Frequency Response and Bode Plots

Shake a system back and forth with a sine wave and, once things settle, the output is a sine wave at the same frequency, just bigger or smaller and shifted in time. A Bode plot shows that gain and shift for every frequency at once.

Natural frequency ωn Damping ratio ζ Input frequency (log)

Gain and phase

For a stable system $G(s)$, an input $\sin\omega t$ gives a steady output $\lvert G(j\omega)\rvert \sin(\omega t + \angle G(j\omega))$. For the second-order system

$$G(j\omega) = \frac{\omega_n^2}{\omega_n^2 - \omega^2 + 2j\zeta\omega_n\omega} \qquad \text{gain in dB} = 20\log_{10}\lvert G\rvert$$

What to notice

Slow shaking passes through unchanged: gain 1, which is 0 dB, and almost no lag. Shake near $\omega_n$ and, with low damping, the output swings much larger than the input: resonance. Shake fast and the system cannot keep up, so the output shrinks and lags by up to 180°. Robot designers read Bode plots to find resonances to avoid, and to check how much phase margin a control loop has before it would oscillate.