Free-pair dynamic vibration absorber ($k=m{\omega }^2$)

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 $ϵ=(k-m{\omega }^2)/(m{\omega }^2)$ sufficiently small, the primary amplitude relative to its free response grows as $(M/m)|ϵ|$. 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.