Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I'm looking for good online resources for 2nd order system dynamics. Any recommendations?

I'm looking for stuff that ideally includes discussion of Q, damping ratio, overshoot, bode plots, for systems with transfer functions of

$$ H_1(s) = \frac{1}{\tau_2^2 s^2 + \tau_1 s + 1} $$

$$ H_2(s) = \frac{\tau_1 s + 1}{\tau_2^2 s^2 + \tau_1 s + 1} $$


$$ H_3(s) = \frac{s}{\tau_2^2 s^2 + \tau_1 s + 1} $$

edit: I've done change of variables to rewrite as

$$ H_1(\sigma) = \frac{1}{\sigma^2 + 2\zeta\sigma + 1} $$

$$ H_2(\sigma) = \frac{2\zeta\sigma + 1}{\sigma^2 + 2\zeta\sigma + 1} $$


$$ H_3(\sigma) / \omega_0 = \frac{\sigma}{\sigma^2 + 2\zeta\sigma + 1} $$

where $\sigma = \tau_2 s = s / \omega_0$, $\omega_0 = 1 / \tau_2$, and $\zeta = \tau_1 / 2\tau_2$ to normalize out the time scaling factor and end up with a transfer function of the single variable $\zeta$ (damping factor) which I think is the conventional treatment.

share|cite|improve this question
Community Wiki. – anonymous Oct 1 '10 at 18:00
done. (.......) – Jason S Oct 1 '10 at 18:20

Searching in Google itself gives you a great deal of information. I got this link which explains has the content which you need.

Next, [redacted: inappropriate link to pirated content] is an excellent source for those who are in need of books online.

share|cite|improve this answer
the problem is there are a lot of mediocre pages out there, and I'm looking for a good one; most of the sources of this type of information just have the transfer function with 1 in the numerator and something in the denominator. (having said that, the one you posted isn't too bad) – Jason S Oct 1 '10 at 18:23

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.