⚙️ undamped forced vibration F₀ · cos(ωt)

steady‑state amplitude & transient animation · mass–spring under sinusoidal force

X = F₀ / (k - m·ω²) _{(ω ≠ ωₙ)}

m mass kg

k stiffness N/m

F₀ force ampl. N

ω driving freq. rad/s

AMPLITUDE X (steady state)

0.000 m

∣F₀/(k−mω²)∣ mechanical reactance

⚡ forcing in phase

📐 governing equation

m·x″ + k·x = F₀·cos(ωt)

particular solution: xₚ = X·cos(ωt)
X = F₀/(k − mω²) (if ω ≠ ωₙ = √(k/m))

displacement x(t) excitation F(t)

ω/ωₙ = 0.83

⚠️ undamped → if ω = ωₙ amplitude grows linearly (resonance).
formula gives infinite; animation shows beating/divergence.