Angular acceleration in 3D

Consider the rotor of a gyroscope. The rotor is attached to a frame that is rotating about a fixed vertical axes with a constant rate of ω1, and with the rotor rotating relative to that frame at a constant rate of ω2. The goal of the problem is to calculate the angular acceleration of the rotor.

The analysis for this is provided below.

This result shows that, although both rotation components for the rotor are constant, the rotor is experiencing a non-zero angular acceleration. How is that possible? Consider the explanation that follows.

Visualizing the motion

  • Shown in left video below is the single rotation (RED) ω1 about the fixed Z-axis.
  • Shown in middle video is the single rotation (BLUE) ω2 about the moving x-axis.
  • Shown in right video is the TOTAL angular velocity vector (MAGENTA) of the rotor, which is the vector sum of the red and blue angular velocity components.

Note that the magnitude of the total angular velocity is a constant; however, since the angular velocity vector changes in direction, the angular acceleration is NOT zero. Here, with the magnitude of ω being constant, then α = dω/dt is perpendicular to ω. In this case, α points in the positive y-direction.