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.
- 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?
- 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?
- 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!