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A helicopter flybar is a kind of gyroscopic stabilizer which acts in feedback to the pitch of the blades. The effect, in words, is that as the helicopter starts to pitch or roll in a particular direction more lift is applied on that side, keeping the helicopter level. These devices are commonly found on the inexpensive RC helicopters which are on sale at all major department stores. An example of the flybar from one of these RC helicopters is shown below.

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Although it isn't obvious from the picture, there is an angle between the flybar and the rotors. On my RC helicpoter at home the angle is roughly $35^\circ$. I believe that this angle has the effect of introducing extra phase in the feedback loop.

My questions are:

  • What is the effect of changing the angle between the flybar and the rotor?
  • How can I mathematically describe the effect of this angle?
  • What would happen if the angle was changed to $90^\circ$? What about $0^\circ$?
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This angle is determined by the lift characteristics of the rotor and the rotational inertia of the whole helicopter. You are exactly correct that the angle changes the phase of the feedback.

For the purpose of discussion let's assume the helicopter has pitched forwards slightly and needs to be corrected backwards, also let's assume a single rotor that spins anti-clockwise when viewed from above.

If we adjust the rotor when it is exactly at the front to give some more lift, the time it takes for the additional lift to overcome the rotational inertia of the whole helicopter would mean that the net effect of the lift was applied at some point to the front-left of the helicopter instead of exactly at the front. The next correction would then be slightly further around, and the next further round again. This would cause the helicopter to oscillate in the same manner that a coin does when you knock it over on the table.

So you pretty much answered it yourself, the angle controls the phase offset between the detected error and where the lift begins to be applied. Remembering that the net effect of the lift will be as if applied between where the rotor is adjusted and where the rotor is reset (after some time the helicopter has moved to correct the error).

Changing the offset angle of the flybar will change the angle at which the correcting force is applied relative to the error. The effect of this will be an oscillation in one direction or the other. Up to 90 degrees from where it should be the flybar will give negative feedback out of phase, so should (at least mathematically) remain in a stable oscillation. Beyond 90 degrees from where it should be the flybar will start to give positive feedback instead of negative feedback, again out of phase until 180 degrees, causing the thing to exponentially spin out of control.

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