 Horizontal-vibration averaged pendulum potential

For a small, rapid horizontal displacement $a\cos \omega t$, the leading fast angular excursion is $\xi =-(a/l)\cos \Theta \cos \omega t$. Correlating that excursion with the driving torque gives the averaged force $\ddot{\Theta }=-(g/l)\sin \Theta +(a^2{\omega }^2/2l^2)\sin \Theta \cos \Theta$. Its effective potential is $-mgl\cos \Theta +m{a}^2{\omega }^2{\cos }^2\Theta /4$. The downward state loses linear stability above $a\omega =\sqrt{2gl}$ and two tilted minima appear. Small $a/l$ and slow dynamics compared with $\omega$ are required; horizontal and vertical support vibration produce different effective potentials.