I have a wheel spinning away from me in the vertical plane. The angular momentum is pointing to the left. The axis of rotation is horizontal, and it is supported on either side by a bearing. Think of a free spinning bike wheel. If I then rotate the whole assembly, bearings and wheel, about the vertical axis, i.e. perpendicular to the axis of rotation, it will induce a torque in the vertical direction. Given what I understand about angular momentum and gyroscopes the wheel would want to resist this and instead rotate in the direction that would align it's horizontal angular momentum vector with the new vertical torque vector. Below is an image of what I am talking about.
The dotted arc shows the direction that the wheel would attempt to go. The Gyro Force exerted on the two bearings is what I am looking for. Also the arrows are not to scale and are unrelated to magnitude, they are only there to show direction. If anything is unclear please let me know and I will gladly attempt to clarify.
How do I calculate the force that would be applied to both bearings by the wheel wanting to turn in the vertical plan perpendicular to it's spinning plane?
If it makes the problem easier we can also assume that I know omega_p which is the pivot speed of the whole setup.
I've searched all over the web for the answer to this question and can't seem to find the answer. I keep coming up with the speed that it would precess or direction of precession. I'm looking for the magnitude of the torque/force that the wheel applies to the bearings.
Here is a copy of a PDF that has gotten the closest to explaining what I am looking for but not quite it since the gyro I am talking about is fully constrained. I may have missed something though: http://veemgyro.com/wp-content/uploads/2015/11/White_Paper_1403-How_Gyros_Create_Stabilizing-Torque.pdf