Natural Frequency from Static Deflection

This is the most useful equation in isolation work, and it looks like a coincidence until you watch the algebra. Start from $f_n = \frac{1}{2\pi}\sqrt{k/m}$. A mount carrying weight $mg$ deflects by $\delta_{st} = mg/k$, so $k/m = g/\delta_{st}$. Substitute, and the mass has vanished: $f_n = \frac{1}{2\pi}\sqrt{g/\delta_{st}}$.

The disappearance is not an accident of the algebra — it is telling you something real. Put a heavier machine on the same mount and it sinks further; the extra sag cancels the extra mass exactly. So an isolator's natural frequency is determined by how far it deflects under its load and by nothing else. That is why isolator catalogues are organised by deflection rather than by spring rate, why the specification on a drawing says "25 mm deflection" instead of "180 kN/m", and why an inspector can verify an installation with a steel rule. In metric shorthand, $f_n \approx 15.76/\sqrt{\delta}$ with $\delta$ in millimetres; in imperial, $f_n \approx 3.13/\sqrt{\delta}$ with $\delta$ in inches.

Three cautions keep the shortcut honest. The deflection is the ISOLATOR'S own deflection under load, not the total sag of the floor and the frame beneath it, and mounts sitting on a springy mezzanine deck give a natural frequency the calculation never sees. The relation assumes the spring is linear across that travel: steel coils very nearly are, rubber-in-shear and cork are not, and an elastomer's dynamic stiffness typically runs 30 to 50% above its static value, which makes the calculated frequency optimistic. And the mount needs travel left over after this deflection to absorb the dynamic motion riding on top of it.

The practical ceiling is around 100 mm. Beyond that a steel coil becomes tall enough that rocking stability, not vertical isolation, sets the design — and the answer stops being a softer spring. It becomes an inertia base, a mass of concrete added to the machine so the same mount deflects further, or air springs, which reach natural frequencies near 1 Hz without the height. Both are standard on sensitive installations, and both are chosen because this equation said the spring alone could not get there.