Damping Ratio (ζ) Calculator

The damping ratio ζ is a dimensionless number measuring how strongly a mass–spring–damper system is damped relative to critical damping: ζ = c / c_c = c / (2·√(k·m)), where c is the damping coefficient [N·s/m], m the mass [kg], and k the spring stiffness [N/m]. ζ = 0 means no damping (it vibrates forever); the larger ζ, the faster oscillations die out.

ζ < 1 = underdamped (vibrates with decaying amplitude); ζ = 1 = critically damped (returns to equilibrium fastest without oscillation); ζ > 1 = overdamped (returns slowly without oscillation). Example: c = 4, m = 1 kg, k = 100 N/m gives ζ = 4/(2·10) = 0.2 (underdamped); c = 20 gives ζ = 1 exactly (critical); c = 40 gives ζ = 2 (overdamped).

The natural frequency is ω_n = √(k/m). For an underdamped system (ζ < 1), the damped frequency is ω_d = ω_n·√(1−ζ²) — always slightly below ω_n. Anchor example: k = 100 N/m, m = 1 kg → ω_n = 10 rad/s (f_n ≈ 1.59155 Hz); with ζ = 0.2 → ω_d = 10·√0.96 ≈ 9.79796 rad/s (f_d ≈ 1.55939 Hz). For ζ ≥ 1, ω_d is undefined because there is no oscillation.

The logarithmic decrement δ = ln(x_n/x_{n+1}) measures the decay rate between two successive peaks, and δ = 2πζ/√(1−ζ²) for ζ < 1. The inverse formula is ζ = δ/√(4π²+δ²). Anchor example: ζ = 0.2 → δ = 2π·0.2/√0.96 ≈ 1.28254983; conversely a measured δ = 1.28254983 → ζ = 0.2.

For the step response of an underdamped second-order system, percent overshoot PO = 100·e^(−πζ/√(1−ζ²)) percent. Example: ζ = 0.2 → PO ≈ 52.66%; ζ = 0.5 → PO ≈ 16.30%. PO is only meaningful for ζ < 1; at ζ ≥ 1 there is no overshoot.

Invert the formula: c = ζ·2·√(k·m). Example: target ζ = 0.5 with m = 1 kg and k = 100 N/m → c = 0.5·2·10 = 10 N·s/m. This calculator reports the critical damping value c_c = 2·√(k·m) for comparison — a dedicated c_c solver (from m & k, or the inverse) is available in the critical-damping tool.

Mass: kg, g, lb (1 lb = 0.45359237 kg). Spring stiffness: N/m, N/mm, lbf/in (1 lbf/in = 175.126835246476 N/m; example 50 lbf/in = 8756.34176 N/m). The damping coefficient c is always in N·s/m (SI). All calculations are converted to SI first, then results are shown in SI units.

This tool computes the RATIO ζ = c/c_c plus classification (under/critical/over), damped frequency, logarithmic decrement, and percent overshoot. The critical-damping tool (#1571) focuses on the critical damping coefficient c_c itself as the sought quantity (c_c from m & k, or k/m from c_c). The two complement each other: here c_c is only intermediate info, not a solver mode.

These formulas apply to linear viscous damping on a single-degree-of-freedom system. Coulomb damping (dry friction), structural/hysteresis damping, or non-linear systems use different models. δ, ζ-from-δ, and PO are only defined for ζ < 1 (oscillation present); ζ ≥ 1 input in those modes will be rejected with a clear message.