en>Mirams: Links to A-stable.

2014-09-16T17:34:04Z

Links to A-stable.

24.176.166.175 at 01:20, 25 February 2013

2013-02-25T01:20:25Z

← Older revision		Revision as of 03:20, 25 February 2013
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	~~53 year old Medical Emission Therapist Calvin Ditsch~~ from ~~Charny~~, ~~usually spends time~~ with ~~hobbies~~ and ~~interests like hang gliding~~, [http://www.sk-~~gaming~~.com/~~blog~~/~~mizanpagla3~~/~~1785219~~- ~~Marvel Puzzle Quest Hack~~] ~~and train spotting~~. ~~Has completed a wonderful around~~ the ~~world trip that included touring the Bourges Cathedral.~~		'''Miller twist rule''' is a mathematical formula created by Don Miller to calculate the optimum rate of twist for a given bullet traveling through a [[Rifling\|rifled]] barrel.<ref name=c1>Miller, Don. ''[http://www.jbmballistics.com/bibliography/articles/miller_stability_2.pdf How Good Are Simple Rules For Estimating Rifling Twist]'', Precision Shooting - June 2009</ref> Miller suggests that while [[Alfred George Greenhill\|Greenhill]]'s formula works well there are better and more precise ways to calculate what the proper twist rate for a bullet should be that are not much more difficult to calculate.

			==Formula==
			[[File:.30-06 Springfield.svg\|thumb\|400px\|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.<ref name="c1" />

			<math>{t}^2 = \frac{30m}{sd^3l(1+l^2)}</math>

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

			Given those definitions we can expand:

			<math>{t} = \frac{T}{d}</math>

			where <math>T</math> = twist in inches per turn, and

			<math>{l} = \frac{L}{d}</math>

			where <math>L</math> = bullet length in inches.

			===Stability factor===
			Using Miller's formula we can also calculate the stability factor assuming we already know the twist. Simply solve for <math>s</math>.

			<math>{s} = \frac{30m}{t^2d^3l(1+l^2)}</math>

			===Twist in inches===
			It is, of course, possible to solve for twist in inches directly by simply solving for <math>T</math>

			<math>{T} = \frac{30m}{sdl(1+l^2)}</math>

			===Notes===
			Note that the constant 30 is Miller roughly approximating the velocity as 2800 ft/sec, standard temperature (59 degrees Fahrenheit), and pressure (750 mm Hg and 78% humidity) in the equation. Miller states that these values are taken from the [[Standard conditions for temperature and pressure\|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 <math>m</math> through the [[moment of inertia]] approximation.

			Finally, note that the denominator in Miller's equation is based upon the relative shape of a modern bullet. <math>l(1+l^2)</math> is roughly the formula for the shape of an American football.

			===Safe values===
			When computing this formula Miller states several ''safe'' values that can be used in place of some of the more difficult to determine variables. For example he states that assuming a [[Mach number\|mach]] number of <math>M</math> = 2.5 (or roughly 2800 ft/sec assuming standard conditions at sea level where 1 Mach is roughly 1116 ft/sec) is a safe value to use in regards to velocity. He also states that when making rough estimates that involve temperature that using <math>s</math> = 2.0.

			==Example==
			If we take the [[Nosler]] [[Spitzer (bullet)\|Spitzer]] [[.30-06 Springfield]] round which is similar to the one pictured above we can easily fill in the variables and calculate the estimated twist rate.<ref>''[http://www.nosler.com/Reloading-Data/30-06-Springfield-180-Grains.aspx Nosler - Up Front]'', Accessed February 2012</ref> Starting with the formula

			<math>t = \sqrt{\frac{30m}{sd^3l(1+l^2)}}</math>

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

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

			Which tells us we have 39.2511937 calibers per turn. We calculate <math>T</math> using <math>t</math> and see that

			<math>T = 39.2511937 * .308 = 12.0893677</math>

			So our rifling should be roughly 12" per turn. The twist, as listed on [[Wikipedia]]'s [[.30-06]] article is 10" inches per turn as an average of .30-06 caliber rifles; thus 12" inches per turn is fairly accurate. The discrepancy seen here also helps explain why certain bullets seem to work better in certain rifles when fired under similar conditions.

			==Comparison to Greenhill's formula==
			Greenhill's formula is actually much more complicated in full form. The [[rule of thumb]] that Greenhill devised based upon his formula is actually what is seen in most places, including [[rifling\|Wikipedia]]. That rule was:

			<math>Twist = \frac{C D^2}{L} \times \sqrt{\frac{SG}{10.9}}</math>

			The actual formula was:<ref>Mosdell, Matthew. ''The Greenhill Formula''. http://www.mamut.net/MarkBrooks/newsdet35.htm (Accessed 2009 AUG 19)</ref>

			<math>S = \frac{s^2 * m^2}{C_M\alpha \div \sin(a) * t * d * v^2}</math>

			where:
			* S = gyroscopic stability
			* s = spin rate in radians per second squared
			* m = polar moment of inertia
			* C<sub>M<sub>α</sub></sub> = [[Stability derivatives\|pitching moment coefficient]]
			* a = angle of attack
			* t = transverse moment of inertia
			* d = air density
			* v = velocity

			Thus, Miller, in essence, took Greenhill's rule of thumb and expanded it slightly while, at the same time, keeping the formula simple enough to calculate without advanced training in mathematics. In order to improve on Greenhill, Miller used mostly empirical data and some basic geometry.

			==Corrective equations==
			Miller, in his work, notes several corrective equations that can be used:

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

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

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

			==See also==
			*[[Rifling]]

			==References==
			{{Reflist}}

			==External links==

			=== Calculators for stability and twist ===
			*[http://kwk.us/twist.html Bowman-Howell Twist Rate Calculator]
			*[http://www.jbmballistics.com/~jbm/cgi-bin/jbmstab-5.0.cgi Miller Formula Calculator]
			*[http://www.jbmballistics.com/~jbm/cgi-bin/jbmdrag-5.0.cgi Drag/Twist Calculator based on Bob McCoy's "McGyro" algorithm]

			<!--- Categories --->

			[[Category:Articles created via the Article Wizard]]
			[[Category:Firearm terminology]]

en>Kri: Wikified

2012-06-19T10:39:22Z

Wikified

New page

53 year old Medical Emission Therapist Calvin Ditsch from Charny, usually spends time with hobbies and interests like hang gliding, [http://www.sk-gaming.com/blog/mizanpagla3/1785219- Marvel Puzzle Quest Hack] and train spotting. Has completed a wonderful around the world trip that included touring the Bourges Cathedral.

L-stability - Revision history

en>Mirams: Links to A-stable.

24.176.166.175 at 01:20, 25 February 2013

en>Kri: Wikified