Damping Ratio Calculator — Formula
## Damping Ratio from Overshoot
The damping ratio determines the character of a second-order system response. It can be extracted from a measured step response by noting the percent overshoot.
### Formula
**zeta = |ln(%OS/100)| / sqrt(pi² + ln²(%OS/100))**
- zeta < 1: underdamped (oscillatory)
- zeta = 1: critically damped (fastest non-oscillatory)
- zeta > 1: overdamped (sluggish, no oscillations)
The damping ratio determines the character of a second-order system response. It can be extracted from a measured step response by noting the percent overshoot.
### Formula
**zeta = |ln(%OS/100)| / sqrt(pi² + ln²(%OS/100))**
- zeta < 1: underdamped (oscillatory)
- zeta = 1: critically damped (fastest non-oscillatory)
- zeta > 1: overdamped (sluggish, no oscillations)
Esempio Risolto
A step response shows 20% overshoot.
- ln(0.20) = -1.6094
- zeta = 1.6094 / sqrt(9.8696 + 2.5902)
- zeta = 1.6094 / sqrt(12.4598) = 1.6094 / 3.5299
- zeta = 0.4559