= Harmonic forcing of a horizontally driven pendulum
{title2=$\ddot\theta+(g/l)\theta=(A\omega^2/l)\cos\omega t$}
For small <angles> about the downward vertical and prescribed pivot <position> $x=A\cos\omega t$, a <simple pendulum> obeys the displayed forced-oscillator equation. Off resonance, its harmonic particular solution is $[A\omega^2/(l(\omega_0^2-\omega^2))]\cos\omega t$, with $\omega_0^2=g/l$. At exact resonance, a pendulum initially vertical with zero angular <velocity> has $\theta=(A\omega_0/(2l))t\sin\omega_0t$ in the undamped linear model. The growing envelope eventually violates the small-angle assumption; <damping> and nonlinear frequency shifts matter in a real pendulum.
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