I have been doing a lot of reading of old posts and of Bill's article and of Varmint Al's analysis.
Seems to me nobody can actually say with 100% scientifically proved certainty just how a barrel actually vibrates under the forces imposed by firing the rifle.
Bill says it flops about the same as a natural vibration with a nice dead area back from the muzzle and a little area near the muzzle that moves up and down. It is this last area that is said to be able to be controlled with a tuner.
Varmint Al analysed the recoil effects etc on the barrel vibration and has presented 5 different vibrations patterns that are all happening at once. One has the entire barrel wobbling up an down such that even if you controlled that little end piece of the barrel with a tuner the whole barrel is still going up and down as well as in the mode that Bill talks about as well as even higher frequency vibrations as well as length increase.
Several posters are also saying that the natural frequency has nothing to do with what happens when the rifle is fired.
How can we be sure of what tuner design will work, how to determine the weight needed and how to adjust it correctly if nobody knows for certain how the barrel is even vibrating !! ??
Several people are talking like they have it totally figured out and why the tuners work. Funny thing is the best scientific analysis that I have seen (Varmint Al's) contradicts this, that is the whole "stopped muzzle" thing. In fact Varmint Al uses al lot of if's, but's and maybe's in his text because it seems he isn't convinced of how a barrel vibrates and how a tuner works even after all his computer modelling and analysis.
I am coming around to thinking that all any tuner does is slow the frequency of the barrel movement and allow some degree of control of the muzzle movement. The muzzle my be stopped in one vibration mode but not in the others ???
Unless ........... perhaps the first mode of vibration is slow enough that any decent load will exit the barrel consistently enough to shoot well, the second can be controlled like Bill says and the 3rd modes of vibration and higher are so fast and small that they make no practical difference anyway. Perhaps there is only one mode of vibration that needs to be controlled even though there are several happening all at once ??????
My head hurts !!!
Bryce