= Horizontal-vibration averaged pendulum potential
{title2=$V_{\rm eff}=-mgl\cos\Theta+ma^2\omega^2\cos^2\Theta/4$}
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 $-mgl\cos\Theta+ma^2\omega^2\cos^2\Theta/4$. 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>.
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