An elementary treatment of the theory of spinning tops and gyroscopic motion, by Harold Crabtree.

34 REPRESENTATION OF ANGULAR VELOCITY

It is clear that the frame mentioned might rotate with a
velocity w2 relative to a second frame which had a total angular
velocity )3, so that the definition can be extended to any number
of simultaneous rotations.
24. Parallelogram of angular velocities. If a rigid body,
of which one point 0 is fixed, has two simultaneous rotations
o1, 02 about axes OA, OB, and if wl, w2 are represented in
direction and magnitude by OA, OB, respectively, then the
resulting rotation, a, is represented in direction and magnitude by OC, the diagonal of the parallelogram of which OA,
OB, form two sides.
N
c.j~ A M
FIG. 16.
Let OA, OB (Fig. 16), represent the two rotations wo, W2 about
OA, OB, respectively, the direction of rotation in each case being
positive viewed from 0, i.e. that of a left-handed screw.
Let us suppose that the angular velocity w1 is relative to the
frame AOB, whilst 2 is the angular velocity of the frame itself.
Complete the figure as shown.
Let C be a point in the body passing through the plane of the
frame AOB and coinciding for the instant with a point C1 in
the plane of the frame.
Then the linear velocity of the point C is (away from the
reader and perpendicular to the paper)
= that relative to C1 +that of C]
=w. CM-W2. CQN
=k. OA. CM-k. OB. CN,
where le is a constant depending on the scale chosen,
=. (2A AOC- 2A BOG)
= zero.. the point C is at rest (as also the point 0), and therefore
the body is rotating about OC, and OC represents the direction
of the axis of rotation.