Lab · Dynamics & Control
Second-Order System
Analyze one system, compare two, or state response requirements and read back the feasible ωn/ζ region. All values come from the same second-order model.
Parameters
Sliders cover the usual working ranges (ωn 0.1–50 rad/s, ζ 0–2). Values outside them can be typed directly.
Equivalent model
- K
- 1
- ωn
- 1.414 rad/s
- ζ
- 0
G(s) = 2 / (s² + 2)
Step response
System characteristics
ζ = 0 · sustained oscillation
| DC gain K = G(0) | 1 |
|---|---|
| Steady-state step value | Does not settle |
| Natural frequency ωn | 1.414 rad/s |
| Damping ratio ζ | 0 |
| Damped natural frequency ωd | 1.414 rad/s |
| Pole 1 | 0 + 1.414j |
| Pole 2 | 0 − 1.414j |
| Percent overshoot | Not applicable |
| Peak time Tp | 2.221 s |
| Approx. 2% settling time | Does not settle |
| Approx. 5% settling time | Does not settle |
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
- ⚠ Note With ζ = 0 the step response oscillates indefinitely: settling time and transient percent overshoot are undefined.
Continue with this system
The current model travels with the link; nothing is re-entered and nothing opens on its own.
Canonical model
G(s) = ωn² / (s² + 2ζωn·s + ωn²)
- ωn — natural frequency [rad/s]
- Undamped oscillation rate; sets the overall time scale of the response.
- ζ — damping ratio [–]
- Dimensionless measure of energy dissipation; sets the shape of the response.
- ζ = 0 — undamped: poles on the imaginary axis; the step response oscillates forever and never settles.
- 0 < ζ < 1 — underdamped: complex conjugate poles; the response overshoots and rings down.
- ζ = 1 — critically damped: repeated real pole at −ωn; fastest approach without overshoot.
- ζ > 1 — overdamped: two distinct real poles; monotonic and slower than critical damping.
What this Lab models
The canonical second-order system Kωn²/(s² + 2ζωn s + ωn²) in three roles: Analyze one system's poles and step response, Compare two systems side by side, or Design — state an overshoot/settling requirement and read back the feasible ωn/ζ region.