill's formula that takes into account imperfections in the ball or things like the Magnus effect, so it's not appropriate to use Greenhill's formula to calculate *how much* twist to use with a round ball.

If people are happy with the twist rates that Greenhill's formula gives them for round balls, I suspect that it's because the twist rate given by (mis)using Greenhill's formula in this manner is actually tighter than what's needed to correct for the imperfections in the ball. And having too much twist (within limits) isn't going to cause any problems.

[B.L.E.]

In order for the Magnus effect to be significant, the ball would have to experience a sideways wind. When you ride a motorcycle 100 mph, what do you experience? A 100 mile head wind on a calm day from the motion of the motorcycle. Throw in a 5 mph crosswind and you experience a 100.1249 mile per hour headwind coming at you at an angle of 2.8624 degrees from dead ahead.

Now for a roundball going 800 miles per hour in a 5 mph crosswind, the ball would experience a 800.01562 mile per hour headwind that's blowing 0.3581 degrees off of dead ahead.

The magnus effect might come into play in long range shooting as the ball shot at a high angle falls back to earth with its original spin orientation.

I have never heard the seasoned muzzleloading bench rest shooters make even a passing mention about having to hold high or low to compensate for a cross wind, and these guys stick all kinds of wind flags on the range and watch them like hawks.

[ScottRiqui]

I agree that the Magnus effect is negligible, at best. I only addressed it because I've had people in the past (erroneously) give it as a reason in favor of using a rifled barrel with a round ball. In actuality, spinning the projectile is what causes the Magnus effect. As you said, the Magnus effect probably isn't going to be noticeable, but it's certainly not an argument in favor of using a rifled barrel.

[pohill]

If rifling has no effect on a roundball, why does my rifled Hawkins .50 shoot accurately at 100 yds but my smoothbore Belgian .54 is, literally, hit or miss at 50 yds?
Also, why does a non-spinning knuckleball move as it does, yet a spinning fastball is straight and accurate?
I'm not challenging, I'm just curious, and possibly I missed your meaning.

[ScottRiqui]

If you read the whole (long and boring) post, I acknowledge in the second-to-last paragraph that rifling *does* improve performance with round balls - just not for any of the reasons that are accounted for in Greenhill's formula.

A knuckleball acts the way it does because of the effect of turbulence on the stitching of the ball - nothing more, nothing less. If the ball were perfectly smooth, you couldn't throw a knuckleball and get the same effect. The spin on a fastball "averages out" the stitching imperfections on the surface of the ball so that it flies true, much like spinning a round ball from a rifled barrel averages out the imperfections in the cast ball. That's the real benefit of rifled barrels with round balls.

Again, Greenhill purposely ignored things like surface imperfections, density variation from poor casting and air resistance when he came up with his formula, so even though such things affect real roundballs in flight, Greenhill's formula doesn't account for them in the first place, so they're not justifications for using his formula for round projectiles.

[pohill]

Interesting stuff. Thanks for the reply.

[suzukibruce]

with a .454 ball in a revolver the ring cut off is pretty significant and turns a round ball into an oval...???? maybe? i don't know i didnt read the long and boring physics lesson, just sounds like common sense to me...

[B.L.E.]

I think the practice of rifling barrels predates the invention of the conical bullet by quite a bit and the optimum twist rate for round ball barrels was probably determined by TLAR (that looks about right), and who won the last shooting match.

[Brazos_Jack]

Greenhill may not be perfect for lead round balls but if followed with c values of 150 to 200, it works VERY well. Further, experiments will find that 150 does work "a little" better than 180-200.

Admitedly early rifling was WAY too fast. They would try something already too fast. The stress of the excessively high angular acceleration would cause the ball to strip the groove. This gave accuracy worse than a smooth bore. Instead of changing one thing at a time, as is good science, early riflesmiths would assume they didn't stableize the ball. They would then start a cycle of faster twist, deeper grooves, and tighter ball fit until they stopped the stripping. By then the ball fit was usually so tight it had to be driven down the bore.

Around 1860, long before Greenhill, a Brit in India with WAY too much time on his hands wrote "The Sporting Rifle and Its Projectile". In it he described his extensive experiments with .69 cal rifles. He found that 1 turn in 8'-8" worked best but rifling as slow as 1 in 12' would suffice for good accuracy. He also found that with such slow rifling he only needed about .004" deep rifling to avoid stripping the grooves and that a greased linen patch around the ball would clean the wide shallow grooves completely each loading.

In 1978 I validated his results by having a .50 cal barrel rifled 1 in 75" with wide shallow (.005") grooves. That rifle was deadly accurate with a 90 grain charge. I tested it with excellent accuracy up to 120 grains but feared continued use of such a charge.

[Noz]

I am not a math whiz nor did I stay in a Howard Johnsons last night. Does not the gyro effect enter into the external ballistics formulations. That was my assumption on the reasoning for rifling a round ball shooter.

RE: twist rates. I built a 50 cal CVA a number of years ago with the express intent of shooting a patched round ball. It happened to have a universal(shoots anything) 1-45 twist. I tried every combination of patch and every powder charge from 120 grs to 40 grs of FFg and FFFg and my best grouping of all of that mess was about 4" at 35 yards. Just before I threw the gun in the trash, I bought a box of TC 285 gr maxi hunters. One hole groups at 35 yards. Increasing the powder charge by 10 grains raised the POI by 1". I settled on a comfortable shooting highly accurate 80 grs of FFg. Off the bench my best group was a hair under 2" at 100 yds.

[ScottRiqui]

Does not the gyro effect enter into the external ballistics formulations? That was my assumption on the reasoning for rifling a round ball shooter.

For a perfectly smooth spherical ball with no density variations, the gyroscopic effect wouldn't be needed. No matter how the ball rotates in flight (pitch, yaw or roll), the profile that it presents to the oncoming air would remain unchanged. Compare that to a longer, pointed bullet where if it yaws or pitches, the nose will no longer be pointed straight ahead and it will eventually begin to tumble if the nose gets too far away from "straight ahead".

What inspired my post in the first place was the total disconnect between the phenomena that are modeled in Greenhill's complete formula and the exterior ballistics of a round ball. On one hand, none of the characteristics that are accounted for in Greenhill's formula apply to round balls - since both the center of pressure and the center of gravity of a perfect round ball are always at the exact center of the ball, a perfect round ball can be neither dynamically stable nor unstable in flight.

On the other hand, the surface imperfections and density variations that make rifling useful for real-world spherical balls are things that are not accounted for or modeled at all in Greenhill's calculations.

In short, the fact that Greenhill's simplified formula "works" for real-world spherical projectiles is nothing more than a happy coincidence. Further, Greenhill's calculations only work for round balls if you use the common simplified approximation (TL = C * d^2). If you were to use his full formula instead of the simplified version, the equations would "blow up" and the result of the formula would be that you should use no twist at all (one twist per infinite inches of barrel length).