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Damped natural frequency

Damping lowers the oscillation frequency slightly compared with the undamped system. For small damping ratios, common in practice, the difference is small, but it grows quickly as ζ\zeta approaches 1.

ωd=ωn1−ζ2\omega_d = \omega_n\sqrt{1-\zeta^2}damped angular frequency

Symbols

ωd\omega_ddamped angular frequencyrad/s
ωn\omega_nundamped natural angular frequencyrad/s
ζ\zetadamping ratio

Example

ωn=20\omega_n = 20 rad/s, ζ=0.2\zeta = 0.2:

ωd=201−0.04≈19.6\omega_d = 20\sqrt{1-0.04} \approx 19.6 rad/s.

For ζ<0.2\zeta < 0.2, ωd≈ωn\omega_d \approx \omega_n – the difference is often negligible.
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Part of Machine Dynamics and Vibrations: Damping.