Damping Ratio → % Overshoot
Translate a second-order damping ratio into peak overshoot, peak time, and settling time.
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The engineering
For a standard underdamped second-order system, the step-response overshoot depends only on the damping ratio ζ — not on ωₙ. This card returns the peak overshoot, and if you supply the natural frequency it adds the peak time and 2% settling time so you can size a controller against a transient spec.
Handy anchors: ζ = 0.707 gives about 4.3% overshoot, ζ = 0.5 gives ~16%, and ζ = 0.3 gives ~37%. If your measured overshoot doesn't match the ζ you dialed in, your plant probably isn't clean second-order — watch for a nearby zero or an extra pole flattening the peak.
Where this math comes from
The percent-overshoot relation falls straight out of the underdamped second-order step response, worked out as control theory matured in the 1940s alongside the servomechanism and fire-control problems of the war effort. The clean exp(−πζ/√(1−ζ²)) form became the standard shorthand engineers memorize.
It reached generations of engineers through Katsuhiko Ogata's Modern Control Engineering (first edition 1970), where the transient-response specifications — rise time, peak time, overshoot, settling time — are laid out as the canonical design targets this card computes.
- 1868James Clerk Maxwell'On Governors' launches the analytic study of feedback stability.
- 1948MIT Radiation LabServomechanism theory formalizes second-order transient specs.
- 1970Katsuhiko OgataModern Control Engineering codifies the overshoot/peak/settling formulas.
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