I've been trying to glean some insight into the motion of a body that is rotating about an axis through the COM of said body whilst traveling in a orbital-like path about a perpendicular axis outside of the body. To paint the scenario a bit more clearly, I use the example of a single propeller airplane negotiating a turn. Depending on the direction of angular momentum vector of the propeller, the negotiation of that turn will cause the plane to pitch up or down.
Recreating this scene with a gyroscope in hand at arm's length and spinning you're body to recreate the turn, the gyro will pitch up/down until the angular momentum vector of the gyro is parallel/antiparallel to the angular momentum vector of turn.
My question is this: if the magnitude of the angular velocity of both the gyro and of the turn were held constant, would the torque (precession)causing the pitching motion remain constant until the the angular momentum vectors are parallel, or would it increase/decrease until the angular momentum vectors are parallel? What is the proper train of thought one could use to derive the equations of motion of that body?