The Greenhill formula is an empirical equation that does a good job of establishing the barrel twist necessary so that a bullet of a given length will be adequately stabilized.
Greenhill made easy:
T=150 x D
---
R
here: T is the twist required (number of inches for one revolu-
tion),D is the bullet diameter (in inches)
R is the bullet length to diameter ratio, (length divided by diameter)
Conversely, to find out what length bullet will be stabilized in a given twist, use:
L=150 x D x D
-----
T (that is, 150 x D squared divided by T)
where L= bullet length
The number 150 is a constant used by Greenhill and works well at velocities in the vicinity of 1500 fps or greater. At 2800 fps the constant can be changed to 180 with good results.
Note that it is bullet LENGTH, not weight that is important. Greenhill works well with all lead/lead-alloys commonly used for bullets.
One must understand that Greenhill is derived from experience as well as from the calculated laws of physics. As such, it is a highly simplified, albeit useful, equation. Greenhill was worked out many years ago and is quoted in the British Textbook of Small Arms (1929). To the best of my knowledge, it was not used for the very low velocities that are representative of pistol bullets. The supposition that the constant decreases with veloc- ity is probably correct but only trials would prove that one way or the other.
There is no "ideal" bullet for any given barrel caliber and twist. Fortunately, many bullet type/velocities will work. Greenhill merely simplifies the starting point and gets one into the ball park. Once in the ballpark there are other factors affecting accuracy that become much more influential than small changes in bullet characteristics and velocities. Except for the extremely exacting requirements of bench rest shooters, and very long range shooters, an "over-stabilized" bullet (rifling twist faster than the minimum required for good stabilization) will shoot as well as a theoretically critically stabilized one.
Barrel twist is usually chosen for the longest bullet that will be fired in that gun. Accordingly, the military has fairly recently changed their .223 barrels to a 7 1/2" twist from a slower one for the newer, longer and heavier bullet that has become standard issue (68 grains, I believe).
For instance, Greenhill shows that the maximum .308 caliber bullet length for a 12" twist, using k=150, to be 1.2 inches. If a bullet longer than this is used, it is possible that it could become unstable. A k of 150 is not magic but has been empirical- ly demonstrated to be a reliable optimum.
Incidentally, groups at 100 yards are not necessarily indicative of long range accuracy, especially if your bullet length to twist relationship approach the critical. The gyroscopic (dynamic) stability can fall off at long range and what worked well at 100-200 yards can go wild at long range. I have (unintentionally) demonstrated this to myself on a number of occasions. It happens quite frequently with handguns because of the greater bullet length and velocity ratios that are commonly encountered.
I have been working with some ultra-light loads in a 7mm Remington Rolling block and loads that shoot about 1 inch at 25 yards go to about 8 FEET at 50 yards -- really enlightening! At these low velocities one can watch the bullet start out straight, then, as it's dynamic stability wanes, go into a dramatic curve. Kicking the velocity (therefore the rpm) up a couple of hundred feet a second increases the accurate range.
I presume that T is the minimal twist? So if my rifle has a twist of 10, and considering the bullet I'd be shooting, Green- hill's formula comes up with a T of 12, then every thing is ok.
Yes.
A T (twist) of 12 means 12" bore length per rifling revolution. In your rifle, the bullet will spin 360 degrees for each 12" of linear rifle barrel that it traverses. A twist of 10 is a faster twist; one revolution for each 10". Within reason, a twist that is faster than necessary will not appreciably effect stability (accuracy). This why the rifling is set up for the longest bullet that is likely to be fired in a particular gun.
What's meant by "how long a bullet will stabilize in a given twist"? You mean how long until it's stable, or how long till it's not stable? What unit is 'L'?
L= bullet length. It is the bullet length that is important - not its weight. A pure lead cylinder would be stabilized at a slower twist than a bullet formed of the same material and weight but with a pointed nose.
If one has a burning interest to pursue the understanding of this very complex subject, he can read the American Rifleman article by Col. E.H. Harrison, pg. 50, July 1962. For those who have the time, facilities and competence, Col. Harrison's article will give all necessary instructions to carry out meaningful experi- ments.
For those who would merely like to read a little more on the subject, the following Cast Bullet Association journal, The Fouling Shot, now called The Cast Bullet, articles are interest- ing and useful.
TWIST PROBLEMS EXPLAINED
TWIST FOR HEAVY .22'S
REFERENCES ON RIFLING TWIST (GREENHILL'S RULE)
THE MATCH POTENTIAL OF 6MM CAST BULLETS
MARLIN'S MICRO-GROOVED 44S
ON BULLET STABILITY
COMPARING CAST BULLET CALIBERS
MORE ABOUT GREENHILL'S FORMULA
GREENHILL MADE EASY
PRACTICAL MATH FOR SHOOTERS
GREENHILL'S FORMULA SIMPLIFIED
84-16 2
DAVIS, W.C. 25-02 17
HARRIS, C.E. 48-03 15
HARRIS, C.E. 48-05 19
LATHAM, GLENN R. 49-25 25
HARRIS, C.E. 49-27 33
BARNISKIS, A.E. 50-08 7
DAVIS, W.C. 50-28 37 Table
DAVIS, W.C. 52-02 20
LATHAM, GLENN R. 53-38 20 TABLE
FLETCHER, J. 73-22 20
These may be obtained from:
Frank Stanard, Director of Services
Cast Bullet Association
Ralland Fortier
4103 Foxcraft Drive
Traverse City, Mich. 49684
616-929-0553
$14.00/yr.
God Bless!
Norm