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      LOGARITHMIC DECREMENT
    

ANALYZER SDOF · FREE VIBRATION

Logarithmic Decrement & Damping Ratio

Extract the damping ratio from measured free-vibration peak amplitudes — enter values manually or upload a time-series CSV.

/ INPUT

INPUT METHOD

PEAK AMPLITUDES

Consecutive positive peak amplitudes, space or comma separated, largest first.

CSV FILE · col 1 = time col 2 = amplitude {{ fileLabel }}

✓  {{ fileName }}

{{ nPeaksSelected }} selected · {{ nPeaksDetected }} detected

Click on the chart to select positive peaks (min 2). Click a selected peak again to deselect it.

SELECTED PEAKS

{{ sp.t }} s {{ sp.x }}

PEAKS USED {{ nPeaks }}

{{ peaksNote }}

LOG. DECREMENT  δ

{{ deltaStr }}

nepers / cycle

DAMPING RATIO  ζ

{{ zetaStr }}

{{ zetaPct }}

DAMPED FREQ  fd

{{ fdStr }}

Hz · CSV only

NATURAL FREQ  fn

{{ fnStr }}

Hz · CSV only

FREE VIBRATION DECAY · {{ chartSubtitle }} CH1 ► {{ xAxisLabel }} δ = {{ deltaStr }}

/ HOW IT WORKS THEORY · FREE VIBRATION

The logarithmic decrement δ is the natural log of the ratio of any two successive same-sign peak amplitudes in free vibration. Spanning the full set of m peaks averages out measurement noise:

+ + + + \[ \delta = \frac{1}{m-1} \ln \frac{x_1}{x_m} \] \[ \zeta = \sqrt{\frac{\delta^2}{4\pi^2 + \delta^2}} \] \[ f_d = \frac{1}{T_d} \qquad\qquad f_n = \frac{f_d}{\sqrt{1 - \zeta^2}} \]

NOTATION

δ Logarithmic decrement — natural log of the peak amplitude ratio, divided by the number of intervals (nepers cycle⁻¹) ζ Viscous damping ratio — fraction of critical damping; dimensionless (typical range 0.01–0.10 for structures) x₁ Amplitude of the first (largest) selected peak x_m Amplitude of the m-th (last) selected peak m Number of selected peaks (m − 1 complete oscillation cycles are spanned) T_d Mean damped period — average time interval between consecutive selected peaks, derived from CSV timestamps (s) f_d Damped natural frequency — rate of the observed damped oscillation, 1 / T_d (Hz; CSV mode only) f_n Undamped natural frequency — f_d corrected for damping: f_d / √(1 − ζ²) (Hz; CSV mode only)

For best accuracy, capture as many peaks as the noise floor allows — the multi-peak form averages out measurement scatter. When a CSV with time data is provided, the damped period T_d is measured directly from peak timestamps, giving f_d and f_n in addition to ζ.

⊕ ENGINEER'S NOTE

Use positive peaks only (all the same sign). Mixing positive and negative half-cycle peaks doubles the apparent δ.

The method assumes a linear SDOF system with viscous damping. A δ that varies from peak to peak signals non-linearity or multiple modes.

For CSV uploads, peaks are detected from positive local maxima of the amplitude column.

ζ typical 0.01–0.10 min 2 peaks