>>14237633I think i figured it out. Before the input, angular momentum is parallel to the spin axis. when input is applied, angular momentum starts moving in the direction of the input axis. this causes it to both rotate about both the spin axis and precess about the input axis. as it begins precesses about the input axis, it the spin axis moves out of the plane of the input axis and angular momentum. this means that angular momentum (and hence the angular velocity) relative to the body has a component along the output axis now, and the body starts nutating about the output axis. if you apply the right forces to the front and back of the wheel to resist the nutation, or the right torque along the output axis, then you can move the angular momentum vector back into the plane of the spin axis and input axis, so that there is no nutation. this required torque must be meant by what is the "instantaneous" "output torque". it also is instantaneous in that it takes an infinitesimal amount of time for the output torque to appear, as the slightest precession about the input axis will move the angular momentum out of the plane without a torque applied. it is not literally at the same time t that the input torque is applied though.
i don't know why i couldn't find an explanation like this. everyone either talks about magic torques out of nowhere or they talk about conservation of energy meaning that some rotations happen in specific combinations magically or that conservation of energy introduces magic combinations of torques. its literally just the angular momentum vector doesn't leave its plane with the input axis while the spin axis does, and the amount along the output axis that it leaves this plane is the output "torque". its a "torque" in the body frame, not the inertial fixed frame.
