English: Step responses for a second order system defined by the transfer function:

${\displaystyle H(s)={\frac {\omega _{n}^{2}}{s^{2}+2\zeta \omega _{n}s+\omega _{n}^{2}}}}$ where ${\displaystyle \zeta }$ is the damping ratio and ${\displaystyle \omega _{n}}$ is the undamped natural frequency. The equations were obtained from here, plotted using maxima and edited in a text editor to insert the Greek alphabets in the plot. The equations are:

1. ${\displaystyle h_{under}(t)=1-{\frac {1}{\beta (\zeta )}}e^{-\zeta \omega _{n}t}\sin \left(\beta (\zeta )\omega _{n}t+\theta (\zeta )\right),\beta (\zeta )={\sqrt {1-\zeta ^{2}}},\theta (\zeta )=\tan ^{-1}\left({\frac {\beta (\zeta )}{\zeta }}\right)}$
2. ${\displaystyle h_{un}(t)=1-\cos \left(\omega _{n}t\right)}$
3. ${\displaystyle h_{crit}(t)=1-e^{-\omega _{n}t}\left(1+\omega _{n}t\right)}$
4. ${\displaystyle h_{over}(t)=1+{\frac {\omega _{n}}{2{\sqrt {\zeta ^{2}-1}}}}\left({\frac {e^{-s_{1}t}}{s_{1}}}-{\frac {e^{-s_{2}t}}{s_{2}}}\right),s_{1}=\left(\zeta +{\sqrt {\zeta ^{2}-1}}\right)\omega _{n},s_{2}=\left(\zeta -{\sqrt {\zeta ^{2}-1}}\right)\omega _{n}}$