Sunday, November 11, 2012

How to Calculate the Twist Rate - II

In our last post, we looked at the venerable Greenhill formula, first developed in 1879 and refined over the years. In 2005, Don Miller suggested an improved formula to calculate the twist rate, in the March 2005 issue of Precision Shooting magazine. We will look at that formula in this post.

Some of the problems with the Greenhill formula was that it was more suited to bullets from another era, where the bullet shapes were more oblong shaped (like an American football) and largely made of lead alone. Modern bullets are longer (e.g. spitzer or boat-tail shape) and made of multiple materials (such as copper and brass jackets, steel core etc.). The corrected Greenhill formula does work better than expected for modern bullet velocities of 2800 feet/sec, but it doesn't work so well for black powder velocities.

What we studied in the previous post about the Greenhill formula was actually its simplified form. The original Greenhill formula was much more complicated and involved calculating such esoteric items as polar moment of inertia, transverse moment of inertia, pitching moment coefficient, angle of attack, air density etc. To work out the original Greenhill formula, one would need a degree in physics and mathematics to make sense of all these terms, as well as access to some high quality scientific instruments to make the measurements needed to calculate all these items. What Miller did was start with the original Greenhill formula and used some empirical data to simplify the calculations so that one didn't need an advanced degree to do the whole calculation and could use basic instruments to do the measurements. The Miller formula is:

     T = Twist rate in inches per turn
     m = Weight of the bullet in grains
     s = Gyroscopic stabilization factor (see below for how this is evaluated)
     d = Bullet diameter in inches.
      l = Bullet length in calibers, which is calculated as L/d, where L = length of bullet in inches.

With the Miller formula, all the information needed for the calculation (length, diameter and weight of the bullet) can be easily obtained from the manufacturer, or can be easily measured by anyone with access to a vernier caliper and a small weighing scale.

Miller notes that his constant 30 in the equation above was taken assuming a standard temperature of 59 degrees fahrenheit, standard pressure of 750 mm. of mercury at 78% humidity, velocity of bullet at 2800 feet/sec and altitude at sea level. He also notes that under the standard conditions, the gyroscopic stabilization factor s in the equation above runs from 1.3 to 2.0 (for military, it runs from 1.5 to 2.0) and he says 1.75 is a good starting value for it. He also states that cold temperatures significantly affect air density and therefore s as well. Hence, he recommends assuming s = 2.0 to account for usage in low temperature environments for preliminary calculations of twist.

With all this in mind, let us perform a sample calculation for a Sierra bullet with the following specifications:
    m = 180 grains
    L = 1.180 inches
    d = 0.308 inches

First we calculate l = L/d, which gives us l = 1.18 / 0.308 = 3.83117

Then we assume s = 2.0 to account for low temperature conditions and calculate the twist rate T, which works out to about T = 12.081. This means that a twist rate of 1 turn in 12 inches ought to work well for this bullet.

In his paper, Miller says that he used experimental data from the US Army's Ballistic Research Lab (BRL) to verify the validity of his formula.


  1. As far as I can see wikipedia is wrong on this and you copied that?

    T^2 = 30m/sdl(1+l^2), you need a square root. Which you obviously used to get the 12.083 figure, but it's not in the formula.

    1. You are correct sir. Mea culpa, I did indeed need to take the square root. I modified the image for the formula to reflect the square root in the expression. Thank you very much for pointing out the error.