The science
Equation of motion. Let θ be the angle of a point bob on a massless rod of length l from the upward vertical, and let the pivot move as yp = a cos ωt. In the frame of the pivot the bob feels gravity plus the pivot's acceleration:
theta'' = [ (g - a w^2 cos(w t)) / l ] sin(theta) - gamma theta'
Averaging (Kapitza, 1951). Splitting θ into a slow part and a small fast wiggle of order a/l and averaging over one cycle gives an effective potential per unit m l²:
U_eff = (g/l) cos(theta) + (a w)^2 / (4 l^2) sin^2(theta)
The second term is a restoring "ponderomotive" term. The upright position θ = 0 becomes a minimum when its curvature is positive:
a^2 w^2 > 2 g l (small oscillations: Omega^2 = a^2 w^2/(2 l^2) - g/l)
Beyond averaging. For small angles the equation is a Mathieu equation y'' + (A − 2q cos 2τ) y = 0 with τ = ωt/2, A = −4g/(lω²), q = 2a/l. The upright state is stable when A > a0(q) = −q²/2 + 7q⁴/128 − 29q⁶/2304 + … The page compares this series, the leading-order Kapitza result and a numerical Floquet monodromy matrix (stable iff |trace| < 2).
Below the threshold the upright rod falls and then tumbles; the drive pumps energy into a non-integrable system and the motion is irregular (chaotic) rather than a clean oscillation. Parametric resonance near f ≈ 2f0/n (f0 = hanging frequency) can also make a hanging pendulum swing wider.
Simplifications: point bob, massless rigid rod, one vertical drive axis, linear damping (real air drag is quadratic), rigid pivot, pure sinusoidal drive. The motor and scotch yoke are drawn to show the mechanism; their inertia is not part of the model. All numbers are illustrative, not measurements of a particular apparatus.
Independent educational model.