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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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Visual explanation

Damped free vibration. SDOF system released from initial displacement u₀; analytical solution — **Figure 23.** Damped free vibrationSource: Chopra (2017), Dynamics of Structures, Ch. 2. Curves computed.

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