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SDOF Forced Harmonic Vibration

F(t) = F₀ sin Ωt · Dynamic Magnification Factor (DMF) · Frequency ratio β = Ω/ωn · Resonance detection · Phase angle φ

System & Loading

Mass m (kg)

Stiffness k (N/m)

Damping c (N·s/m)

Force amplitude F₀ (N)

Excitation Ω (rad/s)

ωn (rad/s)—

Frequency ratio β = Ω/ωn—

Damping ratio ζ—

DMF (Dynamic Magnification Factor)—

Static deformation u_st = F₀/k (m)—

Peak response u_max = DMF × u_st (m)—

Phase angle φ (°)—

Resonance check (|β−1|<0.05)—

—

Eq.(3.9)DMF=1/√[(1−β²)²+(2ζβ)²] · Chopra §3.2

Eq.(3.8)u_p(t) = (F₀/k)·DMF·sin(Ωt−φ)

Eq.(3.11)tan φ = 2ζβ/(1−β²) — phase angle

Verifym=100, k=10000, c=0, F0=500, Ω=5: β=0.5 → DMF=4/3=1.333 ✓

dynamics.js · Chopra §3.2 ·

← Free vibration Newmark-β →

Preliminary educational tool. Total response includes transient term (not plotted). Phase angle sign convention follows Chopra Eq.(3.11).