Assuming a periodic solution of these equations
w, ῶ, σ, = (L, P, Q) exp nti,
(4)
and eliminating L, P, Q, we obtain
(−An2 + Kn + gMa) (g − n2b) − gMn2a2 = 0,
(5)
and the frequency of a vibration in double beats per second is n/2π, where n is a root of this quartic equation.
For upright spinning on a smooth horizontal plane, take b = ∞ and change the sign of a, then
An2 − Kn + gMa = 0,
(6)