The Greenhill Formula
The Greenhill Formula is a simplified method for determining mathematically the amount of spin necessary to stabilize a bullet. It was worked out in 1879 by Sir Alfred George Greenhill who was a Professor of Mathematics at Woolwich and teaching the Advanced British Artillery Officers Class. It was considered satisfactory for bullets having a density of .392 lbs/cubic inch or greater. (Lead has a density of .409 lbs/ cubic inch, and copper has a density of from .318-.325 lbs/cubic inch, depending on the alloy) The formula is Twist required (in calibers) = 150 divided by the length of the bullet (in calibers). It makes no allowance for nose shape, considering round noses and all spitzers and spire points as the same. It does not work for bullets having a density below .392 lb/ cubic inch. All copper or brass solids and most heavy jacketed bullets have average densities below .392 lbs./cubic inch. Notice I said average, as the formula makes no allowance for bullets of variable construction, linearly. The formula was a shortcut and was useful at the time, as most bullets were roundnoses and were lightly jacketed, if jacketed at all. Because the math is simple, the Greenhill Formula has remained in use to this day. Just a few years ago I had an engineer at a major ammunition manufacturing firm quote me the Greenhill Formula as a method for calculating the spin required to stabilize a long, 10 caliber spitzer, 7mm 175 gr. , variable density, hunting bullet. Needless to say, he was not even close. The Greenhill Formula is accurate when used in the context for which it was intended, but many folks who use it today have forgotten, or never learned that context.
The actual formula is much more complicated It is Gyroscopic Stability (GS) = the spin rate (in radians per second, squared) times the polar moment of inertia, squared, divided by the pitching moment coefficient derivative per sine of the angle of attack times the transverse moment of inertia times the air density times the velocity squared. (My keyboard does not have all the correct symbols and that is why I wrote it out). For the bullet to be stable, GS > 1.0. This is actually a short version as the pitching moment coefficient component is a complicated calculation that derives the center of gravity and the center of reverse air pressure. The equation is basically calculating the linear difference between the center of gravity and the center of reverse air pressure on the nose of the bullet. The greater the difference, the greater the spin required to keep the bullet pointed nose forward. It used to take me about three days to calculate one new design by hand. My computer does it in about 20 seconds, now.
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