Damping Regimes Compared

Four systems, same frequency, four personalities. See underdamped ringing, the critically damped sweet spot, and the overdamped crawl side by side.

Damping Regimes ComparedLive

under, critical & overdamped

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Presets

Natural frequency ωₙ

Four responses, same natural frequency, different damping. Underdamped overshoots and rings; critically damped (ζ=1) is the fastest with no overshoot; overdamped crawls to target. Most designs aim near ζ=0.7 for a good speed–overshoot balance. Educational tool.

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Data Inspector

ζ = 0.2 overshoot53%

ζ = 0.707 overshoot4%

Critical dampingζ = 1 (fastest, no overshoot)

Governing equation

\overset{x}{¨} + 2 ζ ω_{n} \overset{x}{˙} + ω_{n}^{2} x = ω_{n}^{2} u, ω_{n} = 3.0, M_{p} = e^{- ζ π / 1 - ζ^{2}}

Reading this result: ωₙ = 3: response timescale is about 1/ωₙ ≈ 0.33 s, with the ζ=0.707 curve settling near 1.9 s. Overshoot depends only on ζ, not on ωₙ.

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How it works

Damping ratio ζ sorts second-order systems into regimes: ζ < 1 overshoots and oscillates, ζ = 1 (critical) reaches target fastest without overshoot, and ζ > 1 (overdamped) is slow and smooth. Designers often target ζ ≈ 0.7 for a balance of speed and modest overshoot. Educational tool.

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Is this underdamped overdamped critically damped tool really free?▾

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The solver uses established numerical methods, but results are for research and educational purposes and should be validated against experiment or professional review before you rely on them.