48 PRECISION SHOOTING — JUNE 2009

BY DON MILLER

SummarySeveral simple rules for esti-

mating twist, some including thestability factor, do not need theshape and mechanical properties of the bullet. Here I compare howwell such rules fit actual experi-mental data for 14 bullets at var-ious velocities (40 examples.) Thenew Miller Twist Rule, presentedin Precision Shooting, March 2005,turns out to be significantly betterthan the others. The Harris Ruleworks very poorly.

IntroductionMost shooters know that a

rifling twist that stabilizes a shortbullet may not stabilize a longerone. The pressure to replace leadbullets with other metals, as inCalifornia, makes this increasinglyimportant. Bullets using popularreplacement metals, such as copperand bronze, are longer for the sameweight than are lead-based ones.Therefore, many rifles may havetoo slow a twist to stabilize suchnon-lead bullets.

The obvious question is whattwist do we need to get sufficientstability in such non-standard bul-let and rifle combinations.

The required twist depends

mostly on the bullet’s length,but also on its shape and velocity.Shape determines its mechani-cal properties, such as its weight,center of gravity, and the twomoments of inertia. Shape alsodetermines the overturning mo-ment coefficient, an aerodynamiccoefficient that depends on thevelocity. Air density is also im-portant (cold weather can make a barely-stable bullet unstable.) All these quantities appear in themathematical formulas that des-cribe stability.

Sufficient gyroscopic stabilityis necessary to keep a bullet fly-ing point forward. However, with too much stability (overstabil-ity), poor quality bullets can give larger group sizes. These issuesare characterized by the stabilityfactor, whose value must be greaterthan 1.0.

The military usually choosesstability factors between 1.5 and2.5. To protect against cold wea-ther, 2.0 is safe, and 1.5 is suitablefor most applications. Bench Rest-ers often opt for 1.3 to minimizethe effects of bullet imperfection,but that can be risky at low tem-perature or if the stability factorwas set for high altitudes. How-ever, stability factors even as highas 3.5 to 4.0 are usually not detri-

mental to small arms shooting,which is at low angles οf elevation(“flat fire”.)

Most twist rules have their ori-gin in mathematical equations forstability, including Greenhill’s orig-inal 1879 rule [G79,M05a,M06].Miller’s 2005 rule [M05,M08a]is directly based on the modern stability equations, and has thestability factor explicit, as do theHarris and Miller-Greenhill rules.See equations (1, 3, 6) on next page.

Although “fast design” compu-ter programs exist for estimating the aerodynamic coefficients andmechanical properties needed forstability calculations, they requiredetails about the construction andshape of the bullet. Unfortunate-ly, these data are available for only a few of the several thousand sporting bullets. The simpler semi-empirical rules we test below ignorethese shape factors, so should helpin analyzing stability and twist forthat large number of sporting bulletswithout such data.

Twist Rules To Be TestedNotation: bullet diameter d in

inches, rifling twist T in inches/turn or t in calibers/turn (t=T/d)(one turn is one complete rotation),bullet length L in inches or l incalibers (l=L/d), bullet weight m

PRECISION SHOOTING — JUNE 2009 49

Continued on next page

in grains, velocity v in ft/sec, andstability factor s (dimensionless). Ichoose Army Standard Metrostandard conditions, and f

vis a

velocity correction. The air densitycorrection f

adepends on tem-

perature, pressure, or alternativelyon altitude above sea level [M05,M08a]. The standard-conditionsaltitude correction is f

a= exp

(3.158 ×10-5×h)(h=altitude in feet)[McC99] and is important. If thetwist is set for a specific stabilityfactor at 7000 ft, but the gun isshot at sea level, the sea-levelstability factor will be about 20%less! Conversely, for the samestability factor, the twist must beabout 10% smaller.

We test the following simplenew and old rules for estimatingrifling twist, equations (1-7) below.

Equations (1) and (2) were firstdescribed in the March 2005Precision Shooting [M05,M08a].They were obtained from themodern stability equation [McC99],with correlations of moments ofinertia and the overturning moment(and its velocity dependence) fromexperimental data in BRL Reports.By (blind stupid) good luck, theshape factors mostly cancel for solidand solid-core bullets. Below thevelocity of sound, 1120 ft/sec, usethe velocity correction f

vfor 1120

ft/sec. An Excel program, by BryanLitz and me, is available for calcu-lating (a) twist given stability factor,(b) stability factor given twist, or (c) maximum length of bullet for a given twist and stability factor.

Equation (3) is less well known,and is due to C. E. Harris. I thankTed Almgren of Sierra Bullets forits formula. Excel programs for itare on the Internet (search for PeterCronhelm).

Equation (4) is the “classical”Greenhill formula found in theBritish Textbook of Small Arms,

(1) ( ) ( )2/1

22

22/1

23

30

1

30vvM f

LdLs

dmf

llds

mdT

+=

+= Miller with velocity (v ) correction

where

(2)3/16/1

2/1

2800,

2800

=

= v

fv

f vv v corrections for twist (T ) and stability factor (s ),

(3) ( ) ( )vs

m

lvs

m

lTvH −

=−

=5705

46.1557

5705

570562.2025.225.2

Harris

(4)9.10

150 density

l

dTGden = “Classical Greenhill with density correction

(5)6/1

28009.10

150

= vdensity

l

dT Gdenv Greenhill with density and Miller v corrections

(6)6/1

28009.10

4.216

= vdensity

sl

dTvMGden Miller-Greenhill with s, density, and v corrections

(7)9.10

5.3 density

l

vdT

vGdenv= Various authors

respectively

“

50 PRECISION SHOOTING — JUNE 2009

How Good Are SimpleRules For EstimatingRifling Twist Continued

1929 [TSA29], with the correctionfor bullet density (in gram/cm3)made explicit. This formula has no velocity correction because itwas based on subsonic flow. It is a good approximation to Green-hill’s much-more-complicated orig-inal formula [M05a,M06]. At 2800ft/sec (Mach number M=2.5), itcorresponds to stability factors of1.5-2.0 [M05a,M06], as also foundby C. E. Harris and Bill Davis.

Equation (5) is equation (4) withthe velocity correction of equation(2).

Equation (6) looks much like theGreenhill formula, but it includes thestability factor s and the velocity

correction of equation (2). It is con-siderably better and was indepen-dently derived from a variant ofequation (1) with a further assump-tion about bullet volume [M08c].

Equation (7) has a number ofsimilar versions from variousauthors with coefficients other than3.5, all of which use a velocitycorrection factor of v1/2. I haveadded the explicit bullet-densitycorrection. I have no original sourcefor equation (7), so I apologize to its original author.

Sources of Input DataThese twist rules were tested on

40 examples for 14 bullets fromvarious sources, almost all fromBallistic Research Lab reports. Such experimentally-measured data,which include weight, twist, sta-bility factor, bullet length, and

velocity, are very hard to find. These40 examples include all the sportingbullets plus some military ball orsolid test projectiles that are readilyavailable from the military litera-ture. These more or less fit into thesolid or solid-core condition for theMiller Rule [M05,M08a]. The rawdata for the .308 Sierra and .50 M33bullets were corrected to zero yawby standard methods [McC99].Their s data were clustered at theircorresponding clustered Mach num-ber values, and both were averaged[M08a]. The 40 examples include 4 data sets at velocities other thanthose presented in my PrecisionShooting Part I article [M08a].These are the three .308 Sierra andthe .224 Federal bullets. Table Icontains the input data. (exp meansexperimental values.)

Greenhill-type rules require bul-let densities. Some bullets weremade of known materials: the 4.32mm ball (a steel test bullet), BRL-1(Cu plated steel), and ANSR-5(Dural aluminum). Some could beestimated from their volume (calcu-lated from their known shapes) andweight (.308 Sierra 168, 175, and190 grain, and the .50 M33.) Theother estimates are somewhat uncer-tain. There are also some uncertain-ties in the description of the gunsand twist for the .224 Federal 68grain in BRL-1630, where only onetwist was stated. The other had to beestimated from T=2πd /spin.

TestsThe test for each example

consists of estimating the twistfrom the various equations (1-7)using measured values of bulletweight, diameter, and length, plus the stability factor and densitywhen necessary. This estimate wascompared with the measured twistfor the test gun. Table II containsthe results.

Table III contains the residuals(errors), where residuals are definedas (estimated — experimental).

PRECISION SHOOTING — JUNE 2009 51

Therefore, a negative residual rep-resents a smaller (faster, quicker,shorter) estimated twist in inches per turn, and a positive one means a larger (slower) twist. Table IVis a summary of statistical quantitiesfor the residuals and percent res-iduals, where the percent residual (not tabulated here) is the residualdivided by the experimental twist.When we reverse calculate thestability factor s from twist, thepercentage error doubles.

Note that equation (1) was origi-nally obtained [M05] using datathat included some of the bulletshere but also some 20 others aswell, all at 2800 ft/sec. The velocitycorrection, equation (2), was thenobtained from data for other vel-ocities. Therefore, the test here isindependent.

The calculated results are givento two decimals for comparisons,although the uncertanties in theestimates are larger than onedecimal.

ResultsTables III and IV show the

following.The Miller Rule, equation (1), is

distinctly the best, with a residualstandard deviation of only 0.3". Furthermore, 95% of the estimatesare within 0.5" (compatible with thestatistical 2 standard deviations). Its percentage residuals are equallyencouraging, with a standard dev-iation of 2.8%. Furthermore, 95%are within 5.2% (again compatiblewith statistics.) The maximumdeviation from experiment is -0.9",and fortunately a minus residualmeans the twist is more conserva-tively estimated, i.e., is faster andtherefore safer against cold weatheror lower altitudes. The worstexamples are the .224 Sierra 55grain and the lowest-velocity SierraInternational Bullet.

Continued on next page

52 PRECISION SHOOTING — JUNE 2009

How Good Are Simple Rules ForEstimating Rifling Twist Continued

The Miller-Greenhill formula, equation (6), is nextbest and surprisingly good. The standard deviation and the range (maximum spread) of both the residuals and percentage residuals are about 50% larger than for the Miller rule, i.e., 0.45" and 5.4%, with 95% ofthese two types of residuals within about 3/4" and 9%respectively. The worst case is the M193 Ball.

The two classical Greenhill cases, equation (4)without velocity correction and equation (5) with the Miller velocity correction, are next best and close to each other. However, their standard deviations areconsiderably larger, and about 41/2 to 5 times larger thanthe Miller Rule’s.

The Harris Rule, equation (3), is even worse andsurprisingly poor. A residual plot (not shown) shows a steady increase with d: with negative values at low d, fairly good results for d from about .270 to .338,and positive values above that. The residual standarddeviation is 7 times larger and the percentage residualstandard deviation is about 5 times larger than thecorresponding Miller ones, and about double the onesfor the Miller-Greenhill case. Although the d depen-dence can be reduced by an empirical power of d in the denominator, its errors are still double those of the Miller-Greenhill Rule, so that correction does not seem useful. Sierra Bullets uses Harris’ Rule forcomparisons not designs, and bases its twist recom-mendations on long experience.

Finally, the worst is the Greenhill with the v1/2

correction factor, equation (7), which was expected tobe poor. Its residual standard deviation is about thesame as Harris’, and the percentage one is about 10%larger than Harris’.

ConclusionsThe analysis above suggests that the best estimation

rule for rifling twist (and in reverse the stability factor)is the Miller Rule, equation (1). This is naturally verygratifying to me. The next best is the Miller-GreenhillRule. Both are easy to use. However, the Miller Rulenot only gives better results, but uses readily-availablebullet weights in contrast to Miller-Greenhill, whichuses the less-available bullet densities.

The Miller Rule is recommended, because so far it’sboth the best simple rule for estimating rifling twists orstability factors of sporting bullets, and therefore a toolfor exploring non-standard bullet-rifle combinations.

Finally, while the Miller Rule agrees very well withexperiment here, such a semi-empirical rule cannotcover all cases. There is anecdotal evidence that (1)

plastic-tip or hollow-base bullets may have largererrors, and (2) the velocity correction above 3600 ft/secmay predict faster twists than needed. Improvementmust await new twist-stability factor data.

Don Miller2862 Waverly WayLivermore, CA 94551don.miller9@comcast.net

References

[G79] A. G. Greenhill, “On the Rotation Required forthe Stability of an Elongated Projectile,” Minutes Proc.Royal Artillery Inst., X (#7), 577–593 (1879).

[M05a] Donald Miller (Donald G. Miller), “Where Did Greenhill’s Twist Rule Come From?” Black PowderCartridge News, Issue 50, Summer, 30-32, 34, 36 (2005).

[M06] Donald G. Miller, “Where did Greenhill’s twistrule come from?” Int. J. Impact Eng. 32, 1786-1799(2006). This article, essentially the same as [M05A],is in a more scientific style with cgs units and alsosummarizes my new twist rule.

[M05] Don Miller (Donald G. Miller), “A New Rulefor Estimating Rifling Twist: An Aid to ChoosingBullets and Rifles,” Precision Shooting, March, 43-48(2005).

[M08a] Don Miller (Donald G. Miller), “The NewTwist Rule: Part I: Tests Against Experimental Data,”Precision Shooting, February, 73-78 (2008).

[McC99] Robert L. McCoy, “Modern Exterior Ballistics:The Launch and Flight Dynamics of Symmetric Pro-jectiles,” Schiffer Publishing, Ltd., Atglen, PA, 1999.

[TSA29] “Textbook of Small Arms, 1929.” HMSO,London, 1929. Reprint Holland Press, 1961.

[M08c] Don Miller (Donald G. Miller), “The NewTwist Rule: Part III: The Basis for a GeneralizedGreenhill’s Rule,” Precision Shooting, April, 70-72(2008).