(30)
This gyroscopic pendulum may be supposed to represent a ship among waves, or a carriage on a monorail, and so affords an explanation of the gyroscopic action essential in the apparatus of Schlick and Brennan.
8. Careful scrutiny shows that the steady motion of a top is not steady absolutely; it reveals a small nutation General motion of the top. superposed, so that a complete investigation requires a return to the equations of unsteady motion, and for the small oscillation to consider them in a penultimate form.
In the general motion of the top the vector OH of resultant angular momentum is no longer compelled to lie in the vertical plane COC′ (fig. 4), but since the axis Oh of the gravity couple is always horizontal, H will describe a curve in a fixed horizontal plane through C. The vector OC′ of angular momentum about the axis will be constant in length, but vary in direction; and OK will be the component angular momentum in the vertical plane COC′, if the planes through C and C′ perpendicular to the lines OC and OC′ intersect in the line KH; and if KH is the component angular momentum perpendicular to the plane COC′, the resultant angular momentum OH has the three components OC′, C′K, KH, represented in Euler’s angles by
KH = A dθ/dt, C′K = A sin θd ψ/dt, OC′ = G′.
(1)
Drawing KM vertical and KN parallel to OC′, then
KM = A dψ/dt, KN = CR − A cos θ dψ/dt = (C − A) R + A dφ/dt
(2)
so that in the spherical top, with C = A, KN = A dφ/dt.