Support-driven unequal-mass spring antiresonance (${\omega }_0^2=k/m$)

For masses $m,2m$ connected in sequence to a moving support by springs $k,2k$, displacement coordinates give $m\mathrm{diag}(1,2)\ddot{\mathbf{x}}+k\left(\begin{array}{cc}3& -2\\ -2& 2\end{array}\right)\mathbf{x}=k(f,0{)}^{\mathsf{T}}$. The squared normal-mode frequencies are $(k/m)(2\pm \sqrt{3})$. For support motion $f=F\sin {\omega }_{0}t$ with ${\omega }_0^2=k/m$, the negligible-damping steady response is $(0,-F\sin {\omega }_{0}t/2)$. The first mass has a response zero between the two resonances. Finite damping generally makes that zero imperfect.