For the motion of the pendulum, including the fly-wheel,
MK2θ = gMH sin θ − M′ = gMH sin θ − Aθ − Kφ cos φ.
(27)
If θ and φ remain small,
Aφ = Kθ, Aφ = K(θ − α),
(28)
(MK2 + A) θ + (K2/A) (θ − α) − gMHθ = 0;
(29)
so that the upright position will be stable if K2 > gMHA, or the rotation energy of the wheel greater than ½A/C times the energy acquired by the pendulum in falling between the vertical and horizontal position; and the vibration will synchronize with a simple pendulum of length
(MK2 + A) / [(K2/gA) − MH].