jhelmuth
09-24-2012, 09:33 PM
I had surmised the 7 as a type-o for a 5. Thanks for clarifying.
I'm a little fuzzy on your assertion regarding how "(s)tability is all about angular momentum..."
The 'amount of' gyroscopic stability required for a given bullet is primarily a function of its length. Weight is a factor, but a minor one in comparison to length. The primary reason why the 75 Amax will _always_ require a faster twist than the 77 SMK is that it's a _much_ longer bullet. If you can shoot the 75 Amax and get adequate gyroscopic stability for a given purpose/function, then the 77 SMK will always have greater gyroscopic stability under the same conditions. Therefore, if the 75 Amax will group and the 77 SMK won't, it ain't because of stability. :)
Actually... the SMK 77gr requires a faster twist than the 75gr AMax (not the other-way-around). Bullet manufacturers are your best source for twist rate requirements as they have more extensive test data and use more highly developed formulas for calculating twist rates than are generally used (such as the Greenhill formula). Bullet stability depends primarily on gyroscopic forces, the spin around the longitudinal axis of the bullet imparted by the twist of the rifling. Once the spinning bullet is pointed in the direction the shooter wants, it tends to travel in a straight line until it is influenced by outside forces such as gravity, wind and impact with the target. Rifling is the spiral or helix grooves inside the barrel of a rifle or handgun. The rifling grooves helix is expressed in a twist rate or number of complete revolutions the grooves make in one inch of barrel length. A 1-in-9 or 1:9 would be one complete turn in 9 inches of barrel length.
How important is twist rate? David Tubb, a winner of several NRA High Power Rifle Championships, was using a .243 rifle with a 1 in 8.5 twist barrel. He wasn’t able to get consistent accuracy until he changed to a rifle barrel with a 1 in 8 twist. The ½" twist change made all the difference between winning or losing the match.
A term we often hear is "overstabilization" of the bullet. This doesn’t happen. Either a bullet is stable or it isn’t. Too little twist will not stabilize the bullet, while too much twist, with a couple of exceptions, does little harm. Faster than optimum twists tend to exaggerate errors in bullet concentricity and may cause wobble. The faster twist also causes the bullet to spin at higher rpm, which can cause bullet blowup or disintegration because of the high centrifugal forces generated. For example, the .220 Swift, at 4,000 fps., spins the 50-grain bullet at 240,000 rpm.
One of the first persons to try to develop a formula for calculating the correct rate of twist for firearms, was George Greenhill, a mathematics lecturer at Emanuel College in Cambridge, England. His formula is based on the rule that the twist required in calibers equals 150 divided by the length of the bullet in calibers. This can be simplified to:
Twist = 150 X D2/L
Where:
D = bullet diameter in inches
L= bullet length in inches
150 = a constant
This formula had limitations, but worked well up to and in the vicinity of about 1,800 f.p.s. For higher velocities most ballistic experts suggest substituting 180 for 150 in the formula. The latest twist formulas use a modified Greenhill formula in which the "150" constant is replaced by a series of equations that allow corrections for muzzle velocity from 1,100 to 4,000 fps. The Greenhill formula is simple and easy to apply and gives a useful approximation to the desired twist. The Greenhill formula was based on a bullet with a specific gravity of 10.9, which is about right for the jacketed lead core bullet. Notice that bullet weight does not directly enter into the equation. For a given caliber, the heavier the bullet the longer the bullet will be. So bullet weight affects bullet length and bullet length is used in the formula.
The Greenhill equation includes no term for muzzle velocity, and several sources suggest replacing the 150 with 180 for muzzle velocities over 2800 fps. Increasing muzzle velocity increases bullet spin, and spin provides the stability. An article in the 11/2001 Single Shot Exchange cites an article by Les Bowman in the 1962 Gun Digest offering an equation which includes muzzle velocity (in fps):
T = 3.5 * V^0.5 * D^2 / L
At 2800 fps, this equation is equivalent to using 185 in the Greenhill equation, and at 1840 fps, this equation is the same as Greenhill's. Ken Howell wrote about twist rate in the 07/1999 issue of Varmint Hunter magazine. He mentioned Greenhill's work began with cannons in 1879. Two quotes Howell took from the Textbook of Small Arms (published in 1929 in Britain) are notable. "In actual practice Greenhill's figure of 150 can be increased safely to 200 and still control the bullet." The classic equation is for solid, lead alloy bullets of specific gravity (SG) 10.9, and "when the density of the bullet is less than that of lead or the density of the resisting medium is greater than that of air, the spin should be increased as the square root of the ratio of the densities." As SG decreases, the gyroscopic inertia of the bullet decreases in proportion, and one needs to increase the spin to compensate.
C.E. Harris, writing in the 08/1983 issue of the American Rifleman, noted Greenhill's formula was developed before spitzer boattail bullets and high velocity cartridges. He used a more modern analysis of gyroscopic stability, in which a factor of 1.4 is minimum and 1.7 is usually good. He found that the numbers given by Greenhill's original formula ranged from 1.5 to 2.0 for military type boattail bullets and were about 2.0 for bullets with either a flat base or short boattails. However, Don Miller has shown this older equation to not be accurate over the full range of bullet shapes and muzzle velocities. Miller's formula is expressed as:
http://upload.wikimedia.org/wikipedia/en/math/7/e/d/7ed2c55716b317ed71cf6dda26f5758a.png
where:
m = bullet weight 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 to...
http://upload.wikimedia.org/wikipedia/en/math/a/6/a/a6a4ca653e461b8fd83507f778827d24.png
where http://upload.wikimedia.org/wikipedia/en/math/b/9/e/b9ece18c950afbfa6b0fdbfa4ff731d3.png = twist in inches per turn, and
http://upload.wikimedia.org/wikipedia/en/math/c/c/f/ccf097652a8716cc8ba28ee99721b83c.png
where http://upload.wikimedia.org/wikipedia/en/math/d/2/0/d20caec3b48a1eef164cb4ca81ba2587.png = 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 http://upload.wikimedia.org/wikipedia/en/math/0/3/c/03c7c0ace395d80182db07ae2c30f034.png.
http://upload.wikimedia.org/wikipedia/en/math/d/9/d/d9d418cdf5ede61f2e114768d74f8f77.png
To measure the "actual" twist of a barrel, use a cleaning rod and a tight patch. Start the patch down the barrel and mark the rod at the muzzle. Push in the rod slowly until it has made one revolution, and then make a second mark on the rod at the muzzle. The distance between marks is the twist of your barrel. Note that this may not be consistent over the entire length of the barrel depending on the type/method and control in machining when the rifling is cut.
Note: The formulas expressed here were taken from searches on the internet and are merely "copies" and not my actual work (such as from Wikipedia, etc.). Please use your own time and energy to research and understand how tist calculations have evolved and how this relates to bullet stability.
I'm a little fuzzy on your assertion regarding how "(s)tability is all about angular momentum..."
The 'amount of' gyroscopic stability required for a given bullet is primarily a function of its length. Weight is a factor, but a minor one in comparison to length. The primary reason why the 75 Amax will _always_ require a faster twist than the 77 SMK is that it's a _much_ longer bullet. If you can shoot the 75 Amax and get adequate gyroscopic stability for a given purpose/function, then the 77 SMK will always have greater gyroscopic stability under the same conditions. Therefore, if the 75 Amax will group and the 77 SMK won't, it ain't because of stability. :)
Actually... the SMK 77gr requires a faster twist than the 75gr AMax (not the other-way-around). Bullet manufacturers are your best source for twist rate requirements as they have more extensive test data and use more highly developed formulas for calculating twist rates than are generally used (such as the Greenhill formula). Bullet stability depends primarily on gyroscopic forces, the spin around the longitudinal axis of the bullet imparted by the twist of the rifling. Once the spinning bullet is pointed in the direction the shooter wants, it tends to travel in a straight line until it is influenced by outside forces such as gravity, wind and impact with the target. Rifling is the spiral or helix grooves inside the barrel of a rifle or handgun. The rifling grooves helix is expressed in a twist rate or number of complete revolutions the grooves make in one inch of barrel length. A 1-in-9 or 1:9 would be one complete turn in 9 inches of barrel length.
How important is twist rate? David Tubb, a winner of several NRA High Power Rifle Championships, was using a .243 rifle with a 1 in 8.5 twist barrel. He wasn’t able to get consistent accuracy until he changed to a rifle barrel with a 1 in 8 twist. The ½" twist change made all the difference between winning or losing the match.
A term we often hear is "overstabilization" of the bullet. This doesn’t happen. Either a bullet is stable or it isn’t. Too little twist will not stabilize the bullet, while too much twist, with a couple of exceptions, does little harm. Faster than optimum twists tend to exaggerate errors in bullet concentricity and may cause wobble. The faster twist also causes the bullet to spin at higher rpm, which can cause bullet blowup or disintegration because of the high centrifugal forces generated. For example, the .220 Swift, at 4,000 fps., spins the 50-grain bullet at 240,000 rpm.
One of the first persons to try to develop a formula for calculating the correct rate of twist for firearms, was George Greenhill, a mathematics lecturer at Emanuel College in Cambridge, England. His formula is based on the rule that the twist required in calibers equals 150 divided by the length of the bullet in calibers. This can be simplified to:
Twist = 150 X D2/L
Where:
D = bullet diameter in inches
L= bullet length in inches
150 = a constant
This formula had limitations, but worked well up to and in the vicinity of about 1,800 f.p.s. For higher velocities most ballistic experts suggest substituting 180 for 150 in the formula. The latest twist formulas use a modified Greenhill formula in which the "150" constant is replaced by a series of equations that allow corrections for muzzle velocity from 1,100 to 4,000 fps. The Greenhill formula is simple and easy to apply and gives a useful approximation to the desired twist. The Greenhill formula was based on a bullet with a specific gravity of 10.9, which is about right for the jacketed lead core bullet. Notice that bullet weight does not directly enter into the equation. For a given caliber, the heavier the bullet the longer the bullet will be. So bullet weight affects bullet length and bullet length is used in the formula.
The Greenhill equation includes no term for muzzle velocity, and several sources suggest replacing the 150 with 180 for muzzle velocities over 2800 fps. Increasing muzzle velocity increases bullet spin, and spin provides the stability. An article in the 11/2001 Single Shot Exchange cites an article by Les Bowman in the 1962 Gun Digest offering an equation which includes muzzle velocity (in fps):
T = 3.5 * V^0.5 * D^2 / L
At 2800 fps, this equation is equivalent to using 185 in the Greenhill equation, and at 1840 fps, this equation is the same as Greenhill's. Ken Howell wrote about twist rate in the 07/1999 issue of Varmint Hunter magazine. He mentioned Greenhill's work began with cannons in 1879. Two quotes Howell took from the Textbook of Small Arms (published in 1929 in Britain) are notable. "In actual practice Greenhill's figure of 150 can be increased safely to 200 and still control the bullet." The classic equation is for solid, lead alloy bullets of specific gravity (SG) 10.9, and "when the density of the bullet is less than that of lead or the density of the resisting medium is greater than that of air, the spin should be increased as the square root of the ratio of the densities." As SG decreases, the gyroscopic inertia of the bullet decreases in proportion, and one needs to increase the spin to compensate.
C.E. Harris, writing in the 08/1983 issue of the American Rifleman, noted Greenhill's formula was developed before spitzer boattail bullets and high velocity cartridges. He used a more modern analysis of gyroscopic stability, in which a factor of 1.4 is minimum and 1.7 is usually good. He found that the numbers given by Greenhill's original formula ranged from 1.5 to 2.0 for military type boattail bullets and were about 2.0 for bullets with either a flat base or short boattails. However, Don Miller has shown this older equation to not be accurate over the full range of bullet shapes and muzzle velocities. Miller's formula is expressed as:
http://upload.wikimedia.org/wikipedia/en/math/7/e/d/7ed2c55716b317ed71cf6dda26f5758a.png
where:
m = bullet weight 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 to...
http://upload.wikimedia.org/wikipedia/en/math/a/6/a/a6a4ca653e461b8fd83507f778827d24.png
where http://upload.wikimedia.org/wikipedia/en/math/b/9/e/b9ece18c950afbfa6b0fdbfa4ff731d3.png = twist in inches per turn, and
http://upload.wikimedia.org/wikipedia/en/math/c/c/f/ccf097652a8716cc8ba28ee99721b83c.png
where http://upload.wikimedia.org/wikipedia/en/math/d/2/0/d20caec3b48a1eef164cb4ca81ba2587.png = 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 http://upload.wikimedia.org/wikipedia/en/math/0/3/c/03c7c0ace395d80182db07ae2c30f034.png.
http://upload.wikimedia.org/wikipedia/en/math/d/9/d/d9d418cdf5ede61f2e114768d74f8f77.png
To measure the "actual" twist of a barrel, use a cleaning rod and a tight patch. Start the patch down the barrel and mark the rod at the muzzle. Push in the rod slowly until it has made one revolution, and then make a second mark on the rod at the muzzle. The distance between marks is the twist of your barrel. Note that this may not be consistent over the entire length of the barrel depending on the type/method and control in machining when the rifling is cut.
Note: The formulas expressed here were taken from searches on the internet and are merely "copies" and not my actual work (such as from Wikipedia, etc.). Please use your own time and energy to research and understand how tist calculations have evolved and how this relates to bullet stability.