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

enter image description here

And its characteristics for an unit step input are enter image description here

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

enter image description here

How to determine damping ratio and natural frequency?

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The standard form for a two-pole low-pass filter is:

$$\frac{K\,\omega_{_0}^2}{s^2+2\,\zeta\,\omega_{_0}\,s+\omega_{_0}^2}$$

\$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.)

If you are interested in more detail, I've provided it here.

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