With great respect and all credit to Varmint Al, and his helpful web page, I would like to portray his work to readers here.

I applaud the better science and the fantastic three dimensioal figures and movies provided by Varmint AL at the link shown

http://www.varmintal.com/aflut.htm
http://24.124.39.10/uploads/modes.zip

A quote from one of these pages::

Full 3-D Finite Element Analysis of a barrel's first few vibration mode shapes and frequencies with LS-DYNA.

This is a 1.25" diameter stainless steel barrel 22" long with the far end fixed and you are looking at the muzzle end and it is tilted down slightly for better viewing. These are the first 9 mode shapes and frequencies. The frequency displayed is in radians/sec. To convert to cycles/second or Hz, divide by 2*Pi. Each bending mode (like movie1) is on one plane, but there was another identical mode in another plane at the same frequency that was not shown to save space. The torsional modes are at a high frequency. Note, modes 1, 2, 3, 6, 7, and 9 are shown in a single plane, but can exist in multiple planes simultaneously. Modes below a frequency of about 500 Hz will not be able to complete one full cycle before the bullet exits the barrel.

Mode Number Frequency
(rad/sec) Frequency
(Hz) Mode Description
1 455.11 72.43 Cantilever Bending
2 2819.4 448.7 1 Node Bending
3 7766.5 1236 2 Node Bending
4 8747 1392 Torsion
5 14348 2284 Axial Extension
6 14888 2369 3 Node Bending
7 23964 3814 4 Node Bending
8 26197 4169 1 Node Torsion
9 34719 5526 5 Node Bending

What this means to me explains the double figure 8 theory has a plausible explanation. there can be two simultaneous vibrations, a greater one cycling through a complet up and down and slightly elliptical around before the bullet leaves the barrel at about 400-500 Hz.

And a simultaneous secondary vibration cycle at about two to three times that frequency. VOILA we have an ellipse with a snowman, inifnity, double figure 8, or actually two small circles or ellipses, within the larger ellipse. of parts of two, or even parts of three...

Anyway the picture is clear to me how this can happen, and it is not from a radial expansion of the barrel, although that may account for the secondary cycles in a smaller orbit.

Table 3. Mode Shapes in the same order as Table 2.







CONCLUSION....

When comparing two barrels equal weight, length, and material and one is solid and one is fluted, the fluted barrel will have greater stiffness and vibrate at a higher frequency.

Fluting a solid barrel will reduce its stiffness, but increase its natural frequency and decrease its end deflection (sag).

Reducing the weight of a barrel by fluting makes a stiffer barrel than reducing the weight by decreasing its diameter.

A shorter barrel of the same section, solid or fluted, will sag less and vibrate at a higher frequncy.


The following is speculation and has not been thoroughly analyzed.

TIMING.... The approximate time a 3300 fps bullet takes to exit the barrel, after it starts to move, assuming a constant acceleration, is 0.0011 seconds. The actual time is greater. The velocity of sound in 416 stainless steel is 14,900 fps and a stress wave has time to propagate up and back the full length of the barrel 4 or more times after ignition and while the bullet is traveling out the barrel. The muzzle end of the barrel has ample time to "know" that something is going on at the breach end before the bullet exits.

VIBRATION PERIOD.... A mode 1 frequency of 100 Hz has a period of vibration of 0.010 seconds. Therefore the time it takes the barrel to make its first upward swing is a fourth of a period or about 0.0025 seconds. This is the approximate time the bullet exits. The mode 2 vibration has a period of 0.0002 seconds and could be at the top of its thirteenth upward swing, at period of 12.25, or about 0.0025 seconds. These two modes could add and amplify the exit angle of the muzzle near the peak of its upward swing, just as the bullet exits.

FINDING THE SWEET SPOT.... When tuning a load to a particular rifle, possibly the sweet spot is not when the bullet exits the barrel at the maximum of the vibration's upward swing but, slightly before the maximum height or exit angle of the upward swing.

UPWARD SWING.... The vertical amplitude of vibration is more heavily excited than the horizontal vibration because the center of gravity of the rifle is located well below the barrel's centerline. The bullet's travel down the barrel causes a vertical turning moment about the rifles center of gravity. The vertical vibration is most important. Also, the barrel is slightly deflected downward due to gravity. When the round is fired, the pressure also tends to straighten the barrel like a bourdon tube in a pressure gage. As the barrel straightens, it over shoot in the upward direction and this adds to the excitation of the Mode 1 vibration. As a side note, the axial vibration mode is also probably heavily excited. This is the mode where the barrel extends and contracts axially. But, this axial mode should only have a negligible affect accuracy.

AVERAGE VELOCITY BULLET.... Possibly the sweet spot occurs when the bullet, with the average velocity, for a particular load, exits the barrel just before the peak of its upward swing.

FASTER BULLET.... A faster bullet will exit the barrel earlier and exit slightly before the average velocity bullet and the angle of the upward swing of the barrel will be slightly less. So the bullet's launch angle is slightly less, but the bullet is going faster and drops less.

SLOWER BULLET.... A slower bullet will exit later and the barrel's vertical swing will be higher and at a steeper angle when the slower bullet exits. The bullet is launched at a higher angle but is slower and will drop slightly more.

This combination, within limits, would print the bullet at about the same vertical location on the target for the normal variations in velocity from round to round.

Again, this is copied from Varmint ALs web page.
What I add to this is the observation that two of these model modes could, indeed probably do, superimpose and scribe several smaller circles within a larger primary mode ellipse, making the figure 8.

Finally, real data! This is what is informative, not conjecture.

Varmint Al has a great website, but it had been a while since I looked at it and I forgot that this info was there.

I found one thing interesting, and it is obvious once you see it. That is the fact that the torsional flex induces diameter changes, in addition to that caused by pressure. This is obvious if you have ever twisted an elastic tube. Do I smell proof of the watermelon seed theory?

Jameister,
Thank you for posting that link. I can see where you're headed on the "figure 8" pattern. While that pattern is a possibility, there are still a lot of "if's" that have to come together for that to happen. For instance you have to keep in mind that while Varmint Al's depiction is shown in a single vertical plane, the vibrations are occuring in many other planes as well.

[ 03-08-2003, 11:53: Message edited by: Chris F ]