 Solution

Keep the same fixed laboratory equilibrium reference, and let the pivot move downward by $z(t)=z_0\cos {\omega }_{0}t$. The dynamic upper-spring extension is now $x_1-z$, soAt ${\omega }_0^2=k/m$, seek a particular solution ${\mathbf{x}}_p=\mathbf{b}\cos {\omega }_{0}t$. The amplitude equation isLet $s_{\pm }=(3\pm \sqrt{5})/2$, ${\omega }_{\pm }=\sqrt{k{s}_{\pm }/m}$ and ${\mathbf{u}}_{\pm }=(1-s_{\pm },1{)}^{\mathsf{T}}$. The unrestricted general motion isFour initial-condition constants are retained because none are specified. The driven particular response is not a resonance: ${\omega }_0$ differs from both eigenfrequencies. Its stationary upper mass is the pivot-driven two-mass spring antiresonance. Relative to the moving pivot, subtract $z(t)$ from both components.