 Pivot-driven two-mass spring antiresonance

Two identical vertical springs connect a driven pivot to two equal masses in sequence. After subtracting the static gravitational equilibrium, the laboratory displacement equation is $m\ddot{\mathbf{x}}+k\left(\begin{array}{cc}2& -1\\ -1& 1\end{array}\right)\mathbf{x}=k(z,0{)}^{\mathsf{T}}$. For $z=z_0\cos {\omega }_{0}t$ and ${\omega }_0^2=k/m$, a particular solution is $\mathbf{x}=(0,-z_0\cos {\omega }_{0}t{)}^{\mathsf{T}}$. Thus the upper mass has zero driven response even though this frequency is not an eigenfrequency. The general solution also contains both free normal modes.