Miller twist rule

A diagram of a .30-06 Springfield showing the bullet diameter (7.85 mm) and length (31.28 mm).

The following formula is one recommended by Miller:[1]

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

where

• m = bullet mass in grains
• s = gyroscopic stability factor (dimensionless)
• d = bullet diameter in inches
• l = bullet length in calibers
• t = twist rate in calibers per turn

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

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

where ${\displaystyle T}$ = twist rate in inches per turn, and

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

where ${\displaystyle L}$ = bullet length in inches.

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]

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

where

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

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

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

${\displaystyle T=39.2511937*.308=12.0893677}$

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:

${\displaystyle Twist={\frac {CD^{2}}{L}}\times {\sqrt {\frac {SG}{10.9}}}}$

The actual formula is:[3]

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

where

• S = gyroscopic stability
• s = twist rate in radians per second
• m = polar moment of inertia
• ${\displaystyle C_{M_{\alpha }}}$ = 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.

Miller notes several corrective equations that can be used:

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

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

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

• Rifling