Usually physics books calculate the precession rate of a gyroscope when its momentum wheel is spinning "fast enough."
I am looking for an equation to calculate the wheel speed whereat a gyroscope "falls" beyond a particular angle towards a table surface (e.g. 45 degrees around an axis parallel to the table top, closer to the table) due to gravity, because the angular momentum of the wheel is not "high enough."
In particular, I would also like be able to compute the time to fall past some angle for some wheel spin speed (though intuitively some speeds will never lend to falling beyond a particular angle).
I'm also generally interested in computing what proportion of gravity's torque gets applied along the axis of the torque versus cross to both the torque and the wheel's angular momentum, depending on the spin speed of the wheel. (Intuitively all of it is applied along the axis of gravity torque due to gravity when wheel speed is 0).
I'll define Gyroscope as the typical spinning wheel attached to a pole placed upon a support post, where the pole is parallel and above some table surface, with c.m. at the rotating wheel, and where gravity would cause precession/rotation about the axis aligned with gravity. This is the most common formulation as found in Feynman's figure 20-5 here.