Sweet spots happen when the harmonic frequencies are whole multiples of the fundamental resonant frequency. There aren't any.
That would be a perfect sweet spot, but in practice all you need is for the vibration frequencies that have significant amplitude at the muzzle at bullet exit to cooperate and then you get a decent sweet spot.
Realistically this probably doesn't happen at a node (a point where the sum of the instantaneous vibration amplitude at the muzzle is at zero or a minimum), but rather at an anti-node. Where the amplitude of the vibration is maximum, that's where the muzzle movement is at minimum velocity. You don't care so much where the muzzle is (as long as it's more or less in the same place every time), the important thing is that it's not moving very fast when the bullet exits.
The article correctly shows that there are a lot of complex things going on, but the visualizations have the amplitudes set up so that they are easy to see and examine, they don't have amplitudes that are scaled relative to each other. The article refers to this briefly when it comments that "The higher frequency modes have
extremely small amplitudes..."
Another point to keep in mind is that the low frequency vibration modes are moving the muzzle very slowly in terms of bullet motion and it would not be especially important to try to tune for a spot where the muzzle velocity (velocity of the muzzle) due to those modes is a minimum because the motion is so slow due to those modes that it won't affect the point of impact significantly even if bullet exit happens at a relatively inopportune time.
So, in his example, Modes 1 & 2 are moving the muzzle relatively slowly relative to the bullet motion, according to his commentary, and the "higher frequency modes" have extremely small amplitudes". He doesn't tell us where the numerical cutoff for "higher frequency" is, but let's say it's only the top two for the sake of argument.
Now we have 4 out of the 8 modes remaining that we need to worry about. Modes 4 and 6 are stretching and twisting modes, which hardly move the muzzle at all in the target plane. That leaves us with Modes 3 and 5. Even assuming that Modes 3 and 5 have similar amplitudes (which is unlikely to be true) the frequency of Mode 5 is about double that of Mode 3--close enough that they could easily be in relatively close sync--close enough that we could reasonably expect a practically significant (though obviously not perfect) sweet spot.
If their amplitudes are quite different, then we could just focus on the one that's larger and ignore the smaller one when we are looking for our sweet spot.
But either way, there's a good chance we'll be able to find a place where the velocity of the muzzle in the target plane is minimum at bullet exit.
In other words, there may be a lot of complex things going on, but it's not just possible, it's actually probable, that only one or two of the vibration frequencies are going to be affecting the muzzle motion/position significantly at the point of bullet exit.
However there can be a place where the barrel can rest where the vibration amplitude is minimal and may be the best place.
Well, as mentioned above, it's probably not the point where the vibration amplitude is minimal but rather where the velocity of the muzzle motion in the target plane is minimal. But that's correct. Most would call that the sweet spot. I don't believe there's really a formal definition of "sweet spot" that implies it's the absolute ideal from a theoretical standpoint. It's a fairly general term that means a point that gives good results.
