My question stems from a loss of tail rotor effectiveness. If a helicopter with a counterclockwise rotating blade like the R22 loses the tail rotor and the fuselage starts a rapid right turn and thus gains momentum, will the control inputs be reversed or will it create a double gyro from the main rotor system and cause control inputs to be normal?
Short answer: no.
What are your flight control inputs?
- Collective Pitch.
- Cyclic adjustments to pitch and roll
- Yaw / anti torque from the pedals, which control the tail rotor.
Let's take them in order.
Collective. If the control inputs were reversed for collective pitch, you'd not be able to get the torque off by lowering collective (which is the usual response to a loss of tail rotor and the attendant upset it causes by loss of the anti torque function of the tail rotor being lost). If the helicopter design reverses what the collective does, in terms of pitch increase or decrease, dependent upon the state of your tail rotor, nobody in their right mind will fly it.
Once again for emphasis: if you lose the tail rotor, you reduce collective pitch to reduce the torque, and thus the requirement for anti-torque (which is why the tail rotor is there in the first place). Check the flight manual for your model of helicopter for further details on that in-flight malfunction / emergency.
Cyclic. The control linkage between the cyclic and the means of tilting the rotor do not change if your tail rotor fails. They apply their changes to the "tilt of the disc" as long as they are connected when you move the cyclic.
What happens as the fuselage beings to rotate (particularly if the rate of rotation increases) is that the pilot is at risk of being disoriented by the rotation and making control inputs that make the situation worse. (See above: first, reduce pitch, and find a place to put the aircraft down per the flight manual). The rotation of the fuselage does not change control linkages, it changes the OODA loop for the pilot: Orient/Observe/Decide/Act is a continuous process (loop) when flying such that you make a control input to make the aircraft do something based on your perceiving a need to change.
The problem with a control input made when you involved with the uncommanded rotation of the fuselage is that the pilot's aim point keeps changing. That isn't a problem of precession, that's a problem of "getting behind the aircraft." It's a cognitive, not a mechanical, problem to overcome so that this doesn't end in tears.
Yaw Pedals. Per your problem statement, they don't work, so reversing control inputs isn't an issue. Your feet are along for the ride. Your brain and your hands get you to the ground safely, or not. The way to deal with yaw is via the torque reaction to main rotor pitch: with power on and no tail rotor, changing collective pitch will tend to make the nose move one way or the other. (Your flight condition: In cruise flight, in a hover, or in one of the many between states that a helicopter can be in will influence that).
The problem with the uncommanded rotation of the fuselage isn't some change in precession, it's the disorientation of the pilot whose reference line keeps changing as the helicopter rotates. The whole point of being the pilot is to control the aircraft. The pilot's objective in this case is to reduce the disorienting rotation by reducing collective pitch and flying the aircraft (lacking your anti-torque inputs) as you normally would. You won't get increased precession (vis a vis the cyclic inputs) due to the anti torque control being lost. Your rotor system has enough inertia if you keep it at flight rpm where at least that isn't your problem. Your problem is in lag, and making control inputs that are out of synch with the (constantly changing) orientation of the helicopter.
Depending upon aircraft type and the situation, you will likely need to secure the engine(s) before final landing so that you don't create a rotating situation right before touchdown if you need to add collective while you still don't have the anti torque from your tail rotor.