You are on the right track. You don't miss the point about the first mode being lower with a heavy barrel. However, I am pointing out the modes that count more in this load tuning discussion are the ones that cause us to resort to heavy barrel sections when we want accuracy and how those affect tuning.
No background in structural dynamics is required to know you would use a heavy barrel for a test rig or target rifle, but why is that if they lower some modes? Why are heavier sections not worse when they lower certain frequencies and how do they relate to being able to see or not see dramatic effects from load tuning?
The low note you are sensing is the first mode shape of the barrel and or it's support which is also very low frequency compared to other modes. When all that weight is a cantilever on the receiver, the simple view is not deceptive. Add weight on a cantilever, the frequency goes down. The first mode of the barrel also goes down.
You were not wrong, but some frequencies (modes) go up with stiffness from increased section (weight), and they can also move down if the weight isn't distributed correctly. This is why barrel flute cuts may or may not help with accuracy and why some barrel contours become popular in the accuracy crowds and others don't. It is also why your first statements were not wrong, heavier barrels lower some mode shapes, especially the low mode ones. The point being, why do high quality match guns seem to be insensitive to load tune variations?
The barrel first mode is usually low enough that the bullet exits within a fraction of that oscillation, however the second mode has enough time to oscillate many cycles.
Oversimplification in the thread means folks are missing the point of load tuning affects and why they are or are not easy to see in some guns and very obvious in others. I am not good at making this simple to explain or easy to understand, sorry. I'm dyslexic and admit I am a horrible writer....
It takes many words, diagrams, and math, to explain ballistic structural dynamics. I know what happens if there are this many words and then we add equations, especially the kind where you have to jump back and forth from frequency domain to time domain in order to have a good discussion.
Once a presentation starts up with phase diagrams, Eigen values, decibels, and Bode plots, folks start getting bleary eyed and head for dream land. Then I pulled the lanyard and the demonstrations got more interesting.... Who doesn't like to see ordinance plowing into targets or would they rather hear a math class on how a barrel and load were tuned.... Most folks just want to see the targets and I don't blame them.
BTW, there were several folks asking for a physical reason about why or even if the barrel mode shapes are mostly vertical. The reason most sporting and competition bolt guns have a strong vertical pattern, is due to two main reasons, 1) the low mode where the CG of the system is below the centerline during recoil, and, 2) the pressure shock inflation of the barrel against the gravity sag.
This description discounts the situation where your barrel touches a stock. All bets are off when that happens for this discussion.
The gun barrel is already sagging down due to gravity. If you consider the shock of the pressure extending the barrel and exciting the first modes of barrel/stock oscillations, the motion of the barrel inflating due to pressure counter acts the barrel sag and that first reaction becomes mostly vertical. The recoil lug and receiver are not symmetrical in bolt guns because they are underneath but even guns with symmetrical barrel supports move vertical on the shot due to that sag and barrel inflation. However, that said... The low mode shapes are not always just a straight line up and down, but many are nearly oval or ellipse with the long axis vertical. Get your tune exactly wrong (or have sloppy charges), and you will see a horizontal dispersion in many systems.
When weight is cantilevered over the length of the stock fore end, with the back of the stock touching a rear bag for example, there is a very low frequency mode associated with the whole gun and the weight of the barrel and receiver on the stock. The barrel first mode can be on the order of roughly 100 Hz.
A bullet is out of the muzzle before this mode gets even a half cycle. You can't really tune out this mode. Not saying we can ignore this mode, but for this discussion you can't do much about it since it is cast into the system. You can't go slow enough to take advantage of it is a different way of saying it.
The response of the first mode of the barrel on the stock doesn't play a strong role with respect to OBT tuning, but it does play into which effects are important on the whole because it is always there and potentially very large. The recoil response of the system on the whole is a huge effect on the difference between the point of aim and the motion of the muzzle as the bullet exits because the whole system is moving before the bullet exits.
It is not what folks want to hear, but it is why shooting technique matters. Rest or hold the gun wrong, and watch what happens next, regardless of how carefully your ammo is crafted.
The same ammo in a heavy system is more likely to be more accurate than that same ammo in a light system, mainly due to the interaction of the first mode with recoil. By the same token, get the load tune right in a light flimsy system (or more importantly get it wrong), and you can see bigger differences on the target. The potential changes here in the light barrel are larger, but the two baselines probably tip in favor of the heavy section to make it a moot point. You pick the heavier gun and barrel if you want accuracy... but nobody like to hump a heavy gun....
With respect to load development and component selection, those low mode frequencies and their effects are not always load tunable. Think about an AR-15 where the barrel extension sits in the upper, and then the whole upper rattles around on pins. As a mental exercise now imagine taking a pencil barrel profile versus a varmint profile in that same gun, and doing a modal analysis.
You will certainly find lots of the mode shapes shifted left on the PSD plot due to the weight of the heavier barrel, however, now think about what you have observed with different AR rigs between light weight pencil barrels versus heavy barrel varmint rigs and match barrels. No structural dynamics background required to guess which mode shapes matter more. Even when we lower some of the frequencies due to the weight of the barrel section, we are jacking other important ones with respect to load tuning. All while sitting in a sloppy pin interface between the upper and the lower.
With the little pencil barrels, the support frequencies shift up but many of those are a fraction of their oscillation compared to the bullet exit time. Keep in mind that some of the frequencies are very low and the others are in multiples of those and some are even in the kilohertz range. Both the thin light flimsy barrels and the thick heavy stiff ones have low first modes. It is those second modes and higher that are different between them, and you don't need to know much about structural dynamics to know that heavy section bull barrels and light section pencil barrels don't do the same things even if they both have a low first mode.
Some choices we make in the gun system set the tone for what the potential group dispersion will be, and as a result only some things can be load tuned. The reason for confusion is the share of the ones that are affected by barrel exit timing and load tune may be very dead or low amplitude on match guns because the barrels are heavy and stiff. The heavy choice lowers some frequencies, but improves recoil issues and stiffens important modes.
If you could hypothetically make a magic barrel that was infinitely rigid and weighed little or nothing but still had a moment inertia, you would still have a low frequency issue from the receiver, stock, and recoil. The recoil modes would force you to add weight to the system to reduce recoil displacement. But think about the share of the target dispersion that could be affected due to the load tuning. Only certain frequencies are within the reach of the load tuning. A key concept is to remember that amplitude is just as important as frequency when all is said and done. YMMV.
But, it's the kind of thing I really like to dig into. Because I'm aware of your education and extensive experience with analysis of such things, I highly respect what you're saying. I know it's difficult to communicate what you are saying without illustrations and the math (though the math is probably well beyond most shooters here). But I feel your effort is probably appreciated by some others than just myself; at least, I hope so. I've saved a copy of this for rereading to try to fully assimilate all that you've pointed out.
