What RPM do most gyros rotate at?
Big variance. A flight instrument rotor spins at 10K to 15K RPM, according to Google. A spacecraft in low Earth orbit is at 1 rotation every 90 minutes. It doesn't matter.
Even with his cardboard disc, the precession demonstrated is 90 degrees - as you have already been told, phase lag is frequently not 90 degrees, with some very large variations depending on rotor design blade inertia etc etc - but a gyro is ALWAYS 90 degrees.
This is not true. You can wiggle a toy gyro back and forth in your fingers, and depending on the stiffness you hold it with (in other words, applying other forces), control the amount of precession that occurs. When you're going down the runway in a taildragger airplane, gyro precession will tend to yaw you to the left while raising the tail, and a lot of money, pride, personal safety, and often preservation of historical artifacats, depends on your application of other forces to control this precession to zero degrees.
Does the angle of gyroscopic precession change with density altitude? Phase lag does change with that variable because of the change in aerodynamic damping - how does that fit with your theory?
It fits in by applying other forces (which vary based on circumstances) in addition to the gyroscopic ones, to the object.
What is your explanation for why phase lag is frequently not 90 degrees when gyro precession is always 90 degrees?
Because there are other forces.
This is like asking "what is your explanation for why my car is drifting to the right while I'm holding the steering wheel straight ahead, when a car with the steering wheel straight always goes straight ahead?"
If all gyros behave exactly the same way, from the cardboard disc in the video through a spinning bicycle wheel to a proper gyro with a number of degrees of freedom designed to do a specific job, why do helicopter rotors not all follow the same rigid ('scuse the pun) reactions if it really is precession?
Because there are other forces.
AC makes a very valid point about rigidity in space - you can't cherry pick the facets of gyro behaviour - if a rotor obeys one, it should obey all - and it doesn't.
I really hesitate to open a second rabbit hole when people are struggling so hard with the concept of more than one force acting on an object, and a resulting motion that's a combination of the motions due to the individual forces - but a rotor does have rigidity in space. When you take away rotor-fuselage drag interactions, aerodynamic forces on the blades, etc. the rotor has the same tendency as all rotating mass. Just think of a spinning rotor in a vacuum chamber.
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Now, I have to ask you again, at a very simple level: what mechanic from a precessing gyro that causes it to precess, is absent from a seemingly precessing rotor?
On a the cardboard gyro, a normal force was applied which deflected the path in that direction, and the new circle had a max offset from the old one 90 degrees later. Yes or no?
On a helicopter rotor, a normal force is applied which deflects the blade path in that direction, and the new circle has a max offset from the old one about 90 degrees later. Yes or no?