Perhaps you should write to the world's helicopter manufacturers and advise them they've all got it wrong.
Absolutely not. Their complex machinery works fine, and where it will have issues is fully described in their manuals.
Angular momentum must of course be conserved, but to claim that a non-rigid rotor system behaves like a gyro violates all sort of assumptions that gyroscopic precession is based on (heedm said the same thing explicitly).
Globally yes, AM is conserved, in terms of the air and the rotor together, but not in the rotor only: it's not conserved within the rotor as the controls are moved, it's being changed. The only assumption being violated is that you have a low-torqued fast-spinning toy gyroscope in mind.
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If the forces on climbing blades explanation gives you a good and helpful picture of how it all works, and as long as you're really sure about the origin of all the 90 degrees-es, and don't swap amplitude for rate of change arbitrarily to make it work, then that's excellent; but does it really give any useful understanding of limits to what can and can't be done when flying it? I fear from some of the replies here that it is rote-learned to pass tests. It's certainly needed by designers to make it all work, as it's the way to get the whirling bits to be the right size to cope with all the lifting and reaction forces. In particular, hinges don't the angular momentum of the blades at all, and I fear that's not what many would say if questioned about it.
All I disagree with in this series of posts is the often seemingly theological tone to the rejection of a role for adding angular momenta together, without emphasizing that any differences from a picture of a toy gyroscope are all there for interesting reasons, and that understanding them could even be beneficial. All the various sorts of rotor designs, and the hockey puck, and a bike wheel, and a hurricane, and a toy gyroscope are all governed by the same principles.
When an explanation based on irrefutable physical principles seems to break down, or someone tells you they doesn't apply to your case, then there's an opportunity for gaining more understanding. The physical principles do apply, but there are other things involved.
I hope people have learned things here. I certainly have, and perhaps shouldn't have jumped straight in with a "rubbish", after "it's not a gyroscope", rather than
interpreting that as "it's not like a toy gyroscope".