Barrel Harmonics Mode Shape Movies

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

POSSIBLY MODE SHAPES ARE NOT SO IMPORTANT.... I ran some FEA calculations of how a barrel reacted to the high gas pressure and recoil. The forced deformations from the high pressure gas and recoil cause the muzzle to change where it is pointing at the target when the bullet exits the muzzlet. Click on the graphic to read the info on Esten's rifle. Esten did extensive testing so that the FEA model could be normalized to his test data.

CONCLUSION.... Maybe the "consensus" was that a rifle barrel vibrated in one or more of the mode shapes when fired. That was because the mode shapes and frequencies were easy to calculate and they did seem to answer some of the questions. From these FEA dynamic pressure calculations, it appears that the recoil and forced deformations are much more important than the natural vibration modes in determining where a barrel is pointing when the bullet exits the muzzle. Then after the bullet exits the muzzle, the rifle barrel vibrates in its various natural frequencies and mode shapes. Put another way, consider a guitar string being plucked. One pulls the string into a position (forced position) then releases it and the string vibrates at is natural frequency. The recoil and bullet motions "pulls" the rifle barrel to a new shape and once the bullet leaves the barrel, then the barrel vibrates. However, the addition of the scope to the model has shown some small high frequency vibrations superimposed on the forced deformations, both of which, slightly alter where the muzzle points before the bullet exits. For lowering the amplitude of the high frequency vibrations, it appears that even an "out of tune" tuner is better than no tuner at all.

This is a 6mm caliber 1.25" diameter full bull 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 in each movie is in radians/sec. To convert to cycles/second or Hz, divide by 2*Pi. Each bending mode (like Mode #1) 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 Movie	Frequency (rad/sec)	Frequency (Hz)	Mode Description
Mode #1	455.11	72.43	Cantilever Bending
Mode #2	2819.4	448.7	1 Node Bending
Mode #3	7766.5	1236	2 Node Bending
Mode #4	8747	1392	Torsion
Mode #5	14348	2284	Axial Extension
Mode #6	14888	2369	3 Node Bending
Mode #7	23964	3814	4 Node Bending
Mode #8	26197	4169	1 Node Torsion
Mode #9	34719	5526	5 Node Bending

Click on a Mode Number to view an animated .gif file of the Mode Shape. The amplitude of each mode shape's deflection is arbitrarily large to more clearly show the deformations.

What is hard to visualize here, is that all of these modes are excited and start from the step function load of firing a round. The higher frequency modes have extremely small amplitudes, but trying to visualize all of these modes at the same time with their different frequencies is mind boggling.

HOW ACCURATE?.... How well do the FEA codes calculate the mode vibration frequencies? Below is a table comparing the modal frequency calculations for the first 5 mode shapes of a steel cantilever beam with a 0.5" square cross section and a length of 20". The equation in Chapter One, from the Shock and Vibration Handbook (Third Edition) should be quite accurate since it uses a "fudge factor" for each mode to normalize the results to test data.

Modal Analysis Accuracy Comparison
Mode Frequencies in Hz
Steel Cantilever Beam 0.5" x 0.5" x 20" long

Mode Number	Shock Handbook Equation	NIKE2D Plane Stress 1280 Elements	LS-DYNA 3-D 1280 Elements	LS-DYNA 3-D 10240 Elements
1	40.2	40.67	39.42	39.90
2	251	254.1	244.0	249.4
3	705	708.3	680.2	695.0
4	1380	1379	1325	1353
5	2280	2260	2173	2218