Resonance Frequency (damped)⚠ unverified
Physics / Vibration · Frequency at which the forced-vibration response peaks
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| wn | ωn | rad/s | 10.0 | Natural frequency |
| zeta | ζ | — | 0.5 | Damping ratio |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| wr | ωr | rad/s | Resonance frequency |
The science & history
Understanding the Parameters
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$\omega_n$, $\zeta$ — more damping lowers and eventually removes the amplitude peak ($\zeta \ge 0.707$: no finite-frequency peak).
-
$\omega_r$ — force transmissibility and velocity resonances peak at different frequencies.
Derivation (Approaching a Proof)
Displacement FRF magnitude $X/F_0$ is maximised vs $\omega$ by $d|H|/d\omega = 0$, giving $\omega_r^{2} = \omega_n^{2}(1-2\zeta^{2})$ for $\zeta < 1/\sqrt{2}$.
History
Resonance avoidance is the first design rule for rotating machinery and seismic isolation.
Related Concepts: Natural Frequency mass-spring, Damped Natural Frequency, Magnification Factor, Damping Ratio
Notes: Registry calculator vibration-resonance-frequency (unverified). Displacement peak of
forced SDOF.