It is possible to damp the amplitude of a harmonic oscillation without changing its frequency. That was my proposition.
Jordan,
I started a new job about 9-months ago that requires me to work on vibration problems. I am by no means an expert but have a very basic understanding of vibrations and mode shapes. Vibration analysis is a very specialized field within mechanical engineering that requires a lot of advanced training/coursework and practical experience... so I am just a hack
In our lab, I have done a few "experimental modal tests". We do these tests to understand the natural frequencies and mode shapes of a structure when it gets excited at these frequencies. Remember the Tacoma Narrows bridge!
We basically use a hammer with loadcell in the tip or a shaker to excite a structure. I've only done the hammer test. We suspend the structure with very soft isolators, excite the structure, and collect accelerometer data all over the structure. This data can then be used to create 3D models showing the mode shapes, like the bridge for example.
These different natural frequencies and "modes" are properties of the structure, and it will behave in this way if the frequency is correct for that mode shape. In our tests, the rigid body mode is the lowest frequency where the whole thing moves in space, via the soft isolators. For a rifle barrel, the first vibration mode is generally believed to be a simple bend, with the muzzle projecting up and down... at a certain frequency.
This first mode "shape" is basically the same for most barrels, but the frequency required to create this shape can be changed by altering the stiffness. We can alter the stiffness by chopping the barrel (stiffer) or adding mass at the muzzle (less stiff). Decreasing the stiffness makes the mode frequency lower.
Some BR dudes talk about "slowing" a barrel down. What they are saying is that they want to lower the mode frequencies using the tuners. Their hypothesis is that the bullet exit needs to be during the upswing, and not during the downswing as Varmint Al illustrates at his website.
I'm going out on a limb here, but I think everyone is still thinking about a barrel "whipping" wildly before the bullet exits the muzzle... or "oscillates". Per Varmint Al, this doesn't happen until AFTER the bullet leaves the barrel. The barrel is getting bent (deformed)... as shown in his animations, then the bullet exits.
At Al's site, he listed ~135 Hz as the first mode frequency. This would be 1/135 = 0.007 seconds for one period. But if you assume that the barrel time is 0.001 seconds (also from Al's site), then the bullet exits well before we have one complete vibration period. The bullet exits at 14% of the periodic cycle. This could be during the upswing, but if the natural frequency and excitation creates a higher frequency vibration the barrel could be past the highest muzzle projection and on its way down. Hence, the BR dudes wanting to "slow" the barrel down.
All that to say that I'm not sure that their goal is to reduce the amplitude, as the different modes can be pretty wild looking. Now the actual mode shape and frequency responsible for the muzzle projection, or excitation frequency, is a different story.
Jason
