# Control System; Finding resonant frequency of 3rd order

We've been requested to find the resonant frequencies of a 3rd order plant in class and were presented to the following method to do so:

My question is, when using the $$\ \omega_r$$ formula, to find both of the resonant freq's do I need to place $$\ \omega_{n1}$$ with $$\ ζ_{1}$$ to get $$\ \omega_{r1}$$ (and $$\ -\omega_{n2}$$ with $$\ ζ_{2}$$ to get $$\ \omega_{r2}$$), or I am supposed to use the same $$\ ζ_{2}$$ (of the Pole) for both, like this:

$$\omega_{r1} = f(\omega_{n1}, ζ_{2})$$

$$\omega_{r2} = f(\omega_{n2}, ζ_2)$$

I hope the question is clear enough, thanks!

• Do you need to use this formula to find $\omega_r$ ? You can find the magnitude in terms of $j \omega$ and check where the derivative of the magnitude is zero (for $ω=ω_r$). – Teo Protoulis Feb 17 '20 at 15:59
• Definitely, though this method can save lots of time if used correctly :) – Yarden2y Feb 17 '20 at 19:40

The resonance is caused by the poles, therefore $$\omega_r = \omega_{n_2} \sqrt{1-2\zeta_2^2} \approx 2.9462$$.
The zeros do not cause a resonance, but an antiresonance. Thus, the antiresonance frequency is located at $$\omega_{ar} = \omega_{n_1} \sqrt{1-2\zeta_1^2}\approx 0.9592$$.