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Just wondering if the cyclic in a real helicopter re-centers like a joystick for a computer?

I'm pretty sure the controls in a fixed wing re-centers due to natural physics if the plane is in dynamically stable flight, without the detent (extra force required to overcome fine-grained controls at the center).

Do the controls in a helicopter re-center in the same way? Not sure because I consider helicopters to be fundamentally unstable. I'm also curious about the center detent (if there is one) in a helicopter.

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Helicopters (I have flown so far) does not have such detent positions for their controls just because, as you have said, the instability and the effects of controls on other controls. To reduce the workload some helicopters have systems which probably changes from type to type. While AB-206 helicopters does not have a system for cyclic centering or holding its position, UH-1 helicopters have force-trim system to hold the cyclic where you have centered or left. AS-532 Cougar ( Eurocopter or Airbus Helicopters now. ) helicopter's cyclic has a cyclic-trim system which returns the cyclic to the position where the pilot set by pushing trim-release button on cyclic. This is all I can tell for now. Not a complete answer for your question but I hope this helps.

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