Damped Natural Frequency⚠ unverified
Physics / Vibration · Compute the damped natural frequency from the damping ratio
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| wn | wn | rad/s | 1.0 | Undamped natural frequency |
| zeta | ζ | — | 1.0 | Damping ratio (dimensionless) |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | wd | rad/s | Damped natural frequency, in radians per second (rad/s). Returns 0.0 if ``wn`` is non-positive or ``zeta`` is greater than or equal to 1.0 |
The science & history
Understanding the Parameters
- $\omega_n$ — from $\sqrt{k/m}$ or a modal analysis.
- $\zeta$ — light damping barely shifts frequency; $\zeta\to 1$ → $\omega_d\to 0$.
- $\omega_d$ — ring-down frequency in free decay (log-decrement tests).
Derivation (Approaching a Proof)
Characteristic equation $s^{2} + 2\zeta\omega_n s + \omega_n^{2} = 0$ has roots $-\zeta\omega_n \pm j\omega_n\sqrt{1-\zeta^{2}}$ when $\zeta<1$. Imaginary part is $\omega_d$.
History
Damped free response is standard SDOF theory and the basis of experimental modal damping estimates.
Related Concepts: Natural Frequency mass-spring, Damping Ratio, Logarithmic Decrement, Resonance Frequency damped
Notes: Registry calculator vibration-damped-natural-frequency (unverified). Underdamped only.