Damped Oscillation

About this calculator

The Damped Oscillation Calculator analyses the damped harmonic oscillator m·x″ + c·x′ + k·x = 0 from the mass, spring constant, damping coefficient, and initial conditions. It classifies the motion as under-damped, critically damped, or over-damped by comparing c with the critical damping 2√(mk), and reports the natural frequency ω₀, damping ratio ζ, decay constant β, and regime-specific values such as the damped period, quality factor, logarithmic decrement, or the two real decay roots, with a plotted displacement curve and envelope.

How to use the Damped Oscillation calculator

1. Enter the quantities you know, using the units shown on each field.
2. Read the computed results — they update instantly as you type.
3. Use the visual explanation to see how the quantities relate.
4. Copy the page URL to share the exact calculation.

Common examples

• m = 1 kg, k = 100 N/m, c = 2 N·s/m → under-damped: ζ = 0.1, ω₀ = 10 rad/s, ω_d ≈ 9.9499 rad/s, Q = 5
• m = 1 kg, k = 100 N/m, c = 20 N·s/m → critically damped (ζ = 1), time constant τ = 0.1 s
• m = 1 kg, k = 100 N/m, c = 30 N·s/m → over-damped: ζ = 1.5, roots r₁ ≈ −3.8197, r₂ ≈ −26.1803 s⁻¹
• m = 2 kg, k = 50 N/m, c = 0 → undamped: oscillates forever at ω₀ = 5 rad/s, period ≈ 1.2566 s