15.1 Simple Harmonic Motion | University Physics Volume 1

Spring Mass System - Definition, Spring Mass System in Parallel and Series Combination
Spring Mass System - Get a complete overview of the spring mass system along with key topics, diagrams, equations and look at various problems with solutions.

Horizontal spring with a mass is a simple model of a barrel with a tuner undergoing elastic deformation (ie bending/ vibrating in a way that doesn't permanently deform the barrel in the horizontal plane). Cantilever bending is better approximation but the math is worse so this will likely just get us a comparative relationship (as this goes up or down this other thing goes up or down).
w= frequency
k= barrel stiffness
m=tuner mass
x=displacement from neutral
PE =potential energy which is equal to the energy that is put into the barrel by the gun shot. Its max value can be considered a constant when using the same load.
KE= kinetic energy put into the barrel by the gun shot. Its max value is equal to PE and can also be considered a constant with the same load.
In the horizontal plane the tuner does not significantly reduce barrel rigidity due to its added weight. Not sure if vertical does.
w=(k/m)^.5
PE=1/2kx^2
KE=1/2mv^2
Horizontal plane:
As the mass of the tuner increases the frequency of vibration goes down and velocity of the vibrations go down. A significant positive. A larger node.
As the barrel gets stiffer the frequency increases. Not good. A smaller node. Lucky for us making a barrel more stiff usually happens in the form of adding mass to the barrel so it partially cancels the increase in frequency by now having more mass to move.
As the barrel gets stiffer the amplitude of the vibration goes down. A good thing less dispersion.
Adding a tuner will slightly reduce the amplitude in the horizontal due to real world conditions providing some dampening.
Vertical: Will have the same effect on frequency. May possibly make amplitude worse (possibly a good thing if you want positive compensation out of a very stiff barrel).
So slightly less amplitude in horizontal (maybe slightly more in vertical) and much lower frequency and muzzle velocity in both combined with repeatable control over the muzzle position on exit. If you can afford the extra weight a quality barrel tuner is a very nice tool. Lowering the frequency and velocity of vibration has a significant positive effect on dispersion of any round fired. Also in any application where the rifle touches the shoulder the hinge point for the whole system is pretty far back so adding a mass at the muzzle has a big effect on how much the muzzle moves because of recoil while the bullet is still in the barrel.
I think the tuner brake is a thing we are going to see on a lot of rifles in the future. A conservative brake already weighs nearly enough to tame narrower profile barrels. So adding a small tuning mass to the brake adds a lot of functionality to that system as long as the brake itself is easy to clean as carbon fowling has mass.
I have not read the new Litz book yet so I can't intelligently comment on his methodology.