Logarithmic Decrement⚠ unverified
Mechanical / Vibration Analysis · Compute the logarithmic decrement from two successive vibration amplitudes
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| x1 | x1 | m | 1.0 | Earlier peak amplitude |
| x2 | x2 | m | 1.0 | Later (successive) peak amplitude |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | δ | — | Logarithmic decrement (dimensionless). Returns 0.0 when the later amplitude is not positive |
The science & history
Understanding the Parameters
- $x_1 > x_2 > 0$ — ratio of successive peaks one period apart.
- $\delta$ — larger $\delta$ → heavier damping. Can also use peaks $n$ cycles apart: $\delta = (1/n)\ln(x_i/x_{i+n})$.
Derivation (Approaching a Proof)
Underdamped free response envelope $\propto e^{-\zeta\omega_n t}$. Over one damped period $T_d = 2\pi/\omega_d$, the amplitude ratio is $e^{\zeta\omega_n T_d}$, so $\delta = \ln(x_1/x_2) = 2\pi\zeta/\sqrt{1-\zeta^{2}}$.
History
Log decrement is the classical free-decay damping measurement method.
Related Concepts: Damping Ratio, Damped Natural Frequency, Quality Factor, Equivalent Viscous Damping
Notes: Registry calculator vibration-logarithmic-decrement (unverified). Two successive peaks.