Calculator · Dynamics & Control
Damped Natural Frequency Calculator
Find the damped natural frequency ωd of a canonical second-order system from ωn and ζ.
Input
Result
| Damping regime | Underdamped (0 < ζ < 1) |
|---|---|
| Damped natural frequency ωd | 8.66 rad/s |
| Undamped natural frequency ωn | 10 rad/s |
The response rings at about 8.66 rad/s, below ωn because damping slows the oscillation.
Diagnostics
- · Info Settling times are exponential-envelope approximations (4/ζωn and 3/ζωn), not simulated crossing times.
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Technical reference
ωd = ωn·√(1 − ζ²), defined only for 0 ≤ ζ < 1
- ωn
- undamped natural frequency — the oscillation rate with no damping (ζ = 0)[rad/s]
- ζ
- damping ratio[–]
- ωd
- damped natural frequency — the actual ringing rate of the underdamped step response[rad/s]
ωd is always ≤ ωn: damping slows the oscillation. At ζ = 0, ωd = ωn.
For ζ ≥ 1 (critically or overdamped) the poles are real and there is no oscillation, so ωd does not exist; the engine reports it as not applicable rather than 0.
Worked example: ωn = 10 rad/s and ζ = 0.5 give ωd ≈ 8.66 rad/s.
Assumptions and limitations
- · Canonical normalized second-order system G(s) = ωn²/(s² + 2ζωn s + ωn²).
- · ωd is defined only for ζ < 1, where the poles are a complex conjugate pair.
- · ωd is an angular frequency in rad/s; divide by 2π for hertz.