Resonant Frequency and Resonant Peak Values

What is Resonant Frequency?

Natural Frequency vs. Resonant Frequency vs. Damped Frequency

Important Points

$ζ$ , $ω_{r}$ , and $M_{r}$

For a second order system...

Resonant Frequency

w_{r} = w_{n} \sqrt{1 - 2 ζ^{2}}

Amplitude Ratio

M_{r} = \frac{1}{2 ζ \sqrt{1 - 2 ζ^{2}}}

As $ζ$ changes:

Note that this is for plotting the Bode plot ( $0.5$ is an approximation; it is $\frac{1}{\sqrt{2}} = 0.707$ )
As $ζ$ increases, the phase transition becomes steeper

Bandwidth

$ω_{B}$ is the $ω$ at -3 $d B$ ( $\frac{1}{\sqrt{2}}$ ) from the low-frequency value

Resources

Understanding Vibration and Resonance