Vibrations & Dynamics · interactive

Frequency Response & Bode Plots

Feed a system a pure sine wave and, once the dust settles, it answers with a sine wave at the same frequency — but scaled and delayed. Sweep across all frequencies and you get the system's fingerprint: the Bode plot. Watch the input and output sinusoids dance, and see exactly where this oscillator amplifies, ignores, or inverts what you give it.

H(iω) = 1 / [ 1 − r² + i·2ζr ]  ,  r = ω/ωₙ
The transfer function of a second-order system. Its magnitude is the gain, its angle is the phase lag — the two curves of a Bode plot.

Input vs. output · same frequency, scaled & phase-shifted

input sin(ωt) steady-state output

Bode magnitude · gain in dB vs. frequency (log)

Bode phase · lag in degrees vs. frequency (log)

Gain at r
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Phase lag
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Peak gain (Q)
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What you're seeing

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System

0.10

ωₙ is fixed as the reference; only the ratio r = ω/ωₙ matters for the shape.

Drive frequency

1.00

Drag to move the marker along the Bode curves and watch the output sinusoid respond.

Three regimes on one plot

At low frequency (r ≪ 1) the system follows the input almost perfectly: gain ≈ 1 (0 dB), phase lag ≈ 0. The mass has all the time in the world to keep up. At resonance (r ≈ 1) the gain peaks — the output can be many times larger than the input — and the phase passes through exactly −90°. At high frequency (r ≫ 1) the mass simply can't respond: the gain rolls off at −40 dB/decade and the output lags by a full −180°, completely inverted. Drag r across all three and watch the input/output sinusoids at the top change scale and slip out of step.

Why decibels and log frequency?

Real systems span enormous ranges of gain and frequency, so engineers plot both on logarithmic scales. In dB (20·log₁₀|H|), multiplying gains becomes adding — so the response of a chain of components is just the sum of their Bode plots. The straight-line asymptotes (flat, then −40 dB/decade past ωₙ) are the famous "Bode sketch" you can draw by hand.

The Q factor

The sharpness of the resonant peak is the quality factor, Q ≈ 1/(2ζ). High Q (low damping) means a tall, narrow peak — a system that's exquisitely selective about frequency, like a tuning fork or a radio filter picking one station. Low Q means a broad, gentle hump that responds to a wide band — a well-damped car suspension. Slide ζ and watch the peak trade height for width, exactly mirroring the resonance you met in the spring–mass–damper lab.

EngineeringCandy · second-order transfer function evaluated live · sweep it, tune it, learn