Underdamped (ζ < 1)
Step Response Characteristics
\[\displaystyle G(s) = \frac{K\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}\]
\[\displaystyle y(t) = K\!\left[1 - e^{-\zeta\omega_n t}\!\left(\cos\omega_d t
+ \frac{\zeta}{\sqrt{1-\zeta^2}}\sin\omega_d t\right)\right]\]
| Characteristic | Formula | Value | Annotation |
|---|---|---|---|
| Steady-state gain | \(y(\infty) = K\) | — | red dashed |
| Damped natural freq. | \(\omega_d = \omega_n\sqrt{1-\zeta^2}\) | — | underdamped only |
| % Peak overshoot | \(M_p = e^{-\pi\zeta/\sqrt{1-\zeta^2}}\) | — | cyan marker |
| Time to peak | \(t_p = \pi / \omega_d\) | — | cyan dashed |
| Rise time (10%→90%) | numerical | — | amber region |
| Settling time (2%) | \(t_s \approx 4/(\zeta\omega_n)\) | — | violet dashed |