Damped systems - deriving equation

Question

I am having some troubles deriving the formula for the roots for different types systems.. I am not quite sure if they are correct (pretty sure they aren't).

$y(s) = \frac{s+2\zeta\omega_n}{s^2 + 2\zeta\omega_ns +\omega_n^2}$

I don't see how i should derive them for different scenarios

In my book it is stated that for an underdamped system $\zeta < 1$ is the roots S1,2 = $-\zeta\omega_n \pm i\omega_n \sqrt{1-\zeta^2}$ which i don't get.

I've derived it to be $\frac{-2\zeta\omega_n \pm \sqrt{4\zeta^2\omega_n^2- 4\omega_n^2}}{2}$

I don't see how the hell they got the other one, and how they are for the other cases!!??

Answer 1

Well, just look at the root!

$$\frac{-2\zeta\omega_n \pm \sqrt{4\zeta^2\omega_n^2- 4\omega_n^2}}{2} =\frac{-2\zeta\omega_n \pm 2\omega_n\sqrt{\zeta^2- 1}}{2}$$ And that is: $$-\zeta\omega_n \pm i\omega_n\sqrt{1-\zeta^2}$$ Elaboration: $$\sqrt{4\zeta^2\omega_n^2- 4\omega_n^2}=\sqrt{4\omega_n^2(\zeta^2-1)}=$$ $$\sqrt{4\omega_n^2}\sqrt{\zeta^2-1}=2\omega_n\sqrt{\zeta^2-1}$$ Different scenarios:

• $\zeta=0$ no dampening, formula $S_{12}=i\omega_n$
• $\zeta<1$ dampening, solution is complex, oscillatory
• $\zeta=1$ critical dampening, the formula will be $S_{12}=-\zeta\omega_n$
• $\zeta>1$ overdamped, two different real roots, will take longer to converge than criticaly dampened, formula $S_{12}=-\zeta\omega_n \pm \omega_n\sqrt{\zeta^2-1}$