Calculator · Dynamics & Control
Damping Ratio Calculator
Find the damping ratio ζ that produces a given percent overshoot in the standard second-order unit-step response.
Input
Peak value above the final value, as a percentage.
Result
| Percent overshoot (input) | 5 % |
|---|---|
| Damping ratio ζ | 0.6901 |
A damping ratio of about 0.6901 gives 5% overshoot. Higher damping reduces overshoot; lower damping increases it.
Explore further
Overshoot alone does not determine ωn; choose a value to see the response in the Lab.
Explore in Second-Order Lab
Opens the Lab in Analyze mode with these parameters.
Technical reference
Mp = exp(−πζ / √(1 − ζ²)) ⇒ ζ = −ln(Mp) / √(π² + ln(Mp)²)
Mp is fractional overshoot, so Mp = OS/100. Worked example: 5% overshoot gives ζ ≈ 0.6901.
Assumptions and limitations
- · Canonical normalized second-order system G(s) = ωn²/(s² + 2ζωn s + ωn²), unit step input.
- · The inversion is exact only in the underdamped regime 0 < ζ < 1.
- · Extra poles, zeros, time delay, saturation or nonlinearity change the real overshoot.