Help with formula to calculate center of gravity offset in a bullet

[Titan]

I was looking for some help concerning the proper formula to calculating center of gravity offset in a purposefully imbalance bullet. If I removed 40mg of copper from the outside jacket of a bullet on side of a bullet that weighted 11631 mg after the material was removed, and the diameter of the bullet was .284 inches, would the offset in inches just be the ratio of the material removed times the radius ? So 40/11631 * .142 resulting in a .00048835 inch offset center of gravity?


I am sure I am oversimplifying the issue because of the bimetallic structure of a jacket bullet and the lead center has a greater density.

And the second question is concerning the correct formula to predict the angular error from a center of gravity offset. I had read that Harold Vaughn's solution presented in chapter 9 of Rifle Accuracy Facts was incorrect as it included time of flight as a variable. I saw a formula quoted here http://riflebarrels.com/a-look-at-bullet-imbalance-and-twist/

Error in moa equals imbalance in inches divided by (.000046 * twist rate inches). Is this approach cited by Daniel Lilja the correct one to predict angular error?

Any help would be appreciated with either of these questions.

 

[dcali]

How you calculate the CG depends exactly how the material is removed. It's all just geometry. The easiest lazy way is to use a CAD package and just draw the flaw you're interested in. Autodesk Fusion 360 is free for non-commercial use, and pretty easy to pick up. The hard way is to use something like http://www.dtic.mil/dtic/tr/fulltext/u2/274936.pdf and piece it together.

I don't have it in front of me, but going from memory, I believe the equation in Rifle Accuracy Facts is indeed incorrect, and was taken from Mann's 1909 book, which was basically a very impressive compilation of amateur experiments. Vaughn should have known better (and I'm sure he would have corrected this if it had been brought to his attention).

Lilja's rule of thumb is too simple to be exactly correct, but he's a smart guy and I have no doubt it's about right for the specific configurations that he had in mind. (I have not verified this to be the case).

The accurate equation can be found in Bob McCoy's Modern Exterior Ballistics, and it is expressed as an angle - not dependent on time of flight at all. Unfortunately, it also depends on aerodynamic coefficients that are typically only able to be estimated.

But "cg jump" or "lateral throwoff" is only half the picture. You've also got to calculate the effect of that cg offset on the moments of inertia and how they'll cause the bullet it tip as it exits the muzzle. This aerodynamic jump is similar to cg jump, but calculated differently - McCoy shows how.

The most productive way to go about using these numbers, if you ask me, is to calculate the bullet's sensitivity to aerodynamic or cg jump. You can boil it down to a single number for a given bullet and twist rate. Details are in McCoy's book.

 

[Titan]

How you calculate the CG depends exactly how the material is removed. It's all just geometry. The easiest lazy way is to use a CAD package and just draw the flaw you're interested in. Autodesk Fusion 360 is free for non-commercial use, and pretty easy to pick up. The hard way is to use something like http://www.dtic.mil/dtic/tr/fulltext/u2/274936.pdf and piece it together.

I don't have it in front of me, but going from memory, I believe the equation in Rifle Accuracy Facts is indeed incorrect, and was taken from Mann's 1909 book, which was basically a very impressive compilation of amateur experiments. Vaughn should have known better (and I'm sure he would have corrected this if it had been brought to his attention).

Lilja's rule of thumb is too simple to be exactly correct, but he's a smart guy and I have no doubt it's about right for the specific configurations that he had in mind. (I have not verified this to be the case).

The accurate equation can be found in Bob McCoy's Modern Exterior Ballistics, and it is expressed as an angle - not dependent on time of flight at all. Unfortunately, it also depends on aerodynamic coefficients that are typically only able to be estimated.

But "cg jump" or "lateral throwoff" is only half the picture. You've also got to calculate the effect of that cg offset on the moments of inertia and how they'll cause the bullet it tip as it exits the muzzle. This aerodynamic jump is similar to cg jump, but calculated differently - McCoy shows how.

The most productive way to go about using these numbers, if you ask me, is to calculate the bullet's sensitivity to aerodynamic or cg jump. You can boil it down to a single number for a given bullet and twist rate. Details are in McCoy's book.

Thanks for pointing me towards McCoy's Modern External Ballistics I will have to go over it and try and figure it out.

 

[dcali]

Be warned - it' REALLY math heavy. But if you're up for that, it's the most complete book on ballistics out there. And make sure to get the 2nd edition. The 1st is riddled with typos.

 

[dmoran]

Not all that long ago, McCoy's Modern External Ballistics 2 was online, downloadable in PDF format for free. Not sure if it still is, or where it was at (sorry). May want to conduct an internet search.
Donovan

 

[Titan]

Be warned - it' REALLY math heavy. But if you're up for that, it's the most complete book on ballistics out there. And make sure to get the 2nd edition. The 1st is riddled with typos.

Well I thanks for the tip and the lead, I will try to get my hands on a 2nd edition, hopefully my head will not explode.

I was trying to find a model to predict angular error from cg offset as I wanted to have an estimate of magnitude of error before I began some live fire testing after sorting projectiles by cg offset. As I was trying to duplicate some of the results reported by Vaughn in his book but with current bullets.

 

[Titan]

Not all that long ago, McCoy's Modern External Ballistics 2 was online, downloadable in PDF format for free. Not sure if it still is, or where it was at (sorry). May want to conduct an internet search.
Donovan

Thank you for the tip!

 

[HuskerP7M8]

First, as Donovan noted, it's quite easy to find downloadable files of McCoy's work online and it only took me seconds to find a copy I already had.

I've always been fascinated by Vaughn's work and I've attempted to duplicate all of his testing, but with a slight twist. I've used RF rather than CF.

In the case of Cg offset, I'll try to illustrate the concept and methodology I've used by presenting the most recent example of the testing I've done in my ballistic tunnel.

Rail-Gun with Stiller 2500X/Muller 4MI barreled action. Shot in my ballistic tunnel on 3/2/16, but I've performed similar testing multiple times over the years.

20 bullets with 0.65 grains removed by drilling a hole to cause a Cg offset.



After mounting the barreled action, I aimed at bull #23 and fired 5 shots using NO shots with a Cg offset induced by drilling holes (Group 1 on the target below).

The scope was adjusted to move the POI to bull #23 and 5 more shots were fired again with NO Cg offset (Group 2).

POA was moved to the center dot on bull #13 and the following methodology was used:

5 shots with the Cg offset indexed in the chamber @ 12:00 were fired and the POI was at 3:00 (Group 3).

5 shots @ 9:00 = 12:00 Impact (Group 4).

5 shots @ 3:00 = 6:00 Impact (Group 5).

5 shots @ 6:00 = 9:00 Impact (Group 6).



A closer look with some notes I made:



An Excel scattershot chart generated from the x,y point coordinates provided by my electronic targeting software. The chart on top shows every point of impact and the chart on bottom is the calculated and exact mathematical center for each group:



Landy

 

[!Peter!]

Not all that long ago, McCoy's Modern External Ballistics 2 was online, downloadable in PDF format for free. Not sure if it still is, or where it was at (sorry). May want to conduct an internet search.
Donovan

Thanks for the tip as well.

 

[!Peter!]

Landy, thanks for sharing the results of your experiments. Much appreciated!

 

[Titan]

I've always been fascinated by Vaughn's work and I've attempted to duplicate all of his testing, but with a slight twist. I've used RF rather than CF.

In the case of Cg offset, I'll try to illustrate the concept and methodology I've used by presenting the most recent example of the testing I've done in my ballistic tunnel.

Rail-Gun with Stiller 2500X/Muller 4MI barreled action. Shot in my ballistic tunnel on 3/2/16, but I've performed similar testing multiple times over the years.

20 bullets with 0.65 grains removed by drilling a hole to cause a Cg offset.


Landy

Landy, your test is very interesting thank you for sharing your results and offering a nice demonstration of the effect I am interest in. I have a few questions if you have the time:
1.) What was the total weight of the rimfire projectile before drilling to remove .65 grains?
2.) Was the formula presented by McCoy's Modern Exterior Ballistics chapter 12 accurate in predicting the results of your life fire testing and observed (approximate .35 inches lateral jump) of the modified bullets?
3.) What was the twist rate of your barrel and the range to target in your tunnel?

 

[Bart B.]

Those plots in post 8 look like the ones in Dr. Mann's book "The Bullets Flight From Powder to Target."

Years ago, a benchrester told me to lay clean bullets on a perfectly level and clean Johansson gauge block. Those that rolled then stopped with the same touch point a few times rested on their heavy side at that place, unbalanced too much for tiny groups. The ones that stayed in place were balanced well enough to shoot in the zeros; under 1/10th inch/MOA at a hundred yards. All else up to it, naturally.

 

[Titan]

Those plots in post 8 look like the ones in Dr. Mann's book "The Bullets Flight From Powder to Target."

Years ago, a benchrester told me to lay clean bullets on a perfectly level and clean Johansson gauge block. Those that rolled then stopped with the same touch point a few times rested on their heavy side at that place, unbalanced too much for tiny groups. The ones that stayed in place were balanced well enough to shoot in the zeros; under 1/10th inch/MOA at a hundred yards. All else up to it, naturally.

I am trying to duplicate the static balance detailed by Vaughn in his book, but with a 3d printed carridge for the bullet as I don't have a mill to cut one from aluminum stock, also I replaced his light source with a diode laser and a mirror on the side of the carridge. The first 90 bullets I ran through it looked like an expected bell curve distribution of cg offset, but I am not convinced my setup is sensitive enough so I am going to try increasing the length of .010 guitar string on both sides of the carridge or try to add more counter weight above the center line maybe I will need to do both. If anyone has built a pendulum like the one Vaughn used, I would appreciate any tips on ideal setup for the dampener like gap between the magnets or ideal aluminum thickness for the vane that passes through the magnets.

 

[dcali]

For those who haven’t, do yourself a favor and read Mann’s book. It’s way out of date, and a good chunk of his conclusions are wrong in some way, but it’s an incredible piece of ballistics history done by a guy who probably would have fit right in on this forum were he alive today. It’s a classic.

 

[Bart B.]

HuskerP7M8's 4 plots centering about 3 hours apart on the clock reminds me of what one of the military teams did testing one lot of 7.62 M118 match ammo rebulleted with Sierra 168 HPMKs, it didn't shoot very accurate and bullet runout tests showed some over .007" next to bullet tip. Runout was measured on case mouth, too, ran about half that next to bullet tip.

The team captain took his 4 best shooters to the range with 20 rounds each all with over .007" runout with the high point marked. Each fired 5 rounds at 200 yards from prone indexed each 3 hours apart in the chamber.

All 4 targets had four 5-shot group centers about 3 hours apart as plotted from shot holes. Some group's overlapped. But it showed why regular M118 ammo wasn't worth writing home about.

 

[HuskerP7M8]

I'm scrambling around trying to get caught up because I'm taking a trip early tomorrow, so I'll make a few comments and answer the questions asked without the benefit of providing the actual data.

All of the RF ammo I use has a projectile with a mass of 40 grains +/- 0.1 and has been shot at 50 yds. Barrel had a 16" twist.

Using Vaughn's equation as well as McCoy's I've been in the ballpark with my calculations and analysis of the total dispersion, but the statistical uncertainties are high enough to preclude making any claims whatsoever of "proving" either equation is spot on. I'm afraid that's up to you and I'll be hoping you can at least reach the 1 Sigma level.
I don't think it will be necessary to offer "proof" at the 4 Sigma level like the army of experimental physicists and super computers do at CERN in Switzerland in proving the "God Particle" exists. LOL

Working with RF allows me to use much larger data sets to decrease statistical uncertainties without the worry of shooting the barrel out during the testing or incurring more expense in ammo costs, but the major problem remains. That being I don't know of a method to construct either a dynamic or static fixture to measure the offset of a RF projectile, and even if I could, it would be of limited or no usefulness because the projectile has to obturate to the bore/chamber during the ignition sequence and will most likely be reformed again as it travels the length of the barrel due to no barrel being perfect.

Fact is, the bore curve that's inherent with nearly every barrel produced will impart enough g-force to bend RF projectiles as they travels thru the bore curve(s) due to the low BHN numbers of most RF ammo. Even the ogive on RF projectiles changes shape and slumps from the acceleration imparted when the BHN is in the range of approximately 7.0-9.0 when you're lucky enough to find a truly straight barrel.

Titan, I sincerely hope you're able to continue with your testing and publish the results. I love stuff like this!

Landy

 

[Titan]

Thank you for the reply and the information about your tunnel test. I will continue ironing out the static balance over the next few days and will post the live fire results and the cg offset data set from the sorting once I am confident in the results observed. So far it has been time consuming processing bullets, I am averaging about two minutes a bullet on the static balance, so the plan is to improve the dampening effect on the balance to speed up the process of taking multiple readings per bullet less time waiting to the balance to settle between rotations hopefully will speed the up process.

 

[Titan]

Titan, I sincerely hope you're able to continue with your testing and publish the results. I love stuff like this!

Landy

As promised the results of sorting via static balance and live fire testing. For the test I sorted 86 180 grain 7mm Berger hybrids using a static balance based off the torsional pendulum detailed in the Harold Vahn’s Rifle Accuracy Facts. These bullets had been sorted by overall length prior to center of gravity sorting and all where +/- .002 of a inch in OAL in an effort to uniform BC.

The distribution was as follows, note the precision of my balance appeared to be able to group bullets into .00006 inch offset lots, I might revisit the design in future to try to get more precision.

12 with no detectable offset

26 with .000061

29 with .000122

14 with .000183

3 with .000305

2 with .000366


Average CG offset in inches was .0001137.


Before I went out to shoot I wanted to see what the predicted difference was going to be based on the only formula I could wrap my head around, the one I found cited by Daniel Lilja: Error in moa equals imbalance in inches divided by (.000046 * twist rate inches). (I tried to understand that formula presented in the McCoy text but I couldn’t figure it out.) I came up with the following for my test rifle with a 1-9 twist barrel.


For the live fire test four 8 shot test groups:

Group 0 no offset this is 0 moa added to the baseline rifle and shooter of .350 moa

Group 1 .000061 offset this .144 moa added to baseline

Group 2 .000122 offset this .294 moa added to baseline

Group 3 .000183 offset this is .444 moa added to baseline


Now I root sum squared this error source with what I guessed at is my baseline accuracy of .350 moa, which yielded the following prediction for the groups:

Group 0 .350 moa

Group 1 .379 moa

Group 2 .457 moa

Group 3 .565 moa


The live fire was conducted a 100 yards with my F open rifle a .284 Shehane, this test was outdoors in the desert 83 F with 4 mph winds and thick mirage.

Group 0 CTC avg of both 4 shot groups .3785

Mean Radius of both 4 shot groups .1555

Group 1 CTC avg of both 4 shot groups .389

Mean Radius of both 4 shot groups .154

Group 2 CTC avg of both 4 shot groups .530

Mean Radius of both 4 shot groups .174

Group 3 . CTC avg of both 4 shot groups .479

Mean Radius of both 4 shot groups .203



While the mean radius correlates to the CG offsets of the groups the test has about half the predicted angular error based on the Lilja formula. The observed effect is weak maybe .1 moa from best to worst case as far as cg offest, however conducting this test outdoors and not with a rifle bolted into a barrel vise in a tunnel might have corrupted some of the results. Also I did use some of the left over cg sorted bullets (groups 0 and 1) during a 20 shot relay in a 600 yard F Class match scored 200-15X CTC 5.19 inches 1.2 inch mean radius, which for me was a pretty good relay, but I file this under anecdotal evidence as you can't really run a control as the condition change too much between relays.

 

[SteveOak]

Not all that long ago, McCoy's Modern External Ballistics 2 was online, downloadable in PDF format for free. Not sure if it still is, or where it was at (sorry). May want to conduct an internet search.
Donovan

Here you go. https://archive.org/details/ModernE...htDynamicsOfSymmetricProjectiles2ndEd.R.McCoy

 

[dcali]

Guys, archive.org is a site that violates copyright on principle. Please consider buying a copy if you like the contents. Bob McCoy was one of America's great ballisticians, and his book has played a large part in the explosion of ballistics knowledge that we are all benefiting from today. His work deserves at least the token of thanks that the royalties provide. Off my soapbox now.