= Free-pair dynamic vibration absorber
{title2=$k=m\omega^2$}
A free mass $M$ driven by $F_0\cos\omega t$ can be coupled by a single <spring> of stiffness $k$ to an auxiliary free mass $m$. With no wall attachment or damping, the periodic displacement amplitude of $M$ is $A=F_0(k-m\omega^2)/\{\omega^2[Mm\omega^2-k(M+m)]\}$. Tuning $k=m\omega^2$ gives <antiresonance>: $A=0$ while the auxiliary mass has signed amplitude $-F_0/k$, supplying an exactly cancelling spring force. A nearby pole occurs at $k=Mm\omega^2/(M+m)$. For relative tuning error $\epsilon=(k-m\omega^2)/(m\omega^2)$ sufficiently small, the primary amplitude relative to its free response grows as $(M/m)|\epsilon|$. A light absorber is therefore narrowband and may require large auxiliary travel. This cancellation dissipates no energy, and undamped free modes persist unless excluded by initial conditions.
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