Miller twist rule is a mathematical formula derived by American physical chemist and historian of science Donald G. Miller (1927-2012) to determine the rate of twist to apply to a given bullet to provide optimum stability using a rifled barrel.[1] Miller suggests that, while Greenhill's formula works well, there are better and more precise methods for determining the proper twist rate that are no more difficult to compute.
The following formula is one recommended by Miller:
{t}2=
| 30m | |
| sd3l(1+l2) |
where
Also, since one "caliber" in this context is one bullet diameter, we have:
{t}=
| T | |
| d |
where
T
{l}=
| L | |
| d |
where
L
Solving Miller's formula for
s
{s}=
| 30m | |
| t2d3l(1+l2) |
Solving the formula for
T
{T}=\sqrt{
| 30m | |
| sdl(1+l2) |
Note that the constant 30 in the formula is Miller's rough approximation of velocity (2800 ft/sec or 853 m/s), standard temperature (59 degrees Fahrenheit or 15 celsius) and pressure (750 mmHg or 1000 hPa, and 78% relative humidity). Miller states that these values are taken from the Army Standard Metro but does note that his values are slightly off. He goes on to point out that the difference should be small enough that it can be ignored.
It should also be noted that the bullet density is missing from Miller's formula despite the fact that Miller himself states his formula expands upon Greenhill's. The bullet density in the equation above is implicit in
m
Finally, note that the denominator of Miller's formula is based upon the relative shape of a modern bullet. The term
l(1+l2)
When computing using this formula, Miller suggests several safe values that can be used for some of the more difficult to determine variables. For example, he states that a mach number of
M
s
Using a Nosler Spitzer bullet in a .30-06 Springfield, which is similar to the one pictured above, and substituting values for the variables, we determine the estimated optimum twist rate.[2]
t=\sqrt{
| 30m | |
| sd3l(1+l2) |
where
t=\sqrt{
| 30*180 | |
| 2.0*.3083*3.83(1+3.832) |
The result indicates an optimum twist rate of 39.2511937 calibers per turn. Determining
T
t
T=39.2511937*.308=12.0893677
Thus the optimum rate of twist for this bullet should be approximately 12 inches per turn. The typical twist of .30-06 caliber rifle barrels is 10 inches per turn, accommodating heavier bullets than in this example. A different twist rate often helps explain why some bullets work better in certain rifles when fired under similar conditions.
Greenhill's formula is much more complicated in full form. The rule of thumb that Greenhill devised based upon his formula is actually what is seen in most writing, including Wikipedia. The rule of thumb is:
Twist=
| CD2 | |
| L |
x \sqrt{
| SG | |
| 10.9 |
The actual formula is:[3]
S=
| s2*m2 | ||||||
|
where
| C | |
| M\alpha |
Thus, Miller essentially took Greenhill's rule of thumb and expanded it slightly, while keeping the formula simple enough to be used by someone with basic math skills. To improve on Greenhill, Miller used mostly empirical data and basic geometry.
Miller notes several corrective equations that can be used:
The velocity (
v
T
| 1/2 | |
| f | |
| v{ |
The velocity (
v
s
fv=[
| v | |
| 2800 |
]1/3
The altitude (
a
fa=
| 3.158x10-5*h | |
| e |
h