If there where no friction at all, would a spinning wheel held up by one end of the axis spin precess forever without falling down?
I just asked another question about the same problem:
Direction of torque precession of a spinning wheel
Since it seems to be a good practice on stackexchange not to ask several questions in one post, I splitted them up into two questions. However if I am wrong, feel free to merge this questions.
It is spinning forever. As you see, change of angular momentum
$$\frac{\text{d}\vec{L}}{\text{d}t} = \vec{\tau}$$
is always perpendicular to angular momentum itself, which means that angular momentum's direction is changed, while its magnitude is constant. Note the mathematical analogy with velocity and acceleration in case of circular rotation with constant velocity:
$$\frac{\text{d}\vec{v}}{\text{d}t} = \vec{a}_\text{cp}$$
In the steady state, yes it stays forever in the same orbit in terms of the axis of rotation for the gyro.
But when the precession is not constant, things are different, and this is because when you set the gyro up in the first place, it starts at full force and no precession. It doesn't instantly gain a precession $\Omega$ or velocity. That is, it has to accelerate.
But where can it get the energy to do so? Correct me if I'm wrong but the precession acceleration gets it energy from the average height of the gyro. So initially the thing drops until the force is balanced between the double axis rotating gyro, and the gravitational force. Just think of the gyro as having a torque, so that it precesses based off a two-axis rotation.
Torque will accelerate its precession, and once its approaches steady state the gyro will have a torque that is balanced, so that the angular momentum is constant.
