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SDOF Free Vibration

Natural frequency ωn, period Tn, critical damping ccr, damping ratio ζ · Damped natural frequency ωd · Free vibration time history plot

System Parameters

Mass m (kg)

Stiffness k (N/m)

Damping coefficient c (N·s/m) — 0 = undamped

Initial displacement u₀ (m)

Initial velocity v₀ (m/s)

Results

Natural period Tn—

Natural frequency ωn (rad/s)—

fn (Hz)—

Critical damping ccr (N·s/m)—

Damping ratio ζ = c/ccr—

Damped frequency ωd (rad/s)—

Damped period Td (s)—

Regime—

Citations

Eq.(2.2)ωn = √(k/m) — Chopra §2.1

Eq.(2.9)ccr = 2mωn — critical damping coefficient

Eq.(2.10)ζ = c/ccr — damping ratio

Eq.(2.12)ωd = ωn√(1−ζ²) — damped natural frequency

Verifym=100, k=10000, c=0 → ωn=10 rad/s, Tn=0.6283 s (Chopra Ex.2.1)

Engine: dynamics.js · Chopra (2012) §2.1–2.2 ·

Forced vibration →

Free Vibration Time History u(t)

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Preliminary educational tool. Results must be verified against textbook solutions before use in engineering practice. Formula references are provided for every computed value.