Damping Ratio From Overshoot⚠ unverified
Physics / Controls · Compute the damping ratio from the percent overshoot
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| overshoot_percent | overshootpercent | — | 1.0 | Peak overshoot, as a percentage |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | ζ | — | Damping ratio (dimensionless). Returns 1.0 when overshoot_percent <= 0 |
The science & history
Understanding the Parameters
- OS — e.g. enter
16for 16% overshoot, not0.16. Require $0 < \mathrm{OS} < 100$. - $\zeta$ — result in $(0,1)$ for underdamped systems with positive overshoot.
Derivation (Approaching a Proof)
From $\mathrm{OS}/100 = e^{-\pi\zeta/\sqrt{1-\zeta^{2}}}$, take natural log and solve for $\zeta$. Let $x = \ln(\mathrm{OS}/100)$ (negative). Then $\zeta = -x / \sqrt{\pi^{2} + x^{2}}$.
History
Inverting the overshoot chart is a standard first step when identifying second-order models from step tests.
Related Concepts: Peak Overshoot, Settling Time 2%, Phase Margin From Damping
Notes: Registry calculator damping-ratio-from-overshoot (unverified). Underdamped assumption.