# Finding roots of undamped and damped time domain response of a 4th order characteristic equation

A fourth order system is expressed in terms of the characteristic equation as $$\s^4 + 8s^3 + 24s^2 + 32s + k = 0\$$

Using the Routh’s array criterion determines the value of the gain parameter k and the corresponding roots for the following two cases:

(i) System time-domain response is undamped oscillations.

(ii) System time-domain response comprises of damped oscillations.

I was able to find the stability range of parameter k as $$\0, but I'm struggling to understand how to find the roots when system gives undamped and damped oscillations.

• Would undamped oscillations be oscillations that maintain the same amplitude? (don't decay or increase). To find those roots you have to find the roots of a 4th order polynomial, that's not simple to do analytically and is easier to do numerically. – jDAQ Jun 11 at 17:29
• I'd guess undamped oscillation would mean same amplitude. But I think solving it numerically is beyond my current scope of learning. One thing I was wondering, if I did find the roots for a particular case of k, let's say, how would I know whether its damped or undamped? Thanks. – S_sg Jun 11 at 19:46
• if you have single poles on the imaginary axis the impulse response will be an "undamped" oscillation, if all the poles are strictly in the open left half-plane then it will be damped. – jDAQ Jun 11 at 19:54
• Try solving the roots (numerically, wolfram alpha is a good place to start) for k=80, you'll notice what @jDAQ mentioned. That would take care of the undamped part, and for 0<k<80 it means that it's stable, thus damped. – a concerned citizen Jun 11 at 20:15
• if you assume there is an imaginary number, $\alpha j$, that solves the characteristic equation you can find for which $\alpha$ and $K$ that holds. Once you have that you can divide the original polynomial by $(s^2+a^2)$ and get a simpler 2nd order polynomial to solve for the remaining roots. – jDAQ Jun 11 at 20:15