Miller twist rule

Miller twist rule is a mathematical formula derived by Don Miller to determine the rate of twist to apply to a given bullet to provide optimum stability using a rifled barrel. 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.

Formula
The following formula is one recommended by Miller:

$${t}^2 = \frac{30m}{sd^3l(1+l^2)}$$

where
 * m = bullet mass in grains (defined as 64.79891 milligrams)
 * s = gyroscopic stability factor (dimensionless)
 * d = bullet diameter in inches
 * l = bullet length in calibers (that is, length in relation to the diameter)
 * t = twist rate in calibers per turn

Also, since one "caliber" in this context is one bullet diameter, we have:

$${t} = \frac{T}{d}$$

where $$T$$ = twist rate in inches per turn, and

$${l} = \frac{L}{d}$$

where $$L$$ = bullet length in inches.

Stability factor
Solving Miller's formula for $$s$$ gives the stability factor for a known bullet and twist rate:

$${s} = \frac{30m}{t^2d^3l(1+l^2)}$$

Twist in inches per turn
Solving the formula for $$T$$ gives the twist rate in inches per turn:

$${T} = \sqrt{\frac{30m}{sdl(1+l^2)}}$$

Safe values
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 $$M$$ = 2.5 (roughly 2800 ft/sec, assuming standard conditions at sea level where 1 Mach is roughly 1116 ft/sec) is a safe value to use for velocity. He also states that rough estimates involving temperature should use $$s$$ = 2.0.

Example
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.

$$t = \sqrt{\frac{30m}{sd^3l(1+l^2)}}$$

where
 * m = 180 grains
 * s = 2.0 (the safe value noted above)
 * d = .308 inches
 * l = 1.180" /.308" = 3.83 calibers

$$t = \sqrt{\frac{30 * 180}{2.0 * .308^3 * 3.83(1+3.83^2)}} = 39.2511937$$

The result indicates an optimum twist rate of 39.2511937 calibers per turn. Determining $$T$$ from $$t$$ we have

$$T = 39.2511937 * .308 = 12.0893677$$

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.

Comparison to Greenhill's formula
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:

$$Twist = \frac{C D^2}{L} \times \sqrt{\frac{SG}{10.9}}$$

The actual formula is:

$$S = \frac{s^2 * m^2}{C_{M_\alpha} \div \sin(a) * t * d * v^2}$$

where
 * S = gyroscopic stability
 * s = twist rate in radians per second
 * m = polar moment of inertia
 * $$C_{M_\alpha}$$ = pitching moment coefficient
 * a = angle of attack
 * t = transverse moment of inertia
 * d = air density
 * v = velocity

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.

Corrective equations
Miller notes several corrective equations that can be used:

The velocity ($$v$$) correction for twist ($$T$$): $$f_v{^{1/2}} = [\frac{v}{2800}]^{1/6}$$

The velocity ($$v$$) correction for stability factor ($$s$$): $$f_v = [\frac{v}{2800}]^{1/3}$$

The altitude ($$a$$) correction under standard conditions: $$f_a = e^{3.158x10^{-5} * h}$$ where $$h$$ is altitude in feet.

Calculators for stability and twist

 * Bowman-Howell Twist Rate Calculator
 * Miller Formula Calculator
 * Drag/Twist Calculator based on Bob McCoy's "McGyro" algorithm