ScottRiqui

May 18, 2010, 07:58 PM

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.

_________________________________________

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.