# Characteristics from transfer function

All! I am trying to figure out characteristics of a second order system The system is (for example)

And its characteristics for an unit step input are

I want to solve this equation by hand and extract damping ratio and natural frequency, but my knowledge is the transfer function of undamped second order system should be in this formula

How to determine damping ratio and natural frequency?

$$\frac{K\,\omega_{_0}^2}{s^2+2\,\zeta\,\omega_{_0}\,s+\omega_{_0}^2}$$
$$\K\$$ is the gain. $$\\zeta\$$ is the damping factor (with $$\Q=\frac{1}{2\,\zeta}\$$, being the ratio of the center frequency to the $$\-3\:\text{dB}\$$ frequency.)
If the denominator is expressed as $$\b_2\,s^2+b_1\,s+b_0\$$, then $$\\omega_{_0}=\sqrt{\frac{b_0}{b_2}}\$$ and $$\\zeta=\frac{b_1}{2\,\sqrt{b_0\,b_2}}\$$. (You may also want to consider what the value of $$\K\$$ must be, too.)