= Force-driven vibration isolation
For a supported <mass> $m$, total <spring constant> $k$, and viscous coefficient $b$, a harmonic applied <force> has response $|x|=F_0/\sqrt{(k-m\omega^2)^2+b^2\omega^2}$. Far above the natural <angular frequency>, light damping gives $|x|\simeq F_0/(m\omega^2)$. For a prescribed displacement bound $x_{\max}$, this estimates $m\gtrsim F_0/(\omega^2x_{\max})$, but the finite spring term can make the estimated equality slightly violate the bound. If $F_0/x_{\max}>b\omega$, the high-mass exact threshold is
$$
m_{\min}=\frac{k+\sqrt{(F_0/x_{\max})^2-b^2\omega^2}}{\omega^2}.
$$
This model describes a specified external <force>. Motion transmitted through a moving support instead has a base-displacement forcing term and a different dependence on damping.
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