Vibrations & Dynamics · Learn
Feed a system a pure sine wave and it answers with a sine wave — same frequency, but scaled and delayed. Sweep every frequency and you get the system's fingerprint: the Bode plot.
For a second-order system, the steady response to a sinusoid is captured by the complex transfer function, with frequency ratio $r=\omega/\omega_n$:
Its magnitude $|H|$ is the gain applied to the input amplitude; its angle $\angle H$ is the phase lag. A Bode plot is just these two, drawn versus frequency.
Real systems span huge ranges, so gain is plotted in decibels ($20\log_{10}|H|$) against log frequency. The payoff: multiplying gains becomes adding, so a chain of components is just the sum of their Bode plots — and the response becomes straight-line asymptotes you can sketch by hand.
| Frequency | Gain | Phase |
|---|---|---|
| $r \ll 1$ (low) | ≈ 1 (0 dB) | ≈ 0° — output tracks input |
| $r \approx 1$ (resonance) | peak ≈ $1/2\zeta$ | −90° |
| $r \gg 1$ (high) | −40 dB/decade rolloff | → −180° (inverted) |
At low frequency the mass keeps up; near $\omega_n$ it resonates; at high frequency it can't respond and the output shrinks and flips.
The sharpness of the resonant peak is the quality factor $Q \approx 1/(2\zeta)$. High $Q$ (low damping) gives a tall, narrow, frequency-selective peak — a tuning fork or a radio filter picking one station. Low $Q$ gives a broad, gentle hump that responds to a wide band — a well-damped car suspension.
Drag the operating point along the Bode curves and watch the input and output sinusoids scale and slip out of phase; lower $\zeta$ and see the peak trade width for height. Open the lab →
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