Does movement along the imaginary axis of a root locus change the natural frequency?

I have two root locus plots (x-real, y-imaginary) of the same oscillating mode: (1) for a base case system and (2) a system with added feedback. They are so similar, almost identical that I am having a hard time interpreting it but the author comments that "the frequency of the mode has been hardly changed". So he concludes it has not been changed significantly but how can he tell? Is it because the root locus has not moved along the imaginary axis?

On a root locus, zeroes and poles are plotted. What could be changed about these zeroes and poles without changing the frequency of the mode?

• What's the open-loop TF? – Chu Nov 23 '15 at 0:55
• I don't know unfortunately. It's from a textbook. I think it is not legal for me to upload the two figures here? – Martin Bakardjiev Nov 29 '15 at 14:25

If a system has a pole at $s = a + ib$, then it has a response that looks like $$r(t) = e^{(a + ib)t} = e^{at}e^{ibt}$$ If these poles have a complex part (ie: $b \neq 0$), then the pair of poles create a real response that looks like $$r(t) = e^{at} \sin(bt) \text{ or } e^{at} \cos(bt)$$
• For a complex root, $\omega_n$ is the length of the vector from the origin to the root, so moving the real part of the root does potentially influence $\omega_n$. By how much depends on the $\zeta$ value. – Chu Nov 23 '15 at 16:39