Calculator · Dynamics & Control
Second-Order System Calculator
Compute the full set of canonical second-order response characteristics from natural frequency ωn and damping ratio ζ.
Input
Results
System characteristics
RegimeUnderdamped
0 < ζ < 1 · decaying oscillation
| Natural frequency ωn | 10 rad/s |
|---|---|
| Damping ratio ζ | 0.5 |
| Damped natural frequency ωd | 8.66 rad/s |
| Pole 1 | -5 + 8.66j |
| Pole 2 | -5 − 8.66j |
| Percent overshoot | 16.3 % |
| Peak time Tp | 0.3628 s |
| Approx. 2% settling time | 0.8 s |
| Approx. 5% settling time | 0.6 s |
Settling times are exponential-envelope approximations (4/ζωn and 3/ζωn), not simulated crossing times. The DC gain K scales the output amplitude only: poles, ωn, ζ, ωd, peak time and percent overshoot do not depend on it.
Diagnostics
- · Info Settling times are exponential-envelope approximations (4/ζωn and 3/ζωn), not simulated crossing times.
Unit-step response preview
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Technical reference
G(s) = ωn² / (s² + 2ζωn·s + ωn²)
Worked example: ωn = 10 rad/s and ζ = 0.5 give ωd ≈ 8.66 rad/s, overshoot ≈ 16.3% and an approximate 2% settling time of 0.8 s.
Assumptions and limitations
- · This page describes the normalized canonical second-order model with unity DC gain and a unit-step input.
- · Real systems with additional poles, zeros, time delay, saturation or nonlinearity will not behave identically.
- · Settling times are exponential-envelope approximations valid for 0 < ζ < 1.