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When is a free float barrel vs a supported barrel more beneficial?

Some airguns have a barrel that "floats" in the action, while others have barrel bands supporting it. Based on what I read, one can generally say that certain airgun models are more accurate with a floating barrel vs supported, but there does not seem to be a rule or clear reason. For example a Daystate Airwolf MCT came without a barrel support/clamp, but later the factory added them on it (supposedly for barrel protection reasons and not accuracy).

So, is this a trial and error situation, and does it matter in the first place? Is it all about the elusive concept of "barrel harmonics" and getting the barrel resonance tuned just right?
 
I am curious in theory....but also because I am considering banding my Airwolf barrel. Right now I pulled the shroud away from the breech block about 1/8 inch so the barrel floats. But it was designed to be flush against the breech. So if the barrel vibrates, the shroud does not touch the breech-block in a random pattern. Yes, ultimately, accuracy is the deciding factor (between using banded or not banded barrels) but then we have to go back to "trial and error" again to find out.
 
It depends on what's supporting the barrel band. The air cylinder is prone to expansion/ contraction due to pressure variations. Stocks can warp and bedding can shift.
As long as the barrel is tight in the action and of reasonable diameter I think you are better off free floating. An oversize barrel band that doesn't actually contact the barrel but can prevent excessive barrel movement due to external pressure or impact is the best of both worlds.
Of course barrel harmonics is a pretty complicated phenomenon so there is no universal rule.
I, too, have a Cricket BP that is extremely accurate, but I have to be careful of POI shifts as the cylinder goes from full to "empty". One thing with this style of bullpup in favor of good accuracy is the rigid coupling of cylinder, barrel, and scope.
 
I have an Air Arms S500 that has a barrel band and it never seems to shift its POI and is very accurate. My Vulcan is supported and so far has wild POI shifts. My Cricket Carbine had some POI shift until I removed the barrel band. My FX Royale has no band and so far is very accurate. I think either way can work, but as was already mentioned, air tube expansion and contraction and stock swelling with changing conditions can potentially move your POI on a barrel with a band.
 
Barrel harmonics a free floated barrel will always "reverberate"\ "vibrate" in the same manor.
Free floated barrel
For example you pull the trigger,gun goes boom, the barrel reverberates up,down,up, and 3/4 of the way back down.

finlay the bullet leaves in a single constant repeatable predictable place you zero to this and hit the target dead on every time.

Supported barrel
You pull the trigger, gun goes boom, the barrel starts to reverberate up,down,up it now makes contact with the barrel support which changes the reverberation.

It starts to send its own competing reverberations which change the original natural reverberations and chaous ensues....

Where the barrel is when the bullet leaves is unpridectiable.


That being said
Bipods that clamp on to barrels = bad idea
Any thing that clamps on to a barrel = bad idea

Watch this and it will explain it all
SNIPER 101 Part 7 - Rifle Vibrations & Harmonics EXPLAINED
https://www.youtube.com/watch?v=h407yVskVeM