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Logarithmic decrement

By measuring how much the amplitude decays from one peak to the next in a free, damped vibration, you can find the damping ratio without knowing $c$ , $k$ or $m$ . The logarithmic decrement $\delta$ is the natural logarithm of the ratio between two successive peaks.

\delta = \ln(x_1/x_2)

logarithmic decrement

\zeta \approx \delta/(2\pi)

approximation for small

\zeta

Symbols

$\delta$	logarithmic decrement
$x_1,\ x_2$	amplitude at two successive peaks	mm

Example

Peaks of 20 mm and 12 mm:

$\delta = \ln(20/12) \approx 0.511$ , so $\zeta \approx 0.511/(2\pi) \approx 0.081$ .

Use several cycles and divide by the count to reduce measurement uncertainty.

Practise damping for free →

← Damped natural frequency · Resonance and amplification →

Part of Machine Dynamics and Vibrations: Damping.