What is the logarithmic decrement?

It is the natural logarithm of the ratio of any two successive peak amplitudes of the same sign in free vibration: δ = ln(x₁/x₂). For n cycles, δ = (1/n) × ln(x₁/xₙ₊₁). A larger δ means faster energy dissipation.

How is the damping ratio calculated from δ?

The formula is ζ = δ / √(4π² + δ²). For lightly damped systems (δ ≪ 1) the approximation ζ ≈ δ / (2π) is accurate to within a few percent.

Do the amplitudes need to be consecutive?

No. You can measure amplitudes n cycles apart to reduce measurement noise. Enter the number of cycles between measurements in the "Cycles apart" field and the calculator applies the 1/n factor automatically.

What range of damping ratios does this cover?

The method is valid for underdamped systems (ζ < 1). Typical structural systems have ζ between 0.001 and 0.15. If your computed ζ exceeds 0.3, verify that the system is truly in free vibration and that the peaks are correctly identified.

Can I enter more than two amplitudes?

Yes. Enter as many peaks as you measured. The tool computes δ for each consecutive pair and also from the first to last peak spanning all cycles, then shows the mean δ and ζ for the full set.

Logarithmic Decrement Calculator

Enter successive peak amplitudes from a free-vibration test to compute logarithmic decrement and viscous damping ratio.

Peak Amplitudes

Enter measured peak amplitudes in order (same sign, e.g. all positive maxima). One value per line.

Amplitudes (x₁, x₂, … xₙ)

One numeric value per line. Minimum 2 values.

Cycles between consecutive measurements

Set to 1 if you measured every cycle. Use a larger integer if you skipped cycles (e.g., measure every 5th peak → enter 5).

Formulas

δ = (1/n) × ln(x₁ / xₙ₊₁)

ζ = δ / √(4π² + δ²)

ζ ≈ δ / (2π) when ζ ≪ 1

n = number of cycles between x₁ and xₙ₊₁.
δ = logarithmic decrement (dimensionless).
ζ = viscous damping ratio (0 < ζ < 1 for underdamped).

Enter amplitude values and click Calculate.

Log. Decrement (δ)

—

from all cycles

Damping Ratio (ζ)

—

viscous

ζ (approximation)

—

δ / (2π), valid when ζ ≪ 1

Mean δ per pair

—

average of consecutive pairs

Amplitude Decay

Cycle-by-Cycle Breakdown

Pair	x₁	x₂	Ratio x₁/x₂	δ (pair)	ζ (pair)

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Summary

The logarithmic decrement (δ) measures how quickly a freely oscillating system loses energy between successive cycles. Enter at least two consecutive peak amplitudes (or one amplitude pair n cycles apart), then calculate the logarithmic decrement and corresponding viscous damping ratio ζ.

How it works

Measure the peak amplitude of the oscillating system at two or more successive cycles.
Enter the amplitudes in order — the tool computes ratios between each consecutive pair.
Specify how many cycles separate the first and last amplitude if you skip cycles.
The logarithmic decrement δ = (1/n) × ln(x₁/xₙ₊₁) is calculated and displayed.
The damping ratio ζ = δ / √(4π² + δ²) is derived from δ.
Review the per-cycle amplitude table to verify measurement consistency.

Use cases

Characterize viscous damping in mechanical structures from free-decay tests.
Validate finite-element model damping assumptions against measured data.
Estimate damping in beams, frames, or machine components after impact excitation.
Compare damping before and after adding vibration-absorbing material.
Screen components during quality control for expected damping levels.
Teach the logarithmic-decrement method in dynamics and vibrations courses.

Frequently Asked Questions

Last updated: 2026-06-18 · Reviewed by Nham Vu