Damped Natural Frequency

Adding a dashpot slows the ringing down. The damped natural frequency is $f_d = f_n\sqrt{1-\zeta^2}$, and the square root is the whole of the story — it is a quarter-circle, flat near $\zeta = 0$ and falling off a cliff as $\zeta$ approaches 1.

Flat near zero is why field practice ignores this equation most of the time. At $\zeta = 0.05$, a generous figure for a machine structure, the damped frequency is 0.125% below the undamped one — one part in 800, far smaller than the uncertainty in the mass or the stiffness that produced $f_n$ in the first place. At $\zeta = 0.10$ it is half a percent. Only past about $\zeta = 0.3$ does the shift become worth carrying, and damping that heavy is rare outside vehicle suspensions and purpose-built dampers.

The useful lesson is the one people get backwards: damping matters enormously to vibration AMPLITUDE and hardly at all to vibration FREQUENCY. Somebody with a resonance problem will often reach for a damper hoping to move the resonance out of the way of a running speed. It will not move. It will only make the peak shorter. Moving a resonance takes a change in mass or stiffness — the two things actually in $f_n$ — and stiffening is usually the practical lever, because adding mass to move a frequency up is impossible and adding it to move a frequency down is expensive.

At $\zeta = 1$ the square root reaches zero and there is no oscillation at all to have a frequency, which is exactly what critical damping means; past 1 the root goes imaginary, and the mathematics is telling you the motion has become two decaying exponentials rather than a decaying sinusoid. The catalog refuses to report a damped frequency at or beyond $\zeta = 1$ for that reason: there is nothing there to report.