What is damping ratio ζ?

Damping ratio ζ (zeta) is the ratio of actual damping to critical damping. ζ = 0 means undamped, ζ = 1 is critically damped, 0 < ζ < 1 is underdamped (oscillatory). Most mechanical systems have ζ between 0.01 and 0.1.

How does the logarithmic decrement method work?

Measure successive peak amplitudes x₁ and x₂ in free vibration. The log decrement δ = ln(x₁/x₂). For n cycles: δ = (1/n)ln(x₁/xₙ₊₁). Then ζ = δ/√(4π² + δ²) ≈ δ/(2π) for small damping.

How does the half-power bandwidth method work?

From a frequency response function, measure the natural frequency fn and the two frequencies f₁, f₂ where amplitude drops to 1/√2 of peak (−3 dB). Then ζ = (f₂ − f₁)/(2fn) = Δf/(2fn).

Which method is more accurate?

Log decrement is better for lightly damped systems (ζ < 0.1) with clear free vibration decay. Half-power bandwidth works well for forced vibration measurements and moderate damping. For very light damping, averaging over many cycles improves log decrement accuracy.

What is the quality factor Q?

Quality factor Q = 1/(2ζ). A high Q indicates low damping and a sharp resonance peak. Q = 50 means ζ = 0.01. Q is commonly used in acoustics, electronics, and structural dynamics.

Free Engineering Tool #066

Damping Ratio Calculator

Determine damping ratio ζ from logarithmic decrement (successive peak amplitudes) or half-power bandwidth method. Supports multi-cycle averaging.

Log DecrementHalf-Power BandwidthAuto-Calc

Method

Peak Amplitude x₁

Peak Amplitude x₂ (after n cycles)

Number of Cycles n (between x₁ and x₂)

Quick presets

Results

Damping Ratio ζ

—

Damping (%)

—

Log Decrement δ / Bandwidth Δf

—

Quality Factor Q

—

Damping Classification

—

Logarithmic Decrement Method

Half-Power Bandwidth Method

Quality Factor

Example — Log Decrement

Given: x₁ = 10 mm, x₂ = 6 mm, n = 1 cycle

δ = (1/1) × ln(10/6) = ln(1.667) = 0.5108

ζ = 0.5108 / √(4π² + 0.5108²) = 0.5108 / 6.304 = 0.0811 (8.11%)

⚠️ Note: The log decrement approximation ζ ≈ δ/(2π) is accurate to within 1% for ζ < 0.1. For higher damping, use the exact formula.

Damping Ratio Calculator | Log Decrement & Half-Power Bandwidth | Free Tool #066

Published by Nikolai Shelkovenko on February 15, 2026 February 15, 2026

Results

Logarithmic Decrement Method

Half-Power Bandwidth Method

Quality Factor