|November 5, 2008, 02:41 AM||#1|
Join Date: November 2, 2008
Location: Deep in the Heart of Texas
Reloading Science and Mathematics
This is called the Greenhill Formula.to determine the optimal rifle twist rate twist for a given bullet in any rifle. The formula is T=150(d/r) for velocities from about 1500 to 2800 fps. Substitute 180 for the 150 value for velocities exceeding 2800 fps. "T" is the twist rate. "D" is the bullet diameter. "R" is the bullet length to diameter ratio (length of bullet divided by diameter).
When V=1500fps-2800fps T=150(D/R)
When V is greater than 2800fps T=180(D/R)
Example with a known:
With this calculation one can find a 180 grain .311 diameter bullet that is 1.19 inches long will work optimally in a rifle barrel with a 1 in 12 twist rate when fired at a velocity of between 1500 fps to 2800 fps.
This does not take into account the RPM of any given bullet but does work as a way to calculate the velocity needed on other size bullets that have another optimum twist rate slower or faster than the given twist rate in the rifle used.
Calculation to find Revolutions Per Minute (RPM):
“V” is the velocity in feet per second. “I” is 12 inches the amount the bullet will travel in one second. “T” is the twist rate. “RPM” is revolutions per minute. Notice the velocity is in feet per second "FPS" so FPS(I/T) is multiplied by 60 to obtain revolutions per minute (RPM).
Example with a known:
The bullet is spinning 132,000 revolutions per minute when fired at 2200 foot per second from a barrel with a 1 in 12 twist rate.
What if the value of T equals a 1 in 15 twist rate?
The 125 grain .311 bullet that is 0.9 inches long takes a slower twist rate of 1 twist in 15 inches than the longer 180-grain bullet of the same diameter to be optimal for accuracy.
Since the given twist rate of the rifle it is being fired from cannot be changed then to make it an accurate stable bullet the RPMs of the 125 grain bullet need to be slowed down in order for it not to become unstable in flight from too fast of a twist rate.
Normally a lighter bullet will be fired at a much faster velocity as is seen in the reloading manuals and because it is lighter and shorter which allows the case to hold more powder also.
In this case when I find the bullet is not optimal to the twist rate of the particular rifling twist rate it is fired from I can achieve greater accuracy by not loading it to a maximum and loading it to a reduced load as a way to slow down the RPMs there by making a bullet that is optimal for a 1 in 15 twist rate work in a barrel with a 1 in 12 twist rate accurately.
This means I have to use another calculation to find what the RPMs would be normal to the smaller bullet in a barrel with a 1 in 15 twist rate to calculate the velocity it needs to be fired from a barrel with a 1 in 12 twist rate to obtain those same RPMs to make it stable and accurate.
Example with known:
Imagine that in a 1 in 12 twist rate barrel
Let’s slow that bullet RPM down some with a slower velocity in the 1 in 12 twist rate barrel
Now we have the same revolutions per minute as in a 1 in 15 twist rate barrel and have a load for the smaller bullet we can calculate to be optimized in a barrel with a 1 in 12 twist rate by using the velocity needed and comparing it to loadings for that particular bullets estimated velocity in reloading manuals.
Kind of seems odd that a 125 grain bullet in that particular caliber would have to be fired at a slower velocity than the heavier bullets to obtain an optimum accuracy for it but believe me when I say it works very well and you will not find a load for that 125 grain bullet in any reloading books published today for the 7.7mm and yet I shot a ½ inch group in the bulls-eye with it at 100 yards just last week using those calculations.
However if I load that same 125-grain bullet to a much faster velocity of 2,700 fps I can see it all over the target with 1-foot groups.
I found the same formula worked with my 7mm-08 concerning the shorter100 grain bullets of which I slowed down to the lowest load in the book still getting about 2,600-2,700fps and saw them hitting one-hole groups from a barrel they should not be optimum in to begin with.
Usually I will load nine rounds for testing with three each of 1/2 grain charge differences and find one of three test loads to give me those 3 shots through one hole groups I am looking for by using the math above and calculating a load for that particular bullet weight for an approximate velocity.
I read a lot of articles about people trying to explain why certain bullets are optimum for certain rifle twist but I have never seen an article explaining much as to why that is.
So I decided to write this little article explaining why that is and how everyone can learn how to make all weights of bullets more accurate in the caliber they are shooting.
Now that I look back on the article I do wonder if it is just like a friend that look at it and then at me and said, “I lost you when you said something about mathematics”. LOL
The other thing I never explained in this article is bullet expansion and how although one can get a certain weight bullet that is not normally optimal for accuracy in a certain rifle twist to become accurate on a target by using mathematics it does not mean that the particular bullet they achieved accuracy with using these formulas will end up with a proper expansion for hunting certain game animals.
Depending on the caliber being used there is a whole other article with more mathematics I would have to write explaining how to calculate energy of any given weight bullet at a given velocity while also calculating the game being hunted weights also.
In short when someone ask me how I make those hand loads that are so accurate while I’m at the rifle range most of the time my answer is just, “Reloading is a science that takes a lot of education in mathematics”.
Yet I did think if I posted this article in this forum that there would be some that would pull out a caliper and measure bullets then find the twist rate of their own rifle barrel while scratching their heads and reading the math and think, “Hummmm…This gives me some new good ideals at the reloading bench now”.
After all you are never going to be able to change the twist rate on your rifle so you might as well do the math if you want to shoot a shorter or longer bullet than what is optimum for the twist rate for that particular rifle accurately.
Just be sure to do the math on the ballistics, expansion and energy for that weight of bullet before you decide to use it for hunting also.
Another part of the math is that a 100-grain bullet is much cheaper than a 180-grain bullet in every case when one just wants to go plinking at the target range for fun of which can turn into not much fun at all if it is not even hitting the target.
|August 19, 2009, 07:07 PM||#2|
Join Date: August 18, 2009
In order for these calculations to help, you must have a known load of good accuracy to base the calculations. If one is starting from square one with no acquired loads for a certain caliber, he should start by finding the optimum twist for the bullet and compare it to the twist of the rifle he will be using. If the optimum twist of the bullet is slower than the rifle (ie. optimum twist for bullet = 1 in 12, barrel twist = 1 in 10) then one should start by selecting a lower velocity from the recommended loads. Rbb50, please correct me if i am wrong.
|August 19, 2009, 08:50 PM||#3|
Join Date: March 4, 2005
Thanks for putting all that in one place.
Welcome to the forum.
No, it's not so simple as you suggested. As the bullet goes slower the drag forces trying to turn if off course are also lessened so that less gyroscopic spin is required and the twist, as you hypothesized it, is still likely to be faster than required.
The Greenhill forumla was developed by George Greenhill for 19th century artillery shells. It happens to work OK for a range of modern supersonic rifle bullets as well. However, you can do better with calculations based on the late Robert McCoy's book, Modern Exterior Ballistics. These are available in free online calculators into which you just plug the numbers. They are at the JBM calculator web site. One of McCoys colleagues, Don Miller, has an updated version of the Greenhill formula that adds velocity, weight, temperature and air pressure into the equation. This calculates the gyroscopic stability factor, s. s is a number that equals 1 when the bullet is on the ragged edge of instability. Under 1, and the bullet tumbles, spins off in arcs, hits the target sideways (keyholes) and otherwise makes general mayhem. Above 1, the bullet is stable, but if it is too close to 1 it won't settle well in its flight bath and its accuracy won't usually be best. Harold Vaughn thinks s=1.4 is about ideal. Don Miller thinks s=1.5 is about ideal. Pretty close to one another.
If you spin the bullet still faster than is required for s=1.4—1.5, s gets larger. This exaggerates errors in the distribution of mass which cause wobble. But modern bullets are made pretty well, and you usually have to spin them awfully fast to get into trouble with so-called "overstabilization". The military usually sets twist so s=2 or a bit more at standard meteorological conditions. This ensures that in extremely dense atmosphere, such as antarctic temperatures, the s still won't drop below 1. Different conditions will change its gyroscopic stability factor, however.
I have an Excel file I put together that uses Don Miller's formula that you can download free for use at my file repository.
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Last edited by Unclenick; August 19, 2009 at 08:56 PM.
|August 23, 2009, 10:31 AM||#4|
Join Date: November 2, 2008
Location: Deep in the Heart of Texas
Your right Marley09
The faster 1 in 10 twist would require to slow the velocity down when selecting a bullet that is optimum for a 1 in 12 twist.
It was the same problem I faced with the 7mm-08 caliber with a 1 in 10 twist when selecting the lighter weight bullets that were optimum for a 1 in 16 twist.
Unclenick has something interesting to read and I can already tell it has to do with bullet expansion within the barrel using certain velocities.
If you have a whole slew of reloading books passed down through generations of family reloaders like I do you can tell most all of them use different length test barrels to obtain their data for the book in all of the different calibers.
A person not thinking about this can and probably would pick up a reloading manual and start using the data taken from a 24 inch test barrel with a 1 in 12 twist to load for a 22 inch barrel with a 1 in 10 twist.
Almost my problem with the 7mm-08 except the test barrel in the book was a 24 inch with a 1 in 9 ½ twist and mine was a 22 inch barrel with a 1 in 10 twist.
I heard a simple solution from my granddad a long time ago.
Shoot the smaller bullets at the slower velocities and the bigger bullets at the faster velocities.
It is true but I just wanted to know exactly why that was true so it involved a whole slew of experimentation and a bunch of math books along the way but if my granddad was alive today to see how accurate that 7mm-08 shoots 100 grain bullet all the way up to 175 grain bullets through the bull’s-eye at 100 yards I already know what he would say, WOW that’s some good shooting there”.
In any case it is better for me to know exactly where that bullet is going to hit out at 100 yards and to know I have the data for all of my calibers set up to work perfectly at that range which allows me to calculate the ballistics and energy out to farther ranges and know exactly where they will hit there.
It is nice to know I can pull out the guns take them to the range or hunting knowing that every round I use is super accurate and will do the job it was manufactured by me to do.
In the end the math pays off when the bullet goes through the bulls-eye or takes down the game I’m shooting at every time.
What really made it start paying off is years of experience shooting asking at least 10,000 questions about reloading from experienced people experienced in doing it and reading tons of books about the subject then using all the math I could find to get the knowledge of exactly how it is suppose to work.
Reloading is a science with a lot of mathematics used to obtain the data you find in those reloading books.
Some skip all the science and math and trek out on a keep shooting until you find the right loads then just stick to that one size bullet that seems to work the best in their twist rate.
I wanted to have all bullet weights work accurately in the same caliber and twist rate and used science and math to figure out how.
Kind of in the way some people will get directions to a place they are driving to then end up in that area confused about how they said turn left but there is no left turn right here I’ll have to call him again and get better directions.
I get the address and look it up on a map before I go and drive right to that location with no problems finding it at all.
My reloading is done in the same way with having all of the information books and tools required to make the science of reloading work perfect every time for all of my guns.
After all if I didn’t have a map book I would be lost just as soon as I drove out of town some place no matter what direction someone had given me.
Same goes for reading those reloading manuals from cover to cover along with finding any other info on reloading I can find and without that I would not have a clue as to how much powder to put in or what size to trim the cases to just in the same way without a map book an address alone would not help me get to that location.
|mathematics , reloading , science|