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.

免費工程工具 #066

阻尼比計算器

根據對數減衰量（連續峰值振幅）或半功率頻寬法計算阻尼比 ζ。支援多週期平均。

對數減衰量半功率頻寬自動計算

方法

峰值振幅 x₁

峰值振幅 x₂ （經過 n 個週期後）

週期數 n （x₁ 與 x₂ 之間）

快速預設

結果

阻尼比 ζ

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阻尼（%）

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對數減衰量 δ / 頻寬 Δf

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品質因數 Q

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阻尼分類

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對數減衰量法

半功率頻寬法

品質因數

範例 — 對數減衰量

已知： x₁ = 10 mm, x₂ = 6 mm, n = 1 個週期

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%)

⚠️ 注意： 對數減衰量近似公式 ζ ≈ δ/(2π) 在 ζ < 0.1 時誤差在 1% 以內。對於較高阻尼，請使用精確公式。