Calculating Reactive Gyroscopic Couple and its Relationship with the Attitude of a Rigid Body

Question

Consider some rigid disk, (as an approximation for a propeller on an aircraft for instance), spinning about an axis through the center of mass of said disk with angular velocity ω. In terms of a cylindrical coordinate system, assume the COM of the disk is at a distance r from the Z-axis, and assume further that angular velocity vector ω of the disk points in the +θ direction initially. In this setup, if one were to impart an additional angular velocity Ω on the disk about the Z-axis, a reactive gyroscopic couple C will manifest along the R-axis causing the orientation of the disk as well as the angular velocity vector ω to pitch up or down (depending on the directions of ω and Ω). Thus, a single propeller aircraft when making a left or right turn will tend to go nose-up or nose-down.

I have a few queries about this motion.

1. How does one obtain the reactive gyro couple vector? The literature suggests that its magnitude should be C=IωΩ, but, as we have two separate rotations about two separate axes, should there not be a separate MOI for the contributions due to ω and Ω? If not, why?
2. Where does the angular momentum/energy causing this gyroscopic couple and rotation about the r-axis come from? (Is the new reactive angular momentum/motion about the r-axis taken from lessening the momentum associated with ω (θ direction), Ω (z-direction), or some combination of the both?
3. How might one calculate the change in orientation of this disk over time due to the couple? (Like the change in the Euler angles of a vector normal to the disk as the couple is applied, for instance).

Thanks in advance!

Answer 1

About your questions:

1. The usual approximation is to take the case where angular velocity of the spinning disk/propellor is far larger than the angular velocity of the yaw, so the contribution of the angular velocity of the yawing motion is negligable in comparison. But yeah, an exhaustive treatment takes both into account.

2. The momentum exchange is discussed by Feynman in section 20-3 of chapter 20 of book I, that section is titled: the gyroscope

3. The following source, by the looks of it, contains an exhausive treatment The overall series:
   MITOpenCourseWare aeronautics
   Specifically the case of gyroscopes:
   3D rigid body dynamics: tops and gyroscopes

Additional remarks:
During WW I many aircraft types had a rotary engine. As you can imagine, the gyroscopic effects were very strong, dangerously so. However, I don't think I have ever read mention of noticable gyroscopic effects from the propellor of today's propellor-driven aircrafts.

It is important to be aware that the onset of gyroscopic precession is about motion. Specifically: to a pilot flying an aircraft with a rotary engine it may well have seemed as if giving rudder resulted in pitching instead. However, precise measurement would show that in order for the pitching to occur a bit of yaw must happen. That is: the rotary engine aircraft responds to yawing motion. It's just that if the pitching response is strong the smaller yawing motion may well go unnoticed. Conversely, initiating a slight pitching motion would result in yaw. So if the pilot wanted to turn he had to push for pitching. Flying those aircrafts must have been very tricky.